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Numerical models of the general circulation in the Strait of Georgia Marinone, Silvio Guido 1994

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NUMERICAL MODELS OF THE GENERALCIRCULATION IN THE STRAIT OF GEORGIABySilvio Guido MarinoneB. Sc. (Oceanology) Universidad Autónoma de Baja California, 1978M. Sc. (Physical Oceanography) CICESE, 1981A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESOCEANOGRAPHYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1994© Silvio Guido Marinone, 1994In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.OceanographyThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date: / /c,23 /flfc/4//ti (/AbstractNumerical models have been used to study the low frequency (< 0.5 cycles per day)currents of the Strait of Georgia. In the central part of the Strait, the observed mean(i.e., time-averaged) residual circulation is characterized by cyclonic eddies in the velocityfield of about eight kilometers in diameter. The observed low frequency currents accountfor approximately half of the total kinetic energy of the fluctuating currents; the otherhalf is associated with diurnal and semidiurnal tidal currents. These low frequencycurrents have about one-third of their energy at the fortnightly and monthly bands. Themagnitudes of the mean and low frequency fluctuating components are about the same.Two different models are used in this thesis to study these currents.The first is a depth-independent numerical model. The model reproduces the locationand scale of the observed depth-averaged mean residual but significantly underestimatesthe magnitude of the velocity. The model low frequency oscillations are basically confinedto the fortnightly band and once again the magnitude of the velocity is significantlyunderestimated.The second is a three-dimensional baroclinic model. This model has energy levels thatare comparable to those of the observed mean residual and low frequency components.The spatial and temporal structure are now reasonably reproduced especially for thefluctuating component.Bearing in mind the limitations of the models in reproducing the observed residual(particularly the depth-averaged model) various diagnostic analyses have been performedwith the aim of revealing the mechanism(s) maintaining the observed residual. It hasbeen found that no matter where the energy comes from (e.g., baroclinic forcing, wind11or tides) the non-linear interactions transfer energy to the low frequency bands.Deep-water renewal in the Strait of Georgia has been successfully modeled. Renewaloccurs in the model, during summer, once a month. The phase and amplitude of renewalevents compare well with those observed in the central Strait of Georgia. A deep waterrenewal event starts at extreme neap tides, the time of minimal mixing, when maximumstratification is reached at Boundary Passage and the denser waters of Juan de FucaStrait penetrate past the sill at Boundary Passage and reach the bottom of the deepbasin in the central Strait of Georgia as a gravity current.111Table of ContentsAbstract 11List of Tables viiiList of Figures xiiAcknowledgements xxix1 Background 11.1 Observations of the Strait of Georgia 11.2 Previous models of the Strait of Georgia 41.3 Objectives of this study 52 A one-layer model of the Strait of Georgia 72.1 The Model..., 72.2 Results 122.2.1 Time Average 122.2.2 Time variability. . 142.2.3 Fourier and harmonic analyses 172.2.4 Momentum and vorticity budgets 232.3 Discussion 332.3.1 Comparison with an analytic model 332.3.2 Comparison with observations 382.4 Summary 45iv47• • 47• . 5355. . . . . . . 5564. . . . . 797984939999111• . . . . . . . . . . . 116• . . . • 1271291291464 Deep Water Renewal in a multi-layer model of the Strait of4.1 Background4.2 Model Setup4,3 Results4.3.1 Evidence of Deep Water Renewal in the model4.3.2 The Deep Water Renewal cycle4.4 The general circulation3 A multi-layer model of the Strait of Georgia3.1 The Model3.2 Model simulations and analysis3.2.1 The general circulation3.2.1.1 Time-Averaged residual3.2.1.2 Time variability3.2.2 Fourier and Harmonic Analysis3.2.2.1 Fourier Analysis3.2.2.2 Harmonic Analysis3.2.2.3 Nonlinear interactions3.2.3 Budgets: momentum, vorticity and energy3.2.3.1 Momentum Budget3.2.3.2 Vorticity Budget3.2.3.3 Energy Budget3.2.3.4 Summary of budgets results .3.3 Discussion3.3.1 Comparison with observations3.4 SummaryGeorgia 148148151155157160185v4.4.1 Comparison with the reference run . . . .4.4.2 Comparison with observations . . . . . .4.5 Summary . .5 Conclusions 206223223224224224225225226C Derivation of the 3D model equationsC,1 Continuity equationC.2 Momentum equationC.2.1 Pressure termC.2.2 Relative accelerationC.2.3 Coriolis accelerationC.2.4 Vertical diffusionC.2.5 Horizontal diffusionC.3 Density equation22T. . . . . . . 228. . . . 228229. 230231231232232185193204Bibliography 214A Equations of Motion 218B Derivation of the 2D model equationsB,1 Continuity equationB,2 Momentum equationsB.2.1 Pressure termB.2.2 Relative accelerationB.2.3 Coriolis accelerationB.2.4 Vertical diffusionB.2,5 Horizontal diffusionviD Bottom Friction Stress Reduction 234E Discrete Vorticity Equation 236F Discrete Energy Equation 242viiList of Tables2.1 Forcing and produced low-frequency constituents. The amplitudes shown(given as a reference only) in the table correspond to the southern side ofJuan de Fuca Strait, the northern side is slightly different. . . 102.2 Percentage of the spatially-averaged variance explained by the given lowfrequency constituent for the regions identified in Figure 2.2b. . .., 202.3 Spatially-averaged mean momentum and vorticity budgets for the regionsshown in Figure 2.2b. The units are cm s2 and 10—10 s2 for momentum and vorticity, respectively 262.4 Region I percentage of the spatially-averaged variance for the given termsin the u-momentum equation 292.5 Same as Table 2.4 for the terms of the vorticity equation 303.1 Tidal constituents used to reconstruct the tidal forcing at the open boundaries. The amplitudes and phases are specified at the two connections tothe Pacific Ocean: Juan de Fuca Strait and Johnstone Strait. The amplitudes shown (given as a reference only) in the table correspond to thesouthern side of Juan de Fuca Strait, the northern side and JohnstoneStrait are slightly different 523.2 Tidal constituents used in the harmonic analysis. “OB” indicates that theconstituent is included in the tidal forcing at the open boundary 86vi”3.3 Percentage of the horizontally-averaged variance, explained by the givenlow frequency constituents for the i and the 8 layers of u, v and p’. Thecolumn Harms stands for the sum of the variance of the N2S, 4Sa, 8Sa,8Ssa, & l6Ssa constituents . . . . . .. 883.4 Ratio of the amplitude of the low frequency tidal constituents, betweenthe reference run (Ref) and runs 5 (R5) and 7 (R7), for the u andfields for the 8 model layers. The reference run has 44 tidal constituentsto reconstruct the tides at the open boundaries, Run 5 has 5 constituents(K1, 0, M2, S2 and N2, the same as used in Chapter 2), and run 7 has7 (same as run 5 plus 1(2 and F1). Only those constituents whose ratiosvary more than 10% averaged over the model domain in the central Straitof Georgia are shown 963.5 Acronyms and summary for terms in the momentum equations for theresidual 3.14 and 3.15 as they are referred to in the text. To facilitatethe identification of the abbreviated names, the last two letters have thefollowing meaning: the penultimate is T or R for tidal or residual, respectively, and the last letter is M for momentum . . . . .. 1063.6 Time-averaged terms from Table 3.5 for the 4 selected gridpoints (StationsI, II, III and IV) for layers 1 and 6 of the model. Units are in 106 m_2 In the last column L, I and S stands for large, intermediate andsmall, respectively, and in general gives the relative size of the terms inthe momentum budget of the model. When the numbers in the table aregiven in bold face it indicates that the classification of that terms as L, Ior S is true for that gridpoint in particular . . . . .. 108ix3.7 Acronyms and summary of terms of the residual vorticity equation 3.22as they are referred to in figures and text. To facilitate the identificationof the abbreviated names, the last two letters have the following meaning:the penultimate is T or R for tidal or residual, respectively, and the lastletter is V for vorticity 1153.8 Time-averaged terms from Table 3.7 for the 4 selected gridpoints (StationsI, II, III and IV) for layers 1 and 6 of the model. Units are in lO _2In the last column L, I and S stands for large, intermediate and small,respectively, and in general, gives the relative size of the terms in thevorticity budget of the model. When the numbers in the table are givenin bold face it indicates that the classification of that term as L, I or S istrue for that gridpoint in particular 1173.9 Acronyms and summary of terms in the residual energy equation 3.25.The last two letters have the following meaning: the penultimate is T orR for tidal or residual and the last letter is E for energy 1233.10 Time-averaged terms from Table 3.9 for the 4 selected gridpoints (StationsI, II, III and IV) for layers 1 and 6 of the model. Units are in 10—8 m2 s.In the last column L, I and S stands for large, intermediate and smaller,and in general, gives the relative size of the terms in the energy budgetof the model. When the numbers in the table are given in bold face,it indicates that the classification of terms as L, I or S is true for thatgridpoint in particular 126x3.11 Summary of the momentum, vorticity and energy budgets Recall thatthe last two letters have the following meaning: the penultimate is T orR for tidal or residual, respectively, and the last letter is M, V or E formomentum, vorticity or energy, respectively. The first two letters are Hor V for horizontal or vertical, respectively, PW for pressure work, CO forCoriolis, S for stretching/shrinking and D for diffusion and A for advection. 1303.12 Cyclesonde data levels used to construct the time series to which the modelresults are compared. Top and Bottom are the levels where the cyclesondeof the different stations were resting between profiles, see text. . . . . 1313.13 Amplitude (cm s’) of the fortnightly and monthly constituents for modeland observations, at stations 1 and 3, for the u and v components ofvelocity. When the amplitudes of the observations and model are withina factor of 3, the numbers appear in bold face in both the data and modelsections . . . 1414.1 Time-averaged density at the bottom for the gridpoints shown in Figure4.11. The average is over the same time period shown in Figure 4.10. Alsogiven are the density differences, the distance between the locations, alongpath A of Figure 4.3, and the density gradient. The density gradient istaken as the difference of o between the two consecutive locations dividedby the distance between them in km 1724.2 Amplitude (cm s’) of the fortnightly and monthly constituents for themodel and observations, at stations 1 and 3, for the u and v components ofvelocity. When the amplitudes of the observations and model are withina factor of 3, the numbers appear in bold face in both data and modelsections 202xiList of Figures1.1 Plan view of the Straits of Georgia and Juan de Fuca: the small rectangleis the area of study in Chapter 2 22.1 The bathymetry in the area of study in meters 92.2 (a) Time-averaged velocity vectors and surface elevation contours (in cm)in the central Strait of Georgia. (b) Position of the different regions referred to in the text as: I a northwestward flowing jet, II a cross channelflow, III a cyclonic eddy and IV a meandering current 132.3 Time series of the residual velocity components for a grid point in each ofthe regions identified in Figure 2.2b 152.4 Spatial structure of the (a) total and (b) de-meaned residual for two different times, approximately 7 days apart 162.5 Power spectra (plotted as f x S(f) with solid lines and as log[f x S(f)]with dashed lines) of the (a) surface elevation, (b) u and (c) v componentsof velocity. The ‘plus’ symbols locate the frequency of the low-frequencytides listed in Table 2.1 182.6 Amplitude (cm) of the Mf constituent for surface elevation 212.7 Spatial structure of the (a) Mf and (b) Msf constituents for velocity 222.8 Percentage of the variance explained by the (a) Mf and (b) Msf constituent for the velocity components 24xli2.9 Budget analysis of the u-momentum equation. The curves correspond toa spatial average over each of the four regions identified in Figure 2.2b.Three plots are shown for each of the four regions. In the top panel thecontributions to the tendency from the pressure gradient (dotted lines),Coriolis (dashed lines) and advection terms (plus sign) are plotted. Inthe middle and bottom panels the result of adding respectively the firsttwo and all three of these terms is shown in thin lines. The bold curve inthe latter two panels is the actual tendency, i.e., the sum of all terms ofthe momentum equation. Note that the middle and bottom panels haveexpanded scales 282.10 As Figure 2.9 for the vorticity equation budget. In the top panel the contributions to the tendency from the advection (plus sign), eddy diffusion(dotted lines) and stretching/shrinking (dashed lines) terms are plotted.In the middle and bottom panels the result of adding respectively the firsttwo and all three of these terms is shown in thin lines. The bold curve inthe latter panels is the actual tendency, i.e., the sum of all terms in thevorticity equation. Note that the scales of regions I and III are differentthan II and IV and the middle and bottom panels have expanded scales 312.11 Comparison of the actual section of Strait of Georgia (dashed line) in thearea of the northeastward flowing jet of Region I in Figure 2.2 with theidealized section of the Butman et al. (1983) tidal rectification model. . 342.12 Scatter diagram of the mean residual against the topographic length scale(h/Vh). The two vertical lines delimit the range of tidal excursions in themodel 36xlii2.13 Amplitude of the Mf constituent at each station as a function of the recordlength (thick curves). Thin (dashed) lines represent the amplitude of theMf constituent with the atmospheric contribution removed when the Mfconstituent is (not) included in the coherence calculations. The stationnumbers refer to the station number of each station shown in Figure 2.14 402.14 Time-averaged velocity from the observations in the central Strait of Georgia for the period June 1984 to January 1985 412.15 Time-averaged current speed at each station as a function of the averagingperiod . 422.16 Mean residual velocity for two-months of integration with different valuesof the eddy viscosity coefficient . 443.1 a) Horizontal layout of the 3-dimensional model. For clarity, the north partof the Strait of Georgia, Puget Sound (PS indicates where it joins Juande Fuca Strait) and Johnstone Strait have been clipped from this figure.The asterisk symbol shows the Fraser River entrance to the Strait. Thenumbers 1-14 show the wind station locations. b) Vertical layer structureof the model. c) Bathymetry, in meters, of the area of study (42 km x 42km). The labels I, II, III and IV are the same points as those in Figure 2.2b, 483.2 Cross-section 0j field at the open boundary at Juan de Fuca Strait for themonths of January and June, 1968 (from Crean and Ages, 1971) 543.3 Time-averaged velocity vectors in the central Strait of Georgia for layers 1to 8 along with the vertically-averaged field (frame number 9). The timeperiod of the average is one year. The depths for layers 1-8 are given inthe lower right-hand corner of each frame. The square insert is the studyregion of Chapter 2 56xiv3.4 Time-averaged wind velocity vectors in the central Strait of Georgia, Theaverage is over the same time period of 1 year as given in Figure 3.3 forthe velocities . . . . . 573.5 As Figure 3.3 without wind forcing 593.6 Difference between the time-averaged velocity field of the reference run(Figure 3.3) and the run without wind forcing (Figure 3.5) 603.7 Time-averaged ot for the different layers. In frame 9 the time-averagedsurface elevation is shown in cm. with a 1 cm contour interval 623.8 Time-averaged pressure (Pa) for the different layers. For plotting purposes, the pressure values of each layer have been obtained by subtractingfrom the total pressure of each layer a pressure value close to the minimumpressure of that layer; these values are shown at the bottom right cornerof the figure 633.9 Time series of the residual velocity components for selected grid pointsfor layer 1 of the model. The time is in Julian days with day 1 beingJanuary 1st, 1984. The points are the same as those shown in Chapter 2(see Figure 3.lc) 653.10 As Figure 3.9 but for layer 3 of the model. Note the change of vertical scale. 663.11 As Figure 3.9 but for layer 5 of the model. Note the change of vertical scale. 673.12 As Figure 3.9 but for layer 7 of the model. Note the change of vertical scale. 683.13 Time series of the wind velocity components for the same set of selectedgridpoints of Figures 3.9-3.12 69xv3.14 Correlation between along channel wind and along channel current velocityfor the four selected gridpoints (Stations I, II, III and IV) and for all layers.The dashed lines are at the correlation value (= 0.29) significant to the95% confidence level for 45 degrees of freedom. The degrees of freedomwas obtained by dividing the record length by the decorrelation time scale.The latter was estimated from the integral of the auto-correlation functionto its first zero crossing (= 8 days) 703.15 Time series of the residual velocity components for selected grid points forlayer 3 of the model for a run without wind forcing. Compare with Figure3.10 733.16 As Figure 3.9 for the vertically-averaged residual velocity components.Note the change of vertical scale 743.17 Horizontal velocity time series of the fluctuating residual, for layer 1. Daysare in Julian days starting 1 January 1984 753.18 Same as Figure 3.17 but for layer 3 . . . 763.19 Same as Figure 3.17 but for layer 5 . . . 773.20 Same as Figure 3.17 but for layer 7 783.21 Spatially-averaged power spectrum of surface elevation . . . 803.22 Spatially-averaged power spectra of the u component of velocity of the 8model layers 813.23 Spatially-averaged power spectra of the v component of velocity of the 8model layers 823.24 Spatially-averaged power spectra of the p’ field of the 8 model layers. 833.25 Power spectra of the 1984 data of a)the Fraser River discharge and b) andc) of the U and V components of the wind, respectively, in the centralStrait of Georgia 85xvi3.26 Percentage of the variance explained by the different tidal constituents forthe surface elevation (‘i) field 893.27 Percentage of the variance explained by the different tidal constituents forthe u component of velocity for layer 3 of the model 903.28 Percentage of the variance explained by the different tidal constituents forthe v component of velocity for layer 3 of the model 913.29 Percentage of the variance explained by the different tidal constituents forthe o field for layer 3 of the model 923.30 Amplitude A of a harmonic produced by the sum of the waves a cos(t—qi)and b cos(wt— 42) for the special case of a = 5 and q5 = 0 as a functionof q. The different curves correspond to different values of 953.31 Time series of bottom v-component of velocity, for a particular grid pointat the center of the study area, for the tidal (thin line: vi) and residual(thick line: v0) 1033.32 Time series of the fluctuating u-momentum terms of layer 7 for StationIII. Thin lines give terms arising from the residual; thick lines give termsarising from the tidal components. The vertical scale is the same for allplots (and is shown in the top one only). From top to bottom the panelsshow horizontal advection, vertical advection, the sum of the Coriolis andpressure gradient terms, horizontal diffusion, vertical diffusion, the sum ofall the residual and all the tidal terms and finally, the sum of all terms.See Table 3.5 for the meaning of the acronyms 110xvii3.33 Time series of the fluctuating vorticity terms for layer 7 of Station IlLThin lines correspond to the terms of the residual and thick lines tothose arising from the tidal components. The vertical scale is the samefor all panels and is only shown in the top one. From top to bottom,the panels show advection, horizontal diffusion, vertical diffusion, stretching/shrinking, the sum of all the residual and all the tidal terms and finally,the sum of all terms. See Table 3.7 for the meaning of the acronyms. . . 1183.34 Time series of the fluctuating energy terms for layer 7 of Station III. Thinlines correspond to the residual and thick lines to the tidal components.The vertical axes are the same for all panels; the scale is shown in thetop one only. From top to bottom the panels show horizontal advection,vertical advection, horizontal diffusion, vertical diffusion, pressure work,the sum of all the residual and all the tidal terms and finally, the sum ofall the terms. See Table 3.9 for the meaning of the acronyms 1283.35 Time-averaged velocity for the model (solid arrows) and for the observations (dashed arrows) in the central Strait of Georgia for the periodJune 1984 to January 1985 corresponding to layers 3 to 8. Cyclesondedata are obtained from vertical-averages over the model layer depths asgiven in Table 3.12. Layer numbers are given as L3 to L8; for cyclesondedata the first and last frames correspond approximately to L3 and L8, Infirst cyclesonde frame, 1 to 4 indicates the station number. Note that inthe frames where the cyclesondes velocities are shown, the model velocitiesare repeated for that particular gridpoint 133xviii3.36 Time series of the u and v velocity components (in cm s’) of the fluctuating residual from the observations (dotted lines) and from the model(solid lines) for stations 1 and 3 in the central Strait of Georgia. Shown atthe top is the third layer (15-30 m) and at the bottom the fourth layer (30-60 m). The velocity from the observations for these layers was obtainedas described in Table 3.12 1353.37 Time series of the u and v velocity components (in cm s’) of the fluctuating residual from the observations (dotted lines) and from the model(solid lines) for stations 1 and 3 in the central Strait of Georgia. Shownat the top is the fifth layer (60-90 m) and at the bottom the sixth layer (90-150 m). The velocity from the observations for these layers was obtainedas described in Table 3.12 1363.38 Time series of the u and v velocity components (in cm s’) of the fluctuating residual from the observations (dotted lines) and from the model(solid lines) for stations 1 and 3 in the central Strait of Georgia. Shownat the top is the seventh layer (150-250 m) and at the bottom the eighthlayer (250-H m). The velocity from the observations for these layers wasobtained as described in Table 3.12 1373.39 Time series of the o from the observations (dashed lines) and from themodel (thick lines) for station 3 in the central Strait of Georgia. Thenumbers at the right and top of each figure stands for the layer number.. 1393.40 Time-averaged velocity vectors in the central Strait of Georgia for layers1 to 8 along with the vertically-averaged field (frame number 9). Thetime period of the average is three-months. The horizontal eddy viscositycoefficient, £‘, of this run is 100 m2 s 144xix3.41 As Figure 3.40 with horizontal eddy viscosity coefficient, 1’, =200 m2 s1.This is simply the reference run averaged over three months 1454.1 Deep water renewal in the central Strait of Georgia during a) summer andb) winter 1494.2 Time varying density forcing at the open boundaries. The function issimply a Cos(at 4a) + bcos(2wat — cb&), where Wa is the annual frequencyand (a, q) and (b, qSb) are the annual and semiannual amplitudes andphases obtained by a least-square fit to the observations of Crean andAges (1971). The fit was done to each layer and then the amplitudes andphases were vertically-averaged. The mean has been removed: the abovedata vary about a mean value which is different for each layer, The thickline corresponds to the values of a and b obtained directly from the fit ofthe actual 1968 data: namely a = 0.36, and b = 0.12 in o unit. Thesevalues for a and b represent the seasonal density variability, vertically-averaged for the year 1968. Using the same phases, the thin line is for a= 1.5 and b = 0.5 in o unit, and this is the forcing density function usedto produce the results in this chapter . 1544.3 Location of Station III and the cross-sections of the model domain whereresults are presented. A connects Juan de Fuca Strait, Haro Strait, Boundary Passage and the central Strait of Georgia and B and C are cross-sections in the central Strait of Georgia traversing from Galiano Island tothe mainland. The square is the area where the results of Chapter 3 wereshown and is included here for reference only 156xx4.4 Time series of the low-frequency o field for the 8 model layers at StationIII (see Figure 4.3). The top line (with the highest density) corresponds tothe bottom layer and the bottom line corresponds to the top layer. Timeis in Julian days for 1984 1584.5 Time series of: a) (U2 + V2), where U and V are the vertically integrated velocities in Boundary Passage (in cms1), showing the mechanicaltidal energy (in cm2 _2), b) o for the bottom layer at Boundary Passage(dashed line) and at Station III in the Strait of Georgia (thin line), andc) along channel velocity component (in cm s1) in the bottom layer atStation III in the Strait of Georgia. The solid vertical lines give the timeof the extreme neap tides (approximately one per month) and the dashedvertical lines give the times when contour plots are presented in subsequentfigures (days 230, 234, 238, 242 and 246) 1594.6 A time series of low pass filtered o contour plots from Juan de FucaStrait (JF) to the central Strait of Georgia (SG) passing through HaroStrait (11S), Boundary Passage (BP) and a secondary sill (2S) along lineA of Figure 4.3. The contours are given at an interval of 1 o unit; contoursfor a below 22 are not shown. The o values of the bottom of the centralStrait of Georgia are given between parentheses at the right and bottom ofeach plot. The position of these times in the tidal cycle is given in Figure4.5: extreme neap day 232, extreme spring day 240, and moderate neap246. The number in the bottom left of the panels indicates the maximumdepth of the panel 162xxi4.7 Contour plot of at from Boundary Passage to the central Strait of Georgiaalong line A of Figure 4.3. Contour interval is 0.1 o unit; Ut contours below23.5 are not shown. The t values of the bottom of the central Strait ofGeorgia and at the bottom of the inner basin before the secondary sill aregiven between parentheses at the bottom right and center of each plot,respectively. The position of these times in the tidal cycle are given inFigure 4.5: extreme neap day 232, extreme spring day 240, and moderateneap day 246. The number in the bottom left of the panels indicates themaximum depth of the panel 1644.8 Contour plot of o from Boundary Passage to the secondary sill alongline A of Figure 4.3. This figure zooms in on the left side of Figure 4.7.The contour interval is 1.0 o unit; here all contours are shown. The ovalues of the bottom of the intermediate basin before the secondary sillare given between parentheses approximately at the center and bottomof each plot. The position of these times in the tidal cycle are given inFigure 4.5: extreme neap day 232, extreme spring day 240, and moderateneap day 246. The number in the bottom left of the panels indicates themaximum depth of the panel, which is the same as Figures 4.6 and 4.7 foreasier comparison 1664.9 Contour plot of aj from secondary sill to the central Strait of Georgiaalong line A of Figure 4.3. This figure zooms in on the right side of Figure4.7. The contours shown are limited to o = 22, 22.5, 23, 23.2, 23.4, 23.5,23.6, 23.7, 23.8 23.9 and 24. The position of these times in the tidal cycleare given in Figure 4.5: extreme neap day 232, extreme spring day 240,and moderate neap day 246. The number in the bottom left of the panelsindicates the maximum depth of the panel 167xxii4.10 Time series of o for the bottom layer at selected gridpoints along the pathA shown in Figure 4.3. The position of each grid point, labeled a to i, isgiven in Figure 4.11. At the top, the tidal energy (in cm2 s2) of Figure4.5 is repeated for reference. The vertical lines represent extreme neap tides. 1694.11 Location of the gridpoints referred to in Figure 4.10. The gridpoints arelocated along path A of Figure 4.3: (a) gives the position in the horizontaldomain and (b) shows the vertical section of path A 1704.12 Time series of contour plots of Froude number (as defined in the text)across Boundary Passage during an extreme neap tide. The plots aredraw every four hours. The insert in each frame shows the tidal height;the vertical scale is shown in the bottom panel only. The position in timeof each frame in the tidal cycle is indicated by a 0 in the insert. Themaximum values of Fr in each frame are 2.6, 1.7, 4.5, 1.7, 3.3 and 7.4.The number at the bottom left of the panels indicates the maximum depthof the panel 1754.13 Time series of contour plots of Froude number (as defined in the text)across Boundary Passage during an extreme spring tide. The plots aredraw every four hours. The insert in each frame shows the tidal height;the vertical scale is shown in the bottom panel only. The position in timeof each frame in the tidal cycle is indicated by a 0 in the insert. Themaximum values of Fr in each frame are 8.3, 10.0, 5.8, 10.0, 6.8, and 7.6.The number at the bottom left of the panels indicates the maximum depthof the panel 176xxiii4.14 Time series of contour plots of o in the southern section B of the centralStrait of Georgia (see Figure 4.3). The contours shown are limited to o= 23.0, 23.5, 23.7, 23.8, 23.9 and 23.95. The position of these times in thetidal cycle is given in Figure 4.5: extreme neap day 232, extreme springday 240, and moderate neap day 246. The number at the bottom left ofthe panels indicates the maximum depth of the panel 1774.15 Time series of contour plots of the along channel velocity component, incm s1, in the southern section B of the central Strait of Georgia (seeFigure 4.3). Thin contours are positive, dashed are negative and thickare zero. Positive velocity is flow to the northwest, up the strait. Theposition of these times in the tidal cycle is given in Figure 4.5: extremeneap day 232, extreme spring day 240, and moderate neap day 246. Thenumber at the bottom left of the panels indicates the maximum depth ofthe panel . 1784.16 Time series of contour plots of o in the central section C of the centralStrait of Georgia (see Figure 4.3). The contours shown are limited to o= 23.0, 23.5, 23.7, 23.8, 23.9 and 23.95. The position of these times in thetidal cycle is given in Figure 4.5: extreme neap day 232, extreme springday 240, and moderate neap day 246. The number at the bottom left ofthe panels indicates the maximum depth of the panel 179xxiv4.17 Time series of contour plots of the along channel velocity component, incm s, in the southern section C of the central Strait of Georgia (seeFigure 4.3). Thin contours are positive, dashed are negative and thickare zero. Positive velocity is flow to the northwest, up the strait. Theposition of these times in the tidal cycle is given in Figure 4.5: extremeneap day 232, extreme spring day 240, and moderate neap day 246. Thenumber at the bottom left of the panels indicates the maximum depth ofthe panel 1804.18 Contour plot (depth vs. time) of: a) the residual u component of velocity(cm s’; positive flow is into the Strait of Georgia) and b) the residualat Boundary Passage. The vertical lines indicate the time of the extremeneap and the extreme spring tides at days 232 and 240, respectively. Thenumber at the bottom left of the panels indicates the maximum depth ofthe panel 1824.19 Contour plot (depth vs. time) of the unfiltered: a) u component ofvelocity (contour interval is 30 cm s’; positive flow is into the Strait ofGeorgia) and b) o at Boundary Passage (contour interval is 1 o unit)during extreme neap tides (day 232 indicated by the vertical line). Thenumber at the bottom left of the panels indicates the maximum depth ofthe panel 1834.20 Contour plot (depth vs. time) of the unfiltered: a) u component ofvelocity (contour interval is 60 cm 5_i; positive flow is into the Strait ofGeorgia) and b) r at Boundary Passage (contour interval is 1 o unit)during spring tides (day 240 indicated by the vertical line). The numberat the bottom left of the panels indicates the maximum depth of the panel. 184xxv4.21 Time-averaged velocity for the 8 layers of the model along with the vertically-integrated field (frame 9) of the run involving deep water renewal. Compare the reference run, Figure 3.3. Depth of layers, in meters, is given inthe lower right hand corner of each frame 1874.22 Difference between the time-averaged velocity field of the deep water renewal run (Figure 4.21) and the reference run (Figure 3.3) 1894.23 Time-averaged o field for the 8 layers of the model for a run involvingdeep water renewal. In frame 9, the time-averaged surface elevation, incm, is shown. Compare the reference run, Figure 3.7. Depth of layers, inmeters, is given in the lower right hand corner of each frame 1904.24 Time series of the residual velocity components, u (thin) and v (thick),in cm s1, for layer 7 of the DWR run at the usual four grid points corresponding to stations I to IV as shown in Figures 3.lc and 4.25. At thetop, the tidal energy (in cm2 _2) of Figure 4.5 is repeated for reference.The vertical lines represent extreme neap tides. Compare to Figure 3.12for the reference run 1914.25 The fluctuating residual of the horizontal velocity for layer 7 for a runinvolving deep water renewal. Days are in Julian days starting 1 January,1984. Compare to Figure 3.20 for the reference run. Note the inflow(renewal) after the extreme neap tide on day 232. In the last panel, theposition of the Stations I, II, III and IV are repeated here for reference. . 192xxvi4.26 Time-averaged velocity for the model (solid arrows) and for the observations (dashed arrows) in the central Strait of Georgia for the periodJune 1984 to January 1985 corresponding to layers 3 to 8. Cyclesondedata are obtained from vertical averages over the model layer depths asgiven in Table 3.12. Layer numbers are given as L3 to L8. In the first cyclesonde frame, 1 to 4 indicates the station number (see Section 3.3.1 forfurther details). Note that in the frames where the cyclesondes velocitiesare shown, the model velocities are repeated for that particular gridpoint. 1954.27 Time series of the u and v velocity components (in cm s1) of the fluctuating residual from the observations (dotted lines) and from the model(solid lines) for stations 1 and 3 in the central Strait of Georgia. Shown atthe top is the third layer (15-30 m) and at the bottom the fourth layer (30-60 m). The velocity from the observations for these layers was obtainedas described in Table 3.12 1964.28 Time series of the u and v velocity components (in cm s’) of the fluctuating residual from the observations (dotted lines) and from the model(solid lines) for stations 1 and 3 in the central Strait of Georgia. Shownat the top is the fifth layer (60-90 m) and at the bottom the sixth layer (90-150 m). The velocity from the observations for these layers was obtainedas described in Table 3.12 1974.29 Time series of the u and v velocity components (in cm s’) of the fluctuating residual from the observations (dotted lines) and from the model(solid lines) for stations 1 and 3 in the central Strait of Georgia. Shownat the top is the seventh layer (150-250 m) and at the bottom the eighthlayer (250-H m). The velocity from the observations for these layers wasobtained as described in Table 3.12 198xxvii4.30 Time series of the o from the observations (dashed lines) and from themodel (thick lines) for station 3 in the central Strait of Georgia. Thenumbers at the upper right hand corner of each frame gives the layer number. 2005.1 Low frequency kinetic energy as a function of depth for stations 1 to 4at the central Strait of Georgia from the observations and from the twoversions of the 3-D model. REF and DWR stand for the reference anddeep water renewal runs studied in Chapters 3 and 4, respectively 211E.1 Horizontal stencil for an Arakawa C-grid. u and v are the horizontalvelocity components and S represents either surface elevation, depth ordensity. 237E.2 Time series of the advective terms of the discretized vorticity equation E.4(thin lines) and that obtained from the curl of the discretized momentumequations (thick lines) for the top and bottom layers of a particular gridpoint.241F.1 Time series of the advective terms of the energy equation for the top andbottom layers. Thick lines are from the discretized energy equation F.2and thin lines from the equation obtained from the discretized momentumequations 245xxviiiAcknowledgementsI started my Ph.D. program in 1989 under the supervision of Dr. John Fyfe. In June1992 he left UBC and Dr. Stephen Pond kindly agreed to continue my supervision. Iwish to express my gratitude to both of them. Their criticism and suggestions, especiallywhile writing, were always very helpful. Dr. Pond’s comments were always like a lessonto me.The members of my research committee, Drs. Paul LeBlond, Paul Harrison and TadMurty helped me in different ways. To each of them I am very grateful.I would also like to thank my friend, Dr. Roger Pieters, for the talks and experiencesshared and especially for proof reading the first draft of my thesis.I am grateful to Centro de Investigación Cientffica y de Educación Superior de Ensenada, B.C., Mexico for maintaining my leave of absence during all these years of study.I am also grateful to Consejo Nacional de Ciencia y Tecnologfa from Mexico for thescholarship granted during most of the time of my studies.This research was supported by the Department of Fisheries and Oceans and theNational Science and Engineering Counciil of Canada.This period of time of my life has been shared in the same graduate student wagonwith my wife, Lucila, her support and encouragement at all times has been always mystrength. Our lives touch land continuously by having four students living with us, Bruno,Paula, Guido and Silvia, our kids. Each of them worth more than a Ph.D. thesis.xxixChapter 1Background1.1 Observations of the Strait of GeorgiaThe Strait of Georgia (Figure 1.1) is an estuarine region between mainland BritishColumbia and Vancouver Island. It is roughly 40 km wide and 200 km long and isconnected to the Pacific Ocean through Johnstone and Juan de Fuca Straits to the northand south, respectively.Considerable effort has gone into the investigation of currents in this topographicallycomplex strait. Studies of currents arising from phenomena such as the Fraser Riverrun-off (Stronach et al., 1988), the tides (Crean et al., 1988a), and atmospheric forcing(Stacey et al., 1986) have revealed that the system responds on a wide range of spatialand temporal scales. In particular, subsurface current measurements have shown highenergy levels at frequencies much lower than those of the diurnal and semidiurnal tides.Residual currents with energy levels as high as those of the tides have been observedon many different occasions in the central Strait of Georgia (e.g., Chang et al., 1976; Yaoet al., 1982, 1985; Stacey et al., 1987). The persistent and high energy nature of thesecurrents, as seen in the observations, suggests their importance to the general circulationof the Strait. These residual motions constitute the subject of this study. To fix thenomenclature, residual current is defined here as that part of the flow left after low-passfiltering the currents with a cutoff period of two days. The residual current is made upof a mean (time-averaged) and a fluctuating (deviation from the mean) component.1Chapter 1. Background 2Figure 1.1: Plan view of the Straits of Georgia and Juan de Fuca: the small rectangle isthe area of study in Chapter 2.Chapter L Background 3Some important characteristics of, and questions about, the residual currents in theStrait of Georgia as reported in previous studies include:Temporal variability: About 37% of the variance is explained by fortnightly, monthlyand semiannual fluctuations arising from the tides either directly or through nonlinearinteraction (Stacey et al., 1987).Vertical variability: The residual currents vary in the vertical with correlation scalesof about 100 m. At times some of the vertical structure is consistent with that of anEkman spiral. Wind forcing is usually present; however, the specific conditions underwhich a wind forced response develops are not known. Evidence of nonlinear interactionbetween wind and tidally forced motions have been reported in Stacey et al. (1987).Horizontal variability: Currents have horizontal correlation scales of about 5 to 10 kmand at times are organized as eddies. The short spatial scales of correlation led LeBlond(1983) to suggest that the low-frequency dynamics may be significantly nonlinear. Thisidea was corroborated by Stacey et al. (1988) by evaluating, from the observations, theimportance of advection in the vorticity equation. Despite these studies it is fair to saythat the mechanism by which these eddies, and the low frequency currents in general,are produced is not known.The observations of Stacey et al. (1987) in the central Strait of Georgia revealed thepresence during summer months of currents near the bottom varying with fortnightlyand monthly periods. These currents have been interpreted by LeBlond et al. (1991) asgravity currents originating at Boundary Passage and introducing at depth water fromJuan de Fuca Strait to the Strait of Georgia thereby producing deep-water renewal.Chapter 1. Background 41.2 Previous models of the Strait of GeorgiaThe Juan de Fuca and Strait of Georgia system has a long modeling history. The focushere will be on Crean et al., (1988a,b) who describe in detail a series of models rangingfrom simple to complex.These models and their contribution are summarized as follows.First was a one-dimensional barotropic model (GF1) which was an initial attempt tosimulate the barotropic tides of the area. The major features of the M2 and K1 tidalconstituents were reasonably reproduced.Second was a combined one- and two-dimensional barotropic model (GF2) whichincluded rotation, the major dynamical component left out of the equations of motionin the previous model. This model had 4 km horizontal resolution. Obviously, betterresults were obtained in the spatial distribution of the major diurnal and semidiurnaltides.The next model (GF3) involved higher resolution (from 4 to 2 km) in Juan de FucaStrait, the southern Strait of Georgia and their connecting passages between the SanJuan islands. The primary object of this model was to improve the prediction of tidalstreams.The final version of these barotropic models was the development of a two-dimensionalmodel (GF7) with 2 km resolution throughout. Such a model provides adequate resolution of the horizontal residual circulation derived from tidal topographic interactions.This latest barotropic version (GF7) is used in Chapter 2 of this thesis.The knowledge obtained from these models led to a successful resolution of the tides inthe system. Inclusion of the earth’s rotation, the better representation of the geometryof the area, the inclusion of non-linear terms in the model equations, as well as theproper distribution of frictional dissipation (through the selection of friction coefficients)Chapter 1. Background 5were key factors to the success. Obviously, these models can only be employed to studyphenomena that are essentially barotropic in character, such as tsunamis or storm surges.The Juan de Fuca and the Strait of Georgia system is basically an estuary with themajor fresh water coming from the Fraser River. Residual circulation related to baroclinicprocesses will obviously be important.The first baroclinic model was an upper layer model of the of Strait of Georgia,namely, a vertically-integrated upper layer driven by a lower layer (GF7) in which thevelocity and pressure gradients due to the barotropic tide are prescribed.The second baroclinic model was a laterally-integrated model (GF5), with variablewidths and depths, adapted from GF1. Residual velocities associated with the overallestuarine circulation and deep water renewal were simulated.The next model was a fully three-dimensional model (GF6) which simulates bothbarotropic and baroclinic processes. This model used the same horizontal grid as GF2with 7 layers and 4 km resolution.Finally there was GF8 (Stronach, 1991). This model combines the 2 km resolution ofGF7 and the dynamics of GF6. This latest baroclinic version is used in Chapter 3 of thisthesis. Studies of residual currents with these models have been limited to short runs ofthe models, a month at most, and examination of the residual has been restricted to onlythe mean part. Longer runs are needed in order to resolve the low frequency currents.1.3 Objectives of this studyFor many reasons but especially because of the short record lengths, experimental observations to date have provided only limited understanding of the dynamics of thesepersistent and energetic residual motions in the central Strait of Georgia. As an alternative to mounting further expensive field experiments, a numerical modeling approachChapter 1. Background 6will be employed here. Results obtained using GF7, a high resolution depth-averagedtwo-dimensional model, where the residual currents are induced exclusively by the interaction of tides and topography and using GF8, a three-dimensional baroclinic model,where the residual currents result from the interaction of tides, winds and runoff will bedescribed.The thesis is organized as follows. Chapter 2 deals with the depth-averaged model(GF7) and Chapter 3 with the three-dimensional model (GF8). In both chapters, themodel results are presented in sections containing the residual circulation, Fourier andHarmonic analyses, and budgets of momentum and related variables, followed by sectionsincluding general discussion as well as comparison with observational data. Chapter 4 isdevoted to a study of the deep water renewal at the Strait of Georgia using the threedimensional model (GF8). In the final chapter, a summary of the overall results will bepresented.Any formulations not central to the arguments are presented in appendices so as notto detract from the readability of this work. For example, Appendix A briefly derivesthe equations of motion from which the two models’ equations are derived.Chapter 2A one-layer model of the Strait of Georgia2.1 The ModelAs a first step in a comprehensive study of the residual currents in the central part ofthe Strait of Georgia, I use the depth-averaged model developed by Crean (1978). Themodel, with its realistic coastline and topography, successfully reproduces the observedtidal heights and currents in the Juan de Fuca/Strait of Georgia system (see Figure 1.1for a map of the region). It remains to be seen how successful the model is in simulatingthe observed residual currents in the central Strait of Georgia. A full description of themodel can be found in Crean et al. (1988a).It is well known that the central Strait of Georgia has strong vertical shears in thecurrents as well as a highly stratified water column; however, in this chapter I focus onthe generation of residual currents by locally generated low frequency currents from tidesand their interaction with topography. The contribution from baroclinic effects is left toChapter 3.The observations in the central part of the Strait of Georgia, as previously mentioned,reveal the existence of horizontal scales of motion in the residual of the order of 5 to 10km (Stacey et al., 1987). Since the model as it stands has only a 2 km horizontal gridspacing, and in this chapter the residual generation comes only from the interaction oftides with topography, I have implemented a nested scheme wherein a 2 km spacing existseverywhere except within the central Strait where a 2/3 km spacing is used instead.7Chapter 2. A one-layer model of the Strait of Georgia 8This fine mesh region (20 km x 20 km), seen as the small rectangle in Figure 1.1, isenlarged in Figure 2.1. (Note that the northward direction is now toward the upperright in Figure 2.1.) The equations are then solved on this finer mesh using as boundaryconditions output from the coarser mesh model. More specifically, the model variablesare interpolated in space and time from the coarser model to the finer model at each timestep. The finer resolution should better resolve the horizontal variability of the residualin the central Strait.The equations used in the model are (for their derivation see Appendix B),ôU ãu2 OUV 2 U=0,(2.1)av auv av2 2(2.2)where the kinematic bottom stresses are given by a quadratic law, namely,- Cd/U2+V..d v,f = 2Q sin p is the Coriolis parameter, (U, V) are the horizontal transport components,i is the velocity vector with components u and v which are related to the transportcomponents as (u, v) = (U, V)/d, d = r + h is the total depth, i and h are the surfaceelevation and bottom depth with respect to the mean sea level, g is the acceleration dueto gravity, and Cd and v are the bottom friction and eddy viscosity coefficients [whosenumerical values are chosen as in Crean et al. (1988a), i.e., Cd = 0.003 over most of thedomain with values at some passes as high as 0.03, and v = 1 x 106 cm2 s9.To assess the effects of nonlinear interactions of the major tidal constituents in themodel Crean et al. (1988a) choose to prescribe at the two open boundaries the fiveChapter 2. A one-layer model of the Strait of Georgia0(1Dor9C//C))fJ10 Uvo.3S.’DV NFigure 2.1: The bathymetry in the area of study in metersChapter 2. A one-layer model of the Strait of Georgia 10Table 2.1: Forcing and produced low-frequency constituents. The amplitudes shown(given as a reference only) in the table correspond to the southern side of Juan de FucaStrait, the northern side is slightly different.Forcing Period Amp. Produced Period OriginConstituent (hours) (cm) Constituent (days)M2 12.42 75 Mf 13.66 K1—OK1 23.93 47 Msf 14.77 M2— S20 25.82 29 Mm 27.55 N2 — M2S2 12.00 21 Msm 31.81 Msf— MmN2 12.66 16 N2S 9.61 N2 — S2Ssa 182.42 Msf — Mflargest constituents, namely the M2, K1, 01, 52 and N2 tides (in order of decreasingmagnitude). I too will focus on the nonlinear interactions arising from these forcingconstituents alone. One anticipates that the interactions between pairs of these tides willgive rise to oscillations in the model at frequencies which are the sums and differencesof the forcing frequencies. Some of the more probable low-frequency components (oftenreferred to as the shallow-water constituents) so generated are shown in Table 2.1.As stated, the observations of the Strait of Georgia residual currents show largefortnightly and monthly signals which I hope will be simulated in the model. The modelshould allow an investigation of the physical origin and relative importance of the variousshallow-water constituents which is beyond the scope of the present day observations forthe region. A one year model integration was performed. The rationale for this longperiod integration (longer, for example, than has ever before been performed with thismodel) is as follows. To avoid violation of the Raleigh criterion (Otnes and Enochson,1972) when performing harmonic or Fourier analyses (as will be done here), the recordChapter 2. A one-layer model of the Strait of Georgia 11length chosen must be greater than the synodic period [= 1/(wi — w2)], where w andw2 are the frequencies of two sinusoids. The resolution of two spectral lines can beaccomplished if w1 — w2 26w, where 6w = (NL)1 and z is the sampling time andN the number of observations. This restriction basically requires the two frequenciesto be separated by a Fourier component which is twice the minimum requirement. Forexample, take the Mf and Msf constituents, their synodic period is 6 months and theycan be separated by Fourier analysis, with hourly sampling, in 8724 hours (363.5 days).Chapter 2. A one-layer model of the Strait of Georgia 122.2 ResultsI begin by showing the mean velocity and surface elevation fields. ilaving presented themean residual, I describe in Subsection 2.2.2 the time-dependence of the residual and inSubsection 2.2.3 its harmonic and Fourier decomposition. The latter analysis is requiredin order to quantify the principal modes of variability in the model. In the spirit ofunderstanding the physical mechanisms behind the modes of variability in the model, Ipresent in Subsection 2.2.4 momentum and vorticity budgets. The relevance of the modelresults to the observed residuals is taken up in the Discussion, in Section 2.3.2.2.2.1 Time AverageFigure 2.2a shows the mean velocity and surface elevation. The maximum modeledmean residual current speed is around 0.7 cm s. Based on a visual inspection ofthe mean velocity fields I identify four fairly distinct regions (shown schematically inFigure 2.2b), namely,Region I: which has a northwestward current flowing along the main axis of the Strait,approximately aligned with the elevation contours;Region II: which, by contrast to Region I, has a current normal to the main axis,approximately perpendicular to the elevation contours;Region III: which has an anticlockwise (cyclonic) rotating eddy, and finally,Region IV: which has a southeastward flowing meandering current.Chapter 2. A one-layer model of the Strait of Georgia 13.— .- . - ‘%\\\ \ \ \ \I.-.-. I%. I \.I \\ \ \iI’’I’’Il 1 4 ‘ ‘ I’/1/ / / -\ - - ... ‘ \////,__\_%S\/ a- — -\ - -. “ \ \/ / / / / / — — —\.. -.. I \/1 / / / / / / - - - ‘— / / I‘4 ‘ \ \ %1 1 ‘ ‘/ / / / _\—.—./ / >%ç S — —‘ ra- —/.-IYFigure 2.2: (a) Time-averaged velocity vectors and surface elevation contours (in cm) inthe central Strait of Georgia. (b) Position of the different regions referred to in the textas: I a northwestward flowing jet, II a cross channel flow, III a cyclonic eddy and IV ameandering current.(a)(b)—H_IIII I’1 cm/sChapter 2. A one-layer model of the Strait of Georgia 14The mean shown in Figure 2.2 was obtained using a full year of model output. Bysystematically varying the averaging period I find that about six-months worth of outputis adequate to achieve statistical steadiness to within about a 10% error (i.e., the meanso-calculated will differ from one calculated using a longer time period by less than 10%).However, in order to distinguish the two possible fortnightly frequencies (see Table 2.1)using Fourier analysis, around one-year’s worth of model output is desirable.2.2.2 Time variabilityTo display the model low-frequency variability I removed all motions with periods ofless than approximately one day by passing the fields three times through a twenty-fivehour running average filter. As discussed by Yao et al. (1982), the filter passes about 50%of the amplitude at 0.3 cycles per day and 95% of the amplitude at 0.08 cycles per daywhile reducing the semidiurnal and diurnal amplitudes to less than 1% of their originalvalues. Figure 2.3 shows the filtered series for the u and v velocity components at a gridpoint in each of the four regions defined in Figure 2.2b. Notice that the magnitude ofthe current fluctuations around the mean are of the same order as the mean itself (a factseen in the observations as well).The u and v time series in Figure 2.3 show a clear fourteen day signal, which is alsopresent in the observations (Stacey et al., 1987). (Each division on the horizontal axescorresponds to fourteen days.) The corresponding time series for the residual elevation(not shown) shows an even more well defined fourteen day oscillation than that seen inthe velocity components. In the next Section, I will show that most of the variability inthe modeled current and elevation series can be accounted for by the Mf constituent.To illustrate the spatial structure of the residual I present in Figure 2.4a two snapshotsof the residual velocity separated by approximately seven days. Note that the eddy simplyChapter 2. A one-layer model of the Strait of Georgia 150.20.0-0.2-0.4-0.6-0.8-1 .00.20.0-0.2-0.4-0.6-0.8-1 .00.60.40,20.0-0.2-0 . 4-0 . 60.60.40.20.0-0.2-0 . 4-0.6u 1cm/si v 1cm/siIJvWAV\/\/vvv11/J\/\JVVA1vv’f\1\1\Aj\h1P\JV\1VVW\JVvV\A”sJWv- —,.- —‘-—-—.-,‘‘I I I I I I I I IIv4. 32. 60. 88. 116. 144. 172. 200. 228. 256. 284. 312. 340.Figure 2.3: Time series of the residual velocity components for a grid point in each ofthe regions identified in Figure 2.2b.Chapter 2. A one-layer model of the Strait of Georgia 16Figure 2.4: Spatial structure of the (a) total and (b) de-meaned residual for two differenttimes, approximately 7 days apart.hour 242 hour 418—7, 7 ‘ \ \ \1 \ \, t ‘i.‘ \ \/1’- ‘ \.‘ ‘.t1(a)(b)1 cm/sII I,7,It •IS -‘1t,_I,,-’I,,-’t It.‘‘‘-I‘I‘I‘II/I/I/ I\ tt1/f/1 ,-—II—I—1’1’//1’/14’‘I//1/114’11Il/I/1III14II1‘IIIIIIII/////>ø5 cm/sChapter 2. A one-layer model of the Strait of Georgia 17spins up and down nearly in situ. In Figure 2.4b I plot the corresponding fluctuatingresidual (i.e., de-meaned). Note that the fluctuating fields have been scaled up by afactor of two relative to those of the total fields. The fluctuating residual has the samespatial structure as the mean but alternates between cyclonic and anticyclonic rotation.2.2.3 Fourier and harmonic analysesAlthough the series seen in Figure 2.3 show a fourteen day oscillation, other periodicities are also evident (note, for example, the six month modulation, especially in RegionII). To quantify these periodicities I performed Fourier and harmonic decomposition. TheFourier analysis is used to identify the harmonics to be used in the subsequent harmonicanalysis.Fourier analysisThe residual time series were subjected to Fourier decomposition. Figure 2.5 showsthe spatial average of the raw spectra over all points of the fine grid. The spectra werecomputed after removing the trend and a cosine taper was applied to the first and last10% of each time series. As solid lines I plot f x S(f) versus log f, where S(f) is thespectrum and f is the frequency; this format is power preserving, i.e., the total area underthe curve is proportional to the total variance of the time series. As dashed lines I plotlog[f x S(f)]. No averaging over frequency has been done and no error estimates are givenbecause the normal statistical estimates of error, which are based on normal distributions,do not apply to tides which have deterministic motions (Stacey et al., 1987).To assist the reader, pluses have been placed at the top of the figure to mark thefrequencies of the Ssa, Msm, Mm, Msf, Mf and N2S shallow water constituents (fromTable 2.1). Recall that these frequencies are the low-frequency differences of the forcing-97 IC4—If)-9.71j0)01 .7Figure 2.5: Power spectra (plotted as f x S(f) with solid lines and as log[f x S(f)] withdashed lines) of the (a) surface elevation, (b) u and (c) v components of velocity. The‘plus’ symbols locate the frequency of the low-frequency tides listed in Table 2.1.Chapter 2. A one-layer model of the Strait of Georgia 18Ssa Msm Mm MsfMf N2S22 i0 + ++ + +(cm/sec )2I_S ‘II SI S Il_ S SII 5 g/ I——— ‘%SI5%_I-4.7Cc)IIll I II I I iI I I1 IS I *I I%5-02x1 0(J02x1 000.00011 800.001 (cph)15 530-7.70.01period (days)Chapter 2. A one-layer model of the Strait of Georgia 19(tidal) frequencies. The spectra reveal a large maximum at a frequency very close tothat associated with the theoretically-predicted Mf tide. For the sake of comparisons,I note that the variance explained by the near-Mf frequency for (u, v, ) is (81, 79,91)%, while for the near-Msf frequency it is (6, 4, 1)%. Much smaller, but nonethelessdistinct, maxima are situated near the other constituents. As well, there are peaks whichare very near to frequencies which are harmonics and combinations of the shallow waterconstituents. For example, the Ssa and Msrn tides (marked with pluses) result from Mf-Msf and Msf-Mm interactions, respectively. The peaks between five and nine daysare frequency multiples of the Mf, Msf and Mm constituents. Similar spectra fromcurrent meters deployed in the region are reported by Stacey et al. (1987).With very few exceptions the velocity component spectra, taken point by point in thedomain (not shown), reveal a clear dominance of the Mf tide. The surface elevationspectra are always dominated by the Mf constituent.Harmonic analysisI now present an account of the relative importance and origin of the dominant modesof variability and, as well, show the spatial structure of each by fitting (in the least-squaressense) the time series of the fluctuating residual to the low-frequency constituents ofTable 2.1. The contribution to the residual from a given harmonic of angular frequencyw (= 2rf) can be written as,‘P(x, y, t) = Re{’(x, y) exp(i[wt — O(x, y)J)}where ‘I’ represents one of the velocity components or surface elevation, i& its corresponding amplitude and 0 its phase.Before considering the spatial structure of the various modes I looked at the spatiallyaveraged variability. Table 2.2 shows the percentage of the variance explained by the fourlargest modes forming the basis of the harmonic analysis. Notice that these four tidesChapter 2. A one-layer model of the Strait of Georgia 20Table 2.2: Percentage of the spatially-averaged variance explained by the given lowfrequency constituent for the regions identified in Figure 2.2b.Variable Constituent I II III IVu Mf 76.1 67.4 67.3 81.3Msf 8.0 18.2 17.3 7.2Mm 11.4 3.8 8.1 7.2N2S 0.8 1.2 1.4 0.3v Mf 82.0 53.6 60.8 83.1Msf 3.2 6.2 3.7 5.2Mm 8.5 11.6 8.1 1.9N2S 0.6 5.6 5.5 1.0q Mf 91.3 91.2 91.2 91.2Msf 1.0 1.1 1.1 1.1Mm 0.8 0.8 0.8 0.8N2S 1.5 1.5 1.5 1.5explain the bulk of the variance. Once again, the dominance of the Mf tide is revealed,especially with respect to the surface elevation.I now turn to the spatial structure of the different modes. Figure 2.6 shows thespatial structure of the Mf surface elevation amplitude. In contradistinction to thevelocity components (shown next), this mode simply oscillates in place with virtually nophase variation in space [i.e., O(x, y) constant].In Figures 2.7a,b I show snapshots (separated, as before, by about seven days) of thevelocity fields corresponding to the Mf and Msf constituents, respectively. Notice theChapter 2. A one-layer model of the Strait of Georgia 213.25Figure 2.6: Amplitude (cm) of the Mf constituent for surface elevation.Chapter 2. A one-layer model of the Strait of Georgia0.5 cm/s22hour 242 hour 418_ — -.I , — -._1111,__t !1/,I,,, -tii,I t \..IIt/I7/,‘A,/ — _‘_‘\I, —/I/ / - _%\1/’’1/’ ‘/1’11’S’/1/1IsI/I1/ //1//1/I’’/11/ I..(a)(b)——— . . . , S... .. .. — I I I S tS % .. — — - , . - . . % \‘ ‘ ‘ — — — — - S ‘ iI ‘ \ ‘/1 ...,____. St .. I I I — — ‘ ‘ ‘S — .. — S \ \—— .s S .‘s—————.- — — . I it, .— *. t it, I — — — — I I / t. •-S. I iii!!S j / / — -. I /S.’’’—S S S I -‘I .1 I I I . . .‘S.’—’ •. — — — - I I1%% — — — I ‘41 * % ‘ ‘/ ,..I,..--S S•__II •1 - — ..— ‘ ‘ ‘5’———— F F IF — .S •— — — — — — S S I 4 5I • — — -. — •. b I.11/• I ? I//IlI / /_ £ /E’J/J/Figure 2.7: Spatial structure of the (a) Mf and (b) Msf constituents for velocity.Chapter 2. A one-layer model of the Strait of Georgia 23very close agreement between the Mf mode (Figure 2.7a) and the fluctuating residual(Figure 2.4b). Also note that the Msf mode (Figure 2.7b) is much weaker and has a morecomplicated structure than the Mf mode. In Figures 2.8a,b I show the variance explainedby each of the velocity component amplitudes for the modes of Figure 2.7. Except inisolated localities the Mf mode explains a very high percentage of the variance. Notethat in those regions where the Msf is competitive with the Mf mode in explaining thevariance, the velocities are generally small.In summary, Fourier and harmonic analyses show that the variability of the residualcurrents in this depth-averaged model is dominated by a fortnightly signal, which Iinterpret as the Mf shallow water constituent.2.2.4 Momentum and vorticity budgetsIn an attempt to identify the physical processes maintaining the residual circulationI evaluate the various terms in the momentum and vorticity equations. The momentumequations (in terms of u and v) are as follows,th7 CduII 2d +V(ud),(2.3)t9v th1 CdvIiiI ‘ 2d +V(vd).The terms on the right-hand sides, reading from left to right, involve advection, Coriolis,pressure gradient, friction and eddy diffusion mechanisms, respectively. The vorticityequation, obtained by cross-differentiating the momentum equations, is,Chapter 2. A one-layer model of the Strait of Georgia(a)(b)U V24Figure 2.8: Percentage of the variance explained by the (a) Mf and (b) Msf constituentfor the velocity components.U VChapter 2. A one-layer model of the Strait of Georgia 25=-€• V- ( + f)V - CV x [iL1 + V x [V2(id)], (2.4)where = — is the relative vorticity. Here, the terms on the right-hand side,reading from left to right, involve advection, column stretching/shrinking, friction andeddy diffusion mechanisms, respectively. In what follows I will use the discrete versionsof these equations as opposed to manipulating the discretized versions of the originalequations. In this special case of an f-plane the results are the same either way (Foremanand Bennett, 1989).I now present the mean momentum and vorticity balances maintaining the meancirculation shown in Figure 2.2. Individual budgets are constructed for each of theregions defined in Figure 2.2b. This analysis is similar to that performed by Greenberg(1983) with the aim of identifying the mechanisms producing residual flows in the Gulfof Maine.Mean budgetsIn Table 2.3 I show mean momentum and vorticity budgets calculated by time-averaging each of the terms evaluated using the total (i.e., unfiltered) fields over thecomplete year of integration and then spatially-averaging over each of the regions. Onefeature common to all of the regions is that bottom friction is negligible, due presumablyto the large depths in the central Strait (see Figure 2.1). Beyond this fact the balance ineach region is quite different so I will characterize them separately.Region I (main-axis jet): In u-momentum the balance is between mean Coriolis andmean pressure gradient terms, so that, fiY—g. In v-momentum, the balance isa more complicated one involving mean Coriolis, pressure gradient and eddy diffusion.In summary, the main-axis jet (where i is small and 11 is large) is in near geostrophicChapter 2. A one-layer model of the Strait of Georgia 26Table 2.3: Spatially-averaged mean momentum and vorticity budgets for the regionsshown in Figure 2.2b. The units are 10 cm s2 and 10—10 s2 for momentum andvorticity, respectively.u momentum advection Coriolis friction eddy pressureI -0.051 -0.405 -0.001 -0.005 0.461II -0.428 -0.108 -0.002 0.060 0.478III -0.212 -0.016 -0.003 0.001 0.229IV -0.350 0.161 -0.003 0.014 0.184v momentumI -0.010 -0.040 0.000 0.027 0.026II -0.331 0.376 -0.005 -0.016 -0.050III -0.099 0.002 -0.004 -0.006 0.102IV -0.045 0.036 -0.008 -0.030 0.052Vorticity advection stret/shr friction eddyI -0.096 -0.026 0.000 0.119II 0.453 0.128 -0.016 -0.583III 0.112 0.012 -0.006 -0.117IV -0.157 0.145 -0.014 0.274Chapter 2. A one-layer model of the Strait of Georgia 27balance, at least as far as the mean circulation is concerned. (This is the only regionwhere mean advection of momentum is relatively unimportant.) The vorticity balance ismaintained predominantly by advection and diffusion of vorticity, with some contributionfrom the shrinking/stretching term.Region II (cross-Strait flow): Here the mean u (v)-momentum balance is between themean advective and pressure (Coriolis) terms in the x (y) direction, The mean vorticitybalance is more or less as in Region I.Region III (eddy): Here the mean momentum balance, in both directions, is betweenthe mean advection and pressure gradient mechanisms (the Coriolis term is negligible).In the mean vorticity balance in this region, and in this region alone, the mean stretching/shrinking term is also negligible because the mean flow is almost nondivergent here.Region IV (meander): Here the mean momentum balance in both directions involvescontributions from all terms (except bottom friction of course) with the exception of themean eddy diffusion term in the u-momentum equation. The mean vorticity balance isalso between all the terms, friction excluded.Having discussed the maintenance of the mean residual I now turn to the question ofwhat physical mechanisms drive the fluctuating part of the residual.Time-variability of the residual budgetsThe budgets which I now describe were obtained by evaluating the various termsusing the total fields and then low-pass filtering the result (the terms have also beende-meaned).With respect to u-momentum (Figure 2.9) I show three panels for each of the fourregions. In the top panel I plot the contributions to the tendency (or local rate of change)from the pressure gradient, Coriolis and advection terms. (The eddy diffusion and bottomfriction terms are very small and are therefore omitted.) In the middle and bottom panelsI show the result of adding the first two and all three of these terms, respectively. TheChapter 2. A one-layer model of the Strait of Georgia 28-0.40,0-0 . 10,10.0-0.1JVVid\JvYYIy.w \Region IIIRegion IVFigure 2.9: Budget analysis of the u-momentum equation. The curves correspond to aspatial average over each of the four regions identified in Figure 2.2b. Three plots areshown for each of the four regions. In the top panel the contributions to the tendency fromthe pressure gradient (dotted lines), Coriolis (dashed lines) and advection terms (plussign) are plotted. In the middle and bottom panels the result of adding respectively thefirst two and all three of these terms is shown in thin lines. The bold curve in the lattertwo panels is the actual tendency, i.e., the sum of all terms of the momentum equation.Note that the middle and bottom panels have expanded scales.0.40.0A A A A M-. Region II0.40.0-0.40.10.0-0. 10.10,0-0.1\Af%E\k,I I I I4, 18, 32. 46. 60. 74.4. 18. 32. 46. 60. 74.days daysChapter 2. A one-layer model of the Strait of Georgia 29Table 2.4: Region I percentage of the spatially-averaged variance for the given terms inthe u-momentum equation.u-momentum Mf Msf Mm N2Sii. Vu 17.6 47.0 32.5 1.6gr 82.4 4.0 5.4 0.3fv 82.4 3.0 8.0 0.694tu 93.6 0.9 0.4 1.0V2(ud) 59.9 18.0 16.4 0.9bold curve in the latter two panels is the tendency itself. Here is a summary of theobservations:1. In each of the regions, with the exception of IV, there is a large degree of cancellation between terms (i.e., two large terms summing to a relatively small tendency). Forexample, in Region I, Coriolis and pressure gradient terms nearly cancel. On the otherhand, in Regions II and III, advection and pressure gradient terms oppose one another.2. In Region I the tendency is driven by a small imbalance between the Coriolis andpressure gradient terms (advection playing a relatively minor role). In the other regionsall three terms contribute to the tendency.3. All the terms in each of the regions reveal a fortnightly signal (note that this signalis virtually absent in the tendencies). In Region I the Mf tide dominates the variabilityin the Coriolis and pressure gradient terms (see Table 2.4). In Regions II-IV the Mf,Msf and Mm constituents contribute about equally to the time-variability (not shown).Chapter 2. A one-layer model of the Strait of Georgia 30Table 2.5: Same as Table 2.4 for the terms of the vorticity equation.Vorticity Mf Msf Mm N2Si• V 30.4 40.3 25.8 1.1(C + f)V ii 75.1 7.6 8.9 0.9CdV x [lIiYj 63.4 19.9 8J. 0.5V x [V2(€d)] 51.8 25.2 18.2 1.2The results for v-momentum are similar and therefore are not shown.With respect to vorticity (Figure 2.10) I also show three panels for each of the fourregions. In the top panel I plot the contributions to the tendency from the advection,eddy diffusion and stretching/shrinking terms. (The bottom friction term is very smalleverywhere and is therefore omitted.) As before, in the middle and bottom panels I showthe result of adding two and all three of these terms, respectively. Here is a summary ofthe observations:1. In each of the regions there is a large degree of cancellation between advection andeddy diffusion terms.2. As with the momentum budget each region reveals a fortnightly signal (note thatthis signal is also in the tendencies). A harmonic analysis (see Table 2.5, for RegionI) shows that the large terms (i.e., advection and diffusion) are a mix of shallow waterconstituents, while the stretching/shrinking term is dominated by the Mf tide.Chapter 2. A one-layer model of the Strait of Georgia 311.00.0-1.00.20.0-0.20.20,0-0.2_i0i’ s2 Region I4. 18, 32. 46. 60. 74.4. 18. 32. 46, 60, 74.days daysFigure 2.10: As Figure 2.9 for the vorticity equation budget. In the top panel thecontributions to the tendency from the advection (plus sign), eddy diffusion (dottedlines) and stretching/shrinking (dashed lines) terms are plotted. In the middle andbottom panels the result of adding respectively the first two and all three of these termsis shown in thin lines. The bold curve in the latter panels is the actual tendency, i.e.,the sum of all terms in the vorticity equation. Note that the scales of regions I and IIIare different than II and IV and the middle and bottom panels have expanded scales.Region III[\ftAAAVsJV’ \J\Øbf3.00.0-3.01.5Region IV\2\—1 .51.50.0-1 .5 I I I I IChapter 2. A one-layer model of the Strait of Georgia 32I close this section by noting that advection plays a major role in the vorticity 4-nainics in this model which is consistent with the observations discussed in Stacey et al.(1988).Chapter 2. A one-layer model of the Strait of Georgia 332.3 DiscussionHere I discuss the model results in comparison with field observations as well as withanalytical results obtained with a simple tidal rectification model. Recall that the residualis made up of a mean and a fluctuating component. Briefly, the main characteristics ofthe model results, for each component, are as follow:1. The mean residual flow is organized into four fairly distinct regions, that is; Region Icontaining a northeastward current flowing in the direction of the main axis of the Straitand approximately aligned with the bottom contours; Region II with a current normalto the main axis and approximately perpendicular to the elevation contours; Region IIIhaving an anticlockwise rotating eddy, and, Region IV with a southwestward meanderingcurrent. The maximum speed is around 0.7 cm s’.2. The fluctuating residual has approximately the same structure as the mean and isdominated by the fortnightly band. In the model the fortnightly band is dominated bythe Mf constituent arising from the nonlinear interaction of the diurnal tides.2.3.1 Comparison with an analytic modelFurther insight into the residual production by the different tidal constituents can beobtained with simple depth-averaged analytical models. Simplified models with idealizedgeometry are available in the literature (e.g., Huthnance, 1973; Loder, 1980; Butman etal., 1983). I use the Butman et al. (1983) bank model which describes the mean flowas well as the amplitude of the low-frequency variation due to the interaction of twotidal components. Of course my topography is more complex than theirs. In Figure 2.11Chapter 2. A one-layer model of the Strait of Georgia 340.100.E-c 200.0300.400.Distance (kms)Figure 2.11: Comparison of the actual section of Strait of Georgia (dashed line) in thearea of the northeastward flowing jet of Region I in Figure 2.2 with the idealized sectionof the Butman et al. (1983) tidal rectification model.real and idealized topographic sections are shown with dashed and continuous lines,respectively. The sloping bank corresponds to Region I, where the numerical modelresidual is basically parallel to the isobaths, one of Butman’s model assumptions.The Butman et al. (1983) equation for the time-independent flow along the bank dueto a given tidal constituent, i, is= _hdujI2(Aj + A)/(4k)and the amplitude of the low-frequency variation due to two tidal constituents, i,j, ishdIuiIIujI2/&4.2h + k2whereA1= w?h(12+ 1)2 {Fk(2 — ) + k(3 — 2) + i[F1k(3 — 4) — Wjhd]},F1=k/(w2h3), k=Cdu1, 5WZWW,0. 4. 8. 12. 16. 20. 24. 28. 32, 36. 40. 44.Chapter 2. A one-layer model of the Strait of Georgia 35the asterisk designates the complex conjugate, f is the Coriolis parameter, Cd is a dragcoefficient, u is the amplitude of the tidal velocity for constituent i, w is the frequencyof the tidal constituent, hd and h3 are the depth of the deep and shallow sides of thebank of monotonic slope .The values for hd, h3 and for the idealized topography of Figure 2.11 are 340 m,200 m and 0.01, respectively. The modeled amplitude of the velocities for K1, 0, M2and S2 for the area are around 4, 2, 7 and 1 cm s1, respectively. With these values,the K1, 0, M2 and S2 contributions to the mean are 0.44, 0.13, 0.36 and 0.01 cmrespectively, for a total of 0.93 cms1. The amplitude of the low-frequency variation dueto the diurnal constituents (i.e., that corresponding to the Mf tide) is 0.15 cm s whilethat due to the semidiurnal ones (i.e., that corresponding to the Msf tide) is only 0.04cm s. These values are quite close to those of the numerical model in Region I. It issignificant that both the mean and the fluctuating components are larger in response tothe diurnal tides than to the semidiurnal ones, even though the M2 semidiurnal tide isthe most energetic component. This apparently contradictory result is explained by thefact that the residual depends on the square of the tidal excursion [i.e., (u/LU)2;see theequation above], which is larger for the diurnal components.The dependence of the residual on the tidal excursion was recognized by Zimmerman(1980) and Loder (1980) who used analytical models (with idealized topographies) andfound that the maximum residual currents occur in areas where the tidal excursion isof the same order as the topographic length scale (h/Vh). Evidently, fluids undergoing large excursions will not sense the topography and hence will lack one of the majorresidual-producing ingredients. From the model output I estimate a range of tidal excursions of between 3 and 11 km depending on the particular forcing constituent usedin the calculation. Figure 2.12 shows the magnitude of the mean residual current as afunction of the topographic length scale for all grid points. I find that the maximumChapter 2. A one-layer model of the Strait of Georgia 360.6U,EU>0.01Topograph Ic length scale (km)1 000Figure 2.12: Scatter diagram of the mean residual against the topographic length scale(h/Vh). The two vertical lines delimit the range of tidal excursions in the model.0000000000000000 0 0 00 000® 0cP 000000010 10080Chapter 2 A one-layer model of the Strait of Georgia 37residual velocities (in this case in Regions I and II) indeed occur approximately wherethe estimated tidal excursion is comparable to the topographic length scale.Before concluding this subsection I will use this analytical model to obtain a crudeestimate of the effect of neglecting certain boundary forcing constituents, as I have inthe numerical simulation. The P1 and K2 constituents which were not included in theforcing (and are the next two important forcing tides) may contribute to the Mf andMsf constituents (via interactions with the 01 and M2 tides respectively) and therebychange the picture of a dominant Mf tidal constituent. In the analytical model whenthese missing forcing tides are included, the resultant Mf and Msf amplitudes are 0.19and 0.07 cm s, respectively, i.e., the Mf constituent continues to dominate, but at thesame time the results indicate that the missing constituents do play a role. [Note thatthis calculation does not include phase differences; therefore there can also be differentresults if the pair of individual contributions to a particular fortnightly constituent is inor out of phase. However, if the two pairs contributing to the Mf are out of phase, theamplitude will be reduced only to 0.11 cm s in this linear model, which is still largerthan the Msf.]Ultimately, the real test of the importance of the missing forcing constituents wouldrequire re-running the model with all of them included. In fact, a one month run withsixty-two constituents at the boundaries was performed by Crean et. al. (1988a). Sincethe integration was only a month long, all of the energy in the fortnightly band wasassumed to lie at the frequency of the Msf shallow water constituent. Performing thesame analysis on just one month of my much longer integration I get approximatelythe same Msf amplitude. This is not a proof that the effect of the missing forcingconstituents would be small in the one year integration but it is suggestive. In Chapter 3the P1 and K2 constituents, among others, will be included and their effect will bequantified there.Chapter 2. A one-layer model of the Strait of Georgia 382.3.2 Comparison with observationsThe comparisons I make here are against vertical-averages from current measurementsobtained from an array of nine stations (1 to 4 are moorings of cyclesondes and 5 to 9are of current meters; current meters were also deployed at stations 2 and 4 part ofthe time) spanning the time interval from June 1984 until January 1985. A completedescription of the data is given in Stacey et al. (1987); however, a brief descriptionon the vertical position on the instruments is appropriate. The current meters werelocated at 100, 200 and 290 m; with only three depths the vertical average may notrepresent the true depth-averaged current. The cyclesondes profile within about 20 m ofthe surface and to a depth of about 300 m; better vertical averages are obtained fromthese instruments; however, there are no data below 300 m and at the surface which alsomay bias the averages. From the raw cyclesonde data, time series of velocity (and 0t)were constructed at discrete depths of 40, 60., ..., 280 m. At 20 and 300 m the datawere obtained while the cyclesonde was resting at the top and bottom of their profilingrange. All of the time series were low-pass filtered with the same twenty-five hour runningaverage filter explained in Section 2.2.2.The area covered by this array roughly corresponds to Region III of Figure 2.2 (i.e.,the eddy). Spectral analyses of the data can be found in Stacey et al. (1987). Since theobservations include forcing contributions from many sources, and not just the tides (asin the model), the observations as they stand, need to be interpreted as an upper bound,in some sense, on the tidally-forced component of the flow.I have made some attempt to remove at least the atmospheric contribution from thefluctuating component of the observations. Following Farmer (1972), Crawford (1980)and Baker (1992), the atmospheric contribution was removed by subtracting from theChapter 2. A one-layer model of the Strait of Georgia 39observations that part of the currents attributed to the wind. This wind driven part ofthe currents was obtained by spectral techniques. Specifically, it was computed from theinverse Fourier transform of the product of the coherence times the Fourier series of thecurrents because in any given band the coherence squared represents that fraction of thevariance in one signal which is related to the other signal regardless of phase (Farmer,1972).Figure 2.13 shows the amplitude of the Mf constituent at each station as a function ofthe record length (thick curves). The nonconvergence of the Mf amplitude with differentrecord length suggest nontidal contributions. The thin lines represent my attempt toremove the atmospheric contribution to the Mf constituent. In general, the resultantamplitude is almost unchanged with the exception of station 3 where a maximum of10% decrease was obtained. As the coherence calculations involve band averaging, ifthe currents have a large Mf relative to the background, the coherence may be higherthan it should be. The dashed lines in Figure 2.13 show the amplitude of the Mfwhen this harmonic is omitted from the coherence calculation to remove the atmosphericcontribution to the Mf constituent. With this last calculation, the resultant amplitude ofstation 3 is now reduced only 5%. Clearly though, because of the relatively short recordlengths it is almost impossible to make any definitive comments on the purely tidally-driven part of the observed currents. Also, for record lengths less than 180 days, theMf and Msf constituents are not separable and it is possible that part of the variationshown in the figure may be due to this problem and, therefore, the best estimate of theMf amplitude will be the one which uses the whole record length for each station.In Figure 2.14 I show the time-averaged velocity field of the vertically-integrateddata. The averaging period used to obtain a given station mean velocity extends to theend of that station’s record. To ensure that these time-averages are stable and not, forexample, biased by the yearly and half yearly components, I have calculated the meanChapter 2. A one-layer model of the Strait of Georgia 401.2 Station90.0 I12-8Cl) 12-60 0I1:20041:2-------------E I21100. 110. 120. 130. 140. 150, 160. 170. 180.Record lengtH (days)Figure 2.13: Amplitude of the Mf constituent at each station as a function of the recordlength (thick curves). Thin (dashed) lines represent the amplitude of the Mf constituentwith the atmospheric contribution removed when the Mf constituent is (not) includedin the coherence calculations. The station numbers refer to the station number of eachstation shown in Figure 2.14.Chapter 2. A one-layer model of the Strait of Georgia 415,J,4 8Na10 cm/sFigure 2.14: Time-averaged velocity from the observations in the central Strait of Georgiafor the period June 1984 to January 1985.current speed at each station using a range of averaging periods (see Figure 2.15). Withthe exception of stations 4, 5 and 7, the variation with the averaging period is not largesuggesting that the mean in Figure 2.14 is reasonably stable.Bearing in mind the limitations discussed above, I now compare the observed andmodel time-averaged velocities in Figure 2.14 and Figure 2.2, respectively. The modelspatial structure with its cyclonic eddy seems consistent with the observations, howeverthe model speeds are more than an order of magnitude too small (0.2 cm s1 verses 5cm s1). A similar order of magnitude discrepancy between model and observed fortnightly and monthly constituent amplitudes exists. Stacey et al. (1988) followed the timeevolution of the fluctuating flow at 100 and 280/290 m by mapping a derived streamfunction, Their results showed the appearance and disappearance of an eddy with a timescale of between ten and twenty days. In particular, the currents at 280/290 m changetheir sense of rotation and back again in about fourteen days. The model eddy fluctuatesChapter 2. A one-layer model of the Strait of Georgia 42stn 123——4-59.0 6 +++7 **80 8 0009 xxx7.0U)4.0 * * x x x x*3.0 + + + + + ++ + + + +2.0 I I I I I I100. 110. 120. 130. 140. 150. 160. 170. 180.Averaging period (days)Figure 2.15: Time-averaged current speed at each station as a function of the averagingperiod.Chapter 2. A one-layer model of the Strait of Georgia 43in a similar manner (see Figure 2.4).I now conclude with a short discussion on the role of imposed eddy diffusion in perhapsover-damping the modeled residual. Consider Figure 2.16 which shows the mean residualfor four two-month trials with v = (0.5,0.2,0.05,0) x 106 cm2 s1, respectively. Theviscosity coefficient used in the control run was v = 1 x 106 cm2 s, the original valueused by Crean et. al. (1988a). As expected one finds more energetic residuals fordecreasing viscosity (Tee, 1976). The question here is by how much can I reduce theviscosity and thereby increase the residual currents before seriously degrading the modelperformance (with respect to the total flow, say)? An inspection of these figures, andothers of the total flow (not shown), suggest that a five-fold decrease in the viscosityfrom the control value, and hence a doubling of the mean residual currents (with littlestructural change) can be accommodated before any significant degradation of the totalflow in the region is seen. This doubling of the mean residual current takes us a littlecloser to the observational “bound” provided by Figure 2.14. I conclude by noting theneed for a more careful analysis of the effect of ad hoc diffusion, as well as atmosphericforcing, on the residual currents in this two-dimensional model.Needless to say it is far too optimistic to expect a purely tidally-driven model toreproduce observed residuals given the presence of the other forcing mechanisms existingin the real world. Despite this limitation I believe that an understanding of this relativelysimple model is a useful stepping-stone to understanding more complex models, as wellas the observations.Chapter 2. A one-layer model of the Strait of Georgia= 0.5 ,< 106——0——-;z.%%\III,,/11111 II’’’’’1111111111169’’%39 39 5. 3. •4 39 ‘S111 1 3%’I/I—i’1/’ -. —.‘--9,—‘ Sv = 0.2 106I I I S S I3. 3. I I I I — —,•%\ 3. III I I•II-—9.———9’•I,——I’—I//Il 11541 JII’’551/19.4 jjg’1I11I413.1ll9°,Is 99I 3.\3.\%’3.’.’ 1 1%3.%’. I I I 3.5 S I 3 ‘ ‘ 3./_t/ Iç :44Figure 2.16: Mean residual velocity for two-months of integration with different valuesof the eddy viscosity coefficient.v = 0.05 x io6 v =02 cm/sChapter 2. A one-layer model of the Strait of Georgia 452.4 SummaryI have studied the residual circulation in the central Strait of Georgia with a spatially-nested depth-averaged numerical model. The residual was obtained by applying a low-pass filter to the modeled currents. The mean and fluctuating residual components werefound to have maximum values of around 0.7 and 0.3 cm s1, respectively.The modeled mean residual forms a cyclonic eddy in the middle of the Strait. Theflow around the eddy has a northeastward jet on the mainland side, a cross channel flowon the southern flank of the model domain, and a southwestward meander on the islandside, The fluctuating residual has the same spatial structure as the mean but alternatesbetween cyclonic and anticyclonic rotation in about fourteen days. The time variability isdominated by the fortnightly Mf constituent originating from the nonlinear interactionof the diurnal components K1 and 01.I performed a budget analysis of the momentum and vorticity equations. The relativeimportance of the various terms depends on location: in some areas the momentumbalance involves a simple geostrophic balance and in other areas all terms are important(except bottom friction which is small due to the large depths in the central Strait).Stacey et al. (1991) reported, however, an approximately geostrophic balance in theobservations in a region where the model is not geostrophic. The advection of vorticityin the model is mainly balanced by the lateral eddy diffusion.The simple analytical tidal rectification model of Butman et al. (1983) shows someagreement with my numerical results despite its simplifying assumptions. The analyticalmodel was useful in interpreting the dominance of the Mf constituent in terms of tidalexcursion. Field measurements show temporal and spatial scales in accord with themodeled residual, but the magnitude of the modeled velocities were an order of magnitudeChapter 2. A one-layer model of the Strait of Georgia 46too small.It appears that the residual currents in the central Strait of Georgia are too complexto be accurately reproduced with this depth-averaged model. The observations show agreat deal of vertical structure not taken into account here. Wright and Loder (1985)found that the inclusion of vertical structure amplifies the magnitude of the residual.Moreover, previous short three-dimensional runs (Crean et al., 1988a) indicate largerresiduals than can be obtained with a depth-averaged model.I anticipate that a fully three-dimensional model will significantly improve matters.That study is presented in Chapter 3, where more realistic forcing from the tides, runoffand winds are included in the model, with better results, i.e., the model results are closerto the observations.Chapter 3A multi-layer model of the Strait of Georgia3d The ModelThe three-dimensional model was developed by Backhaus (1983, 1985) and then adaptedto the Strait of Georgia and Juan de Fuca system by Stronach (1991) and is referred toas GF8, It is a layer model and I use 8 layers with nominal thicknesses of 5, 10, 15, 30,30, 60, 100, and 150 m, respectively. The horizontal grid is the same as that of the twodimensional model used in Chapter 2 (Figure 3.1). The number of layers in the modeldomain depends on the local depth and the last layer will have a variable thickness tofurther accommodate the topography at any given place. The thickness of the last layeris defined as the difference between the total depth and the sum of the thickness of themaximum number of layers that do not exceed the local depth. The thickness of the firstlayer is 5 m plus the surface elevation.In this part of the study, I will continue to focus in the central part of the Straitof Georgia. I will keep the original model resolution without using any nesting scheme.However, I will increase the study area to cover the central Strait of Georgia from coastto coast: this increase of area allows a better appreciation of the stirring produced bythe topography, both at the bottom and laterally. Figure 3.1 shows the area on whichthe results of this chapter are based along with the area of the results used in Chapter 2.The governing equations of the model are a set of vertically-averaged equations for47Chapter 3 A multi-layer model of the Strait of Georgia 48Figure 3.1: a) Horizontal layout of the 3-dimensional model. For clarity, the north partof the Strait of Georgia, Puget Sound (PS indicates where it joins Juan de Fuca Strait)and Johnstone Strait have been clipped from this figure. The asterisk symbol shows theFraser River entrance to the Strait. The numbers 1-14 show the wind station locations.b) Vertical layer structure of the model. c) Bathymetry, in meters, of the area of study(42 km x 42 km). The labels I, II, III and IV are the same points as those in Figure2.2b.Chapter 3. A multi-layer model of the Strait of Georgia 49each layer. [For a derivation of the following layer averaged equations from the continuouspartial differential equations see Appendix C.]The layer-averaged equations for momentum areOu 1 Ouuh Ouvh wu) 0,1 1 Op 02 02+ h(3.1)9v 1 Ouvui 9vvui 1(wv) O, 1 Op 02 02 IXrOx + 1+ h +fu+g-+--—v(-j+--j)v—--j--=O,for continuityOuh OvhWd = + + Wzu, (3.2)Ox Oyfor density conservationOp’+Oup’+Ovp’+(wp’)— 3 3at Ox Oy h —The hydrostatic equationOP= —pg, (3.4)and the overall continuity011 OU OV (3.5)closes the system of equations of the model. The variables are: = (u, v) is the horizontalvelocity, w is the vertical velocity, (U, V) = (u, v)h are the transport for each layer, Iiis the layer thickness which is equal to ii the nominal thickness except in the first layerwhere h = h +,is the surface elevation, p = p + p’ is the total density, p isa reference density (constant) and p’ is the density variation (in the rest of the thesiswhen I refer to density, I actually refer to the p’ or o field), P is the total pressure(= (,1 z)p*g + p(x, y, z, 1)), p is the baroclinic pressure (= f p’gdz), f is the CoriolisChapter 3. A multi-layer model of the Strait of Georgia 50parameter, g is the acceleration due to gravity, z(...) is the difference of (..) takenbetween the upper (zu) and lower (zd) surfaces of a layer.The horizontal eddy viscosity coefficient is i and the vertical stresses are defined by“-4-T =azwhere A is the vertical eddy coefficient,aA— Oz 36v1+/3v/and Ri is the Richardson number [= _/II2]. The values of the coefficients a and /9as well as the horizontal eddy viscosity i are here kept fixed to those values to which themodel was calibrated (a = 10 m2, /9 = 1 and 1’ = 200 m2 s; see Stronach, 1991) Thevalues that A takes range from an imposed minimum of 1 x 1O m2 to maximumvalues between 1 and 2 m2 s’. On average over the whole domain, A oscillates around0.02 m2Note that equation 3.3 for density has no explicit horizontal or vertical diffusivity.However there are several sources of density diffusion in the model. First, if the watercolumn becomes statically unstable, the density field is explicitly mixed in the watercolumn to remove the density inversion. Second, in the density equation, horizontaladvection is done by the method of characteristics (upwind formulation) and verticaladvection is done with a Lax-Wendroff scheme; both of which contain numerical diffusionwhich takes the place of eddy diffusion (Crean et al., 1988a, Stronach, 1991). Using themethods of Roache (1972) it is found that the numerical diffusion of the Lax-Wendroffscheme is wZt, where w0 is a characteristic vertical velocity and /.t is the time step;= 600 s here. Thus rough estimates of the vertical mixing due to numerical diffusioncan be given. Average values for the bottom of the central Strait of Georgia results in avertical mixing of ‘-. 6.0 x iO m2 s1. For Juan de Fuca Strait and Boundary PassageChapter 3. A multi-layer model of the Strait of Georgia 51average values are 2.4 x i0 m2 s1 and 1.5 x 10—2 m2s1, respectively. (Maximumvalues can be an order of magnitude larger at extreme tides.) Note that vertical diffusionplays an especially important role in mixing in the Juan de Fuca Strait and BoundaryPassage region and in determining density decay from the deep part of the central Straitof Georgia: both are important to deep water renewal events. In Chapter 4, where asimulation of deep water renewal is performed, explicit vertical diffusion of density isincluded in the model.The model is forced by the three major forcing agents in the system, namely tide,wind and runoff (LeBlond, 1983).Tidal Forcing: As in the two dimensional model, the tidal forcing comes from theopen boundaries, where the tidal elevation is reconstructed from tidal harmonics. Thereference run uses 44 tidal constituents which are shown in Table 3,1.Wind Forcing: The model was also forced by winds at the sea surface. Wind datawere obtained from the Atmospheric Environment Service (AES) for 14 stations aroundthe Strait of Georgia and Juan de Fuca Strait (see Figure 3.la). The data are hourly and,for each time step, the data are interpolated from the 14 stations to the whole modeldomain. The wind forcing enters as a boundary condition at the surface (see A.14 inAppendix A). To calculate the wind stress a drag coefficient at the air/water interfaceof 1.5 x iO is used (Stronach, 1991).Runoff: Fraser River and Squamish discharge are included in the model. Daily datawere obtained from the Inland Waters Directorate of Water Survey of Canada.The initial conditions for the density field are taken from observations of Crean andAges (1971) which is a data set of 12 monthly cruises in the Strait of Georgia and Juan deFuca Strait for 1968. For inflow the density is specified at the open boundaries. For thereference run (Section 3.2 below), densities at the open boundaries are held constant andcome from the same set of observations. In Figure 3.2 the density at the open boundaryChapter 3. A multi-layer model of the Strait of Georgia 52Table 3.1: Tidal constituents used to reconstruct the tidal forcing at the open boundaries.The amplitudes and phases are specified at the two connections to the Pacific Ocean:Juan de Fuca Strait and Johnstone Strait. The amplitudes shown (given as a referenceonly) in the table correspond to the southern side of Juan de Fuca Strait, the northernside and Johnstone Strait are slightly different.Forcing Period Amplitude Forcing Period AmplitudeConstituent (h) (cm) Constituent (h) (cm)Zo oo 0.0 SK3 7.99 0.5M2 12.42 78.0 27.85 0.4K1 23.93 49.1 p 26.72 0.901 25.82 30.1 24.13 0.852 12.00 22.4 Xi 23.87 0.4N2 12.66 15.9 4i 23.80 0.8P1 24.07 15.4 Ui 23.21 0.5K2 11.97 5.6 501 22.42 0.524.00 2.2 M6 4.14 0.6Sa 8766.23 14.2 OP2 12.46 0.2Ssa 4382.91 1.3 MKS2 12.39 0.3Mm 661.31 1.6 12.22 0.2Msf 354.37 1.8 11.98 0.2Mf 327.86 1.4 MSN2 11.79 0.126.87 5.6 MO3 8.39 0.9T2 12.02 1.4 M3 8.28 0.223.10 2.5 SO3 8.19 0.6OO 22.31 1.5 MS4 6.10 0.5P2 12.87 1.8 MK4 6.09 0.212.63 3.0 S4 6.00 0.2NO1 24.83 2.4L2 12.19 1.2MK3 8.18 1.4M4 6.21 1.1Chapter 3. A multi-layer model of the Strait of Georgia 53at Juan de Fuca Strait is shown for the months of June and January (1968). In Chapter 4the density at the open boundary will be given an appropriate annual variability.3.2 Model simulations and analysisThe results of this chapter correspond to a model run of one year starting June 1 of1984. This starting date was chosen to allow direct comparison with observations whichstarted in mid June of 1984. As mentioned, 44 tidal constituents are used to reconstructthe tidal elevations at the open boundaries. Recall that only 5 forcing constituents wereused in the 2-D model (Chapter 2) which accounted for 95% of the variance of thefull 44 constituents used here in the 3-D model. Thus, a little more energy is put intothe 3-D model and there is direct forcing at low frequencies as well. The wind and runoffcome also from actual observations during this period of time. An initial density fieldis set up corresponding to the month of June (1968 data must be used but hopefullythey are reasonably typical and after some time the density field will adjust to the 1984forcing). The densities at the open boundaries are held fixed. Eddy and drag coefficientsare chosen to be those from Stronach (1991), i.e., Cd at the bottom is the same as thatfor the 2-D model (0.003 over most of the domain with values at some passes as high as0.03). The above will be referred to as the reference run.In the next chapter, allowance will be made for a time-dependent density field at theopen boundary with which I hope to induce deep water renewal. Deep water renewal ispresent in the Strait of Georgia (LeBlond et. al., 1991) and will undoubtedly affect andinduce low frequency currents.Chapter 3. A multi-layer model of the Strait of Georgia 54January (1968)0.24,565. -E 25.0______________________25.5130. -26.00195. -USA CANADA260.June (1968)0.____65,25 0130, -CI)fl 195. -USA CANADA260. I I I4. 8. 12. 16. 20, 24.Distance (km)Figure 3.2: Cross-section o field at the open boundary at Juan de Fuca Strait for themonths of January and June, 1968 (from Crean and Ages, 1971).Chapter 3. A multi-layer model of the Strait of Georgia 553.2.1 The general circulation3.2.1.1 Time-Averaged residualIn this section I will show the mean velocity, density and surface elevation as well asthe total pressure fields for the different layers of the reference run. Figure 3.3 showsthe horizontal velocity fields for each of the 8 layers along with the vertically-averagedvelocity field. Figure 3.3 shows a great deal of both horizontal and vertical structure.The energy levels are much higher than the 2-D case and the velocities are now of thesame order of magnitude as observed. (Further comparisons with the observations willbe given at the end of the chapter.)In the first layer, the river discharge and the wind stress dominate the flow. The riverdischarge enters the strait from right to left near the bottom of the frame and then isdeflected up-strait by the Coriolis effect, a pattern documented by Stronach et al. (1988).(To appreciate the Fraser River channel in the model see frame 2 in Figure 3.3.) Theeffect of the wind stress is to push this upper-layer as a slab in the wind direction. Thiskind of response has been reported by Buckley and Pond (1976) in Howe Sound. Thetime-averaged wind is in the up-strait direction with velocities of about 1 m s as can beseen in Figure 3.4. The model velocities in the top layer of the water column decreasesfrom around 50 cm s’ at the river mouth, to around 7 cm s at the central area andto about 4 cm s further away from the river mouth which corresponds to about 4% ofthe wind velocities.The flow in layers 2 and 3 is basically down-strait with velocities between 2-5 cm s*From the second to the sixth layer, in the bottom part of each frame, part of a cycloniceddy appears to be present; it also appears in the vertically-averaged field. Layers 4 and5 contain a coastal up-strait ‘jet’ on the east side of the strait with velocities of about3-4 cm s1. Layer 6 contains an anti-cyclonic eddy in the central part of the domain andChapter 3. A multi-layer model of the Strait of Georgia 56I, , / / / lilT.L,‘,,,,,, ø__,,/ fIf //, , - , ,, I I I . ..I__.II....%’%. t1/’-FII SI1•lIl••,_,; f 4 1 4 ‘ ‘ 4 •• •II I4414 ,• —I I I I —) ‘ ,. ‘%•/1••II \\\\\-•‘—.% 4 •.—____I’%___ -•.__-.__ f -I‘. \ IFigure 3.3: Time-averaged velocity vectors in the central Strait of Georgia for layers 1to 8 along with the vertically-averaged field (frame number 9). The time period of theaverage is one year. The depths for layers 1-8 are given in the lower right-hand corner ofeach frame. The square insert is the study region of Chapter 2.1 2 3_Itt tII-——4 5tIS7 8II’’.‘I’,411’’’ 14 I I • I % LIJ— / 4 I • • • • S——I / ‘I,•j1 IIIl__I ..-_•• 41 • l/’ —I915 cm/sChapter 3. A multi-layer model of the Strait of Georgia 57__I__I,‘ I/ / /_j / / / / / I, / / / / t I‘, / / / 11,,, /1/11I,,,‘I!,/1!?till Stit t 1 1’ ‘1luI I I I 1’1/1111‘‘till/1/Itt/ /7//It/— /1//Il1 1 1 1 1 1 / / / -01111111/ / ,——,I t 1 1 7 1 1 1 / / .‘ —111111111 / / /11111 lilt I / /11111 It 14t tilt ililt I I I \t1thhiiIlt\\\ttuttitii\\1ihhhlStlt1\ti1lhtSttltltlt\I I 4t 4t 4t St t riI iiTI ‘t ‘1 liii itttill lii iii t111111?? III I I I I I I I/ I t I I I I Ii2 rn/sFigure 3.4: Time-averaged wind velocity vectors in the central Strait of Georgia. Theaverage is over the same time period of 1 year as given in Figure 3.3 for the velocities.Chapter 3. A multi-layer model of the Strait of Georgia 58the cyclonic eddy at the bottom part now is almost complete and shifted to the northwith respect to the second to the fifth layers. Layer 6 makes a transition to the last 2layers which basically have flow down-strait.To see how much and how many layers (i.e., how deep) the wind influences in the time-averaged circulation, a model run without wind forcing was performed. In Figure 3.5 thetime-averaged horizontal velocity fields of this run for all 8 layers along with the depth-averaged field are shown. The difference between the two runs is evident in the first threelayers. In the top layer, the flow structure remains almost unchanged but with a changein the current speeds which is most easily seen in the upper part of the panel. Althoughthe river discharge dominates the flow in the top layer there are wind effects too. Theflows in layers 2 and 3 have larger changes in their structure with changes in directionof almost 450, In layers 4 and 5 the structures do not change but the speeds show somedifferences. As expected, the rest of the water column (layers 6, 7 and 8) show littlewind effect. Figure 3.6 shows the actual difference between the two runs (reference runminus run without wind forcing) for all layers and for the depth-averaged field. Againthe difference is large in layers 1, 2 and 3, but now in layer 5 a large difference is clearlyrevealed. So, the wind influence in the model time-averaged circulation is felt as far as90 m depth (5 layers). The difference between the depth-averaged fields show similarvelocities to the depth-averaged field of the reference run. In layer 1 the difference fieldshows a couple of ‘coastal jets’ flowing northwest (southeast) on the east (west) coast ofthe strait with a cyclonic eddy at the west side.Consistent with the empirical orthogonal function results of Stacey et al. (1987),where the first 3 vertical modes dominate most of the variance (see their Table 5b), itappears that the time-averaged circulation behaves as a four-layer system. [This resultfor the vertical structure of the time-averaged currents gives some confidence in the useof eight layers, as a first approximation to solve the vertical structure of the central StraitChapter 3. A multi-layer model of the Strait of Georgia 5914.. .4,‘-——4.I,._‘‘‘14I•.jII‘14111%’—4,,.15 cm/s2 3\\ ‘ ‘ ‘ 4 • 4 ‘ I It I4’’., tI•4l??I4’4’,4’671/I1//lithu% 44Is ‘.‘4’ ‘.4—’.’I. . • •I,. .iil—==1,8I ‘ ‘‘ ‘S44% 44%44 I 4 ‘ 4 LIJ.•I ‘‘‘•‘\h’ii’•’ ‘F’IIlj II_I,,‘‘1•j I I/”” —I’,9I;:Figure 3.5: As Figure 3.3 without wind forcing.Chapter 3. A multi-layer model of the Strait of Georgia 60•,)“‘t1’uir11/ t ;lI r-//i ,/I//!I 1Lfr////,,//////t!//////irf/” ?//////,,I f/F///,_,i/a, f////f,,__1’-“‘4°I”. .-5A1i’/1Tt.4% ‘fTllIs.11,,t I,Ito30-60I T:: E!EEE— — —• I • — — — Si a a . —a a•I.a. 5—15Figure 3.6: Difference between the time-averaged velocity field of the reference run (Figure 3.3) and the run without wind forcing (Figure 3.5).5I• .. a a a a aa a a.liii 0’’II,,,’,.I_lila a’I__I’’,I,,,,, ..I—...I-,,, - -lit/i’I,,,,’—._JI I / — — —L._.s i I —J441/,4 4 t I I I a:H:::j 60-903• ,• . . . I,__________,_ /////,,,• . ,_,,•_l,l/,,,,,t//• -ti/f ‘f,,,,,li//f ‘,,,,___I/I/f ‘,,,______,a////,____1,,,,• I.1? 1111/11g.,. ,f1? liii,..t ?tiis15—3090—150_JlI ii I’’ 4’i ( -I—————, I1__11q‘‘‘‘‘‘i.Jr4II1II481814‘aII ‘Ia \\\‘ ‘•.‘ a S ‘ s —/ I a•%—\ ‘%• .. —. —- —— _ — - . _•-.•-._S.a-..•.- —•‘——‘-.-•‘kf —-•.—..—— \—, —-.l///”’\ ‘1JI/’ ; :>15 cm/s7 8 9150-25 250—CChapter 3. A multi-layer model of the Strait of Georgia 61of Georgia. Of course for a numerical model, the higher the resolution the better thesolution is likely to be, but as usual increasing the number of layers will cost computertime and disk space.]Now I comment on the period of time required to obtain a statistically stable meanfields in this model. As in Chapter 2 the averaging period was systematically varied andit was found that about half a year is necessary to achieve statistical steadiness within10% difference.The time-averaged density field for each layer is shown in Figure 3.7. The verticaldensity gradient is large between the 1st and 3rd layers and decreases considerably inthe rest of the water column. Obviously, the densities of the first two layers are heavilyinfluenced by the river runoff and, because of the absence of explicit density diffusion,the resulting density field of the upper layers may be too low. Actually, as mentionedbefore, there is some numerical diffusion and the water density of these layers is raisedto more reasonable values. The runoff enters near the bottom right of the figures andthe resulting fresh water is mainly confined to the east side of the strait. In the thirdand fourth layers, the situation is reversed and denser waters are found to the east. Inthe fifth layer, the structure suggests penetration of denser waters from the south. Thesixth and eight layers show inflows of denser waters from both south and north. Theseventh layer has lighter water in the southeast and northwest and heavier water in thesouthwest and northeast.Finally, the time-averaged total pressure for the different layers is given in Figure 3.8.The horizontal structure of the isobars corresponds, in general, to that of a geostrophicflow in all layers. That is, the flow in each layer is almost parallel to the isobars with highpressure to the right of the flow. Note in layer 6 that the high pressure corresponds to theanti-cyclonic eddy observed in the velocity field shown in Figure 3.3. I note that Stacey etal. (1991) found, in the observed currents, that a geostrophic mean state is a reasonable1 2 3Figure 3.7: Time-averaged t field for the different layers. In frame 9 the time-averagedsurface elevation is shown in cm. with a 1 cm contour interval.Chapter 3. A multi-layer model of the Strait of Georgia 62Chapter 3. A multi-layer model of the Strait of Georgia 631 2 3P(1) = 26900P(2) = 101800P(3) = 227000P(4) = 452000P(5) = 752000P(6) = 1203800P(7) = 2006100P(8) = 3260100Figure 3.8: Time-averaged pressure (Pa) for the different layers. For plotting purposes,the pressure values of each layer have been obtained by subtracting from the total pressureof each layer a pressure value close to the minimum pressure of that layer; these valuesare shown at the bottom right corner of the figure.Pressure subtracted (Pa)Chapter 3. A multi-layer model of the Strait of Georgia 64approximation in the central Strait of Georgia. So, both model and observations suggestthat the time-averaged currents are mainly in geostrophic balance. In Section 3.2.3 acomplete analysis of the dynamics of the system will be undertaken. There I will showthat ageostrophic effects are important.3.2.1.2 Time variabilityTo focus the attention on the low-frequency variability in the model, I remove all motionswith periods of less than approximately one day by passing the fields three times througha filter which performs a twenty-five hour running average. Figures 3.9 to 3.12 show thetotal residual corresponding to four selected gridpoints for the indicated layers. (Theseselected points are the same as those used in Chapter 2: see Figure 2.2 and 3.lc.) Thetime variability is not as simple as in the case of the 2-D model; however, it resembles theobservations to a much greater extent. A visual inspection of Figures 3.9 to 3.12 readilyidentifies strong monthly as well as fortnightly variations instead of the predominatelyfortnightly ones of the 2-D case. As in the observations, the amplitude of the variabilityis equal or greater than the magnitude of the the mean itself. Typical speeds ranges from20 cm s in the top layers to 3 cm s1 in the bottom layer.It was found that the time-averaged model currents are influenced by the wind stressto depths of 90 m. The fluctuating residual also shows the wind influence. In Figure 3.13a time series of the wind velocity components are given for the same set of selectedgridpoints as those for the time series shown in Figures 3.9 to 3.12.For the fluctuating residual it can be found, by visual inspection, that the wind effectpenetrates at least as far as the third layer for all stations except station I, where theinfluence of the Fraser runoff is greater. In Figure 3.14, I show a simple linear correlationbetween the velocity components along the strait and the wind for the different layersof the model. As expected from Figures 3.9-3.13, the correlation is higher and persistsChapter 3. A multi-layer model of the Strait of Georgia 65(cm/s) v (cm/s) layer’ 140-40.0. i i i180. 195. 210. 225. 240. 255. 270. 285. 300. 315, 330, 345. 360.daysFigure 3.9: Time series of the residual velocity components for selected grid points forlayer 1 of the model. The time is in Julian days with day 1 being January 1st, 1984. Thepoints are the same as those shown in Chapter 2 (see Figure 3.lc).Chapter 3 A multi-layer model of the Strait of Georgia 6615.u (cm/s) v (crn/s) layer 3—1515-15.J‘\—cJ—c’g /7 I15.—15. ‘*180. 195. 210. 225. 240. 255. 270. 285. 300. 315. 330, 345. 360.daysA AlJ1 A iv00IIIII0Figure 3.10: As Figure 3.9 but for layer 3 of the model. Note the change of vertical scale.Chapter 3. A multi-layer model of the Strait of Georgia8.—88—8.(cm/s) v (cm/s) layer 5678.—8,flS j& /LN1180. 195. 210. 225. 240. 255. 270. 285. 300. 315. 330. 345. 360.days000IIIIII0zi-:II Iy: v‘EFigure 3.11: As Figure 3.9 but for layer 5 of the model. Note the change of vertical scale.Chapter 3. A multi-layer model of the Strait of Georgia 688,0.—8.8.0.—8.8.u (crn/s) v (cm/s) layer 7III180. 195. 210. 225. 240. 255. 270. 285. 300. 315. 330. 345, 360.-8.8.0.—8.wIVdaysFigure 3.12: As Figure 3.9 but for layer 7 of the model. Note the change of vertical scale.Chapter 3. A multi-layer model of the Strait of Georgia 6910.-10.—10.U (m/s) V (m/s)w wII10.• V \tv’ V 1 ‘ fl/W”VI I I I I IdaysIIIFigure 3.13: Time series of the wind velocity components for the same set of selectedgridpoints of Figures 3.9-3,12.010.I10,-10.-10.0. A—-t AA A Ar’ I IV180. 195, 210. 225. 240. 255. 270. 285. 300. 315. 330, 345. 360.—C4)a0Figure 3.14: Correlation between along channel wind and along channel current velocityfor the four selected gridpoints (Stations I, II, III and IV) and for all layers. The dashedlines are at the correlation value (= 0.29) significant to the 95% confidence level for45 degrees of freedom. The degrees of freedom was obtained by dividing the recordlength by the decorrelation time scale. The latter was estimated from the integral of theauto-correlation function to its first zero crossing (= 8 days).Chapter 3. A multi-layer model of the Strait of Georgia 70+ + + Sta.ton I* * * Station IIo o o Station IIIx x x Station IV0.50.1 00.E150.200,250.300.350.-0.5 0.0 0.5 1 0Chapter 3. A multi-layer model of the Strait of Georgia 71deeper at stations II, III and IV. For all stations the correlation is largest in the top layerand then decreases with depth and reverses sign. This vertical structure or behaviordepends on location. Note the position of stations I to IV relative to the Fraser Riverwaters in the Strait of Georgia (see Figure 3.1). Stations I and II are closer but station Iis closer to the east coast where the fresh water influence is larger because of the Corioliseffect. Stations III and IV are in the central part of the strait with station IV furthestfrom the fresh water influence. These facts imply that station I has more fresh water inthe top layer making it more buoyant, consequently, the wind influence will not penetrateas far into the water column. On the other hand, at station IV a deeper penetration ofthe wind forcing occurs because the density difference between layers is smaller. Thesesituations are reflected in the correlations shown in the figure. Station IV has significantpositive correlations at the first three layers, station III in the first two with the thirdlayer still positively correlated but not significant anymore. At station II, the first twolayers are also well correlated with the wind, but with smaller values (because buoyancyis larger at station II than at station III). At station I, the first layer is well correlatedwith the wind but with a slightly smaller value.Independent of having a significant correlation at particular levels, the pattern thatthe correlation as a function of depth shows, is more indicative of a large influence of thewind not only in the surface layers, but in the interior of the water column as well. Thecorrelations goes from significant positive values at the surface to significant negativevalues at depths of about 100 m, and then tends towards zero at larger depths. Also thetime-averaged currents in layer 5, which is between 60-90 m, show a large wind influence(see Figure 3.6), that is, both mean and fluctuating residual show a wind influence asdeep as 90 m. Stacey et al. (1986, 1987) found a good fit between the classic Ekmanspiral model and the first-mode empirical orthogonal function at their station 1, whichis close to station IV, down to a depth of 140 m. Note that the correlation at stationsChapter 3. A multi-layer model of the Strait of Georgia 72III and IV is also significant at layer 6 which goes to 150 m.The time series of the total residual for individual gridpoints without wind forcing aresimilar to those of the reference run but smoother (Figures 3.9 to 3.12 being much noiserwhere wind forcing is included). As an example, Figure 3.15 shows the total residual forlayer 3. Note the difference, due to the wind, with Figure 3.10 specially at stations IIIand IV.For sake of comparison with the 2-D model, Figure 3.16 shows the time series of thevertically-integrated currents for the 4 stations. From Figure 3.16 it is apparent thatthere is significant cancellation when the water column is vertically integrated. (Thisbehavior is also evident from the plan view of the vertically integrated velocity in frame9 of Figure 3.3.) Blumberg (1978) found that this result must be obtained becausecontinuity must be obeyed. This result indicates that the barotropic component is smallin the model but not as small as the results from the 2-D model.The horizontal structure of the fluctuating residual is given in Figures 3.17-3.20 witha time series of 29 days for layers 1, 3, 5 and 7. The time difference between framesis 2 days. In general three different patterns of circulation can be detected in the flowat different times. At times the flow just goes north or south (see for example day 226for layer 1 and day 242 for layer 7). When the flow is about to change direction thevelocity field sometimes develops a coastal jet (see for example day 238 for layer 3 andday 236 for layer 5). At other times, cyclonic or anticyclonic eddies appear (see forexample day 252 for layer 1 and day 250 for layer 3). Eddies tend to appear every fifteenor thirty days. The eddies tend to be the size of the model domain as can be observedin Figures 3.17-3.20. At times these eddies can be detected close to coastal features (seefor example day 242 for layer 3 and day 246 for layer 7) while at other times a coupleof counter-rotating eddies are formed (see for example day 238 for layer 1). Note thatthe velocity scale of the fluctuating residual is different from that of the mean residualChapter 3 A multi-layer model of the Strait of Georgia 7315.u (cm/si v (cm/s) layer 3daysFigure 3.15: Time series of the residual velocity components for selected grid points forlayer 3 of the model for a run without wind forcing. Compare with Figure 3.10.‘ \J \%J z/ ‘%-‘ kd’ %‘ ‘f0-15.‘\\PSPcJAJIIIIIIIvI I I I I I I I I I I I180. 195. 210. 225. 240. 255. 270. 285. 300. 315. 330, 345. 360.-3. I I I I I I I I I I I180. 195. 210. 225. 240. 255. 270. 285. 300. 315, 330. 345. 360.daysFigure 3.16: As Figure 3.9 for the vertically-averaged residual velocity components. NoteChapter 3. A multi-layer model of the Strait of Georgia 74u (cm/s) v (cm/s)3.0 I-3,3.0. II-3.3.0. I I-3, I3,0. Ivthe change of vertical scale.Chapter 3. A multi-layer model of the Strait of Georgia 75day 224 day 226 day 228 day 230Ir1l / / t. /L:/ / , •- /1//•I /S_j,jLayer20 cm/s1Figure 3.17: Horizontal velocity time series of the fluctuating residual, for layer 1. Daysare in Julian days starting 1 January 1984.I I?/.i-’l/J/, /Vv”_i./1t1f,,• -1day 238day 242;L i/’/, .__f / - —. — —rr_Ij — ‘ — •—U , ,5%— ‘ II—--Jday 246I///I“-S—Chapter 3. A multi-layer model of the Strait of Georgia 76______________day_228‘ t a• r-:_.i ‘ • —r_L1 - / ‘ —JI, / / — — - -rt / / — -7J /,,__ a -day_236ni• Efl — / iirt’ /“// /_yç’/Va/.-/ //*-4-a- a- -day 224_I1_•_,L,, , I -, , I.day 230day 232I I— , I I ,I — — II —. •_—,.;I /S I I;r::;rrJdJ_Jday 226I a,,.•rJ”•I I I I• ‘ ‘ : :I S / —— — —11/ /day 234ni• L_91- — - I• - — - - I/ / —Ji / /Iday 242-•• — t4.JJ I a /I. -.—-4day 250day 238n IIr—///// /H1///v//‘.-— I I/I I • .1/day 2444 6:r.%-.r’//-day 248I //1/I,,1/,tt_.. p1/,/ ,/,_/1/,../II•ri : ://i- _,//- 1f-.H’•?//wLayer 320 cm/sT1 L1 - -/ i • . I1, fit..,1Figure 3.18: Same as Figure 3.17 but for layer 3.Chapter 3. A multi-layer model of the Strait of Georgiaday 224 day 226 day 228, I9 I S S Sday 236day 230Layer>20 cm/s775Figure 3.19: Same as Figure 3.17 but for layer 5.L1,, fI . I 7 7, 9, 9, 1 t 71J, F__,,fH , — — -. — /I I / - r-day 232L_•I_% t9 9 I Ir1 / ‘ ‘ 9‘ _ — — — — ,r’ — — — — —I,, /___•__9•_•day 238day 240LL\t,.It / , , , -_rT/ / / - - -r, / / - - - -Ht / / / — - -I f,__#H;hirT— LL I77 / / ,9, -rI / / - - -/ , FH’ / / ‘ 9, 9,I,,IL1El’ / ‘_1-Ti / i •r7 /!19___ri—’I •-- ‘6day 244 day 246• - - ———a-.——is //__o——- —day 248I I — — — —cia 252r2rrChapter 3. A multi-layer model of the Strait of Georgiaday 224 day 226 day 228day 236day 244day 23078Layer 720 cm/sr*Thday 246•1’’’I. ‘ ‘ ‘‘ II / IIday 252Figure 3.20: Same as Figure 3.17 but for layer 7.Chapter 3. A multi-layer model of the Strait of Georgia 79shown in Figure 3.3; the amplitude of the fluctuating residual is equal or larger than thetime-averaged residual.The wind effect on the time variability of the residual can also be observed in thelatter time series. For example, in layer 1 (Figure 3.17) the flow is mainly down-straiton day 224 and two days after the flow is up-strait! The reason is that between days 220and 225 a down-strait wind was blowing and then on day 226 the wind direction wasup-strait (see Figure 3.13). The flow at the upper layer has been driven by the wind.The mean residual is characterized by coastal jets and eddies of the size of the domainboth changing direction and sense of rotation with depth. The horizontal structure ofthe fluctuating and mean residual are similar. The maximum speed for the total residualranges from 5 cm s at the bottom to 40 cm s at the surface, The time-averagedresidual speeds range from 3 cm s at the bottom to 10 cm s1 at the surface andaway from the river discharge. The oscillations of the fluctuating residual are dominatedby periodicities of about 15 and 30 days.3.2.2 Fourier and Harmonic AnalysisTo quantify the time variability of the complex time series of the fluctuating residual(e.g., Figure 3.9), Fourier and Harmonic analyses were performed as in Chapter 2.3.2.2.1 Fourier AnalysisFigures 3.21, 3.22, 3.23, and 3.24 show the spatial average of the raw spectra of i, u, vand p’ for each layer of the model. The spectra were computed after removing the trendand a cosine taper was applied to the first and last 10% of each time series. The spectraare plotted with the same format as in Chapter 2, i.e., f x S(f) versus log f which ispower preserving. The ‘plus’ symbols locate the frequencies of the more important lowfrequency tides. (Note the different vertical scales for the different layers.) The spatialChapter 3. A multi-layer model of the Strait of Georgia 803Ssa MsmMm MsfMf N2SrCrn AI L0.0001 0.001 0.01cph180 30 1 5period (days)Figure 3.21: Spatially-averaged power spectrum of surface elevation.average was performed on 6 x 6 square of grid points in the central area of the domain.The spectra of the individual grid points within the square are not very different fromthose shown in the figures; the standard deviations are too small to show on the plot.As in the 2-D model, the spectra reveal large maxima at frequencies close to thetidal frequencies. It was observed in the previous section that fortnightly and monthlyvariability determined, to a great extent, the time variability of the fluctuating residual;the spectra here confirm this fact by showing the larger peaks at Fourier frequencies closeto the main fortnightly and monthly tidal constituents.It is also observed that there is a large percentage of energy spread over a broad bandof frequencies (‘..‘0.001-0.01 cph or 5-’35 days) in the velocity field, especially in the vcomponent. The Fourier frequencies containing energy over this band are frequenciesclose to overtide frequencies (where the frequency is an exact multiple of one of thetidal frequencies) or close to compound tidal frequencies (where the frequency is anexact sum or difference of two tidal constituents). These frequencies, overtides andcompound tides, are combinations of the tidal frequencies by which the model has beenforced and they can originate in the model through nonlinear interactions. Also, thesefrequencies of tidal origin interact with processes of different frequencies, the wind forcedChapter 3. A multi-layer model of the Strait of Georgia 81-2 Ssa Msm Mm MsfMf N2S Layer2x10 2 + ++ ++ +(cm/s) (1)2I I I I I I I _1 --1 xl 0(2)0 I I I I - ••I I1 xl(3)0 I I I I I iii I I I I2x1(4)0 I I I I I I I I _L L I I i.iJlxlO(-I_ (5)0.----1 xl 00 I I I I I I I_I .. -(6)5x10 -(7)0 - I I I I I I I I — I I5x10 -(8)0 I I I I I0.0001 0.001 0.01cphperiod (de.ys)Figure 3.22: Spatially-averaged power spectra of the u component of velocity of the 8model layers.Chapter 3. A multi-layer model of the Strait of Georgia 82-2 Ssa Msm Mm MsfMf N2S Layer2x10 2+ ++ + +(cm/s) (1)0 I I I _J_ L”r”L IlxlO -lxlO -(3)0 I I I I I I I _I____.___...._._....’”’..,..__._.L__ \.i..__....__. i I I2x103 -(4)0 I I I L t L iilxlO -0 I(5)lxlO -0 I I6)- II5x10I I I0,0001 0.001 0.01cph1’5period (days)Figure 3.23: Spatially-averaged power spectra of the v component of velocity of the 8model layers.Chapter 3. A multi-layer model of the Strait of Georgia042x1 003x1 003x1067 xl 007x109x1 09x1 0083Mem Mm MsfMf N2S Layer•‘‘.I IAAA.( 1 )(2)(3)(4)(5)(6)(7)(8)0.001 0,01cph30 156x1Sc a2’f’(cm/s)++ ++ +—5(I)- I-,—- I II I I J’•••’—._ I 1\i I I I I II I — I I I I I Irr0.0001‘ —I-— I I__L %4I’’\/I/\2%’\J ,......J— I Jieoperiod (d&ys)Figure 3.24: Spatially-averaged power spectra of the p’ field of the 8 model layers.Chapter 3. A multi-layer model of the Strait of Georgia 84motion in particular, and the nonlinear interactions continue to spread energy at differentfrequencies; the mechanism will result in the kind of broad band energy spectra of thecurrents observed in the Strait of Georgia. This result is indicative of the turbulentnature of the flow in the Strait.The density variability is restricted to fewer harmonics than the velocity field. Thefirst layer is heavily influenced by the Fraser runoff and the wind; Figure 3.25 shows thepower spectra of both. The spectrum of the density field in layer one (top of Figure 3.24)shows a large peak around 2 x i0 cph (a period of 20 days). The runoff has no signalnear the peak observed in the surface layer of 2 x i0 cph. In fact 95% of the runoffvariance is explained by the first 6 Fourier components (which are around the annualand semiannual frequencies). However, the wind has some energy close to the frequencyof 2 x iO cph. Perhaps as the wind pushs the fresh water from the Fraser River upand down the east coast of the strait, it generates density fluctuations in the top layerat these frequencies.In general the variance decreases with depth for all variables, but the decrease isgreatest for density.3.2.2.2 Harmonic AnalysisAs in Section 2.2.3 harmonic analysis was performed to give a more focused account ofthe relative importance and origin of the different modes of variability. Here the sameanalysis is repeated but with an increased number of constituents. Table 3.2 gives thedifferent constituents utilized; this set of constituents was determined with the help ofthe raw spectra from the previous section.Table 3.3 shows the percentage of the horizontally averaged variance, for the moreenergetic modes of Table 3.2, for ‘i, and for each layer of u, v, and p’. The dominant modesare the Msf and Mm for the velocity components, the Sa and Ssa for the density fieldChapter 3. A multi-layer model of the Strait of Georgia1 700Ss a+Msm Mm MsfMf N2S++ +85(a)0.0002 -0.00100.0000(m/s) 2(b)cph(c)180 30 15period (days)Figure 3.25: Power spectra of the 1984 data of a)the Fraser River discharge and b) and c)of the U, and V, components of the wind, respectively, in the central Strait of Georgia.0.0000(m/s) 20.0001 0.001 0.01Chapter 3. A multi-layer model of the Strait of Georgia 86Table 3.2: Tidal constituents used in the harmonic analysis. “OB” indicates that theconstituent is included in the tidal forcing at the open boundary.Name Period Origin(days)Z0Mf 13.66 K1—0,K2M&OBMsf 14.77 S2 — M2, Pi—01 & OBN2S 9.61 S2 — N2Mm 27.55 MN&OBMsm 31.81 Msf — MmMmSa 25.62 Mm — SaSa 365.25 K1—S, S — P1 & OBSsa 182.67 K1— Fi, K2 — S2, Mf — Msf, Harmonic of Sa & OB4Sa 91.29 Harmonic of Sa8Sa 45.65 Harmonic of Sa8Ssa 22.82 Harmonic of Ssal6Ssa 11.41 Harmonic of SsaChapter 3. A multi-layer model of the Strait of Georgia 87and the Sa for the surface elevation field.A vertical average of the total variance from Table 32 results in 50, 46, 66 and 94%of the variance for u, v, p’ and i, respectively. Stacey et al. (1987) found that for alltheir stations and all depths, the Ssa, Msm, Mm, Msf and Mf account for 37% ofthe variance for both the u and v components of velocity. Looking only to those sameconstituents used by Stacey et al. (1987), the model results correspond to 39% for u and35% for v, which average to 37%! If this result holds, and is not a fortuitous result ofthe model with its current resolution and parameterization, then this is an importantresult because the model is giving total energy levels close to those observed with aboutthe same amount explained by the low frequency tidal constituents. Further comparisonbetween model and observations will be presented at the end of the chapter.Now I will show the horizontal distribution of the variance of the more importanttidal constituents, in particular, the two fortnightly (Mf and Msf), the three monthly(Mm, Msm and MmSa), the annual (Sa) and the semiannual (Ssa) constituents. Forthe velocity components (u, v) and the density field, the spatial structure is shown forlayer three only. This layer is one of the layers where a large percentage of the varianceis explained by the low-frequency tidal constituents (see Table 3.3). It is representativeof the spatial distribution of the variance in the other layers.Figure 3.26 shows the percentage of the variance of the surface elevation field. Mostof the variance of i is accounted for by the annual component, Sa, The second largestcomponent is the Msf fortnightly constituent but with less than 10% of the variance.The Ssa and Mf constituents have comparable amplitudes and together are of the sameorder of the Msf. Note that Sa in the surface elevation is not tidal but due to the riverinput. (Also the river discharge dominates layer 1 of the density field; not shown.)Figure 3.27 and figure 3.28 show the percentage of the variance of the u and v components of velocity at layer 3, respectively. The dominant modes are the Msf and MmChapter 3. A multi-layer model of the Strait of Georgia 88Table 3.3: Percentage of the horizontally-averaged variance, explained by the given lowfrequency constituents for the i and the 8 layers of u, v and p’. The column Harmsstands for the sum of the variance of the N2S,4Sa, 8Sa, 8Ssa, & l6Ssa constituents.layer Mf I Msf Mm MmSa Msm j Sa Ssa Harms Totalu field1 0.8 7.1 2.5 1.3 2.1 12.7 3.3 4.3 34.12 2.6 15.5 9.7 2.4 3.9 7.4 2.8 2.7 47.03 2.7 24.7 26.8 2.4 3.0 3.8 3.4 2.7 69.54 3.0 16.9 30.7 2.1 1.7 3.2 5.5 4.3 67.45 3.2 11.6 14.9 3.9 4.7 4.8 3.1 5.0 51.26 3.1 14.0 26.5 2.8 3.0 3.2 6.4 3,3 62.37 2.7 11.1 12.9 3.9 1.9 1.2 3.6 4.7 42.08 4.4 7.7 9.6 3.8 1.2 1.0 1.3 4.1 33.1v field1 0.9 8.1 2.3 2.2 3.6 5.0 3.1 4.4 29.62 2.0 16.4 8.9 3.4 4.7 3.2 1.9 4.0 44.53 2.7 17.0 21.9 4.1 4.7 3.2 3.4 3.4 60.44 2.1 12.8 28.1 2.3 1.8 4.3 7.8 4.9 64.15 1.4 4.8 8.0 4.2 6.5 4.1 4.4 4.8 38.26 2.3 11.5 24.8 3.9 4.7 3.2 5.4 3.2 59.07 2.0 10.4 16.9 4.1 2.4 0.9 3.4 4.0 44.18 3.1 5.4 9.8 3.6 1.4 0.4 1.4 3.5 28.6p’ field1 0.4 1.3 0.4 0.7 1.3 63.7 5.5 2.3 75.62 1.0 7.5 6.9 2.7 3.7 19.4 7.7 4.7 53.63 1.4 10.9 18.9 5.2 5.7 7.9 10.7 5.3 66.04 0.5 6.0 19.5 2.9 3.1 10.1 23.8 5.7 71.65 0.7 4.0 17.4 0.8 0.4 7.9 19.4 7.7 58.36 0.3 0.9 2.6 0.9 1.1 28.0 9.0 8.2 51.07 0.5 1.0 1.9 0.7 0.2 48.2 15.2 6.6 74.38 0.2 0.2 0.3 0.3 0.1 59.5 14.1 4.3 79.0q field— [ 5.3 1.6 0.1 0.4 77.8 4.2 [ 0.8 93.9Chapter 3. A multi-layer model of the Strait of Georgia 89Mr MsfMmSasurfaceKce 1 evat onFigure 3.26: Percentage of the variance explained by the different tidal constituents forthe surface elevation () field.Mm MsmSsa SaChapter 3. A multi-layer model of the Strait of Georgia 90Mr MsfMmSau at layer 3Figure 3.27: Percentage of the variance explained by the different tidal constituents forthe u component of velocity for layer 3 of the model.Mm MsmSsa SaChapter 3. A multi-layer model of the Strait of GeorgiaMf MsfMmSav at layer 391Figure 3.28: Percentage of the variance explained by the different tidal constituents forthe v component of velocity for layer 3 of the model.Mm MsmSs a SaChapter 3. A multi-layer model of the Strait of Georgia 92Mf MsfMmSadensity at layer 3Figure 3.29: Percentage of the variance explained by the different tidal constituents forthe o field for layer 3 of the model.Mm MsmSs a SaChapter 3. A multi-layer model of the Strait of Georgia 93with a tendency to be larger on the mainland side of the Strait. The rest of the modesshare about the same proportion of the rest of the total variance. For the v component,the semiannual constituent explains about 10% of the variance in the central part of thedomain.Figure 3.29 shows the percentage of the variance of the density field in layer 3. Herethe variance is more spread between the modes, the Msf, Mm and Ssa constituentsbeing somewhat larger the others.The previous figures have shown that the amplitude of the different tidal constituentshas considerable spatial variability (probably/mainly in response to the changing bathymetry).3.2.2.3 Nonlinear interactionsAs stated in Chapter 2, the effect of excluding several constituents in the forcing at theopen boundaries is addressed here. In Chapter 2, the tidal forcing came through onlythe 5 dominant constituents (see Table 2.1) and therefore the low frequency currentswere generated locally, as there was not direct low frequency forcing. The main mode ofvariability was found to be the Mf constituent and it was larger than the Msf by anorder of magnitude.Two additional runs of the 3-D model were performed with the same conditions asthe reference run: the only difference being the number of constituents used at the openboundary to reconstruct the tides.These runs are labeled Run 5 (R5) and Run 7 (R7). Run 5 has only the 5 constituentsused in Chapter 2. Run 7 has these same 5 constituents plus the next two more importantconstituents, P1 and K2. The difference between Runs 5 and 7 will show how much of thefortnightly Msf and Mf constituents are produced by the nonlinear interaction betweenP1-01 and K2-M2, respectively. When these runs are compared to the reference run, oneChapter 3. A multi-layer model of the Strait of Georgia 94can see how much of the low frequency current is locally generated. (Recall that thereference run also has low frequency forcing, see Table 3.1.)Recall that the Mf constituent arises from nonlinear interaction of K1-0 and K2-Mand the Msf from the M2-S andP1-0. In Chapter 2, I gave indirect evidence ofK2-Mand P1-0 contributions with a linear model (Butman et al., 1983), and found that it issmall; given the large dominance of the Mf in the 2-D model, the picture did not changethere. However, in that linear model analysis, phase differences in the nonlinear interactions were not taken into account and as a result constructive or destructive interferencebetween the modes was not accounted for.Obviously, the result depends on the phase. Suppose that B is a wave generatedby K1-0 with amplitude and phase (a, q), and C a wave generated by K2-M withamplitude and phase (b, q5). A simple addition result inA = A cos(wt— 4) = B + C = a cos(wt — q) + b cos(wt — q2)and with a =42—c51_____A = 2a cos( 2 cos(wt — 2The resulting amplitude is modulated by the cos of the difference in phases between thetwo waves times twice the amplitude. Figure 3.30 shows the amplitude A obtained byharmonic analysis of a time series produced by adding the two waves with a = 5, q = 0and b = 0, 1, 2, 3, 4 and 5 as a function of q!?2. It is easy to see in the figure how theamplitude A decreases as the two waves go out of phase.Here, using the 3-D numerical model, it was found that, the results of having 5, 7 orall 44 constituents do not differ much. Table 3.4 gives the ratio of the amplitude of thelow frequency tidal constituents for the u and fields that vary more than 10%, amongthe different runs, averaged in the horizontal domain. (The ratios for the v componentare very similar to that of u and hence have not been shown.)Chapter 3. A multi-layer model of the Strait of Georgia 95A cos(wt—’) = 5 cos(wt) + b cos(wt—Ø2)10. ----_..S—S4 —S.___% %%• • . _ ••%%, 5%-S S••%•%I I I I I0. 30. 60. 90, 120. 150. 180.Figure 3.30: Amplitude A of a harmonic produced by the sum of the waves a cos(wt—and b cos(wt— c2) for the special case of a = 5 and 4 = 0 as a function of 2’ Thedifferent curves correspond to different values of b.Chapter 3. A multi-layer model of the Strait of Georgia 96Table 3.4: Ratio of the amplitude of the low frequency tidal constituents, between thereference run (Ref) and runs 5 (R5) and 7 (R7), for the u and ii fields for the 8 modellayers. The reference run has 44 tidal constituents to reconstruct the tides at the openboundaries. Run 5 has 5 constituents (K1, O, M2, S2 and N2) and run 7 has 7 (sameas run 5 plus K2 and F1). Only those constituents whose ratios vary more than 10%averaged over the model domain in the central Strait of Georgia are shown.Ref/R5field & layer Mf Msf Mm Msm Ssau field 1 0.69 1.15 1.06 1.03 1.082 0.64 1.29 1.17 1.28 1.213 0.62 1.24 1.14 1.74 1.524 0.71 1.15 L13 0.57 2.375 0.70 1.07 1.25 1.14 1.526 0.70 1.25 1.14 1.73 3.647 0.67 1.50 1.03 0.88 1.898 0.73 1.59 1.19 0.73 1.13field - 1.45 1.53 1.47 1.11 2.02Ref/R7u field 1 1.07 1.04 1.08 1.03 0.972 0.97 1.08 1.17 1.22 0.993 1.04 1.07 1.16 L81 0.944 1.20 1.02 1.15 0.74 0.945 1.00 0.98 1.23 1.20 1.106 1.00 1.06 1.19 1.62 1.057 0.84 1.14 1.12 1.07 1.068 0.90 1.13 1.19 1.00 1.29,field- 1.50 1.78 1.65 1.24 1.02R7/R5u field 1 0.65 1.11 0.98 1.00 1.102 0.66 1.20 1.00 1.05 1.233 0.60 1.16 0.98 0.96 1.614 0.60 1.12 0.98 0.77 2.535 0.70 1.09 1.02 0.95 1.386 0.70 1.17 0.96 1.07 3.457 0.79 1.32 0.92 0.82 1.788 0.81 1.41 1.00 0.73 0.88,field- 0.96 0.86 0.89 0.89 1.99Chapter 3. A multi-layer model of the Strait of Georgia 97The largest changes occur for the Mf, Msf and Ssa constituents.The surface elevation () response of the latter constituents, with the exception ofthe Ssa, is simply that when the low frequency components are excluded, the amplitudeof the low frequency constituents in the Strait are reduced. At the open boundary, theamplitudes of the low frequency constituents are 1.4, 1.8, and 1.6 cm for Mf, Msf andMm, respectively. The amplitude in the Strait of these constituents when the direct lowfrequency forcing is excluded is about the same. So when both effects act together theyadd up to give a larger amplitude. The Ssa response is different from the response justdescribed. The Ssa amplitude is about equal between the reference run (Ref) and R7.This constituent arises in the model from various interactions, these arein R5: from (K1-0)— (S2-M)only,in R7: as in R5 plus K1-P,K2-S,in Ref: as in R7 plus Mf-Msf, harmonic of Sa and Ssa.The amplitude of the Ssa in the open boundary is 1.3 cm. The average amplitude in thecentral Strait of Georgia is ‘-. 1.8 for both Ref and R7 and 0.9 for R5. These resultingamplitudes suggest that: 1) the local generation of Ssa is important, 2) the inclusion ofthe K2 and P1 constituents results in a larger amplitude of Ssa, and 3) the low frequencyforcing from the open boundary is not important, even though the amplitude at theboundary is as large as the response inside the Strait.The velocity field responds differently, especially with respect to the Mf constituent.The difference is with the run that excludes the K2 and P1 tidal constituents. When theyare included, their contribution is somewhat out of phase with respect to that generatedby the I(-O interaction, and the result is a smaller Mf in the Strait. This can be seenin the ratios of R’T/RS in Table 3.4. The direct effect is negligible as suggested by theratio of Ref/R7 in Table 3.4, because Run 7 has all the nonlinear interaction that canoccur from the diurnal and semidiurnal tides. The effect on the Ssa is about the sameChapter 3. A multi-layer model of the Strait of Georgia 98for the velocity field as for the surface elevation field explained above. The Msf, Mmand Msm are basically always larger for the reference run.Summarizing, most of the residual energy is confined to the most important lowfrequency tidal constituents at the fortnightly and monthly bands. The more importantare the Mm and Msf constituents. They arise basically from nonlinear interactions ofthe diurnal and semidiurnal tides.Chapter 3. A multi-layer model of the Strait of Georgia 993.2.3 Budgets: momentum, vorticity and energyIn this section, the dynamics of the residual motion in the central Strait of Georgiaare analyzed by evaluating the various terms of the momentum, vorticity and energyequations.3.2.3.1 Momentum BudgetThe momentum equations, vertically-integrated over each layer, are= —Vjj (€uii) — + fv — + t’Vu + z(r), (3.7)and= —kV (€3vui) — — fu — + t’Vv + (3.8)where P = gp*( — z) + p, VH and V are the horizontal divergence and Laplacian,respectively and z(...) is the difference of (...) taken between the upper and lowersurfaces of a layer. Note that these equations are defined for a given layer and so eachvariable should be marked with an appropriate subscript ‘j’to indicate which layer isbeing considered: u, v3, h, wj For convenience these subscripts are omitted. Theterms on the right-hand side of the equations, reading from left to right, are horizontaland vertical advection, Coriolis acceleration, pressure gradient, horizontal and verticaleddy diffusion, respectively.Two approaches are taken to evaluate the different terms: both must give the samefinal results, but as will be seen below, the second gives more information.The first approach (1) is the same as done in Chapter 2 where the different terms inthe equations are evaluated using the total fields and then a low-pass filter is applied tothe resulting time series. If the filter is denoted by (...), then the x-momentum equationbecomesChapter 3. A multi-layer model of the Strait of Georgia 100= (u) - Z(w) + 7 -- + Vu + (r), (3.9)and the y-momentum equation becomes—V (i3v) —— j— + t’Vv + (3.10)This first method will oniy be used as a check of the second.In the second method, (2), the fields are split into low and high frequency components,namely,= so + i, (3.11)where p is any of the dynamical variables, y = represents the low frequency components (i.e., the residual), and y represents the high frequency components (i.e., thetides), where=0. (3.12)Now substitute (3.11) into (3.7) and apply the filter, (...), term by term.The local rate of change isat — at — atThe Coriolis acceleration isfv=fY = fvo.The pressure gradient isloP iaP 1OP0 10(gp*p)— Ox — — p* Ox — í Ox —The horizontal diffusion isi’Vu = tiVi =Chapter 3 A multi-layer model of the Strait of Georgia 101The horizontal advection is1 1 Ouuh 9uvh 1 auih OviYh—VH.(vuh)=—.-[ a + a ox + ôwhere the resulting velocity products will simplify, with the use of 3.12, toY=(uo+u1)(vg+v = uovo+W+j3-I-uj5T= uovo+üYj,and iii = u0 + iiji. Thus the horizontal advection term becomes1 1. O(uouo+ij)ii O(uovo+ij)ii—-VH• (vuh)= —-[ +h h OxFor the vertical advection and diffusion terms it is convenient to write D = h’. However,recall that each of the variables is labeled with a subscript ‘j’indicating which verticallevel is being considered. For all levels but the first (i.e., top), h3 is constant and thus= D03. However, as the first layer contains the time dependent surface height,D1 = D01 + D131.The vertical advection for the first layer then becomes,—(w) = —DuIX(w) = —(D0 + Di)(uo + ui)i(wo + wj) =— [Douoz.(wo) + DouiL(wi) + DiuiLi(wo) + DiuoZ(wi) + DiuiL(wj)]. (3.13)The vertical advection for the interior and bottom layers are obtained by simply settingD1 = 0 in the above expression.Each of the 5 terms of —L(w) on the RHS of 3.13 for the first layer were evaluatedusing the model results, and it was found that the DiuiLS(wo) and DiuoL\(wi) termswere small. Thus—*L(w) —[Douo/.S(wo) + Douiz(wi) + Diui(wi)J.Chapter 3 A multi-layer model of the Strait of Georgia 102Finally the vertical diffusion becomes= Dz(r) = (D0 + Di)(ro +r1) =DoLS.(ro) + DoLS(ri) + DiLS.(ri).Again for the interior and bottom layers, D1 = 0. Note that LSri) is retained, since thestresses depend on the layer, i.e.,toplayer:Pw9uinterior layers: =LIZbottom layer: = = Cdu%/u2 + v2,and, as a result their separation into low and high frequency components needs moreattention.For the top layer, the separation is made directly; namely (r3) = r8o +r1, and in thiscase j = 0.For the interior layer the vertical eddy viscosity coefficient is also time dependent (as itis a function of the flow, see equation 3.6). Then the low and high frequency separationis as follows,_________________/9u c9u0= A0—--— +A1—-—,LIZ ()Z LIZso9u1= A,o——, and r,1 = A1—-—.LIz azThe bottom stress has the form (see Appendix D)2u+v? u1v —TbCd[ uo+—-—vo+uiIviI]lvii lviiand the low and high frequency components are split as follows:2u + v u1vTbrO = Cd[ no + —-—vo], and Tbxi = Cauiivii.lvii lviiChapter 3. A multi-layer model of the Strait of Georgia 103[rn/secl160. 170. 180. 190. 200.daysFigure 3.31: Time series of bottom v-component of velocity, for a particular grid pointat the center of the study area, for the tidal (thin line: vi) and residual (thick line: vo).To obtain expressions for the bottom stress an assumption was used that the low frequency velocities were smaller than the high frequency velocities at the bottom (seeAppendix D). The assumption is not unreasonable as can be seen in Figure 3.31 whichshows, the bottom velocity for the tidal current and for the residual, for a grid point atthe center of the study area. It can be seen that the amplitude of the residual is reallymuch smaller than the tidal currents.Collecting all terms, the x-momentum equation for the residual becomes8u0 1 9uououi ãuovoiz 1 (gp* + p0)+ 9—L(wo) +— 1[ôUiulh+ôU1Vh— [u1(wi) + DiujL(w1)]+ z(ri) + DiL(ri). (3.14)Similarly, the y-momentum equation for the residual becomesChapter 3. A multi-layer model of the Strait of Georgia 1040v 1 Ouovoi Ovovoli 1 ö(gp*0p)—-----[ + ]—fuo——h ox t9y72 VOAI \ Al+VVjjVO — -x--Wo) + Ty0— 1[OUivlh+o5ih]— [vii(wi) + Divi(wi)] + (r1)+ Diz(ri). (3.15)The reason why this second method gives more information now becomes clear. Recallthe first method where a filter was applied term by term to the momentum equationsresulting in equations 3.9 and 3.10. Thus the residual (filtered) flow Oii/Ot is maintainedby forces (pressure gradients, advection, etc.) which are represented by terms containingthe total fields (u, v, F, etc.). However in the second method, where the fields were splitinto low and high frequency components, there are terms representing the forces drivingthe residual, Oi/Ot = Ouo/Ot, that contain the residual fields only (lines 1 and 2 ofequations 3.14 and 3.15) and terms with the time averages of the tidal or high frequencyfields only (line 3 of equations 3.14 and 3.15). The later terms (resulting from highfrequency forcing) are independent of yo and are identified in the literature (e.g., Nihouland Ronday, 1975) as tidal stresses. They enter the equation for the residual velocitiesas an effective force. Thus in the second method forcing due to low and high frequenciesare represented separately, resulting in a clearer understanding of the dynamics.When the second method is checked against the first method by evaluating the momentum terms, the sum of the residual and tidal components of each of the differentforces in the second method is checked against the corresponding force in the first. Forexample, in the x-direction the horizontal advection of the first method must be equal tothe sum of the horizontal advection of the residual plus the horizontal advection of theChapter 3. A multi-layer model of the Strait of Georgia 105tides, namely,1 1 ‘. 1-VH (iJuh) = -V• [hvovo] + ‘c-V [hvivi].1st method 2nd methodTable 3.5 gives acronyms for each of the terms of equations 3.14 and 3.15.Mean Momentum budgetsThe terms appearing in Table 3.5 were evaluated for the reference run, for all layersand for several gridpoints, over the complete year of integration, and these terms werethen time-averaged. While the particular balance among terms depends on location andis quite complicated, the following classification and features can, in general be noted:The largest terms are those involved in the geostrophic balance [pressure gradient (PGRM) and Coriolis (CORM)] and the horizontal advection terms of the tidalcomponents (HATM). Usually the geostrophic terms are larger and their imbalance iscompensated by the HATM.The intermediate terms playing a role in the mean balances are the vertical advection by tides (VATM) and the horizontal diffusion of the residual (HDRM). These termsare typically one order of magnitude smaller than the dominant terms.The smallest terms are the vertical diffusion by the tides (VDTM) and the verticaladvection of the residual (VARM).The horizontal advection of the residual (HARM) and the vertical diffusion of theresidual appear at various times to be intermediate, small or between the two.Table 3.6 shows the mean balances of the terms of the u-momentum equation, 3.14 forthe 4 gridpoints shown in Figure 3.1 for two layers (1 and 6) of the model. (Similar resultsare obtained for the v-equation.) These results are time-averaged and they represent theChapter 3. A multi-layer model of the Strait of Georgia 106Table 3,5: Acronyms and summary for terms in the momentum equations for the residual3.14 and 3.15 as they are referred to in the text. To facilitate the identification of theabbreviated names, the last two letters have the following meaning: the penultimate isT or R for tidal or residual, respectively, and the last letter is M for momentum.Terms eqns. 3.14 and 3.15 Identification—V [uii3oiio] HARM: horizontal advection of residual momentum—[I] HATM: horizontal advection of tidal momentum—-(wo) VARM: vertical advection of residual momentum—[j€iz(wi) + Diiz(wi)] VATM: vertical advection of tidal momentum—f x CORM : Coriolis of residual momentum— 4V(gp*io + p0) PGRM: residual pressure gradient+z)VWo) HDRM: horizontal diffusion of residual momentumVDRM: vertical diffusion of residual momentum+Iz() +D1z(f) VDTM: vertical diffusion of tidal momentumChapter 3. A multi-layer model of the Strait of Georgia 107typical balance of forces in most of the different levels of the model. The last column ofTable 3.6 classifies the terms as large (L), intermediate (I) or small (S). However, thisclassification does not hold for all the gridpoints and layers. When the classificationdoes hold, the corresponding number in Table 3.6 appears in bold face. The last row ofTable 3.6 gives the sum of all terms for each region and layer. Note that the size of thesum is not insignificant, but in general it is smaller than 10% of the largest terms; insome cases it is 20% of the largest terms (see layer 1 in region II and layer 6 in regionIV).The integration time of the mean budgets is one year and the sum of the terms shouldbe zero. However, as shown in Table 3.6, this is not true in the model; furthermore, as willbe shown below, the mean budgets of vorticity and energy show larger imbalances. Theseimbalances are occurring in these diagnostic calculations only, and does not mean thatthe model does not reach equilibrium. The velocity, vorticity and energy fields oscillatearound a mean value (not shown). Note that the budget’s results are given for individuallayers and gridpoints. The imbalances found are common in this kind of evaluation (e.g.Hall, 1986). Because of the complexity of the space and time variability of the flows, thebudget analysis methods often involve considerable averaging (large areas or full domain).These methods may not characterize the dynamical behavior of a subregion of the flowitself (Harrison and Robinson, 1978). To investigate if the results presented in Table 3.6for momentum (and the corresponding for vorticity and energy below) are meaningful, Ispatially averaged all the different terms in the equations (not shown) around the usualfour gridpoints; the imbalance in the equations was reduced as the area of averaging wasincreased and the general picture on the relative size of the terms in the equations didnot change (i.e., the classification of L, I or S was the same).Chapter 3. A multi-layer model of the Strait of Georgia 108Table 3.6: Time-averaged terms from Table 3.5 for the 4 selected gridpoints (StationsI, II, III and IV) for layers 1 and 6 of the model. Units are in 10—6 m s2. In the lastcolumn L, I and S stands for large, intermediate and small, respectively, and in generalgives the relative size of the terms in the momentum budget of the model. When thenumbers in the table are given in bold face it indicates that the classification of thatterms as L, I or S is true for that gridpoint in particular.Layer 1 Layer 6Terms I II III IV I II III IV SizeHARM 0.42 0.19 0.24 0.28 0.03 0.04 0.03 0.03 I or SHATM 0.27 1.51 0.59 -0.04 -0.10 0.36 0.31 0.38 LVARM -0.02 0.00 0.01 0.00 0.00 -0.01 0.00 0.00 SVATM -0.03 -0.65 -0.28 -0.02 0.17 -0.03 -0.06 -0.03 ICORM 3.04 1.71 3.29 3.32 -1.16 -0,75 -0.55 0.23 LPGRM -3.16 -1.46 -2.93 -2.79 1.00 0.41 0.19 -0.64 LHDRM 0.05 -0.26 -0.12 -0.08 0.08 0.03 0.02 -0.09 IVDRM -0.62 -0.88 -0.90 -0.79 0.00 0.00 0.00 0.00 I or SVDTM 0.07 0.17 0.15 0.11 0.00 0.00 0.00 0.00 S> 0.01 0.33 0.05 0.01 0.01 0.05 -0.06 -0.11Chapter 3. A multi-layer model of the Strait of Georgia 109Time-Variability of the Residual Momentum budgetsTo explore the time-variability of the momentum terms each term was first evaluatedand then de-meaned. The general classification of the mean (time-averaged) terms givenin Table 3.5 also applies to the fluctuating terms. As an example, I show in Figure 3.32a time series of the momentum terms of equation 3.14 in the x direction for the 7thlayer for the gridpoint corresponding to station III (for reference see also Table 3.5).The figure shows thin lines for the terms involving the residual and thick lines for thetidal components. In the first and second panels, the horizontal and vertical advectionare determined by the tidal momentum component (HATM and VATM: thick lines).The third panel gives the sum of the Coriolis and pressure gradient terms (CORM +PGRM) which are the largest terms and tend to cancel; their sum is usually about 5times smaller than the individual terms. In the fourth panel the horizontal diffusionof residual momentum (HDRM) is shown; note the relative size of this term which hasbeen classified as intermediate (I). The fifth panel shows the vertical diffusion of bothresidual and tidal momentum (VDRM, VDTM) and they are very small. The sixthpanel showns the sum of all the residual components and the sum of all tidal componentsseparately. Note that the sum of the residual terms is basically determined by thegeostrophic imbalance and the sum of the tidal components by the horizontal advectionof tidal momentum (HATM). The bottom panel shows the sum of all terms, i.e., local rateof change of the residual momentum (= auo/at). Note that it is basically determined bythe geostrophic imbalance and the horizontal advection of the tidal momentum (HATM)with some horizontal diffusion of the residual playing a role (HDRM),Chapter 3. A multi-layer model of the Strait of Georgia 1106x107 [rn/s2 ]HARM & H ATM-6x107VARM & VATM_______Co R M + PG R MU fl D M —-----_ — ——---I 1 L_) F’ I —.—---._...____._———-.--.VDRM & VDTMSum RMRM TM+ termsI I I I I I225. 229. 233. 237. 241 245, 249. 253.daysFigure 3.32: Time series of the fluctuating u-momentum terms of layer 7 for Station III.Thin lines give terms arising from the residual; thick lines give terms arising from thetidal components. The vertical scale is the same for all plots (and is shown in the top oneonly). From top to bottom the panels show horizontal advection, vertical advection, thesum of the Coriolis and pressure gradient terms, horizontal diffusion, vertical diffusion,the sum of all the residual and all the tidal terms and finally, the sum of all terms. SeeTable 3.5 for the meaning of the acronyms.Chapter 3. A multi-layer model of the Strait of Georgia 1113.2.3.2 Vorticity BudgetThe vorticity equation for the residual is obtained either 1) by cross-differentiating themomentum equations of the total flow (3.7) and (3.8), splitting the fields in low and highfrequencies and then apply the temporal filter Jj or 2) by direct cross-differentiationof the momentum equations of the residual (3.14) and (3.15). I will take the first route.Taking the curl of (3.7) and (3.8) the vorticity equation becomes= —fVH i + z)Vk + —— -[VH (iv&)] + -[kVH. (iuui)] — + -[(w)], (3.16)where = — is the relative vorticity. Recall that all variables are labeled with asubscript ‘j’for each layer. Also recall that for the top layer h = + and for theinterior and bottom layers k = h,. The terms in the second row of (3.16), come fromthe non-linear advection terms and they require further work to express them in termsof the recognized physical mechanisms. Substituting the continuity equationãuh ãvhin (3.16), the non-linear terms now read01 01— —[-VH. (vvh)] + —[VH (vuh)]—(zih)]. (3.17)For the interior and bottom layers h = h and the terms of (3.17) reads,[kVH. (ivii)] + (iuiz)] + —[VH (zii)j — [VH (I)} =—--[VH. (vi) — VH• (ii)j + -[kVH. (i3uui) — VH• ()j =Chapter 3. A multi-layer model of the Strait of Georgia 112—VHv) + -(6. VHu) = —VH 6— 6 VH. (3.18)For the top layer h = h + ,, h = constant and the nonlinear terms, (3.17), become,4[VH.(6v)- (((h+)VHV+VVH)]+-[VH (6u)_( )((h +)VH .6+6. VH)} =——[v.VHV— v.VH]x (h+i)+—[6 VHU — V. Vq]Oy (h+)(3.19)where0 v -. 0 U -.Z=— VVHI1.0x(h+’q) e9y(h+q)In the model this extra term, Z, turns out to always be 2 or 3 orders of magnitudesmaller than the other terms and therefore is neglected in all of the analysis. With thisassumption, the nonlinear terms are identical for all layers.Substituting (3.18) into (3.16) the vorticity equation then becomes= —6• VHC— (f + C)VH .6+ u’V + {-[L(r)] — -[L(rT)]}. (3.20)Here, the terms on the RHS involve advection, column stretching/shrinking, horizontal diffusion and vertical diffusion of vorticity, respectively.A point of concern must be addressed here. The numerical model solves the momentum equations 3.1. However the equations that govern the numerical model are thediscretized momentum equations rather than the continuous momentum equations. Nowcomes the question of how to evaluate the equations for other formulations of the problem, such as the vorticity or energy equations. One way would be to take the continuousChapter 3. A multi-layer model of the Strait of Georgia 113momentum equations, derive the continuous vorticity equation, discretize the vorticityequation and then evaluate the resulting terms. A second way would be to use thediscretized momentum equations (which, after all, govern the numerical model) and usethese to derive a discretized vorticity equation. Depending on the grid and the discretization, these two versions may or may not be consistent. For example, for a homogeneousocean with a /-plane approximation, using the Arakawa B grid (Mesinger and Arakawa,1976), Foreman and Bennett (1989) showed that the second method, deriving the discretevorticity equation from the discrete momentum equation contains a term related to thevariable Coriolis parameter that has no counterpart in the discretized vorticity equationderived from the continuous momentum equations. In the case of differences, the correct version comes from those obtained directly from the discrete momentum equations(Haltiner and Williams, 1979).In order to evaluate the vorticity equation term by term, the third and fourth termsof (3.20) are evaluated directly from the discretized momentum equations, but the firstand second terms are evaluated from the discrete vorticity equation derived from thecontinuous momentum equations. In Appendix E I show that the discrete version of (3.20)is effectively consistent with the discretized vorticity equation derived directly from thediscretized momentum equations if the nonlinear terms are expressed in flux form asj. VH—CVH•i= —VH (vO.For the interior layers it is exact. For the top and bottom layers there are inconsistencies but it is shown in Appendix E that the difference between the two formulations isinsignificant. The first formulation, resulting in 3.20 is preferred because of the physicalmeaning of the terms expressed in this form.Now, the residual vorticity equation is obtained by splitting the fields in 3.20 into lowand high frequency components and then apply the low pass filter, (.,.); this givesChapter 3. A multi-layer model of the Strait of Georgia 114O(Coi)=+iV1(C0+ Ci) + -[(D0 + Di)(ro +r1)] — + Di)(ro + Ti)J, (3.21)which simplifies to!=—oVH(’o)—(f+Co)VH•io+V(Co) + _[Do/(ro)] — -[Doz(r0)j—— 1VH V1+ + Di)z(r1)]— + Di)L(r2,j j. (3.22)As in the momentum equation, the above process makes clear what mechanisms aredriving the residual vorticity: whether it be the residual vorticity or the rectified tidalforcing. To identify the different terms in the text, the acronyms given in Table 3.7 willbe used.Mean Vorticity budgetsAs in momentum, the different terms of 3.22 were evaluated for the reference run andthen time-averaged over the complete year of integration. The largest terms are easilyidentifiable. However for the intermediate and small terms the balance is a bit morecomplicated than in the case of momentum. The following classification can be made:The large terms are the advection of tidal vorticity (ATV), column stretching ofresidual vorticity (SRV) and horizontal diffusion of residual vorticity (HDRV).The intermediate terms are the column stretching of tidal vorticity (STV) and theadvection of residual vorticity (ARV).The small term is usually the vertical diffusion of tidal vorticity (VDTV).Chapter 3. A multi-layer model of the Strait of Georgia 115Table 3.7: Acronyms and summary of terms of the residual vorticity equation 3.22 asthey are referred to in figures and text. To facilitate the identification of the abbreviatednames, the last two letters have the following meaning: the penultimate is T or R fortidal or residual, respectively, and the last letter is V for vorticity,Term from eqn 3.22 Identification—o Vji(o) ARV: advection of residual vorticity—VH(1) ATV: advection of tidal vorticity+t’V(Co) HDRV: horizontal diffusion of residual vorticity+[Doi..(ro)] —[Doi.(r1o)] VDRV: vertical diffusion residual stresses+[(D0+ Di)LS.(ri)]—[(D0+D1)L(r] VDTV: vertical diffusion tidal stresses—C1VH STV: stretching/shrinking of residual vorticity—(f + Co)VH SRV: stretching/shrinking of residual vorticityChapter 3. A multi-layer model of the Strait of Georgia 116The vertical diffusion of residual vorticity (VDRV) varies between large and small. Itis usually large at the upper layers and small in the rest of the water column.As in the mean momentum budgets, Table 3.8 shows the results for the 4 gridpoints(see Figure 3.1) of the time-averaged terms for layers 1 and 6 of the model, which arerepresentative of the vorticity balances in the different levels of the model. Table 3.8shows the dominance of the advection of tidal vorticity (ATV), a consequence of thetidal stress forcing (the HATM terms) as found in the budgets of momentum, Note thatthe table shows fewer bold numbers than the corresponding table for momentum, in otherwords, the classification of L, I or S is not as straight forward as it is for momentum,especially at stations II and IV for layer 1.Time-Variability of the Residual Vorticity budgetsEach term of the vorticity equation was evaluated and then de-meaned. The classification of the mean terms above also applies to the fluctuating terms. As an example,Figure 3.33 shows the time series for various terms of the vorticity equation 3.22 (seealso Table 3.7) for a gridpoint corresponding to station III for the 7th layer. The figureshows thin and thick lines for terms involving residual and tidal components, respectively.Note that the advection is dominated by the tides, while the stretching/shrinking mechanism is dominated by the residual component. Basically these two terms plus horizontaldiffusion determine the local rate of change of vorticity.3.2.3.3 Energy BudgetFinally, as part of the budget analysis, the energy balance is examined. In general, thekinetic energy equation is obtained by forming the scalar product of the velocity withthe momentum equations (3.7 and 3.8): i• becomes , where K = (u2 + v2) is theChapter 3. A multi-layer model of the Strait of Georgia 117Table 3.8: Time-averaged terms from Table 3.7 for the 4 selected gridpoints (Stations I,II, III and IV) for layers 1 and 6 of the model. Units are in 10_li s2. In the last columnL, I and S stands for large, intermediate and small, respectively, and in general, gives therelative size of the terms in the vorticity budget of the model. When the numbers in thetable are given in bold face it indicates that the classification of that term as L, I or S istrue for that gridpoint in particular.Layer 1 Layer 6Terms I II III IV I II III IV SizeARV -2.69 -13.39 -7.98 -0.02 -1.07 -0.71 -0.42 -0.54 IATV -7.36 -3.12 3.64 2.45 -1.38 8.56 149 2.36 LHDRV -12.01 -4.38 -4.79 -5.11 -8.58 -8.71 0.21 2,42 LVDRV 22.51 20.00 14.25 1.72 -0.04 -0.03 0.00 0.01 L, I or SVDTV -0.85 -1.99 -0.21 0.39 0.00 0.00 0.00 0.00 5SRV -0.27 1.76 -2.59 0.24 7.00 -0.98 -0.92 -4.00 LSTV 0.63 2.13 0.00 0.15 0.21 1.32 0.04 0.05 I-0.05 1.00 2.32 -0.17 -3.86 -0.54 0.39 0.31Chapter 3. A multi-layer model of the Strait of Georgia 118r —2LS7xl 0_’”ARV&ATV_____7 x 10h1H DR VVDRV & VDTVSRV&STVSum RV terms& TV terms____RV TV+ terms I II225. 229. 233. 237. 241 . 245. 249, 253.daysFigure 3.33: Time series of the fluctuating vorticity terms for layer 7 of Station III. Thinlines correspond to the terms of the residual and thick lines to those arising from thetidal components. The vertical scale is the same for all panels and is only shown in thetop one. From top to bottom, the panels show advection, horizontal diffusion, verticaldiffusion, stretching/shrinking, the sum of all the residual and all the tidal terms andfinally, the sum of all terms. See Table 3.7 for the meaning of the acronyms.Chapter 3. A multi-layer model of the Strait of Georgia 119kinetic energy per unit mass.Now recall from the section on vorticity budgets that there are two ways to derivethe energy equation: (1) from the continuous momentum equations one can derive thecontinuous energy equations and then discretize the energy equation, or (2) from thediscretized momentum equations one can derive the discretized energy equation. Asdiscussed previously the second method is preferred (Haltiner and Williams, 1980) andtherefore this second method will be followed throughout the rest of this section. As inthe case of vorticity, the differences between methods are very small; see Appendix F.However, writing out the discretized forms of the equations in the text below is cumbersome; therefore in the text that follows, the equations will be written in continuous form,though in the actual analysis (and coding) the equations were used in their discretizedform, beginning with the discretized momentum which is consistent with method (2).In what follows the kinetic energy equations for both total and residual flow will beexamined. First, the total kinetic energy will be examined in order to identify the variousterms, The total kinetic energy equation is obtained by forming the scalar product ofthe velocity ‘iY with the original model equations 3.7 and 3.8, the result isaKu 9uuh 9uvui v ãuvui OvvI 2K+ a + ]— —--L(w)horizontal advection vertical advectionuô(gp*+p) va(gp*7l+p)axpressure terms+ i)(uVu + vVrv)+ + (r). (3.23)horizontal diffusionvertical diffusionBefore calculating the kinetic energy of the residual flow (i.e., K0) the terms of (3.23)can be expressed in a more compact and standard form.Chapter 3 A multi-layer model of the Strait of Georgia 120Consider first the advective terms. Expanding the derivatives and using the continuityequation for (w) (eqn. 3.2), the advective terms become,u Ouh -.Ou Ovh .‘Ou—-[u-—— + uh— +u + vh—]h Ox Ox Oy Oyv Ouh ‘Ov 9vh ‘Ov u2+v Ouh Ovh—[v+uh—+v—-+vh—]+ (—+——)=h Ox Ox Oy Oy h Ox OyOu Ou Ov liv 0 0 -—uu— — uv— — vu— — vv— = —(u— + v—)K = —v VHK,Ox Oy Ox Oy Ox OyIn other words, the horizontal and vertical advection from the momentum equations addup to become the horizontal advection of kinetic energy in each layer. Note that thehorizontal advection of kinetic energy is equal to the total flux of kinetic energy [i.e.,(iK)} because wK = 0 within each layer. Nevertheless, in the coming analysis,the results from the advective terms will be presented in two parts: the kinetic energyfrom the horizontal advective terms separated from the kinetic energy from the verticaladvective terms. However, keep in mind that these two add up to the total flux of kineticenergy.Consider next the pressure terms. The terms can be simply grouped as-‘I—— Vjj(gp + p)and represent the increase or decrease in kinetic energy as given by the product of velocityand the force due to the pressure gradients (Kundu, 1990). The results will be presentedin this form. Note however this term can also be written as the work done by the pressureforce (now in three dimensions) plus a conversion to potential energy, namely,V WO WV(gp*7+p) — __(gp* +p) + __(gp*:r +p) =V wOV(gp* +p)+ __(gp* +p) =Chapter 3. A multi-layer model of the Strait of Georgia 121VV(gp + p)—because of the hydrostatic relation [i.e., P =— z) + p) = _p*g + p =—gp = _g(p* + p’) = = —gp’]. Using the continuity equation, V. V = 0, the aboveexpression can be re-written as1 -.__(V.[V(gp*i+p)j+ wgp’ ).pressure work transfer to potential energyThe horizontal and vertical eddy diffusion terms are left in the form indicated at (3.23).Collecting the previous terms, the compact form of (3.23) is= VHK — - . jj(gp* +p) + (uVu + vV,v) + . z). (3.24)The kinetic energy equation for the residual is obtained by forming the scalar productof io with each of the terms in the momentum equations (3.14) and (3.15). The resultingequation is,atu0 au0uh öu0vh v0 au0vh av0vh 2K0ax + a a + a ]T(wo)*—----•VH(gp qo+po)p+i’(uoV,uo + voVvo) +-uO[azLlulh + aJ1— V [aulvlh+ax ay ax ay—uo[uiL(w1)+ Diui(wi)j — vo[viz(wi) +D1vz(w)]+-L(ri) + z(r1)+ uoDii.(ri) + voDiI(T1)+ E, (3.25)Chapter 3 A multi-layer model of the Strait of Georgia 122where K0 (ug + vg) and € represents the energy change due to to the work done by theCoriolis force, which appears only due to the discretization. Recall that for conveniencethe equations are written in continuous form though the actual analysis and coding wasdone with the discretized equations: for the continuous case, is exactly zero and in thediscretized case, e is small as discussed in Appendix F.Note that if the low-pass filter, (...), had being applied directly to the total kineticenergy equation 3.23, one would have obtained the energy of the residual plus the rectifiedenergy of the tides, which is (u + v?)/2( 0). For example, consider first the residualequation: the rate of change of the residual kinetic energy due to horizontal diffusion inthe x-momentum equation is1jU0“ 2_____2u0 [—- = VVHUO I = UOVVHUO.However, if this form is obtained from the equation involving the total fields, the resultIs2 t9u/2 2u [-- — 1) HU 1 — UV HUand then applying the low frequency filter this becomes,2_________2 —2____________= uIVu = U0a U1 =u0IiVu +u1t’Vu,In other words, using the total fields and filtering, results in expression for energy of bothresidual and tides.Equation (3.25) will be used for the results below and the different terms of (3.25)will be referred to with the acronyms given in Table 3.9. Nevertheless, as in the case ofvorticity, various terms of (3.25) can be simplified, which is given for reference here. Theadvective terms in (3.25) can be simplified as was done for the advective terms in (3.23):they add up to the horizontal advection of K0 by the residual and similarly, an advectiveChapter 3. A multi-layer model of the Strait of Georgia 123Table 3.9: Acronyms and summary of terms in the residual energy equation 3.25. Thelast two letters have the following meaning: the penultimate is T or R for tidal or residualand the last letter is E for energy.Term from equation 3.25 Identification—uo[-(uouoh) + uovoh)]/h HARE: Horizontal advection—vo[(uovoh) + (vovoh)j/h of energy by the residuals—u0[(ph) + -(uj3jh)]/h HATE: Horizontal advection—vo[(Ujh) + (5iwh)j/h of energy by the tides—2Koh’i(wo) VARE: Vertical advection ofenergy by the residuals—uo[ii’ui(wi) + Diui(wi)] VATE: Vertical advection—vo[ii’vi(wi) + DiviLS(wi)] of energy by the tides+i(uoVuo + voVvo) HDRE: Horizontal diffusionof energy by the residual+iioh1 VDRE: Vertical diffusionof energy by the residual+uoui’(r3,i)+ voir’z(r1) VDTE: Vertical diffusion+uoDiz(ri) + voDii(ri) of energy by the tidesVH(gpo + po) PWRE: Pressure workChapter 3. A multi-layer model of the Strait of Georgia 124term involving interaction between the tides and residuals. To do this, the continuityequation (3.2) was split into residual and tides. Splitting the fields in (3.2) and applying(...) gives______ôuh c9vhz(wo + wi) = —-—- — —a---, (3.26)Using h >> 1o gives,9uoiz ôv0h ãu1i1 OiYjj/.(wo) = —__—____—_____— . (3.27)Oy_ _______only top layerSubtracting (3.27) from (3.26), the tidal continuity equation is— öuiiz Ov1ui a(uo+u1) ô(vo+vi)iiL(w1)— —_____— —______— . (3.28)Oy_____only top layerNote the extra terms for the top layer. Substituting (3.27) and (3.28) into the advectiveterms of (3.25) and after some algebra becomesa a ——(uo-- + vo—)Ko +KoVH’(vli)only top layer— uo(iii . VH)u1 — vo(iii . VH)v1 + (uoui + vovl)[DVH (iY011)+ DlVH (€ii)j. (3.29)only top layerThe first term represents the residual advection of residual kinetic energy, K0. The thirdterm is an interaction between tides and the residual resulting from the product of theresidual velocity i3o and the momentum advection of the tidal flow, i.e., from the termsthat are part of the tidal stress (HATM and VATM). The second and fourth terms, whichappear only at the top layer, come from continuity and are also part of the tidal stress.Once again, recall that the results will be shown in reference to equation (3.25); seealso Table 3.9.Chapter 3. A multi-layer model of the Strait of Georgia 125Mean Energy budgetsParallel to the description of results of momentum and vorticity budgets, each termof equation (3.25) was evaluated for the full year for each layer at stations I-TV, and theneach was time-averaged. Again, the results depend on location and these four stationscan be considered representative and it is possible to make the following classification:The largest terms are the horizontal advection of tidal energy (HATE), the pressurework (PWRE) and the horizontal diffusion of residual energy (HDRE).The intermediate terms are the vertical advection of tidal energy (VATE) and thehorizontal advection of residual energy (HARE).The smallest terms are the vertical advection of residual energy (VARE) and thevertical diffusion of energy by the tides (VDTE).As in the mean vorticity budgets, the vertical diffusion of residual energy (VDRE)can be of any size. It is usually large at the upper layers and small in the rest of thewater column.The results for the four stations are given in Table 3.10. Boldface numbers in thetable indicate agreement with the above classification. Note that Table 3.10 shows thesame proportion of bold numbers as the corresponding table for momentum, but theimbalance is larger for energy. The classification of L, I or S holds better for layer 6 thanfor layer 1.Time-Variability of the Energy budgetsThe time-variability of the terms of the kinetic energy equation were evaluated andthen de-meaned. The classification of the mean terms just above also applies to thefluctuating terms. As an example, Figure 3.34 shows time series of the terms of theChapter 3. A multi-layer model of the Strait of Georgia 126Table 3.10: Time-averaged terms from Table 3.9 for the 4 selected gridpoints (StationsI, II, III and IV) for layers 1 and 6 of the model. Units are in 10_8 m2 s3. In the lastcolumn L, I and S stands for large, intermediate and smaller, and in general, gives therelative size of the terms in the energy budget of the model. When the numbers in thetable are given in bold face, it indicates that the classification of terms as L, I or S istrue for that gridpoint in particular.Layer 1 Layer 6Terms I II III IV I II III IV SizeHARE 0.56 4.07 3.07 3.40 -0.04 0.00 0,02 0.03 IHATE -2.67 8.04 3.65 1.59 -0.13 -0.53 -0.04 0.39 LVARE 0.02 0.00 0.22 -1.04 0.01 -0.02 0.00 0.01 SVATE 1.49 -2.85 -1.53 -0.86 0.06 0.09 -0.02 -0.03 IHDRE -1.17 -3.47 -1.30 -3.85 -0.40 -0.79 -0.08 -0.28 LVDRE 10.52 8.87 12.60 24.00 0.00 0.00 0.00 0.00 L, I or SVDTE -0.21 0.78 0.83 0.68 0.00 0.00 0.00 0.00 SPWRE -5.51 -11.40 -12.08 -14.47 0.48 1.32 0.18 -0.09 L3.03 4.03 5.46 9.45 -0.01 0.08 0.05 0.02Chapter 3. A multi-layer model of the Strait of Georgia 127energy equation 3.25 for the 7th layer of station III. The figure shows thin and thicklines for residual and tidal terms, respectively. Note that the advection is dominatedby the tidal component, HATE, and is basically balanced by the pressure work. Thesetwo terms, in this particular case, determines the local rate of change of kinetic energy.While horizontal diffusion of residual energy is usually a large term, for this layer andgridpoint it is small.The vertical diffusion (both residual and tidal components) were found to be small orintermediate in size. At the surface, the residual component can be large depending onthe wind energy input. At times it is large. At the bottom, the friction term is usuallyvery small and so is the vertical diffusion. (Recall that vertical diffusion is defined as thedifference in the stresses between the upper and lower levels of each layer.)3.2.3.4 Summary of budgets resultsThese budget analyses indicate that the dynamics of the central Strait of Georgia arehighly nonlinear. The equations for the residual momentum, vorticity and energy showthat the residual flows are mainly forced by the nonlinear advective terms. This resultwas expected as Stacey et al. (1988) found in their analysis of the observations in thecentral Strait of Georgia that advection in the vorticity equation dominates the balance.Also, even though the Strait of Georgia is an enclosed coastal sea, the geostrophic balanceis still predominant, which is in agreement with the observations as reported by Stacey etal. (1991). Table (3.11) collects together the different terms of the momentum, vorticityand kinetic energy equations for the residual according to their relative sizes in theirrespective equations. With the separation of high and low frequency components it wasfound that the nonlinear interaction of the high frequency tides is the main force on theresidual; the column labeled ‘large’ in Table 3.11 contains the advection terms of thetidal components, i.e., HATM, ATV and HATE.Chapter 3. A multi-layer model of the Strait of Georgia 128r 2,31,5x108 Lm /sHARE & HATE—1 .5x10VARE & VATEHDREVDRE & VDTEP WE-—Sum RE___ ________-D T —I L. erms —I I I I225. 229, 233. 237. 241 . 245. 249. 253.daysFigure 3.34: Time series of the fluctuating energy terms for layer 7 of Station III. Thinlines correspond to the residual and thick lines to the tidal components. The verticalaxes are the same for all panels; the scale is shown in the top one only. From top tobottom the panels show horizontal advection, vertical advection, horizontal diffusion,vertical diffusion, pressure work, the sum of all the residual and all the tidal terms andfinally, the sum of all the terms. See Table 3.9 for the meaning of the acronyms.Chapter 3. A multi-layer model of the Strait of Georgia 129This result is consistent with the results of the different runs of the 3-D model shownin Section (3.2.2.3) on ‘Nonlinear Interactions’. There the amplitude of the low frequencytidal components was found to be determined basically by forcing the model with thediurnal and semidiurnal constituents only.The mean budgets of momentum and especially vorticity and kinetic energy showsome imbalances at individual gridpoints. A simple spatial average reduces the imbalanceand the role played by the different terms in the equations remains unchanged.3.3 DiscussionHere I compare the model results against the observations in the central Strait of Georgia.The observational data set was briefly described in Chapter 2.Briefly, the main characteristics of the model residual currents are as follows:- The mean residual is characterized by coastal jets and eddies on scales comparable tothe size of the domain. These jets and eddies change both direction and sense ofrotation with depth. The horizontal structure of the fluctuating residual is similar.- The maximum speeds for the total residual ranges from 5 cm s at the bottomto 40 cm at the surface. The speeds of the mean residual range from 3cm s at the bottom to 10 cm s1 at the surface and away from the riverdischarge. The fluctuating residual is dominated by periodicities of the fortnightlyand monthly bands.3.3.1 Comparison with observationsBefore comparisons are made, a short description of the data sets and the data processingwill be given. Recall from Section 2.3.1 that I have time series of velocity and o fromChapter 3. A multi-layer model of the Strait of Georgia 130Table 3.11: Summary of the momentum, vorticity and energy budgets. Recall thatthe last two letters have the following meaning: the penultimate is T or R for tidal orresidual, respectively, and the last letter is M, V or E for momentum, vorticity or energy,respectively. The first two letters are H or V for horizontal or vertical, respectively, PWfor pressure work, CO for Coriolis, S for stretching/shrinking and D for diffusion and Afor advection,Equation TERMSLargest Intermediate SmallestMomentum HATM, CORM, VATM, HDRM VDTM, VARMPGRMVorticity ATV, SRV, STy, ARV VDTVHDRVEnergy HATE, HDRE, HARE, VATE VARE, VDTEPWREChapter 3. A multi-layer model of the Strait of Georgia 131Table 3.12: Cyclesonde data levels used to construct the time series to which the modelresults are compared. Top and Bottom are the levels where the cyclesonde of the differentstations were resting between profiles, see text.Model Model Datalayer depth range cyclesonde levels(m) (m)3 15-30 Top4 30-60 40, 605 60-90 60, 806 90-150 100, 120, 1407 150-250 160, 180, 200, 220, 2408 250-H 260, 280, Bottoman array of nine stations; stations 1 to 4 are cyclesondes and stations 5 to 9 are ofcurrent meters (see Figures 2.14 and 3.35 for the location of the stations). From the rawcyclesonde data, time series of velocity (and oj) were constructed at discrete depths of40, 60., ..., 280 m. At 20 and 300 m the data were obtained while the cyclesonde wasresting at the top and bottom of their profiling range. For simplicity, I will compare themodel results with the cyclesonde data only.In order to compare the model results with the observations, the cyclesonde data werevertically-integrated to approximately the same layer thickness of the model. The levelsused to group the cyclesonde data into the layers of the model are shown in Table 3.12.As can be noted in the table, the first two layers of the model, representing the top 15 mof the water column, can not be compared with the observations. The third model layeris compared directly with the cyclesonde data from its top level. The depth of this toplevel of the cyclesondes sampling sampling range is 20, 18, 27 and 17 meters for stationsChapter 3. A multi-layer model of the Strait of Georgia 1321, 2, 3 and 4, respectively. The maximum depth of the cyclesonde also differ for the 4stations but they are all close to 300 m: 300, 294, 296 and 297 meters for stations 1 to4, respectively. The model gridpoints which correspond to the cyclesonde stations havenominal depths of 368, 342, 315 and 346 m. The total depth of the water column atthe cyclesonde stations are 368, 349, 305 and 366 m below datum. Obviously, as thereare no cyclesonde data below 300 m, some bias will occur in the vertical average of thecyclesonde data for the bottom part of the water column.During summer, the Strait of Georgia experiences deep water replacement once amonth. This phenomenon implies inflow at depth from Juan de Fuca Strait (flow upto the Strait of Georgia) and surface outflow (flow down the Strait of Georgia). Thisphenomenon is not present in the model as the mechanisms that produce it are notincluded yet; they will be included and results concerning the modeling of deep waterrenewal are left to the next chapter. Bearing in mind that no deep water renewal is presentin the model, it will be harder to match the currents from the model with the observations.However, the model results should show the basic character of the circulation in thecentral Strait of Georgia as all the other main forcing agents -tides, winds and runoff(LeBlond, 1983)- are included in this stage of the model.Time-averaged currentsFigure 3.35 shows the time-averaged currents, in dashed arrows, for the four cyclesondes for the complete period of observation (June 1984 to January 1985) for layers 3 to8 of the model. The maximum speeds observed for the different layers range between 6and 14 cm s1. In the third and fourth layers, namely, the top of the cyclesonde rangeand the 30-60 m layers, the flow is mainly in the down-strait direction. In the fifth layer(60-90 m), the observations show a cyclonic eddy which persists all the way to the bottomlayer. The fact that this eddy persists throughout of the water column reflects the strongbarotropic component of the observations; see Figure 2.14 of Chapter 2.CYCLES0NDE15 cm/sFigure 3.35: Time-averaged velocity for the model (solid arrows) and for the observations (dashed arrows) in the central Strait of Georgia for the period June 1984 toJanuary 1985 corresponding to layers 3 to 8. Cyclesonde data are obtained from vertical-averages over the model layer depths as given in Table 3.12. Layer numbers are givenas L3 to L8; for cyclesonde data the first and last frames correspond approximately toL3 and L8. In first cyclesonde frame, 1 to 4 indicates the station number. Note that inthe frames where the cyclesondes velocities are shown, the model velocities are repeatedfor that particular gridpoint.Chapter 3. A multi-layer model of the Strait of Georgia 133M00EL690-150L7I 50-250LB250-HChapter 3. A multi-layer model of the Strait of Georgia 134The time-averaged velocities of the model and the observations compare poorly, however, there is some agreement as will be noted below. The magnitude of the velocities islarger for the observations than for the model with a few exceptions. However, relativeto the 2-D model results of Chapter 2 (Figure 2.2) the time-averaged currents of the 3-Dmodel are now closer to the observations.Within the small box where both the 2-D study was done and the observations weretaken, the maximum 3-D model mean residual ranges between 4 and 1.5 cm s’ betweenlayers 3 to 8 compared to 14 and 6 cm s’ of the observations (layers 1 and 2 of the 3-Dmodel have larger velocities but they can not be compared to the observations). Thedirection of the flow agrees only at some locations.Therefore, the 3-D model performance with respect to the time-averaged residualis poor at this stage. Better results will be reported in Chapter 4. However, someimprovement with respect to the 2-D model has been achieved.Fluctuating currentsFigures 3.36, 3.37 and 3.38 show time series of the velocity components for stations 1and 3 for layers 3 to 8 of the model superimposed on those obtained for the correspondinglevels from the observations (see Table 3.12).Here is a summary of the comparisons:1,- Again, the observed velocities are larger than the modeled velocities. At timesthe difference is very large but the amplitude of the oscillations are, in general, the sameorder of magnitude.2.- The amplitude of the velocity field decreases more rapidly with depth for the modelthan for the observations. There is a better agreement at the upper layers, especially atstation 3 in the third layer for the v component.3.- The v component of velocity agrees better between model and observations thanthe u component. See for example station 1 at 60-90 m and 90-150 m. The second partChapter 3. A multi-layer model of the Strait of GeorgiaI’1iNI \ /_f3V’%iI I I I225 240 255 270 285 300 225 240 255 270 285 300days daysFigure 3.36: Time series of the u and v velocity components (in cm s’) of the fluctuatingresidual from the observations (dotted lines) and from the model (solid lines) forstations 1 and 3 in the central Strait of Georgia. Shown at the top is the third layer (15-30m) and at the bottom the fourth layer (30-60 m). The velocity from the observations forthese layers was obtained as described in Table 3.12.2,0,-20.Layer 3: DeptH 15-30 mr,j-.- --‘S—cT,-, V135Stn-v 1320. --20,U VLayer 4: DeptH 30—60 m20.0,-20.20.-201”:..,V \jfrS, ‘-I I IChapter 3. A multi-layer model of the Strait of Georgia 136Layer 5: Depth 60-90 m Stn20,-0. 1iJ7- 1—20 I I20. ---20.ILayer 6: Depth 90-15020.-0.—1—20. I I20, --0 3-20. I I I I I I I I225 240 255 270 285 300 225 240 255 270 285 300days daysFigure 3.37: Time series of the u and v velocity components (in cm s’) of the fluctuatingresidual from the observations (dotted lines) and from the model (solid lines) forstations 1 and 3 in the central Strait of Georgia. Shown at the top is the fifth layer (60-90m) and at the bottom the sixth layer (90-150 m). The velocity from the observations forthese layers was obtained as described in Table 3.12.‘‘.—•ç:—-;:;-ç;;/ \;/1I I I IV -\__.-_•_‘.J ;,——s.———.c’—/ v-, •c:—-Vth 250—H m1A/—L-;_ 3I I I IFigure 3.38: Time series of the u and v velocity components (in cm s’) of the fluctuatingresidual from the observations (dotted lines) and from the model (solid lines) forstations 1 and 3 in the central Strait of Georgia. Shown at the top is the seventh layer(150-250 m) and at the bottom the eighth layer (250-H rn). The velocity from theobservations for these layers was obtained as described in Table 3.12.Chapter 3. A multi-layer model of the Strait of GeorgiaLayer 7: Depth 150-250 m20._f\ . / AAS’C%, V —137Stn1—20.20.0.-20.20.0,-20.20.0.—20.ULayer 8: Dep225 240 255 270 265 300days225 240 255 270 285 300daysChapter 3. A multi-layer model of the Strait of Georgia 138of the v component time series of the model and the observations are similar; meanwhilethe u model component is very small and oscillates almost out of phase as compared tothe data. Note at station 3 for the 15-30 m layer, the u component of the model showsgood agreement with the data but fluctuates about 5 days out of phase.4.- The model velocity fluctuations are smoother than the observations. The observations show, at times, strong bursts or peaks which are not present in the model. In thebottom layer, the peaks in the velocity are associated with gravity currents involved indeep water renewal of the central Strait of Georgia. As mentioned, this phenomenon isnot present in the model and, obviously, no agreement is expected here, however betteragreement will be shown in the next chapter when deep water replacement is modeled.Figure 3.39 shows the o field for the 6 layers for station 3. (The values of o do notdiffer much from station to station in both model and observations.) The figure showsthat the fluctuations in o between model and observations in layer 3 are close in bothphase and amplitude and in layer 4, the fluctuations are close in amplitude but, at times,they are a few days out of phase. However, the absolute values are off by almost oneot unit in some cases. At deeper levels, the absolute values are closer but the modeledvalues of o show almost none of the fluctuations which are present in the observations.Once again, recall that deep water renewal is present in this set of observations and thevalues of ot in the Strait will be strongly influenced by this phenomenon not present atthis stage of the model.The discrepancy in the absolute values between model and observations can be attributed to a variety of causes, the main one being that the density equation is notproperly modeled, namely, that there is no systematic diffusion of density (though thereis numerical diffusion). Another factor, which turns out to be important, because themodel is sensitive to it, is the initial density field. I repeated the reference run but with adifferent initial density field: in particular I started the model with the Crean and AgesChapter 3. A multi-layer model of the Strait of GeorgiaI I I I————— ________\6225 240 255 270 285 300daysFigure 3.39: Time series of the ot from the observations (dashed lines) and from themodel (thick lines) for station 3 in the central Strait of Georgia. The numbers at theright and top of each figure stands for the layer number.139layerlayer323.221 .720.224.022.521 .026 024.523.057 8I I I225 240 255 270 285 300daysChapter 3. A multi-layer model of the Strait of Georgia 140data corresponding to the month of January, instead of June (see Figure 3.2), and theresulting densities oscillated in a similar fashion around different mean values. The difference between the two runs of these mean o values are as large as the difference foundbetween the reference run and the observations, suggesting that part of the discrepancyof the model with respect to the observations can also be attributed to initial densityconditions.Harmonic AnalysisFinally, I want to show how the low-frequency tidal constituents compare with theobservations at the same locations. In Section 3.2.2.2, it was mentioned that, on average,the low-frequency tidal constituents explain about the same percentage of the variancein the model as in the observations. Here I want to show how these tidal constituentscompare to the observations in the central Strait of Georgia.Table 3.13 shows the amplitude of the fortnightly and monthly constituents for bothvelocity components, for the observations at stations 1 and 3 and for the correspondinggridpoints of the model. In general, the amplitudes of the constituents are larger from theobservations than from the model. Recall that the amplitudes of the tidal constituents inthe 2-D model were an order of magnitude or more smaller than the vertically-averageddata from the observations. With the 3-D model, the amplitudes are generally withinthe same order of magnitude. To stress the fact that the amplitudes are of the sameorder of magnitude, the amplitudes for both the model and the data are in bold facewhen the amplitudes of the constituents are within a factor of 3. In fact, over half theamplitudes are within this factor of 3 (only of the amplitudes are within a factor of 2).The agreement is slightly better for station 3 than for station 1. For station 3, 29 of the48 amplitudes are within the factor of 3 compared to 25 of the 48 for station 1. (Thenumber 48 results from 6 layers x 4 constituents x 2 velocity components.)With respect to the different velocity components, u or v, the agreement is slightlyChapter 3. A multi-layer model of the Strait of Georgia 141Table 3.13: Amplitude (cm s’) of the fortnightly and monthly constituents for modeland observations, at stations 1 and 3, for the u and v components of velocity. When theamplitudes of the observations and model are within a factor of 3, the numbers appearin bold face in both the data and model sections.STATION 1 STATION 3u CYCLESONDElayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 1.27 1.70 2.78 1.91 3 0.77 0.66 0.21 2.544 1.24 1.36 1.56 0.32 4 1.62 0.25 0,86 1.975 1.40 1.16 1.58 0.46 5 1.73 1.26 1.53 1.486 1,41 1.14 0.65 0.95 6 1.10 0.85 1.28 1.057 0.33 0.67 0.75 0.79 7 0.17 0.64 0.23 1.148 0.05 1.35 1.54 1.60 8 0.76 2.46 2.60 2.89u MODELlayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 0.33 1.39 1.82 0.70 3 1.07 3.17 3.31 1.024 0.18 0.62 0.69 0.16 4 0.37 1.21 1.46 0.225 0.15 0.22 0.44 0.48 5 0.57 0.44 1.28 0.666 0.12 0.05 0.25 0.24 6 0.71 0.86 1.65 0.467 0.15 0.08 0.03 0.09 7 0.15 0.38 0.46 0.108 0.38 0.30 0.52 0.20 8 0.43 0.28 0.55 0.20v CYCLESONDElayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 1.32 5.68 3.22 2.20 3 1.67 2.99 2.89 3.674 0.93 3.66 0.81 0.65 4 0.50 1.62 1.77 3.305 0.93 2.51 0.49 1.00 5 1.06 0.75 1.84 1.696 0.77 0.70 1.31 1.22 6 1.38 0.37 1.43 1.607 0.88 0.69 1.43 0.85 7 0.74 0.32 0.94 0.988 0.85 0.89 2.12 0.77 8 1.63 2.22 3.06 3.35v MODELlayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 0.48 0.55 1.81 1.26 3 0.61 1.45 2.12 0.834 0.39 0.74 0.53 0.65 4 0.26 0.83 1.19 0.335 0.40 0.30 0.60 1.42 5 0.42 0.11 0.38 0.726 0.54 0.09 1.23 0.74 6 0.39 0.44 1.33 0.537 0.23 0.18 1.35 0.54 7 0.15 0.34 0.98 0.338 0.39 0.27 0.54 0.23 8 0.32 0.24 0.50 0.20Chapter 3. A multi-layer model of the Strait of Georgia 142better for v with 29 of the 48 amplitudes within the factor of 3 against 25 of the 48 forU.With respect to the different constituents, at both stations only the Mm constituentcompares consistently throughout the water column, for both u and v components. Atstation 1, the Mf and Msm model amplitudes are comparable to the amplitudes fromthe observed data for station 1 for the v component. The largest disagreement is for theMsf at station 1 for the v component.Finally, it is worthwhile to note that the observations show that the amplitude of theMf constituent is often as important as the Msf constituent. The latter has usuallybeen said to dominate the fortnightly band, however, it seems that in the central Strait ofGeorgia the Mf is equally important. This result was reported for first time in Chapter 2and in Marinone and Fyfe (1992).In summary, the fluctuating residual of the model has fortnightly and monthly constituents with amplitudes of about the same order of magnitude as the observations.From Table 3.13, only 4 of the 48 amplitudes for each of the u and v components of themodel have amplitudes smaller by an order of magnitude than the observations. Again,the results of Chapter 4 will be better.It is worthwhile to note that, in general, the along-strait velocities agree better thanthe cross-strait velocities. This fact is true for both components of the residual, for themean and for the fluctuations. The model is under estimating the u velocity componentin the central Strait of Georgia.Now I conclude with a short discussion on the role of numerical diffusion in theunderestimation of the velocities in the model as compared to the observations. Stronach(1991) tested the model with values of the horizontal eddy viscosity coefficient, 1’, of 100,Chapter 3. A multi-layer model of the Strait of Georgia 143200 and 300 m2 s1 and found significant changes in the flow field. He concluded thatthis 3-D model has a strong dependence on £‘. (This 3-D model requires horizontaleddy viscosity to maintain numerical stability, the 2-D model does not: results with v= 0 were shown in Figure 2.16). As expected the energy of the tidal current decreaseswith increasing eddy viscosity. Based on comparisons between the modeled and predictedsurface elevations, Stronach (1991) reports that E = 200 m2 s1 produced the best results.In order to see the effect of t’ on the residual, three months of the reference run wasrepeated with a variety of horizontal eddy viscosities, i, in the range of 100 to 200 m2s1.It was found that as 1.’ decreased, the model runs became shorter because the model, atsome point, developed instabilities which lead to non-convergence of the solution. Thisbehavior was not unexpected as the model generated more energetic residuals as theviscosity was decreased.Figure 3.40 shows the mean residual for the three-month trial with i = 100 m2 s1.For this value of 1’, the model breaks down after 92 days. The speeds, at some locations,are 50% larger than in the reference run for the same period of averaging. The three-month time averaged velocity of the reference run is shown in Figure 3.41. (Recall thatthe reference run has £‘ = 200 m2 s’.) The basic flow direction between both runsremains unchanged. The most significant difference is in layer 6 where an anticycloniceddy, which in the reference run (Figure 3.41) is weakly formed, is now fully developedat the west coast of the Strait (Figure 3.40). Therefore, the velocity field seems to becloser to the observations with lower values of the horizontal eddy viscosity. However, tomaintain numerical stability for long time integrations, the model requires a z) of at least175 m2 s’. The model results for 1’ = 175 m2 s1 are almost identical to those of thereference run which uses 1’ = 200 m2 s1.Chapter 3. A multi-layer model of the Strait of Georgia9\ \ .% _,,55‘5•_\• •‘SI’ ‘5 S 5% b % * 4%V 50 0 &4 ‘‘558514•05I —— —— —- 4 I I ‘-# — — — a’ /•/ / 1 1 —11442 34 5 615 cm/sFigure 3.40: Time-averaged velocity vectors in the central Strait of Georgia for layers 1to 8 along with the vertically-averaged field (frame number 9). The time period of theaverage is three-months, The horizontal eddy viscosity coefficient, z), of this run is 100m2 s1.Chapter 3. A multi-layer model of the Strait of Georgia 145— — — p I 1 1I. _-.—‘pr’’—ttl ‘I-7/__I’’’ ‘“‘ Z’a,——. P—————.: ‘4/1/6I-- 8 I 8 IlI I a e i i I ILi.. . a a 8 8 8 II e 4 4 4 4 IJa •.aa444‘ I I 4 4 4 4 4 4__j. . ‘ I I I I I I 4 4 8I..., IIIII!Jj 8.__J,—-— 1/1/1/41 4I. —/1/i/il_I. ‘-- 1////ff I‘‘.-—Figure 3.41: As Figure 3.40 with horizontal eddy viscosity coefficient, 1’, =200 m2 s1.This is simply the reference run averaged over three months.1 2________3OhLi I 4 ). •.‘ —. ———•— — — —I. • ‘ -————.———.——— —-.Nh ---— -• . .- _•__•••••_%%••__ —\___4i1’’ /AW-iJ1:;:: 1r. 15-304 5J•j ‘-• II __.I. 1 4 8 S —J I...’’ •‘I I,—r1.. .Ia—--’.1--S 8i•,8,,__,,JS a4f/, I II j7‘S.-.a 4%%544a a a aaI 0 •%\ —Isle 18 IS. •j4#ts sasil •.1I - ‘as,9I a a a a a • aI’a_____ __ ____4:;; ::::::::, ••%_••1 is a . -15 cm/sChapter 3. A multi-layer model of the Strait of Georgia 1463.4 SummaryI have studied the residual circulation in the central Strait of Georgia with a 3-D eight-layer model. The residual was obtained by applying a low-pass filter to the modeledfields. The mean and fluctuating residuals are characterized by coastal jets and eddies.The maximum speeds for the total residual ranges from 5 cm s at the bottom to40 cm s at the surface. The speeds of the mean residual ranges from 3 cm sat the bottom to 10 cm s1 at the surface and away from the river discharge. Theoscillations for the fluctuating residual are dominated by periodicities of the fortnightlyand monthly bands. The fortnightly band is dominated by the Msf tidal constituentand the monthly band by the Mm tidal constituent.Budget analysis of the momentum, vorticity and kinetic energy equations show thatthe relative importance of the various terms depends on location. It was found thatthe first order balance always involve the pressure gradients and the Coriolis terms inthe momentum equation. However, the residual flow is not in pure geostrophic balancethough close, and the ageostrophic imbalance is maintained by the advective terms.The advective terms are always part of the main balance in the budgets of momentum,vorticity and kinetic energy indicating the highly nonlinear or turbulent character of thedynamics of the central Strait of Georgia. From these budget analyses it was foundthat high frequency forcing (through the nonlinear terms) is forcing the residual. Thehigh frequency forcing is mainly tidai, but winds contribute as well. The energy of thehigh frequencies motions is being transferred to the low frequency bands by nonlinearinteractions.It was found that the wind forcing also plays an important role as a forcing agentof the residual currents in the central Strait of Georgia. Without wind, the flow is verytidal and the oscillations are very smooth. With wind, the residual currents look moreChapter 3. A multi-layer model of the Strait of Georgia 147realistic.The results of the 3-D model are a large improvement over the results of the 2-Dmodel studied in Chapter 2. The orders of magnitude of the velocity for mean andfluctuating components of the residual, as well as the amplitude of the low-frequencytidal constituents, are now comparable to, though still somewhat lower (especially themean and, in general, at deeper layers) than, those of the observations.Part of the observed low frequency currents in the Strait of Georgia are due to deepwater renewal. At this stage of study, the residual currents in the 3-D model, do notreproduce this phenomenon. The presence of deep water renewal will significantly affectthe residual currents in the Strait of Georgia and the next chapter deals with this process.Chapter 4Deep Water Renewal in a multi-layer model of the Strait of Georgia4.1 BackgroundAny study of low-frequency circulation must address the phenomenon of deep waterrenewal (DWR), a challenge set out by LeBlond (1983). Replacement of water in the deepbasin ( 400 m) of the central Strait of Georgia was first explained by Waldichuck (1957)based on historical data (mainly synoptic surveys from December 1949 to April 1953).Samuels (1979) and Doherty (1987) provided partial confirmation of his ideas throughexamination of further monthly surveys [the Crean and Ages (1971) data]. LeBlond etal. (1991), including the cyclesonde records of velocity and density at the central Strait,were able to provide a more detailed picture.Their conclusions can be summarized as follows:In summer (Figure 4.la) intrusions of cold, saltier and relatively low oxygen water fromJuan de Fuca Strait occur as pulses of gravity currents originating at Boundary Passageduring periods of minimum tidal mixing.In winter (Figure 4.lb) the combined effects of atmospheric cooling and tidal mixingin the southern Strait of Georgia produce cold, fresh (relative to the summer water)and high oxygen waters. This water spreads northward and sinks to intermediate andsometimes deep levels.So far, this study has not attempted to give any results concerning deep water renewal.Even though density is calculated as a prognostic quantity in the model (GF8), the148Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 149Figure 4.1: Deep water renewal in the central Strait of Georgia during a) summer andb) winter.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 150physical mechanisms necessary to produce DWR are not fully represented in the model.(In fact there is no DWR in the reference run of the model, Chapter 3.)In order to get deep water renewal in 0F8, two components were added to the model:(1) systematic vertical mixing of density and (2) an increase in density at the openboundary condition. These two components are, in fact related. Vertical mixing acts(among other things) to decrease the density of the water in the deep basin as salt isfluxed up into the lower density layers higher up in the water column. Increasing thedensity at the open boundary increases the density that will become available to the sill.Both decreasing the density in the basin and increasing the density at the sill increasesthe likelihood of gravity currents developing from the sill to the basin and hence, from thissomewhat simplified picture, increases the likelihood of renewal. Clearly, the frequencyof the renewals will depend on the availability of dense water and on mixing within thebasin (Farmer and Freeland, 1983).It is anticipated that these results will only be indicative because the actual forcingof density at the open boundary is not known in detail and different forcing scenarios ofdensity produce obviously different responses inside the Strait of Georgia. The forcingat the open boundary is determined by the dynamics and variability of the waters of theadjacent continental shelf. Take for example the following scenarios:- In summer, DWR in the Strait of Georgia is produced by the intrusion of deep, denserwaters coming from Juan de Fuca Strait which in turn comes from waters upwelled alongthe west coast of Vancouver Island. However, this summer upwelling depends on thenorthwesterly wind which can last from days to weeks and hence the dense upwelledwater can be present or absent at any given time.- In winter, the external forcing can be quite complex and in fact can conspire againstDWR. At this time the predominant southeasterly winds contribute to a northward flowalong the west coast of Vancouver Island with downwelling: the pycnocline along theChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 151coast is pushed downwards. Also the northward flow will transport Columbia Riverwaters northward and these relatively less dense waters can, at times, enter Juan deFuca Strait and reverse the estuarine circulation (Crean et al., 1988a). Thus renewal intothe Strait of Georgia would be blocked because DWR at this time depends on cooling ofthe surface waters and these waters can now be too fresh for the cooling to increase thedensity sufficiently for these waters to sink.4.2 Model SetupThe 3-D model was run with the same conditions as the reference run described inChapter 3 with respect to forcing by tides, winds and runoff. However, in order toproduce DWR in the Strait of Georgia the following two mechanisms were included inthe model:1) Explicit vertical diffusion was included in the density equation 3.3 in order to providea mechanism for the bottom waters of the Strait of Georgia to become less dense.2) The model was forced with a seasonal density variation at the open boundary of Juande Fuca Strait in order to provide dense enough waters for renewal.Obviously, it will only be possible to simulate the summer scenario of DWR in theStrait of Georgia. Winter atmospheric cooling of the surface waters has not been implemented.Now the two mechanisms included in the model are discussed.1) First consider vertical diffusion of density. Note that in the reference run therewas no systematic vertical mixing of density in the model. Explicit vertical mixing isonly done to remove density inversions. There is also numerical diffusion in the verticaladvection of density: a Lax-Wendroff scheme is used which has numerical diffusion onthe order of wLSt where w0 is a characteristic vertical velocity. In Juan de Fuca Strait,Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 152Boundary Passage and the central Strait of Georgia, the model vertical velocities are1.0, 2.0 and 0.2 cm s1 at maximum, respectively, and 0.2, 0.5 and 0.1 cm s’1 onaverage, respectively: thus numerical diffusion is 6.0 x 10_2, 2.4 x 10—’ and 2.4 x 1Om2 s1 at maximum, respectively, and 2.4 x i0, 1.5 x 102 and 6.0 x 1O m2 s onaverage, respectively. Note that numerical diffusion is low in the central Strait of Georgiawhile in the Juan de Fuca Strait and Boundary Passage numerical diffusion is large.This difference has direct consequences on the model behavior. A high degree of mixingby numerical diffusion in Juan de Fuca Strait and at Boundary Passage indicates thatdensities higher than those observed will probably be required at the model boundary inorder to get deep water replacement. The density had to be raised above the observations.However, there was almost no decrease in density in the deep basin of the central Straitof Georgia without explicit diffusion. In the basin, numerical diffusion was not enoughto lower the densities and match the observations between replacements. Thus, it wasnecessary to include explicit diffusion, even though DWR can be induced by simplyincreasing the density at the open boundary, in order to match the observed decrease ofdensity in the deep basin. In order to provide explicit vertical diffusion of density, verticaldiffusion identical to that used in the momentum equation was added everywhere to thedensity equation.Trials were done varying a (Equation 3.6) and a value of a = 23 was chosen forboth momentum and density. (In the reference run a = 10 for momentum only.) Note,however, that the actual values for A,, are not significantly different than those of the reference run, even though the coefficient a is more than twice the original value, reflectingthe nonlinear character of A,,.2) Second, consider the density at the open boundary, and look first at the seasonalvariability. The seasonal variability of the density field was obtained from the Creanand Ages (1971) data (December 1967-December 1968) taken near the mouth of JuanChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 153de Fuca Strait. The data were interpolated in the vertical to the mid-layer depths of themodel and then an annual and semiannual fit by least-squares was performed to obtaina value for the amplitude and phase of the seasonal density variation in each layer. Thedata were then fit to the function a cos(wat— cia) + bcos(2wat—çj, where “-pa is theannual frequency, 4a and b& are the annual and semiannual phases and a and b are theannual and semiannual amplitudes in each layer. The amplitude and phases obtained aresimilar among layers; the annual amplitude is about 3 times larger than the semiannualamplitude. The annual amplitudes vary between 0.3 and 0.7 o unit while the semiannualamplitudes range from 0.1 and 0.2 o unit; the smallest amplitudes for both componentsare in the first two layers (the top 15 m of the water column) and the largest in the middleof the water column (from 15 to 150 m). From initial trials it was found that the densityvalues required to induce renewal in the model turn out to be higher than observed. Itwas necessary to increase the amplitudes of the seasonal functions in all layers. Theresults from these initial trials suggested that the amplitudes of the seasonal densityforcing could be simplified by using the same amplitudes throughout the water column;thus a simple vertical average of the amplitudes and phases were used and reduced theseasonal cycle forcing of density to two parameters, the amplitudes a and b, but kepttheir same ratio, namely a/b = 3. Figure 4.2 shows the time variation about the mean ofthe boundary density (thick line) obtained from the 1968 field data (vertically-averagingthe amplitude and phases obtained from each layer). Figure 4.2 also shows the curvewhich have the same phases, q and and the same amplitude ratio a/b as the fielddata, but with a larger amplitude. Trials were done varying a (and hence b = a/3) anda value of a = 1.5 and b = 0.5 in o unit were chosen, corresponding to the thin line inFigure 4.2. This temporal function was applied equally to all grid points at the openboundary.The final parameter for the density at the open boundary is its mean value. The meanChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 1542.01.00.0-1.0-2.0MonthsI I I I I I I I I I30 60 90 120 150 180 210 240 270 300 330 360DaysFigure 4.2: Time varying density forcing at the open boundaries. The function is simplya C05(Wat— q’a) + bCO5(2at— 4b), where wa is the annual frequency and (a, q$) and (b, b)are the annual and semiannual amplitudes and phases obtained by a least-square fit tothe observations of Crean and Ages (1971). The fit was done to each layer and thenthe amplitudes and phases were vertically-averaged. The mean has been removed: theabove data vary about a mean value which is different for each layer. The thick linecorresponds to the values of a and b obtained directly from the fit of the actual 1968data: namely a = 0.36, and b = 0.12 in o unit. These values for a and b represent theseasonal density variability, vertically-averaged for the year 1968. Using the same phases,the thin line is for a = 1.5 and b = 0.5 in o unit, and this is the forcing density functionused to produce the results in this chapter.J F M A M J J A S 0 N 0Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 155density field is simply the same initial field shown in Figure 3.2 for the month of June piusa fraction of that field. The fraction within each layer is constant but different amonglayers; namely at the open boundary, the mean density ot(x, y, z) = 0t(X, y, z)[1 + P(z)];trials were done varying P(z) and values of P = 0.17 for the first four layers and P =0.21 for the last three layers were chosen (the open boundary has only seven layers).The model run here will have the same forcing by tides, wind and runoff and all thesame parameters as the reference run of Chapter 3. As discussed the only differencewill be: 1) that vertical diffusion is included in the density equation (Equation 3.3), 2a)that a time dependent density forcing is included at the open boundary (Figure 4.2) 2b)and that the mean density at the boundary is raised by O.2 on average. For convenience,this run will be referred to as the deep water renewal run: DWR run.4.3 ResultsAs before, all fields presented have been passed through the same low-pass filter; onlyresidual components will be shown unless otherwise stated.The results will be presented at Station III and along three different cross-sections inthe Strait of Georgia, shown in Figure 4.3. Line A connects the deepest gridpoint of eachcross section (median line) from Juan de Fuca Strait to the central Strait of Georgia,passing through Haro Strait and Boundary Passage. Lines B and C are two sections inthe central Strait of Georgia from Galiano Island to the mainland; section C is roughly inthe center of the study area where model results have previously been shown (Figure 3.1)and section B is in the south of this same area.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 156Figure 4.3: Location of Station III and the cross-sections of the model domain whereresults are presented. A connects Juan de Fuca Strait, Haro Strait, Boundary Passageand the central Strait of Georgia and B and C are cross-sections in the central Strait ofGeorgia traversing from Galiano Island to the mainland. The square is the area wherethe results of Chapter 3 were shown and is included here for reference only.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 1574.3.1 Evidence of Deep Water Renewal in the modelA DWR event or cycle has an initial inflow of dense water to the bottom of the Strait ofGeorgia, indicated by an increase in density there, followed by a decrease in the densityat the bottom by mixing/diffusion until the next exchange happens. This behavior istypical of DWR; Farmer and Freeland (1983) note that the density of deep water showsa saw-tooth behavior as a function of time in which a jump in density is followed by agradual decay towards equilibrium conditions. As will be presented next, this behavioris found in the time series of the model.First a time series of the low frequency variations of the o field at Station III forthe 8 model layers are shown in Figure 4.4. The variance of ot decreases with depth, afact consistent with previous observations (LeBlond, 1983). Large monthly oscillationsoccur throughout the water column and fortnightly oscillations are evident in the top 4layers. The decay of density is stronger in the upper layers where the fresh water fromthe Fraser River has more influence.As can be seen in Figure 4.4, DWR events occur approximately monthly. The increasein o occurs over 3 to 5 days every month and is then followed by a slow decrease, Themaximum values of o occur a few days after the extreme neap tides as will be shownnext.Figure 4.5 shows the following time series. Frame a) gives the mechanical tidal energyof the barotropic currents [= (U2+V2), where U = f u dz, and V = f v dz] at BoundaryPassage. Frame b) gives o for the bottom layer of Boundary Passage and at StationIII in the Strait of Georgia. Frame c) gives the along channel velocity for the bottomlayer at Station III in the Strait of Georgia. At neap tides, the mechanical tidal energyhas minima (Figure 4.5a), and therefore mixing is at a minimum. Consider the extremeneap tide on day 232. (Day 232 is given by the first solid vertical line in Figure 4.5.)Chapter 4 Deep Water Renewal in a multi-layer model of the Strait of GeorgiaDaysFigure 4.4: Time series of the low-frequency at field for the 8 model layers at Station III(see Figure 4.3). The top line (with the highest density) corresponds to the bottom layerand the bottom line corresponds to the top layer. Time is in Julian days for 1984.158Station IIIbottom layer25,24.23.22.2120.19.18.p layer180 210 240. 270 300.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 159— I I i (b)I I I q %,, ——•‘i”i ..% — —I,—/. !.‘ I I, S..jI I I I I, \._._--. I I I. I I I I II I I I I — ——. I ,)----4-’I I I I (c)I I I I II-1 iiI I III I I I I225. 240. 255. 270. 285. 300.daysFigure 4.5: Time series of: a) (U2 + V2), where U and V are the vertically integratedvelocities in Boundary Passage (in cm s1), showing the mechanical tidal energy (incm2s2), b) o for the bottom layer at Boundary Passage (dashed line) and at Station IIIin the Strait of Georgia (thin line), and c) along channel velocity component (in cm s’)in the bottom layer at Station III in the Strait of Georgia. The solid vertical lines give thetime of the extreme neap tides (approximately one per month) and the dashed verticallines give the times when contour plots are presented in subsequent figures (days 230,234, 238, 242 and 246).- ED (N ..O(V) (V) (Y)600. - EN400. II I I I II I I I I(a)/0.27.023.520.-20.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 160The maximum density at the bottom of Boundary Passage occurs 2-3 days later (seedashed curve, Figure 4.5b). The maximum density at Station III in the Strait of Georgiaoccurs 3-4 days after the extreme neap (see solid curve, Figure 4.5b). Clearly there is anorthward propagation of dense water. The northward flow of water can be appreciatedfrom Figure 4,5c with the appearance of relatively large positive velocities at the bottomat Station III in the Strait of Georgia during the inflow event. This behavior is exactlythe same as described by LeBlond et al. (1991), namely, after a minimum of mechanicalenergy is reached, the water column is able to stratify and a two-layer exchange can beestablished allowing dense waters to penetrate past the sill at Boundary Passage and tosink into the bottom of the Strait of Georgia. This two-layer exchange will be examinedbelow.4.3.2 The Deep Water Renewal cycleThe intrusion of waters from Juan de Fuca Strait to the Strait of GeorgiaHere I am going to show xz contour plots of the low pass filtered o corresponding toa section along line A of Figure 4.3; this section follows the deepest path from Juan deFuca Strait to the central Strait of Georgia. Four different plots will be shown. The firstplot will show the full section from Juan de Fuca Strait to the central Strait of Georgiawith the aim of identifying the arrival of waters from the source region at the centralStrait of Georgia. The second plot will zoom in showing the section from BoundaryPassage, which has been identified as the main gate of renewal water (LeBlond et al,,1991), to the central Strait of Georgia, the area of study. The third and fourth figureswill zoom further by showing the first and second half of the previous figure. Note thatthe intermediate sill between Boundary Passage and the central Strait of Georgia willhere be called the secondary sill.Figure 4.6 shows the time series of o contour plots from Juan de Fuca Strait to theChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 161central Strait of Georgia. The corresponding day for each plot is given in the bottom leftof each panel and the position of each day in the tidal cycle is given in Figure 4.5. Thetime series goes from Julian day 230 to day 246 at increments of four days, showing thetime evolution of density through one neap-spring tidal cycle. It starts two days beforethe extreme neap tide, which occurs at day 232, then passes through the extreme springtide, at day 240, and finishes two days before the next moderate neap tide.Figure 4.6 shows the appreciable density contrast between Juan de Fuca Strait watersand those of the Strait of Georgia. The Juan de Fuca waters are much higher in density,responding to the forcing from the Pacific. (Note that the densities of Juan de Fuca Straitare higher than those actually observed because of the artificially increased density atthe boundary as discussed earlier.) Next note that this strong density difference betweenJuan de Fuca Strait and the Strait of Georgia is at all times centered at Boundary Passagewhere the isopycnal slopes are maximal. The large slopes of the isopycnals indicate ahigh degree of vertical mixing in the water column. On the Juan de Fuca Strait side, theisopycnals slope down towards the Strait of Georgia and advance back and forth throughthe neap-spring tidal cycle. This same behavior was reported by Crean et al, (1988a; seetheir Figures 1.4, 1.5 and 1.11). The movement of denser isopycnals towards the Straitof Georgia during neap tides indicates the attempt of the dense Pacific waters to intrudethrough Juan de Fuca Strait to reach the Strait of Georgia. On the Strait of Georgiaside, the isopycnals are more horizontal indicating that there is not much mixing, which isexpected as the basin is larger and thus, by continuity, the tidal velocities and hence themixing must be smaller. Also, the input of fresh water from the Fraser River stabilizesthe water column.Sweeping through the neap-spring tidal cycle, Figure 4.6 shows a maximum stratification at day 234: the stratification is highest just after neap tide when mechanicalenergy is at a minimum, mixing is low and high density water is able to pass BoundaryChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 162Figure 4.6: A time series of low pass filtered o contour plots from Juan de Fuca Strait(JF) to the central Strait of Georgia (SG) passing through Haro Strait (HS), BoundaryPassage (BP) and a secondary sill (2S) along line A of Figure 4.3. The contours are givenat an interval of 1 u unit; contours for o below 22 are not shown. The o values of thebottom of the central Strait of Georgia are given between parentheses at the right andbottom of each plot. The position of these times in the tidal cycle is given in Figure 4,5:extreme neap day 232, extreme spring day 240, and moderate neap 246. The number inthe bottom left of the panels indicates the maximum depth of the panel.335 mChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 163Passage. Note that there is a secondary sill, labeled 2S in the figure, where some blockingalso occurs. However, once dense water fills the intermediate basin between BoundaryPassage and the secondary sill, a gravity current of dense water begins to move down intothe basin of the Strait of Georgia: see days 234 and 238 in Figure 4.6. After this time,mixing begins to increase again and the denser waters retreat back until, at day 242, thecycle begins again with high density isopycnals moving forward. However, at day 246,the neap tide is moderate and not enough dense water traverses Boundary Passage toreach the deep basin of the central Strait of Georgia. Thus deep renewal occurs on amonthly basis rather than fortnightly but there are fortnightly variations at shallowerdepths (Figure 4.4). All of these features can be better appreciated in the followingfigures where parts of Figure 4.6 will be enlarged.The section from Boundary Passage to the central Strait of Georgia is shown inFigure 4.7, now with contours given at intervals of 0.1 o unit. The first feature to noteagain is the role of the secondary sill which is located in the center of the plot. Thissecondary sill blocks waters dense enough to traverse the sill at Boundary Passage andproduces a delay in the travel of the dense waters toward the deep basin. Now, theevents of DWR can be followed; on day 230, tidal mixing is almost at a minimum as thetidal cycle approaches an extreme neap (see Figure 4.5) and the level of stratification isrelatively high as can be seen from the spacing of the isopycnals. The bottom of the basinbefore the secondary sill and the Strait of Georgia o are 24.1 and 23.81, respectively.Stratification increases as the isopycnals get closer to each other by day 234 (two daysafter neap tides) and where the isopycnal of 24.3 has now been raised to the level of thesecondary sill and passed over it. In the Strait of Georgia a mass of water with o = 23.9moves toward the bottom. On day 238 waters with o = 24.1 are passing the secondarysill and moving toward the central Strait of Georgia. However mixing has started toincrease as spring tides are approached and more mechanical energy is present. Still onChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 164Figure 4.7: Contour plot of o from Boundary Passage to the central Strait of Georgiaalong line A of Figure 4.3. Contour interval is 0.1 at unit; o contours below 23.5 are notshown. The o values of the bottom of the central Strait of Georgia and at the bottomof the inner basin before the secondary sill are given between parentheses at the bottomright and center of each plot, respectively. The position of these times in the tidal cycleare given in Figure 4.5: extreme neap day 232, extreme spring day 240, and moderateneap day 246. The number in the bottom left of the panels indicates the maximum depthof the panel.23;9335 m23.5—rJ___23 23.67day 246335 mChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 165day 238, water with o > 23.9 is filling the bottom of the Strait of Georgia. On day242 just after spring tide, mixing is at a maximum and the waters from Juan de FucaStrait are blocked (see also Figure 4.6) and the density of the deep water in the Strait ofGeorgia is decreasing.Note that in the model the blocking of Juan de Fuca waters is not only occurring atthe Boundary Passage sill (as suggested by LeBlond et al., 1991) but also at ilaro Straitand the secondary sill. At the next neap tide, some water is able to traverse the sill atBoundary Passage and even pass the secondary sill but it is not dense enough to sinkto the bottom of the central Strait of Georgia and reaches only intermediate levels (day248, not shown).Figure 4.7 contains two distinct density regimes; one from Boundary Passage to thesecondary sill has a sharp density interface and a large top to bottom density differenceand the other from the secondary sill to the central Strait of Georgia has a more gradualand smaller top to bottom density change. This section will be further enlarged in thefollowing figures. Figure 4.8 zooms in on the left side of Figure 4.7, namely the areafrom Boundary Passage to the secondary sill and Figure 4.9 zooms in on the right side ofFigure 4.7, namely the area from the secondary sill to the central Strait of Georgia. Notethat both figures show the secondary sill. Figure 4.8 allows for a better appreciation ofhow the denser waters of Juan de Fuca Strait traverse Boundary Passage, fill the smallintermediate basin and then pass the secondary sill to form a gravity current into thedeep basin in the central Strait of Georgia. Starting again on day 230, two days beforean extreme neap tide, water with at ‘- 26 is found at the bottom of Boundary Passageand water with a 24 is found in the inner basin. Then, advancing to day 234, whichis two days after the neap tide, the stratification begins to increase (the isopycnals getcloser together), higher densities are found at the bottom levels and waters with o 24are starting to pass the secondary sill. Now from day 238, as the extreme spring tide atCDaqCD0CD, ç,CD0CDCDCDCDH—•0’Pr,)e--,-lrp,_.0ei- I-Q-0—j0oq.__<0l),00e-CDCD0CD-0CDECDj.cDClo00OCD0,—.—.HCDi-CD0CDoPq-0—.•oCDe+CDeCDCD0,CDCD‘CDr•CDSCD,eCD.q—.CDCD<—••,-I-_CD—.0CDCDCD.-“CD -•-CDCDa-’-..CDCDCDCl)CD CD ICDtCi)hrj>CDCD CiDQCDcc’Ci)___Ci)e-s-CDnCDCD-t a‘-C,.q0 c’P‘NPCD—“-C-nt30Ci)CDe<c.$CD‘CDCi)O CC, : p,t’3t3C.A3,yq0—••lj_Ct_z_H—*t%)Ci)CD—•C-n‘-C)CDCDCDCD>,t\)C,HCDe+::0e+eCDjoC0C3CDcnCD CD I-.eq. I CD CD 0 I. C)Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 168day 240 is approached, the dense waters retreat and disconnect from the central Strait ofGeorgia. Thus the waters that traverse Boundary Passage still have to survive mixing inthis inner basin of the Strait of Georgia, filling this small basin with density high enoughthat when it passes the secondary sill, the density will be higher than the deeper watersof the central Strait of Georgia in order to create a gravity current.Figure 4.9 shows the DWR event from the secondary sill to within the central Straitof Georgia. Here on day 230, water with crt = 23.9 is passing the secondary sill. Thenext frame, two days after an extreme neap, shows a wedge of denser water passingthe secondary sill and water with o 23.9 is advancing toward the central Strait ofGeorgia. On day 238, a retreat of dense water begins near the sill, but the water with o23.9 that already passed the sill continues to advance to the deeper part of the Straitof Georgia. From day 242, the inner basin is totally disconnected from the denser watersand diffusion starts to lower the density at the bottom of the central Strait of Georgia.In the previous contours plots, the blocking of dense water from the Pacific could beobserved from the way the contour lines would slope and end at the bottom, signifyinghorizontal gradients. Another way to see blocking is to plot a time series of density fordifferent locations.Figure 4.10 shows a time series of o for the bottom layer at selected gridpoints fromJuan de Fuca Strait to the central Strait of Georgia. The gridpoints are labeled a to i andtheir positions are given in Figure 4.11 along with the vertical section from Juan de FucaStrait to the central Strait of Georgia following line A in Figure 4.3. In Juan de FucaStrait (a) the waters are consistently much denser than in the deep basin of the Strait ofGeorgia (i). From the open boundary to the eastern part of Juan de Fuca Strait (a) thedensities are about the same, which means that the waters from the Pacific enter withoutmuch modification through this part of the Strait. However, these waters experience aconsiderable drop in density from (a) to (b), namely, a considerable amount of mixing isChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 169600.400.200.0,33.3129.27,25.23.225.daysFigure 4.10: Time series of i for the bottom layer at selected gridpoints along the pathA shown in Figure 4.3. The position of each grid point, labeled a to i, is given in Figure4.11. At the top, the tidal energy (in cm2 _2) of Figure 4.5 is repeated for reference.The vertical lines represent extreme neap tides.abI—240. 255. 270. 285, 300.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 170Figure 4.11: Location of the gridpoints referred to in Figure 4.10. The gridpoints arelocated along path A of Figure 4.3: (a) gives the position in the horizontal domain and(b) shows the vertical section of path A.ModelDomabChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 171occurring between eastern Juan de Fuca Strait (a) and Haro Strait (b): there is a drop of3 o unit in this region. This behavior is consistent with that shown in the contouredvertical sections of salinity along Juan de Fuca Strait by Crean et al. (1988) (see theirFigure 1.11). The water passing over the Boundary Passage sill is basically defined inilaro Strait; from Haro Strait (b) to the west side of Boundary Passage (c) the densitiesat the bottom are very similar, which indicates that these waters are the same and thereis little mixing between Haro Strait and the west side of Boundary Passage sill in themodel. The waters that pass the sill at Boundary Passage (d) experience a larger dropin density at the west side (c)-(d) of the sill than on the east side (d)-(e) because themixing at the top of the sill (d) is large; the model velocities are the largest there, andthe water that reaches the sill is subject to mixing with the fresher surface waters flowingout of the Strait of Georgia.Once the waters reach the sill at Boundary Passage and make their way into thesmall inner basin between the sill at Boundary Passage and the secondary sill, theyagain experience some mixing as can be appreciated from lines (e) and (f) of Figure 4.10.Next at the secondary sill (g), as at the sill at Boundary Passage, more mixing occurs atthe south side of the secondary sill (f)-(g) than occurs on the north (g)-(h). Then thereis a final, but small amount of mixing as the water moves from the secondary sill to thebottom of the Strait of Georgia.Table 4.1 shows the time-averaged bottom layer densities for the selected gridpointsof Figure 4.11, along with the density differences, the distance between locations and thedensity gradient ãot/ôx. Even though a large drop in the water density is experiencedbefore the denser water arrives at Boundary Passage (a)-(c), it is at Boundary Passagewhere the largest mixing occurs over a short distance. It can also be seen from the table,that the water density just passing the secondary sill is almost the same as that of thecentral Strait of Georgia. Note also that there is a very small difference between theChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 172Table 4.1: Time-averaged density at the bottom for the gridpoints shown in Figure 4.11.The average is over the same time period shown in Figure 4.10. Also given are thedensity differences, the distance between the locations, along path A of Figure 4.3, andthe density gradient. The density gradient is taken as the difference of at between thetwo consecutive locations divided by the distance between them in km.Location label Fig. 4.11 o distance (km)Juan de Fuca a 32.393.62 59 0.061Haro Strait b 28.770.23 34 0.007west of Boundary Passage c 28.542.57 6 0.428Boundary Passage d 25.970.80 4 0.200east of Boundary Passage e 25.170.65 23 0.028south of Secondary Sill f 24.520.30 5 0.060Secondary Sill g 24.220.13 6 0.022north of Secondary Sill h 24.090.07 28 0.003central Strait of Georgia i 24.02Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 173secondary sill and the deep basin: on average the density is larger at the secondary sill(Table 4.1) but from Figure 4.10 it can be seen that at times the density in the basin ishigher than at the sill; density for exchange is only available when tidal mixing is at aminimum at neap tides.As a final note, consider the blocking from spring to neap tides. If blocking is definedas the density difference between locations then there is little change in blocking fromspring to neap tides: the curves in Figure 4.10 run relatively parallel to each other. Inother words there is a large blocking during both spring and neap tides. However, aroundneap tides mixing is reduced, the vertical stratification increases slightly and there is aslight increase in density at the bottom of all locations. This increase in density is smallcompared to the density differences due to blocking. Thus renewal occurs from a smallchange in the large blocking: renewal being thus a small and sensitive effect.It is clear that the degree of mixing that occurs in a particular place is dependent onthe relative size of the inertial and buoyancy forces. A Froude number is defined hereas the square root of the inverse gradient Richardson number (Kundu, 1990), namely,Fr = /()2/N2 = i//, where N=is the Brunt-Vaisala frequency. Fr iscalculated from the total, unfiltered fields, For continuously stratified parallel flowsthe inequality Ri > guarantees stability (LeBlond and Mysak, 1978); thus Fr < 2is a sufficient condition for stability. However, note that the condition Fr > 2 doesnot mean the flow is necessarily unstable; a flow with Fr > 2 will be more likely tobecome unstable leading to turbulence and mixing. From the expression for Fr, smallFr indicates high stratification or small shear, and hence less mixing. Large Fr indicateslow stratification or high shear and, thus, more mixing. Here I am using the Fr as anindicator of the relative size between the inertial and buoyancy forces to show that atneap tides, not only a minimum of the mechanical energy is reached, but that neap tidesalso represent a minimum in mixing. Figures 4.12 and 4.13 show Fr at Boundary PassageChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 174for different stages of the tides (every four hours) during an extreme neap (day 232) andan extreme spring (day 242) tide, Fr changes from low to high values during each tidalcycle, nevertheless during spring tides it is substantially larger than during neap tides.Thus mixing is larger during spring tides than during neap tides.The arrival of water at the central Strait of GeorgiaHere I will show the structure of 0t and the along channel velocity component atthe cross sections B and C in the central Strait of Georgia (see Figure 4.3) for thesame five days as in Figures 4.6 to 4.9. For sections B and C, Figures 4.14 and 4.16show contour plots of o and Figures 4.15 and 4.17 show contour plots for the v (alongchannel) component of velocity. Recall that section B is south of section C, andtherefore the penetration of waters from Juan de Fuca Strait occurs through this sectionfirst. For section B, the figures show that the penetration of denser waters from thesouth occurs with maximum strength between days 234 and 238 and occurs mainlythrough the mainland side of the strait, reflecting the effect of the earth’s rotation on thecirculation in the Strait. This inflow with reverse flow in the upper layers has actuallybeen observed by Schumacher et al. (1978) with current meter observations duringFebruary and March of 1975. During the extreme spring days 238-242, the bottomdensity decreases and the bottom current slows down, and the horizontal structure of obecomes more homogeneous across the strait.Further north, in section C, the cross-section is larger and the flow weakens relative tosection B. The penetration of water to the deep basin is through the center of the section,Largest velocities occur on day 234 and on day 238 the density reaches a maximum inthe deep basin (crt 23.96) and there is a relatively large area with a northward velocity.When the tidal cycle shifts to spring tides, the water inflow slows down and the waterdensity decreases.CD‘CD——.CD —ctoi-*CDp •‘CDet-CDCDp,0 D>1.ra.e-t-Z—.C±dCo SCDdcaCDes-COCDCD3—.CD-t—s-ictCDp,—5•’ DCDCnCDCDCDCD cc,qCDCD It—A•CDCCDCDrIt CD•s-ID-CDCD-jI-’.,,CDC1HCDI-I,(i‘•0-CD1CDCD5cqCO,nc-t-><QCDCDti0-CDetCoIDCDdt eses-I-’•Ces-CD0-CDCD CD s-CDCD—CD—es_O0-CDCDCD0-‘I•iCop,CD—•s-It<CDestDoCD0’s-o-‘es-Coes-s-s-IS:•-T C’Li,U,aI CD CD S I.-. CD CCbe0c4-..0ofr+I4I.-. oI-’,•Ieq-•-pp,—IDC+CCDçI,00Chapter 4 Deep Water Renewal in a multi-layer model of the Strait of GeorgiaSection B177Figure 4.14: Time series of contour plots of o in the southern section B of the centralStrait of Georgia (see Figure 4.3). The contours shown are limited to o 23.0, 23.5,23.7, 23.8, 23.9 and 23.95. The position of these times in the tidal cycle is given in Figure4.5: extreme neap day 232, extreme spring day 240, and moderate neap day 246. Thenumber at the bottom left of the panels indicates the maximum depth of the panel.______—23.0 ——23.5 It_1,__335 m day 23023.023.523.7 23.8 23.9 23.95335 m day 23423.023.523.7 23.8 23.9 23.95335 m day 238L1—234ZZE335 m day 242‘3. -335 m day 246Chapter 4 Deep Water Renewal in a multi-layer model of the Strait of Georgia 178Figure 4.15: Time series of contour plots of the along channel velocity component, incm s1, in the southern section B of the central Strait of Georgia (see Figure 4.3). Thincontours are positive, dashed are negative and thick are zero. Positive velocity is flowto the northwest, up the strait. The position of these times in the tidal cycle is given inFigure 4,5: extreme neap day 232, extreme spring day 240, and moderate neap day 246.The number at the bottom left of the panels indicates the maximum depth of the panel.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 179Section CI2S.—23.50270335 m day 23023.-—23.70335 m—4dayI _335 m day 238Figure 4.16: Time series of contour plots of o in the central section C of the centralStrait of Georgia (see Figure 4.3). The contours shown are limited to t 23.0, 23.5,23.7, 23.8, 23.9 and 23.95. The position of these times in the tidal cycle is given in Figure4.5: extreme neap day 232, extreme spring day 240, and moderate neap day 246. Thenumber at the bottom left of the panels indicates the maximum depth of the panel.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 180Figure 4.17: Time series of contour plots of the along channel velocity component, incm s1, in the southern section C of the central Strait of Georgia (see Figure 4.3). Thincontours are positive, dashed are negative and thick are zero. Positive velocity is flowto the northwest, up the strait. The position of these times in the tidal cycle is given inFigure 4.5: extreme neap day 232, extreme spring day 240, and moderate neap day 246.The number at the bottom left of the panels indicates the maximum depth of the panel.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 181The two-layer exchange flowDeep water renewal occurs in the Strait of Georgia during extreme neap tides (once amonth). It is during these times, when tidal mixing is low and the water column is highlystratified, that an effective two-layer exchange through Boundary Passage is established(LeBlond, et al., 1991). Here I will show how the two-layer system appears in the model.First, time series of o and u at Boundary Passage are shown in Figure 4.18. Timeprogresses from day 230 to day 242, capturing both an extreme neap and an extremespring tide. The figure indicates that the residual flow is always in the form of two-layerswith water flowing west (towards Juan de Fuca Strait) in the first 30 m (top three modellayers) and water flowing east (towards the Strait of Georgia) below 30 m. This flowpattern is consistent with the inflow associated with DWR and the estuarine characterof the Juan de Fuca/Strait of Georgia system; indeed, runoff is maximum in summer.However, during spring tides, the two-layer flow weakens: the maximum speed of theoutflow (upper layers) decreases from 30 to 23 cm s and the maximum speed of theinflow (lower layers) decreases from 48 to 31 cm s1. Also the difference between themaximum and minimum values of o during neap tides is 26.4 - 20,8 = 5.6 crt unit andduring spring tides the difference drops to 25.8 - 22.2 = 3.6 t unit. These differenceslead to smaller Fr numbers during neap tides than during spring tides (see Figures 4.12and 4.13).In Figures 4.19 and 4.20 the total, unfiltered fields of o and u in Boundary Passageare shown for two 4 day periods. The two figures run two days before and two days afterextreme neap tides on day 232 and extreme spring tides on day 240. During neap tides,(Figure 4.19) the two-layer flow is still appreciable and the inflow of water in the lowerlayers still occurs which ultimately leads to DWR. Only a few times does the barotropiccomponent dominate with all the flow throughout the water column going in or out. Onthe other hand, during spring tides (Figure 4.20), the barotropic tides (dominated byChapter 4 Deep Water Renewal in a multi-layer model of the Strait of Georgia 182Neap (a) Spring— — — — — — — ——2____________0•0.0—30125 ni_____(b)_____21l2Sni I__ __230. 232, 234. 236. 238, 240. 242.daysFigure 4.18: Contour plot (depth vs. time) of: a) the residual u component of velocity(cm s1; positive flow is into the Strait of Georgia) and b) the residual o at BoundaryPassage. The vertical lines indicate the time of the extreme neap and the extreme springtides at days 232 and 240, respectively. The number at the bottom left of the panelsindicates the maximum depth of the panel.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 183230Neapdays(a)Figure 4.19: Contour plot (depth vs. time) of the unfiltered: a) u component of velocity(contour interval is 30 cm s’; positive flow is into the Strait of Georgia) and b) o atBoundary Passage (contour interval is 1 o unit) during extreme neap tides (day 232indicated by the vertical line). The number at the bottom left of the panels indicates themaximum depth of the panel.231 232. 233. 234.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 184238Spr I ngdays(a)Figure 4.20: Contour plot (depth vs. time) of the unfiltered: a) u component of velocity(contour interval is 60 cm s’; positive flow is into the Strait of Georgia) and b) o atBoundary Passage (contour interval is 1 oj unit) during spring tides (day 240 indicatedby the vertical line). The number at the bottom left of the panels indicates the maximumdepth of the panel.239. 240. 241. 242.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 185the diurnal constituents) are larger than the residual and in every cycle the flow goescompletely in and out (with, of course, some shear).Therefore, the denser waters of Juan de Fuca Strait pass over the sill at BoundaryPassage all the time, however during neap tides a stronger two-layer system is establishedand denser water is able to pass and pour into the Strait of Georgia and reach the deepbasin of the central Strait of Georgia. Meanwhile, during spring tides, the barotropictides dominate; the two-layer system almost disappears, and the water that passes thesill during the flood phase of the tides is not dense enough to generate renewal.The water that arrives at the bottom of the Strait of Georgia is water that hasexperienced extensive mixing throughout Juan de Fuca Strait. As shown, blocking occursat several places, with a major part always occurring at the sill of Boundary Passage,but it is smaller during neap tides when renewal occurs.Summarizing, the model reproduces the summer Deep Water Renewal phenomenonin the Strait of Georgia. DWR occurs once a month during extreme neap tides whenmechanical energy is minimum and the water column reaches a maximum of stratificationat Boundary Passage allowing the denser waters of Juan de Fuca Strait to pass the silland reach the bottom of the central Strait of Georgia. However, it is fair to say thatthese results are obtained at the expense of higher densities than observed at the openboundary which must be imposed to overcome the excessive mixing in Juan de FucaStrait and perhaps Boundary Passage and allow DWR to occur.4.4 The general circulation4.4.1 Comparison with the reference runNow that DWR is occurring in the model, I will briefly re-examine the general circulationof the central Strait of Georgia, comparing it to the general circulation of the referenceChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 186run in Chapter 3 where renewal did not occur.Time-Averaged residualIt turns out that with deep water renewal, the general circulation for the reference runshown in Figure 3.3 in Chapter 3 changes significantly. Deep water renewal in the Straitof Georgia in summer involves inflow from the south through the bottom layers. Thereference run has a mean flow in the two bottom layers going south (see Figure 3.3). Forcomparison, I time-averaged the results of this DWR run for the full year of integrationand the result is shown in Figure 4.21. The figure shows that the general circulation ofthe run with DWR is quite different. A common difference throughout the water columnis that the run involving DWR has larger speeds than the reference run: I will highlightonly the differences in direction. The first layer remains basically unchanged. The secondlayer maintains a tendency for flow northwards at the northeast side as in the referencerun but the flow is also now in the northward direction on the northwest side which isopposite to the reference run. The third layer has a strong coastal flow down-strait whichfollows the coast more closely in the DWR than the reference run and the DWR run has aweak up-strait flow (forming an anticyclonic eddy) on the Vancouver Island side which isabsent in the reference run. The fourth layer is down-strait everywhere in the DWR runwhich is completely different from the reference run where an up-strait flow appears onthe mainland side. The fifth layer of the DWR run has a strong down-strait flow on theisland side and an up-strait flow on the mainland side which is present in the referencerun. From the sixth layer to the bottom, the flow is basically up-strait, which meansthat the DWR events, even though they appear only once a month and in summer only,have a strong effect on the time-averaged circulation. Finally, a spatial feature commonto both runs is the presence of part of a cyclonic eddy in several layers (from the 4thto the 7th layer in the DWR run and from the 2nd to the 6th layer in the referencerun) which is reflected also in the depth-averaged field. Figure 4.22 shows the actualChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia1187I’ ‘‘III’ift!tttLI//k,t,,4,,,?_________%t / F_H7;,90—15015 cm/sFigure 4.21: Time-averaged velocity for the 8 layers of the model along with the vertically-integrated field (frame 9) of the run involving deep water renewal. Compare thereference run, Figure 3.3. Depth of layers, in meters, is given in the lower right handcorner of each frame.27 9• t• 8II9488t It ¶ ¶It‘it’s..11188Ii,,,...I1F# / /_ IIChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 188difference between the two runs (DWR run minus reference run) for all layers and forthe depth-averaged field. As a result of larger velocities in the DWR run, the differenceis large everywhere. With the exception of layer 1, the figure showing the difference isnot very different from Figure 4.21.Figure 4.23 shows the time-averaged density field for the run involving DWR for thefull year of integration. The first feature to note in the top two layers is how the explicitdiffusion increases the density compared to the reference run where explicit diffusion wasabsent. The low density fresh water is now almost confined to the mouth of the FraserRiver, whereas in the reference run, the density was low throughout the surface layers.For the same reason, the vertical gradient of density is not as large as in the referencerun. Another feature is that the isopycnals are now more aligned with the flow reflectinga larger influence of the density field as a driving force (through the baroclinic pressuregradient). In the first two layers, higher densities are found to the north of the centralStrait. Layers 3 and 4 have higher densities on the south and northwestern sides. Layers5, 6 and 8 have higher densities on the mainland side. Layer 7 has higher densities onthe island side.Time variabilityTo illustrate the time variability for this run involving Deep Water Renewal, Figure 4.24 shows the total residual for layer 7 at the usual four selected gridpoints. StationsIII and IV reveal the events of DWR at extreme neap tides (starting at day 203) withpositive velocity indicating inflow from the south. Note that stations I and II are outof phase due to recirculation in the central part of the Strait as can be appreciated bylooking at the horizontal structure of the currents: Figure 4.25 shows the fluctuatingresidual of the horizontal currents for layer 7 over 28 days (compare to Figure 3.20 forthe reference run). The strong currents occur during DWR, which is few days after neaptides as was observed in Figure 4.5. Velocities reach values as high as 28 cm s1 on dayChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 189Figure 4.22: Difference between the time-averaged velocity field of the deep water renewalrun (Figure 4.21) and the reference run (Figure 3.3).4 5 615 cm/sChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 1901 2Figure 4.23: Time-averaged o field for the 8 layers of the model for a run involvingdeep water renewal. In frame 9, the time-averaged surface elevation, in cm, is shown.Compare the reference run, Figure 3.7. Depth of layers, in meters, is given in the lowerright hand corner of each frame.3Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of GeorgiaI I I I I I I I I I IdaysFigure 4.24: Time series of the residual velocity components, u (thin) and v (thick),in cm s1, for layer 7 of the DWR run at the usual four grid points corresponding tostations I to IV as shown in Figures 3.lc and 4.25. At the top, the tidal energy (incm2 s2) of Figure 4.5 is repeated for reference. The vertical lines represent extremeneap tides. Compare to Figure 3.12 for the reference run.800,400.0.12.0.-12.12.-12.12,0.-12.12.—12191IIIIIIIv180. 195. 210. 225. 240, 255, 270. 285. 300. 315. 330. 345. 360.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 192day 224 day 226 day 228 day 230Layer20 cm/sFigure 4.25: The fluctuating residual of the horizontal velocity for layer 7 for a runinvolving deep water renewal. Days are in Julian days starting 1 January, 1984. Compareto Figure 3.20 for the reference run. Note the inflow (renewal) after the extreme neaptide on day 232. In the last panel, the position of the Stations I, II, III and IV arerepeated here for reference.Position Sta.tionseIvIII I7Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 193234; these are absent in the reference run. Note that the temporal and spatial structureof the bottom layer (not shown) is very similar to layer 7 shown here.4.4.2 Comparison with observationsHere the focus will be on a comparison of the model results of the deep water renewal runagainst the observations in the central Strait of Georgia with the same data and formatas used in Chapter 3.Briefly, the main characteristics of the model results of this chapter are as follows.- Deep water renewal occurs once a month during summer months. Dense waters fromthe Juan de Fuca Strait traverse Boundary Passage at extreme neap tides whenmechanical energy, and hence mixing, is at a minimum. Stratification is at amaximum which also contributes to the low mixing.- The total residual flow is characterized by strong coastal jets and eddies on scalescomparable to the size of the domain. These jets and eddies change both directionand sense of rotation with depth.- The maximum speeds for the total residual reaches values of 20 cm s1 at the topand bottom layers and 12 cm s1 at intermediate layers. The speeds of the meanresidual ranges from 4 cm s at the bottom to 20 cm s1 at the surface andaway from the river discharge.- The oscillations for the fluctuating residual are dominated by periodicities of the fortnightly, monthly and semiannual bands.Even though the model run of this chapter is basically a process study, because thedensity forcing at the open boundary is raised artificially, the results are encouragingChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 194because comparisons with the observations in the central Strait of Georgia are in generalbetter than those obtained with the reference run of Chapter 3. In particular, the modelvelocities are larger and closer to the observations.Time-averaged residual currentsFigure 4.26 shows the currents for both the model (solid arrows) and for the cyclesondes (dashed arrows) averaged over the period of the observations (June 1984 to January1985). The cyclesonde data have been averaged over depths corresponding to layers 3 to8 of the model (see Table 3.12).Now the time-average velocities of the deep water renewal run compare better to theobservations than do those of the reference run (Figure 3.3). However, there are stillsome disagreements.The magnitudes of the velocities are more comparable now. Within the small rectangle, the maximum of the mean velocity from the DWR run ranges between 7.5 and 2,5cm s1 compared to 14 and 6 cm s for the observations for layers 3 to 8. With someexceptions, the direction of the flow also shows better agreement.Therefore, with respect to the time-averaged residual, the deep water renewal run,with the seasonally varying density forcing and with vertical diffusion of density included,can be viewed as an improved model simulation of the residual currents of the centralStrait of Georgia.Fluctuating residual currentsFigures 4.27, 4.28 and 4.29 show for stations 1 and 3, time series of the velocitycomponents of the fluctuating residual for layers 3 to 8 of the model and observations.Again, the format is the same as Chapter 3, so both the deep water renewal run and thereference run can be compared.Here is a summary of the comparisons:CYCLES0NDEChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 195M00EL690-150L71 50-250LB250-H 15 cm/sFigure 4.26: Time-averaged velocity for the model (solid arrows) and for the observations (dashed arrows) in the central Strait of Georgia for the period June 1984 toJanuary 1985 corresponding to layers 3 to 8. Cyclesonde data are obtained from verticalaverages over the model layer depths as given in Table 3.12. Layer numbers are given asL3 to L8. In the first cyclesonde frame, 1 to 4 indicates the station number (see Section3.3.1 for further details). Note that in the frames where the cyclesondes velocities areshown, the model velocities are repeated for that particular gridpoint.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 196Layer 3: Depth 15-30 m Stn20,/ i’-20.f.p17N/%.N”fLayer 4: Depth 30-60 m20. -13—20 I I I I I I I I225 240 255 270 285 300 225 240 255 270 285 300days daysFigure 4.27: Time series of the u and v velocity components (in cm s’) of the fluctuatingresidual from the observations (dotted lines) and from the model (solid lines) forstations 1 and 3 in the central Strait of Georgia. Shown at the top is the third layer (15-30m) and at the bottom the fourth layer (30-60 m). The velocity from the observations forthese layers was obtained as described in Table 3.12.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 197Layer 5: Depth 60-90 m Stn20.i0. —.-,- ..‘1—20. I I20, - -IN0. y\1#7. j-_ 3-20. I IU VLayer 6: Depth 90—150 m20,_________________________7Th.yç X--\_-20. I________________________20. - -A0—20. I I I I I I225 240 255 270 285 300 225 240 255 270 285 300days daysFigure 4.28: Time series of the u and v velocity components (in cm s’) of the fluctuatingresidual from the observations (dotted lines) and from the model (solid lines) forstations 1 and 3 in the central Strait of Georgia. Shown at the top is the fifth layer (60-90m) and at the bottom the sixth layer (90-150 m). The velocity from the observations forthese layers was obtained as described in Table 3.12.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 198— \...r-- -I I I225 240 255 270 285 300 225 240 255 270 285 300days daysFigure 4.29: Time series of the u and v velocity components (in cm _1) of the fluctuatingresidual from the observations (dotted lines) and from the model (solid lines) forstations 1 and 3 in the central Strait of Georgia. Shown at the top is the seventh layer(150-250 m) and at the bottom the eighth layer (250-H m). The velocity from theobservations for these layers was obtained as described in Table 3.12.Layer 7 DeptH 150-250 m Stni\/: / ‘r 1-mfl\20.0,-20.20.0.-20.20.0.-20.20.0-20.-‘/ ____7_U VLayer 8: DeptH 250—H m-!90\J\!1& 13Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 1991.- The fluctuating residuals in the model are comparable in magnitude to the fluctuating residuals in the observations. At times the difference is still large but the amplitudeof the oscillations are, in general, the same order of magnitude, Sometimes the modelcurrents look like a smoothed version of the observations (see for example u for station3, layers 3 and 4 in Figure 4.27, or v at station 1 for layer 8 in Figure 4.29) but at othertimes they are out of phase (see for example u for station 1 for layer 3 in Figure 4.27).2.- In the reference run, the amplitudes of the velocity field decreased very rapidlywith depth. Now, the model also shows large currents in the bottom layers making thevertical structure of the whole water column more comparable to the observed currents.3.- In general, neither the u component nor the v component show better agreementthan the other (in the reference run, the v component agrees with the observation betterthan the u component). Sometimes both components agree well, as noted before andsometimes both disagree.4.- It was noted in the previous chapter that the model velocity fluctuations aresmoother than the observations. This is still the case. However, the strong bursts invelocity that the observations showed, at times, are now present in the model. Forexample, in the bottom layer, the peaks in the velocity associated with gravity currentsinvolved in deep water renewal are now reproduced (see layer 8 Figure 4.29).Figure 4.30 shows the o field for the 6 layers at station 3. The figure shows that thefluctuations in o of the model and observations in layers 7 and 8 are very close both inphase and amplitude. Layer 3 is somewhat out of phase and the density is lower in themodel. Layer 4 is oscillating a bit more in phase and the absolute values are closer butare still lower in the model. Layers 5 and 6 are varying also slightly out (or in) phaseand the absolute value is closer: now it is larger in the model.Note that for layers 3 and 4, the model underestimates the density mean values; layers5 and 6 have larger densities in the model and finally layers 7 and 8 have the same meanChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 200I I I225 240 255 270 285 300daysI I225 240 255 270 285 300daysFigure 4.30: Time series of the o from the observations (dashed lines) and from themodel (thick lines) for station 3 in the central Strait of Georgia. The numbers at theupper right hand corner of each frame gives the layer number.layer3layer23.221 .720,224.022.521 .026.024.523,0I7 8Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 201density values. The discrepancy in the absolute values between model and observationsmay be dependent on the eddy viscosity diffusion coefficient. Different values of a in theformulation of A,, result in different mean values of density in the different layers. TheDWR run uses a value a = 23. This value was selected through a series of trials withvarious values of a by comparing, as a reference, the model density field of the bottomlayer in the Strait of Georgia with that of the observations, since modeling of deep waterrenewal was the goal.Harmonic AnalysisFinally, I want to show how the low-frequency tidal constituents compare with theobservations at the same locations when deep water renewal is present. Recall that thisphenomenon is controlled by the tides and occurs once per month (in summer months).Table 4.2 shows the amplitude of the fortnightly and monthly constituents for bothvelocity components, for the observations at stations 1 and 3 and for the correspondinggridpoints of the model. In general the amplitudes of the constituents are larger in theobservations than those obtained from the model. After having found larger residualflows in the deep water renewal run, it might be expected that amplitudes of the tidalconstituents would be larger as well, especially the monthly constituents. This is, in fact,the case. The amplitudes of the constituents from the model are almost always of thesame order of magnitude as those from the observations; in fact only 2 of the 48 amplitudes of the u component of velocity have amplitudes smaller by an order of magnitudethan the observations and only 3 for the v component. (The number of amplitudes withamplitudes smaller by an order of magnitude in the reference run, Table 3.13, were 4 of48 for each of u and v components.) Again to stress the fact that the amplitudes are ofthe same order of magnitude, the amplitudes for both model and data are in bold facewhen the amplitudes of the constituents are within a factor of 3. Over two-thirds of theamplitudes are within this factor of 3 (compared to one-half for the reference run) andChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 202Table 4.2: Amplitude (cms1) of the fortnightly and monthly constituents for the modeland observations, at stations 1 and 3, for the u and v components of velocity. When theamplitudes of the observations and model are within a factor of 3, the numbers appearin bold face in both data and model sections.STATION 1 STATION 3u CYCLESONDElayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 1.27 1.70 2.78 1.91 3 0.77 0.66 0.21 2.544 1.24 1.36 1.56 0.32 4 1.62 0.25 0.86 1.975 1.40 1.16 1.58 0.46 5 1.73 1.26 1.53 1.486 1.41 1.14 0.65 0.95 6 1.10 085 1.28 1.057 0.33 0.67 0.75 0.79 7 0.17 0.64 0.23 1.148 0.05 1.35 1.54 1.60 8 0.76 2.46 2.60 2.89u MODELlayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 0.53 1.62 1.93 0.95 3 1.06 2.20 2.53 1.184 0.40 0.47 0.80 0.65 4 0.81 2.23 2.20 0,995 0.29 0.76 0.93 0.30 5 0.56 2.22 2.28 0.106 0,05 0.49 0.93 0.54 6 0.22 1.16 1.51 0.497 0.03 0.33 0.79 0.45 7 0.18 0.86 0.60 0.208 0.08 0.6’T 0.90 0.45 8 0.20 1.23 1.87 0,22v CYCLESONDElayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 1.32 5.68 3.22 2.20 3 1.67 2.99 2.89 3.674 0.93 3.66 0.81 0.65 4 0.50 1.62 1.77 3.305 0.93 2.51 0.49 1.00 5 1.06 0.75 1.84 1.696 0.77 0.70 1.31 1.22 6 1.38 0.37 1.43 1.607 0.88 0.69 1.43 0.85 7 0.74 0.32 0,94 0.988 0.85 0.89 2.12 0.77 8 1.63 2.22 3.06 3.35v MODELlayer Mf Msf Mm Msm layer Mf Msf Mm Msm3 0.55 1.92 1.78 0.86 3 0.72 1.89 1.60 0.364 0.39 0.35 1.75 1.29 4 0.27 1.12 1.91 0.875 0.17 1.05 1.81 0.89 5 0.13 0.95 1.34 0.496 0.21 1.15 1.70 0.66 6 0.17 0.99 1.47 0.427 0.24 1.02 1.78 0.86 7 0.16 0.89 1.54 0.598 0.23 1.15 2.52 0.78 8 0.19 0.71 2.02 0.73Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 203of the amplitudes are within a factor of 2 (compared to one-third for the referencerun). The agreement is now better for station 1 than for station 3. For station 1, 38of the 48 amplitudes are within the factor of 3 compared to 30 of the 48 amplitudes forstation 3. (Recall, the number 48 results from 6 layers x 4 tidal constituents x 2 velocitycomponents.)With respect to the different velocity components, u or v, there is basically no difference in the number of amplitudes in agreement to within a factor of 3.With respect to the different constituents, the Mf model amplitude constituent forthe v component at both stations has the major disagreement. The Msm disagrees alsofor v at station 3. The rest of the constituents compares well within this factor of threethroughout the water column. At station 1 for the u component, the major disagreementis for the Mf constituent at 3 of the 6 layers. Agreement within this factor of threethroughout the water column is found for the Msf and M.sm at station 1 for the ucomponent, for the Msm at station 1 for the v component, and for the Mm at station3 for the v component.In summary, the fluctuating residual of the model has fortnightly and monthly constituents with amplitudes of about the same order of magnitude as the observations. Thelarge monthly signal is produced by the strong gravity currents involved in the deep waterrenewal process. This strong inflow, by continuity, is reflected in all the water column.Thus renewal has a large impact on the general circulation throughout the water column.It is worthwhile to note that, in general, the along-strait velocities agree better thanthe cross-strait velocities. As in Chapter 3, if horizontal diffusion were reduced, themodel amplitudes would probably increase.Chapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 20445 SummaryIn this chapter, I have studied the residual circulation related to the deep water renewalof the central Strait of Georgia with a 3-D eight-layer model. In contrast to the studydone in Chapter 3, the model was run with a seasonally varying density forcing at theopen boundary and with explicit vertical diffusion of density.Deep water renewal into the Strait of Georgia was successfully modeled for the firsttime. This modeling study supports the recent interpretation of the mechanism of therenewal of waters of the Strait of Georgia given by LeBlond et al. (1991) for the summermonths. The intrusion of dense water from Juan de Fuca Strait is controlled by thetides. The intrusion of these water from Juan de Fuca Strait is an ongoing process bothin space and time. Spatially, the mixing of waters leading to deep water renewal beginsat the eastern part of Juan de Fuca Strait and Haro Strait and ends at the secondary sill,located between Boundary Passage and the central Strait of Georgia: the largest mixingoccurs at Boundary Passage. Temporally, mixing and blocking always happen, however,a maximum of stratification occurs at extreme neap tides when the tidal mixing due tothe tides is at a minimum and the denser waters from the Pacific are able to traverse thesill at Boundary Passage and the secondary sill, and these waters remain dense enough togenerate a gravity current of dense water moving down into the deep basin in the centralStrait of Georgia.Even though the model run of this chapter is basically a process study, because thedensity forcing at the open boundary is artificially raised, the results are encouragingbecause the comparison with the observations in the central Strait of Georgia are ingeneral, better than those obtained with the reference run of Chapter 3. In particular,the residual velocities in the model are larger and closer to the observations.The mean and fluctuating residuals from the deep water renewal run are characterizedChapter 4. Deep Water Renewal in a multi-layer model of the Strait of Georgia 205by stronger coastal jets and eddies than those of the reference run of the previous chapter;more energy was put into the system. The maximum speeds for the total residual rangesbetween 12 and 20 cm s1 in the different layers. The oscillations for the fluctuatingresidual are dominated by periodicities of the fortnightly, monthly and semiannual bands.The fortnightly band is dominated by the Msf tidal constituent and the monthly bandby the Mm tidal constituent. The Mm constituent is large and consistently comparableto the observations. This later fact is a consequence of the deep water renewal eventswhich occur once a month.Chapter 5ConclusionsI studied the residual currents in the central Strait of Georgia by means of numericalmodels. Residual currents were defined in this study as that part of the flow left afterlow-pass filtering the currents with a cutoff period of about two days. The residual currentwas divided into a mean (time-averaged) and a fluctuating (time-dependent) component,which gave the variations about the mean. Both observations and the model results arepresented as these two components.The main characteristics of the observed residual currents were:o The mean residual showed a cyclonic eddy which persisted through most of thewater column.• The fluctuating residual also showed eddy activity throughout the water column.The eddies alternated between cyclonic and anticyclonic rotation in time and inthe vertical.• The fluctuating residual accounted for approximately half of the total kinetic energyof the fluctuations (the other half is associated with diurnal and semidiurnal tidalcurrents); about one-third of this low frequency kinetic energy was contained inthe low frequency tidal constituents (at the fortnightly, monthly and semiannualbands).• The residual currents had scales of correlation of about 100 m in the vertical andabout 5 to 10 km in the horizontal.206Chapter 5. Conclusions 207Two numerical models were used in this thesis to study these currents.The first was a 2-D vertically-integrated model and Chapter 2 dealt with this partof the study. The model was forced only by tides and as a result the residuals weregenerated only by interaction of the tides and topography. The main results from thismodel were:• The mean residual formed a cyclonic eddy at the middle of the Strait. However,this eddy was significantly smaller in velocity than observed.• The fluctuating residual had the same spatial structure as the mean and alternatesbetween cyclonic and anticyclonic rotation. As in the mean, the amplitudes of thevelocities were smaller than observed.• Most of the time variability was confined to the fortnightly Mf constituent whichoriginates by nonlinear interaction of the diurnal components K1 and 0.• Budget analysis of the momentum and vorticity equations showed that the dynamics of the residual were nonlinear. In some areas the momentum balance involvesa simple geostrophic balance and in other areas all terms in the equations wereimportant.In the 2-D model, the residual energy levels were smaller by an order of magnitudeor more than the observed residuals; a depth-averaged model is not sufficient tounderstand and reproduce the residual currents of the Strait of Georgia.The second model was a 3-D baroclinic model. The model was forced by the tide,wind and runoff and the residuals were generated by interaction among all these forcingagents. Two different versions of this 3-D model were run. In the first version, the densityof the water at the open boundaries was held constant, and the residual was studied asdescribed in Chapter 3. The main results of this version of the model were:Chapter 5. Conclusions 208• The mean model residual was characterized by coastal jets and eddies of the sizeof the domain and alternating in direction with depth.• The fluctuating model residual had the same spatial structure as the mean andalternates between cyclonic and anticyclonic rotation.• Wind forcing affected the mean and fluctuating residual down through a large partof the water column. The inclusion of the wind forcing at the sea surface causedthe fluctuating residual to be more similar to the real residual. Without wind,the time variations were very smoothed. Thus winds are essential for propersimulation of the residual.• The time variability of the fluctuating residual was spread through a broad energyband. About 50% of the low frequency variance in the model was contained infrequencies which were of tidal origin. The largest tidal constituents were theMsf, Mm and Ssa components.• Budget analysis of the momentum, vorticity and energy equations showed, again,that the dynamics of the residual were nonlinear. The momentum balance showedthat the residual, to first order, was close to a geostrophic balance. The smallimbalance between the pressure gradient and the Coriolis terms was mainly due tothe advective terms.The energy levels of the residual in the 3-D model were now within the same order ofmagnitude as observed. Thus baroclinicity is an essential part of the dynamicsof the residual in the Strait of Georgia.In the second version of the 3-D model, the density of the waters at the open boundarywere raised and were given a seasonal variability and diffusion was included in the densityequation of the model. These changes allowed clear simulation of deep water renewal inChapter 5. Conclusions 209the Strait of Georgia for the first time. Renewal and the resulting residual currents weredescribed in Chapter 4. The main results of this version of the model were:• Deep water renewal in the Strait of Georgia occurred, during summer, once a month.The phase and amplitude of renewal events are comparable to those observed inthe cyclesonde records from the central Strait of Georgia.• Blocking of dense Pacific waters occurred throughout Juan de Fuca Strait all theway up to an intermediate sill between Boundary Passage and the central Strait ofGeorgia. However, most of the blocking happened at the sill of Boundary Passagewhere intense mixing occurred during spring tides.• A deep water renewal event started at extreme neap tides, the period of minimummixing, when a maximum of stratification was reached at Boundary Passage andthe denser waters of Juan de Fuca Strait penetrated past the sill at BoundaryPassage and reached the bottom of the deep basin at the central Strait of Georgiaas a gravity current.The deep water renewal process introduced energy into the Strait of Georgiaand the amplitudes of the residuals in general were larger particularly indeeper levels. The velocities of the model, with this version, were now closer to theobservations. The amplitude of the low-frequency tidal constituents were closer to thoseobtained from the observations, particularly for the monthly Mm constituent.The sources of energy for the low frequencies were the tides, winds, runoff and deepwater renewal which, by nonlinear interactions, spread energy over a broad band offrequencies. The model captured energy at some frequencies, basically at the frequenciesthat arose from direct interactions of two forcing phenomena, for example, the M2 andN2 semidiurnal tidal constituents fed energy to the Mm constituent. While the lowChapter 5. Conclusions 210frequency kinetic energy of the model was closer to the observations at some locations, itwas still smaller at others. Figure 5.1 shows the low frequency kinetic energy for stations1 to 4 for both model runs (the 3-D reference and DWR runs) and for the observations inthe central Strait of Georgia. By showing the two versions of the model, I want to stressthe fact that the second version of the model, namely the DWR run was in fact a moreenergetic simulation than the reference run. Note also how close the energy levels wereto those observed at station 4 throughout the water column: the same was not true atthe other stations, especially station 3. (The total kinetic energy for the same locationsfrom the 2-D model is only 0.02 erg/gr!)The results of both 2-D and 3-D models showed the important role that nonlinearinteractions plays in the dynamics of the Juan de Fuca Strait/Strait of Georgia system.The 3-D model results were better, especially when density was included as a time dependent forcing and explicit diffusion was added to the density equation. The energy levelsin the central Strait of Georgia were comparable to the observations and, in general, thetime and spatial variability agreed with observations,Finally, for the first time, deep water renewal has been clearly simulated.Future workThe following discussion will highlight some recommended areas in which the modelperformance could likely be improved. The GF8 model proved to be a good tool for thestudy not only of the tides, but of the residual currents in the Juan de Fuca Strait andStrait of Georgia system. However, the model is still underestimating the total kineticenergy of the residual currents and small scale eddies are not yet well resolved.Better horizontal resolution should improve the simulations. Of course for a numericalmodel, the higher the resolution the better the solution is likely to be, not only becauseof smaller errors in the numerical schemes, but also because of the improved bathymetryChapter 5. Conclusions0.— 100.E-c0a,200,300.0.100,E-c0a,rD 200.300.*** Observationsxxx Models REF Run000 Model& DWR Run0. 10. 20. 30. 40. 0. 10. 20. 30.Total Kinetic Energy (ergs/gr)211Figure 5.1: Low frequency kinetic energy as a function of depth for stations 1 to 4 atthe central Strait of Georgia from the observations and from the two versions of the 3-Dmodel. REF and DWR stand for the reference and deep water renewal runs studied inChapters 3 and 4, respectively.40.Chapter 5. Conclusions 212that results. Accurate representation of the bathymetry is important because topographicsteering is a significant part of the dynamics. With higher resolution, eddies of smallerscales might also be resolved. Note also that with a higher resolution, the horizontaleddy viscosities could be reduced and, therefore less energy would be artificially takenout of the system and may give more energetic and realistic results.The number of layers may also be increased. It was found that the model residualcurrents behaved as a four-layer system. The use of eight layers may be barely adequate.Also, higher vertical resolution will reduce numerical diffusion.Given the importance of the nonlinearities to the dynamics of the system, it wouldbe desirable to implement advective schemes of higher order, in both the momentumand density equations. This change would result in a more numerically stable modeland would allow for the reduction of high horizontal eddy viscosity coefficients whichare currently needed to maintain numerical stability. Thus, better results in terms ofamplitude could be expected.The parameterization of eddy viscosities is still a very difficult aspect, which is notparticular to this model only. The horizontal eddy viscosity was held constant; a sheardependent horizontal coefficient may be required for better simulation of the currents,especially at large tides. The current vertical mixing is somewhat ad hoc: improvementsare possible here as well.Wind forcing was found to play an important role as a forcing agent of the residualcurrents in the central Strait of Georgia. At this moment, the wind field is taken froma few coastal stations and then interpolated to the model domain. Obviously, a betterresolution of this forcing is recommended. Perhaps a wind model coupled to GF8 wouldbe better.The baroclinic forcing at the open boundaries is a problem. At the moment (andperhaps ever), density observations are not sufficient to prescribe this forcing at the JuanChapter 5. Conclusions 213de Fuca Strait entrance. One solution for this boundary condition problem is to forcethe model with a continental shelf model.Bibliography[1] Backhaus, J. 0., 1983: A semi-implicit scheme for the shallow water equationsfor application to shelf sea modelling. Cont. 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LeBlond, 1986: A wind-forced Ekman spiral asa good statistical fit to low-frequency currents in a coastal strait. Science, 233,470-472.[42] Stacey, M. W., S. Pond and P. H. LeBlond, 1988: An objective analysis of thelow-frequency currents in the Strait of Georgia. Atmosphere-Ocean, 26, 1-15.[43] Stacey, M. W., S. Pond and P. H. LeBlond, 1991: Flow dynamics in the Straitof Georgia, British Columbia. Atmosphere-Ocean, 29, 1-13.[44] Stacey, M. W., S. Pond, P. H. LeBlond, H. J. Freeland and D, M. Farmer, 1987:An analysis of the low-frequency current fluctuations in the Strait of Georgia,from June 1984 until January 1985. J. Phys. Oceanogr., 17, 326-342.[45] Stronach, J. A., 1991: The development of GF8: A three-dimensional numericalmodel of the Straits of Georgia and Juan de Fuca. Seaconsult Marine ResearchLtd., Vancouver, B.C., 212 pp.[46] Stronach, J., P. Crean and T. Murty, 1988: Mathematical modelling of the FraserRiver plume. Can. J. Water Poll. Res., 23, 179-212.[47] Tee, K. T., 1976: Tide induced residual current, a 2-D nonlinear numerical tidalmodel. J. Mar. Res., 34, 603-628.[48] Waldichuck, M. 1957: Physical oceanography of the Strait of Georgia, BritishColumbia. J. Fish. Res. Board Can., 14, 321-486.[49] Wright, D. G. and J. W. Loder, 1985: A depth-dependent study of the topographic rectification of tidal currents. Geophys. Astrophys. Fluid Dynamics, 31,169-220.[50] Yao, T., S. Pond and L. A. Mysak, 1982: Low-frequency subsurface current anddensity fluctuations in the Strait of Georgia. Atmosphere-Ocean, 20, 340-356.[51] Yao, T., S. Pond and L. A. Mysak, 1985: Profiles of low-frequency subsurfacecurrent fluctuations in the Strait of Georgia during 1981 and 1982, J. Geophys.Res., 90, 7189-7198.[52] Zimmerman, J. T. F., 1980: Vorticity transfer by tidal currents over an irregulartopography. J. Mar. Res., 38, 601-630.Appendix AEquations of MotionThe purpose of this appendix is to present the Reynolds or turbulent equations of motionand give a brief outline of where they come from. From these equations the modelequations of this thesis presented in the subsequent appendices are derived. A completederivation of the Reynolds or turbulent equations can be found in such text books asProudman (1953), LeBlond and Mysak (1978), Pedlosky (1979), Gill (1982), Pond andPickard (1983) and Kundu (199O)The equations of motion are the expression of Newton’s second law of motion for afluid continuum in a rotating frame;dV -p--+p2c xV=—VP+pV4+pVq!T+F, (Ad)where = (+V), i is the velocity vector with u, v, w components to the east, northand upward directions respectively, ! is the earth’s angular velocity (= 7.29 lO5rad/s),P is the pressure, 4 is the earth’s gravitational potential [ = Vq = (0,0, —g); g = 9.8m s2], cT is the tidal potential, and P is in principle any force, but in fact is thefrictional force of the fluid. [The tide producing forces are taken as known and omittedhere; furthermore, for the Fuca-Georgia system, Crean et al, (1988a) showed that thetide-generating potential contributes little to the overall energy balance and thereforeis also not included in the numerical models either. This is an approximation in themodels.]218Appendix A. Equations of Motion 219For a Newtonian fluid (such as the ocean), F is expressed as-.1 -.,v2v+ pV(V V), (A.2)where p is the molecular viscosity (Pedlosky, 1979). Assuming that the flow is incompressible, that is,(A.3)P is simplified in (A.2) and substituting in (A.1) the Navier-Stokes equations for anincompressible fluid are obtained, i.e.,ai — — 1+ V . VV + 2l x V = --VP + Vq + vV2V (A.4)where v is the kinematic eddy viscosity. These equations describe the motions of viscousfluids for all scales, from very large scales, such as ocean currents, to very small scale,such as capillary waves, with an enormously broad spectrum of turbulent motions thattransfers energy from the larger to the smaller scales where the kinetic energy is ultimatelydissipated by molecular viscosity. The transfer of energy comes from the non-linear termsof the equations where all scales interact.Obviously the description in detail of all scales of motion is intractable. The commonway to overcome the problem follows the Reynolds’ procedure that consists in splittingthe velocity, pressure and density fields in some mean [mean in some sense to be defineddepending on the problem] plus fluctuating or turbulent components, i.e., i = +P = i + F’ and p = + p’; then substitution into (A.4) and taking the average theequations transforms to8* -- -s;- — — ---+VVV+2QxV=-VP+VqS+vVV-V’ ’ (A.5)Appendix A. Equations of Motion 220where = 1/p. These are the Reynolds equations and the last term of the RHS is theReynolds stress and represents the effect of the velocity fluctuations or turbulence on themean flow. For example, in the x-direction they look asOu’u’ Ou’v’ 8u’w’a öz’ (A.6)where V = 0 has been used.The next step is to represent the Reynolds stresses in terms of the mean flow; inanalogy with the molecular viscosity, they are supposed to be related to the mean flowgradients by some turbulent eddy viscosity coefficients as-= - = Ah; -= (A7)The coefficients Ah and A,, are called the horizontal and vertical turbulent or eddy viscosity coefficients respectively.Omitting the overbar the turbulent momentum equations for the mean can be expressed as:9u äuu ôuv Ouw 1ÔP U Uu U U Uu— + + + — fv + —— — —(Ah—) — —(Ah—) — —(A,,—) = 0,Ut Ux Uy Oz p Ux Ux Ux Uy Uy Oz Oz(A.8)Uv Uvu Ovv Uvw 1UF U Uv U Uv U Uvat ax ay az p ay ax ax cy oy az azthe mass conservation expressed as:(A.9)the density conservation expressed as:Appendix A. Equations of Motion 221+ V• (p9) =0, (A.10)and the hydrostatic equation expressed as:!=pg. (A.11)These are the governing equations on which the models used in this thesis are based on.Boundary conditionsIt is necessary to know not only the governing equations, but also the appropriateconditions to apply at boundaries.The condition of no normal flow across solid boundaries is(A.12)where 1 is the normal velocity to the wall.The kinematic surface and seabed boundary conditions to be satisfied are:th7 8 0w = + (u- + z =(AJ3)8 8w = —(u— + v—)h; z = —H.Ox OyThe surface and bottom stress boundary conditions are given by:A-==(A.14)= = Cd?3\/u2 + v2,where the subindices s and b stand for surface and bottom, i = (U, V) is the horizontalwind velocity vector, = (u, v) is the horizontal velocity vector, Cd is a drag coefficientAppendix A. Equations of Motion 222(different for the air/water interface and for the sea bottom), and Pa and Pw are the airand water densities respectively.Appendix BDerivation of the 2D model equationsAn outline of the derivation of the 2-D model equations is presented here. Heaps (1978)and LeBlond and Mysak (1978) can be consulted for detailed study.The equations are obtained by vertical integration of the equations (A.8) and (A.9)from the sea surface z = i(x, y, t) to the sea bottom z = —H(x, y) for a homogeneousocean (p = constant).B.1 Continuity equationThe vertical integration of the continuity equation (A.9) is,‘ ôu ôv ow “ Ou Ov+ + = J[— + + w(x, y, i, t) — w(x, y, —H, t) = 0.Using Leibnitz’s rule, the above expression can be written asth1_____1-Hth1 O(—H)+--— — [v—j, + [v 1—H+w(x, y, i, ) — w(x, y, —H, t) = 0. (B.1)where the transports U and V are defined asf’lU = I udz=ud,J—HV= I vdz=vd,i—H223Appendix B. Derivation of the 2D model equations 224assuming that the velocities are depth independent and d= ‘ + h. With the use of theboundary conditions (A.13), (B.1) reduces to the 2D continuity equation, i.e.,o,j OU B2B.2 Momentum equationsFor simplicity only the x-component equation of the momentum equation (A.8) is derivedhere.B.2.1 Pressure termRecalling that pressure obeys the hydrostatic law (A.11), i.e.,op= -pg,then the pressure in any point z isP(z) = F(q) + pg( — z). (B.3)Neglecting the atmospheric pressure (F() = 0), the pressure gradient reduces toOP(z) th7, —pg-;;--, ( ,)ax axwhich is independent of z and its vertical integration is readily evaluated, giving thepressure force of equations (A.8) asJ_H’Ox= gd. (B.5)B2.2 Relative accelerationThese terms, integrated from to —H are17 Ou 9uu Ouv OuwJ-HOtOxOyOZAppendix B. Derivation of the 2D model equations 225Again, using Leibnitz’s rule9Uö UU a1_____+—(-—) — (uu),— + (uu)_H0 UV th7+—(——) — (uv),— + (uv)_H+(uw) — (uw)=_H.Grouping again the terms evaluated at z=-H0(h) 0(h)— (uv)ã__ — (urn) = 0,and at the sea surface—u — (uu)— — (uv)— + (urn) = 0,because of the boundary conditions. So, the relative accelerations terms results inau auu auv+ -(--) + -(---). (B.6)B2.3 Coriolis accelerationf(_fv)dz= —fV. (B.7)B.2.,4 Vertical diffusionThe vertical diffusion term, a a Ouf_H——(A--) = —(A—), + (A)_H = Tsar + Tbx (B.8)where r3 and rb1, are the surface (wind) and bottom stresses specified in (A.14) respectively, They enter in the dynamics of the system as body forces affecting the entire watercolumn.Appendix B. Derivation of the 2D model equations 226B.2.5 Horizontal diffusionFinally, the integral of the horizontal diffusion terms,j —[-(Ah) + -(Ah)]dZ = f [(Fxx) + _(Fry)]dz =—_ j’1 Fdz — — F(i) —-Fdz — Fry(—H) —is usually simplified to- v(—+= -vVU (B.9)where v is again an eddy viscosity coefficient appropriate to the two-dimensional depth-averaged situation (Robinson, 1983). [These eddy viscosity coefficients are properties ofthe flow, not of the fluid, so their estimates vary widely. Also, in some cases they are anadjustable parameter used to fit observations (Pond and Pickard, 1983). In numericalmodels the values sometimes are adjusted to maintain numerical stability.]Grouping the terms (B.5), (B.6), (B.7), (B.8) and (B.9), the depth-averaged equationof motion for the x-direction is obtained, i.e.,+-(-) + -() — fV + gd — r8x + TbX — vV2U = 0, (B.10)and similarly for the y-direction,(B.11)The velocities are obtained from the transports simply by doing u = U/d and v = V/d.In this version of the model, wind stress is not included and the stresses are givensimply as = 0 and =Appendix CDerivation of the 3D model equationsA summary of the 3-D model equations is presented here. Backhaus (1983, 1985) andStronach (1991) can be consulted for a more detailed derivation. As in the 2D model,the equations (A.8), (A.9) and (A.1O) are vertically integrated, but now over a discretenumber of layers instead of over the entire water column. As in the vertical integration,Liebnitz’s rule is used to bring the gradients in front of the integrals, we will considerthree different cases of layers, top, middle and bottom layers, i.e.,h= r(x,y,t)—z; j=1z,_i—z; j=2,...,J—1h= z_i+H(x,y); j=J(x,y)where J is the total number of layers and z are the discrete levels of the model to bechosen at will, and H(x, y) is the bottom depth. It is assumed that there is no verticalvariation of density and horizontal velocity within each layer. The vertical velocity varieslinearly within each layer in order to satisfy the continuity equation (A9).In what follows, the integrals will be represented as.)dzwhere zu is the upper level constant except in the first layer, where it is equal to ‘i, andzd is the lower level constant, except in the bottom layer, where it is equal to —H, anddz is the discrete interval h3.227Appendix C. Derivation of the 3D model equations 228C.1 Continuity equationThe integral of the continuity equation is,zu öu ôv t9wI (—+—+—)dz=Jzd ox Oj Oz0 zu zu Ozu Ozu Ozd Ozd— j udz + vdz + w(zu) — w(zd) — (u-h-— + + (u-— +The derivatives of zu and zd will vanish except for the top and bottom layers, respectively. Thus we have for theTop layer:a a Ozu Ozu—(uh) + (vh) = —w(zu) + w(zd) + (uã + v--).Bottom layer:0 Ozd Ozd—(uh) + -(vh) = —w(zu) + w(zd) + (u-h— + vã_)zd.Middle layers:-(uh) + -(vh) = -w(zu) + w(zd).And, upon recalling the boundary conditions (A.13)w(zd)—topa a—(uh) + —(vh) = w(zd) — w(zu); middle (Cd)—w(zu); bottomC.2 Momentum equationNow we integrate the x-component of the momentum equation (A.8).Appendix C. Derivation of the 3D model equations 229C02.1. Pressure termAs the model includes density variations, these variations will give rise to baroclinicmotions through baroclinic pressure gradients. We will split explicitly the contributionsto the pressure force into two components, namely, barotropic and baroclinic. [Thebarotropic part is independent of z and gives rise to the surface pressure gradient term.]The term that we want to evaluate is,zu 1 oP—--—dz,Jzd UXand requires the pressure to be known. The Boussinesq approximation is used to evaluatethe baroclinic part of the above expression. The density is split into two components asfollows: p = p” + p’(x, y, z, 1), i.e., a constant, p, and its variations, p’. The hydrostaticpressure (A.11) at depth zd isP(zd)=pgdz=p*gz + j’ p’gdz= (7, — zd)p*g +p(x,y,zd,t)after neglecting the atmospheric pressure. The horizontal pressure gradient in the xdirection is thenop o Op=p*g_+_,and its vertical integrationf =(zu_zd)p*gL+f. (C.2)Again using Leibnitz’s rule, the baroclinic part readsP7 Op 0 in Ozd 07, 8 ,n OzdI —=-— pdz+p(zd)——p(7,)--=--i pdz+p(zd)---Jzd OX OX Jzd OX OX oX Jzd OXagain neglecting the atmospheric pressure. Once the density variations are obtained, theabove expression is easily evaluated (remember that p’ is depth independent within eachlayer). The extra term contributes only at the bottom layer, i.e.,= —-.Appendix C. Derivation of the 3D model equations 230In the numerical model, the pressure gradient is evaluated as follows: as p’ is constantin each layer, the baroclinic pressure is linear within the layer, i.e., p = a — bz, so0 fZU ,zu a dJ pdz = iZd (a — bz)dz = — zd)[a 2 b] }i.e., for any layera , aOx Jzd pdz = —{(zu — Zd)Pjmid_lcjyerand adding the term —p(—H), the result is0 j pdz = (zu + H) (C.3)Ox -H mid-layerfor the bottom layer. Both can be expressed for all layers asa ap-—pdz=h--- (C.4)OX Jzd OX mid—layerSo, the pressure gradient terms reads asJ --dz = h3g+ (C.5)zd pOx Ox pOxC.2.2 Relative accelerationThe relative acceleration terms are1zu Ou Ouu Ouv OuwI [—+—+—-+—--—]dz=Jzd Ot Ox Oy Oz0 g-zu 0 zu jzu— I udz +— I uudz +— I uvdz + (uw) — (uw)dOt Jzd Ox Jzd JzdOzd Ozd Ozd Ozu Ozu Ozu+Ud[ + u-b--— + — u[—-- — u-a--_ — v-ã-_]ztL.These terms, again for the different layers, results in—uw(zd); tophu + —h,uu + -huv + uw(zu) — uw(zd); middleuw(zu); bottomAppendix 0. Derivation of the 3D model equations 231with h3 = zu — zd. For the top layer, the first 3 terms can be written as(9 0 0 0 (9 (9hi[-u + + —uv] + u[-hi + u—h1+ v—hi]a a a= hi[u + + —uv] + uw(zu).a ia. a= + + —uvhi)j + uw(zu),where hi =—= z1. So, defining îi = h, j = 2, , J, the final expressions forthe relative terms area ia 10+ z——h3+ + i(uw) (C6)whereuw(zu) — uw(zd); topL(uw) = uw(zu) — uw(zd); middleuw(zu); bottomC02.3 Coriolis accelerationThe Coriolis term is simplyfvdz = fv(zu— zd) = fvh (C.7)C.2.4 Vertical diffusionThe vertical diffusion term is easily integrated as— =— [(A)1d = + Tzd = (C.8)Appendix C. Derivation of the 3D model equations 232C2.5 Horizontal diffusionFinally, the integral of the horizontal diffusion terms,j:u—[_(Ah) + -(Ah)]dz= j [(Fzr) + (Fry)]dz =Fdz + F(zd)-_d — Fxa(zu)j_ —— j Fxdz +—is usually simplified toô2u ã2u 2— + = —h3vVu (C.9)where i is again an eddy viscosity coefficient appropriate to the situation.Grouping (C.5), (C.4), (C.6), (C.7), (C.8) and (C.9) and dividing by h the xmomentum equation readsa+ x—[--uuh + —uvh,] — fv ++ . — t(tau)/h3— i’Vu + h;’L(uw) = 0. (C.1O)f Xm:d—layerC.3 Density equationThe last equation is that of density. Integrating (A.10) between zu and zd as beforerzu -J [-i- + V (p’V)]dz = 0,zd vtand using Leibnitz’s ruleU zu zup’dz + j— j up’dz + .L vp’dz + (p’tv)zu — (p’W)zd +Ozd Ozd 9zd , Ozu Uzu Uzu—0.Evaluating the integrals the above expression becomesa a a+ —h3up’ + —hvp + (P’W)zu — (p’W)zd +,Uzd Uzd Uzd Uzu Uzu Uzup + + Vp’_ã — p’ — tLp’ — vp’ä = 0Appendix C. Derivation of the 3D model equations 233Expanding the derivatives and with the constriction of no density flux through theair/sea and sea/bottom interfaces, the layer density equations reduces to—p’w(zd); top+ + i—vp’] + p’w(zu) — p’w(zd); middlep’w(zu); bottom(Cdl)Note that all the above layer equations can be written in one single equation andsatisfy the boundary conditions if w is set to zero at the bottom and to at the surfaceand p’w is also set to zero at the surface. The model equations reduce to the followingversions for:density (C.ll)+ —up’ + —Vp’ + —[p’(w)] = 0, (C.12)continuity (C.1)—(uh) + —(vh) = —i(w) (C.13)hydrostatics (A.11)= —p’g, (C.14)and momentum (C.l0), in the x-direction,Ou 18 8 lOp— + ---[—uuh + —uvh,] — fv + g— + —Ot Ox Oy Ox fi Ox mid-layer—I(T)/h — iVu + LSuw)/1i = 0 (C.15)and, similar, for the y-direction,Ov 18 0 th7 lOp— + -x---[—uvhj + —vvh] + fu + g— + ——9t Ox P 8Ymid-layer_(ry)/hj — EVv + (vw)/h3= 0. (C.16)Appendix DBottom Friction Stress ReductionThe bottom friction stress terms are reduced or simplified here in terms of the lowand high frequency flow. Hunter (1975) and Heaps (1978) can be consulted for similaranalysis.The bottom stresses are defined as A.14= = Cd ws/u2 + V2, T = Tby = Cd vv’u2 + v2. (D.1)Splitting the velocity into low and high frequency motions as u = uo + u1 and v =v0 + v1, substituting in D.1 for the x-component= Cd (u0 +u1)(ug + 2u0u1 + u?) + (v + 2v0v1 + v?)=Cd (uo+ui)/ug+v+u?+v?+2(uoui+vovi).Assuming that ii <<i7 (which is correct in the model, see Figure 3.31) thenT = Cd (uo + ui)iju? + v? + 2(uou1 + vovi)= Cd (UO + ui)IiI1 + (D.2)Now— 2(uoui + vovi)2 <<1,lviwhich allows ‘FT 1 + e/2 then (D.2) can be written— uoui + voviTTCd(Uo+Ul)IVI(1 2 )V1234Appendix D. Bottom Friction Stress Reduction 235(uo+ui) IiI2 +(uo+ui) (uou1+v vi)=Cd -.IV1u0 + u1 RiI2 + uu1 + uovov1 + uo4 +u1v0=Cd -.I vi Iand neglecting terms involving products of the residuals (3rd and 4th) compared withthose involving products of the high frequencies (1st, 2nd and 5th) the final bottom stressis:2u+v? u1vuo+——vo+uiIviIJIV1I lviiand similarly for y_______u?+2v? u1v -TbCd[ vo+—--—uo+viIviI].lvii lviiAppendix EDiscrete Vorticity EquationIn this appendix I show that the vorticity equation as derived from the discretized momentum model equations is consistent with the discretized version of the continuousvorticity equation when the advective and stretching/shrinking terms are grouped in fluxform as(E.1)The vorticity equation (3.20) using (E.1) then reads= —V• () — fVH €+ t’VC + {I—[z(ry)] — (E.2)The discretization is made on the Arakawa C grid (Figure E.1). The difference andaverage operators appropriate to the grid are denoted as== [(. .— (. . .)k]/zSi6(.. .)a = [(...)ik — (...)+1k]/Zl. (E.3)Using the difference and average operators just defined, equation (E.2) is approximated as—(k—1C’k_l)—8y(_1kC1k)—f(7,’_1+6y371k) + £‘(66 + 6y5yXic—(E4)236Appendix E. Discrete Vorticity Equation 237Vi—i ,IcU S UI .k—1Vi.kFigure E.1: Horizontal stencil for an Arakawa C-grid. u and v are the horizontal velocitycomponents and S represents either surface elevation, depth or density.andcik = — (E5)Now, the discretized momentum equations [(3.7) and(3.8)] are= —6Qizk_l)2—6Y(—1k—1k) + + y(_1kv_1k)1 + f1k— —(gpo’7ik + Pik) +(6Suk_1 + 66U_1k) + =-(rk), (E.6)P ikand5tVk= —6y(_1k)2— k_1vik_1) + [6r(]k_1Uk_l) + 6y(i_1kv_lk)1 — f’7’’_—6(gpo77ik + Pik) + +6SV_1k) +4-z(rk), (E.7)P ikwhere the continuity equation-yLWik =—5X(hlk_lutk_1) —5y(h_lkvj_1k) (E.8)Appendix E. Discrete Vorticity Equation 238has been used.Taking the curl of these equations, the horizontal and vertical diffusion terms as wellas the Coriolis terms have direct or clear continuous counterparts, (see E.4), however, itis not straightforward for the advective terms, where we have to shown that+‘5,{y(’_1k)2+ k—1k—1) — —[8x(ik_1uk 1) + v(’_1kv_1k)i}_6{5(r)2 +6v(—1k—1k) — += 6x(ik..1k_1) + Y(.1k_1k) (E.9)in order for both expressions to be equal after discretization is made. The LHS of E.9comes from the curl of the discretized momentum equations, while the RHS comes fromthe discretized vorticity equation.For the interior layers, the previous expression simplifies a great deal as h is constant foreach layer. Expanding the averages in the first term of the LHS of (E.9) gives (h=const.)+ Vk)2 +6[(uk—i + U+1k_1)(Vk_1 + VIk)]VIk—--—[(uk_1 + Uj+lk_1) + S(V_1k + Vjc)]},now expanding the finite difference operator inside the braces+16X{(vi_1k+vik)2 +tL+1k_1)(Vk_1+Vk)]Vjk—-1[(uk + U+1k) + (V_1k + Vk) — (ulk_1 + U+1k_1) — (vk + Vj+lk)]}which can be grouped as+ Vk)(V1_1k — VIk) + (vk + Vj+lk)(Vjk — Vj+lk)+(uk + tt+1k)(Vk+1 — Vk) + (uk_1 + U+1k_1)(Vk — Vk_1)}.Appendix E. Discrete Vorticity Equation 239Now applying the 6, operator, the previous expression reads14zl2{(v_1k+1 + vk+1)(v_1k+1 — Vk+1) + (vlk+1 + Vj+1k+1)(Vk+1 — Vi+lk+1)—(vI_lk + Vk)(V_1k — Vk) — (vk + Vj+lk)(Vjk — Vj+lk)+(uk+1 + Uj+1k+1)(Vk+2 — Vjk+1) + (ulk + Uj+lk)(Vjk+1 — Vk)—(ulk + U+1k)(Vk+1 — Vk) — (uk_1 + Uj1k_1)(VjJg — Vk_1)} =14Ll2 {(vI_lk+1 + Vjk+1)(V_1k+1 — Vk+1) + (vIk+1 + Vj+1k+1)(Vk+1 — V1+lk+1)—(v_1k + VIk)(Vj_lk — VIk) — (vk + V+1k)(Vjk — V1+lk)+ (uk+1 + u1+1k+1)(vk+2 — vk+1) — (uk_1 + ?Li+1k_1)(Vik — Vk_1)}. (EJO)The terms involving only v velocities can be rearranged asV_1k+1 — V+1k+1 — V,_lk + V+lk =(V_1k + V_1k+1)(V_1k÷1 — V_lj4j(V+ k + V+1k+1)(V+1k — V+1k+1)and substituting in (E.1O)14zl2{(V_1k + V1_lk+1)(Vj_lk+1 — V_1k) + (V+1k + V+1k+1)(V+1k — V+1k+1)+ (uk+1 + u1÷1k+1)(Vk+2 — Vk+1) + (uk_1 + U+1k_1)(Vjk_1 — Vk)} (E.11)Repeating the same steps for the second terms of the LHS of (E.9) we get14/l2 {(u+1k+1 + Uk+1)(U+1k+1 — Uk+1) + (uk_1 + U+1k_1)(Uk_1 — U+1k.1)+ (v+1k + V+1k+1)(u+1k — uI+2k) + (V1_lk + Vj_1k+1)(Uk — U_1k)}. (E.12)Now adding (E.11) and (E.12), we get14zl2 {(uk+1 + Uj+lk+1)(Vjk+2 — Vjk÷1 + Uj+lk+1 — Uk+1)Appendix E. Discrete Vorticity Equation 240—(uIk_1 + Uj+lk_1)(Vjk — Vjk_1 + Uj+lk_1 — Uk_1)+(t’I_lk + V_1k+1)(V_1k+1 — Vj_lk + Uk— U1—lk)—(V+1k + V+1k+1)(V+1k+1— Vj+lk + Uj+2k — U+1k)}.Expressing this with the operators (E.3), we finally get— k_1Cik—1 + _1ki—1k — Vj+lkCi+lk} =‘k_1jk_1) + (..1kCIk) (E. 13)which is the RHS of (E.9).The algebra is to extensive to repeat the analysis for the top and bottom layers wereextra terms appear due to the variable thickness of those layers which introduces inconsistencies between both versions. The difference, however, is very small. As an example,Figure E.2 shows the advective terms, for several time steps, as obtained from the discrete vorticity equation, in thin line, and from the curl of the discretized momentumequations, in thick line, for the top and bottom layer for a particular grid point. It canbe seen that the difference is very small.Appendix E. Discrete Vorticity Equation 2412.E-09 -0.E÷00ARBottom layerAA-2.E-092.E-090.E+00-2.E-091 600.Time stepsFigure E.2: Time series of the advective terms of the discretized vorticity equation E.4(thin lines) and that obtained from the curl of the discretized momentum equations (thicklines) for the top and bottom layers of a particular gridpoint.ATop layer0. 400. 800. 1200.Appendix FDiscrete Energy EquationIn this appendix, I show that the energy equation as derived from the discretized momentum model equations is not consistent with that derived from the discretized versionof the continuous energy equation.The velocity components of the model are staggered and the finite difference representation of the kinetic energy isKk = + Uk + V_lk + vi). (F.1)The model kinetic energy equation is formed by the sum of the products of the xmomentum equation (E.6) at the u-points (i, k) and (i, k—i) by ujk and ujk_1 respectively,and the y-momentum equation (E.7) at the v-points (i, k) and (i — 1, k) by vik and Vj_1,krespectively.The former average or sum is the main reason for the discrepancies with respect to theenergy equation obtained by discretizing the continuous energy equation. The differenceof the different terms is always very small in the model and because of the extensivealgebra involved, the expressions of that difference is outlined for those terms where thealgebra is not so cumbersome. Some plots will be presented to show the contrast ordifference between the two.The discretized version of the energy equation (3.23) is=:k—lrc, L ‘% £f—XY Z.— 1U)xLik_1Uik_11tik—1) + OyjU_lk_l Vj_lkItI_lk242Appendix F. Discrete Energy Equation 243_.x__[6x(ujk_luxYik— lhk_l) + y(v_1kvi_1khI_lk)]2KIk— ZX(w2k)—a;___________—y____________k—1_____— *6(gp*q + P)k_1— *c5y(gp*q + P)_1kPik Pik1[—r (j—a; I—a; ‘—y (—y juyvyU_1j_-TV_i,yxva;V_ i vyuyVj_j+ + (F2)Now, consider the baroclinic pressure terms. From the discrete momentum equationsI added (or averaged) the contributions from the u-points and v-points as(u1k_16p1k_1 + UikSxPik) + (v1_1k5p_1k + VikyPik) =aUik_loxPjk_1 + Vi_lkOyPj_lk ,i.e., the average of the product of the velocity times the pressure gradients. From thedirectly discretized energy equation the same terms are—x I —y £—YUk_1vxPk_1 m V_1kuyP_1k,i.e., the product of the average of velocity times the average of the pressure gradients,which in principle would not be equal to the average of the products. The actual differencein the model turned to be very small.The nonlinear terms also give a small difference. (Because I want to be sure that thecontributions of the horizontal and vertical advection terms of the momentum equationsadd to the horizontal advection of horizontal kinetic energy as I interpreted these termsfrom the continuous equations in the text.) The algebra is not included here but thedifference is very small; the largest difference is in the bottom layer. As an example, inFigure F.1 I show with a thick line, the result of the discretization of the advective termAppendix F. Discrete Energy Equation 244VHK from the discretization of the energy equation, and in thin line the sum of theadvective energy terms as constructed directly from the discrete momentum equationsfor the top and bottom layers for a grid point where the difference was found to be large.As can be appreciated, the difference is almost nil.The other terms of the equations (discretized energy vs. energy from discretizedmomentum) behave similarly, with the exception of the terms coming from the earth’srotation. In the continuous equations the Coriolis force cannot change the kinetic energybecause f1 x ii = 0 everywhere. However, in our model this is not the case as can beseen when we add the terms as we did in with the baroclinic pressure gradient, i.e.,_ry?Ljk_1fV_lk_l + UjkfV_lk — Vj_1kfU_lk_l — VjkfUk_l =+ V_1k + Vk1 + vik) + uk(v_1k + Vj_lk+1 + Vik + vik + 1)V_1k(U_1k_1 + Uj_lk + uik — 1 + uk) — v1k(uk_1 + Uk + Uj+lk_1 + U+1k)] 0.This term, in equation 3.25, is a measure of the error due to the discretization of theequations. It can be as large as the smallest terms of the energy budget, i.e., is not veryimportant but it plays some role moving energy in the model.Note that if the Coriolis term contribution is evaluated directly at the center of thegrid, which would be the case for the discretized energy equation, the Coriolis contributionto the energy change is—T—y f—Y —x —J U_1V_1 — .‘V1_jU_ —i.e., the Coriolis work is zero.Appendix F. Discrete Energy Equation2,E-06Ø.E+00-2E-06245M A A AA A AkRA A- -- ‘VI- -—SJ VV Jv -v ‘ y V -I I I0. 400. 800. 1200. 1600.Time stepsFigure F.1: Time series of the advective terms of the energy equation for the top andbottom layers. Thick lines are from the discretized energy equation F.2 and thin linesfrom the equation obtained from the discretized momentum equations.Bottom layerTop layer2,E-06 -0,Ei-00-2.E-06

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