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A multi-layer urban canopy model for neighbourhoods with trees Krayenhoff, Eric Scott 2014

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A MULTI-LAYER URBAN CANOPY MODEL FOR NEIGHBOURHOODS WITH TREES by  Eric Scott Krayenhoff  B.Sc. (Honours), The University of British Columbia, 2002 M.Sc., University of Western Ontario, 2005  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Geography)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December 2014  © Eric Scott Krayenhoff, 2014 ii  Abstract  Over 50% of the world’s population lives in cities, many of which are hot, polluted, and expanding. The design of cities impacts local meteorology and climate, which affect building energy use and the comfort and health of urban residents. Numerical models that incorporate the relevant urban elements and physical processes can predict these effects and guide management strategies. Addition of vegetation is a key design strategy for moderation of local urban climate, and many cities already boast extensive vegetation. Relative to shorter vegetation, urban trees have unique effects on local climate and pollutant dispersion: they provide shade and shelter, interacting with buildings and streets to alter climate and wind flow. Urban canopy models (UCMs) predict neighbourhood-scale (102 – 104 m) energy exchange and climate of atmospheric layers between and above the buildings. Few UCMs represent the urban canopy with multiple layers, which permit more flexible and process-based representation of canopy physics. Most UCMs neglect vegetation, or incorporate it with a separate model, neglecting interaction between vegetation and built elements. This dissertation develops BEP-Tree, the first multi-layer urban canopy model that explicitly includes trees and their interaction with buildings. It consists of an existing multi-layer UCM, a foliage energy balance model, and two major developments: firstly, a model that distributes solar and infrared radiation amongst tree foliage, road, roof, and wall elements at multiple heights, accounting for radiation ‘trapping’ and mutual shading; secondly, parameterization of building and tree foliage effects on flow, including generation and dissipation of turbulence, drag on the mean wind, and explicit consideration of sheltering. The combined model permits a range of building and tree configurations, and makes possible advanced assessment of impacts of trees on urban climate, air quality, human comfort and building energy loads. BEP-Tree is compared with measurements from the Sunset neighbourhood in Vancouver, Canada. Urban trees channel sensible heat into latent heat (evaporation), shift surface-atmosphere energy exchange upwards, slow canopy wind, and dissipate turbulence, especially if taller than nearby buildings. Effects of trees on canopy thermal climate depend on representation iii  of neighbourhood-scale foliage clumping in radiation versus dynamical processes, and further theoretical advances are required. iv  Preface  Chapter 1:  I wrote the chapter. Dr. A. Christen redrew Fig. 1.1.   Chapters 2 and 3 (and Appendix A):  A version of these chapters has been published: Krayenhoff ES, Christen A, Martilli A, Oke TR (2014) A multi-layer radiation model for urban neighbourhoods with trees. Boundary-Layer Meteorology 151, 139-178.  I designed the model with Drs. A. Christen and T. R. Oke. I designed the model tests and application with help from Drs. A. Christen, A. Martilli, and T. R. Oke. I coded the model, tested it, and prepared the chapters. Dr. Christen redrew Figs. 2.2, 2.3 and 3.1. Dr. Oke contributed substantially to the writing of Sect. 3.2.2.2.   Chapter 4 (and Appendices B and C):  A version of this chapter has been submitted for publication. Krayenhoff ES, Santiago JL, Martilli A, Christen A, Oke TR. Parameterization of drag and turbulence for urban neighbourhoods with trees.  I designed the CFD model experiments with Drs. A. Martilli and J.-L. Santiago. Dr. Santiago performed the CFD simulations. I performed the analyses and prepared the chapter. E. Leinberger (Geography, UBC) redrew Figs. 4.4-4.7 and all figures in Appendix B. Figs. 4.2 and 4.3 were provided by Dr. Santiago. Fig. C2 was drawn by Dr. Santiago.   Chapter 5:  I combined the models to form BEP-Tree with help from Dr. A. Martilli. I designed the model tests with input from Drs. A. Christen and T. R. Oke. I performed model testing and analysis, and wrote the chapter.   Chapter 6:  I wrote the chapter. v  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Tables ..................................................................................................................................x List of Figures .............................................................................................................................. xii List of Symbols ........................................................................................................................... xix Acknowledgements .................................................................................................................. xxiii Chapter 1: Introduction ................................................................................................................1 1.1 Micrometeorology of canopies: relevant theory ............................................................. 1 1.1.1 Flow, turbulence, and dispersion ................................................................................ 2 1.1.2 Radiation exchange ..................................................................................................... 2 1.2 Process-based modeling of urban climate and meteorology ........................................... 3 1.2.1 Urban canopy modeling .............................................................................................. 4 1.3 Modeling urban vegetation and the urban forest ............................................................ 6 1.3.1 Building-tree interaction ............................................................................................. 9 1.4 Overall research objectives ........................................................................................... 10 Chapter 2: Multi-layer radiation model for urban neighbourhoods with trees ....................12 2.1 Radiation exchange modeling ....................................................................................... 12 2.1.1 Ray tracing and radiosity approaches ....................................................................... 12 2.1.2 Monte-Carlo ray tracing for view factor determination ............................................ 13 2.2 Urban radiation model with integrated trees ................................................................. 14 vi  2.2.1 Two-dimensional urban geometry ............................................................................ 17 2.2.2 Incident direct shortwave radiation ........................................................................... 21 2.2.2.1 Interception by buildings .................................................................................. 22 2.2.2.2 Interception by tree foliage ............................................................................... 24 2.2.2.3 Interception by ground ...................................................................................... 25 2.2.2.4 Total incident direct shortwave radiation ......................................................... 25 2.2.3 View factors for longwave and reflected shortwave radiation ................................. 27 2.2.3.1 Ray starting point and direction ........................................................................ 27 2.2.3.2 Ray-environment interaction ............................................................................ 29 2.2.3.3 Diffuse exchange between patches ................................................................... 31 2.2.4 Infinite reflections and total patch absorption .......................................................... 32 2.2.4.1 Total incident and absorbed shortwave radiation ............................................. 33 2.2.4.2 Total incident and absorbed longwave radiation .............................................. 34 2.3 Summary ....................................................................................................................... 37 Chapter 3: Testing and application of the multi-layer radiation model .................................38 3.1 Model approximations .................................................................................................. 38 3.2 Model testing ................................................................................................................ 39 3.2.1 System responses—shortwave radiation ................................................................... 40 3.2.1.1 Solar zenith angle .............................................................................................. 40 3.2.1.2 Tree foliage height, density, and clumping ....................................................... 44 3.2.1.3 Partitioning between urban elements ................................................................ 47 3.2.1.4 Photosynthetically-active and near-infrared bands ........................................... 49 3.2.2 System responses—longwave radiation ................................................................... 50 vii  3.2.2.1 Net longwave exchange: vertical distribution................................................... 51 3.2.2.2 Net longwave exchange: ground and canyon fluxes......................................... 52 3.2.2.3 Longwave divergence and canopy air heating rate ........................................... 55 3.3 Summary ....................................................................................................................... 58 Chapter 4: Parameterization of drag and turbulence for urban neighbourhoods with trees60 4.1 Objectives and degrees of freedom ............................................................................... 62 4.2 Numerical models: description and evaluation ............................................................. 64 4.2.1 Three-dimensional RANS k- model and parameterization of foliage effects ......... 66 4.2.2 One-dimensional column k-l model .......................................................................... 70 4.2.3 Dispersive processes ................................................................................................. 75 4.2.4 Comparison of the column model against the CFD model ....................................... 76 4.3 Impacts of tree foliage on flow: important processes ................................................... 77 4.3.1 Intermediate building plan area density .................................................................... 78 4.3.2 Low and high building plan area densities ............................................................... 83 4.3.3 Source and sink terms for new parameterization ...................................................... 85 4.4 Parameterization of tree foliage and building impacts on flow .................................... 87 4.4.1 Parameterization of building impacts: revised Santiago and Martilli (2010) parameters ............................................................................................................................. 88 4.4.2 Testing of urban canopy parameterization of building and tree impacts on flow .... 90 4.4.3 Extension of the parameterization: multiple building heights and clumping ........... 98 4.5 Summary and conclusions ............................................................................................ 99 Chapter 5: BEP-Tree: A multi-layer urban canopy model for neighbourhoods with trees102 5.1 BEP-Tree model design .............................................................................................. 102 viii  5.1.1 Column model for non-neutral urban surface layer ................................................ 103 5.1.2 Temperature and water vapour source terms: built surfaces, tree foliage, ambient air 106 5.1.3 Foliage and built surface energy balances and surface temperatures ..................... 109 5.1.4 Diffuse shortwave radiation .................................................................................... 110 5.2 Model evaluation: Sunset neighbourhood .................................................................. 111 5.2.1 Simulation development ......................................................................................... 111 5.2.2 Model-observation comparison .............................................................................. 116 5.2.3 Sources of model-observation disagreement .......................................................... 124 5.3 Effects of trees on canopy dynamics and climate ....................................................... 126 5.4 Summary and conclusions .......................................................................................... 131 Chapter 6: Summary and conclusions .....................................................................................134 6.1 Multi-layer radiation model for urban neighbourhoods with trees ............................. 134 6.2 Parameterization of drag and turbulence for urban neighbourhoods with trees ......... 136 6.3 Multi-layer urban canopy model for urban neighbourhoods with trees: BEP-Tree ... 138 6.4 Limitations and future work........................................................................................ 141 6.5 Conclusion .................................................................................................................. 144 References ...................................................................................................................................145 Appendices ..................................................................................................................................157 Appendix A Radiation model sensitivity to computational parameters ................................. 157 A.1 Direct solar irradiance: number of rays and ray step size ....................................... 157 A.2 Inter-patch view factors: number of rays ................................................................ 159 A.3 Building wall and foliage layer resolution .............................................................. 162 ix  A.4 Computational parameter recommendations .......................................................... 165 Appendix B Comparison of the 1-D column model with CFD model results ........................ 167 B.1 Intermediate building plan area density .................................................................. 167 B.2 Low and high building plan area densities ............................................................. 170 B.3 Sensitivity to parameter C5 in the CFD ................................................................. 172 Appendix C Evaluation of the CFD model with tree foliage parameterization ...................... 174 C.1 Flow through cubic arrays ...................................................................................... 174 C.2 Continuous forest .................................................................................................... 174         C.2.1 Simulation design .................................................................................................. 175         C.2.2 Results ................................................................................................................... 175 C.3 Flow at the edge of a forest ..................................................................................... 175         C.3.1 Simulation design .................................................................................................. 177         C.3.2 Results ................................................................................................................... 178 C.4 Overall assessment .................................................................................................. 181 C.5 The standard k- parameterization of vegetation canopy turbulence ...................... 181  x  List of Tables  Table 2.1 Inputs for neighbourhood radiation model with tree foliage. ...................................... 16  Table 3.1 Solar angle, and building and tree foliage characteristics for the shortwave scenarios. Solar irradiance is 85% direct and 15% diffuse except for scenario 2a.1. Solar azimuth is perpendicular to the canyon ( = 90°) except for scenario 2b.5. †See Figure 3.1. *Neighbourhood-average LAI multiplied by the clumping coefficient. ‡Effective 2-D solar zenith angle is 45°, and azimuth relative to street direction is  = 25°. .................................................. 42  Table 4.1 Terms investigated and the equations in which they appear. Also included is the naming convention used in subsequent figures and in the text, and a description of each term. Terms 1-3 are sink terms in the momentum equation. Terms 4-7 are source/sink terms in the turbulent kinetic energy equation. Terms 8 and 9 affect the length scales, which directly impact both  and  balances. Impacts of buildings are captured by terms 1, 5, and 9, effects of tree foliage by terms 2, 6, 7, and 8, and ‘interaction’ between buildings and trees by terms 3, 4, and 8. ............................................................................................................................................. 79  Table 4.2  RMSD of  and  between the CFD model and the column model with updated Santiago and Martilli (2010) parameters, as a function of building density (P) and for scenarios without tree foliage. ...................................................................................................................... 90  Table 4.3 RMSD of  and  with the proposed parameterization as a function of building density and tree foliage height for foliage area densities 0.50 m2 m-3 and LD = 0.13 m2 m-3. In brackets: percent change of RMSD from the Santiago and Martilli (2010) building-only parameterization due to the addition of foliage terms 2, 6 and 7 (Table 4.1), i.e. the foliage-related terms of the proposed parameterization. ........................................................................... 93  Table 4.4 RMSD of  and  normalized by vertical mean of  and , respectively, over the appropriate atmospheric layer. All other features are as in Table 4.3. ........................... 94  Table 4.5 Percent change of column model RMSD with addition of impacts of tree foliage on length scales (i.e., term 8 in Table 4.1) in addition to terms 2, 6 and 7; values are in addition to those in Table 4.4. ......................................................................................................................... 95  Table 5.1 Input parameters for multi-layer urban canopy model with trees. ............................. 116 u ku ku ku k u kxi   Table 5.2 Mean Absolute Error (MAE), Mean Bias Error (MBE) and Root Mean Square Error (RMSE) of several variables measured and modeled half-hourly at Vancouver Sunset for 0430 LST May 19 – 0400 LST May 20,  2011, and for 0430 LST July 20 – 0400 LST July 21,  2008 (n = 48 for each simulation). ....................................................................................................... 119  xii  List of Figures  Figure 1.1 Indirect interaction (via the atmospheric model) with the tile approach (a) as compared to direct built-vegetation interaction with integrated urban vegetation modeling (b). UCM = Urban Climate Model. SVAT = Soil-Vegetation-Atmosphere Transfer scheme. ............. 8  Figure 2.1 The order in which computational procedures are effected in the radiation model (numbers), their computational flow (thin grey arrows), numerical methods (red, in brackets) and outputs (yellow), and their frequency of application in simulations using a mesoscale/urban canopy model (left). *See Table 2.1. ............................................................................................ 15  Figure 2.2 Two-dimensional view of the conceptualization of the urban surface that underlies the model geometry: buildings are infinitely long (into the page), of equal width and equally-spaced, and their heights are specified by a height frequency distribution and buildings of different heights are randomly ordered in the horizontal. Foliage layers of different densities are present above and between buildings and at random horizontal locations. An example building density profile and foliage distribution in terms of a leaf area density (LD) profile is shown. ..... 18  Figure 2.3 A portion of the model domain for the example urban neighbourhood with trees illustrated in Fig. 3, highlighting aspects of the two-dimensional model geometry (a). The two possible cases when a ray passes an upstream or downstream building corner in a single ray step (b), e.g. as indicated by “I” and “II” in (a). Depending whether a ray’s effective zenith e is greater than  (ray 2) or less than  (ray 1) different patches intercept the ray. .......................... 19  Figure 3.1 Model geometries and solar zenith angles for shortwave radiation system response tests. “B” = building, “V” = vegetation (tree) foliage, numbers refer to foliage layer, and subscripts refer to building (e.g. “V2b”) or canyon (e.g. “V2c”) columns. Vertical resolution is 1 m for all simulations. Roof, ground and wall albedos are 0.15, 0.15 and 0.25, respectively; foliage reflection and transmission coefficients sum to 0.50. ....................................................... 41  Figure 3.2 Cumulative percent (from top of canopy) of shortwave irradiance absorbed, after infinite reflections, as a function of: (a) solar zenith angle; (b) tree foliage density and clumping; (c) tree foliage height; and (d) shortwave frequency band. Each data point corresponds to the total absorption both above and at the corresponding level (z). Overall neighbourhood albedo () for each scenario appears in the legend. Specifics of the urban configuration and foliage characteristics for each scenario appear in Table 3.1. (a) The ‘diffuse’ simulation assumes all incoming shortwave radiation is diffuse and isotropic. (b) LAI values are neighbourhood averages; LAI = 1.0 corresponds to leaf area density LD = 0.375 m2 m-3. Solar zenith angle is 45° for all but the final scenario. (c) Solar zenith angle is 20°. (d) Solar zenith angle is 45° and xiii  diffuse is 15% of incoming. Foliage reflection and transmission coefficients combine to be 0.50 for broadband, 0.20 for PAR, and 0.80 for NIR. *Effective 2-D solar zenith angle is 45°, actual solar zenith angle is 67.1°, and solar azimuth is 25° from the canyon orientation, i.e.  = 25° (as opposed to 90° for the remainder of the scenarios). ..................................................................... 44  Figure 3.3 Neighbourhood albedo as a function of foliage layer base height for solar zenith angles 20° and 45°. A 12 m wide canyon is simulated with 6 m tall buildings (H/W = 0.5) and a 4 m thick tree foliage layer in the canyon column (LD = 0.375 m2 m-3,  = 0.5). Horizontal lines indicate the albedo without the foliage layer for each scenario. ................................................... 46  Figure 3.4 Partitioning of total incoming shortwave irradiance between different elements of the urban canopy—buildings (roofs, walls), ground, tree foliage, and reflected radiation—for select cases based on scenarios from Figures 3.2a-c. Scenarios in Columns A-D have 50% B1 buildings and 50% B2 buildings, whereas Columns E and F have only B1 buildings (Fig. 3.1). Foliage layers are: V1c (Low), V3c (High), V2c (scenarios E and F). Solar zenith angle is  = 45° for scenarios B-F. ............................................................................................................................... 48  Figure 3.5 Cumulative vertical profile (from top) of total neighbourhood net longwave flux for an H/W = 0.5 canyon with and without a 4 m thick, LAI = 1.0 (LD = 0.375 m2 m-3,  = 0.75) tree foliage layer at different heights in the canyon. All values are means of 20-member ensembles. 52  Figure 3.6 The effect of adding a layer of trees into an urban canyon. Ensemble mean L* on the floor of a range of canyons with different cross-sectional geometries (H/W). The tree layer is spread uniformly across the canyon column, in a layer between 2 - 6 m above the floor. Two foliage cases are considered: (a) a layer of foliage with 0.375 m2 m-3 (LAI = 1.0) and clumping  = 0.75; (b) a layer of 0.750 m2 m-3 (LAI = 2.0) also with  = 0.75. Also, L* of the walls (combined) and the whole canyon (both per m2 plan area of canyon) of non-treed canyons. Error bars for floor values are ensemble standard deviations, whereas those for the whole canyon denote overall uncertainty derived from the standard deviations of individual patches. Individual data points are output by the present model. Black lines are results for canyons with no trees from the TUF-2D model (Krayenhoff and Voogt, 2007). Dotted red line is from the current model for the no-tree case but with the effects of air layers in the canyon included. Lines are linear interpolations between discrete results at H/W = 0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00 and 2.50................................................................................................................................................ 53  Figure 3.7 Ensemble mean heating rate of the 1-2 m canyon air layer (“1.5 m”), and of the complete canyon air column (“canyon”), as a function of canyon height-to-width ratio. Also shown is the impact of adding a LAI = 1.0 foliage layer in the upper half of the canyon (v-u), or immediately above the canyon (v-a), for the H/W = 0.5 scenario. ............................................... 56 xiv   Figure 3.8 Height-normalized profiles of ensemble mean heating rate for select canyon H/W, computed at 1 m intervals. Error bars are ensemble standard deviations. Black diamonds: ten-1 m layers of air (no buildings or foliage); purple squares and light blue circles (o): H/W = 0.5 with foliage in upper half of the canyon, and above the canyon, respectively (“v-u” and “v-a” scenarios from Fig. 3.7). ............................................................................................................... 57  Figure 4.1 The Santiago and Martilli (2010) methodology. Higher fidelity models and measurements are used to inform or evaluate simpler models (grey boxes/brown arrows). The objective is to determine spatial-mean flow profiles (yellow boxes) with a column model informed by fully-parameterized inputs (green box), i.e., an independent urban canopy parameterization. Thick blue arrows indicate model output, thin blue arrows indicate model input. Orange numbers indicate the section in which the model or process (i.e., evaluation or model input) is described or used. ................................................................................................ 61  Figure 4.2 The five tree foliage height scenarios that are simulated, as well as the case without trees (a), for P = 0.25: (b) Tree1 (0 - 8 m); (c) Tree2 (4 - 12 m); (d) Tree3 (8 - 16 m); (e) Tree4 (12 - 20 m); (f) Tree5 (16 - 24 m). Building height H is 16 m and foliage layer thickness is 8 m. Leaf area density varies between the following for each scenario: 0.06, 0.13, 0.25, and 0.50 m2 m-3. Forcing wind is from the left and perpendicular to the building faces (staggered block array)........................................................................................................................................................ 65  Figure 4.3 An example foliage height (Tree2) for the P = 0.11 (a) and P = 0.44 (b) building densities.  All other features are the same as Fig. 4.2. .................................................................. 66  Figure 4.4 Change in column model RMSD compared to the CFD model for spatially-averaged streamwise wind velocity (a) and spatially-averaged turbulent kinetic energy (b) with the removal of each of seven building/tree foliage induced terms (Table 4.1). Median and maximum change in RMSD across the 20 treed scenarios at P = 0.25 are calculated for two atmospheric layers (z = 0 – H, z = H – 2 H). Horizontal lines indicate the actual RMSD, i.e., the RMSD that equates to a 100% increase in RMSD. .......................................................................................... 80  Figure 4.5 Profiles of spatially-averaged mean streamwise velocity (a) and turbulent kinetic energy (b) from the CFD, and from the column model with the cumulative introduction of several foliage-induced terms at building density P = 0.25. All scenarios have foliage density LD = 0.50 m2 m-3, and the Tree3, Tree4 and Tree5 scenarios are presented. The Bdrag simulation includes no foliage-related terms, but only building-related terms 1, 5, and 9 in Table 4.1. Grey shading indicates the building canopy, green shading the tree foliage. ........................................ 82  xv  Figure 4.6 Change in RMSD of the column model relative to the CFD model over z = 0 – 2 H and for spatially-averaged wind velocity (a, c, e) and turbulent kinetic energy (b, d, f), with the removal of each of eight building/tree foliage induced terms/modifications (Table 4.1). Several building plan area fractions (P = 0.00 [Forest], 0.06, 0.11, 0.44), tree layer heights, and foliage densities, are displayed. ................................................................................................................ 84  Figure 4.7 Profiles of spatially-averaged streamwise velocity (a, b) and spatially-averaged turbulent kinetic energy (c, d) from CFD output (symbols) and the column model with the new parameterization (lines) for P = 0.25. Panels (a) and (c) show variation with foliage density for Tree5; panels (b) and (d) show variation with foliage height for LD = 0.25 m2 m-3. .................... 92  Figure 5.1 View from Sunset tower toward the southwest on May 17, 2009 (no images are available for this time period in 2008). Grass and trees are still green. ...................................... 112  Figure 5.2 View from Sunset tower toward the southwest on July 23, 2008. Grass is mostly dry, while trees are still green. ........................................................................................................... 114  Figure 5.3 Profiles of fractional building (i.e., roof) area and foliage (leaf) area as a function of height within 500 m of the Vancouver Sunset tower, as determined by LiDAR. ...................... 115  Figure 5.4 Observed and modeled energy balance at Vancouver Sunset tower, 0300 May 19 – 0500 May 20, 2011. Dotted lines are from the corresponding simulation without trees. ........... 117  Figure 5.5 Observed and modeled energy balance at Vancouver Sunset tower, 0300 July 20 – 0500 July 21, 2008. Dotted lines are from the corresponding simulation without trees. ........... 118  Figure 5.6 Observed and modeled upward radiation fluxes at Vancouver Sunset tower, May 19, 2011. Dotted lines are from the corresponding simulation without trees. .................................. 120  Figure 5.7 Observed and modeled upward radiation fluxes at Vancouver Sunset tower, July 20, 2008. Dotted lines are from the corresponding simulation without trees. .................................. 121  Figure 5.8 Observed and modeled near-surface air temperature (a) and specific humidity (b), and forcing level friction velocity (c), at Vancouver Sunset tower, May 19, 2011. Dotted lines are from the corresdponding simulation without trees. *These are measurement heights. Model heights are 1 m, 1 m, and 26 m, for ,  q, and u*, respectively. .................................................. 122  Figure 5.9 Observed and modeled near-surface air temperature (a) and forcing level friction velocity (b), at Vancouver Sunset tower, July 20, 2008. Dotted lines are from the corresponding xvi  simulation without trees. *These are measurement heights. Model heights are 1 m, and 26 m, for ,  and u*, respectively. ............................................................................................................... 125  Figure 5.10 Modeled energy balance plus upward longwave for “Oakridge” case (lines) vs. simulation without trees (dotted). ............................................................................................... 127  Figure 5.11 Profiles of modeled wind speed, turbulent kinetic energy, and potential temperature at four times, for the “Oakridge” case (lines) vs. simulation without trees (dashes). ................. 128  Figure 5.12 Modeled surface temperatures of built elements for “Oakridge” case (lines) vs. simulation without trees (dashes). R = roofs, Wl = east facing wall, Wr = west-facing wall, G = canyon floor. ............................................................................................................................... 130  Figure 5.13 Modelled effect of trees on 1.0 m Sunset air temperature at 1500 LST July 20, 2008, as a function of foliage clumping applied to radiative vs. dynamic processes (rad and dyn, respectively). ............................................................................................................................... 131  Figure A.1  Maximum over all scenarios of the summed patch-level differences of direct solar irradiance receipt relative to the highly accurate (HA) simulation, as a function of number of rays (ni) and input ray step size (sin). Maximum difference is computed over a suite of simulations with canyon resolution of 10 and varying canyon geometry and solar angle. The log of the maximum computation time over all scenarios relative to the HA simulation is given by the dashed white lines (e.g., “-2” = 1% of the computation time, “-4” = 0.01% of the computation time). 158  Figure A.2  Maximum patch-level view factor difference from the highly accurate (HA) simulation across all patches and all configurations, as a function of number of rays per patch, nk (symbols). The maximum patch difference over an ensemble of 20 simulations is used for each configuration. Also, the maximum patch-level view factor error as a function of number of rays per patch for two configurations (lines). The maximum is over an ensemble of 20 simulations for an H/W = 1.0 canyon with a vertical resolution of one patch and an H/W = 2.0 canyon with resolution 10 patches, both without foliage. ............................................................................... 161  Figure A.3  Shortwave albedo () of a horizontally homogenous layer of foliage as a function of vertical resolution for three different solar zenith angles (). The foliage layer exists from 10-20 m with a leaf area density of 0.30 m2 m-3 and no clumping ( = 1.0). Leaves absorb 50% of intercepted radiation and reflect and transmit the remainder equally. Ground albedo is 0.25. Incident shortwave is assumed to be wholly direct beam. .......................................................... 163  xvii  Figure A.4  Shortwave albedo of a H/W = 2.0 canyon as a function of the vertical resolution of the walls. Three solar zenith angles () with azimuthal angle perpendicular to the canyon axis are modeled. Incident shortwave is assumed to be wholly direct beam. Wall and canyon floor albedos are 0.25. ......................................................................................................................... 165  Figure B.1  RMSD of spatially-averaged streamwise velocity (a), turbulent kinetic energy (b), and Reynolds stress (c) between the column and CFD models for all scenarios with building density P = 0.25. RMSD is for two atmostpheric layers: canopy (0 < z ≤ H) and above-canopy (H < z ≤ 2H). Leaf area density, from left to right for each foliage height, is 0.06 m2 m-3, 0.13 m2 m-3, 0.25 m2 m-3, and 0.50 m2 m-3. The y-axis scale is magnified for . u = 0.45 m s-1. ... 168  Figure B.2  Profiles of spatially-averaged streamwise velocity (a) and turbulent kinetic energy (b) from the CFD (symbols) and column (lines with corresponding colours) models, for foliage height scenarios Tree1, Tree3, and Tree5 (see domain visualizations above plots), each with low and high leaf area densities, as indicated in the legend. Building density is P = 0.25. Results from both models for the foliage-free (No Trees) case are plotted in each panel. ..................... 171  Figure B.3  Profiles of spatially-averaged streamwise velocity (a) and turbulent kinetic energy (b) from the CFD (symbols) and column (lines with corresponding colours) models, for three building densities (P), with foliage density LD = 0.50 m2 m-3 and varying foliage height. The “Forest” scenario has foliage for 0 ≤ z/H ≤ H/2, and no buildings. ............................................ 173  Figure C.1  Comparison of CFD modeled mean horizontal wind speed ( u ), turbulent kinetic (k) and Reynolds stress ( ''wu ) profiles for a ‘continuous forest’ with wind tunnel results from Brunet et al. (1984) and CFD results from Foudhil et al. (2005). Turbulent kinetic energy is normalized using the free stream friction velocity (u*). h is tree foliage canopy height. ........... 176  Figure C.2  Side view (i.e. x-z plane) of the forest-clearing model configuration in the wind tunnel (Raupach et al. 1987) and in the CFD. Green areas indicate modeled foliage, and vertical dashed lines indicate locations of profile comparisons. h is tree foliage canopy height. ........... 177  Figure C.3  Comparison of modeled wind profiles upwind and downwind of the upwind edge of the forest (x / h = 0.0) against wind tunnel results from Raupach et al. (1987) for three values of the parameter C5. Also included are k- CFD results from Foudhil et al. (2005) and LES results from Dupont and Brunet (2008) for the same case. The dashed red line indicates the C5 = 1.26 wind profile at x / h = -8.5 (i.e., approximately the center of upwind clearing). Wind speeds are normalized by u, the streamwise velocity at z / h = 2.0 and x / h = -8.5, for each case. ............. 179 ''wuxviii   Figure C.4  Comparison of modeled turbulent kinetic energy profiles upwind and downwind of the upwind edge of the forest (x / h = 0.0) with wind tunnel* results from Raupach et al. (1987), for three values of the parameter C5. Also included are k- results from Foudhil et al. (2005) and LES results from Dupont and Brunet (2008)* for the same case. Turbulent kinetic energy k is normalized by the square of the wind speed at z / h = 2.0 and x / h = -8.5 (i.e., u2) for each case. * Turbulent kinetic energy (k) is calculated as  2225.0 wvuk   , assuming v = w. ................ 180  xix  List of Symbols  Description        Symbol  Roman  Area of urban surface element P at level/layer iz relative to total AP(iz) horizontal area Sectional building density (m2 area facing wind per m3 outdoor BD air) Volumetric heat capacity of roofs, walls, roads   CR, CW, CG Sectional drag coefficient for buildings    CDB Sectional drag coefficient for tree foliage    CDV Coefficient for the destruction of turbulent dissipation rate () C5 Heat capacity of dry air      cP Displacement height       d Vapour pressure deficit      D Characteristic leaf dimension      dl Emitted or scattered radiation flux density from level iz of  DP(iz) urban surface element P Emitted or scattered radiation flux density incident on level jz of DPQ(iz,jz) urban surface element Q and originating from level iz of urban surface element P Vapour pressure       ea Saturation vapour pressure      es Ratio of three-dimensional to two-dimensional ray step size for fi ray i Strength of ray i attenuated by the ground (road) during ray step j Gi,j Acceleration due to gravity      g Conductance across leaf boundary-layer for heat (one side)  gHa Radiative conductance for leaves     gr Average surface conductance of leaves for humidity (both sides) gv Conductance across leaf boundary-layer for humidity (one side) gva Stomatal conductance of leaf      gvs Height of a uniform tree foliage canopy    h Height of a uniform building canopy/Mean building height  H Height-to-width ratio of a canyon     H/W Ray index (direct shortwave)      i Layer/level index       iz xx  Ray step index       j Layer/level index       jz Ray index (radiation model: view factors)    k Turbulent kinetic energy (column & combined models)  k Thermal conductivity of roofs, walls, roads    kR, kW, kG Shortwave upwelling from the surface    K Extinction coefficient for vegetation foliage    Kbs Incident diffuse solar irradiance on an unobstructed horizontal  surface element Incident direct solar irradiance on an unobstructed horizontal  surface element Turbulent diffusion coefficient/Turbulent viscosity   Km Total solar irradiance on surface element P at level iz after  KP(iz) reflections Direct solar irradiance on surface element P at level iz   Net longwave flux density of a (combination of) surface elements L* Longwave upwelling from the surface    L Leaf area index (neighbourhood-average)    LAI Leaf area density (in layer iz of canyon, building column)  LD, LDC (iz), LDB (iz) Length scale for turbulent dissipation    l Length scale for turbulent diffusion     lk Total incident longwave radiation flux density on surface  LP(iz) element P at level iz after reflections Incident diffuse longwave radiation flux density on an   unobstructed horizontal surface element Urban stability length scale      Lurb Number of rays (direct shortwave)     ni  Maximum number of ray steps (over all ni or nk)   nj Number of rays (view factors)     nk Number of highest layer containing building or foliage  nz Strength of ray i intercepted by surface element P at level iz Pi,j(iz) during step j Atmospheric pressure       p Generic variables representing surface elements R (roof),  P, Q Wl (left wall), Wr (right wall), Vc (canyon column foliage), Vb (buillding column foliage), G (ground), and S (sky) Turbulent (effective) Prandtl number     Pr Specific humidity       q Net radiation flux density      Q* difK dirK)(izK dirPskyLxxi  Turbulent latent heat flux density     QE Conduction heat flux density      QG Turbulent sensible heat flux density     QH Strength of ray i attenuated by roof at level iz during ray step j Ri,j(iz) Strength of ray i at step j      ri,j Actual ray step size       s Strength of ray k escaping to the sky during ray step j  Sk,j Input ray step size       sin Source/sink terms in the column model for k, , q   sk, s, sq  Source/sink terms in the CFD model for u, k,    Su, Sk, S Temperature of surface element P at level iz    TP(iz) Wind velocity in x-direction, or streamwise wind velocity  u (also ui) Wind speed (three-dimensional)     U Friction velocity       u* Scaling velocity based on horizontal pressure gradient   u Strength of ray i attenuated by tree foliage in layer iz and in  Vci,j(iz), Vbi,j(iz) canyon, building column during ray step j Wind velocity in y-direction      v (also uj) Wind velocity in z-direction      w (also uk) Strength of ray i attenuated by left, right wall at layer iz during  Wli,j(iz), Wri,j(iz) ray step j Widths of the canyon and buildings columns, respectively  xc, xb Thicknesses of roof, wall and road layers    xR, xW, xG Vertical layer thickness      z Height in the model domain      z Roughness lengths for roofs and roads    z0mR, z0mG Height of top of highest layer containing building or foliage zmax  Greek  Neighbourhood-scale albedo       Reflection coefficient over a specific wavelength band    Albedo of surface element or patch P    P Discrete variable (= 0 or 1) to select which canyon wall is sunlit  The angle between the solar azimuth and the street direction  Viscous dissipation rate of turbulent kinetic energy    Emissivity of surface element or patch P    P Azimuth angle of ray k relative to its emitting patch   sfck  Probability of a building roof level iz       (iz) xxii  Effective probability of a building roof level iz   e (iz) Probability of wall patch iz, or of building height ≥ iz  (iz) Ratio of wall area facing ambient wind to plan area   F Plan area fraction of buildings (roofs)     P Latent heat of vaporization for water      Viscosity of air        Fraction of volume that is outdoor air (not buildings)  L Potential temperature (combined model)     Solar zenith angle (radiation model)      Effective two-dimensional solar zenith angle   e Zenith angle of ray k relative to its emitting patch   sfck  Reference potential temperature     o Density of air         Neighbourhood-scale clumping coefficient for tree foliage  Random variables for ray k       k k Actual view factor from level iz of surface element P to level jz PQ(iz,jz) of surface element Q Area-weighted view factor from level iz of surface element P to PQ(iz,jz) level jz of surface element Q    xxiii  Acknowledgements  I am thoroughly indebted to my co-supervisor, Professor Tim Oke. His mentorship and support over the course of my doctoral work have been invaluable and without peer. I am grateful for his exceptional generosity, including the numerous opportunities he afforded me to engage with the international scientific community, in the form of conferences, workshops, and visits to colleagues. His tremendous passion for the field of urban climatology, for high-quality research and communication, and his broad-spectrum understanding and insight, have taught me a remarkable amount and spurred me on at every stage. I was most fortunate to have Professor Andreas Christen join as my co-supervisor shortly after I arrived at UBC. His creative genius consistently propelled my research in new and exciting directions, and his attention to detail and rigour improved the dissertation immensely. Moreover, his partnership in our work together created a genuinely enjoyable and challenging scientific engagement, and it was ultimately the perfect mix of guidance and space to explore. I am grateful for his unerring support over the years. Committee member Dr. Alberto Martilli (CIEMAT, Spain) quickly became, in many respects, a third supervisor. His clarity and expertise expedited my research considerably, and rendered several portions of it feasible. I learned a tremendous amount from his thorough understanding of the physics of the atmosphere, and grew considerably as a scientist with his mentorship. I was very fortunate to spend five months, spread across three visits, with Dr. Martilli at CIEMAT in Madrid, Spain, to complete various components of the dissertation. I am grateful for his generosity, hospitality, and friendship. I was fortunate to have Professors Douw Steyn (Atmospheric Science, UBC) and James Voogt (Geography, Western University) as the final two members of my committee. The breadth of their expertise and guidance at key moments significantly improved the dissertation. Dr. Jose-Luis Santiago (CIEMAT, Spain) very kindly spent many days performing dozens of CFD simulations and made the drag and turbulence research presented in Ch. 4 possible, for xxiv  which I am indebted to him. I am also grateful to Andres Simon (CIEMAT, Spain), who coupled two of the models that form the basis of BEP-Tree, and whose parallel stream of research informed and enhanced mine in many ways. Many others at CIEMAT were exceptionally helpful or otherwise helped make my stays enjoyable; in particular, Fernando, Vittoria, and Lorenzo. Many other academics and staff have been of tremendous assistance. To name but a few: Professors Phil Austin, Andy Black, Jing Chen, and Ian McKendry; Dr. Fred Meier; Sandy Lapsky, Vincent Kujala, Eric Leinberger, Al Teng. I am deeply grateful for the friendship, intellectual engagement, and injections of levity from so many exceptional colleagues at UBC Geography, in particular Ben, Jason, Iain, Nick, and Markus. I give heartfelt thanks to my family, for their continuous and unquestioning support and love during this long journey, and to my ancestors, for all they have gifted me. Numerous friends have been exceptionally supportive and helpful, in particular during the final stages of this work: Karen, Ana Elia, Jen, Daylen, Eileen, Danielle, Steph, Steve, Daniel, Cammie. This research was made possible by a Doctoral Fellowship (Discovery Grant to Professor Oke), a Canada Graduate Scholarship, and additional financial support (Discovery Grant to Professor Christen), from the Natural Sciences and Engineering Research Council of Canada (NSERC). Support was also provided by UBC in the form of Pacific Century Graduate Scholarships, a Li Tze Fong Memorial Fellowship, a Four Year Fellowship, a PhD Tuition Award, and an International Research Mobility Award. Much of Ch. 4 was completed at CIEMAT, Madrid, Spain, and was partially-funded by Spanish Ministry for Economy and Competitiveness Project CGL2011-26173 (granted to Drs. Alberto Martilli and Jose-Luis Santiago).1  Chapter 1: Introduction  Global climate is changing and the world is urbanizing: 54% of the world’s population now lives in urban areas (United Nations, 2014). How cities are designed impacts their immediate meteorology and climate, which in turn affects how much energy they use and how comfortable and healthy their residents are. This dissertation presents development and evaluation of an urban atmospheric modeling tool to predict the impacts of urban design on local climate, meteorology and dispersion. The best-known urban effect on local climate is the urban heat island (UHI), the characteristic warmth of cities relative to the surrounding countryside. In many regions the UHI exacerbates air pollution, energy use (for cooling), and heat-induced discomfort and mortality. One of the most effective antidotes to the UHI, and to urban heat more generally, is addition of vegetation to urban landscapes. Vegetated areas alter the surface energy balance and moderate local climate. Relative to shorter vegetation, urban trees have unique effects on local climate and dispersion; due to their size, they provide shade and shelter, interacting with buildings and streets to alter local climate and wind flow. The urban atmospheric modeling tool developed here is the first to include urban trees and their interactions with the built form in a physically-robust, process-based manner. Next, relevant theory underlying development of this model is briefly reviewed.  1.1 Micrometeorology of canopies: relevant theory Scientific understanding of a phenomenon typically proceeds from description of its qualities, to quantification (measurement) of key variables and determination of statistical relations between them, to development of theoretical understandings which underpin process-based model development and predictive capabilities (e.g., Oke, 1982). While all stages still coexist, urban climate research has fully entered the predictive stage over the past two decades, largely due to development of theory related to canopy micrometeorology (Oke, 2006). What follows is a brief review of theory relevant to the two primary realms of investigation in this dissertation: flow and turbulent exchange in canopies, and radiation exchange in canopies.  2  1.1.1 Flow, turbulence, and dispersion The presence and character of a three-dimensional urban canopy radically alters flow and dispersion processes (Britter and Hanna, 2003; Belcher, 2005). Flow in urban areas exhibits significant three-dimensionality up to ≈2 H or higher, where H is the mean building height, a region termed the roughness sublayer (Raupach et al., 1991; Rotach, 1999; Barlow and Coceal, 2009). In other words, microscale advection is the norm due to the extreme heterogeneity (Roth, 2000). Buildings and other obstacles such as trees exert form drag and lift forces, slowing and deviating the flow, and they occupy significant volume in the canopy, displacing flow upwards. They also generate wake-scale turbulence and hence diffusion (Roth, 2000), particularly near roof-level, often forming a ‘shear layer’ for neighbourhoods composed of buildings with flatter roofs and little height variation (Belcher, 2005). The flow in forested canopies exhibits an inflected wind profile, form drag in the canopy layer, wake production of turbulence (though at smaller scales than for buildings due to the prevalence of smaller elements, e.g., leaves), and unlike built obstacles, waving production (Finnigan, 2000). Due to the proximity of the plant elements, sheltering reduces the drag efficiency of individual elements (Thom, 1971). Furthermore, vegetation canopies may be classified as being ‘dense’ relative to ‘sparser’ urban canopies, due to their larger frontal areas relative to the canopy air volume and consequently reduced wind speeds (Belcher 2005). Forest and urban canopies are not amenable to description by standard surface layer theory, e.g. Monin-Obukhov similarity, at least within and immediately above the canopy (e.g. Christen et al., 2009; Finnigan 2000). Standard ‘flat surface’ approaches may only be appropriate if flow within the canopy is not of interest, and then only for flow above the roughness sublayer in the inertial sublayer (or above the canopy, at a minimum). Even for such cases, there is evidence that correct representation of urban boundary-layers requires appropriate accounting of the vertically-distributed effects of the canopy on flow (Uno et al., 1989; Martilli, 2002; Rotach et al., 2004).  1.1.2 Radiation exchange Theoretical approaches to radiation exchange typically differ between urban and forest canopies, primarily due to differing size, distribution and orientation of absorbing and reflecting 3  elements. As previously mentioned, forest canopies are typically denser than urban canopies. Furthermore, built elements tend to be larger and with fewer, more uniform orientations; typically they are represented as interacting elements of a two-dimensional urban canyon or a regular array of three-dimensional buildings: roads, and building walls and roofs (e.g., Masson, 2000; Martilli et al. 2002; Kanda et al. 2005a). Conversely, leaves of tree foliage present a particular set of angular distributions depending on species, typically in a select range of heights above ground and with some degree of clumping (i.e., non-random spatial distribution). Exposure of plant elements to direct shortwave irradiance declines exponentially according to Beer’s Law, depending on leaf area density, angle distribution, clumping, and radiative properties (Campbell and Norman, 1998). In built areas, geometric relations are typically used to determine which surfaces are sunlit, and how much longwave and diffuse shortwave each surface is exposed to (e.g., Masson, 2000; Martilli et al. 2002; Kanda et al. 2005a). In both built and forested canopies, the average height of shortwave absorption is shifted upward from the ground (i.e., the ground receives more shade), and ground sky view is decreased, ‘trapping’ radiation and stabilizing the surface thermal climate. Finally, leaves typically transmit shortwave radiation, whereas only select building materials (e.g., glass) do so, and usually only to the building interior (and therefore out of the purview of most canopy models). Relative to a ‘flat’ land cover, such as a bare field, canopies introduce two complexities: exchange of reflected and emitted radiation between multiple elements, and interception of shortwave and longwave irradiance at multiple levels or spatially-distinct locations (especially direct shortwave). It is typically critical that these processes be represented, and methods for doing so, for both spectrums (longwave and shortwave) and surface covers (built and forested), are discussed in detail in Ch. 2.  1.2 Process-based modeling of urban climate and meteorology Process-based modeling of urban-atmosphere energy exchanges has been underway since the 1960s. However, only in the past 15 years have models capable of parameterizing the micrometeorology in the urban canopy (the airspace below average roof height), and capturing the unique thermal regimes of different urban surfaces, been developed (e.g., Masson, 2000; 4  Kusaka et al. 2001; Martilli et al. 2002). These models, when coupled to mesoscale atmospheric models, allow prediction of urban climates in the context of larger atmospheric processes, as well as inclusion of urban effects on atmospheric phenomena. Simulations of urban meteorology, climate, and pollutant dispersion must account for a broad range of scales, from individual elements at the city surface (<102 m) to transport across, into and out of the complete urban airshed (≈105-106 m). Due to computational limitations, grid resolution of atmospheric (mesoscale) models has an upper limit of O (103 m) (issues of turbulence parameterization aside, e.g., see Wyngaard, 2004). This resolution is too coarse to explicitly compute the flow around individual features such as buildings and trees, e.g., with a Computational Fluid Dynamics (CFD) approach. Moreover, computation of radiation exchange, turbulent heat exchange, and heat storage at individual urban facets is too costly to perform for a city, or even a large neighbourhood. Hence, these features cannot be modeled explicitly, and are ‘subgrid-scale’ processes. Instead, their effects on the mean flow must be parameterized at the neighbourhood, or local, scale (102 – 104 m). Parameterization of surface-atmosphere interaction at the local scale provides surface boundary conditions to atmospheric models. ‘Flat-surface’ representations of urban areas, typified by adjustment of surface roughness, reflectivity and moisture availability in surface energy balance- and roughness-based models, are insufficient for several purposes: prediction of canopy climate and dispersion, most obviously, but also prediction of the above-canopy atmosphere, e.g., wind and boundary-layer formation (Martilli, 2002), as discussed at the end of Sect. 1.1.1. Hence, more complex models that explicitly represent the urban canopy layer are required.  1.2.1 Urban canopy modeling Process-based numerical models of urban canopy meteorology and climate – urban canopy models, or UCMs – have been designed in recent years to predict the time-averaged effects of canopy micrometeorology and to be coupled with mesoscale atmospheric models. Typically they are based on the archetypal ‘urban canyon’ (Nunez and Oke, 1977). According to Masson (2006), UCMs possess several qualities that simple roughness, empirically-based, or modified vegetation exchange schemes do not: explicit building shape, radiative interactions 5  between roads and walls, and distinct energy exchanges for horizontal (road and roof) and vertical (wall) facets via conduction, convection and radiation. They typically account for momentum absorption by the urban surfaces. Some UCMs include latent heat fluxes and basic urban hydrology (e.g., Masson, 2000), but very few integrate vegetation (Lee and Park, 2008; Lemonsu et al. 2012). Urban canopy models can be broadly divided into single-layer (e.g., Masson, 2000; Kusaka et al. 2001; Kanda et al. 2005b; Harman and Belcher, 2006; Oleson et al. 2008) and multi-layer (e.g. Martilli et al. 2002; Dupont et al. 2004; Kondo et al. 2005) models. Single-layer models have only one atmospheric layer in the urban canopy, i.e., between the buildings. This means that they estimate one air temperature, wind speed and humidity, and these values must be assumed representative of the whole urban canopy. Multi-layer models compute meteorological variables for several vertical layers within the canopy which allows for reduced empiricism in the canopy physics, inclusion of building (and tree foliage) height distributions, and more detailed prediction of the street level climate and dispersion; however, multi-layer models are more computationally-demanding and challenging to couple with atmospheric models. Many UCMs use highly-parameterized and empirically-based means of calculating canopy wind profiles, momentum flux and the exchange of scalars (e.g., Masson, 2000; Kusaka et al. 2001), while some add additional theoretical detail such as vertical diffusion of momentum (e.g., Coceal and Belcher, 2004; Kondo et al. 2005) or theory regarding the exchange of scalars (e.g., Harman and Belcher, 2006). What is referred to here as an Urban Canopy Parameterization (UCP) is subsumed in several multi-layer urban canopy models. A UCP typically includes a 1.5 order k-l turbulence closure: coefficients for vertical turbulent exchange are calculated based on turbulent kinetic energy (k), and a prognostic (predictive) equation for k is solved. UCPs are based on the representation of canopy-induced processes in the equations of vertical exchange of momentum (wind) and turbulent kinetic energy, such as drag on the mean flow, and generation and dissipation of turbulent kinetic energy (Martilli et al. 2002; Dupont et al. 2004; Hamdi and Masson, 2008; Santiago and Martilli, 2010). Therefore, UCPs are multi-layer by definition. A UCP for mean flow and turbulence in urban neighbourhoods with trees is developed in Ch. 4. Urban canopy models simplify micrometeorological processes in numerous other ways. Sensible heat exchange from area-limited facets is based on theory developed for extensive, homogeneous 6  grass plains, with notable exceptions (e.g., Kanda et al. 2005b). Roads are not discretized, i.e., they are treated as one facet, and hence their energy exchanges are assumed homogeneous. The same is true for walls in single-layer models. Radiation exchange is frequently limited to just one reflection between urban facets. Diversity of urban form, such as courtyards and alleys, as well as small scale features such as fences, bushes, bins, chimneys and balconies, are typically ignored, as are many effects generated by people and vehicles. Moreover, trees, urban vegetation, and urban hydrology are ignored in most models; instead, soil-vegetation-atmosphere transfer (SVAT) models are run separately but concurrently to represent any non-urban features, i.e. the ‘tile’ approach. Overall, UCMs represent a balance between computational efficiency, and accuracy and completeness. They operate at a scale and with a complexity that meets the requirements for surface boundary conditions of mesoscale atmospheric models applied at the scale of an urban airshed, given presently available computational resources. This dissertation focuses on inclusion of the radiative, dynamic, and climatic impacts of urban trees in UCMs, as a key advance beyond the ‘tile’ approach. Hence, their importance, their relation to the built form, and their inclusion in UCMs, are considered next.  1.3 Modeling urban vegetation and the urban forest Vegetation is common in cities worldwide, and its inclusion in models is critical for proper simulation of the neighbourhood-scale energy balance (Grimmond et al. 2011), street-level climate (Shashua-Bar and Hoffman, 2000), and air pollutant dispersion (Gromke and Ruck, 2007; Vos et al. 2013). Urban vegetation may even be critical to prevention of exacerbated urban-rural temperature differences during heat waves (Li and Bou-Zeid, 2013). More broadly, it is an important design tool in urban environmental management (Bowler et al. 2010; Oke, 1989). Soil and vegetation store water and slow its release, thereby moderating climate by buffering the rate of evaporation. Trees in particular also offer shade and shelter to pedestrians and buildings, and modify near-surface turbulent and radiative exchanges. Furthermore, they increase absorption and deposition of pollutants (Litschke and Kuttler, 2008), emit biogenic volatile compounds (a temperature-dependent process; Calfapietra et al. 2013), and affect pollutant dispersion by exerting drag on the flow and reducing exchange between the canopy and above-canopy (Gromke and Ruck, 2009; Vos et al. 2013). 7  Parameterizations of canopy flow, radiation exchange, and energy balance exist for vegetation (forest) stands, and typically pre-date those for urban areas. Simple mixing length models for vegetation canopy flow first originated with Cionco (1965), and these ideas were later extended to urban canopies (Macdonald, 2000; Coceal and Belcher, 2004). CUPID (Norman, 1979) includes an advanced treatment of radiation exchange in the canopy with a first order turbulence closure. While such first-order models have limitations, they are simple and useful for a variety of purposes (Katul and Albertson, 1998). Katul et al. (2004) find that 1.5 order k-l models perform as well as second order closure schemes across multiple vegetation canopy datasets, and they require fewer closure constants and associated degrees of freedom. In their words, such models are “attractive for linking the biosphere to the atmosphere in large-scale atmospheric models or multi-layer soil-vegetation-atmosphere transfer schemes within heterogeneous landscapes.” Higher-order models have fewer limitations, but are more computationally expensive, and have greater requirements for closure coefficients and hence more degrees of freedom. Pyles et al. (2000), for example, include a third order turbulence parameterization and the radiative and thermal effects of the canopy and the surface. According to Chen et al. (2012), a primary challenge in numerical modeling of the urban climate and energy balance is the representation of vegetation-related processes in urban canopy models, as opposed to including them using a ‘tile’ scheme such that they may interact only indirectly, via an atmospheric model (e.g., Lemonsu et al. 2004, Kawai et al. 2009). With a tile approach, vegetation-building interactions are not included (Fig. 1.1).  In essence, the direct interactions between vegetation and the ‘built’ fabric (e.g., buildings, streets) in cities must be better understood and modeled. These interactions are more complex, and are expected to be more significant, for trees than for shorter vegetation. To date, two process-based UCMs have integrated explicit building-vegetation interaction. Lemonsu et al. (2012) integrate low vegetation (e.g., grass) into a multi-layer version of the Town Energy Balance model (Hamdi and Masson, 2008), incorporating building-vegetation radiative interaction, and demonstrate improved simulation of canyon microclimate. An urban canyon model developed by Lee and Park (2008) and Lee (2011) also includes the effects of trees; however, it is a highly-parameterized single-layer model that permits tree foliage only in the canyon space. Like Lee and Park (2008), Dupont et al. (2004) include flow effects of 8  both buildings and trees and their absolute effects on each other; however, trees and buildings do not interact radiatively in a rigorous fashion. At this point a more thorough discussion of building-tree interaction is warranted.      Figure 1.1 Indirect interaction (via the atmospheric model) with the tile approach (a) as compared to direct built-vegetation interaction with integrated urban vegetation modeling (b). UCM = Urban Climate Model. SVAT = Soil-Vegetation-Atmosphere Transfer scheme.   9  1.3.1 Building-tree interaction Trees are expected to interact with buildings primarily in terms of radiation exchange (e.g., shading) and flow dynamics (e.g., sheltering). Buildings shade trees and other buildings, and trees shade buildings and other trees. Diffuse radiation is exchanged between buildings, between and within trees, and between buildings and trees, enhancing ‘radiation trapping’. Trees and buildings shelter each other and both modify their shared turbulent and thermal environments; however, the neighbourhood configurations and scales of analysis for which these effects are significant remains an open question. The challenge is to represent these varied and complex interactions with an internally consistent numerical model that is relatively simple and computationally undemanding.  Radiative interaction between trees and built elements of the urban surface (roofs, roads, walls) takes several forms. As previously noted, trees shade urban elements from shortwave (solar) irradiance, and vice versa. Trees also block exchanges of longwave (infrared) radiation between urban elements and the sky, warming the urban surfaces with the replacement of exposure to the ‘cool’ sky with ‘warm’ trees. These effects can only be captured with an integrated approach, i.e., one where trees and buildings are interspersed in the same model domain. Furthermore, an ‘infinite’ number of reflections of both shortwave and longwave take place between all urban elements, including trees, on a continual basis. Hence, trees are continuously coupled with all other urban elements within line of sight visibility. Building-tree interaction in the context of urban canopy flow may be conceptualized in two ways. Trees slow the wind and hence reduce the absolute drag force exerted by buildings, and this effect simply requires the inclusion of drag terms for both buildings and trees in the solution of the canopy momentum balance (i.e., an integrated approach instead of a tile approach). Trees also affect the relative impact of buildings on the flow: the efficiency with which buildings remove momentum from the flow, i.e., the building drag coefficient. The importance of integration of urban trees in urban canopy models, and of building-tree interaction, is assessed in subsequent chapters.  10  1.4 Overall research objectives Results from the International Urban Energy Balance Models Comparison Project (Grimmond et al. 2010, 2011) indicate that inclusion of vegetation in urban climate models is critical to accurately model the neighbourhood-scale energy balance. Heat stress, building energy and air pollution dispersion outcomes are likely even more strongly impacted by urban trees. Effects of trees on these canopy and building scale outcomes can only be assessed with an integrated model, such that trees and built surfaces coexist in the same model domain; a tile approach is unable to represent many of the impacts of trees on their urban surrounds. Hence, the objectives of this research are:  1. To create a numerical model for the impacts of urban trees on neighbourhood scale radiation exchange that: a. is multi-layer and sufficiently computationally efficient to combine with an urban canopy model; b. integrates trees and built surfaces and explicitly accounts for building-tree interaction (Ch. 2); c. can be used to assess radiative impacts of urban canyon designs with tree foliage above or between buildings, and those without tree foliage (Ch. 3); 2. To create a multi-layer model for the impact of urban trees on neighbourhood-average wind and turbulence that integrates trees and built surfaces and explicitly accounts for building-tree interaction (Ch. 4); 3. To combine both models with a multi-layer urban canopy model and evaluate the combined model’s ability to simulate the effects of trees on flow, energy balance and climate in an urban neighbourhood (Ch. 5).  A multi-layer UCM is chosen in objective 3 for two reasons. First, multi-layer UCMs are flexible in terms of the number of atmospheric and building (and tree) layers modeled. Second, due to this flexibility of model domain, model physics must be scalable, and hence more process-based and less empirically-based. Overall, multi-layer canopy models are well-equipped to represent vertical distributions of tree foliage (e.g., Pyles et al. 2000) and built elements (e.g., 11  Martilli et al. 2002) and their combined impacts on canopy-layer climate and dispersion. Until now, no model has been capable of both. The present contribution develops a multi-layer urban canopy model capable of predicting the local-scale effects of trees and built surfaces on canopy and above-canopy energy exchanges, climate, and flow (dispersion). The new model consists of new multi-layer exchange models for shortwave and longwave radiation exchange, and momentum drag and turbulence generation and dissipation, coupled to the BEP multi-layer urban canopy model (Martilli et al., 2002). The new UCM, BEP-Tree, explicitly includes building-tree interaction and retains significant flexibility in terms of the layout of building and tree foliage elements. Furthermore, it represents a significant advance beyond ‘tile’ approaches to the inclusion of vegetation, and contributes to the full inclusion of vegetation in multi-layer urban canopy models. 12  Chapter 2: Multi-layer radiation model for urban neighbourhoods with trees  Trees and buildings exchange radiation in the shortwave and longwave spectral bands. Buildings shade trees and other buildings, and trees shade buildings and other trees. Diffuse radiation is exchanged between buildings, between and within trees, and between buildings and trees, enhancing ‘radiation trapping’. This chapter develops a numerical model to represent these varied and complex interactions for both shortwave and longwave radiation. This new radiation model forms the radiation portion of the BEP-Tree urban canopy model in Ch. 5. The new model is relatively simple, computationally efficient, and internally consistent. Its geometry is based on the urban canyon, which it represents with multiple layers in the vertical. Building-tree interaction is represented explicitly, and significant flexibility is retained in terms of the layout of building and tree foliage elements. The model formulation is presented in Sect. 2.2.  2.1 Radiation exchange modeling This section provides a brief background of the methods applied in Sect. 2.2 to model radiation exchange: ray tracing, Monte Carlo simulation and view factor computation.  2.1.1 Ray tracing and radiosity approaches  Ray tracing is a common technique in complex radiative transfer modeling where the action of individual bundles of energy (i.e., rays) are assessed separately rather than solving for the overall radiation exchange concomitantly (Siegel and Howell, 2002). Radiosity solutions, by contrast, assume Lambertian emission/reflection and construct a matrix of view factors (or shape or configuration factors), each of which represents the fraction of radiation emitted or reflected from one surface patch that is incident on another. However, as scenario complexity increases computational loads increase rapidly and view factor determination often becomes unachievable. With ray tracing, relations describing the interaction of radiation bundles with the environment may be applied to geometries of arbitrary complexity, and with only modest increases of computational cost. Another advantage of ray tracing is that individual ray paths are explicit, and 13  hence radiation interaction with optically active (e.g., translucent) media and non-Lambertian surfaces is possible. To date, UCMs do not use ray tracing, and hence they are obliged to substantially simplify the geometries of real neighbourhoods (e.g., infinite street length, constant street width) so that view factors can be determined analytically. Conversely, microscale urban models have typically used ray tracing due to their more complex surface representation, often in combination with view factors (Krayenhoff and Voogt, 2007; Asawa et al. 2008).  2.1.2 Monte-Carlo ray tracing for view factor determination The term ‘Monte Carlo’ refers to a grouping of stochastic methods that make use of repeated random sampling to estimate certain average characteristics of a process. They require a minimum of assumptions regarding the physics of the processes of interest and are not limited by the number of coupled degrees of freedom (i.e., problem complexity) in the same ways as analytical or other numerical approaches. In fact, increasing problem complexity leads to similar increases in complexity of a Monte Carlo simulation, whereas conventional approaches increase in complexity much more rapidly (Siegel and Howell, 2002). However, Monte Carlo approaches are computationally demanding as a rule. Indeed, Monte Carlo approaches are most advantageous when complex scenarios are being considered. As the number of samples of a process (e.g., directional ray emission) increases, convergence toward a solution is approached. The challenge is to strike a balance between a sufficiently accurate solution for the application and the number of samples required to achieve this (i.e., computation time). Monte Carlo ray tracing (MCRT) refers to the tracking of radiation bundles through an arrangement of objects, where the start position, direction of emission, and interaction with objects (e.g. absorption and [direction of] scattering) are subject to random sampling from probability density functions that characterize the behaviour of each component. The benefits of MCRT include its robustness, simplicity, and flexibility of implementation. Computational requirements are substantial; however, increasingly fast and cheap computer power is mitigating this drawback. Furthermore, MCRT is ideally suited to parallelization, that is, to the division of a computational task into smaller tasks which can be undertaken concurrently with different processors. MCRT has previously been used in several urban radiation modeling studies (Aida and Gotoh, 1982; Kobayashi and Takamura, 1994; Montavez et al. 2000; Kondo et al. 2001). 14   MCRT has been adapted to compute view factors between objects. A large number of rays are sent in random directions (or random-weighted directions, according to the appropriate probability density function) from random positions on an object and the first object to intercept each ray is recorded. The view factor from one object to another is then simply the fraction of rays leaving the first object that is intercepted by the second. Accurate estimation of view factors demands careful selection of ray numbers, ray start locations and ray departure angles (Walker et al. 2010). Effectively, computationally-demanding MCRT is used only once to determine the view factors for a given environment. Thereafter, an efficient matrix solution makes use of these stored view factors to compute radiation exchange between elements of the environment.  2.2 Urban radiation model with integrated trees  The current model builds on the two-dimensional multi-layer ‘canyon’ geometry of Martilli et al. (2002). It largely reformulates the model physics and computational methods in order to incorporate several processes, most significantly the radiative interactions and effects of tree foliage. The new model also addresses the weaknesses in the Martilli et al. (2002) formulation identified by Schubert et al. (2012); it incorporates sky-derived diffuse shortwave radiation, permits full radiative interaction of lower roofs with canyon (and now foliage) elements, and fully accounts for the radiative effects of fractional building coverage at each height (i.e., it conserves energy). Additionally, absorption and re-emission of longwave by constituents of ambient air, primarily water vapour, is also included. As implied in Sect. 2.1, a range of computational methods are exploited in order to permit urban configurations of variable complexity while optimizing both accuracy and computation time. Ray tracing tracks direct shortwave radiation as it descends through the domain, impinging on different elements of the urban system (Sect. 2.2.2). A Monte Carlo ray tracing implementation (Sect. 2.2.3) computes view factors between building and tree foliage elements for diffuse shortwave and longwave reflection and emission (these calculations are performed only once at the beginning of a model run, and may be stored for future use). A system of linear equations is then solved to model an ‘infinite’ number of reflections between these elements using the stored view factors (Sect. 2.2.4); as such, reflections are not computed explicitly via ray tracing. Hence, MRCT-determined view factors determine the exchange between all tree 15  foliage and built elements that are visible to each other, including interception and scattering of diffuse and reflected shortwave radiation, and of emitted and scattered longwave radiation. Fig. 2.1 outlines the different model components and their computational methods, as well as their outputs and frequency of application. Table 2.1 lists the inputs required to run the model; despite its flexibility the model requires relatively few input parameters.      Figure 2.1 The order in which computational procedures are effected in the radiation model (numbers), their computational flow (thin grey arrows), numerical methods (red, in brackets) and outputs (yellow), and their frequency of application in simulations using a mesoscale/urban canopy model (left). *See Table 2.1.   16  Description Symbol Default (range) Units Computational parameters Number of rays (view factors) nk 10,000 - Number of rays (direct shortwave) ni 1000 - Input ray step size sin 0.1 - Neighbourhood geometry Vertical layer thickness z - m Width of buildings xb - m Width of canyons xc - m Heights of buildings - - m Probability of each building height (iz) (0.0 – 1.0) - Heights with tree foliage - - m Leaf area density at each height in the canyon & building columns LDC (iz), LDB (iz) - m2 m-3 Foliage clumping coefficient  (0.0 – 1.0) - Radiative parameters Albedo of roofs R (0.0 – 1.0) - Albedo of ground (street) G (0.0 – 1.0) - Albedo of walls W (0.0 – 1.0) - Sum of leaf reflection + transmission coefficients V (0.0 – 1.0) - Emissivity of roofs R (0.0 – 1.0) - Emissivity of ground (street) G (0.0 – 1.0) - Emissivity of walls W (0.0 – 1.0) - Emissivity of leaves V (0.0 – 1.0) - Radiation environment Solar zenith angle  (0 – 180) ° Solar azimuth angle  (0 – 360) ° Street direction - (0 – 360) ° Direct shortwave irradiance Kdir - W m-2 Diffuse shortwave irradiance Kdif - W m-2 Longwave irradiance from the sky Lsky - W m-2 Surface thermal state Roof surface temperature TR(iz) - K Ground surface temperature TG - K Wall surface temperature TW(iz) - K Leaf surface temperature in the canyon & building column TVc(iz), TVb(iz) - K  Table 2.1 Inputs for neighbourhood radiation model with tree foliage.  While the radiation model operates in two dimensions, it accounts for three-dimensionality in two ways. Firstly, the actual three-dimensional path of each ray is selected by Monte-Carlo simulation and subsequently mapped onto the two-dimensional model domain. 17  Secondly, the ratio of three-dimensional to two-dimensional path length of each ray is accounted for during travel through layers containing foliage and/or “air” (e.g., see Sects. 2.2.2.2, 2.2.3.2 and 3.2.1.2). The Bouguer-Lambert-Beer law is used to model radiation interception by layers of foliage (Campbell and Norman, 1998; Sinoquet et al. 2001) and air. All elements are assumed to be Lambertian and hence emit and reflect radiation diffusely. The model is designed for both shortwave (≈0.4-3.0 m) and longwave (≈3.0-100.0 m) radiation wavelength bands. For the purposes of the present model, all wavelengths (or wavelength ranges) in the shortwave spectrum exhibit the same basic behaviour, and the same is true for longwave. An obvious example of the spectral difference between shortwave and longwave is that the former frequently has a direct beam component, but longwave does not. Furthermore, longwave radiation is absorbed by canopy air constituents and emitted by all non-sky elements in the model, whereas shortwave radiation is not. Most of the subsequent discussion assumes that shortwave and longwave are “broadband,” that is, they encompass the ranges as defined above; however, provided wavelength-specific reflection and transmission coefficients the model can be applied to scenarios with narrower bands (e.g., see Sect. 3.2.1.4). Detailed model geometry is presented in the next section, followed by ray tracing of direct solar radiation in Sect. 2.2.2, view factor determination by Monte Carlo ray tracing in Sect. 2.2.3, and finally the solution of infinite reflections by matrix inversion in Sect. 2.2.4.  2.2.1 Two-dimensional urban geometry Urban areas are conceptualized in the model as very long urban canyons with equally-spaced buildings of equal width, randomly-ordered and present according to a building height probability distribution (Fig. 2.2). Note that this conceptualization may underestimate overall wall area relative to real cities by ignoring walls that face along-canyon, such as exist when buildings forming one side of a canyon are of different heights or not immediately adjacent. Nevertheless, the current model geometry is then derived from the conceptualization in Fig. 2.2, and consists of layers divided by levels in the vertical, and an infinite succession of alternating ‘building’ and ‘canyon’ columns in the horizontal (Fig. 2.3a). From a ray tracing perspective, this is equivalent to the assumption of periodic boundary conditions at the lateral boundaries. Six types of ‘elements’ interact radiatively in the model: roof, wall, ground (or road, street, canyon 18  floor), tree foliage, ‘air’, and ‘sky’. ‘Element’ is used to refer to a type of facet (e.g. a wall or roof), whereas a ‘patch’ describes an individual unit that interacts radiatively (i.e., a roof at one level, the street floor, one layer of a wall, or one layer of tree foliage or air). Walls, tree foliage and ambient air are divided into layers, whereas roofs exist at one or more levels. There are nz layers, each with identical thickness z, where layer nz contains the highest layer of building and/or foliage. Layer nz+1 contains only air. Ground and ‘sky’ are each present at one level only.     Figure 2.2 Two-dimensional view of the conceptualization of the urban surface that underlies the model geometry: buildings are infinitely long (into the page), of equal width and equally-spaced, and their heights are specified by a height frequency distribution and buildings of different heights are randomly ordered in the horizontal. Foliage layers of different densities are present above and between buildings and at random horizontal locations. An example building density profile and foliage distribution in terms of a leaf area density (LD) profile is shown.    The spatially-averaged building density (probability) at each height in a neighbourhood is represented as a profile of building density (transparency) in the two-dimensional model (as illustrated in Fig. 2.2). For example, consider a building height distribution in a residential area as follows: 50% of buildings are 5.0 m tall, 25% are 7.5 m tall, and the remaining 25% are 10.0 m tall (i.e., Figs. 2.2 and 2.3a, with z = 2.5 m). In this case the building column in the 2-D model will contain walls that intercept 25% of incident radiation above 7.5 m, 50% of incident radiation for 5.0-7.5 m, and 100% of incident radiation below 5.0 m. Meanwhile, roofs at 7.5 m  19   Figure 2.3 A portion of the model domain for the example urban neighbourhood with trees illustrated in Fig. 3, highlighting aspects of the two-dimensional model geometry (a). The two possible cases when a ray passes an upstream or downstream building corner in a single ray step (b), e.g. as indicated by “I” and “II” in (a). Depending whether a ray’s effective zenith e is greater than  (ray 2) or less than  (ray 1) different patches intercept the ray. 20   and at 10.0 m each intercept 25% of incident radiation, while roofs at 5.0 m intercept 50% of incident radiation. Finally, it is important to note that the building column may have any percentage of its ‘roofs’ at a height of z = 0. This permits cases without buildings to be represented, such as forest or forest-clearing scenarios. The adjacent ‘canyon’ column layers contain air. It also may contain tree foliage, according to an input profile of leaf area density in the canyon space, LDC(iz), in m2 of single-sided leaf area per m3, which is also used to approximate the effects of woody parts (see Sect. 2.2.3.1). Likewise, layers in the building column with building fraction less than 1.0 contain air and may contain tree foliage according to a different input profile LDB(iz). One building column and one canyon column together define one ‘urban unit’, and their widths are xb and xc, respectively (Fig. 2.3a). Effectively, the urban unit is the computational domain and the horizontal boundary conditions are cyclic. Tree foliage is treated as a surface ‘patch’ that is divided into small pieces (e.g., leaves) of different orientation angles and dispersed throughout a layer. A spherical leaf angle distribution is chosen because it eliminates directionality and is a reasonable approximation to real plant canopies (Campbell and Norman, 1998). Unlike previous urban vegetation models (e.g., Lee and Park, 2008) tree foliage may be present in the building column above rooftops, or on ‘green roofs’, and in both columns above the tallest buildings, a common occurrence in many lower-density residential neighbourhoods. The ambient air between and above the buildings is considered to contain gases that are transparent to solar radiation but that absorb and emit in the thermal infrared wavelengths. This effect has been postulated as an important cause of the thermal differentiation of near-surface urban climates from their rural counterparts, but only during calm meteorological conditions (e.g., Nunez and Oke, 1976).  During ray tracing, the urban unit is replicated in the horizontal due to the partially transparent nature of the foliage, air, and any building layers that are present with less than 100% probability. Depending on their direction of travel, rays may pass horizontally through many iterations of the cyclic two-column urban unit prior to complete attenuation by elements of the urban surface or ‘escape’ to the sky. Rays incident at each patch are aggregated over all 21  iterations of the urban unit in the whole domain to obtain the total view factor, or flux, for each patch.  2.2.2 Incident direct shortwave radiation  The instantaneous distribution of direct solar irradiance over the model geometry is solved by means of ray tracing. A user-defined number of rays (ni)1, each representing an equivalent bundle of shortwave radiation energy, are evenly-placed over the width of one urban unit at a constant height zmax + 1.5 s cos(e), where s is the actual ray increment in metres, e the effective 2-D solar zenith angle, and zmax the top of the tallest tree layer or the tallest roof level, whichever is higher. The rays proceed down through the domain with step size s until they reach ground level (z < 0) after nj steps, where:  ),,min( bcin xxzss  ,         (2.1)  and sin is the step size input by the user (sensitivity to sin is presented in Appendix A). Their path is given by the effective 2-D solar zenith angle:    max,1 ,)sin()tan(tanmax ee ,        (2.2)  where  is the solar zenith angle,  is the angle between the solar azimuth and the street direction (Martilli et al. 2002, Eq. A12), and a maximum value of  is imposed in order to avoid infinite iterations during periods with low solar elevation:     max1max,100tan zxx cbe,        (2.3)                                                  1 An assessment of the number of rays ni (and the input ray step size sin) required to achieve good results while minimizing computation time is performed in Appendix A. As a general recommendation, accurate results with moderate computation time are achieved for ni = 1000 and sin = 0.1. 22   with the factor of 100 deemed sufficiently large to avoid significant influence on the results for realistic scenarios, but small enough to prevent excessive computation times.  2.2.2.1 Interception by buildings As the rays travel downwards they impinge on wall and roof patches, and travel through foliage layers. The canyon can be oriented in any direction, and hence a canyon of nonspecific orientation is considered throughout (recall that actual 3-D paths of all rays are mapped onto the 2-D domain, depending on canyon orientation). In fact, multiple canyons, each with a unique orientation and geometry, can be modeled in order to faithfully represent a particular urban canopy or neighbourhood. An individual ray may be intercepted by the same surface patch more than once due to the semi-transparency of the building surfaces and foliage layers and the periodicity of the horizontal boundary conditions (hence the necessity of the index j below). When a ray i crosses from the canyon to the building column in layer iz during ray step j, it contributes:  jijiji rizrizWl ,1,, )1()(           (2.4a)   jijiji rizrizWr ,1,, )1(1)(         (2.4b)  to the total ray strength incident on layer iz of the left (Wl) and right (Wr) canyon walls, where  = 1 for <  and  = 0 for >  (since both walls cannot be sunlit at the same time), and(iz+1) is the probability of walls in layer iz, or the probability of having a building with height equal to or greater than level iz+1 (Martilli et al. 2002). The ray in turn loses an equal amount (ri,j) from its previous fractional energy, or strength, ri,j-1 (non-dimensional), leaving it with ray strength:  jijiji rrr ,1,,    .          (2.5)  When a ray i does not cross from the canyon to the building column during ray step j, or if it does so in layer iz but (iz + 1) = 0, we have simply: 23   0)()( ,,,  jijiji rizWrizWl .        (2.6)  When a ray i crosses level iz in the building column, it loses the following ray strength to roof level iz:  jicolumnbuildingejiji rizrizR ,1,, |)()(   ,       (2.7)  where e(iz) is the effective probability of a building roof at level iz:   1)1(,01)1(,)1(1)()(izforizforizizize ,       (2.8)  and (iz) is the actual probability of a building roof at level iz (Martilli et al. 2002), and may take any value ≤1.0 here. This effective building height probability e(iz) allows the model geometry (Fig. 2.3a) to preserve the energy partitioning inherent in the 3-D conceptualization of the urban surface in Fig. 2.2. In effect, Eq. 2.8 accounts for the fact that a ray in the model building column at a height z only exists in those columns in the three-dimensional conceptualization (Fig. 2.2) where building height is less than z, otherwise it would be inside a fully opaque building. When a single ray step crosses both a column boundary and a vertical level with (iz+1) > 0 the ray has passed a building corner at the ‘downstream’ (Fig. 2.3b, I) or ‘upstream’ (Fig. 2.3b, II) edges. To determine whether the ray passes above or below the corner, the ‘zenith’ angle of the line segment joining the ray’s final position to the corner ( in Fig. 2.3b) is compared to the effective solar zenith angle (e). The result determines whether the roof intercepts the ray, and if it is an upstream corner, which wall patch intercepts the ray. With these simple tests a high degree of accuracy is achieved with fewer rays.  24  2.2.2.2 Interception by tree foliage Tree foliage may be present in either the canyon or the building column. In the latter case it is only present to the extent that the building is transparent in that layer (i.e., [1 - (iz+1)]) Tree foliage intercepts rays at each ray step according to the Bouguer-Lambert-Beer law (i.e., the turbid medium model):     jiiDCbsjiji rfsizLKrizVc ,1,, )(exp1)(  ,     (2.9a)    jiiDBbsjiji rfsizLKrizVb ,1,, )(exp1)(  ,     (2.9b)  where “c” and “b” indicate the canyon and building columns, respectively, s is the 2-D ray step size (m), fi is the ratio of 3-D (actual) to 2-D (model) distance travelled by the ray, LDC(iz) and LDB(iz) are the leaf area densities (m2 m-3) at layer iz in the canyon and building columns, respectively (i.e., leaf-area density is a property of the foliage at a given height in a given column, rather than a neighbourhood-average), and Kbs is the foliage extinction coefficient (dimensionless). This is a widely-used model of radiation interception by vegetation foliage (e.g., Sinoquet et al. 2001). is a clumping factor that takes the value 1.0 for random foliage distributions, and smaller for more clumped distributions (Campbell and Norman, 1998); it encompasses the effects of clumping at several scales, e.g. from shoots to crowns to the between-crown scale, and it is a global parameter in that it applies to all foliage in the model.  Clumping is the norm rather than the exception in forests (Kucharik et al. 1999), and it is required in order for Eq. 2.9 to apply to non-random foliage distributions. Urban foliage distributions tend to have greater clumping at larger scales (e.g., crown and larger), even when considering neighbourhood-averages; hence, smaller values of  than are typical for forests are generally required. An example that may have relevance to typical urban forests is an open savannah ecosystem with substantial clumping at crown scales and larger. Ryu et al. (2010) determined the clumping coefficient in such an ecosystem to be 0.49±0.10. Better estimation of urban clumping may be possible based on the work of Chen et al. (2008), who develop an analytic model of the effects on clumping of heterogeneous canopies of tree spacing, tree width, crown depth, solar zenith angle, leaf angle distribution and leaf area density. 25  For vegetation with a spherical leaf angle distribution the foliage extinction coefficient Kbs for beam radiation density travelling a certain distance in the vertical and impinging on a horizontal level, depends on solar zenith angle (Campbell and Norman, 1998):  )cos(21)(  bsK.          (2.10)  This is considered a good first approximation for all types of canopies (Chen et al. 1997), and includes interception by both sides of the leaves. However, Eq. 2.10 is directionally-independent if leaf angles are assumed to be spherically and randomly distributed, and the leaves are approximately randomly distributed in space (i.e.,  ≈ 1.0). For these conditions, the extinction coefficient is constant at Kbs = 0.5 in the direction of radiation (ray) travel.  2.2.2.3 Interception by ground Those rays that reach ground level (z = 0) contribute their remaining strength to the ground, or street:  jijiji rrG ,1,,   .          (2.11)  This is always true in the canyon column. In the building column, however, rays only reach the ground for scenarios where some ‘rooftops’ are at ground level, i.e., (2) < 1.0.  2.2.2.4 Total incident direct shortwave radiation Finally, incident direct shortwave flux density to each surface patch is computed as:  )()()( 1 1,izAizPniKizKPniinjjjidirdirP  ,        (2.12)  26  where is the incoming direct shortwave irradiance on an unobstructed horizontal surface element, and P is replaced by Wl, Wr, R, Vc, and Vb for patches of wall (left and right), roof, and tree foliage in the canyon and building column, respectively, and nj is the maximum number of ray steps taken by any of the ni rays. Incident direct shortwave flux density to the ground (G) is calculated analogously, but only at one level (no iz index). The AP terms are the area of each patch relative to the total horizontal area of the urban unit:  cbWrWl xxizzizAizA  )1()()(,        (2.13a) cbbR xxizxizA  )()( ,          (2.13b) cbcG xxxA ,          (2.13c) cbcDCVc xxxzizLizA  )(2)(,        (2.13d)  cbbDBVb xxizxzizLizA  )1(1)(2)(,       (2.13e)  where the area of tree foliage (AVc or AVb) is taken as the ‘active’ portion of the complete two-sided leaf area, in which the effective leaf area density (e.g., )(izLDC ) is used in place of the actual leaf area density at each level. Hence, the foliage in a given layer interacts radiatively as a single surface patch, but for radiation interception purposes one that is broken into many small pieces (i.e., leaves) of random 3-D orientation. For clumped foliage ( < 1.0) some of these leaves effectively do not emit radiation towards, or intercept radiation from, the surroundings, as they are situated at the interior of ‘foliage clumps’. At this point, direct shortwave radiation receipt prior to reflections is determined. Subsequently, diffuse receipt is computed using view factors, followed by infinite reflections between all surface patches. Furthermore, emission and reflection of longwave radiation is dirK27  calculated based on these same view factors. Hence, view factors between all interacting patches must be known in order to proceed.  2.2.3 View factors for longwave and reflected shortwave radiation  View factors between all surface patches are computed once with Monte Carlo ray tracing (MCRT), and are then used to calculate diffuse radiation exchange for the remainder of a mesoscale or urban canopy model simulation. As a result, the computational expense associated with ray tracing occurs only once at the start of the simulation, or, in principle, occurs only once for a given neighbourhood configuration. In other words, the calculation of view factors by MCRT is part of the pre-processing of the land surface description only, and is not involved in the prognostic portion of the model.  2.2.3.1 Ray starting point and direction A user-defined number of rays nk are ‘emitted’ from each patch and view factors are determined based on the first patch to intercept each ray. Rays begin at a random point along a surface patch, or at a random location in a layer of foliage or air. A ray leaving a solid patch (roof, ground, wall) may take any direction into the hemisphere adjacent to the surface. The zenithal angle of its path is chosen from a weighted random distribution because all surface elements are assumed to be isotropic, or Lambertian, emitters and reflectors:       2 21cos1cossin111 kkksfck ,      (2.14)  where  k is a random number between 0 and 1, and its azimuthal angle is simply:  ksfck  2 ,           (2.15)  where  k is another random number between 0 and 1 (Siegel and Howell, 2002). The weighted distribution of zenith angle (Eq. 2.14) results from the combined effects of increasing solid angle 28  per zenithal increment d [as sin()], and decreasing intensity in W sr-1 [as cos( sfc ), i.e. Lambert’s cosine law], as  increases. Rays leaving a layer of tree foliage or ambient air are assumed isotropic and therefore choose from a full sphere (4 solid angle) with equal probability, and require 2 nk rays to preserve ray density per solid angle. Hence, scattering of shortwave and longwave radiation by leaves is assumed to have no preferred direction. In other words, forward scattering (transmission) and backward scattering (reflection) are assumed equal, a classic simplification for shortwave radiation incident on leaves (Sinoquet and Bonhomme, 1992). In reality, coniferous shoots (assemblages of needles) typically exhibit some enhanced backward scattering relative to forward scattering (Smolander and Stenberg, 2003), and many deciduous leaves exhibit specular reflection and reduced transmission of shortwave at high (grazing) incidence angles. However, it is challenging to account for these phenomena without explicit ray tracing of the reflection and transmission processes. In the deciduous case, there is little effect on the absorption coefficient of leaves at these grazing angles, as decreased transmission is approximately offset by enhanced reflection (Monteith and Unsworth, 2008).  There are two further points regarding the assumption of isotropic scattering from foliage layers. Woody portions of trees do not transmit radiation, which is problematic in terms of the assumption of directionally-uniform scattering. However, they represent the minority of the radiative impacts of trees for evergreens and deciduous leaf-on scenarios. Furthermore, tree leaves are typically positioned so as to preferentially mask (e.g., shade) the woody elements (Kucharik et al. 1998). Deciduous leaf-off scenarios may suffer from larger relative errors, but the absolute impacts of bare trees on radiation exchange tend to be smaller than for leaf-on scenarios. Similarly, while leaves transmit and reflect similar amounts of shortwave irradiance, they do not transmit longwave radiation. Hence, the assumption of isotropic scattering from foliage layers is incorrect for these wavelengths. However, the errors induced by this assumption will be minimal for two reasons: a) reflection coefficients of leaves are small in the longwave spectrum (≈0.01-0.06; Oke, 1987); and b) foliage layers receive comparable intensities of longwave radiation from every direction, unlike shortwave radiation. Fortunately, isotropic emission of longwave radiation from tree foliage is a reasonable assumption. 29  View factors from the sky to each patch (as opposed to sky view factors, which are view factors from each patch to the sky) are required to compute receipt of incoming longwave and diffuse shortwave. A patch’s sky view factor is equal to the area-weighted view factor (i.e., Eq. 2.20) from the sky to the patch, assuming an isotropic sky radiance distribution. Several models take advantage of this fact (e.g. Masson, 2000; Kusaka et al. 2001; Krayenhoff and Voogt 2007). However, this relation does not hold for vegetation or for anisotropic sky radiance distributions. Hence, explicit computation of view factors from the sky to each patch is included. Rays leaving the ‘sky’ begin at a random horizontal location in each of several equal intervals along the width of the urban unit, and at a user-defined height above the tallest roofs or trees (≥ 1.1 [zmax + z]); results are insensitive to both the height and interval width for isotropic sky radiance. For simplicity, each ray selects from a downward-facing hemisphere (2 solid angle) using Eqs. 2.14 and 2.15. This approach is also known as the isotropic model of sky diffuse radiation. It is a reasonably good assumption for longwave radiation with overcast conditions but less so for clear conditions (Monteith and Unsworth, 2008). It is also a significant assumption for shortwave diffuse radiation, particularly for clear conditions. More complex sky radiance distributions could easily be incorporated and are left as future work.  2.2.3.2 Ray-environment interaction Once a starting point and direction are determined for a ray k originating from a patch P, it remains to compute its incidence upon, and dissipation by, other surface patches. Downward travelling rays use the same set of relations as for incident direct shortwave (Sect. 2.2.2; replace i index with k). The only difference is that each ray now has a unique direction of travel. Eq. 2.9 with Kbs = 0.5 remains valid for foliage layers, due to the assumption of a spherical leaf angle distribution. Upward travelling rays cannot impinge on roofs or ground, and hence use a subset of these same relations, namely Eqs. 2.4 and 2.9. When an upward travelling ray k takes a step j such that it exceeds height zmax + s the remainder of its energy is ‘lost’ to the sky:  jkjkjk rrS ,1,,   ,         (2.16)  30  and contributes to the total reflected shortwave radiation (albedo), or to the upward-directed longwave radiation. If air is chosen to be nontransparent to longwave then separate sets of view factors are computed for longwave and shortwave exchange, and for each wavelength band in the longwave spectrum if more than one is modeled. In these cases ‘air’ intercepts rays at each ray step according to the Bouguer-Lambert-Beer law:    jkairabskjkjk rkfsrizA ,1,, exp)(   ,      (2.17)  where s is the 2-D ray step size (m), fk is the ratio of 3-D (actual) to 2-D (model) distance traveled by the ray, abs is the density of absorbing molecules (≈v, the water vapour density, for most cases), and kair is the mass extinction cross section of the absorbing molecules (m2 kg-1). Like vegetation, ‘air’ is only present in layer iz in the building column to the extent that the building at that level is transparent (i.e., [1 - (iz+1)])The effective surface “areas” of absorbing and emitting gas molecules AAc and AAb as a fraction of the total horizontal area of the urban unit are:   cbcairabsAc xxxzkizA  4)(,        (2.18a)   cbbairabsAb xxizxzkizA  )1(14)( ,      (2.18b)  where abs and kair are assumed constant with height. These effective surface “areas” of gas molecules per unit volume (4∙abs∙kair; for multiplication by T4 via the area-weighted view factor [Eq. 2.20]) are a consequence of Kirchoff’s law, and the inclusion of the factor of 4 accounts for the 4π solid angle that is available for emission from each gas molecule. Strictly, this relation is valid for monochromatic radiation absorbed and emitted by a medium in thermal equilibrium (Sparrow and Cess, 1978). There are some limitations to the application of Equations 2.17 and 2.18. View factors are not updated during an urban canopy model simulation, and hence an implicit assumption is 31  that abs does not change significantly. This is a good approximation for slowly varying atmospheric conditions typical of many fair weather situations. Furthermore, the Bouguer-Lambert-Beer law is valid, strictly, for attenuation of monochromatic radiation. Determination of a unique value for kair for broadband exchange is a further challenge, as it varies with composition and density of absorbing molecules, as well as air temperature. This issue is discussed further in Sect 3.2.2.3.Finally, recall that tree foliage is treated as a ‘surface patch’ that is divided into small pieces (i.e., leaves) of random orientation dispersed throughout a layer. All of the leaves in each layer are assumed to have the same properties and states: view factors, surface temperatures, etc. This has important implications for the resolution with which a tree canopy is represented, and is explored further in Appendix A. A property that emerges naturally from ray tracing with this geometry is that each layer of tree foliage, in contrast to other facets in the model, has a ‘self view factor’. That is, a portion of its emitted or scattered radiation is incident on itself. In other words, a ray that starts its journey in the middle of a foliage layer has a certain likelihood of being intercepted by neighbouring leaves in the same layer. The same applies to layers of ambient air, however their rate of radiation extinction tends to be substantially smaller.  2.2.3.3 Diffuse exchange between patches Diffuse exchange (i.e., reflection/scattering and emission) between patches during the simulation is computed as follows:  )(),(),( izDjzizjzizD PPQPQ  ,        (2.19)  where DP(iz) is the emitted or reflected/scattered flux density from the izth level (or patch) of surface element P, PQ(iz,jz) is the area-weighted view factor from the izth level of surface P to the jzth level of surface element Q, and DPQ (iz, jz) is the emitted or reflected flux density incident on the jzth level of surface element Q that originates from the izth level of surface element P. Area weighted view factors are calculated as follows:  32  )()(),(),( jzAizAjzizjzizQPPQPQ ,        (2.20)  where AP and AQ are given by Eqs. 2.13a-e, and PQ (iz, jz) is previously determined from MCRT as the fraction of ray energy leaving the izth level of surface element P that is incident on, and intercepted by, the jzth level of surface element Q:  PoflayerlevelizfromoriginatekraysnkknjjjkPQthjzQnkjziz /1 1, )(1),(  ,    (2.21)  where “P” and “Q” are each one of Wl or Wr, R, Vc or Vb, G, or S, and the Qk,j(jz) are computed as in Eqs. 2.4, 2.7, 2.9, 2.11, and 2.16, respectively. Hence, PQ (iz, jz) is the fraction of ray energy (i.e., emitted/reflected radiation) leaving P(iz) that is incident on Q(jz). It is important to note here that diffuse exchange between tree foliage layers and other urban elements, between different foliage layers, and within a foliage layer, occurs by means of view factors calculated by MCRT (Eq. 2.21). Hence, tree foliage emits longwave radiation (Eqs. 2.25-2.27), and intercepts (and subsequently absorbs or scatters) longwave that is scattered or emitted by other model elements (Eq. 2.9 and Sect. 2.2.3.2), in addition to intercepting, absorbing and scattering direct and diffuse shortwave radiation.  2.2.4 Infinite reflections and total patch absorption  There are a maximum of 2 nz wall patches, nz + 1 roof levels, one ground level, 2 nz foliage layers, and 2 nz + 2 layers of air. Hence, a system of 7 nz + 3 equations and 7 nz + 3 unknowns is solved by matrix inversion to determine both total incident longwave radiation, at each surface patch after infinite reflections. There are only 5 nz + 1 equations and unknowns for shortwave radiation because air layers are assumed transparent at these wavelengths. The following equations apply to a canyon of any orientation.  33  2.2.4.1 Total incident and absorbed shortwave radiation The total shortwave radiation incident on the left wall after infinite reflections is given by:              vegetationcolumnbuildingnzjzVbVVbWvegetationcolumncanyonnzjzVcVVcWstreetfloorGGGWroofnzjzRRRWwalleastnzjzWrWWWskydifSWdirWlWjzKizjzjzKizjzKizjzKizjzjzKizjzKizizKizK,1,1)(1,1,1)(),()(),()()(),()(),()()()(,   (2.22)  where Kdif is diffuse shortwave from the sky, and the P are the albedos of the different patches P. Total shortwave incident on the right wall after infinite reflections is given by switching “l” and “r” and “” with “(1 - )” in Eq. 2.22. The equation for total shortwave radiation incident on canyon column tree foliage layers after reflections is:               vegetationcolumnbuildingnzjzVbVVbVcvegetationcolumncanyonnzjzVcVVcVcstreetfloorGGGVcroofnzjzRRRVcwallseastandwestnzjzWrWlWWVcskydifSVcdirVVcjzKizjzjzKizjzKizjzKizjzjzKjzKizjzKizizKizK,1,1)(1,1,1)(),()(),()()(),()()(),()()()(  (2.23)  The equation for building column tree foliage layers is identical except all instances of “Vc” are swapped for “Vb”, and vice versa. Roofs are all assumed horizontal (flat, non-pitched roofs), and hence do not interact with other roofs or the ground:  34           vegetationcolumnbuildingnzjzVbVVbRvegetationcolumncanyonnzjzVcVVcRwallseastandwestnzjzWrWlWWRskydifSRdirRRjzKizjzjzKizjzjzKjzKizjzKizizKizK,1,1,1)(),()(),()()(),()()()(   (2.24)  Street floor total incident shortwave is determined identically to roofs with the “R” subscript replaced by “G” (i.e., different view factors), and the elimination of the iz index as there is only one ground level. Finally, total absorbed shortwave radiation at each patch is (1-P) KP(iz), where P is the albedo of surface element P, and P is one of Wl, Wr, Vc, Vb, R, or G (both walls have the same albedo, likewise for both sets of foliage layers). All built (flat) surfaces reflect radiation diffusely. However, V for the foliage layers represent the fraction of shortwave that is reflected and transmitted, i.e. radiation incident on leaves is assumed to scatter forward and backward equally (Sect. 2.2.3.1). Strictly, the photosynthetically-active (PAR) and near infrared (NIR) portions of the shortwave spectrum should be modeled separately as their vegetation scattering coefficients differ greatly (Campbell and Norman, 1998). This can be achieved by running the solar reflection component of the model (Eqs. 2.20-2.22) twice, once for PAR and once for NIR (see Sect. 3.2.1.4).  2.2.4.2 Total incident and absorbed longwave radiation All view factors for longwave exchange are identical to their shortwave counterparts unless longwave absorption by ‘air’ is included. Furthermore, view factors from the sky to the urban elements will differ if different sky radiance models are used for the two wavebands. The total longwave radiation incident on the left wall after infinite reflections is given by:  35                          aircolumnbuildingnzjzAAbWaircolumncanyonnzjzAAcWroofnzjzRRRRRWstreetfloorGGGGGWvegetationcolumnbuildingnzjzVbVVbVVbWvegetationcolumncanyonnzjzVcVVcVVcWwalleastnzjzWrWWrWWWskyskySWWljzTizjzjzTizjzjzTjzLizjzTLizjzTjzLizjzjzTjzLizjzjzTjzLizjzLizizL1,141,141,14)(4,14,14,14)(),()(),()()(1),(1)()()(1),()()(1),()()(1),()()(,  (2.25)  where Lsky is longwave irradiance from the sky, and P is emissivity from patch P. Total longwave incident on the right wall after infinite reflections is given by the analogous equation (switch “l” and “r”). No reflection terms for the air layers appear in Eq. 2.25 and subsequent equations as it is assumed that absorbing molecules have longwave emissivities of 1.0. Furthermore, canyon and building column air are assumed to have identical temperature. For tree foliage layers the equation is as follows:  36                            aircolumnbuildingnzjzAAbVcaircolumncanyonnzjzAAcVcroofnzjzRRRRRVcstreetfloorGGGGGVcvegetationcolumnbuildingnzjzVbVVbVVbVcvegetationcolumncanyonnzjzVcVVcVVcVcwallseastandwestnzjzWrWlWWrWlWWVcskyskySVcVcjzTizjzjzTizjzjzTjzLizjzTLizjzTjzLizjzjzTjzLizjzjzTjzTjzLjzLizjzLizizL1,141,141,14)(4,14,14,144)(),()(),()()(1),(1)()()(1),()()(1),()()()()(1),()()(   (2.26)  As for shortwave, the equation for building column tree foliage layers is identical except all instances of “Vc” are swapped for “Vb”, and vice versa. The equation for longwave incident on canyon column air (LAc) is identical to Eq. 2.26, with all instances of “Vc” swapped for “Ac” with the exception of the “canyon column vegetation” term, where only “VcVc” is swapped for “VcAc”. The equation for the building column air LAb is then simply a direct swap of all instances of “Ac” for “Ab”. Again, roofs do not interact with other roofs or the ground:                    aircolumnbuildingnzjzAAbRaircolumncanyonnzjzAAcRvegetationcolumnbuildingnzjzVbVVbVVbRvegetationcolumncanyonnzjzVcVVcVVcRwallseastandwestnzjzWrWlWWrWlWWRskyskySRRjzTizjzjzTizjzjzTjzLizjzjzTjzLizjzjzTjzTjzLjzLizjzLizizL1,141,14,14,14,144)(),()(),()()(1),()()(1),()()()()(1),()()(   (2.27) 37   The equation for street floor total incident longwave is identical to Eq. 2.27 with the “R” subscript replaced by “G” (i.e., different view factors), and the iz index eliminated as there is only one ground level.  Total absorbed longwave radiation at each patch is P LP(iz), where P is one of Wl, Wr, Vc, Vb, R, Ac, Ab, or G (both walls have the same emissivity, likewise for both sets of foliage layers and both air layers [A = 1.0]).   2.3 Summary A multi-layer urban radiation model with trees is developed that explicitly computes building-tree interaction. Furthermore, it does so using standard radiative transfer methods: (Monte Carlo) ray tracing for direct shortwave and view factor determination, and the Bouguer-Lambert-Beer law for attenuation by tree foliage layers. The model is flexible—for trees, any heights, thicknesses, foliage densities and clumping are permitted, and for buildings, any heights and height probability distributions are allowed. The use of ray tracing renders the model less dependent on the complexity of the model geometry, while the initial calculation of inter-element view factors permits a computationally speedier matrix solution to diffuse exchange for the remainder of a mesoscale or urban canopy model simulation. Incident shortwave radiation absorption is computed according to Sect. 2.2.2; view factors for receipt of diffuse shortwave and emitted longwave, and exchange of reflected shortawave and longwave, are determined as described in Sect. 2.2.3; and final absorption of shortwave and longwave by all model elements, and total shortwave and longwave escaping to the atmosphere, is solved in Sect. 2.2.4. This new ‘treed’ multi-layer radiation model is tested in Ch. 3, and it forms part of a new multi-layer urban canopy model for urban neighbourhoods with trees in Ch. 5.     38  Chapter 3: Testing and application of the multi-layer radiation model  In this chapter, spatial distributions of radiation receipt and absorption output by the model proposed in Chapter 2 are assessed as a function of scenario (building geometry, vegetation distribution, solar angle). Model parameters (numbers of rays, ray step size, vertical resolution) are considered in Appendix A. Model approximations are considered first, followed by sensitivity tests of the model for both shortwave and longwave. No observations of radiation profiles in urban canyons with trees exist at present; however, the model’s computation of longwave exchange for urban canyons without trees is compared against a high-resolution radiation code based on different radiation theory (TUF-2D; Krayenhoff and Voogt, 2007).  3.1 Model approximations  The physics of the proposed model are robust given the following assumptions:  1. All surface elements are diffuse (Lambertian) emitters and reflectors;  2. The Bouguer-Lambert-Beer law for radiation transmission with a spherical leaf angle distribution and clumping index is a good approximation for spatially heterogeneous distributions of urban tree foliage;  3. Wide-band (e.g., broadband longwave) radiation interception by gaseous species in ambient air may be approximated by the Bouguer-Lambert-Beer law;  4. Trees scatter both shortwave and longwave bands equally in forward and backward directions (see Sect. 2.2.3.1).  Importantly, there are several assumptions implicit in the 2-D model geometry itself, many of which are common in urban canopy modeling. The proposed model geometry is based on the following assumptions and approximations: 39   1. The combination of two or more two-dimensional ‘infinite’ canyon geometries of appropriate orientation permits the faithful reproduction of the neighbourhood-average radiation distribution (by height and surface element type) of real three-dimensional geometries; a. The effects of small scale structures or variation (e.g., building width variation, pitched roofs, chimneys, balconies, windows, cars, etc.) can be neglected; b. The infinitely long canyon is a good representation of all non-built spaces in an urban canopy (backyards, courtyards, laneways, intersections, etc.);  2. Model resolution in the vertical is essential for the representation of urban canopy processes for most neighbourhoods, and it is more important than model resolution in the horizontal (i.e., further division of the ‘canyon’ and ‘building’ columns);  3. Tree foliage is approximately evenly-distributed across canyon spaces, at the crown-scale, when averaged over neighbourhoods.  3.2 Model testing  The ‘validation’ of the current model over all realistic scenarios is extremely difficult. Moreover, given the highly complex nature of real urban geometries, in particular those that include trees, to the authors’ knowledge appropriate observational datasets are not currently available for evaluation purposes. Therefore, only three types of testing are performed on the model: a) energy conservation tests on all model components (not shown here for brevity’s sake, but have been conducted and the model conserves energy within the bounds of numerical error); b) sensitivity tests to determine model parameters that yield sufficient accuracy while minimizing computation time (Appendix A); and c) system response tests to demonstrate that modeled distributions of radiation exchange for different arrangements of built and foliage elements as a function of solar angle agree with expected neighbourhood-scale responses (Sects. 3.2.1 and 3.2.2).  40  3.2.1 System responses—shortwave radiation  Several scenarios are presented to demonstrate: a) the impacts of tree foliage of different heights and densities on these radiation distributions; and b) that the model distributes shortwave radiation in a realistic manner for a range of geometries and solar zenith angles. Distribution of direct and diffuse shortwave irradiance over the modeled urban surface, and subsequent (infinite) reflections, are both modeled in order to allocate absorbed shortwave radiation in each scenario. Fig. 3.1 outlines the radiative parameters and the range of urban geometries considered in these scenarios, while Table 3.1 details the specific solar angle, building arrangement, and density and location of tree foliage in each scenario. The simulations in this section meet or exceed the minimum recommendations detailed in Appendix A in terms of the computational parameters (number of rays, ray step size) and wall and foliage vertical resolution. The reader’s attention is drawn in particular to the assessment of the importance of sufficient wall and foliage layer resolution in Appendix A. The results have implications for urban canopy radiation models generally, and suggest that single-layer radiation models in particular may introduce substantial errors.  3.2.1.1 Solar zenith angle The impact of solar zenith angle on the vertical distribution of shortwave absorption in the canopy is examined for a canyon with moderate H/W and two building heights (Fig. 3.2a). Three solar zenith angles are simulated, and a fourth scenario assumes all incoming shortwave is diffuse and isotropic. Most obvious is the increased absorption at 0 m, 6 m and 12 m, corresponding to the ground and the two roofs, respectively (Fig. 3.2a; note that a flatter slope indicates greater absorption in a given layer). These horizontal surface elements result in a comparatively larger total (3-D) area per 1 m vertical interval, and hence greater absorption. Further, lower walls (0-6 m) absorb more than upper walls (6-12 m) because they are present with double the frequency—all buildings have walls from 0-6 m, whereas only the taller B2 buildings have walls from 6-12 m. When  = 70° the lower portions of the walls absorb less radiation because they are shaded by upstream buildings. Finally, roof-level interval absorption (at 6 m and 12 m) is approximately the same for all scenarios because they are present with the 41  same frequency. Note that cumulative absorption at ground level fails to reach 100% by an amount that corresponds to the albedo for each scenario.      Figure 3.1 Model geometries and solar zenith angles for shortwave radiation system response tests. “B” = building, “V” = vegetation (tree) foliage, numbers refer to foliage layer, and subscripts refer to building (e.g. “V2b”) or canyon (e.g. “V2c”) columns. Vertical resolution is 1 m for all simulations. Roof, ground and wall albedos are 0.15, 0.15 and 0.25, respectively; foliage reflection and transmission coefficients sum to 0.50.     42  Scenario Solar zenith angle () Building height & probability† Foliage layer height & column† Effective Leaf Area Index* Fig. 3.2a: Solar zenith angle 2a.1 diffuse only B1 (50%), B2 (50%) - 0.0 2a.2 20° B1 (50%), B2 (50%) - 0.0 2a.3 45° B1 (50%), B2 (50%) - 0.0 2a.4 70° B1 (50%), B2 (50%) - 0.0 Fig. 3.2b: Tree foliage density & clumping 2b.1 45° B1 (100%) - 0.0 2b.2 45° B1 (100%) V2c 0.5 2b.3 45° B1 (100%) V2c 1.0 2b.4 45° B1 (100%) V2c 1.5 2b.5 67.1°‡ B1 (100%) V2c 0.5 Fig. 3.2c: Tree foliage height 2c.1 (=6a.2)  20° B1 (50%), B2 (50%) - 0.0 2c.2 20° B1 (50%), B2 (50%) V1c 0.5 2c.3 20° B1 (50%), B2 (50%) V2c 0.5 2c.4 20° B1 (50%), B2 (50%) V3c 0.5 2c.5 20° B1 (50%), B2 (50%) V3c, V3b 0.5 Fig. 3.3: Albedo and tree foliage height 3.1 (=2b.1) 45° B1 (100%) - 0.0 3.2 45° B1 (100%) variable 0.5 Fig. 3.4: Partitioning between urban elements 4.1 (=2a.2) 20° B1 (50%), B2 (50%) - 0.0 4.2 (=2a.3) 45° B1 (50%), B2 (50%) - 0.0 4.3 45° B1 (50%), B2 (50%) V1c 0.5 4.4 45° B1 (50%), B2 (50%) V3c 0.5 4.5 (=2b.2) 45° B1 (100%) V2c 0.5 4.6 (=2b.4) 45° B1 (100%) V2c 1.5 Fig. 3.2d: Photosynthetically-active and near-infrared bands 2d.1-2d.4 (=2b.2) 45° B1 (100%) V2c 0.5  Table 3.1 Solar angle, and building and tree foliage characteristics for the shortwave scenarios. Solar irradiance is 85% direct and 15% diffuse except for scenario 2a.1. Solar azimuth is perpendicular to the canyon ( = 90°) except for scenario 2b.5. †See Figure 3.1. *Neighbourhood-average LAI multiplied by the clumping coefficient. ‡Effective 2-D solar zenith angle is 45°, and azimuth relative to street direction is  = 25°. 43   44  Figure 3.2 (previous page) Cumulative percent (from top of canopy) of shortwave irradiance absorbed, after infinite reflections, as a function of: (a) solar zenith angle; (b) tree foliage density and clumping; (c) tree foliage height; and (d) shortwave frequency band. Each data point corresponds to the total absorption both above and at the corresponding level (z). Overall neighbourhood albedo () for each scenario appears in the legend. Specifics of the urban configuration and foliage characteristics for each scenario appear in Table 3.1. (a) The ‘diffuse’ simulation assumes all incoming shortwave radiation is diffuse and isotropic. (b) LAI values are neighbourhood averages; LAI = 1.0 corresponds to leaf area density LD = 0.375 m2 m-3. Solar zenith angle is 45° for all but the final scenario. (c) Solar zenith angle is 20°. (d) Solar zenith angle is 45° and diffuse is 15% of incoming. Foliage reflection and transmission coefficients combine to be 0.50 for broadband, 0.20 for PAR, and 0.80 for NIR. *Effective 2-D solar zenith angle is 45°, actual solar zenith angle is 67.1°, and solar azimuth is 25° from the canyon orientation, i.e.  = 25° (as opposed to 90° for the remainder of the scenarios).   As solar zenith angle increases, the median height of shortwave absorption (≈45th percentile) shifts higher in the canopy, from ≈1 m for  = 20° to ≈5.5 m for  = 45° to ≈6.5 m for  = 70°, notably elevating the overall albedo (Fig. 3.2a). This corresponds to an increase of wall absorption at the expense of canyon floor absorption as the Sun moves lower in the sky. Notably, the diffuse-only scenario distributes energy in the vertical similarly to scenarios with low to moderate solar zenith angle, and with equivalent overall albedo; however, unlike the other scenarios it distributes radiation evenly to both walls (not shown). Overall the model reproduces known behaviour, such as the increase of neighbourhood albedo with solar zenith angle (Aida, 1982; Christen and Vogt, 2004; Kanda et al. 2005a).   3.2.1.2 Tree foliage height, density, and clumping  The impacts of tree foliage density on shortwave radiation absorption are explored for a canyon of H/W = 0.5 with a 4 m thick layer of tree foliage above the canyon space. The leaf area index (LAI) and clumping of this foliage layer are varied and the resulting scenarios are compared to a non-vegetated case, where LAI is the neighbourhood-average value. Foliage characteristics are chosen to represent cases with moderate (LAI = 1.0) and high (LAI = 3.0) urban tree coverages and relatively ‘patchy’ foliage distributions ( = 0.5). Results are shown for a solar zenith angle of 45° (Fig. 3.2b).  Tree foliage shifts shortwave absorption from roofs, ground, and walls up to the vegetated layer, substantially increasing the average height of shortwave absorption (Fig. 3.2b). An increase in foliage density (LAI) and a decrease in foliage clumping (i.e., greater ) both 45  enhance this effect. In fact, the effective LAI (= LAI ; Chen et al. 1997) determines the interception efficiency of the foliage; each increase of the effective LAI, from 0.5 (Scenario 2b.2), to 1.0 ( = 1.0; Scenario 2b.3), to 1.5 (LAI = 3.0; Scenario 2b.4), results in additional shortwave absorption, but with ‘diminishing returns’. The overall albedo () varies little between scenarios 2b.2 to 2b.4, but drops more noticeably for the scenario without tree foliage (2b.1). The final scenario (2b.5) illustrates the importance of accounting for the three-dimensional path lengths of rays travelling through foliage. Contrary to the other scenarios, which have the solar beam perpendicular to the canyon direction ( = 90°), the solar zenith is increased to  = 67.1° and the solar azimuth relative to the canyon axis is reduced to  = 25°, which preserves the effective (2-D) solar zenith angle at e = 45°. The absorption by the 7-11 m tree foliage layer increases by 39%, reducing absorption by 23% for walls and by ≈10% for roofs and ground, relative to Scenario 2b.2. Furthermore, overall albedo increases significantly (by ≈0.02). A strictly 2-D model would not capture this important effect, which exists for all solar azimuth angles other than  = 90° (or 270°).  The height of tree foliage also affects the vertical distribution of shortwave radiation absorption (Fig. 3.2c). A 4 m thick layer of foliage corresponding to a neighbourhood LAI of 1.0 is placed with its base at heights of 1 m (Low), 7 m (Mid) and 13 m (High) in the canyon with two building heights from Fig. 3.2a. The additional absorption attributable to foliage at each level is evident in Fig. 3.2c, as is the corresponding reduction of canyon-floor absorption relative to the vegetation-free scenario. As expected, elevation of the foliage layer increases the average height of shortwave absorption. Distributing the same neighbourhood LAI evenly in the horizontal (“V3c, V3b”, or Scenario 2c.5) has a small effect, e.g., a 10% reduction in roof absorption and a similar percent increase in wall absorption. However, this impact on rooftop absorption doubles for a solar zenith angle of 20°, for example. The variation of overall albedo is intriguing: the addition of foliage in the lowest part of the canyon has very little effect, whereas foliage layers at heights V2 and V3 clearly increases overall albedo. A more straightforward and complete example of this effect is shown in Fig. 3.3. The same canyon of width xc = 12 m is chosen, but with buildings uniformly 6 m tall (H/W = 0.5), and with a 4 m foliage layer characterized by LD = 0.375 m2 m-3 and  = 0.5 (neighbourhood LAI = 1.0) present at varying heights in and above the canyon. Interestingly, introduction of the 46  foliage layer adjacent to the ground (“0” on the x-axis in Fig. 3.3) reduces the overall albedo as compared to the non-vegetated case for small solar zenith angle ( = 20°). Overall, neighbourhood albedo increases approximately linearly as the foliage layer ascends from ground level, peaking when the middle of the foliage layer is even with the roof level and descending to an asymptote thereafter (Fig. 3.3). The exact character of this effect likely depends on the albedos of the foliage, walls, and ground, and also on the built structure and foliage density and distribution. The main point is that such a phenomenon can only be captured with integrated modeling of building and trees; it cannot be captured with the tile approach (Fig. 1.1) or a single-layer approach to the integration of urban trees in UCMs.     Figure 3.3 Neighbourhood albedo as a function of foliage layer base height for solar zenith angles 20° and 45°. A 12 m wide canyon is simulated with 6 m tall buildings (H/W = 0.5) and a 4 m thick tree foliage layer in the canyon column (LD = 0.375 m2 m-3,  = 0.5). Horizontal lines indicate the albedo without the foliage layer for each scenario.  47   Overall the model produces physically reasonable modifications to vertical profiles of shortwave absorption as a function of foliage layer density, height, and clumping. It is clear that the height and density of tree foliage significantly modulates the vertical distribution of solar radiation absorption in urban canopies. Tree foliage absorbs a greater fraction of shortwave irradiance at higher leaf area densities, and with reduced clumping and greater exposure to the direct beam (e.g. foliage located above buildings), and for higher solar zenith angles.  3.2.1.3 Partitioning between urban elements  The vertical distribution of shortwave absorption (Figs. 3.2a-c) gives information on the heights that are experiencing maximum radiation loading, and therefore that are most likely to exhibit either significant energy exchange with the atmosphere, or storage. However, the distribution of solar absorption between urban elements (roofs, walls, tree foliage, ground) is also likely to be critical in predicting building energy load and/or pedestrian thermal comfort. Furthermore, current UCMs, with or without the tile approach, are all incapable of capturing the effects of tree foliage height and density on partitioning of shortwave absorption between urban elements.  Fig. 3.4 displays the partitioning between urban elements for select cases based on scenarios from Figs. 3.2a-c. Scenarios A and B demonstrate the shift of absorption from the ground to the building walls with increasing solar zenith angle (e.g., Fig. 3.2a) for a vegetation-free canyon with two building heights (H/W = 0.5, 1.0), each present with equal frequency. The addition of a 4 m thick layer of 0.375 m2 m-3 ( = 0.5) tree foliage in the lower canyon decreases the wall and road absorption but does not affect the roofs (Scenario C). The equivalent foliage layer above the canyon (Scenario D) absorbs about twice as much shortwave as that in Scenario C, shielding roofs and walls to a greater extent. Scenarios E and F in Fig. 3.4 retain the foliage layer above the canyon but include only one building height, at 6 m or H/W = 0.5 (i.e., as in Fig. 3.2b). Scenario E shows increased ground-level absorption and reduced wall absorption compared to Scenario D due to the loss of the taller (12 m) buildings, and consequent loss of wall area (by 33%). Finally, the effect of increasing foliage density above the canyon is substantial (Scenario F vs. Scenario E), with tree foliage now absorbing almost half of the incoming shortwave. 48  Overall it is wall and ground-level absorption that vary most profoundly across these scenarios, in particular with solar zenith angle, relative wall area, and leaf area density. Roof absorption varies to a lesser degree, mainly in response to elevated, dense foliage. The presence of trees shifts solar radiation absorption away from built surface elements, in general, and away from different built surfaces depending on foliage height and solar zenith angle. The interplay of these effects is impossible to capture with the tile approach or highly parameterized canyon vegetation radiation models. To the extent that the current model geometry represents the distributions of radiation exchange of real urban canopies, it is clearly capable of evaluating the effects of tree heights and foliage densities on neighbourhood-average building and ground-level (e.g., pedestrian) radiative loads for different solar angles.   Figure 3.4 Partitioning of total incoming shortwave irradiance between different elements of the urban canopy—buildings (roofs, walls), ground, tree foliage, and reflected radiation—for select cases based on scenarios from Figures 3.2a-c. Scenarios in Columns A-D have 50% B1 buildings and 50% B2 buildings, whereas Columns E and F have only B1 buildings (Fig. 3.1). Foliage layers are: V1c (Low), V3c (High), V2c (scenarios E and F). Solar zenith angle is  = 45° for scenarios B-F. 49  3.2.1.4 Photosynthetically-active and near-infrared bands  The simulations in Sects. 3.2.1.1-3.2.1.3 have treated shortwave radiation as a single “broad” band. However, vegetation exhibits radically different absorption and scattering behaviour over different portions of the shortwave spectrum, whereas most building materials display less wavelength-dependent variation. Vegetation absorbs the majority of incident ultraviolet radiation (UV; ≈0.29-0.40 m) and photosynthetically-active radiation (PAR; ≈0.40-0.70 m), whereas it scatters the majority of near-infrared radiation (NIR; ≈0.70-2.80 m; Monteith and Unsworth, 2008; Escobedo et al. 2011). For clear-sky conditions UV typically accounts for ≈5%, PAR for ≈50%, and NIR for ≈45% of solar irradiance (e.g., Escobedo et al. 2011). As PAR and UV have similar leaf reflectances, they are lumped together in the following simulations, and the relative importance of separating PAR+UV from NIR is explored (hereafter, “PAR+UV” is referred to as “PAR” only).  Scenario 2b.2 (LAI = 1.0,  = 0.5), with a 4 m thick vegetation layer above a 6 m tall canyon and a solar zenith of 45°, is re-run with scattering/reflection in PAR and NIR bands treated independently. Combined foliage reflection and transmission coefficients are assumed to be 0.50, 0.20, and 0.80, for broadband, PAR, and NIR, respectively (Monteith and Unsworth, 2008). Corresponding foliage absorption coefficients are 0.50, 0.80, and 0.20, respectively. Reflection coefficients of all ‘urban’ materials are assumed invariant between the wavelength bands. PAR and NIR are each assumed to compose 50% of the broadband shortwave flux density and the vertical distribution of absorption is computed in each case. The total broadband radiation absorption as the sum of the PAR and NIR scenarios is also calculated (“PAR + NIR” in Fig. 3.2d).  The most obvious feature in Fig. 3.2d is the enhancement of PAR absorption and decrease of NIR absorption in the 7-11 m foliage layer, relative to the broadband case. There is a small reciprocal reduction (increase) of PAR (NIR) absorption at the roof, wall and ground surface elements. Meanwhile, the cumulative profile of absorption of shortwave energy with and without the PAR/NIR split (“PAR + NIR” and “Broadband,” respectively) is similar. Individual patch absorption is also similar; median relative differences are 1% for built surface patches and 6% for foliage layers. The overall neighbourhood albedo differs by 0.006 between these approaches, an effect that increases for higher solar zenith angle (0.010 for  = 70°) or higher 50  foliage density (0.023 for LAI = 3.0), but decreases for lower solar zenith angle (0.005 for  = 20°) or foliage layers in the canyon (0.003 for foliage layer at 1-5 m). These results suggest that separation of PAR and NIR can be moderately important in terms of overall shortwave receipt for scenarios with denser foliage and/or elevated foliage relative to the buildings. It is likely to be critical for accurate estimation of PAR absorption by the foliage, and therefore to the calculation of stomatal resistance and consequently for modeling of transpiration and latent heat flux. For both scenario 2b.2 and the scenario with the foliage layer placed deep in the canyon (i.e., at 1-5 m), overall foliage absorption of broadband is remarkably constant at ≈70% of PAR absorption for the solar zenith angles tested. Hence, substitution of broadband shortwave for PAR in terms of foliage absorption would incur a ≈30% error. Separation of the two bands adds little to the computation time, because interception of direct beam PAR, NIR and broadband are identical. Hence, the direct solar scheme need only be run once each time the radiation routine is called, but the shortwave reflection matrix must be solved an additional time. The matrix solution represents a minority of the shortwave radiation computation time.  3.2.2 System responses—longwave radiation Longwave radiation exchange differs from shortwave in that matter at terrestrial temperatures not only absorbs in this wavelength range, but also emits. Hence this section explores the impacts of sky view reductions due to buildings and trees on the net longwave exchanges (L*) of built surfaces, foliage, and air, i.e., the net rate of gain or loss of thermal radiation energy. Simulations with varying canyon H/W and foliage heights and densities are performed. Temperatures of all urban facets and foliage are set to 28 °C (isothermal) and downwelling longwave is 320 W m-2, approximating an early evening cooling scenario during midlatitude summer. Surface and foliage emissivities are 0.95. Canyon and building widths are 12 m and 6 m, respectively, and vertical resolution is 1 m. Computational parameters follow recommendations from Appendix A. Means and standard deviations of 20-member ensembles are presented because all longwave radiation exchanges depend on view factors derived from MCRT, whereas in the shortwave spectrum the majority of the exchange is determined by the receipt of direct shortwave, which does not have a 51  stochastic component. Means of canyon aggregate values have uncertainty estimates based on the standard rules of uncertainty approximation (derived from the patch-level ensemble standard deviations). Simulation results necessarily represent a “snapshot” in time with imposed surface temperatures because the energy balances of patches are not solved by the model.  3.2.2.1 Net longwave exchange: vertical distribution  Cumulative vertical profiles of net longwave radiation loss are first assessed, similar to the shortwave scenarios in Sect. 3.2.1. Specifically, the effects of adding a 4 m thick layer of moderately-high density foliage at different heights in a H/W = 0.5 canyon are simulated. The neighbourhood total net longwave flux density differs by less than 2% for all scenarios (Fig. 3.5), as the presence of tree foliage in or above the canyon does not significantly alter neighbourhood total net longwave flux density for the isothermal conditions tested here. Differences would likely to be much larger for the common situation where built surface element temperatures are significantly higher than foliage and air temperatures.  Prior to the addition of tree foliage, the canyon floor has the largest negative L* (i.e., z = 0-1 m), followed by the roof, and finally the walls (‘No Trees’ in Fig. 3.5). The addition of the foliage layer in the upper two thirds of the canyon (2-6 m) results in significantly greater longwave loss in the upper part of the canyon (≈45% of the total, up from ≈15%) and reduced longwave loss by the canyon floor (road), as the foliage ‘traps’ thermal radiation in the lower canyon. If the foliage layer is shifted upwards to 4-8 m, and then fully above the canyon at 6-10 m, the centre of longwave loss shifts progressively upwards from the upper canyon and roofs. The foliage layer loses 35%, 41%, and 43% of total neighbourhood net longwave at heights of 2-6 m, 4-8 m, and 6-10 m, respectively. The corresponding rooftop fraction of L* losses decreases, from 33%, to 30%, and finally to 25% as the tree foliage exits the canopy (6-10 m). The model predicts, as expected, that the overall effect of tree foliage with sufficient exposure to the sky (e.g., for lower density canyons) is to shift the location of active energy loss upwards and therefore to moderate the nocturnal climate deeper in the canopy. The next two sections provide more complete examples of the effects of building density and vegetation on canopy climate.  52    Figure 3.5 Cumulative vertical profile (from top) of total neighbourhood net longwave flux for an H/W = 0.5 canyon with and without a 4 m thick, LAI = 1.0 (LD = 0.375 m2 m-3,  = 0.75) tree foliage layer at different heights in the canyon. All values are means of 20-member ensembles.   3.2.2.2 Net longwave exchange: ground and canyon fluxes  In this section the model is applied to investigate a well-known relation in urban climatology: the effect of canyon geometry on longwave exchange in the urban canopy. The response of the net longwave balance to canyon height-to-width ratio (H/W) is computed for the canyon floor alone, the canyon walls alone, and for the complete canyon, to elucidate the role of longwave exchange in urban heat island genesis. Foliage addition is also assessed.  Greater canyon H/W decreases sky view factor of the canyon floor (streets). This reduces the magnitude of net longwave losses from the ground surface leading to warmer street surface temperatures (Oke, 1981; Oke et al. 1991). This relation is reproduced correctly in Fig. 3.6, both by the current model and by the TUF-2D model of Krayenhoff and Voogt (2007).  53    Figure 3.6 The effect of adding a layer of trees into an urban canyon. Ensemble mean L* on the floor of a range of canyons with different cross-sectional geometries (H/W). The tree layer is spread uniformly across the canyon column, in a layer between 2 - 6 m above the floor. Two foliage cases are considered: (a) a layer of foliage with 0.375 m2 m-3 (LAI = 1.0) and clumping  = 0.75; (b) a layer of 0.750 m2 m-3 (LAI = 2.0) also with  = 0.75. Also, L* of the walls (combined) and the whole canyon (both per m2 plan area of canyon) of non-treed canyons. Error bars for floor values are ensemble standard deviations, whereas those for the whole canyon denote overall uncertainty derived from the standard deviations of individual patches. Individual data points are output by the present model. Black lines are results for canyons with no trees from the TUF-2D model (Krayenhoff and Voogt, 2007). Dotted red line is from the current model for the no-tree case but with the effects of air layers in the canyon included. Lines are linear interpolations between discrete results at H/W = 0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00 and 2.50.     This phenomenon has been postulated to be a leading cause of the nocturnal (canopy-layer) urban heat island. Simple extension of the relation between sky view of individual surfaces and their resulting longwave balance and surface temperature, to the case of screen-level air temperature or air temperature in the urban canopy as a whole, is not appropriate. For 54  example, some numerical models incorrectly apply reduced sky view factors to whole canyons, even the whole urban surface. This results in artificially small magnitudes of urban L* (e.g., Atkinson, 2003; Grossman-Clarke et al. 2005).  The canyon floor and walls2 each individually have a reduced sky view factor, and therefore reduced exchange with the cold sky. But together, as a canyon system, they exhibit L* (per unit plan area of canyon) of similar or greater magnitude to that of a flat surface, according to both models (‘canyon’ in Fig. 3.6). This is because the surface area of the walls increases with H/W, increasing their total net longwave loss per plan area of canyon, and offsetting the reduced canyon floor L* loss (‘walls’ in Fig. 3.6). Given that building walls and street surfaces tend to remain warmer than flat ‘rural’ surfaces (as opposed to being equal in temperature, as assumed here), it is apparent that canyon L* magnitude typically exceeds that of more rural areas. Therefore, the only way that neighbourhood-scale average L* could be less than typical rural values is if a sufficiently large fraction of the area is covered with materials possessing low emissivity and/or low thermal admittance (e.g., roofs). The urban structure on its own is not sufficient for this purpose. Clearly, therefore, the nocturnal urban canopy-layer heat island (in the air) is unlikely to relate to reduced loss of longwave radiation by the urban canopy as a whole. However, it very well may relate to the proximity of the screen-level measurement height to surfaces with reduced loss of longwave radiation (i.e., the canyon floor and lower walls). It might also relate to the increased surface area of urban canopies, which are convoluted relative to rural areas and are thus more available to retain daytime heat, among other factors. It must also be appreciated that the air temperature depends on the heat balance of the air itself not just of nearby surfaces.  The addition of moderately-high tree foliage density in the canyon significantly reduces canyon floor longwave losses (Fig. 3.6). Reductions of net longwave radiation loss are ≈50% and ≈75% for the LAI = 1.0 and LAI = 2.0 scenarios, respectively, irrespective of H/W. Hence, the presence of trees reduces the cooling rate of the canyon floor, as expected, an effect that is more pronounced in an absolute sense for lower H/W. For the particular conditions considered here,                                                  2 Note that L* of the walls in Fig. 3.6 is total wall net longwave flux per plan area of canyon, so as to sum with the “floor” flux per plan area to equal that of the canyon. Hence, the L* at the wall surfaces themselves, on average, is obtained by dividing this value by H/W, and it is thus decreasing with H/W, as expected. 55  the addition of the neighbourhood-average foliage density of LAI = 1.0 has an impact equivalent to an increase in H/W of 0.75-1.25, and this effect increases to H/W ≥1.5 for the LAI = 2.0 scenario. As for non-treed cases, the total L* of the whole canyon including tree foliage (or the whole neighbourhood for scenarios for which foliage protrudes above the canyon), varies little with H/W (not shown). Canyon floor L* at H/W = 0.0 (no buildings) illustrates the versatility of the model. It has a value of -139.0 W m-2 without trees, which drops in magnitude to -72.2 W m-2 and -42.7 W m-2 with the application of canyon column leaf area densities of 0.375 m2m-3 and 0.750 m2m-3, respectively, between 2-6 m. This amounts to a forest-clearing scenario, and the L* of the ground under the forest portions (canyon column) is included for consistency with the built scenarios (i.e., H/W > 0.0) in Fig. 3.6. L* in the clearings (i.e., the ground level in the building column) has a greater magnitude, as expected: -96.1 W m-2 for LAI = 1.0 and -80.3 W m-2 for LAI = 2.0. Finally, extension of the canyon column tree foliage layer into the building column would lead to a continuously forested scenario; hence, the model is potentially able to represent the range of scenarios between forest and completely urban. To summarize, the new model suggests that both urban structure and tree foliage can substantially decrease the magnitude of net longwave exchange of individual facets, such as the canyon floor. However, the net longwave exchange of canyons or urban neighbourhoods as a whole does not vary significantly with H/W or added foliage for the isothermal conditions modeled here. The latter is likely to depend more on the (distribution of) materials and thermal states of the individual urban elements.  3.2.2.3 Longwave divergence and canopy air heating rate  Canopy air heating rate is now assessed for canyons with varying H/W. Impacts on heating rate of adding foliage in the upper half of the canyon, and above the canyon, are also tested. All parameters are identical to those in Sects. 3.2.2.1 and 3.2.2.2 except the following: canyon width is now 20 m; air density is 1.2 kg m-3; water vapour density v is set to 0.015 kg m-3 (relative humidity ≈ 53%); and mass extinction cross section kair (see Eq. 2.17) is set to 0.1 m2 kg-1. Mass extinction cross section for broadband longwave is approximate as it was found to vary with water vapour path length (kg m-2); the laboratory data of Staley and Jurica (1970) 56  suggest 0.1-0.3 m2 kg-1, whereas MODTRAN simulations for typical urban canopy sightlines (10-100 m) with a range of temperatures and relative humidity combinations indicate 0.06-0.12 m2 kg-1, with higher overall ‘effective’ extinction when aerosol is included.    Figure 3.7 Ensemble mean heating rate of the 1-2 m canyon air layer (“1.5 m”), and of the complete canyon air column (“canyon”), as a function of canyon height-to-width ratio. Also shown is the impact of adding a LAI = 1.0 foliage layer in the upper half of the canyon (v-u), or immediately above the canyon (v-a), for the H/W = 0.5 scenario.   It is typically assumed that nocturnal screen-level air temperature variation in cities is controlled by canyon floor cooling as a result of surface-air coupling (Sect. 3.2.2.2). To date, all urban canopy models assume this coupling is only turbulent. Here, cooling of canyon air layers is assessed, where cooling is due to longwave divergence only.  57    Figure 3.8 Height-normalized profiles of ensemble mean heating rate for select canyon H/W, computed at 1 m intervals. Error bars are ensemble standard deviations. Black diamonds: ten-1 m layers of air (no buildings or foliage); purple squares and light blue circles (o): H/W = 0.5 with foliage in upper half of the canyon, and above the canyon, respectively (“v-u” and “v-a” scenarios from Fig. 3.7).   Results suggests that radiative coupling can indeed be significant for air temperature (Fig. 3.7), but much less so for cooling of surfaces (e.g., dotted red line in Fig. 3.6: road L* when air layers are considered). Canyon total and 1.5 m cooling rates diverge for H/W = 0.00-0.75, after which the two trends are equivalent (Fig. 3.7). Addition of foliage layers to the H/W = 0.5 canyon (“v-u”, “v-a” in Fig. 3.7) lowers the magnitude of the cooling rate of the canyon air by about 70 %; cooling rate at 1.5 m for a forest-only case with LAI = 2.0 is almost identical to these ‘forested’ urban cases with H/W = 0.5 and LAI = 1.0 (not shown).  Select vertical profiles of the cooling rates in Fig. 3.7 are plotted to assess the model in more detail (Fig. 3.8). For the current scenario without buildings or trees (only air), the model 58  predicts a consistent cooling rate with height of -1.31±0.08 K hr-1 over the 10 m above ground. As H/W increases cooling rate decreases substantially in magnitude in the lower part of the canyon, whereas cooling rates as a function of absolute distance from the canyon top do not differ significantly between the different H/W. The introduction of LD = 0.4 m2 m-3 (LAI = 1.0) foliage above the H/W = 0.5 canyon produces a relatively consistent cooling rate with height. The same foliage layer in the upper half of the canyon shows dramatically more cooling as the canyon top is approached and the sky view factor of the air layers increases. For both foliage additions, cooling rate in the lower canyon is about -0.20 K hr-1, similar to the H/W = 2.0 canyon.  Modulation of longwave divergence by buildings and elevated foliage significantly decreases the radiative cooling rate magnitude of canopy air layers and has the potential to play a role in canopy-layer urban heat island genesis. However, results in Figs. 3.7 and 3.8 are intended primarily as a demonstration of the model’s ability. Robust quantification of cooling rates due to longwave divergence awaits better determination of the broadband mass extinction coefficient for water vapour, or a multi-band treatment of longwave with spectrally appropriate extinction coefficients. Nevertheless, the response of the cooling rates to canyon geometry and added tree foliage appears realistic.  3.3 Summary Model simulations are performed to ensure energy conservation, and subsequently to demonstrate appropriate system responses to varying geometries, foliage characteristics, and solar zenith angle. The effects of different heights and densities of tree foliage on the vertical distribution of shortwave radiation absorption are demonstrated. Furthermore, the partitioning of shortwave absorption between buildings, vegetation and ground is presented. Denser foliage layers exhibiting less clumping absorb more shortwave at the expense of building and ground surfaces, in particular for higher solar zenith angles or when foliage layers are elevated relative to the buildings. Splitting broadband shortwave into separate PAR and NIR bands has moderate implications for total shortwave absorption for some scenarios, but is essential for coupling with energy/water balance models (e.g., for accurate assessment of PAR to model stomatal resistances). 59  System response scenarios in the longwave spectrum focus on the net exchange. Simulated canyon floor (road), wall and whole canyon net longwave flux densities (L*) as a function of canyon H/W are in agreement with an independent radiation model (TUF-2D). The magnitude of canyon floor L* is shown to decrease with deeper canyons and with the addition of canyon foliage. For the isothermal conditions modeled here the L* of the whole canyon system (and of the whole neighbourhood—canyon + roofs) does not vary significantly with canyon H/W and tree foliage in the canyon. Canyon air water vapour density is prescribed and its mass extinction coefficient estimated in order to demonstrate the model’s ability to compute urban longwave divergence. Greater H/W and tree foliage density both substantially reduce longwave divergence and cooling rate magnitude. The model indicates that nocturnal air temperature is substantially impacted by longwave radiation divergence, a process that is not currently represented in urban canopy models.   60  Chapter 4: Parameterization of drag and turbulence for urban neighbourhoods with trees  Neighbourhood-scale numerical models of urban meteorology and climate represent canopy wind, turbulence, and fluxes of momentum and scalars with varying degrees of complexity. Many state-of-the-art multi-layer models include an Urban Canopy Parameterization (UCP), typically a 1.5 order k-l turbulence closure and representation of canopy-induced processes in the equations of vertical exchange of momentum and turbulent kinetic energy, such as drag, and generation and dissipation of turbulent kinetic energy (Martilli et al., 2002; Dupont et al., 2004; Hamdi and Masson, 2008; Santiago and Martilli, 2010). Vegetated urban canopies are typically modeled by ‘tiling’ urban and soil-vegetation surface models, such that they interact independently with the atmospheric model in each grid square. With this approach, built and natural surface exchanges only influence each other indirectly via the atmospheric model, and potentially important vegetation-building interactions, and tree-building interactions in particular, are not included. Building-tree interaction in the context of urban canopy flow may be conceptualized in two ways. Trees slow the wind and hence reduce the absolute drag force exerted by buildings, and this effect does not require parameterization but simply the inclusion of drag terms for both buildings and trees in the solution of the canopy momentum balance (i.e., an integrated approach instead of a tile approach). Trees also affect the relative impact of buildings on the flow: the efficiency with which buildings remove momentum from the flow, which is proportional to the ratio between the drag force and the square of the mean wind velocity at the same height and is also referred to as the sectional building drag coefficient (CDB). An analogous phenomenon, albeit operating at a smaller scale, is the sheltering between plant elements identified by Thom (1971) and confirmed by Brunet et al. (1994). This latter sense of building-tree interaction is investigated here. If this effect is significant, its parameterization for any combination of building and foliage arrangement and densities is likely to be complex.   61    Figure 4.1 The Santiago and Martilli (2010) methodology. Higher fidelity models and measurements are used to inform or evaluate simpler models (grey boxes/brown arrows). The objective is to determine spatial-mean flow profiles (yellow boxes) with a column model informed by fully-parameterized inputs (green box), i.e., an independent urban canopy parameterization. Thick blue arrows indicate model output, thin blue arrows indicate model input. Orange numbers indicate the section in which the model or process (i.e., evaluation or model input) is described or used.    The situation for turbulence production in building and tree wakes is identical, because it depends not only on mean velocity at each level for both tree and buildings, but also on their respective drag coefficients (i.e., turbulent kinetic energy production by element wakes is proportional to the rate of removal of mean momentum by element drag). Turbulence length scales, however, are ‘overall’ properties of the flow, unlike drag, which may be represented as separate source terms for buildings and trees. A key question is whether trees affect turbulent 62  length scales in addition to the substantial effects of buildings demonstrated by Santiago and Martilli (2010). This chapter aims to represent the combined effects of trees and buildings on the spatially-averaged mean flow with a relatively simple parameterization. These effects are assessed in the three-dimensional (3-D) flow and subsequently parameterized in one-dimension (1-D). Airflow within and above vegetation and especially urban canopies is fully 3-D, but in many applications the horizontal spatial variability cannot be resolved. Hence a filter, consisting of horizontal averaging over an area much larger than the size of the obstacles, is applied in order to reduce the canopy-atmosphere exchanges to one-dimension (vertical). Effects of 3-D processes must be accounted for in 1-D models in order to capture the vertical variation of the flow, hence appropriate horizontal averaging is critical (Raupach and Shaw 1982). The present chapter is an extension of previous work to parameterize 3-D flow processes in 1-D vertical diffusion models for urban canopies (e.g., Santiago and Martilli 2010), and the evaluation of foliage-related impacts on flow, and their interaction with building-related impacts, are new contributions. The overall aim is to develop a UCP for urban neighbourhoods with trees for incorporation in the full BEP-Tree urban canopy model (Ch. 5).  4.1 Objectives and degrees of freedom Considering the complexity of flow through urban arrays with tree foliage, theoretical approaches such as those developed for forest and building-only cases (e.g. Cionco, 1965; Belcher et al., 2003) are less plausible. Here, a parameterization is developed based on results from a 3-D computational fluid dynamics (CFD) model. The methodology of Santiago and Martilli (2010) forms the backbone of the current chapter (Fig. 4.1). The CFD model is first evaluated against wind tunnel measurements to ensure model robustness and accuracy (Appendix C; Simon-Moral et al., 2014). Subsequently, obstacle-resolving CFD simulations of neutral flow through canopies of blocks (buildings) with various distributions and densities of porous media (tree foliage) are conducted, and the spatial-average impacts on the flow of these building-tree combinations are assessed. Based on the CFD results, a parameterization of the vertically-distributed impacts of trees on mean flow and turbulence in urban canopies for neutral conditions is formulated, explicitly considering building-tree interaction. The new CFD-derived 63  parameterization of drag and turbulence contributes to the full inclusion of vegetation in multi-layer urban canopy models and neighbourhood-scale dispersion models. A large range of building-tree configurations could be considered in the design of the CFD simulations. Relatively large simplifications are made here, viz.: regular arrays of cubic blocks (buildings) with foliage layers of thickness H / 2 (where H is building height), and foliage homogeneously distributed in the horizontal (ignoring clumping, for the moment) across the non-built area at each height. Even foliage distribution is chosen for simplicity; note that discontinuous foliage distributions give somewhat different results. Moreover, wind direction is maintained perpendicular to windward building faces for all simulations; effects of varying wind direction have been explored for non-treed configurations (Santiago et al. 2013a), and they result in additional complications such as lift forces that are beyond the current scope. Other findings suggest that the aerodynamic effects of trees depend on wind direction relative to the building layout, as well as on building configuration (Buccolieri et al., 2011). Here, only building plan area density, tree foliage layer height and leaf area density are varied, resulting in three degrees of freedom.  A methodology to formulate the one-dimensional UCP for urban canopies with trees is developed. First, the ability of the 1-D column model to reproduce spatially-averaged CFD results is evaluated (Sect. 4.2.4, Appendix B). Subsequently, the new parameterization for impacts of urban trees on flow is developed from, and contextualized by, the four primary objectives of this chapter:  A) to assess the relative importance of the (source/sink) terms added to the momentum and turbulent kinetic energy budgets in the column model to represent the effects of urban tree foliage on flow profiles, as compared to the 3-D CFD model (Sect. 4.3);  B) to determine if trees and buildings can be treated independently, or if their relative impacts on the flow (e.g., their efficiencies as momentum sinks, i.e., their drag coefficients) are affected by each other’s presence (Sect. 4.3);  64  C) to assess accuracy of the resulting parameterization, which is based on those terms found in Sect. 4.3 to be important to the reproduction of the spatially-averaged flow profiles in urban neighbourhoods with trees (Sect. 4.4);  D) to determine for which subset(s) of scenarios tree foliage-related terms are of consequence beyond the building-related terms (Sect. 4.4).  Sect. 4.2 discusses the equations of fluid flow that form the basis of the 3-D CFD model and the column model (i.e., the UCP). Sect. 4.3 address objectives A and B, and Sect. 4.4 addresses objectives C and D. Sect. 4.4 also updates parameters for the Santiago and Martilli (2010) parameterization of spatially-averaged flow through non-treed urban neighbourhoods. Sect. 4.5 concludes with a summary and some conclusions.   4.2 Numerical models: description and evaluation A 1-D column model with k-l closure is parameterized based on results from a 3-D Reynolds-Averaged Navier-Stokes (RANS) CFD model with standard k- turbulence closure. Hence, a 3-D CFD model that accurately represents the primary effects of the key canopy processes on vertical profiles of flow is essential. When applied, 3-D CFD models inhabit a spectrum of ‘realism’ along two axes: the degree to which the flow is accurately represented, and the degree to which the surface boundary conditions - the urban structure and land cover which influence the flow - approximate those found in real neighbourhoods. Compared to real neighbourhoods, direct numerical simulation (DNS), and large-eddy simulation (LES) to a significant extent, result in a realistic representation of one realization of flow conditions. However, they are computationally-expensive. Reynolds-Averaged Navier-Stokes (RANS) implementations are much computationally cheaper, yet can produce reasonable results compared to full-scale, wind tunnel and DNS results for flow through canopies of cubes (Santiago et al., 2007; Santiago et al., 2008) and canopies of trees (Foudhil et al., 2005; Dalpe and Masson, 2009) in terms of the vertical profiles of spatial mean flow properties, and also in terms of vertical profiles adjusting to surface heterogeneity (Foudhil et al., 2005). It is known that standard k- RANS implementations tend to overpredict the drag force 65  and turbulent kinetic energy production at the front of cubes (Santiago et al., 2008). Nevertheless, the objective here is to accurately reproduce the spatial mean vertical profiles of flow properties over a wide range of scenarios with varying spatial distributions of built and foliage elements; hence, RANS is the more feasible choice.    Figure 4.2 The five tree foliage height scenarios that are simulated, as well as the case without trees (a), for P = 0.25: (b) Tree1 (0 - 8 m); (c) Tree2 (4 - 12 m); (d) Tree3 (8 - 16 m); (e) Tree4 (12 - 20 m); (f) Tree5 (16 - 24 m). Building height H is 16 m and foliage layer thickness is 8 m. Leaf area density varies between the following for each scenario: 0.06, 0.13, 0.25, and 0.50 m2 m-3. Forcing wind is from the left and perpendicular to the building faces (staggered block array).   The other ‘axis of realism’ begins with the complexity of real neighbourhoods, which contain streets, alleys, yards, fences, poles, vehicles, balconies, chimneys, people, etc. Some numerical and wind tunnel experiments have attempted to represent ‘real’ neighbourhoods to a greater degree (e.g. Kastner-Klein and Rotach, 2004; Xie et al, 2008; Kanda et al., 2013). However, most urban flow studies are simplified into arrays of cuboids (e.g., Macdonald et al., 1998; Coceal et al., 2006, Santiago et al., 2007) or the archetypal urban canyon in two-dimensions (e.g., Sini et al., 1996). The degree to which flow results for such simplified representations of the urban surface (i.e., that ignore all aspects of the surface structure with the exception of some ‘average’ building geometry and spacing) can be extrapolated to the flows 66  through real urban neighbourhoods remains an open question (Martilli et al., 2013). Nevertheless, for clarity and generality, the simulations performed here are for a simple array of cubic blocks in staggered formation (Figs. 4.2 and 4.3), with tree foliage evenly distributed in the horizontal in the plan area collocated with the canopy space. It remains an open question whether the use of a more complex flow modeling approach (e.g., LES) with simplified surface representation (e.g., arrays of cubes) significantly improves the ‘realism’ of the spatially-averaged flow. However, the primary reason for choosing a RANS approach here is its computational feasibility. Cheng et al. (2003) found that LES required 2-3 orders of magnitude more computational power than RANS with standard k- closure; hence, RANS is considered the more feasible option here.    Figure 4.3 An example foliage height (Tree2) for the P = 0.11 (a) and P = 0.44 (b) building densities.  All other features are the same as Fig. 4.2.   In conclusion, a 3-D RANS flow model with standard k- turbulence closure is chosen for three main reasons: (a) there is an established history of use for flows through building and tree canopies, (b) it’s relative computational efficiency (so as to investigate multiple scenarios), and (c) its ability to accurately reproduce profiles of spatially-averaged flow properties from measurements and high fidelity flow models.  4.2.1 Three-dimensional RANS k- model and parameterization of foliage effects  The CFD model solves the steady-state Reynolds-Averaged Navier Stokes (RANS) equations in three-dimensions with the standard k- model as turbulence closure. Hence, 67  prognostic equations for both the turbulent kinetic energy (k) and its rate of dissipation () are solved, and model simulations proceed until a steady-state is achieved. A description of the CFD equations for the non-treed cases, and the source terms required to represent buildings, can be found in Santiago et al. (2007). The commercial code STAR-CCM+ is used (CD-adapco, 2012).  The domain height is 4H, where H is building (cube) height. Domain length and width are each the sum of two building widths and two street widths. Symmetry conditions are imposed in the spanwise direction, and periodic boundary conditions are imposed in the streamwise direction. Simulations are forced by a horizontal pressure gradient (). From it, a scaling velocity  /4 Hu   can be derived, where  is the air density (kg m-3). At the domain top, zero normal derivatives are prescribed. A cartesian grid is used, which resolves each cube with 16 cells in each dimension. Grid independence is tested by doubling the number of cells in each dimension to 32, and 16 cells across each cube face is determined to be sufficient (not shown). Further details are available in Santiago and Martilli (2010) and Santiago et al. (2008).  Source and sink terms arise in, or are added to, the momentum, turbulent kinetic energy and turbulent kinetic energy dissipation rate equations to represent the effects of tree foliage on the 3-D flow, following Santiago et al. (2013b). This particular combination of source and sink terms originates for flows through plant canopies (Green, 1992; Liu et al., 1996). Its applicability to foliage in urban environments requires further testing, in particular because it has been developed in the context of a single-band turbulence model and coefficients of the source and sink terms are determined for flows without buildings, and buildings modify the turbulence spectrum. Nevertheless, there are precedents for its use in urban environments (e.g., Dupont et al. 2004; Santiago et al. 2013b). Moreover, investigation of its merit is beyond the scope of the current analysis. Form drag due to foliage results from spatial-averaging the Navier-Stokes equations at the scale of the numerical grid, which is not of sufficient resolution to resolve leaves and branches. Form drag is parameterized as follows (Green, 1992; Liu et al. 1996):  iDVDu uUCLS i ,           (4.1)  68  where ui is the wind velocity in direction i, 3,12iiuU, both in m s-1, LD is the leaf area density ((m2 of one-sided leaf area per m3 of outdoor air volume), and CDV is the sectional drag coefficient for the foliage (dimensionless). The drag coefficient of forest foliage has been found to range between 0.15 and 0.30 (e.g., Li et al., 1985; Massman, 1987), and the optimized value of 0.20 determined by Katul and Albertson (1998) is chosen here. CDV acts as a blunt, covering parameter, and its value typically includes the impacts of sheltering at several scales (typically shoot-crown), an effect related to foliage clumping. Furthermore, nuanced effects on CDV such as leaf fluttering and streamlining, which depend on wind speed (e.g., Walter et al., 2012), are not explicitly included. Eq. 4.1 represents a sink of momentum due to foliage-atmosphere interaction that originates from averaging of the momentum equation. This sink of mean momentum implies a source of turbulence due to extraction of mean kinetic energy from the flow (assuming total conversion of mean kinetic energy to turbulent kinetic energy), which is a source term in the turbulent kinetic energy (k) equation and is parameterized as:  3UCLS DVDk  .          (4.2)  This term is often called wake production. However, while vegetation clearly generates turbulent energy, at a rate approximated by Eq. 4.2, it does so at the scale of the drag elements (i.e., leaves and branches). For vegetation, these ‘wake’ scales are small and hence turbulence generated in this fashion dissipates more rapidly to heat than turbulence generated by the shear of the mean flow or the building wakes (Raupach and Shaw, 1982). Observations demonstrate that turbulent kinetic energy in vegetation canopies derives primarily from the downwards transport of shear-generated k at or above the canopy top, and the same has been observed in a dense urban canopy for z < 0.7 H (Christen et al., 2009). Evidence of wake-generated k in canopies (as predicted by Eq. 4.2, above) is not apparent, presumably because it dissipates rapidly (Raupach and Shaw 1982, Meyers and Baldocchi 1991). In fact, despite the rapid generation of wake turbulence, the typical effect of vegetation is to reduce overall turbulence levels (Green, 1992; Green et al., 1995), as larger turbulent eddies are chopped up by the small foliar drag elements. The 69  representation of this ‘short circuiting’ of the turbulent energy cascade is not possible with a one-band model of turbulent energy (Raupach and Shaw, 1982). Nevertheless, Green (1992) proposed a parameterization of this enhanced dissipation of turbulence generated by foliage element drag (but not by building drag) with an addition to the prognostic equation for k. Wilson (1988) presents heuristic logic that supports addition of this term. With this addition the source term for tree foliage instead reads:   IIdIpDVDk UkUCLS  3,        (4.3)  where coefficients p = 1.0 (no direct conversion to heat) and d = 6.5 based on Sanz (2003). Liu et al. (1996) found that this additional sink term (i.e., d k U) was critical for the correct reproduction of experimentally-measured k in a forest canopy. An alternative approach would be to add a source term in the -equation (see discussion in Appendix C), or split the turbulent spectrum into two or more wavebands (e.g., Wilson, 1988). However, Eq. 4.3 is followed here, and the corresponding source terms in the -equation are (Sanz, 2003):      UCUkCCLS dDVD 534,       (4.4)  where Ce4 = Ce5 = 1.26 and are based on analytical expressions from Sanz (2003) and values used by Dalpe and Masson (2009). Several authors find C5 < C4 provides better comparisons against wind tunnel results (e.g. Liu et al., 1996, Foudhil et al., 2005), and this as well as the merit of the parameterization as a whole is further discussed in Appendix C. Results of the 3-D CFD RANS model with tree foliage implementation as described here are used to determine values for the coefficients necessary for the parameterizations in the column model (Sect. 4.2.2).  70  4.2.2 One-dimensional column k-l model A mesoscale model (or any model that represents urban areas at the neighbourhood-scale) requires vertical profiles of the effects of buildings and trees on the spatially-averaged mean flow. Hence, the overall objective is to accurately represent the effects of a variety of simple arrangements of buildings and trees on the spatially-averaged vertical profiles of flow properties. This requires that the interactions between buildings and trees be considered. Two averaging processes operate on the momentum equation in a mesoscale model. First, all variables are time- or ensemble-averaged, to separate turbulent features from the mean flow, i.e. Reynolds decomposition. Second, all variables are spatially-averaged in the horizontal, typically at the local/neighbourhood scale (100-10,000 m), to filter out features smaller than the scale of the mesoscale grid. Neglecting Coriolis and buoyancy effects, the equation for the evolution of the spatial mean of the time-averaged flow, after these averaging procedures is (Raupach and Shaw, 1982):   VIIVIiViIViIIIjjiIIjjiIjiji uuxpxpxuuxuuxuutu ~~11~~''22  (4.5)  where the overbars denote the time mean and angle brackets the spatial mean, 'u  is departure of the instantaneous horizontal velocity at a fixed point from the time mean (i.e. turbulent fluctuation),  is the departure of the time-averaged velocity from the spatial mean (i.e. dispersive fluctuation), p is the pressure, and  is the air density (assumed to be constant). Martilli and Santiago (2007) describe the averaging technique in more detail, and formally define the turbulent and dispersive fluctuations. Terms on the right side of Eq. 4.5 are: the advection of mean velocity by the mean flow (I); the divergence of the spatially-averaged turbulent flux of momentum (II); the divergence of the spatially-averaged dispersive flux of momentum (III); the spatially-averaged acceleration due to the large-scale pressure gradient (IV), and the spatially-averaged acceleration due to dispersive pressure variation (form drag; V); the divergence of the mean viscous flux of momentum (VI); and the spatial-average of dispersive viscous dissipation (viscous drag; VII). Assuming horizontal homogeneity of the spatially-averaged time mean flow u~71  (and hence, zero mean vertical velocity due to the assumed incompressibility), the mean advection and viscous dissipation terms are zero, and non-vertical divergences of turbulent and dispersive momentum fluxes are also zero, leading to a one-dimensional representation of the canopy:    VIIVIVIIIIIuxpxpzwuzwutu ~~11~~''2     (4.6)  This closely corresponds to the 1-D vertical diffusion model of turbulent flow in and above a canopy described in Santiago and Martilli (2010), with the exception that they neglect the final term (viscous drag) because it is very small (about 5% of form drag) in 3-D CFD simulations with ‘smooth’ surfaces at realistic atmospheric Reynolds numbers. The impact of neglecting the dispersive flux (term III in Eq. 4.6) is discussed in Sect. 4.2.3.  The turbulent flux of momentum in Eq. 4.6 is parameterized using a K-theory approach:  ,          (4.7)  where Km is the turbulent diffusion coefficient for momentum, computed with a k-l closure:  ,          (4.8)   where Ck is a model constant, and lk is a length scale. The evolution of the spatial-mean k is calculated via a prognostic equation, the complete version of which is (Raupach and Shaw, 1982):  zuKwu m ''21klCK kkm 72  VIIjijiVIiiVjjIVjjIIIjjIIjijiIjjxuuuuuxupxukxukxuuuxkutk~''''''1~~''''2.   (4.9)  Terms on the right side of Eq.  4.9 are, from left to right: advection by the mean wind (I); shear production at the horizontal averaging scale (II); divergences of the spatial mean turbulent transport of k (III); dispersive transport of k (IV); and pressure transport of k (V); viscous dissipation (VI); and wake production (shear production by dispersive motions, i.e., at scales smaller than the horizontal averaging; VII). It should be noted that Eq. 4.9 implies the existence of analogous equations for mean kinetic energy and dispersive kinetic energy. Assuming horizontal homogeneity and hence no mean vertical wind at the averaging scale, parameterizing the turbulent flux of k in the shear production and turbulent transport terms with a K-theory approach (e.g., Eq. 4.7), neglecting the pressure transport term, and representing the viscous dissipation rate with , we are left with:  VIIIkVIIjijiVIIVIIImIImsxuuuzwkzkKzzvzuKtk    ~''~~22,    (4.10)  where an additional sink term (VIII) has been added to account for the dissipative effects of tree foliage (i.e. term II in Eq. 4.3; term VII in Eq. 4.10 is term I in Eq. 4.3). Term IV (dispersive transport) is neglected by Santiago and Martilli (2010) and here as well, and this is discussed in Sect. 4.2.3. 73  The spatial mean of dissipation rate of turbulent kinetic energy () in Eq. 4.10 is not determined by a prognostic equation as in the 3-D CFD model (Section 4.2.1), but is modeled more simply as:  bvlkC23 ,          (4.11)  where bvl  is a dissipation length scale, that in the present context has been modified by the presence of buildings (“b”) and tree foliage (“v”), and C is a model constant (dimensionless). While several values of Ck (i.e., Eq. 4.8) and C have been proposed in the literature, lk Ck and l/C are parameterized instead of lk and l, and hence results are independent of Ck and C (Santiago and Martilli, 2010).  To solve prognostic equations 4.6 and 4.10 several additional terms require parameterization and inclusion. In the momentum equation (Eq. 4.6), the drag due to buildings and tree foliage is parameterized as follows (Foudhil et al., 2005; Santiago and Martilli, 2010):  iDVDDBvDViuUCLCBxp 21~1,       (4.12)  where BD is sectional building area density (m2 of area facing the wind per m3 of outdoor air volume), LD is the neighbourhood-average leaf area density (m2 m-3), U is the three-dimensional wind speed, CDV is the drag coefficient for tree foliage (=0.2, as in the CFD model), and the numbers “1” and “2” (etc. in subsequent equations) will be referred to in Table 4.1 and Sect. 4.3. CDV in the CFD model (Sect. 4.2.1) controls foliage drag at the grid scale (≈1 m), whereas here in the column model it informs neighbourhood-average drag at a particular height. Moreover, clumping of urban tree foliage at crown-neighbourhood scales is typically substantial. This is not a concern in the emulation of CFD results, which are based on homogeneously-distributed foliage across the domain (neighbourhood); however, clumping at these larger scales must be 74  accounted for in application to real neighbourhoods. Marcolla et al. (2003) propose one method for doing so: use the foliage clumping coefficient as defined for radiative transfer through canopies to determine an ‘effective’ LD’. CDBv is the sectional drag coefficient for buildings when foliage is present, where:  DBDBv CC4,3 ,           (4.13)  where CDB is the sectional drag coefficient for buildings without any tree foliage in the domain, and  represents the effect of the foliage on the building drag coefficient for a particular building and foliage configuration. Hence, interaction between buildings and trees is accounted for in the building drag coefficient, because the foliage drag coefficient is fixed. This interaction between buildings and trees is a relative effect. That is, simply by including a drag term for buildings and another for tree foliage, they each impact the absolute effect the other has on the flow. However, they do not impact each other’s drag coefficient, or drag ‘efficiency’. The question investigated here is whether the presence of tree foliage impacts the sectional drag coefficient of buildings, that is, the drag force that they exert relative to the inertial force in the same atmospheric layer.  As in the CFD model, the loss of momentum (and therefore mean kinetic energy) due to building and foliage drag implies a reciprocal production of turbulent kinetic energy. Furthermore, the more rapid dissipation of the fine ‘wake’ scale turbulence produced by tree foliage that is incorporated in the CFD (Eq. 4.3) must also be included here. Therefore, terms VII and VIII in the k-equation (Eq. 4.10) become:    7365~'' UkCLUCLCBsxuuu DVDdDVDDBvDVIIIkVIIjiji  .   (4.14)   Finally, the turbulent length scale (lk, in Eq. 4.8) is derived from the dissipation length scale (lbv, in Eq. 4.11). Following Santiago and Martilli (2010):  75   ClClC bvkk ,          (4.15)  where C is a model constant equal to 0.09, and lbv /C is derived from the CFD simulation (or a parameterization) and is the dissipation length scale (divided by C) in the presence of a built canopy and tree foliage:  89ClCl bv ,          (4.16)  where l / C is calculated by the CFD, or parameterized, for the equivalent case without buildings or tree foliage,  is dimensionless and represents the effect of buildings on the length scale for a particular configuration, and the dimensionless factor represents the effect of the foliage on the length scale for a particular building and foliage configuration. Note lb = lbv when = 1. 4.2.3 Dispersive processes  The 1-D column model approach in Sect. 4.2.2 cannot directly account for the dispersive processes present in the 3-D CFD model; the only option is to derive a parameterization in 1-D for their effects. Three primary dispersive process are in operation in the CFD model: dispersive transport of mean momentum wu ~~  (Eq. 4.6), dispersive transport of turbulent kinetic energy wk ~~  (Eq. 4.10), and dispersive shear production (wake production), which is parameterized in the column model by terms 5 and 6 in Eq. 4.14. Given the variability of dispersive processes with urban configuration and incident wind direction, a theoretical approach to the representation of dispersive transport terms is unlikely at present. Thus, a statistical or fitted approach would be required, derived from a comprehensive suite of 3-D model data, presumably, and this is well beyond the scope of the present research. Our present objective is the inclusion of the effects of tree foliage on flow in urban neighbourhoods, and we therefore opt to proceed with our analysis 76  as if dispersive transport processes were negligible. This is in line with previous studies (e.g. Santiago and Martilli, 2010). Dispersive transport processes are in fact non-negligible for some scenarios studied here (e.g., Appendix B.1) and for many real urban configurations. Hence, the subsequent analysis seeks to represent the effects of urban tree foliage on spatially-averaged flow with neglect of dispersive motions in the column model as a starting point. 4.2.4 Comparison of the column model against the CFD model  The fidelity with which the 1-D column model reproduces vertical profiles of spatially-averaged flow from the 3-D CFD model is assessed for a range of urban block scenarios with varying heights and densities of tree foliage (Fig. 4.1). A staggered formation of cubic blocks, 16 m (and 16 cells) on a side, is used in the CFD model (Figs. 4.2 and 4.3), as in Santiago and Martilli (2010). The staggered formation is chosen because the aligned formation with wind direction identical to the street direction results in ‘channeling’ flow (e.g., Simon-Moral et al., 2014), which yields tree foliage impacts that are highly specific to this particular combination of building configuration and wind direction. The analysis is performed for plan area density of buildings P = 0.25 (Fig. 4.2), and the results are subsequently extended over the full range of P with the aid of additional simulations for P = 0.00, 0.06, 0.11, and 0.44 (e.g., Fig. 4.3). Given that neighbourhood scale (spatially-averaged) results are desired, tree foliage is represented in a simplified manner. It consists of 8 m thick layers, homogeneously distributed across (or above) the canopy air space with foliage tops at heights of 8 m, 12 m, 16 m, 20 m, and 24 m, and leaf area densities of 0.06, 0.13, 0.25, and 0.50 m2 m-3 (e.g., Fig. 4.2). This range of tree foliage heights permits analysis of the distinct effects of tree foliage on the flow at different heights relative to the buildings, e.g. deep in the canopy vs. the canopy-top shear layer vs. above the built canopy. Foliage is represented as a porous medium and different plant structures (e.g. leaves vs. branches) are not distinguished (Eqs. 12 and 14). Flow profiles from the column model are compared against spatially-averaged profiles from the CFD model for all combinations of urban plan area built densities, foliage heights, and foliage densities, including the foliage-free cases. Exact profiles of the building drag coefficient (CDBv) and the dissipation length scale (lv) for each scenario are extracted from the CFD and input to the column model. CDBv at each height is based on pressure drop between windward and 77  leeward faces of the buildings, and the spatially-averaged wind speed, at the same height. Hence, the column model receives the best possible information from the CFD, and differences between the models are due to differences in formulation (e.g. k-l vs. k- turbulence scheme) and to the 3-D vs. 1-D representation of the flow (i.e., dispersive motions). Root mean square difference (RMSD) between spatially-averaged profiles from the CFD and the column model is calculated for the profiles of spatially-averaged streamwise velocity , turbulent kinetic energy , and Reynolds stress ''wu . RMSD is chosen as the comparative statistic because it more effectively includes rare but extreme differences in the overall measure of difference as compared to a ‘hit rate’ approach (e.g., Schlünzen et al., 2004). RMSD is calculated over two height ranges: the canopy layer (0 < z ≤ H), where vertical and horizontal transport of scalars is critical; and (approximately) the above-canopy roughness sublayer (H < z ≤ 2H), so as to capture the largest vertical exchanges (see Appendix B). RMSD varies depending on the size of samples compared; however, RMSD values compared below all occur over the same number of sample points, avoiding this drawback. RMSD is normalized by u for , and by its square for  and , and therefore represents the difference normalized by flow forcing. Note that Reynolds stress  is always well-reproduced (Appendix B) and is not analyzed further. Average differences between the models are less than or equal to those reported by Santiago and Martilli (2010) for non-treed scenarios; differences are even smaller for cases with tree foliage above the buildings (see Appendix B). Hence, while there is opportunity for improvement, it is concluded that the column model performs adequately in terms of reproducing spatial-mean flow profiles.  4.3 Impacts of tree foliage on flow: important processes In this section objectives A and B are addressed: source terms required to represent spatially-averaged flow through simplified urban neighbourhoods with trees are determined, and the potential to treat impacts of buildings and trees on the flow independently is studied. To address these objectives, the suite of scenarios in Sects. 4.2.4 and B.1 are re-run multiple times, u ku k ''wu''wu78  and in each run select terms, as identified by numbers ‘2’ through ‘8’ in Eqs. 12, 13, 14, and 16, are individually removed from the column model equations. Each term that is removed represents a particular impact of buildings or tree foliage on the flow. Therefore, with removal of each term the relative importance of the corresponding process (e.g., drag on mean flow, turbulence production or dissipation) is assessed, and the most critical processes are determined. These terms are further described in Table 4.1. In this section, profiles of variables required to compute Bdrag-u and Blength-u,k (i.e., building drag coefficient CDB, and length scale modification due to buildings lb, respectively) are derived from the CFD simulation for each scenario for input to the column model. In Sect. 4.4, by contrast, an updated version of the Santiago and Martilli (2010) parameterization of these variables is applied (Fig. 4.1). Each term is deemed ‘significant’ if its removal from the column model equations increases RMSD of wind speed and/or turbulent kinetic energy with the CFD model (see Appendix B), normalized by u and u2, respectively, by ≈100% or more. A threshold of 100% indicates that a term is only ‘significant’ if it’s removal causes a model difference at least as large as differences caused by neglect of all processes in the 1-D model relative to the 3-D model, namely dispersive fluxes and processes affecting dissipation rate. The purpose of these simulations is to determine which building- and foliage-related terms are essential to reproduce in the column model the spatially-averaged flow profiles computed by the CFD model. Analysis is first performed for all scenarios with a building plan area density (P) of 0.25, and in Sect. 4.3.2 it is extended to the full range of P.  4.3.1 Intermediate building plan area density  Fig. 4.4a presents, for two height ranges, the maximum and median change in RMSD between the column and 3-D models across all 20 scenarios with P = 0.25 from Sect. 4.2.4, i.e., all combinations of the following: foliage tops at heights of 8 m, 12 m, 16 m, 20 m, and 24 m, and leaf area densities of 0.06, 0.13, 0.25, and 0.50 m2 m-3. These RMSD changes result from the removal of each term that represents an impact of the buildings or the tree foliage on spatially-averaged streamwise velocity. For removal of the terms Bdragv-u and Bprodv-u,  in Eq. 4.13 is determined from the CFD simulations.   79   Term Equation Name Description 1 4.12 Bdrag-u Drag due to buildings (  equation). 2 4.12 Vdrag-u Drag due to tree foliage (  equation). 3 4.13 Bdragv-u Modification to building drag due to presence of tree foliage (  equation). 4 4.13 Bprodv-k Modification to production of turbulence by building drag due to presence of tree foliage (  equation). 5 4.14 Bprod-k Production of turbulence by building drag (  equation). 6 4.14 Vprod-k Production of turbulence by tree foliage drag (  equation). 7 4.14 Vdiss-k Enhanced dissipation of turbulence due to the small (wake) scales produced by the presence of tree foliage (  equation). 8 4.16 Blengthv-u,k Modification to length scales for case building-only case due to the presence of tree foliage (  and  equations). 9 4.16 Blength-u,k Modification to length scales due to the presence of buildings (  and  equations).  Table 4.1 Terms investigated and the equations in which they appear. Also included is the naming convention used in subsequent figures and in the text, and a description of each term. Terms 1-3 are sink terms in the momentum equation. Terms 4-7 are source/sink terms in the turbulent kinetic energy equation. Terms 8 and 9 affect the length scales, which directly impact both  and  balances. Impacts of buildings are captured by terms 1, 5, and 9, effects of tree foliage by terms 2, 6, 7, and 8, and ‘interaction’ between buildings and trees by terms 3, 4, and 8.   Drag force due to tree foliage (Vdrag-u) is the most important term in Fig. 4.4a; it significantly impacts flow both above and within the canopy. Production of turbulent kinetic energy by the buildings (Bprod-k) is modestly important for the wind above the canopy (H < z ≤ 2H), but not significant as defined above. RMSD increase due to removal of any other terms is much smaller than the RMSD between the column model and the CFD determined in Sect. B.1, as indicated by the colour-coded horizontal lines in Fig. 4.4a. Hence, only Vdrag-u is significant uuukkkku ku ku k80  for spatially-averaged streamwise velocity for the range of scenarios considered here. Notably, modification of the building drag and the length scales due to the presence of foliage (‘interaction’ terms Bdragv-u and Blengthv-u,k) are insignificant (Fig. 4.4a).     Figure 4.4 Change in column model RMSD compared to the CFD model for spatially-averaged streamwise wind velocity (a) and spatially-averaged turbulent kinetic energy (b) with the removal of each of seven building/tree foliage induced terms (Table 4.1). Median and maximum change in RMSD across the 20 treed scenarios at P = 0.25 are calculated for two atmospheric layers (z = 0 – H, z = H – 2 H). Horizontal lines indicate the actual RMSD, i.e., the RMSD that equates to a 100% increase in RMSD.   The foliage drag term in the momentum equation (Vdrag-u) is also the foliage-related term with the largest impact on spatially-averaged turbulent kinetic energy (Fig. 4.4b), particularly for scenarios where foliage protrudes above the buildings (i.e., Tree4, Tree5). The corresponding production term in the turbulent kinetic energy equation (Vprod-k), while significant above-canopy for some cases, produces a much smaller impact. The enhanced dissipation term that represents short-circuiting of the turbulent cascade (Vdiss-k) is of similar magnitude to the RMSD between the column and CFD models in the median (Fig. 4.4b), 81  suggesting that it should be included. Disregard for production of turbulence by building drag (Bprod-k) slightly improves results for turbulent kinetic energy, contrary to the case for streamwise velocity. Yet, Bprod-k is implied by the corresponding drag term (Bdrag-u) in the momentum equation (see Eqs. 4.1 and 4.2); moreover, its retention improves the prediction of spatially-averaged streamwise wind (Fig. 4.4a) and maintains consistency with Santiago and Martilli (2010). Although not shown in Fig. 4.4, drag force induced by buildings (Bdrag-u) is the most important term, and, depending on the variable ( u  or k ), atmospheric layer (0 < z ≤ H, or H < z ≤ 2H), and statistic (maximum or median), the increase in RMSD induced by its absence is a factor 3-100 times larger than the next most significant term (Vdrag-u). Likewise, modification of length scales by the buildings is also of high import (Santiago and Martilli, 2010) and is included in all simulations.   Accounting for both streamwise velocity and turbulent kinetic energy, modification of both the length scales (Blengthv-u,k/term 8), and the building drag coefficient in both u  and k  equations (Bdragv-u and Bprodv-k/terms 3 and 4), due to the presence of tree foliage, are unimportant, and so these three terms can be neglected for P = 0.25. Hence, ‘interactions’ between buildings and trees are unimportant to the spatial mean flow, at least for the range of scenarios tested (P = 0.25, variable tree height and density), and effects of buildings and trees on flow may be treated independently. Similarly, terms related to production of turbulence are of lesser importance, but their existence is implied by addition of the all-important drag terms in the momentum equation.  Foliage-related terms are now added one at a time, in order from most to least impactful in terms of profiles of spatially-averaged velocity and turbulent kinetic energy. This helps to assess the essential combination of terms required to represent building-foliage combinations. The turbulent kinetic energy production term due to trees (Vprod-k/term 6) is appended to its corresponding momentum drag term (Vdrag-u/term 2). Together they are referred to as ‘Vdrag’. Furthermore, all building terms (i.e., Bdrag-u and Bprod-k/terms 1 and 5, and Blength/term 9) are included in all simulations, so as to focus on those terms required to represent tree foliage in urban scenarios. 82    Figure 4.5 Profiles of spatially-averaged mean streamwise velocity (a) and turbulent kinetic energy (b) from the CFD, and from the column model with the cumulative introduction of several foliage-induced terms at building density P = 0.25. All scenarios have foliage density LD = 0.50 m2 m-3, and the Tree3, Tree4 and Tree5 scenarios are presented. The Bdrag simulation includes no foliage-related terms, but only building-related terms 1, 5, and 9 in Table 4.1. Grey shading indicates the building canopy, green shading the tree foliage.    Profiles of spatially-averaged streamwise velocity are shown in Fig. 4.5a for three scenarios where foliage is expected to be more significant (i.e., taller, dense foliage): Tree3, Tree4, and Tree5 with LD = 0.50 m2 m-3. The primary impact on streamwise velocity, particularly for foliage that protrudes above the buildings (Tree4, Tree5), is the foliage drag and turbulence 83  production (+Vdrag). These same terms also reduce turbulent kinetic energy, and therefore vertical transport, from ground level up to the top of the foliage layer, and do so to a greater extent the more foliage protrudes above the buildings (Fig. 4.5b). Enhanced dissipation generated by foliage (+Vdiss) slightly enhances the above-canopy wind (Fig. 4.5a) by reducing turbulent kinetic energy within the foliage layer (Fig. 4.5b), thereby diminishing turbulent momentum flux down into the canopy. Modification of the length scales due to the presence of tree foliage (+Blengthv) modestly reduces both the wind and turbulent kinetic energy above foliage-top for Tree4 and Tree5 but has little effect for foliage in the upper half of the canopy (Tree3). Finally, modification of the building drag and turbulence production, due to the presence of tree foliage (+Bdragv), serves to only slightly increase streamwise velocity near the canopy floor and has virtually no impact on turbulent kinetic energy. These final two modifications (+Blengthv, +Bdragv) do make minor adjustments to the flow profiles, in particular for cases with tree foliage protruding above the rooftops. However, they are dwarfed by impacts on flow profiles generated by foliage drag and turbulence production (+Vdrag), to profiles of both streamwise velocity and turbulent kinetic energy, and by the enhanced dissipation (+Vdiss) on the spatially-average turbulent kinetic energy profiles.  4.3.2 Low and high building plan area densities Foliage drag and turbulence production (Vdrag-u & Vprod-k), and enhanced dissipation of turbulent kinetic energy by foliage (Vdiss-k), are found to significantly influence flow for P = 0.25. To determine whether any additional foliage-related source terms are significant at other built densities, select foliage scenarios from Sect. 4.3.1 are re-run for building plan area densities P = 0.00, 0.06, 0.11, and 0.44. Bdrag-u is included in the analysis, for comparison, and because it may be less important at lower P. Modification of length scales due to buildings (Blength-u,k) is included in all simulations. The highest leaf area density (LD = 0.50 m2 m-3) is evaluated for most tree layer heights at each P, as denser foliage is most likely to exert a significant impact on the flow. A more realistic neighbourhood average LD of 0.13 m2 m-3 is also evaluated for select scenarios. Particular attention is paid to scenarios where trees and buildings are approximately the same height (Tree2, Tree3, Tree4), where ‘interaction’ between the two is most likely. The following 84  scenarios are not studied: foliage deep within dense building arrangements (Tree1 at P = 0.44), where it will have little effect, or foliage well above sparse building densities (Tree5 at P = 0.06), where it will dominate.     Figure 4.6 Change in RMSD of the column model relative to the CFD model over z = 0 – 2 H and for spatially-averaged wind velocity (a, c, e) and turbulent kinetic energy (b, d, f), with the removal of each of eight building/tree foliage induced terms/modifications (Table 4.1). Several building plan area fractions (P = 0.00 [Forest], 0.06, 0.11, 0.44), tree layer heights, and foliage densities, are displayed.   The important terms are immediately clear from Fig. 4.6. Drag due to buildings and foliage (Bdrag and Vdrag) clearly dominate in most cases for both spatially-averaged streamwise 85  velocity and spatially-averaged turbulent kinetic energy. Enhanced dissipation (Vdiss) and modification of length scales (Blengthv) by the foliage are both of some importance, particularly at the lower building densities, and for turbulent kinetic energy and streamwise velocity, respectively. Turbulence production by building and foliage wakes (Bprod and Vprod), as well as interaction terms Bdragv and Bprodv, are of minimal importance across the full range of scenarios. Blengthv is the only term identified here in addition to those in Sect. 4.3.1 as being important to the correct reproduction of CFD flow profiles for urban canopies with trees, and even for lower, more realistic neighbourhood-average foliage densities such as LD = 0.13 m2 m-3 (Fig. 4.6). This effect on length scales by foliage is primarily of consequence for P = 0.06; it is already modest for P = 0.11. Furthermore, Blengthv, being a foliage-building interaction term, is complex in its variation with building and foliage density and distribution. In effect, its representation in the 1-D column model would require parameterization based on statistical fitting, i.e., based on relations derived from a suite of CFD simulations. This is judged to be outside the scope of the present contribution, and Blengthv is omitted from the parameterization. The impacts of this omission are explored in Sect. 4.4. Hence, considering results for both streamwise velocity and turbulent kinetic energy in Fig. 4.6, the results obtained for P = 0.25 hold for all P ≥ 0.11.  4.3.3 Source and sink terms for new parameterization Based on the results in Sects. 4.3.1 and 4.3.2, only four terms in Table 4.1 are required to parameterize the effects of buildings and foliage on the flow (strictly, for P ≥ 0.11):  A) Bdrag-u in the momentum equation (Eq. 4.12):    1 uUCB DBvD ;  B) Vdrag-u in the momentum equation (Eq. 4.12):    2 uUCL DVDp ;  86  C) Vdiss-k in the turbulent kinetic energy equation (Eq. 4.14):    7 UkCL DVDd ; and  D) Blength-u,k, with consequences for both momentum and turbulent kinetic energy equations (Eq. 4.16): 9ClCl bv .  Additionally, two other terms are marginally important but are implied by the corresponding drag terms in the momentum equation (Raupach and Shaw, 1982), and so are included in the turbulent kinetic energy equation in order to conserve total (mean plus turbulent) kinetic energy:  E) Bprod-k in the turbulent kinetic energy equation (Eq. 4.14): 53UCB DBvD ; and   F) Vprod-k in the turbulent kinetic energy equation (Eq. 4.14):   63UCL DVDp .   Modification of length scales due to the presence of foliage, Blengthv, is of some importance for P < 0.11 (approximately). However, it is neglected for reasons discussed in Sect. 4.3.2, and impacts of this omission are evaluated during parameterization testing (Sect. 4.4.2). Overall, important effects due to buildings and tree foliage, represented in the model as source terms, are the slowing of mean wind by both buildings and trees (via drag terms), enhanced dissipation of turbulence by buildings (via length scale reduction) and tree foliage (via short-circuit term), and the reduction of vertical turbulent transport in and immediately above the building canopy (via length scale reduction). Drag by buildings and foliage generates pronounced wind gradients above the canopy, and therefore shear production, indirectly. Impacts of foliage on length scales are ignored here but are of moderate importance for select scenarios: i.e., those with higher foliage densities, and which are either unaccompanied by buildings or 87  whose foliage tops are below or approximately coincident with building height for low building densities. Most significantly, terms that represent the interactions between buildings and trees in the current formulation, Bdragv-u and Bprodv-k, are unimportant across all scenarios simulated here. Hence, buildings and trees may be treated independently in terms of their effects on the spatially-averaged flow (e.g., Dupont et al., 2004); for example, while drag force due to buildings is affected by the presence of tree foliage, because mean wind is slowed by foliage, building drag coefficient need not be modified due to the presence of tree foliage. Thus, parameterization of the effects of urban neighbourhoods with trees on flow can be greatly simplified. Finally, results presented in this section arise in the context of the simple representation in the CFD of neighbourhood configuration and building and foliage impacts, and of the choice of wind direction perpendicular to building faces. Furthermore, they are impacted by neglect in the column model of the effects of dispersive motions in the CFD (e.g. Section 4.2.3). Hence, the relative importance of each term may be impacted by these approximations.  4.4 Parameterization of tree foliage and building impacts on flow To address objectives C and D in Sect. 4.1, the parameterization of the effects of buildings and trees on the flow determined in Sect. 4.3 is tested against CFD model results, and the range of scenarios for which foliage-related terms 2, 6, and 7 (Table 4.1) are required is determined. The column model is run with only the six source terms or modifications denoted by A-F in Sect. 4.3.3: Bdrag-u and Vdrag-u in the momentum equation (Eq. 4.12), Vdiss-k, Bprod-k, and Vprod-k in the turbulent kinetic energy equation (Eq. 4.14), and Blength-u,k, modification of the length scales due to the presence of buildings (impacting both equations). The RMSD is determined for all scenarios, and profiles of spatially-averaged quantities are compared for select cases. Building-induced length scale modifications ( in Eq.) and drag coefficients (CDB) are parameterized in this section, contrary to Sect. 4.3 and in Appendix B, where building-induced length scale modifications and drag coefficients are output directly from the CFD (Fig. 4.1). Hence, this section tests the efficacy with which the 1-D model acts as an urban canopy 88  parameterization for neighbourhoods with trees. In particular, its ability to reproduce spatially-averaged flow profiles with the new parameterization of urban tree foliage is evaluated. RMSD between the column model and the CFD model in this section arises from model differences (1-D vs. 3-D, k-l vs. k-), as in Sect. 4.3 and Appendix B, and additionally from neglect of foliage-building interaction and foliage impacts on length scales, and from parameterization of building-related terms CDB and lb (i.e., imperfect inputs), which are derived from Santiago and Martilli (2010). However, coefficients in their relations for drag coefficients and length scales for non-treed neighbourhoods require updating.  4.4.1 Parameterization of building impacts: revised Santiago and Martilli (2010) parameters CDB (Eq. 4.13; Cdeq in Santiago and Martilli, 2010) and lb (Eq. 4.16) are computed using the Santiago and Martilli (2010) parameterization with updated parameters. The Santiago and Martilli (2010) simulations are re-done with the stricter convergence criteria possible due to increased computational power. Slight changes in vertical profiles of the mean wind and turbulent kinetic energy are observed, as well as more important differences in distribution of pressure over obstacle faces. Parameters are updated based on current CFD results for block arrays without porous media (i.e., without trees) at the following densities: P = 0.06, 0.11, 0.16, 0.25, 0.33, and 0.44. The updated parameterization of building drag coefficient and length scales, for scenarios without tree foliage (i.e., block arrays), is as follows.  Drag coefficient (CDB):  33.0,67.3)(   forCDB         (4.17a) 33.0,30.7)( 62.0   forCDB          (4.17b)  where  is either P or F, the building frontal area density.  Length scales (lb):  89    0.1/,)(1  HzfordHCl b        (4.18a)   5.1/0.1,)(1  HzfordzCl b        (4.18b)   5.1/,)(22  HzfordzCl b         (4.18c)  where 1 = 1.95 and 2 = 1.07 are the revised values. While 2 does not vary with , 1 is a weak function of . However, determination of this relationship is left as future work, and 1 is assumed constant here, as in Santiago and Martilli (2010) and Simon-Moral et al. (2014). 1 is determined by minimizing both the median and maximum RMSD of spatially-averaged streamwise velocity between H and 2H over all . RMSD of spatially-averaged streamwise velocity, turbulent kinetic energy, and Reynolds stress below H, and turbulent kinetic energy and Reynolds stress between H and 2H, are not sensitive to the choice of 1. Note that Simon-Moral et al. (2014) find similar values for aligned arrays of cubes: 1 = 2.19 and 2 = 1.20. Hence, length scales are also relatively insensitive to building configuration.  Displacement height (d):  15.0)(  Hd ,           (4.19)  This matches the formulation obtained by Simon-Moral et al. (2014) for aligned arrays of cubes; this suggests, as they mention, that displacement height is not sensitive to configuration for regular arrays of cubes, at least for wind direction normal to cube faces.  Finally,  21212 )(15.1)(  dHd  .        (4.20) 90   The variable  in Eqs. 4.17-4.20 is intentionally not identified as either P or F. Because P = F for all scenarios studied here, it is not clear which is more relevant in each equation. However, it is most likely that displacement height is a function of P, whereas drag coefficient depends more strongly on F.   P RMSD of  RMSD of  z = 0 - H z = H - 2H z = 0 - H z = H - 2H 0.06 0.14 0.79 0.42 0.10 0.11 0.11 0.68 0.40 0.11 0.16 0.15 0.35 0.42 0.12 0.25 0.36 0.26 0.48 0.12 0.33 0.34 0.44 0.46 0.11 0.44 0.20 0.62 0.45 0.11      mean 0.21 0.52 0.44 0.11 median 0.17 0.53 0.44 0.11 maximum 0.36 0.79 0.48 0.12  Table 4.2  RMSD of  and  between the CFD model and the column model with updated Santiago and Martilli (2010) parameters, as a function of building density (P) and for scenarios without tree foliage.   4.4.2 Testing of urban canopy parameterization of building and tree impacts on flow  This section presents testing of the new parameterization of building and tree impacts on spatially-averaged flow. It also includes an assessment of the relative importance of the tree foliage-related terms across a range of scenarios; that is, when is the current parameterization required over and above the Santiago and Martilli (2010) parameterization for building-only scenarios? The column model is run for all scenarios in Sect. 4.3, and RMSD compared to CFD model results is determined. Only the six source terms identified in Sect. 4.3.3 are included to represent the effects of buildings and tree foliage; building-tree interaction terms (terms 3 & 4 in Table 4.1) and effects of trees on the length scales (term 8) are not included. These six source u ku k91  terms comprise the proposed parameterization (Sect. 4.3.3), which is compared with CFD results. Building impacts on drag and length scales are parameterized according to Santiago and Martilli (2010) with updated coefficients from Sect. 4.4.1. RMSD ≤ 0.5 is considered acceptable, and RMSD > 1.0 is considered to be poor performance; recall that RMSD in all cases is for   and k  normalized by u and u2, respectively.  The updated parameterization of building terms in Sect. 4.4.1 is first assessed (Table 4.2). Mean RMSD across all P scenarios for spatially-averaged streamwise velocity is 0.21 in the canopy and 0.52 above, as compared to the mean RMSD values of 0.37 in the canopy and 0.71 above reported in Santiago and Martilli (2010). Corresponding values for spatially-averaged turbulent kinetic energy are 0.44 and 0.11, which are comparable to those in Santiago and Martilli (2010): 0.46 and 0.19. Hence, the new parameterization coefficients for building drag and length scales improve the ability of the column model to predict streamwise velocity, whereas turbulent kinetic energy is predicted with similar accuracy. Subsequently, scenarios from Sect. 4.3.1, with variable foliage height and density at P = 0.25, are simulated with the new parameterization implemented in the column model. For the range of scenarios considered, spatially-averaged streamwise velocity and turbulent kinetic energy vary more with foliage height than with foliage density, and this is reproduced by the parameterization (Fig. 4.7). As noted in previous sections, foliage reduces RMSD in the canopy, in particular when it is located above the buildings, and when it has greater density. The parameterization captures this phenomenon, which is particularly apparent in the upper canopy for k  (Fig. 4.7c, d). Largest errors for both  and k  tend to appear in the building shear zone (i.e., z ≈ H), or well above the canopy (Fig. 4.7). Simulations for the complete range of combinations of building densities, foliage heights and foliage densities sampled in Sects. 4.3.1 and 4.3.2, with the new parameterization in the column model, yield RMSD relative to the CFD model of ≤0.43 for spatially-averaged turbulent kinetic energy, and ≤0.40 for spatially-averaged streamwise velocity in the canopy (Table 4.3). The column model is not able to reproduce above-canopy streamwise velocity as well, and largest errors result for foliage layers within the building canopy (i.e., Tree1, Tree2, Tree3) at lower building densities (P = 0.11, and especially 0.06) or higher building densities (P = 0.44). uu92  This phenomenon is reduced for P = 0.25 (Fig. 4.7).  Notable also is that lower foliage densities can exacerbate RMSD (Table 4.3).      Figure 4.7 Profiles of spatially-averaged streamwise velocity (a, b) and spatially-averaged turbulent kinetic energy (c, d) from CFD output (symbols) and the column model with the new parameterization (lines) for P = 0.25. Panels (a) and (c) show variation with foliage density for Tree5; panels (b) and (d) show variation with foliage height for LD = 0.25 m2 m-3.    93   Foliage height LD (m2 m-3) RMSD of  (RMSD of ) RMSD of  (RMSD of ) 0 ≤ z ≤ H H ≤ z ≤ 2H 0 ≤ z ≤ H H ≤ z ≤ 2H 0.00 Tree1 0.50 0.19 (-100%) 0.78 (-99%) 0.15 (-87%) 0.18 (-90%) 0.06 Tree2 0.13 0.33 (-79%) 1.49 (-34%) 0.31 (-73%) 0.19 (-41%) 0.06 Tree3 0.13 0.15 (-92%) 1.43 (-57%) 0.14 (-90%) 0.23 (-60%) 0.06 Tree1 0.50 0.40 (-78%) 1.41 (-27%) 0.37 (-64%) 0.18 (-32%) 0.06 Tree2 0.50 0.28 (-88%) 1.24 (-49%) 0.22 (-85%) 0.17 (-51%) 0.06 Tree3 0.50 0.09 (-97%) 1.03 (-72%) 0.08 (-96%) 0.21 (-68%) 0.06 Tree4 0.50 0.06 (-97%) 0.53 (-89%) 0.12 (-94%) 0.18 (-86%) 0.11 Tree2 0.13 0.23 (-65%) 0.54 (-24%) 0.32 (-51%) 0.14 (-23%) 0.11 Tree3 0.13 0.11 (-88%) 0.97 (-45%) 0.13 (-87%) 0.13 (-64%) 0.11 Tree2 0.50 0.30 (-73%) 0.44 (-44%) 0.25 (-72%) 0.13 (-36%) 0.11 Tree3 0.50 0.08 (-94%) 0.75 (-65%) 0.07 (-95%) 0.13 (-72%) 0.11 Tree4 0.50 0.06 (-96%) 0.59 (-85%) 0.09 (-94%) 0.19 (-84%) 0.11 Tree5 0.50 0.07 (-95%) 0.17 (-96%) 0.09 (-94%) 0.10 (-94%) 0.25 Tree1 0.13 0.33 (-15%) 0.42 -(0%) 0.44 (+4%) 0.11 (0%) 0.25 Tree2 0.13 0.39 (-10%) 0.47 (+1%) 0.42 (-7%) 0.11 (0%) 0.25 Tree3 0.13 0.33 (-16%) 0.27 (+23%) 0.19 (-72%) 0.05 (0%) 0.25 Tree4 0.13 0.14 (-65%) 0.54 (-79%) 0.14 (-84%) 0.12 (-67%) 0.25 Tree5 0.13 0.15 (-67%) 0.39 (-90%) 0.15 (-83%) 0.20 (-76%) 0.25 Tree1 0.50 0.23 (-39%) 0.43 (0%) 0.40 (-1%) 0.11 (0%) 0.25 Tree2 0.50 0.33 (-28%) 0.49 (+3%) 0.35 (-28%) 0.10 (0%) 0.25 Tree3 0.50 0.10 (-80%) 0.24 (+198%) 0.03 (-96%) 0.01 (0%) 0.25 Tree4 0.50 0.09 (-88%) 0.66 (-81%) 0.05 (-96%) 0.21 (-78%) 0.25 Tree5 0.50 0.09 (-87%) 0.30 (-93%) 0.02 (-98%) 0.16 (-88%) 0.44 Tree3 0.13 0.18 (-18%) 0.75 (-12%) 0.28 (-55%) 0.08 (-32%) 0.44 Tree2 0.50 0.19 (-20%) 0.84 (0%) 0.43 (-3%) 0.10 (0%) 0.44 Tree3 0.50 0.11 (-63%) 0.81 (-13%) 0.08 (-90%) 0.05 (-53%) 0.44 Tree4 0.50 0.11 (-76%) 0.66 (-85%) 0.05 (-95%) 0.24 (-76%) 0.44 Tree5 0.50 0.09 (-80%) 0.43 (-92%) 0.04 (-96%) 0.24 (-84%)  Maximum RMSD at each P for LD = 0.500 m2 m-3 P      0.06 - 0.50 Tree1: 0.40 Tree1: 1.41 Tree1: 0.37 Tree3: 0.21 0.11 - 0.50 Tree2: 0.30 Tree3: 0.75 Tree2: 0.25 Tree4: 0.19 0.25 - 0.50 Tree2: 0.33 Tree4: 0.66 Tree1: 0.40 Tree4: 0.21 0.44 - 0.50 Tree2: 0.19 Tree2: 0.84 Tree2: 0.43 Tree4/5: 0.24  Table 4.3 RMSD of  and  with the proposed parameterization as a function of building density and tree foliage height for foliage area densities 0.50 m2 m-3 and LD = 0.13 m2 m-3. In brackets: percent change of RMSD from the Santiago and Martilli (2010) building-only parameterization due to the addition of foliage terms 2, 6 and 7 (Table 4.1), i.e. the foliage-related terms of the proposed parameterization. u u k ku k94  PFoliage height LD (m2 m-3) RMSD of /zu RMSD of k / zk 0 ≤ z ≤ H H ≤ z ≤ 2H 0 ≤ z ≤ H H ≤ z ≤ 2H 0.00 Tree1 0.50 0.10 0.20 0.16 0.10 0.06 Tree2 0.13 0.18 0.29 0.33 0.09 0.06 Tree3 0.13 0.10 0.34 0.23 0.13 0.06 Tree1 0.50 0.24 0.26 0.36 0.09 0.06 Tree2 0.50 0.23 0.25 0.36 0.09 0.06 Tree3 0.50 0.09 0.28 0.37 0.13 0.06 Tree4 0.50 0.06 0.22 0.73 0.16 0.11 Tree2 0.13 0.16 0.09 0.38 0.07 0.11 Tree3 0.13 0.08 0.20 0.22 0.07 0.11 Tree2 0.50 0.30 0.08 0.42 0.06 0.11 Tree3 0.50 0.10 0.17 0.35 0.07 0.11 Tree4 0.50 0.07 0.24 0.64 0.17 0.11 Tree5 0.50 0.06 0.10 0.60 0.13 0.25 Tree1 0.13 0.45 0.06 0.64 0.06 0.25 Tree2 0.13 0.60 0.07 0.68 0.05 0.25 Tree3 0.13 0.54 0.04 0.41 0.02 0.25 Tree4 0.13 0.21 0.14 0.42 0.08 0.25 Tree5 0.13 0.24 0.15 0.49 0.18 0.25 Tree1 0.50 0.33 0.06 0.67 0.05 0.25 Tree2 0.50 0.57 0.07 0.75 0.05 0.25 Tree3 0.50 0.20 0.04 0.18 0.01 0.25 Tree4 0.50 0.17 0.24 0.41 0.17 0.25 Tree5 0.50 0.15 0.17 0.19 0.21 0.44 Tree3 0.13 0.43 0.09 0.69 0.04 0.44 Tree2 0.50 0.45 0.10 1.07 0.05 0.44 Tree3 0.50 0.29 0.10 0.42 0.02 0.44 Tree4 0.50 0.31 0.21 0.61 0.18 0.44 Tree5 0.50 0.25 0.22 0.40 0.29  Maximum RMSD normalized by vertical mean, at each P, for LD = 0.500 m2 m-3 P      0.06 - 0.50 Tree1: 0.24 Tree3: 0.28 Tree4: 0.73 Tree4: 0.16 0.11 - 0.50 Tree2: 0.30 Tree4: 0.24 Tree4: 0.64 Tree4: 0.17 0.25 - 0.50 Tree2: 0.57 Tree4: 0.24 Tree2: 0.75 Tree5: 0.21 0.44 - 0.50 Tree2: 0.45 Tree5: 0.22 Tree2: 1.07 Tree5: 0.29   Table 4.4 RMSD of  and  normalized by vertical mean of  and , respectively, over the appropriate atmospheric layer. All other features are as in Table 4.3.  uu k u k95  PFoliage height Foliage density (m2 m-3) RMSD of  RMSD of  0 ≤ z ≤ H H ≤ z ≤ 2H 0 ≤ z ≤ H H ≤ z ≤ 2H 0.00 Tree1 0.50 -0% 0% 6% -3% 0.06 Tree2 0.13 -5% -38% -5% -11% 0.06 Tree3 0.13 -2% -32% -4% -26% 0.06 Tree1 0.50 -3% -34% 0% -2% 0.06 Tree2 0.50 1% -29% -2% -9% 0.06 Tree3 0.50 1% -24% -1% -25% 0.06 Tree4 0.50 -0% -10% -1% -13% 0.11 Tree2 0.13 2% 10% 4% 6% 0.11 Tree3 0.13 -1% -27% -1% -10% 0.11 Tree2 0.50 6% 20% 5% 11% 0.11 Tree3 0.50 1% -21% -1% -12% 0.11 Tree4 0.50 -1% -14% -2% -13% 0.11 Tree5 0.50 -1% -2% -3% -3% 0.25 Tree1 0.13 -2% -31% -1% 2% 0.25 Tree2 0.13 -2% -36% 1% 4% 0.25 Tree3 0.13 -1% -63% 2% 8% 0.25 Tree4 0.13 -12% -13% -9% -4% 0.25 Tree5 0.13 -15% -2% -13% -7% 0.25 Tree1 0.50 -3% -25% -2% 2% 0.25 Tree2 0.50 0% -31% 2% 6% 0.25 Tree3 0.50 -2% -38% 0% 8% 0.25 Tree4 0.50 -4% -13% -1% -9% 0.25 Tree5 0.50 -1% -3% 0% -4% 0.44 Tree3 0.13 -10% -50% 1% 19% 0.44 Tree2 0.50 -7% -23% -3% 6% 0.44 Tree3 0.50 -12% -58% 1% 47% 0.44 Tree4 0.50 -6% -5% -1% -4% 0.44 Tree5 0.50 -3% -1% -2% -4%  Maximum RMSD decrease at each P for LD = 0.500 m2 m-3 P      0.06 - 0.50 -3% Tree1: -34% -2% Tree3: -25% 0.11 - 0.50 -1% Tree3: -21% -3% Tree4: -13% 0.25 - 0.50 -4% Tree3: -38% -2% Tree4: -9% 0.44 - 0.50 Tree3: -12% Tree3: -58% -3% -4%  Table 4.5 Percent change of column model RMSD with addition of impacts of tree foliage on length scales (i.e., term 8 in Table 4.1) in addition to terms 2, 6 and 7; values are in addition to those in Table 4.4.  u k96   RMSD of spatially-averaged streamwise velocity and turbulent kinetic energy, further normalized by the vertical mean of spatially-averaged u  and k , respectively, and over the appropriate atmospheric layer (0 < z ≤ H, or H < z ≤ 2H), tells a different story. Because streamwise velocity increases with z, larger RMSD above the canopy is due in part to increased magnitude of u . Table 4.4 demonstrates that RMSD of streamwise velocity normalized by its layer mean is in fact of similar magnitude (≈20%) in both atmospheric layers, and slightly higher above-canopy for low P, and larger within-canopy for high P. RMSD of spatially-averaged turbulent kinetic energy normalized by its vertical mean is low above the canopy and substantially higher within the building canopy, in large part because k  is very small there due to the dampening effect of foliage. Overall, model performance relative to magnitude of mean and turbulent flow is good, with the exception of within-canopy turbulent kinetic energy (and within-canopy velocity for specific cases). These larger relative differences from the CFD are the result of very small values of u  and k  in the canopy, and may be less relevant to transport and dispersion. Bracketed values in Table 4.3 are percent changes of RMSD that result from inclusion of tree foliage-related terms in the proposed parameterization (terms 2, 6, and 7), in addition to the building-related terms of Santiago and Martilli (2010) and Sect. 4.4.1. For most scenarios they reduce RMSD of both  and k  within and above the canopy to a fraction of the value without the foliage terms; median reductions are 89% in the canopy and 71% above the canopy for the LD = 0.50 m2 m-3 scenarios. Select scenarios with foliage in the canopy are not greatly improved by the proposed parameterization, in particular those with foliage in the canopy at greater building density (P = 0.44). Overall, the proposed parameterization has a significant impact, even when foliage density is reduced from 0.50 m2 m-3 to 0.13 m2 m-3. Foliage in the canopy for P = 0.25 and 0.44 is the only subset of scenarios for which foliage terms are less consequential. Therefore, foliage components of the proposed parameterization are most critical for foliage that protrudes above the building canopy, and for foliage in the canopy for P < 0.25 (approximately). Low density tree foliage protruding above the buildings is always important; for example, median decrease in RMSD of  and k  over uu97  the two atmospheric layers, for Tree4 and Tree5 scenarios with P = 0.25 and LD = 0.06 m2 m-3, is 80% when foliage-related terms are included in addition to building-related terms. This is akin to the disproportionate effect on drag and k  production of tall, isolated buildings (Xie et al., 2008): low density foliage above the mean drag element height (the buildings) exerts a disproportionate influence, and additional foliage density results in ‘diminishing returns’ in terms of total drag by the elevated foliage.  If the impact of tree foliage on the length scales (Blengthv) is extracted from the CFD and applied to modify the length scales in the current parameterization (i.e., Blengthv is applied to the Santiago and Martilli [2010] parameterization in Sect. 4.4.1, i.e., Eq. 4.18), there is a further reduction of column model error relative to the CFD, on average (Table 4.5). Reduced RMSD of the above-canopy wind is particularly apparent for specific scenarios. Nevertheless, Blengthv is less effective than the proposed foliage parameterization (terms 2, 6, and 7) in terms of reducing RMSD for all but three scenarios: Tree1 and Tree2 for P = 0.06, and Tree3 for P = 0.44. Hence, foliage within the building canopy for sparse building canopies, and foliage tops at the same height as the building canopy for dense building arrangements, render foliage impacts on length scale important. However, adding Blengthv also increases RMSD for specific scenarios (Table 4.5). Furthermore, Blengthv as implemented here is extracted from the CFD, and any parameterization would not be expected to perform as well. Overall, addition of Blengthv increases median RMSD reduction from the building-only (Santiago and Martilli, 2010) parameterization from 89% to 90% within the canopy, and from 71% to 86% above the canopy, as compared to the proposed parameterization of impacts of tree foliage (i.e., only terms 2, 6, and 7 in Table 4.1). Notably, if Bdragv and Bprodv (terms 3 and 4 in Table 4.1, respectively) are extracted from the CFD and applied to the parameterized drag coefficients there is a small decrease in RMSD for low P with canopy foliage and a larger increase for P = 0.44 with canopy foliage; overall, RMSD is more often and substantially increased than decreased amongst scenarios considered in this section (not shown). As such, interaction terms Bdragv and Bprodv do not lower column model RMSD when implemented for the present suite of scenarios.  Considering these results (Tables 4.3, 4.4, and 4.5), neglect of foliage impacts on length scales, and on building drag coefficients in particular, is deemed justified. Parsimonious 98  parameterization of Blengthv is problematic given the range of potential building-foliage configurations. Moreover, its importance only approaches or exceeds the proposed parameterization of foliage impacts on the flow for very specific cases. Hence, parameterization of this term is left for future work. Here it is simply noted that the parameterization may perform less well when foliage is in the building canopy and P is approximately ≤0.11; foliage in the building canopy is relatively unimportant for P ≥ 0.25. Overall, the tree foliage-related terms included in the new parameterization (Sect. 4.3.3) are simple to include, they improve results in virtually all scenarios tested, often substantially, and hence they should be included in all scenarios that include tree foliage.  4.4.3 Extension of the parameterization: multiple building heights and clumping To apply the new parameterization for neighbourhoods with multiple building heights, building drag coefficient (CDB) at each height is determined from Eq. 4.17 based on the actual building density at that height. Length scales are determined based on Eq. 4.18 assuming H is the mean building height. This approach assumes that drag is accurately treated as a sectional phenomenon, that sectional drag coefficients determined from flow through regular arrays of cubes are generalizable, and that length scales are determined by mean building height regardless of building height distribution. While unlikely to be fully robust, these assumptions provide a first approach to modeling spatially-averaged flow for complex building geometries, and future work should evaluate their robustness. Tree foliage distributions in urban areas are typically clumped at the crown-neighbourhood scale, not distributed randomly or evenly as assumed here. Use of an effective LD, which is smaller than the actual LD, is a potential solution. Marcolla et al. (2003) suggest that effective LD for use in parameterization of drag may be based on the clumping coefficient used for radiation interception by foliage (e.g., Ch. 2). These methods are elaborated on in Ch. 5, and they should be further evaluated and explored in future work. Moreover, buildings may be ‘clumped’ at the neighbourhood-scale, an issue which has so far not been addressed in terms of urban canopy modelling.  99  4.5 Summary and conclusions Obstacle-resolving Computational Fluid Dynamics (CFD) simulations of neutral flow through canopies of blocks (buildings) that have various distributions and densities of porous media (tree foliage), are conducted, and the spatially-averaged impacts on the flow of these building-tree combinations are assessed. The CFD model, with standard k- turbulence scheme and standard parameterization of foliage effects on flow, is evaluated against two sets of wind tunnel measurements (Appendix C). The accuracy with which a one-dimensional (column) model with k-l turbulence scheme represents the spatially-averaged CFD results is assessed (Sect. 4.2.4, Appendix B). Representation of individual effects of trees and buildings in the column model are evaluated in terms of relative importance (Sect. 4.3), and the resulting parameterization is evaluated against the CFD results (Sect. 4.4).  This chapter presents a methodology for determining the source and sink terms required in the momentum and turbulent kinetic energy equations to represent the spatially-averaged impacts of tree foliage on flow in urban areas. It builds on the work of Santiago and Martilli (2010) for neighbourhoods without trees, and updated parameters for their parameterization of flow and turbulent exchange for non-treed urban neighbourhoods are presented in Sect. 4.4.1. Considering the effects of both buildings and trees on flow, terms deemed important and included in the proposed parameterization are the drag terms due to buildings and tree foliage in the momentum equation (and, although much less important, corresponding production terms in the turbulent kinetic energy equation for energy conservation), enhanced dissipation of turbulent kinetic energy by the small tree foliage wakes in the turbulent kinetic energy equation, and the modification of length scales due to buildings. The most notable finding is that trees do not significantly affect the efficiency with which buildings exert drag on the flow and produce turbulence, i.e., building sectional drag coefficients do not require modification due to the presence of tree foliage to accurately predict spatially-averaged flow profiles in and above treed urban canopies. In other words, the impact of buildings on the flow relative to flow forcing is not affected by the presence of tree foliage; therefore, impacts of trees and buildings on the spatially-averaged flow can be represented independently in the prognostic equations for momentum and turbulent kinetic energy. Hence, sheltering between buildings and trees is not significant, using a definition of sheltering analogous to that of 100  Thom (1971) for the plant element scale. However, the presence of trees significantly affects the absolute value of the drag force exerted by buildings, and vice versa, for a given driving pressure gradient. Hence, tiling urban and natural surface-atmosphere exchange schemes is inadequate, and integrated treatment of flow dynamics for urban neighbourhoods with trees is essential. Impacts of tree foliage on length scales are also neglected in the new parameterization; for most of the scenarios considered, these effects are dwarfed by the terms included in the parameterization. Neglect of foliage impacts on length scales causes significant errors for select cases, in particular for scenarios with foliage that is below or vertically coincident with sparse buildings (P ≤ 0.11). Hence, this is a limitation of the new parameterization, and resolution of this matter is left to future work. Overall, results indicate that tree foliage with spatial-average leaf area density ≥0.06 m2 m-3 (the lowest density tested here) significantly impacts the spatially-averaged mean and/or turbulent flow for a range of building densities 0.06 < P ≤ 0.44 if foliage protrudes above buildings. It also significantly affects the spatially-averaged flow if foliage is at or below building canopy height for P ≤ 0.11, and possibly for 0.11 < P < 0.25. The proposed parameterization of tree foliage impacts on the flow (Sect. 4.3.3) should be included for these scenarios in addition to the Santiago and Martilli (2010) parameterization for building-only neighbourhoods. Because it is easy to include, it is recommended that it be included in all simulations that include tree foliage. The new parameterization and column model, i.e. the whole urban canopy parameterization, will be incorporated in an urban canopy model for urban neighbourhoods with trees in Ch. 5. Conclusions presented herein are for neutral conditions, and derive from a specific wind direction (perpendicular to cube faces), and a specific range of configurations: staggered cubic arrays with interspersed and homogeneously-distributed foliage layers of thickness H/2, where H is building height. While recent work shows significantly different flow effects for different wind directions (Santiago et al. 2013a, Buccolieri et al. 2011), as well as for more realistic urban morphologies as compared to the uniform height block arrays used here (Kanda et al., 2013), the focus of the present chapter is an important step toward inclusion of trees in the parameterization of neighbourhood-scale urban flow (e.g., for mesoscale modeling purposes). Moreover, tree foliage serves to reduce the specificity of dispersive motions that are introduced by constraining 101  modeled urban configurations to arrays of blocks. Regardless, future studies with flow models of greater accuracy and/or thermal stratification and/or more realistic urban surface configurations and/or wind direction variation can draw on the methodology presented here. Additionally, results are valid for scenarios for which 1.0DVD CL m2 m-3, since the maximum values of LD and CDV chosen are 0.50 m2 m-3 (i.e., LAI = 2.24-3.75, depending on P) and 0.2, respectively. Neighbourhood-average LD is unlikely to rise substantially above 0.50 m2 m-3 for most neighbourhoods; urban tree canopy cover rarely exceeds 50% (e.g., Nowak et al., 1996), and LD for most tree species is less than 1-2 m2 m-3. CDV of 0.20 is a commonly cited drag coefficient for trees, and tree foliage will exert a smaller impact on the flow for any values <0.20.  Nevertheless, it is unlikely that modestly higher values of DVD CL  would produce results that depart radically from those presented here. Overall, results presented herein are specific to these conditions and also contingent on the accuracy of the CFD model and the parameterization of foliage impacts (i.e., construction of the source and sink terms) in both the CFD and the column model (Sect. 4.2). Nonetheless, this is the first CFD analysis of the interaction between bluff obstacles and porous media in terms of the spatial-average mean flow and turbulence, and it provides justification for a simple approach to the representation of tree foliage impacts on flow in urban environments.  102  Chapter 5: BEP-Tree: A multi-layer urban canopy model for neighbourhoods with trees  The radiative and dynamic model developments of chapters 2 and 4 are combined with a multi-layer urban canopy model to assess their functionality in a full simulation of neighbourhood-scale flow and exchange of heat and humidity. The combined model, called BEP-Tree, is the first multi-layer model of urban energy exchange and flow at the neighbourhood scale that includes trees and their effects on the presence of buildings. Model design is first described in Sect. 5.1, including the host urban model and treatment of the tree foliage energy balance. Subsequently, the full model is compared to available measurements at the Sunset location in suburban Vancouver (Sect. 5.2). Finally, in Sect. 5.3 a simple sensitivity experiment is undertaken to demonstrate the effects of trees on overall energy balance, climate and dynamic characteristics.  5.1 BEP-Tree model design  Radiative and dynamic model developments for the inclusion of urban tree foliage, in chapters 2 and 4, respectively, are designed to integrate with most urban canopy models. They are particularly suited to multi-layer models, and the radiation model is based on the urban canyon. Here, these model developments are incorporated into the latest version of the Building Effect Parameterization (BEP), originally by Martilli et al. (2002). BEP is a multi-layer, neighbourhood-scale, urban meteorological and dispersion model based on the urban canyon. The version of BEP used here has been updated to include CFD-derived building drag and length-scale effects (Santiago and Martilli, 2010). It has also been combined with a column model of vertical turbulent exchange in the urban atmosphere, similar to Ch. 4, except that it now includes conservation equations for heat and humidity, and buoyant production of turbulent kinetic energy. In this chapter, the following model processes are also added: thermal effects on turbulent length scales, and tree foliage energy balance and temperature, including leaf sensible and latent heat fluxes.  103  5.1.1 Column model for non-neutral urban surface layer  The one-dimensional column model with k-l turbulence closure in Sect. 4.2.2 forms the dynamical basis for the complete urban canopy model with trees. As in Ch. 4, equations for vertical turbulent transport of horizontal-mean, Reynolds-averaged velocity (Eq. 4.6) and turbulent kinetic energy (Eq. 4.10) are solved. Both x- and y-components of velocity (u, and v, respectively) are solved independently. Source and sink terms representing drag and turbulence for urban canopies with trees, as distinguished in Sect. 4.3.3, are included.  The conservation equation for the x-component of velocity (Eq. 4.6), assuming incompressibility, horizontal homogeneity, and no mean vertical velocity, becomes:       VIImdVDVDDBvDIImVIIVIIuUfcuUCLCBzuKzuxpzwutu~~1''2,   (5.1)  where term II is divergence of vertical turbulent momentum flux, term V is form drag due to the pressure differential across tree foliage elements and buildings, term VII is skin drag due to roads and roofs; U = (u2 + v2) 0.5 is mean wind speed, cd = k2 / (ln(z/z0))2 is a coefficient of skin drag for horizontal surfaces for neutral stability, and fm is a corresponding stability function. Both cd and fm depend on surface roughness and height above the surface, and fm additionally depends on Richardson number. Term II is parameterized following Eq. 4.7 (K-theory), and term V is parameterized as in Eq. 4.12, where F is used to calculate form drag (Eq. 4.17), whereas P is input to compute displacement height and hence length scales (Eqs. 4.19 and 4.18). Term VII is parameterized by Eq. 13 of Martilli et al. (2002), which follows the Monin-Obukhov similarity approach of Louis (1979); hence, thermal effects are explicitly included in the computation of skin drag. Divergence of dispersive flux and acceleration due to large-scale pressure gradient are both ignored; that is, neighbourhood-average horizontal pressure gradient below forcing height is assumed zero. The balance equation for the y-component of velocity (v) is identical, with x 104  replaced by y, and all instances of u replaced by v; built (BD) and foliage (LD) area densities are assumed directionally uniform. As in Ch. 4, angle brackets indicate a horizontal-average, overbars indicate Reynolds (ensemble) averaging, and tildes indicate dispersive fluctuations. Buoyant effects are irrelevant due to the assumption of horizontal homogeneity, and therefore of zero mean vertical velocity. Effective leaf area density LD is used for foliage drag in Eq. 5.1, as per Marcolla et al. (2003), where  is total neighbourhood foliage clumping (i.e., includes all scales). Turbulent kinetic energy, k, is required to determine the vertical turbulent exchange coefficient (Km) via Eq. 4.8. The conservation equation for spatially-averaged mean k is Eq. 4.10, with buoyant production (term IX) included, and dispersive flux of k removed:    XkIXmoVIIIDVDdVIIDVDDBvDVIIIImIImszPrKgUkCLUCLCBzkPrKzzvzuKtk        322, (5.2)  where  is potential temperature and o is reference potential temperature, and g is the acceleration due to gravity. Term II is shear production; term III is vertical turbulent transport; term VI is dissipation; term VII is wake production due to building and foliage drag (Eq. 4.14); and term VIII is enhanced dissipation due to the small scale of foliage elements (Eqs. 4.3 and 4.14). Term X is shear production by roofs and roads, which is calculated as in Martilli et al. (2002). The turbulent Prandtl number Pr, the ratio of eddy diffusivities for momentum and heat, was measured to be ≈0.50 by Raupach et al. (1996) in the above-canopy roughness sublayer. Values as low as 0.25 have been found in the ‘urban’ canopy for non-neutral CFD simulations; Pr appears to be highly dependent on the particular combination of flow and heating, and hence could be a significant source of uncertainty (A. Martilli, personal communication). Pr = 0.25 is chosen here as the one act of model ‘tuning’, because higher values yield too little venting from 105  the canopy and air temperatures become unrealistic. Pr may be this low on average, or this low value may compensate for processes that are not represented, or not fully captured, such as dispersive fluxes and thermal modification of turbulent length scales.  Length scales for calculation of dissipation  (Eq. 4.11) and turbulent viscosity Km (Eq. 4.8) are also affected by stability. This effect is observed in recent CFD simulations for arrays of cubes with ‘realistic’ distributions of sensible heat flux imposed for several times throughout the day (Santiago et al. 2014). An urban length scale Lurb, analogous to the Obukhov length scale, is defined:  PHourbcQguL3*,          (5.3)  where u* and QH are the Reynolds stress and sensible heat flux just above the roughness sublayer, respectively,  is air density, and cP the heat capacity of air (J kg-1 K-1). From the CFD data, modification of length scales for neutral conditions (Eq. 4.18) due to thermal effects is determined here as follows:  0.1/,0.3,min5.01)()(0.1/,0.3,min3.02.01)()(HzforLHzlzlHzforLHHzzlzlurbbbTurbbbT,   (5.4)  where H is the mean building height. There are no available CFD simulations for H / Lurb > 3.0 and so the parameterization does not extrapolate beyond that point. The present treatment of thermal effects on mixing via the Prandtl number and modification of length scales is a first approach, and more robust treatment of this issue is left as future work.  To complete the 1-D micrometeorological model, equations for vertical turbulent transport of spatial- and ensemble-average potential temperature, specific humidity and a passive tracer are solved in addition to those for velocity and turbulent kinetic energy. Assuming 106  horizontal homogeneity and zero mean vertical velocity due to the assumed incompressibility, the equation for conservation of potential temperature is:  AVbVcWrWlGRm ssssssszPrKzt  ,    (5.5)  where the s are the source terms due to sensible heat exchange with roofs, roads, walls (left and right sides of canyon), tree foliage (canyon and building columns), and radiative divergence (sA), respectively. Distributed and boundary sources are mixed in Eq. 5.5 because buildings and trees are not resolved (neighbourhood averaging). The conservation equation for moisture has fewer source terms than that for heat, because built surfaces are not moisture sources in the current model; addition of urban hydrology to the model (e.g., Jarvi et al. 2011; Ramamurthy and Bou-Zeid, 2014) is left as future work. The conservation equation for specific humidity is:  VbqVcqm sszqPrKztq  ,        (5.6)  where sqVc and sqVb are sources of moisture from canyon column and building column vegetation, respectively. The equation for the passive tracer is the same as Eqs. 5.5 and 5.6, only with source term location and magnitude defined by the user.  5.1.2 Temperature and water vapour source terms: built surfaces, tree foliage, ambient air  Sources terms for heat and moisture in Eqs. 5.5 and 5.6, respectively, are now elucidated.  Sensible heat fluxes from roofs (sR) and roads (sG) are determined by the Louis (1979) formulation of Monin-Obukhov Similarity Theory, whereas sensible heat flux from walls (sWl, sWr) follows a stability-independent bulk transfer formulation that depends on wind speed (see Martilli et al. 2002, Eqs. 15 and 16). 107  Radiative divergence (sA) is obtained by conversion to heating rate of longwave interception by constituents in the ambient air (e.g., water vapour):      columnbuildingAbPcolumncanyonAcPLPairabsA LLcks 114 ,      (5.7)  where LAc and LAb are the absorbed flux densities of longwave radiation in the canyon and building air layers (Sect. 2.2.4.2), abs is the density of absorbing molecules and kair is their mass extinction cross section (see Sect. 2.2.3.2), while  is the fraction of buildings at a given height (see Ch. 2), and L = (1 – P) + P (1 - ) is the fraction of total volume that is outdoor air. Sources of heat from tree foliage are:            columnbuildingAVbPDBcolumncanyonAVcPDCLPPMHaVbVc LLccgss 112   (5.8)  where the factor 2 indicates that sensible heat flux occurs at both sides of the leaf, Vc and Vb are tree foliage (leaf) potential temperatures in canyon and building columns, respectively, and A is air potential temperature, cPM is the molar heat capacity for air (mol s-1 m-2), and gHa is the conductance for heat in mol m-2 s-1 across the boundary-layer on one side of a leaf (Campbell and Norman, 1998):  lHa dUg 135.04.1 ,         (5.9)  where 0.135 is empirically-derived for laminar forced convection, the factor 1.4 is for turbulent outdoor environments, and velocity local to the leaf is ideal but not available, hence spatially-averaged wind speed U is used. The characteristic dimension of the leaves dl is 0.72 multiplied by average leaf width, which is set by the user. 108  Sources of moisture in the current model are limited to tree foliage:                      columnbuildingAVbPDBcolumncanyonAVcPDCLvMcolumnbuildingaVbsPDBcolumncanyonaVcsPDCLvMVbqVcqpDTTsLpDTTsLgeTeLeTeLpgss11)(1)(1,             (5.10)  where the linearization is according to Campbell and Norman (1998), originally by Penman (1948). M is molar latent heat of vaporization (J mol-1),  is latent heat of vaporization in J kg-1, es is saturated vapour pressure, ea is actual vapour pressure, p is atmospheric pressure, D is the vapour deficit of the atmosphere, s = / p, where  = des(T) / dT, and gv is the average surface and boundary-layer conductance for humidity for the whole leaf (mol m-2 s-1):  vaadvsvaadvsvaabvsvaabvsv ggggggggg ,         (5.11)  where gvs is the molar stomatal conductance, “ab” and “ad” refer to the abaxial and adaxial sides of leaves, respectively. The molar boundary conductance for humidity, gva, is assumed to be the same for both sides of the leaf, and its empirically-derived coefficient differs only slightly from that of gHa:  lva dUg 147.04.1 .          (5.12)  Plant species that have stomata on both sides are termed amphistomatous, whereas those with stomata on only one side are hypostomatous. Most urban trees are hypostomatous, and so gv is 109  computed with only the first term on the right side of Eq. 5.11 in the current model unless otherwise specified. Note that foliage clumping coefficient () does not appear in Eqs. 5.8 or 5.10 because all leaves are assumed to contribute equally to turbulent heat exchange, despite the fact that some are more radiatively active than others (i.e., those at the interior of foliage clumps). A more complex and robust approach, and an opportunity for future work, is to calculate leaf energy balance and therefore source terms (Eqs. 5.8 or 5.10) for sunlit and shaded leaves separately (e.g., Baldocchi, 1997).  5.1.3 Foliage and built surface energy balances and surface temperatures  Surface energy balance of each roof and wall layer, and of the roads, is identical to Martilli et al. (2002), and does not include latent heat flux. Shortwave and longwave exchange computed by the radiation scheme in Ch. 2 drive the energy balance at each surface:      01 4  GPHPPPPPP QQTLK  ,      (5.13)  where P is one of R (roofs), Wl (left wall), Wr (right wall), or G (ground), and total incident shortwave KP and longwave LP after infinite reflections are determined as in Sect. 2.2.4.1. Conduction heat flux (QG) is determined by Fourier diffusion (Martilli et al., 2002; their Eq. A1), and sensible heat flux is determined as discussed at the beginning of Sect. 5.1.2. Leaf energy balance at a given height in a given column (canyon, or building) is expressed as follows, neglecting storage heat flux:      0221 4  EVHVVVVVV QQTLK  ,     (5.14)  where 2  KV (2  LV) is incident shortwave (longwave) flux density, averaged over one side of all leaves, after reflections (Sect. 2.2.4.1; see Eqs. 2.13d/e, 2.23, 2.26), V is foliage scattering coefficient (reflection plus transmission; see Sect. 2.2.3.1), V is foliage emissivity, QHV is flux density of sensible heat from both sides of foliage: 110    AVHaPMHV TTgcQ  2 ,         (5.15)  and QEV is flux density of latent heat flux from foliage:        pDTTsgpeTegQ AVvMaVsvMEV )(,     (5.16)  where the second, approximate version is linearized (Campbell and Norman, 1998; their Eq. 14.4). The vapour conductance gv (Eq. 5.11) accounts for evaporation from one vs. both sides of leaves, depending on stomatal distribution. A linearized, approximate form of the energy balance equation (Campbell and Norman, 1998; their Eq. 14.6) is rearranged to solve for the leaf or foliage temperature at each timestep, in each layer and model column:       *1** 4pDcgTLKsTT PMHrAVVVVAV,     (5.17)  where * = gHr / gv, gHr = gHa + gr is the convective-radiative conductance, and the radiative conductance gr derives from the linearization of the longwave emission term:   AVrPMAVVV TTgcTT  44  .       (5.18)  5.1.4 Diffuse shortwave radiation The new radiation model described in Ch. 2 requires diffuse shortwave flux as an input, which the original BEP model of Martilli et al. (2002) did not. Therefore, a shortwave scheme based on the Bird and Hulstrom (1981) model, as presented in Iqbal (1983), is added to robustly parameterize the incident direct and diffuse shortwave radiation. While simple, it demonstrates excellent performance relative to both observations and more complex models (Gueymard and 111  Myers, 2008), and it has been used successfully by Krayenhoff and Voogt (2007), Krayenhoff and Voogt (2010), and Stewart et al. (2014).  5.2 Model evaluation: Sunset neighbourhood  Measurements on a 28.8 m tower (49.2261 °N, 123.0784 °W, WGS-84) in the Sunset neighbourhood of Vancouver (Christen et al. 2013) during the EPiCC measurement campaign (Christen et al. 2010; Christen et al. 2011; Crawford and Christen, 2014a; Jarvi et al. 2011; Voogt et al. 2010) are used to evaluate the model. Sunset is a residential neighbourhood with north-south and east-west gridded streets, and it is classified as ‘Open Low-rise’ in the Local Climate Zone scheme of Stewart and Oke (2012). It is chosen because it has a significant leaf area index (LAI = 0.39 m2 m-2; Liss et al. 2010), about half of which is located above the mean building height (H ≈ 5 m) – hence, trees are expected to be impactful. Moreover, LiDAR mapping of the area has generated detailed information about the vertical distributions of built volume (van der Laan et al. 2011) and tree foliage (Goodwin et al. 2009; R. Tooke, personal communication). As discussed in Ch. 3, no comprehensive measurements of flow and radiation exchange in urban canopies with substantial tree cover are known to exist. Even if they did exist, suitability of urban canopy layer measurements of radiation, and of flow in particular, for comparison with neighbourhood-average models, is questionable. Some measurements of air temperature and humidity in treed urban canopies do exist, and they are more likely to be representative of the surrounding local-scale zone (e.g., Stewart and Oke, 2012).  5.2.1 Simulation development  Two periods are chosen for simulation: May 18-19, 2011, a cool period with high solar insolation (Fig. 5.1), and July 19-20, 2008, a period with similarly-high solar insolation but hotter, drier conditions—so dry that the grass contributes negligible evaporative heat flux (Fig. 5.2). In each case, the first day is the model spinup period, and the second is used for model-observation comparison. These days are chosen for three reasons: all instruments were functioning, i.e., a full dataset is available; there had not been any rain for at least two full days and therefore energy balance effects of water stored in urban materials, which are not currently represented in the model, are reduced; and finally, they are all fair weather days, apart from a 112  few clouds on May 18. Simulations begin at 0400 Local Solar Time (LST) on May 18 and July 19, just prior to sunrise, and finish at 0500 LST May 20 and July 21, just after sunrise, respectively.    Figure 5.1 View from Sunset tower toward the southwest on May 17, 2009 (no images are available for this time period in 2008). Grass and trees are still green.   The plan area fractions in the Sunset neighbourhood are composed of 29% buildings (P), 34% vegetation, and 37% impervious within a 500 m radius from the micrometeorological tower (Liss et al. 2010). About 12% of plan area of vegetation is tree crowns, and a further 22% is low vegetation, e.g. grass. The surface underneath the trees is initially assumed to be fully impervious (road, sidewalk, etc.). LiDAR visualizations of the Sunset neighbourhood indicate that most of 113  the tree foliage is located between, not above, rooftops. Hence, all foliage is assumed to be in the canyon column in terms of the radiation scheme (Figs. 2.2 and 2.3). Unfortunately the LiDAR flight was performed in February, during leaf-off conditions. Nevertheless, total leaf area index (LAI) for leaf-on conditions was determined to be 0.39 m2 m-2 from ground surveys and aerial mapping combined with allometric relations (see Liss et al. 2010 for details). Hence, a neighbourhood-average profile of plant area density (including conifers and deciduous branches) is computed from LiDAR returns for leaf-off conditions and scaled up to yield a leaf area density (LD) profile that integrates to LAI = 0.39 m2 m-2 (Fig. 5.3). Note that peak LD in Fig. 5.3 is 0.07 m2 m-3. Further details regarding the LiDAR dataset are available in Egli (2014). The profile of building (i.e., roof) fraction, as determined by LiDAR (van der Laan et al. 2012), is also plotted in Fig. 5.3. Total neighbourhood clumping  is determined from 1 m by 1 m gridded LiDAR data for the 500m by 500m area around the tower. Using nearest neighbour theory (e.g., Chandrasekhar, 1943), clumping is related to leaf area density and nearest neighbour distance between particles (leaves). Average leaf area is assumed to be 0.005 m2. Nearest neighbour for each leaf is assumed to be in the same LiDAR cell, and no clumping is assumed within each 1 m3 cell (small scale clumping is ignored). This method returns an average clumping coefficient for tree foliage of  = 0.34. By comparison, a foliage clumping coefficient typical for deciduous trees is ≈0.90 (Campbell and Norman, 1998). Further discussion of tree crown scale vs. neighbourhood-scale clumping is found in Sect. 6.3. Because a model for variation of stomatal conductance with environmental conditions is not included, stomata are assumed to be fully open (maximum conductance) throughout the simulation, with gvs = 0.330 mol m-2 s-1. A note about foliage density and distribution is warranted at this point. Foliage densities chosen to test the radiation (Ch. 3) and turbulent exchange (Ch. 4) models are at upper end of, or exceed, typical neighbourhood-average values. Layers of thickness 4 m and canopy leaf area density 0.19 – 1.13 m2 m-3 (neighbourhood LAI = 0.5 – 3.0 m2 m-2), and 8 m thick layers with canopy leaf area density 0.06-0.50 m2 m-3 (neighbourhood mean LAI = 0.4 – 3.0 m2 m-2), are explored in the radiation (Ch. 3) and dynamic (Ch. 4) chapters, respectively. The rationale is that if the parameterizations function for these more ‘extreme’ cases they will function for scenarios with less tree foliage. Moreover, clumping coefficient is assumed to be 0.50-1.00 in those earlier 114  chapters. Hence, Sunset neighbourhood, with LAI = 0.39 m2 m-2, peak LD = 0.07 m2 m-3, and  = 0.34, is likely to demonstrate somewhat less impact.    Figure 5.2 View from Sunset tower toward the southwest on July 23, 2008. Grass is mostly dry, while trees are still green.    Building geometric and material properties are derived from several sources. Frontal area density (F) is estimated to be about half of P based on photographs of the neighbourhood and associated building archetypes in van der Laan et al. (2012) (Table 5.1). Radiative and thermal properties in Table 5.1 are those used for the Open Lowrise Local Climate Zone by Stewart et al. (2014), with a few tweaks based on values from a detailed analysis for the Norridge residential 115  neighbourhood in Chicago by Krayenhoff and Voogt (2010), and roof and wall material inventories in van der Laan et al. (2012). North-south and east-west canyons are simulated, and their interactions with the canopy atmosphere are equally-weighted.     Figure 5.3 Profiles of fractional building (i.e., roof) area and foliage (leaf) area as a function of height within 500 m of the Vancouver Sunset tower, as determined by LiDAR.    In terms of the radiation scheme, broadband shortwave is calculated for simplicity, instead of splitting into PAR, NIR, and UV bands. Since no model for stomatal conductance is included there is no advantage to computing PAR. Also, longwave divergence in canopy air is neglected for simplicity (i.e., kair = 0), and to focus on the effects of trees.  The model timestep is 60 s, and the direct solar ray tracing scheme is run every 5 minutes. All forcing and evaluation data are converted to Local Solar Time (LST). 116   Parameter Symbol Value Units Geometric    Building plan area density P 0.29 - Building frontal area density F 0.15 -     Radiative    Albedo (roof, wall, ground, foliage) R, W, G, V 0.13, 0.25, 0.14, 0.50* - Emissivity (roof, wall, ground, foliage) R, W, G, V 0.91, 0.90, 0.95, 0.95 -     Thermal    Conductivity (roof, wall, ground) kR, kW, kG 1.00**, 1.25**, 0.60 W m-1 K-1 Heat capacity (roof, wall, ground) CR, CW, CG 1.44**, 2.05**, 1.47 MJ m-3 K-1 Thickness (roof, wall, ground) xR, xW, xG 0.15, 0.20, 0.50 m Leaf width dl 0.05 m     Dynamic    Foliage drag coefficient CDV 0.2 - Facet roughness length (roof, road) z0mR, z0mG 0.02, 0.02 m  Table 5.1 Input parameters for multi-layer urban canopy model with trees. *This is the foliage scattering coefficient, assumed to be half reflection and half transmission (see Sect. 2.2.3.1). **A layer of thickness x = 0.06 m adjacent to the innermost layer is designated ‘insulation’. In this layer kR = kW = 0.10 W m-1 K-1 and CR = CW = 0.10 MJ m-3 K-1.   5.2.2 Model-observation comparison  Urban canopy models require a host of parameters, many of which are uncertain due to challenges with characterizing neighbourhood heterogeneity and parameterizing canopy and 117  roughness sublayer physics. By varying model parameters within their bounds of uncertainty, a favourable comparison with observations can usually be obtained, but little may be learned. A different approach is taken here. Reported model results are based on the best a priori estimate of the physics (processes) and associated parameters required. There are two exceptions: turbulent Prandtl number Pr is reduced to 0.25 to vent sufficient heat from the canopy, as discussed in Sect. 5.1.1; and low vegetation (grass) is not included in the model, because an appropriate parameterization does not currently exist in BEP. Sources of model-observation disagreements are briefly explored in Sect. 5.2.3.       Figure 5.4 Observed and modeled energy balance at Vancouver Sunset tower, 0300 May 19 – 0500 May 20, 2011. Dotted lines are from the corresponding simulation without trees.  118      Figure 5.5 Observed and modeled energy balance at Vancouver Sunset tower, 0300 July 20 – 0500 July 21, 2008. Dotted lines are from the corresponding simulation without trees.   Measurements at the top of Sunset tower available for model comparison include: turbulent fluxes of heat, humidity and momentum, upward fluxes of longwave and shortwave radiation, wind speed and direction, and air temperature and specific humidity. An ultrasonic anemometer-thermometer and an infrared gas analyzer are located 40 cm apart in the horizontal, at 28.8 m above ground level (Crawford et al. 2013). Unfortunately there are few measurements in the urban canopy; only air temperature and specific humidity at 1.2 m in a representative backyard (not at the base of the tower). More generally, there are very few canopy measurements for treed urban canopies in the literature, as discussed in Ch. 2 in the context of evaluation of the 119  new multi-layer radiation model. Furthermore, the neighbourhood representativeness of point measurements in a single urban canyon is unclear.  Variable Units MAE MBE RMSE May 19, 2011 Q* W m-2 30.2 -11.5 36.1 QH W m-2 35.0 -7.1 47.2 QE W m-2 33.1 -26.2 48.7 QG W m-2 53.7 21.5 65.7 K W m-2 5.8 4.4 9.0 L W m-2 24.0 9.3 27.9 TA at z = 1 m K 2.5 2.1 3.5 q at z = 1 m g kg-1 0.48 -0.41 0.64 u* at z = 29 m m s-1 0.13 -0.04 0.15 July 20, 2008 Q* W m-2 23.6 3.0 26.4 QH W m-2 41.6 -3.0 53.0 QE W m-2 15.1 2.3 21.5 QG W m-2 58.7 3.9 70.5 K W m-2 3.3 -1.0 4.7 L W m-2 16.3 -4.1 17.6 TA at z = 1 m K 2.6 2.6 3.5 u* at z = 29 m m s-1 0.06 -0.02 0.08  Table 5.2 Mean Absolute Error (MAE), Mean Bias Error (MBE) and Root Mean Square Error (RMSE) of several variables measured and modeled half-hourly at Vancouver Sunset for 0430 LST May 19 – 0400 LST May 20,  2011, and for 0430 LST July 20 – 0400 LST July 21,  2008 (n = 48 for each simulation).    Model-observation agreement in terms of overall energy exchange between the urban surface and the atmosphere is remarkably good for both simulations, considering all parameters are decided a priori (Figs. 5.4 & 5.5, Table 5.2). Modeled sensible heat flux (QH) is somewhat 120  too large at midday and too negative at night, and daytime results improve with addition of trees. Total conduction heat flux (QG) in all built facets appears to follow the residual of the observed energy balance (Figs. 5.4 & 5.5); however, it is smaller during the midday period and consistently larger than the residual during afternoon and nighttime, resulting in a substantial mean difference (Table 5.2). Yet, the residual of the measurements incorporates measurement errors from all other fluxes, and may not be a reliable estimate of QG. Daytime overestimation of QH and underestimation of QG, and the reverse at night, are probably related, and likely result from low Prandtl number and/or insufficiently high thermal admittance of canopy materials. Net radiation (Q*) is generally well-reproduced by the model, and addition of trees to the model slightly improves daytime results.      Figure 5.6 Observed and modeled upward radiation fluxes at Vancouver Sunset tower, May 19, 2011. Dotted lines are from the corresponding simulation without trees. 121      Figure 5.7 Observed and modeled upward radiation fluxes at Vancouver Sunset tower, July 20, 2008. Dotted lines are from the corresponding simulation without trees.   Latent heat flux (QE) is underestimated by the model on May 19 (Fig. 5.4), but is remarkably well-reproduced on July 20 (Fig. 5.5, Table 5.2). This difference is probably due to soil and grass not being represented in the model, and to their 22% (approx.) share of plan area being designated as ‘road’ in the model; recall that dry conditions render the soil & grass insignificant sources of QE on July 20 (Fig. 5.2), but not on May 19 (Fig. 5.1). Also notable is the correct timing of the peak of all fluxes in Figs. 5.4 and 5.5; the exception is QE, which peaks several hours late in the model on May 19, probably because grass lawns are active in evapotranspiration around midday but not included in the model. Modelled QE peaks at 1500 122  LST for both dates, which is correct for July 20. Modelled leaf energy balance on both dates (not shown) indicates QE from trees in this afternoon period becomes driven as much by negative QH  Figure 5.8 Observed and modeled near-surface air temperature (a) and specific humidity (b), and forcing level friction velocity (c), at Vancouver Sunset tower, May 19, 2011. Dotted lines are from the corresdponding simulation without trees. *These are measurement heights. Model heights are 1 m, 1 m, and 26 m, for ,  q, and u*, respectively.   123  due to hot urban air temperatures, as by Q*. In this sense, trees not only impact urban climate and built surfaces, but urban micrometeorology and thermal state impacts energy exchange at the leaf surfaces. Overall, addition of trees improves all modelled fluxes during the daytime and has little impact on overall fluxes at night (Figs. 5.4 & 5.5). In particular, trees increase daytime QE, probably at the expense of QH. Total reflected shortwave K is well-predicted, while shortwave and longwave irradiance, K and L, are model inputs; hence, overprediction of daytime L is the primary source of Q* underestimation, at least for May 19 (Table 5.2, Figs. 5.6 & 5.7). Daytime L is modestly improved by the addition of trees (Figs. 5.6 & 5.7). K depends only on model geometry, radiative parameters, and radiation module physics. L additionally depends on the energy balances of all the surfaces; therefore, its prediction is more complex. Its overprediction by day and underprediction by night suggests that modeled surfaces are too warm by day and too cool by night. Again, inclusion of grass over 31% of canyon floor area would reduce daytime L and, if sufficiently irrigated, would increase nighttime L (by including the water storage capacity of the soil). Alternatively, greater thermal admittance of the canyon floor (road) would have a similar effect on L by reducing daytime and increasing nighttime surface temperatures. Evidence for the former explanation is provided by the fact that K and L prediction for July 20, 2008 – with dry soil & grass contributing less heat storage capacity and minimal transpiration – is quantitatively much improved (e.g., Figs. 5.6 & 5.7, Table 5.2). Overprediction of daytime near-surface canopy air temperature (Figs. 5.8a, 5.9a) is further evidence that modeled canopy surface temperatures, canyon floor temperatures in particular, are too hot. Air temperature at ≈1 m in the urban canopy is typically closely related to canyon floor temperature, especially for a relatively open canyon (H/W = 0.21 in the model). Again, grass would help remedy this model-observation difference, especially for the May 19 case. Moreover, 1-m air temperature observations are conducted in grass-covered yards rather than above roads, and hence it is unsurprising that 1-m air temperature above modeled canyon floor (roads) does not agree during daytime. Lack of venting of the canopy may also be a culprit, due to dispersive fluxes, enhanced turbulent length scales, horizontal advection or other processes not currently represented in the model, and this is left as future work. Provided sufficient irrigation or precipitation, grass would also increase daytime specific humidity, which 124  is underpredicted by the model during May 19 daytime (Fig. 5.8b). Notably, nocturnal temperature and humidity are very well predicted. Friction velocity at the top of the domain is reasonably well-predicted for both cases, with some daytime underprediction on May 19 (Figs. 5.8c & 5.9b). Overall, the new multi-layer model with integrated trees performs well compared to available measurements in the Vancouver Sunset neighbourhood for both a cool period in late spring and a hot dry mid-summer period, especially given the a priori specification of parameters. Its primary weakness appears to be insufficient evacuation of heat from the lower canopy, despite the low Prandtl number Pr = 0.25 chosen; however, this model-observation difference may result largely from lower canopy air temperature measurement above grassy lawns while the model represents the canopy floor as road (asphalt). The next section explores potential reasons for model-observation disagreement.  5.2.3 Sources of model-observation disagreement Overprediction of daytime L (Fig. 5.6) and canopy air temperature (Fig. 5.8), underestimation of daytime QG, and underestimation of nighttime L, suggest that canyon floor (road) thermal admittance may be too low, at least on May 19, 2011, potentially due to neglect of grass & wet soil. Thermal conductivity of the canyon floor is doubled such that it has approximately the same magnitude as walls and roofs. This increases the thermal admittance of the road by a factor of 1.4. Results for May 19, 2011 are modestly sensitive to this change in thermal admittance (not shown). Errors in Table 5.2 change little, with the exception that mean bias in L is virtually eliminated, and absolute errors of L, Q*, and TA at 1 m are reduced by 10-25%. Hence, canyon floor thermal admittance (e.g., thermal storage impacts of wet soil and grass) appears to account for some of the model-observations difference. Model-observation differences may also derive from characteristics of, and errors in, the measurements. In addition to instrument error, measurement source areas for radiation vs. turbulent fluxes differ and vary in size and location in the neighbourhood with time. Moreover, near-surface measurements may not be representative of the neighbourhood-scale the model aims to represent. In essence, modest model-observation differences are to be expected and there is always an irreducible error of measurement. For the available observation data, the model performs well, with some evidence of too much daytime heat accumulation in the lower canopy. 125  Neglect of grass may play a role, in particular for the May 19, 2011 case when grass has access to moisture. Uncertainty around the Prandtl number, and coefficients and parameters that affect mixing more generally, may also be a factor.    Figure 5.9 Observed and modeled near-surface air temperature (a) and forcing level friction velocity (b), at Vancouver Sunset tower, July 20, 2008. Dotted lines are from the corresponding simulation without trees. *These are measurement heights. Model heights are 1 m, and 26 m, for ,  and u*, respectively.   126  5.3 Effects of trees on canopy dynamics and climate  Effects of trees on canopy flow and climate are explored further in this section. The Oakridge residential neighbourhood in Vancouver (49.2306 °N, 123.1329 °W), which hosted another flux tower during the EPiCC campaign, is similar to the Sunset neighbourhood in most respects; its houses are slightly larger and more spaced, and most importantly, it has about twice as many trees (Liss et al. 2010; van der Laan et al. 2012). For the purposes of the present simulations, the May 19, 2011 scenario in Sect. 5.2.2 is re-simulated with leaf area density (LD) at each level 2.2 times its value for Sunset, such that LAI = 0.86, the value for Oakridge (A. Christen, personal communication). Hence, these simulations are an approximation of the Oakridge neighbourhood, for which micrometeorological measurements are only available for July-August of 2008 and 2009, and they are compared to the scenario without trees in Sect. 5.2.2. Alternatively, these two simulations indicate the impacts of approximately doubling the number of trees in the Sunset neighbourhood vs. cutting down all existing trees. While forcing data for these simulations does not perfectly match surface cover in the model, it is assumed that this does not significantly hinder accuracy of these sensitivity experiments.  Perhaps the clearest effect of tree foliage addition is the increase of QE (Fig. 5.10). The two primary means of evacuating heat from the neighbourhood in a 1-D framework, L and QH, are decreased, in particular the latter. Increased QE and radiation trapping both play a role, as might reduction of canopy mixing, brought about by trees. Average potential temperature in and above the canopy is decreased on a daily average when trees are added, by ≈0.3 K in the canopy and less above the canopy. Cooling is more pronounced in the nocturnal period, and some warming due to trees is predicted during the midday period (e.g., Fig. 5.11c). Hence, for this specific scenario, street trees would serve to alleviate heat stress related to air temperature (e.g., convective exchange with the body), on a neighbourhood-average and diurnal average, with all other factors equal; overall effects on heat stress may differ due to changes to the radiation, wind, and moisture environment associated with trees. Reduction in  at z = 1 m is 0.4-0.5 K at 0300 and 0700 LST and 0.0 K at 1500 LST (Fig. 5.11c). The model predicts a warming of 0.3 K at 1200 LST. These values decrease in magnitude by ≈ 60% at mean building height (z ≈ 5 m) and even further above the canyon.  127     Figure 5.10 Modeled energy balance plus upward longwave for “Oakridge” case (lines) vs. simulation without trees (dotted).    Diurnal-average wind speed and turbulence in the canopy are decreased by addition of trees by 25-30% for wind, and by 35-40% for turbulent kinetic energy (Fig. 5.11a, b). This has negative implications for venting from the canopy and dispersal of scalars (e.g. air pollution) more generally, and many similar findings exist in the literature (e.g., Gromke and Ruck, 2009; Vos et al. 2013). Consequently, heat will be less efficiently vented from the canopy, and this effect is compensated for by tree shading (and associated reduced storage) to produce the canopy cooling previously discussed (e.g., Fig. 5.11c). 128    Figure 5.11 Profiles of modeled wind speed, turbulent kinetic energy, and potential temperature at four times, for the “Oakridge” case (lines) vs. simulation without trees (dashes).   Overall, a substantial coverage of tree foliage, such as that presented by the Oakridge neighbourhood, has noticeable effects on energy balance, climate, and dynamics in and above the 129  canopy. All this, despite the fact that the effects of trees counteract each other, at least during daytime. For example, tree foliage above the urban canopy reduces canopy surface temperatures by intercepting their main driver – shortwave irradiance – and it increases these same surface temperatures by blocking the cold sky and trapping longwave radiation. By exerting drag on the flow and dissipating turbulence, tree foliage diminishes penetration of momentum and turbulence into the canopy (Fig. 5.11 a, b), and therefore decreases the sensible (and latent) heat flux from canopy surfaces. In chapters 3 and 4, the effects of trees were explored in more isolated fashion, and so counteracted each other less. Nevertheless, the Oakridge tree coverage generates mean diurnal surface temperature differences of -2.7 K, -3.3 K, and -1.1 K, for roofs, walls, and canyon floor, respectively, relative to the same neighbourhood devoid of trees (Fig. 5.12). The net effect is to reduce mean radiant temperature in the canopy and pedestrian heat stress. The impact of trees on building energy load is similarly a cooling effect, for the summertime conditions studied, and will further depend on building construction and other factors. By coupling the current urban canopy model with a building energy model (e.g., Salamanca et al. 2010, which is already included in BEP) effects of trees on building energy loads can be quantitatively explored. To further explore these counteracting effects of trees on the canopy climate, a suite of 16 simulations with variable rad and dyn (each = 0.25, 0.50, 0.75, 1.00) are performed for the July 20, 2008 scenario, where rad is the total clumping (i.e., over all scales, in particular the neighbourhood scale) factor applied to radiation exchange (e.g., interception, emission), and dyn is the total clumping applied to dynamic processes (e.g., drag, turbulence production/dissipation). rad may not equal dyn, because radiative and dynamic effects of trees are different in nature; radiation incident on trees typically has a much more vertical component, whereas wind is mostly horizontal, and hence will be more sensitive to horizontal spatial distribution of foliage. In terms of how they function in the model, rad and dyn simply modulate the magnitude of the radiative vs. dynamic effects of trees.  Trees warm the canopy layer air at 1500 LST (peak diurnal temperature) if neighbourhood-scale foliage clumping coefficient as applied dynamic effects is greater than that applied to their radiative effects (i.e., if trees are more clustered in terms of radiative processes than in terms of dynamic processes; Fig. 5.13). Trees cool the canopy air if the opposite is true. 130  Notably, impacts on temperature are of similar magnitude for rad and dyn, or in other words, radiation and dynamic effects of trees are of approximately the same magnitude for the conditions at 1500 LST, July 20, 2008. During nighttime for this particular case, variation depicted in Fig. 5.13 differs because forcing wind speed is reduced, and because longwave exchange becomes more important, though shortwave interception by foliage remains an important factor due to storage in urban canopy materials (not shown).   Figure 5.12 Modeled surface temperatures of built elements for “Oakridge” case (lines) vs. simulation without trees (dashes). R = roofs, Wl = east facing wall, Wr = west-facing wall, G = canyon floor.   Clumping as discussed here includes all scales up to the neighbourhood-scale and applies to specific groups of processes (e.g., radiative, or dynamic). As such, it is no longer simply a property of the foliage, but depends on the processes in question and their relation to the horizontal distribution of foliage. It is likely that rad is larger than dyn, for reasons discussed at 131  the end of the previous paragraph (see also Sect. 6.3). Canopy-layer thermal climate is a critical output of urban canopy models and of significant importance for planning and management purposes. Thus, determination of relative magnitudes of rad and dyn is a critical issue to address in order for BEP-Tree, and ultimately other neighbourhood-averaged urban canopy models with trees, to be useful.    Figure 5.13 Modelled effect of trees on 1.0 m Sunset air temperature at 1500 LST July 20, 2008, as a function of foliage clumping applied to radiative vs. dynamic processes (rad and dyn, respectively).    5.4 Summary and conclusions  A multi-layer urban canopy model with integrated trees, BEP-Tree, is created by combining a column model for vertical diffusion (Santiago and Martilli, 2010), the BEP urban 132  canopy model (Martilli et al., 2002), and the new radiation (Ch. 2) and drag and turbulence (Ch. 4) modules developed in this dissertation. Additional features such as vertical turbulent transport of heat and humidity, buoyant effects on turbulence, the foliage energy balance, and a solar scheme capable of calculating sky-derived diffuse, are added. BEP-Tree is the first multi-layer urban canopy model to fully integrate trees and account for radiative and dynamic interactions between trees and the built surface.  The new model is evaluated against measured fluxes of radiation and turbulent exchange in the inertial sublayer, as well as measurements of air temperature and humidity in the lower canopy layer. It performs well overall for two cases: a hot, dry summer case, and a cooler late spring case. It primarily differs from measurements by overpredicting air temperature in the lower canopy, and to a lesser extent outgoing longwave (and hence surface temperatures), during daytime. A plan area coverage of 22% of soil and grass is not included in the model, because no such module is included in BEP-Tree at present, and this may play a role, especially given that low-level air temperature measurements were conducted above these grassy areas. Insufficient canyon floor thermal admittance appears to play a role (and is in some ways a surrogate for wet soil and grass). Insufficient venting of the canopy space due to missing processes in the model is also a possible reason for these model-observations discrepancies.  Overall, the model suggests trees are relatively important in terms of the overall energy balance. It suggests they are important in terms of canopy climate, overall energy balance, canopy flow dynamics and surface temperatures, with potentially important implications for heat stress, building thermal loads, weather forecasting, and air pollution dispersal. There are a few possible reasons why trees do not exert even more impact in the model. First, vegetation density is relatively low compared to built density, even in the Oakridge neighbourhood. Second, as discussed in the previous section, different foliage impacts counteract each other. Third, previous chapters suggest impacts of vegetation on flow (Ch. 4) tend to be more substantial than impacts of radiation exchange on fluxes (Ch. 2), for a given foliage density. This is in part due to the clumping factor  used to account for the non-random, clustered nature of most foliage that is included in the radiation scheme, but does not play a role dynamically in Ch. 4. However, Marcolla et al. (2003) suggest that clumping should be accounted for in terms of the flow as well. Indeed, the assumption of random distribution of 133  foliage at higher vertical levels in the model where in the real neighbourhood only sparse distributions of trees exist, will vastly overestimate their impact on the flow. Hence, the Marcolla et al. (2003) approach has been followed in this chapter, and foliage clumping has been included as a factor multiplying the leaf area density LD in foliage equations for drag (Eq. 4.12) and turbulent kinetic energy (Eq. 4.14). As such, dynamic and radiative impacts of foliage in these simulations represent the impact of foliage with clumping as determined for Sunset:  = 0.34. The representation of impacts of sparse (i.e. highly clumped) foliage on flow is an area for future research (e.g., Egli, 2014), and further discussion of this issue is found in Sect. 6.3.        134  Chapter 6: Summary and conclusions   A neighbourhood-scale, multi-layer urban canopy model is developed that integrates trees and explicitly computes building-tree interaction. It represents a significant step beyond the ‘tile’ approach to simulation of urban vegetation. This new model includes two major developments: a multi-layer model for radiation exchange in neighbourhoods with trees (Ch. 2), and a multi-layer model for the effects of buildings and foliage on mean and turbulent flow in urban neighbourhoods (Ch. 4). In Ch. 5, these two models are combined with the BEP multi-layer urban canopy model (Martilli et al. 2002) and a column model for vertical turbulent exchange in the urban canopy and surface layer to create BEP-Tree. Furthermore, foliage energy balance and other minor developments, such as increased turbulent length scales during unstable conditions, are included. The combined model is evaluated against measurements from the Sunset neighbourhood of Vancouver, Canada, and sensitivity experiments are conducted to explore the energetic and dynamic impacts of urban trees.  This chapter is organized as follows: a summary of each major development is included next, followed by a discussion of opportunities for future work, and finally, some concluding perspectives are offered.  6.1 Multi-layer radiation model for urban neighbourhoods with trees A multi-layer urban radiation model is developed that fully integrates trees and explicitly computes building-tree interaction (Ch. 2). Furthermore, it does so using standard radiative transfer methods: ray tracing for direct shortwave, Monte-Carlo ray tracing for view-factor determination (once at the beginning), and the Bouguer-Lambert-Beer law for attenuation by tree foliage layers. The model is flexible—for trees, any heights, thicknesses, foliage densities and clumping are permitted, and for buildings, any heights and height probability distributions are allowed. The use of ray tracing renders the model less dependent on the complexity of the model geometry, while the initial calculation of inter-element view factors permits a computationally speedier matrix solution to diffuse exchange for the remainder of a mesoscale or urban canopy model simulation. 135  The radiation model is tested for energy conservation. Subsequently, the trade-off between accuracy and computational expense, represented by choice of ray step size and numbers of rays during ray tracing, is explored in Appendix A. In Ch. 3, model simulations are performed to demonstrate appropriate system responses to varying building geometries, foliage characteristics, and solar zenith angle. The effects of different heights and densities of tree foliage on the vertical distribution of shortwave radiation absorption are demonstrated. Furthermore, the partitioning of shortwave absorption between buildings, vegetation and ground is presented. Denser foliage layers exhibiting less clumping absorb more shortwave at the expense of building and ground surfaces, in particular for higher solar zenith angles or when foliage layers are elevated relative to the buildings. Splitting broadband shortwave into separate PAR and NIR bands has moderate implications for total shortwave absorption for some scenarios, but will be important for coupling with energy or water balance models that predict stomatal resistance. System response scenarios in the longwave spectrum focus on the net exchange. Simulated canyon floor (road), wall and whole canyon net longwave flux densities (L*) as a function of canyon H/W are in agreement with an independent radiation model (TUF-2D; Krayenhoff and Voogt, 2007). Magnitude of canyon floor L* is shown to decrease with deeper canyons and with the addition of canyon foliage. For the isothermal conditions modeled in Ch. 3, the L* of the whole canyon system (and of the whole neighbourhood—canyon + roofs) does not vary significantly with canyon H/W and tree foliage in the canyon. The new ‘treed’ urban radiation model is based on the urban canyon (Nunez and Oke, 1977). It may be used to calculate radiation exchange for a particular canyon, accounting for building height and foliage density distributions local to the specific canyon. However, it is intended primarily for use at the local scale, in which case input building height distributions and foliage density distributions are neighbourhood averages, as are output model fields. The new model is designed to simulate any combination of shortwave and longwave radiation bands, and to be portable to any urban surface model based on an urban canyon. The model is sufficiently flexible that it may be extended to forest and forest-clearing scenarios in addition to urban cases with (and without) trees. 136  Radiation typically drives energy exchange at the surface. The new multi-layer radiation model captures the distribution of energetic forcing, both in the vertical and by surface element type, for urban neighbourhoods with trees (e.g., Ch. 3). To capture the full range of energetic exchange processes, it requires coupling with models for canopy dynamics, energy balances of foliage and built surfaces, as well as urban hydrology and physiology of urban trees. This is undertaken in Ch. 5.  6.2 Parameterization of drag and turbulence for urban neighbourhoods with trees Urban canopy parameterizations designed to be coupled with mesoscale models must predict the average effects on the neighbourhood-scale flow at each height of obstacles in and above the canopy, without resolving the obstacles. To assess these neighbourhood-scale effects on the flow for simplified geometries, the results of microscale simulations are one-dimensionalized by horizontal-averaging. In Ch. 4, obstacle-resolving Computational Fluid Dynamics (CFD) simulations of neutral flow through canopies of blocks (buildings) that have various distributions and densities of porous media (tree foliage), are conducted, and the spatially-averaged impacts on the flow in these building-tree combinations are assessed. The CFD model, with standard k- turbulence scheme and standard parameterization of foliage effects on flow, is evaluated against two sets of wind-tunnel measurements (Appendix C). The accuracy with which a one-dimensional (column) model with k-l turbulence scheme represents the spatially-averaged CFD results is then assessed (Sect. 4.2.4, Appendix B).  Ch. 4 presents a methodology for determining the source and sink terms required in the momentum and turbulent kinetic energy equations to represent the spatially-averaged impacts of tree foliage on flow in urban areas. Considering the effects of both buildings and trees on flow, terms deemed important and included in the proposed parameterization are: drag terms due to buildings and tree foliage in the momentum equation (and, although much less important, corresponding production terms in the turbulent kinetic energy equation for energy conservation); enhanced dissipation of turbulent kinetic energy by the small tree foliage wakes in the turbulent kinetic energy equation; and the modification of length scales due to buildings. The most notable finding is that trees do not significantly affect the efficiency with which buildings exert drag on the flow and produce turbulence, i.e., building sectional drag coefficients 137  do not require modification due to the presence of tree foliage to accurately predict profiles of spatially-averaged flow in and above treed urban canopies. Hence, sheltering between buildings and trees is not significant, using a definition of sheltering analogous to that of Thom (1971) for the plant element scale. However, the presence of trees significantly affects the absolute value of the drag force exerted by buildings, and vice versa, which requires an integrated treatment of flow dynamics for urban neighbourhoods with trees. Overall, results indicate that tree foliage with spatial-average leaf area density ≥0.06 m2 m-3 (the lowest density tested in Ch. 4) significantly impacts the spatially-averaged mean and/or turbulent flow for a range of building densities 0.06 < P ≤ 0.44 if foliage protrudes above buildings. It also significantly affects the spatially-averaged flow if foliage is at or below building canopy height for P ≤ 0.11, and possibly for 0.11 < P < 0.25. The proposed parameterization of tree foliage impacts on the flow (Sect. 4.3.3) should be included for these scenarios, in addition to the Santiago and Martilli (2010) parameterization for building-only neighbourhoods. Because it is easy to do so, it is recommended that it be included in all simulations that include tree foliage. Updated parameters for the Santiago and Martilli (2010) parameterization for non-treed urban neighbourhoods are presented in Sect. 4.4.1. The new parameterization for drag and turbulence in urban neighbourhood with trees is applicable to models of urban canopy flow that solve prognostic equations for horizontally-averaged mean wind and turbulent kinetic energy, e.g., column models with 1.5-order turbulence closures, and models that include the ‘standard’ representation of foliage effects on turbulent flow (e.g., Green, 1992; Liu et al., 1996; Sanz, 2003). Furthermore, it is a critical step in the development of a comprehensive urban canopy model for urban neighbourhoods with trees (Ch. 5). A model for radiation exchange in urban neighbourhoods with trees was presented in Ch. 2. Momentum forcing is required in order to solve the energy balances of tree foliage and built elements. Implementation of the proposed parameterization for drag and turbulent kinetic energy in a k-l column model is a simple way to capture the principal impacts of trees and buildings on neighbourhood-average flow and turbulence.  138  6.3 Multi-layer urban canopy model for urban neighbourhoods with trees: BEP-Tree Distribution of radiation forcing amongst different surface elements at varying heights is determined by the radiation model in Ch. 2. In Ch. 4, a model for vertical turbulent exchange and vertical distribution of momentum and scalars, the atmospheric drivers of energy balance partitioning, is developed. Both models rigorously account for effects of both tree foliage and built elements, as well as interaction between them where significant. Note that the radiation model is based on the 2-D urban canyon, whereas the 1-D model of flow and turbulent exchange is parameterized based on CFD results of flow through a 3-D building array. The reason is as follows: a 3-D multi-layer radiation model with trees based on ray tracing is currently too computationally expensive, whereas a 2-D urban canyon CFD simulation does not permit determination of sectional building drag because canopy flow is blocked in the canopy (i.e., the canopy can no longer be treated as a porous medium). In Ch. 5, a flexible and robust multi-layer ‘treed’ urban canopy energy balance model with trees, BEP-Tree, is developed. To do so, the new radiation and drag/turbulence models are combined with the BEP multi-layer urban canopy model (Martilli et al., 2002), and with the column model in Ch. 4, updated for non-neutral conditions. In doing so, the following processes are added: vertical exchange of potential temperature, specific humidity, and a passive tracer; buoyant production of turbulent kinetic energy; and energy balances of built surfaces. Additionally, foliage (leaf) energy balances are solved, thermal effects on turbulent length scales are parameterized, heating rate due to longwave divergence through air is calculated, and a radiation scheme is added to model diffuse and direct fractions of shortwave irradiance.  The model is evaluated for the Vancouver Sunset neighbourhood (P = 0.29; mean building height ≈ 5 m; LAI = 0.39 m2 m-2) during two fair-weather days: May 19, 2011, and July 20, 2008 (hot and dry). Available measurements include: surface-layer turbulent fluxes of momentum, heat, and humidity; upward shortwave and longwave fluxes; near-surface canopy air temperature and specific humidity. The model performs well, especially considering model parameters are chosen a priori and mostly from stock sources, primarily the “Open Lowrise” zone in the Local Climate Zone classification (Stewart et al., 2014). Overprediction of daytime upward longwave flux density and near-surface air temperature, and underprediction of latent heat flux on May 19, are the primary model-observation differences. Underestimation of canyon 139  floor thermal admittance plays a role, and parameters related to thermal effects on mixing (e.g., turbulent Prandtl number) probably do as well. Errors may also be attributable to the most significant land cover (and associated processes) neglected in the model: soil and low vegetation (grass), which accounts for ≈22% of plan area in the Sunset neighbourhood. In particular, overall latent heat flux is well-reproduced when grass is dry and transpires minimally (July 20), but is underestimated on May 19 when evapotranspiration from grass and soil are more substantial. Most other neighbourhood features and processes are rigorously accounted for. For example, detailed vertical distributions of buildings and tree foliage based on a LiDAR dataset are included, permitting detailed computation of their vertically-distributed impacts on flow, and on radiation and energy exchange. However, uncertainties also remain regarding the representation of tree foliage clumping (see below).   Effects of trees on the neighbourhood energy balance, and canopy climate and dynamics, are explored for the nearby Oakridge neighbourhood in Vancouver. It is similar to the Sunset neighbourhood; a primary difference is that it has about twice the number of trees. The chief impacts of trees, relative to a simulation without trees, are the following: increase in latent heat flux and reduction of outgoing longwave and especially sensible heat flux, and diurnal-average decreases of mean wind and turbulence in the canopy by ≈25-40%, of air temperature by ≈0.3 K, and of canopy surfaces by 1.0-3.5 K. Implications for thermal comfort and building energy are that trees reduce air temperature-related heat stress, on a neighbourhood-average and diurnal basis, but may produce some warming around midday. Overall effects of trees on thermal climate arise from a trade off between their reduction of solar irradiance penetration into the canopy (and associated storage in urban materials), versus their prevention of heat escape from the canopy by turbulent transfer and longwave flux. This trade-off at the neighbourhood-scale requires further work, especially given uncertainties around neighbourhood-scale foliage clumping. A preliminary analysis demonstrates that canopy thermal climate is sensitive to clumping of foliage as it applies to radiative versus dynamic processes. The model also indicates that trees reduce canopy dispersion, at least during daytime, a finding for which there is much support in the literature.  Tree foliage impact may be underpredicted in Ch. 5 due to clumping, which exists at two scales that require discussion in the context of modeling real urban neighbourhoods: individual 140  tree crowns exhibit clumping at several scales (i.e., non-random leaf distributions), and neighbourhoods have non-random distributions of trees. In BEP-Tree (Ch. 5), all scales of clumping are included for all processes, including foliage drag and turbulence and radiation exchange. However, it is probable that the non-random distribution of tree crowns primarily alters flow-related impacts of trees. The hypothesis here is that urban tree crowns affect each other more substantially in terms of flow dynamics than radiation exchange; wind flow is typically more horizontal, whereas radiation has a greater vertical component. Hence, this hypothesis predicts that urban trees are more likely to shelter each other than shade each other. Therefore, the total neighbourhood clumping coefficient of 0.34 for Sunset and Oakridge neighbourhoods may underestimate foliage radiation interception and emission, as well as turbulent fluxes of heat and humidity from leaves. As a comparison, foliage clumping for row crops is about 0.70-0.80, and it reaches 0.90-0.95 for some deciduous tree species (Campbell and Norman, 1998). This uncertainty may affect some of the preliminary findings, in particular the neighbourhood-average canopy air temperature effects of foliage addition. This issue requires further investigation. It is possible that neighbourhood-scale fluxes can be represented almost as well with a tile approach as with the integrated approach taken here. Clearly, however, foliage effects on canopy flow and climate appear to require an integrated approach; for example, wind, turbulent kinetic energy and potential temperature in Fig. 5.11 vary in a way that is not simply driven by forcing height variability – the only way a tiled vegetation scheme can communicate with an urban canopy model, if both are coupled to an atmospheric model. Moreover, trees in BEP-Tree are exposed to urban environmental conditions (e.g., temperature, incident radiation, etc) that a tile approach would be unable to capture. An interesting example of the importance of an integrated approach to urban trees, as opposed to the tile approach, relates to air pollution. Biogenic Volatile Organic Compounds (BVOCs) are emitted by trees and increase tropospheric ozone formation. The rate of BVOC emissions depends strongly on temperature, among other factors. An integrated model resolves the actual radiation and turbulent environments to which trees are exposed, and hence computes a more accurate leaf temperature, and hence BVOC emission, than a tile model.  141  6.4 Limitations and future work  There is a wide array of useful future work to consider. First, limitations and further model developments and testing are discussed, in terms of the radiation scheme, the parameterization for the dynamics, and the complete model. Subsequently, future model experiments are discussed, and several theoretical and practical issues are touched on. The most significant limitation of the multi-layer radiation model (Ch. 2) is its computational requirements, which are mitigated by the single-use of Monte-Carlo ray tracing at the start of a simulation to determine a matrix of view factors, and by limiting the frequency of direct shortwave ray tracing calculations. Given the restrictions on model geometry, a faster algorithm than ray tracing probably exists for calculation of direct shortwave irradiance distribution. This would be a very helpful model development, especially in support of its inclusion in mesoscale model simulations for large cities. Several other improvements to the radiation model may be weighed against the additional computational costs of their introduction. While the model permits high resolution in the vertical, it does not do so in the horizontal. If, for example, urban trees are usually located towards the edges of canyons, it may be productive to introduce some additional discretization in the horizontal, or possibly extend the model into three-dimensions. Another improvement to be weighed against additional computation costs is the inclusion of additional wavebands, for example PAR and NIR (and ultra-violet) in the shortwave spectrum and analogous divisions in the longwave; this capability already exists in the model, however, this refinement may be less useful unless sufficient information is available on material and foliage radiative parameters in specific wavebands. Currently sky radiance is isotropic in both shortwave and longwave bands; the introduction of more realistic distributions where possible (e.g., for clear-sky longwave) is the simplest and most computationally feasible development. Finally, and perhaps most obviously, the model requires testing against ray tracing methods in real three-dimensional urban configurations, and/or full-scale measurements of radiative exchange in treed canyons, once datasets become available. The parameterization of drag and turbulence neglects impacts of tree foliage on length scales, causing significant errors for select cases, in particular for scenarios with foliage that is below or vertically coincident with sparse buildings (P ≤ 0.11). Resolution of this matter, e.g. 142  via development of a parameterization of tree foliage effects on length scales, would be helpful. Moreover, dispersive transport terms are neglected (Sect. 4.2.3), and formulation of a robust and general method for inclusion into a 1-D model of their impacts would be a significant advance; however, given the range of urban configurations, this may not be feasible. As well, the parameterization derives from a specific wind direction (perpendicular to windward building faces) and a limited range of configurations: staggered cubic arrays with interspersed and evenly-distributed foliage layers of thickness H/2, where H is building height. Development of a similar parameterization based on flow models of greater accuracy and/or more realistic urban surface configurations (e.g., variable building heights) and/or a greater range of wind directions would be useful, and could draw on the methodology presented here. Parameterization of drag and turbulence for neighbourhoods with variable building height is a critical next step. At present, multiple building heights are represented as an array of buildings with uniform height equal to the mean building height, with the exception that building drag coefficient at each height is determined based on the actual building density at that height. This approach assumes that drag is accurately treated as a sectional phenomenon, that sectional drag coefficients determined from flow through regular arrays of cubes are generalizable, and that length scales are determined by mean building height regardless of building height distribution. While unlikely to be fully robust, these assumptions provide a first approach to modeling spatially-averaged flow for complex building geometries, and future work should evaluate their robustness. Furthermore, impacts of sparse (i.e., highly clumped) foliage and buildings on both turbulent and radiative exchanges at the neighbourhood-scale requires additional careful thought and investigation, as previously discussed for tree foliage.  The complete urban canopy model with trees, BEP-Tree (Ch. 5), requires basic representation of soil and low vegetation. Furthermore, addition of urban hydrology to the model (e.g., Ramamurthy and Bou-Zeid, 2014) would increase functionality, since urban materials can store and release significant amounts of water, altering the energy balance and thermal climate. A model for stomatal conductance would permit more realistic representation of tree transpiration throughout the day. Calculation of leaf energy balance and therefore source terms (Eqs. 5.8 and 5.10) separately for sunlit and shaded leaves may improve accuracy. The treatment 143  of thermal effects on mixing via the Prandtl number and modification of length scales taken in Ch. 5 is a first approach, and more robust treatment of these effects should be developed. Coupling BEP-Tree with the Building Energy Model (BEM) of Salamanca et al. (2010) would permit investigation of the effects of trees on building energy use; BEM is already coupled to BEP. A simple experiment would be addition of a passive tracer source, e.g. at street level to mimic vehicle tailpipes, which would permit investigation of effects of trees on dispersion. Another interesting exploration would be to calculate radiative divergence of air in a fully coupled model, and assess how important it is relative to other components of the heat balance of the air layers in and above the canopy. The tradeoff between accuracy and computational expense in terms of the frequency with which the direct solar radiation scheme is called should be determined. How often can it be run and with how many rays/which ray step size such that it is still feasible in a mesoscale meteorological simulation for a large city, and so that it still produces reasonable results? Furthermore, a more complete determination of which scenarios and for which purposes an integrated model such as BEP-Tree is required over and above the tile approach, would be a helpful contribution. Robust evaluation of radiation (Ch. 2), flow (Ch. 4), and complete urban canopy (Ch. 5) models requires more comprehensive measurements in urban canopy layers with trees. Specifically, temperatures and radiation exchanges of canopy surfaces, and profiles of wind and turbulence within treed urban canopies, would be immensely valuable to modellers. Because UCMs represent spatial averages, whereas most observations are point measurements, some degree of neighbourhood representativeness, e.g. sampling at multiple locations within a neighbourhood, would be optimal. Such results would greatly enhance model fidelity, but at present they do not exist. In the interim, further evaluation in the Sunset and Oakridge neighbourhoods of Vancouver would be useful, particularly during periods with additional measurements of distributions of surface or air temperatures (e.g., Crawford and Christen, 2014b; Adderley, 2013)  144  6.5 Conclusion New multi-layer canopy models for radiation exchange, flow and turbulence, energy balance and thermal climate, are developed to better represent urban neighbourhoods with trees in atmospheric modeling. These models fully integrate trees and built surfaces in the same model domain, and represent built-tree interactions where significant. The radiation and flow models are designed to be easily integrated within other common models and approaches. They are also capable of representing a range of canopies, from dense forest to dense urban and every combination in between. They are based on the relevant physical processes and exceptionally-flexible, permitting any set of building and foliage densities as a function of height. The complete urban canopy model, BEP-Tree, is the first multi-layer urban canopy model with fully integrated trees. This coupled, next-generation model stands to benefit local-scale and mesoscale modeling of vegetated urban neighbourhoods. It permits advanced assessment of impacts of trees on the urban climate, air quality, human comfort and building energy loads.   145  References  Adderley, C. D., 2013: The effect of preferential view direction on measured urban surface temperature. 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Boundary-Layer Meteorol 120:377–412. 157  Appendices  Appendix A  Radiation model sensitivity to computational parameters The radiation model accounts for radiation absorption, scattering and emission by foliage in the canyon column and above rooftops in the building column. For simplicity, parameter sensitivity tests are shown for geometries with tree foliage in the canyon column only. Canyons (i.e., walls and vegetation) in Sects. A.1 and A.2 are resolved with 10 layers unless otherwise stated, as this is a typical number of layers needed to resolve many urban geometries. For a given set of computational parameters, model accuracy is slightly higher (lower) for lower (higher) wall/foliage resolution (not shown). The importance of sufficient resolution for walls and foliage layers is elucidated in Sect. A.3, and this has implications for the vertical resolution of multi-layer urban canopy models generally, and in terms of the compromises inherent in the use of single-layer models.  A.1 Direct solar irradiance: number of rays and ray step size  The trade-off between accuracy and computational expense as a function of number of rays (ni) and input ray step size (sin) in the direct shortwave ray tracing module (Sect. 2.2.2) is explored. A key point is that actual model ray step size depends on layer thickness z in addition to sin (Eq. 2.1), whereas the number of rays does not. A canyon with one building height and tree foliage in the canyon column (neighbourhood LAI = 0.50), evenly distributed from the ground to the rooftop level, is chosen. For each ni, sin pair the model is run for 25 scenarios in order to sample a range of urban geometries and solar zenith angles. These 25 scenarios are composed of every combination of the following: five canyon height-to-width (H/W) ratios: 0.25, 0.50, 1.00, 2.00, and 4.00, all having street (xc) and building (xb) widths of 10 m; and five solar zenith angles: 5°, 25°, 45°, 65°, and 85°. Solar azimuth is perpendicular to the canyon in all cases. The walls and canyon foliage are resolved with 10 layers. Hence, as H/W varies from 0.25 to 4.00, vertical layer thickness z varies from 0.25 m to 4.00 m. All 25 scenarios are also 158  simulated with a suite of “highly accurate” (HA) simulations (ni = 20,000 and sin = 0.0005), which serve as the standards of comparison. Differences from the HA simulations are first examined in aggregate. Fig. A.1 displays the maximum over all scenarios of the summed patch-level absolute difference of solar receipt; that is, the total difference relative to incident radiation where positive and negative patch-level differences do not cancel. For ray step sizes less than 0.1 it is clear that the number of rays is of primary importance, suggesting that below a threshold of sin ≈ 0.1 the relation between actual ray step size and layer thickness (Eq. 2.1) gives sufficient accuracy. There is only further improvement with smaller step size for large numbers of rays (ni > 1000).    Figure A.1  Maximum over all scenarios of the summed patch-level differences of direct solar irradiance receipt relative to the highly accurate (HA) simulation, as a function of number of rays (ni) and input ray step size (sin). Maximum difference is computed over a suite of simulations with canyon resolution of 10 and varying canyon geometry and solar angle. The log of the maximum computation time over all scenarios relative to the HA simulation is given by the dashed white lines (e.g., “-2” = 1% of the computation time, “-4” = 0.01% of the computation time). 159   The maximum computational time taken by the direct solar ray-tracing scheme across all scenarios, relative to the HA simulations for each ni-sin combination, is overlaid on Fig. A.1. Maximum overall difference from the HA simulation is ≈1% for ni = 500 and sin = 0.1, while the maximum computation time across all scenarios at this accuracy is 0.03% of the HA simulation. At the level of the individual patch (e.g., individual roof levels, foliage layers, etc), a maximum difference at all patches of ≈1% is achieved with sin = 0.1 and an increase in the number of rays to 1000 (not shown). Using 2000 rays with ray step 0.01 reduces maximum overall difference to ≈0.3%, but this comes with a 20-fold increase in computation time. Conversely, decreasing ni to ≈200 with sin = 0.25 keeps both mean relative patch-level (not shown) and maximum overall differences (Fig. A.1) below 5% and reduces the computational load to about 10-6 % of the HA simulation, or 10% of the ni = 1000, sin = 0.1 simulations in the median. Overall, small patch-level and overall error (i.e., < 1%) are achieved with reasonable computation time for 1000 rays with an input ray step size of 0.1; in fact, the product of the maximum overall difference and the maximum computation time is minimized for ni ≈ 1000, sin ≈ 0.1. On an Intel CPU, T7200 @ 2.00 GHz, the median and maximum computation time for this accuracy over the 25 scenarios is 0.06 s and 0.48 s, respectively. For ni = 200 with sin = 0.25 the median and maximum computation times drop to 0.01 s and 0.05 s, respectively. To achieve accuracy significantly better than 1% difference, both ni and sin must be increased several-fold with resulting computational increases. However, to reduce the computational expense, the solar scheme need only be run every few timesteps when coupled with a mesoscale model.  A.2 Inter-patch view factors: number of rays  View factors between patches are computed only once at the beginning of an urban canopy or mesoscale model simulation, and therefore higher accuracy trumps lower computational time in terms of parameter selection for Monte Carlo ray tracing (MCRT; Sect. 2.2.3). Tests show the model conserves energy—the sum of view factors from each patch to all other patches equals 1.0 to a high degree of precision in all cases. The model also reproduces 160  view factor reciprocity between patches (Sparrow and Cess, 1978). That is, for all pairs of patches (e.g., surface element P at level/layer iz, surface element Q at jz) the following relation is satisfied to a sufficient degree of precision (≈0.001 view factor difference or less), provided parameter recommendations are followed:  )(),()(),( jzAizjzizAjziz QQPPPQ  .       (A.1)  Subsequently, modeled view factors between patches are evaluated against ‘exact’ values for non-vegetated canyons. View factors are compared to those given by the analytical formulae of Sparrow and Cess (1978) for an H/W = 1.0 canyon with one wall patch (“H/W = 1.0” in Fig. A.2), and to the results of the highly-accurate numerical radiation model TUF-2D (Krayenhoff and Voogt, 2007) for an H/W = 2.0 canyon with 10 wall patches (“H/W = 2.0” in Fig. A.2). An input ray step size (sin) of 0.01 is chosen, and several values of nk, the number of rays emitted from each patch during the determination of view factors by MCRT, are investigated. The randomness in the selection of ray start location and direction during view factor computation (Sect. 2.2.3.1) means that ensemble values better approximate average model performance. It is found that the running mean of all view factors converges to within 2% relative error for ensembles of 20 or larger. Hence, maximum error over all view factors (road-wall, wall-road, wall-wall) in all 20 ensemble members is assumed to give the error a reasonable upper bound for each nk value. Maximum view factor error in Fig. A.2 is below 0.01 for all patches for nk ≥ 10,000 for both canyons, and errors appear to drop more slowly with further increases in nk. In order to include foliage view factors in the evaluation, model outputs are compared to the ensemble means of a new suite of highly accurate (HA) simulations with nk = 50,000 and sin = 0.01. This is done for the same canyon LAI, canyon resolution and H/W as for direct solar receipt (Sect. A.1). However, larger nk values are tested because rays now travel in all directions and therefore more rays are required for comparable accuracy, and furthermore computational time is a lower priority than for direct solar radiation (which must be computed throughout a mesoscale model run). Input ray step size is fixed at 0.01, because view factor error is found to be insensitive to further decreases of sin. Ensembles of 20 simulations are performed for each scenario and nk value. The maximum ensemble difference from the HA ensemble mean is then 161  determined for each view factor (i.e. each pair of patches). Subsequently, the maximum of these ensemble maximum view factor differences is calculated, yielding a maximum over 11,000 view factor calculations for each data point in the “All” curve in Fig. A.2 (i.e., 550 unique view factors multiplied by 20 ensemble members each). This represents an extremely strict test of the model’s ability to compute view factors.     Figure A.2  Maximum patch-level view factor difference from the highly accurate (HA) simulation across all patches and all configurations, as a function of number of rays per patch, nk (symbols). The maximum patch difference over an ensemble of 20 simulations is used for each configuration. Also, the maximum patch-level view factor error as a function of number of rays per patch for two configurations (lines). The maximum is over an ensemble of 20 simulations for an H/W = 1.0 canyon with a vertical resolution of one patch and an H/W = 2.0 canyon with resolution 10 patches, both without foliage.    162  The analysis is completed three times: with all possible view factors between the road, all wall patches and all foliage layers (“All” in Fig. A.2); with only those view factors to and/or from foliage patches (“Foliage”), and with only road and wall view factors for an identical canyon without foliage (“Built”). Clearly, view factors between foliage and other elements do not result in maximum errors, and if anything, help to reduce maximum errors; however, recall that 2 nk rays are fired from each vegetation layer (Sect. 2.2.3.1). This result may vary somewhat with the resolution and also with the vegetation foliage density. The relatively small input ray step size (sin = 0.01) likely also plays a role in the small error induced by foliage; as sin increases, particularly beyond 0.1, errors in vegetation view factors can grow rapidly (not shown). The tests in Fig. A.2 generally indicate a marked drop in maximum view factor error/difference at nk ≈ 10,000 with smaller decreases beyond that. Hence, nk ≈ 10,000 with sin = 0.01 appears sufficient to achieve excellent accuracy, without excessive computational effort over a range of configurations; for some geometries smaller nk and larger sin may also yield excellent results.  A.3 Building wall and foliage layer resolution  The number of layers with which wall and foliage layers are represented becomes important when scattering and reflection are included. It was not as important for computation of view factors or direct solar radiation receipt, particularly in the case of vegetation because the Bouguer-Lambert-Beer law is independent of resolution for vertical uniform leaf area density profiles. That is, a constant density foliage layer will have approximately the same direct solar receipt and area-weighted view factors (i.e. Eq. 2.20) regardless of the number of layers with which is represented.  Wall resolution is limited to unity by definition for single-layer canopy models. Meanwhile, close attention is rarely paid to wall resolution during the application of multi-layer canopy models. With the addition of vegetation foliage it is critical that the impacts of vertical resolution on model accuracy are assessed. Two tests are performed here, both directed toward shortwave albedo. They assess the impacts of vertical resolution over a range of solar zenith 163  angles for tree foliage only (e.g. a forest), and for a vegetation-free canyon with solar zenith angle chosen to illuminate only part of one wall.    Figure A.3  Shortwave albedo () of a horizontally homogenous layer of foliage as a function of vertical resolution for three different solar zenith angles (). The foliage layer exists from 10-20 m with a leaf area density of 0.30 m2 m-3 and no clumping ( = 1.0). Leaves absorb 50% of intercepted radiation and reflect and transmit the remainder equally. Ground albedo is 0.25. Incident shortwave is assumed to be wholly direct beam.    The overall albedo for a continuous canopy of trees (neighbourhood-average LAI = 3.0 with equal foliage density in both model columns, i.e. LDC = LDB), with broadband leaf reflection and transmission coefficients of 0.25, demonstrates the importance of sufficient resolution, especially for higher solar zenith angles (Fig. A.3). For the two lower solar zenith angles (0° and 30°) the albedo converges to within 0.01 of the correct albedo when foliage resolution is ≥2, and 164  to within 1% when foliage resolution ≥4. For the higher solar zenith angle (60°) corresponding resolutions of 3 and 6 are required, respectively. Hence, higher solar zenith angles may require higher resolution; however, solar irradiance tends to be low at these angles and so errors will be less consequential. Clearly it is critical to have at least two layers; a minimum of 4 is recommended here and 6 yields excellent results for all zenith angles tested. Fewer layers may be necessary for layers with lower foliage density. Wall resolution is an important consideration for multi-layer models. The effects of wall resolution on canyon albedo for a canyon with a height-to-width ratio (H/W) of 2.0 and wall (Wl, Wr) and road (G) albedos of 0.25 are presented in Fig. A.4. Three solar zenith angles are chosen such that only the upper half ( = 45.0°), upper third ( = 56.3°), or upper fifth ( = 68.2°) of one wall is sunlit (sky-derived diffuse is neglected). One wall patch only (e.g., a single-layer model) results in errors of 0.025-0.052 (29-46%) in canyon albedo relative to a high resolution simulation by TUF-2D (Krayenhoff and Voogt, 2007), and is clearly insufficient to give good results. This is because the model cannot distinguish how much of the wall is sunlit; the same total flux of radiation spread equally over the entire wall (i.e.,  = 26.6°) would result in the same canyon albedo. Two, three, and four wall patches are required for absolute error in canyon albedo to drop below 0.01 for  = 45.0°,  = 56.3° and  = 68.2°, respectively. As a smaller portion of the wall is sunlit a greater wall resolution is required to contain the error in the canyon albedo; however, this is offset by the decrease in total shortwave irradiance entering the canyon at these higher solar zenith angles. The other obvious feature in Fig. A.4 is the oscillation with increasing resolution. This can be explained as follows: the sunlit-shaded boundary coincides with a wall patch boundary for resolutions that are multiples of 2, 3 and 5 for  = 45.0°,  = 56.3° and  = 68.2°, respectively, but falls somewhere in the middle of a wall layer in all other cases. Such is the nature of a discrete grid representing a continuous process. Despite the continued oscillation, relative differences from the ‘exact’ solution given by TUF-2D drop consistently below 1% for a wall resolution of between 6 and 9 for the cases modeled here. It is clear that resolution is a significant consideration for both wall and foliage layers. A minimum of two layers appears to avoid the very worst of the errors, and further increases in resolution yield declining benefits. Nevertheless, a minimum of four layers for both foliage and 165  walls is recommended, with higher resolutions being recommended in particular for canyons with higher H/W and foliage layers with greater LAI. A previous study encounters similar tradeoffs and recommends similar minimums for urban facet resolution (Krayenhoff and Voogt, 2007).    Figure A.4  Shortwave albedo of a H/W = 2.0 canyon as a function of the vertical resolution of the walls. Three solar zenith angles () with azimuthal angle perpendicular to the canyon axis are modeled. Incident shortwave is assumed to be wholly direct beam. Wall and canyon floor albedos are 0.25.    A.4 Computational parameter recommendations Computational parameter recommendations to achieve sufficient accuracy while minimizing computation time are summarized here based on tests performed in Sects. A.1 to 166  A.3. To keep patch (whole canyon) errors below 1% it is recommended that the solar radiation model have a minimum of 500 (1000) rays and a maximum input ray step size of 0.1, or 10% of the smallest patch size (e.g., wall patch length). View factor computation, which occurs only once for a given urban configuration, requires a minimum of 10,000 rays and a maximum ray step size of 1% of the smallest patch size to ensure maximum view factor errors between individual patches are ≈0.01 or less. Based on the scenarios tested here a minimum resolution of walls and vegetation layers of 4 is recommended; lower (higher) resolution may be appropriate for lower (higher) foliage density and/or canyon H/W than is used here.                167  Appendix B  Comparison of the 1-D column model with CFD model results   The fidelity with which a 1-D column model with k-l turbulence closure reproduces profiles of spatially-averaged flow as simulated by a 3-D CFD model with standard k- closure, is assessed for all urban block scenarios (i.e., with and without tree foliage). The suite of urban configurations and the evaluation methodology are described in Sect. 2.4. Configurations with building plan density P = 0.25 are first evaluated, and analysis is subsequently extended to other P.  B.1 Intermediate building plan area density RMSD for the profiles of spatially-averaged streamwise velocity , spatially-averaged turbulent kinetic energy , and spatially-averaged Reynolds stress ''wu , are plotted for all scenarios in Fig. B.1, where the spatial-average is over the outdoor atmosphere only. RMSD is normalized by the scaling wind velocity (u) for , and by its square for  and , and therefore represents the difference normalized by flow forcing. Reynolds stress is always well-reproduced by the column model (i.e., normalized RMSD < 0.05; Fig. B.1c), and hence profiles of  are not a focus of the subsequent analysis. RMSD for both  and  is substantially reduced when the tree foliage layer protrudes above the building tops (i.e., Tree4 and Tree5; Figs. B.1a and B.1b). This is primarily true for  above the canopy and  in the canopy. The major part of this effect is not simply reduction of the magnitude of  and , as RMSD values normalized by local averages of  and , respectively, remain substantially lower for Tree4 and Tree5 relative to the other scenarios (not shown). Furthermore, this is true for all tree foliage densities, suggesting that even small leaf area densities (e.g. 0.06 m2 m-3) above the roof height can substantially reduce column model error relative to the no-tree case.  uku k ''wu''wuu kuku k u k168    Figure B.1  RMSD of spatially-averaged streamwise velocity (a), turbulent kinetic energy (b), and Reynolds stress (c) between the column and CFD models for all scenarios with building density P = 0.25. RMSD is for two atmostpheric layers: canopy (0 < z ≤ H) and above-canopy (H < z ≤ 2H). Leaf area density, from left to right for each foliage height, is 0.06 m2 m-3, 0.13 m2 m-3, 0.25 m2 m-3, and 0.50 m2 m-3. The y-axis scale is magnified for . u = 0.45 m s-1.    These tendencies are related to the impact of tree foliage on dispersive transport of momentum in the canopy in the CFD (primarily downwards), which is not represented in the column model. When an LD of 0.06 m2 m-3 is added in the lower half of the building canopy (Tree1), dispersive transport is virtually unchanged, whereas it decreases by a factor of ≈4 when ''wu169  this same layer is added above the canopy (Tree5). It decreases further by a factor of ≈3 as leaf area density increases to 0.50 m2 m-3 for Tree5, whereas it only decreases by 25% for the same increase of LD in the Tree1 case. Hence, tree foliage deep in the canopy has little effect on the dispersive (‘subgrid’) flow, regardless of its density, whereas density is more important for tree foliage above the building canopy. Clearly, the height of the tree foliage relative to the building tops is a critical variable in the determination of the effects of both elements on the flow. Note that these results are contingent on homogeneously distributed foliage, and foliage clumping at crown-neighbourhood scales would presumably diminish the dampening effect of foliage on dispersive motions. Profiles of  and  in Fig. B.2 confirm that agreement between the models is better with foliage above the canopy (Tree5), and for dense foliage in the upper part of the canopy (Tree3, LD = 0.50 m2 m-3) than for cases without trees or foliage deeper in the canopy, particularly for turbulent kinetic energy in the canopy. In fact, higher foliage density at all levels tends to reduce the RMSD for both variables (Figs. B.1a and B.1b). Again this is particularly true for foliage in the upper portion of the canopy (Tree3), where the combination of higher foliage densities and rooftops begin to create a new ‘surface’ at z = H. Fig. B.2 also confirms that even small densities above the canopy (i.e., Tree5) strongly influence the profiles of mean flow. For Reynolds stress, conversely, location of tree foliage is of less importance, whereas increasing foliage density still reduces model differences (Fig. B.1c). The column model is better able to reproduce the wind profile within the canopy, rather than above it. This is largely due to the lower wind speeds at these levels (Fig. B.1a), although agreement above the canopy is also generally good (Fig. B.2a). The column model has more difficulty representing turbulent kinetic energy in the canopy for both the non-treed scenario, as in Santiago and Martilli (2010), and scenarios with foliage in the canopy (Fig. B.2b). Trees introduce several additional processes/terms in the canopy, some of which are not included in the column model formulation but which directly affect turbulent kinetic energy (e.g., terms in the -equation, Eq. 4.4). Tree foliage reduces both wind speed and turbulent kinetic energy for the selected cases; the higher the tree foliage relative to the buildings the greater are these effects. The column model is able to reproduce the CFD profiles of spatially-averaged streamwise velocity and u k170  turbulent kinetic energy, for scenarios with tree foliage, as well or better than for the cases without trees (i.e., those of Santiago and Martilli, 2010). RMSD is less than half of u, or its square u2, for  and , respectively, both in the canopy and above it. RMSD falls to about 10% of these forcing values for scenarios with tree foliage extending higher than the rooftops; a likely explanation is that foliage renders the flow less 3-D and more amenable to prediction in a 1-D framework. In other words, dense tree foliage that extends above the buildings reduces the importance of dispersive processes to a greater degree. Median RMSD over all simulations (with and without tree foliage) at P = 0.25 are 0.13 and 0.31 for  in and above the canopy, respectively, and 0.25 and 0.23 for  in and above the canopy, respectively. Overall, the 1-D column model performs similar to or better than a column model with the Santiago and Martilli (2010) building-only parameterization, relative to the CFD for all scenarios, for P = 0.25.    B.2 Low and high building plan area densities  The column-CFD comparison is extended to other built densities. Building plan area densities (P) of 0.00, 0.06, 0.11, and 0.44 are simulated with tree foliage height and density variation as in Sect. 4.3.2. Column-CFD differences for these built densities closely resemble those at P = 0.25 for ,  and  (not shown). As for P = 0.25, RMSD is larger above the canopy for  and in the canopy for , but overall the spatially-averaged CFD profiles are well-reproduced by the column model (Fig. B.3). Trees consistently reduce RMSD with the exception of some instances at P = 0.44. RMSD at these other built densities is of similar magnitude to, or smaller than, that for P = 0.25 (not shown). Average RMSD is less than or equal to that reported by Santiago and Martilli (2010) for the scenarios without trees. Hence, u ku ku k ''wuu k171  while there is opportunity for improvement with respect to canopy  and above-canopy , it is concluded that the column model performs sufficiently well for all P.     Figure B.2  Profiles of spatially-averaged streamwise velocity (a) and turbulent kinetic energy (b) from the CFD (symbols) and column (lines with corresponding colours) models, for foliage height scenarios Tree1, Tree3, and Tree5 (see domain visualizations above plots), each with low and high leaf area densities, as indicated in the legend. Building density is P = 0.25. Results from both models for the foliage-free (No Trees) case are plotted in each panel.    k u172  B.3 Sensitivity to parameter C5 in the CFD The Ce5 parameter determines the sink of dissipation rate () in the CFD model, and spatially-averaged flow results are quite sensitive to this parameter. There is evidence that lower values of this parameter (relative to the theoretical value of 1.26 computed based on Sanz [2003], used here as the default) may be more accurate, at least relative to select wind tunnel measurements (see Appendix C). Hence, select scenarios in Figs. B.1 and 4.4 are reproduced with C5 = 1.00 and C5 = 1.10 in the CFD model: Tree2, LD = 0.50 m2 m-3; Tree4, LD = 0.50 m2 m-3; Tree4, LD = 0.06 m2 m-3. The column model is also re-run for each case and again with each source term in Table 4.1 removed, with modified sectional drag coefficients and length scales output from the CFD. The ability of the column model to reproduce the CFD profiles of , , and ''wu  is effectively identical. RMSD between the column and CFD models changes by less than 0.08, 0.03, and 0.02, respectively. As such, we conclude that the correspondence of the column model with the CFD model is not significantly affected by the choice of C5 in the CFD model over the range Ce5 = 1.00-126. Furthermore, the same terms identified in Sect. 4.3.3 are significant (not shown), and hence it is concluded that terms deemed important for inclusion in the new parameterization are not affected by the choice of C5 in the CFD model over the range Ce5 = 1.00-126.          uk173      Figure B.3  Profiles of spatially-averaged streamwise velocity (a) and turbulent kinetic energy (b) from the CFD (symbols) and column (lines with corresponding colours) models, for three building densities (P), with foliage density LD = 0.50 m2 m-3 and varying foliage height. The “Forest” scenario has foliage for 0 ≤ z/H ≤ H/2, and no buildings.     174  Appendix C  Evaluation of the CFD model with tree foliage parameterization  The ability of the 3-D CFD model (i.e., Sect. 4.2.1) to reproduce measured or DNS profiles of spatially-averaged flow is ensured prior to its use to inform and parameterize the 1-D column model. Parameterization of drag and turbulent kinetic energy generation and dissipation due to vegetation (i.e., Sect. 4.2.1) requires evaluation, despite being very similar to previous work (e.g., Liu et al., 1996; Foudhil et al., 2005; Dalpe and Masson 2009; Santiago et al. 2013b). No measurements or DNS/LES results for canopies composed of both bluff (i.e. buildings) and porous (i.e. trees) obstacles are currently available, to the authors’ knowledge, that would be appropriate for evaluation of the CFD model; hence, it is evaluated against wind tunnel measurements of flow through vegetation canopies. This is done for forest and forest-clearing scenarios; first, however, evaluation of the code for building-only arrays is addressed.  C.1 Flow through cubic arrays Evaluation of CFD flow through arrays of cubic obstacles, with a different computational package but identical model formulation, has been achieved by Santiago et al. (2007) against wind tunnel results, by Santiago et al. (2008) against DNS results, and by Santiago et al. (2010) against LES and scale model results. Evaluation of the current computational package has been conducted by Simon-Moral et al. (2014) for aligned arrays of cubes.  C.2 Continuous forest Brunet et al. (1994) measured ‘steady state’ wind tunnel flow profiles in and above a canopy of model vegetation sufficiently far downstream to avoid effects from the upwind edge.  Measurement of flow profiles of mean horizontal wind speed, mean shear and standard deviations of horizontal and vertical wind was conducted downstream at x = 4 m. This data set was used to evaluate and tune the CFD model of Foudhil et al. (2005); further details are available in their article.  175  C.2.1 Simulation design The non-dimensional quantity CDV LD H is preserved in the current CFD implementation and canopy height is increased to a realistic full-scale value of 10 m. To compensate, CDV is reduced to 0.2 and LD to 0.159 m2 m-3, both realistic values. Furthermore, the x-axis is scaled identically to the canopy height , which increases from 0.047 m to 10 m. At the inlet boundary, the free-stream velocity is set to 10.2 m s−1, and TKE and its dissipation rate are estimated following Foudhil et al. (2005). Domain height is 60 m and there are 20 computational levels in the canopy.  C.2.2 Results The CFD model is found to be sensitive to parameter C5 in Eq. 4.4. Hence, results using ‘tuned’ values of this parameter (similar to Foudhil et al., 2005) are presented in addition to the value of 1.26 which is based on theoretical arguments (Sect. 4.2.1). Sensitivity to C5 is apparent in Fig. C.1, where a modest decrease in C5, of ≈10-30%, yields noticeably different results. This parameter controls the destruction of dissipation due to the foliage – lower C5 yields a higher dissipation rate and less k in the canopy (and elsewhere depending on vertical transport). In general, the model captures the qualitative trend of the observations, and with C5 = 1.00 it further reproduces the wind tunnel results in a quantitative sense, with the exception of the above-canopy k. The Foudhil et al. (2005) CFD captures the wind tunnel data well, which is not surprising because it uses a value of C5 that is tuned to the same data set. However, given that the foliage parameterization implemented in both our CFD and that of Foudhil et al. (2005), does not represent the effects on the flow of ‘waving’, we do not expect the CFD to compare well with the wind tunnel results, especially the k profiles. Hence, a C5 parameter that is tuned to the results may not yield better results for more general cases.  C.3 Flow at the edge of a forest Raupach et al. (1987) performed wind tunnel experiments in several forest-clearing configurations. Wind and turbulence statistics were measured. Their experiment with alternating 176  patches of clearing and forest, each of length 21 h, is chosen to test the CFD, where h is the canopy height (Fig. C.2). This data set has been used to evaluate several other RANS and LES implementations: the k-l model of Wilson and Flesch (1999), the k- model of Foudhil et al. (2005), and the LES models of Yang et al. (2006) and Dupont and Brunet (2008).     Figure C.1  Comparison of CFD modeled mean horizontal wind speed ( u ), turbulent kinetic (k) and Reynolds stress ( ''wu ) profiles for a ‘continuous forest’ with wind tunnel results from Brunet et al. (1984) and CFD results from Foudhil et al. (2005). Turbulent kinetic energy is normalized using the free stream friction velocity (u*). h is tree foliage canopy height.   The ability of the model to reproduce adjustment of flow profiles after a clearing-to-forest transition is evaluated. In one sense it is essential that the model be able to represent discontinuous foliage, because tree foliage distributions in cities tend to be patchy. On the other hand, the assumption made throughout the current contribution is that tree foliage is uniformly-distributed across canopy spaces (Sect. 4.1), and hence foliage-no foliage transitions are less frequent, because foliage layer edges correspond to building/obstacle walls for foliage layers 177  situated within the canopy. In this sense, the continuous forest evaluation (Sect. C.2) is more appropriate. Nevertheless, foliage still presents vertical boundaries, and horizontal boundaries in two cases (see ‘Tree4’, ‘Tree5’ in the simulations in Sect. 4.3 and Appendix B; Figs. 4.2 & 4.3).    Figure C.2  Side view (i.e. x-z plane) of the forest-clearing model configuration in the wind tunnel (Raupach et al. 1987) and in the CFD. Green areas indicate modeled foliage, and vertical dashed lines indicate locations of profile comparisons. h is tree foliage canopy height.   C.3.1 Simulation design In the CFD, a regular mesh of 840 x 2 x 200 cells is used. Only two cells are simulated in the y-direction, and 200 in the vertical. In the z direction, the vegetation canopy is resolved with 10 cells, and hence the domain top is at 20 h. Other simulations with higher domain tops (i.e., at 40 h, 100 h) are also simulated and similar results are obtained. Other mesh resolutions were also used (e.g., a canopy resolution of 5 cells), and results were again very similar. Hence, it is concluded that results are not very sensitive to the choice of grid layout within these ranges. The forest is modeled using a Leaf Area Index (LAI) of 2.0 (Dupont and Brunet, 2008), evenly distributed in the vertical for z = 0 - h. The sectional drag coefficient due to vegetation foliage (CDV) is set to 0.2, such that CDV LAI = 0.4 as for previous modeling of these wind-tunnel measurements (Yang et al., 2006). The inlet profiles used are as follows:  178      The values chosen for u*, κ, z0/h and C are 0.5, 0.42, 0.0032 and 0.09, respectively. Vegetation (tree) foliage is accounted for in the momentum, turbulent kinetic energy, and dissipation equations as discussed in Sect. 2.1.  C.3.2 Results Raupach et al. (1987) measured vertical profiles of mean horizontal velocity , standard deviation of horizontal velocity (u), standard deviation of vertical velocity (w) and momentum flux ( ) at different locations (x/h = –8.5, 0.0, 2.1, 4.3, 6.4 and 10.6, as indicated by the dashed lines in Fig. C.2). x/h = 0.0 indicates the upwind edge of the canopy. From these experimental data turbulent kinetic energy (k) was computed assuming v =w (Wilson and Flesh, 1999). Vertical profiles of u and k are compared between the CFD model and the wind tunnel. The horizontal wind velocity u is normalized by mean horizontal velocity at x/h = –8.5 and z/h = 2 (i.e., u) and k is normalized by u 2. Note that u is influenced to a small extent by the upwind vegetation canopy. CFD results for several values of C5 are computed to explore the sensitivity of the results to this critical parameter. Results from an LES implementation (Dupont and Brunet, 2008) and another RANS CFD model (Foudhil et al., 2005) are included. The wind speed generally compares favourably for all values of C5 and adjusts the profile appropriately relative to the upwind clearing (Fig. C.3). The current CFD performs similarly to, or perhaps better than, the Dupont and Brunet (2008) LES and Foudhil et al. (2005) k- models, with the exception of the superior performance of the LES well above the canopy (as z/h goes to 2.0). It slightly underestimates wind speed above the canopy, and slightly under/over-0lnu zU z     2ukC3uz u' 'u w179  estimates the canopy wind speed for C5 = 1.00  and C5 = 1.26, respectively. Interestingly, all of the numerical models underestimate the wind speed immediately after entering the canopy at x/H = 2.1.     Figure C.1  Comparison of modeled wind profiles upwind and downwind of the upwind edge of the forest (x / h = 0.0) against wind tunnel results from Raupach et al. (1987) for three values of the parameter C5. Also included are k- CFD results from Foudhil et al. (2005) and LES results from Dupont and Brunet (2008) for the same case. The dashed red line indicates the C5 = 1.26 wind profile at x / h = -8.5 (i.e., approximately the center of upwind clearing). Wind speeds are normalized by u, the streamwise velocity at z / h = 2.0 and x / h = -8.5, for each case.   The turbulent kinetic energy k is less well-reproduced by the models (Fig. C.4), and it is more sensitive to the vegetation parameterization (i.e., Eqs. 4.3 and 4.4). The current CFD with C5 = 1.26 provides perhaps the best results of all models in the clearing (x/h = -8.5) and well downstream of the canopy edge (x/h = 10.6). Immediately downwind of the canopy edge (i.e., x/h = 2.1) none of the models perform well, although the LES performs qualitatively better than the k- models. Overall, the current model with C5 = 1.10 performs at least as well as the LES 180  and perhaps the best of all the models. C5 = 1.26, however, overestimates k, in particular within the canopy, downwind of the forest edge, presumably because larger C5 suppresses dissipation. Overall, the current CFD model performs well with C5 = 1.10, and certainly no worse than the Foudhil et al. (2005) CFD or the Dupont and Brunet (2008) LES. Performance with both C5 = 1.00 and C5 = 1.26 is reasonable. Canopy edge flow is a difficult process to simulate numerically, even with LES models (Yang et al., 2006; Dupont and Brunet, 2008).        Figure C.2  Comparison of modeled turbulent kinetic energy profiles upwind and downwind of the upwind edge of the forest (x / h = 0.0) with wind tunnel* results from Raupach et al. (1987), for three values of the parameter C5. Also included are k- results from Foudhil et al. (2005) and LES results from Dupont and Brunet (2008)* for the same case. Turbulent kinetic energy k is normalized by the square of the wind speed at z / h = 2.0 and x / h = -8.5 (i.e., u2) for each case. * Turbulent kinetic energy (k) is calculated as  2225.0 wvuk   , assuming v = w.   181  C.4 Overall assessment The evaluation of the model against the “continuous forest” case yields an optimal C5 of  ≈1.00, whereas that of the forest-clearing case gives a value of  ≈1.10, both slightly lower than the theoretical value of 1.26 calculated based on Sanz (2003). A survey of the application of this tree foliage parameterization for CFD flow through vegetation foliage (i.e. Sect. 2.1), unearths a wide range of values for C5 (0.4 – 1.5), as well as for other parameters involved in the parameterization of the impacts of foliage on k and  (Green, 1992; Liu et al., 1996; Foudhil et al., 2005; Endalew et al., 2009; Dalpe and Masson, 2009; Rosenfeld et al., 2010).  As such, it appears likely that C5 is case dependent, or more correctly stated, the parameterization as a whole requires a suite of parameters that is case dependent. Clearly this is not optimal, but its rectification, such as the development of a new source term in the -equation to represent the spectral shortcut due to vegetation, is beyond the scope of the present work. Given this situation, we opt to retain the original theoretical value of C5 = 1.26 for the combined building-tree foliage simulations in Sect. 3 and Appendix B. There is no indication that another dataset would not yield another ‘tuned’ value of C5 entirely. As a precaution, the analysis is re-run for select scenarios with values of C5 of 1.10 and 1.00 in Sect. 3 and Appendix B in order to assess the robustness of the results obtained using C5 = 1.26. Moreover, the primary aim of this contribution is to present a methodology for the development of a UCP for urban canopies with trees, and the final results are subject to the many simplifications and assumptions inherent in the use of a CFD model with highly simplified urban configuration, and in particular, the adoption of the now ‘standard’ vegetation source and sink terms in the k and  equations (i.e., Eqs. 4.3 and 4.4).  C.5 The standard k- parameterization of vegetation canopy turbulence A significant issue with the parameterization of the effects of foliage on the k and  budgets (i.e., Eqs. 4.3 and 4.4) is that the flow, especially k and , are highly sensitive to these source and sink terms, in particular to C5 (Foudhil et al., 2005). In our tests, for example, a 10% reduction of C5 doubled the peak of  (not shown), whereas a 30% reduction virtually eliminated all turbulence in the vegetation canopy space. One approach is to use C5 as a tuning parameter, 182  as in Foudhil et al. (2005). However, its tuning to a particular data set may not yield a set of coefficients that are capable of reproducing flows through vegetation more generally, in particular if the tuning is conducted against wind tunnel measurements with artificial ‘foliage’ (Liu et al., 1996; Foudhil et al., 2005). A recent study by Silva Lopes et al. (2013) calibrated the coefficients in Eqs. 4.3 and 4.4 against LES for both homogeneous forest and forest-clearing scenarios and determined the following optimal values: p = 0.0, d = 4.0, C4 = 0.0, C5 = 0.9. These values suggest that wake turbulence due to flow around leaves is converted immediately to heat, and hence only the short-circuit effect (diminishment of larger eddies) needs to be accounted for in terms of the k and  budgets. Although foliage wake production was included in the present CFD formulation, it is notable that it played a very minor role in the reproduction of the spatially-averaged profiles. Conversely, the enhanced dissipation term was significant. Introduction of the d k U term in Eq. 4.3, to represent the rapid dissipation of wake-scale turbulence generated by foliage, is not an optimal approach. Its introduction implies a corresponding term in the -equation (C5d  U), which includes the coefficient C5 to which the model is very sensitive. A more parsimonious approach would not modify the k-equation source term (i.e., retain only the wake production term), but simply modify the existing source term in the  -equation (i.e. C4). Or for greater flexibility, one additional source term could be added to the -equation (just as d k U is added to the k-equation in the ‘standard’ parameterization), to represent the enhanced dissipation of the wake turbulence generated by the  LD CDV U3 term in Eq. 4.3. This latter approach would reduce by one the number of terms derived heuristically. Such a development is beyond the scope of the present work; therefore, the ‘standard’ approach as described in Eq. 4.3 and 4.4 (Green, 1992; Liu et al., 1996; Sanz, 2003) is followed, with careful attention paid to the value of C5. More generally, the single-band (bulk spectral) approach taken here may not have a set of optimal parameters p, d, C4, C5, or they may be dependent on the scale and distribution of the foliage elements, for example. Transfer of energy from mean to turbulent flow cannot be differentiated between building drag and foliage drag, despite the large differences in scale. The spectral shortcut from large to small turbulent scales by foliage elements can only be represented as enhanced dissipation of turbulent kinetic energy. Two or greater band approaches are a possible alternative (e.g., Wilson, 1988), and they assume a clear separation of turbulent scales. 

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