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Implications of non-uniform wind stress on lake circulation : with application to Quesnel Lake, B.C. North, Ryan P. 2006

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Implications of Non-Uniform Wind Stress on Lai Circulation With Application to Quesnel Lake, B.C. by R y a n P . Nor th B . S c , Queen's University, 2002 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( C i v i l Engineering) T h e Univers i ty of B r i t i s h C o l u m b i a January 2006 © R y a n P . N o r t h 2006 1 1 Abstract In hydrodynamic modeling of lakes, w i n d stress is often assumed uniform over an entire water body. T h i s is not always an accurate representation of the w i n d stress since it does not consider wind dis t r ibut ion inhomogeneities or the effect of wind-waves. These factors modify wind stress at the air-water interface and mix ing induced in the surface waters. Th i s thesis focuses on the influence of wind-waves on wind stress dis t r ibut ion and discusses the implicat ions for wind-driven circulat ion in stratified water bodies. Significant wave height throughout the basin is estimated using fetch and du-rat ion l imi ted equations. W i n d stress is then calculated from wind speed and a modified coefficient of drag. The latter is related to the significant wave height v i a a surface roughness formulation. T h i s accounts for variations in wind stress due to wave fetch and durat ion. W i n d stress estimated wi th and without the influence of wind-waves is applied to an analyt ical solution for wind-driven circulat ion in a simplified lake. The comparison demonstrates the significance of wind-waves on lake circulat ion. Results indicate an increase i n internal seiching and surface layer currents w i th the inclusion of the surface roughness height. Further testing of the influence of wind-waves is conducted on Quesnel Lake, B . C . Comparisons using an idealized wind (uniform and steady) once again indicates an increase in the internal c irculat ion when accounting for surface roughness. The importance of properly quantifying the w ind speed and direction is also discussed. I l l Contents Abstract i i Contents i i i List of Tables . v List of Figures v i List of Symbols ix Acknowledgements x i 1 Introduction 1 2 Literature Review 4 2.1 Spat ia l and Temporal Var ia t ion of W i n d Speed 5 2.2 Lake Wind-Waves : . . , 7 2.2.1 Computer Mode l ing of Wind-Waves 8 2.2.2 Fetch and Dura t ion L i m i t e d Wind -Wave Equat ions . . . . . . 10 2.2.3 Simplif icat ion for Smal l to M e d i u m Sized Lakes 15 2.3 Surface Roughness Height 16 2.4 Drag Coefficient 19 2.5 Hydrodynamics of Lakes - W i n d Induced C i rcu la t ion 20 2.6 Hydrodynamic Mode l ing Improvements . 23 3 Application to a Linear Hydrodynamic Model 25 3.1 Es t imat ing the Wave Parameters 25 3.2 Hydrodynamics • • • 27 3.3 Analys is 29 3.3.1 W i n d Stress Dis t r ibu t ion 30 3.3.2 Bas in Ci rcu la t ion 31 3.4 Summary , 32 iv 4 Application to Quesnel Lake 34 4.1 Quesnel Lake 36 4.2 Numer ica l Simulations 37 4.3 Es t ima t ing Wave Parameters 40 4.4 Analys is 41 4.4.1 Surface Roughness Height Compar ison 42 4.4.2 Bas in Length and Fetch 52 4.5 Summary 53 5 Conclusions and Recommendations 55 Bibliography 59 Appendix A - Analysis of Heaps and Ramsbottom (1966) 64 Appendix B - Wind-Wave Equations 73 List of Tables 2.1 Equations for Fe tch-Limi ted Growing Seas. Where tX)U [s] is the durat ion of the w i n d after which waves become fetch-limited, X [m] is the fetch, Uio [ rns - 1 ] is the wind speed, Hs [m] is the significant wave height, Tp [s] is the peak period which corresponds to the frequency of the wave energy spectrum's peak, i i * [ rn s - 1 ] is the friction velocity and g [ms~2] is the acceleration due to gravity VI List of Figures 2.1 Compar ison of Fe tch-Limi ted W i n d - W a v e Equat ions. C E M : F r o m the Coasta l Engineering M a n u a l (Resio et al, 2002), using fetch ( C E M ) ; J O N S W A P : A s outl ined in Wuest and Lorke (2003) and Csanady (2001) based on fetch ( J O N S W A P ) and wave age ( J O N -S W A P (cp/C/io)); and Carter: F r o m Carter (1982). The solid lines are for a 2 m / s wind speed and the horizontal dashed lines are for a 5 m/ s wind speed 13 2.2 W i n d - D r i v e n Ci rcu la t ion in a Lake wi th one (a.,b.,c.) and two layers (d.). A s the wind begins to blow (a. to b.) water builds up at the downwind end (b.). T h i s results in internal c irculat ion and surface seiching ( c ) . For the two layer case the circulat ion occurs in both layers and seiching occurs at the surface and interface (d.) 21 3.1 F low diagram showing progression of wave development. Where tx,u [s] is the durat ion of the wind , X [m] is the fetch, Uio [ms"1} is the wind speed, Hs [m] is the significant wave height, Tp [s] is the peak period which corresponds to the frequency of the wave energy spectrum's peak, [ m s _ 1 ] is the friction velocity and g [ms~2] is the acceleration due to gravity (Resio et al, 2002). F low chart adapted from Carter (1982) 26 3.2 Layout of the basin used in Heaps and Ramsbottom (1966) 30 3.3 Drag Coefficient comparison w i t h and without accounting for surface roughness for w i n d speed dis t r ibut ion given by Equa t ion 3.2. E a c h plot is for a part icular m a x i m u m wind speed 31 3.4 The impact of considering wind-waves on internal seiching. Shown are the a) ampli tude and b) offset of periodic seiching, rj2, at the downwind end of the basin, and c) the ampli tude of the depth aver-aged horizontal velocity, u\, at the centre of the basin 33 4.1 Quesnel Lake temperature recordings at Loca t ion M 2 (West Basin) and M 8 (Lake Junction) as shown in Figure 4.2. T h e West Bas in shows strong temperature fluctuations in comparison wi th the rela-tively ca lm conditions in the ma in por t ion of the lake 35 V l l 4.2 M a p of Quesnel Lake wi th major rivers indicated by arrows and the town of L ike ly w i th a dot. The West Bas in , Junct ion and Nor th , East and West A r m s are labeled. Numbered locations on the lake mark the locat ion of thermistor chains. East-West and Nor th-South Thalwegs are also provided. Modif ica t ion of the map found in Potts (2004) 36 4.3 Plots of max imum fetch, max imum basin length, average friction velocity and seiching ampli tude wi th wind direction using Polar (a.) and Cartesian (b.) coordinates 42 4.4 Simulated isotherms of Quesnel Lake over a 7 day period, for a w ind speed of 5 m / s from 270° at T - C h a i n 2 (M2) and 8 ( M 8 ) . Compar ison of results from a wind stress using a constant C i n ( R E G ) , th in line, and a wind stress which accounts for z0 ( C E M ) , thick line. The upper contour line represents a temperature of 1 5 ° C followed by 10°C, 7 ° C , and 5 ° C (if visible) 43 4.5 Compar ison of basin scale movements w i t h and without accounting for surface roughness. Panels represent progression of isotherms ( °C) along Quesnel Lake's East-West thalweg for w ind speed of 5 m / s from 270°. The top panels are snap shots after 1 day, the middle panels after 3 days and the bot tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have 45 4.6 Compar ison of basin scale movements w i th and without accounting for surface roughness. Panels represent progression of isotherms (°C) along Quesnel Lake's South-North thalweg for w ind speed of 5 m / s from 270°. The top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have 46 4.7 Simulated isotherms of Quesnel Lake over a 7 day period, for a w ind speed of 5 m / s from 250° at T - C h a i n 2 (M2) and 8 (M8) . Compar ison of results from a wind stress using a constant C\Q ( R E G ) , th in line, and a w ind stress which accounts for z0 ( C E M ) , thick line. T h e upper contour line represents a temperature of 1 5 ° C followed by 10°C, 7°C, and 5 ° C (if visible) 47 4.8 Compar ison of basin scale movements w i th and without accounting for surface roughness. Panels represent progression of isotherms (°C) along Quesnel Lake's East-West thalweg for w i n d speed of 5 m / s from 250° . T h e top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have 47 V l l l 4.9 Compar ison of basin scale movements w i th and without accounting for surface roughness. Panels represent progression of isotherms (°C) along Quesnel Lake's South-Nor th thalweg for wind speed of 5 m / s from 250°. The top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have 48 4.10 Simulated isotherms of Quesnel Lake over a 7 day period, for a wind speed of 5 m / s from 340° at T - C h a i n 2 (M2) and 8 (M8) . Compar ison of results from a wind stress using a constant C i n ( R E G ) , th in line, and a wind stress which accounts for za ( C E M ) , thick line. The upper contour line represents a temperature of 1 5 ° C followed by 10°C, 7 ° C , and 5 ° C (if visible) 49 4.11 Compar ison of basin scale movements wi th and without accounting for surface roughness. Panels represent progression of isotherms ( °C) along Quesnel Lake's East-West thalweg for wind speed of 5 m / s from 340°. The top panels are snap shots after 1 day, the middle panels after 3 days and the bot tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have 50 4.12 Compar ison of basin scale movements w i th and without accounting for surface roughness. Panels represent progression of isotherms ( ° C ) along Quesnel Lake's South-Nor th thalweg for wind speed of 5 m/ s from 340°. The top panels are snap shots after 1 day, the middle panels after 3 days and the bot tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have 51 4.13 Compar ison of Drag Coefficient along the a.) East-West and b.) South-Nor th thalwegs. Values are taken at the completion of the simulation and account for surface roughness height due to wind-wave conditions ( C E M simulations. In comparison the R E G simulations used a D r a g Coefficient of 1.4 x 10~ 3 52 A . l Layout of the basin used in Heaps and Ramsbottom (1966) . 64 I X L i s t of Symbol s The following is a list of symbols used in this thesis. Symbols from the L a t i n and Greek alphabets are listed separately. Latin Characters A Constant used in determining surface roughness height An The m a x i m u m ampli tude of the shear stress, kgm~ls~2 An The average shear stress ampli tude, kgm~ls~2 B Constant used i n determining surface roughness height C The internal wave speed, ms~l C i o Drag coefficient corresponding to a wind speed U\Q C i n e A constant drag coefficient C\oi T h e estimated in i t ia l drag coefficient Cz D r a g coefficient corresponding to a wind speed Uz D The effective depth, m H The total lake depth, m Hs T h e significant wave height, m H(t) The Heayside step function K\ The vert ical eddy viscosity for the surface layer K2 T h e vert ical eddy viscosity for the bot tom layer L T h e basin length, m Lp The wavelength corresponding to the peak period, m T The wave period, s ^ The peak period which corresponds to the frequency of the p wave energy spectrum's peak, s Uio W i n d speed measured 10 m above the water surface, ms~l Umax T h e m a x i m u m ampli tude of the wind speed, m s - 1 T T W i n d speed measured a height z above the water surface, - 1 ms ^ The wind fetch. T h e distance of water over which the wind has blown, m cp The phase speed of the dominant waves, m / s g The gravitat ional acceleration [9.81 ms~2] g' The reduced gravity, ms~2 h\ T h e depth of the surface layer, m A positive integer which provides spatial variance in the x-direction for the shear stress X The wind durat ion. The t ime elapsed from the beginning of a w ind event, s tx<u The durat ion after which waves become fetch-limited, s u\ The horizontal velocity in the surface layer, m s " 1 U2 The horizontal velocity in the bo t tom layer, ms~l i t * The friction velocity, ms~1 w\ The vert ical velocity in the surface layer, ms"1 W2 The vert ical velocity in the bo t tom layer, ms~l x The distance downwind of the shore, m z The height above the water surface, m zch The Charnock parameter typical ly [ 0.0015] z0 T h e Surface roughness height, m Greek Characters Pi T h e wave speed of the surface seiche, ms~l Pi T h e wave speed of the internal seiche, m s - 1 A t T h e model t ime step, s Ax T h e grid spacing in the x or y direction, m T h e displacement of the surface relative to the unforced state, m T h e displacement of the thermocline relative to the unforced m , , state, m K von Karman ' s constant [0.4] pi The surface water density, kgms~2 P2 T h e bo t tom layer water density, kgms~2 Pa Densi ty of air, kgms~2 p0 The density of water, kgms~2 W i n d stress applied at the surface of the water body, kgm~ls~2 Tmax T h e m a x i m u m wind stress, equivalent to An, kgm~ls~2 W i n d stress applied through the forcing of water i n the sur-T S B L face boundary layer, kgm~1s~2 W i n d stress applied to the formation, maintenance and ac-w a v e celeration of wind-waves, kgm~1s~2 X I Acknowledgements I would like to thank two groups of people, those who have assisted academically and those who have provided personal support. I wish to thank my family first, Sonja, M o m , Dad , Becky and Travis, a l l of who have helped me in so many ways. I wish also to thank my advisors D r . Bernard Lava l and D r . Michae l Isaacson for their support, advice, and most impor tant ly willingness to let me perform my research as I felt necessary. I would also like to thank my co-workers or classmates who made the whole experience enjoyable. Chapter 1 Introduction i Increasing awareness of the potential impacts of global cl imate change have led to uncertainties concerning the longevity of the world's freshwater supply. These uncertainties demonstrate the increased need for understanding the world's bodies of freshwater. In lakes chemical, biological and physical processes are important and interdependent aspects. In part icular properly interpreting lake hydrodynamics (movements of water) enables forecasting of lake mix ing and transport. These predictions are applicable to a wide range of issues including pol lu t ion control, fish spawning and future climate change impacts. Ga in ing insight into lake hydrodynamics begins w i th in-si tu observations and data collection. A p p l y i n g the obtained temperature and meteorological da ta to nu-merical models is often the next step in understanding lake hydrodynamics . A 3 -D Hydrodynamic model is able to provide water temperatures and currents throughout the lake basin w i t h spatial and temporal variat ion. The surrounding environmental conditions (e.g. longwave and shortwave radiat ion, inflows and outflows and wind over the waters surface) are considered and a prediction of the physical processes occurring wi th in the lake are provided. In addit ion, understanding the hydrody-namics is essential for predictions of pollutant transport and dispersion as well as determining meteorological impacts on the lake ecosystem. Due to computat ional l imitat ions, lake processes are often simplified or general-ized when applied through a numerical model . In hydrodynamic modeling of lakes, w ind stress is one of the main forcings of lake circulat ion, yet is often assumed to be 2 uniform. W i n d stress is created as the wind blows across the water surface resulting in a surface flux of momentum. Measur ing the w i n d stress is difficult and is often empir ical ly determined from the easier to measure wind speed i n combinat ion w i t h a dimensionless drag coefficient, typical ly formulated as: T = CzPaU2z [kgm-h-2] (1.1) where r is the wind stress applied at the surface of the water body, Cz [—] is the constant drag coefficient corresponding to a w i n d speed Uz [ m s - 1 ] , measured a height z [m] above the water surface (usually 10 m), and pa [kgm~3] is the density of air. F r o m this point forward it w i l l be assumed that the wind speed is measured at a height of 10 m, and w i l l be referred to as ( T i n -T y p i c a l l y the drag coefficient is assumed constant. In addit ion, due to l imi ted wind data a uniform wind speed is often applied across the entire water surface. Thus the resulting wind stress is also assumed uniform. Th i s assumption of unifor-mi ty does not adequately represent the fact that the wind stress is applied to the water not only through the forcing of water in the surface boundary layer ( S B L , range of a meter to several tens of meters), TSBL, but also through the formation, maintenance and acceleration of wind-waves, rwave. Conservation of momentum at the interface requires the total wind stress equal the sum of the momentum fluxes in the water: T = TwaVe + TSBL [kg/ms2] (1.2) which highlights the waves as a second path for momentum transfer from the air to sea. Thus, the wind stress is a function of both the wind speed and the presence and state of wind-waves. Wind-waves can be represented through the surface roughness height, z0 [ m s - 1 ] ; z0 is the roughness the wind feels as it blows across the water 3 surface, and for lakes is mostly a function of wind-waves. Surface roughness height in tu rn can be related to the drag coefficient when the momentum equation is applied at steady-state w i th a logari thmic velocity profile leading to a log-law equation (Jones and Toba, 2001): where K is von Karman ' s constant taken as 0.4 and z [m] is the height above the surface. A s wind speeds are measured at 10 m the drag coefficient w i l l be referred to as C io -T h e main focus of this thesis is to investigate the representation of wind stress for hydrodynamic model applications. Comparisons of modeling wi th and without consideration of the surface roughness height w i l l highlight the importance of wind-wave effects. The following section provides a review of past research related to this investigation. Account ing for surface roughness height is then applied to a simple 2-D model to test its appl icabi l i ty (Chapter 3), followed by application to a test case (Quesnel Lake, Br i t i sh Columbia , Canada) using a 3D hydrodynamic lake model (Chapter 4). Results and implicat ions are summarized and discussed i n Chapter 5. (1.3) Chapter 2 Literature Review 4 Currents, surface movements and interactions between internal layers are a l l exam-ples of the physical processes occurring in lakes. These processes are driven by long and shortwave radiation, inflows and outflows (surface and ground water), and wind blowing across the air /water interface. A s a l l the aspects of lake hydrodynamics cannot be discussed here, only those relevant to the thesis are considered, namely wind-wave interaction at the ai r /water interface and wind-driven lake circulat ion. The reader is directed to the recent work of Wuest and Lorke (2003) for a more in depth review of small-scale lake hydrodynamics and suggestions for further reading. In addit ion, the works of Potts (2004) and James (2004) are recommended for an i n depth review of research conducted on the test case, Quesnel Lake. Wind-d r iven circulat ion develops from wind blowing across the water surface resulting in a momentum flux across the ai r /water interface creating a wind stress. Proper quantification of wind stress leads to the determination of the magnitude of momentum transfer into the S B L and the resultant wind-induced circulat ion. Methods of determining wind stress are discussed in Sections 2.1 to 2.4 for cases wi th and without consideration of wind-wave effects. In i t ia l ly proper determination of a spatial ly and temporal ly varying wind speed is discussed (Section 2.1). Methods of determining wind-wave characteristics (Section 2.2) and surface roughness height (Section 2.3) are then compared. F ina l ly , the drag coefficient is reviewed for several constant cases and a wind-wave dependent case (Section 2.4). W i t h an acceptable wind stress determined the resultant wind-induced circulat ion is described (Section 5 2.5) and a brief review of s imilar improvements to hydrodynamic models is provided (Section 2.6). 2.1 Spatial and Temporal Variation of Wind Speed Accurate representation of wind stress to determine wind-induced lake circulat ion requires characterization of spatial and temporal variat ion of the wind speed dis-t r ibut ion. In 3D hydrodynamical simulations of Lake Kinnere t , Israel, Laval et al. (2003) found that accounting for a spatial ly varying w i n d speed improved their abi l i ty to simulate basin scale seiching. In addit ion, the authors felt that further improvements to the wind stress calculations could be achieved if spatial variabi l i ty were applied to the drag coefficient. Rueda et al. (2005) studied a mult i-basin lake and found significant variabi l i ty in wind speed due main ly to the surrounding topog-raphy. Account ing for spatial variations improved the modeled exchange rates and residence times of the basins. Sarkkula (1991) numerical ly modeled wind-induced flow fields in a shallow lake using a uniform and non-uniform wind stress. W h e n compared wi th field measurements on the lake, only a non-uniform wind field pro-vided quali tat ively and quanti tat ively accurate results of lake circulat ion. W i t h o u t spatial variat ion of wind speed, the model d id not produce the observed upwind flow along the shallow shore. Similar to lakes, estuaries are also subjected to local generation of wind-waves which play an important role in transport and mix ing (Smith et ai, 2001). Smith et al. (2001) investigated wind-wave development across a large estuary and determined four complications to wave development in estuar-ies. These were: (1) surrounding topography; (2) t ida l ly varying depth; (3) t idal ly varying currents; and (4) non-stationarity due to the changes i n water depth over the t ime required for waves to propagate across the estuary. A l l of these examples demonstrate the var iabi l i ty in wind speed across a body of water and wi th time. 6 Determining the spatial dis t r ibut ion of the wind speed and direction is ideally accomplished by recording wind speed and direction across the entire lake surface. Unfortunately, this is rarely feasible and wind data is instead obtained from one or two locations on the lake or a nearby location. L i m i t e d wind data often results in the assumption of uniform wind across the lake. If da ta is obtainable at several locations, spatial averaging of the wind speed may be sufficient to provide accurate s imulat ion of lake hydrodynamics. In the previously mentioned work of Laval et al. (2003), the authors compared hydrodynamic simulations of Lake Kinnere t using uniform wind speed based on 1) a horizontally averaged wind stress from several w ind measurement stations and 2) measurements at the lake centre. B o t h wind fields reproduced isotherm displacement, however only the former accurately mod-eled internal wave motions (Laval et al., 2003). In addit ion, proper s imulat ion of surface layer motion required a spatial ly varying w i n d field (Laval et al, 2003). In the current work a spat ial ly varying wind stress is applied based on spatial varia-tions in wind-wave state and a uniform wind speed. It is hypothesized that this method w i l l provide similar accuracy to the method of Laval et al. (2003) for a lake w i t h l imi ted wind field data. Tempora l d is t r ibut ion of w ind speed and direction is also a highly variable and important consideration. In part icular for lakes, the variat ion in w ind speed and direction over t ime is highly discontinuous. Findikakis and Law (1999) successfully simulated the mix ing of reservoir surface layers using the net energy input of the wind for each t ime step by averaging the wind speed cubed and drag coefficient over the t ime step. T h i s form of temporal averaging reduces computat ional expense without sacrificing the importance of w ind variabil i ty. A s discussed in the following sections a uniform wind speed w i l l be applied in the current work. Due to a lack of a available wind data the wind field cannot 7 be characterized adequately enough for a realistic s imulat ion. A s such the results are only applicable for comparison purposes. After determining the wind speed dis tr ibut ion, the resultant wind-wave field can be calculated from one of the various methods discussed in the following section. 2.2 Lake Wind-Waves Accurate representation of wind stress requires calculat ion of the surface roughness, which in turn requires determination of wind-wave characteristics. Es t imat ion of wind-wave characteristics began during the Second W o r l d W a r when landing opera-tions required forecasts of the sea-state. Research began wi th the work of Sverdrup and Munk (1947) who developed the idea of significant wave height, Hs, a measure from crest to the trough of the one-third highest waves (Csanady, 2001). Sverdrup and Munk (1947) found that Hs and the wave period, T, could be related to wind speed, fetch and durat ion. Considering a locat ion on the lake surface, the fetch is the over-water distance the wind has traveled to reach that location. T h e durat ion is the length of t ime the wind has blown at that location. Based on the wind speed, fetch, and durat ion one can estimate the significant wave height and period at any point on a body of water. Bretschneider (1952) expanded on the work of Sver-drup and Munk (1947) and developed empir ical ly based fetch- and durat ion-l imited equations for wave forecasting. The equations show that for a given wind speed a wind-wave reaches a point at which it w i l l stop growing, known as fully-developed conditions. The l imi t ing factor can either be the fetch or the durat ion. If the du-rat ion is short and the fetch is long, the durat ion w i l l l imi t the wind-wave growth. If the durat ion is long and the fetch short, the fetch is the l imi t ing factor. Fetch and durat ion l imits are related to the point at which the wind stops transferring momentum to the wind-wave as the wave has become saturated. Ini t ia l ly it was 8 assumed that once the wave speed reached that of the wind speed the waves would become fully-developed. However, it has been shown that fully-developed waves oc-cur after the phase speed of the waves has surpassed that of the wind (Young, 1999). A s it is believed that energy from the atmosphere cannot be transfered to waves propagating faster than the wind , nonlinear interactions are thought to play a role (Young, 1999). T h e ratio of wave speed to wind speed is known as the wave age, which represents the stage of growth of the wind-waves (Csanady, 2001). Waves become fully developed in the wave age range of 1.14-1.20 (Donelan et al, 1992). The work of Sverdrup and Munk (1947) and Bretschneider (1952) led to the Sverdrup-Munk-Bretschneider ( S M B ) equations for fetch- and durat ion-l imited waves Since then several other modified or new versions of equations have been developed. Through the use of computers, numerical modeling has improved wind-wave estima-t ion greatly. Majo r developments in computer modeling of wind-waves are discussed i n the following section. 2.2.1 Computer Modeling of Wind-Waves Determinat ion of wind-waves through fetch- and durat ion-l imited equations is very useful and sufficiently accurate for many circumstances. However, it is in reality a simplification or estimation of the actual conditions, as many other factors beyond fetch and durat ion affect wind-wave development. Examples of these factors include refraction, diffraction, shoaling, reflection, breaking due to depth and swell prop-agation, among others. The use of computer modeling allows for more accurate simulations of wind-wave growth. Over the past half-century numerous methods of numerically modeling waves have been proposed. Some of the more recent and accepted versions are outl ined below. The 1970's saw the development of what are known as first-generation mod-9 els (e.g. G loba l Spectral Ocean Wave M o d e l , Clancy et ai, 1986; Spectral Ocean Wave M o d e l , Pierson, 1982; and Ocean D a t a Gather ing Program, Cardone et ai, 1976). These models were based on the assumption that nonlinear interactions could be ignored. After the exceptional work of the Joint N o r t h Sea Wave Project ( J O N S W A P ) (Hasselmann et al., 1973) it was realized that these interactions are required for proper modeling of growing wind seas. T h i s led to second-generation models which used the wave energy spectrum determined by J O N S W A P to pa-rameterize the nonlinear interactions (e.g. W I S W A V E , Hubertz, 1992; S H A L W V , Hughes and Jensen, 1986; and A D W A V E , Resio, 1981). A n in depth analysis of various first- and second-generation models by the S W A M P group (SWAMP, 1985) concluded that the models were only applicable to l imi ted wind fields and were unreliable in extreme conditions. A s a result third-generation models were devel-oped which expl ic i t ly represent the physical processes relevant to wave evolution and give a full two-dimensional description of the sea state (Komen et ai, 1994). Ris (1997) refers to the model W A M (Wave Model ) (WAMDI, 1988) as the pro-totype third-generation model . Its concept has been the start ing point for several other third-generation models developed for the deep water conditions of oceanic and shelf seas, as well as the shallow water conditions along the coastline (e.g. W A V E W A T C H , Tolman, 1991; P H I D I A S , Van Vledder et al, 1994; T O M A W A C , Benoit and Becq, 1996; and S W A N , Ris, 1997;). A s for the future, according to Liu et al. (2002), the abi l i ty to improve third-generation models does not currently ex-ist. These authors determined that wind-wave models which represent the sea-state w i t h a wave energy spectrum have reached their peak in accuracy. Thus, contin-ued improvement of wave models would require the removal of the energy spectrum method as a l imi t ing factor. A s discussed wind-wave models provide more accurate results then fetch- and 10 durat ion-l imited wind-wave equations. However, when considering computat ional t ime and effort the simple equations are much more affordable. In combinat ion w i t h the computat ional cost of running a 3D hydrodynamical model , even a first-generation wind-wave model is not feasible for the extent of this project. Instead, fetch- and durat ion-l imited equations w i l l be used to estimate the wind-wave condi-tions for simplified conditions. The results w i l l provide a good indicator of whether future implementat ion of a full wind-wave model is worthwhile. The major fetch-and durat ion-l imited methods are discussed and compared in the following section. 2.2.2 Fetch and Duration Limited Wind-Wave Equations Over the past half century the original work of Sverdrup and Munk (1947) and Bretschneider (1952) has been continuously altered. Th i s is often based on a new data set and has resulted in some confusion as to which variat ion is appropriate. In light of this, several methods were compared for conditions similar to those found at Quesnel Lake. The methods are: 1. T h e Coas ta l Engineering M a n u a l ( C E M ) (Resio et al, 2002); 2. T h e Joint N o r t h Sea Wave Project ( J O N S W A P ) as outl ined in Wuest and Lorke (2003) and Csanady (2001); and 3. F r o m the work of Carter (1982) which is also based on the work of J O N S W A P . For each of these methods equations for fetch-limited growing seas are listed in Table 2.1. A s described in the following section equations for other sea states and the durat ion-l imited case are not considered. Other works not considered are that of Donelan (1980) and SPM (1984). The former was considered dated in comparison wi th the other works, and the latter has been updated in the C E M (Resio et al, 2002). The fetch- and durat ion l imi ted equations are restricted to the following three situations (Resio et al, 2002): 1. A w i n d blows in a constant direction for sufficient t ime to achieve steady-state, fetch-limited waves; 11 Table 2.1: Equat ions for Fe tch-Limi ted Growing Seas. Where tX}U [s] is the durat ion of the wind after which waves become fetch-limited, X [m] is the fetch, U~IQ [ms~1] is the wind speed, Hs [m] is the significant wave height, Tp [s] is the peak period which corresponds to the frequency of the wave energy spectrum's peak, u* [ m s - 1 ] is the friction velocity and g [ms~2] is the acceleration due to gravity. C E M = 7 7 . 2 3 ^ ^ Hs = 0.0413 Tp = 0.651 ( § ) " (^)(2.1) J O N S W A P not discussed # s = 0.051 ( ^ ) * T p = 0.885 ( § ) * ( ^ ) (2-2) C A R T E R tx>u = 3 3 . 3 7 1 ^ - i 7 s = 5.1545 x 1Q-*X*U10 Tp = 0 . 0 7 1 3 X a 3 C / 1 ° 0 4 (2,3) 2. A wind's speed increases very quickly i n an area without nearby boundaries and the waves become durat ion-l imited. T h i s rarely occurs, especially in lakes where fetch-limited conditions dominate, as discussed in Section 2.2.3; 3. Waves reach a fully-developed state. One of the major differences i n the three methods considered is that C E M and J O N S W A P require the input of a shear velocity, «*, defined as: (2.4) where p\ [kgm~3] is the surface water density. B o t h the C E M and J O N S W A P methods require the shear velocity and thus the drag coefficient, C i o (a function of r, Equa t ion 1.1). A s the intention is to determine a value fo r .Cio, the input value must be an estimate. Carter 's equations do not require this estimate and thus in i t ia l ly appeared ideal for determining a C io based on surface roughness, z0. 12 W h e n considering the fetch-limited case, wind-wave field properties are gener-al ly expressed as a function of fetch, X. However, it is also possible to express these properties as a function of wave age, Cp/Uio (Wuest and Lorke, 2003; Csanady, 2001). Wave age is the non-dimensional form of the peak phase speed or gravity-wave celerity, Cp. It can also have the form cp/u*, but the UIQ form w i l l be used here. A s the wind blows and creates waves the wave speed increases. A s only a con-stant wind speed is considered for fetch-limited conditions, U\o remains constant. Thus, as the wave evolves cp approaches U\o and eventually surpasses it . A s men-tioned earlier, once CP/U\Q reaches 1.14-1.20 the waves have become fully developed (Donelan et al., 1992). Th i s is approximately equivalent to a value of 31 for cp/u*. T h e dimensionless fetch in the various fetch and durat ion l imi ted methods can be replaced by wave age using the approximat ion UIQ = 27.5u* in the form (Csanady, where g [ms~2] is gravitat ional acceleration. For comparison purposes the wave-age variat ion was applied to the C E M and J O N S W A P method. The variat ion was not applied to the method of Carter as the lack of dependence on an in i t ia l C\o is the reason it was considered. W i n d speeds of 2, 5 and 10 m / s were used to compare (a.) C i o and (b.) cp/U\o against fetch. Results for 2 and 5 m / s wind speeds are shown in Figure 2.1. The 10 m / s wind speed is not shown as it closely resembles the results of the 5 m / s wind speed.The fetch ranges from 5 to 35 k m which, along wi th the wind speeds, are the general ranges applicable to Quesnel Lake. O n l y the fetch-limited case is compared. There is very l i t t le variat ion in the results of the C E M method when using fetch ( C E M ) or wave age ( C E M ( c p / [ / i o ) ) , thus the wave age method is not shown in Figure 2.1. Compar ison of the J O N S W A P fetch ( J O N S W A P ) and wave age 2001): (2.5) 1 3 10' Fetch, km ~ 10' Fetch, km —CEM — JONSWAP—JONSWAP(cp/U10)—Carter Wind Speeds: — 2 m/s --5 m/s Figure 2.1: Compar ison of Fe tch-Limi ted W i n d - W a v e Equat ions. C E M : F r o m the Coas ta l Engineering M a n u a l (Resio et ai, 2002), using fetch ( C E M ) ; J O N S W A P : A s outl ined in Wuest and Lorke (2003) and Csanady (2001) based on fetch ( J O N -S W A P ) and wave age ( J O N S W A P ( c p / [ / 1 0 ) ) ; and Carter : F r o m Carter (1982). The solid lines are for a 2 m / s wind speed and the horizontal dashed lines are for a 5 m / s w i n d speed. ( J O N S W A P ( c p / £ / i o ) ) method shows a significant difference. Th i s difference is most evident i n Figure 2.1-b which shows increased wave age values for the J O N S W A P ( c p / C / i o ) method as compared to the others. The high wave age values stem from high wave heights and peak periods. The plot shows that J O N S W A P ( C P / U \ Q ) predicts fully-developed wind-waves for a l l conditions as cP/U\a is close to or greater than 1.17. These results are abnormal as it is nearly impossible for the waves to reach a fully-developed state so quickly for the range of wind speeds. There appears to be a problem w i t h the methodology or its applicat ion. A s such, the J O N S W A P ( c p / C / i o ) method w i l l not be considered hereafter. The remaining methods of determining wind-wave characteristics show some similarit ies. In both plots the C E M and J O N S W A P results have negative slopes showing a decrease in C\Q w i t h fetch. Carter 's results have a slope showing next to no C i o variat ion wi th fetch. T h e slope is slightly positive giving an increase i n C i o w i t h fetch. A positive slope disagrees w i t h the generally accepted belief that 14 the wind stress w i l l be higher near the shore (smaller fetch, younger wave) and w i l l decrease as the wave speed approaches the wind speed (larger fetch, older wave) (Wuest and Lorke, 2003). The younger, underdeveloped waves w i l l be rougher and produce more turbulence leading to increased surface roughness and wind stress (Wuest and Lorke, 2003). The older waves w i l l be larger but calmer and less steep providing less vert ical surface for the wind to push on resulting in reduced wind stress (Wuest and Lorke, 2003). T h e magnitude of C i o for bo th C E M methods is higher than J O N S W A P and Carter 's . The latter two produced similar magnitudes which is likely due to both being based on the same empir ical data. The results can also be compared to recommended values of constant C\Q. Imberger and Patterson (1990) suggested values of 1.0 x 10~ 3 for 2 m/ s and 1.2 x 1 0 - 3 for 5 to 10 m/s . The C E M (Resio et al, 2002) uses the calculation C10 = 0.001.*(1.1+0.035.*U 1 0) g iving 1.2 x 1 0 - 3 for 2 m/s , 1.3 x 1 0 - 3 for 5 m / s and 1.4 x 10~ 3 for 10 m/s . A s compared w i t h constant C\o values the C E M results are significantly higher (> 2.0 x 10~ 3 for 1 0 m s - 1 ) while the J O N S W A P and Carter equations have a similar median but higher max imum and m i n i m u m values. Investigation of Figure 2.1-b shows similar trends to Figure 2.1-a. For low wind speeds, a l l results indicate that the fully-developed wave condit ion is reached (cp/Uio > 1.14 - 2.0). However, the J O N S W A P and Carter results imply that this condi t ion is reached at very low fetches, while C E M barely reaches this con-di t ion. A l l methods agree (except J O N S W A P ( c p / t / i o ) ) that for a m a x i m u m fetch of 35 k m the fully-developed condi t ion is not reached at w i n d speeds of 5 m / s and higher. T h i s result fits well w i th the observation that waves remain young in small to medium sized lakes (Wuest and Lorke, 2003). T h i s is due to l imi ted fetches which result in increased wave-induced complexities as quasi-steady conditions observed in the open ocean are not reached (Wuest and Lorke, 2003). The rate of increase 15 in wave age w i t h fetch is similar for a l l methods except Carter where the rate is slightly reduced. Th i s is similar to the smal l slope in Figure 2.1-a indicat ing the Carter wave field is less affected by fetch in comparison w i t h C E M and J O N S W A P . Further analysis was conducted to investigate the importance of the estimated in i t i a l drag coefficient, C i o i , for calculation of u*. App l i ca t ion of a range of Cia values showed l i t t le to no variation in the resultant C i o values as a function of z0. Thus, while in i t ia l ly this was considered a downfall of the C E M and J O N S W A P equations, it appears to be inconsequential. Based on the C E M method's wider acceptance and applicat ion it w i l l be used for this investigation. 2.2.3 Simplification for Small to Medium Sized Lakes T h e compared methods o f wind-wave estimation are a l l based on data from the ocean or large lakes. A s a result, a l l of the conditions the methods use may not be applicable for small to medium sized lakes. Figure 2.1 is used for determining possible simplifications to wind-wave estimation. Figure 2.1-b (ignoring the J O N S W A P ( c p / £ / i o ) results) shows very few instances of fully-developed waves. The relatively short fetches l imi t the abi l i ty of waves to reach the quasi-steady conditions observed in the open ocean (Wuest and Lorke, 2003). Instead the wave field remains much more complex wi th waves out of equi-l i b r ium wi th the wind field (Wuest and Lorke, 2003). Th i s results i n shorter wave-lengths, greater steepness and higher frequency than is often seen in the open ocean. These characteristics indicate that the waves break more frequently and remain un-derdeveloped (Wuest and Lorke, 2003). Accord ing to Figure 2.1-b waves only be-come fully-developed at larger fetches (> 35A;m) for very low windspeeds ( « 2m/s). A t theses low wind speeds there is less momentum to be transferred to the waves from the wind . Thus, the waves become saturated, reach their peak heights and 16 approach the speed of the wind faster then at higher wind speeds. T h i s results in fully-developed wave conditions at relatively short fetches. Based on the wind-wave methods compared, fully-developed conditions are reached at: 9-13 k m for a 2 m / s wind speed; 58-85 k m for a 5 m/ s w ind speed; and 232-406 k m for a 10 m / s w ind speed. The results indicate fully-developed wind-wave conditions are only reached at very low wind speeds for smal l to medium sized lakes. A s the vast majori ty of the wind speeds and fetches considered result i n growing wind-wave conditions, the smal l percentage of fully-developed waves may be ignored. Another consideration of smal l to medium sized lakes is the effect of their l imi ted fetch on the fetch- and durat ion-l imited state. Due to the proximi ty of the shore-line the amount of t ime waves are durat ion-l imited is very small . F r o m the C E M and Carter methods, the typical length of t ime that waves are durat ion l imi ted is approximately 2-3 hours for a 5 k m fetch, and 5-10 hours for a 35 k m fetch. For a typica l model run of 1 week or 168 hours, durat ion-l imited conditions exist for 1-2% of this time for a 5 k m fetch and 3-6 % for a 35 k m fetch. W h i l e such constant w i n d conditions are unlikely to occur in nature they are applicable for the simulations considered here. These results suggest it may be worthwhile to only consider the fetch-limited state. Use of the above mentioned simplifications w i l l reduce the computat ional t ime of the numerical model without significantly impact ing the accuracy of the surface roughness height. A review of methods for determining surface roughness height follows. 2.3 Surface Roughness Height There has been a great deal of research over the past two decades at tempting to quantify the roughness a w ind feels blowing across a water surface. It is generally 17 accepted that the drag coefficient is dependent on two factors, the wind speed across the water surface and the surface roughness height (e.g., Equat ions 1.1 and 1.3). Ca lcu la t ing the surface roughness height is s t i l l an issue of significant debate. One of the first proposed methods was by Charnock (1955) who used dimensional analysis to show that surface roughness is proport ional to ul/g. Th i s resulted i n the relationship: where zch is the Charnock parameter typica l ly assumed to be a constant equal to Current ly a great deal of research is focused on relating the Charnock param-eter to inverse wave age (e.g. Donelan, 1990; Toba et al, 1990; Smith et al., 1992; Donelan et al., 1993; Johnson et al., 1998). The inverse wave age is a dimension-less relationship between the phase speed of the dominant waves, cp[ms~l] and the wind speed or the friction velocity. Thus , zQ w i l l vary over time wi th the ratio of the wind speed to the wave speed (Jones and Toba, 2001). Th i s relationship is usually wri t ten in the form of a power law: where A and B are constants which, depending on the data set, can range from 0.02 to 2.87 and -0.5 to 1.7, respectively (e.g. Toba et al, 1990; Maat et al, 1991; Smith et al, 1992; Johnson et al, 1998; Drennan et al, 2003; Lange et al, 2004). Th i s large range of constants was at t r ibuted to self-correlation by Lange et al. (2004), as the friction velocity occurs on both sides of the equation. Due to the self-correlation Lange et al. (2004) felt that only by using mult iple data sets resulting in a wide range of wave age, could a physical relation between the Charnock parameter and (2.6) 0.0015. (2.7) 18 wave age be determined. W h i l e relating surface roughness to wave age appears to be popular, there has yet to be a single formulation that fits al l available data sets (Taylor and Yelland, 2001). Another method of est imating z0 relates surface roughness to wave steepness, H g L p 1 [—] (e.g. Taylor and Yelland, 2001; Guan and Xie, 2004). Here L p is the wavelength defined as: gT2 LP = ~ ^ N (2-8) Wave steepness is a dimensionless function defined as the ratio of wave height to wavelength. Taylor and Yelland (2001) used several data sets to relate wave steep-ness to surface roughness in the form of a power law given by: where A and B are constants determined to be 1200 and 4.5 respectively (Taylor and Yelland, 2001). The authors showed that the relationship worked well for various conditions including the open sea, wide open lakes, as well as medium sized, narrow lakes. T h e relationship d id not perform well for fetches less than 2 k m or for very young waves (U\QC~1 > 3) in a dataset from Lake Ontar io . In addit ion, the problem of self-correlation has once again been introduced wi th H s occurr ing on both sides of Equa t ion 2.9. W h i l e both methods of est imating the surface roughness height have benefits, the test case considered here is that of a long and narrow lake. Thus, based on the wave steepness relationship's (Equat ion 2.9) proven effectiveness in medium sized lakes it w i l l be applied to our test case. Once the surface roughness has been calculated wi th Equa t ion 2.9 the drag coefficient is determined from Equa t ion 1.3. The following section review methods of determining a drag coefficient without 19 considering wind-waves. 2.4 Drag Coefficient T h e y are two main methods of determining the drag coefficient applied in Equa t ion 1.1. The more recent method, as discussed previously, determines a drag coefficient as a function of the wind-wave state. However, the t radi t ional method uses a constant drag coefficient which may or may not vary wi th w ind speed. Three variations of the constant drag coefficient are considered here. The first variat ion is used by the hydrodynamic model E L C O M , which assumes a constant C i o of 1.4 x 1CT 3 . Accord ing to the C E M C i o is a function of w ind speed in the form (Resio et al., 2002): Cio = 0.001(1.1 + 0.035/7i 0) [ m s - 1 ] (2.10) Imberger and Patterson (1990) also used a constant drag coefficient which varies w i t h w ind speed: • for Uw < 4 m s " 1 Cw = 1.0 x 1 0 " 3 ; • for 4 m / s < Cio < H m s - 1 C i o = 1-2 x 10~ 3 , as per Large and Pond (1981); • for 11 m / s < Uw < 25 m s " 1 C i o = 0.49 + 0.065C/i O , as per Large and Pond (1981). To this point ideal methods have been outl ined for determination of a drag coefficient and resultant w ind stress. Notab ly use of the C E M fetch-limited, growing seas wind-wave equations (Table 2.1, Resio et al. (2002)) and the wave steepness based surface roughness height equation (Equat ion 2.9, Taylor and Yelland (2001)). For the test cases in Sections 3 and 4 these equations w i l l be used to determine a drag coefficient and wind stress as a function of wind-wave state and w i l l be compared 20 w i t h the three constant drag coefficient cases outl ined above. Th i s comparison w i l l provide the most accurate and feasible method of determining the wind stress act ing on the lake surface and the resultant wind-induced circulat ion. The mechanics of this circulat ion are reviewed in the following section. 2.5 Hydrodynamics of Lakes - Wind Induced Circulation The general idealized concept of wind-induced circulat ion for narrow, deep lakes is shown in Figure 2.2 (see also Append ix A ) . Star t ing wi th a calm lake (Figure 2.2-a) a uniform wind begins to blow across the surface (Figure 2.2-b). The momentum transferred to the S B L forces the surface water downwind, creating a bui ldup of water at the downwind shore (Figure 2.2-b). Th i s forcing results i n two major physical processes. The first is the circulat ion created along the wind 's axis. A s the water piles up at the downwind end of the basin it is forced by the shoreline constraint toward the bo t tom of the lake. Reaching the bot tom boundary forces the water to flow i n the upwind direction. A s it reaches the upwind shore the flow is forced upward and returns to the S B L . Th i s flow in combinat ion w i t h the currents created by the wind in the S B L , results in the circulat ion pattern shown in Figure 2.2-c. T h e second physical process is a result of the downwind bui ldup of water overshooting its equi l ibr ium posit ion due to the unsteady startup. Steady-state equi l ibr ium results from the pressure force of the sloped water surface balancing the wind shear. The steady-state slope of the water surface is related to the uniform wind speed i n the form (Heaps and Ramsbottom, 1966): 21 Figure 2.2: W i n d - D r i v e n Ci rcu la t ion i n a Lake w i t h one (a.,b.,c.) and two layers (d.). A s the w i n d begins to blow (a. to b.) water builds up at the downwind end (b.). T h i s results in internal circulat ion and surface seiching (a ) . For the two layer case the ci rculat ion occurs in both layers and seiching occurs at the surface and interface (d.). where 771 is the displacement of the surface relative to the unforced state, x is the distance downwind of the shore, H is the lake depth, g is the acceleration due to gravity and is the friction velocity. Once the equi l ib r ium posit ion is passed the rise of the downwind water surface begins to slow and eventually stops. The water surface then reverses direction and moves toward the same equi l ibr ium posit ion. A s momentum pushes it past equi l ib r ium again it begins to slow, stops and returns toward equi l ibr ium. The oscil lation of the surface is called seiching and is a basin-scale standing wave. For a basin of length L and depth H, the seiche has a wavelength of 2L and a fundamental period of T = 2L/c, where c is the phase speed of the s tanding wave, equal to \fg~H. Assuming a steady wind , the ampli tude of the seiching w i l l decrease due to bo t tom friction and the water surface w i l l eventually reach an equi l ibr ium posit ion. Should the w i n d stop or change magnitude a new equi l ibr ium must be reached and the seiching recommences. For density stratified lakes (Figure 2.2-d) the circulat ion described above is re-22 stricted to the surface layer as momentum is not transferred by shear across the pycnocline (the density interface between the surface and bot tom layer). However, the bu i ld up of water in the surface layer pushes down the pycnocline and results i n internal basin-scale standing waves along the pycnocline. For steady-state con-ditions the slope of the free-surface is balanced by the pycnocline. T h e slope of the pycnocline at steady-state wi th uniform wind stress is (Heaps and Ramsbottom, 1966): dr]2 u 2 (2.12) dx g'hi L 1 where 772 [m] is the displacement of the interface relative to the unforced state, hi [ra] is the depth of the surface layer and g' [ms~2] is the reduced gravity. The latter is a function of the density difference between layers (Heaps and Ramsbottom, 1966): g' = gBl^Plt [ m a - 2 ] ( 2 . 1 3 ) Po where pi and P2 [kgm~3] are the densities of the surface and bot tom layer respec-tively, and po is a reference water density, usually taken as 1000 kgm~3. Resultant currents in the bot tom layer are in the opposite direction of the surface layer (see Figure 2.2-d). A s the density differences across the surface interface (air to surface layer) is greater than the density difference across the thermocline (surface layer to bo t tom layer), the slope of the thermocline w i l l be significantly larger. The larger slope is accompanied by an increased seiche ampli tude and period. The following section outlines some of the recent improvements used i n modeling interface movements and basin circulat ion. 23 2.6 Hydrodynamic Modeling Improvements Through detailed analysis of wave development and coupling w i t h a hydrodynamic model, both internal movements and surface gravity wave estimation is greatly improved. Previous work by Sarkkula (1991) investigated the effect of relating wind stress to w ind speed and surface roughness. A spatial ly varying wind speed was used to accurately model current flow in a shallow lake. The authors used a 2-D depth integrated flow model in combinat ion wi th a fetch dependent w i n d stress field. The latter gave variations in w ind speed and wave height w i th fetch. Appt et al. (2004) applied E L C O M to Upper Lake Constance and accounted for surface roughness height w i th a constant estimated value of 1.15 x 1 0 - 4 m. T h e y determined that their choice of surface roughness was acceptable for their purpose. The majori ty of coupling of models occurs w i th ocean circulat ion and wave models. Greater understanding of the ocean-atmosphere interaction has shown the importance of the interdependence. Thus, in order to properly model the circulat ion in the ocean, the boundary interactions must be modeled, and vice-versa. Huang et al. (2002) used the Pr inceton Oceanic M o d e l ( P O M ) to look at the surface w i n d effects on t ida l inlets of an estuary. They were able to simulate time-dependent wind and showed that the surface wind had significant effects on the volume fluxes of water being exchanged across the inlet between the ocean and the estuary. T h e P O M was also coupled wi th the third-generation wave model, W A V E W A T C H - I I by Moon (2005). The authors found improvements in wind-wave predictions as well as circulat ion and water temperatures. In a similar coupling Welsh et al. (1999) used the C H 3 D marine circulat ion model (Chapman et al, 1996) and W A M wind-wave model for improving predictions of wave heights, currents and water elevations. F u l l ocean and atmospheric model coupling is also being conducted. Proper ly modeling the large-scale interactions between the ocean and the atmosphere provides more accurate weather and climate change prediction. Chapter 3 25 Applicat ion to a Linear Hydrodynamic Mode l A n analyt ical solution for circulat ion and seiching i n a long, narrow density-stratified lake is used to investigate the impact of surface roughness height on lake hydrody-namics. The solution is driven by the surface wind stress making it ideal for such a comparison. T h e process of determining r from the wind-wave state and Uio is described in Section 3.1 followed by a description of the analyt ical solution (Section 3.2). In Section 3.3, the analyt ical solution is used to il lustrate the effect wind-waves have on basin scale circulat ion through the comparison of four test cases. One test case applies the constant drag coefficient used in E L C O M and two other test cases apply the wind speed based drag coefficients described i n Section 2.4. T h e final test case uses a drag coefficient as a function of wind speed and surface roughness height. 3.1 Estimating the Wave Parameters Wind-wave properties can be estimated using duration- and fetch-based empir ical ly derived formulas. Here, w ind durat ion, t [s], refers to the t ime elapsed from the beginning of a w i n d event, and wind fetch, X [m], refers to the distance the wind has blown over water. Figure 3.1 shows the general process of determining the significant wave height and peak period (Resio et ai, 2002). Initially, as the wind starts to blow, w ind speed and durat ion determine the size of the waves and their period. 26 D u r i n g this period, the waves are referred to as durat ion-l imited. Once the wind durat ion exceeds tXiU = 7 7 . 2 3 X 0 6 7 / ( C ^ V 3 3 ) seconds (Resio et al, 2002) wave properties are controlled by wind speed and fetch, and the waves are considered fetch-limited. B o t h duration- and fetch-limited waves wi l l evolve (growing seas) unt i l they are fully-developed at which point they have reached the m a x i m u m height and period for a given wind speed. T h e transi t ion from growing to fully-developed waves occurs when the dimensionless wave height and peak per iod exceed 2.115 x 10 2 and 2.398 x 10 2 respectively (Figure 3.1) (Resio et ai, 2002). Once waves are fully-developed their height and period are only a function of wind speed. 8", u: >2.J15xl02 > 2.398xl02 Y e s F u l l y D e v e l o p e d F e t c h L i m i t e d YNOI t > 77.23-Y e s 3 X™ V G r o w i n g S e a N o D u r a t i o n L i m i t e d Figure 3.1: F low diagram showing progression of wave development. Where tXtU [s] is the durat ion of the wind, X [m] is the fetch, d o is the wind speed, Hs [m] is the significant wave height, Tp [s] is the peak period which corresponds to the frequency of the wave energy spectrum's peak, u* \ms~1} is the friction velocity and g [ms~2] is the acceleration due to gravity (Resio et al., 2002). F l o w chart adapted from Carter (1982). Typ ica l l y relations found i n the Coasta l Engineering M a n u a l ( C E M ) (Resio et al., 2002) are used to estimate wave properties for the various sea-states de-scribed above. These relations are based on the empir ical ly derived equations of Hasselmann et al. (1973) which require a drag coefficient as input . Since our a im is to calculate the drag coefficient as a function of wave properties, requiring an in i t ia l C io ; was considered a drawback. However, analysis of the variabi l i ty due to the in i t i a l Cio, showed l i t t le impact on the determination of C i o as a function of surface roughness. Thus, C E M ' s method w i l l be applied, based on it 's wide acceptance, 27 applicat ion and the comparison conducted in Section 2.2.2. In the future a compar-ison of the other methods for est imating wave properties applied to a hydrodynamic model may be useful. T h e test cases used to illustrate the impact of wind waves on lake circulat ion (see Section 3.3) are dominated by fetch-limited, growing seas and for s impl ic i ty only relations for this case are presented. In addit ion, depth effects are not considered as only deep-water conditions exist. Deep-water conditions occur when the depth over wavelength ratio exceeds approximately 0.5. The dimensionless significant wave height and period for deep-water, fetch-limited, growing seas are given by Equat ion 2.1. T h e friction velocity, u* [ms"1} is defined as: u* = (j-^J2 [ms-1] (3.1) where r is determined i n Equa t ion 1.1 and requires an in i t i a l drag coefficient, C\on which must be estimated. A n estimation of Ciot from the C E M based on wind speed w i l l be used (Resio et a/., 2002). F r o m the peak period the wavelength, Lp is determined from Equa t ion 2.8. W i t h the determination of wind-wave characteristics the surface roughness height (Equat ion 2.9), drag coefficient (Equat ion 1.3) and wind stress (Equat ion 1.1) are calculated. In the following sections, wind stress is used as a boundary condi t ion i n an analyt ical solution for mot ion i n an enclosed basin. T h e solut ion w i t h and without the effect of surface roughness is compared. 3.2 Hydrodynamics Heaps and Ramsbottom (1966) analyt ical ly solved the linear, hydrostatic, Navier Stokes equations for motion in a long, narrow, two-layered lake. The i r solution (see 28 A p p e n d i x A ) is used here to illustrate the importance of considering wind-waves when estimating the effect of wind stress on the motion in a lake. T h e two lake layers are each homogeneous wi th a small density difference between the two. T h e lake is subjected to an instantaneous rise in wind stress from zero to a constant value, which is not necessarily uniform along the length of the basin. Free-slip bo t tom and side boundary conditions along wi th a no-slip condit ion at the interface are assumed. T h e analyt ical solution is based on several assumptions, most notably that there are only two-dimensions in the x (along the w i n d direction) and z (vertical) planes, non-linear relations are ignored and there is a constant vertical eddy viscosity in each layer. A s shown i n the basin layout of Figure 3.2, the solved equations estimate the displacement of the surface ( 7 7 1 ) and interface ( 7 7 2 ) , as well as the horizontal ( i t i and U2) and vertical velocities (w\ and W2) for the surface and bo t tom layers. However, only 772 and u\ w i l l be analyzed as the rest of the parameters provide l i t t le extra insight into the mot ion of the lake. A s a test-case the solved equations are applied to a 6.6 k m basin w i th a total depth of 36 m and a surface layer depth of 15m. T h i s lake configuration was chosen to allow verification of results as it is s imilar to both the configuration used by Heaps and Ramsbottom (1966), and one of the datasets used by Taylor and Yelland (2001) in proving the val idi ty of Equa t ion 2.9. For the test-case, several w ind speeds w i t h a sinusoidal d is t r ibut ion in the x-direct ion were applied to the surface of the basin. Th i s w ind dis t r ibut ion is required to satisfy the end-wall boundary conditions i n the Heaps and Ramsbo t tom solution. It also imitates the w ind speed dis t r ibut ion on a lake where vegetation might result i n reduced wind speeds near the shore. The wind dis t r ibut ion is of the form: (3.2) 29 where Umax [ m s - 1 ] is the m a x i m u m ampli tude of the wind speed at the centre of the lake. The wind stress in Equa t ion 1.1 is determined from the wind speed dis t r ibut ion of Equa t ion 3.2 and the non-uniform drag coefficient (Equat ion 1.3). The wind stress is applied in the form: r(x,t) = H(t)C10(x)paU?0(x) [kgm-ls-2] (3.3) where H(t) is the Heavyside step function. Th i s wind stress provides the surface boundary condit ion for the analyt ical solution to predict interface displacements and horizontal velocity in the surface layer. T h e steady-state interface displacement amplitudes can be estimated from (Heaps and Ramsbottom, 1966): 'a2. L r , / m = H <3-4> where g' [ms~2] is the reduced gravity (Equat ion 2.13), h\ [m] is the surface layer depth and L [m] is the basin length. Equa t ion 3.4 shows the dependence of seich-ing amplitudes on the basin length and shear velocity (or surface wind speed and roughness height). The applicat ion of the linear solution and results are discussed in the following section. 3.3 Analysis A drag coefficient as a function of wind speed and surface roughness height was compared wi th the three constant drag coefficients outl ined in Section 2.4. A l l test cases were applied to the analyt ical ly solved equations of Heaps and Ramsbottom (1966) and the resultant circulations were compared. Results are summarized and discussed in the following sections. 30 z 1 } _______ p. -— - u , p2 1 • U , x=0 x=L Figure 3.2: Layout of the basin used in Heaps and Ramsbottom (1966) 3.3.1 Wind Stress Distribution D r a g coefficient distributions are shown in Figure 3.3 for peak wind speed values of 2, 5 and 10 m/s . The plots compare three drag coefficients (one constant and two varying wi th w ind speed) and one drag coefficient as a function of wind speed and surface roughness. O n l y the case of fetch-limited, growing seas is presented. The overall shape of the plots is due to w ind stress being proport ional to w ind speed squared. Figure 3.3 shows a general trend of increasing C i o wi th wind speed for a l l cases except the constant C io - A t the low wind speeds the surface roughness case has a lower wind stress then the constant cases (Figure 3.3-a). A s the wind speed increases the difference between the two sets of cases is reduced (Figure 3.3-b) and eventually at higher wind speeds the surface roughness case has significantly higher drag coefficients (Figure 3.3-c). T h e effect this variat ion in wind stress has on interface displacements and circulat ion is discussed in the following section. 31 2 3 4 Fetch, km 2 3 4 Fetch, km — CEMGg —— Constant C 1 0 per ELCOM file echo_elcom.txt Constant C l ( ) per Imberger and Patterson (1990) Constant C per CEM (Resio et al., 2002) Fetch, km Figure 3.3: Drag Coefficient comparison w i t h and without accounting for surface roughness for wind speed dis t r ibut ion given by Equa t ion 3.2. Each plot is for a part icular m a x i m u m wind speed. 3.3.2 Basin Circulation The analyt ical solution has an undamped, basin-scale oscil latory component as well as a t ime independent offset which describes the interface displacements. For the velocities there is only an undamped, oscillatory component. The variat ion in the offset and ampli tude of the periodic internal interface displacements is shown for varying wind speeds in Figures 3.4-a and 3.4-b, respectively. B o t h the surface roughness case and the three constant C i o cases are plotted. Lower wind speeds result i n s imilar offsets and amplitudes for the surface roughness case and for a l l constant C i o cases. A s the wind speed increases the interface displacements increase at a higher rate for the surface roughness case than for the constant C i o cases. Variat ions in 772 are due only to variations in u* for the four compared drag co-efficients. A s compared 772 values are equivalent at low w i n d speeds it is understood 32 that values of it* are also similar. As the wind speed increases, there is a signifi-cantly larger increase in 772 and thus u* for the za case. The increasing values of u* are therefore due to both the wind speed increase and surface roughness effects. The amplitude of the depth-averaged, horizontal velocity at the centre of the basin is shown in Figure 3.4-c. The variation of the horizontal velocity with wind speed displays the same trend as the interface displacement. As the wind speed increases the horizontal velocities increase. Initially the surface roughness case has similar velocities to the constant Cio cases. However, at higher wind speeds the constant Cio case's velocities are significantly lower than for the surface roughness case. For all test cases, as the wind stress increases, there are larger amplitudes and offsets of interface displacements, as well as larger horizontal velocities. The comparison shows that accounting for surface roughness amplifies this relationship at higher wind speeds. 3.4 Summary A method that accounts for the variation in wind velocity and surface roughness height in determining a drag coefficient for calculation of the surface wind stress is presented. This is accomplished through the determination of fetch-limited wave characteristics which leads to the calculation of the surface roughness height. The resultant wind stress is then applied to an analytical solution of the circulation in a long, narrow, enclosed basin. Comparison with several constant drag coefficient cases shows the effect accounting for wind-waves has when determining wind stress and the resultant basin circulation. This effect is significant as it leads to changes in seiching amplitudes at the surface and interface, as well as changes to the horizontal and vertical velocities of the surface and bottom layer . In general, for larger wind speeds (> 5m/s) the amplitudes and velocities were significantly greater when 3 3 0 2 4 6 8 10 Wind Speed, m/s Figure 3 . 4 : T h e impact of considering wind-waves on internal seiching. Shown are the a) ampli tude and b) offset of periodic seiching, 772, at the downwind end of the basin, and c) the ampli tude of the depth averaged horizontal velocity, t z i , at the centre of the basin. accounting for surface roughness height then for the constant C i o case. T h i s linear solution makes it apparent that proper model ing of the wind speed dis t r ibut ion and surface gravity wave conditions is l ikely important in properly understanding transport in lakes. The importance is investigated further in the following section, where a surface roughness dependent w ind stress is appl ied to a 3 D hydrodynamic s imulat ion of Quesnel Lake. Chapter 4 Applicat ion to Quesnel Lake 34 Quesnel lake is located in the interior of B r i t i s h Columbia , Canada at the western base of the Car iboo mountains. A t 506 m deep it is the deepest lake in B r i t i s h C o l u m b i a and one of the ten deepest in the world. The lake sits on the interior plateau wi th a north and east a rm entering the Car iboo range providing highly variable terrain and climate. A s part of the Fraser River system the lake once re-ceived up to 30% of the system's B C sockeye salmon (Thompson, 1945). D a m m i n g and other factors i n the early part of the century reduced salmon numbers drasti-cally in Quesnel Lake. Efforts to replenish the stock have been successful as numbers are on the rise and nearing the past records (Fisheries and Canada, 2001). In order to ensure the continuing success of the salmon, Quesnel Lake continues to be moni-tored by Fisheries and Oceans Canada . In addi t ion investigations into the physical, chemical and biological processes wi th in the lake are being conducted. Specific to the physical processes is an understanding of the circulat ion and mix ing which occurs wi th in the lake. It is hypothesized that these processes may explain abnor-ma l temperature fluctuations observed in Quesnel River (personal communicat ion wi th John Morr i s ion) . The temperature increases can lead to stress and disease in the salmon populat ion, negatively impact ing spawning (Royal, 1966). Inspection of recorded water temperature data throughout the lake has shown significant variat ion between act ivi ty in the West Bas in in comparison wi th the rest of the lake. Figure 4.1 shows thermistor chain data at locations M 2 , wi th in the West Bas in , and M 8 , at the Junct ion of the lake's arms (see Figure 4.2). The internal movements at the 35 two locations are significantly different. The highly active fluctuations in the West M2 - Field Data - 2003 M 8 - F i e l d Data - 2003 10 20 30 08/02 08/03 08/04 08/05 08/06 08/07 08/08 08/02 08/03 08/04 08/05 08/06 08/07 08/08 Figure 4.1: Quesnel Lake temperature recordings at L o c a t i o n M 2 (West Basin) and M 8 (Lake Junction) as shown in Figure 4.2. T h e West Bas in shows strong temperature fluctuations in comparison wi th the relatively ca lm conditions i n the main por t ion of the lake. Bas in are l ikely the cause of the abnormal temperatures in Quesnel River (personal communicat ion w i t h John Morr is ion) . In order to prove this theory the physical processes behind the West Bas in act ivi ty must be understood. To gain this under-standing a hydrodynamic model such as E L C O M (Estuaries and Lake Computer Model ) is used to simulate internal lake processes. Recorded water temperatures throughout the lake provide in i t i a l conditions for s imulat ion runs i n addi t ion to verification and comparison wi th the model results. To improve the accuracy of the model an addit ional module has been added to determine the w ind stress as a function of w ind speed and the surface wind-wave conditions which create surface roughness. E L C O M normally calculates the w i n d stress as a function of w ind speed and assumes a constant value for the surface roughness height and drag coefficient. Improved modeling of the lake processes should provide insight into the rela-tionship between Quesnel Lake and the Quesnel River . In addi t ion to ensuring the m a x i m u m survival rate of the lake's fish, the recorded data and numerical modeling w i l l be useful i n current and future climate change research. T h e following section summarizes the characteristics considered when simulat ing Quesnel Lake's processes wi th a hydrodynamic model . T h e hydrodynamic model 36 E L C O M is then discussed followed by the determination of wind-wave character-istics and its applicat ion to the wind stress. Results of s imulat ion runs are then compared and discussed. 4.1 Quesnel Lake Quesnel Lake is a long narrow fjord-type lake w i t h one a r m reaching to the west, one to the nor th and one to the east as shown in Figure 4.2. T h e East and West A r m s combined span 81 k m while the N o r t h A r m reaches 36 k m . The lake has three major inflows, Horsefly River being the largest and dra in ing a 2750 k m 2 area of the Interior Pla teau. T h e lake empties into Quesnel R ive r through the West Bas in which is separated from the rest of the West A r m by a s i l l approximately 20 to 30 m i n depth. The bathymetry of the lake is characterized by its depth w i t h a Figure 4.2: M a p of Quesnel Lake wi th major rivers indicated by arrows and the town of L i k e l y w i th a dot. The West Bas in , Junc t ion and N o r t h , East and West A r m s are labeled. Numbered locations on the lake mark the locat ion of thermistor chains. East-West and Nor th-South Thalwegs are also provided. Modi f ica t ion of the map found in Potts (2004). mean value of 157 m. T h e majori ty of the lake can be considered deep for wind-52 °N 35.00' 37 wave purposes. The steep shoreline results in l i t t le modification of the wave-field by the bathymetry thus min imiz ing wave breaking and shoaling to a narrow surf zone. Due to the layout of the lake (see Figure 4.2) the lake consists of relatively short fetches. There is a potential for larger fetches if the wind is funneled along the length of the arms by the surrounding topography. However, the wind-waves w i l l not be funneled in the same manner. Instead i f the wind changes direction around a bend in the lake, the wind-waves w i l l be subjected to wind from a new direction, equivalent to start ing at a fetch of zero. T h e resulting fetches have a m a x i m u m of less than 30 k m , which places Quesnel Lake in the range of smal l to medium sized lakes. Analys is of fetch- and durat ion-l imited wind-wave equations in Section 2.2.3 found simplifications applicable to smal l to medium sized lakes. These included ignoring the durat ion-l imited state and the fully-developed condit ion for w ind speeds greater than or equal to 5 m s ' 1 . A s Quesnel Lake has similar fetches and wind speeds to those compared, the assumption that wind-waves w i l l remain fetch-limited and growing for the vast majori ty of the analysis is acceptable. M a k i n g these assumptions should decrease the computer processing t ime without affecting the accuracy of the results. 4.2 Numerical Simulations The Es tuary and Lake Computer M o d e l ( E L C O M ) is applied to Quesnel Lake to simulate the wind-forced circulat ion and temperature stratification. T h e model has successfully simulated circulat ion in several stratified lakes (e.g. Hodges et al., 2000; Laval et al, 2003; Appt et al, 2004) and was not specifically calibrated for Quesnel Lake. The 3D model uses hydrostatic and Boussinesq assumptions to solve the Reynolds-averaged, Navier-Stokes and scalar transport equations separating mix ing 38 of scalars and momentum from advection (Hodges et ai, 2000; Hodges, 2000). The model is driven by a uniform stead wind , impulsively started and run unt i l a steady-state is reached. A uniform gr id w i t h 200 m spacing is used for both the x and y axis. In the vert ical plane grid spacing varied wi th depth ranging between 1 m near the surface and 50 m close to the bot tom. Th i s gr id spacing provides increased accuracy near the surface for proper representation of the thermocline. A gradual increase in gr id spacing without abrupt changes ensures accuracy and stabi l i ty of the model. A free-slip condit ion is applied to the bot tom and side land-boundaries. Based on availabil i ty of temperature data the simulations w i l l start from rest at 1200 hours on the 213th day of 2003 and run for a week to ensure steady-state is reached. Determinat ion of the appropriate t ime step for stratified flows is based on the baroclinic Courant-Friedrichs-Lewy, CFL^ condit ion (Hodges, 2000). To ensure E L C O M ' s s tabil i ty the time step must satisfy the condit ion of (Hodges, 2000): ( 9 ' D ) ^ < V 2 (4.1) where A i [s] is the time step, Ax [m] is the grid spacing in the x or y direction, C [ms~l] is the internal wave speed, g' [ms~2] is the reduced gravity due to stratifi-cation, D[m] is the depth and \fgTD is an approximation of the internal wave speed. A t ime step of 60 seconds was chosen to easily meet the CFLb condit ion w i t h a value of approximately 0.65 in both the x and y direction. The E L C O M simulat ion is simplified to focus on the wind-driven circulat ion by neglecting inflows and outflows of surface and ground water. A c t u a l thermodynamic values are applied for in i t ia l water temperature, air temperature, solar radiat ion, relative humidi ty and cloud cover for a per iod of 7 days i n Augus t of 2003. Stress at the surface boundary due to wind is modeled as a momentum source distr ibuted evenly over the surface wind-mixed layer. T w o methods of calculat ing 39 wind stress are applied and compared in an attempt to improve model accuracy. The current method that E L C O M uses calculates the wind stress through a bulk formulation as a function of w i n d velocity only (Hodges, 2000): T = PadojUwlUw [kgm-ls-2] (4.2) where pa is the density of air (1.25 kgm~3), C\oc is a constant drag coefficient (1.40 x 10~ 3 ) , and Uio is the wind velocity measured 10 m above the water surface. The second method aims to improve on the previous by calculat ing wind stress as a function of wind velocity and the presence and state of wind-waves. The wind-waves (see Section 4.3) create a roughness on the surface of the water that affects the magnitude of the wind stress. Several methods of determining the surface roughness are discussed in detail in Section 2.3. T h e method used here calculates the roughness as the surface roughness height, zQ [m], i n the form of a power law given by Equat ion 2.9 (Taylor and Yelland, 2001). T h e surface roughness affects the wind stress through modification of the drag coefficient, C\Q. A p p l y i n g the momentum equation at steady-state wi th a logari thmic velocity profile leads to a log law equation, given by Equat ion 1.3 (Jones and Toba, 2001). Th i s method of determining wind stress uses a wind velocity and drag coefficient which may vary spatially and temporally. Th i s ensures accurate representation of the wind stress across the entire lake surface and through time. A s seen in Equa t ion 2.9, z0 requires the determination of the wind-wave characteristics Hs and Lp. Ca lcu la t ion of these characteristics is explained in the following section. 40 4.3 Estimating Wave Parameters Wave parameters are estimated using duration- and fetch-based empir ical ly derived equations. The C E M equations (Resio et ai, 2002) w i l l be applied to Quesnel Lake based on the comparison conducted in Section 2.2.2. T h e C E M equations wi l l be applied using the fetch based method as opposed to the wave age based method (see Section 2.2.2). A s discussed in the previous section, w i t h the exception of low wind speeds ( < 5 m s - 1 ) only fetch-limited growing seas w i l l be considered. For a complete description of wave analysis methodology see Append ix B . A s a complex wave model w i l l not be applied, several factors affecting wind-wave generation and propagation w i l l not be considered. These factors include alteration of wave direction and height due to currents, depth, breaking, refraction and diffraction. A s described in Section 4.1 the lake being modeled is extremely deep wi th only a small surf zone where refraction, diffraction and alteration due to depth (including breaking) might occur. It is also assumed that the wind is measured over the water and that a neutral boundary layer exists at the water surface. W i n d measured over the water requires no conversion of over-land to over-water w ind speed. A neutral boundary layer implies that there is no temperature difference between the air and the water at the surface. Often non-neutral unstable (water temperature is higher than air) or stable (air temperature is higher than water) conditions are encountered i n nature. Due to the varying meteorological conditions across the lake being studied (see Section 4.1) obtaining this variat ion in temperatures is very difficult. Determinat ion of the wind-wave characteristics allows for the calculat ion of sur-face roughness height and wind stress as explained in Section 4.2. Once the wind stress is determined it is used for forcing E L C O M simulations of Quesnel Lake. Simulations are analysed and discussed in the following sections. 41 4.4 Ana lys i s A n impulsively started steady and uniform 5 / m s _ 1 w ind is applied to Quesnel Lake representing simplified average wind conditions. Westerly w ind directions resulting i n m a x i m u m internal seiching in the West Bas in w i l l be applied along wi th a wind direction of min ima l fetch and basin length for comparison purposes. T h e internal seiching amplitudes were estimated from relationships applied in Section 3.2; where the ampli tude of the internal seiche is a function of basin length and friction velocity, estimated by Equa t ion 3.4 (Heaps and Ramsbottom, 1966). The friction velocity in turn is a function of wind speed and fetch through the surface roughness height. Us ing Polar and Cartesian coordinates Figure 4.3 shows the variat ion wi th wind direction of m a x i m u m fetch, max imum basin length, average friction velocity and seiching ampli tude for a 5 / m s _ 1 wind . The Polar plot (Figure 4.3-a) gives a general idea of the variat ion w i t h wind direction in relation to the basin layout. Us ing Cartesian coordinates (Figure 4.3-b) provides a more detailed view of directional effects, in part icular for u*. The plots show 772 influenced strongly by L but also impacted by variations in u*. Based on the dominant seiching amplitudes, the wind directions of interest are wi th in the range of 210° to 270°. Considering this range and the max imum fetch values, w ind directions of 250° and 270° were chosen. These represent the upper basin lengths as well as the m a x i m u m fetches. In addi t ion a wind direction 340° was chosen to represent the case of m i n i m u m basin length, fetch and surface roughness effect. Wind-wave conditions are determined throughout the lake and over t ime from which a surface roughness and drag coefficient are calculated. T h e latter is used to calculate surface wind stress. Results comparing the effect of surface roughness height and basin length on E L C O M simulations of Quesnel Lake are analyzed and discussed in the following sections. 4 2 0 0 50 100 150 200 250 300 350 1 8 0 Wind Direction. 10 degree Bins —i• Max Basin Length, (km) — - Mean Ustar. (m/slxlC Max Fetch. {km)x5 mm Internal Seiching Amplitude, (m)xlO Figure 4.3: Plots of m a x i m u m fetch, m a x i m u m basin length, average friction velocity and seiching amplitude wi th w ind direction using Polar (a.) and Cartesian (b.) coordinates. 4.4.1 Surface Roughness Height Comparison Simulat ion results for Quesnel Lake are compared for runs w i t h a C i o which consid-ers wind-wave state ( C E M ) and for runs wi th a constant drag coefficient of 1.4 x 10~ 3 ( R E G ) . Temperature profiles for each wind direction are provided and discussed. Thermistor chains 2 (M2) and 8 (M8) are used for plot t ing temperature at a specific location over the 7 day simulat ion run (see Figure 4.2). M 2 is located in the West Bas in and for a l l w ind directions is essentially at the upwind end of the basin. M 8 is at the centre of the Junc t ion where the three arms intersect, placing at mid-basin length for al l w ind directions. In addi t ion to the thermistor chain plots, snapshots of temperature profiles are presented for an East-West and South-Nor th thalweg (see Figure 4.2). The East-West thalweg is approximately aligned wi th a w ind direction of 270° while the South-Nor th thalweg is more closely aligned to the 210° w ind direction. P lo ts after 24 hours, 3 days and 7 days are provided. T h e thermocline for the simulation runs is located between 1 0 ° C and 16°C . A wind direction of 270° 13 is analyzed first followed by the 250° and 340° w ind directions. Wind from 270° Modeled temperatures at M 2 and M 8 are shown in Figure 4.4 for a wind direction of 270°. M 2 (upper panel), located upwind shows a deep water circulat ing toward the surface due to the setup caused by the constant wind . A s the w ind blows i n a westerly direction the sloping thermocline results in upwelling at the upwind end of the basin. T h e upwelling is seen i n the isotherms gradual movement toward the lake's surface for both C E M and R E G simulations. Over the second half of the s imulat ion run there is an increased rate of upwelling for C E M as compared to R E G . The amount of upwelling is l ikely greater due to the increased surface roughness and wind stress of the C E M simulat ion. Loca t ion M 8 results (lower panel Figure 4.4) show lit t le movement of the thermocline and no upwelling. There is some variat ion between the results of C E M and R E G , however these are minor in comparison w i t h the M 2 results. 08/02 08/03 OB/04 08/05 08/06 08/07 08/08 M8 - 2003 - Wind From 270 Deg. 10 1 1 1 1 1 J 1 1 — 08/02 08/03 08/04 OB/OS OB/06 08/07 08/08 Figure 4.4: Simulated isotherms of Quesnel Lake over a 7 day period, for a w ind speed of 5 m / s from 270° at T - C h a i n 2 (M2) and 8 (M8) . Compar ison of results from a wind stress using a constant C i o ( R E G ) , th in line, and a w ind stress which accounts for z0 ( C E M ) , thick line. The upper contour line represents a temperature of 1 5 ° C followed by 10°C , 7 ° C , and 5 ° C (if visible). 44 T h e East-West thalweg for a w ind direction of 270° is shown in Figure 4.5. After one day there is l i t t le change or difference between the C E M and R E G results. B o t h have negatively sloping thermoclines in the downwind direction as the wind pushes the water this way. For both C E M and R E G , upwind upwelling is observed in the lower surface layer after three days. A t the end of the s imulat ion the R E G results show upwelling of 10°C water while the C E M has upwelling of the deeper 6 ° C water. In addit ion, the C E M upwelling reaches further downwind than the R E G upwelling. T h e slope of both thermoclines is comparable and reduced as compared to the 24 hour results. The change in thermocline slope indicates it is reaching an equi l ibr ium posit ion. The sharp spike in the temperature profiles, at 20 k m downwind, is caused by the si l l at the entrance of the West Bas in . W h i l e this spike eventually disappears from the R E G results it is s t i l l prominent for the C E M panel creating a very sharp interface. T h e South-North thalweg for a w ind direction of 270° is shown in Figure 4.6. In general there is very l i t t le movement over the 7 day period. The downwind end does show some pinching of the thermocline after 3 and 7 days for both the C E M and R E G results. Th i s pinching is l ikely due to the downwind flow and bui ldup of water and results in a smal l downwind slope of the upper thermocline. Below this level there is l i t t le movement. W i n d f r o m 250° Temperatures at M 2 and M 8 are shown in Figure 4.7 for a w ind direction of 250° . Results at location M 2 (upper panel) show evidence of upwelling for both C E M and R E G simulations. Upwel l ing occurs quicker and for colder and deeper water for C E M than for R E G . Th i s relates to increased movement of the thermocline due to C E M ' s greater wind stress wi th surface roughness height. A t location M 8 45 Quesnel Lake, EW Thalweg-REG 02-Aug-03 (12:00) Quesnel Lake, EW Thalweg - CEM 02-Aug-03 (12:00) 20 40 60 80 ( k m l 100 0 20 40 60 80 l k m l 100 Figure 4.5: Compar ison of basin scale movements wi th and without accounting for surface roughness. Panels represent progression of isotherms (°C) along Quesnel Lake's East-West thalweg for wind speed of 5 m / s from 270° . T h e top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have. (bot tom panel) there is l i t t le observable movement w i th the exception of some minor perturbations. A s a result there is no difference between C E M and R E G results. The East-West thalweg for a wind direction of 250° is shown in F igure 4.8. Upwel l ing at the upwind end of the basin begins to occur for bo th C E M and R E G after 3 days. After 7 days the magnitude and downwind extent of upwell ing is significantly higher for C E M results. B o t h C E M and R E G show water from below the thermocline reaching the lake surface. Downwind pinching of the thermocline is s imilar and downwind sloping of the thermocline is evident at a l l stages, w i th a reduced slope after 7 days. Once again the effect of the s i l l is more evident for C E M than for R E G creating a sharp interface at the thermocline. T h e South-Nor th thalweg for a wind direction of 250° is shown in F igure 4.9. Hi Quesnel Lake, NS Thalweg - REG Q2-Aug-03 (12:00) Quesnel Lake, NS Thalweg - CEM 02-Aug-03 (12:00) 0 • I 10 10-20 30 K7-*1S -- i d 15 20 25 3 0 | k m ) 04-Aug-03 112:00) , 5 20 25 3 0 l K m | 04-Aug-03 (12:00) - i s -— —--—« I Q IS 20 25 30 1 K m ' Ofl-flug-03 (12=00) 15 20 25 3 0 l K m l OB-Aug-03 (12:00) -7«V 10 15 20 25 30'*' 10 15 20 25 30 , K : Figure 4.6: Compar ison of basin scale movements w i th and wi thout accounting for surface roughness. Panels represent progression of isotherms ( °C) along Quesnel Lake's South-Nor th thalweg for w ind speed of 5 m / s from 270° . The top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have. Over the 7 day period there is l i t t le variation between C E M and R E G results. B o t h show some pinching at the upwind and downwind ends of the basin, but in general show li t t le evidence of significant movement or thermocline slope. Wind from 340° Temperatures at M 2 and M 8 are shown in Figure 4.10 for a w i n d direction of 340° . Loca t ion M 2 (upper panel) experiences a wave-like mot ion w i t h the onset of the wind as the isotherms in i t ia l ly lower then rise again toward the surface. After the wave passes there is l i t t le movement at a l l depths. Compar i son of C E M and R E G results shows l i t t le variat ion over the length of the s imulat ion. S imi lar to M 2 results, locat ion M 8 (lower panel) shows l i t t le movement of the isotherms or 47 o 1 1 / — ' 08/02 I 08/03 i ^ - • ~~ i 08/04 08/05 08/06 M8 - J003 - Wind From 250 Deg. 08/07 08/08 l l 1 1 1 1 1 i i I i I i i 08/02 08/03 08/04 08/05 08/06 08/07 08/08 Figure 4.7: Simulated isotherms of Quesnel Lake over a 7 day period, for a wind speed of 5 m / s from 250° at T - C h a i n 2 (M2) and 8 (M8) . Compar ison of results from a wind stress using a constant C i o ( R E G ) , t h in line, and a w ind stress which accounts for z0 ( C E M ) , thick line. The upper contour line represents a temperature of 15°C followed by 10°C, 7 ° C , and 5 ° C (if visible). Quesnel Lake, EW Thalweg-REG 02-Aug-03 {12.-00) Quesnel Lake, EW Thalweg - CEM 02-Aug-oj (12:00) 0 20 40 60 80 l k m l 100 0 20 40 60 80 , k m l 100 Figure 4.8: Compar i son of basin scale movements w i th and wi thout accounting for surface roughness. Panels represent progression of isotherms ( ° C ) along Quesnel Lake's East-West thalweg for w ind speed of 5 m / s from 250° . T h e top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have. 48 Quesnel Lake, NS Thalweg - REG 02-Aug-01 (12:00) Quesnel Lake, NS Thalweg - CEM 0 02-Aug-03 (12:00) 15 • • _ 10 15 20 25 04-Aug-03 (12:00) 04-Aug-03 (12:00) - W -15 20 25 OB-Aug-03 (12:00) OB-Aug-03 (12:00) 10 15 20 25 Figure 4.9: Compar ison of basin scale movements w i th and wi thout accounting for surface roughness. Panels represent progression of isotherms ( °C) along Quesnel Lake's South-Nor th thalweg for wind speed of 5 m / s from 250° . The top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have. variat ion between between C E M and R E G simulations. T h i s lack of movement is l ikely due to the min ima l basin length and fetch resulting i n low w i n d stress. The East-West thalweg for a wind direction of 340° is shown i n Figure 4.11. Movement of isotherms in and below the thermocline are nearly identical for C E M and R E G over the 7 day run. A t these depths both simulations show only minor movement, the largest occurr ing near the s i l l . T h e downwind slope and pinching of the thermocline are also similar. The surface isotherms of C E M show increased movement in comparison wi th R E G . Notab ly the upwelling of C E M ' s 15°C water, that also occurs near the s i l l . W h i l e the basin length is m in ima l , it is apparent that the increased drag coefficient of C E M has affected the w i n d stress and subsequent internal movements. The magnitude of the variat ion is not the extent of previous 49 M2 - 2003 - Wind From 340 Oeg. Figure 4.10: Simulated isotherms of Quesnel Lake over a 7 day period, for a wind speed of 5 m / s from 340° at T - C h a i n 2 (M2) and 8 (M8) . Compar i son of results from a wind stress using a constant C 1 0 ( R E G ) , t h in line, and a w ind stress which accounts for z0 ( C E M ) , thick line. The upper contour line represents a temperature of 1 5 ° C followed by 10°C , 7°C, and 5 ° C (if visible) . wind directions due to the reduced affect of basin length. Thus this highlights the impact accounting for surface roughness has on internal seiching. The South-Nor th thalweg for a wind direction of 340° is shown i n Figure 4.12. Once again there is l i t t le variat ion between the C E M and R E G results. B o t h sim-ulations show pinching of the thermocline at the nor th end of the lake as well as a thermocline sloping i n this direction. Th i s indicates that the general Northwest direction of the Nor th A r m results in the south end of the A r m (Lake Junct ion) being the upwind end of the basin. Thus the surface water is flowing to the north end of the a rm causing the downward movement of the thermocline here. A s the basin length and fetch is min ima l for this fetch, the internal movements are min ima l and are not affected by accounting for surface roughness height. D i s c u s s i o n Simulat ion results indicate that the inclusion of a surface roughness height impacts basin scale circulat ion. Increased movement and upwelling are evident i n the West 50 Quesnel Lake, EW Thalweg-REG 02-Aug-03 H2:00) Quesnel Lake, EW Thalweg - CEM 02-Aug-03 0 20 40 60 80 l k m ' 100 0 20 40 60 80 l k m l 100 Figure 4.11: Compar ison of basin scale movements w i th and without accounting for surface roughness. Panels represent progression of isotherms ( ° C ) along Quesnel Lake's East-West thalweg for w ind speed of 5 m / s from 340° . T h e top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have. Bas in w i t h smaller effects apparent in the Lake Junct ion and N o r t h A r m . A s both C E M and R E G were subjected to the same uniform and steady wind speed, when comparing for a set wind direction only the drag coefficient differs. The effect of surface roughness height on the drag coefficient is seen in Figure 4.13 which plots the variat ion in drag coefficient along the East-West (Figure 4.13-a) and South-Nor th (Figure 4.13-b) thalweg. The figure shows varying values of C i o along the thalweg due to variations in fetch and wind-wave conditions. T h e large variations of C i o along the East-West thalweg (Figure 4.13-a) are due to sharp changes i n basin direction resulting in changes in fetch. W i n d directions of 270° and 250° have the highest C i o values as they are better aligned wi th the lake geometry, creating larger fetch. T h e increased fetch results in increased wind-wave ac t iv i ty and surface 51 Quesnel Lake, NS Thalweg - REG 02-Aug-03 112:00) Quesnel Lake, NS Thalweg - CEM 02-Aug-03 (12:00) 01 ' ' ' ' • • r 01 ' ' ' ' ' ' r 0 5 10 15 20 25 3 C , k m ) 0 5 10 15 20 25 3 0 , k m ) OB-Aug-03 (12:00) OB-Aug-03 (12:00) 0 5 10 15 20 25 3 0 l k m l 0 5 10 15 20 25 3 0 ( k m l Figure 4.12: Compar ison of basin scale movements w i th and without accounting for surface roughness. Panels represent progression of isotherms ( ° C ) along Quesnel Lake's South-Nor th thalweg for wind speed of 5 m / s from 340° . T h e top panels are snap shots after 1 day, the middle panels after 3 days and the bo t tom panels after 7 days. Panels on the left ( R E G ) have not considered surface roughness height while panels on the right ( C E M ) have. roughness height. There is also a significant variat ion of C i o along the thalwegs. For the East-West case values start high and decrease towards mid-length before increasing again. The South-North thalweg shows gradually increasing C i o along the entire length. Compar ing average C i o values in Figure 4.13 shows significant differences be-tween the 270° and 250° simulations and the 340° s imulat ion. T h e latter has a lower average C i o due to its min ima l fetch. The 270° and 250° simulation's larger C i o values result in larger w ind stress and internal seiching as shown in the previous sections. Cau t ion however should be observed when interpreting these results due to the various assumptions stated previously. In addi t ion relating C i o to w i n d stress does not account for the wind stress direction which impacts the internal seiching 52 Figure 4.13: Compar ison of Drag Coefficient along the a.) East-West and b.) South-N o r t h thalwegs. Values are taken at the complet ion of the s imulat ion and account for surface roughness height due to wind-wave conditions ( C E M simulations. In comparison the R E G simulations used a Drag Coefficient of 1.4 x 1 0 - 3 . along wi th the magnitude of shear. Rela t ing the results in Figure 4.13 to basin movements depicted in the previous sections explains the variations seen between C E M and R E G results. For the East-West case the high drag coefficients at the west end contr ibuted to the increased occurrence of upwelling. A l o n g the South-Nor th thalweg for w i n d directions of 250° and 270° significant downwind bui ldup of surface water was expected. However the high downwind C i o values explain why this bui ldup was min imized , as the majori ty of the thalweg was not subjected to significant w i n d stress. Instead the wind stress peaked at the downwind end and had min ima l affect on the internal movements. A l l of these results highlight the impact surface roughness height has on internal seiching. 4.4.2 Basin Length and Fetch A s outl ined previously the major contributors to basin scale movement are both the shear velocity (wind speed and surface roughness height) and the basin length. The variat ion of basin scale movements due to the latter can be determined through comparison of C E M results provided in the previous section. 53 Considering only the East-West thalweg provides a good indicat ion of the impact basin length has on internal seiching. Bas in lengths are similar for 270° (Figure 4.5) and 250° (Figure 4.8) w ind directions but significantly shorter for 340° . The effect of the shorter basin is immediately apparent as the latter w ind direction results in very l i t t le basin scale motion. T h e extent of the basin length's impact on internal lake mot ion is clearly larger then the effect of surface roughness height. It is s t i l l apparent that surface roughness height is an important factor when simulat ing internal lake circulat ion, however the importance of the basin length highlights the need to properly determine the wind direction for accurate results. 4.5 Summary Three wind direction scenarios were simulated for Quesnel Lake using the hydrody-namic model E L C O M . T w o methods of wind stress calculat ion were compared to determine the impact of surface roughness height on wind-induced lake circulat ion. A l l three wind directions showed variations between constant C i o results and results using a surface roughness height. In general, accounting for surface roughness height results i n larger wind stress values and increased movement of the thermocline. Va l ida t ion of the results is difficult due to the use of an ideal w ind condit ion. It was assumed that the wind speed stayed constant at 5 m / s and blew in the same direction for 7 consecutive days. In reality both the wind speed and direction vary significantly over smaller periods. Thus, detailed comparison of model results w i t h field data is not possible. A general comparison however is s t i l l useful and insightful. Figure 4.1 plots the recorded temperature data at station M 2 (within the West Basin) and station M 8 (at the junct ion of the three arms). The most noticeable aspect of the field data is the dis t inct ion between the ac t iv i ty observed 54 in the West Bas in and at the Lake Junc t ion (M8) . Osciallat ions of the thermocline are significantly larger and at a higher frequency in the West Bas in (M2) than at the Lake Junct ion. In addi t ion the 10° C water nearly reaches the surface in the West Bas in but remains well below a depth of 10 m at the Lake Junct ion . Simulated temperature data for w ind directions of 270° and 250° shows several similarities to the field data. B o t h simulated wind directions show significant differ-ences in internal act ivi ty between the West Bas in and the Lake Junct ion . Frequent thermocline movement and upwelling occurs at M 2 while M 8 shows comparatively l i t t le movement. In addi t ion 1 0 ° C water reaches the surface in the West Bas in yet remains below 10 m at the Lake Junct ion . Results for a 340° w ind direction show li t t le internal movement i n the West Bas in or at the Lake Junct ion . The lack of movement is due to the smal l wind stress and m i n i m u m basin length resulting for a 340° wind . Compar ing wi th field results indicate the l ikel ihood of less frequent occurrence of w ind from this direction and internal movements dominated by wind directions of m a x i m u m basin length and wind stress. A l l s imulat ion results have noticeably smoother movements of the isotherms as compared wi th the field data. T h i s is l ikely due to a combinat ion of the simulations uniform wind speed and direction as well as some smoothing by the model . The simulated lake movement shows a gradual bu i ld up to an equi l ibr ium slope. A s seen i n the field data, a realistic non-uniform and unsteady wind results in numerous internal movements continually affecting one another. 55 C h a p t e r 5 Conclusions and Recommendations Increasing awareness of the potential impacts of global warming has highlighted the importance of and risk to, our world's freshwater supply. Through comprehensive field research programs combined wi th accurate prediction tools, a more complete understanding of freshwater processes can be determined. In particular, the abi l -i ty of numerical modeling to simulate lake circulat ion has provided significant in -sight into lake hydrodynamics and future climate change predictions. Cont inua l improvement of the numerical model ing process is essential to ensure an accurate representation of current and future scenarios. A n attempt to improve the deter-minat ion of wind-driven lake circulat ion investigated the importance of the lake's surface roughness in calculat ing the wind stress. A s the wind is a major dr iv ing force behind lake circulat ion, the transfer of momentum from the air to the water must be properly interpreted. Through the determination of the wind-wave field a surface roughness height is calculated that reflects the roughness the wind feels as it blows across the water surface. The wind stress then becomes a function of the wind speed and a drag coefficient which in turn varies w i th wind speed and wind-wave conditions (surface roughness height). In the past the drag coefficient was considered to be constant or s imply a function of w ind speed. In contrast, the new method provides a spat ial ly and potential ly temporal ly varying drag coefficient and resultant w ind stress. T h e new method of determining wind stress was applied to an analyt ical solution 56 for circulat ion and seiching in a long, narrow density-stratified lake. A s compared wi th t radi t ional methods of w ind stress determination (constant drag coefficient or varying wi th w ind speed) the new method resulted in significant differences in internal seiching amplitudes and surface layer currents. A s only a linear 2D c i rcu-lat ion solution was applied, the results are only applicable as a guideline. T h e y are useful in indicat ing the potential importance of surface roughness height i n more complex and realistic scenarios. Based on the indications of this result the new wind stress method was applied to a more complex hydrodynamical model . The chosen model, the Es tuary and Lake Computer M o d e l ( E L C O M ) currently assumes a con-stant drag coefficient in calculation of the w ind stress. To test the effect of surface roughness height a constant and uniform w i n d speed was applied to Quesnel Lake using both the new and current w ind stress methods. For cases of m a x i m u m fetch and basin length, the new wind stress method resulted in increased upwelling of deeper, colder water at the upwind end of the basin in comparison wi th a constant drag coefficient method. T h i s effect was highlighted by applying a w ind direction of min ima l fetch and basin length. W h i l e to a smaller degree then for the maxi -m u m case, variations between the two methods were evident. T h i s result suggests that w i th basin length effects minimized, the impact of surface roughness height on internal seiching is s t i l l significant. Combin ing results indicates accounting for surface roughness height has a significant impact on lake circulation! W h i l e the methods applied for calculat ion of a surface roughness height depen-dent w ind stress were simplified the results are useful in several ways. The method-ology in conjunction wi th a complex hydrodynamical model such as E L C O M can be applied to simple basin situations. App l i ca t i on to a lake such as Quesnel Lake may not be feasible due to the complexities introduced by topographic features and spatial and temporal variations in wind speed and direction. However the method 57 is effective for small to medium sized lakes subjected to uniform wind speeds. The results also provide an indicator of future improvements to increase the accuracy of current hydrodynamic models. The two major areas of improving the determination of surface wind stress are the improvement of the applied wind speed and the determination of the wind-wave field. For the latter the method applied here used empirically-based fetch-and duration-limited estimations of wind-wave conditions. The method is based on ocean data and is meant for initial estimations for Coastal Engineering design. The equations provide only an indicator of wave heights for small to medium sized lakes which are not subjected to the steadier conditions of the open ocean. As a result the accuracy of the wind-wave values and wind stress calculations were restricted. Application of a more complex wind-wave model would improve the methodology greatly. Notably, accounting for wave propagation and the effect of changing wind direction would be ideal for basins subject to topographical wind effects and unsteady wind speed and direction. Wind speed and direction improvement can be accomplished through applied temporal and spatial variation. In a basin such as Quesnel Lake, the wind vector varies significantly from one end of the basin to other. However, accurately measur-ing these variations is rarely feasible in lake studies. Instead other methods must be applied to provide accurate wind representation. These methods may include increased focus on wind monitoring, wind modeling from limited data or new tech-nologies such as satellite mapping. As discussed in Section 2.1 spatial and temporal averaging may be ideal middle ground for determining an accurate wind field from minimal data. With improved wind representation the importance of the surface roughness height will be easier to determine through comparison with field data. Comparison of several wind directions applied to Quesnel Lake provided insight 58 into the impact of basin length on internal seiching. It was evident that wind direc-tions oriented along the lakes axis of m a x i m u m basin length resulted in max imum movement of the thermocline as predicted by the linear equations in Section 3 .2 . This highlights the importance of chosen wind direction when simulat ing lake hy-drodynamics. For example, when investigating potential flooding due to basin-scale seiching of the surface, the wind direction is l ikely chosen based on wind magni-tude or frequency. Thus , a record of previous wind occurrences would be analyzed to determine the max imum wind speed wi th corresponding wind direction; or the most frequently occurring wind direction w i t h corresponding wind speed. However, it is the basin length which influences the surface seiching to a larger extent than wind speed. Therefore, to calculate the m a x i m u m lake response, the wind direction should be chosen based on m a x i m u m basin length. 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Pierson, W . , The spectral ocean wave model ( S W O M ) , a northern hemisphere com-puter model for specifying and forecasting ocean wave spectra, F i n a l Repor t N o . D T N S R D C - 8 2 / 1 1 , U . S . Naval Oceanography C o m m a n d Detachment, Ashevil le , N C , 1982. Potts , D . , The heat budget of Quesnel Lake, B r i t i s h Columbia , Master 's thesis, Univers i ty of B r i t i s h Co lumbia , 2004. Resio, D . , The estimation of wind-wave generation in a discrete spectral model, J . Geophys. Res., 11, 510-525, 1981. Resio, D . , S. Bratos, and E . Thompson, Coastal Engineering Manual, chap. 2, Meteorology and Wave Cl imate , Pa r t II, Hydrodynamics , pp. 1110-2-1100, U S A r m y Corps of Engineers, Washington, D C , 2002. 62 Ris , R . , Spectral modell ing of w ind waves in coastal areas, P h . D . thesis, Delft Univers i ty of Technology, 1997, delft Univers i ty Press. Roya l , L . A . , Problems i n rehabi l i ta t ing the Quesnel sockeye run and their possible solution, New Westminster, B . C , International Pacific Salmon Fisheries Commis-sion: 85 pp, 1966. Rueda, F . , S. Schladow, S. M o n i s m i t h , and M . Stacey, O n the effects of topography on wind and the generation of currents in a large mult i -basin lake, Hydrobiologia, 532, 139-151, 2005. Sarkkula , J . , Measur ing and model l ing wind induced flow in shallow lakes, in Hydrology of Natural and Manmade Lakes, Proceedings of the Vienna Symposium, vol . 206, 1991. Smi th , M . J . , C . L . Stevens, R . M . Gorman , J . A . Mcgregor, and C . G . Neilson, Wind-wave development across a large shallow inter t idal estuary: a case study of Manukau Harbour, New Zealand, New Zealand Journal of Marine and Freshwater Research, 35, 985-1000, 2001. Smi th , S., Journal of Geophysical Research, 93, 15,467-15,472, 1988. Smi th , S., et al . , Sea surface wind stress and drag coefficients: T h e H E X O S results, Boundary-Layer Meteoroi, 60, 109-142, 1992. Spigel, R . , and J . Imberger, The classification of mixed layer dynamics in lakes of small to medium size, J. Phys. Oceanogr., 10, 1104-1121, 1980. S P M , Shore Protection Manual, 4 ed., U S A r m y Engineer Waterways Exper iment Stat ion, Vicksburg , M S , 1984. Sverdrup, H . , and W . M u n k , W i n d , sea and swell - theory of relations for fore-casting, H . O . P u b . N o . 601, U . S . N a v y Hydrographical Office, Washington, D . C . , 1947. S W A M P , Ocean Wave Modelling, P l enum Press, New Y o r k and London , 1985. Taylor , P. K . , and M . J . Yel land , The dependence of sea surface roughness on the height and steepness of the waves, J. Phys. Oceanogr., 32, 572-90, 2001. Thompson, W . , Effect of the obstruction at Hell ' s Gate on the sockeye salmon of the Fraser River , International Pacific Salmon Fisheries Commiss ion, Bu l l e t i n 1, 1945. Toba, Y . , N . Iida, H . Kawamura , N . Ebuch i , and I. Jones, Wave dependence on sea-surface wind stress, J. Phys. Oceanogr., 20, 705-721, 1990. 63 Tolman, H . , A th i rd generation model for w ind waves on slowly varying unsteady and inhomgenous depths and currents, Journal of Physical Oceanography, 21, 782-797, 1991. V a n Vledder , G . P. , J . de Ronde, and M . Stive, Performance of a spectral wind-wave model in shallow water, i n Proc. 24th Int. Conf. Coastal Engineering, pp. 761-774, 1994. W A M D I , The W A M M o d e l - a th i rd generation ocean wave prediction model, Journal of Physical Oceanography, 18, 1775-1810, 1988. Welsh, D . , R . Wang, P . Sadayappan, and K . Bedford, Coup l ing of marine and circu-lat ion and wind-wave models on parallel platforms, E R D C M S R C P E T Technical Report 99-22, U . S . A r m y Engineer Research and Development Center, Vicksburg , M S . , 1999. Wuest, A . , and A . Lorke, Small-scale hydrodynamics i n lakes, Annual Review Fluid Mechanics, 35, 375-412, 2003. Young , I., Wind Generated Ocean Waves, Elsevier Science L t d , Oxford, U K , 1999. 64 Appendix A Analysis of Heaps and Ramsbottom (1966) The methodology and derivations behind the work of Heaps and Ramsbottom (1966) are summarized in this section. The analyt ical solution is based on the basin con-figuration shown in Figure A . l . z p , p 2 '2 1 • U j x=0 x=L Figure A . l : Layout of the basin used i n Heaps and Ramsbottom (1966) T h e assumptions are as follows: • Two-dimensional in the x and z planes; • Coriol is forcing is neglected; • L inear i ty is assumed; • Horizonta l shear stresses in the x-direction are neglected; 65 • Based on Hydrostat ic , Reynolds Averaged Navier Stokes equations; • T h e basin has two homogeneous layers w i t h p\ and K\ and pi and K 2 constant in the top and bot tom layers respectively. Here, p is the water density and K is the vertical eddy viscosity. Based on these assumptions the hydrodynamical equations reduce to: Dw 1 dp r^d2w ., , m=--pd-z-9 + Kd^ ^ du dv dw dx dy dz The M o m e n t u m equation for incompressibil i ty can be further reduced, in the x and z-plane, to: 9u _ _ l 9 £ , J^d^u dt p dx " r dz7 ^ ^ In the surface layer (1), the z-momentum equation is integrated from some depth z (above the bot tom layer (2)) to the surface, 771: J ^dz = J -pigdz (A.5) p{vi) - v{z) = -pig(m - z) (A.6) 66 p(vi) - P(Z) = -PMVI ~ Z) (A-7) In the bot tom layer (2), the z-momentum equation is integrated from some depth z (in Layer 2) to the surface, rji. T h i s required integration in parts, from z to (-hi + 772) and from (-hi + 772) to r]x: r j i n\ -hi+V2 Vi J -T^dz = J -pigdz = J -p2gdz - J -pigdz (A .8) z z z —hi+r/2 PiVi) - P{z) = -P2d{-hx + m - z) - pig{rn + hi - rj2) (A.9) p(z) = 75(771) + p2g(-hi + 772 - z) + pig(r)i + hi - 772) (A.10) For Layer 1, knowing = 0, take the derivative of E q n : 9: _______ = __L _____ -_ _ _____ (A 111 Pi dx pi dx dx T h i s solution (Eqn: 13) can then be substituted into the x-momentum equation: dux _ 1 dp d2m - d T - ~ 1 d x - + K l ^ ( A - 1 2 ) dui _ _3?7i , d2 Simi la r ly for Layer 2: .__.___ = ___.5____._ p±g _ _!_ 1 (- A 1 4) P2 dx p2 dx p2 \ dx dx ' 67 du2 _ 1 dp d2U2 . dt p2 dx 2 dz2 du2 pi drj! ( p i \ dr)2 d 2 u 2 ~ d t = - - 7 i 9 ^ - { 1 ~ 7 2 ) g ^ + K*~d# ( A ' 1 6 ) T h e assumed Boundary Condi t ions are as follows: 1. No-Sl ip Boundary conditions on the sidewalls. m = u2 = 0 @ x = 0,1 (A.17) 2. A Free-Slip bot tom condit ion resulting in zero-bottom friction. -p- = 0@ z = -(h1 + h2) (A.18) az 3. The internal friction at the surface of discontinuity is zero. T h i s implies that at the interface the water i n the top layer is able to slide freely over the water in the second layer. du\ du2 ^ - ~ = -^- = 0@z = - h 1 + r h - - h 1 (A.19) 4. The wind stress across the surface of the lake is represented by the shear velocity u*. Us ing the Heavyside step Funct ion the wind turns on at t > 0. for t > 0, ul = - f = v§% 9* c°Pfw @ z = m t n 0 ' * Po dx po ( A_20) for t<0, u2 = 0 @ z = r n ^ 0 68 • where the shear stress r is represented by: s . . /"mrx\ ., _ . r = i f ( t J A n a i n ( — ( A - 2 1 ) • where H(t ) is the Heavyside step function, A n is the m a x i m u m ampl i -tude of the shear stress and n is a positive integer which can provide spat ial variance i n the x-direct ion for the shear stress. T h i s variance is accomplished by superimposing several sine functions. For purposes of i l lustrat ion, n is set to 1 resulting in a simple wind stress dis t r ibut ion in the shape of a sine curve along the length of the lake. Thus, r = 0 at x=0 , l and rmax = An at x = 1/2. Therefore at the centre of the lake (x = 1/2), u2 = Solution A p p l y i n g the boundary conditions listed above the hydrodynamic equations for surface and internal seiching and horizontal velocities in the surface and bot tom layer are derived. The results is as follows: -I An + lAr. [nnpihig mrpihig(Pf - Pi) {01 ~ gh2)cos nirPit {02 ~ gh2)cos (A.22) m = lAr, + h2glAn -cos-nnftit ~ j f ° ' -nir(32t cos nixx _nir{p2 - px)hxg ' nirp2((J2 - ffi) \P2 (A.23) B o t h the surface and internal seiching are composed of three components: steady-state and seiche of periods (i = 1,2). The steady-state component is the slope of interface as a result of the wind stress. T h e seiching components are the interface 69 oscillations varying wi th t ime. Steady State The steady state component for movement at the surface is: m -lAr. [UTTX COS ( I (A.24) B y setting n = l , and substi tut ing an average wind amplitude, An = 2 ^ , E q n : 26 becomes: m (A.25) hig 2pi The solution shown i n E q n : 27 is the same as that found in Spigel and Imberger (1980) as: V = -ujL Hg 2 where ui = ^ * P Simi lar ly for the internal movement at the interface: mr(p2 - p\)hig 'nnx cos Using the same assumptions as for the surface layer: (A.26) (A.27) I Ar, m = ghi{p2 - Pi) 2 and input t ing pag' = g(p2 — p\) gives the solution: (A.28) A n I g'hip02 (A.29) 70 Once again this is similar to the solution found in Spigel and Imberger (1980): Time Varying Components The other two components of motion are t ime varying seiche of period (i = 1,2). One component is associated wi th Baroc l in ic oscillations and the other w i th Barotropic . Barotropic Oscillations For the Surface Layer, there is an oscil lation of the form: lAn(/32-gh2) nirPit /nrrxN / A Q I \ = n n ^ M f i - f i r — ™ (—) ( A ' 3 1 ) where cos (^j^) is the d is t r ibut ion along the x-axis due to the w ind shear and cosni*^x 1 represents the period of the oscil lat ion. Th i s is usually wri t ten in the form, cos2^ which gives a period of: 21 • T = — (A.32) S imi la r ly for the bo t tom layer the oscil lation is of the form: h2glAn nnPit /nnx\ V 2 = n n P 2 p 2 ( p 2 - P 2 2 ) C O S ~ T C O S ( A - 3 3 ) and also has a period of: T = 4 " (A.34) n/3i Baroclinic Oscillation For the Surface Layer, there is an oscil lation of the form: 71 where cos ( ^ f 2 ) is the dis t r ibut ion along the x-axis due to the wind shear and c o s n 7 r ^ i represents the period, usually wri t ten i n the form, c o s ^ p . Thus the period is, T = ~ (A.36) S imi la r ly for the bot tom layer the oscil lation is of the form: h2glAn nixf32t /nirx^ ^2 = —M^cos—;—cos ~T~ (A.37) and also has a per iod of: T = \ (A.38) n/3 2 . Compar ing r\\ and r\2 Assuming p\ « p2 the following simplifications can be made: P\ = gQn + h2) ^2 hx+h2 \ L P2 ) For the Barot ropic mode of oscil lation this gives a period for both r / i and r\2 of: (A.39) T = — 2 1 (A.40) ny/g(hi + h2) Equa t ing rji and rj2 for the simplified Barot ropic mode yields: 72 A s generally, h 2 » hi, E q n : 43 states that v?i = 772 • Simi la r ly for the Baroc l in ic mode after simplification the periods become: r _ _ { i _ _ _ a + ' ) r - ( A . 4 2 ) n [9 P2 - Pi \h2 hi) J Equa t ing 771 and 772 for the Baroc l in ic mode yields: ? » = - ( l - ! £ ) - » - (A .43) Vi \ hij p 2 - pi Here p 2 ^ p i = ^ and ^ « 1000, which implies 772 >> 771. In general, the solution shows that the Barot ropic components of the surface and internal seiching are similar in magnitude. However, the Baroc l in ic component is magnitudes larger for the internal seiching then for the surface. 73 Appendix B Wind-Wave Equations The process of determining the wind-wave conditions and the resultant surface roughness height, drag coefficient and wind stress is described in the following sec-tions. The process is a set of M a t l a b files known as a toolbox. The toolbox is made up of several files called from the main running file or from wi th in another file. Required as input are the bathymetry file used in E L C O M (bathymetry.dat), the file which is used to run E L C O M (run_elcom.dat) and a file to run this tool-box (run_get_wind_speed.dat). The latter file is of a specific format s imilar to the run.elcom file. A sample of a l l input files is provided wi th the toolbox. T h e toolbox begins by opening and reading the E L C O M bathymetry file. F r o m this file it obtains information about the bathymetry including depths, gr id spacing and orientation. The run file for the toolbox is then loaded and the wind speeds, anemometer heights and wind directions are uploaded. F i n a l l y the run.elcom file is opened to determine the length of the E L C O M run, the number of time-steps and the durat ion of each time-step. Us ing the bathymetry and wind direction information the m a x i m u m fetch for the given wind speed is determined. The program steps along the grid in the downwind direction from the grid point of interest un t i l a land value is detected. The distance to the shore in the x and y directions is obtained and the fetch calculated. T h e error is approximately the size of a gr id space due to the accuracy of the bathymetry file. It is also possible to determine the fetch for every direction if the wind direction varies w i th time. However, consideration of memory l imitat ions is necessary and 74 min ima l gr id points may be necessary. After determining the fetch the durat ion the wind has blown for each time-step is calculated. If the wind has not yet reached the grid point then the durat ion is considered to be zero. T h e core file is then called to calculate the wind-wave field and wind stress. The wind speeds measured at a height of 10 m are determined based on the anemome-ter height and the given wind speeds. Th i s is accomplished through the A i r / S e a Toolbox which references the work of Large and Pond (1981) and Smith (1988). Us ing the corrected wind speed the wind-wave characteristics are estimated using fetch- and durat ion-l imited equations. These equations are taken from the Coastal Engineering Manua l , C E M (Resio et ai, 2002). The toolbox begins by calculat ing the friction velocity, u* from (Resio et al, 2002): u* = (CWIU^)1/2[m/s] ( B . l ) where U\o [m/s] is the wind speed and C i o [-] is an in i t i a l drag coefficient. The latter is estimated from (Resio et ai, 2002): C i o i = 0.001(1.1 + 0.035<7io)[-] ( B - 2 ) To determine if a wave is fetch- or durat ion-l imited tXiU is calculated as: X 0 - 6 7 * x , . = 7 7 . 2 3 a 3 4 3 3 [see] (B.3) uio 9 where g is gravitat ional acceleration [9.81 m/ s 2 ] and i X ) U is the durat ion required for a wave crossing a fetch X [m] under a w ind speed U\Q to become fetch l imi ted . Thus , i f the durat ion of the wind is greater than tXtU the wave has become fetch l imi ted, else it remains durat ion l imi ted. For durat ion-l imited growing seas the durat ion is converted to an equivalent 75 dimensionless fetch as: - 5 . 2 3 x 1 0 - ( £ ) " * [ - ] (B.4) where t [sec] is the durat ion of the wind . T h e dimensionless significant wave height for both fetch- and durat ion-l imited deep-water, growing seas are calculated from (Resio et al, 2002): gHs = 4-13x 10"2 (§) V2 H (B-5) & = 0.651 (94)1 ^  [-] (B.6) u* \u% J where Hs [m] is the significant wave height and Tp [sec] is the peak per iod which corresponds to the frequency of the wave energy spectrum's peak (Resio et al, 2002). Once the values of Hs and Tv are determined the l imi t of fully-developed wave state is imposed. For the fully developed case, Hs and Tp are a function of w ind speed only: ^ = 2.115 x 10 2 [ - ] (B.7) & = 2.398 x 10 2 [ - ] (B.8) u* If the dimensionless Hs and Tp calculated in Equat ions B .5 and B .6 exceed the values i n Equat ions B . 7 and B .8 , the latter values are used as the waves have become fully-developed. Us ing the resultant Tp values the wavelengths, Lp [m], are determined from: o T 2 LP = Y-[m) (B.9) 76 F r o m the wavelengths and significant wave heights the surface roughness height, z0 is calculated as: (B.10) where A and B are constants determined to be 1200 and 4.5 respectively (Taylor and Yelland, 2001). The surface roughness height is then used to calculate a new drag coefficient: C w = K2(ln^j , [ - ] . ( B . l l ) where K is von Karman ' s constant taken as 0.4 and z [m] is the height above the water surface, taken as 10 m. Final ly , the wind stress, r [kg/ms2] is calculated from: r = doPaU^kg/ms2}. (B.12) where pa is the density of air. T h e calculated wind stress cannot be easily used as an input for E L C O M , as the model calculates its own wind stress from a given U\Q and a constant C i o of 1.40 x 1 0 - 3 . Thus, using this constant C i o and the calculated r , a 'true' wind speed value is determined from Equat ion B.12. The 'true' w ind speed values are wri t ten to an A S C I I file in the format shown here: 77 1 'comment l ine' 'number of wet gr id points') 'number of data per gr id point ' 'number of timesteps' T I M E 'year' 'day' 'hour' 'row of gr id point ' ' column of grid point ' 'wind speed' 'wind direction' - repeated for every gr id point and every timestep -The created A S C I I file must then be converted to a U N F file using C O N V 2 D . l i n u x . Once the U N F file is created it may be called as an input file by run_elcom.dat file w i th the tag 'windfield_in_file' (see E L C O M User M a n u a l for more details). 

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