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The effect of discharge variability on the heat budget and tributary mixing dynamics of a proglacial… Knudson, Justin M. 2012

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The effect of discharge variability on the heat budget and tributary mixing dynamics of a proglacial river by Justin M. Knudson B.Sc., Montana State University, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE STUDIES (Geography) The University Of British Columbia (Vancouver) September 2012 © Justin M. Knudson, 2012 Abstract A distinctive characteristic of proglacial streams is unsteady streamflow associated with diurnal ice melt. The role of discharge variability on downstream tempera- tures is not well understood. This study addressed the influence of diurnal dis- charge fluctuations on temperature by quantifying longitudinal heat advection and unsteady flow effects in a heat budget model for a proglacial stream in the Coast Mountains of British Columbia, Canada. Given advection has not been quanti- fied in previous modeling studies, the dominant role of advection over surface heat fluxes found here was surprising. Advection was expected to have a considerable cooling effect due to the flow contributions from cold meltwater. This effect was confirmed while discharge generally increased; however, advection also exhibited a diurnal warming phase of similar magnitude and duration as the cooling phase, while flow generally decreased. The role of discharge variability on transverse mixing dynamics found in previous studies has been inconsistent. Here, transverse mixing lengths tended to be longer with greater tributary flow relative to the main channel. These findings need to be confirmed with further research. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation for the study . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Characteristics of alpine glacial catchments . . . . . . . . 2 1.1.2 Stream temperature response to climate change . . . . . . 5 1.2 Processes influencing stream temperature . . . . . . . . . . . . . 7 1.2.1 Vertical energy exchanges . . . . . . . . . . . . . . . . . 7 1.2.2 Surface-subsurface interactions . . . . . . . . . . . . . . 8 1.2.3 Longitudinal advection/dispersion . . . . . . . . . . . . . 8 1.2.4 Unsteady flow . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.5 Tributary mixing . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Research objectives and thesis structure . . . . . . . . . . . . . . 11 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Study area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Stream temperature . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 Meteorological data . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Parameters for modelling net radiation . . . . . . . . . . . 17 2.2.4 Streamflow . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.5 Electrical conductivity . . . . . . . . . . . . . . . . . . . 18 iii 2.3 Analysis and modelling . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Transverse mixing . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Reach-scale heat budget model . . . . . . . . . . . . . . . 22 2.3.3 Net radiation model . . . . . . . . . . . . . . . . . . . . . 24 2.3.4 Convective heat exchanges . . . . . . . . . . . . . . . . . 27 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Overview of the field season . . . . . . . . . . . . . . . . . . . . 29 3.2 Longitudinal variations in water temperature . . . . . . . . . . . . 36 3.3 Transverse mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.1 Observed transverse mixing in Lillooet River . . . . . . . 39 3.3.2 Modelled transverse mixing . . . . . . . . . . . . . . . . 44 3.3.3 Effect of stream discharge . . . . . . . . . . . . . . . . . 45 3.4 Surface-atmosphere energy exchanges . . . . . . . . . . . . . . . 47 3.4.1 Radiative exchanges . . . . . . . . . . . . . . . . . . . . 47 3.4.2 Net surface-atmosphere energy exchange . . . . . . . . . 53 3.5 Reach-scale heat budgets . . . . . . . . . . . . . . . . . . . . . . 54 3.5.1 Clear-sky period . . . . . . . . . . . . . . . . . . . . . . 55 3.5.2 Cloudy periods . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.3 Precipitation event . . . . . . . . . . . . . . . . . . . . . 63 3.5.4 Landslide event . . . . . . . . . . . . . . . . . . . . . . . 66 3.5.5 Spatial variation . . . . . . . . . . . . . . . . . . . . . . 66 4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.1 Reach-scale heat budget considerations . . . . . . . . . . . . . . 72 4.1.1 Longitudinal heat advection . . . . . . . . . . . . . . . . 72 4.1.2 Unsteady flow effects . . . . . . . . . . . . . . . . . . . . 75 4.1.3 Surface energy exchanges . . . . . . . . . . . . . . . . . 75 4.2 Transverse mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1 Key findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Future research recommendations . . . . . . . . . . . . . . . . . 80 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A Supporting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.0.1 Shade function, gt . . . . . . . . . . . . . . . . . . . . . 90 A.0.2 Sky view factor, fsky . . . . . . . . . . . . . . . . . . . . 92 iv List of Tables Table 2.1 Instrument specifications . . . . . . . . . . . . . . . . . . . . 19 Table 3.1 Daily summary statistics of water temperature at each sub-reach from July 24 to October 15, 2010. sd = standard deviation. . . 32 Table 3.2 Transverse mixing length statistics for the given degree of mix- ing (Pm) at 10% increments. s.d. = standard deviation . . . . . 39 Table 3.3 Predicted transverse mixing lengths using Rutherfords (1994) models for given channel form and tracer input location. Pre- dicted ranges reflect given transverse dispersion coefficient ranges. Mean observed Lz80 was 3036 m. . . . . . . . . . . . . . . . . 44 Table 3.4 Transverse mixing distances and dispersion coefficients at times of the given Qratio statistic. . . . . . . . . . . . . . . . . . . . 46 Table 3.5 Summary of parameters used in the heat budget model for each reach: upper Lillooet River (UL), lower Lillooet River (LL), and Ryan River (RR). . . . . . . . . . . . . . . . . . . . . . . 52 Table 3.6 Heat budget model performance for given site and time period (month/day). MBE is mean bias error, RMSE is root mean squared error, NRMSE is RMSE normalized by the range of observed values and expressed as a percentage, and Em is the Nash-Sutcliffe model efficiency. . . . . . . . . . . . . . . . . 61 v List of Figures Figure 2.1 The study area with the Lillooet River catchment delineated from the lower extent of the monitoring area. . . . . . . . . . 14 Figure 2.2 The monitoring area included 27.6 km of Lillooet R., one ma- jor tributary (Ryan R.), and one minor tributary (Miller Ck.). Stream reaches were upper Lillooet R. (UL), lower Lillooet R. (LL), and Ryan River (RR). . . . . . . . . . . . . . . . . . . 15 Figure 2.3 Longitudinal bank temperature profiles on 13 Aug at 17:50 along Lillooet River downstream of the tributary confluence. . 21 Figure 3.1 Historical mean monthly air temperature from 1969-2006 and 2010 measured at PACS. Mean monthly air temperature for August and September, 2010, measured on-site (OSMS), is shown with solid black points. . . . . . . . . . . . . . . . . . 30 Figure 3.2 Monthly total precipitation for 2010 and mean monthly total precipitation for 1969-2006. Data were missing for May and June, 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Figure 3.3 Discharge for 2010 measured near LL with historical maxi- mum, mean, and minimum daily discharge from 1914-2010. . 32 Figure 3.4 From top to bottom: incident solar radiation, air temperature, vapor pressure, and wind speed measured at OSMS. . . . . . 33 Figure 3.5 Instantaneous water temperature at (top to bottom) UL, LL, RR, and MC. . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Figure 3.6 Maximum, mean, and minimum daily air temperature at OSMS (top) and daily mean water temperature at UL, LL, RR, and MC (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Figure 3.7 Daily water temperature ranges at each reach. . . . . . . . . . 36 Figure 3.8 Longitudinal temperature gradients in Lillooet River on July 25 showing the end of the cooling phase (blue) in the morning followed by the warming phase (red). . . . . . . . . . . . . . 37 vi Figure 3.9 Longitudinal temperature gradients in Lillooet River showing the end of the warming phase (red) late on July 25 followed by the cooling phase (blue) through the morning of July 26. . . . 38 Figure 3.10 Time-series of left (blue) and right (red) bank water temper- ature along Lillooet River at four downstream locations from the Ryan River confluence. Temperature difference (black) be- tween the banks at each location is shown on the bottom of each panel with axis on right. . . . . . . . . . . . . . . . . . 40 Figure 3.11 Mean transverse mixing lengths for the percent of lateral mix- ing at 10% increments. . . . . . . . . . . . . . . . . . . . . . 41 Figure 3.12 Longitudinal bank temperature profiles on 13 Aug at 17:50 along Lillooet River downstream of the tributary confluence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 3.13 Electrical conductivity (EC) was surveyed by eight lateral tran- sects approximately 250 m apart, extending downstream of the Ryan River confluence 2.1 km. . . . . . . . . . . . . . . . . . 43 Figure 3.14 Observed and modelled mean transverse mixing lengths for given degree of mixing. . . . . . . . . . . . . . . . . . . . . 45 Figure 3.15 Transverse mixing lengths at the 80% mixing level for given ratios of tributary to mainstem discharge. . . . . . . . . . . . 47 Figure 3.16 Measured albedo (top) and suspended sediment concentration (bottom) for Lillooet River, Ryan River, and Miller Creek on August 25, 2011, with calculated solar zenith angle (center). . 49 Figure 3.17 Atmospheric emissivity (εa) was calculated over the monitor- ing period. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Figure 3.18 Shading function gt for each reach, August 14-17. . . . . . . 51 Figure 3.19 Radiative exchanges averaged over the upper Lillooet River reach (UL): incident and outgoing shortwave (K) and long- wave (L) radiation (top), net all-wave radiation (Q∗) (bottom). 52 Figure 3.20 Net all-wave radiation (Q∗) for upper Lillooet River (UL), lower Lillooet River (LL), and Ryan River (RR). . . . . . . . . . . 53 Figure 3.21 Latent and sensible heat flux at UL. . . . . . . . . . . . . . . 53 Figure 3.22 Net surface heat exchange (H) with net radiation, latent heat, and sensible heat flux components at UL. . . . . . . . . . . . 54 Figure 3.23 Heat budget (top), discharge (center), and water temperature (bottom) during clear-sky conditions, July 24 to July 27, 2010 at UL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 3.24 Predicted (gray) and observed (black) temporal stream temper- ature change and their difference (red) at UL. . . . . . . . . . 57 vii Figure 3.25 Heat budget (top), discharge (center), and water temperature (bottom) from July 30 to Aug. 2, 2010 at UL. Conditions were clear on July 30, switching to overcast July 31, and partly- cloudy Aug. 1. . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 3.26 Heat budget (top), discharge (center), and water temperature (bottom) from Aug. 3 to Aug. 6, 2010 at UL. Aug. 4 was consistently overcast while Aug. 3 and Aug. 5 were partly- cloudy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 3.27 Heat budget (top), discharge (center), and water temperature (bottom) for the partly cloudy-overcast period from Sept. 5 to Sept. 8, 2010 at UL. . . . . . . . . . . . . . . . . . . . . . . 60 Figure 3.28 Heat budget (top), discharge (center), and water temperature (bottom) for Sept. 18 to Sept. 21, 2010 at UL. A 30 mm rain event occured Sept. 19 - 20. . . . . . . . . . . . . . . . . . . 64 Figure 3.29 Heat budget (top), discharge (center), and water temperature (bottom) for Aug. 5-8, 2010 at UL. An outburst flood occured on Aug. 7 from the release of a lake created from a landslide on Aug. 6 in a tributary catchment of Lillooet River. . . . . . 65 Figure 3.30 Heat budget for UL (top) and LL (bottom) during mostly clear skies from Aug. 14-17. . . . . . . . . . . . . . . . . . . . . . 68 Figure 3.31 Heat budget (top), discharge (center), and water temperature (bottom) during mostly clear skies from Aug. 14-17 at UL. . . 69 Figure 3.32 Heat budget (top), discharge (center), and water temperature (bottom) during mostly clear skies from Aug. 14-17 at LL. . . 70 viii Acknowledgements This work could not have been completed without the help from many people. Above all, my gratitude goes out to my supervisor, Dan Moore, for his guidance, attentiveness, patience, funny stories, and never-ending support. Many thanks go to Jason Leach for being my go-to problem solver, Joe Shea for weather station training, and Jennifer Guay and Dave Hutchinson for data support. Assistance with field work was provided by Matthew Chuang, Luisa Muenter, Eli Heyman, Pascal Szeftel, Jason Leach, Peggy Donnelly, Marc Edwards, and Saskia Hoevelmann. I appreciate the help from Brett Eaton, Ian McKendry, and the many colleagues, faculty, and staff of the Geography Department. Last but certainly not least, I am forever grateful to my loving family and friends for their enduring encouragement. ix Chapter 1 Introduction This chapter introduces the topic of study, highlights additional research needed, and outlines the objectives of this research. 1.1 Motivation for the study Stream temperature is an important fundamental control on the general well-being of lotic ecosystems through its influence on water quality and a range of phys- ical, chemical and biological processes (Milner and Petts, 1994; Beschta, 1997; Coutant, 1977; Gu and Li, 2002; Smith and Lavis, 1975; Gooseff et al., 2005; Cad- bury et al., 2008). For example, dissolved oxygen concentration decreases with warmer temperatures; furthermore, oxygen consumption increases with the higher biological activity associated with warmer temperatures (Caissie, 2006). Stream temperature impacts the mortality, stress, and energy reserves of salmonids, even over diel time scales (Thomas et al., 1986), and specific temperature ranges are required at each life stage (Coutant, 1977; Lee and Rinne, 1980; Burgner, 1991; Nelitz et al., 2007). Salmonids have been observed congregating at local thermal refugia within rivers that would otherwise exceed their lethal temperature, such as at cooler tributary junctions (Torgersen et al., 1999; Ebersole et al., 2003). Al- though glacial-fed streams have often been viewed as having harsh, unstable habi- tat conditions and low biodiversity, a growing body of research is showing that glacial contributions to stream flow can create diverse habitat which benefits many 1 plant and animal species, including important fisheries and invertebrate commu- nities (Milner et al., 2001, 2009; Moore et al., 2009). Glacial meltwater benefits aquatic ecosystems by augmenting streamflow and moderating stream temperature during hot, dry weather when non-glacial streams are often stressed. Such bene- fits can outweigh the negative effects of instability and higher turbidity, especially where lakes allow suspended sediment to settle (Moore et al., 2009). Stream temperature can be affected by environmental changes, such as for- est harvesting or water management (Moore et al., 2005a; Caissie, 2006). The province of British Columbia, Canada recently passed legislation to protect ther- mal habitat for cool water species which requires the designation of temperature- sensitive streams to guide in forest harvesting restrictions (Nelitz et al., 2007). There is additional concern about the influence of climate change on stream tem- perature and aquatic habitat. Hydrological regimes influenced by meltwater have shown particular sensitivity to climatic warming (Moore and Demuth, 2001). Glacial systems exhibit complex spatiotemporal interactions between climate, glacier mass balance, meltwater generation and transport, and stream temperature. A better un- derstanding of these systems is needed in order to properly assess potential impacts of climate change and other environmental disturbances on stream temperature. 1.1.1 Characteristics of alpine glacial catchments As stream thermal fluxes are dependent on channel geometry and riparian micro- climate, it is important to understand the common progression of channel mor- phology and vegetation characteristics following glacier retreat and, therefore, dis- played spatially downstream of the glacier margin. Shortly after ice retreat (near the glacier margin), the stream sediment budget is generally transport-limited, and the lack of colonized vegetation creates a highly unstable, braided channel net- work (Milner and Petts, 1994; Sidle and Milner, 1989). However, confined valleys are typical of headwater areas and commonly support only a single channel until the valley broadens and allows lateral expansion into a braided channel network (Uehlinger et al., 2003). Thermal heterogeneity in confined streams is dominated by longitudinal temperature gradients, but a braided stream network exhibits sig- nificant lateral thermal heterogeneity as well (Malard et al., 2001; Uehlinger et al., 2 2003). With increasing time since glacier retreat, coinciding with increasing dis- tance downstream and lower elevation, bars and banks are strengthened by en- croaching vegetation. The channel network then narrows and progresses into a wandering dominant channel which is laterally-mobile, of irregular sinuosity, and may have a set of relatively stable secondary channels (Milner and Petts, 1994). Eventually the stream evolves toward an equilibrium between sediment supply and transport, larger trees provide greater bank and bar stability, fallen trees create deep pools, and a single, stable meandering channel of higher sinuosity forms (Sidle and Milner, 1989; Milner and Petts, 1994). The general progression is thus a contin- uum from a braided channel network to a single meandering channel, but this can be disrupted by variable valley confinement, tributary confluences, and the pres- ence of lakes (Milner and Petts, 1994; Uehlinger et al., 2003). Stream temperature is also controlled in part by streamflow, and glacial cover in a catchment has a strong influence on climate-streamflow relations (Moore et al., 2009). Glaciers of southwestern British Columbia have generally retreated since the mid-nineteenth century, with occasional periods of minor advance (Osborn and Luckman, 1988; Moore and Demuth, 2001). Streamflow responded through two apparent long-term phases: (1) glacier volume was lost mostly through thinning and meltwater generation increased, and (2) meltwater generation eventually de- clined when more ice area was lost (Moore and Demuth, 2001). Most glacial streams in southern BC are in the second phase (Fleming and Clarke, 2003; Stahl and Moore, 2006). Given that stream temperature is typically negatively correlated with discharge, streamflow declines associated with glacier retreat should have a general long-term warming influence on stream temperature. Moore (2006) linked lower glacier coverage to higher median monthly stream temperature, with an av- erage increase of 1.2 °C for July and August per 10% decrease in glacier cover. Net mass balance and glacial streamflow respond to decadal-scale climatic vari- ability. A shift from the cold to warm phase of the Pacific Decadal Oscillation (PDO) in 1976 initiated a persistent period of more negative net mass balance and more rapid terminal retreat at Place Glacier of the southern Coast Mountains, BC. However, after accounting for the effects of winter mass balance and monthly air temperature in a multiple regression analysis, Moore and Demuth (2001) found that mean August discharge at Place Glacier declined after this PDO shift. This is 3 consistent with the idea that for given hydroclimatological conditions, inter-annual discharge should generally decrease with further glacier retreat due to the lower glacier surface area available to generate meltwater. Glacial streams are generally characterized by low winter flow when precipita- tion is mostly stored as snow. From spring to mid-summer, streamflow is enhanced by snowmelt runoff (hereafter, the nival freshet), which typically terminates early to mid-summer in North America (Milner and Petts, 1994; Fleming, 2005; Moore et al., 2009). The annual hydrograph for glacial streams differs from snowmelt- dominated streams by having an extended period of enhanced flow beyond the nival freshet charged by the continuous supply of glacial meltwater (Milner and Petts, 1994; Fleming, 2005; Moore et al., 2009). Annual peak flow is likely to oc- cur later as the proportion of ice-to-snowmelt increases (Milner and Petts, 1994). The prolonged meltwater contribution moderates downstream temperature at a time when non-glacial streams are often thermally stressed (Milner et al., 2001; Moore et al., 2009), which has been explained by higher thermal capacities from higher flow volumes and a higher cool meltwater component of total flow (Brown et al., 2005; Cadbury et al., 2008). For warmer summers, or for summers following lower winter glacier mass balance, streamflow tends to be higher, peak earlier in the sea- son, and have greater diurnal variation. This was explained by earlier ablation and a larger exposed area of lower-albedo ice, which enhances meltwater production and transport relative to higher-albedo snow (Moore and Demuth, 2001). On a sea- sonal scale, given the negative correlation of winter mass balance to late-summer meltwater contribution and the general negative relation between streamflow and water temperature, a lower winter mass balance should relate to higher streamflow, which in turn should relate to lower summer stream temperature. In southwest BC, glacial influences are typically most distinct in August, after the majority of non-glacial snow has melted and before the cooler and more rainy weather of au- tumn (Moore and Demuth, 2001). Streamflow generally declines in autumn due to decreasing meltwater contributions, but is often punctuated by rain events. These relations show that temperature variability in proglacial streams depends on the timing and magnitude of meltwater discharge, which in turn depends on seasonal mass balance and climate history. Within the seasonal melt period is a daily melt pattern, resulting in a character- 4 istic diurnal pulse in streamflow (Milner and Petts, 1994; Smith et al., 2001; Brown et al., 2003, 2005). Near the glacier terminus, the wave peak tends to occur late in the afternoon, reflecting the lag time required for the transport of meltwater to the stream channel (Milner and Petts, 1994). The diurnal peak occurs relatively earlier and fluctuations are greater following winters of lower mass balance or during pe- riods of warmer weather (Fountain, 1996; Moore and Demuth, 2001). Stream tem- perature likewise shows marked diurnal variation; however, interactions between flow and temperature in glacial streams have not been well studied (Uehlinger et al., 2003). Stream temperature near the glacier terminus is usually near 0 °C with little variability (Milner and Petts, 1994; Uehlinger et al., 2003; Cadbury et al., 2008; Moore et al., 2009). With increasing downstream distance, temperatures are gen- erally higher and exhibit greater diurnal variability, trends which have been de- scribed by atmospheric heating (primarily through solar radiation) trumping the cooling effects of meltwater (Milner and Petts, 1994; Malard et al., 2001; Cadbury et al., 2008; Moore et al., 2009). Evidence for this was higher air-water temper- ature correlations with increasing distance below a glacier (Cadbury et al., 2008). Higher flows during warmer weather have been linked to lower longitudinal ther- mal gradients and lower diurnal variability, resulting from the apparent cooling effect of meltwater. The cooling effect has been inferred in a qualitative manner to be caused by higher thermal capacity (from higher water volume) and a higher frac- tion of meltwater to other, warmer water sources (e.g. tributaries and groundwater) (Cadbury et al., 2008). Such trends do suggest that meltwater promotes lower tem- perature and attenuates variability, but they do not improve our understanding of heat exchange processes related to discharge variability. The specific influences of diurnal meltwater fluctuations on stream temperature remain uncharacterized, at least from a deterministic, quantitative standpoint. 1.1.2 Stream temperature response to climate change There is increasing concern about the effects of climate change on stream tem- perature (Caissie, 2006; Moore et al., 2009). Glacial-fed streams may be par- ticularly vulnerable due to the additional linkages between climate, glacier mass balance, glacier retreat, meltwater generation and streamflow, and stream temper- 5 ature. For any hydrological regime, expected effects of climatic warming include increased incident longwave radiation, increased sensible heat flux from the at- mosphere, and warmer groundwater discharge (Meisner et al., 1988; Milner et al., 2009). In glacierized catchments, continued climatic warming and glacier retreat should continue to cause a decrease in meltwater generation. Given the moderating effect of meltwater, streamflow declines associated with glacier retreat should gen- erate higher stream temperatures (Moore et al., 2009). In addition, lower flow vol- ume (heat capacity) would cause greater temperature response to heat exchanges through the water surface and bed, and exposure time of water parcels to surface energy exchanges would increase with ice retreat and stream lengthening. Estimating stream temperature response to climatic variability has been the focus of several studies, although physical processes have largely been ignored. Most previous studies have used correlations between stream and air temperature, in conjunction with projected air temperatures from general circulation models, to predict stream temperature response to climatic warming (Eaton and Scheller, 1996; Mohseni et al., 2003). In glacial systems, however, the air-water temper- ature correlation is often poor due to meltwater generation also being positively correlated with air temperature. Furthermore, empirical models are not suited to extrapolation beyond the range of data used to fit them, and the model is not transferable to different locations (Johnson, 2003). Even within the same stream, for example, the air-water temperature correlation can change markedly between reaches (Brown et al., 2005). Understanding the physical processes controlling temperature variability is fundamental for prediction of ecological response to en- vironmental changes (Brown et al., 2005). Statistical models are insufficient for understanding physical processes as they do not imply causative effects (Johnson, 2003; Brown et al., 2005). Process-based heat budget models satisfy these limita- tions of statistical models and are more robust for determining causation. 6 1.2 Processes influencing stream temperature 1.2.1 Vertical energy exchanges Energy exchanged in the vertical dimension of the water column, that is, through the water surface and stream bed, will be referred to as vertical heat flux. In de- creasing order of their general influence, surface fluxes include solar radiation, inci- dent/outgoing longwave radiation, latent heat flux, and sensible heat flux (Caissie, 2006; Morin and Couillard, 1990; Sinokrot and Stefan, 1993). Solar radiation typi- cally dominates vertical heat flux during clear weather, except in highly shaded en- vironments, and is highly sensitive to cloud cover (Brown, 1969; Webb and Zhang, 1997a; Evans et al., 1998; Johnson, 2003; Caissie, 2006; Hannah et al., 2008). In- cident longwave radiation depends on air temperature and atmospheric emissivity, which varies with cloud cover, water vapor content and air temperature. Latent and sensible heat fluxes (convective fluxes) are usually secondary terms for cold and cool streams and are a function of wind speed (Evans et al., 1998). Bed heat con- duction is often relatively important in shallow streams but is typically negligible in larger, deeper streams (Brown, 1969; Cadbury et al., 2008; Evans et al., 1998; Hannah et al., 2004; Poole and Berman, 2001; Webb and Zhang, 1997b, 1999; Sinokrot and Stefan, 1993). Fluid frictional heating/dissipation can be important during winter (Webb and Zhang, 1997b) or for steep (>5%) streams under low solar radiation conditions (Meier et al., 2003; Moore et al., 2005a). Shading and stream surface albedo are important controls on net shortwave ra- diation (Leach and Moore, 2010; Richards and Moore, 2011; Webb and Zhang, 1997b; Johnson, 2003; Hannah et al., 2008). Characterizing shade is, therefore, critical in reach-scale heat budget analysis. Stream shading has been estimated by a variety of methods, but perhaps the most rigorous and accurate method em- ploys hemispherical sky/vegetation photographs taken along the reach (Leach and Moore, 2010). However, this approach can be logistically problematic for longer reaches or deeper rivers. Albedo varies with several parameters, including solar angle, suspended sediment concentration (SSC), water surface roughness, water surface foam, aeration of the water, and the diffuse fraction of global radiation (Nunez et al., 1972; Payne, 1972; Whitlock et al., 1982; Han, 1997; Jin et al., 2004; 7 Richards and Moore, 2011). Evans et al. (1998) found surface albedo was highest in the morning and evening and was lowest near midday. Previous heat budget analyses have typically ignored albedo variation and assumed a constant value not based on site-specific measurements, usually between 0.05 and 0.1 (Richards and Moore, 2011). Glacier-fed streams have high SSC levels during summer and likely have higher albedo than assumed in most studies. Field measurements of stream albedo should be made for process-based modeling, especially for streams having higher turbidity. 1.2.2 Surface-subsurface interactions Groundwater and hyporheic discharge are potentially important heat sources. Their influences are dependent on relative flow contribution and temperature. Ground- water advective flux has been shown to be an important control on temperature in small streams (Story et al., 2003; Leach and Moore, 2010). In larger-order streams, the reach-scale effect is relatively minor (Poole and Berman, 2001). Hyporheic heat exchange is a type of advective energy exchange associated with water ex- change between stream bed sediment and the water column. In a small proglacial stream (up to 10 m3s−1) surface and hyporheic water differed by 12.1 °C over a dis- tance of only 80 cm, suggesting hyporheic flux to be an important control (Malard et al., 2001). Burkholder et al. (2008) calculated the overall cooling effect from hyporheic discharge to be negligible at the reach-scale in a larger river, but pointed out that local hyporheic discharges can create a spatial mosaic of temperature vari- ability. Groundwater and hyporheic advective fluxes can be important in smaller streams but their relative influence should diminish with higher stream flow. 1.2.3 Longitudinal advection/dispersion The focus of nearly all previous energy budget studies has been on vertical energy exchanges, with some attention to hydrological processes such as groundwater- surface water interactions. Longitudinal dispersion may be important to consider in cases involving localized injections of heat (such as power plant cooling wa- ter), but it has typically been excluded for situations where localized injections do not occur (Chikita et al., 2009; Hockey et al., 1982). No known studies have 8 specifically investigated longitudinal heat advection as a component in the energy balance of glacial-fed streams. Some modelling studies have implicitly included advection by employing a Lagrangian frame of reference (Hockey et al., 1982; Chikita et al., 2009). Some studies have included the effects of longitudinal heat advection within a numerical model, but did not quantify their magnitude (Sinokrot and Stefan, 1993; Meier et al., 2003). The influence of longitudinal heat advection on stream temperature in relation to other heat fluxes is thus poorly known over a range of hydrological regimes. 1.2.4 Unsteady flow The influences from changes in within-reach heat storage and advective flux di- vergence should be considered when modelling stream temperature over unsteady flow conditions. The former effect is associated with heat capacity changes. The latter effect is associated with wave celerity, which differs from stream flow veloc- ity, and longitudinal heat advection changes with passage of the diurnal discharge wave. These influences, which collectively will be referred to as unsteady flow effects, have been implicitly calculated in numerical modeling studies but have not been explicitly quantified as components of a reach-scale heat budget (Chaudhry et al., 1983; Meier et al., 2003). 1.2.5 Tributary mixing Stream temperature is influenced by advective inflows such as tributaries. These inflows can be especially important in glaciated catchments where tributary sources often differ substantially in temperature, such as a proglacial stream joining a lake- fed or groundwater-fed stream. Temperatures in the mixing zone are variable in time and all three spatial dimensions. Longitudinal mixing is rapid and only a con- cern when pulse injections of a tracer are involved (such as power plant thermal effluent) (Rutherford, 1994). Vertical mixing can be important for tracer sources which enter the water column heterogeneously in the vertical dimension, such as where tributary beds are discordant (Biron et al., 2004). The motion of turbulent eddies arising from bed friction is dominantly vertical, causing vertical mixing to be much more rapid than transverse mixing, which is the slowest of the three and 9 therefore often of most concern (Rutherford, 1994). The laterally unmixed zone can provide a diversity of habitat conditions, such as thermal refugia for salmonids (Kaya et al., 1977; Berman and Quinn, 1991). Research has shown considerable variation in the river length required for complete transverse mixing (Lz) to occur (Lane et al., 2008; Mackay, 1972). Mixing lengths are commonly between 100 and 300 multiples of channel width (Gaudet and Roy, 1995; Rutherford, 1994; Fischer, 1979), although mixing lengths have often been much lower (Day, 1977; Biron et al., 2004; Lane et al., 2008; Rutherford, 1994). There can be considerable varia- tion in Lz within a site due to changing conditions. For example, Lane et al. (2008) reported a range in Lz between 8 and 400 channel widths and ascribed the variation to changes in flow dynamics between the two tributaries, with bed discordance as an additional control. In addition to protecting critical thermal habitat, understand- ing the drivers of transverse mixing variability also finds important application in pollution transport studies. Transverse mixing occurs by a complex array of interacting processes. The fundamental processes acting in all directions are advection and diffusion. Molec- ular (Fickian) diffusion is relatively negligible on its own, but transverse turbulent diffusion is considerable and commonly occurs when turbulent eddies, formed by bed friction, undergo rotation and move along the transverse axis. This results in increased transverse tracer gradients and more rapid diffusion (Rutherford, 1994). Transverse dispersion is the dominant transverse mixing mechanism. It is caused by vertical variation in transverse velocity of secondary currents, a condition com- monly generated by helical secondary currents and lateral movement of the thal- weg. Helical secondary currents are common at river bends, in which water moves in opposite lateral directions at the surface and at the bed, increasing local tracer gradients and diffusion rates and greatly enhances transverse mixing rates (Hen- derson, 1966; Rutherford, 1994). Lateral movement of the thalweg can occur from bathymetric non-uniformities. Bed friction from lateral movement of the thalweg causes vertical variation in transverse velocity in the same manner bed friction of the primary current creates vertical variation in longitudinal velocity. However, transverse mixing from lateral thalweg movement is of far less magnitude than that of helical currents (Rutherford, 1994). Transverse mixing is largely controlled by dispersion processes, but stream flow may also be an important factor influencing 10 transverse mixing. There have been conflicting findings on the influence of discharge variability on transverse mixing dynamics. Higher transverse mixing rates (kz) have been related to both lower (Gaudet and Roy, 1995; Biron et al., 2004) and higher total discharge (Chu and Babarutsi, 1988). The influence of fractional discharge (smaller tributary flow/larger tributary flow) on kz has also been ambiguous. Lane et al. (2008) and Biron et al. (2004) reported more rapid mixing with higher fractional discharge, but Gaudet and Roy (1995) found that mixing rates were not affected by fractional dis- charge changes. These findings may have been confounded by other factors known to influence tributary mixing, such as tributary bed discordance, junction angle, and sinuosity (Gaudet and Roy, 1995; Best, 1988; Boxall and Guymer, 2003). In a laboratory flume experiment, Mosley (1976) found that opposing helical flow structures formed at a confluence and maintained flow separation between the wa- ters of the two channels. The strength of flow separation increased with fractional discharge approaching unity. Rhoads and Kenworthy (1995) confirmed the occur- rence of flow separation downstream of a natural junction when momentum ratio was less than one but found a single helical flow cell (such as at a bend) when the momentum ratio exceeded one. Best (1988) also illustrated the formation of mutual flow deflection and flow separation of confluent channels, although flow separation strengthened with increasing fractional discharge (even beyond one). The down- stream persistence of flow separation and its influence on mixing dynamics have not been determined, but flow separation suggests that lateral secondary currents (such as at bends) would be limited to half the channel width, thus limiting kz, until flow separation structures break down. Flow separation potentially has important implications for explaining the effects of stream flow variability on tributary mix- ing. 1.3 Research objectives and thesis structure Attention has recently been drawn to stream temperature dynamics in glacierized catchments because of their sensitivity to climatic variability (Hannah et al., 2007; Cadbury et al., 2008; Moore et al., 2009). The role of meltwater flux on stream tem- perature dynamics remains a key gap in our understanding of the thermal regime 11 of glacial streams. Higher heat capacity and greater relative discharge contribution from cold meltwater have been qualitatively used to explain the apparent cooling effect of advected glacial meltwater (Cadbury et al., 2008; Brown et al., 2005), and our understanding of the influence of meltwater flux on temperature thus remains mainly theoretical. The need to further investigate and understand stream tem- perature responses to hydrological fluxes in glacier-fed streams has been directly expressed (Uehlinger et al., 2003; Brown et al., 2005; Cadbury et al., 2008; Moore et al., 2009). This study will address the role of diurnally varying discharge on stream temperature changes by specifically quantifying longitudinal heat advection and the effects of unsteady streamflow as components in a reach-scale heat budget model. The effects of these components have not previously been specified quanti- tatively in deterministic heat budget models (Mohseni and Stefan, 1999; Sinokrot and Stefan, 1993; Hockey et al., 1982), and particularly not for glacial streams Chikita et al. (2009). This study will also be unique by applying the heat budget to a reach considerably farther downstream than previous glacial stream temperature studies to investigate potential influences of the added length and travel time. Tributary mixing dynamics are highly complex, particularly in natural settings such as river confluences, as they depend on relationships and interactions between several different flow mechanisms and physical factors. There have been conflict- ing findings on relationships between discharge variability and transverse mixing. However, some studies were based on only a small number of observations (Biron et al., 2004; Lane et al., 2008). This study aims to elucidate the influence of vari- able streamflow on transverse mixing lengths below the confluence of two glacial- fed tributaries. This setting should provide a wide range of flows and fractional discharge values to attain a high number of observations to meet this objective. The remainder of the thesis is organized as follows. Chapter 2 describes the study area, field measurements and monitoring, and the methods of data analysis. Chapter 3 presents collected data and the results of data analysis. Chapter 4 dis- cusses the results in the context of the research objectives stated above. Chapter 5 summarizes the key findings from the study, draws conclusions, and identifies topics for further research. 12 Chapter 2 Methods 2.1 Study area The Lillooet River lies within the Pacific Ranges physiographic region of the south- ern Coast Mountains of British Columbia, Canada. The upper Lillooet River flows nearby the town of Pemberton, B.C., about 150 km north of Vancouver, B.C, and drains approximately 3150 km2 before flowing into Lillooet Lake 90 km down- stream (Prent and Hickin, 2001). The upper basin is characterized by rugged to- pography, with local relief up to 2800 m and alpine summits reaching elevations of 3000 m (Mathews and Pratt, 1986; Friele et al., 2005). The river originates at the Lillooet Icefield and flows into Silt Lake (1.4 km long) 2.5 km downstream (Figure 2.1). The 55 km stretch from Silt Lake to Railroad Creek has mostly a natural braided channel morphology with some reaches confined by valley walls. Much of the remaining 35 km stretch to Lillooet Lake (a large fjord-like lake) has been straightened and confined by dike networks, resulting in substantial reduction of the original sinuosity, and can be classified as wandering to meandering (Friele et al., 2005; Desloges and Church, 1987). This study focused on this lower reach. At the inlet to Lillooet Lake, glacier coverage of the Lillooet River catchment has been estimated at about 15% (Prent and Hickin, 2001; Friele et al., 2005). Snow and ice meltwater account for the long freshet from approximately May to September. Lillooet River discharge varies considerably on a diurnal cycle during the melt season, with a typical range from 7 to 35% of the daily mean discharge 13 > > > >> > > >> >> 440000 460000 480000 500000 520000 540000 555 000 0 556 000 0 557 000 0 558 000 0 559 000 0 560 000 0 561 000 0 562 000 0 563 000 0 0 20 4010 km Meager C k Monitoring Area Rai lroa d Ck Ryan River Lillooet River Lillooet Glacier Lillooet Lake> water temp. logger Ryan R. basin Miller Ck. basin glacier Silt Lake British ! Vancouver 0 250 500125 km Columbia elev. (m) Value 2900 2600 2100 1600 1100 600 100 UTM zone 10 U Miller Creek Figure 2.1: The study area with the Lillooet River catchment delineated from the lower extent of the monitoring area. (Prent and Hickin, 2001). The Meager Creek area of the upper Lillooet River basin has a long history of large landslide events caused by failure of the weak hydrothermally altered vol- canic rock making up the Mount Meager volcanic complex. Four slides in the twentieth century alone were in excess of 1×106 m3, one of which produced a de- bris flow that reached the mouth of Meager Creek and caused flood surges along Lillooet River (Friele et al., 2005). 14 Lillooet River UL LL RR Q1 WSC Gage Q2 Q3 Open Site Met Station! Water Temperature Logger Q1-3 > Stream ReachUL Stream Gaging Station Lillooet River Ryan River 0 2 41 km > > > > > >> > > > > >> >> > > >>> > ! 500000 500000 504000 504000 508000 508000 512000 512000 516000 516000 520000 520000 557 600 0 557 600 0 558 000 0 558 000 0 558 400 0 558 400 0 558 800 0 558 800 0 559 200 0 559 200 0 UTM zone 10 U Pemberton Miller Ck Figure 2.2: The monitoring area included 27.6 km of Lillooet R., one ma- jor tributary (Ryan R.), and one minor tributary (Miller Ck.). Stream reaches were upper Lillooet R. (UL), lower Lillooet R. (LL), and Ryan River (RR). 15 2.2 Data collection On-site monitoring and manual data collection were carried out between July and October, 2010, with some supplementary data collected in August, 2011. The monitoring area spanned 27.6 km of Lillooet River, from a bridge 5 km downstream of the Railroad Creek confluence (upper bridge) to the Highway 99 bridge 13 km upstream from Lillooet Lake (lower bridge), (Figure 2.1). At the lower bridge, the Lillooet catchment is approximately 2120 km2 and 19% glacierized. Two tributaries were monitored just upstream of their junctions to Lillooet River. Ryan River joins Lillooet R. 8.0 km upstream from the lower bridge and has a catchment area of 413.4 km2 with 18% glacier cover. Miller Creek joins Lil- looet River 475 m below the Ryan River junction and has a catchment area of 73.1 km2 with 22% glacier cover. Upstream extents of both tributaries were at bridge crossings on Pemberton Meadows Road. 2.2.1 Stream temperature Water temperature (Tw) was sampled at 5-min intervals using Onset TidbiT v2 data loggers anchored to the stream bed (please refer to Table 2.1 for all instrument specifications). It was assumed that the flow was turbulent enough to cause neg- ligible vertical temperature differences. Sampling locations for Tw are shown in Figure 2.2. Sampling was most concentrated along both banks of Lillooet River below the Ryan River confluence to facilitate the investigation of transverse mix- ing of unsteady flows. Other sampling locations were at points which defined upper and lower bounds of stream reaches intended for heat budget modelling. 2.2.2 Meteorological data Meteorological data were collected at an automated weather station installed 30 m from the Lillooet River in an open field. Incident shortwave and longwave radiation were measured with a Kipp & Zonen CM6B pyranometer and a Kipp & Zonen CGR3 pyrgeometer, respectively. Ambient air temperature and relative humidity were measured with a Rotronic HC-S3 sensor outfitted with a multi-plated shield to reflect solar radiation while allowing easy air passage. Wind speed was measured with an R.M. Young 5103-10 anemometer. Instruments were scanned every 1 s and 16 averages were recorded every 1 min with a Campbell Scientific CR10X datalogger. 2.2.3 Parameters for modelling net radiation To calculate stream albedo, incident and reflected solar radiation were measured manually over the course of the day on Aug 25, 2011. Incident solar radiation was measured using a Kipp & Zonen CM6B pyranometer mounted on a tripod on shore. A second Kipp & Zonen CM6B pyranometer was mounted on a 2.5-m-long pole using a specially designed gimbel joint that kept the instrument level while being held inverted. Measurements were made at a height of 0.25 m above the stream to limit the pyranometer’s view of non-water surfaces while also limiting the effect of the instrument’s shadow. The pyranometers were scanned every 1 s and means were recorded every 10 s with a Campbell Scientific CR10X datalogger. The 10 s means were then averaged over the duration of each manual measurent, which ranged from 3-5 min. A water sample was collected with a DH-48 sampler at the location of each albedo measurement. These samples were taken to a laboratory in the U.B.C. Department of Geography to determine suspended sediment concentra- tions (SSC) by filtering the sample through a pre-weighed 0.45 micron filter, then drying and weighing the filter. A hemispherical photograph at the Open Site Meteorological Station (OSMS) was used to determine the site’s view factor for use in modelling the longwave radiation incident at the stream surface (details in Appendix) following Oke (1987) and Leach and Moore (2010). A Nikon Coolpix 4500 4.0 mega pixel digital camera was fitted with a Nikon ”fisheye” FC-E8 lens, mounted on a tripod with lens facing upward, levelled at the same height and location as the pyrgeometer, and oriented north to capture hemispherical images of the sky and horizon. The images were analysed using Gap Light Analyser (Frazer et al., 1999). For modelling stream shading and calculating view factors for the stream sur- face, riparian tree heights were measured at several riparian access areas using a clinometer and tape. Trees were selected for measuring based on the intent to rep- resent the spatial average over the monitoring area. Subsequent to the monitoring period was it realized that the stream surface area beneath vegetation overhanging the stream may be important to consider in the reach-scale view factor calculation 17 and the solar shading function. The estimate of vertical distance from the wa- ter surface to vegetation overhanging the stream was based on field observations and on-site photographs. Reach-averaged lateral vegetation overhang, longitudi- nal vegetation presence, plan-view channel orientation, and channel width were determined for each reach using Geoeye-1 satellite images (0.41 m panchromatic resolution). 2.2.4 Streamflow Streamflow was gauged at the upper boundaries of the upstream Lillooet River reach (UL) and the Ryan River reach (RR) to provide discharge and hydraulic ge- ometry data (Figure 2.2). Data for the downstream boundary of the lower Lillooet River reach (LL) were provided by The Water Survey of Canada from the Lillooet River Near Pemberton gauging station (station number 08MG005). Stream flow was calculated by the standard velocity-area method. Water velocity and depth were measured at locations across the stream at each gauging site by suspending a Price-type current meter from a bridge with a winch and crane. Stage was recorded every 5 min in slackwater areas near the flow measurement sites using a Van Es- sen Diver pressure transducer anchored to the stream bed, and barometric pressure was accounted for with measurements of a second Diver on shore. Mean depth, width, and water velocity were computed for each flow measurement, which were later used to produce hydraulic geometry vs. discharge rating curves for each flow measurement site. 2.2.5 Electrical conductivity Electrical conductivity (EC) was surveyed from an inflatable boat with an outboard motor on Aug 15, 2010 to provide a second method to determine transverse mixing for comparison with the estimates based on water temperature. EC measurements were taken with a WTW TetraCon 325 conductivity probe at five locations across the river channel: near each bank, center, left-center, and right-center. The survey began near the confluence of the Lillooet and Ryan rivers, and eight transverse routes were repeated at roughly 250 m intervals downstream. The longitudinal extent, which was approximately 2km, was limited by accessibility constraints. 18 Table 2.1: Instrument specifications. Parameter Sensor Range Accuracy Tw Onset TidbiT v2 -20 to 70 °C ± 0.2 °C stage (cm) Van Essen Diver DI243 2900 cm ± 0.1% Ta Rotronic HC-S3 -30 to +60 °C ± 0.2 °C RH Rotronic HC-S3 0 to 100% ± 1.5% @ 23 °C u RM Young 05103 0 to 100 ms−1 ± 0.3 ms−1 K↓ Kipp & Zonen CM6B 300 to 2800 ηm < 5% L↓ Kipp & Zonen CGR3 4500 to 42000 ηm < 5% EC WTW TetraCon 325 1µS/cm to 500 mS/cm ± 1.5% 2.3 Analysis and modelling 2.3.1 Transverse mixing Lateral mixing was investigated downstream of the confluences of Ryan River and Miller Creek primarily by using temperature differences between the left and right banks of Lillooet River. Whereas mixing studies commonly have utilized chemi- cal tracers and visual indicators (Day, 1977; Gaudet and Roy, 1995; Rathbun and Rostad, 2004; Lane et al., 2008), water temperature can be used when the waters being mixed differ sufficiently in temperature (Mackay, 1972). Because mixing was hypothesized to be most rapid near the confluence, tem- perature loggers were concentrated toward the confluences with Ryan River and Miller Creek in order to provide higher resolution of the degree of mixing. Longi- tudinal temperature profiles (in the x direction) at a 10 m resolution were computed for each bank of Lillooet River by linear interpolation between logger stations. At any x location and any time t, the degree to which the waters had mixed was indi- cated by the temperature difference between the banks relative to the initial bank temperature difference. The deposition of suspended sediment in Lillooet River caused several tem- perature loggers to become buried at various times during the monitoring period. Even slight burial compromised the measurements. The period from August 12 - 20 had the highest spatial resolution of valid water temperature data below the 19 Ryan-Lillooet confluence and will henceforth be referred to as the mixing study period. A subset of data from this period was selected in which initial tempera- ture differences between the merging rivers (∆Ti) were ≥ 1°C. Application of this criterion resulted in removal of about half the data but was necessary to minimize the effect of instrument accuracy (temperature logger accuracy = 0.2 °C) in de- termining longitudinal distances for transverse mixing completeness (Pm) of 80% and lower. The transverse mixing completeness was defined as a percentage of the initial bank temperature difference at the confluence: Pm = ( 1− ∆Tx ∆Ti ) ×100% (2.1) where Pm is the percent completeness of transverse mixing, ∆Tx is the bank temper- ature difference at longitudinal distance x downstream of the confluence, and ∆Ti is the initial bank temperature difference at the confluence. The distance for a par- ticular Pm will be referred to as the transverse mixing length,Lz, with an extension for Pm (e.g. Lz80 for Pm = 80). Figure 2.3 illustrates the method used to estimate mixing lengths for August 13, 2010, at 17:50. At this time, ∆Ti was 1.07 °C (gray line), ∆Tx was 0.63°C where ∆Ti decreased by 40% at 780 m (gray dotted line), and ∆Tx was 0.21°C where ∆Ti decreased by 80% at 2550 m (black dotted line). Rutherford (1994) integrated results from several published studies to derive empirical methods for estimating Lz and the transverse dispersion coefficient (kz). Rutherford’s methods will be used here as a check on Lz estimated by the method described above and to make estimates of Lz beyond the limits of instrument mea- surement accuracy. The transverse dispersion coefficient kz, which represents the rate of lateral mixing, has been found to range considerably between different rivers within reaches of similar channel morphology. However, when kz was scaled by mean depth and mean shear velocity for gently meandering reaches, a general range was found so that 0.3 < kz H ·U∗ < 0.9 (2.2) where kz is the transverse dispersion coefficient (m2s−1), H is mean depth (m) and 20 ll l l l l l l 0 2000 4000 6000 8000 9 10 11 12 13 14 Distance downstream from confluence (m) T w  (°C ) l Lillooet R. left bank Lillooet R. right bank initial difference difference at 40% mixed difference at 80% mixed Figure 2.3: Longitudinal bank temperature profiles on 13 Aug at 17:50 along Lillooet River downstream of the tributary confluence. U∗ is mean shear velocity (ms−1), defined as U∗ = √ g ·R · s≈ √ g ·H · s (2.3) where g is gravitaional acceleration (ms−2), s is longitudinal channel slope (dimen- sionless), and R is hydraulic radius (m) (Rutherford, 1994). For shallow channels, the hydraulic radius is approximately equal to the mean depth (Dingman, 2002; Rutherford, 1994). Using mean H and U∗ from Lillooet River over the mixing 21 study period, 0.21 < kz < 0.64. (2.4) Rutherford presented curves for different transverse locations of tracer input show- ing how Pm increases with non-dimensional distance downstream. Mixing lengths were extracted from these curves to compare with estimates from Lillooet River. Non-dimensional distance was defined by Rutherford (1994) as x∗ = Lz · kz v ·b2 (2.5) where x∗ is non-dimensional distance, Lz is transverse mixing distance for a certain Pm (m), v is mean velocity (ms−1), and b is mean channel width (m). 2.3.2 Reach-scale heat budget model The heat budget model (Eq. 2.6) was applied to three stream reaches within the study area: the Lillooet River in the upstream end of the study area (UL), the Lillooet River in the downstream end of the study area (LL), and Ryan River near the Lillooet River confluence (RR). Due to accessibility constraints, the reaches were not of equal length. A summary of reach properties is provided in Table 3.5. For unsteady flow, the Eulerian heat budget for a laterally well-mixed stream with no tributaries or groundwater discharge can be expressed as follows (Chaudhry et al., 1983; Meier et al., 2003): ∂ (AT ) ∂ t + ∂ (QT ) ∂x − ∂ ∂x [ AE ∂T ∂x ] = Q∗+Qh+Qe+Qb ρCp (2.6) where A is the cross-sectional area of the stream (m2), E is the longitudinal disper- sion coefficient, Qh and Qe are the sensible and latent heat fluxes (Wm−2), Qb is heat conduction from the bed (Wm−2), ρ is the density of water (kg m−3), and Cp is the specific heat capacity of water (J kg K−1). Cp was assigned a constant 4191 J kg K−1, which is the specific heat capacity of water at 10 °C (Dingman, 2002) (water temperatures of the proglacial streams of this study were typically between 8 °C and 12 °C.) Because longitudinal dispersion is typically a second-order term for streams 22 without point sources of heat (Gu et al., 1998; Sinokrot and Stefan, 1993), it will not be considered here. In addition, the bed heat flux can be considered a second- order influence for larger streams (Gu et al., 1998). Given these simplifications, and expanding the derivatives using the product rule, the governing equation can be expressed as: ∂T ∂ t = Q∗+Qh+Qe ρCpD − v∂T ∂x − T A (∂A ∂ t + ∂Q ∂x ) (2.7) The first term on the right-hand side represents the temperature change associated with exchange of energy across the water surface. The specification of net radia- tion, sensible, and latent heat fluxes are described in the next two sections. The second and third terms on the right-hand side represent the effects of ad- vection and unsteady flow, respectively. Cross-sectional area, depth, width and velocity were computed from stream gauging data. Continuous time series of these quantities were derived by fitting power-law relations to predict them from recorded discharge. Derivatives were estimated using finite differences. Heat advection occurs where a thermal gradient is present between the upper and lower bounds of a reach. The advection term is preceeded by a minus sign to maintain the standard thermal gradient calculation, which was used to estimate the space derivative of water tem- perature: ∂T ∂x ≈ ∆T ∆x = Tds−Tus L (2.8) where the middle term is the reach-averaged thermal gradient (°C/m), T is water temperature with subscripts ds and us the downstream and upstream ends of the reach, and L is reach length (m). Therefore, when the upstream water temperature is less than the downstream water temperature and the thermal gradient is positive, heat advection has a cooling effect; heat advection has a warming effect when the thermal gradient is negative. The time derivative of cross-sectional area was estimated as ∂A ∂ t ≈ ∆A ∆t = Ai+1−Ai−1 ti+1− ti−1 (2.9) 23 where the subscript i denotes the time step of the heat budget model calculation. The space derivative of discharge is difficult to determine without having mul- tiple gauging sites with no intervening tributaries, which was not feasible in this study. However, an order-of-magnitude estimate was computed as follows, based on the assumption that the wave celerity can be approximated by the mean velocity of flow: ∂Q ∂x ≈ 1 v ∂Q ∂ t (2.10) 2.3.3 Net radiation model Incident shortwave and longwave radiation measurements made at the Open Site Meteorological Station (OSMS) were used in conjunction with measured riparian vegetation dimensions to develop a reach-scale net radiation model for each stream reach. The general approach was adapted from Moore et al. (2005b). Net radiation Q∗ can be expressed as Q∗ = K∗+L∗ (2.11) where K∗ is net short wave radiation (Wm−2) and L∗ is net long wave radiation (Wm−2). The remaining portion of this section explains the calculation of K∗ and L∗. Net short wave radiation Net short wave radiation K∗ is computed as: K∗ = K↓(1−α) (2.12) where K↓ is incident short wave radiation (Wm−2) and α is stream surface albedo. Incident shortwave radiation K↓ was calculated over a reach as K↓ = gtKd + fvKs (2.13) where gt is the fraction of the stream surface that is not shaded by riparian vege- 24 tation at time t, Kd is direct solar radiation (Wm−2), Ks is diffuse solar radiation (Wm−2), and fv is the sky view-factor. Global solar radiation, which included both direct and diffuse components, was measured at OSMS as a surrogate for incident solar radiation above the riparian foliage canopy. Neither component was specifi- cally measured on site. Therefore, the components were partitioned by calculating the diffuse fraction according to the procedure presented by Erbs et al. (1982) and as used by Leach and Moore (2010) for computing insolation at a stream surface. The procedures for estimating gt and fv are described in detail in the Appendix. Stream surface albedo measurements have typically ranged between 0.05 and 0.1 for low-gradient and valley-bottom streams (Evans et al., 1998; ?; Leach and Moore, 2010), but the albedo of water is known to vary with factors such as the in- cidence angle of direct solar radiation, the ratio of diffuse to global solar radiation, suspended sediment concentration, and aeration (Han, 1997; Richards and Moore, 2011). In this study, albedo was specified based on measurements at Lillooet and Ryan rivers. In addition, the data were used to investigate the dependence of albedo on solar zenith angle and SSC. Net longwave radiation Longwave radiation emitted by the sensor was included in the longwave radia- tion measurement (Lmeas) at OSMS and required post-processing to remove. The Kipp&Zonen CGR3 manual (Kipp and Zonen, 2009) provided an equation to cal- culate the sensor temperature: Tsens = (a+b[ln(v)]+ c[ln(v)]3)−1 (2.14) where Tsens is the temperature of the CGR3 sensor (K), v is sensor output voltage (µv), a = 1.03× 10−3, b = 2.39× 10−4, and c = 1.57× 10−7. The sensor was considered a black-body and assumed to have an emissivity of 1. The incident longwave radiation at OSMS was then calculated as L↓OSMS = Lmeas − σ T 4sens (2.15) where σ is the Stefan-Boltzmann constant (5.67 × 10−8 Wm−2K−4). 25 Measured incident longwave radiation at the open site meteorological station was used to calculate atmospheric emissivity (εa). This method of estimating εa implicitly accounts for the presence of clouds and was valid during day and night. The incident longwave radiation at OSMS can be expressed as follows: L↓OSMS = [εa fOSMS+ ε f (1− fOSMS)]σ(Ta+273.2)4 (2.16) where fOSMS is the sky view factor at OSMS, ε f is emissivity of foliage, and Ta is air temperature (°C). The emissivity of deciduous trees is between 0.97 - 0.98 (Oke, 1987). For this study ε f was assumed constant at 0.97, and the foliage temperature was assumed equal to the air temperature measured at OSMS. Given values for fOSMS and ε f , the atmospheric emissivity can be computed by solving Eq. 2.16 for εa. Hemispherical sky photos at OSMS were analysed with Gap Light Analyser software (Frazer et al., 1999) to calculate fOSMS by the method presented by Moore et al. (2005b): fOSMS = 1 pi ∫ 2pi 0 ∫ pi/2 0 g∗(θ ,ψ)cosθ sinθ dθ dψ (2.17) where θ and ψ are solar zenith and aziumuth angles, respectively, and g∗(θ ,ψ) is the canopy gap fraction at the hemispherical image position θ , ψ . The image was overlaid with a 5◦ zenith by 5◦ azimuth grid and gap fractions were computed for each grid square. The double integral was approximated by summing all gap fractions. The longwave radiation reaching the stream surface can be expressed as L↓ = [εa fv+ ε f (1− fv)]σ(Ta+273.2)4 (2.18) where fv is the sky view factor for a particular reach (Eq. A.15). The longwave radiation emitted by the stream surface can be calculated as L↑ = εwσ(Tw+273.2)4 (2.19) where εw is the emissivity of the stream and Tw is the stream temperature measured at the nearest Tidbit logger (°C). The stream emissivity was assumed to be 0.95, 26 which is the mean of the water emissivity range provided by Oke (1987). Net longwave radiation was then calculated as L∗ = εwL↓−L↑ (2.20) where εw represents the fraction of incoming longwave radiation absorbed by the stream. 2.3.4 Convective heat exchanges The latent heat flux was estimated using a Penman-type equation presented by Moore et al. (2005b) which was adapted from Webb and Zhang (1997a): Qe = 285.9(0.132+0.413u)(ea− ew) (2.21) where u is measured wind speed (m s−1), ea and ew are vapor pressures of the am- bient air and the hypothetical thin layer of air above the water surface, repectively (kPa), and the constants account for the latent heat of vaporization, the specific weight of water, various unit conversions, and empirical constants of the Penman equation. The actual vapour pressure of the ambient air above the stream (ea) was calcu- lated as ea = e∗ ( RH 100 ) (2.22) where RH is relative humidity measured at OSMS and e∗ is saturation vapor pres- sure (kPa), which was calculated as a function of air temperature from an empirical relation presented by Dingman (2002): e∗ = 0.611 · exp ( 17.3 ·T T +273 ) (2.23) where T is air temperature (°C). For hydrologic computations, this relation is suf- ficiently accurate compared to the true relation between e∗ and air temperature represented in the more complex Goff-Gratch Equation (Dingman, 2002). In cal- culating ea, T in Eq.2.23 was the ambient air temperature recorded at OSMS. 27 The hypothetical thin layer of air immediately above the water surface was assumed to have a temperature equal to the water temperature and to be saturated with water vapor. Therefore, the vapour pressure within this boundary (ew) was calculated directly from Eq. 2.23 with T equal to the water temperature measured at the nearest Tidbit logger. Following Moore et al. (2005b), sensible heat flux (Qh) was calculated as Qh = Qeγ ( Ta−Tw ea− ew ) (2.24) where γ is the psychrometric constant (kPa/°C). The psychrometric constant is not a true constant, but varies with atmospheric pressure according to Brunt (1952): γ = Ca ·P rM ·λ (2.25) where Ca is the specific heat of moist air = 1.013 kJ kg−1 °C−1, P is atmospheric pressure (kPa), rM is the ratio of the molecular weight of water vapor to the molec- ular weight of dry air = 0.622, and λ is the latent heat of vaporization = 2.45 MJ kg−1. Following Allen et al. (1998), atmospheric pressure was calculated as a function of air temperature and elevation: P = P0 ( Ta−Γ · z Ta )g/(Γ·R) (2.26) where P is atmospheric pressure (kPa), P0 is standard atmospheric pressure = 101.3 kPa, Ta is air temperature (K), Γ is the moist lapse rate = 0.0065 K m−1, z is the site elevation = 220 m, and R is the specific gas constant = 287 J kg−1 K−1. 28 Chapter 3 Results 3.1 Overview of the field season Figure 3.1 shows mean monthly air temperatures from 1969-2006 near Pember- ton, BC. Also shown are mean monthly air temperatures for 2010 from PACS and August/September 2010 from the open site meteorological station (OSMS). Mean monthly air temperatures at PACS for the 2010 field season were about 1 °C above normal in July, close to historical averages in August and September, and slightly above average in October. Air temperature at OSMS was about 1.5 °C lower than at PACS in August and September, which was expected given its higher elevation, closer proximity to Lillooet Glacier, and its location adjacent to the river. Precipitation preceding the study period (January to April) was cumulatively below normal (Figure 3.2). Precipitation in July was well below normal and Au- gust saw only about half the normal precipitation. Precipitation in September was roughly double the normal amount, while less than half the historic average fell in October. Maximum, mean, and minimum mean daily discharge from 1914-2010, along with 2010 mean daily discharge at the Lillooet River WSC gauging station, are shown in Figure 3.3. Flow in early July peaked above average. This peak was not associated with a rain event detectable at PACS and was likely the result of higher than normal air temperature in July and high rates of meltwater generation. Flow was near the long-term mean in late July and early August, after which flow de- 29 ll l l l l l l l l l l − 10 0 10 20 30 M ea n M on th ly Ai r T e m pe ra tu re  ( ° C ) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec l l l l l l l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l 1969−2006 Max. 1969−2006 Mean 1969−2006 Min. 2010 2010 Open Site Figure 3.1: Historical mean monthly air temperature from 1969-2006 and 2010 measured at PACS. Mean monthly air temperature for August and September, 2010, measured on-site (OSMS), is shown with solid black points. clined more rapidly than the long-term mean until the rain events of late September through November. Incident solar radiation, air temperature, vapor pressure, and wind speed mea- sured at OSMS from late July to mid-October are shown in Figure 3.4. Late July and mid-August had periods of about one week of daily maximum air temperatures over 30 °C. After mid-August, temperatures in general declined gradually. Incident solar radiation in July and August commonly exceeded 800 Wm−2 and gradually declined through September and October. Vapour pressure generally declined af- ter a broad peak in early August, but late September and October had periods of several days in which vapour pressure was elevated. Wind speed, measured as one- minute averages, was typically less than 1 ms−1 with spikes between 1 and 2 ms−1. There was no discernible seasonal pattern of wind speed, except for a lull in early 30 Pr ec ip ita tio n (m m) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0 50 10 0 15 0 20 0 1969−2006 2010 Figure 3.2: Monthly total precipitation for 2010 and mean monthly total pre- cipitation for 1969-2006. Data were missing for May and June, 2010 . October. Water temperatures at the four reaches (UL = upstream Lillooet River, LL = lower Lillooet River, RR = Ryan River, and MC = Miller Creek) ranged between 5 °C and 13 °C, varied on a diurnal cycle, and generally decreased gradually after mid-August (Figure 3.5). Daily mean water temperature is shown for each reach in Figure 3.6 (bottom panel). The temperature pattern at each reach generally fol- lowed the pattern of daily mean air temperature (top panel), but with lower magni- tudes. Average daily statistics for each site are displayed in Table 3.1. Mean daily temperatures were highest at LL and lowest at MC. Daily variability (expressed as standard deviations) was highest at MC and UL, and LL had the least variability. 31 Jan Mar May Jul Sep Nov Jan 0 50 0 10 00 15 00 M ea n D ai ly Di sc ha rg e ( m 3 s − 1 ) Maximum Minimum Mean 2010 Figure 3.3: Discharge for 2010 measured near LL with historical maximum, mean, and minimum daily discharge from 1914-2010. Table 3.1: Daily summary statistics of water temperature at each sub-reach from July 24 to October 15, 2010. sd = standard deviation. Water temperature (°C) daily stat. UL LL RR MC mean 9.70 9.99 9.44 7.79 sd 1.12 0.83 0.87 1.14 maximum 11.39 11.36 10.94 9.67 minimum 8.07 8.76 8.20 6.31 range 3.32 2.60 2.74 3.36 32 0 20 0 60 0 K ↓ (W m − 2 ) 0 5 15 25 35 T a  (°C ) 0 5 10 15 20 25 e a  (m b) 0. 0 0. 5 1. 0 1. 5 2. 0 w  (m /s) Aug Sep Oct Figure 3.4: From top to bottom: incident solar radiation, air temperature, va- por pressure, and wind speed measured at OSMS. 33 6 8 10 12 14 Upper Lillooet 6 8 10 12 14 Lower Lillooet 6 8 10 12 14 Ryan 6 8 10 12 14 Miller T w (°C ) Aug Sep Oct Figure 3.5: Instantaneous water temperature at (top to bottom) UL, LL, RR, and MC. 34 0 5 10 20 30 T a (°C ) Max. Mean Min. 6 7 8 9 10 11 T w (°C ) Upper Lower Ryan Miller Aug Sep Oct Figure 3.6: Maximum, mean, and minimum daily air temperature at OSMS (top) and daily mean water temperature at UL, LL, RR, and MC (bot- tom). 35 0 1 2 3 4 5 T w   R an ge  (° C) Upper Lillooet Lower Lillooet Ryan Miller Aug Sep Figure 3.7: Daily water temperature ranges at each reach. 3.2 Longitudinal variations in water temperature Figure 3.8 shows temperature profiles for Lillooet River at two-hour time intervals from 07:00 - 21:00 on July 25 (all times reported are in Pacific Standard Time). The upper two stations are the upper and lower boundaries of reach UL. The furthest downstream station is the lower boundary of reach LL (the logger at the upper boundary of LL was not yet installed). The 07:00 and 09:00 profiles represent the end of the cooling phase at the uppermost site and the remaining profiles were during the following warming phase. By 11:00, UL had entered a warming phase 36 l l l l l 0 5000 10000 15000 20000 25000 30000 6 7 8 9 10 11 12 13 Distance downstream (m) T w  (°C ) l 07:00 09:00 11:00 13:00 15:00 17:00 19:00 21:00 July 25 07:00 to 21:00 Figure 3.8: Longitudinal temperature gradients in Lillooet River on July 25 showing the end of the cooling phase (blue) in the morning followed by the warming phase (red). while LL still experienced cooling. Longitudinal temperature gradients at 11:00 were negative at UL and positive at LL; as a result, longitudinal heat advection was having a warming effect at UL and cooling effect at LL. A rather uniform longitudinal temperature gradient existed throughout the entire study segment of Lillooet River from 13:00 to 19:00. UL warmed slightly between 19:00 and 21:00 while considerable warming still occurred downstream. 37 0 5000 10000 15000 20000 25000 30000 6 7 8 9 10 11 12 13 Distance downstream (m) T w  (°C ) l l l l l l l l l l l l 19:00, July 25 21:00 23:00 01:00, July 26 03:00 05:00 07:00 09:00 July 25 19:00 to July 26 09:00 Figure 3.9: Longitudinal temperature gradients in Lillooet River showing the end of the warming phase (red) late on July 25 followed by the cooling phase (blue) through the morning of July 26. 38 Table 3.2: Transverse mixing length statistics for the given degree of mixing (Pm) at 10% increments. s.d. = standard deviation Pm mean (m) s.d. (m) min, max (m) 10 473 105 n/a, 820 20 529 126 n/a, 1170 30 602 193 n/a, 1570 40 740 319 400, 1980 50 1110 464 430, 2490 60 1717 460 460, 3010 70 2332 421 500, 3510 80 3036 343 2150, 4560 90 4456 835 2970, n/a By 23:00, UL had entered a cooling phase while the lower section of Lillooet River was still warming, reflecting the passage of a temperature wave along the study segment (Figure 3.9). By 01:00 on July 26, UL had a positive thermal gra- dient and was thus subjected to advective cooling while LL was still warming, still had a negative thermal gradient and experienced advective warming. The entire study segment was cooling from 03:00 to 09:00 (July 26). However, the longitudi- nal temperature gradient was non-uniform during the cooling phase. 3.3 Transverse mixing 3.3.1 Observed transverse mixing in Lillooet River This section focuses on the period from August 12 - 20, 2010. Figure 3.10 shows time-series of left and right bank water temperatures, interpolated linearly from ob- served temperatures, at four locations downstream of the Ryan River confluence, along with the temperature difference between the banks. The temperature differ- ence near the confluence ranged from 0 °C to 1.8 °C. The difference between left and right banks decreased downstream and was near zero most of the time at 6 km downstream of the confluence. Transverse mixing progressed most rapidly immediately below the confluence 39 n8 10 12 14 150 m downstream 0 2 n 8 10 12 14 2 km downstream 0 2 n 8 10 12 14 4 km downstream 0 2 n 8 10 12 14 6 km downstream 0 2 T w  (°C ) ∆T w  (°C ) Aug 13 Aug 15 Aug 17 Aug 19 Figure 3.10: Time-series of left (blue) and right (red) bank water temperature along Lillooet River at four downstream locations from the Ryan River confluence. Temperature difference (black) between the banks at each location is shown on the bottom of each panel with axis on right. 40 ll l l l l l l l 1000 2000 3000 4000 20 40 60 80 Mixing Length  Lz (m) Pe rc e n t M ix in g Figure 3.11: Mean transverse mixing lengths for the percent of lateral mixing at 10% increments. and was continually less rapid downstream. An example for a specific time is shown in Figure 3.12. Roughly half the initial temperature difference between the left and right banks was closed within a distance of 1 km from the confluence. The mixing length at that time for Pm = 80% was 2.55 km, which was over three times that for Pm = 40% at 0.78 km. This behaviour is also illustrated in Figure 3.11. For example, the mean value of Lz80 (3036 m) was more than four times that of Lz40 (740 m). Table 3.2 summarizes the values of Figure 3.11 with additional statistics. 41 ll l l l l l l 0 2000 4000 6000 8000 9 10 11 12 13 14 Distance downstream from confluence (m) T w  (°C ) l Lillooet R. left bank Lillooet R. right bank initial difference difference at 40% mixed difference at 80% mixed Figure 3.12: Longitudinal bank temperature profiles on 13 Aug at 17:50 along Lillooet River downstream of the tributary confluence. Electrical conductivity (EC) measurements confirmed the general pattern of lateral mixing. Figure 3.13 presents results from the first of two watercraft surveys that measured EC across Lillooet River downstream of Ryan River junction on Aug. 26, 2010 between 12:25 and 16:45. Surveys extended only to 2.1 km below the confluence due to logistical constraints. EC above the confluence was nearly uniform at 44 µ S/cm. Just below the confluence, EC on the right bank (the side of the junction) dropped to 23 µ S/cm. After 2.1 km, right bank EC was 35 µ 42 l l l l l l l l 0 500 1000 1500 2000 0 20 40 60 Distance downstream (m) EC  ( µ S/ cm ) l l l l l l l l l l l l l l l l l l left bank left center center right center right bank left − right bank Figure 3.13: Electrical conductivity (EC) was surveyed by eight lateral tran- sects approximately 250 m apart, extending downstream of the Ryan River confluence 2.1 km. S/cm and the left bank remained near 44 µ S/cm, which was a 40% closure of the EC difference near the junction. This was similar to the upper range of Lz40 of 2.0 km observed over the mixing study period using temperature differences. The second survey was conducted immediately after the first survey with nearly identical results. 43 Table 3.3: Predicted transverse mixing lengths using Rutherfords (1994) models for given channel form and tracer input location. Predicted ranges reflect given transverse dispersion coefficient ranges. Mean observed Lz80 was 3036 m. Channel form Input location Predicted Lz80 (m) Gently meandering bank 3970 to 11900 Gently meandering 1/3 3080 to 9250 Gently meandering 1/2 980 to 2940 Sharp bends/constrictions bank 1205 to 3565 3.3.2 Modelled transverse mixing Transverse mixing lengths calculated from the model presented by Rutherford (1994) (Section 2.3.1) for various channel morphology and tracer input locations are given in Table 3.3. Mean observed Lz80 was 3036 m, less than the range pre- dicted by the model for bank source tracer input in gently meandering channels. Considering the momentum of a tributary, however, the effective input location across the receiving channel’s width may be some distance from the bank. Mean observed Lz80 was within the range predicted by Rutherford’s model for tracer in- put between one-third and one-half the width of the receiving channel in gently meandering channels. Though Lillooet River can be characterized as mostly gen- tly meandering, there are some sharp bends below the confluence. Mean observed Lz80 was within the range predicted by Rutherford’s model for bank source tracer input in channels having sharp bends. Instrument accuracy limitations prevented determining transverse mixing lengths higher than the 80% mixing level. Complete transverse mixing lengths were there- fore estimated for comparison to other studies, which often report this metric. Mean observed Lz80 was used to calculate the mean transverse dispersion coef- ficient (kz), which was used to estimate Lz at the 98% mixing level. Mean ˆLz98 = 5420 m (69 multiples of average river width) and ˆLz98 ranged from 3870 m to 8300 m (49 to 105 river widths). Figure 3.14 compares Lz predictions from Rutherford’s model (calibrated at Pm = 80%) to observed values for a range of mixing completeness levels. Near the con- 44 ll l l l l l l l l 0 2000 4000 6000 8000 0 20 40 60 80 10 0 Mixing Length  Lz (m) Pe rc e n t M ix in g l l l l l l l l l l l measured modelled Figure 3.14: Observed and modelled mean transverse mixing lengths for given degree of mixing. fluence, observed transverse mixing was more rapid than modelled. Where mixing neared completion downstream, observed mixing was less rapid than modelled. 3.3.3 Effect of stream discharge Variation in Lz was in part explained by the fraction of tributary to mainstem streamflow (Qratio), calculated as 45 Table 3.4: Transverse mixing distances and dispersion coefficients at times of the given Qratio statistic. Statistic Qratio Lz80 (m) Lz98(m) kz instantaneous min 0.46 2580 4670 0.98 instantaneous max 0.94 3760 6809 0.67 mean daily min 0.57 2710 4900 0.94 mean daily max 0.86 3480 6300 0.73 Qratio = Qtributary Qmainstem (3.1) where Qtributary and Qmainstem refer to discharge in Ryan River and Lillooet River, respectively (m3s−1). Miller Creek’s flow was negligible in comparison to the other rivers and not considered in Eq. 3.1. Lz tended to increase with increasing Qratio. Figure 3.15 shows this relation for Pm = 80%. The best-fit line for this plot took the form ˆLz80 = 1481+2237 ·Qratio (3.2) where ˆLz80 is the predicted value. The fitted relation has R2 = 0.43 and p < 0.001. The relation between Ryan River flow and Lz was not as strong (R2 = 0.27), and there was no significant relation between Lillooet River flow and Lz. As shown in Table 3.4, Qratio increased 51% on average over the course of a day, from an average daily minimum of 0.57 to a maximum of 0.86. Lz80 corre- spondingly increased by 28%, on average, between daily minimum and maximum Qratio. For modelled Lz98, the increase was 29% between daily minimum and max- imum Qratio. Estimated kz correspondingly decreased by 22%. 46 l l l l l l ll ll lll l l lll l ll ll ll l lll l l ll ll l l l l l l l l l l l l lll l lll l l l l l l l l ll l l l l lll l ll ll ll llll l lll lll l l l ll ll l ll ll l l l l ll ll l ll l l l l l l ll l ll ll ll ll l l l l lll ll l l l ll l l ll ll l l ll l ll l l l l l l ll l lll l l l l ll l ll l l l ll ll l l l l lllll l ll lll l l ll l l l l l ll ll l l lll l l ll l ll l lll l l ll l l ll l l ll ll lll l l l ll ll l ll l ll l l l l l l l l l l l l l l l l l l lll l ll l lll l l l l l l l llll l llll l l l l l l ll lll l l l l l l l ll l l ll ll ll l l l l l l l l l ll ll l l lll ll l l l l ll l lll l l l l l l l lll l l l l l l l llll l ll l ll l l l l l l l l l l l l l l l l l l l l l ll ll l lll ll l l ll l ll l l lll l l l l ll l lll llll lll l l llll ll l l l ll ll l l l l l l l l l l l l l ll l l ll ll l ll l l l l l l l l l l ll l l l ll l ll l ll l ll l l l lll l l l l ll l ll ll l l ll l lll l l l ll ll l l ll l ll l l l l l l l ll l l ll l l l l l ll l l l ll l ll l l l l ll l lll l l ll l l ll l l l l ll llllll l l lll lll 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 20 00 25 00 30 00 35 00 40 00 45 00 50 00 QRyan QLillooet L z 80  (m ) Figure 3.15: Transverse mixing lengths at the 80% mixing level for given ratios of tributary to mainstem discharge. 3.4 Surface-atmosphere energy exchanges 3.4.1 Radiative exchanges The material in this section first focuses on the quantification of key factors that control the radiation budget, including albedo, sky view factor, shading factor and atmospheric emissivity. The calculated net radiation for each reach are then pre- sented. 47 Albedo Measured albedo varied with solar zenith angle θ (0◦ overhead to 90◦ at the hori- zon), although there was scatter in the relation, especially for Lillooet River (Figure 3.16). At a given time (and θ ), albedo was highest for Lillooet River, followed by Ryan River and Miller Creek (Figure 3.16). Albedo measurements were not pos- sible for θ above 75◦, due to shading from banks, or below 40◦, as this was the lower limit of θ for this time and latitude. The effect of the upper θ limit on en- ergy balance calculations should be minor since the streams were largely in shade under those conditions. Suspended sediment concentration (SSC) was highest in Lillooet River, fol- lowed by Ryan River and Miller Creek, with medians of 831, 71, and 34 mg L−1, respectively (Figure 3.16). A regression of albedo as a function of zenith angle, SSC, and their interaction was fitted over all observations, and took the form α̂ = aθ +bSSC+ c(θ ·SSC) (3.3) where α̂ is the predicted albedo, θ is solar zenith angle, SSC is suspended sediment concentration (mg L−1), and the coefficients are a = 0.00116, b = not significant at α = 0.05, and c= 1.04×10−6, with p-values = 0.00015, 0.416, 0.0385, respectively (R2 = 0.83, p << 0.0001, standard error of the estimate = 0.008). The means (and medians) of albedo measurements were 0.10 and 0.08 for Lil- looet River and Ryan River, respectively. These values were used as constants in calculating net shortwave radiation, as described below. Sky view factor and atmospheric emissivity The Upper Lillooet reach had the highest percentage of riparian canopy presence of the three reaches and the lowest sky-view factor ( fv) (Table 3.5). The Lower Lillooet reach had less canopy presence and a wider channel, and thus a higher fv, than UL. The Ryan River reach had the lowest riparian canopy presence and the lowest channel width, and thus the highest fv of the three reaches. View factors were assumed constant through the study period, which should be reasonable given that the study period began after spring leaf-out and before leaf-fall in the autumn. 48 l l l ll l lll l l 0. 04 0. 08 0. 12 Al be do Al be do l Lillooet Ryan Miller ll l l l l l ll ll l ll ll ll lll ll ll 40 45 50 55 60 65 70 Ze ni th  A ng le lll lll ll l ll 10 10 0 10 00  SS C (m g/L ) 07:00 09:00 11:00 13:00 15:00 17:00 19:00 Figure 3.16: Measured albedo (top) and suspended sediment concentration (bottom) for Lillooet River, Ryan River, and Miller Creek on August 25, 2011, with calculated solar zenith angle (center). 49 0. 70 0. 80 0. 90 1. 00 ε a Aug 01 Aug 15 Sep 01 Figure 3.17: Atmospheric emissivity (εa) was calculated over the monitoring period. Atmospheric emissivity, computed using (Eq. 2.16), varied between 0.69 and 1 over the study period and had a mean of 0.90 (Figure 3.17). Values of εa were typically lower during the day than at night, which may be expected from cloud formation. Shade function, gt Table 3.5 summarizes mean gt over the study period for each reach (means include zero values during night or complete shade). Each site reached similar maxima near 0.95 but their non-zero minima and daily patterns differed (Figure 3.18). Net radiation, Q∗ Figure 3.19 shows the components of net shortwave and longwave radiation with resulting net all-wave radiation for UL. Figure 3.20 shows Q∗ for each reach. RR consistently had the highest daytime values and UL the lowest. On average, Q∗ at RR was 20% higher than at UL, while LL received 12% more net radiation than UL. 50 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 UL 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 LL 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 RR g t 14Aug 15Aug 16Aug 17Aug Figure 3.18: Shading function gt for each reach, August 14-17. 51 Table 3.5: Summary of parameters used in the heat budget model for each reach: upper Lillooet River (UL), lower Lillooet River (LL), and Ryan River (RR). item UL LL RR reach length (m) 1767 2757 2079 channel width (m) 61.6 78.7 45.1 orientation (◦) 322 298.3 303.5 canopy presence (%) 78.2 49.9 21.6 canopy height (m) 28.5 28.5 28.5 albedo, α 0.10 0.10 0.08 sky view-factor, fv 0.666 0.818 0.861 shade function, gt (mean) 0.51 0.71 0.82 0 40 0 80 0 K ↓ K ↑ L ↓ L ↑ 0 40 0 80 0 Q* R ad ia tio n ( W m − 2 ) 08/01 08/15 09/01 Figure 3.19: Radiative exchanges averaged over the upper Lillooet River reach (UL): incident and outgoing shortwave (K) and longwave (L) radiation (top), net all-wave radiation (Q∗) (bottom). 52 0 40 0 80 0 Q *  ( W m − 2 ) UL LL RR 08/01 08/15 09/01 Figure 3.20: Net all-wave radiation (Q∗) for upper Lillooet River (UL), lower Lillooet River (LL), and Ryan River (RR). − 10 0 0 50 10 0 H ea t F lu x ( W m − 2 ) Qe Qh Aug 01 Aug 15 Sep 01 Figure 3.21: Latent and sensible heat flux at UL. 3.4.2 Net surface-atmosphere energy exchange Latent heat flux (Qe) was 7% of Q∗ at UL, on average, and ranged between -80 Wm−2 and 61 Wm−2 (Figure 3.21). Negative values indicate heat loss by evapo- ration and positive values indicate heat gain by condensation. Sensible heat flux (Qh) was typically higher than Qe and ranged from -20 Wm−2 to 127 Wm−2. Qh was 15% of Q∗ at UL, on average. In Figure 3.22, sensible and latent heat flux are plotted with net radiation at UL, the sum of which 53 0 20 0 60 0 10 00 H ea t F lu x ( W m − 2 ) HQ* Qe Qh Aug 01 Aug 15 Sep 01 Figure 3.22: Net surface heat exchange (H) with net radiation, latent heat, and sensible heat flux components at UL. is the total energy flux across the water surface, which will be referred to as vertical energy flux. 3.5 Reach-scale heat budgets Eq. 2.7 provided the model used to predict rates of change of stream temperature with time for each site. The terms on the right-hand side of Eq. 2.7 represent the contributions to temperature change associated with vertical energy exchange across the surface, longitudinal heat advection, heat storage change with stream- flow, and effects of flow divergence. For brevity, these terms may be referred to as vertical, advection, δA/δ t, and δQ/δx terms, respectively. The latter two can be grouped to represent the effects of unsteady flow. This section provides heat budget model results for UL, which had the longest period of record, for clear-sky, partly cloudy, mostly cloudy, and rainy periods. The landslide event was also modelled. Weather conditions were categorized according to the shortwave radiation record at OSMS and the precipitation record at PACS. The following section focuses on heat budget calculations for UL to illustrate the contrast between upstream and downstream (LL) sites. 54 3.5.1 Clear-sky period Figure 3.23 shows UL heat budget model results (top panel) for the clear-sky period July 24 to July 27 along with discharge and water temperature. Vertical heat flux was mostly in phase with advection, and both terms had relatively smooth arching patterns during this clear-sky period. The vertical and advection terms were out of phase with the unsteady flow terms δA/δ t and δQ/δx. Daytime positive ver- tical heat flux had a magnitude of about half that of advection. At night, vertical heat flux was near zero and advective cooling reached slightly greater magnitudes than advective warming during the day. Heat advection also had a greater mag- nitude than the unsteady flow terms and was thus the dominant control on stream temperature dynamics. The unsteady flow terms were opposite in phase with each other due to the dif- ferent directions of their relations with the temporal rate of discharge (Q) change, thus offsetting their individual effects (Figure 3.23). The positive phase of the δQ/δx term had a similar magnitude to the positive phase of the vertical term (but differed in timing). At any time, however, the magnitude of the δQ/δx term was only slightly greater than that of the δA/δ t term of opposite sign. The result was a combined influence near zero (green line), although during rising flow the com- bined terms had a slight warming influence and a slight cooling influence during decreasing flow. Discharge (middle panel of Figure 3.23) displayed a mirror-like phase rela- tion with advection. Daily maximum discharge during this period occurred near the time of minimum heat advection (and minimum δT/δ t), and daily minimum discharge occurred near the time of maximum advection (and maximum δT/δ t). The sign of the advection term depends on the longitudinal temperature gra- dient, which alternated diurnally and varied spatiotemporally in the Lillooet River study reach. Positive longitudinal temperature gradients (cooler upstream) typ- ically occured during nighttime cooling phases and negative thermal gradients (warmer upstream) during daytime warming phases. There was a lag time in tem- perature gradient shifts between UL and LL, which related to their timing of ad- vection phases. The heat budget model performed quite well over this period at UL (Nash- 55 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 15 0 16 0 17 0 18 0 19 0 20 0 7 9 11 13 T w  ( ° C ) 24Jul 25Jul 26Jul 27Jul Figure 3.23: Heat budget (top), discharge (center), and water temperature (bottom) during clear-sky conditions, July 24 to July 27, 2010 at UL. 56 dT /d t ( ° C hr − 1 ) − 1 − 0. 5 0 0. 5 1 predicted observed predicted−observed 24Jul 25Jul 26Jul 27Jul Figure 3.24: Predicted (gray) and observed (black) temporal stream temper- ature change and their difference (red) at UL. Sutcliffe model efficiency (Em) = 0.96, Table 3.6). Figure 3.24 highlights the dif- ference (red) between predicted and observed values in time. The model tended to over-predict, with a mean bias error (MBE ) of 0.04 °C/hr. The largest errors oc- curred on the rising and falling limbs of δT/δ t each day, in which over-predictions reached up to 0.26 °C/hr on either limb. Advection alone closed the heat budget on the rising limb overnight until advection began to level off in the morning. At that time, the model began over-predicting by a combination of advection and ver- tical heat flux. The combined unstready flow terms were near zero. This error persisted until around 11:00, when the heat budget was closed by all components. Overprediction occurred again in the evening when vertical flux was near zero. At this time, advection was greater than observed δT/δ t and the combined unsteady terms were slightly positive. The error lessened as δT/δ t approached the daily minimum and the heat budget was again closed by advection alone. 3.5.2 Cloudy periods The heat budget components were sensitive to changes in atmospheric conditions (Figure 3.25). Daytime cloud cover can be identified in periods when the verti- 57 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 15 0 16 0 17 0 18 0 19 0 20 0 7 9 11 13 T w  ( ° C ) 30Jul 31Jul 01Aug 02Aug Figure 3.25: Heat budget (top), discharge (center), and water temperature (bottom) from July 30 to Aug. 2, 2010 at UL. Conditions were clear on July 30, switching to overcast July 31, and partly-cloudy Aug. 1. 58 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 15 0 16 0 17 0 18 0 19 0 20 0 7 9 11 13 T w  ( ° C ) 03Aug 04Aug 05Aug 06Aug Figure 3.26: Heat budget (top), discharge (center), and water temperature (bottom) from Aug. 3 to Aug. 6, 2010 at UL. Aug. 4 was consis- tently overcast while Aug. 3 and Aug. 5 were partly-cloudy. 59 − 1. 0 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 70 80 90 10 0 11 0 12 0 7 9 11 13 T w  ( ° C ) 05Sep 06Sep 07Sep 08Sep Figure 3.27: Heat budget (top), discharge (center), and water temperature (bottom) for the partly cloudy-overcast period from Sept. 5 to Sept. 8, 2010 at UL. 60 Table 3.6: Heat budget model performance for given site and time period (month/day). MBE is mean bias error, RMSE is root mean squared er- ror, NRMSE is RMSE normalized by the range of observed values and expressed as a percentage, and Em is the Nash-Sutcliffe model efficiency. site dates conditions MBE RMSE NRMSE (%) Em UL 7/24-7/27 clear 0.0381 0.096 6.8 0.961 UL 7/30-8/02 clear to cloudy 0.0690 0.111 7.5 0.926 UL 8/03-8/06 partly cloudy 0.0784 0.110 8.6 0.892 UL 8/05-8/08 (landslide) 0.0863 0.195 7.1 0.695 UL 9/05-9/08 mostly cloudy -0.0284 0.157 10.9 0.673 UL 9/18-9/21 rain 0.0195 0.096 11.3 0.600 LL 8/14-8/17 clear 0.022 0.053 6.9 0.876 UL 8/14-8/17 clear 0.079 0.150 13.2 0.833 cal component does not display a smooth arch such as it did in Figure 3.23 for clear conditions. Partly-cloudy conditions were associated with irregular patterns (spikes and dips) in the vertical component of the heat budget, as incident solar radiation fluctuated from the passing shade of clouds, (e.g., on August 1). During those partial-cloud periods, the advection component also displayed an irregular pattern, which contrasted to the smooth arch during clear conditions. The irregu- lar advection pattern typically transitioned into a smooth arch the night following partly-cloudy daytime conditions (Figure 3.26). Similar to clear conditions, advec- tion during partly-cloudy conditions was generally in phase with, and of greater magnitude than, the vertical component. Advective warming and cooling rates during partly-cloudy conditions had mag- nitudes similar to those during clear skies although, at times, advective warming rates actually reached higher values and advective cooling did not reach values as low as those seen during clear weather. The timing of the response of heat advec- tion to cloud cover changes was inconsistent at times. An immediate response occurred on July 31, when clear conditions suddenly became overcast (Figure 3.25). Advection dropped simultaneously with the vertical term (it interestingly rebounded shortly after, despite the vertical term remaining low). A delayed re- sponse occurred on August 5, when a major dip in advection was not coincident 61 with a dip in the vertical component (Figure 3.26). The most recent dip in vertical heat exchange had occurred several hours prior. The heat budget model performed well in predicting these temperature responses, in spite of the apparent inconsis- tency in the timing relation between irregularities in the advection and vertical components. With increasing cloud cover, the vertical and advection components decreased. On September 6, the vertical component remained low during heavy overcast con- ditions (Figure 3.27). Advective warming was greatly subdued, and there were periods of several hours in which advection was negligible. There was still a con- siderable period of advective cooling, although shorter and lower magnitude than during clearer conditions, with similar onset timing in late evening, but ending in the early, rather than late, morning hours. This shortened advective cooling phase was associated with warmer absolute nighttime water temperatures (bottom panel of Figure 3.27). Thus, in general for a variety of conditions, the magnitudes of advective warming and cooling, and also the cooling phase timespan, related with the magnitude of the daytime vertical component. Similar to clear conditions, during partly-cloudy conditions, the advection term generally increased (decreased) while discharge decreased (increased) (Figures 3.25 and 3.26), but this association was not as apparent when cloud cover was heavy (Figure 3.27), and did not exist during the rain event (Figure 3.28. Overcast conditions were associated with a substantial decline in streamflow, displaying ir- regularities and a less well defined diurnal meltwater wave (center panel of Figure 3.27). With the vertical and advective components having low magnitudes, the rel- ative influence from unsteady flow had more importance than during clearer con- ditions, noticeably during rapid flow changes within the hydrograph irregularities. However, the combined unsteady flow effect was still minor. Predicted temperature changes were generally less accurate with greater cloud cover (Table 3.6). The model performed well during partly-cloudy conditions (Em = 0.89), although less well than during clear conditions (Em = 0.96), despite rapid fluctuations in vertical and advective heat exchanges. It tended to over-predict more than during clear conditions, (MBE = 0.078, 0.038 °C/hr, respectively). Model performance was poorer in the afternoon and evening as flows increased than at other times, when flows were decreasing (Figure 3.26). Advection alone was often 62 greater than observed dT/dt while the other terms were near zero. Advection alone closed the heat budget over much of the night-time cooling phase (other terms were near zero); at other times, closure was more or less met by advection and vertical components. Predicting the irregularities in temperature changes was particularly inaccurate under heavy overcast conditions (Em = 0.67). Predictions tended to be, however, only slightly under-estimated on average, with MBE = -0.028 °C/hr. Although the patterns of the heat budget components varied with weather con- ditions, the advection component remained the dominant component throughout the conditions encountered. During partly-cloudy to light overcast conditions, ad- vection was in-phase with the vertical term and out of phase with discharge, which was consistent with clear conditions. With heavy overcast conditions, advection displayed different phase timing and much lower magnitudes in both warming and cooling influences, even having several-hour periods of negligible effect. These results showed that the heat budget was quite sensitive to changes in atmospheric conditions, often responding in an irregular manner and at times in a manner that was difficult for the model to predict. 3.5.3 Precipitation event A significant rain event of over 30 mm occured September 19 - 20, resulting in a two-fold discharge increase (Figure 3.28). Water temperature during the event was moderated compared to clear sky conditions, with less diurnal variation. There was a less well defined advection trough following the positive vertical flux on Septem- ber 19 than on the previous overcast day, limiting cooling. During elevated flow on September 20, warming was much lower than usual. The advection term displayed an extremely sporadic pattern during elevated flow, often switching between having a warming or cooling influence, and advection alone nearly closed the heat budget. The unsteady flow terms reached much higher magnitudes (negative and positive) within the rising and falling limbs of the storm hydrograph than seen during the typical diurnal flow patterns of clearer weather. Model closure over the rain event was similar to the cloudy period, despite the storm flow (Em = 0.60), and predictions were slightly high (MBE = 0.02 °C/hr). The greater positive magnitude of the combined unsteady flow components within 63 − 1. 0 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 80 10 0 12 0 14 0 16 0 18 0 7 9 11 13 T w  ( ° C ) 18Sep 19Sep 20Sep 21Sep Figure 3.28: Heat budget (top), discharge (center), and water temperature (bottom) for Sept. 18 to Sept. 21, 2010 at UL. A 30 mm rain event occured Sept. 19 - 20. 64 − 3 − 2 − 1 0 1 2 3 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 50 10 0 15 0 20 0 25 0 30 0 7 9 11 13 T w  ( ° C ) 05Aug 06Aug 07Aug 08Aug Figure 3.29: Heat budget (top), discharge (center), and water temperature (bottom) for Aug. 5-8, 2010 at UL. An outburst flood occured on Aug. 7 from the release of a lake created from a landslide on Aug. 6 in a tributary catchment of Lillooet River. the rising limb of the storm hydrograph created additional over-prediction (Figure 3.28). 65 3.5.4 Landslide event August 5 showed a typical δT/δ t (top) and temperature (bottom) pattern for cloudy weather (Figure 3.29). Discharge declined sharply at 08:00 on August 6, several hours after a landslide dammed the upstream tributary of Meager Creek and par- tially dammed Lillooet River (Figure 3.29). The landslide dam on Meager Creek was breached early on August 7 and the flood water from the draining lake reached UL around 03:00, creating a discharge spike from 160 to 290 m3/s in about 2 hours. Water temperature over the event was irregular (bottom plot). August 6 showed an irregular δT/δ t pattern similar to that observed during the September 19 - 20 rain event despite the sudden drop in discharge, with advection alone basically closing the heat budget. However, δT/δ t increased suddenly on August 7 to 1.92 °C/hr, over twice the maximum warming rate observed during clear-skies (0.74 °C/hr). This was precisely the same time that discharge surged. The magnitude of the unsteady flow terms increased drastically during the drop and spike in discharge, and their combined influence was of a similar magnitude to the vertical term. The spike in δT/δ t and discharge was short-lived. Both returned to typical cloudy-sky patterns within hours. The model provided reasonably good fits throughout this extreme event (Em = 0.70), better than during the September 19 - 20 rain event (Em = 0.60), and predic- tions tended to be high (MBE = 0.086 °C/hr). Advection alone did a reasonable job of closing the heat budget over this event. When the flood arrived on August 7, advection spiked to the highest values observed over the entire study period (2.46 °C/hr), although this slightly over-estimated the observed warming rate. When the flood receded, advection plummeted to the lowest value over the study period (- 1.04 °C/hr), which slightly under-estimated the observed cooling rate. The fit was relatively good and made only slightly poorer with the addition of the unsteady flow terms over the sudden temperature spike associated with advection from the lake water. 3.5.5 Spatial variation Modelled heat budgets differed between UL and LL, as shown in Figure 3.30 dur- ing a mostly clear-sky period from August 14 - 17. The two sites are compared for 66 this period because LL did not have complete data for the July 24 - 27 clear-sky period. The fit at UL for August 14-17 was not as good as for July 24 - 27 in regard to magnitude during the warming phase; however, the cooling phase predic- tions were accurate and the predicted timing of warming/cooling phase shifts were similar to observed. In addition, relations in timing and magnitude between heat budget terms were similar to the clear-sky period for UL. Model performance was reasonably good at both sites: Em for August 14-17 at UL was 0.83 and at LL was 0.88 (Table 3.6). Warming and cooling rates reached considerably higher magnitudes and di- urnal phase shifts occured earlier at UL than LL over this period. The observed maximum warming rate at UL was 78% higher than at LL (0.71 °C/hr and 0.40 °C/hr, respectively) while maximum cooling at UL was 13% higher than at LL (-0.42 °C/hr and -0.37 °C/hr, respectively). Warming began at UL each day near 09:30 and at LL near 11:30. Cooling began at UL each day near 19:00 and at LL near 23:30. Therefore, the duration of observed warming (and cooling) at LL was 12 hours and at UL was only 9.5 hours. The greater warming rates at UL com- pensated for the shorter warming phase and resulted in similar diurnal maximum temperature to LL. The cooling rates at UL were only slightly greater than at LL, but these rates, along with longer cooling phase duration at UL, resulted in lower diurnal minimum temperature than at LL. Vertical heat flux timing was nearly identical between the two sites, but the magnitude at LL was slightly greater than at UL, consistent with LL having less shade (gt). The timing of heat advection, as well as magnitude, differed substan- tially between the sites. Advective warming began at UL near 09:30 each day, which was the same time as observed overall warming and slightly after the onset of positive vertical fluxes. At LL, advective warming began near 14:00 each day, about 5.5 hours after the onset of positive vertical heat flux and about 2.5 hours after observed warming began. Like at UL, advective cooling at LL began concur- rently with the start of observed cooling after the vertical term declined. However, advective cooling at LL began 4.5 hours later than at UL. Thus, durations of advec- tive warming at each site were about 9.5 hours and durations of advective cooling at each site were about 14.5 hours over this clear-sky period, but advection phase shift timing at LL lagged UL by about 4.5 hours. 67 − 1. 0 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed − 1. 0 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) 14Aug 15Aug 16Aug 17Aug Figure 3.30: Heat budget for UL (top) and LL (bottom) during mostly clear skies from Aug. 14-17. 68 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 13 0 14 0 15 0 16 0 17 0 18 0 7 9 11 13 T w  ( ° C ) 14Aug 15Aug 16Aug 17Aug Figure 3.31: Heat budget (top), discharge (center), and water temperature (bottom) during mostly clear skies from Aug. 14-17 at UL. 69 − 1. 0 − 0. 5 0. 0 0. 5 1. 0 dT /d t ( ° C hr − 1 ) vertical component advection component dQ/dx component dA/dt component dQ/dx+dA/dt components predicted observed Q  (m 3 s − 1 ) 25 0 26 0 27 0 28 0 7 9 11 13 T w  ( ° C ) 14Aug 15Aug 16Aug 17Aug Figure 3.32: Heat budget (top), discharge (center), and water temperature (bottom) during mostly clear skies from Aug. 14-17 at LL. 70 Discharge had a mirror-like phase relation with advection at both sites (Figure 3.31 and 3.32): peak advection occurred within discharge troughs and minimum advection occured within discharge peaks. This was true despite the considerable difference in timing of advection variations between UL and LL, and further sug- gests a general relation between discharge and advection variability. At both sites, as discharge waned, advective heat flux increased, reaching its maximum at nearly the same time as the discharge minimum. As discharge increased, advective heat flux tended to decline or remain negative, reaching its lowest values near the time of maximum discharge. Peak diurnal discharge lagged peak vertical flux by about 12 hours at UL and 18 hours at LL. At UL, discharge peaked around 02:00 and was decreasing when vertical and advective warming began. At LL, high flows extended from about 05:00 to 12:00 – halfway through the vertical flux warming phase – and advective warming did not begin until flows began decreasing. 71 Chapter 4 Discussion This chapter discusses the results presented in the previous chapter in relation to the research objectives outlined in Chapter 1. It begins with the heat budget model and the roles of longitudinal heat advection and unsteady streamflow in governing temporal changes in temperature. Physical processes are discussed and inferences are made into the underlying causes of spatiotemporal stream temperature variabil- ity and the relation of the diurnal meltwater wave to heat advection. Then, tributary mixing and transverse mixing lengths are discussed with reference to the findings of other mixing studies. Inferences are made regarding potential physical processes governing the somewhat surprising relation found between relative discharge and transverse mixing lengths. Conclusions and suggestions for future research are presented in the next chapter. 4.1 Reach-scale heat budget considerations 4.1.1 Longitudinal heat advection A main objective of this study was to quantify the influence of longitudinal heat advection. Longitudinal heat advection was found to be the dominant control of stream temperature changes under all weather conditions. This was unexpected, given that Caissie et al. (2007) assumed longitudinal heat advection was negligible and other modelling studies did not specifically quantify it (Hockey et al., 1982; 72 Chaudhry et al., 1983; Sinokrot and Stefan, 1993; Meier et al., 2003; Chikita et al., 2009). While advection has been included in a number of numerical models, its influence has not specifically been quantified in the reach-scale heat budget using field observations. The results here demonstrate that longitudinal heat advection must be included in deterministic models. Previous studies for glacierized catchments have pointed to the cooling effects of glacial meltwater to help explain the fact that observed stream temperatures in glacier-fed streams tend to be lower than in streams without contributions from glacier runoff (Brown et al., 2005; Cadbury et al., 2008). Here, heat advection was found to have not only a cooling effect, but also an equally substantial warming effect. The cooling effect was expected, given previous findings of warmer water with downstream distance from the glacier terminus. Given these previous find- ings, the warming influence from advection, and especially the substantial magni- tude, was surprising. The warming influence from advection found here may be in part a function of the distance of the stream reach from the glacier terminus. Most previous proglacial stream temperature research was conducted within a few kilometers of the glacier terminus, whereas the reaches of this study were over 50 km from the glacier. The additional travel time and exposure to vertical heat sources for the sites studied here, in combination with the diurnal fluctuations in streamflow, account for the observed magnitude and variations in longitudinal heat advection, as explained in more detail below. Under clear-sky conditions, advection warming/cooling phases were related to the falling/rising limbs of the diurnal discharge wave, respectively. This relation can be explained by changes in heat capacity within the diurnal meltwater wave. At UL, streamflow was in a state of decline from approximately 03:00 to 18:00 (Figure 3.31). When discharge decreases through time at a given location, discharge must be lower at a point upstream. When positive vertical heat flux became available near 09:00, the lower heat capacity upstream, due to lower water volume, allowed greater warming and higher temperature than at the downstream boundary. Thus at UL, positive heat advection commenced at a similar time as positive vertical heat flux (there was a slight lag) with the key requirement that discharge was in a decreasing phase. The onset of advective warming at LL did not occur until discharge began decreasing at 13:00 (Figure 3.32), which strongly suggests a cause 73 and effect relation. Hence, advective warming commenced at both UL and LL when two conditions apparently allowed it: (1) discharge was declining while at the same time (2) there was positive vertical heat exchange. The timing of the cooling phase of advection was also related to discharge at both reaches. At each reach, advective cooling began shortly after discharge began increasing. This lag time was similar at each reach (about 2 hours) and may be an artefact of the complex relations between thermal history and travel time within the system, or perhaps simply due to the thermal ”memory” associated with wa- ter’s high heat capacity. Advective cooling during increasing flow can be explained by spatially varying thermal capacities similar, but opposite, to advective warming. With increasing flow, heat capacity was greater upstream and temperature response to earlier vertical heating was lower than downstream. The cooler upstream tem- perature thus made and advective cooling effect. The relation of advection to discharge (explained above) was supported by the consistency of the relation between both reaches, despite differences between the two reaches in timing relations between advection, vertical heat flux, and discharge phases. Longitudinal heat advection patterns in a non-glacial stream were markedly different than Lillooet River. Fishtrap Creek is a non-glacial stream in the inte- rior of British Columbia and was analyzed for clear weather in August, for which streamflow was approaching steady-state conditions (Leach, 2012). Advection was minimal at night and became negative (a cooling influence) during parts of the daytime, with magnitudes of only about half that of the vertical term. There was no significant warming effect from advection, as opposed to Lillooet River. In addition, the advective cooling phase at Lillooet River occurred largely at night and displayed a more consistent, arch-shaped pattern, and advection magnitudes of warming and cooling phases were typically twice that of the vertical compo- nent. With streamflow variability being a key difference between these streams in August, the notion that the advection patterns found in Lillooet River are mainly due to diurnal discharge variability is supported by the different advection patterns between these streams. 74 4.1.2 Unsteady flow effects The stream temperature response to the effects of the combined unsteady flow com- ponents were relatively minor in contrast to the other terms of the energy balance 3.23. The unsteady flow terms were dependent on the rate of discharge change. At UL, streamflow typically increased more rapidly than it decreased, and the un- steady flow effect was relatively greater with the arrival of the meltwater wave than during decreasing flow. However, even during increasing flow the effects were rel- atively negligible in contrast to advection and the vertical term. At times of rapid streamflow change, such as with irregular meltwater generation associated with variable cloud cover, the heavy rain event, and the dam breach associated with the landslide event, the combined unsteady components were of the same order of magnitude as the other heat budget components (vertical and advective fluxes). 4.1.3 Surface energy exchanges Characterizing reach-scale net radiation is a challenge in energy balance studies and important to estimate accurately, as it is typically the dominant component of the vertical energy exchanges. Here, a simple geometric relation between solar po- sition and average stream reach characteristics, such as geographical orientation, riparian canopy height and occurrence, and channel geometry, was used to deter- mine the amount of shade casted upon the stream surface at each time step. This approach apparently worked well during clear conditions, according to the good heat budget model performance. Stream surface albedo was found to vary between the Lillooet River and its tributaries as a function of suspended sediment concentration (SSC) and also with solar zenith angle. The measured albedos were approximately double the values of 0.03 to 0.05 that are often assumed in modelling studies (Caissie et al., 2007; Benyahya et al., 2011) or have been measured over low-turbidity streams (Evans et al., 1998; Leach and Moore, 2010), but are consistent with the values reported by Chikita et al. (2009) for a highly turbid proglacial river in Alaska. The values are also similar to those reported by Richards and Moore (2011) for a proglacial stream at lower flows, when aeration was relatively low. These results suggest that stream temperature models applied to proglacial rivers should use albedos that are 75 appropriate for turbid water. 4.2 Transverse mixing There was considerable variability found in transverse mixing lengths (Lz). The range of complete mixing lengths ( ˆLz98) in multiples of channel width was from 49 to 105, with a mean of 69. This mean was lower than the 100 to 300 channel widths that have been previously reported (Rutherford, 1994). Although the studied reach of Lillooet River mostly had gentle meanders, there were some sharper bends, which could explain the relatively more rapid mixing. However, some studies have found much shorter mixing lengths (e.g. Day, 1977; Lane et al., 2008). The variability in mixing lengths observed in this study was partially explained by the influence of the discharge ratio between the tributary and mainstem (Qr). Previous studies have found conflicting relations between Lz and flow, both total and relative tributary flow. Lane et al. (2008) found Lz was lower during higher Qr, but Gaudet and Roy (1995) found no relation between Lz and Qr. This study found Lz to be positively related to Qr. This relation was somewhat unexpected. It is well established that faster trans- verse mixing rates (shorter Lz) result from tracer being input further toward the center of the receiving channel’s width (Rutherford, 1994). For the case of increas- ing Qr, the higher flow of the tributary would carry greater momentum, forcing tributary water further out into the main channel and essentially making the initial mixing location (analogous to the tracer input location) further toward the center of the channel. Rutherford’s model, however, was not developed specifically for river confluences and may not represent the effects of mixing dynamics occurring at river junctions. The positive Lz - Qr relation may be in part explained by flow mechanisms occurring at the confluence. In a flume experiment of confluent channels, Mosley (1976) found that sediment from each channel was separately transported down- stream in the merged channel. This was caused by streamflow separation of water from the two channels, which was maintained by the action of two helical flow cells, one from each channel, which rotated in opposite directions. The strength of the two opposing helical flow cells was greatest when Qr was nearest unity, and at 76 low or high Qr the flow cells did not form. It could be inferred that flow separation caused by opposing helical flow cells would limit transverse mixing rates. It was not determined how far downstream the helical flow cells persisted. Such opposing helical flow cells, if they do in fact have a considerable downstream persistence, could help explain the relationship observed in this study of longer Lz with higher Qr. The absence of these flow cells in tracer experiments could explain the appar- ent inconsistency (explained above) that tracer mixing was more rapid when put in closer to the center-width. The existence and persistence of near- field flow mech- anisms at natural river junctions, such as opposing helical flow cells, and impacts on local and downstream transverse mixing rates needs further attention. 77 Chapter 5 Conclusions This chapter summarizes the key findings in relation to the research objectives and provides recommendations for future research. 5.1 Key findings A main objective of this study was to quantify the influences of longitudinal heat advection and unsteady flow on stream temperature changes. Longitudinal heat advection was the dominant stream temperature control under all weather condi- tions. This was unexpected, given it has been assumed negligible (Caissie et al., 2007) and has not been specifically quantified in other modeling studies (Hockey et al., 1982; Chaudhry et al., 1983; Sinokrot and Stefan, 1993; Meier et al., 2003; Chikita et al., 2009). Given previous findings of cooler water upstream in glacial- fed streams (Cadbury et al., 2008; Brown et al., 2005), the cooling effect from lon- gitudinal heat advection was expected; however, it was surprising that the warming influence was equally important in both magnitude and duration. The timing of cooling/warming advective phases was explained by timing relations between lon- gitudinal temperature gradient shifts and the passage of the diurnal meltwater wave. This explanation was supported by the fact that advective flux patterns in a steady- flow, non-glacial stream are markedly different from those in Lillooet River. These findings demonstrate that longitudinal heat advection must be included in deter- ministic models for glacier-fed streams where unsteady flow conditions dominate. 78 The influences of the individual heat budget terms representing unsteady flow conditions were considerable. However, their combined effect was mostly negli- gible, with the exception of periods in which discharge changed rapidly, such as during the rain and landslide events. At times, variable cloud cover caused rapid flow changes which made the combined effect of the unsteady flow terms a simi- lar magnitude to vertical or advective fluxes. The thermal effects of unsteady flow should be considered in modelling studies in which flow changes are rapid, such as the case of hydro-peaking releases from dams (Toffolon et al., 2010). Solar radiation dominated vertical heat fluxes during clear weather conditions. Field measurements demonstrated that albedo depends on both solar zenith angle and suspended sediment concentration. Albedo values were generally in the range 0.08 to 0.1, similar to values observed at other turbid glacier-fed streams (Chikita et al., 2009; Richards and Moore, 2011) and approximately double those measured over low-turbidity streams (Evans et al., 1998; Leach and Moore, 2010). This study confirms that a surface albedo of 0.03 to 0.05, which is typically assumed in many previous studies (e.g. Caissie et al., 2007; Benyahya et al., 2011; Magnusson et al., 2012) is inappropriate for turbid proglacial streams. Transverse mixing length (Lz) at the confluence showed considerable variabil- ity. The tributary to mainstem discharge ratio (Qr) was positively related to Lz and explained approximately half of the variation. This result disagrees with previous studies that found more rapid transverse mixing with higher Qr (Biron et al., 2004; Lane et al., 2008) and no relation between Lz and Qr (Gaudet and Roy, 1995). This also contrasts with tracer studies which have found longer Lz with tracer input closer to the bank (Rutherford, 1994). Higher Qr, which is associated with longer Lz, would force the tributary water further from the mainstem bank. These appar- ent inconsistencies are not necessarily surprising, given the multitude of factors and processes involved in mixing at and downstream of natural river confluences. The well-documented opposing helical flow cells, which can form at confluences and limit transverse mixing by separating the flows from each channel, could help explain the positive Qr - Lz relationship found in this study, as these structures strengthen with increasing Qr approaching unity (Mosley, 1976; Best, 1988). Below-confluence (total) discharge had no significant relation to Lz, which is inconsistent with previous studies that found more rapid mixing with lower total 79 discharge (Gaudet and Roy, 1995; Biron et al., 2004) and with higher total dis- charge (Chu and Babarutsi, 1988). 5.2 Future research recommendations Given the importance of stream albedo in controlling the absorption of solar radi- ation, a study should be conducted to measure albedo for a broader range of sky conditions, solar zenith angles and suspended sediment concentrations. The results of such a study could be used to develop an improved parameterization of stream surface albedo to be incorporated into physically based models. This study illustrated the importance of including longitudinal heat advection, in addition to vertical heat flux, in process-based temperature models for glacier- fed streams. The diurnal phase relations between vertical flux, advective flux, and streamflow shown here were for two reaches over 50 km from the glacier margin and may not represent the patterns elsewhere along the river. Future work should investigate these phase relationships in a more spatially distributed sequence of study reaches downstream of the glacier margin. The phase timing relation between meltwater flux and advective heat flux was similar at both reaches. Considering that the absolute times differed between the sites, there would appear to be a causative effect of the passage of the meltwater wave on heat advection. Such inferred processes can be supported if similar pat- terns are found in other catchments. Therefore, similar work should be conducted in other glacier-fed streams. The influence of longitudinal heat advection should be investigated for other hydrological regimes as well. It would be useful to combine field investigations with numerical modelling. This study suggests that, with further climatic warming, glacier retreat, and streamflow decline, longitudinal heat advection (cooling and warming phases) should decline in magnitude due to lower diurnal discharge (heat capacity) variability. Stream temperature responses to climate change should be addressed by coupling the stream heat budget with models representing the additional process linkages between climate, glacier mass balance, glacier retreat, and meltwater generation. The positive relation found between relative tributary flow and transverse mix- ing distance needs to be confirmed with further experiments in natural and labo- 80 ratory confluences. The underlying processes controlling transverse mixing, and their variation with discharge, need further investigation. The role of near-field flow structures, such as opposing helical flow cells, on transverse mixing dynam- ics is an area for further research. 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Parameters used in calculating gt which were assumed constant over all reaches were vegeta- tion height, lateral extent of vegetation overhanging the stream, and height from stream surface to vegetation overhang. Where present, riparian vegetation was as- sumed to be opaque to solar radiation. The shading potential of the banks was considered separately from that of the riparian vegetation. For the bank on the side in which the solar azimuth comes from, the length of shade extending on the stream surface along the line of solar azimuth from the vegetation edge (not the bank) was sazi = Yf / tanβ (A.1) 90 where Yf is height of foliage above stream surface and β is solar angle (angle from ground to sun). The length of shade from the bank and perpendicular to the bank was s = sazi(cosω)+ z f (A.2) where ω is the acute plan-view angle between the solar azimuth line and the cross- sectional line, z f is the additional lateral length of shade from foliage overhanging the stream. Since the corridor is straight and y f is constant, s is the same along the bank and solving gt can be simplified to a one-dimensional approach across the channel. The percent longitudinal occurence of riparian vegetation was estimated as Pf = 1 l n ∑ i=0 x f (A.3) where n is the number tree stands along the bank, x f is the longitudinal length of a stand, and l is reach length, can then be used as a coefficient for Eq. A.2 so that s̄ = Pf [ sazi(cosω)+ z f ] (A.4) where s̄ is reach-averaged perpendicular length of shade from the bank on the shady side of the stream. Pf is unique for each bank and is selected based on solar posi- tion. For the side of the stream opposite the solar azimuth direction, stream surface shade would only stem from overhanging vegetation. The calculation of the shade length is similar to that of the shady side, except the length of sun (not shade) is first found, which is subtracted from the overhang distance. The length of shade under the overhang (s2), on the stream surface, perpendicular to the bank, and compensated for the reach by Pf is s̄2 = Pf [z f − (y f / tanβ · cosω)] (A.5) where y f is the vertical distance between the stream surface and overhanging veg- etation. 91 The average riparian canopy gap function over a reach is then gt = (w̄− s̄− s̄2)/w̄ (A.6) where w̄ is reach-averaged channel width. A.0.2 Sky view factor, fsky Calculation of sky view-factor fsky was based on the same assumptions as used to compute the canopy gap fraction. However, directionality in respect to solar position is not relevant for calculating fsky, which can be treated as a constant for each reach over the study period. The longitudinal structure of a reach was simplified to two cases: (1) trees on both banks and (2) no trees on either bank. Case 1: Trees on both banks The riparian corridor is again assumed infinitely long and symmetrical, with opaque vertical walls of trees on either bank which overhang the stream. With longitudinal uniformity in Case 1, the problem can be approached from a cross-sectional view broken into three segments: (i) under the left bank overhanging vegetation, (ii) open sky, and (iii) under the right bank overhanging vegetation. For symmetrical corridors, the sky view factors for segment (i) and segment (iii) are identical. Segment i or iii: under overhang For any z location across the stream under the overhang, the sky view factor φz is φz = cosβ1− cosβ2 (A.7) where β1 and β2 are the angles from z to the edge of the nearby overhang and the top edge of the far wall, respectively. The sky view factor φz can be integrated across segment (i) to calculate the segment’s average sky view factor φ̂i: φ̄i = 1 c− zo ∫ c zo (cosβ1− cosβ2)dz (A.8) Eq. A.8 is solved as 92 φ̄i = 1 c { [(d− zo)2+h2]1/2 − [(d− c)2+h2]1/2 −[(c− zo)2+ y2]1/2 + y } (A.9) where c is the lateral position of riparian overhang (the near corridor wall), d is lat- eral position to far corridor wall, h is corridor wall height above the stream surface, y is overhang height above the stream surface, and zo is the lateral length where β1 = β2, calculated as zo = (h · c)− (y ·d) h− y (A.10) For zo < 0, which is typical for wide channels, zo can be dropped from Eq. A.8. Segment ii: between corridor walls For any z location across the stream between the overhanging vegetation on either bank, the sky view factor φx12 is φx12 = 0.5(cosθ1+ cosθ2) (A.11) where θ1 and θ2 are the angles from z to the top edge of either corridor wall. The sky view factor φx12 can be integrated across section (2) to calculate the the segment’s average sky view factor ¯φ12: φ12 = 1 d− c ∫ d c 0.5(cosθ1+ cosθ2)dz (A.12) Eq. A.12 is solved as φ12 = 1 d− c { [ (d− c)2+h2 ]1/2−h } (A.13) Segments 1 and 2 for Case 1 were combined over the average channel width as φ1 = 2(c−b) φ11 + (d− c) φ12 2(c−b)+d− c (A.14) 93 Case 2: No vegetation on either bank The view factor for Case 2 (φ2) was calculated using the same form as Eq. A.13 for the open segment of Case 1 but substituting channel width for (d−c) and bank height for h. Case 1 and 2 Finally, the overall average view factor for a reach ( fv) was found by combining Case 1 and Case 2 according to the tree presence along each bank of the reach: fv = (Pf ) φ1 + (1−Pf ) φ2 (A.15) where Pf is the average longitudinal tree presence of both banks. View factors vary with seasonal vegetation structure. View factors were assumed constant through the study period since vegetation changes were negligible until late October. 94

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