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Hydraulic analysis of outflow winds in Howe Sound, British Columbia Finnigan, Timothy D. 1994

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HYDRAULIC ANALYSIS OF OUTFLOW WINDS IN HOWE SOUND, BRITISH COLUMBIA by TIMOTHY D. FINNIGAN B.A.Sc., The University of British Columbia, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August 1994 © Timothy D. Finnigan  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  C,vi/  The University of British Columbia Vancouver, Canada Date  DE-6 (2188)  3a  Abstract  Previous studies (Jackson, 1993) suggest an outflow wind, which flows below an inversion in a well defined layer through Howe Sound, may exhibit hydraulic behaviour. Strong outflow winds in Howe Sound are simulated in the laboratory using a single fluid layer in a small scale one-dimensional physical model. Model results are presented and compared with observations recorded in Howe Sound during a severe outflow wind event in December, 1992. Field observations affirm the findings of the physical modelling with both indicating the presence and location of controls and hydraulic jumps in the wind layer. Hydraulic behaviour is found to change as the synoptic pressure gradient and the flow rate increase. An additional comparison is made with output from the computer model, Hydmod of Jackson and Steyn (1994b). Numerical simulations, configured for the conditions present in Howe Sound during the December, 1992 event, indicate channel hydraulics (and thus spatial wind speed variation) closely resembling the physical model and field results.  Outflow winds are studied in more detail through a series of experiments conducted with a three-dimensional physical model which is geometrically and kinematically similar to the prototype, Howe Sound. The results reveal the structure of the wind layer over a wide range of possible field conditions.  Hydraulic features, which do not behave in a  traditionally one-dimensional manner, are identified. The 3D model results, although more detailed, verify the findings of the 1D modelling in general. Together the results provide a predictive tool for determining hazardous zones of extreme wind during an outflow event.  II  Table of Contents Abstract  ii  Table of Contents  iii  List of Tables  v  List of Figures  vi  Foreword  viii  Acknowledgment  ix  Introduction  1  Introduction  1  Outflow Winds  2  Hydraulic Theory  5  Previous Work  8  Gap Winds  9  Downslope Winds and the Hydraulics of Layered Flows  11  One Dimensional Modelling and Field Investigation Physical Model Study  16 17  Experimental Methods  17  Results  19 Results for flow rate A  20  Results for flow rate B  21  Field Program  21  Field Experiment  22  Field Data Interpretation  24  Results for Period 1  25  Results for Period 2  26  Comparison of Physical Model, Hydmod and Field Results  27  Period 1 (flow rate A)  27  Period 2 (flow rate B)  29  Further Comparison with Hydmod Output Discussion  111  29 31  Three Dimensional Modeffing  .33  Description  33  Design Considerations  34  Model Design and Construction  43  Data Aquisition  45  Data Aquisition System  45  Depth Data Aquisition  46  Velocity Data Aquisition  49  Results  50 Important Parameters  50  Model Results  51  Discussion of Cases 1 Through 8  52  Discussion and Conclusions  56  Comparison of 1D Modelling and 3D Modelling Comparison of field results with 3D model results  Summary and Conclusions  56 57 59  List of Symbols  62  Bibliography  63  Appendix A  66  Appendix B  70  iv  List of Tables  Table 1  page Values of parameters as observed during Period 1 and Period 2 which  72  were used in Hydmod comparisons. 2  Scale numbers resulting from Froude number similarity between model  72  and prototype. 3  3D Model parameters and corresponding values scaled to prototype  73  dimensions. 4  Model parameter settings and some important physical aspects of the  73  results for the eight simulated cases. 5  Case 6 of 3D model results compared with Period 1 of field study  74  results. 6  Case 7 of 3D model results compared with Period 2 of field study results.  v  74  List of Figures  Figure  Page  1  Schematic representation of outflow wind system.  75  2  Geographical location and important features of region surrounding  76  Howe Sound. 3  Howe Sound; locations and topography.  77  4  Schematic drawing of experimental apparatus used for 1D physical  78  modelling of outflow winds. 5  Channel axis shown with respect to the model of Howe Sound (a) and  79  Howe Sound (b). 6  (a) Depth, (b) wind speed, and (c) Froude number along Howe Sound: as  80  predicted by the physical model for flow rate A ; as measured during the December, 1992 outflow event for Period 1; and as produced by Hydmod for Period 1. 7  (a) Depth, (b) wind speed, and (c) Froude number along Howe Sound: as  81  predicted by the physical model for flow rate B; as measured during the December, 1992 outflow event for Period 2; and as produced by Hydmod for Period 2. 8  Composite chart of atmospheric pressures recorded at 5 stations in Howe  82  Sound over a 9 day period. 9  Relative pressures, with respect to that at station 1, at each of the five  83  field stations in Howe Sound for (a) Period 1 and (b) Period 2. 10  Photograph of 3D model topography.  84  11  Cross sectional sketch of 3D model topography construction.  85  vi  Figure 12  Page Schematic view from above of the video apparatus cart and rail system  86  (a) and schematic side view of complete 3D model equipment set up (b). 13  Photograph of 3D model.  87  14  Definition sketch for depth data aquisition.  88  15  Plots for validation of video techniques for depth data aquisition.  89  16  Reference diagram for 3D model results (a) and Topography of region  90  covered by model results (b). 17  3D Model. Case 1  18  3D Model. Case 2  19  3D Model. Case 3  20  3D Model. Case 4  21  3D Model. Case 5  22  3D Model. Case 6  23  3D Model. Case 7  24  3D Model. Case 8  25  Microbarograph pen tracings from 28 December 1992 for stations at  -  -  -  -  -  -  -  -  Q=L, hf=L, dP/dx=L  91  Q=H, hf=L, dP/dx=L  92  Q=H, hf=L, dP/dx=H  93  Q=L, hf=L, dP/dx=H  94  Q=L, hf=H, dP/dx=L  95  Q=H, hf=H, dP/dx=L  96  Q=H, hf =H, dP/d =H  97  Q=L, hf=H, dP/dx=H  98  Porteau Cove (a) and Lions Bay (b).  vii  99  Foreword  The work reported in chapter 2 of this thesis forms the contents of a paper entitled, “Hydraulic physical modelling and observations of a severe gap wind”. This paper was accepted for publication in Monthly Weather Review (AMS journal) in April 1994 and is currently in press. The authors are as follows; Finnigan, T.D., Vine, J.A., Jackson, P.L., Allen, S.E., Lawrence, G.A. and Steyn, D.G. The first author, Finnigan (also author of this thesis) wrote the paper in its entirety including composition of the text, analysis of all data, interpretation of results and drafting of figures. Finnigan also conducted the field investigation. Vine conducted physical model experiments and contributed the data. Jackson developed the computer model and helped with the simulations. Jackson also provided the interpretation of the synoptic weather conditions during the recorded outflow event of December 1992. Allen, Lawrence and Steyn supervised the work, contributed many helpful suggestions and corrected drafts of the manuscript.  viii  Acknowledgment  Several people contributed their efforts during the two year period this research was conducted.  The author wishes to thank Dr. S. Allen (Oceanography) and Dr. G.  Lawrence (Civil Engineering), who supervised the research and provided guidance from start to finish. Dr. D. Steyn (Geography) supplied the microbarographs and participated in several useful discussions about the field investigation.  Dr. P. Jackson of The  University of Western Ontario, who developed the Hydmod model, helped with the computer simulations and the interpretation of the synoptic conditions for the field work. Kurt Nielson of the Civil Engineering shop assembled the video apparatus for the 3D model and helped with many technical aspects of the model itself.  I would also like to thank Sewell’s Marina (Horseshoe Bay), Lion’s Bay Marina, the Cunneyworths (Porteau Cove), Britannia Beach Arts and Crafts and the Squamish Terminals for allowing microbarographs to be stationed on their premises during December 1992 and January 1993. The Pacific Region of the Atmospheric Environment Service provided the supplemental field data from Pam Rocks. The research was supported by grants from the Atmospheric Environment Service of Environment Canada and the Natural Science and Engineering Research Council.  ix  Chapter 1 1.1.  -  Introduction  Introduction  Local meso-scale 1 windstorms occur in many parts of the world with various synoptic conditions responsible for their creation and unique local features influencing their behaviour. Downslope winds result when air is forced over topography and accelerated down the lee side of a mountain, possibly being enhanced by vertical propagation of wave energy and hydraulic effects. Mountain valley winds are common and often diurnal in nature with cool air drainage at night and warm, daytime heating induced, upslope flow in the day. Gap winds can be described as flow of relatively dense low lying air through natural channels. These flows, which are generated when a synoptic pressure gradient is aligned with the channel, are often constrained above by an inversion 2 and surmounted by a stable layer. The term gap wind was first coined by Reed (1931) who thus defined the flow of low lying air through “gaps” in a mountain barrier when an across-barrier pressure gradient is present.  In some locations the interaction of  topography and regional or meso-scale pressure gradients can produce very complex and sometimes violent gap winds.  Gap winds are most pronounced where a mountain range separates two climatically different regions. Coastal mountain ranges are special cases where a topographical barrier separates drastically different oceanic and interior climatic zones. When air bodies of different temperature, and therefore density, develop on either side of the barrier, the situation is favorable for gap winds to occur. Since the air in the coastal  1 of or relating to a meteorological phenomenon approximately 1 to 100 kilometers in horizontal extent. A thin layer across which the temperature increases dramatically in the upward direction. This is a 2 reversal of the normal atmospheric temperature gradient.  1  region is moderated by the ocean, extreme conditions in the interior will produce a pressure gradient resulting in a flow of air.  During hot weather, daytime warming induced vertical convection inland will produce lower pressure at the surface relative to that in the cooler ocean region. This results in a pressure gradient perpendicular to the coast and landward flow (seabreeze or inflow) which, when funneled through mountain valleys, is accelerated sometimes producing brisk winds.  The result is much more extreme during cold winter weather when the situation is reversed. If cold (relatively dense) air occupies the interior and warm air remains in the coastal region the pressure gradient is directed inland perpendicular to the mountain barrier.  The resulting seaward flow of cold air remains close to the land surface  displacing the lighter air above as it flows in a well defined layer towards the ocean. The air movement is restricted by the mountains resulting in low lying flow through valleys  and mountain passes. Seeking the most direct route to relieve the pressure, the air descends the lee slope of mountain valleys, accelerating under the additional influence of gravity, and enters the coastal region through inlets and fjords that dissect the coast. The scenario is depicted simply in Figure 1 and is described in more detail by Jackson (1993).  1.1.1 Outflow Winds  Strong gap winds are encountered in the valleys and inlets of coastal mountainous regions where cold weather is prevalent. During the winter months the British Columbia coast is geographically suited to extreme gap winds. Here such phenomena are referred to as outflow  winds; a term which implies the flow of air from the interior of the province out  3 O utflow wind is an example of a gap wind phenomena. The term outflow wind best describes the wind system under study and will therefore be used preferentially throughout this thesis.  2  towards the coast. These winds occur when an Arctic outbreak forces cold air south into the interior plateau region of British Columbia (see Figure 2). The cold air, being pooledu between the Coast and relatively dense, deepens over a period of days becoming tt Rocky mountain ranges. The Coast mountains act as a partial barrier separating cold, dry interior air from warm air on the coast. The resulting across-barrier pressure gradient often produces outflow winds in valleys and inlets along the coast.  The present study focuses on the outflow winds, or as locally known Squamish winds, that occur in Howe Sound which is a fjord located in the southwest corner of the British Columbia mainland (see Figures 2 and 3). This inlet is typical of many along the coast of British Columbia which also experience strong outflow winds during the winter months. The topography of the channel drastically varies over short distances. Rugged mountains interspersed along the channel give it a tortuous shape with many abrupt expansions and contractions. Steep mountain faces rise dramatically from the sea to heights of 1600 m in some parts of the Sound and combine with islands to influence and control the flow of air.  Outflow events in Howe Sound occur during the winter with durations as short as 8 to 10 hours but often lasting for 4 to 5 days. Windspeeds commonly reach 20 m/s with gusts to 30 or 40 rn/s (1 rn/s  =  1.94 knots). The wind layer is generally less than 1000 m deep at  all locations along the channel (Jackson, 1993). On average, outflow winds occur on 4 to 5 days in each of December and January (Schaeffer, 1975).  The cold temperatures that accompany these winds and their unpredictability make them a serious hazard. During outflow events power loss and substantial property damage are common. Navigation in the region, whether by land, sea or air, is difficult and can be dangerous.  Extreme wind conditions throughout the Sound during events are  3  detrimental; however, the single most dangerous aspect of the outflow winds may be their spatial variability. The wind flows in a complicated layer through the channel. In several locations velocities change abruptly over short distances. This is largely due to hydraulic effects (discussed below) and flow separation. Localized regions of very intense wind will develop during an outflow event. Problems arise when people expect conditions to be similar throughout the Sound. Traveling into or building in one of these high wind pockets will generally cause unexpected problems.  The severity of problems and extent of dangerous conditions associated with the strong winds can be expected to increase in local areas of maximum windspeed.  The  improvement of predictive capabilities for the above mentioned aspects of the outflow winds is part of the motivation for this study. The primary goal is to identify the regions of most intense wind and predict velocities throughout Howe Sound. The analysis is done for a range of typical conditions found in Howe Sound during an outflow event.  The research reported in this thesis is centered around the application of hydraulic theory to describe the flow during outflow events. There are two parts comprising the work; a one-dimensional (1D) study which combines physical modelling, computer modelling and a field investigation; and a three-dimensional (3D), fully representative, physical model study.  The remainder of this chapter includes a description of hydraulic theory, as it is applied to the wind system under study (section 1.2), and a review of previous research on related subjects (section 1.3). The 1D modelling and the field investigation are decribed in chapter 2 and the 3D modelling is described in chapter 3. Results are discussed and conclusions drawn in chapter 4.  4  1.2.  Hydraulic Theory  Single layer hydraulic theory was originally developed for engineering applications in open channel flow (see Henderson, 1966). For a single-layer flow, the ratio of convective velocity, u, to surface wave speed, c = (gh) , is known as the Froude number, 2 ” 1  F=(g;:)1I2  (1)  where h is fluid depth and, g, gravitational acceleration. The flow is termed subcritical when F < 1, critical when F =1 and supercritical when F> 1.  In open channels flow is controlled by channel features that determine a depth-discharge relationship (Henderson, 1966). Such features (local contractions or changes in surface elevation) are called hydraulic controls, or simply controls, and the flow changes from subcritical to supercritical as it passes through them. At a control the flow is critical. Where a region of subcritical flow exists some distance downstream of a control the supercritical flow must somehow connect with the subcritical region. For supercritical flow gravity waves can not propagate upstream transmitting information about the subcritical region downstream. Therefore the flow undergoes a spontaneous transition from super- to subcritical flow through what is called a hydraulic jump. Enhanced turbulence intensity and energy loss accompany the hydraulic jump as the flow abruptly decreases in speed and increases in depth.  The extension to multiple fluid layers has made the hydraulic theory useful in the study of geophysical flows. Outflow wind in Howe Sound is suitable for the application of hydraulics since it is comprised of a stratified two-layer system with the cold wind layer flowing beneath an essentially infinitely thick warmer layer. The two layers are generally 5  separated by a distinct interface in the form of an inversion. While there may be regions where the interface is relatively thick and fluid is exchanged between layers it is reasonable and quite accurate to idealize the system, for analysis purposes, as two distinct layers (Jackson, 1993).  The hydraulic theory of layered flows makes the following assumptions: the fluids are inviscid, the pressure is hydrostatic, and within each layer the density is constant and the velocity varies only in the flow direction. For two-layer flows solutions of the hydraulic equations, as described by Armi (1986) and Lawrence (1990), yield characteristic velocities for both external (free surface) waves and internal (interfacial) waves. Lawrence (1990) investigated the solutions and presented exact expressions for internal and external Froude numbers based on celerities of infinitesimal long waves. Lawrence (1990) shows that, if we assume the relative density difference between the layers is small (Boussinesq approximation),  (P2  )/P <<1, 1 —P 2  (2)  then the internal Froude number for a two-layer flow is expressed as, F  +u u 1 h 2  (g’hkh ( 2 1  -  2 F))”  3  where,  (4)  6  is the stability Froude number, h = h 1  +  2 the total depth of fluid, g’ = (p h 2  —  1 /p p 2 )g the  reduced gravity and the subscript 1 refers to the upper layer while the subscript 2 refers to the lower layer. When F >1 internal phase speeds are imaginary and internal hydraulic theory no longer applies.  The internal Froude number, F , determines the internal 1 >1  supercritical  <1  subcritical  1 = 1 the flow is internally critical criticality of two-layer flow, i.e., when F  In the context of outflow winds the internal Froude number describes the hydraulic behaviour of the wind layer. If we assume the upper layer, h 1 h  1 then with u h 2  >>  >>  2 and therefore that h  1 and F —*0, which is the case during outflow events, equation u  (3) reduces to,  1 F  ’ 2 (g’Ijj”  (5)  which is exactly analogous to the single-layer Froude number (equation (1)) with g replaced by g’. This indicates that the lower layer of the outflow wind system behaves hydraulically like a single layer of fluid reacting under a reduced gravitational force (i.e. g’). When the upper layer is much thicker than the lower layer, the approximation (5) is valid and the upper layer thickness is unimportant. As described in chapter 2 and in more detail in chapter 3, this simplification allows us to accurately model the two-layer wind system using only a single layer of fluid.  Following Lawrences (1990) discussion, the external Froude number is,  FE  ’ 112 (gh)  7  (6)  where the flow weighted mean velocity ü  (uhj  =  +u h )/h. 2  This Froude number, which  is expectedly similar to that of single layer flow, is of little significance in the outflow wind scenario since the upper layer is essentially infinitely thick and FE  0.  The criticality and hydraulic features determined, for single-layer flow, by equation (1) are determined in precisely the same way, for the two-layer wind system, by the internal Froude number (5). In the vicinity of controls and hydraulic jumps the wind velocity changes abruptly over short distances, and with time, making them serious hazards. Supercritical flow that spans the regions between controls and hydraulic jumps is characterized by high windspeeds and should also be identified as a hazard. The research described in chapters 2 and 3 is aimed at determining where these flow features occur, what regions of Howe Sound are occupied by sub- and supercritical flow, and what the resulting velocities are.  1.3.  Previous Work  Of all the mesoscale wind phenomena discussed in literature, the two most relevent to the study of outflow winds are gap winds and downslope winds. These wind phenomena are closely related and might only be distinguished from each other by their geographical influences. We refer to gap winds as flow through topography (mountain channels) and to downslope winds as flow over topography (mountain ridges). In terms of hydraulics, gap winds are generally controlled by changes in channel width while downslope winds are controlled by changes in elevation.  In many cases, effects of both types are  combined. Both gap winds and downslope winds can be generated by similar synoptic conditions. Below we discuss separately previous work related to both wind phenomena. As discussed above in section 1.2, some wind systems are well described by hydraulic theory.  In the following discussion, where possible, emphasis is given to studies 8  concerned with the hydraulics of layered flows. Most of the previous hydraulic studies have dealt with flow over obstacles rather than flow through channels. Since these studies relate closely with downslope winds we discuss them in combination with downslope wind studies.  1.3.1 Gap Winds  Gap winds in the fjords and valleys along the west coast of North America have been observed from Alaska (Bond and Mackim, 1993) to as far south as Oregon (Cameron and Carpenter, 1936). Over the last century many occurrences have been documented and all reported to be triggered by similar synoptic conditions; high pressure inland forcing low lying cold air through mountain barriers towards low pressure regions on the coast. Observational studies have been reported by several researchers (Lackman and Overland, 1989; Mass and Aibright, 1985) and wind record data has been analyzed over many years by Schaeffer (1975), revealing the statistical properties of outflow events in southwestern British Columbia.  Some attempts have been made to explain the structure and dynamic behaviour of specific events. Overland and Walter (1981) described two distinct gap wind cases in the Strait of Juan de Fuca (see Figure 2) in February 1980. They identified the synoptic pressure gradients responsible for driving the flows and, through research aircraft and a dense network of surface stations, revealed the structure of the wind layer. Major characteristics of the wind field were accounted for by the combined effect of the synoptic pressure gradients and local topography. Notable features of both cases were an inversion layer which capped the flows and an abrupt transition resembling a hydraulic jump.  9  An example of a gap wind case from another part of the world is the well known “Mistral” which flows through the Rhône Valley in France. Pettre (1982) conducted a month long observational study of this violent wind with high spatial resolution and fine temporal scale. The results emphasized the prominent role played by an inversion layer in the air flow dynamics. Pettre adopted a two-layer model of the system and compared observations with a hydraulic computer simulation.  The simulated transition to  supercritical flow and subsequent hydraulic jump coincided with observed conditions in the field.  In particular, the existence of an observed hydraulic jump was well  documented.  Some theoretical studies such as that by Mackim  et a!. (1990) have had success in  describing the causes and dynamics of gap winds. No physical model studies specifically pertaining to gap winds, prior to the present study, have been found. Numerical model studies are also rare. Recently Jackson and Steyn (1 994a) reported observations of a moderate outflow event in Howe Sound and compared these with output from a threedimensional mesoscale numerical model. Although their results agreed in general, model flows underestimated actual windspeeds in the channel and small scale flow features were not captured. Jackson and Steyn suggested that the flow is strongly influenced by local topography and a hydraulic analysis of their model output supported this. In a subsequent paper Jackson and Steyn (1994b) described a simple 1-dimensional hydraulic computer model which was more successful at predicting observations. Here they extended the classical hydraulic theory by adding the influence of synoptic pressure gradient in the form of a slope. This model, called Hydmod, is used in the present study to generate output for comparison with field observations and physical model results. It will be discussed further in chapter 2.  10  An in depth review of the phenomenological studies on gap winds and related mesoscale winds, reported before 1992, was done by Jackson and Steyn (1993, 1994a) and will not be repeated here. It is useful, however, to summarize some of the basic characteristics of outflow winds on the west coast of North America. In general, the wind layer is between 50 and 1500 m thick, it is capped by an inversion layer which varies in thickness and stability depending on windspeed and the difference between the lower and upper layer temperatures, average wind speeds of 20 m/s with gusts of up to 50 mIs are not uncommon.  1.3.2 Downslope Winds and the Hydraulics of Layered Flows  Downslope winds have received much attention and are the subject of numerous experimental and theoretical studies. Although fundamentally different in nature from outflow winds, downslope winds possess some related characteristics and can be informative.  The utility of the hydraulic theory has long been recognized in the  description of the general nature of geophysical flows encountering topography. Despite its relative simplicity, this theory is of great value since it retains the essential non linearity of the flow. The one-dimensional hydraulic formulation has allowed the development of relatively simple theoretical and experimental models which have shown success in describing otherwise very complex flows.  The original work on downslope winds was by Long (1953) who performed laboratory experiments simulating the “Bishop Wave” phenomenon, which is thought to resemble an atmospheric hydraulic jump. The flows were produced by towing an obstacle through a rectangular channel filled with three layers of stationary immiscible fluid. Each interface between layers represented an atmospheric temperature inversion.  11  Long (1954)  performed similar, but more refined experiments and reported the upstream motions that are generated over the obstacle.  Many experimental efforts have since been made to explain the flow of stratified fluids over topography. These are reviewed by Baines and Davies (1980) who discuss in detail single-layer, two-layer and continuously stratified flows. In every case the flows are produced in the same manner as by Long (1953); by towing two-dimensional idealized representations of mountain ridges along channels filled with fluids at rest. More recently Baines (1984) elaborated on the experimental base by performing more towing-tank experiments over a wide range of parameters. By varying the obstacle height and speed, and the relative layer depths, Baines investigated the factors governing the nature and magnitude of upstream disturbances (e.g. turbulent bores and rarefactions) in the general flow of stratified fluid over topography. Although the unsteady upstream effects always present in towing-tank experiments may be important in certain atmospheric flows over mountain ridges, they are not often observed and do not play an important role in the behaviour of outflow winds.  Lawrence (1993) reported results of a theoretical and experimental study of steady twolayer flow over a fixed two-dimensional obstacle. His approach avoided the problem of upstream effects inherent to towing tank experiments. The hydraulic theory was used to develop a classification scheme, which differed from that based on the previous towing tank experiments, to predict the flow regime for various experimental parameters. The fixed obstacle experiemnts enabled Lawrence to study approach-controlled flows in detail and thereby show that non-hydrostatic forces, rather than a hydraulic drop (Baines, 1984), are important in their understanding.  12  In the above studies, the flow behaviour over the obstacle (mountain ridge) could be analogous to that within the realm of outflow winds when an isolated ridge or island is encountered.  Downslope windstorms are relatively common and have been studied extensively for over 30 years. However, their dynamics are still not well understood. Three different theoretical mechanisms have been proposed to explain strong downslope winds. The first is based on linear theory of internal gravity waves in a continuously stratified, semiinfinite fluid. The second is based on numerical integrations of the governing equations. Some numerical results suggest that as wave amplitudes aloft are increased and breaking occurs, lee slope flow plunges beneath the mixed region (Smith, 1985, Rottman and Smith, 1989). The third mechanism is based on the hydraulic theory (Long, 1954; Houghton and Kasahara, 1968). A transition to supercritical flow over the mountain ridge produces strong supercritical flow down the lee slope and a subsequent hydraulic jump. This approach has been quite successful in explaining observed downslope windstorms.  The disadvantage of hydraulic theory is that it requires flows to be  structured in discrete layers. This only allows one internal wave mode (per layer) whereas continuously statified flows support many.  In a paper by Durran (1986) the role played by hydraulics and vertically propagating internal waves on the development of large-amplitude waves was investigated. Numerical simulations were conducted to investigate the various amplification mechanisms responsible for the waves.  When the static stability has a two-layer  structure, with linear stratification within each layer, the lower layer behaves analogously to that of a single layer of fluid. A direct comparison of hydraulically interpreted numerical results with observations from the Boulder 1972 windstorm (Bower and  13  Durran, 1986) reveal that an inversion plays a key role in the development of a supercritical region and following hydraulic jump.  When flow over three dimensional topography is considered the analyses become substantially more complex. Experimental studies are often used to determine some criteria for flow behaviour for certain conditions. Hunt and Snyder (1980) describe the flow over a bell-shaped hill which was placed first in a large towing tank containing stratified saline solutions with uniform stable density gradients and second in an unstratified wind tunnel. Flow visualization techniques were used to obtain mostly qualitative information about the flows. Snyder et al. (1985) elaborated on this by introducing a dividing streamline concept and performing experiments which could be compared with field observations. Through an energy argument the dividing streamline was defined as that which separates streamlines which pass over the hill from those which go around. The concept was applied to a variety of shapes and orientations of hills with different upwind density and velocity profiles.  In a paper describing shallow water flow over isolated topography, Schär and Smith (1993) used the fundamentally one dimensional hydraulic theory to characterize different flow regimes. For subcritical upstream flow encountering a three dimensional hill, the following three regimes occur: (1) fore-aft symmetry, essentially inviscid dynamics, and entirely subcritical conditions; (2) transition to supercritical flow and the occurrence of a hydraulic jump over the lee slope; and (3) the inability of the flow to climb the mountain top resulting in flow separation.  Numerical simulations were performed and the  controlling parameters determined. This paper has particular relevance to the study of outflow winds where an essentially shallow fluid encounters semi-isolated mountains and islands. Schär and Smith borrow from the field of gas dynamics and draw an analogy  14  with shallow water flow. Shear discontinuities and oblique shocks are important features in the flows described and may play a role in the spatial variation of outflow winds.  15  Chapter 2  -  One Dimensional Modelling and Field Investigation  In this chapter results from a 1D physical model study of extreme outflow winds in Howe Sound are presented (also found in Finnigan et. al., 1994). The results are compared with observations from a field study and with output from the hydraulic computer model (Hydmod) of Jackson and Steyn (1994b). The field data acquired are from a severe outflow event in Howe Sound which commenced on 27 December 1992 and persisted throughout the following four days while an Arctic airmass and anticyclone resided in the interior of British Columbia. Channel hydraulics, predicted by the physical model study, provided guidance in establishing the field research program.  Hydraulic theory, as described in chapter 1, is used throughout this chapter to interpret the flows. The physical model and Hydmod are referred to as one-dimensional because, for both of these, the main channel of Howe Sound is idealized as rectangular in crosssection, with the channel width, flow depth, flow velocity and Froude number varying only in the along channel, or x, direction.  We are able to show, by comparing field study results with model predictions, that distinct hydraulic profiles persist in Howe Sound for long periods of time during an outflow event. The term hydraulic profile refers to the variation in depth and densimetric Froude number in the along-channel direction. The identification of these hydraulic profiles allows the specification of which flow regime (sub- or supercritical) occupies specific sections of Howe Sound, thereby indicating the areas of most intense wind.  16  2.1.  Physical Model Study  This study was designed specifically to investigate the hydraulics of an extreme outflow wind in Howe Sound. The following section describes the experimental arrangement and the results are discussed in section 2.1.2. The hydraulics of the wind layer, as predicted by the model, are presented for two flow rates.  2.1.1 Experimental Methods  In order to determine the underlying dynamics, some simplifying assumptions were made in the development of the model study program. Although some of these assumptions lead to characteristics of the model being quite different from those of the prototype (Howe Sound), the topographical aspects of the prototype that most influence the hydraulics of the flow were retained.  The laboratory model system consists of a Plexiglas channel, a six inch wide flume and a video camera with which to document the flow (see Figure 4). As shown in section 1.2, the equations for a single inviscid incompressible homogeneous layer of fluid flowing beneath another homogeneous fluid, which is effectively infinitely deep, are identical to those for a single fluid alone if g’is replaced by g. It is readily shown that this analogy holds independently of any mean horizontal motion in the deep upper layer, and hence a wide range of two-layer flows may be modelled with a single layer (Benton, 1954, Baines, 1980). Lackman and Overland (1989) suggest that gap wind is primarily a boundary layer phenomenon with little influence from opposing winds above. These arguments allowed a one-layer model to be used in the present study with water representing the outfiowing wind layer. Effects due to the upper atmospheric layer and mixing and friction between the layers were neglected.  17  Although the complicated topography of the actual channel is likely to produce an equally complicated flow, we are only interested here in the layer averaged behaviour that is governed by the basic hydraulics of the flow. This idealization permits the design of a model with a straight central axis in the flow direction across which there is symmetry (Figure 4). A further simplification was made by assuming that the slope of the channel walls does not greatly influence the hydraulics. This assumption allowed the model to have vertical side walls. For a channel with vertical walls and flat bottom the flow is controlled by changes in width. The depth of the wind layer during an extreme event was expected to be approximately 1000 m (Jackson, 1993) so the width variation along Howe Sound was taken to be the width at an elevation of 500 m.  To accommodate the large passage between Anvil Island and the west coast of the Sound (123°17’W, 49°35’N  -  Figure 3) the model was simply widened out to the walls of the  flume. Smaller passages between the three main islands were assumed to be insignificant relative to the main channel and were incorporated into the model in the form of widening.  The prototype to model horizontal length ratio was 68000 and the vertical length ratio was 16600. Froude number equivalence between model and prototype was used to match the flows and to scale model results to prototype dimensions.  The model was placed in the flume with the camera’s viewing axis positioned perpendicular to the flow direction as shown in Figure 4. While being videotaped for later analysis, water was allowed to flow through the model continuously. The recorded video images were digitally processed to extract depth measurements so as to compile surface profiles of the flow. The mean flow velocity upstream of the model was  18  measured using a propeller type flow meter. This information was used to calculate the Froude number throughout the channel, determining the locations of controls and hydraulic jumps and defining the regions of sub- and supercritical flow. Further details of the experimental procedure may be found in Vine (1992).  Uncertainty in model results can be attributed to various sources. Hydraulic theory assumes the flow velocity is parallel to the channel axis and constant over each crosssection. This assumption neglects boundary layers and secondary flows. The model results aie obtained from measurements along the channel axis which are then assumed to be constant across the channel width. In some parts of the channel, depth varied slightly across the channel width, indicating velocities not parallel with the channel axis. As well, the Reynolds number of the prototype flow is generally several orders of magnitude larger than that of the model flows, which does not affect the results as far as hydraulic properties are concerned, but may produce different velocity distributions.  2.1.2 Results  Model runs were performed for several flow rates. The lowest flow rates did not force any channel features to act as hydraulic controls, leaving the flow subcritical throughout the channel. As flow rates were increased, certain features of the channel began to control the flow. The hydraulic profile changed when critical flow was achieved at a particular location in the channel (i.e. u =  A curvi-linear channel axis through Howe Sound is shown along with its straightened model counterpart in Figure 5. In each of the figures (6 and 7) which present the spatial variation of quantities along Howe Sound, the length scale represents the distance along the channel axis in the direction of flow. Results from the physical modelling, field  19  experiment and Hydmod output appear together in Figure 6 and Figure 7 in order to facilitate direct comparison. Each set of results is introduced separately before being compared in section 2.3.  2.1 .2a Results for flow rate A  For a model flow rate of 0.032 3 m / s (hereafter referred to as flow rate A) the predicted wind layer elevation (or depth) is shown in Figure 6(a). The horizontal model length scale has been converted to that of the prototype to allow for direct comparison with field measurements (Period 1, described below). Flow is from left to right with the 0 km location being the upstream end of the channel and the 60 km location being downstream of the channel terminus. Wind speed variations along the channel are shown in Figure 6(b) where high wind speeds coincide with regions of supercritical flow. Figure 6(c) shows the Froude number as it varies along the model channel for flow rate A.  Upon entering the channel, flow is accelerated by the contracting walls and reaches the critical point (F = 1) at about km 17. It appears that supercritical flow exists for a short distance beyond the control section before a sudden expansion (km 25) in the channel induces a hydraulic jump (km 20) transforming the flow from supercritical to subcritical. The flow again becomes supercritical near km 35 following the contraction imposed by Anvil and Gambier Islands. This controlling feature is much stronger than the first as is evident in the high values of F. Another expansion beyond Gambier Island (km 40) forces the occurrence of a hydraulic jump at about km 48. The narrow passage between Bowen Island (km 50) and the east side of the channel forces a transition to supercritical flow which extends beyond the channel terminus where a hydraulic jump reconnects the flow with subcritical conditions in the Strait of Georgia.  20  2.1 .2b Results for flow rate B  Results for a higher model flow rate of 0.043 m /s (hereafter referred to as flow rate B) 3 appear in Figure 7. An increase in flow rate from A to B produced similar results to those explained above but with some notable differences. The same features act to control the flow but the hydraulic profile is expectedly characteristic of a stronger flow. Supercritical flow is first reached further upstream (km 14) and extends for several kilometers before the expansion near km 25 induces a hydraulic jump. The position of the jump at km 22 is further downstream than that for flow rate A. The higher flow rate has extended the supercritical flow region here by moving the critical point upstream and the hydraulic jump downstream.  The flow rate increase from A to B does not seem to enlarge the supercritical region between km 35 and km 48. It is likely that the positions of the control and subsequent hydraulic jump that encompass this region are fixed by the channel topography (for flow rates A and B) and essentially confine the supercritical region. The flow exits the channel in much the same way as for flow rate A.  2.2.  Field Program  2.2.1 Synoptic Weather Conditions for the December, 1992 Event in Howe Sound The evolution of synoptic scale weather patterns creates the atmospheric boundary conditions within which gap winds occur. The synoptic conditions in the December, 1992 case were typical of other gap winds (Jackson, 1993). An upper level ridge, lying north-south across the Aleutian Islands (Figure 2), increased in amplitude during 26 27 -  December 1992. Meanwhile, an upper level cold low and an associated 998 mb sea level low developed in a trough to the east and moved southward down the British Columbia 21  coast to a quasi-stationary position 900 km southwest of Vancouver Island by 0400 LST 27 December. This pattern resulted in east to northeasterly flow aloft over the coastal zone.  Linked with the upper level ridge, a 1060 mb surface high pressure zone,  associated with very cold Arctic air, formed over Alaska and moved to a quasi-stationary position over central Yukon Territory by 0400 LST 27 December. Associated with these developments, an Arctic front moved southwards across Howe Sound during the day on 28 December. Behind the Arctic front, a zone of very large horizontal sea-level pressure gradient, oriented perpendicular to the coast, resulted in strong low-level gap winds through the valleys and fjords dissecting the coast range. Sometime during the night of 28 December a large emergency ferry wharf located near Porteau Cove (Figure 3) was damaged by the storm resulting in approximately $250 000 damage. The strong pressure gradient and resulting winds began to weaken after 29 December when the upper level ridge-trough pattern decreased in amplitude; the Yukon high moved southeastwards in British Columbia, but weakened, and the Arctic front moved further offshore.  2.2.2 Field Experiment  To further document the existence of hydraulic controls in Howe Sound and attempt to identify their locations, a field investigation was undertaken. The field research program was initiated at the beginning of the 1992/93 winter season and the extreme outflow event described above occurred in December, 1992. Measured pressure variations between selected sea-level stations along the Sound during the event show distinct along channel hydraulic profiles. The interpretation of the pressure data allows determination of controls in the channel.  Surface pressures during the strong outflow event that commenced on 27 December 1992, were recorded by microbarographs placed at five stations along Howe Sound. The  22  pressure variation with time, recorded at each location, indicates the relative thickness of the outfiowing layer at each station. Pressure differences between stations indicate a change in depth of the outfiowing layer. The mean pressure due to the entire atmosphere is assumed to be approximately equal at all stations (which lie within a 50 km range) before the onset of outflow. When an outflow event occurs, the added pressure, due to denser air in the outfiowing layer, varies between stations indicating changes in layer depth along the channel. Since the pressure measurements are static, as opposed to dynamic, and were recorded from within enclosed shelters, the wind velocity is not thought to affect them.  Jackson (1993) shows that the flow is confined, to some degree, to the main eastern channel of Howe Sound, with Bowen Island, Gambier Island and Anvil Island forming a partial barrier to the flow (see Figure 3). For this reason instruments were placed along the eastern shore of the main channel. An effort was made to place the instruments at locations on the shore as close as possible to the central axis of the channel. Since the model (and hydraulics in general) produces cross-sectionally averaged quantities, the field results are assumed to be representative of the cross-sectional average.  The instrument locations relative to the model are indicated in Figure 5(a) as points numbered 1 through 5. These locations are situated between predicted control sections and correspond to actual locations in Howe Sound where the instruments were placed (shown in Figure 5(b) and Figure 3). The physical model results were used to predict the points of hydraulic control and indicate roughly where to position the instruments. A difference in recorded pressure across a control section during an outflow event indicates a change in depth (between those stations) and a possible transition between flow regimes. The transition may occur either smoothly from sub- to supercritical flow or rapidly as a hydraulic jump from super- to subcritical. A decrease in pressure in the  23  direction of flow would indicate the former while an increase would indicate the latter. This information leads to the determination of which flow regime occurs at each of the five field stations.  2.2.3 Field Data Interpretation  Figure 8 is a composite of the pressure recordings of all five stations over a period of nine and a half days where hour 0 coincides with 0000 LST 25 December 1992 (raw data may be found in Finnigan, 1993). Each pressure trace corresponds to one of the stations shown in Figure 8. At each station the pressure initially varies only with the large scale synoptic field.  This variation is seen in Figure 8 up until about hour 52.  On 27  December at about hour 60, winds, and relative pressure deviation among stations, increased in the channel. These pressure differences represent depth changes in the wind that develop as the flow accelerates and decelerates through the channel topography. The flow is unsteady with the pressure differences (depth differences) increasing during the onset and varying through the duration of the outflow event. Finally as the wind subsides (hour 180), the pressure differences among stations decrease and pressure coincides with the mean regional pressure.  Data acquired from an automatic weather station, located at Pam Rocks (see Figure 3) in the middle of the channel near station 4, were used to confirm the pressures recorded at station 4. As well as pressure, the Pam Rocks station recorded wind velocity and temperature. From the velocity, temperature in the wind layer, temperature in the atmosphere above the wind layer (estimated from recorded temperatures just before onset of outflow) and relative pressure information, it is possible to estimate the absolute depth of the wind layer at station 4. The calculations involved in this estimate are outlined in Appendix A. Converting the relative pressure at each station to relative depth (assuming  24  hydrostatic pressure variation), and using the known depth at station 4, the depths at all stations can be determined at any time. From the information at station 4, the volumetric flow rate in the wind layer can be calculated. By estimating the average width of the channel at each station from topographic maps (width at half depth), and assuming the volumetric flow rate to be constant throughout the channel, the Froude number, F, can be determined at each station at any particular time during the outflow event (see Appendix A for details on the calculation). If a change in depth between stations is accompanied by a change in flow state (indicated by a change in F from >1 to <1 or vice versa), then it is assumed that a control or hydraulic jump exists between the stations.  Through the course of the four day outflow, two distinct along channel hydraulic profiles are evident. The first lasts from the onset of the outflow until the flow is well established. The second, being forced by a higher horizontal pressure gradient and characterized by an increased flow rate, persists for approximately 12 hours before the wind begins to subside. The period of time occupied by each is indicated on Figure 8 as Period] and Period 2, respectively. Both hydraulic profiles are described below.  The acquisition of the field data and the process by which the acquired data were transformed into the final results introduce some uncertainty. The instruments contribute some error to the raw data (± 0.25 hrs, ± 0.2 mb) but the main sources of overall error are due to the process of approximating channel dimensions and averaging the results over time.  2.2.3a Results for Period]  Following the initial increase, the wind layer develops a steady hydraulic profile which persists for 12 hours: from hour 60 to hour 72 during the day of 27 December. This time  25  frame is indicated in Figure 8 as Period 1. In order to more clearly see the changes between stations, the field results are shown again in Figure 9 where the difference in pressure from station 1 (P1) is plotted for each station. For station 1 the difference is of course zero resulting in a straight line. During Period 1 (Figure 9(a)) the pressure and hence depth decreases between stations 1 and 2 and is only slightly higher at station 2 than at station 3. Beyond station 3 the pressure drops substantially at station 4 before increasing again at station 5.  The winds during Period 1 are thought to be associated with a transient stage in the outflow event. This stage represents the onset of the flow and is characterized by lighter winds with stronger winds occurring once the flow is fully established. This low flow rate stage of the outflow approximately corresponds to the low flow rate run (A) of the model results. The average depth, wind speed and Froude number calculated for this stage of the flow are plotted (as solid square points at each of the five stations) along with the results for the physical model flow rate A in Figure 6.  Between hour 72 and hour 88 data at stations 3 and 4 were lost due to instrument failures.  2.2.3b Results for Period 2  As the winds increased during the evening of 28 December, the along channel hydraulic profile changed to a new state. This is indicated as Period 2 in Figure 8 and Figure 9(b). Referring to Figure 9(b) the pressure now drops substantially between stations 1 and 2 before increasing again at station 3. Following station 3 the pressure drops again between stations 3 and 4 and now remains low at station 5.  26  This period of the event is characterized by higher winds and more closely resembles the physical model results for the higher flow rate (B). The average depth, wind speed and Froude numbers for this period are plotted (as solid square points at each of the five stations) on Figure 7 along with the model results for flow rate B.  2.3.  Comparison of Physical Model, Hydmod and Field Results  Since the field results only indicate the conditions at five points along the channel it is possible that some aspects of the hydraulics are not revealed by them. However, by comparing the field results directly with hydraulically similar model results (i.e. approximately equivalent Froude numbers) the conditions throughout the channel can be inferred.  2.3.1 Period 1 (flow rate A)  During the onset of an extreme gap wind event, before the winds have reached full strength, the hydraulics of the wind in the main channel of Howe Sound are expected to resemble what is shown in Figure 6. The field results confirm model predictions of subcritical flow at the channel entrance (station 1). The flow accelerates as it progresses downstream toward station 2, as indicated by increasing Froude number and decreasing depth. Field results indicate that the flow is subcritical at stations 2 and 3, as do physical model results. However, the numerical model indicates that a control and subsequent hydraulic jump can develop between stations 2 and 3 if the flow rate is high enough. The resulting region of supercritical flow between stations 2 and 3 will expand upstream with increasing flow rate (as will be discussed below) and eventually encompass station 2. The combined effect of the expansion in the channel near km 25 (just upstream of Anvil  27  Island) and flow blocking due to Anvil Island results in subcritical flow limiting the extent of the supercritical region upstream.  Beyond station 3 the flow is accelerated through the contraction imposed by Anvil Island and passes through a control point near km 35, before reverting to subcritical flow in a hydraulic jump near km 48. The physical model and field results agree in this region, although the Froude number at station 4 is less than that predicted by the model. Substantial error associated with the estimation of Froude numbers from field results (estimation of average depth, width and velocity) and modelling inadequacies can account for this discrepancy. The lower Froude number reported from the field results may be due to the existence of the side channels between the islands which were ignored in the model. Despite minor discrepancies, the model and field results confirm the general hydraulic behaviour in this region. The supercritical region which reaches its maximum expanse (at low flow rates) between km 35 and km 48 is confined on both ends by regions of subcritical flow.  Beyond station 5 no field results are available. The model predicts that, downstream of station 5, the flow is controlled again as it passes through the contraction between Bowen Island and the protrusion on the east side of the channel (Figure 3). From field results, a simple calculation shows that the width reduction between station 5 and this point may force transition (see Appendix B for the calculation). The flow exits the channel supercritically and must then reconnect to subcritical conditions in the Strait of Georgia through a hydraulic jump downstream. The hydraulic jump was observed in the model, downstream of the channel terminus, but was not documented.  28  2.3.2 Period 2 (flow rate B)  For the higher flow rate some of the hydraulic characteristics of the model flow are readily confirmed by the field results, while others require some interpretation. Throughout most of the channel the situation is much the same as described above. Flow enters the channel subcritically and is accelerated (Figure 7). However, with increased flow rate, the control that was between stations 2 and 3 in the above discussion has advanced upstream beyond station 2. The model now indicates supercritical flow at station 2 and the field results confirm this. The flow returns to subcritical before station 3 as the expansion (km 25) forces transition through a hydraulic jump.  As is the case for the lower flow rate, the flow is controlled near Anvil Island (km 35) before reaching station 4. Through the region spanned by stations 4 and 5, the model and field results differ in some respects. The field results indicate supercritical flow at station 5, whereas the model results indicate subcritical flow (although near critical) there. Since both model flows produce subcritical conditions at station 5, it is possible that the model strongly confines the region of supercritical flow upstream of station 5, while in reality the hydraulic jump (Figure 7, km 47) may be washed downstream and possibly out of the channel. The extension downstream of this supercritical region would explain the findings at station 5.  2.3.3 Further Comparison with Hydmod Output  For further comparison, the hydraulic computer model, Hydmod, was run for the conditions in Howe Sound during Period 1 and Period 2.  Input parameters were  calculated from measurements obtained during the December 1992 event. The average along channel synoptic pressure gradient, dP / dx, for each period was calculated from  29  direct observations at two government stations (Pemberton, 67 km upstream from Squamish and Pam Rocks). Table 1 lists the values of each input parameter for both periods. With these input data, Hydmod ‘steps’ along the channel determining points of possible control. The energy equation is solved over finite steps, in an upstream direction for subcritical flow, and in a downstream direction for supercritical flow. The sections between control points are added together to form a complete solution over the length of the channel.  The hydraulic jump equation (momentum) is used to determine the  conditions at a hydraulic jump in reaches where there is supercritical flow from an upstream control, and subcritical flow from a downstream control. The reader is referred to Jackson (1993) and Jackson and Steyn (1994b), who created the model, for mathematical details.  Hydmod results appear as a thin line, along with the physical model and field results, in Figure 6 and Figure 7. Near the channel entrance, the supercritical flow is predicted to jump to subcritical. Although not within the range of the physical model or field results, this jump may actually occur with the wind descending the mountain slope upstream of Squamish in the supercritical regime before entering a flat expansion which could induce a hydraulic jump. Referring to Figure 6 and Figure 7 Hydmod predicts that the channel hydraulics are governed by the same controls identified by the physical modelling and field observations. The two models show slight differences in the exact location of these controls and the ensuing hydraulic jumps. This is due to the vastly different nature of the two models and the different representations of channel topography and friction coefficients (among other things) that each relies on. One notable difference is the large supercritical region predicted by Hydmod between km 12 and km 25 in Figure 6 which does not appear in the physical model results for flow rate A. In Figure 7 Hydmod indicates the flow is controlled near km 12 but passes through a hydraulic jump and subsequent control before reverting to subcritical flow near km 24. This additional  30  control and jump is not predicted by the physical model and can not be detected by the field measurements. Upstream depths differ substantially between the two models which is likely due to each models deficiency in simulating the upstream boundary conditions. The physical model appears to overestimate the upstream depth for both flows (Figure 6 and Figure 7). This may be because the fluid enters the model from a narrow flume rather than entering from an essentially infinite reservoir. Despite the differences, the Hydmod results serve to confirm the existence of the hydraulic profiles as predicted by the physical model and as suggested by field observations.  2.4.  Discussion  Physical modelling of gap winds in Howe Sound, followed by field measurements recorded during an actual outflow event, led to an understanding of the hydraulic behaviour of the wind layer for two distinct flow rates: one representing lighter winds typical of the onset period of an outflow event, the other representing the fully established flow. Comparison of the physical model and field results confirmed the model findings at specific points and thereby allowed the inference of prototype behaviour from model predictions at other locations in the channel. For the lower flow rate (Period 1, flow rate A) short regions of supercritical flow were observed. With an increased flow rate (Period 2, flow rate B) these regions were observed to expand and occupy more of the channel. In some cases, fixed control points limited the expansion of the supercritical regions, effectively confining them between regions of subcritical flow.  The two modelling exercises serve to reinforce the findings of the observational program and allow the specification of which hydraulic regime is prevalent at locations along Howe Sound during an outflow wind event.  Supercritical flow and corresponding  extreme wind conditions are defined in Figure 6 and Figure 7. Model results show some  31  deviation from field results for fully established flow. Insufficient similarity between the physical model and the prototype near the channel terminus and beyond may explain the fixed control that exists in the model, but is not observed in the field. In chapter 3 we describe another physical model study that improves on that described above by incorporating geometric and kinematic similarity and proper simulation of the boundary conditions.  32  Chapter 3 3.1.  -  Three Dimensional Modeffing  Description  The 1D study described in chapter 2 considered flow through topographically simplified representations of Howe Sound and was intended to provide one dimensional, cross sectionally averaged results along the main channel.  Predicted hydraulic profiles  indicated approximately where extreme conditions occur in the Sound and where hydraulic jumps are expected to form. The relatively simple study, involving many assumptions, can be considered to only approximate actual conditions.  We now wish to address many of the questions left unanswered by the 1D study. To gain a more realistic understanding of the outflow winds in Howe Sound, and extend the knowledge gained by the 1D study, we developed a more elaborate physical model study with fewer simplifying assumptions. In particular, the three-dimensional (3D) physical model described in this chapter includes the following which were not considered in the 1D study:  •  proper simulation of boundary conditions  •  effects due to channel sinuosity and elevation changes  •  flow over and around, and effects due to, islands in the channel  •  variation in wind across, as well as along, the channel  •  energy losses due to form drag and skin friction drag  •  geometric and kinematic similarity  •  realistic reproduction of Howe Sound topography  •  simulated pressure gradients  33  A single layer of water was again used to simulate the outfiowing wind layer and mixing and friction between the layers were ignored. To document the flows, a unique method of data collection, using video/image analysis, was developed. This method provides a comprehensive and realistic set of predictive results.  The complicated hydraulic  characteristics of the outflow wind system were accurately reproduced, and conditions were predicted in detail throughout the Sound, for several different synoptic cases.  3.1.1 Design Considerations  a. Similarity  Obtaining useful results from hydraulic physical models requires careful planning in the design stage. If a model is to be similar to the large scale original, also called the prototype, then it is not enough that the solid boundaries be geometrically similar in the two systems; it is also necessary that the two flow patterns be similar. If this is true then it is said that kinematic similarity holds between the two systems. Without going into detail on the laws of similitude we simply state that if viscosity, surface tension and compressibility effects are negligible then only Froude number,F, is important. However, to ensure the absence of viscous effects in the model special attention must be paid to the Reynolds number. Therefore, we invoke Froude number similarity between model and prototype and use the Reynolds number for guidance in designing a physical model with the same flow conditions as found in the field.  b. Scale Ratios  Representative model flows are based on the constancy of F between model and prototype. The detailed interpretation of model measurements then requires that scale  34  numbers be available for translating model values of various quantities into corresponding prototype values. With F defined as in (1) we require, F = 1 F,  (7)  where the subscripts m and p refer to model and prototype respectively. Since the single layered model flows are influenced by gravity, g, and the two-layered prototype flows are influenced by reduced gravity, g’, substitution from (1) and (5) into (7) leads to,  It/h 11 gh  u Urn  (8)  or, by indicating the ratio of prototype:model quantities with subscript r, /  ,  \1/2  Ur=:[hr1 ‘\g  )  (9)  where g’ must be estimated or known from field observations.  A list of relevant scale numbers, based on Froude number similarity and the relation (9), is given in Table 2, where the vertical and horizontal scales (hr and Lr respectively) have been defined separately to accommodate depth exaggeration (discussed below). In the case of a geometrically similar model hr = Lr With the length scale numbers known, the remaining scale numbers allow model measurements to be translated into prototype values.  35  c. Reynolds Number and Model Distortion  Howe Sound prototype flows, like many geophysical flows, have a high Reynolds Number and are not influenced by viscous effects. Here we define Reynolds Number based on fluid depth, i.e. (10)  where a represents fluid speed, h, fluid depth and v, dynamic viscosity. With this definition, and considering the results of chapter 2, Re is on the order of 106 for all prototype flows. It is impossible to produce this magnitude of Re in laboratory flows. Low Reynolds numbers are not a concern as long as the model operates in the fully rough turbulent region at all important locations, which depends on both Re and the model boundary roughness. For typical roughnesses, the flow will generally be fully rough for Re>1500 (Gibson 1934, Yalin 1989). If this requirement is satisfied then the effects of viscosity will be negligible and model flows can be assumed to represent prototype flows. Although turbulence scales will differ between model and prototype, they are not important as we are studying flow features much larger than turbulent structure and are not concerned with diffusion or mixing.  Reynolds number dictates to some extent the minimum size of an acceptable model. It is the depth of flow that is the limiting factor. As a general rule in channel flow modelling the flow at all locations should be at least 2 cm deep and fully turbulent (Yalin, 1989). Considering typical depths of prototype flows this translates into a scale model with very large horizontal extent. However, it is common practice in river flow modelling to vertically exaggerate, or distort, model dimensions to obtain sufficient flow depth and  36  ensure high enough Reynolds number, while maintaining horizontal scales which are reasonably easy to work with.  A distortion factor may be defined by,  e=--,  which is equal to the scale of slopes, Sr and is always  (11)  1. The model described here has  a distortion factor of e = 0.57 which is well above the acceptable lower limit of 0.25, as suggested by Nicollet (1989). The reasons for choosing this particular value stem from building supplies and construction limitations which are discussed below. Distortion will cause secondary flows, velocity distributions and mixing to differ between model and prototype. This is not a concern as we are interested only in mean velocities and hydraulic behaviour which are not affected.  d. Boundaiy Conditions and Model Size  Although the results of chapter 2 indicated subcritical flow at the upstream end of the channel near Squamish, we have no way of prescribing or ensuring this. To properly simulate the conditions at this location the model was extended up through the Cheakamus Valley to the Pemberton Valley which acts as a reservoir for the cold air. This guarantees that the boundary conditions near Squamish are simulated correctly. The entire model topography is pictured in Figure 10.  Upon steady state, the flow some distance downstream of the channel terminus, in the Strait of Georgia, is subcritical as it must connect with the mass of cold air built up there.  37  The subcritical downstream boundary condition is set by a weir, at a model location corresponding to approximately 20 km beyond the actual mouth of the channel.  The entire model covers a region that is actually 128 km long and 40 km wide. Lab space limitations and restrictions due to building materials led to model dimensions of 270 cm long and 84 cm wide which translate into a horizontal length scale number of Lr = 48000. The process of forming the topography, which is described in more detail below, limited the choice of vertical scale factors which would produce deep enough flows. A layering method was used to reproduce the contour intervals from topographic maps. The maps used had 500 ft intervals and the material used to represent each interval in the model had a thickness of 5.5 mm. hr = 500fl/5.5mm =27548.  This led to a vertical scale number of  These values together give the distortion factor,  e = hr /Lr = 0.57, as mentioned above.  e. Frictional Effects  Energy losses in the flow, resulting from both form drag and skin friction drag, can influence the flow and affect the location of hydraulic jumps. Form drag, induced by complicated channel boundaries, is simulated in the model by replicating the channel features present in the prototype. To properly simulate skin friction, or surface drag, the the surface roughnesses of the prototype and model channels must be considered. It is common to use the Manning-Strickler resistance formula (Henderson, 1966),  (12)  where R is the hydraulic radius (cross sectional area/wetted perimeter) and K obeys  38  the following empirical relationship,  , 6 K=26/d”  (13)  with d representing the size of bed roughness elements.  For wide channels (12) becomes,  __h2,/3c12 U 1/6  ‘  l4  with h representing the mean fluid depth. This simplification is quite approximate at some locations in our system where the channel is narrow and the fluid relatively deep. Comparing this with the Chézy (Darcy-Weisbach) equation,  (15)  gives the roughness coefficient 2 as a function of the relative roughness hid:  * which is true for 5 < h/d  <  =  2.9353.).  (16)  500. This states that the dimensionless roughness coefficient  2. will be the same in prototype and model if the hid ratio is the same. The validity of this argument requires that the model operates in the hydraulically rough region (Henderson (1966), pp. 98 and 492).  39  Since the model is vertically distorted we must consider the effect this has on surface friction. The distortion modifies the shape of cross sections, involving a variation of the hydraulic radius. Considering the Chézy equation in scale number form,  ,  r  SrRrhrRrRr u Lr hr Lr’  (17)  the roughness scale number, Ajr, depends on the hydraulic radius, which appears to make complete similarity impossible. However, if the channel is assumed wide enough to approximate R by h then ‘r = e, the distortion coefficient. The above relation (12) gives the scale number equation,  (18)  or dr/hr = e 3 which means that the relative size of the roughness elements varies like the third power of the distortion coefficient. The greater the model is distorted, the greater the exaggeration of the roughness elements.  We have prescribed the distortion coefficient (e = 0.57) as described above by building the model with specific dimensions. Surface roughness elements in the field were estimated, from visual observations, at 5 to 10 m on land, due to tree tops and large rocky protrusions. Similarly, visual estimates made by BC Ferries crew (Whailin, 1993) during a severe outflow event, give values of 1 to 2 m on water, due to enhanced sea state. Considering an average prototype flow depth of 800m, the approximate relative  40  roughness in the field is: =  =  5 to 10 metres 800 metres 1 to 2 metres 800 metres  =  0.006 to 0.013 on land (19) 0.0012 to 0.0025 over water  Now using equation (18) the relative roughness values for the model are: tin, ti,,, 1 h  —  0.00625 to 0.0125 3 e 0.00125 to 0.0025 3 e  0.O3toO.06 onland =  0.006 to 0.012 over water  (20)  With these values the appropriate sizes of roughness elements were chosen for the model surface. The average model flow depth is approximately 3 cm which gives roughness element sizes of about 1 to 2 mm over land and 0.2 to 0.4 mm over water. The particles were applied to the model surface and some attention was paid to the spacing in an effort to approximate the concentrations found in the field. In general, trees have a relatively dense concentration in the area. Wave heights and wavelengths vary throughout the channel but were given average values from visual estimates in the field.  The simulation of surface friction in the model is approximate. More elaborate model studies often include lengthier processes which include trial and error application of surface elements and careful calibrations which require detailed records of field conditions (Nicollet, 1989). Since form drag induced by the complex topography of the region, which is replicated in the model, is the dominant source of energy losses, the surface friction simulation methods used here are sufficiently accurate.  41  e. Pressure Gradient  Outflow winds in Howe Sound are driven by synoptic pressure gradients which result when cold air resides inland of the Coast mountain range and relatively warm air occupies the coastal region. Generally, a significant component of the pressure gradient lies parallel to the channel axis, which is oriented almost perpendicular to the mountain barrier. Since we can not simulate an external pressure gradient in the laboratory we impose a physically equivalent gravitational force by sloping the model in the downstream direction, parallel to the channel axis. The model was fitted with a hinge at the downstream end and a jacking mechanism at the upstream end so that the slope could be varied.  If the external pressure gradient, dP/dx, is defined as a slope, S, and made positive for increasing pressure along the channel axis in the upstream, or positive x direction, then  S, = (g’p) dP/dx.  (21)  As stated above in equation (11) the distortion coefficient is equal to the scale of slopes, i.e. e = Sr = S, /S,,,. Therefore, with observed values of the quantities in (21) (examples from the December 1992 event appear in Table 1) we are able to predict a range of suitable model slopes which will simulate pressure gradients expected in the field.  42  3.2.  Model Design and Construction  a. Topography  The complicated topography of the Howe Sound region was reproduced, on a small scale, in the 3D model described here. This was done in an effort to recreate the flow conditions present throughout the Sound during extreme outflow events. A rectangular region that encompasses all of Howe Sound and the Cheakamus Valley was selected and compiled from Government of Canada topographic maps with 500 ft contour intervals. The maps were enlarged so that the region to be modelled was exactly the intended size of the model. From these, templates for each contour level were made and then used to cut the shape of each individual contour out of sheets of cork material 5.5mm thick. The contour interval on the maps and the choice of material thickness sets the vertical length scale number, and therefore, the distortion coefficient.  The cork material was successively stacked up and glued together starting at the 0 contour and progressively building up to the mountain peaks. A clay filler material was then applied to the model to fill in the “steps” left by the stacking process (see Figure 11). This method produces a model that resolves the detail of the actual topography to 500ft. Smaller features, consumed by the smoothing process between steps, would likely only affect the flow through surface friction, which is included through the application of roughness elements as described above.  b. Video Apparatus  The aquisition of data from the model flows was achieved through video techniques. Since the model topography is complicated and not transparent the video must view the  43  flows from above rather than through the model from the side. This constraint required an apparatus, on which a camera and light source could be mounted, which would move around the model, above the flow domain, in a precise manner. As described below in section 3.3, the camera moves up the length of the model stopping every 2 cm to record. It then moves across one view width and repeats the procedure. To accommodate this movement an aluminum frame was constructed, on which rails were mounted and suspended approximately 1.75 m above the model. A small cart, fixed with wheels, was made to run on the rails and carry the video equipment the length of the model. The cart was fitted with a mechanism to allow the camera and lighting equipment to be moved sideways. Figure 12(a) shows the rail and cart system and its motions. A mechanism was fitted to the cart which allows the operator to externally move the cart, fix its position and track its location.  The system is shown schematically in Figure 12 and in a  photograph in Figure 13.  c. Fluid Supply  The single layer of water used to model the outflow winds was dyed with flourescein to enhance video images and improve data accuracy. To conserve dye a recirculating fluid supply system was required. The simple circuit, as shown in Figure 12(b), includes a centrifugal pump, flow meter, valve, diffuser, the model and a retrieval tank. Water is pumped up through the valve, which controls the total discharge; through the diffuser and over a weir, which provides an even flow; through the model; then over the downstream weir and through a drain into the retrieval tank.  44  3.3.  Data Aquisition  3.3.1 Data Aquisition System  Video image analysis techniques were chosen to record and analyze the flows because of the relative ease with which large amounts of information can be obtained and processed. These techniques are non-invasive, more thorough, and require less time to obtain data than manual instrument techniques (i.e. point gauge, pitot tube etc.). Because the model design imposed certain limitations, where data aquisition is concerned, a unique method of recording the flows had to be developed. Physical complexity of the model required that all video images be recorded from above. The aim was to collect data throughout the entire flow domain both along the channel and across it. This required that the video -  system (VS) be mobile. A system was designed to move above the model collecting information, which would later be compiled to provide complete coverage of the flow area.  The VS acquires information that provides depth and velocity data throughout the flow. Both require a separate system operation and are thus recorded individually. First the VS moves through the flow domain and records information for depth and then it is reconfigured and moves through again recording information for velocity. Once video for a particular flow has been recorded, images are digitally captured and analyzed on a computer. The method of image capture and the image to data translation are different for both depth and velocity. Depth and velocity data are referred to the same spatial coordinates so they can both be specified at coinciding points throughout the flow field. A 2cm grid, with a depth value and a velocity value at each node, was selected as a reasonable compromise between high data resolution and collection time.  45  Details of the VS design are best described by discussing its function in recording depth and velocity images. Below we describe, separately for depth and velocity, how the VS operates and the transformation of the captured images into data.  3.3.2 Depth Data Aquisition  Depth information is recorded by a video camera directed down at an angle and focused on a projected light sheet which illuminates a vertical cross-section through the flow, roughly perpendicular to the flow direction. Figure 12(b) shows the configuration of camera and light source and how the system is positioned over the model. The camera is angled sharply down (approximately  500  from horizontal) in order to view the light sheet  behind islands and in deep valleys. As the camera and light sheet move throughout the flow domain, stopping momentarily at predetermined locations, a mapping of the flow depth is compiled.  The camera zoom is adjusted to a narrow field of view (15 cm wide) which covers only a portion of the model width but is necessary to record images free from wide angle distortion and with image depths substantial enough to provide accurate results. This camera setting requires that the entire flow domain be swathed out in longitudinal parallel strips. The viewing system moves the length of the model pausing at several locations, each 2 cm apart, for about 30 seconds each. The system is then moved transversely a distance of one view width and another longitudinal pass is made. The procedure is repeated until the entire flow domain has been covered (3 passes). When the recorded video is later played back, images of flow cross-sections at each location where the camera paused, are captured and analyzed with a computer. Since the flow is turbulent, with rapid small fluctuations at the surface, any instantaneous “snap shot” of a flow cross section has a highly distorted bottom profile. This is due to refraction of light as it passes  46  through the fluid surface on its way to the camera. The problem is remedied by capturing several sequential images (about 15) of the same cross-section and digitally averaging them together which eliminates most of the time dependent noise.  The method of viewing requires that depth measurements taken from the captured images be translated to obtain the correct or “true” depth. As depicted in Figure 12(b) the camera looks obliquely at the vertical cross-section in the flow and therefore sees a skewed representation of the actual depth (i.e. a projection along the camera’s viewing axis). Figure 14 indicates schematically how the depth is viewed by the camera. It is assumed that the camera is far enough away that at any point along a cross-section the light rays leaving the water surface and reaching the camera are parallel to the camera viewing axis which lies at an angle,  ‘,  from the plane of the light sheet. Referring to Figure 14 the  true depth of fluid, d, appears to the camera as the projection, d’, which is perpendicular to the viewing axis.  In addition to the correction that must be made to compensate for the camera angle, two more factors must be considered. The first involves refraction of light at the free surface. The true depth at any point along a cross-section can be considered as the vertical distance between the light ray reflected up to the camera from the free surface and that reflected from the bottom. The ray reflected from the bottom is refracted at the free surface and bends toward the horizontal as shown in Figure 14. According to Snell’s Law,  nsinO=n’sinO’,  (22)  where 0 and 0’ are defined in Figure 14 and n and n’ are the indices of refraction of water and air respectively.  47  The second additional factor to consider is the width of the light sheet reflecting from the free surface.  This width,w, adds wcos8’ to the projected depth which must be  subtracted to obtain the true depth.  The three corrections discussed above are embodied in one relation,  d’—wcosO’  d  = (cos  which  gives  8’)  tan(sin’  (  (23) sin  the true depth and where all quantities on the right side of the equation are  known.  Each captured depth image is processed with thresholding and line detection routines to reveal the surface and bottom. The two resulting lines are then automatically discretized into data points and converted to projected depths which are transformed to actual depths with equation (23). The results of one depth transformation are shown as a solid line in Figure 15(a). These results indicate the variation in depth across one 10 cm wide flow cross-section as measured using the video image analysis techniques described above. To verify the method, direct measurements were taken across the same flow cross-section using a manual point gauge system. Depths, recorded manually at 5 mm intervals, appear as solid squares in Figure 15(a). The same comparison for a different position (and flow regime) in the model is shown in Figure 15(b). The close agreement serves to validate the video techniques.  48  3.3.3 Velocity Data Aquisition  For velocity data, the camera is directed straight down and focused on a small region of the flow surface which is broadly illuminated from above. Positively buoyant plastic beads are introduced into the flow upstream and videotaped as they flow through the camera’s view. The beads, floating at or near the surface, are used to indicate velocities which are assumed to approximate the mean velocity throughout the depth. The velocity actually varies slightly over the depth due to boundary layer effects and secondary circulations; however; we are concerned here with variations on a larger scale so surface velocities adequately represent the mean.  The VS moves throughout the flow domain in the same manner as described above except that it now focuses on rectangular portions of the surface rather than crosssections. The region of the model viewed at each camera location corresponds to a known portion of the 2cm grid described above. Within each image the grid contains 40 points which coincide with points from the depth sections so that data can later be combined to give Froude number values.  At each position the camera is allowed to record passing particles for a number of seconds. Multiple images (about 20) from each location are captured on a computer. The camera frame rate determines the length of time each frame, or captured image, represents. In our case the camera was set to 1/30 second. The moving particles in these images appear as streaks which represent the distance traveled by the particle in 1/30 second. Each image may have only a few or several streaks depending on the density of particles at the instant of image capture. In order to obtain measurements at all desired grid points, streaks are measured from the several captured images, at each camera location. The streaks are manually traced on the computer screen and automatically  49  scaled and stored. Approximately 160 streaks are measured for each location where only 40 grid points require values. All streak measurements within a 20mm by 20mm box around each grid point are averaged together and the result is associated with the appropriate grid point. The location of a streak is identified by its midpoint. The procedure is repeated throughout the model domain resulting in a data set representing the entire flow.  3.4.  Results  Using the data aquisition methods described above, results were obtained for eight different model flows. Velocity, depth and Froude number, presented for each flow, reveal the subcritical and supercritical regions and some of the dynamic features present in the wind layer. A range of conditions was simulated, including those most likely to occur in the field. The modelling program was based on the conditions present in Howe Sound during the December 1992 event (see section 2.2.1), but was broadened to encompass other possible flows.  3.4.1 Important Parameters  Model flows are governed by three parameters; total discharge,  Q, downstream depth,  hf. and external (or synoptic) pressure gradient, dP/dx. Each of these is prescribed, for an individual model flow, as described in the above sections. A valve sets the discharge, a weir the downstream depth and a model slope the pressure gradient. The settings are scaled from model to prototype in the manner described above in section 3.1.1. Two typical values for each parameter were chosen and all possible combinations of the three parameters were simulated. The values chosen appear in Table 3 and the eight resulting combinations in Table 4.  50  3.4.2 Model Results  As described in section 3.3 the acquired data for the 3D model provides a mapping of conditions throughout Howe Sound.  Although velocities may be largely three  dimensional in some locations we are concerned with horizontal variations on a relatively large scale and have therefore presented only the horizontal component of the depth averaged velocity. In particular, it is the hydraulic behaviour of the wind system that is of primary interest.  Results are referred graphically to the Howe Sound region which is shown in Figure 16. This reference figure may be used to find locations which appear in the results figures (17-24), which are not extensively labeled to avoid confusion. The spatial dimensions on Figure 16 provide a coordinate system for reference and the locations of field stations for the December 1992 outflow event are shown.  The results appear, for the eight cases modelled, in Figures 17 through 24. In each figure the first panel shows the velocity distribution, the second panel shows the depth distribution and the third panel shows the Froude number distribution. Velocities are represented by vectors with length proportional to magnitude (velocities less than 1 m/s are not shown). Depth is represented by contours with a 200m interval. The 0 m contour, or layer boundary, is not shown because the method of data aquisition was unable to accurately resolve depths down to Om. Froude number is represented by contours at F = 1, 3, and 5. The F = 1 contours enclose regions of supercritical flow with the contour itself representing critical flow and indicating the location of controls and hydraulic jumps.  51  The hydraulics of three dimensional flows differs in nature from the simple, cross sectionally averaged, 1D flow case. Schar and Smith (1993) investigated the hydraulic structure of flows encountering 3D topography and characterized some important features. In the case of a complicated channel like Howe Sound, such features as controls and hydraulic jumps are not likely to span the entire width of the channel wherever they occur. Often a portion of the flow width will be controlled while a region alongside it remains subcritical. The supercritical region may be flanked on one or both sides by regions of subcritical flow. The boundary between the two flows may be termed a shear discontinuity since the flow will have different depth, speed and even direction on either side.  3.4.3 Discussion of Cases 1 Through 8  The two values for each of the three model parameters appear in Table 3 where it is indicated that the lower setting of each value will be referred to by L and the higher setting by H. This convention appears in Table 4 which outlines the parameter settings for each case and lists some physical characteristics from the results. The overall effect of each parameter on the wind system is suggested by these results.  The most important information in Table 4 is contained in the column listing the total supercritical area. The regions occupied by supercritical flow will have distinctly higher, more consistent and unidirectional winds and can therefore be labeled “danger zones”. Referring to Table 4 it is apparent that the expanse of supercritical flow is governed mainly by hf and dP/dr. The coupled effect of these parameters is reflected in a positive influence by dP/dx and a negative influence by hf. The discharge, lesser effect and its influence seems to depend on the other parameters.  52  Q, has a  The set of 3D model results was designed to encompass possible prototype flows. Therefore some of the extremely weak or strong cases modelled (i.e. Cases 3, 4, 5) may not actually occur in nature.  The topographical influence on the flow is now discussed for each case in succession (Figure 17 through 24). Locations will be referred by their Cartesian (x,y) coordinates. Case 1  -  Q=L, hf=L, dP/dx=L  -  Figure 17  Although mostly subcritical, the upstream region of the channel near Squamish (8,48), is hydraulically controlled by local elevation changes resulting in “patches” of supercritical flow. The flow is controlled by the promontory near Britannia (9,38) but only across a portion of the channel. This partial control results in a shear discontinuity and a slowly rotating clockwise eddy on the opposite side of the channel (not apparent in the figure but observed). A hydraulic jump occurs before the flow is again controlled, this time across the entire channel, by the contraction near (8,34). The abrupt widening of the channel near (8,30) combined with upstream blocking from Anvil Island forces a hydraulic jump near (8,27). The shape and location of Anvil Island guide most of the flow through the main eastern channel where it is controlled by the “throat” between Anvil Island and the eastern shore. Some flow manages to enter the west arm (7,28) where an eddy forms. As was proposed by Jackson (1993) the flow is largely confined to the main channel by the islands (Anvil, Gambier, Bowen). A hydraulic jump occurs near (12,16) before the flow is again controlled, forming a small region of supercritical flow, near the headlands at Horseshoe Bay (14,12). Case 2  -  Q=H, hf=L, dP/dx=L  -  Figure 18  The increase in discharge increases the depth of flow. Although it seems to decrease the supercritical region near Horseshoe Bay it has an overall effect of increasing the expanse  53  of supercritical flow. The deeper flow is now able to go over, rather than around, larger portions of Gambier and Bowen islands and it is controlled as it does so. Most of the flow is controlled in the same manner as for Case 1. Case 3  -  Q=H, hf=L, dP/dx=H  -  Figure 19  The increase in pressure gradient causes a substantial increase in supercritical flow. The supercritical region near Britannia (8,38), however, seems fixed in size as a hydraulic jump is forced on the lee side of the promontory. A large supercritical region now spans most of the length of the lower channel, extending out beyond the terminus before reverting to subcritical conditions through an apparent undulating hydraulic jump. Case 4  -  Q=L, hf=L, dP/dx=H  -  Figure 20  As expected a decrease in discharge causes only minor changes to the flow. The undulating hydraulic jump does not form and the region of supercritical flow is slightly smaller. Case 5  -  Q=L, hf =H, dP/dx =L  -  Figure 21  This combination produces a flow pattern which is different from those discussed above. The flow is substantially deeper and slower and although Anvil Island directs the flow into the main channel the other islands do not confine it. The flow separates from the east side of the channel (10,26) and proceeds in a direct path over the islands towards the Strait of Georgia. In the separated region (13,18) a slow moving counter-clockwise eddy is formed. The flow is predominantly subcritical but small regions of supercritical flow exist where local elevation changes induce transition.  54  Case 6  -  Q=L, hf=H, dP/dx=L  -  Figure 22  The increase in discharge from the previous case delays the separation from the eastern side of the channel. The flow is again confined mainly to the main channel but a small separated region exists near (14,18). Relative to the first four cases the flow is slow and deep and although supercritical regions near (8,30) and (8,38) are present the downstream region (8,22) is somewhat smaller than in the first four cases. Case 7  -  Q=H, hf=H, dP/dx=H  -  Figure 23  This case is similar to that of cases 1 and 2 but is characteristic of a stronger flow with a larger area occupied by supercritical flow. The three prominent regions of supercritical flow are again present but are slightly larger. Case 8  -  Q=L, hf =H, dP/dx =H  -  Figure 24  The change in discharge has the opposite effect in this strong flow case as it did in the weak flow cases 1 and 2. With high forcing, the response to a decrease in discharge is a an increase in the amount of supercritical flow. Controls occur further upstream due to the decrease in depth that accompanies a decrease in discharge.  The two main  downstream supercritical regions have joined as in cases 3 and 4. A hydraulic jump limits the extent of this region and it does not extend out of the channel as before.  55  Chapter 4 4.1.  -  Discussion and Conclusions  Comparison of 1D Modeffing and 3D Modelling  The fundamentally different approach taken in the 1D and 3D studies makes them difficult to compare. Since the 3D study was based, in some ways, on the findings of the 1D study, it may be useful to simply compare the hydraulic characteristics of similar flows. For the two flows studied in the 1D experiments of chapter 2, the two 3D flows that most resemble them are Case 6 and Case 7.  Through a qualitative comparison of the results presented in Figure 6(c) with those in Figure 22(c) it is possible to draw some conclusions. For a lower pressure gradient, the flow is controlled near Britannia Beach but remains supercritical only for a short distance. The 1D physical model does not predict the subsequent supercritical region which is produced by the 3D model (centered at 8, 29  -  Figure 22(c)). The Hydmod results  appearing in Figure 6(c) do however indicate a supercritical region in this vicinity with a hydraulic jump at km 25 which corresponds to that indicated in the 3D results (9,25), Figure 22(c). The flow is then controlled through the contraction at Anvil Island and a subsequent hydraulic jump occurs some distance downstream before Horseshoe Bay.  The hydraulic jump which occurs near Porteau Cove in the 3D model for Cases 1, 2, 6 and 7 (near 9,25) is also present in the Hydmod results of Figure 6(c) (km 25) and both the physical model and Hydmod results of Figure 7(c) (km 24). This jump occurs due to the abrupt expansion near (8,30), Figure 16 and the upstream blocking effects of Anvil Island. Although the 1D results indicate a point location for the jump and the 3D results indicate a line location, the actual hydraulic jump may occupy a region 1 or more km in length. The existence of this jump in the field is supported by the microbarograph 56  tracings recorded during the December 1992 event. Figure 24 shows a portion of the pen tracing from Porteau Cove and the corresponding tracing from Lions Bay. The erratic pen movements of the Porteau Cove tracing suggest strong low level turbulence due to the presence of a hydraulic jump. The Lions Bay tracing is relatively stable which suggests a very steady, characteristically supercritical flow.  Hydraulics of the higher pressure gradient scenario are shown in both Figure 7(c) and Figure 23(c). These indicate that the flow is again controlled near Britannia Beach and may subsequently go through a hydraulic jump, control, hydraulic jump sequence before reaching Anvil Island. This is supported by both the Hydmod results of Figure 7(c) and the 3D model results of Figure 23(c). The flow is again controlled and may remain supercritical throughout the channel or until another hydraulic jump is induced downstream of Lions Bay.  4.1.1 Comparison of field results with 3D model results  a. Low pressure gradient, Case 6 Period] -  Values of depth and Froude number, obtained from the 3D model for Case 6, are shown in Table 5. These values were taken from the five locations in the model domain which correspond to the field study locations as described in chapter 2. The field results for Period 1 are also shown in Table 5 for comparison. Of the 8 3D model runs, Case 6 results correspond best with Period] field results. However, by refering to Table 5, it is immediately apparent that the overall depth of the two flows differs substantially. This is because some of the controlling parameters differ. Refering to Table 1 (field study parameters) and Table 3 (3D model parameters), the pressure gradient is the same for  57  both flows but the discharge and depth settings are substantially greater for the field results. This makes the two flows difficult to compare.  Initially the 3D model was tested at flow rates corresponding to those found for the field results for Period 1. However, these flow rates produced unrealistically strong model flows. This suggests that the method used in chapter 2 to calculate depth and discharge from pressure data, may produce values that are too high. The process of estimating channel widths and averaging over an idealized rectangular channel may make the field results subject to some uncertainty. The 3D model indicates that outflow winds in Howe Sound are complicated in nature and often vary in velocity, depth and flow regime across the channel width. Therefore, although the 1D study of chapter 2 is useful for predicting roughly the location of hydraulic controls and jumps, it does not capture some of the smaller scale hydraulic features of the flow. As well, comparison of a single point in a highly three dimensional flow with a corresponding cross-sectionally averaged value of the Froude number, is not likely to be informative. Table 5 shows that the two flows have a similar trend in the along-channel direction. This substantiates the findings in some of the locations but does not allow for a detailed comparison.  b. High pressure gradient, Case 7 Period 2 -  In the same manner as described above, the 3D model results for a higher pressure gradient, Case 6, are compared with Period 2 field results in Table 6. The depth difference between flows is again apparent. The same parameter differences as described above, and the fundamental difference in the data types (i.e. single point as opposed to cross-sectional average), are responsible for the lack of overall agreement.  58  4.2.  Summary and Conclusions  a. Summary  The work reported in this thesis is comprised of two separate but complementary studies; firstly a 1D study which combines results from physical modelling, field observations and computer modelling, and secondly a more accurate 3D physical model study. The 1D study was intended to identify the hydraulic characteristics of the outflow wind system in Howe Sound. The results are simple yet useful in predicting roughly where extreme conditions will occur during an outflow event.  The 3D physical model study broadens the knowledge gained by the 1D study by providing a more accurate and comprehensive set of results. The wind system is studied in detail through a series of experiments conducted with a three-dimensional physical model which is geometrically and kinematically similar to the prototype, Howe Sound. Simulations covered a range of parameters representative of likely field conditions. Hydraulic features, which do not behave in a traditionally two-dimensional manner, are identified.  b. Conclusions  As discussed above, a quantitative comparison of 1D model and field results with 3D results is difficult. It is, however, informative to briefly identify common predictions of both studies. We can specify conditions in Howe Sound, and possible locations of controls and hydraulic jumps, during an outflow event.  59  In general the flow is subcritical in the vicinity of Squamish and is accelerated through a control over the promontory just upstream of Britannia Beach. This control may not extend across the entire channel, resulting in a shear discontinuity and subcritical conditions near the west shore. A hydraulic jump and subsequent control may or may not form just downstream of Britannia Beach.  Regardless, the abrupt expansion just  upstream of Anvil Island and the blocking effects due to Anvil Island always force a strong hydraulic jump which piles fluid up on the steep north side of the island. The flow is controlled through the contraction inposed by Anvil Island and the west side of the Sound and generally remains supercritical until a hydraulic jump occurs, across the main channel, at the north end of Bowen Island. The exact location of this jump can vary considerably.  For stronger flows, a separated region can form along the west side of the lower Sound, as apparent in Figures 21 and 22. The large supercritical region downstream of Anvil Island can, in some instances (Figures 7, 19 and 20), extend over portions of Bowen and Gambier Islands and out into the Strait of Georgia.  Some practical suggestions, regarding important weather dependent functions in Howe Sound, can be made. The Atmospheric Environment Service weather station which is located at Pam Rocks, just south of Anvil Island, is well situated to measure some of the more intense winds in the channel during an outflow event. The region surrounding Pam Rocks is generally occupied by strong supercritical flow, thereby producing wind reports which indicate a “worst case scenario”. The BC Ferries route, between Horseshoe Bay (on the east side of the mouth of Howe Sound) and Langdale (on the opposite side), follows a path around the north side of Bowen Island. During a severe event, this region is occupied by supercritical flow. The reports from Whailin (1993) indicate that winds increase substantially as the boat nears the northern tip of Bowen Island. This claim is  60  supported by the 3D model results which, in most cases (Figures 17, 18, 19, 20, 23, 24), indicate that the boat passes through a hydraulic jump into supercritical flow. Although it makes the trip further, an alternative route south of Bowen Island may be advantageous.  The 3D model results, although more detailed, verify the findings of the 1D study in general. The two sets of results effectively compare in a qualitative manner and show where controls and hydraulic jumps, and therefore supercritical regions, are likely to occur. Together the results provide a predictive tool for determining hazardous zones of extreme wind during an outflow event.  c. Recommendations forfurther research  The comparison between field results and 3D model results, discussed above in section 4.1.1, could be potentially very informative. Unfortunately the nature of the two sets of results makes them difficult to compare. The methods of determining average values for velocity, depth and Froude number from the raw field data completely eliminates the 3D characteristics of the actual flow. The difficulties in the interpretaion of the field data, as described in chapter 2, arose due to the lack of velocity data at each field station. Therefore, a further field investigation is recommended which should involve measuring static pressure as well as velocity at several locations throughout the wind field.  61  List of Symbols u c h g F p 1 F  fluid layer velocity long wave celerity layer thickness acceleration due to gravity Froude number fluid density internal Froude number stability Froude number  r’2 A  Appendices depth difference across hydraulic jump pressure p temperature T universal gas constant R p change in pressure across hydraulic jump q discharge per unit width mean channel width b specific energy E  hf  total fluid depth reduced gravity Subscripts upstream of hydraulic jump 1 external Froude number downstream of hydraulic jump 2 along channel coordinate pressure gradient horizontal length Reynolds number dynamic viscosity distortion coefficient slope hydraulic radius Manning roughness coefficient roughness element size dimensionless roughness coefficient pressure slope incident angle refracted angle index of refraction light sheet width total fluid discharge downstream fluid depth  Subscripts 1 2 m P r  upper fluid layer lower fluid layer model prototype prototype:model ratio  h = h, g’ F  +  2 h  xE  dP I dx L Re V  e S R K d  s 9’ n w  Q  62  Bibliography Armi, L., 1986. The hydraulics of two flowing layers of different densities. J. Fluid Mech., 163, 27. Baines, P.G., 1984. A unified description of two-layer flow over topography, J. Fluid Mech., 146, 127-167. Baines, P.G. and Davies, P.A., 1980. Laboratory studies of topographic effects in rotating and/or stratified fluids. In Orographic Effects in Planetary Flows, chap. 8, pp. 233-299. GARP Publication no. 23, WMOIICSU. Benton, G.S., 1954. The occurrence of critical flow and hydraulic jumps in a multilayered fluid system, J. Met., 11, 139. Bond, N.A. and Mackim, S.A., 1993. Aircraft observations of offshore-directed flow near Wide Bay, Alaska, Mon. Wea. Rev., 121, 150-161. Brinkmann, W.A.R., 1974. Strong downslope winds at Boulder, Colorado. Mon. Wea. Rev., 102, 592-602. Cameron, D.C. and Carpenter, A.B., 1936. Destructive easterly gales in the Columbia River Gorge, December 1935, Mon. Wea. Rev., 64, 264-267. Durran, D.R., 1986. Another look at downslope windstorms. Part I: the development of analogs to supercritical flow in an infinitely deep, continuously stratified fluid, J. Atmos. Sci., 43, 2527-2543. Finnigan, T.D., Vine, J.A., Jackson, P.L., Allen, S.E., Lawrence, G.A., Steyn, D.G., 1994. Hydraulic physical modeling and observations of a severe gap wind, in press, Mon. Wea. Rev. Finnigan, T.D., 1993. Microbarograph pressure recordings from a severe outflow event  in Howe Sound in December, 1992. U.B .C. Oceanography data report #60. Gibson, A.H., 1934. Hydraulic and its Applications, Constable and Company Ltd. London. Henderson, F.M., 1966. Open Channel Flow, Macmillan, New York. 63  Houghton, D.D and Kasahara, A., 1968. Non-linear shallow fluid flow over an isolated ridge, Commun. Pure AppL Math., 21, 1-23. Hunt, J.C.R. and Snyder, W.H., 1980. Experiments on stably and neutrally stratified flow over a model three-dimensional hill, J. Fluid Mech., 96, 67 1-704. Jackson, P.L. and Steyn, D.G., 1994a. Gap winds in a fjord, part I: Observations and numerical simulation, in press, Mon. Wea. Rev. Jackson, P.L. and Steyn, D.G., 1994b. Gap winds in a fjord, part II: Hydraulic analog, in press, Mon. Wea. Rev. Jackson, P.L., 1993. Gap winds in a fjord: Howe Sound, British Columbia, University of British Columbia, Ph.D. thesis. Lackman, G.M. and Overland, J.E., 1989. Atmospheric structure and momentum balance during a gap-wind event in Shelikof Strait, Alaska, Mon. Wea. Rev., 117, 18 171833. Lawrence, G.A., 1990. On the hydraulics of Boussinesq and non-Boussinesq two-layer flows, J. Fluid Mech., 215,457-480. Lawrence, G.A., 1993. The hydraulics of steady two-layer flow over a fixed obstacle, J. Fluid Mech., 254, 605-633. Long, R.R., 1953. A laboratory model resembling the “Bishop-Wave” Phenomenon, Bull. Am. Meteorol .Soc., 34, 205-211. Long, R.R., 1954. Some aspects of the flow of stratified fluid systems, Tellus, 6, 97-115. Macklin, S.E., Bond, N.A. and Walker, J.P., 1990. Structure of a low-level jet over lower Cook Inlet, Alaska, Mon. Wea. Rev., 118, 2568-2578. Mass, C. F. and Aibright, M.D., 1985. A severe windstorm in the lee of the Cascade mountains of Washington State, Mon. Wea. Rev., 113, 1261-128 1. Nicollet, G., 1989. River Models, In Recent Advances in Hydraulic Physical Modelling, chap. 2, R. Martins (Ed.), Kluwer Academic Publishers, 39-63.  64  Overland, J.E. and Walter, B.A., 1981. Gap winds in the strait of Juan de Fuca, Mon. Wea. Rev., 109,2221-2233. Pettre, P., 1982. On the problem of violent valley winds, .1. Atmos. Sci., 39,542-554. Reed, T.R., 1931. Gap winds in the strait of Juan de Fuca, Mon. Wea. Rev., 109, 23832393. Rottman, J.W. and Smith, R.B., 1989. A laboratory model of severe downslope winds, Tellus, 41A, 401-415. Schaeffer, G., 1975. Climatology. The Squamish River Estuary Status ofEnvironmental Knowledge to 1974, L.M. Hoos and C.L. Vold, Eds., Environment Canada. Schär, C. and Smith, R.B., 1993. Shallow-water flow past isolated topography. Part I: Vorticity production and wake formation, J. Atmos. Sci., 50, 1373-1400 Smith, R.B., 1985. On severe downslope winds, J. Atmos. Sci., 42, 2597-2603. Snyder, W.H., Thompson, R.S., Eskridge, R.E., Lawson, R.E., Castro, I.P., Lee, J.T., Hunt, J.C.R. and Ogawa, Y. The structure of strongly stratified flow over hills: dividing-streamline concept, J. Fluid Mech., 152, 249-288. Vine, J.A., 1992. Squamish Winds Research Project. Report EFM 92/02, Environmental Fluid Mechanics Group, Department of Civil Engineering, University of British Columbia. Whailin, C., 1993. BC Ferries Corporation, personal communication. Yalin, M.S., 1989. Fundamentals of Hydraulic Physical Modelling, In Recent Advances in Hydraulic Physical Modelling, chap. 1, R. Martins (Ed.), Kiuwer Academic Publishers, 1-37.  65  Appendix A: Calculation of Froude number, wind layer depth and velocity at each field station  i.) Description of the method  In order to examine the hydraulics of the wind layer in Howe Sound during the December, 1992 event it was necessary to convert the recorded absolute pressures and relative pressure changes along the channel to absolute layer depths. The absolute layer depth at one field station had to be determined in order to use the recorded relative pressure change between each station (assuming hydrostatic pressure distribution) to calculate absolute depth at each station.  The Atmospheric Environment Servic&s (AES) weather station at Pam Rocks is situated near our recording station at Lion’s Bay (station 4). Hourly pressure data acquired at Pam Rocks during the event were used to confirm our data at station 4. As well as pressure, the Pam Rocks station also records wind speed and air temperature. These data, along with our pressure data at stations 4 and downstream at station 5, were used to estimate the absolute layer depth at station 4 and therefore at all the stations.  The procedure involves the assumption that during Period 1 of the event the increase in pressure between stations 4 and 5 is due to the presence of a hydraulic jump. The steady pressure difference between the stations, of approximately I mb during Period 1 (Figure 9(a)), validates this assumption.  66  ii.) Method of solution  The hydraulic jump equation, as described in Henderson (1966) and expressed in terms of the upstream conditions, can be used to predict the depth of flow just downstream of a hydraulic jump. In its usual form, which assumes the channel is rectangular in section and neglects friction and synoptic pressure gradients, the equation appears as, (Al)  where F 1 is the Froude number upstream of the jump and h 1 and h 2 are the upstream and downstream depths, respectively. Expressing F 2 as u/g’h 1 1 and h 2 as h 1  +  Ah, where  Ah represents the change in depth across the hydraulic jump, and solving (Al) explicitly for h , leads to the following form of the hydraulic jump equation: 1 I  2  Ii’  2 2N  —I 3Ah—2- 1+111 3zh—2- I g’)  gJ  4  —8&? ,  (A2)  where the reduced gravity, g=(P2P1)g  P2  (A3)  is a function of the upper and lower layer densities which we will call p 1 and p , 2 respectively. Equation (A2) yields the depth of flow upstream of a hydraulic jump in terms of the upstream fluid velocity,  , 1 is  the depth change across the jump, zTh, and the  reduced gravity, g’.  67  The density can be determined from the ideal equation of state,  p=—--,  (A4)  where P is the pressure and T is the temperature in the particular layer and R is the gas constant for air. The data from Pam Rocks provides the temperature, T , in the lower 1 layer, whereas the temperature in the upper layer, T , is approximated by surface 2 temperature readings taken shortly before the onset of the event (before a lower layer of intruding cold air occupied the region). Only pressure in each layer is then needed to determine the densities and therefore the reduced gravity.  The data at station 4 gives us the pressure at the ground surface which, considering the relatively small vertical extent of the lower layer, approximates the pressure, P , in the 1 lower layer. Pressure in the upper layer (near the interface) is determined, assuming the pressure varies hydrostatically, from  = P — 2 g . 1 ’h P p  (A5)  This relation involves quantities described by the previous equations, which in turn rely on it.  The only remaining unknown in (A2) is Ah. This quantity is determined from the relative pressure change across a hydraulic jump, zIP, as recorded between stations 4 and 5, i.e.  g’ 1 p  68  (A6)  Now all quantities in (A2) have been determined and the system consisting of (A2), (A3), (A4), (A5) and (A6) can be solved iteritively for h . 1  iii.) Calculation of layer depth, Froude number and velocity at each station  Using the hydrostatic equation in the form of (A6) the layer depths at each station are found from the relative pressure data, once the absolute depth at station 4 is determined.  The Froude number can be expressed in terms of the flow rate per unit width, q, rather than the velocity, in the following manner,  F2=4_..  gh  The total flow rate,  (A7)  Q = qb = uhb,where b is the average width of the channel, calculated  at station 4 using the known velocity, depth and width there, is assumed constant throughout the channel. Since the average channel width, b, can be measured from topographic maps and the value of  Q is known at each station location, the value of q is  also known. This, coupled with the known depth at each station, gives the Froude number and the velocity at each station.  69  Appendix B: Calculation of transition to supercritical flow downstream of station 5 for Period 1  For the model results with flow rate A, the flow is predicted to transit from sub- to supercritical downstream of the location corresponding to the field station 5. The field results indicate subcritical flow at station 5 (F = 0.9), but no results are available downstream to directly confirm the model predicted transition.  A simple calculation, using the known conditions at station 5 and the extent of further contraction in the channel beyond station 5, reveals that transition is likely immediately downstream as predicted by the model. Here, we use the concept of specific energy, from hydraulic theory, to relate the flow between station 5 and the point of minimum channel width just downstream. Specific energy may be defined at a particular location in the flow by,  (Bi)  where synoptic pressure gradient is neglected and h is the depth of flow, q the flow rate per unit width and g’ the reduced gravity. Neglecting frictional losses between station 5 (which we’ll refer to as location 1) and the point of minimum width (location 2) the specific energy remains constant between the two points.  The specific energy at 1 is calculated to be E 1  =  1010 m which is also the value at 2. The  average channel width at 2 is reduced to approximately 60% of that at 1.  70  Therefore, the flow rate per unit width at 2 is,  =  0.6  =  22 x io  (B2)  —,  where the numerical values are determined from the calculations described in Appendix A. Inserting these values into (B 1) for position 2 and solving the resulting cubic equation 2 yields two alternative depths at position 2: one for subcritical flow and one for for h supercritical flow (see Henderson (1966)). The root corresponding to the supercritical depth of flow for the equivalent specific energy at 1 has the value y 2 lower than the depth at 1, y 1  =  =  673 m, which is  763 m. This state may be reached if the flow passes  through the point of minimum specific energy (critical point). We have now determined both q 2 and y 2 which, using (A7), gives the Froude number at location 2. The value obtained is F 2  =  1.6 indicating supercritical flow at that location.  The above calculation, from the field results, affirms the model prediction that a transition may occur from subcritical flow at station 5 to supercritical flow downstream before the channel terminus.  71  Input Variable synoptic pressure gradient  - dpi dx (Pa rn) 1  initial height - h (m) total discharge  is) 3 - Q (m  Period 1  Period 2  0.0121  0.0168  1200  1200  6.5 x  7.4 x 10  lower pot. temp.  - 8 (K)  272  265  upper pot. temp.  -  (K)  281  281  (-)  0.02 land  0.02 land  0.01 water  0.01 water  °2  drag coefficient - C  Table 1: Values of parameters as observed during Period] and Period 2 which were used in Hydmod comparisons.  Vertical Length  hr  = hr  Horizontal Length  Lr  = Lr  Mass  Mr  = Pr’’r  Velocity  Ur  =  2 2 h’ (g’/g)”  Time  Tr = LrU;’  =  112 Lrh; (gig’) 112  Discharge  Qr  Force  Fr = MrLrT; 2  Pressure  UrLrhr  = FrL 2  = (g’/g) 2 Lrh7 ” 1 2 = (g’/g)prLrh = (g’/g)prhr  Table 2: Scale ratios resulting from Froude number similarity between model and prototype. 72  Model Parameter Settin s Discharge (GPM)  Weir Height (mm)  Model Slope  Low (L)  6  32  0.039  High (H)  9  41  0.053  Corresponding Prototype Values Is) 3 Q (m  hf (m)  dP/dx (Pa/m)  Low (L)  1.72 x i0 7  965  0.0 123  High(H)  7 2.58x10  1212  0.0165  Table 3: 3D Model parameters and corresponding values scaled to prototype dimensions.  dP/dx  Max. Velocity (mis)  Max. Depth (m)  Max. Froude Number  Total Supercritical Area (km ) 2  L  L  22.6  757  3.6  116  H  L  L  25.1  943  3.9  123  3 4  H  L  H  28.6  844  6.0  257  L  L  H  25.5  696  4.8  242  5  L  H  L  16.8  1046  7.0  41  6  H  H  L  19.8  1044  5.3  53  7  H L  H H  H  25.8  937  8.2  162  H  24.4  886  11.1  178  Case  Q  hf  1  L  2  8  Table 4: Model parameter settings and some important physical aspects of the results for the eight simulated cases. L refers to the lower setting and H to the higher.  73  3D Model-Case 6  Field Results-Period]  station  depth (m)  Froude number  depth (m)  Froude number  1  560  0.79 (sub)  1290  0.55 (sub)  2  605  0.64 (sub)  980  0.63 (sub)  3  275  1.4  (sup)  980  0.55 (sub)  4  270  0.22 (sub)  598  1.14 (sup)  5  680  0.05 (sub)  763  0.95 (sub)  Table 5: Case 6 of 3D model results compared with Period] of field study results.  3D Model-Case 7  Field Results-Period 2  station  depth (m)  Froude number  depth (m)  Froude number  1  519  0.32 (sub)  1289  0.63 (sub)  2  222  1.23 (sup)  857  1.18 (sup)  3  292  1.3  (sup)  1117  0.55 (sub)  4  399  2.33 (sup)  598  1.58 (sup)  5  325  0.44 (sub)  598  2.10 (sup)  Table 6: Case 7 of 3D model results compared with Period 2 of field study results.  74  Figure 1: Schematic representation of outflow wind system. Isobars indicate the synoptic pressure gradient which drives the flow. Low lying cold air flows through a partial mountain barrier from the interior region towards the oceanic zone. 75  Figure 2: Geographical location and important features of region surrounding Howe Sound. 76  STRAIT OF GEORGIA  Figure 3: Howe Sound; locations and topography. Instruments positioned in Howe Sound are numbered 1 through 5 in the direction of flow. 77  20cm!  -;;\  U,’.  section  I  I 72cm  I model channel I  I I  plan  1 5 cm  video camera  Figure 4: Schematic drawing of experimental apparatus used for 2D physical modeling of outflow winds. The line marked with the symbol V indicates a simulated water level in the vertical section. 78  0  3  :I .  60km  (b) Howe Sound  (a) Model  Figure 5: Channel axis shown with respect to the model of Howe Sound (a) and Howe Sound (b). Distance increases in the direction of flow and the field station locations are labeled 1 through 5. 79  I  2500200015004-I  Q)  D  -  5000•  Co  a) a) Co  a,  E a)  t5 0 I-I-  Distance along channel N-S (km)  Figure 6: (a) Depth, (b) wind speed, and (c) Froude number along Howe Sound: as predicted by the physical model for flow rate A (thick line); as measured during the December, 1992 outflow event for Period 1 (squares); and as produced by the numerical model Hydmod (thin line) for Period]. Flow is from left to right and a point of hydraulic control exists wherever the F = 1 line is crossed from less than 1 to greater than 1 (i.e. sub- to supercritical flow). 80  E  250020001  gg: 5000-  U)  E G) U, -  1.  a)  10  4  -D  E  a) -  0 LL  0  I  I  I  10 40 20 30 50 Distance along channel N-S (km)  60  Figure 7: (a) Depth, (b) wind speed, and (c) Froude number along Howe Sound: as predicted by the physical model for flow rate B (thick line); as measured during the December, 1992 outflow event for Period 2 (squares); and as produced by the numerical model Hydmod (thin line) for Period 2. Flow is from left to right. 81  1040Period 2  1030  101 j :ed1  1000990  • Dec25 Dec26 Dec27 Dec28 Dec29 Dec 30 Dec 31 I  0  24  48  72  Jan 1  Jan 2  11I1III11111111I11I1I1  96  Jan 3  I.’.  120 144 168 192 216 240  Hours  Figure 8: Composite chart of atmospheric pressures recorded at 5 stations in Howe Sound over a 9 day period. Strong gap winds occurred from 27 December to 01 January. Individual pressure tracings are shown more clearly in Figure 9. 82  Period 1  0—  i=1  -6— -7  -6  (a) 60  I  62  64  -7  I  66  68  Hours  Period 2  0-  70  72  1=1  —  (b) I  96  98  I  100 102 104 106 108  Hours  Figure 9: Relative pressures, with respect to that at station 1 (P1), at each of the five field stations in Howe Sound (i= 1 through 5) for (a) Period] and (b) Period 2. 83  Figure 10: Photograph of 3D model topography. View from downstream end. 84  00  0  I-  0  CD  C)  0  _•  0  -  0  CD  C)  C)  0  CD  0  0 cq  Q  CD  0  CD  Z  -  0  0  —  0  0 C) G CDCD  CD  I-..  I-.  CD  CD  Tj  3 3  CD CD  cd  -  C, 0  rit  A  flow direction  rail  (a)  (b)  Figure 12: Schematic view from above of the video apparatus cart and rail system (a) and schematic side view of complete 3D model equipment set up (b). 86  Figure 13: Photograph of 3D model and video apparatus. 87  light sheet  w  Figure 14: Definition sketch for depth data aquisition. Fluid surface is indicated by V. 88  distance across section (cm) (a)  020406080100  distance across section (cm)  (b)  Figure 15: Plots for validation of video techniques for depth data aquisition. Figure shows depth across a 10cm flow section as measured using the video process (solid line) and a manual point gauge (solid squares) for (a) a subcritical section, and (b) a partially supercritical section. 89  “S  28 -  -  Anyd Island  -  ‘‘-  26 (km)  26 (km)  -  -  Pam Rocks  Gambiec  Islaac  L  18 -16 14 5• HorseshQE Bay  -  -  -  -  -  -  - - Bawen -  tsland  -  -12 ‘10 -8  -  ‘6  0  2  I  4  6  -4  and over 9ø m  S TRAIT OF GEORGIA I  •  I  ‘2 •  I  -o  I  8 10121416  0  2  4  6  I  I  I  8 10121416  (km)  (km)  (a)  (b)  Figure 16: Reference diagram for 3D model results (a) and topography of region covered by model results (b). 90  :  441 \  \ ‘  .14 .,1  j 1 lS  ‘‘I,  -  -  •  •  +  f I I  •  ) •.-•  2 =  —  IIIl  •  I  •  I 4 I,  •  I  41’’  I I I 44 1 4 I I I  j  4  I  6  fo&oI  —8 —4  4  41  (a) 0  ‘.  II  (b)  I I ‘ 1 8 10121416 0 ‘‘  .  2  I  4  •  6  I  I  •  8 10121416 0  2  I  4  ‘  6  I  •  -  Q=L, h=L, dP/dx=L  (a) Horizontal velocity vectors. Reference arrow represents 30 m/s windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=l (critical flow), F=3 and F=5. 91  I  ‘  I  8 10121416  distance (km)  30  Figure 17: 3D Model. Case 1  -2  (c)  ‘ii 1,  •  ‘Sf  f  f.  ‘,,  .•!,  i’  — .5  •  —  .•  -  -  -  ti.  • v  •  .‘  .  •  .  S  ,  .  , •  .  . ‘  •  .  •  •  .  •-...  S •  •  .  • .  .  -10  ‘I  Ik kf 5  .,, ‘S  6  —4  S  5’.  .5,  -6  .  •  S.,  —8  Q  •  S  —  —12  ‘fi.  .  5,  o 2 4  -14  \  S  5  a)  ‘  if  8 10121416  02  I 4  6  I I 8 101214160  ‘  2  I 4  ‘  6  —0 I ‘ I ‘ I 8 10121416  distance (km)  = 30  Figure 18: 3D Model. Case 2  -2  (c)  (b)  -  Q=H, hf=L, dP/dx=L  (a) Horizontal velocity vectors. Reference arrow represents 30 mIs windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=1 (critical flow), F=3 and F=5. 92  .:.E1  a a  /  ‘  ( -\  \  \  \\  ‘.14  \\  SS  I  0  /.:/ ‘f*  I  a  4  1  S  ‘  ‘  S  ‘  I / / / ‘  1  ‘‘-/  j  I  f\ \  \ k  t  t I  1 Itl// ‘l.:t/,j?..I SSf//S’.535EI.  •14 +  -12 r—  -8  s  6  fS  SSS’fV1,S5l,. SSSS  10  \  55.55555  ,5  \‘  oo  (b)  (a) 02 —-  =  p 4  I  6  I I 8 10121416 0  •  2  I  4  6  I  ‘  I  8 101214160  (c) ‘  I  2 4  6  I  ‘  -  Q=H, hf=L, dP/dx=H  (a) Horizontal velocity vectors. Reference arrow represents 30 mIs windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=l (critical flow), F=3 and F:=5. 93  I  ‘  I  8 10121416  distance (km)  30  Figure 19: 3D Model. Case 3  6OO  -4  C  Si  ‘2 -0  ‘  \\‘  ‘I 4 _...lI-l4  i4  :J  1  44  \‘  oj  k\  Uiflfth  ‘:  +1  ‘-S  S  1  -  •  414444 1\\\ 4 1 / t114 \ 4  I/’y’ I LI III ‘fI’ ‘I I •:-‘.  ——I •  —  •  •1IlI,._ •  I  I  I  I  It  I’  I,,  I’  (a ‘  I  0 2 4 =  ‘  6  ‘—  •—  I  4 4  41 4 1  ‘141 I  •14 11 4  -12 •10  I  -8  .41 I  II I 4 •1  6  I  4’’  4’’ 4’’ II 4441  -4  I ‘ I I 8 101214160  2  I 4  I I I 8 101214160 ‘  6  2  I 4  6  I I I 8 10121416  distance (km)  30  Figure 20: 3D Model. Case 4  2  (c)  (b  -  Q=L, hf=L, dP/dx=H  (a) Horizontal velocity vectors. Reference arrow represents 30 m/s windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=1 (critical flow), F=3 and F=5. 94  -0  ‘—I, ‘‘‘/,  I’  f14f  I,,,  f  • :w.  400  I  I I —  •  ‘ 2 =  I  I  II  10  II  I  If  I.  I  -8 0  I..  I  If  III  I  (a) 0  I  1k  I III  1111  I 4  II  ‘  6  I  1Q00  (b)  I I I— I 8 10121416 0  ‘  2  6  I I 8 101214160  -4  (c) ‘  2  I 4  ‘  6  ‘ I I I 8 10121416  distance (km)  30  Figure 21: 3D Model. Case 5  0  I 4  6  800’  -  Q=L, hf=H, dP/dx=L  (a) Horizontal velocity vectors. Reference arrow represents 30 m/s windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=l (critical flow), F=3 and F=5. 95  2 -0  50 0  it  i1 i’’ ‘11 I  ‘  /  I  I ‘ i  fl\  Ik\  ‘I  ‘I ‘I  ‘‘‘(‘‘S  5’,’’’’  •  -12 10  5’  -8 ‘6  ‘‘S  ‘ ‘ ‘S  \  (a) 0  2 =  I  4  6  I  I  H  ‘S  -4  (I \ \1 II  I  8 101214160  ‘  2  4  6  I  I  I  8 10121416  •  I  0 2 4  I  6  I  -  Q=H, h=H, dP/dx=L  (a) Horizontal velocity vectors. Reference arrow represents 30 m/s windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=1 (critical flow), F=3 and F=5. 96  I  ‘  I  8 10121416  distance (km)  30  Figure 22: 3D Model. Case 6  I  2  (c)  1 OOO I  -0  I,,’.-.  ‘1 14  41  \\  ‘  \  I.  \  ‘I —‘I  -  I  ‘1 .1, ‘It •.l li..14•  I  4.  \\  ‘‘I  --  \\  l  -  -  ‘/ II’ 4 . ‘4 4.  I  -,  I  .4 I  -—‘  -  I  3  I,  •!‘‘  I  4’.  ‘  4  I  I  ‘  4  o  2 =  I 4  6  oo  I  I I 1 4 4 1 I 4 ‘‘I’’’’  I 4 4 • ‘44444’ .44444’  -8 6 -4  J6O9  Ill  ‘‘‘4444’ I I III 4’  (a)  -  1 4 4 4  ‘‘114  ‘  -  44441  .444.  •  I  8OO-  I ‘ I I 8 10121416 0  ‘ 2  6  I I I I 8 101214160  ‘  2  I 4  6  I ‘ I ‘ 8 10121416  distance (km)  30  Figure 23: 3D Model. Case 7  I 4  -2  (c)  -  Q=H, hf=H, dP/dx=H  (a) Horizontal velocity vectors. Reference arrow represents 30 mIs windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=l (critical flow), F=3 and F=5. 97  -0  52  50  -. ‘1/ lii, ‘‘‘S  ‘  ‘  \\\  \ \ \.  14 I  -.  4  -‘  1—  ‘I’ I  ‘4  -  %\‘  \\\\  —16  iii..\\\\\\. • •-5 \ I I I ‘III ‘ \ I I I -‘ --  14 —12  I\\,  .10  ‘-.‘\‘ I  -  \  -8  6 I III I I  (a)  III  024 —p-  =  8OOrZ\  6  I ‘[ 8 10 121416 0  I  2  I  6  I J ‘ J 8 101214160  I  2  J  4  I  6  -  Q=L, hf =H, dP/dx =H  (a) Horizontal velocity vectors. Reference arrow represents 30 m/s windspeed. (b) Depth contours. Interval is 200 m and the 0 m contour is not shown. (c) Froude number contours for F=1 (critical flow), F=3 and F=5. 98  J  I  J  8 10121416  distance (km)  30  Figure 24: 3D Model. Case 8  J 4  2  (c)  -o  ç2() ‘V.  4H-  \  .\  N  ..  _  .  ...  \  rv  .,....  —  \__  —  (a)  -  (b)  Figure 25: Microbarograph pen tracings from 28 December 1992 for stations at Porteau Cove (a) and Lions Bay (b). The relatively erratic pen behaviour at Porteau Cove supports the existence of a strong hydraulic jump that is present in most model simulations. 99  


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