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The influence of the earth’s rotation on the wind-driven flow in Hecate Strait, British Columbia Hannah, Charles Gordon 1992

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THE INFLUENCE OF THE EARTH’S ROTATION ON THE WIND-DRIVEN FLOWIN HECATE STRAIT, BRITISH COLUMBIAByCharles Gordon HannahB. A. Sc. (Engineering Physics) University of British Columbia, 1985A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1992© Charles Gordon Hannah, 1992Signature(s) removed to protect privacyIn 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.PhysicsThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:2Signature(s) removed to protect privacyAbstractA regional model of the depth-averaged currents in Hecate Strait, British Columbiahas been developed: the Hecate Strait Model. When driven by local winds the modelsimulates the winter transport fluctuations and captures the character of the observedcirculation patterns. The modelled currents are consistent with the historical view of thewinter circulation and contain the counter-current observed in southern Hecate Strait.The counter-current is due to topographic steering: the tendency, in a rotating fluid, forthe flow to follow the local depth contours. The model results suggest a new interpretationof the observed currents in southern Hecate Strait, which has implications for particletransport.The influence of the earth’s rotation on the water transport through Hecate Strait isinvestigated using the concept of rotation-limited-flux. The effect of rotation is to reducethe flux through the strait compared with a non-rotating strait. Numerical experimentswith the Hecate Strait Model show that the earth’s rotation reduces the steady statetransport by a factor of three. The relationship between the steady-state transport andthe Coriolis parameter is consistent with rotation-limited-flux.11Table of ContentsAbstractList of TablesList of FiguresAcknowledgement11ixxl’xxiv1 Introduction1.1 Motivation for this Work1.1.1 Fisheries1.1.2 Oil Spills1.2 Objectives1.3 Plan of the Thesis1333452 Physical Oceanography of the Queen Charlotte Islands2.1 Geography and Bathymetry . .2.2 Tides2.2.1 Residual Tidal Currents2.3 Temperature, Salinity, Density, and2.4 Winds2.5 CirculationFresh WaterRegion 88111116181920212.5.1 Hecate Strait 1983-19842.5.2 Queen Charlotte Sound 19821114 Low Frequency Flow in Sea Straits4.1 Cross-strait Geostrophic Balance4.2 Frictional Adjustment4.3 Rotation-limited-flux4.3.1 Derivation4.3.2 Wind and Pressure Driven Sea Level Responses4.3.3 Comparison with Observations in Hecate StraitHecate Strait ModelThe Depth-Averaged, Non-Linear Shallow Water Equations.5.1.1 Topographic Steering5.1.2 Geostrophic Balance5.2 Numerical Formulation5.2.1 Discrete Equations: Arakawa-Lamb, 19812.5.3 Dixon Entrance2.5.4 Outer Shelf and the Deep Ocean2.6 Summary3 Hecate Strait: Data for Model Comparison3.1 Calculation of Wind Stress3.2 Wind and Current Fluctuations3.2.1 Spatial Patterns3.2.2 Water Transport Through the Strait3.2.3 Transport and the Mode 1 Currents3.3 Drifters3.4 Summary2427293030323538434554565658596266685 The5.1737477777979iv5.2.2 Time Stepping: Leap-frog Scheme5.3 Bathymetry, Grid Size and Time Step.5.4 Energy Dissipation Mechanisms5.5 Lateral Boundary Conditions5.5.1 Side Walls5.5.2 Open Boundary Conditions5.5.3 Implementation of the Open5.6 Atmospheric Forcing Fields .5.6.1 Wind Stress5.6.2 Pressure Gradients5.7 Computer Program6 Evolution from Rest to Steady State in the6.1 Frictional Adjustment6.2 The Basic Experiment: Spin-up6.3 Selecting Friction Parameters6.4 Adjustment Time-Scales in Hecate Strait6.4.1 Basin Resonance6.5 Circulation Patterns: A Preview6.6 Force Balance7 Rotational Limitations on the Water Transport7.1 Winter 19847.2 Friction, Coriolis Parameter, and Transport7.3 Frequency Response7.4 SummaryBoundary ConditionsHecate Strait ModelThrough Hecate Strait80818385858586898990909192949698103103106108108112120122V8 Wind-Driven Flow Patterns in Hecate Strait8.1 Basic Patterns8.1.1 Vorticity Balance8.1.2 Effect of Rotation8.1.3 Chatham Sound Diversion8.2 Comparison with Observations in Hecate Strait8.2.1 Currents: Winter 19848.2.2 Drifters8.3 Regional Circulation Patterns8.3.1 Southern Hecate Strait8.4 Summary125126130• . 134• . 1371381391441501551579 Conclusion9.1 Future WorkBibliography 164A List of symbols 170B Empirical Orthogonal Function Analysis 172C Butterworth Filter: End Effects 176D Hecate Strait Model: Details11.1 Numerical formulationD.1.1 Arakawa and Lamb, 1981D.1.2 DiscussionD.1.3 Leap-frog Scheme159161177177178182183viD.1.4 The Robert FilterD.2 Model BathymetryD.2.1 Creating the BathymetryD.3 Drifter AlgorithmD.3.1 Trajectory ComputationsD.4 Lateral Boundary ConditionsD.4.1 Side Walls and Boundary LayersD.4.2 Open Boundary ConditionsD.4.3 Cross-shelf BoundaryD.4.4 Off-shore BoundaryE Tests in Rectangular DomainsE.1 Flat BottomE.1.1 Uniform Along-Shore WindE.1.2 Bell-Shaped WindE.2 Sloping ShelfE.3 Frictional Adjustment on a Sloping ShelfE.4 StoriesF Hecate Strait Model: Tests and ExperimentsF.1 Open Boundary Condition TestsF.2 Adjustment Time-Scales in Hecate StraitF.3 Steady State Velocity FieldsF.3.1 Bottom FrictionF.3.2 BathymetryF.3.3 Spatially Varying Wind214214217225225225229184185187189190191191193196196198198200203203210213viiG Friction, Coriolis Parameter and Transport 2310.1 Steady State Transport 2310.2 Adjustment Time 2340.3 Discussion 235viiiList of Tables3.1 Statistics for the current EOF analysis for the winter of 1984. The A arethe first three eigenvalues. Scaled refers to whether each time series wasscaled by its standard deviation before EOF analysis 363.2 Characteristics of time series in winter, 20 Jan to 4 Apr, 1984. The last twocolumns give maximum correlations and phase leads in hours of a giventime series with transport through W-line. EOF 1 is the lowest (first)mode of wind from the the empirical orthogonal function analysis. Tableic in Crawford et al.(1988) 393.3 Correlation of transport with various wind time series for the period 25Jan - 30 Mar 1984. The maximum correlation rmax was obtained for thelead given (lead of the wind with respect to the transport). Wind timeseries are the along-shore component of wind speed (W), wind stress usinga constant drag coefficient (ri), and wind stress using Smith (1988) (r2) 424.4 Estimates of parameters for Hecate Strait 664.5 Sub-surface pressure EOF modes in Hecate Strait. Each mode has twocolumns: the dimensionless eigenvector, and the eigenvector with units ofcm of water. The eigenvalues are dimensionless. Fall 1983: 28 Septemberto 20 December 1984. Winter 1984: 20 January to 4 April 1984. (CourtesyW.R. Crawford.) 706.6 Standard regional model parameters 96ix6.7 Linear friction. Steady state transport (Sv) for a range of k and i. Along-shore wind stress r = 0.1 Pa 966.8 Quadratic friction. Steady state transport (Sv) for a range of Cd and u0.Along-shore wind stress r = 0.1 Pa. Rayleigh friction ji = 3.0. iO s_i 977.9 Model verification experiments 1097.10 Model verification. Comparison of the simulated and observed transportthrough Hecate Strait for the period 25 Jan- 30 March 1984. The correlations are the maximum linear correlation coefficients of the transport timeseries with the along-shore wind stress and with the observed transport. 1118.11 The first two eigenvalues from the EOF analysis of the simulated andobserved velocities for the period 25 January- 30 March 1984. Eacheigenvalue, A, is represented by two values; its magnitude and the fractionof total energy that this represents (in brackets) 139A.12 List of symbols used. 171D.13 Amplitude response of the time domain filter for sinusoidal input. Forperiods of an hour of more the amplification factor is very close to 1 andis written in the form 1 — , where is a small number. /t = 10 s and7 = 0.01 185E.14 Parameters for open boundary condition tests with a flat bottomed ocean 199E.15 Parameters for open boundary condition tests with a sloping shelf . . . . 207xE. 16 Frictional adjustment. Comparison of theoretical frictional adjustmenttimes and the steady state velocity with the model results. The modelresults were fit to v = v(1 — e_t/t0). Fits were computed using timeseries of length 167 hours. Wind stress r = 0.1 Pa. Density p = 1025 kgm3 212F.17 Steady state transport in domains of different size with an along-shorewind (r = 0.1 Pa) and the standard friction parameters. Domain sizes aregiven in grid points. The open boundary conditions (OBC) are the flowrelaxation scheme (FRS) and closed boundaries. The results for standard,big, and long were the same for FRS zone widths of 20 and 40 grid cells. 215G.18 Steady-state transport (Sv) through Hecate Strait as a function of Coriolisparameter. The wind stress r = 0.1 Pa, fo = 1.1 x iO s1, and i =3 x iO s1 2310.19 Localized wind experiment. Estimating the parameter W/L from thesteady-state transport. The wind stress r = 0.1 Pa, fo = 1.1 xs’, and ,a = 3 x iO s1 2320.20 Uniform wind experiment. Estimating the parameter W/L from the steadystate transport. The wind stress r = 0.1 Pa, fo = 1.1 x iO srn’, andh*=70m 2320.21 Localized wind experiment. Estimating the parameter W/L from the spinup time constant data. The wind stress r = 0.1 Pa, fo 1.1 X iO s’,and t = x iO s1 235xiList of Figures1.1 The Queen Charlotte Islands region 22.2 Chart of the Queen Charlotte Islands region. Depth contours are in metres.Courtesy of M.G.G. Foreman 102.3 Co-range and co-phase values for the semi-diurnal tide. Tidal range (broken line) in metres; tidal phase (solid line) in degrees. Difference of 29°corresponds to time difference of 1 h. (Fig. 14.5 in Thomson, 1981). . 122.4 Observed semi-diurnal tidal streams in the upper 50 m depth. Solid linesand arrows give orientation of major flow and ebb directions: dashed linesgive minor flow directions. Scale measures speed relative to the midpointof each axis. (Fig. 22 in Thomson, 1989) 132.5 Barotropic residual tidal velocities. (Courtesy of M.C.C. Foreman, 1992.) 152.6 The seasonal cycle of density at a station in central Hecate Strait, 1960-62.(Fig. 84 in Dodimead, 1980) 172.7 The seasonal cycle of density at a station on the northern flank of MiddleBank in Queen Charlotte Sound, 1954-55. (Fig. 82 in Dodimead, 1980) 172.8 Average currents and winds in Hecate Strait. Solid arrows represent currents within 50 m of the surface, dashed lines, currents at intermediatedepths and dotted lines, currents within 15 m of the bottom. Wide shadedarrows represent winds. (a) summer, 16 May to 9 September 1983, (b) fall,28 September to 20 December 1983, (c) winter, 20 January to 4 April 1984.(Fig. 2 in Crawford et al., 1988) 22xii2.9 Currents and winds in Queen Charlotte Sound for the period 4 June - 15September 1982. Vectors of average currents, and winds at Cape St. Jamesand Mclnnes Island are the solid lines and the principal axes of varianceare the dotted lines, a) near-surface currents and winds, b) near-bottomcurrents. (Fig. 7 in Crawford et al., 1985) 232.10 Smoothed drifter tracks from Dixon Entrance summer 1984. Tidal currentsare averaged out. The solid circles mark the position of the drifters atthe onset of a storm on 21 June 1984, with winds of 15 m s1 from thesoutheast. (Fig. 3 in Crawford and Creisman, 1987.) 252.11 Average currents and winds in Dixon Entrance for the period 22 April-19August 1984. Solid arrows represent currents within 50 m of the surface,dashed lines, currents at intermediate depths and dotted lines, currentswithin 15 m of the bottom. The wide arrows at D7S and Langara Islandrepresent winds. (Fig. 5 in Crawford and Greisman, 1987.) 262.12 Prevailing surface currents in the North Pacific Ocean. Double arrowsare intense boundary currents, speeds typically 1-2 m/s: over most of therest of the region speeds are less than 0.25-0.50 m/s. Arrows correspond toprevailing winter time flow off the west coast of North America. (Fig. 13.17in Thomson, 1981) 283.13 Dependence of drag coefficient Gd on wind speed for winds measured at10 m and neutral atmospheric stability (Smith, 1988) 313.14 Locations of current meters, pressure gauges and anemometers in HecateStrait from Jan - April 1984. Fig. 1 from Crawford et al. (1985, withpermission) 33xlii3.15 Vectors of the empirical orthogonal functions for the period 20 January-4April, 1984. Fig. 8 from Crawford et al., 1990 373.16 The observed transport and along-shore wind stress fluctuations in HecateStrait from 25 Jan— 30 Mar 1984. Both time series have been normalizedby subtracting the mean and dividing by the standard deviation 423.17 Comparison of the transport and mode 1 currents for the period 25 Jan -31 Mar 1984. Both time series have been normalized by subtracting themean and dividing by the standard deviation 443.18 Path of satellite tracked drifter for 9 Sept — 29 Nov 1983. The winds atCape St. James are shown in the lower left corner (1 day low-pass filtered).The free end of the wind vectors point in the direction the wind is blowingto. (Adapted from Fig.13 Hannah et al., 1991.) 463.19 Drifter tracks near Cape St. James for the period 12—20 March 1986.(Adapted from Fig. 12a in Thomson and Wilson, 1987.) 483.20 Drifter tracks in southern Hecate Strait from 8—29 July 1990. The windswere from the NW or N. (Adapted from Fig. 14 in Hannah et al., 1991.) 493.21 Paths of two satellite tracked drifters from 19 January to 20 March 1991.The starting locations are marked by the two black dots east of Cape St.James. The winds were generally from the SE 513.22 Drifters in northern Hecate Strait from July 1991. The abrupt change indirection displayed by drifters a35, c22, and b40 corresponds to the changein wind direction from NE to SE on July 10. Drifter b32 was launchedafter the wind changed direction. The arrows on the drifter paths mark 2days elapsed time 52xiv3.23 Two drifters in northern Hecate Strait from 11 to 13 July 1991. The drogueon drifter b32 was centred at 10 m, the drogue on b34 at 3.5 m. The twodrifters were launched at the same time and b34 (shallow) travelled twiceas far as b32 (deep). The arrows mark 1 day elapsed time 534.24 Schematic diagram of Hecate Strait 574.25 Plan view of the strait. i and 72 are the far-field sea level elevations. Theother are the sea levels at the respective corners. The corner labellingscheme is for the northern hemisphere 574.26 Frictional adjustment with sinusoidal forcing. The solid line is the velocityand the dotted line is the wind stress. The frictional scale ) = /6. . . 604.27 The sea level structure for the wind-driven mode, N 674.28 The sea level structure for the pressure-driven mode, N 674.29 Spatial patterns of the observed sub-surface pressure fluctuations in HecateStrait. Plotted are the first two modes of the empirical orthogonal functionanalysis for the fall of 1983 and the winter of 1984. The modal amplitudesare in cm of water. The arrow in Hecate Strait is the direction of thegeostrophic current implied by the cross-strait pressure difference. Thedata are listed in Table 4.5 695.30 Sketch of the ocean defining the axis system 755.31 A sea strait 785.32 The model domain 826.33 Frictional adjustment in a simple channel. Spin-up time as function of thewater depth for k = 0.5 . iO s and (a) i = 0; (b) u = 3 . 10_8 s’; (c)= 3 . iO s’; (d) = 3. 10_6 s—I 93xv6.34 Frictional adjustment in a simple channel. Steady state velocity as a function of the water depth for k O.5• iO s1 and (a) = 0; (b) t = 3. iOs’; (c) jt 3. iO s; (d) 3. 10 s 936.35 The standard model domain. The cross-shore flow relaxation zones arenot shown; they extend 100 km off the top and bottom of the figure. Theline on the left-hand side is the off-shore open boundary 956.36 Spin-up test. Time series of transport through Hecate Strait for an impulsively started wind. (a) R-line, (b) W-line, (c) M-line. The dotted line in(b) is the transport when the wind was started smoothly (see text). . 996.37 Spin-up test. Time series of sea levels in Hecate Strait. a) Atli and b)Beauchemin 1006.38 Spin-up test. Time series of sea levels on the outer coast. a) Cape Muzon,b) W. QCI, and c) Cape Scott 1006.39 Spin-up test. Cross strait pressure difference Beauchemin minus Atli. . 1006.40 Spin-up test. Along-shore pressure difference Cape Scott minus Cape Muzon 1016.41 Spin-up test. Comparison of transport through R-line with a two component adjustment 1016.42 The high frequency oscillation at W-line 1046.43 The power spectrum of the oscillation at W-line 1046.44 Steady-state transports with a uniform SE wind. The wind direction isindicated with an arrow on Graham Island. The 200 m contour is theheavy dark line. Only 1/4 of the vectors are plotted. The vectors weresuppressed for depths greater than 500 m 1056.45 The along-strait force balance at R-line after two days of steady alongshore SE wind 107xvi7.46 The wind stress time series used to drive the verification experiments. Thevectors are oriented to conform to the orientation of the model domain;up represents a SE wind. The winds were measured at W4S 1097.47 Transport through W-line from 25 Jan - 30 Mar 1984. Comparison ofsimulation E501a (solid) and the observations (dotted) 1107.48 Time series of along-shore wind stress (dotted) and simulated transport(solid) in Hecate Strait from 25 Jan- 30 Mar 1984. Both time series havebeen normalized 1107.49 Cross-strait sea level difference Beauchemin minus Atli for the period 25Jan- 30 Mar 1984. Comparison of E501a (solid) with the observations(dotted) 1137.50 Steady state transport as a function of Coriolis parameter for a wind localized over Hecate Strait. The Coriolis parameter has been normalizedwith fo = 1.1 X iO s 1157.51 Steady-state transport as a function of Coriolis parameter for a uniformalong-shore wind stress (triangle) and for a wind stress localized overHecate Strait. In both cases k = 0.5 iO s1. The Coriolis parameter has been normalized with fo 1.1 X iO s’ 1157.52 Transport spin-up for an along-shore wind localized over Hecate Strait fork = 0.5 . iO s1. (a) f = 0 and (b) f = 11 . iO s1 1187.53 Relaxation time as a function of Coriolis parameter for a wind localizedover Hecate Strait. a) k = 0.5 . iO s1, b) k = 2.0 . iO s1 1187.54 Amplitude spectrum computed from the response of the transport throughHecate Strait to oscillating winds (data points). The solid line is an approximate spectrum (see text) 121xvii7.55 Phase spectrum computed from the response of the transport throughHecate Strait to oscillating winds 1217.56 Transport time series for an oscillating along-shore wind with period 6days. The solid line is the transport, the dotted is the wind forcing. Bothtime series have been normalized by the amplitude of the respective sinewaves (0.15 Sv, 0.1 Pa) 1237.57 The transport spin-up in Hecate Strait for a non-rotating (solid) and rotating (dashed) version of the model. For the non-rotating case k = 1.3 iOm s1 and for the rotating case k = O.5• iO m s1. The bottom frictionfor the non-rotating case was chosen so that the steady-state transportswere similar 1238.58 Evolution of the velocity field with a steady SE wind. The wind directionis indicated with an arrow on Graham Island. The 200 m contour is theheavy dark line. Only 1/4 of the vectors are plotted 1278.59 The velocity field and transport vector field at Day 8, with a steady SEwind - typical winter storm winds. On the shelf the velocities have reacheda steady state. In the deep ocean the velocities are still adjusting, but thisdoes not affect the velocity field on the shelf. The 200 m contour is shown 1288.60 Steady-state velocity fields for two different wind directions 1318.61 Steady-state velocity field for a SE wind and a non-rotating earth (f = 0) 1358.62 Steady-state velocity field when: (a) northern Hecate Strait is blocked off;(b) the constriction at the northern end of Hecate Strait is removed (seetext). This was for a SE wind and a rotating earth, f = 1.1 x iO s_i 1368.63 Close up of the northeast corner of Hecate Strait 137xviii8.64 Model verification. Comparison of the simulated (E501a) and observedcurrents in Hecate Strait from 25 January to 30 March 1984. The depth-averaged observations are shown with open arrow heads; the simulatedcurrents with solid arrow heads. The model was driven with winds measured near W04 1408.65 Time series of modal amplitudes of the EOF-mode-1 velocities. Comparison of simulation E501a (solid) with the observations (dotted) in HecateStrait from 25 January to 30 March 1984 1438.66 Comparison of an observed drifter trajectory and simulated drifters inMoresby Trough from 12 to 18 July 1990. The observed drifter (solid)started at the cross. Simulated drifters (dashed) were started at dailyintervals (circles) along the observed drifter trajectory. The simulateddrifter trajectories are 4 days long. In cases where the simulated drifterdid not move very far an extra drifter was started 1 grid point away. Thewinds were measured at a weather buoy located 80 km below the bottomof the figure (see text). The wind stress time series is shown in the lowerleft corner. The free end of the wind vectors indicates the direction thewind is blowing to. The 200 m contour is shown for reference 1458.67 Comparison of an observed drifter trajectory and simulated drifters nearCape St. James from 21 to 26 July 1990. See Fig. 8.66 for details. . 1468.68 Observed and simulated drifter trajectories in northern Hecate Strait from11 to 19 July 1991. Drifter b32 was launched shortly after the windschanged direction on July 10. The wind stress time series shown in thelower left corner was measured at the * in the center of the strait. Thefree end of the wind vectors indicates the direction the wind is blowing to 148xix8.69 Observed and simulated drifter trajectories in northern Hecate Strait from8 to 10 July 1991. All the trajectories are 2 days long. During this time thewind was steady and from the NW. The end of the trajectories correspondsto the change in wind direction near July 10. The wind stress time seriesshown in the lower left corner was measured at the * in the center of thestrait 1498.70 Simulated drifters in the northern half of the model domain for 25 Jan to30 Mar 1984. The starting location of each drifter is marked by a cross.The drifter trajectories are marked with arrows at 20 d intervals 1518.71 Simulated drifters in Queen Charlotte Sound for 25 Jan to 30 Mar 1984.The starting location of each drifter is marked by a cross. The driftertrajectories are marked with arrows at 20 d intervals 1528.72 Simulated drifters near Cape St. James. The starting location of eachdrifters is marked by a cross. The drifter trajectories are marked witharrows at 20 d intervals 1538.73 Sketch of the circulation in Hecate Strait and Queen Charlotte Soundunder a SE wind 156C.74 The end effect of the Butterworth filter used to filter the time series data.The filter was an 8-th order filter with 1/2 power at 40 h. The time stepwas 15 m. The solid line is the filter output and the dashed line is theinput 176D.75 The C-grid 179D.76 Boundaries in the C-grid 192xxE.77 Model domain for the open boundary condition tests. The flow relaxationzones are marked by shading 199E.78 Uniform along-shore wind experiment. Time series of along-shore velocityat the coast (site B). The time series at sites A,C are identical 201E.79 Uniform along-shore wind experiment. The steady state a) sea level contours (cm), and b) velocity vectors 202E.80 Bell-shaped wind experiment. Sea level after 48 hours. The contour interval is 0.5 cm 204E.81 Bell-shaped wind experiment. Time series of along-shore velocity at sitesA, B, C. Data from the test domain is plotted as solid lines and data fromthe extended domain is plotted as dotted lines 205E.82 Bell-shaped wind experiment. Time series of excess mass. Data from thetest domain is plotted as solid lines and data from the extended domainis plotted as dotted lines 206E.83 Cross-section of the sloping shelf. Topography taken from regional model. 207E.84 Sloping shelf experiment. Time series of along-shore velocity at threedifferent depths. The depth h = 77 m corresponds to location (i,j) =(25,36). The depths h = 155 m and h = 1786 m correspond to (i,j) =(20,36) and (10,36) respectively 208E.85 Sloping shelf experiment. a) sea level field (cm), and b) velocity field, after6.5 days 209xxiE.86 Sloping shelf experiment. Along-shore velocity as a function of along-shore position. The velocity field was sampled at 3 distances off-shore,corresponding to depths of 77 m, 155 m, 1786 m. The model was forcedwith a uniform along-shore wind r = 0.1 Pa and the Rayleigh friction= 0. The velocity field was sampled after 6.5 days of integration Theflow relaxation zones occupy along-shore locations 1 to 11 and 61 to 71.There are no kinks in the velocity field but there is a slight downwardslope in the in the down-wind direction (from left to right) 211F.87 Comparison of the transport through W-line for two different open boundary conditions: (a) flow relaxation method (E321,solid); and (b) closed(E320, dashed). The domain is the long domain 215F.88 The velocity field after 6 days near the northern cross-shore boundary.The top of the figure is the open boundary. All of the vectors are shown. 216F.89 The steady state velocity field forced by an along-shore wind localized overHecate Strait. The maximum wind stress is 0.1 Pa 219F.90 The steady state velocity field forced by an along-shore wind localized overcentral Queen Charlotte Sound. The maximum wind stress is 0.1 Pa. . . 220F.91 Comparison of the transport spin-up for three along-shore wind scenarios:(a) spatially uniform, (b) localized over Hecate Strait and (c) localizedover Queen Charlotte Sound. The maximum wind stress is 0.1 Pa. . . . 221F.92 Curve fit to the transport time series for a wind localized over HecateStrait 222F.93 Curve fit to the time series of sea level at Cape Scott minus sea level atCape Muzon. The wind forcing was the standard spatially uniform wind. 222xxiiF.94 The sea level field for a spatially uniform along-shore wind (r = 0.1 Pa).The contours are labelled in cm 223F.95 Quadratic friction experiment. The steady state velocity field for a spatially uniform along-shore wind. The drag coefficient 0d = 2.5 x i0 andthe background velocity u0 = 0/0. The wind stress r = 0.1 Pa 226F.96 Spatially varying linear friction experiment. The rms tidal velocity field(cm/s) and the resulting steady state velocity field from the Hecate StraitModel. The wind stress was spatially uniform with r = 0.1 Pa 227F.97 Velocity field with altered bathymetry. The wind stress was spatially uniform with r = 0.1 Pa. Standard linear friction was used 2280.98 Uniform wind experiment. Steady-state transport as a function of Coriolisparameter for three values of k and it = 2.7 x 10 s1. The wind stressr = 0.1 Pa 233xxiiiAcknowledgementA graduate student’s most important decision is the first one, the choice of supervisor.I thank Paul LeBlond for making my choice seem inspired. His insight, guidance, andsupport were crucial to the success of this work. I owe a special debt of gratitude toWilliam Crawford at the Institute of Ocean Sciences (lOS), Sidney, B.C., who providedthe observational data discussed in this thesis, and was responsible for much of myeducation as an oceanographer. Thanks Bill. The members of my committee have eachmade an important contribution to my education, I thank Boye Ahlborn, Luis Sobrinoand Paul Budgell.None of this would have been possible without the support of my family. Thanks toSusan for her support, encouragement, and faith throughout; to Elizabeth and Theodorefor the daily reminders of the important things in life.I have been blessed with good friends, they have contributed to my education andkept me sane. I thank Warren Lee, Mike St. John, the Sunday morning quantum group,the members of FOMI, and the oceanography softball team. A special thanks to theTuesday morning hockey players.During my extended visit to the Institute of Ocean Sciences, it was my pleasure towork with many dedicated people. Thanks to the Tides and Currents Section, especiallyAllan, Anne W., Anne B., Keith, and Tony for letting me abuse their computer system;and to Mike Woodward for the experience of Queen Charlotte Sound in January. Mythanks to Greg Holloway and Mike Foreman for the time spent.I gratefully acknowledge the financial support of the Panel for Energy Research andDevelopment (PERD) Grant 67146, NSERC Grant A7490 and a UBC graduate fellowship. Computer time, the life blood of numerical models, was generously provided byWilliam Hsieh at UBC, by the Institute of Ocean Sciences, and by the Ontario Centrefor Large Scale Computing.xxivChapter 1IntroductionThe subject of this thesis is the wind-driven ocean currents on the continental shelf aroundthe Queen Charlotte Islands. This work is one of a group of projects whose ultimate goalis the prediction of the movement of particles in the waters of British Columbia’s northcoast. Particles of interest include fish larvae, nutrients, sediments and pollutants. Thework reported here concentrates on the contribution of the wind-driven currents to thecirculation: the low frequency component of the flow, with time-scales of days to weeks.The daily oscillations of the tides are ignored.The focus of this thesis is the winter circulation in Hecate Strait (Fig. 1.1). Thehistorical view was that of a broad current flowing north from Queen Charlotte Soundinto Dixon Entrance. Only recently has this description been modified to include acounter-current in south-central Hecate Strait (Crawford et al., 1988; Crawford et al.,1990). The central theme of this work is the influence of the earth’s rotation on thewind-driven flow. Two aspects are singled out: 1) the role of rotation in the existence ofthe counter-current, and 2) the role of rotation in limiting the flux of water through thestrait.The primary tool is a numerical model of the vertically integrated, or depth-averaged,currents in the Queen Charlotte Islands region. The observations indicate that a depthaveraged, wind-driven model is most applicable to Hecate Strait in the winter. Thismodel is called the Hecate Strait Model throughout this thesis. The Hecate Strait Modelis used to probe the dynamics and to suggest interpretations of the observations. A1Chapter 1. Introduction 252 N50 N54 NFigure 1.1: The Queen Charlotte Islands region.Chapter 1. Introduction 3practical question addressed is: Does the Hecate Strait model provide a useful guide tothe circulation patterns in the region?1.1 Motivation for this Work1.1.1 FisheriesDemonstrating a causal relationship between fluctuations in ocean currents or temperature and fluctuations in the number of fish caught in Hecate Strait is very difficult. Thisis due in part to the inherent difficulty of estimating the size of the fish stock and in partto our limited knowledge of the ocean. In Hecate Strait there is evidence that fluctuationsin the year-class-strengths of English sole (Parophrys vetulus) and Pacific cod (Gadusmacrocephalus) are linked to inter-annual fluctuations in water transport through HecateStrait in the first quarter of the year (Ketchen, 1956; Tyler and Westrheim, 1986; Tylerand Crawford, 1991).The regional model results presented in this thesis are being used in a project aimedat illuminating the relationship between the physical oceanography and the recruitmentrates of the fish. The project involves hind-casting the transport and current patternsin Hecate Strait for the past 40 years and combining this information with a biologicalmodel to investigate the transport of English sole larvae from the spawning grounds insouthern Hecate Strait to the nursery areas in northwestern Hecate Strait (Walters etal., 1992).1.1.2 Oil SpillsThe Hecate Strait Model began as one component of a larger project intended to addressthe physical oceanographic aspect of the recommendations of the West Coast OffshoreExploration Environment Assessment Panel (April 1986, p.96):Chapter 1. Introduction 4The Panel recommends that the Department of Fisheries and Oceans, incooperation with other agencies, develop a comprehensive research programdesigned to reduce data gaps necessary to develop a credible model of theimpact of an oil blowout on important fish species at their various life stages.The overall objective of the project is to improve the prediction of oil spill trajectoriesin northern British Columbia waters. Oil spills are fundamentally a surface phenomenonand the Hecate Strait Model looks at the currents averaged over the water column. Therefore this model will not form the basis for operational predication of oil-spill trajectories.The goal is to gain insight into the gross features of the circulation patterns and to provide a framework for interpreting near-surface drifter observations. Future modelling willbuild on this work. In an indirect fashion, the application of the regional model to studythe early stages of the life of English sole in Hecate Strait also helps fulfill the mandate.1.2 ObjectivesThe principal objective of this thesis is to describe the influence of the earth’s rotationon the wind-driven flow in Hecate Strait. The primary tool used is the Hecate Straitmodel, a numerical model of the depth-averaged currents in the Queen Charlotte Islandsregion. The creation of the model required:• The development of a numerical model of the depth-averaged currents on the continental shelf surrounding the Queen Charlotte Islands.• Verification of the model by comparison with current meter measurements in HecateStrait from 25 January to 30 March 1984.• Comparison of simulated and observed drifter trajectories for drifter deploymentsin Hecate Strait during July 1990 and July 1991.Chapter 1. Introduction 5Comparison of the overall circulation pattern in the Hecate Strait Model with theobserved flow patterns to determine whether the model provides a useful tool forunderstanding the circulation.Understanding the role of rotation in limiting the flux through Hecate Strait requiresmore than the Hecate Strait Model. A simple analytic model of flow through sea straitsis needed to illustrate the important physical concepts. A secondary goal of this thesis isthe extension of an existing conceptual model of flow in straits of finite length to includelocal wind forcing. The extended version is called the rotation-limited-flux model.Considerable time is spent on details in order to create confidence in the numericalresults. Testing a numerical model requires more than comparing model output withobservations and then tuning parameters. For valid physical insight the model resultsmust depend on the physics, not on the details of the numerics. This concern for detailis one of the reasons that numerical modelling becomes an end in itself. Writing thisthesis has required balancing the opinion that Cod is in the details, with the sentimentof Richard Hamming: ‘The purpose of computing is insight, not numbers.’1.3 Plan of the ThesisThe thesis is organized as follows.Chapter 2 contains a review of the physical oceanography of the Queen CharlotteIslands region. The observations indicate that a depth-averaged numerical model is mostapplicable to Hecate Strait in the winter. Thus the name of the regional numerical model:the Hecate Strait Model.Chapter 3 is a detailed discussion of the observations used to test the Hecate StraitModel. Analysis of the winds, currents, and water transport through Hecate Strait forthe period January to April 1984 is reviewed and a previously unnoticed relationshipChapter 1. Introduction 6between the transport fluctuations and the spatial pattern of the velocity fluctuations isdiscovered. Drifter trajectories from all seasons are reviewed. The observations indicatethat the local wind is an important forcing mechanism in Hecate Strait.Chapter 4 introduces some simple ideas about low frequency flow in sea straits. Therotation-limited-flux model provides a description of how the interaction between theearth’s rotation (Coriolis force) and the finite length of the strait limits the flux throughthe strait. The original work here is the extension of the rotation-limited-flux modelto include wind forcing and the application to Hecate Strait. The concept of rotation-limited-flux is used to interpret the relationship between the flux through Hecate Strait,the bottom friction, and the Coriolis force as seen in the Hecate Strait Model results(Chapter 7).Chapter 5 provides an over-view of the Hecate Strait Model. The equations of motion,the shallow water equations, are discussed and the concept of topographic steering isintroduced. Topographic steering, the tendency for the low frequency flow to follow thedepth contours, is used extensively in the discussion of the flow patterns in Chapter 8.The numerical scheme for integrating the shallow water equations is discussed, as are theopen boundary conditions and the bottom friction representation. What is important isnot the details, but a sense of how the model works.Chapter 6 builds basic understanding of the behaviour of the Hecate Strait Modelusing the model’s evolution from rest to a steady state as a test case. Important topics arethe adjustments time-scales and the sensitivity of the steady-state water transport (flux)through Hecate Strait to the friction parameters. This discussion lays the foundation forunderstanding the simulations of the winter of 1984.Chapter 7 investigates the rotational limitations on the water transport throughHecate Strait. The Hecate Strait Model is shown to provide good simulation of theobserved transport for the period 20 January to 30 March 1984. The model is thenChapter 1. Introduction 7used to examine the relationship between the transport, bottom friction and the Coriolisforce. Rotation-limited-flux provides an explanation for two features of the Hecate StraitModel: 1) the steady state transport increases by a factor of 3 when the rotation of theearth is ignored, and 2) the transport is not sensitive to the friction parameters.The flow pattern, or circulation, in Hecate Strait is investigated in Chapter 8. TheHecate Strait Model captures the character of the observed flow pattern. The modelledflow is dominated by the interaction between the Coriolis force and topography: topographic steering. A new interpretation of the the observed current patterns in southernHecate Strait is suggested.The technical details and the results of preliminary tests of the Hecate Strait Model,including the open boundary conditions, are contained in the appendices.Chapter 2Physical Oceanography of the Queen Charlotte Islands RegionThis chapter contains a review of the physical oceanography of the Queen CharlotteIslands region. The review looks to the application of a wind-driven, depth-averagedmodel of the circulation. Such a model assumes that the density and velocity do not varysignificantly with depth and that the wind is the primary forcing mechanism. Circulationdue to the density structure of the ocean is ignored as are the tides and the effects offresh water input. The review of the observations in this chapter and the next, indicatesthat such a model is most applicable to Hecate Strait in the winter. In particular, theanalysis of the moored current meter records in Hecate Strait for the winter of 1983/84indicates that the barotropic response is important. The mean currents are discussed inSection 2.5.1 and an empirical orthogonal function analysis of the fluctuations is presentedin Chapter 3.Thomson (1981, 1989) has written reviews of the physical oceanography of the QueenCharlotte Islands region for a general audience. Much of the material presented in thischapter can be found there.2.1 Geography and BathymetryThe continental shelf around Queen Charlotte Islands is generally divided into four regions: Queen Charlotte Sound, Hecate Strait, Dixon Entrance, and the narrow shelf alongthe west coast (Figures 1.1 and 2.2). These divisions are useful because each region hasits own distinct bathymetry and experiences different atmospheric and oceanographic8Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 9forcing. On the other hand the bathymetric features which separate the regions are thesource of many of the interesting current patterns. The region must be considered as awhole so that important features are not lost in the boundaries.The bathymetric features of interest here (Figures 1.1 and 2.2) are:• Moresby Trough (or Gully) which cuts across the shelf from Cape St. James to themainland and then turns north to form the axis of Hecate Strait.• Dogfish Bank, the extensive shallow region in north-western Hecate Strait. Thisis the nursery area for Pacific cod and English sole. The larval advection problem mentioned in the Introduction involves the transport of the larvae from thespawning grounds in southern Hecate Strait to Dogfish Bank.• the steep escarpment that runs northeast from Rose Spit and provides the dividingline between Dixon Entrance and Hecate Strait.• Learmonth Bank, the shallow ridge which dominates the mouth of Dixon Entrance.• the shallow banks and deep troughs that comprise Queen Charlotte Sound.• the narrow continental shelf on the west coast of the Queen Charlotte Islands. Atthe southern end the shelf is less than 5 km wide (depth < 200 m) and depths of2500 m are reached within 30 km of shore.For the purpose of this thesis, Hecate Strait extends from Moresby Trough to the escarpment at the north end.There is a wealth of detail along the coastline with numerous islands, rocks, fjords,and channels. The fine detail is not included in the numerical model. The re-connectionof Clarence Strait with the Pacific Ocean after its narrow, winding journey through theAlaska Panhandle is not part of the model.Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 1054 N52 N50 N126 WFigure 2.2: Chart of the Queen Charlotte Islands region. Depth contours are in metres.Courtesy of M.G.C. Foreman.134 W 130 WChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 112.2 TidesThis thesis concentrates on the wind-driven circulation and does not consider the tides.Nevertheless the tides are the most energetic phenomenon in the region and they deserveto be discussed if only to appreciate what is being ignored.Figure 2.3 is the co-tidal chart for the semi-diurnal (M2) tide. Notice that the phase ofthe tide is almost constant along the outer coast of the Queen Charlotte Islands and thatthe tides from the two ocean entrances meet in northern Hecate Strait. It takes about30 minutes for the tide to reach northeastern Hecate Strait from the open ocean, andanother 15 minutes for the combined tide to swing across the strait and reach GrahamIsland (Thomson, 1981). The maximum tidal elevations are in northern Hecate Strait.The tidal currents are rotary in the open waters of Queen Charlotte Sound and becomerectilinear in Hecate Strait (Figure 2.4). The maximum surface tidal currents are of theorder of 50 cm/s (1 knot) during spring tide and about half this during neap tide.The tidal currents can affect the long term transport of particles in two ways: 1)through the generation of residual tidal currents, and 2) by moving particles from onecurrent regime to another over part of a tidal cycle. In regions where the wind-driven currents exhibit large horizontal shear the second effect makes particle trajectories sensitiveto the phase of the tide.2.2.1 Residual Tidal CurrentsA residual tidal current is the net current found when the current is averaged over a tidalcycle. These residuals can be thought of in two separate ways: Eulerian residuals, thenon-zero mean current measured at a moored current meter, and Lagrangian residuals,the net displacement of a fluid parcel. The Lagrangian residual is what is measured by adrifter and is the important parameter for tracking particles in the water. Residual tidalChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 12Figure 2.3: Co-range and co-phase values for the semi-diurnal tide. Tidal range (brokenline) in metres; tidal phase (solid line) in degrees. Difference of 29° corresponds to timedifference of 1 h. (Fig. 14.5 in Thomson, 1981).Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 13Figure 2.4: Observed semi-diurnal tidal streams in the upper 50 m depth. Solid linesand arrows give orientation of major flow and ebb directions: dashed lines give minorflow directions. Scale measures speed relative to the midpoint of each axis. (Fig. 22 inThomson, 1989)Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 14currents have many causes including the non-linear interaction of tidal constituents, theinteraction of the tidal currents with topography (tidal rectification), bottom friction,and the spatial variability of the tidal current field. The last one will not contribute tothe Eulerian residual.One region where residual tidal currents have been thought to be important is DixonEntrance. The existence of the Rose Spit Eddy (or Dixon Entrance Eddy) was one ofthe predictions of the large hydraulic model Project Hecate (Bell and Boston, 1963; Bell,1963). This prediction prompted a field program to observe the feature (Crean 1967).Recent work by Bowman et al. (1992) suggests that the presence of the eddy in the modelwas an artifact due to the way the model was forced at the open boundaries. The RoseSpit Eddy is not seen in the tidal residuals of the non-linear tidal model of Foreman etal. (1992). The eddy does exist and is discussed in section 2.5.3 as part of the circulationin Dixon Entrance. This is a warning to interpret all model results with extreme care.The results of Foreman et al. (1992; Fig. 2.5) show significant residual tidal currentsat Cape St. James, over Learmonth Bank, over the escarpment that separates HecateStrait from Dixon Entrance, and in the shallow water of northern and western HecateStrait. The results near Cape St. James are in general agreement with the observationsand analytical results of Thomson and Wilson (1987). Extreme caution is called for wheninterpreting the wind-driven model results in the vicinity of Cape St. James.The southward currents on the western side of Hecate Strait will have an impact onthe advection of larvae onto Dogfish Bank. The contribution of these residual currentsto the net transport through Hecate Strait is 0.03 x 106 m3 s to the south (Ian Jardine,pers. comm., 1991). This represents 50% of the observed mean transport during thesummer of 1983 but only 10% of the observed mean transport in the fall of 1983 and thewinter of 1984 (Crawford et al., 1988).Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 15Figure 2.5: Barotropic residual tidal velocities. (Courtesy of M.G.G. Foreman, 1992.)Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 162.3 Temperature, Salinity, Density, and Fresh WaterDodimead (1980) described the temperature, salinity, and density structure in QueenCharlotte Sound and Hecate Strait in terms of two states and the transitions betweenthem. Summer conditions are characterized as consisting of a thin surface-mixed layeroverlying a marked thermocline (a large vertical temperature gradient) and pycnocline(a large vertical density gradient). Below this the water has a lower temperature, highersalinity and higher density. Summer conditions usually prevail in July and August.Winter conditions are characterized by a thick mixed layer which reaches its maximumdepth in December. Isothermal conditions can extend to depths of 150-200 m (Dodimead1980). The deepening of the mixed layer is due to surface cooling, increased wind mixing,and downwelling. Due to runoff from the coastal mountains, the water is generally fresher(less saline) on the eastern side of the strait at all times of the year (Dodimead, 1980;Thomson, 1981).The seasonal change in the density structure in Hecate Strait is dramatic (Fig. 2.6).The strongly stratified surface layer completely disappears in the winter. The seasonalcycles iii Queen Charlotte Sound (Fig. 2.7) and Dixon Entrance (Crean, 1967; Thomson,1989) are less dramatic. The stratification is significantly reduced in winter but neverdisappears.The north coast receives an enormous amount of rainfall, but runoff from winterrainfall is thought to have only a marginal influence on the surface currents in HecateStrait and Queen Charlotte Sound (Thomson, 1981). The major sources of fresh water arethe spring/summer freshets of the Nass and Skeena rivers which discharge into ChathamSound at the eastern end of Dixon Entrance.The influence of freshwater discharge is most strongly felt in Dixon Entrancein late spring and summer when runoff from the Nass and Skeena rivers tendsChapter 2. Physical Oceanography of the Queen Charlotte Islands Regiona)a)ON 0 .1 F MA M .J J A SON 0 J F MA1960 1961 196217Figure 2.6: The seasonal cycle of density at(Fig. 84 in Dodimead, 1980)a station in central Hecate Strait, 1960-62.0 0 o /(268)— DENSITY (oApproiinnteBottom Deph J -I I I I t.__I I I I I_I IM J J A S 0 N 0 J F M A M J1954 1955Figure 2.7: The seasonal cycle of density at a station on the northern flank of MiddleBank in Queen Charlotte Sound, 1954-55. (Fig. 82 in Dodimead, 1980)oApproiooteC50 •0 0 025100IIcLw0150-c26.520C)250,Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 18to flow seaward within a comparatively warm brackish layer 10-20 m thickthat hugs the northern side of the channel. In contrast, cooler, saltier oceanicwater generally prevails over the southern half of the channel at this time.(Thomson, 1981)An example of the impact of the fresh water input on the circulation was observed inJuly 1991. During strong SE winds three drifters were carried through Brown Passage,south of Dundas Island, into Chatham Sound. Two of the drifters had drogues centred at3 m and one at 10 m. The two shallow drifters stopped when they hit the front createdby the Skeena River discharge. The deeper drifter carried on into Chatham Sound andoff to the north (W.R. Crawford, pers. comm., 1991).2.4 WindsThe general features of the winds in the Queen Charlotte Islands region are quite simple.Winter storms are generally more severe and more frequent than surmner storms and thewinds are generally from the southeast. Summer storms can be either from the southeastor the northwest. The large scale weather patterns from the Pacific Ocean are heavilymodified by the coastal mountains. In Hecate Strait the coastal mountain ranges channelthe wind to southwest/northeast directions. In Dixon Entrance the winds are more nearlyeast/west. At Cape St. James the wind is less constrained but most of the wind energyis directed parallel to the coastal mountains.Information on the spatial distribution of wind speed is sketchy. Thomson (1981)notes that• the frequency and intensity of southeasterlies is greater over the northernsector of the seaway than over the southern sector, whereas the frequency andintensities of northeasterlies is greater in the south than in the north.Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 19This is not sufficient to construct forcing fields for a numerical model. Compilationof reliable wind fields from the atmospheric pressure observations and the lighthousewinds is a major undertaking and to my knowledge has never been done for the QueenCharlotte Islands region. Lighthouse winds are considered unreliable estimators of thewind speed over the ocean. They are generally biased by blockage from trees and hillsand are considered to be overly influenced by the coastal boundary layer. Only in the lastyear or so has the Atmospheric Environment Service (AES) had enough weather buoysin operation to even consider defining a spatially variable wind field from over-the-oceanmeasurements.The winds are discussed with the current observations in the next section and inChapter 3.2.5 CirculationThe first program of current meter observations devoted to obtaining a clear picture ofthe circulation in Queen Charlotte Sound, Hecate Strait and Dixon Entrance was carriedout from 1982 to 1985. Current meter observations were made in Queen Charlotte Soundduring the summer of 1982 (Crawford et al., 1985) in Hecate Strait from May 1983 toApril 1984 (Crawford et al., 1988), and in Dixon Entrance from May 1984 to May 1985(Crawford and Greisman, 1987; Bowman et al, 1992).Before that time the bulk of the current meter studies concentrated on measuringtidal currents for navigation purposes. These studies were usually limited to the summermonths because it is extremely difficult to work in these areas in the fall and winter.The circulation was studied using hydrographic measurements (temperature, salinity,oxygen) with current patterns inferred from the density fields. The enormous amountof hydrographic data collected during the 1950’s and 1960’s is reviewed by DodimeadChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 20(1980). The circulation as understood before the 1982/85 observations is discussed byCrean (1967), Dodimead (1980), and Thomson (1981, 1989). The accepted view of thewind-driven circulation before 1981 is summarized concisely by Bell and Boston (1962):Observations have shown that during the winter period of strong southeastwinds, light surface waters accumulate along the coast, displacing the deepestwaters to an offshore position. During the summer period of weak northwestwinds, the accumulated surface waters move offshore allowing the return ofthe deep waters. The winter accumulation is accompanied by a northwardflowing coastal current which appears to be continuous from California to theGulf of Alaska. During the summer, this current is small or absent.The winter downwelling and displacement of the deep water has the added effect ofdecreasing the stratification of the water column- making it more homogeneous. Inthe summer the replacement of the deep water increases the stratification of the watercolumn.2.5.1 Hecate Strait 1983-1984The 1983/84 observations in Hecate Strait do not contradict the historical picture so muchas add a new twist. An unwary reader might think that the description given implies thatthe water flows between Hecate Strait and Queen Charlotte Sound in the direction ofthe wind. The observed mean winds and mean currents in Hecate Strait (Fig. 2.8) showthat in southern Hecate Strait (M-line) the water does not simply flow down-wind. Inthe fall and winter, the observations clearly show flow along the axis of Moresby Troughagainst the prevailing wind. The mass transport through the northern part of HecateStrait (W-line or R-line) is the small difference between the large northern and southerntransports through the M-line (Crawford et al., 1988). In the words of Crawford, TylerChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 21and Thomson (1990, hereafter CTT9O)The historical view of the winter circulation, which described a simple currentflowing through Hecate Strait into Dixon Entrance, is clearly not correct.For scientists concerned with tracking fish larvae this twist changes everything. CTT9Oremark that without some sort of recirculation pattern, particles deposited in the southern end of Hecate Strait in the winter should be swept into Dixon Entrance in about 25days (typical current of 10 cm/s, length of strait 250 km). The larval stage of Pacific codand English sole, which spawn in the winter, ranges from 6-10 weeks depending on thetemperature (CTT9O, Walters et al., (1992), Ketchen, 1956) so they would be carriedthrough the strait before they settle onto the bottom. CTT9O note that there is no reasonto believe that Hecate Strait is restocked from other regions since ‘the species dominantin Hecate Strait give way to other species in Dixon Entrance and the Alaska Panhandle.’The conditions in Hecate Strait in the winter are well approximated by a depth-averaged wind-driven model: the density stratification disappears in the winter; at mostmeter locations the currents are reasonably approximated by a single depth-averaged current, and are generally confined to a single quadrant; the currents are predominately winddriven (CTT9O). The relationship between the winds and the currents is investigatedfurther in Chapter 3.2.5.2 Queen Charlotte Sound 1982The current meter observations in Queen Charlotte Sound for the summer of 1982 aresummarized in Fig 2.9. At most locations the magnitude of the fluctuations (representedby the principal axes of variance) are comparable to the means. A sensible pattern doesnot leap off the page. The counter-clockwise flow around the Goose Island Bank (metersG01-G07) may be due to tidal rectification (Freeland et al., 1984).-4--n:—.CJ’t)CDe--CDc:,cr0 (-3 Ct, CD CD•CDC4J)•q--0 CDQ.Ct(0..cCD0.(),-0•0CD(a)summer(b)fall(c)winterCD—CDCDabChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 23U)C)c)C÷3CC)U)÷3C)QC)0U)C)Figure 2.9: Currents and winds in Queen Charlotte Sound for the period 4 June - 15September 1982. Vectors of average currents, and winds at Cape St. James and MclnnesIsland are the solid lines and the principal axes of variance are the dotted lines, a)near-surface currents and winds, b) near-bottom currents. (Fig. 7 in Crawford et al.,1985)Chapter 2. Physical Oceanography of the Queen charlotte Islands Region 24Crawford et al. (1985) found that while the wind observations were well representedby the mean and first EOF mode, the currents did not partition neatly into a few modes.This suggests that a simple wind-driven model will not reproduce the currents in QueenCharlotte Sound. Empirical Orthogonal Function (EOF) analysis is discussed in Appendix B.2.5.3 Dixon EntranceObservations from the summer of 1984 (Fig. 2.10 and Fig 2.11) suggest two eddies, one inthe eastern end near Rose Spit and a second near the mouth of Dixon Entrance (perhapscentred over Learmonth Bank). In the summer of 1991 a drifter actually completed twocircuits of the Rose Spit Eddy, thus confirming the existence of a closed loop (W.R.Crawford pers. comm.).Unlike the situation in Hecate Strait, the current pattern in Dixon Entrance does notchange with the seasons. The mean currents from October 1984 to May 1985 (Bowmanet al., 1992) look like the summer currents in Fig. 2.11. The change in the wind regimewas reflected only in the currents at R05, in Hecate Strait, which were southward in thesummer and northward in the winter. The circulation in Dixon Entrance is discussed atlength in Crean (1967), Thomson (1981), Crawford and Creisman (1987), and Bowmanet al. (1992). The consensus is that the Rose Spit Eddy is driven by tidal rectificationand thermohaline circulation. The wind forcing is of secondary importance.A movie of the low-pass filtered current observations shows that when a strong southeast wind starts up in autumn the eddy is temporarily obliterated by a pulse of waterfrom Hecate Strait. The eddy quickly re-establishes itself when the winds subside (W.R.Crawford pers. comm., 1990).A depth-averaged wind-driven model is not a good approximation to the dynamics inDixon Entrance. The water is always stratified and the wind is not the primary forcingChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 25Figure 2.10: Smoothed drifter tracks from Dixon Entrance summer 1984. Tidal currentsare averaged out. The solid circles mark the position of the drifters at the onset of astorm on 21 June 1984, with winds of 15 m s1 from the southeast. (Fig. 3 in Crawfordand Greisman, 1987.)Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 26Figure 2.11: Average currents and winds in Dixon Entrance for the period 22 April-19August 1984. Solid arrows represent currents within 50 m of the surface, dashed lines,currents at intermediate depths and dotted lines, currents within 15 m of the bottom.The wide arrows at D7S and Langara Island represent winds. (Fig. 5 in Crawford andGreisman, 1987.)Chapter 2. Physical Oceanography of the Queen Charlotte Islands Region 27mechanism.2.5.4 Outer Shelf and the Deep OceanThe currents along the shelf break appear to be wind-driven. From southern VancouverIsland to Cape St. James the monthly average currents are generally in the direction ofthe prevailing wind: to the northwest in the winter and to the southeast in the summer(Freeland et al., 1984). Along the west coast of the Queen Charlotte Islands, a warmnorthward flowing surface current, probably wind-driven, has been observed in the winter (Thomson and Emery, 1986). However the water is strongly stratified and poorlyapproximated by a depth-averaged model. Nevertheless, it is important to include thewest coast of the Queen Charlottes and the open ocean in order to provide the outsideconnection between the Queen Charlotte Sound and Dixon Entrance.The circulation and water properties in the open ocean are believed to have onlya minor impact on the circulation on the shelf. The Queen Charlotte Islands sit in abifurcation zone of the offshore currents (Fig. 2.12). As a result, the mean ocean currentsoff the British Columbia coast are generally weak and confused (Thomson, LeBlond andEmery, 1990). There is no feature such as the Gulf Stream to provide a source of energeticeddies or persistent along-shore pressure gradients. Very little is known about the impactof the intrusion of open ocean water masses on the shelf circulation. Besides the weakforcing, the steep shelf break has an insulating effect, partially decoupling the shelfcirculation from the deep ocean (Wang, 1982).Shelf wavesShelf waves, or coastal trapped waves, are sub-inertial waves which travel parallel tothe coastline and are trapped against the coast by topographic gradients (Allen 1980,Mysak 1980). In the northern hemisphere they travel with the coast to the right. TheyBERING rSEA.14F —GYRE A-_r ,.• %•1—-A’1’ ç2——SubarctJc Currentwest Wind DrIflSUBARCTIC BOUNDARyChapter 2. Physical Oceanography of the Queen Charlotte Islands Region 28C 180Ci,,\50 WESTERNSUBARCTICYREIIrJ*PAPA— CWT.I,Figure 2.12: Prevailing surface currents in the North Pacific Ocean. Double arrows areintense boundary currents, speeds typically 1-2 m/s: over most of the rest of the regionspeeds are less than 0.25-0.50 rn/s. Arrows correspond to prevailing winter time flow offthe west coast of North America. (Fig. 13.17 in Thomson, 1981)Chapter 2. Physical Oceanography of the Queen Gharlotte Islands Region 29provide a mechanism by which events to the south can influence the circulation in theQueen Charlotte Islands region. At this time it is not known whether shelf waves havean impact on the circulation around the Queen Charlotte Islands. Shelf waves havebeen observed off the coast of Washington and Oregon (Allen, 1980) and along the westcoast of Vancouver Island (Crawford and Thomson, 1984). Halliwell and Allen (1984,1987) found evidence for forced wave propagation in their studies of the response of sealevel to wind forcing along the west coast of North America (Baja California to PrinceRupert). To date there is no firm evidence for shelf wave propagation in the current meterrecords north of Brooks Peninsula on the west coast of Vancouver Island (Crawford andThomson, 1984; Yao et al., 1984).2.6 SummaryWinter conditions in Hecate Strait are reasonably approximated by a depth-averaged,wind-driven model. In the other regions significant density stratification is present in allseasons and in Dixon Entrance wind forcing appears to be of secondary importance. Theobservations in Hecate Strait are discussed in detail in Chapter 3.At this time there is no compelling evidence that the shelf circulation is dominatedby forcing from the deep ocean or by remotely generated shelf waves. Residual tidalcurrents are an important local effect, especially near Cape St. James and along theescarpment at the northern end of Hecate Strait. The model studies reported in thisthesis concentrate on the effects of local wind forcing.When the water column is stratified, a depth-averaged model is missing an importantpart of the dynamics. This does not mean that the model results can not be used tosuggest interpretations of the observations. However one should be careful.Chapter 3Hecate Strait: Data for Model ComparisonThe period chosen for direct comparison of the model results with current meter observations was January to April 1984. The winter conditions in Hecate Strait make awind-driven, depth-averaged model a reasonable approximation: the winds are strongand the vertical stratification is weak. The winter season is when Pacific cod and Englishsole spawn and the winter currents are an important component of the larval advectionproblem discussed in the Introduction.Drifter trajectories provide information about the circulation that is especially relevant to the study of larval advection and oil spills. Winter drifter studies are rare andthere are none from the winter of 1984. Observations from all seasons are discussed.The chapter is divided into 3 sections: calculation of wind stress from the observedwinds; the winds and the currents; and drifter trajectories.3.1 Calculation of Wind StressThe wind stress is the tangential stress imposed on the ocean surface by the wind. Itrepresents momentum transfer from the atmosphere to the ocean. In numerical oceanmodels the wind stress is calculated from a bulk formula of the formT Pa0d14’i47 (3.1)where r is the surface wind stress, Pa IS the density of air, Cd is the drag coefficient andW is the wind velocity. In general Cd depends on the sea state, the surface roughness, the30Chapter 3. Hecate Strait: Data for Model Comparison 312.521.5c1C.)ci)0C.)I000 5 10 15 20 25 30 35Wind Speed (mis)Figure 3.13: Dependence of drag coefficient Cd on wind speed for winds measured at 10m and neutral atmospheric stability (Smith, 1988).stability of the air over the ocean, the relative humidity and the height of the anemometer.For use with the regional model I have assumed that the atmosphere is neutrally stableand that the drag coefficient depends only on the wind speed. The dependence of thedrag coefficient on wind speed measured at 10 m (Smith, 1988) is shown in Fig. 3.13.Conversion of winds measured at heights other than 10 m is done by interpolation on thetables in Smith (1988). The drag coefficient does not take into account the differencesin effective drag when the wind is blowing with the waves and across the waves (Largeand Pond, (1981). Incorporating such effects into a numerical model would require eithera parameterization based on changes in the wind direction or that the model containedinformation about the wave heights and sea state.The drag coefficient Cd increases when the atmosphere is unstable and decreaseswhen the atmosphere is stable. The stability is measured by the sea-air potential virtualtemperature difference, an adjusted version of the difference between the sea surfacetemperature and the air temperature at lOm (see Smith 1988 for details). For a windChapter 3. Hecate Strait: Data for Model Comparison 32speed of 10 m s1 and sea-air potential virtual temperature differences of (-2, 0, 2) 0 C,the drag coefficients are Cd = (1.21, 1.30, 1.35)x iO. The stability of the atmospherecan make a difference, although it is ignored in this thesis.Most of the wind velocity data for the test period was from land stations— airportsand lighthouses. There was only one ocean-based anemometer: W4S located near currentmeter W04 in Fig. 3.14. The results of Smith (1988) are valid only for wind measurementsmade over the ocean. The time series used to drive the regional model for the winter of1984 comparison was computed from observations at W4S. For the drifter comparisonsthe wind observations were taken from the nearest Atmospheric Environment Serviceweather buoy.There does not exist a systematic method for converting winds measured at a lighthouse station to surface wind stress over the ocean. An approximate wind stress canbe computed by using (3.1) and choosing a reasonable value for the drag coefficient.This means that when lighthouse winds are used to drive a numerical model there is anarbitrary scale factor- the drag coefficient.There are logistical problems related to trying to compute surface wind stress fromobserved winds. Lighthouse wind measurements are usually biased by blockage due totrees or a hill. Winds measured from moored anemometers can have problems when thewaves get high and the anemometer is alternating exposed and blocked. These effectsare not accounted for here.3.2 Wind and Current FluctuationsThe discussion of the wind and current fluctuations in Hecate Strait is organized asfollows. The first section looks at the dominant spatial patterns (mode 1) of the windand current fluctuations. The main results are that the mode 1 currents and windsChapter 3. Hecate Strait: Data for Model Comparison 33Figure 3.14: Locations of current meters, pressure gauges and anemometers in HecateStrait from Jan - April 1984. Fig. 1 from Crawford et al. (1985, with permission).Chapter 3. Hecate Strait: Data for Model Comparison 34represent fluctuations in magnitude (not direction) about the mean currents and winds.The next section looks at the relationship between the transport fluctuations and thewind and the cross-strait pressure gradients. In this context transport is the net flow ratethrough Hecate Strait. The two sections are based on the work of Crawford, Huggett andWoodward (1988, hereafter CHW88) and Crawford, Tyler and Thomson (1990, hereafterCTT9O). In the final section I propose that the mode 1 current fluctuations and thetransport fluctuations are closely related. This has implications for the use of transporttime series in the study of larval advection and fisheries stock-recruitment prediction(Tyler and Crawford, 1991).The collection of the current meter data is discussed by CHW88. The geographiclocations are shown in Fig. 3.14. Meters W05, W06, R06, R07 failed during the Jan toApr 1984 deployment and are not included in any of the analysis.The tidal and inertial currents were removed from the observations with a 2-d lowpass filter. The filter used was an 8th order Butterworth low-pass filter with half powerat 40 hr and 98% transmission at 48 hr. The price paid for this elegant filter (steeptransition zone and minimum ringing) is a wide footprint. Five days must be trimmedfrom each end of every time series to eliminate end effects (Appendix C).The work of CHW88 and CTT9O used the complete data set from 20 January to 4April 1984. Many of these results are discussed in this section. The comparison of theobserved and modelled currents in this thesis was carried out for the period 25 Januaryto 30 March 1984. The current meter data was re-constructed from the original datafiles and much of the analysis repeated for both the 20 Jan to 4 Apr time period andthe 25 Jan to 30 Mar period. Changes in the results of the analysis due to the shortertime series were minor and these changes are noted. The character of the results do notchange. Only selected wind time series were re-constructed.All the times series mentioned in this section were filtered with the same filter as theChapter 3. Hecate Strait: Data for Model Comparison 35currents. The discussion is concerned with time scales of days to weeks.For ease of plotting the time series I have used Julian day notation on the time axis.The Julian day is the ordinal number of the day of the year: 1 January is day 1 and noonon 1 January is day 1.5. The period 25 January to 30 March 1984 corresponds to days25 to 91, 1984.3.2.1 Spatial PatternsIn their analysis of the winter 1984 current meter observations in Hecate Strait, CTT9Omade three observations: 1) there was a flow in Moresby Trough against the dominantwind direction; 2) the mean currents and mode 1 current fluctuations were generally inthe same direction; and 3) the mean and mode 1 current fluctuations were wind driven.The mode 1 currents are the spatial pattern associated with the first mode foundby applying empirical orthogonal function (EOF) analysis to the data. In brief, EOFanalysis finds coherent patterns in the fluctuations of the data. The data is reduced toa set of modes, where each mode has three components: an eigenvalue, an eigenvector,and a time series of modal amplitude. The eigenvalue is a measure of the total energycontained in the mode and is a measure of the mode’s overall importance. The eigenvectorcontains the spatial character of the mode. The time series of modal amplitude recordsthe fluctuations of the mode. EOF analysis is a statistical technique and interpretationof the patterns in terms of physical processes is a separate issue. The details of EOFanalysis are discussed in Appendix B.The pattern of the EOF mode 1 currents and winds for the winter of 1984 are shownin Figure 3.15. The wind and current modes were computed separately. The mode1 currents contain 23% of the energy and the mode 1 winds contain 49% (CTT9O).When interpreting the patterns, the magnitude of a vector is the RMS amplitude of thefluctuations at that point (Appendix B). The whole pattern fluctuates in phase, withChapter 3. Hecate Strait: Data for Model Comparison 36Dates 1984 comments scaled # modes A1 A2 A320 Jan - 1 Apr CTT9O yes 58 23 % — —25 Jan - 31 Mar without R05 yes 58 23% 11% 10%25 Jan - 31 Mar with R05 yes 60 25% 10% 9%25 Jan - 31 Mar with R05 no 60 25 % 10 % 9%25 Jan - 31 Mar depth averaged no 26 39% 12% 11%25 Jan - 31 Mar depth averaged yes 26 31% 12% 11%Table 3.1: Statistics for the current EOF analysis for the winter of 1984. The A arethe first three eigenvalues. Scaled refers to whether each time series was scaled by itsstandard deviation before EOF analysis.the currents at each location directed along the line of the vector. The vectors do notrotate. The time history of the fluctuations is given by the associated time series of modalamplitude. I repeat, the wind and current modes shown in Figure 3.15 were computedseparately and have different modal time series.The current vectors generally lie in the same direction as the mean currents (Fig.2.8c),thus they represent fluctuations in magnitude not direction. The major exception are atM03 where the mean flow is to the west and the mode 1 fluctuations are NE/SW, and atW03 where the mean is to the north, along the strait, and the fluctuations are directedacross the strait. The fluctuations are required to be in phase, thus when the northwardflow at MOl and M02 increases so does the southwesterly flow at M04.The mode 1 winds are an along-shore pattern and the vectors are generally in thesame direction as the means. It is not completely unreasonable to approximate the mode1 winds by a single vector.CTT9O concluded that both the mean and mode 1 currents were predominately winddriven. Their conclusion was based on 1) the high coherence between the mode 1 windsand the mode 1 currents (greater than 0.75 for frequencies less than 0.5 cycles/day); 2)the similarity of the mean and mode 1 currents; and 3) the change in the mean currentsfrom summer to winter when the winds change direction and character (Fig. 2.8).Chapter 3. Hecate Strait: Data for Model Comparison 37Figure 3.15: Vectors of the empirical orthogonal functions for the period 20 January-4April, 1984. Fig. 8 from Crawford et al., 1990.Chapter 3. Hecate Strait: Data for Model Comparison 38For comparison with the model, I depth-averaged the observations before the EOFanalysis. The depth-averaging caused an increase in the relative importance of mode1. Table 3.1 shows how the EOF statistics changed for minor changes in the analysis.The eigenvalues (A) are given in terms of the percentage of the total energy representedby that mode. There are only small changes associated with reducing the length of thetime series and with adding the data from current meter R05 (not included in CTT9O’sanalysis). The column labelled ‘scaled’ refers to whether the time series were dividedby their standard deviation before analysis. Scaling the time series means that oneenergetic time series can not dominate the analysis. Table 3.1 shows that scaling hasonly a small effect on the eigenvalues. The spatial patterns, when plotted in dimensionalform (Appendix B), were not greatly affected by the scaling. The EOF mode 1 spatialpattern computed from the depth-averaged currents has the same character as the patternshown in Fig. 3.15 The pattern is plotted in Fig. 8.64 where it is compared with the modelresults.3.2.2 Water Transport Through the StraitThe transport is the volume of water moving perpendicular to a line across Hecate Strait.In a steady-state situation the transport is the amount of water moving from one end ofthe strait to the other. The standard oceanographic unit of transport is the Sverdriip(Sv): 1 Sv = 106 m3 s1.The computation of the transport from the current meter observations is describedby CHW88. The most reliable estimates were made at W-line (Fig. 3.14). The transportfluctuations (W-line) are compared with the along-shore wind stress fluctuations (W4S)in Fig. 3.16. The time series are obviously related, the peaks line up, but they are notidentical. The time series were normalized by subtracting the mean and dividing bythe standard deviation. The linear trend was also removed. The mean and standardChapter 3. Hecate Strait: Data for Model Comparison 39FluctuationsMean St. Dcv. Correlationmajdir. magn. dir. maj. mm. leadVINDS (deg) (m s) (deg) (m s1) (m s) magn. (h)Cape St James 19 3.4 329 5.5 3.9 0.70 12Sandspit 7 3.1 322 4.4 1.9 0.55 12Bonilla Island 336 6.3 329 6.5 2.9 0.64 6Etheldalsland 330 2.1 320 2.5 1.9 0.74 6Lawyerlsland 340 3.5 327 2.5 1.4 0.46 6Lucylsland 331 4.7 318 5.5 1.6 0.41 6Triple Island 341 5.6 333 5.9 3.4 0.65 6W4S 342 5.3 326 5.7 2.8 0.58 6Mean FluctuationsSPATIAL-AVERAGE WINDEOFI—— 0.71 6AIR PRESSURE (kPa) (kPa)[PRA + Mcl]/2— SA 0.21 0.23 0.69 6SUBSURFACE PRESSUREBeauchemjn—Atlj(B—A) 0.39 0.44 0.91 0[PR + BB]/2— QCC—5.6 0.72 0.83 0PrinceRupert(PR) 141. 0.69 0.85 0Q.CharlotteC.(QCC) 141. 0.51 0.15 12BellaBella(BB) 131. 0.67 0.78 0UPWELLINGIND1CES (lokgs’ m) (10kgs m’)Bakun5l°N—7.6 10.1—0.78 12Bakun54°N—11.3 11.5—0.65 12TRANSPORT (106 m3 s’) (106 m3 s’)R-line 0.29 0.30 0.89 0W-Iine 0.33 0.31 1.00 0Table 3.2: Characteristics of time series in winter, 20 Jan to 4 Apr, 1984. The lasttwo columns give maximum correlations and phase leads in hours of a given time serieswith transport through W-Iine. EOF 1 is the lowest (first) mode of wind from the theempirical orthogonal function analysis. Table ic in Crawford et al.(1988).Chapter 3. Hecate Strait: Data for Model Comparison 40deviation of the transport and wind stress time series were (0.36 Sv, 0.30 Sv) and (0.13Pa, 0.17 Pa) respectively.The EOF mode 1 wind pattern (Fig. 3.15) represents along-shore fluctuations in thewind field. The results in Table 3.2 support the idea that the major wind fluctuationsare parallel to the coast. At all stations the major axis of variance is directed along-strait (roughly 340°T). At most stations the mean wind is also directed along the strait(roughly 340°T). The exception is Cape St. James where the mean wind has a largecross-strait component.The linear correlation between the transport (as measured at W-line) and the variouswind time series range from very good at Ethelda Bay (r=0.74) to not very good at LucyIsland (r=0.41). The correlation with the wind EOF 1 time series was better than withmost of the individual wind time series. This suggests that the basin wide winds aremore important to the transport than the wind at any one location (CHW88).The calculation of the transport based on discrete current meters is potentially limitedby the density of current meters. High frequency spatial variations in the velocity field cancause problems. The transport estimated by the single current meter R05 in R-line agreedvery well with the estimate from W-line (Table 3.2). This increases the confidence in theestimate. CHW88 regarded the estimated transport through M-line as untrustworthybecause it was the small difference between large northward and southward flows (seeFig. 2.8 and Fig. 3.15).The aim of CHW88 was to test time series that could be used as surrogates forthe transport in studies of fisheries stock recruitment in Hecate Strait. The cross-straitpressure difference B-A (Beauchemin Channel minus Atli Inlet) was a very good indicatorof transport (Table 3.2). The high correlation between the transport and the crossstrait pressure difference is an indication that the transport in the strait is in geostrophicbalance (balance between the Coriolis force and the cross-strait pressure gradient). ThereChapter 3. Hecate Strait: Data for Mode] Comparison 41is speculation that the cross-strait pressure difference is a better indicator of transportthan that estimated from the current meters (W.R. Crawford, pers. comm.).The sub-surface pressure gauges at Beauchemin and Atli were in place for only a fewyears. Among the time series with long historical records, adjusted sea level at PrinceRupert (PR) and Bella Bella (BB) were good indicators as were some of the wind timeseries.Part of the motivation for the regional model was the hindcasting of the oceanographicconditions (transport) in Hecate Strait. An important question for this work is, doesforcing the model with local winds improve the ability of the wind to predict the transportthrough the strait? Since the model is driven with wind stress, an assessment of the modelrequires correlation of transport with wind stress. Otherwise an improvement attributedto the model could simply be due to the use of wind stress instead of wind speed.The correlation between the transport and the along-shore component of the windspeed (W) has been repeated for the model test period (25 Jan to 30 Mar 1984) for 4lighthouse stations and W4S (Table 3.3). The correlation increased markedly at CapeSt. James (0.70 to 0.78) due to the shorter time series. The first and last 5 days wereparticularly poorly correlated. The correlations at the other stations stayed much thesame. The changes in the lead of the wind over the transport were due to the changeto hourly time series (from six hourly in CHW88). The correlation analysis was donefor all stations using wind stress time series calculated from (3.1) and a constant dragcoefficient (r1). The final calculation (r2) computed the wind stress at the ocean-basedanemometer W4S using the method of Smith (1988). The stress calculations were donebefore the time series were filtered and the along-shore component computed.The correlations were not strongly peaked. In most cases the correlation coefficientrmax was reduced by less than 0.02 up to four hours on either side of the maximum.The analysis of CHW88 indicated that a sample correlation of 0.7 was drawn from aC)V0.EVN0Station type rmax lead (h)Cape St. James W 0.78 9r1 0.78 9Bonilla Island W 0.68 5T1 0.74 3Ethelda Island W 0.77 5r1 0.77 5Triple Island W 0.67 4Ti 0.69 4W4S W 0.62 8T1 0.72 8T2 0.70 7Table 3.3: Correlation of transport with various wind time series for the period 25 Jan -30 Mar 1984. The maximum correlation rmax was obtained for the lead given (lead of thewind with respect to the transport). Wind time series are the along-shore component ofwind speed (W), wind stress using a constant drag coefficient (T1), and wind stress usingSmith (1988) (T2).Chapter 3. Hecate Strait: Data for Model Comparison 42420-2-495Figure 3.16: The observed transport and along-shore wind stress fluctuations in HecateStrait from 25 Jan— 30 Mar 1984. Both time series have been normalized by subtractingthe mean and dividing by the standard deviation.25 35 45 55 65 75 85JuNan day 1984Chapter 3. Hecate Strait: Data for Model Comparison 43population correlation in the range 0.5 - 0.8 (at 95% confidence limits). One should notplace too much faith in the exact values of the correlation coefficients or the lags.Using r1 rather than W did not increase rmax at Cape St. James, Ethelda Island, andTriple Island. The use of r1 gave a slight increase at Bonilla. The wind stress estimatesi and r2 markedly increased rmar at W4S (from 0.6 to 0.7). r2 did not improve thecorrelation over r. The best correlations at each station came up to the same level. Thesquare of the correlation coefficient is a measure of how well one time series explains theother. Based on r2, the local winds explain roughly 50% of the energy in the transportfluctuations. It is interesting to note that using the wind stress estimates had the largesteffect at the two stations in central Hecate Strait.3.2.3 Transport and the Mode 1 CurrentsTyler and Crawford (1991) provided evidence that the transport fluctuations in HecateStrait are an important predictor of the fluctuations in the year-class strength of Pacificcod. No specific mechanism for the influence was proposed. CTT9O suggested that themean and mode 1 current patterns were important for the advection of larvae in HecateStrait. In hindsight an obvious question is, are the transport fluctuations and the mode1 current pattern related?The time series of the EOF mode 1 current fluctuations is compared with the transport fluctuations in Figure 3.17. To allow comparison the mean was removed from thetransport and both time series were scaled by their standard deviation. They are almostthe same time series. The correlation between the transport and the mode 1 currents iscloser than the correlation between the transport and local winds (rmax = 0.9, comparedwith rma = 0.7 — 0.8 for the winds). The current pattern shown in Fig. 3.15 is associated with the movement of water from one end of Hecate Strait to the other. For use inFig. 3.17 the current meter data was normalized before the EOF modes were computed.Chapter 3. Hecate Strait: Data for Model Comparison4a,0EcaVC,N-2-44495Figure 3.17: Comparison of the transport and mode 1 currents for the period 25 Jan - 31Mar 1984. Both time series have been normalized by subtracting the mean and dividingby the standard deviation.25 35 45 55 65 75 85Juliari day 1984Chapter 3. Hecate Strait: Data for Model Comparison 45When the currents are not normalized before analysis, mode 1 current fluctuations andthe transport fluctuations are identical.The relationship between transport and the mode 1 currents implies that the wintertransport fluctuations are associated with a particular current pattern. This providessome reason to believe that transport fluctuations are related to advection of fish eggsand larvae. On the other hand, the 75% of the energy in the current fluctuations notassociated with mode 1 is not related to transport fluctuations. This places a hard limiton predicting advection patterns from transport (or surrogate transport) series. Thisrelationship has not been tested for the summer and fall data sets.3.3 DriftersMost of the drifter observations have taken place in the summer, when the weatherallows drifters to be recovered and redeployed. The drifters are intended to track thenear surface waters. In general the drogue is a 5-m-high holey sock centered at 10 m. Afew have drogues centered at 3 m.For several years the drifter path shown in Fig. 3.18 represented most of the drifterdata in Hecate Strait. The drifter was seeded to the east of Cape St James with theexpectation that it would drift west past the Cape and get caught in an eddy. Instead itremained trapped in Queen Charlotte Sound for 6 weeks before escaping north throughHecate Strait and into Dixon Entrance. The winds during this first period (Septemberand early October 1983) were variable. This drifter illustrates that the trapping of driftersin Queen Charlotte Sound near Cape St James is possible.In late October the drifter travelled north along Ilecate Strait under the influence ofstrong winds from the SE. The drifter entered Dixon Entrance at its eastern end andwent in and out of Clarence Strait before moving west along Dixon Entrance and on intoChapter 3. Hecate Strait: Data for Model Comparison 46Figure 3.18: Path of satellite tracked drifter for 9 Sept — 29 Nov 1983. The winds atCape St. James are shown in the lower left corner (1 day low-pass filtered). The free endof the wind vectors point in the direction the wind is blowing to. (Adapted from Fig.13Hannah et al., 1991.)Chapter 3. Hecate Strait: Data for Model Comparison 47the Pacific.The drifters in Fig. 3.19 were part of an investigation into tidal rectification aroundCape St James in March 1986 (Thomson and Wilson, 1987). The winds were generallyfrom the south. The two drifters (1190, 1195) seeded near the coast on either side ofthe island travelled along the coastline in the direction of the wind. Drifters 1191 and1193 escaped from Queen Charlotte Sound and illustrate the strong outflow near CapeSt. James.An important component of the overall project of which this thesis forms a part isan extensive program of near surface drifter studies. The first of these was conducted insouthern Hecate Strait in the summer of 1990. The drifters in Fig. 3.20 are representativeresults from this deployment. The winds were generally from the northwest. The drifterswere drogued at 10 m. Of the four drifters shown, three were seeded in the same area atdifferent times and illustrate the variability in the flow. Drifters A and C illustrate thecirculation up Moresby Trough against the wind and around the bank. These tracks arecompatible with the summer mean currents in Fig 2.8. Drifter D, seeded later, went inthe opposite direction. It moved to the north and then to the southwest where it followeda path similar to drifter B, which was seeded closer to Cape St James. These two drifters(B,D) travelled past Cape St James and across the shelf break and then looped backinto the mouth of Queen Charlotte Sound. These two drifters indicate significant surfaceflows in and out of the mouth of Queen Charlotte Sound near Cape St James.Strong outflow past Cape St James is often observed during NW winds. The outflowis believed to be driven by tidal rectification (Thomson and Wilson 1987) and estuarineoutflow. The summer 1990 drifters confirm that the outflow is confined to an area veryclose to the Cape. Gray Rock is oniy 15 km southeast of Cape St. James and 10 of 12drifters which escaped past Cape St James in July 1990 went between the Cape and GrayRock. Drifters B and D were the only two to escape south of Gray Rock. The dynamicalChapter 3. Hecate Strait: Data for Model Comparison 48Figure 3.19: Drifter tracks near Cape St. James for the period 12—20 March 1986.(Adapted from Fig. 12a in Thomson and Wilson, 1987.)131°WChapter 3. Hecate Strait: Data for Model Comparison 49Figure 3.20: Drifter tracks in southern Hecate Strait from 8—29 July 1990. The windswere from the NW or N. (Adapted from Fig. 14 in Hannah et al., 1991.)Chapter 3. Hecate Strait: Data for Model Comparison 50processes believed responsible for this outflow are not part of this model. Modelling thisoutflow feature is further complicated by its small spatial scale (15 km).The tidal excursion at Cape St. James can be up to 10—15 km which makes driftertrajectories near the cape strongly dependent on tidal phase as the drifters approach it.The two drifters shown in Fig. 3.21 along with the ones from March 1986 (Fig. 3.19)represent all of the winter drifter data in Hecate Strait. These satellite tracked drifterswere seeded in January 1991 to provide some test data for the regional model. One drifterfloated out of the mouth of Queen Charlotte Sound and travelled up the outside of theQueen Charlotte Island. This drifter made a short excursion into the mouth of DixonEntrance. The other drifter stayed around Middle Bank and Moresby Trough for a longtime until it grounded in a small inlet.The drifter program in July 1991 concentrated on northern Hecate Strait. This regionis most likely to be well described by a wind-driven, depth-averaged model as the wateris shallow and well mixed even in the summer.Fig. 3.22 shows selected drifter tracks that illustrate the character of the flow. Thedramatic change in direction of three drifters (a35, b40, c22) corresponds to the changein direction of the wind around July 11. These drifters were all drogued at 3 m. Thedrifters in the deeper water actually turned around slightly later than the drifters in theshallow water (W.R. Crawford pers. comm., 1992). The fourth drifter, b32 (drogued at10 m) was launched just after the wind shift.Drifter c22 illustrates that the drifters tend to move along the steep escarpmentextending NE from Rose Spit and that movement from Hecate Strait into Dixon Entrancetends to be confined to the region around Dundas Island. This behaviour was observedin other drifters as well.The two drifters in Fig. 3.23 started at almost the same location and the same time.Drifter A was drogued at 3.5 m and B at 10 m. These drifters were put in place on JulyFigure 3.21: Paths of two satellite tracked drifters from 19 January to 20 March 1991.The starting locations are marked by the two black dots east of Cape St. James. Thewinds were generally from the SE.Chapter 3. Hecate Strait: Data for Model Comparison 51flChapter 3. Hecate Strait: Data for Model Comparison 52Figure 3.22: Drifters in northern Hecate Strait from July 1991. The abrupt changein direction displayed by drifters a35, c22, and b40 corresponds to the change in winddirection from NE to SE on July 10. Drifter b32 was launched after the wind changeddirection. The arrows on the drifter paths mark 2 days elapsed time.Chapter 3. Hecate Strait: Data for Model Comparison 53Figure 3.23: Two drifters in northern Hecate Strait from 11 to 13 July 1991. The drogueon drifter b32 was centred at 10 m, the drogue on b34 at 3.5 m. The two drifters werelaunched at the same time and b34 (shallow) travelled twice as far as b32 (deep). Thearrows mark 1 day elapsed time.Chapter 3. Hecate Strait: Data for Mode] Comparison 5411 just after the wind shifted to the southeast (Fig. 3.22). The difference in distancetravelled over two the days was quite dramatic. The complete trajectory of drifter Bis shown in Fig. 3.22 (drifter b32). These two drifters illustrate that there was verticalshear even through the water column was well mixed (W.R. Crawford, pers. comm.).3.4 SummaryIn Hecate Strait during the winter the local winds provide the primary mechanism forforcing velocity fluctuations with time scales of days to weeks. For the period Januaryto April 1984 observation period, the 2-d low-pass filtered transport fluctuations werehighly correlated with the local wind (filtered in the same manner). The mean velocitypattern is also believed to be wind-driven. In northern Hecate Strait the near-surfacedrifters reversed direction when the wind reversed.The observations indicate that a model of the depth-averaged currents, forced bylocal winds, should simulate the transport fluctuations in Hecate Strait and capture thecharacter of the current fluctuations. For the test period 20 January to 30 March 1984the model should exhibit the following behaviour.• The model should provide a good simulation of the transport fluctuations. Thecorrelation between the simulated transport and the observed transport should begreater than r 0.7, the maximum value of the correlation coefficient correlationbetween the winds at W4S (used to driven the model) and the observed transport.• The model currents should capture the character of the mean and EOF mode 1current pattern. In particular there should be a counter-current in the middle ofM-line. The currents in the middle of M-line should be directed in the oppositedirection to the currents at the edges of M-line and at R-line. This should be trueof the mean currents and the EOF-mode-1 currents.Chapter 3. Hecate Strait: Data for Model Comparison 55• The simulated transport fluctuations and EOF-mode-1 current fluctuations shouldbe highly correlated, as they are in the observations.It is difficult for a depth-averaged model to provide accurate simulation of driftertrajectories when even the well mixed water in northern Hecate Strait exhibits significant vertical current shear. However comparison of individual trajectories will be donefor selected drifters from July 1990 and July 1991. A more general comparison of thesimulated drifters with the general character of the observed drifters will also be carriedout.Chapter 4Low Frequency Flow in Sea StraitsThe purpose of this chapter is to develop the rotation-limited-flux model and apply itto Hecate Strait. Before delving into rotation-limited-flux, it is useful to introduce twoimportant concepts: geostrophic balance and frictional adjustment. While the discussiontakes place in the context of channel flow, both have wider application.The sea strait under consideration is illustrated in Fig. 4.24 and Fig. 4.25. Thestrait has a flat bottom, uniform sides and the along-strait velocity is uniform in boththe horizontal and vertical directions. The cross-strait velocity is ignored. The slopeof the sea surface is exaggerated to emphasize the fact that the sea surface is not levelin rotating channel flow: it slopes in both the cross-strait and along-strait directions.Two additional assumptions are made. Attention is restricted to low frequencies so thatthe assumption of cross-strait geostrophic balance can be made, and the velocities areassumed small enough so that linear models are appropriate.4.1 Cross-strait Geostrophic BalanceOne of the important force balances in oceanography is geostrophic balance: the balancebetween the Coriolis force and the pressure gradient. When studying the low frequencyflow in sea straits, geostrophic balance is assumed to hold in the cross-strait direction.To first order, the cross-strait sea-surface slope balances the Coriolis force arising from56T773 15f_____L_____Figure 4.25: Plan view of the strait. ‘h and 772 are the far-field sea level elevations. Theother Tb are the sea levels at the respective corners. The corner labelling scheme is forthe northern hemisphere.Chapter 4. Low Frequency Flow in Sea Straits 57Figure 4.24: Schematic diagram of Hecate Strait.77i 772Chapter 4. Low Frequency Flow in Sea Straits 58the along-strait velocity:fu = (4.2)where f is the Coriolis parameter, u the along-strait velocity, g the acceleration due togravity, the departure of the sea surface from its mean level, and y the cross-straitcoordinate. The cross-strait velocity is ignored. For a fluid of constant density, the seasurface gradient and the pressure gradient are equivalent: V(p/p) = gV, where p isthe pressure and p the density of water. The identity of the sea surface and pressuregradients is assumed throughout this thesis.Cross-strait geostrophic balance means that flow through the strait affects the sealevel at the coast. When looking in the direction of flow, the sea surface slopes up tothe right and down to the left (in the northern hemisphere). In the southern hemispheref < 0 and the sea surface slopes up to the left and down to the right. Cross-straitgeostrophy is the reason for the correlation between the transport through Hecate Straitand the cross-strait pressure difference (Chapter 3).4.2 Frictional AdjustmentConsider an infinite channel with uniform width and depth where the earth’s rotationis ignored. The assumption of infinite length allows one to ignore along-strait pressuregradients. Let the depth-averaged, horizontally uniform, along-strait velocity be u andassume that the wind stress r is parallel to the axis of the strait (Fig. 4.25). In the absenceof along-strait pressure gradients, the dynamic balance is between the acceleration, thewind stress, and the bottom friction; a relationship of the form:+ An = - (4.3)where A is a linear friction coefficient, h is the depth of the water, and p is the densityof the water. This equation has some simple solutions.Chapter 4. Low Frequency Flow in Sea Straits 59Let the fluid start at rest, u(O) = 0. The solution for a constant wind stress r0 appliedat t = 0 isu(t) = i(1 — e_At) (44)This is exponential relaxation to the steady state velocity ‘u = ro/ph) with e-foldingtime )1• The time scale )J is referred to as the relaxation time, the frictional adjustment time, the spin-up time, and the spin-down time. The solution (4.4) is also thesolution for the velocity at the coast in a semi-infinite ocean, where the coastal boundarycondition (u=0 at the coast) suppresses the Coriolis terms (Roed and Cooper, 1987).The solution for an oscillating wind r = ro sin(wt) is/ U00 / WU00—/ut) = sinL’t— q5) + e+ (w/A)2 1+ (/A)2where tan=The solution for To/ph = 1, w = 1, ) = w/6 is shown in Fig. 4.26.The parameters were chosen so that the two time scales were about the same (1T 2’ir/). The initial overshoot is clearly visible.The simplest model of flow in rotating straits assumes that frictional adjustment(4.3) holds in the along-strait direction and that geostrophic balance (4.2) holds in thecross-strait direction. What is ignored here is the along-strait gradient. The next sectionthe along-strait pressure gradient in introduced. The frictional adjustment solution isrecovered in the non-rotating limit or the limit of W/L —* 0.4.3 Rotation-limited-fluxRotation-limited-flux is a conceptual model of the influence of the finite length of thestrait on the low-frequency flow. The work presented here is based on the original derivation by Garrett and Toulany (1982, hereafter GT) and the extension to include the localwind forcing by Hannah (1992a). The low frequency sea level response can be characterized in terms of two modes: a pressure-forced mode where the sea surface slopes downChapter 4. Low Frequency Flow in Sea Straitsa)-oE602101-2Figure 4.26: Frictional adjustment with sinusoidal forcing. The solid line is the velocityand the dotted line is the wind stress. The frictional scale ) = w/6.0 1 2 3 4 5Chapter 4. Low Frequency Flow in Sea Straits 61in the direction of flow, and a wind-forced mode where the sea surface slopes up in thedirection of flow. Both modes are assumed to be in geostrophic balance in the cross-straitdirection, and therefore have cross-strait sea surface slopes. Iii the literature, the idea ofrotation-limited-flux is also referred to as geostrophic control.Hannah (1992a) compared the rotation-limited-flux model with an analytical modelof flow in straits of finite length applied to Bass Strait (Middleton, 1991; Middleton andViera (1991), hereafter MV). Their model deals explicitly with the ocean/strait couplingand the scattering of the Kelvin waves at the ends of the strait. For fluctuations with aperiod of 240 hours, the pressure-forced and wind-forced sea level responses derived fromrotation-limited-flux were shown to be identical to those of MV.The agreement between the rotation-limited-flux model and the results of MV arisesfrom the following. When the width of the strait is less than one external Rossby radius(j7i/f), the low frequency (long wavelength) Kelvin waves have a uniform sea surfaceslope over the entire strait. When this is true, the primary assumption made in thederivation of rotation-limited-flux, that the derivatives in the equations of motion can bereplaced by differences, is valid. In those cases, the rotation-limited-flux model providesuseful insight into the low-frequency dynamics of the strait.This section contains a brief derivation and a short discussion of the application toHecate Strait. The details of the application to Bass Strait and the comparison with theresults of MV are part of a paper in press (Hannah, 1992a). The original work, arisingfrom this thesis, is the extension of the results of GT to include direct wind forcing, thedemonstration that the rotation-limited-flux model is a valid long-wave approximationto the more sophisticated model of MV, and the application to Hecate Strait.Chapter 4. Low Frequency Flow in Sea Straits 624.3.1 DerivationConsider the same strait as before (Fig. 4.25) but now include the Coriolis force andboth cross-strait and along-strait pressure (sea-level) gradients. The strait has lengthL, width W and uniform depth h. Assume that the strait connects two infinite oceans.The coordinate origin is the lower left corner (labelled 774). Bottom friction is againparameterized by )u, where ) = r/h, and r is a friction coefficient. Following GT andMV, cross-strait geostrophic balance is assumed to hold and in the along-strait directionthe local acceleration is balanced by the pressure gradient, friction, and wind stress. Themodel equations arefu = —g (4.6)ãu th7= (4.7)where F = T/(ph). The cross-strait velocity is ignored.The cross-strait pressure difference is approximated by774—773= 776 — 775 = —(f/g)uT’V (4.8)and the along-strait pressure difference isLOu L L (4.9)got g gThe equation for the along-strait velocity, found by subtracting (4.8) from (4.9) andsolving for u is+ ( + fW/L)u = (774 — + F (4.10)This has the same form as the frictional solution (4.3). The strait responds to a suddenlyimposed forcing with time-scale )1, where = \ + fW/L. The effect of the finitelength of the strait disappears in the limit fW/L —* 0.Chapter 4. Low Frequency Flow in Sea Straits 63Consider the forced response to a single driving frequency, such that u,,and F varyas It is useful to separate the flow into 2 parts: the pressure-driven flow and thewind-driven flow. Let u = Up + Uw, where=. (4.11)p zw+;\+fW/L= iw + + fW/L(4.12)The sea level difference 774— 175 is an external forcing term, like the wind stress. Itrepresents the pressure forcing due to the sea level difference between the two basins. Alleffects related to the difference in sea level between the two basins are associated withUp. The effects of the local winds are contained in Uw. The ratio of the velocities issimply the ratio of the forcing terms.In the limit w,\ <<fW/L, the along-strait velocity can be written— i) + FU = Up + Uw= fW/L (4.13)This is the maximum flow rate through the strait. The effect of friction and time variationis to further reduce the flux. This is the origin of the name rotation-limited-flux.The steady-state limit (w = 0) of the wind mode is equivalent to that derived byBaines et al. (1991) for use in Bass Strait (their equation 14) and is subject to all thelimitations discussed therein. The pressure mode corresponds to the result of GT, when174 and take on the appropriate far field values: ‘175 —* 772, 774 —f 77i. The validity ofthis identification is part of the ocean/strait coupling. When dealing with observationsone is not always able to distinguish between say 17 and and one is forced to use theavailable data.Discussion of the ocean/strait coupling in the context of rotation-limited-flux (geostrophic control) can be found in GT, Toulany and Garrett (1984), Wright (1987), Rocha andClarke (1987), Tang (1990), Pratt (1991), and Hannah (1992a). The coupling requiredChapter 4. Low Frequency Flow in Sea Straits 64by rotation-limited-flux is consistent with that derived by MV. Therefore the rotation-limited-flux model is a consistent approximation to the more complete theory.A consistent solution in the southern hemisphere (f < 0) can be derived by recognizing that and are the upstream points with respect to Kelvin wave propagation.Leave the coordinate system untouched (origin in the lower left corner) but interchange14 and and interchange ‘p15 and 76. With this labelling convention the along-straitvelocity can be written—ii) + Fu=up+uw= iw++fW/L (4.14)which is valid for both hemispheres.The sea level structure associated with each mode can be derived by assuming that thesea level varies linearly within the strait. Following the notation in MV, let wind-drivensea level be iw(x, y, t)iw(x,y,t) = aiNw(x,y)wherea1 = rLet/(pgh)The sea level structure of the wind mode (in the northern hemisphere) can be written asN = aiong(x/L) + Acr03s(y/W)where 7along is the along-strait sea level difference due to the local wind and-7acrossis the corresponding cross-strait sea level difference. The along-strait sea level differencefound by substituting (4.12) into (4.9) isI iw+\1a1ong (l7s— i3)w a1 1— . + +(4.15)which may be simplified by defining(‘15 — 713)W a1Ke8Chapter 4. Low Frequency Flow in Sea Straits 65where0 = tan’( )K = F[(F + )2 + w2]_h/2F=fW/L1’ is the rotation-limited-flux parameter.The cross-strait sea level difference (substitute (4.12) into (4.8)) is equal to the along-strait pressure difference. This means that the diagonal pressure term (14 — 1)w isidentically zero. The no pressure forcing condition defines the coupling of the strait tothe ocean for the wind mode and is consistent with the interpretation of Up and UW aspressure and wind modes. The sea level structure of the wind-driven mode isN = I(e°(x/L — y/W) (4.16)The steady-state limit of the sea level response (aiNw) is equivalent to that derived byflames et al. (1991) (their equation 13).The sea level for the pressure mode is derived in a similar manner. Let(x,y,t) =ci2Np(x,y)where a2= ( — s)ei1t. The sea level structure of the pressure mode isNp = 1 — (1 — Ke°)(x/L)— Ke8(y/W) (4.17)The expressions for the sea level response structures in the southern hemisphere can berecovered with the substitution y = 1 — y/W (Hannah, 1992a). The necessity for this isa consequence of the labelling convention for the southern hemisphere.Chapter 4. Low Frequency Flow in Sea Straits 664.3.2 Wind and Pressure Driven Sea Level ResponsesFor the purposes of illustration consider the application of rotation-limited-flux to HecateStrait. Parameter estimates are given in Table 4.4: the frequency corresponds to a periodof 6 days. The parameter F is greater than and )., but all three are of the same orderof magnitude. For these parameters, the steady-state velocity is reduced by a factor of 4compared with the non-rotating case:—I- F) = 1/4.parameter valueCoriolis parameter f 1 x 10 sfrequency w 1 x 10 swidth W 50 kmlength L 150 kmdepth h 50 mfriction coefficient r 5 x i0 m sRossby radius /7/f 220 kmW/L 0.3=r/h lx 105sF = IfIW/L 3 x i0 sK 0.70 14°Table 4.4: Estimates of parameters for Hecate StraitThe phase lag 0 is reasonably small and the sea level response modes can be discussedin steady state terms. For the wind mode (Fig. 4.27) the wind is blowing from left toright and the sea surface is tilted up in the direction of flow. This set-up is the definingcharacteristic of the wind mode. The pressure mode (Fig. 4.28) is characterized by tiltingdown in the direction of flow. By assumption, both modes are in geostrophic balance inthe cross-strait direction. When looking in the direction of flow, they slope up to theright in the northern hemisphere and up to the left in the southern hemisphere.The set-up character of the wind mode can be seen by considering the steady-stateChapter 4. Low Frequency Flow in Sea Straits 67LFigure 4.27: The sea level structure for the wind-driven mode, N.N0N—+NFigure 4.28: The sea level structure for the pressure-driven mode, Np.Chapter 4. Low Frequency Flow in Sea Straits 68limit of the along-strait pressure difference (4.15)( — = a1 (i— 1 + F/A) (4.18)The first term is the pressure gradient, or set-up, that would arise if the down-wind endof the strait was closed. The second term is the reduction in the slope necessary to letthe appropriate amount of water through. When the set-up is suppressed, the effectof rotation-limited-flux completely disappears and the frictional adjustment problem isrecovered.The along-strait structure can be hidden by combining the modes so that the along-strait gradients cancel and the combined sea level field looks like simple geostrophy. Inthe low frequency limit the condition is a1/a2 = A/I’. For Hecate Strait this wouldrequire that the pressure forcing be 5 times the wind forcing.Hecate Strait is less than one Rossby radius wide (Table 4.4), so the rotation-limited-flux model should provide a useful description of the low-frequency sea level fluctuations.4.3.3 Comparison with Observations in Hecate StraitThe work on rotation-limited-flux was motivated by the sub-surface pressure observationsshown in Fig. 4.29. The data was filtered with the same 2-d low-pass filter as the currentsand the time series were normalized before the EOF modes were computed. In both thefall and winter data, the first mode contained roughly 80-90% of the variance due toits large vertical excursion, and the second mode contained roughly 8%. In Table 4.5each mode has two numbers listed for each station. The first is the component of theeigenvector corresponding to that station. This dimensionless number is a measure of thestation’s contribution to that EOF mode. For both seasons, all the stations contributeequally to mode 1 but some stations do not contribute to mode 2. The second numberis the dimensional component plotted in Fig. 4.29 (see Appendix B for details).chapter 4. Low Frequency Flow in Sea StraitsPressure gauge locations69Figure 4.29: Spatial patterns of the observed sub-surface pressure fluctuations in HecateStrait. Plotted are the first two modes of the empirical orthogonal function analysisfor the fall of 1983 and the winter of 1984. The modal amplitudes are in cm of water.The arrow in Hecate Strait is the direction of the geostrophic current implied by thecross-strait pressure difference. The data are listed in Table 4.5.\( Prince RupertWelcome HarbourAtliHeater BeaucheminHarbourBella BellaHakaiFaIl 83 Winter 84Mode 1Mode 2Chapter 4. Low Frequency Flow in Sea Straits 70Fall 1983 Winter 1984Station mode 1 mode 2 mode 1 mode 2cm cm cm cmPrince Rupert 0.37 6.7 0.44 3.0 0.45 6.5 0.48 3.1Welcome 0.38 7.3 0.41 3.0 0.45 6.5 0.48 3.1Atli 0.41 3.9 -0.77 -3.3 0.41 3.9 -0.77 -3.3Beauchemin 0.39 8.3 0.18 1.4 0.47 7.3 0.05 0.3Heater Harbour 0.35 4.9 -0.58 -3.1 — — — —Bella Bella 0.39 9.1 -0.005 -0.04 0.45 6.3 0.20 1.2Hakai 0.40 8.2 0.02 0.2 — — ——[ eigenvalues 6.3 0.9 ] 4.4 0.89Table 4.5: Sub-surface pressure EOF modes in Hecate Strait. Each mode has twocolunms: the dimensionless eigenvector, and the eigenvector with units of cm of water. The eigenvalues are dimensionless. Fall 1983: 28 September to 20 December 1984.Winter 1984: 20 January to 4 April 1984. (Courtesy W.R. Crawford.)After considerable studying of Fig. 4.29, two points emerge. For a given season, themodes have: 1) comparable cross-strait pressure differences and 2) comparable along-strait pressure differences. If one assumes that a cross-strait pressure differences impliesan along-strait current in geostrophic balance, then the modes represent comparablecurrents. Mode 1 has the along-strait gradient in the direction of the implied currentand mode 2 has the along-strait gradient opposing the implied current.Mode 1 can be interpreted as a pressure-driven rotation-limited-flux mode. The aimof the work discussed in Section 4.3 was to learn if mode 2, with the along-strait pressuregradient opposing the flow, could be interpreted as a wind-driven mode. There are otherpossible explanations for the along-strait gradients. They could be artifacts imposed bythe requirement that the modes be orthogonal, or they could be due to complex localeffects.When applied to Hecate Strait, rotation-limited-flux makes the following predictionsabout the qualitative structure of the sub-surface pressure modes. (The sea levels inFig. 4.27 and Fig. 4.28 are equivalent to the observed sub-surface pressure fluctuations.)Chapter 4. Low Frequency Flow in Sea Straits 71The pressure mode (EOF mode 1) should have:• an along-strait pressure gradient sloping down in the direction of the implied current.• high correlation with the sub-surface pressure fluctuations at all the observationlocations in Hecate Strait.• the same sign at all locations. The sub-surface pressures rise and fall together.The wind mode (EOF mode 2) should have:• an along-strait pressure gradient opposing the implied current.• a sign change across the strait. When the sub-surface pressure goes up on one sideof the strait it should go down on the other.• high correlation with the sub-surface pressure fluctuations at Atli and Heater Harbour (equivalent to location W in Fig.4.27).• high correlation with the sub-surface pressure fluctuations at Welcome Harbour(location L in Fig.4.27).• low correlation with the sub-surface pressure fluctuations at Beauchemin (equivalent to location 0); note the location of the zero line in Fig.4.27.The observed modes have all these properties. Bella Bella and Hakai are outside ofHecate Strait.The rotation-limited-flux model also predicts that both modes should be poorly correlated with the sub-surface pressure fluctuations in the north-west corner of Hecate Strait.This prediction can not be checked because there are no observations in that area.Chapter 4. Low Frequency Flow in Sea Straits 72Direct use of rotation-limited-flux as a model of the transport fluctuations in HecateStrait raises some problems:• Estimating the pressure forcing requires observations in the northwest corner inorder to compute 774—?7, and there are no observations there. In the HecateStrait Model, the pressure forcing in the strait arises naturally as a consequence ofmovement of the sea surface over the whole Queen Charlotte Islands region.• The estimate of the flux, Q = Whu, requires estimates of W and h. Given the strongbathymetric variations in Hecate Strait, the rotation-limited-flux model would contain 4 adjustable parameters: 14/, L, h, and ). This is more adjustable parametersthan are in the Hecate Strait Model.Further application of rotation-limited-flux to Hecate Strait is limited to interpreting theresults of the Hecate Strait Model.Chapter 5The Hecate Strait ModelThe Hecate Strait Model is the first non-tidal model of the region. The decision made wasto start with simple physics and create a model where the model was simpler than thereal world and the assumptions were obvious rather than obscured by layers of numerics.To this end, the equations of motion were considered in their depth-averaged form whichinvolves two restrictive assumptions:• Density gradients do not contribute to the dynamics: the density field is replacedby the mean density.• Vertical shear is ignored: the velocity vectors of the fluid elements in a column ofwater are replaced by a single vector.These are important restrictions and they severely limit the practical application of themodel. On the other hand, the restrictions make the problem tractable by reducing thesize of parameter space. The model results can be used as a guide for more sophisticatedmodels.This chapter starts with a discussion of the shallow water equations which form thebasis for the Hecate Strait Model. This is followed by a brief look at the finite differencingscheme. The balance of the chapter is devoted to an overview of the technical aspects ofthe model. Many of the details are contained in Appendix D.The computer program that implements the numerical model is a modification ofsoftware provided by W.P. Budgell (Buckley and Budgell, 1988). My contributions to73Chapter 5. The Hecate Strait Model 74the software development are discussed in Section 5.7.5.1 The Depth-Averaged, Non-Linear Shallow Water EquationsThe numerical model is based on the shallow water equations and these consist of statements of conservation of mass and momentum. The equations are used in depth-averagedor vertically integrated form. The coordinate system is defined in Fig. 5.30. The horizontal velocity vector has components (u, v) and the departure of the sea level fromthe average height (H), the sea level anomaly, is denoted .Conservation of mass, after integration over the fluid depth, is writtenV (iYh) = (5.19)where h = H + i is the total depth and t is the time. A net influx of fluid into a controlvolume is accompanied by a rise in sea level.Conservation of momentum in the horizontal, after depth-averaging, is written as—- + v Vv = —fk x v gVi + G (5.20)where f is the Coriolis parameter, k is the unit vector in the vertical direction, g is thegravitational constant, and G(x, y, t) represents the forcing and dissipation. Equation(5.20) is simply Newton’s second law, with the acceleration of a fluid parcel being equalto the sum of the forces, which include the Coriolis term, pressure gradients, friction andbody forces. The approximations, pararneterizations, and outright fudges are restrictedto the terms that make upThe derivation of the shallow water equations from the Navier-Stokes equations in arotating reference frame can be found in standard texts such as Pedlosky (1979). Thefundamental assumption is that the horizontal scale of the motion is much greater thanthe vertical scale and the equations consider nearly horizontal motion: thus the shallowChapter 5. The Hecate Strait Model 75zz=Of/2y,v (x,y,t)g‘VFigure 5.30: Sketch of the ocean defining the axis system.Chapter 5. The Hecate Strait Model 76water equations. Based on these scaling arguments, conservation of momentum is separated into vertical and horizontal components. In the vertical the assumption aboutthe scales of the motion leads to the hydrostatic approximation, which reduces the vertical momentum equation to a balance between the vertical pressure gradient and thegravitational potential= —pg (5.21)where p is the density, p is the pressure and z the vertical coordinate (positive up). Thehydrostatic approximation has been used to express the pressure in terms of the sea levelanomaly in the horizontal momentum equation (5.20).The forcing and dissipation C takes the form(5.22)where (x,y,t) is the wind stress and A(x,y,t) is the bottom friction. These surfacestresses appear as body forces rather than boundary conditions because the flow is integrated over the water column. The last two terms represent internal frictional forces.The friction terms are discussed later in this chapter.For the purposes of this thesis an f-plane model is applied, whereby the Coriolisparameter f is assumed constant over the model domainf = 2singS = 1.15 x i0s (5.23)where = 7.29 x iO s, the rotation rate of the earth, and = 52.5°N the latitude.Equations (5.19) and (5.20) are the depth-averaged, non-linear, shallow water equationson an f-plane.Chapter 5. The Hecate Strait Model 775.1.1 Topographic SteeringAnother view of the motion is found by taking the curl of (5.20) to form the vorticityequation:(5.24)where the relative vorticity = . V x is the vertical component of the full relativevorticity vector. Equation (5.24) can be reorganized to yield(5.25)where the material derivative isD 8-n =(+.V)and the potential vorticity is defined byq= (C+f)h1.Equation (5.25) is a statement of conservation of potential vorticity.In the limit V x G 0 (no friction, no surface stresses), << H, and << fconservation of potential vorticity reduces to=0 (5.26)In the absence of other things, low frequency flow follows contours of f/H. When f isconstant, the flow is parallel to the depth contours. This is called topographic steering.5.1.2 Geostrophic BalanceThe equations used to develop the conceptual models are simplifications of the shallowwater equations. Consider the strait in Fig. 5.31, the same strait used in Chapter 4.Chapter 5. The Hecate Strait Model 78T‘13 ‘15‘1’ ‘12Figure 5.31: A sea straitLet (zt, v) be the along-strait and cross-strait velocities respectively. Let the wind bedirected along-strait. Assume that the only friction is linear bottom friction and ignorethe non-linear term. The horizontal momentum equation reduces to8u r— fv =—g— + — — (5.27)8v 071+ fu = —g-- — (5.28)Under the further assumption that there is no significant cross-strait velocity (5.28)reduces tofu = (5.29)Equation 5.29 is an expression of geostrophic balance. The Coriolis force is balanced bythe pressure gradient created by the sloping sea level. Looking in the direction of flowthe sea level must be higher on the right-hand- side (in the northern hemisphere). Thisrelationship, which is valid for low frequency flow in sea straits, is the reason for the highcorrelation observed between the transport through Hecate Strait and the cross-straitpressure difference (Chapter 3).Equations 5.29 and 5.27 were used in the development of geostrophic control in Chapter 4.Chapter 5. The Hecate Strait Model 795.2 Numerical FormulationThe numerical integration of the shallow water equations requires conversion of the continuous partial differential equations into discrete form. The technique chosen here isfinite differences. There are many possible finite difference versions of the shallow waterequations and each version has its own properties and problems (Mesinger and Arakawa,1976; Haltiner and Williams, 1980). The scheme chosen follows Arakawa and Lamb(1981) for the spatial differencing and the leap-frog scheme for the time differencing.5.2.1 Discrete Equations: Arakawa-Lamb, 1981The scheme of Arakawa and Lamb (1981, hereafter AL81) is based on the idea that potential vorticity should be explicitly conserved in the finite difference form of the equations(5.20) and (5.19). The AL81 scheme guarantees conservation of mass, energy, momentum,and potential vorticity for the shallow water equations over arbitrarily steep topography.These conservation properties can hold oniy in the absence of the corresponding sourcesand sinks and the spatial discretization is specified for the unforced, frictionless, shallow-water-equations: G = 0 in (5.20). The scheme is second order accurate in space. Detailsare given in Appendix D.An important property of AL81 is the conservation of the domain integral of potentialenstrophy, hq2/2. The inaccuracies of finite difference versions of the non-linear shallowwater equations tend to manifest themselves through the collection of energy at thesmallest resolvable spatial scale: 2x , where L\x is the grid size. AL81 attempts tocontrol the transfer of energy to these small scales by conserving the domain integral ofpotential enstrophy. Restricting this spurious energy transfer means that 1) the modelresults will not change significantly when the grid resolution changes and 2) large viscousterms are not needed to soak up the small scale energy which would otherwise make theChapter 5. The Hecate Strait Model 80model unstable.5.2.2 Time Stepping: Leap-frog SchemeArakawa and Lamb (1981) specify the differencing of the spatial derivatives but not thetime derivatives. The time stepping is done here using the leap-frog scheme. The frictionterms are dealt with implicitly to avoid a well known instability (Mesinger and Arakawa,1976).Stability considerations require that the fastest wave can not cross a grid cell in lessthan one time step: the Cauchy- Fredricks-Levi or CFL criteria. For the numerical representation of the shallow water equations in two dimensions using the leapfrog scheme,the time step zt must satisfy2/2ghm (5.30)where zs is the grid spacing and hrnax is the maximum depth in the model domain(Haltirier and Williams, 1980).The leap-frog scheme introduces a computational mode which is filtered using aRobert (1966) filter. The filter has the forms(n) = s(n) + 7[s(n + 1) — 2s(n) + s(n — 1)] (5.31)where s is the variable being filtered, n is the time index and the over-bar denotes afiltered value. The filter is applied to all three prognostic variables (u, v, ) at everygrid point and every time step. A filter parameter of-y = 0.01 was used (Foreman andBennett, 1988; Buckley and Budgell, 1988). The properties of this filter are discussed byAsselin (1972). There is a brief discussion in Appendix D.Chapter 5. The Hecate Strait Model 815.3 Bathymetry, Grid Size and Time StepThe model domain is 450 km x 995 km with a grid spacing of 5 km (90 x 199 grid points).The domain is centered at 52.5°N, 131°W and rotated so that the shelf break is roughlyperpendicular to the cross-shelf open boundaries. The map shown in Fig. 5.32 covers 90x 150 grid points, the model domain extends 20 grid points (100 km) north and south ofthe area shown.Although the ocean depths reach 3500 m in the model domain, the maximum depthin the model bathymetry is 2525 m. This permits a time step of /t = 10 s and savescomputation time. The depth limitation reduces the maximum Kelvin wave speed from185 m s1 to 157 m s’. This might limit the usefulness of this bathymetry as part ofa tidal model since the meeting of the two arms of the tide in northern Hecate Strait isimportant. The effect on the results presented here is expected to be minimal.Accurate representation of the bathymetry, or ocean topography, is important because topographic steering (the tendency of the flow to follow the depth contours) is animportant part of the dynamics. The representation of the bathymetry is limited by thegrid resolution. The choice of grid resolution requires balancing two competing considerations; the need for accurate bathymetry, and the need to limit the computational load.The computational load is linearly proportional to the number of grid points N and thenumber of time steps. Doubling the resolution increases the computational load by 8.There are 4 times as many grid points and twice as many time steps. The bathymetry,grid size, and time step are related through the CFL criteria (5.30). The grid size of5 km was a compromise between 10 km (not enough resolution) and 3 km (too muchcomputation).Along-shore bathymetric variations were removed near the cross-shore open boundaries to suppress the generation of topographic waves near the cross-shore boundariesChapter 5. The Hecate Strait Model 82Figure 5.32: The model domainChapter 5. The Hecate Strait Model 83and to accommodate the open boundary conditions discussed in Section 5.5. The uniform along-shore bathymetry extends 20 grid points (100 km) beyond the northern andsouthern ends of Fig. 5.32.5.4 Energy Dissipation MechanismsEnergy dissipation was parameterized in terms of the depth-averaged velocity. The formchosen wasFf = + + vV2 (5.32)These are the last three terms in G (5.22) and they represent bottom friction, Rayleighfriction and eddy viscosity respectively. The primary mechanism for removing energyin the model is bottom friction. Additional dissipative effects due to the Robert (1966)filter are discussed in Appendix D.Four bottom friction parameterizations were considered at early stages in the modeltrials:Fb = CdQu + v2)1/2 (5.33)= kv (5.34)= (aho/h)iY (5.35)= CdVrms(X,Y)’ii (5.36)These are referred to as quadratic friction, linear friction, h2 linear friction, and spatiallyvarying linear friction respectively. The first two forms are quite standard, the last twoare not in common use.Drag coefficients for quadratic friction (5.33) in depth-averaged models are usually inthe range Cd 2 — 4 x iO (Murty, 1984). The spatially-invariant background velocityu0 attempts to represent the effect of the missing tidal velocities by a single number.Chapter 5. The Hecate Strait Model 84The linear friction coefficient (5.34) can be derived by linearizing the quadratic frictionwith respect to a background velocity, k = Cduo. Tidal models indicate that the rootmean square (rms) barotropic tidal velocities on the shelf range from 10 cm s1 to over 40cm s1 (M.G.G. Foreman, lOS, pers. comm.). This gives k in the range (0.3—1.6) x 10m srn’.The h—2 form (5.35) was discussed by Simons (1980), where it was suggested thatah0 0.1 m2 s1 in Lake Ontario. In this case, the use of (5.35) represents an attemptto have friction in the shallow water regions dominate the friction in the entire model.The choice of name comes from the fact that in the model equations Fb/h has an h2dependence.The form (5.36) was an attempt to improve the spatial distribution of bottom frictionby using a spatially-variable tidal velocity when computing the linear friction coefficient.The rms tidal velocities (Vrms(X, y)) from the model described in Foreman et al. (1992)were used.Linear friction is the standard bottom friction used in this thesis. The other forms areused to test the sensitivity of the model results to the bottom friction parameterization.The use of linear friction makes it easier to use the conceptual models presented inChapter 4 to interpret the regional model results.The depth dependence of the bottom friction means that it provides very little dissipation in deep water. The Rayleigh friction was introduced to provide dissipation in theoffshore region. The parameter u is important because it controls both the adjustmenttime and the steady-state velocity in the deep water.The horizontal eddy viscosity v was originally introduced to remove small scale retrograde shelf waves in a model of the Beaufort Sea (Buckley and Budgell, 1988). Theeddy viscosity does not seem to affect the results in the Queen Charlotte Islands region.As reported by Hanllah et al. (1991), tests with and without the eddy viscosity showedChapter 5. The Hecate Strait Model 85that: 1) the model is stable when the eddy viscosity is removed (ii = 0), and 2) the modelresults are not sensitive to the value of ii. The model usually runs with the horizontaleddy viscosity v = 10 m2 s.5.5 Lateral Boundary ConditionsRegional ocean models have two types of lateral boundaries: side walls and open boundaries. Side walls are where the land and ocean meet. Open boundaries are arbitrary linesdefining the limits of the regional model domain.5.5.1 Side WallsThe side wall boundary condition used is: no flow through the boundary and free slipalong the boundary. This is implemented by setting the normal velocity and the relativevorticity, C equal to zero at the boundary. Neither the tangential velocity nor the sea levelneed to be specified at the land/ocean boundary (Appendix D). The reader should beaware that the real ocean does not have side walls, only a sloping bottom that intersectsits surface.5.5.2 Open Boundary ConditionsAn open boundary is the arbitrary line drawn in the ocean that separates the known(the model domain) from the unknown (the rest of the ocean). Difficulties arise becausethe equations of motion contain spatial derivatives that require information from outside the domain when computing the solution at the open boundary. One way aroundthis difficulty is to prescribe the solution along the boundary. This is the only wayto get information into the model domain about events that occur outside the domain.This technique requires good information/observations about the solution along the openChapter 5. The Hecate Strait Model 86boundary. In coastal tidal models and fjord models this is the only viable solution becausethe external forcing is of primary importance. In other cases one can make the approximation that all important events happen inside the domain— local forcing dominates.In these cases . . . the purpose of the open boundary condition is to allow disturbancesoriginating inside the model domain to leave without disturbing or deteriorating the solution in the interior.’ (Roed and Cooper,1987; hereafter RC87). In other words thenumerical model must compute a solution along the open boundary which is transparentto all disturbances that reach the open boundary.There is no general solution to the open boundary condition problem. As observed byRoed and Cooper (1987),’ . . . the choice of OBC depends upon the particular applicationthat modeler has in mind.’ In a regional model one cannot prove that an open boundarycondition is fully satisfactory. The best that one can do is to demonstrate that it worksin simple test cases and then carefully watch the numerical solution near the boundaries.Paranoia is a virtue.For this model I assumed that the local winds provide the only important forcingmechanism. The only purpose of the open boundary conditions was to let disturbancesout. External influences such as tides and shelf waves were ignored.5.5.3 Implementation of the Open Boundary ConditionsThe open boundary of the Queen Charlotte Island region can be thought of being composed of 3 pieces; northern and southern cross-shelf open boundaries and an offshoreopen boundary (Fig. 5.32).Cross-shelf boundaryThe cross-shore open boundary conditions used in this model are based on the flowrelaxation scheme of Martinsen and Engedahl (1987). In this scheme the solution in theChapter 5. The Hecate Strait Model 87interior of the domain is relaxed to a specified (exterior) solution at the open boundaryin a relaxation zone. The cross-shore relaxation zones are not shown in Fig. 5.32. Eachzone extends 100 km (20 grid points) beyond the edge of the figure. The discussion hererefers to a single cross-shelf open boundary. The second one is dealt with in a similarmanner.The exterior solution chosen was a wind-forced solution: the solution to the equations of motion with all the along-shore gradients removed. This solution, which canbe computed at the open boundary, does not permit the propagation of waves in thealong-shore direction. The underlying assumption is that the shelf continues to infinitywith the same topography. The exterior solution in the relaxation zone was computedduring the model run using the same wind forcing as the rest of the model.Relaxing the solution in the interior of the domain to the exterior solution means thatwaves propagating in the along-shore direction are damped and that there are no abruptchanges in mass fluxes near the open boundary. In the absence of wind the boundarycondition reduces to a sponge layer: the exterior solution becomes a no-motion solutionwith u = 0, v = 0, and = 0. As one might expect because the Ekman flux is handledcorrectly this open boundary condition is most successful when there is strong windforcing.The relaxation was done as follows. At all points not in the relaxation zone, themodel solution equals the interior solution. At all points along the open boundary, themodel solution equals the exterior solution. The model solution must then vary smoothlyacross the relaxation zone. At each time step and at each point in the relaxation zone aprognostic variable such as the along-shore velocity v is updated according tov = av1 + (1 — a)v2 (5.37)where v1 is the exterior solution, v2 is the interior solution, and the relaxation parameter aChapter 5. The Hecate Strait Model 88varies oniy in the along-shore direction. At the open boundary a =1 and at 20 grid pointsfrom the open boundary a = 0. Thus the relaxed variable v equals the exterior solutionat the open boundary and the interior solution at the interior side of the relaxation zone.All of the experiments reported use a of the forma(i) = 1— tanh[(20 — i)/4], i = 1,. . . ,20 (5.38)where i =1 in the interior and i=20 at the open boundary.The choice of external solution means that the wind forcing is the only mechanism forforcing water through the cross-shore open boundaries. The other possible mechanismfor forcing water through the boundaries is an along-shore pressure gradient. Since theexternal solution does not allow along-shore gradients, they are not part of the solutionat the boundary. Techniques for handling outflow due to persistent pressure gradientsnear the open boundary are being developed (Lars Petter Roed, pers. comm., 1991).Tests of the open boundary conditions are discussed in Appendix EOff-shore BoundaryIn their experiments ME87 use a relaxation zone along the off-shore boundary which issimilar to the cross-shore one discussed. I am not convinced that a reasonable schemeexists for dealing with the fact that the cross-shore and off-shore relaxation zone intersectin the corners of the domain. Recall that in the cross-shore relaxation zone the solutionhas no along-shore gradients. In the off-shore relaxation zone the solution has no crossshore gradients. In the corners these two solutions have to be combined.The model used something simpler. The offshore boundary condition is a clampedsea level condition: i = 0 along the offshore boundary. This is the condition used in theboundary condition studies of Chapman (1985) and Roed and Cooper (1987).Chapter 5. The Hecate Strait Model 895.6 Atmospheric Forcing Fields5.6.1 Wind StressFor the model simulations the wind stress field was constant in space and varied in time.The wind stress forcing was computed using winds measured by a sea-based anemometerclose to the area of interest. For example the model verification tests in Hecate Straitwere driven by winds measured at W4S in the middle of the strait.The conversion of raw wind speeds to model wind stress was discussed in Chapter 3.For the simulations the wind stress time series were passed through a 1-day low passfilter to suppress inertial oscillations and other high frequency motion in the model. Thewind stress in the model was updated hourly.The fact that the conversion from wind speed to stress is non- linear means that itis important that the filtering be done after the conversion, not before. For the winterof 1984, filtering the wind velocities before converting to stress resulted in a four-folddecrease in the peak wind stress compared with filtering after. The more the winds areaveraged before the conversion, the lower the peak wind stress. Using a larger effectivedrag coefficiellt does not recover the peaks.The decision not to use lighthouse winds for calculating the wind stress to drive themodel has an important consequence. It reduces the number of tunable parameters inthe model. Recall that Smith (1988) has no free parameters whereas using lighthousewinds involves at least one, the drag coefficient. When both the wind stress and bottomfriction are adjustable there is room for endless adjustment of parameters to try andimprove the agreement between the model results and the observations. When only thebottom friction is adjustable the room for parameter adjustment is drastically reduced.Chapter 5. The Hecate Strait Model 905.6.2 Pressure GradientsAtmospheric pressure gradients were not used to force the model. In general, openboundary conditions do not react well to forcing with atmospheric pressure gradients.I have not developed a robust method of handling pressure gradients near the openboundaries. The effect of atmospheric pressure gradients on the circulation in this regionhas not been investigated.5.7 Computer ProgramThe original version of the model software was provided by W.P. Budgell. The modelwas developed for use in the Beaufort Sea (Buckley and Budgell, 1988). The softwarehas been extensively rewritten to make the program easier to use, easier to modify, andto fix a few bugs. These changes have been largely confined to the input, output anddiagnostic routines. The code that implements the Arakawa and Lamb (1981) algorithmis untouched.The code for the open boundaries was rewritten to implement the flow relaxationboundary conditions. The drifter module was modified to implement different time stepping algorithms and to check for drifters crossing land/ocean boundaries. The vorticitybalance module was rewritten to improve the estimate. The input and output moduleshave been rewritten.A combined user’s manual and model documentation (Hannah, 1992b{37}) has beenwritten to provide a new user with enough information to run the Hecate Strait Modeland make minor changes to the software.Chapter 6Evolution from Rest to Steady State in the Hecate Strait ModelThis chapter contains a short discussion of the behaviour of the Hecate Strait model asit evolves from a rest to a steady state: the model spin-up. The two important resultsare:• The spin-up of the transport through Hecate Strait is dominated by 2 time-scales:a fast adjustment (4 h) and a slow adjustment (48 h). The fast adjustment is theresponse of the strait to the local wind forcing. The slow adjustment is the responseto the along-shore pressure gradient which results from the spin-up of the rest ofthe model domain.• The steady-state transport through the strait is only weakly dependent on thevalues of the friction parameters.Other topics discussed are frictional adjustment, the selection of the friction parametersfor the simulations in the next chapter, and a preview of the circulation patterns. Theresults presented here form the basis for the discussion in the next chapter, and the readershould at least browse through this chapter before continuing.For the purpose of this thesis the phrase numerical experiment means that the modelwas forced with very simple forcing; a sinusoidal wind, for example. The word simulationmeans that model was forced by a time series computed from observed winds. Thepurpose of a numerical experiment is to learn about the inner workings of the model.The purpose of a simulation is to compare the model results with observations.91Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 926.1 Frictional AdjustmentNumerical experiments reported in Appendix E show that frictional adjustment (4.4)u = u(1 — et)provides a good description of the along-shore velocity in a simple rectangular domainwith uniform along-shore topography and realistic cross-shelf topography. In the HecateStrait Model, the concept of frictional adjustment is important for three reasons: 1) itprovides an upper bound on the adjustment time scale A’, 2) the time-scales in theopen-ocean portion of the model domain are determined by friction, and 3) the conceptcan be generalized to include effects such as geostrophic control (Chapter 4).Linear friction (5.34) is the standard form for bottom friction in the Hecate StraitModel. From Chapter 4, the relaxation constant has the form= k/h +where k and u are constants and the total water depth h is variable. The dependenceof the steady state velocity (u = r/ph\) and frictional adjustment time (‘X’) ontotal water depth h and Rayleigh friction ii are shown in Fig. 6.33 and Fig. 6.34 (fork O.5 iO s’). The general behaviour is that the frictional adjustment time increasesand the steady-state velocity decreases as the depth of the water increases. Notice thewide range of adjustment time as a function of depth. For = 3. iO s1 the adjustmenttime ranges from 1 day for h = 50 m to 21 days for h = 2000 m.In the limit t = 0, the steady state velocity is independent of the water depth andthe spin-up time increases linearly. In the limit k = 0, both the steady state transport(vh) and the spin-up time are independent of water depth and the velocity decreaseswith water depth. For k 0 and t 0 the behaviour is a mixture of the two limitingcases.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 93I E I30C,)C20- . _——— -./_ ————.c ——.2I — —d0 ‘ I I I0 500 1000 1500 2000 2500Water depth (m)Figure 6.33: Frictional adjustment in a simple channel. Spin-up time as function of thewater depth for k 0.5 i0 s and (a) p = 0; (b) p = 3 10—8 s’; (c) p = 3 . iOs; (d) p = 3. 10_6 s1.25I :10 I0 500 1000 1500 2000 2500Water depth (m)Figure 6.34: Frictional adjustment in a simple channel. Steady state velocity as a functionof the water depth for k= 0.510 s1 and (a) p = 0; (b) p = 3.108 s1; (c) p = 310s1 (d) p = 3 10_6 s_i.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 94The along-shore velocity and the adjustment time are very sensitive to the value ofthe Rayleigh friction. The Rayleigh friction is a one-parameter model of the deep ocean.It was not intended to affect the shelf circulation. The dependence of the regional modelresults on both k and u is investigated in the next section.6.2 The Basic Experiment: Spin-upA useful way to study the model is to watch it evolve from rest to a steady state: a spinup experiment. In the basic spin-up experiment the ocean starts at rest (u=0, v=0, i=0)and a constant wind stress is applied at time t=0. The initial impulse excites a broadrange of frequencies and allows observation of the relaxation (or adjustment) processesand the natural modes of the system (resonances). The spin-up experiment is the basicexperiment conducted in this chapter. In cases where an impulsive start is not desired,the wind is started smoothly with a cosine rampI ro(0.01 + 0.495(1 — cos(t/to)) t <t0= (6.39)ITo tt0where t0 equals one or two days.Most of the experiments in this chapter were forced with a uniform along-shore windstress of r0 = 0.1 Pa: a wind speed of roughly 8 m s1. Other directions are consideredin later chapters. The wind stress was tapered linearly to zero at the off-shore boundaryover the outer 25 grid points. The standard model parameters are listed in Table 6.6.Bottom friction is linear (5.34) unless explicitly noted. The Hecate Strait Model domainand the observation locations are shown in Fig. 6.35.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 95Figure 6.35: The standard model domain. The cross-shore flow relaxation zones are notshown; they extend 100 km off the top and bottom of the figure. The line on the left-handside is the off-shore open boundary.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 96L domain length 199 grid pointsW domain width 90 grid pointsLs grid size 5 kmhmax maximum depth 2525 mzt model time step 10 sf Coriolis parameter 1.15 iO s’k friction parameter 0.5• iO s1Rayleigh friction 3.0• 1D’ sv eddy viscosity 10 m2 sfilter parameter 0.01p density of sea water 1030 kg m3_____relaxation zone width 20 grid pointsTable 6.6: Standard regional model parameters.6.3 Selecting Friction ParametersThe friction and dissipation mechanisms discussed in Chapter 5 are the only adjustableparameters in the model. The experiments reported here provide a basis for selectingparameter values. The selection criterion was that the steady-state transport be roughly0.3 Sv for an along-shore wind stress of r = 0.1 Pa, in rough agreement with the observations (Chapter 3). The parameter values chosen are used in the simulations in thenext chapter. More important than the actual values selected is the fact that the modeltransport is only weakly dependent on the friction parameter values.k (s’) (s’)0 310 3.10_60.3. iO 0.34 0.33 0.240.5• iO 0.27 0.26 0.221.0. i0 0.19 0.19 0.17Table 6.7: Linear friction. Steady state transport (Sv) for a range of k and i. Along-shorewind stress r = 0.1 Pa.The first parameter that needed selection was the Rayleigh friction t. Table 6.7 showsthat the steady-state transport is insensitive to changes in ,u for it < . io s1. TheChapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 97Rayleigh friction takes the value ji = . i07 s1 from this point on.The transport is weakly dependent on the linear bottom friction coefficient k. Cuttingk in half increases the transport by roughly 30%. For a steady-state transport of roughly0.3 Sv, the linear friction parameter should be in the range 0.3 . i0 < k < 0.5 . iO(assuming it = . io s’). The value k = 0.5. io— m s1 was chosen as the standardvalue.Cd u0 (cm s’)0 20 401.3. i0 0.36 0.31 0.252.5. iO 0.31 0.25 -5.0. iO 0.26 - -Table 6.8: Quadratic friction. Steady state transport (Sv) for a range of Cd and u0.Along-shore wind stress r = 0.1 Pa. Rayleigh friction = 3.0. i0’ s1.For quadratic friction (5.33), an effective linear friction coefficient at each point wascomputed at each time step fromk*= Cd(u +where Cd and UO are constants and i5 is the local velocity vector. Table 6.8 shows that arange of Cd and u0 yields the required transport. The values chosen for the simulationwere Cd = 2.5• iO and n0 = 0.For the spatially varying linear friction (5.36) the linear friction coefficient was computed at each point fromCdVrms(X,Y)where Vrms were the rms tidal velocities from a barotropic tidal model (M.G.G. Foreman, pers. comm., 1991). This frictional form is discussed further in Chapter 8 andAppendix F. For Cd = (1.3,2.5,5.0). iO the steady state transports were 0.42 Sv, 0.32Sv, and 0.23 Sv, respectively. A value of Cd = 2.5 i0 was selected.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 98The steady-state circulation pattern is previewed at the end of this chapter. Asdiscussed in Chapter 8, the pattern is not sensitive to the choice of the functional formof the bottom friction.6.4 Adjustment Time-Scales in Hecate StraitThis section describes the evolution of the model transport and coastal sea level duringspin-up. The results are used to estimate the characteristic relaxation or adjustmenttime-scales in Hecate Strait. The standard model domain and parameters were used.The transport through Hecate Strait as measured at R, W, and M lines is shownin Fig. 6.36. At all three sections the spin-up was dominated by two time scales: oneof the order of several hours, the other of a few days. At W and M lines there was athird time scale, a high frequency oscillation. At the end of this section the oscillation isinterpreted as a basin resonance. The oscillation disappeared when the wind stress wasstarted smoothly using the cosine ramp with t0 = 1 d.After 9 days the transport had stopped growing but the sea level was still rising inHecate Strait (Fig. 6.37) and on the outer coast (Fig. 6.38) . The relationship betweenalong-shore velocity and cross-shore pressure gradient (geostrophic balance) makes thecoastal sea level sensitive to the open ocean currents. From Fig. 6.33 the frictionaladjustment time for a depth of 2000 rn is 21 days. The sea level was still rising becausethe open ocean currents were still growing. The dynamical variable of interest is sea-levelgradients not absolute sea level. The cross-strait pressure (sea level) difference in HecateStrait (Fig. 6.39) had the same spin-up characteristics as the transport.A plausible explanation for the two adjustment time-scales in Hecate Strait is thatthe fast time-scale is the response to the local wind forcing and the slow time-scale is theresponse to an along-shore pressure gradient set up by processes outside of Hecate Strait.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 990.30::r0.15(b):::‘“I I I I I I0.15(c)0.00 I I I I I I0.0 10.0Time (days)Figure 6.36: Spin-up test. Time series of transport through Hecate Strait for an impulsively started wind. (a) R-line, (b) W-line, (c) M-line. The dotted line in (b) is thetransport when the wind was started smoothly (see text).0) Time (days)Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 10020. -Ui:.0.0 5.0 10.0Time (day5)Figure 6.37: Spin-up test. Time series of sea levels in Hecate Strait. a) Atli and b)Beauchemin.20.U; 10.>0)0.Time (days)Figure 6.38: Spin-up test. Time series of sea levels on the outer coast. a) Cape Muzon,b) W. QCI, and c) Cape Scott.EU4.0I 2.0fj 0.0 I I I tabC0.0 10.00.0 5.0 10.0Figure 6.39: Spin-up test. Cross strait pressure difference Beauchemin minus Atli.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 101EU—1.0 I I I I I I0.0 5.0 10.0Time (days)Figure 6.40: Spin-up test. Along-shore pressure difference Cape Scott minus Cape Muzon.0.00 I I I I ILF- 0.0 5.0 10.0Time (days)Figure 6.41: Spin-up test. Comparison of transport through R-line with a two componentadjustment.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 102The sea level difference between Cape Scott and Cape Muzon (Fig. 6.40), a measure ofthe along-shore pressure gradient, has an adjustment time scale similar to that seen inthe transport (Fig. 6.36). The analysis and experiments reported in Appendix F supportthis explanation and suggest the following simple relationship for the transport QQ(t) = 0.13(1 — e’t)+ 0.13B(1 — e_2(t_5)) (6.40)where the locally wind-driven response has an e-folding time of j’ = 4 h, the pressureforced response has an e-folding time of )‘ = 48 h, and /3 0 if t < 0.5 d and j3 = 1otherwise. The pressure-forced component is suppressed for the first 1/2 day becausecomputing a robust estimate of the e-folding time scale from the sea level differenceCape Scott minus Cape Muzon required trimming the first 1/2 day from the time series,suggesting a time delay. This simple relationship is in good agreement with the transportin the regional model (Fig. 6.41). The only characteristic not reproduced is the bumpduring the first day. The amplitudes of the two components were set equal based on along look at the spin-up in Fig. 6.36. Other choices are certainly possible.Given the assumption that the adjustment time scales are determined by friction, onecan compute the depth that is controlling the flow from= k/h + ,uUsing the values of k and i (Table 6.6) yields h1 = 7 m and h2 = 90 m from and )2respectively. Since there are no significant regions in Hecate Strait, or anywhere in themodel, with depths less than 10 m, this suggests that something other than friction isdetermining the value of q. The contribution of the Rayleigh friction t to the adjustmenttime-scales is insignificant.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 1036.4.1 Basin ResonanceThe high frequency oscillation seen in the basic spin-up experiment can be separated fromthe underlying adjustment processes spin-up by subtracting the transport at W-line fromthat at R-line. Figure 6.42 shows a decaying oscillation with approximately three cyclesper day. The power spectrum (Fig. 6.43) reveals that the dominant mode has a frequencyof = 0.133 ± 0.004 cph (cycles per hour) and a period of 7.5 + 0.5 h. This is very closeto the resonance mode found by Foreman et al. (1992) with = 0.128 cph. This isthe half wavelength mode of a resonance set up between the shelf break in the mouth ofQueen Charlotte Sound and the shelf break in the mouth of Dixon Entrance, analogousto a double open-ended organ pipe. The peak at w = 0.258 cpd is the one wavelengthmode.6.5 Circulation Patterns: A PreviewThe through-strait transports calculated in this chapter have an associated transportvector field = h, where is the velocity field. The pattern of the steady-statetransport field (Fig. 6.44) develops quickly. In the spin-up experiments, the underlyingvelocity pattern is established between Day two and three (Chapter 8).The important characteristics of the steady-state transport pattern in Hecate Straitare:• The pattern is constrained by the topography.• There is an up-wind counter flow in the south-central strait opposite to the directionof net transport. The flow is along the north slope of Moresby Trough.• At the southern end, the net northward transport is the small difference of largenorthward transports at the edges and a large southward transport in the center.>(I)-4-aC0a(1)CCI—La)1.000.0 5.0Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 1040 .020.00-0.02Time (days)Figure 6.42: The high frequency oscillation at W-line.2.00.00.0 0.1 0.2 0.3 0.4Frequency (cycles/Hour)Figure 6.43: The power spectrum of the oscillation at W-line.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 105Figure 6.44: Steady-state transports with a uniform SE wind. The wind direction isindicated with an arrow on Graham Island. The 200 m contour is the heavy dark line.Only 1/4 of the vectors are plotted. The vectors were suppressed for depths greater than500 m.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 106• At the northern end, the transport is concentrated in the deep water on the easternside.• There are regions of low transport in the northwest corner and in the center of thestrait.The validity of the steady-state picture as a guide to the time-varying model results isdiscussed in Chapter 8.6.6 Force BalanceForce balances were computed in Hecate Strait during the spin-up experiments. Thesecomputations show that the primary cross-strait force balance is between the pressuregradient and the Coriolis force (geostrophic balance). The wind stress enters the balanceat the northern end, where the axis of the strait makes a large angle with the gridlines. The along-strait force balance at southern end (M-line) is between the wind stress,pressure gradient and the Coriolis force. The friction is negligible except in the shallowwater at the edges. At the northern end (R-line), by Day 2, the acceleration termis important only in the deep water on the eastern (right hand) side. Over the restof the line, the wind stress is balanced by the bottom friction and the pressure gradient(Fig. 6.45). In the southern end of the strait, the along-strait and cross-strait componentsof the forces were resolved along the grid lines. At R-line, in the northern end of thestrait, the forces were resolved parallel and perpendicular to the local velocity. Thisremoves the Coriolis term from the along-strait force balance in Fig. 6.45.Chapter 6. Evolution from Rest to Steady State in the Hecate Strait Model 107.4,’_____________________: Iv -100— - —- accelerationEpress. grad.? 5 -50 —0--- wind stressX bottom stress0 0.__- -+- residual0_____0depthzero50_ ______ _G)E-10 I I 10060 65 70 75x-coordinateFigure 6.45: The along-strait force balance at R-line after two days of steady along-shoreSE wind.Chapter 7Rotational Limitations on the Water Transport Through Hecate StraitThis chapter concentrates on the transport of water through Hecate Strait, startingwith a comparison of the simulated and observed transport time series for the winterof 1984. The Hecate Strait Model transport is shown to be a good representation ofthe observed transport. The model is then used to explore the relationship betweenthe friction parameters, the Coriolis parameter, and the transport. The purpose is tounderstand why the magnitude of the transport fluctuations in the simulations are notvery sensitive to the friction parameters. Rotation-limited-flux provides an explanation.7.1 Winter 1984In this section the observed transport through Hecate Strait is compared with simulatedresults for the period 25 Jan to 30 Mar 1984 (Day 25 to 91). The simulations were forcedby the wind stress time series shown in Fig. 7.46. The wind was predominantly along-shore and from the SE. The computation of the wind stress from the observed winds atW4S was discussed in Chapter 3 and Chapter 5. The cosine ramp (Chapter 6) was usedfor the first two days to avoid the high frequency oscillations seen in the test cases.The model has done a reasonable job of simulating the observed transport. Comparison of the observed time series with that from a simulation using the standard modelparameters (E501a in Table 7.9) shows that the peaks line up but that there is disagreement about the magnitude of many of them (Fig 7.47). There were two periods ofpersistent poor simulation: Day 30 to 40 and Day 75 to 85. There were also some short108Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 109I I I I1 31 6t 91 121’Day , 1984Figure 7.46: The wind stress time series used to drive the verification experiments. Thevectors are oriented to conform to the orientation of the model domain; up represents aSE wind. The winds were measured at W4S.Experiment Friction type Parameter valuesE501a linear k 0.5• i0 sE501b linear k = 1.0. iO s’E502a quadratic Cd = 1.3 iO, u0 = 0E502b quadratic Cd = 2.5• i0, uo = 0E503 rms Cd = 2.5 iOTable 7.9: Model verification experiments.events that were poorly simulated; near Day 62, for example.A series of simulations were conducted to test the sensitivity of the modelled transportto the bottom friction (Table 7.9). The statistics computed from the transport timeseries were not very sensitive to the bottom friction (Table 7.10). The mean transportvaried, but the magnitude of the fluctuations did not change very much. The time seriesthemselves were very similar.The observed transport is more highly correlated with the simulated transport (rmax =0.77) than with the along-shore wind stress (rmax = 0.70). This shows that using thewinds to drive the model provides a better estimate of the transport than using the windsalone. Using the model brings the predictive ability of the winds measured at W4S up toChapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 1100.5Cl)0C, -0.5Figure 7.47: Transport through W-line from 25 Jan -simulation E501a (solid) and the observations (dotted).00EC0CuciLL30 Mar 1984. Comparison ofFigure 7.48: Time series of along-shore wind stress (dotted) and simulated transport(solid) in Hecate Strait from 25 Jan- 30 Mar 1984. Both time series have been normalized.1—125 35 45 55 65 75 85Julian day 198495420-2Juflan day 1984Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 111Experiment Transport Correlationmean (Sv) st. dev. (Sv) wind ohs. transportE501a (linear) 0.30 0.26 0.90 0.78E501b (linear) 0.23 0.22 0.92 0.77E502a (quadratic) 0.37 0.26 0.90 0.78E502b (quadratic) 0.33 0.24 0.91 0.77E503 (RMS) 0.44 0.27 0.85 0.76observations 0.36 0.30 0.70 —Table 7.10: Model verification. Comparison of the simulated and observed transportthrough Hecate Strait for the period 25 Jan- 30 March 1984. The correlations are themaximum linear correlation coefficients of the transport time series with the along-shorewind stress and with the observed transport.the level of the best of the lighthouse winds (Table 3.3). A word of warning: the analysisof CHW88 indicates that for the observed transport, the sample correlation is drawnfrom a population correlation in the range 0.53—0.82 (95% confidence limits). Therefore,the improved correlation is not statistically significant.The model underestimates the magnitude of the observed transport fluctuations (standard deviation). The results in Table 7.10 indicate that it is difficult to increase themagnitude of the fluctuations by adjusting the bottom friction. This is supported bythe results in Chapter 6. The fluctuation could be brought up to the level of the observations by increasing the wind stress estimate by 10%. One could use the uncertaintiesinherent in the calculation of the wind stress from the wind speed to justify such an adhoc increase. Nevertheless, this would not improve the overall simulation since the modelalready over-estimates the transport near Days 30 and 63.Improvements in the transport simulation require improved atmospheric forcing. Comparison of the along-shore wind stress fluctuations with the transport fluctuations (Figure 7.48) shows that the transport tracks the wind stress very closely. This is the meaningof a correlation coefficient r = 0.9. There is nothing in the model dynamics that allowsthe transport to depart significantly from the wind forcing. Comparison of Fig. 7.47 andChapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 112Fig. 7.48 shows that the model performed well when the winds were strong (Day 40 to60) and not as well when the winds were weak (Day 25 to 40). This suggests a non-localeffect, perhaps an along-shore atmospheric pressure gradient or a wind field not wellrepresented by a single vector.In CHW88 three surrogate transport series were constructed from the sub-surfacepressure observations: Beauchemin minus Atli (B-A), Prince Rupert, and the linearcombination (Prince Rupert plus Bella Bella)/2 minus Queen Charlotte City (PBQ). Inthe simulations, B-A and PBQ were well correlated with the transport (rmax > 0.95).Prince Rupert was a poor predictor, rmax 0.71. The poor performance of Prince Rupertin the model is related to the absolute sea-level problem discussed in Chapter 6.The comparison of the observed and simulated cross-strait sea level difference is shownin Fig. 7.49. The model under-estimates the magnitude of many of the large peaks. If,as mentioned in Chapter 3, the cross-strait sea level difference Beauchemin minus Atli(B-A) is a better indicator of the transport through Hecate Strait than that computedfrom the currents, then the model is under-estimating the peak transport.7.2 Friction, Coriolis Parameter, and TransportRotation-limited-flux provides a plausible explanation for the observation that the simulated transport through Hecate Strait is relatively insensitive to changes in the bottomfriction parameter (Section 7.1 and Chapter 6). The rotation-limited-flux equation forthe along-strait velocity (4.14)—i) + FU= iw + + fW/L(7.41)suggests that the steady-state transport through a flat-bottomed channel, Qo = Whu,can be writtenQo =+ W/L(7.42)EC.)C.)ci)L..a)-a)>a)ccia)Cl)Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 113151050-525 95Figure 7.49: Cross-strait sea level difference Beauchemin minus Atli for the period 25Jan - 30 Mar 1984. Comparison of E501a (solid) with the observations (dotted).35 45 55 65 75 85Julian day 1984Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 114where A represents the forcing terms. The estimates from Table 4.4 of A = 1 . 10s and fW/L = 3 . 10 s1 indicate that the the effect of rotation is to decrease thesteady-state transport by a factor of 4 A/(A + fW/L = 1/4). Further the steady-statetransport should not vary linearly with A’. In fact the transport should have a weakdependence on A. For time varying flow the presence of the ic term in (7.41) furtherweakens the dependence on A.To test the idea that rotation-limited-flux is important in Hecate Strait a series ofspin-up experiments were conducted. In these experiments the spin-up of the watertransport through the strait was monitored for different values of the Coriolis parameterf and the bottom friction coefficient k. The standard model domain (Fig. 6.35) andlinear bottom friction were used throughout.The discussion is split into two parts. In the first part the steady-state results areused to estimate the parameter W/L which plays an important role in rotation-limited-flux. A reasonable estimate of W/L 0.2 is obtained. In the second part estimates ofthe spin-up time scale are used to estimate both W/L and A. The analysis of the spin-updata is more sensitive to the limitations of rotation-limited-flux than the steady-statedata.The relationship between the steady-state transport in the model and the Coriolisparameter shown in Fig. 7.50 is in agreement with (7.42). The transport increases asf — 0 and decreases as f —+ cc. For large f the effect of the difference in the bottomfriction coefficient disappears and the two curves converge. These experiments were donewith an along-shore wind localized over Hecate Strait. As shown later in Fig. 7.52, thetransport spin-up exhibits only one time scale when the model is force by such a wind.This represents the local wind-driven response; the forcing due to along-shore pressuregradients, 774—775 in (7.41) is minimized. One process has been isolated.Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 1150.6—— k=0.5E-30.5 --+--k=2.OE-30.40.3o I I0 1 2 3 4 5 6 7f/f 0Figure 7.50: Steady state transport as a function of Coriolis parameter for a wind localized over Hecate Strait. The Coriolis parameter has been normalized with fo = 1.1 X iOs—i.0.7A uniform wind0.6 • local wind0.5I__________f/tOFigure 7.51: Steady-state transport as a function of Coriolis parameter for a uniformalong-shore wind stress (triangle) and for a wind stress localized over Hecate Strait.In both cases k = 0.5 . iO The. Coriolis parameter has been normalized withfo = 1.1 X iO s.Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 116Estimating W/L from Fig. 7.50 is non-trivial. In (7.42) A, A and W/L are all unknowns and curve fitting to estimate all three leads to unstable estimates since thevariables are highly correlated: increases in A can be balanced by increases in A andW/L. A stable form for curve fitting isB1 + Cf/f0(7.43)where B = A/A and C = Wf0/LA. The smooth curves in Fig. 7.50 are the best fits of thedata to an equation of this form. The data and fit parameters are listed in Appendix 0.Unfortunately the parameter W/L = CA/f0 cannot be estimated directly from the fitbecause A is unknown. There are three unknowns A, W/L and A and only two fittedparameters.One way to estimate A is to make an independent estimate of the effective depth.From the transport vector field in Fig. 6.44 a depth of h* = 70 m is a reasonable estimateof the mean depth corresponding to the region of maximum transport in northern HecateStrait. Using this value and the known form of the friction in the model (A = k/h + t)yields estimates of W/L in the range 0.2 — 0.3. The estimates vary linearly with (h*)_l.The experiments in Fig. 7.50 used a localized wind to limit the forcing in the straitto local wind forcing only. The character of the results did not change appreciablywhen the uniform wind was re-introduced. The steady state transports for the uniformalong-shore wind and the localized along-shore wind are compared in Fig 7.51. To afirst approximation the curves are offset by a constant. As described in Appendix 0 anestimate of W/L = 0.15 was obtained from the uniform wind data.These dynamical estimates of W/L 0.2 are reasonable. From Fig. 6.35 the width ofHecate Strait is 70 km and the length is roughly 250 km; W/L = 0.3. The steady-statetransport vector field (Fig. 6.44) suggests that the effective width and length are quitedifferent from the geographic width and length. A width of 20 km and length of 120Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 117km yields W/L = 0.2. Both estimates are in reasonable agreement with the dynamicalestimates.Rotation-limited-flux provides a reasonable explanation for the fact that the transport through Hecate Strait is relatively insensitive to changes in the friction parameters.Estimates for the ratio W/L range from 0.15 to 0.3 depending on the friction parameter values and the method of forcing the strait. The rest of this section is devoted toextracting estimates of W/L and A from the spin-up data. The discussion of the resultspoints out the limitations of rotation-limited-flux in a complex environment. The readercan skip to the end of the section without missing any important concepts.The derivation of rotation-limited-flux suggests that the response of the transport toan implusively started wind should resemble frictional adjustment. Equation (4.10) canbe written(7.44)where A = A + fW/L and F is the forcing term. For the experiments with the localizedwind, the spin-up time series should be well approximated by a curve of the formQ(t)= Qo(1 — e_A1t)where Q is the transport through Hecate Strait. The two spin-up examples shown inFig. 7.52 are well approximated by (7.45).The form of ) suggests a method for estimating W/L and A. W/L is the slope of thebest fit straight line through a plot of ) versus f, and A is the y-intercept. The spin-updata from the localized wind experiments (Fig. 7.53) yields estimates of W/L equal to0.5 and 0.6 for k = 0.5 . iO s1 and k = 2.0 . iO s1, respectively. These resultsimply a wider channel than the previous estimates. To bring the previous estimate ofW/L 0.2 in line with the new estimate would require an effective depth of 20 to 30 m.This is much too shallow.Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 1180.60 (a)—4—’ —fL 0.30 -o 1’(b)vtS 000 I I I I I I I IL0.0 5.0 10.0Time (days)Figure 7.52: Transport spin-up for an along-shore wind localized over Hecate Strait fork = 0.5 iO s1. (a) f = 0 and (b) f = 1.1. i0 s1.12 I I—— k=0.5E-3 / -10--+--k=2.OE-38-7Cl)0 —C)—, .49 //0 I I0 0.5 1 1.5 2 2.5 3f/f 0Figure 7.53: Relaxation time as a function of Coriolis parameter for a wind localized overHecate Strait. a) k = 0.5• iO s’, b) k = 2.0• i0 s1.Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 119Effective depths for the bottom friction can be obtained from the estimates of A usingthe known form of the bottom friction,\ = k/h+, and u = 3. i0s1. For k = 0.5• lOthe adjustment time A = 0.4 d’ and the effective depth h* = 100 m. Similarly, fork = 2.0 . 1O s1, A = 2.3 d’ and h* = 76 m. These estimates of the effective depthare consistent with the ad hoc estimate of h* = 70 m. No purpose would be served byusing these effective depths to refine the estimates of W/L made from the steady-statedata. For both data sets in Fig. 7.53 the estimate of A, and thus h*, is sensitive to whichdata points are used to compute the best fit straight line. In both cases the estimate ofA decreased when the point at f = 0 was not included in the fit. For k = 0.5. iO sthe estimated A was less than t, which implies a negative effective depth. The estimateof the slope, and thus W/L, was robust.The differences in the estimates of W/L obtained with the two methods is an indication that rotation-limited-flux has limitations when applied to a real channel. Thebiggest limitation is the fact that Hecate Strait is not a flat-bottom channel. As f/fo 0the flow is not constrained to follow contours of f/h since it is no longer true that f>> C,where is the relative vorticity. I believe that the fact that the bathymetric gradientscut diagonally across the strait causes the effective geometry of the strait to change asf/fo —* 0. A close look at the data in Fig. 7.50 suggests that the curves flatten out forsmall f/fo. For both curves, a straight line through the data with f = 0 and f = 0.1removed had a similar slope but a much smaller y-intercept.At the other end of the spectrum, there were problems estimating the time constantas f increased and the transport decreased. The noise in the spin-up which was was onlya minor annoyance in Chapter 6 when the steady-state transport was 0.26 Sv becamea major factor as the steady-state transport decreased. This affected all the results forf/fo 1 and made the estimate for f/fo = 6 worthless.Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 120Given the acknowledged limitations of rotation-limited-flux, the analysis was not pursued any further. Appendix G.3 contains a brief discussion of the validity of the assumption of a flat bottom.7.3 Frequency ResponseThe experiments with an impulsively started wind provide one view of the model’s behaviour. A complementary view is provided by the frequency response: the model’sresponse to sinusoidal forcing.The frequency response was computed using a spatially uniform wind with sinusoidal time variation r(x,y,t) = T0sinwt, where r0 = 0.1 Pa. The test variable was thetransport through Hecate Strait. The amplitude spectrum of the response (Fig. 7.54)is well approximated at any frequency by the transport in the basic spin-up experiment(Fig. 6.36) at time t = T/4 where T is the period. The flat portion of the amplitudespectrum (Fig. 7.54) between 0.1 < < 0.7 cpd (periods between 1.5 and 10 days)corresponds to the knee or flat-spot in the spin-up response from day 0.5 to 1.5.The lag of the transport behind the wind (Fig. 7.55) has a range of 5 to 15 h overmost of the spectrum. This is consistent with the lags of 5 to 12 h obtained from thecorrelation analysis of the winds and the observed transport in Chapter 3, where theyare reported as the lead of the wind over the transport.In Chapter 6 the basic transport spin-up time series was shown to be reasonablyapproximated by the sum of two friction adjustment processes with time scales = 4 hand = 48 h. In analogy with the frictional adjustment solution (4.5), an approximateamplitude spectrum Q(w) was computed fromQ2(,) = A2 + B2 + 2ABcos(1— 2) (7.46)>C,)DaEFigure 7.54: Amplitude spectrumHecate Strait to oscillating windstrum (see text).computed from the response of the transport through(data points). The solid line is an approximate spec-Figure 7.55: Phase spectrum computed from the response of the transport through HecateStrait to oscillating winds.Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 1210.30.250.20.150.10.0500 0.5 1 1.5 2 2.5Frequency (cpd)C)C,00302520151050I I-I I0 0.5 1 1.5 2 2.5Frequency (cpd)Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 122where tan =A=+ (w/1)2B=(K2)+ (/)and the K are the amplitudes of the two components. The amplitude spectrum for thecase== 0.13 is shown in Fig. 7.54. The approximate spectrum looks like asmooth curve through the regional model’s spectrum, it lacks the broad flat section. Forthe frequency range of interest, periods in the range 1 to 20 days, the knee in the spin-up(Fig. 6.36) is important.The idea that the amplitude spectrum should be well approximated by a frictionaladjustment style process has its basis in Fig. 7.56. The model’s transport response to anoscillating wind looks very much like the frictional adjustment solution with sinusoidalwind forcing shown in Fig. 4.26.The amplitude spectrum was not sensitive to the value of the friction parametersfor periods in the range 3 - 9 days. The most noticeable effect of larger friction was toincrease the phase lag and to reduce the height of the transient peak in Fig. 7.56.7.4 SummaryThe rotation of the earth reduces the transport through Hecate Strait compared witha non-rotating earth. The effect is consistent with the rotation-limited-flux model withparameter W/L 0.2. For the numerical experiments discussed, the rotation of theearth (f = 1.1 . iO s’) reduced the steady-state transport by a factor between 2 and5 depending on the friction parameter and the method of forcing the strait.The character of the response of Hecate Strait to an impulsive wind changes when therotation of the earth is neglected. After half a day the transport spin-up curves for theChapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 123a)0EFigure 7.56: Transport time series for an oscillating along-shore wind with period 6 days.The solid line is the transport, the dotted is the wind forcing. Both time series have beennormalized by the amplitude of the respective sine waves (0.15 Sv, 0.1 Pa).Figure 7.57: The transport spin-up in Hecate Strait for a non-rotating (solid) and rotating(dashed) version of the model. For the non-rotating case k = 1.3 iO m s and forthe rotating case k = 0.5• iO m srn’. The bottom friction for the non-rotating case waschosen so that the steady-state transports were similar.1 .510.50-0.5—1-1.50 3 6 9Time (d)12 15 180.30-1-aC 0 1500U)0.00CF- 0.0 5.0Time (days)10.0Chapter 7. Rotational Limitations on the Water Transport Through Hecate Strait 124rotating earth and non-rotating earth (Fig. 7.57) are very different. The major differenceis that the non-rotating response has one time scale, while the rotating case has two. Thebottom friction values were selected so that the steady-state transports were similar.In the Hecate Strait Model, the water is forced through Hecate Strait not only bythe local wind forcing but also by the along-shore pressure gradients established by thecurrents in other parts of the model. When the rotation of the earth is turned off,topographic steering and geostrophic balance disappear from the dynamics. This affectsthe circulation both inside and outside Hecate Strait.The bottom friction in a non-rotating model can be chosen so that the non-rotatingand rotating models give similar results for short time scales. The results in this chaptershow that the bottom friction will be larger in the non-rotating case and that the resultswill be sensitive to the friction parameter value. The use of quadratic friction in a nonrotating model may give very different results from linear friction. In the rotating versionof the Hecate Strait Model, the results were only weakly dependent on the bottom frictionparameters.Chapter 8Wind-Driven Flow Patterns in Hecate StraitThis chapter presents a comparison of the observed flow patterns with the patterns fromthe Hecate Strait Model. The model results support a new interpretation of the observedcurrents in southern Hecate Strait. The discussion is divided into three sections: 1)the basic flow patterns from the Hecate Strait Model, 2) direct comparison of modelsimulations with the observations, and 3) the regional circulation patterns.The discussion of the basic flow patterns (circulation) starts with a look at the spin-up of the velocity field under a SE wind. The observed counter-current in south-centralHecate Strait is seen to playa major role in the model circulation. Then the model is usedto show how the character of the velocity pattern changes when the rotation of the earthis ignored. Finally, the pattern in southern Hecate Strait is shown to be independent ofthe details of the flow in northern Hecate Strait.Comparison of the observed and simulated velocities from 25 January to 30 March1984 shows that the Hecate Strait Model captures many of the features of the observations. In particular, the counter-current in southern Hecate Strait is present in the modelresults. Comparison of observed and simulated drifter trajectories from July 1990 andJuly 1991 demonstrates that the model can provide useful information about trajectorydirections, but the simulated drifters significantly under-estimate speed of near-surfacedrifters. The model does not provide good information in the vicinity of Cape St. James.Drifter trajectories from the winter 1984 simulation show that the steady-state winddriven flow pattern provides a useful picture of the overall flow pattern. These drifters125Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 126emphasize that the model currents are strongly constrained by the topography (topographic steering). The model results indicate that the counter-current is part of a patternthat (under SE winds) takes water flowing north along the eastern side of Hecate Straitto the western side of the strait.8.1 Basic PatternsThe evolution of the velocity field during a spin-up experiment exhibits a fast responseand a slow response, analogous to the transport (Fig. 6.36). The first part of the domain to respond to the wind is the shallow water in northern and western Hecate Strait(Fig. 8.58a; Day 1). This pattern corresponds to the fast transport response. By the endof the Day 2 (Fig. 8.58b), one can see the beginning of a southwest flowing counter-currentin south-central Hecate Strait and a current developing along the coast from northernVancouver Island to northern Hecate Strait. The fully developed pattern (Fig. 8.59a;Day 8) has the same character as that seen on Day 2. The pattern has simply becomemore intense. In interest of clarity only one vector in four has been plotted. This is trueof all the plots of the velocity and transport vector fields.The northward transport of water through Hecate Strait is complicated by the southwest flowing counter-current along the north flank of Moresby Trough (represented bythe 200 m contour in south-central Hecate Strait). Water which starts out flowing northalong the mainland coast in Queen Charlotte Sound can either continue to hug the coastand continue north through Hecate Strait and into Dixon Entrance or it can turn tothe SW and follow the north flank of Moresby Trough towards Cape St. James. Onceat the cape, the water can either escape to the open ocean or remain in Hecate Straitand flow north along the western side of Hecate Strait. The currents along the easternand western sides of the strait eventually recombine in the northeastern corner of HecateChapter 8. Wind-Driven Flow Patterns in Hecate Strait 127777‘7/f7,,,7,,,. ‘( II,“It ,,,,1 7 /,A,,ft / ,—,17?,,.71tIf,.fItII,II17.. ..____,..— ____(a) Day 1 (b) Day 2Figure 8.58: Evolution of the velocity field with a steady SE wind. The wind directionis indicated with an arrow on Graham Island. The 200 m contour is the heavy dark line.Only 1/4 of the vectors are plotted.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 128Figure 8.59: The velocity field and transport vector field at Day 8, with a steady SE wind- typical winter storm winds. On the shelf the velocities have reached a steady state. Inthe deep ocean the velocities are still adjusting, but this does not affect the velocity fieldon the shelf. The 200 m contour is shown.(a) velocity field (b) transport fieldChapter 8. Wind-Driven Flow Patterns in Hecate Strait 129Strait.The relationship between the flow pattern and the topography is clear: the steady-state currents follow the topography. This is topographic steering, where the conservationof potential vorticity creates a tendency for the low-frequency flow to follow the localisobaths (lines of constant depth; equation 5.26 in Chapter 5). The steady-state velocitypattern (Fig. 8.59a) is the basic pattern that the reader should remember. The velocitiesin the northern part of the Hecate Strait are in the direction of the wind and there is alarge flow against the wind in the south-central portion of the strait. This is in agreementwith the observations (Chapter 3).The steady-state velocity pattern and the steady-state transport pattern (Fig. 8.59b)are very similar. Topographic steering is the dominant factor in both patterns. Themost noticeable difference is in the shallow water of northwestern Hecate Strait wherethe large velocities transport very little water.In Queen Charlotte Sound the near-shore flow is parallel to the coast and there arelarge flows into and out of the sound near Cape Scott. In the central portion of thesound the currents are parallel to the local isobaths. The pattern is consistent withthat of the principal axes of variance of the observed near-surface current fluctuations(Fig. 2.9). Topographic steering provides an organizing principle for the currents in thecentral sound which were described by Crawford et al. (1985) as ‘weak and disorganized’.In Dixon Entrance there is no evidence of the eddies seen in the observations. Thestrong flow along the northern side is consistent with the historical view of the winddriven flow (Chapter 2). The model velocity pattern in Dixon Entrance is sensitive tothe details of the topography at the mouth of Dixon Entrance, especially at LangaraIsland (Appendix F).On the western side of Hecate Strait, about half way along the Queen CharlotteIslands, there are several very large velocities (Fig. 8.59a). A simulation of the winterChapter 8. Wind-Driven Flow Patterns in Hecate Strait 1301984 with k = 0.3• iO s1 failed when these currents exceeded 8 m s1. Linear frictionhas limitations. Notice also the large velocity vector at the one grid point that connectsChatham Sound and Hecate Strait (the northeast corner of the strait).The steady-state pattern for an along-shore wind from the NW is shown in Fig. 8.60a.Reversing the wind reverses the pattern. The pattern for a cross-shore wind is shownin Fig. 8.60b. The pattern for a wind direction between the along-shore and cross-shoredirections is a linear combination of the basic along-shore and cross-shore patterns.The circulation patterns are robust. The character of the patterns did not changewhen quadratic friction and spatially varying linear friction were used. The magnitudeof the vectors and minor details changed when the parameter values were changed butthe pattern remained. Velocity fields corresponding to quadratic friction and spatiallyvarying linear friction are shown in Appendix F.Given that the model does not represent a stratified fluid very well the circulation inQueen Charlotte Sound and Dixon Entrance was not investigated in more detail.8.1.1 Vorticity BalanceDuring the spin-up run, diagnostic vorticity balances were calculated at three sectionsacross Hecate Strait: at M-line in the south, at W-line in the middle, and at R-line inthe north. The results of these calculations are discussed here.Recall from Chapter 5 that the vorticity balance can be written-+(h).V()=Vx (_Pf) (8.47)where r is the wind stress and Pf represents the friction and dissipation terms discussedin Chapter 5.4.The term (Yh) V(f/h) is the vortex stretching term. In the Hecate Strait Model,where f is a constant, it represents the movement of water across depth contours. WhenChapter 8. Wind-Driven Flow Patterns in Hecate Strait(a) along-shore wind131(b) cross-shore windFigure 8.60: Steady-state velocity fields for two different wind directions.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 132a parcel of water is moved from one depth to another, the vortex column is stretched orcompressed. When the vortex stretching term is small, the water is flowing parallel tothe local depth contours; thus topographic steering.When a wind stress is imposed on a motionless fluid, the water starts to move downwind without regard for the topography. This generates a large vortex stretching term.In the initial stages the balance is between the time rate of change of the relative vorticityand the vortex stretching(flh) . V(f/h) (8.48)with a small contribution from the curl of the wind stress. Relative vorticity (shear) isgenerated and the flow reorganizes itself to minimize vortex stretching. In most regionsof Hecate Strait the reorganization is successful and the flow is parallel to the localdepth contours. In Hecate Strait, the regions of cross-isobath flow tend to occur wherethe constraints imposed by the coastline and the bathymetry force the water to changedepth; at the edges of the strait and over most of the northern Hecate Strait. Thesteady-state vorticity balance in these areas is(vh) . V() V x ( - Pf) (8.49)with a contribution from the advection of relative vorticity. The curl of the bottom stressis an important factor in the vorticity balance.Before moving on to the specifics of the vorticity balance, three useful results:• The surface wind stress r was spatially uniform. However the 1/h dependencecreates a wind stress curl. In the spin-up experiments, = (O,r’) and the windstress curl wasr9hOnly the cross-strait depth gradients couple with the along-strait wind stress togenerate vorticity.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 133• The curl of the bottom friction makes an important contribution. For linear bottomfrictionkvY kC k h 0hVx(T)=_ ua;._vThus the bottom friction contributes through both the relative vorticity and theinteraction of the velocity with the bottom slope.• The term (f/h)(th/Ot) was always very small and is ignored in this discussion.At R-line the steady-state vorticity balance was established within the first day. Thebalance was between vortex stretching and the curl of the wind stress and bottom stress(8.49). There was a small contribution from the advection of relative vorticity. The timerate of change 8C/Ot was very small after the first 12 hours of integration.The evolution of the vorticity balance at M-line and W-line was slower than at R-line.The balance between the time rate of change and the vortex stretching (8.48) dominatedfor the first two days of the spin-up, with a small contribution from the curl of the windstress. By Day 4 the two dominant terms had decreased by an order of magnitude andthe vorticity balance required all the terms, except (f/h)(th7/ôt).By Day 8, the vorticity balance involved all the terms. The magnitudes were verysmall. For example, the maximum value of 8/8t, which occurred in the middle of M-line,was1Os2This represented an increase in the shear between two neighbouring grid points of 0.05 cmper day. While the vorticity was still changing in one small region, the pattern wasessentially the steady-state pattern. The primary regions of cross-isobath flow (vortexstretching) were the eastern and western edges of the strait. In these regions, both thecurl of the wind stress and the curl of the friction terms were important factors in thevorticity balance.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 134In northern Hecate Strait, the flow pattern with the velocity roughly parallel to theisobaths is established very quickly. After a short initial start-up period, the fluid parcelsdo not feel the strong topographic gradients because they are constrained to flow parallelto depth contours This might explain why rotation-limited-flux (with its flat bottom)provided a good explanation for the the steady-state transport results but not for thespin-up time results (Chapter 7 and Appendix G). In the initial spin-up, the vorticitydynamics play a large role as the flow is forced to align with the depth contours. After theflow is aligned, the importance of the sloping bottom diminishes and rotation-limited-fluxbecomes a useful concept.8.1.2 Effect of RotationThe steady-state circulation patterns are clearly governed by topographic steering. However it has been suggested that the counter-current along the north flank of MoresbyTrough is not due to topographic steering but due to a back-pressure induced by theconstriction at the northern end of the strait: that the narrowing and shallowing of thestrait causes water to pile up at the northern end of the strait, and the resulting pressuregradient forces water to flow back down the strait and around Cape St. James. Thenumerical model is well suited to investigating this possibility.The steady-state velocity pattern when rotation is absent (Fig. 8.61) is qualitativelydifferent from the basic rotating pattern (Fig. 8.59a). The non-rotating pattern consistsof broad sweeping currents which flow across the topographic features, not around them:topographic steering is absent when there is no rotation. The broad current out of QueenCharlotte Sound and around Cape St. James may be due to the constriction in northernHecate Strait. However this current is not the concentrated counter-current seen in thebasic pattern.To further test the idea that the counter-current is independent of the details ofChapter 8. Wind-Driven Flow Patterns in Hecate Strait 135Figure 8.61: Steady-state velocity field for a SE wind and a non-rotating earth (f = 0).Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 136Figure 8.62: Steady-state velocity field when: (a) northern Hecate Strait is blocked off;(b) the constriction at the northern end of Hecate Strait is removed (see text). This wasfor a SE wind and a rotating earth, f 1.1 x iO s.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 13754 NFigure 8.63: Close up of the northeast corner of Hecate Strait.the flow in northern Hecate Strait, two experiments were conducted with altered modelbathymetry. In the first, the northern end of Hecate Strait was blocked off (Fig. 8.62a). Inthe second, the northern end of Hecate Strait was widened and all the depths in northernHecate Strait and Dixon Entrance were set to 80 m (Fig. 8.62b). In both cases thepattern in southern Hecate Strait and Queen Charlotte Sound were not affected by thechanges in northern Hecate Strait. The steady-state pattern is dominated by topographicsteering and the counter-current in Moresby Trough is independent of the details of theflow at the northern end of Hecate Strait.8.1.3 Chatham Sound DiversionBrown Passage, which connects Chatham Sound to Hecate Strait, looks like a widechannel (Fig. 8.63) but it is actually quite shallow and the portion deeper than 10 m isnarrow. During preliminary experiments I noticed that during SE winds most driftersmoving north along Hecate Strait made a hard right turn into Brown Passage. The131 W‘1Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 138drifters then proceeded north along Chatham Sound and into Dixon Entrance. I assumedthat this was wrong and closed the Brown Passage down to 1 grid point.Narrowing Brown passage did not change the drifter tracks. Further investigationfound that 30-50% of the transport through Hecate Strait was diverted through BrownPassage and into Chatham Sound (recall the large velocity vector in Fig. 8.59). Thesmaller the friction, the larger the diversion. A related problem was experienced byCrean et al., 1988 in the Strait of Georgia. They found that large friction parameterswere needed to get reasonable tidal velocities in narrow poorly-resolved passages. Theyintroduced a spatially varying friction coefficient to deal with the problem.At the time of the writing of this thesis it is not known whether a significant fractionof the transport through Hecate Strait is actually diverted through Chatham Sound on itsway into Dixon Entrance. Analysis of the observations from the 1990-1992 field programshould provide some insight. The drifters from the July 1991 drifter deployment (Fig. 3.22and Fig. 3.23) indicate that there is surface flow into Chatham Sound via Brown Passageduring SE winds.8.2 Comparison with Observations in Hecate StraitIn this section the model is tested by comparison of model simulations with the observations in Hecate Strait. In the first part the mean and EOF mode 1 velocity patterns arecompared for the period 25 January to 30 March 1984. This is the same period as thetransport comparison in Chapter 7. In the second part simulated drifter trajectories arecompared with observations from July 1990 and July 1991.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 1398.2.1 Currents: Winter 1984The simulations of the winter of 1984 discussed here are the same simulations that wereused in the transport comparison in Chapter 7. The model results plotted in the figures inthis section were taken from simulation E501a which used linear friction (k = 0.5 x iOs’) and the standard parameters (Table 6.6). The results from the other simulations(Table 8.11) were very similar and are not shown.experiment A1 (cm2 s2) A2 (cm2 s2)E501a (linear) 374 (0.67) 82 (0.15)E502a (quadratic) 359 (0.72) 56 (0.11)E503 (RMS) 395 (0.69) 110 (0.19)depth averagedobservations 633 (0.39) 192 (0.12)observationsreported in 1231 (0.28) 605 (0.14)Chapter 3Table 8.11: The first two eigenvalues from the EOF analysis of the simulated and observedvelocities for the period 25 January- 30 March 1984. Each eigenvalue, A, is representedby two values; its magnitude and the fraction of total energy that this represents (inbrackets).The simulated mean currents have the same character as the observations (Fig. 8.64a).The flow is to the SW in the middle of M-line, to the NE at R05 and W04, and confused inthe middle of W—line. Notable exceptions are the strong cross-isobath velocities observedat M03 and the small mean velocity observed at M06. The simulated velocities at W02and W03 are much too small and go the wrong way. At R05 the observed velocity has astrong cross-isobath component, absent in the simulated velocity.The model current meters were placed at the grid point closest to the actual meter.The differences in location can be seen by the location of the tails of the vectors in(Fig. 8.64).The time series were not scaled by their standard deviation before EOF analysis.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 140Figure 8.64: Model verification. Comparison of the simulated (E501a) and observedcurrents in Hecate Strait from 25 January to 30 March 1984. The depth-averaged observations are shown with open arrow heads; the simulated currents with solid arrow heads.The model was driven with winds measured near W04.(a) mean currents (b) mode 1 currentsChapter 8. Wind-Driven Flow Patterns in Hecate Strait 141Therefore the eigenvalues reported in Table 8.11 have units of cm2 s2 and are a directmeasure of the energy. The eigenvalues from the simulations are smaller than those fromthe observations. This is consistent with the results in Chapter 7 where the simulatedtransport fluctuations were too small. In the simulations mode 1 (A1) represents 70%of the energy, whereas in the observations it represents 30% to 40%. The higher modescontain much more energy in the observations than they do in the simulations. Thisreminds us that there are significant processes in the ocean that are not represented inthis model (Chapter 2).The simulated EOF mode 1 velocity pattern captures the character of the observedpattern (Fig. 8.64b; note the scale change between the mean and mode 1 currents).The model reproduces the gyre structure in M-line with the velocity fluctuations inthe deep water in the opposite direction to those in the shallower water and in theopposite direction to the fluctuations at R05. There are problems, however. At M-linethe fluctuations in the deep water are too small compared with those in the shallowwater. The fluctuations at M06 are much too large. At R05 the velocity is too smalland seems overly constrained by the topography. The velocities in the middle of W-lineexhibit the same problem as the mean velocities— too small and in the wrong direction.The velocities at W-line are sensitive to the details of the friction. The velocities inthe middle of W-line (W2E, W02, W04) are at the northern end of the counter-currentin Moresby Trough. When the friction coefficient is increased, these small southwardvelocities become small northward velocities (see Fig. 6 in Hannah et al., 1991).At M-line the major problem is that the velocities in the deep water are too smallwith respect to the velocities in the shallow water. This is especially true in the mode 1velocities. It is conceivable that the observations do not accurately represent the depthaveraged flow. In the middle of M-line (MOl to M05) the upper current meter was 50 mbelow the surface. At M1E and M06 the upper meters were at 21 m and 25 m respectively.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 142A significant fraction of the water in the upper 50 m might move in the direction of thewind. This would decrease the depth-averaged velocity in the region where the waterflows against the wind. Blaming the observations is a time-honoured excuse of numericalmodellers.The velocity patterns from the other experiments reported in Table 8.11 are very similar to those shown. The eddy viscosity had a negligible effect on the results. Removingit resulted in an overall energy level increase of less than 2%.The time series of modal amplitudes for the observed and simulated EOF-mode-1velocities are compared in Fig. 8.65. The differences between the modal amplitudes aremirrored in the differences between the observed and simulated transports (Fig. 7.47).Recall that in the observations, the mode 1 velocities represented the transport of waterthrough the strait (Chapter 3). The same is true of the simulation results.The observed and simulated EOF mode 2 time series (not shown) bear no resemblanceto each other.DiscussionChanging the friction parameters has more impact on the velocities in the shallow waterthan in the deep water. Therefore the friction could be increased to improve the patternacross the M-line and in the middle of W-line. Such an increase in friction would decreasethe transport and the velocity at R05. This could be compensated by increasing the windstress estimate. Trying to force the model to fit the observations by this ad hoc techniquehas drawbacks. Nothing within the range of the model dynamics will change M06 froma strong along-strait velocity to a weak on-shore velocity. Trying to reduce the velocityat M06 (h=100 m) while increasing the velocity at R05 (h= 55 m) is unlikely to besuccessful.Using the rms tidal velocities to compute a spatially varying friction coefficient wasc.EC’,0EChapter 8. Wind-Driven Flow Patterns in Hecate Strait 14380400-408025Figure 8.65: Time series of modal amplitudes of the EOF-mode-1 velocities. Comparisonof simulation E501a (solid) with the observations (dotted) in Hecate Strait from 25January to 30 March 1984.35 45 55 65 75 85 95Julian day 1984Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 144an attempt to redistribute the friction in a physically realistic manner. The lack ofsignificant improvement in the results indicates that friction is not the key to improvedsimulations.These results indicate that minor changes in the bottom friction formulation areunlikely to improve the simulations. One possible improvement is the scheme of Hunterand Hearn (1990) which removes the constraint that the mean velocity and the bottomstress be parallel. This might be important and the results would be instructive.8.2.2 DriftersTests showed that the simulated drifters travelled much more slowly than the real near-surface drifters. To demonstrate that the simulated drifters had the proper behaviouralbeit a bit slow, the following was done. Simulated drifters were started at regularintervals along the observed drifter track. The simulated drifters were then followedfor several days to illustrate the model behaviour. In many cases the simulated driftertrajectories were very sensitive to the exact location of the drifter and extra drifters areused to show this.July 1990The July 1990 drifter study took place in southern Hecate Strait. Two simulations areshown. The observed drifters are near-surface drifters with drogues centred at 10 m. Theobserved trajectories contain the tidal motions, while the simulated ones do not. Thewinds used in these simulations were measured at an Atmospheric Environment Service(AES) weather buoy located 150 km SE of Cape St. James.In Figure 8.66 the observed drifter started near the mouth of Moresby Trough andproceeded in a clockwise loop around the shallow bank to the south (Middle Bank).Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 145Figure 8.66: Comparison of an observed drifter trajectory and simulated drifters inMoresby Trough from 12 to 18 July 1990. The observed drifter (solid) started at thecross. Simulated drifters (dashed) were started at daily intervals (circles) along the observed drifter trajectory. The simulated drifter trajectories are 4 days long. In caseswhere the simulated drifter did not move very far an extra drifter was started 1 gridpoint away. The winds were measured at a weather buoy located 80 km below the bottom of the figure (see text). The wind stress time series is shown in the lower left corner.The free end of the wind vectors indicates the direction the wind is blowing to. The 200m contour is shown for reference.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 146Figure 8.67: Comparison of an observed drifter trajectory and simulated drifters nearCape St. James from 21 to 26 July 1990. See Fig. 8.66 for details.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 147The simulated drifters were started at daily intervals and were followed for 4 days. Theobserved drifter trajectory was 6 days long (12 July to 18 July).In Figure 8.67 the observed drifter started to the east of Cape St. James and travelledwest past the cape before it looped back into Moresby Trough. The observed trajectorywas 5 days long and the simulated drifters were followed for 4 days. The model did notallow the drifters to move out past the cape. However it did bring the drifter at day 2back into Moresby Trough.In both cases, the simulated drifters were much too slow, generally 1/4 to 1/10 thespeed of the observed drifters. Nevertheless the simulated drifters were good indicatorsof the direction of travel of the near-surface drifters, except in the vicinity of CapeJames. The results shown are representative of simulations of other drifters from thesame deployment.July 1991The July 1991 drifter study took place in northern Hecate Strait. Two simulations areshown. Again the observed trajectories contain the tidal motions, while the simulatedones do not. The winds used in these simulations were measured at an AES weatherbuoy whose location is shown in the figures with the drifters.The observed drifter in Fig. 8.68 was launched just after the wind abruptly shiftedfrom northwesterly to southeasterly. The drifter was driven north along Hecate Straitand then turned into Chatham Sound. The trajectory was 8 days long. The drifter hada drogue centred at 10 m and it drifted at half the speed of a drifter launched at thesame place and time with a drogue centred at 3.5 m (see Fig. 3.23).The simulated drifters, started at daily intervals and followed for 4 days, do a goodjob of tracking the observed drifter when the winds were strong. The velocities areabout 1/4 the observed values. When the wind died down (about the time the drifterChapter 8. Wind-Driven Flow Patterns in Hecate Strait 148Figure 8.68: Observed and simulated drifter trajectories in northern Hecate Strait from11 to 19 July 1991. Drifter b32 was launched shortly after the winds changed directionon July 10. The wind stress time series shown in the lower left corner was measured atthe * in the center of the strait. The free end of the wind vectors indicates the directionthe wind is blowing to.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 149Figure 8.69: Observed and simulated drifter trajectories in northern Hecate Strait from8 to 10 July 1991. All the trajectories are 2 days long. During this time the wind wassteady and from the NW. The end of the trajectories corresponds to the change in winddirection near July 10. The wind stress time series shown in the lower left corner wasmeasured at the * in the center of the strait.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 150entered Chatham Sound) the simulated drifters did not move far enough to give a senseof direction.In the second test four drifters were launched in a line across northern Hecate Strait(within 2 hours of each other). The drogues were centred at 3.5 m. All the trajectoriesare two days long (observed and simulated) and no extra drifters were used (Fig. 8.69).The simulated drifters travel in the right direction and at about 1/2 to 3/4 the speed ofthe observed drifters.A marked feature of the observed drifters from this deployment was that the drifterstended not to cross the steep escarpment that separates Hecate Strait from Dixon Entrance (see drifter c22 in Fig. 3.22). Drifters tended to work their way along the escarpment and cross at the eastern end. This is also a feature of the simulated drifters.Simulation of particular drifters was unsatisfactory.8.3 Regional Circulation PatternsThe drifter trajectories from the winter of 1984 simulation (E501a) provide a clear iilustration of the model’s circulation pattern. The near-shore drifters in Fig. 8.70 andFig. 8.71 trace out the historical winter flow pattern. Under winter (SE) winds, the water flows into Queen Charlotte Sound at Cape Scott and then follows the coastline norththrough Hecate Strait and Dixon Entrance and then into the Pacific Ocean. During thesimulation the winds were consistently from the SE.The effect of topographic steering is clearly illustrated by the drifters in Queen Charlotte Sound (Fig. 8.71). Simulated drifters do not cross the 200 m contour: offshoredrifters stay offshore and inshore drifters stay inshore. None of the drifters show anytendency to flow out of the mouth of Queen Charlotte Sound as one was observed to doin the winter of 1990.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 151Figure 8.70: Simulated drifters in the northern half of the model domain for 20 Jan to30 Mar 1984. The starting location of each drifter is marked by a cross. The driftertrajectories are marked with arrows at 20 d intervals.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 132Figure 8.71: Simulated drifters in Queen Charlotte Sound for 20 Jan to 31 Mar 1984.The starting location of each drifter is marked by a cross. The drifter trajectories aremarked with arrows at 20 d intervals.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 153Figure 8.72: Simulated drifters near Cape St. James. The starting location of eachdrifters is marked by a cross. The drifter trajectories are marked with arrows at 20 dintervals.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 154The counter-current in south-central Hecate Strait shows clearly with the drifterstravelling to the SW along the north slope of Moresby Trough. Along the mainlandcoast there is a fine line between those drifters which continue to hug the coast and thosewhich turn to the SW towards Cape St. James.The drifter trajectories have some details in common with the observed trajectories:• Drifters in northeast Dixon Entrance make a ioop into and out of Clarence Straiton their way to the Pacific (compare with Fig. 3.18).• Under SE winds, many drifters travelling from Hecate Strait to Dixon Entrance doso by detouring through Chatham Sound rather than floating directly into DixonEntrance.• The drifter in Fig. 8.70 that goes into and out of the mouth of Dixon Entrancenear Langara Island is similar to that seen from the winter of 1990 (compare withFig. 3.21).The fate of a drifter caught in the counter-current depends on the details of the flow asthe drifter nears Cape St. James. The drifter trajectories shown in Fig. 8.72 illustrate thevariability of the simulated flow near Cape St. James. The starting locations representone of the regions where English sole are believed to spawn. Drifters were started every5 days at each location and selected ones plotted. All the drifters started at location Aescaped past Cape St. James into the open ocean. All the drifters started at locationC travelled close to Cape St. James and then moved north along Hecate Strait andinto Dixon Entrance. The fate of the drifters started at location B was less predictable.Some drifters escaped past Cape St. James and some remained in Hecate Strait (andtravelled into Dixon Entrance). The details of the flow at Cape St. James are crucial indetermining the fate of drifters travelling SW along Moresby Trough.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 155In this simulation the 200 m contour defines a region that separates those locationswhere the drifters always escaped past Cape St. James and those locations where thedrifters remain in Hecate Strait. The exact position of this critical region depends on thebottom friction parameterization.The overall circulation pattern revealed by the drifters in Fig. 8.70 and Fig. 8.71 isconsistent with the pattern of the steady-state currents in Fig. 8.59. The time variationof the winds affects the details such as the fate of individual drifters near Cape St.James but not the overall circulation pattern. This is consistent with the idea that theEOF-mode-1 currents represent fluctuations in magnitude (not direction) about the meanvelocities. The features discussed in this section were evident in all the simulations anddo not depend on the friction parameterization.All of the drifter trajectories shown in this section were computed during the simulation runs except for the drifters in Fig. 8.71 which were computed using the dailycurrents saved during the simulation. This technique is discussed in Appendix D andAppendix F.8.3.1 Southern Hecate StraitThe model results presented above suggest the following winter circulation pattern insouthern Hecate Strait. Water from Queen Charlotte Sound enters Hecate Strait alongthe mainland coast on the eastern side. Some of the water hugs the coastline and continues north. The rest of the water flows southwest along Moresby Trough to Cape St.James where some water escapes around the cape into the open ocean and some flowsnorth along the western side of Hecate Strait (Fig. 8.73). This pattern is consistent withthe observations.The new idea is the S-shaped path from the eastern side of Hecate Strait to CapeSt. James and then north along the western side of the strait. The S-shaped path has156Figure 8.73: Sketch of the circulation in Hecate Strait and Queen Charlotte Sound undera SE wind.Chapter 8. Wind-Driven Flow Patterns in Hecate StraitChapter 8. Wind-Driven Flow Patterns in Hecate Strait 157implications for the larval advection problem and differs from the interpretation of themean and mode 1 current patterns proposed by Crawford, Tyler and Thomson (1990).They speculated that the observed current pattern at M-line represented the north end ofan eddy that closed somewhere to the south. This eddy was invoked as a mechanism to‘...recirculate a significant fraction of the larvae in the Strait, increasing their residencetime sufficiently to allow settling out, thus enhancing recruitment.’ The S-shaped pathwould also increase the residence time of the larvae. The proposed path is also supportedby a remark made by Crawford, Tyler and Thomson (1990):Fish eggs and larvae in the Strait tend to be concentrated on the western side,putting them in a position to be retained by the eddy. They seem equallyabundant in the northern and southern halves rather than concentrated tothe south where the eddy seems most evident (Mason et al., 1981c).The pattern proposed in Fig. 8.73 would concentrate larvae in a narrow band alongthe west side of Hecate Strait. If the S-shaped circulation pattern is correct then theadvection of English sole larvae from the spawning grounds in southern Hecate Straitto the nursery in the northwest corner depends on the details of the flow near Cape St.James.8.4 SummaryThe Hecate Strait Model, a depth-averaged model driven by local winds, is a useful modelof the flow patterns in Hecate Strait. The simulations showed that:• The model does a good job of simulating the winter transport and mode 1 velocityfluctuations in Hecate Strait.Chapter 8. Wind-Driven Flow Patterns in Hecate Strait 158• The model currents capture the character of the spatial patterns of the observedmean and mode 1 velocities.• The model drifters generally give a good indication of near-surface drifter directionbut under-predict the velocity.• The steady-state current pattern (Fig. 8.59) provides good prediction of the modeldrifter trajectories and is a reasonable guide to the circulation patterns in theregion.• The simulated drifters do not perform well near Cape St. James.• The results were best when the wind was strong, which indicates that other forcingbecomes important when the winds are weak.The model results in southern Hecate Strait are summarized in Fig. 8.73. The pattern is consistent with the observations and differs from a previous interpretation of theobservations.Chapter 9ConclusionA regional model of the depth-averaged currents was successfully applied to the wintercirculation in Hecate Strait, British Columbia. Simulations using the Hecate Strait Modeldemonstrated:• good agreement between the observed and simulated water transport through HecateStrait for January to March 1984.• qualitative agreement between the observed and simulated current patterns for thesame period.• reasonable prediction of the direction of travel of near-surface drifters, but not theirspeeds.The physical insights provided by this study indicate that the earth’s rotation hasa significant impact on the water transport in Hecate Strait. The interaction of therotation with the finite length of the strait creates an extra impedance, which behaveslike a linear friction term. The simulated transport through Hecate Strait was foundto be consistent with the rotation-limited-flux model. In numerical experiments with anon-rotating earth, the linear bottom friction had to be increased by roughly three timesto be equivalent to the effects of rotation.The circulation pattern in the Hecate Strait Model depended strongly on the rotationof the earth. This was interpreted as the effect of topographic steering — the tendency forthe low-frequency flow in a rotating fluid to be parallel to the local isobaths. In particular159Chapter 9. Conclusion 160the counter-current in south-central Hecate Strait was a consequence of topographicsteering and independent of the details of the flow in northern Hecate Strait.The analysis of the observations and the model results led to three insights of lastingvalue:• For the winter of 1984, the observed fluctuations in water transport through HecateStrait were highly correlated with the EOF-mode-1 currents. In fact, the EOFmode-i current pattern is the pattern that moves water from one end of HecateStrait to the other. The remaining 75% of the energy in the current fluctuations isrelated to redistribution of water within Hecate Strait.• The counter-flow in south-central Hecate Strait is part of an S-shaped flow patternthat moves water from the eastern side of Hecate Strait to the western side, ratherthan the northern edge of an eddy. This has implications for the larval advection problem and makes the details of the flow near Cape St. James extremelyimportant.• The rotation-limited-flux model is a valid long-wave approximation to a more complete theory. This means that: 1) rotation-limited-flux provides useful insight intothe low-frequency sea-level fluctuations in sea straits; and 2) the rotation of theearth is important even for straits where the width is less than one external Rossbyradius (W <This study was limited in scope. The Hecate Strait Model was the first circulationmodel of the region and the assumption was made that the water velocity and densitywere vertically homogeneous. This restricted the application of the model to HecateStrait. Even within Hecate Strait these assumptions severely limit the application of themodel to practical problems such as oil spill trajectory prediction and larval advectionproblems.Chapter 9. Conclusion 161Despite these restrictions, the Hecate Strait Model has been a useful tool for understanding the circulation in the Queen Charlotte Islands region. Predictions such as theproposed circulation pattern in southern Hecate Strait were used as working hypothesesfor planning the 1990 to 1992 field program. The steady state velocity pattern was usedin planning the July 1990 and July 1991 drifter programs. The inevitable discrepanciesthat will arise between the new observations and the predictions of the Hecate StraitModel will provide fertile ground for future modelling studies.9.1 Future WorkThere is room to improve the Hecate Strait Model. The areas for model improvementfall into three main categories: 1) atmospheric forcing, 2) open boundary information,and 3) model dynamics. The first two categories reflect the fact that the circulation inHecate Strait is affected by processes outside the strait. The greatest improvement in thetransport simulations would come from including these non-local effects. Improvementin the simulated currents requires improved model dynamics.Improvements in atmospheric forcing can be easily incorporated into the model. Themajor hurdle is the construction of reliable forcing fields. At this time good informationon the spatial variability of the winds is not available. The construction of a meso-scaleclimatology would be very useful.The effects of the baroclinic and thermohaline circulation are beyond the reach of thisnumerical model. Nevertheless information about the vertical shear could be included bythe addition of a spectral model in the vertical (Davies, 1985 and 1987; Forristall, 1974).This would provide useful information about velocity profiles and provide insight intothe reasons for the differences between the simulated and observed currents at M-line.The near-bottom currents could be used to improve the bottom stress estimate.Chapter 9. Conclusion 162Improving the open boundary information is a very large task. The open boundaryconditions could be modified to include the tides, but I am not sure that doing so wouldimprove our understanding of the circulation. Incorporating remotely generated shelfwaves into the regional model requires improved information along the open boundaries.Before this is done, there are many interesting process studies that could be done toillustrate the effect of shelf waves on the circulation. Accurate representation of the shelfwaves probably requires a model that supports density variations, but useful studies canbe done with the depth-averaged model.The most difficult area for improvement is the impact of the deep ocean circulationon the model region. Very little is known about the impact of water mass intrusion,along-shore pressure gradients, and eddies. These are beyond the scope of the modelpresented here.Improvement and modification to the Hecate Strait Model may not be the best routefor future work. A suite of finite element models have been adapted to the north coastregion by M.G.G. Foreman at the Institute of Ocean Sciences (lOS), Sidney, B.C. Themodels range from a barotropic tidal model (Foreman et al., 1992) to a three-dimensionaldiagnostic model (the density field is fixed). A complete three-dimensional model is beingdeveloped. These models permit the investigation of the circulation using a range ofphysics and complexity while maintaining a consistent numerical framework. As well, afull three-dimensional model based on the Princeton model (Bhimberg and Mellor, 1987)is being developed by P.C. Cummins at lOS.This thesis has concentrated on the response to local wind forcing. Very little isknown about how events outside the Queen Charlotte Islands region affect the circulationin Hecate Strait. Shelf waves are an important mechanism for transmitting informationabout remote events. The understanding of the shelf circulation in the Queen CharlotteIslands region would benefit from understanding the impact of remotely generated shelfChapter 9. Conclusion 163waves. In particular: 1) Is Brooks Peninsula a barrier to shelf wave propagation? and 2)What happens as the shelf wave passes by the mouth of Queen Charlotte Sound? Thedepth-averaged regional model could be used for such a study. However more reliableresults would be obtained from a model that allowed some density stratification, whichwould improve the shelf wave dispersion relation.BibliographyJ.S. Allan, J.A. Barth, and P.A. Newberger, 1990: On intermediate models forbarotropic continental shelf and slope flow fields. Part II: Comparison of numericalmodel solutions in doubly periodic domains. Journal of Physical Oceanography, 20,1044—1076.J.S. Allen, 1980: Models of wind-driven currents on the continental shelf. Ann. Rev.of Fluid Mech., 12, 389—433.A. Arakawa and V.R. Lamb, 1981: A potential enstrophy and energy conservingscheme for the shallow water equations. Monthly Weather Review, 109, 18—36.R. Asselin, 1972: Frequency filter for time integrations. Monthly Weather Review,100, 487—490.P.G. Baines, C. Hubbert, and S. Power, 1991: Fluid transport through Bass Strait.Continental Shelf Research, 11, 269—293.W.H. Bell, 1963: Surface current studies in the Hecate Model. MS Rep. Ser.(Oceanogr. and Limnol.) 159, Fisheries Research Board Canada.W.H. Bell and N. Boston, 1962: The Hecate Model. MS Rep. Ser. (Oceanogr. andLimnol.) 110, Fisheries Research Board Canada.W.H. Bell and N. Boston, 1963: Tidal calibration in the Hecate Model. J. Fish.Res. Board Canada, 20, 1197—1212.J.R. Bennett and A.H. Clites, 1987: Accuracy of trajectory calculation in a finite-difference circulation model. Journal of Computational Physics, 68, 272—282.A.F. Blumberg and G.L Mellor, 1987: A description of a three-dimensional coastalocean circulation model. In Three-Dimensional Coastal Ocean Models, pages 1—16.American Geophysical Union.M.J. Bowman, A.W. Visser, and W.R. Crawford, 1992: The Rose SpitEddy in Dixon Entrance: Evidence for its existence and underlying dynamics.ATMOSPHERE-OCEAN, 30, 70-93.J.R. Buckley and W.P. Budgell, 1988: Meteorologically induced currents in theBeaufort sea. Unpublished manuscript. Copy in lOS library.164Bibliography 165D.C. Chapman, 1985: Numerical treatment of cross-shelf open boundaries in abarotropic coastal ocean model. Journal of Physical Oceanography, 15, 1060—1075.W.R. Crawford and P. Greisman, 1987: Investigation of permanent eddies in DixonEntrance, British Columbia. Continental Shelf Research, 7, 851—870.W.R. Crawford, W.S. Huggett, and M.J. Woodward, 1988: Water transport throughHecate Strait, British Columbia. ATMOSPHERE-OCEAN, 26, 301—320.W.R. Crawford, W.S. Huggett, M.J. Woodward, and P.E. Daniel, 1985: Summercirculation of the waters in Queen Charlotte Sound. ATMOSPHERE-OCEAN, 23,393—413.W.R. Crawford and R.E. Thomson, 1984: Diurnal period continental shelf wavesalong Vancouver Island: A comparison of observations and numerical models. Journal of Physical Oceanography, 14, 1629—1646.W.R. Crawford, A.V. Tyler, and R.E. Thomson, 1990: A possible eddy retentionmechanism for ichthyoplankton in Hecate Strait. Can. J. Fish. Aquat. Sci., 47,1356—1363.P. B. 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Thomson, 1984: Currents along the Pacificcoast of Canada. ATMOSPHERE-OCEAN, 22, 151—172.C.J.R. Garrett and B. Toulany, 1982: Sea level variability due to meteorologicalforcing in the northeast Gulf of St. Lawrence. Journal of Geophysical Research, 87,1968—1978.G.E. Halliwell, Jr. and J.S. Allen, 1984: Large scale sea level response to atmosphericforcing along the west coast of North America, summer 1973. Journal of PhysicalOceanography, 14, 864—886.G.E. Halliwell, Jr. and J.S. Allen, 1987: Wave number-frequency domain propertiesof coastal sea level response to along-shore wind stress along the west coast of NorthAmerica, summer 1980-1984. Journal of Geophysical Research, 92, 11761—11788.G.J. Haltiner and R.T. Williams, 1980: Numerical Prediction and Dynamic Meteorology. Wiley, 477 pp.C.G. Hannah, 1992a: Geostrophic control with wind forcing, Application to BassStrait. Journal of Physical Oceanography, accepted.C.G. Hannah, 1992b: The Hecate Strait Model, Users guide and documentation.Unpublished manuscript. Contact W.R. Crawford at the Institute of Ocean Sciences,Sidney, BC.C.G. Hannah, P.H. LeBlond, W.R. Crawford, and W.P. Budgell, 1991: Winddriven depth-averaged circulation in Queen Charlotte Sound and Hecate Strait.ATMOSPHERE-OCEAN, 29, 712-736.Bibliography 167J.R. Hunter and C.J. Ream, 1991: On obtaining the steady-state solutions of thelinearized three-dimensional hydrodynamic equations. Appl. Math. Modelling, 15,200—208.M. Israeli and S.A. Orszag, 1981: Approximation of radiation boundary conditions.Journal of Computational Physics, 41, 115—135.B.M. Jamart and J. User, 1986: Numerical boundary layers and spurious residualflows. Journal of Geophysical Research, 91(C9), 10621—10631.K.S. Ketchen, 1956: Factors influencing the survival of lemon sole (Parophrysvetulus) in Hecate Strait, British Columbia. Fish. Res. Bd. Canada, 13, 647—694.W.G. Large and S. Pond, 1981: Open ocean momentum flux measurements inmoderate to strong winds. Journal of Physical Oceanography, 11, 324—336.E.A. Martinsen and H. Engedahi, 1987: Implementation and testing of a lateralboundary scheme as an open boundary condition in a barotropic ocean model.Coastal Engineering, 11, 603—627.C.R. Mechoso, A. Kitoh, S. Moorthi, and A. Arakawa, 1987: Numerical simulationsof the atmospheric response to a sea surface temperature anomaly over the equatorialeastern Pacific ocean. Monthly Weather Review, 115, 2936—2956.F. Mesinger and A. Arakawa, 1976: Numerical methods used in atmospheric models.GARP Publication Series No. 17.J.F. Middleton, 1991: Coastal trapped wave scattering into and out of straits andbays. Journal of Physical Oceanography, 21, 681—694.J.F. Middleton and F. Viera, 1991: The forcing of low frequency motion withinBass Strait. Journal of Physical Oceanography, 21, 695—694.T.S. Murty, 1984: Storm Surges: meteorological ocean tides. Can. Bull. Fish. Aquat.Sci. 212. Ottawa, 879 pp.L.A. Mysak, 1980: Recent advances in shelf wave dynamics. Reviews of Geophysicsand Space Physics, 18, 21 1—241.G.D. Newton, 1985: Computer programs for common map projections. Bulletin1642, U.S. Geological Survey, 33 pp.I. Orlanski, 1976: A simple boundary condition for unbounded hyperbolic flows.Journal of Computational Physics, 21, 25 1—269.Bibliography 168J. Pedlosky, 1979: Geophysical Fluid Dynamics. Springer-Verlag, 624 pp.L. J. Pratt, 1991: Geostrophic versus critical control in straits. Journal of PhysicalOceanography, 21, 728—732.R.W. Preisendorfer, 1988: Principal Component Analysis in Meteorology andOceanography. Elsevier, 425 pp.A.J. Robert, 1966: The integration of a low order spectral form of the primitivemeteorological equations. Journal of the Meteorological Society of Japan, Ser. 2, 44,237—245.C.A. Rocha and A.J. Clarke, 1987: Interaction of ocean tides through a narrowsingle strait and multiple narrow straits. Journal of Physical Oceanography, 17,2203—2218.L.P. Roed and C. Cooper, 1987: A study of various open boundary conditionsfor wind-forced flow barotropic numerical ocean models. In J.C.J. Nihoul and B.N.Jamart, editors, Three dimensional models of marine and estuarine dynamics, pages305—335. Elsevier.L.P. Roed and O.M. Smedstad, 1984: Open boundary conditions for forced wavesin a rotating fluid. SIAM J. Sci. Stat. Comput., 5, 414—426.R. Shapiro, 1970: Smoothing, filtering, and boundary effects. Reviews of Geophysicsand Space Physics, 8, 359—387.T.J. Simons, 1980: Circulation models of lakes and inland seas. Can. Bull. Fish.Aquat. Sci. 203. Ottawa.S.D. Smith, 1988: Coefficients for sea surface wind stress, heat flux, and wind profilesas a function of wind speed and temperature. Journal of Geophysical Research, 93,15467—15472.B. Tang, 1990: Fluctuating flow through straits of variable depth. Journal ofPhysical Oceanography, 20, 1077—1086.R.E. Thomson, 1981: The Oceanography of the British Columbia coast. Can. Spec.Pubi. Fish. Aquat. Sci., Bull 56. Ottawa.R.E. Thomson, 1989: The Queen Charlotte Islands physical oceanography. InG.G.E Scudder and N. Cessler, editors, The Outer Shores, pages 27—63.R.E. Thomson and W.J. Emery, 1986: The Haida current. Journal of GeophysicalResearch, 91, 845—861.Bibliography 169R.E. Thomson, P.R. LeBlond, and W.J. Emery, 1990: Analysis of deep-droguedsatellite tracked drifter measurements in the Northeast Pacific. ATMOSPHERE-OCEAN, 28, 409—443.R.E. Thomson and R.E. Wilson, 1987: Coastal countercurrent and mesoscale eddyformation by tidal rectification near an oceanic cape. Journal of Physical Oceanography, 17, 2096—2126.B. Toulany and C.J.R. Garrett, 1984: Geostrophic control of fluctuating barotropicflow through straits. Journal of Physical Oceanography, 14, 649—655.A.V. Tyler and W.R. Crawford, 1991: Modeling of recruitment patterns in Pacificcod (Gadus macrocephalus) in Hecate Strait, British Columbia. Can. J. Fish. Aquat.Sci., 48, 2240—2249.A.V. Tyler and S.J. Westrheim, 1986: Effects of transport, temperature, and stocksize on recruitment of Pacific cod (Gadus macrocephalus). In INPFC Bull. 7, pages175—190.C.J. Walters, C.G. Hannah, and K. Thomson, 1992: A microcomputer program forsimulating effects of physical transport processes on fish larvae. Fisheries Oceanography, 1, 11—19.D-P Wang, 1982: Effects of continental slope on the mean shelf circulation. Journalof Physical Oceanography, 12, 1524—1526.West Coast Offshore Exploration and Environmental Assessment Panel. OffshoreHydrocarbon Exploration: Report and Recommendations of the West Coast OffshoreExploration and Environmental Assessment Panel, April 1986.D.G. Wright, 1987: Comments on Geostrophic control of fluctuating barotropic flowthrough straits. Journal of Physical Oceanography, 17, 2375—2377.T. Yao, H.J. Freeland, and L.A. Mysak, 1984: A comparison of low-frequencycurrent observations off British Columbia with coastal-trapped wave theory. Journalof Physical Oceanography, 14, 22—34.Appendix AList of symbols170Appendix A. List of symbols 171f Coriolis parameterA bottom frictiong acceleration of gravityforcing and dissipationH mean depthh total depth, h = H + ik friction parameterIc unit vector in vertical directionK kinetic energyL length of the strait; length of domainq potential vorticity, (f + )/hr linear correlation coefficientt timehorizontal velocity vector(u,v) components of the horizontal velocity vectorW width of the strait; width of domainx, y horizontal coordinatesz vertical coordinate7 time domain filter parametersthe relative vorticity, V xdeparture from mean sea level, the sea level anomalyfriction coefficientA eigenvalueRayleigh frictionv eddy viscosityp density of sea watera standard deviationr wind stresslatitudegravitational potentialw frequencyrotation rate of the earth, 7.29 x iO sV horizontal gradient operatorLs grid sizeLt time step sizeTable A.12: List of symbols used.Appendix BEmpirical Orthogonal Function AnalysisEmpirical orthogonal function (EOF) analysis is a statistical technique that attemptsto reduce large amounts of data to some understandable numbers and patterns. Thetechnique looks for patterns in the fluctuations of the data — it is analysis of variance.EOF analysis is a statistical technique, therefore interpretation of the patterns in termsof physical processes is a separate issue. Preisendorfer (1988) provides an excellent introduction to this field.Consider a set of M observation stations, which each contribute a time series of lengthN. Assume that the time series are synchronous. For the moment assume that the timeseries are scalars. The data is reduced to a set of M modes, where each mode has 3components; an eigenvalue, an eigenvector, and a time series. The eigenvalue measuresthe amount of the variance of the original data that is contained in the mode and is ameasure of the mode’s importance. The eigenvector contains the spatial character of themode: a spatial pattern which fluctuates in phase. The ith element of the eigenvector isa measure of the contribution of time series i to that mode. The time series of modalamplitude records the time history of the mode.The original time series can be reconstructed from the eigenvectors and the timeseries. If most of the variance is contained in a few dominant modes, then the originaldata is well represented by a few modes. Sometimes the patterns can be shown to havephysical significance and insight is gained.The calculation is very simple. Consider M time series each of length N, organized as172Appendix B. Empirical Orthogonal Function Analysis 173a matrix z’(t, x), ; t = 1,. .. , N; x = 1,.. . , M. The time series are the columns of z’. Thevariable t is an integer label for the elements of the time series. Time itself ranges from 0to (N — 1)tt, where iS.t is the sampling interval. The variable x is an integer label for thetime series. x is used to suggest that the time series are recorded at separate locations— this need not be so. The first step is to compute and remove the mean of each timeseries,z(t, x) = z’(t, x)—(B.50)Nz(x) = N-’ z’(t, x) (B.51)Deterministic trends can also be removed at this point. Linear trends were removedin the analysis reported in this thesis. The scatter matrix (in this case the covariancematrix) is defined asNS(x, x’) = z(t, x)z(t, x’) (B.52)k= 1The spatial patterns are the eigenvectors of the scatter matrix (e(x), i = 1,.. . M). Themodes are ranked by their eigenvalues (A, i = 1,... , M). Each eigenvector has oneelement for each time series and the number of eigenvectors is equal to the number oftime series.The computation of the EOF modes can be done without removing the mean of eachtime series. However, then one is not analysing the variance (or fluctuations) in the data.The time series of the amplitudes of the modes are computed as followsMa(t) = z(t, x)e(x) (B.53)The original data can be reconstructed as follows.Mz(t,x) = a(t)e(x) (B.54)When most of the variance is contained in the first few modes, the sum can be truncatedand the important character of the data is retained.Appendix B. Empirical Orthogonal Function Analysis 174When the eigenvalues are distinct (no two equal) then the eigenvectors are orthonormal and complete. In the case of one or more sets of multiple eigenvalues one canconstruct a complete orthonormal basis set. The existence of a complete, orthogonal setof basis vectors guarantees that the calculation of the amplitudes makes sense and thatthe reconstruction of the data works (Preisendorfer, 1988).For analysis where the time series have different units (e.g. velocity and pressure) itis necessary to remove the scaling problem caused by the choice of units. A standardchoice is to divide each time series by its standard deviation. The scatter matrix is thenformed from the normalized time series. The rest of the analysis proceeds in the samemanner. In this case the eigenvalues and the time series are dimensionless.Dividing the time series by their standard deviation is a useful way to prevent one ortwo energetic time series from dominating the analysis.For two-dimensional vectors the obvious thing is to do the analysis using complexnumbers. Unfortunately, information is lost in the complex formalism (Preisendorfer,1988, p 182-186). In the real-valued case the eigenvectors can be multiplied by ±1. Inthe complex case the eigenvectors can be multiplied by any complex number with length1 (e°). Thus the pattern can not only be reversed, it can be rotated by any angle — theorientation of the pattern in space is arbitrary. We are interested in the relation of thevelocities to the topography, therefore orientation is important.The preferred method is to treat each vector component as a separate time series(Preisendorfer, 1988; Crawford et al., 1990). The analysis is done in the same mannerand then the vectors reconstructed at the end. The eigenvectors are only indeterminantby a factor of ±1. For ease of analysis and interpretation, the two components of thevector occupy adjacent locations in the data array (k vectors generate 2k time series).For interpretation and plotting it is useful for the eigenvectors to have the same unitsas the original time series. To do this the eigenvector is scaled by the square root of theAppendix B. Empirical Orthogonal Function Analysis 175eigenvalue. If the time series have been normalized then the value at each location mustalso be scaled by the standard deviation of the appropriate time series. For scalars thedimensional value of eigenvector i at location x isb(x) =A2ae(x) (B.55)If the time series were normalized by dividing by the standard deviation then o is thestandard deviation of time series x, otherwise o = 1.For the vector data the dimensional velocity (ui, v) of mode i at location x isu(x) = (B.56)v(x) = + 1) (B.57)This vector represents the root mean square velocity fluctuations (associated with modei) about the mean current at location x. A large vector represents large fluctuations (intime) along the line of the vector. A valid pattern (eigenvector) is obtained by reversingall the vectors. This is the meaning of the +1 indeterminancy of the eigenvectors.The dimensionless mode has a useful interpretation. If all the components of aneigenvector are of similar magnitude then all the time series are contributing to the mode.Components much smaller than the others indicate time series that are not contributingto the mode.Time series a(t) can reveal cyclic or episodic behaviour. The time series can be usedto correlate with other time series of interest in a search for a physical interpretation ofthe mode. CTT9O use the close correlation of the time series of the mode 1 winds withthe mode 1 currents to claim that the mode 1 current pattern is wind driven.A dimensional vector can be recovered by scaling each element of the vector by theappropriate standard deviation.Appendix Ca,DEFigure C.74: The end effect of the Butterworth filter used to filter the time series data.The filter was an 8-th order filter with 1/2 power at 40 h. The time step was 15 m. Thesolid line is the filter output and the dashed line is the input.Butterworth Filter: End EffectsFrom Fig. C.74 it is clear that about 5 days must be trimmed off the ends of the recordsto remove the end effect of the filter. The 5 day trimming was evident for both 15 minuteand hourly time series.1 5010050050-100-150outputinputr0 5 10 15I I . • . . I . . • • I . . . • I • . . I I • i •20 25Time (days)30176Appendix DHecate Strait Model: DetailsD.1 Numerical formulationThe scheme of Arakawa and Lamb (1981, hereafter AL81) is based on the idea that potential vorticity should be explicitly conserved in the finite difference form of the shallowwater equations (5.20) and (5.19). AL81 guarantees potential vorticity conservation forthe non-linear shallow water equations (SWE) over arbitrarily steep topography, as wellas conservation of energy, mass and momentum.Many finite difference schemes conserve mass, energy and momentum. Conservationof potential vorticity in the non-linear equations over steep topography is more difficult.Steep topography means large changes in depth over a few grid points, as is found nearCape St. James and Rose Spit.The conservation of energy, mass, momentum, potential vorticity and potential enstrophy hold only in the absence of the corresponding sources and sinks. With 0,(5.20) and (5.19) can be written in vector invariant form as= —qk x — V(K + ) (D.58)(D.59)where = v h is the volume transport, K = 1/2v2 the kinetic energy per unit mass andis the gravitational potential energy. All the symbol definitions are collected togetherin Appendix A.177Appendix D. Hecate Strait Model: Details 178For our purposes = g, where is the sea level anomaly. However in the derivationof AL81 = g(h + h5), where h is the total fluid depth and h3 is the distance from anarbitrary reference surface to the bottom of the ocean. Recall that h(x, y,t) = H(x, y) +y, t) where H is the mean depth. It turns out that H+h3 is a constant and V’ = gV,.The reference level formulation is used to get a consistent form for energy conservation.AL81 provide a spatial discretization for the unforced, inviscid shallow water equations. That is a scheme for differencing the right-hand-sides of (D.58) and (D.59). Thetime stepping method is left to the user. The regional model uses the leap-frog scheme,which is discussed later.D.1.1 Arakawa and Lamb, 1981I present here the details of the Arakawa and Lamb (1981) scheme for the spatial differencing of the inviscid, unforced shallow equations. The wind stress and friction termsare discussed at the end.The spatial differencing is done on the C-grid, a staggered grid, where the variablesu, v, i, and q are defined at different points (Fig. D.75). The properties of this grid andothers are discussed in Mesinger and Arakawa (1976). The C-grid is the standard gridused in numerical models of the continental shelf.Let the grid resolution be d. The grid is defined at the q points (D.75), that is thegrid indices take integer values at the q point (qj,j). Following this scheme the othervariables are uj112,v112,j and h+1j2,i.The continuity equation (D.59), defined at the h points, is written as:+ (V.= 0 (D.60)where the finite difference form of the transport divergence is(V. )i+1/2,j+1/2 = — U,j112 +v1i2,+ —v112,] (D.61)Appendix D. Hecate Strait Model: DetailsFigure D.75: The C-grid.The volume transports, defined at the u and v points, are* rLulUj,,÷l/2 = [1 ji,j+1/2= [hv}1, (D.63)and the corresponding depths are defined in terms of the depths given at the h points:= (h_112,+h+112,)/ (D.64)= (h1112,+h+112,_)/ (D.65)The differencing of the momentum equations (D.58) was done in the most generalform possible by writing all the spatial derivatives in terms of the primitive variables andarbitrary weights. The weights are determined by applying the constraints (conservationof mass, energy, potential vorticity, and potential enstrophy). In the end the weights at a179q v q v qu h u h. .Uj+ 1j+1/2j-1 /2j—1q V q VUqhVqU h uq •v q1+1/2 1+1i—i i-i /2Appendix D. Hecate Strait Model: Details 180given grid point are linear combinations of the potential vorticity at neighbouring points.The scheme is second order accurate in space.The momentum equations are defined at the appropriate velocity points0 = — (D.66)——— Si,j+1,2v÷l,2+ i+1/2,j+/2Uj,j+1/2 —+ d’[(K + )i+1/2,j+1/2 — (K + )i_1/2,j÷1/2J0 = + i++112u11 (D.67)+6i,j+1/2Uj,j/2+—i+l,j_l/2tLi+i,i_i/+i+,jv —i_1/2,jv+ d’[(K + )i+1/2,j+/2 — (K + )i+1/2,j_1/21The momentum weighting coefficients are:i+1/2,j+1/2 = [qi+i,j+i + qi,j+i — qi,j— q+i,1 (D.68)i+1/2,j+1/2 = [—qi+i,j+i + q,ji + qi,j — q÷i,] (D.69)= + qj,j+i + + q÷1,j (D.70)i,j+1/2 = [qi,j+i + + qji,j + 2q,] (D.71)= + qji,j+i + 2q_1,+ (D.72)61,j+1/2 = + 2qj,j + qi,j + 2qj,J (D.73)Notice that/3i,j+1/2 = ‘Yi—1,j+1/2 and Yi,j+1/2 = so that there are oniy 4 sets ofmomentum weighting coefficients to compute.Appendix D. Hecate Strait Model: Details 181The kinetic energy, defined at the h points, is:= + U 11 +v112, +v112,÷) (D.74)and the gravitational potential F isi+1/2,j-f 1/2 = gii-f-/2,j÷i/2 (D.75)The potential vorticity q and relative vorticity are defined as-1qi,j= (D.76)= d [u,._112 — Qti,j+1/2 + vi+l/2,j — (D.77)The depth at the q point, l4, is simply the average of the depths at the neighbouringdepth points1= + h_112,2 + h_112,.. +h112,3_) (D.78)As defined the scheme does not contain forcing or friction. They were implementedas follows. The components of the wind stress vector were placed on the same staggeredgrid as the velocities. Since there was no differencing involved they were simply includedon the right-hand-sides of on the RHS of (D.66) and (D.67). All the friction coefficients(bottom friction, Rayleigh friction and eddy viscosity) were defined at the depth pointsand appropriate averages taken to get the values at the u and v points.Atmospheric pressure gradients were not used in this thesis. Nevertheless the modeldoes contain provision for atmospheric pressure gradients. They are defined at the hpoints and included with .The time stepping was done by the standard leapfrog scheme. This means that all themomentum weighting coefficients, pressure gradients, and wind forcing were evaluatedat the n time level. The friction terms were dealt with implicitly because the leapfrogscheme is unstable with respect to friction terms. This suffices to define the numericalscheme to advance the velocities from the n — 1 time level to the n + 1 time level.Appendix D. Hecate Strait Model: Details 182D.1.2 DiscussionThe AL81 scheme has not been heavily used in either the oceanographic or atmosphericcommunities. It has been used in atmospheric studies of flow over topography (Dempsey,1988; Deardorif, 1984) and a version with fourth-order accuracy has been used in theUCLA general circulation model (Mechoso, 1987). I know of no examples of the use ofthe AL81 scheme in a regional oceanographic model. The most recent application wasas the ground truth in a comparison of approximations to the shallow water equations(Allen et al., 1990).The perceived drawbacks are the complexity of the differencing scheme and the heaviercomputational load. I admit that the scheme looks complicated, however it is relativelyeasy to turn into computer code since all the variables are explicitly specified. The heaviercomputational load may or may not be a factor. The leapfrog version of the model isslower than a standard version of the shallow water equations. However the computercode optimizes very well and respectable results are obtained on the newer workstations.A version written by W.P. Budgell using a semi-implicit scheme for the time steppinginvolves no time penalty at all.The most serious drawback is related to the way AL81 achieves its conservationproperties. As described by Foreman (1987) the scheme has a large footprint. Whencomputing the derivatives at u,112 (D.66), information is drawn from an area 3 gridpoints wide (i index) and 2 grid points high (j index). Consider the sequence: 1) theoriginal depths are averaged to obtain 2) M is used to compute q, 3) q is used tocompute the weighting coefficients, 4) the weighting coefficients are used to computederivatives. The conservation of potential vorticity in the presence of steep topographyis achieved by averaging the depths.Now consider what happens when land is introduced. The presence of an islandAppendix D. Hecate Strait Model: Details 183directly influences the computation up to 3 grid points away. What is q on land? Thiswas dealt with as follows. Define a minimum depth hmin. At a land point q = f/hmin.A value hmin = 5 m was used. Experiments by W.P. Budgell and by myself have shownthat the results are not sensitive to the value of hmim in the range 1 to 20 m. In mostatmospheric problems the topography does not break the surface and the problem doesnot arise.An annoying feature arises when computing diagnostic vorticity balances. Precisebalances of the formaq 1 —+v•Vq=VxG (D.79)can be obtained. However, dividing the left-hand-side into the classical form representedby (5.24) is much less precise. The model thinks of q as a single entity not as separatecomponents.Care is required when linearizing the momentum equations. In the vector invariantform (D.58) the non-linear term (. ViY) contributes in two places: the kinetic energyK and through the relative vorticity which appears in q. This is not a big deal it justrequires some care.D.1.3 Leap-frog SchemeArakawa and Lamb (1981) specify the differencing of the spatial derivatives but not thetime stepping method. The model discussed in this thesis used the leap-frog scheme. Asa simple example consider the equation= F(u,t); u = u(t) (D.80)Think of moving everything except the time derivative to the right hand of (D.58). Usingthe leap-frog scheme (D.80) is differenced as— = 21tF’ (D.81)Appendix D. Hecate Strait Model: Details 184where n is the time index, t = nL\t. F is the known value of F(u, t) at time step n.This provides enough information to advance ‘u from the n — 1 time level to the n + 1time level. The leapfrog scheme is second order accurate in time (Haltiner and Williams,1980).The leap frog scheme has a small problem. The underlying differential equation (5.20)is first order in time. The difference equation is essentially second order in time (it nses3 time levels). This introduces a second solution called the computational (or parasitic)mode. There are many ways to deal with the computational mode. The method chosenhere is the Robert (1966) filter.D.1.4 The Robert FilterThe Robert (1966) filter is a very weak three-point time domain filter. It is applied toall three prognostic variables u, v, at every grid point and every time step. The filterlooks likeu(n) = u(n) + -y[u(n + 1) — 2u(n) + u(n — 1)] (D.82)where n is the time index and the over-bar represents a filtered value. The properties ofthe filter are discussed in the context of the leapfrog scheme by Asselin (1972). Minorcomplications are introduced because the filter is recursive — old filtered values are usedto compute new values. Asselin showed that for small filter parameter,-y < 1/3, theamplitude response of the filter is very close to that of the standard centered filteru(n) = u(n) + -y[u(n + 1) — 2u(n) + u(n — 1)] (D.83)Table D.13 shows the amplification factor for several different frequencies with a filterparameter of -y = 0.01 and a time step of L\t = 10 s. The amplification factor is veryclose to one, and the damping is very small, for time scales longer than a day.Appendix D. Hecate Strait Model: Details 185period T /t/T amplificationper zt per day20 s 0.5 0.995 1.5 x60 s 0.17 0.9993 3.0 x iO1 hr 2.8 x 1O — 1 — 1.6 x i01 day 1.2 x 1O — 1— 3 x 10_67 day 1.6 x iO — 1— 6 x 10_828 days 4.1 x 10—6— 1 — 4 x 10’°Table D.13: Amplitude response of the time domain filter for sinusoidal input. Forperiods of an hour of more the amplification factor is very close to 1 and is written inthe form 1 — , where f is a small number. Lt = 10 s and = 0.01.D.2 Model BathymetryAccurate representation of the bathymetry, or ocean topography, is important becausetopographic steering is an important part of the dynamics. The representation of thebathymetry is limited by the grid resolution. The choice of grid resolution requiresbalancing two competing considerations; the need for accurate bathymetry, and the needto limit the computational load. The computational load is linearly proportional to thenumber of grid points N and the number of time steps. Doubling the resolution increasesthe computational load by 8. There are 4 times as many grid points and the time step ishalved. The bathymetry, grid size, and time step are related through the CFL criteria.How large does the model domain have to be? At the very least the model domainmust extend from northern Vancouver Island to the northern side of Dixon Entrance(Fig. 1.1), and from the mainland coast to an offshore boundary west coast of the QueenCharlottes. The question of the size of the model domain becomes a question of locationof the open boundaries (one offshore boundary and two cross-shore boundaries).My philosophy on open boundaries is simple — move them as far away as possible.The grid has grown with the increase in computer power. My initial design decision wasthat I was willing to spend roughly half the computational effort on the open PacificAppendix D. Hecate Strait Model: Details 186west of the Queen Charlottes. In the end roughly 3/4 of the grid is outside the region ofinterest defined by Cape Scott to Dixon Entrance and the mainland to the west coast ofthe Queen Charlottes.The grid resolution chosen was 5 km. This represents a compromise between 10km (not enough resolution) and 3 km (too much computation). The standard modeldomain is 90 x 199 grid points. The whole domain has been rotated by 30 degrees from anorth-south orientation to conform to the general trend of the coastline (Fig. 5.32). Thisreduces the number of wasted land points in the NE corner and makes the shelf-breakroughly perpendicular to the cross-shelf open boundaries. The latter feature reduces theamount of energy incident on the open boundaries at oblique angles: recall that most ofthe flow is constrained to flow parallel to the isobaths.After the best possible depths (Section D.2.1) were obtained the whole domain was filtered to remove the two-grid-space variations (often called 2Lx noise) from the bathymetry(Shapiro, 1970). The rationale being, if you cannot resolve it, remove it! As mentionedearlier, any non-linear, finite difference model tends to accumulate energy at the smallestresolvable scale 2Lx. The presence of 2Lx noise in the bathymetry enhances this cascadeof energy to small scales. The AL81 scheme restricts the spurious energy cascade butI did not think I was doing the model any favours by including unresolved bathymetricfeatures.I made one exception to this rule. Learmonth Bank which dominates the mouth ofDixon Entrance is a ridge 1 grid cell wide and 4 long and was removed in the filtering.In this case I put my faith in Arakawa and Lamb and the bank was put back in by hand.The effect of Learmonth Bank on the model circulation in Dixon Entrance is discussedin the next chapter. A feature which was not put back was the long narrow seamountjust to the northwest of Cape St. James.Smoothing the bathymetry is still an open question. On one hand the AL81 scheme isAppendix D. Hecate Strait Model: Details 187designed to deal with the problem of unresolved topography and the analysis of Foreman(1987) showed that the scheme actually smooths the topography. On the other hand Ido not think that the model results would benefit from a lot of small scale bathymetricvariations.As a final step, along-shore bathymetric variations were removed within 40 grid pointsof the cross-shore open boundaries to suppress the generation of topographic waves nearthe cross-shore boundaries and accommodate the open boundary conditions.As part of the open boundary condition tests, larger domains are used. The depthsin the extended regions were filled by extending the boundary values of the standarddomain.D.2.1 Creating the BathyrnetryThe sources for the bathymetry data and the procedure used to massage the differentsources onto a common grid is discussed.The sources for the bathymetry were• Canadian coastal waters — Energy, Mines and Resources Canada (EMR) topographic data base (nominally 1 km resolution) and Canadian Hydrographic Servicecharts.• American coastal waters — National Oceanic Atmospheric Administration (NOAA)charts.• Offshore waters — taken from the ETOPO5 data base which was compiled by theNational Geophysical Data Center in Boulder Colorado and is available on thecentral computer at the Institute of Ocean Sciences,. Sidney, BC. The resolution isquite good; the 1/12 degree resolution is roughly 5 km x 10 km at 52° N. However,the depths in the coastal waters are inaccurate.Appendix D. Hecate Strait Model: Details 188• Coastline — taken from the high resolution coastline contained in the World DataBase 2. The data base was compiled by the CIA and is available on the centralcomputer at the Institute of Ocean Sciences, Sidney, BC.The latitude and longitude coordinates were converted to cartesian coordinates usinga Universal Transverse Mercator (UTM) projection with the central meridian at 131°W. The UTM projection has a maximum scale error of 1/2500 within ±3° of the centralmeridian. This covers most of the area of interest. The conversion algorithms were takenfrom Newton (1985).There are three coordinate systems to consider in what follows: 1) latitude, longitude;2) UTM; and 3) model coordinates. Recall that the model domain was rotated by 30°from north.-south.Step 1: Choose a model grid.Choose the grid resolution (5 km) and the model domain size (90x199). Choosethe central location about which to rotate the model domain (52.5° N, 131° W).Choose the model coordinate which corresponds to the central point (96, 56). Thelast was chosen after much experiment.Step 2: Generate a coastline.Map the grid points defined in Step 1 to latitude, longitude coordinates. Thisrequires two coordinate transformations. Plot the high resolution coastline and thegrid points on the same chart. Choose the digitized coast line by hand. Manyof the large fjords and inlets were included even through they do not affect thecirculation.Step 3: Get the EMR and ETOPO5 data onto the same grid.The two digitized bathymetry data bases were mapped onto a 1 km x 1 km UTMAppendix D. Hecate Strait Model: Details 189grid using the nearest neighbour technique. The ETOPO5 values were only usedin the deep water and American open coastal waters. It was not used where therewas valid EMR data.Step 4: Reduce the 1 km data to 5 km.The dense part of the array was reduced to 5 km resolution by averaging over the9 nearest neighbours. Only valid water points were used in the average. Locationswhere the coast-line data conflicted with the depth data were flagged and dealtwith by hand. Some mythical islands in the EMR data were discovered this way.The sparse part of the array was mapped on to the grid as follows: if one of the 4nearest 1 km grid points has a value use it. Otherwise flag it and fill by hand.Step 5: Alaska.The ETOPO5 data is inaccurate in coastal waters and the EMR data goes onlyto the Canada/US border. The Alaskan coastal waters were digitized in a similarmanner to the coastline. The grid points were plotted to chart scale and the NOAAcharts digitized by hand and eye.Step 6: Final check.The grid was plotted to chart scale and the depths compared directly with recentCanadian Hydrographic Service charts. Discrepancies were noted and changes madeto the model bathymetry.D.3 Drifter AlgorithmComputing simulated drifter trajectories on the C-grid presents a problem in addition tothe usual problems of interpolation in space and time. The model’s velocity componentsare not defined at the same points on the grid (Fig. D.75). The velocities must beAppendix D. Hecate Strait Model: Details 190interpolated just to create a valid field. The algorithm used to define the velocity field(Bennett and Clites, 1987) has the following properties:• The velocities are interpolated onto the q points (Fig. D.75).• Only wet points are used in the interpolation.• Bilinear interpolation is used to compute the velocity field between q points.This approach ensures that that drifters do not get trapped in corners or slow downas they approach the coast (caused by using zero velocities for the land points in theinterpolation). The algorithm has an interesting quirk; drifters can travel up to onegrid-space onshore from an apparent land boundary. This is not serious except on thoseoccasions when a drifter crosses a narrow (one grid-space wide) island or peninsula. Thefix consists of checking the location of every drifter at every time step to see if it is onland. Grounded drifters are either moved back into the ocean or removed. An elegantfix does not seem to exist (David L. Schwab, GLERL, pers. comm., 1990).The drifters move with the depth-averaged flow. No windage (impact of the winddirectly on the drifter) has been added because the objective is to study the movementof the water not to track the wind.D.3.1 Trajectory ComputationsDuring the model runs the drifter trajectories were computed using the full Bennettand Clites (1987) algorithm, with the checks to see if drifters had grounded. A schemeusing a fourth-order Runga-Kutta algorithm has also been used, but it used too muchcomputation time.Often it was only in hindsight that proper drifter placement was evident. In order toallow rapid evaluation of drifter trajectories the following procedure was developed. ForAppendix D. Hecate Strait Model: Details 191each experiment the model domain was seeded with a reasonable number of drifters indifferent areas and these drifters were tracked during the model run. During the modelrun, the current data at all grid points was saved at regular intervals (usually once amodel day). After the model run was completed, a twin experiment was conducted withthe new drifter trajectories being computed from the saved current records. If the twintrajectories were very close then additional drifter trajectories were computed using onlythe saved currents. The time integration of the drifter paths was done using a fourthorder Runga-Kutta algorithm.The technique worked best when the winds were from one direction. Daily currentsand even twice-daily currents were found to be insufficient to resolve the changes in thecurrents due to the rapid changes in wind direction seen in the summer simulations.For this thesis the sampled-current-technique was used to find out what the drifterswere doing. The proper simulation was then repeated with drifter locations and starttimes determined from the experiments using the the sampled-current-technique. Thisworked very well. The only drifters shown in this thesis that were computed using thesampled-current technique are the drifters in Fig. 8.71, which illustrate the flow in QueenCharlotte Sound.D.4 Lateral Boundary ConditionsD.4.1 Side Walls and Boundary LayersThe model reported on in this thesis uses free slip boundary conditions at side walls.Figure D.76 shows a land/ocean boundary on the C-grid. Free slip boundaries are anatural on the C-grid. Only the normal velocity and the vorticity (q) appear on theland/sea boundary. A free slip boundary is implemented by setting the normal velocityequal to zero at the boundary and setting the relative vorticity ( equal to zero theAppendix D. Hecate Strait Model: Details 192Figure D.76: Boundaries in the C-grid.boundary. Setting ç = 0 means no shear at the boundary. Neither the tangential velocitynor the sea level need to be specified at the land/ocean boundary.The proper fluid dynamics boundary condition for the interface between a fluid and asolid is no flow through the interface and no slip along the interface. In the Hecate StraitModel the no slip part was ignored. Tbe no-slip boundary condition is most applicablewhen either the eddy viscosity is an important part of the model dynamics or the modelresolves both the spatial scales and the important physics in the boundary layer. Notresolving the natural width of the boundary layer means that the viscous region gets tooV qu h u h u h. . . .Vv q v q V q- -. . I I I- - -hIV q v q V• I • • •h u h u h• I I I Iv q V q V qAppendix D. Hecate Strait Model: Details 193wide. The Hecate Strait Model ignores the coastal boundary layer and uses the free-slipcondition at the side walls.Even no-flow through the boundary can cause problems. In 3-D models the questionarises as to whether the no-flow condition should be imposed at each vertical grid pointor only in the depth-average sense. The two choices give different solutions. Imposingno-flow through the boundary at each grid point creates a strong coastal jet (Jamart andOzer, 1986). Nothing is ever simple.D.4.2 Open Boundary ConditionsAn open boundary is the arbitrary line drawn in the ocean that separates the known (themodel domain) from the unknown (the rest of the ocean). For this thesis I assumed thatthe local winds provide the only important forcing mechanism. External influences suchas tides and shelf waves were ignored. Therefore the only purpose of the open boundaryconditions was to let disturbances out.The use of open boundary conditions in regional models is one of the black arts ofnumerical modelling. In general all open boundary conditions are bad. The true art liesin picking the one that is least bad for the given application.The number of open boundary conditions that have been developed is enormous.Most boundary conditions can be grouped into the following categories:• no flow normal to the open boundary• clamping the sea level• zero gradient condition• sponge layers• radiation conditionsAppendix D. Hecate Strait Model: Details 194• relaxation schemes‘No flow normal to the open boundary’ means what it says. Turn the ocean into alake by putting a wall around the domain, which reflects waves back into the domain.Clamping the sea level means that the sea level fluctuation () is set to zero everywherealong the open boundary: water can flow through, but sea level variations are reflected.The zero gradient condition means that there is no gradient in sea level across theopen boundary. This simple condition is effective in many situations (Roed and Cooper,1987; hereafter RC87).Sponge layers are regions of high energy dissipation that soak up energy propagatingtowards the boundary. They can be used to reduce the reflection from closed or clampedboundaries or combined with radiation conditions (Isreali and Orzag, 1981).Radiation conditions are the most commonly used family of boundary condition inregional modelling. They assume that the dominant process near the open boundary iswave propagation and attempt to make the boundary transparent to waves propagatingout of the domain. The success of the radiation condition at the boundary depends onthe ability to accurately compute the dispersion relation of the outgoing waves. Thisis easily done in one dimension with systems that only have one family of waves (onedispersion relation). The presence of waves with different dispersion relations createsproblems, as do waves hitting the boundary at oblique incidence.In relaxation schemes the solution in the interior of the domain is forced to a specified(exterior) solution at the open boundary in a relaxation zone. If the exterior solution is‘no motion’ then the relaxation zone becomes a sponge layer. The trick with relaxationschemes is choosing the exterior solution (Martinsen and Engedahl 1987).Hybrid conditions can be formed by combinations of the above.Intercomparisons of open boundary conditions in depth-averaged models were carriedAppendix D. Hecate Strait Model: Details 195out by RC87 and by Chapman (1985). The comparisons were done with an ideal shelf,a clamped offshore boundary and the OBC to be tested used on the cross-shore openboundaries. RC87 concluded that radiation conditions can work well when the forcingnear the boundary is weak, and sponges can work well when the forcing near the boundaryis weak or variable. Both types have difficulties when the forcing near the boundary isstrong and persistent.Sponge layers do not work well with strong forcing near the boundary because thesponge inhibits mass flow through the boundary. If the wind forcing near the boundaryis strong or persistent then the low mass flux corrupts the solution in the interior.The problem with the radiation condition is more subtle. In general radiation conditions are created by extracting free wave solutions from the equations of motion andassuming that all disturbances satisfy this condition. Most radiation conditions are basedon a variant of the one dimensional wave equationq + cb = 0where c is the wave speed. This can work quite well. Strong forcing near the openboundary creates problems when the forced solution is incompatible with the radiationcondition. Small scale waves are generated which can contaminate the solution in theinterior (Enquist and Madja, 1977, see Remark #2). Satisfying the compatibility condition requires that the wind forcing be tapered to zero near the boundary which cancause a new set of difficulties. Strong topographic variations near the open boundarycan cause similar problems.Roed and Smedstad (1984) attempted to get around the compatibility problem byseparating the solution near the boundary into a forced solution and a free wave solutionand dealing with them separately. The results have been mixed (RC87).The flow relaxation scheme proposed by Martinsen and Engedahl (ME87) approachesAppendix D. Hecate Strait Model: Details 196the problem from a different perspective. They assume that the dominant process nearthe open boundary is wind forcing. Thus the important quantities are the along-shore andcross-shore mass fluxes. They chose the exterior solution to be the model equations withthe along-shore gradients removed, there are no abrupt changes in mass fluxes near theopen boundary and waves propagating in the along-shore direction are damped. In theabsence of wind the boundary condition reduces to a sponge layer; the exterior solutionbecomes a no-motion solution (u = 0, v = 0, i = 0). As one might expect, because theEkman flux is handled correctly this open boundary condition is most successful whenthere is strong wind forcing. This method is used here (Chapter 5).D.4.3 Cross-shelf BoundarySeveral radiation conditions including Orlanski (1976) and Roed and Smedstad (1984)were tested on the cross-shelf boundary. The results were consistently plagued withaccumulation of energy trapped along the northern boundary (downstream with respectto propagation of Kelvin waves). The results became unstable with time-varying winds.The flow relaxation scheme of Martinsen and Engedahi (ME87) has had very fewproblems. Tests and anecdotes are reported in Appendix F.D.4.4 Off-shore BoundaryThe offshore boundary condition chosen was a clamped sea level condition. The reasonsare discussed in Chapter 5. Because of concerns about reflection off this boundary asponge layer was created along the off-shore boundary. The sponge layer uses higheddy viscosity to dampen the short wavelengths and depth-independent linear frictionto dampen the long wavelengths. The sponge region was 30 grid poillts wide and theviscosity and friction were ramped up with a parabolic profile. Tests showed no differenceAppendix D. Hecate Strait Model: Details 197in the solutions with and without the sponge layer. The sponge was not used in any ofthe experiments reported in this thesis.Appendix ETests in Rectangular DomainsThis appendix reports on tests of the cross-shore open boundary conditions in simpletests cases. Results of the tests of the open boundary conditions in the full Hecate StraitModel are reported in Appendix F.The cross-shore open boundary condition used was the flow relaxation scheme of Martinsen and Engedahi (1987). This scheme was discussed in Chapter 5 and Appendix D.The offshore boundary condition was the clamped sea level condition: = 0 along theoff-shore boundary. The sponge layer was not used.The test reported here are tests in an idealized domain: a rectangular box. The firsttwo tests are done with a flat-bottomed box and replicate the tests of Roed and Cooper(1987), hereafter RC87. The final set of tests were done using the same box but with asloping shelf and other parameters characteristic of the regional model. The sloping shelfprovides at test of the model and the open boundary conditions in an environment similarto the Hecate Strait Model. The test case is used to illustrate frictional adjustment on asloping shelf and to test the model since the frictional adjustment solution is known.E.1 Flat BottomThe model domain was a section of coastline in the semi-infinite ocean x < 0 (Fig E.77).For this appendix the x-axis, the index i, and velocity u denote cross-shelf quantities,while the y-axis, the index j, and velocity v denote along-shore quantities. The modelparameters for the fiat-bottom tests are list in Table E.14198Appendix E. Tests in Rectangular Domains 199clampedboundaryyxFigure E.77: Model domain for the open boundary condition tests. The flow relaxationzones are marked by cross hatching.j=71j=61j=36j=1 1= 1—iLwLsfkHdomain lengthdomain widthgrid sizeCoriolis parameterfriction parameterRayleigh frictiontime incrementequilibrium depthrelaxation zone width1000 km500 km20km1.2 x iO s2.4 x iO s0300 S50 m200 km10 grid cellsTable E.14: Parameters for open boundary condition tests with a flat bottomed oceanAppendix E. Tests in Rectangular Domains 200E.1.1 Uniform Along-Shore WindThe first test uses a uniform along-shore wind to test the open boundary conditions withpersistent forcing at the boundaries. The wind stress is a positive along-shore wind stresswhich decays off-shore.= 0= Toexp(ax)where the maximum wind stress r0 = 0.1 Pa, the decay scale a1 = 200 km, and x <0 inside the model domain. The wind stress at the off-shore boundary is 8% of themaximum. The fluid starts at rest and the wind starts impulsively at t = 0.The linearized equations are used to correspond with Roed and Cooper (1987), hereafter RC87. Interestingly there is no complete solution to this problem (see RC87 fordetails). At the coast, the along-shore velocityv(0,y,t) = v(1 — e_kt/H)and the steady state velocityv(x,y) = vewhere v = ro/(pHk) = 4.06 cm/s and e-folding time scale (k/H)’ = 5.8 hr. TheRayleigh friction was set to zero to correspond with RC87. In Chapter 4 the along-shorevelocity was denoted u, here it is v to correspond with the choice of co-ordinate axes.The model handles this case very well. The time series at sites A, B, C are identicaland correspond to the analytical solution (Fig. E.78). Figure E.79a,b show the steadystate sea levels and currents over the whole domain. There are no surprises; the sea levelcontours and the velocity vectors are parallel to the coast.Appendix E. Tests in Rectangular Domains 201U,EC.)>sC.)00)>a)0Cl)C0Time (Hours)Figure E.78: Uniform along-shore wind experiment. Time series of along-shore velocityat the coast (site B). The time series at sites A,C are identical.I I I I543210y = ml*(1 - exp(m0*m2))Value Errorml 4.0580291679 0.000610722m2 0.16806247188 0.000195251Chisq 0.0026643024356 NAR 0.99996282945 NAI I 10 20 40 60 80 100Appendix E. Tests in Rectangular Domains 202Figure E.79: Uniform along-shore wind experiment. The steady state a) sea level contours(cm), and b) velocity vectors.I ISCALE:4 CM/SC) C)cDcD>-E-7-ICxw*00P1???It? ttt ft I Itft tiff?! ittI tIlt ft ft ttlilt? ttttttitt tt tttit? ff1 It,tttt tttttttuft tt ttt1111ff it tttttt tiff tti tttII,, ft itt tttiti ft ft 1 Ittt,i ft it? tttttttft fit tttIt It? tftt tttt tiff!! tt’ttttift it I ttttttttt tilt ttittiff ill ttt,tttftt tilttIlt? ft ft tttttttff it ttttII?? ft fit ItCCD0(0CCCCJPC”J(a) sea level contours (b) velocity fieldAppendix E. Tests in Rectangular Domains 203E.1.2 Bell-Shaped WindA wind stress with a bell-shaped or Gaussian along-shore profile was used to test theopen boundary conditions (OBC) with strong forcing in the interior and weak forcing atthe boundaries.The wind is turned on at t = 0 and off at t = 48 hr.I roexp(—a2y)exp(ax) fort 48 hrTy =0 fort>48hrwhere the maximum wind stress r0 = 0.1 Pa, the decay scale a = 200 km, and TZ = 0.The wind stress at the cross-shore boundaries is 0.2% of the maximum. Figure E.80shows the sea levels in the test domain after 40 hours. These are in good agreement withRC87.Lacking an analytic solution for this case, I followed RC87 and added 4000 km to eachend of the domain. This guarantees that waves reflecting ofF the end of the domain cannot interfere with the solution before the end of integration (96 hours). This is referred toas the extended domain. Figure E.81 shows the time series of the along-shore velocities.The results are in excellent agreement with the extended domain case (and RC87). SiteB exhibits the wind forced solution and the spin-down is a mirror image of the spin-up.At the downstream site one can see the Kelvin waves generated by the wind’s start-upand shut-down pass by at roughly 9 and 57 hours (Fig. E.81, site C). There is a smalldiscrepancy in the velocities at site C after hour 10 and this is seen in the excess mass(Fig. E.82). The excess mass is the average value of the sea level i over the domain.E.2 Sloping ShelfThis test case uses the same rectangular domain as before, however the cross-shelf topography is taken from the northern end of the regional model domain (Fig. E.83). TheAppendix E. Tests in Rectangular DomainsPCJ>-0C’\J(-)ccCxw*204Figure E.80: Bell-shaped wind experiment. Sea level after 48 hours. The contour intervalis 0.5 cm.N(i—.0////.(‘Li)//-IAppendix E. Tests in Rectangular DomainsC’,E0>‘C)00>ci)0COa)t02052.10-0.0 20 40 60 80 100Time (hours)Figure E.81: Bell-shaped wind experiment. Time series of along-shore velocity at sitesA, B, C. Data from the test domain is plotted as solid lines and data from the extendeddomain is plotted as dotted lines.Appendix E. Tests in Rectangular Domains 2060.30. 2•0.10U)U)EU)U)a)-0.1-0.2-0.Figure E.82: Bell-shaped wind experiment. Time series of excess mass. Data from thetest domain is plotted as solid lines and data from the extended domain is plotted asdotted lines.Time (hours)Appendix E. Tests in Rectangular Domains 207topography does not change in the along-shelf direction. The non-linear equations areused and the other physical parameters (including grid size and friction) are taken fromthe Hecate Strait Model (Table E.15). Once again the x-axis, the index i, and velocity u denote cross-shelf quantities, while the y-axis, the index j, and velocity v denotealong-shore quantities (Fig E.77).L domain length 250 kmW domain width 100 kmLS.s grid size 5 kmf Coriolis parameter 1.1 x iO sk friction parameter 0.5 x iO’ si Rayleigh friction variableLi t time increment 12 srelaxation zone width 50 km10 grid cellsTable E.15: Parameters for open boundary condition tests with a sloping shelf050010001 5002000250026Figure E.83: Cross-section of the sloping shelf. Topography taken from regional model.E-cci,01 6 11 16 21cross-shelf location-I02C)>sC)0a)>a)0C,)C)0Appendix E. Tests in Rectangular Domains 208201510500 200Figure E.84: Sloping shelf experiment. Time series of along-shore velocity at threedifferent depths. The depth h = 77 m corresponds to location (i,j) = (25,36). Thedepths h = 155 m and h = 1786 m correspond to (i,j) = (20,36) and (10,36) respectively.50 100 150Time (hours)(b) velocity fieldFigure E.85: Sloping shelf experiment, a) sea level field (cm), and b) velocity field, after6.5 days.Appendix E. Tests in Rectangular DomainsIIIi I it?209SCRLE:25 CM/S0 0C)c(0ci)bcN -flitliltlii ItillI •I I IIt Itii,,tiltliltII ItI IttliltI it?I it?liltItitIt I ItIlttillliltIi??tiltIt Ititt,lit Ililttilttill.1 I IItIIIiItliiiIII?tiltflitliiitiltit itIt ittilttiltIt??tillitt ITilttilttiltIt itIt it‘litit itIll?tilllilttIltitt?lilttIltIt itIII ItilltilttiltIll titt?itt?itt?tit?it??it??tillitt?itt?itt?i ititit utitt?it I itI fiti ititii’?itittt t ittiltitt?itt?it? ititt ittittttiltitt itt ititit tttitt ttt itititt ititt it(0GD 0 D‘IC\JIi 0 000xw*(a) sea level contoursAppendix E. Tests in Rectangular Domains 210The wind field is a uniform along-shore wind. In this case there is no offshore decay,the imposed wind stress is the same everywhere. The fluid starts from a state of rest.The along-shore velocity time series at three different depths are shown in Fig. E.84.The time series show the increase in frictional adjustment time as the water gets deeper.After 196 hours of integration the two deeper locations have not reached the steady state(discussed further in the next section). The time series were taken from the followingthree locations: 1) at the coast, (i,j) = (25,36) and h = 77 m; 2) 5 grid points off-shore,(i,j) = (20,36) and h = 155 m; 3)15 grid points offshore, (i,j) = (10,36) and h = 1786 m.The sea level and velocity fields are shown in Fig E.85. The jet structure is simplydue the long adjustment times in the deep water. There are no surprises and no obviouskinks in the fields. Figure E.86 is a plot of along-shore velocity as function of along-shorelocation at three distances off-shore. There is a very small decrease in the along-shorevelocity in the down-wind direction.E.3 Frictional Adjustment on a Sloping ShelfThe sloping shelf test case provides an opportunity to test the model, particularly our understanding of the friction terms. The spin-up time series in Fig. E.84 look like frictionaladjustment. This can be made more rigorous by comparing the steady-state velocitiesand e-folding times with the theoretical values.A series of experiments were conducted for several different values of the Rayleighfriction coefficient i. The along-shore velocities were monitored at the same three locations as before (depths of 71 m, 155 m, and 1786 m) and the best fit curves of theformv(t) = v(1 — e_t/t0)were computed. The values of v and to = are compared with the theoretical valuesAppendix E. Tests in Rectangular Domains 211— I I I I I I —20-- h=77mEh=155m>s15--0>G)0 --U)05- -h = 1786 m0- I I I I I I -1 11 21 31 41 51 61 71Along-shore location (grid cells)Figure E.86: Sloping shelf experiment. Along-shore velocity as a function of along-shoreposition. The velocity field was sampled at 3 distances off-shore, corresponding to depthsof 77 m, 155 m, 1786 m. The model was forced with a uniform along-shore wind r = 0.1Pa and the Rayleigh friction i = 0. The velocity field was sampled after 6.5 days ofintegration The flow relaxation zones occupy along-shore locations 1 to 11 and 61 to 71.There are no kinks in the velocity field but there is a slight downward slope in the in thedown-wind direction (from left to right).Appendix E. Tests in Rectangular Domains 212______theory model resultp (s’) h (m) v (cm s1) t (hr) v (cm s1) t (hr)0.0 71 19.5 39.4 19.5 39.5155 19.5 86.1 19.5 86.91786 19.5 922— 1000.3.0• 10_8 71 19.4 39.3 19.4 39.4155 19.3 85.3 19.4 86.21786 17.6 896— 9093.0• iO 71 18.7 37.8 18.7 37.8155 17.8 78.8 17.8 79.51786 9.4 479 — 4833.0• 10_6 71 13.7 27.7 13.7 27.6155 10.1 44.6 10.1 45.01786 1.7 84.7— 85.5Table E.16: Frictional adjustment. Comparison of theoretical frictional adjustment timesand the steady state velocity with the model results. The model results were fit tov = v,(1 — 6_tIto). Fits were computed using time series of length 167 hours. Windstress r = 0.1 Pa. Density p = 1025 kg m3.in Table E.16. The theoretical values were computed fromTV00 —ph..\For h = 1786 m, the time scale t0 was so long that the steady state velocity and theadjustment time-scale could not be estimated at the same time. For this depth steadystate velocity v00 was set to the theoretical value and only t0 was estimated.The good agreement between theory and the numerical experiments indicates thatthere are no hidden dissipation mechanisms in the model. The results also suggest thatfrictional adjustment provides a good model of the spin-up of the water in the open oceanportion of the Hecate Strait Model domain.Appendix E. Tests in Rectangular Domains 213E.4 StoriesThe reason for the going overboard here is because of my experience with open boundaryconditions. Things always go wrong and one or two figures do not tell the whole story.My first experience with this was when I was testing the Orlanski radiation condition.In a simple uniform wind test case I found that the sea levels looked fine but the velocityfield reversed in the last few grid points near the outer boundary. After talking to manypeople working in the field I found that this is a common problem. When testing openboundary conditions, nobody ever publishes the velocity field.My second war story relates to the flow relaxation boundary condition. After coding and initial testing it seemed to be working correctly. Eighteen months later, whiletracking down a small problem near the northern open boundary, I found a bug. Theexterior solution at the current time step was being relaxed with the interior solution atthe previous time step. The solutions are only 10 s apart so this should not matter verymuch, right? Wrong!. The time step is chosen so that the fastest wave almost crossesa grid cell in one time step (almost but not quite). Combining different time levels inthe relaxation scheme is a big problem. The problem was masked in the RC87 test casesbecause of the large friction, but it showed up in the real topography case as a large kinkin the along-shore velocity in the relaxation zone.Appendix FHecate Strait Model: Tests and ExperimentsF.1 Open Boundary Condition TestsIn any limited-area or regional model, there is concern about the effect that the limiteddomain size has on the solution. In the Hecate Strait Model it is a fact of life that theinterior knows that the boundaries are there. For the standard domain (Fig. 6.35) thetravel time of shallow water waves from one end to the other and back is 3.5 h in the deepwater. The external Rossby radius in 2000 m of water is 1300 km, and the distance fromCape St. James to the off-shore open boundary is 200 km (40 grid points). Therefore, aKelvin wave travelling along the shelf break has a significant amplitude at the off-shoreopen boundary. The boundaries can not be moved far enough away.Experiments were conducted for a range of domain sizes within computational reach.The larger domains were obtained by extending the depths at the boundaries. The resultsshow that the flow relaxation boundary conditions (FRS) at the cross-shore boundarieswork well (Table F.17). The steady-state transport changed slightly with the increasein domain size. There were significant changes when the boundaries were closed. Thedifference between the solution using the FRS and using closed boundaries manifesteditself in the long time scale part of the spin-up, the behaviour after the first day ortwo (Figure F.87). The initial spin-up was identical. The time series for the standarddomain (E310) lies on top of that for the long domain (E321) for the FRS open boundarycondition. The standard parameters were used (Table 6.6).214Appendix F. Hecate Strait Model: Tests and Experiments 215expt domain name size OBC Transport (Sv)E302b small 90 x 160 FRS 0.26 (.259)E310 standard 90 x 199 FRS 0.26 (.262)E320 long 90 x 499 closed 0.21 (0.209)E321 long 90 x 499 FRS 0.26 (0.263)E300b big 199 x 317 closed 0.20 (0.196)E300c big 199 x 317 FRS 0.27 (0.265)Table F.17: Steady state transport in domains of different size with an along-shore wind(r = 0.1 Pa) and the standard friction parameters. Domain sizes are given in grid points.The open boundary conditions (OBC) are the flow relaxation scheme (FRS) and closedboundaries. The results for standard, big, and long were the same for FRS zone widthsof 20 and 40 grid cells.-; 0.300.00 I I I I I I0.0 5.0 10.0Time (days)Figure F.87: Comparison of the transport through W-line for two different open boundaryconditions: (a) flow relaxation method (E321,solid); and (b) closed (E320, dashed). Thedomain is the long domain.Appendix F. Hecate Strait Model: Tests and Experiments 216,‘F‘Fp-‘‘p• pt.’‘ _%‘__ %%, ,I —— —_‘,.._.._., ?t::::: ::zz:4/I‘.V•%_1%’. ‘I‘-1/1s•••”_—IJ . -—a—— — .— I —Figure F.88: The velocity field after 6 days near the northern cross-shore boundary. Thetop of the figure is the open boundary. All of the vectors are shown.SCRLE:25 CM/SInterior edge of theflow relaxation zoneAppendix F. Hecate Strait Model: Tests and Experiments 217Two flow relaxation zone widths were tried: 20 and 40 grid cells. For the standardand larger domains there were no differences in the steady state results. For the standarddomain, the extra FRS zone width was acquired by letting the FRS move further into themodel domain: the overall domain size remained the same (90x199). The only significanteffect of the larger relaxation zone was a reduction in the amplitude of the high frequencyoscillation. This indicated that the open boundary condition was affecting the solutionin Hecate Strait. Unless otherwise stated the flow relaxation zone width used in thisthesis was 20 grid cells.The impact of the open boundary conditions is illustrated in Fig. F.88. In the relaxation zone, the jet structure off the west coast of the Queen Charlotte Islands is mergedwith the wind forced solution at the open boundary in dramatic fashion. The velocitypattern in the mouth of Dixon Entrance and around Forrester Island did not changewhen the open boundary was moved 750 km downstream (E321). However the jog in thevelocity field did move downstream.I was concerned about reflections from the clamped off-shore boundary. To reducethe reflections a 30-grid-point-wide sponge layer was created along the off-shore boundary. This was done by increasing the Rayleigh friction and the eddy viscosity near theboundary. To date I have not been able to detect any difference in the solutions on theshelf with and without the sponge layer. The sponge layer was not used in this thesis.For now accept that open boundary conditions are not having an enormous impacton the shelf circulation.F.2 Adjustment Time-Scales in Hecate StraitAs discussed in Chapter 6, a plausible explanation for the two adjustment time-scales inHecate Strait is that the fast time-scale is the response to the local wind forcing and theAppendix F. Hecate Strait Model: Tests and Experiments 218slow time-scale is the response to an along-shore pressure gradient set up by processesoutside of Hecate Strait. The purpose of this section is to take a closer look at this idea.The steady-state velocity field for a wind localized over Hecate Strait is shown inFig. F.89. The wind forcing was an along-shore wind that tapered to zero at the edges ofthe strait. The iso-lines of the wind stress are shown. The velocity pattern, after 8 days,looks like the pattern from the uniform wind experiment at Day 1 (Fig. 8.58a; note thevelocity scale change). There are large currents in northern and western Hecate Straitand not much anywhere else. A tiny southwesterly flow along the north side of MoresbyTrough is visible.The picture changes dramatically when the model is forced with a wind localizedover central Queen Charlotte Sound (Fig. F.90). In southern Hecate Strait the patternis similar to the uniform wind pattern, and not at all like the velocity pattern with thewind localized over Hecate Strait. A very similar pattern is obtained when the model isforced by an along-shore wind localized near the southern open boundary (not shown).The combination of Fig. F.89 and Fig. F.90 contains all of the important elementsof the velocity pattern with a uniform wind (Fig. 8.59). On this basis, I proposed thatthe flow pattern is Hecate Strait is due to both local wind forcing and remote pressureforcing. The pattern due to local wind forcing is Fig. F.89, which also corresponds toDay 1 in Fig. 8.58. The pattern due to pressure forcing is Fig. F.90.The transport time series for the uniform wind and the two localized winds are shownin Fig. F.91. The initial response from the wind localized over Hecate Strait has thesame response time as the uniform uniform wind. A curve of the formQ(t) = Qo(1 — et)was fit to the transport time series for the wind localized over Hecate Strait (Fig. F.92).From the legend in the figure, the relaxation constant is m2 = 6.06 d’. An e-foldingAppendix F. Hecate Strait Model: Tests and Experiments 219Figure F.89: The steady state velocity field forced by an along-shore wind localized overHecate Strait. The maximum wind stress is 0.1 Pa.Appendix F. Hecate Strait Model: Tests and ExperimentsI...220Figure F.90: The steady state velocity field forced by an along-shore wind localized overcentral Queen Charlotte Sound. The maximum wind stress is 0.1 Pa.Appendix F. Hecate Strait Model: Tests and Experiments 221-; 0.300.15bC 000 I I I I I I0.0 5.0 10.0Time (days)Figure F.91: Comparison of the transport spin-up for three along-shore wind scenarios: (a) spatially uniform, (b) localized over Hecate Strait and (c) localized over QueenCharlotte Sound. The maximum wind stress is 0.1 Pa.Appendix F. Hecate Strait Model: Tests and Experiments 2220.30.250.20.150.10.0500 2 4 6 8Time (days)Figure F.92: Curve fit to the transport time series for a wind localized over Hecate Strait.Time (days)Figure F.93: Curve fit to the time series of sea level at Cape Scott minus sea level atCape Muzon. The wind forcing was the standard spatially uniform wind.y = m1*(1exp(m0*m2))>Cl)00U)I—Value Errorml 0.11478710546 0.000404143m2 6.0592005401 0.312192Chisq 0.0017662664074 NAR 0.9604257843 NAI I [______________1032.50a, 201.5a)10.5a,U)-0.5dmlm2ChisqRy = m1*(lexp(.(m00.5)*m2))...Value Error2.5402688387 0.02556810.51876412227 0.01802031.6049806395 NA0.98222752743 NA0 2 4 6 8 10Appendix F. Hecate Strait Model: Tests and Experiments 223Figure F.94: The sea level field for a spatially uniform along-shore wind (r = 0.1 Pa).The contours are labelled in cm.Appendix F. Hecate Strait Model: Tests and Experiments 224time of )j’ = (m2) = 4 h was used in Chapter 6.For the pressure forcing, the e-folding time was computed from the time series ofthe sea level difference between Cape Scott and Cape Muzon (from the uniform windexperiment reported in Chapter 6). A curve of the formb(t) = bo(1 —was fit to the time series. The first 0.5 days were trimmed from the record beforeanalysis: thus t0 = 0.5 d. The fit is shown in Fig. F.93. From the legend the relaxationtime constant is m2 = 0.52 d’. An e-folding time of ‘ = (m2)’ = 48 h was used inChapter 6.The reason for trimming the sea level difference time series is evident from the plot ofthe complete time series in Fig. 6.40. For the first half day the high frequency oscillationdominates the record. The exponential relaxation does not appear until Day 1 or so.Curve fits to the complete record yielded very poor results.The sea level field for an along-shore spatially uniform wind is shown in Fig. F.94.Notice that the on-shore Ekman transport has raised the sea level everywhere. The sealevel difference between Cape Scott and Cape Muzon is roughly 2 cm. The pressure, or sealevel forcing, in Hecate Strait might be as much as 3 or 4 cm. Recall from Chapter 4 thatthe pressure forcing term—775 roughly corresponds to the sea level difference betweenBeauchemin and Rose Spit. Rose Spit is the northeast corner of the Queen CharlotteIslands and Beauchemin is located in the southeast end of Hecate Strait (under the 0 inthe 11.0).Appendix F. Hecate Strait Model: Tests and Experiments 225F.3 Steady State Velocity FieldsF.3.1 Bottom FrictionThis section takes a second look the steady-state velocity field. Of interest is whetherthe patterns seen in Fig. 8.59 are robust to changes in the bottom friction formulationand the model topography.The steady state velocity field resulting from quadratic bottom friction (5.33) is shownin Fig. F.95 for the case Cd = 2.5 x i0 and u0 = 0.0. The rms tidal velocity field usedto the compute the spatially varying linear friction coefficient is shown in Fig. F.96. Ateach point, the linear friction coefficient was computed fromk*(x,y)= CdVrmswhere Vrms is the value of the rms velocity field at location (x,y) and Cd is the dragcoefficient. The resulting steady state velocity field (Day 8) is shown for the case Cd =2.5 x iO. The choice of the drag coefficients was discussed in Chapter 6.In both cases the patterns are not significantly different from the linear friction pattern(Fig. 8.59). This indicates that bottom friction does not play an important role indetermining the flow patterns.F.3.2 BathymetryThe generation of the coastline and bathyrnetry required many subjective decisions withrespect to narrow features. To assess the impact of these decisions, the following changeswere made to the bathymetry• All the fjords and inlets were removed and the narrow passages closed.• Rose Spit and Langara Island were cut back.Appendix F. Hecate Strait Model: Tests and Experiments 226Figure F.95: Quadratic friction experiment. The steady state velocity field for a spatiallyuniform along-shore wind. The drag coefficient Cd = 2.5 x iO and the backgroundvelocity UO = 0/0. The wind stress r = 0.1 Pa.Appendix F. Hecate Strait Model: Tests and Experiments 227Figure F.96: Spatially varying linear friction experiment. The rms tidal velocity field(cm/s) and the resulting steady state velocity field from the Hecate Strait Model. Thewind stress was spatially uniform with r = 0.1 Pa.Appendix F. Hecate Strait Model: Tests and Experiments 228Figure F.97: Velocity field with altered bathymetry. The wind stress was spatially uniform with r = 0.1 Pa. Standard linear friction was used.Appendix F. Hecate Strait Model: Tests and Experiments 229• Learmonth Bank was removed.The steady-state circulation pattern is shown in Fig. F.97. In Queen Charlotte Soundand Hecate Strait the changes are confined to within a few grid points of the bathymetrychanges. Closing Brown Passage resulted in larger velocities along the west side of DundasIsland and a 10% reduction in the steady state transport through Hecate Strait.In Dixon Entrance there are structural changes. Cutting back Langara Island andletting water flow around the corner has changed the currents along the southern sideof Dixon Entrance. The constraints of the depth-averaged dynamics makes the flow inDixon Entrance sensitive to the depths near Langara Island. One pattern is obtained ifthe water can follow the isobaths around the corner, and another if it cannot.With the exception of Dixon Entrance, the overall circulation pattern is not overlysensitive to minor changes in the bathymetry. The fjords and inlets are aestheticallypleasing but not crucial to the model circulation.V.3.3 Spatially Varying WindAs reported in this thesis, the simulations conducted with the Hecate Strait Model usedspatially uniform winds. As was discussed in Chapter 2, the spatial variability of thewinds in the Queen Charlotte Islands region is not well defined. The discussion here islimited to results obtained with the Hecate Strait Model.I conducted experiments with steady along-shore winds that were uniform in thealong-shore direction and varied smoothly in the cross-strait direction. When comparedto the uniform wind case, the velocity at any point varied qualitatively in the same senseas the change in the local wind stress. The transport through Hecate Strait was mostsensitive to the value of the wind stress on the eastern side of the strait (where the volumetransport is).Appendix F. Hecate Strait Model: Tests and Experiments 230The storm systems that affect the Queen Charlotte Islands regions are generally largescale systems with a length scale of the order of 1000 km; this can be seen by watching theatmospheric pressure charts in the local newspapers. When a moving circular storm witha radius of 1000 km was used in the model, the results in Hecate Strait were very similarto the results obtained by assuming a spatially uniform wind with a value based on theproper value in the middle of Hecate Strait. The real storm systems are not circular,but the large scale means that in general the winds in Hecate Strait are a reasonableindicator of the winds in Queen Charlotte Sound and Dixon Entrance.This does not mean that the spatial variability of the winds is not important. Aswas shown in Section F.2, a wind localized over Queen Charlotte Sound can force waterthrough Hecate Strait. The experiments reported there are examples of extreme spatialvariability in the wind stress field. The velocity at any point depends on the local pressuregradient as well as the wind stress. As was seen in Section F.2 and Chapter 6, the alongshore pressure gradient depends on the wind stress in Queen Charlotte Sound and DixonEntrance.Based on these results, I believe that the character of the results presented in this thesis does not depend on the assumption of spatially uniform winds. A more sophisticatedmodel would address the issue of the spatial variability of the wind stress field.Appendix GFriction, Coriolis Parameter and TransportG.1 Steady State TransportThe results of numerical experiments examining the relationship between the bottomfriction, the Coriolis parameter and the steady state transport through Hecate Strait arelisted in Table G.18. The data is plotted in Fig. 7.50 and Fig. 7.51. The localized windis an along-shore wind localized over Hecate Strait, with a maximum wind stress of 0.1Pa (see Appendix F). The uniform wind has a constant wind stress of 0.1 Pa over thewhole model domain (except near the off-shore boundary— Chapter 6).f/fo localized wind uniform windk = 0.5 x i0 s k = 2.0 x iO s k = 0.5 x 1O s0 0.55 0.141 0.680.1 0.46 0.139 620.5 0.20 0.11 0.380.75 0.15——1.0 0.115 0.08 0.261.5 0.08——2.0 0.06 0.51 0.196.0 0.02 0.02 0.11For the localized wind experiments, the parameter W/L was estimated as follows. Byanalogy with rotation-limited-flux, a curve of the form= 1 + c,r,ij0 (G.84)Table 0.18: Steady-state transport (Sv) through Hecate Strait as a function of Coriolisparameter. The wind stress r = 0.1 Pa, fo = 1.1 x 10” s’, and 1u = 3 x iO s1.B231Appendix G. Friction, Coriolis Parameter and Transport 232k B C h*A W/Ll0 s Sv rn 10_6 s CA/f00.5 0.55 + 0.02 3.4 ± 0.4 60 8.6 0.27 ± 0.0370 7.4 0.23 ± 0.0280 6.5 0.20 ± 0.022.0 0.147 + 0.004 0.81 + 0.1 60 30. 0.24 + 0.02170 29 0.21 ± 0.0280 25 0.19 + 0.02Table G.19: Localized wind experiment. Estimating the parameter W/L from thesteady-state transport. The wind stress r = 0.1 Pa, fo = 1.1 x 10” s1, and i = x iOs—i.was fit to the data defining the steady-state transport as a function of Coriolis parameter.The functional form is discussed in Chapter 7. This was done for both values of k. Theeffective relaxation constant was computed from A = k/h + t, where k and are knownand the effective depth was estimated from the charts and the steady-state transportvector field (Fig. 6.44): h* = 70 m was deemed reasonable. The results are listed inTable G.19.k B C A W/LiO s 10 s Sv 10_6 s CA/f0use data at f = 00.5 0.3 0.63 + 0.01 2.0 + 0.1 7.0 0.13do not use data_at_f = 00.5 0.3 0.69 + 0.01 2.5 + 0.1 7.0 0.160.55 2.7 0.53 + 0.01 2.3 + 0.1 10.0 0.211.1 2.7 0.29 + 0.01 1.2 + 0.1 18.0 0.172.2 2.7 0.15 ± 0.01 0.55 + 0.1 34.0 0.16Table G.20: Uniform wind experiment. Estimating the parameter W/L from thesteady-state transport. The wind stress r = 0.1 Pa, fo = 1.1 x iO s’, and h* = 70 m.A slightly modified version of rotation-limited-flux was used for fitting to the uniformwind data. The form (G.84) was replaced withB B= 1+ Cf/f0 + (G.85)Appendix G. Friction, Coriolis Parameter and Transport 2330.5k=0.55E-3I--.-- k=1.1E-3. 0.4 k=2.2E-3aU) 0.3•—“ \a,0.2 -0>s00.1Co0 I I I I0 1 2 3 4 5 6 7f/f 0Figure G.98: Uniform wind experiment. Steady-state transport as a function of Coriolisparameter for three values of k and = 2.7 x 1O_6 s’. The wind stress r = 0.1 Pa.Appendix G. Friction, Coriolis Parameter and Transport 234The modification arose because it was evident that (G.84) did not provide a good fit:the steady-state transports did not go to zero fast enough. To account for this a fit ofthe formQ 1+Cf/f (G.86)was tried where B, C, and D are all fit parameters. In experiments with several differentvalues of the wind stress, I noticed that D scaled with the wind stress and that D = B/10was a good approximation.The estimates of W/L in Table G.20 were computed using using h* = 70 m. Thethree sets of experiments with t = 2.7 x 10_6 s (10 times the standard value) werecomputed using an older version of the model and the steady-state transport at f = 0was never computed. For comparison, the values of B, C, and W/L were computed fromthe standard experiment without using the data point at f = 0. The results are verysimilar. Notice that the background component B/10 represents 1/4 to 1/3 of the totalsteady state transport when f/fo = 1.The data from the standard version is listed in Table G.18 and plotted in Fig. 7.51.The three curves from the experiments with = 2.7 x 10_6 s’ are plotted in Fig. 0.98.G.2 Adjustment TimeThe parameters for the straight line fits through the adjustment time data shown Fig 7.53are listed in Table 0.21. The value of 14/IL is 2 to 3 times the value computed from thesteady state transport. The value of the intercept changed much more than the slope,when the fit was recomputed without using the data at f = 0,Appendix G. Friction, Coriolis Parameter and Transport 235k intercept slope W/Li0 s 10 s d’ d 1 slope/f0use data at f = 00.5 0.3 0.42 5.7 0.602.0 0.3 2.3 4.4 0 46do not use data at_f = 00.5 0.3 0.25 5.9 0.622.0 0.3 2.0 4.5 0 46Table 0.21: Localized wind experiment. Estimating the parameter W/L from the spin-uptime constant data. The wind stress r = 0.1 Pa, fo = 1.1 x iO s1, and t = 3 x iOs—i.G.3 DiscussionHecate Strait does not have a flat bottom and this raises concerns about the validity ofthe rotation-limited-flux model. In Chapter 8 I showed that topographic steering causesthe flow in Hecate Strait to follow the local depth contours. In northern Hecate Strait,the velocity pattern is established during the first 12 hours. After that the flow is generally parallel to the local depth contours and only the magnitude of the velocity vectorschanges. Once the pattern is established, the depth variations are not an impediment tothe flow.Tang (1990) showed that the presence of along-strait depth gradients required theintroduction of the concept of the effective length of the strait. In general the effectivelength is less than the physical length.I have tried to avoid the issue of the correct value for the length of the strait. I usedthe Hecate Strait Model to compute a reasonable value for the important parameterW/L.In this thesis, I make the following claims:• Rotation-limited-flux provides a useful qualitative description of the observed spatial pattern of subsurface pressure fluctuations in Hecate Strait.Appendix 0. Friction, Coriolis Parameter and Transport 236• In the Hecate Strait Model, rotation-limited-flux provides a useful guide to understanding the dependence of the steady-state transport on the Coriolis parameterand the bottom friction.• The Hecate Strait Model results are consistent with the ratio W/L 0.2. Theanalysis of the steady-state results in this appendix gives values in the range 0.1 <W/L 0.3.When the model is pushed beyond these areas, it starts to break down. In particular,rotation-limited-flux does not provide a useful tool for analyzing the adjustment timedata in the spin-up experiments conducted with the Hecate Strait Model.I believe that the limitations of rotation-limited-flux arise from the fact that HecateStrait does not have a flat bottom and that the vorticity dynamics play an importantrole during the first 12 hours, when the flow pattern in northern Hecate Strait is beingestablished.

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