simulation of barotropic and baroclinic tides off …...tanc 5 , (2) 12 n222v where f is the local...

762 VOLUME 27J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

q 1997 American Meteorological Society

Simulation of Barotropic and Baroclinic Tides off Northern British Columbia

PATRICK F. CUMMINS

Institute of Ocean Sciences, Sidney, British Columbia, Canada

LIE-YAUW OEY

Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey

(Manuscript received 1 May 1996, in final form 2 October 1996)

ABSTRACT

The tidal response of northern British Columbia coastal waters is studied through simulations with a three-dimensional, prognostic, primitive equation model. The model is forced at the boundaries with the leadingsemidiurnal and diurnal constituents and experiments with stratified and homogeneous fluid are compared. Thebarotropic response shows good agreement with previously published studies of tides in the region. A comparisonwith tide gauge measurements indicates that average relative rms differences between observations and the modelsurface elevation field are less than 5% for the largest constituents.

An internal tide is generated in cases where the model is initialized with a vertical stratification. Diagnosticcalculations of the baroclinic energy flux are used to identify regions of generation and propagation of internaltidal energy. With a representative summer stratification, the integrated offshore flux is about 0.5 gigawatts,higher than previously estimated from theoretical models. Comparisons between observed and modeled M2

current ellipses are discussed for several moorings and demonstrate the significant influence of the internal tide.

1. Introduction

Motions at tidal frequency in the coastal ocean areoften composed of an astronomically driven surface(barotropic) tide and an internal (baroclinic) tide, arisingfrom interaction of the barotropically forced, stratifiedfluid with bathymetry. In contrast to the barotropic tide,the baroclinic component contributes relatively little tooscillations of the sea surface; its signature appears pri-marily in tidal currents and internal isopycnal motions.Numerical models have been applied in several in-stances to study the response of a coastal region to spec-ified tidal forcing. To date, however, models of opencoastlines have been concerned almost exclusively withthe barotropic tide. (See Foreman et al. 1995 for a re-view.) While baroclinic models in three dimensionshave been applied to embayments and channels of lim-ited extent (Matsuyama 1985; Wang 1989), to ourknowledge, baroclinic tidal studies of open coastlineshave been limited to applications of two-dimensional,cross-shelf models (e.g., Craig 1988; Sherwin and Tay-lor 1990). Recently, Holloway (1996) applied an essen-tially two-dimensional version of the Princeton OceanModel to the Australian North West Shelf. Results, par-

Corresponding author address: Dr. Patrick F. Cummins, Instituteof Ocean Sciences, P.O. Box 6000, 9860 West Saanich Rd. Sidney,BC V8L 4B2, Canada.E-mail: [email protected]

ticularly M2 baroclinic energy fluxes, were highly sen-sitive to specification of the cross-shelf topographic sec-tion.

Tides along the northern coast of British Columbiahave been the subject of a number of previous modelingstudies. Flather (1987, hereinafter F87) forced a finitedifference barotropic model of the northeast Pacific withthe M2 and K1 tidal constituents. Foreman et al. (1993,hereinafter FHWB93) presented results from a baro-tropic finite element model with eight constituents fora somewhat smaller region. Extensive comparisons withsea level data in these two papers showed generallyexcellent agreement between observations and the mod-el surface elevation field. This work was extended re-cently by Ballantyne et al. (1996, hereinafter BFCJ96)who employ a three-dimensional diagnostic finite ele-ment model. The performance of the three-dimensionalmodel with respect to sea level is comparable to thebarotropic models. In addition, detailed comparisons oftidal current ellipses with observations through the wa-ter column were presented for various locations. For theM2 constituent, significant discrepancies were found thatmay be attributable to the presence of baroclinic tides.In each of the aforementioned studies, the nature of themodel precluded the possibility of an internal tidal re-sponse.

In the present paper, tidal simulations of the northerncoastal waters of British Columbia with a fully prog-nostic three-dimensional model are discussed. The nu-

MAY 1997 763C U M M I N S A N D O E Y

merical formulation is based on the version of thePrinceton Ocean Model (G. Mellor 1993, personal com-munication) used by Oey and Chen (1992) to simulateflows over the northeast Atlantic shelves. The model isdriven at the boundaries with the principal diurnal andsemidiurnal constituents. Numerical experiments withhomogeneous and with vertically stratified fluid arecompared. With homogeneous fluid, the solution is sim-ilar to that of BFCJ96. The barotropic tide is well rep-resented, but current ellipses in many locations bearlittle resemblance to observations. Inclusion of a verticalstratification allows an internal tide to be generated,significantly affecting current ellipses for the semidi-urnal constituents. This leads to an improved represen-tation of M2 currents in several regions over the shelfand slope. Calculations of the baroclinic energy flux areused to identify regions of internal tide generation andpropagation.

In the next section, a brief review of the regionaloceanography and geography is provided. This is fol-lowed by a section containing a description of the modelconfiguration and outlining the numerical experiments.Section 4 discusses the barotropic tidal response of themodel, while the baroclinic response is described insection 5. A summary and suggestions for future workare given in the concluding section.

2. Regional oceanography

A thorough review of the physical oceanography ofnorthern coast of British Columbia is available in Thom-son (1989). Our purpose here is to draw attention to thefeatures of the region that are pertinent to discussion ofthe numerical simulations. The coastal geography andbathymetry of the region are illustrated in Figs. 1a and1b. The bathymetry is characterized by a broad, O(100km wide) shelf, which is partially sheltered from theopen ocean by the presence of Moresby and GrahamIslands, known collectively as the Queen Charlotte Is-lands. The shelf is narrow on the west side of theseislands with depths dropping to 2500 m over a distanceno greater than 30 km from the coast. Three connectedseas separate the Queen Charlottes from the coastalmainland. Queen Charlotte Sound is open to the Pacificon its western side and marked by three relatively shal-low banks. Proceeding from south to north, these areCook Bank, Goose Island Bank and Middle Bank. Tothe north, Hecate Strait is a broad, shallow passage withan elongated submarine valley on its eastern side. Owingto the presence of Dogfish Banks, Hecate Strait is veryshallow (less than 30-m depth) on its western side. Thethird significant sea is Dixon Entrance, a passage of O(60 km) width connecting northern Hecate Strait to theopen Pacific. Learmonth Bank is a shallow barrier foundnear the opening of Dixon Entrance.

Barotropic tides of the semidiurnal and diurnal fre-quency bands propagate approximately as large-scaleKelvin waves in a cyclonic sense around amphidromes

in the North Pacific. Amplitudes of both bands are sig-nificant in the northeast Pacific, and hence tides alongthe coast of British Columbia are classified as mixed,predominantly semidiurnal. Mean tidal ranges are about2 m in deep water, increasing to about 3.5 m near theentrances of Queen Charlotte Sound and Dixon Entranceand to about 5 m near Prince Rupert. The tides aresubject to a spring–neap cycle, which arises through aninterference of M2 and S2, the dominant semidiurnalconstituents, and also K1 and O1, the largest diurnalconstituents.

In the presence of a sloping bottom, barotropic tidescan force vertical oscillations of a stratifed fluid andthus provide a source of baroclinic energy. In two di-mensions (x, z), the dynamics can be expressed by anequation governing the streamfunction (Huthnance1989) c,

Q 12c 2 tan cc 5 z , (1)xx zz 2 2 1 2(1 2 v /N ) H

xx

where Q is the barotropic cross-shelf volume flux, Hthe bottom depth, and v the tidal frequency; N 5 (2grz/r0)1/2 is the buoyancy frequency with r the density, r0

a reference density, and g the gravitational constant.Provided that (v2 . f2), (1) is hyperbolic with propa-gation along rays whose slope is given by

1/22 2v 2 ftanc 5 , (2)

2 21 2N 2 v

where f is the local Coriolis parameter. The bottom slopea is supercritical if a/tanc . 1, and propagation towarddeep water is anticipated. Conversely, a subcritical slopeis associated with onshore propagation.

Using typical temperature and salinity data from thecontinental slope region to determine N (section 3, Fig.3b), the ratio a/tanc is found to easily exceed unity overthe slope. Stratification in the upper 100–200 m of thewater column undergoes a seasonal modulation, asso-ciated with coastal runoff and surface warming duringsummer months and vertical mixing during winter. Be-low 200 m such variations are weak and internal tidegeneration with offshore propagation thus appears pos-sible throughout the year over the slope. Although thebarotropic forcing is constant, the internal tide may nev-ertheless be intermittent as physical processes affectingthe stratification (e.g., mixing or wind-induced down-welling) can modulate generation or propagation. Drak-opoulos and Marsden (1993) examined evidence for in-ternal tides off the west coast of Vancouver Island, tothe southeast of our study area. They observed offshorepropagation and a seasonal modulation, with maximumintensity during summer months.

3. Numerical model

The numerical model used in this study is based onthe Princton Ocean Model (G. Mellor 1993, personal


FIG. 1a. Map of the northern British Columbia and southern Alaska coasts showing place names, the distribution oftide gauges sites (triangles), and locations of current meter moorings (circles).

communication), and a detailed description is given inOey and Chen (1992). Briefly, the model solves finitedifference analogs of the primitive equations, with anonlinear equation of state relating density to the tem-perature and salinity. The level-2.5 turbulent kinetic en-ergy closure scheme of Mellor and Yamada (1982) isused to determine vertical mixing coefficients for mo-mentum and tracer (temperature and salinity) fields.Horizontal mixing coefficients are modeled with theSmagorinsky formulation for diffusion. Bottom fric-tional stresses are modeled by a quadratic drag law witha drag coefficient formulated to yield a logarithmic bot-tom boundary layer. Buoyancy and momentum fluxesthrough the surface have been set to zero.

The numerical algorithm is described by Mellor inthe user’s guide (1993) for the Princeton Ocean Model.The model employs a C grid in the horizontal and aterrain following sigma coordinate system in the ver-tical. A mode-splitting technique is applied for integra-tion of external and internal modes with different timesteps. The integration proceeds according to an explicitleapfrog time stepping scheme, except for terms in-volving vertical diffusion of momentum and that aretreated implicitly. A weak time filter is included to pre-vent development of a computational mode with theleapfrog scheme.

Through a forced gravity wave radiation condition(see Oey and Chen 1992; F87), the model is driven at


FIG. 1b. Bathymetry of the region with depth contours given in meters.

open boundaries with the four leading tidal constituents.Boundary elevation data for these constituents are ob-tained from the global tidal model of Schwiderski (1979,1981a–c). Depth-averaged tidal velocities required atopen boundaries for the radiation condition are obtainedfrom an iterative procedure similar to one outlined inF87. Boundary conditions differ from those of Oey andChen (1992) in certain respects. An Orlanski radiationcondition has been applied to the internal mode veloc-ities, temperature, and salinity at the open boundaries.

Barotropic tidal motions are driven entirely by os-cillations at open boundaries; the earth tide, load tide,and tide-generating potential have been neglected. Witha coastal model of similar extent, FHWB93 concludedthat these forcings produce only slight modifications tosemidiurnal constituents.

a. Model grid

The governing equations of the model are solved overa Cartesian grid with a uniform resolution of 5 km in

the horizontal (x, y) coordinates. The universal trans-verse mercator map projection, used by Hannah et al.(1991), relates the (x, y) coordinate of each grid pointto latitude–longitude coordinates. A rotation of 308 isapplied to align the y coordinate of the model approx-imately with the shelf break, thus reducing the numberof land points. The Coriolis parameter is set to a constantappropriate to 538N. In most cases, there are 21 sigmalevels in the vertical with the following distribution: sk

5 (0, 20.003, 20.006, 20.015, 20.03, 20.05, 20.07,20.09, 20.11, 20.13, 20.165, 20.2, 20.3, 20.4,20.5, 20.6, 20.7, 20.8, 20.9, 20.95, 21), where sk

5 0 at the surface and sk 5 21 at the bottom. Velocityand tracer fields are defined at midpoint between theselevels.

The model topography was constructed by averagingdepth data within a 5 km radius at each grid point. Thedepth data for most of the domain were obtained froma high resolution (1 km) topographic database availablefor the British Columbia coast. It was necessary to sup-plement this with ETOPO5 topography for the northern


FIG. 2. Model domain with contours of bottom topography. Thehorizontal arrow indicates the position of the cross-shelf sectionsshown in Figs. 10 and 11.

TABLE 1. Numerical experiments mentioned in this study.

ExperimentStratification

profile Comments

14v14u14s14pRT1RT2RT3

NoneSummerWinterSummerSummerSummerSummer

Homogeneous fluid case

Diagnostic caseM2 onlyM2 only, 41 levelsM2 only, 2.5-km grid

reaches of Dixon Entrance and for regions located welloffshore of the continental slope. No smoothing wasdone to the resulting topographic field except for anarea adjacent to the offshore boundary (0 , X , 150km), where a smoother was applied, eliminating thepresence of Bowie and Union seamounts from the mod-el. The resulting model domain is shown in Fig. 2 withcontours of bottom topography. The domain is open onthe two cross-shelf boundaries and on the offshoreboundary. Two narrow straits, Johnstone Strait to thesouth and Clarence Strait to the north (Fig. 1) have beenclosed off.

b. Experiments

The numerical experiments of this study are listed inTable 1. Most of the discussion is focused on a com-parison of the first two cases, but mention is also madeof some additional runs. The first case, denoted 14v, isa control run in which a homogeneous unstratified fluidis specified. The baroclinic response of the model wasexamined with a second experiment, denoted 14u. Herethe temperature and salinity are initialized as horizon-

tally homogeneous vertically stratified fields represen-tative of summer stratification. Figure 3a illustrates thevertical variation of temperature and salinity used forexperiment 14u. The vertical profile is specified withan analytic form composed of a linear region near thesurface and an exponential form below. This profile,although idealized, provides a reasonable approximationto the density stratification typically observed over thecontinental slope and outer shelf during summer con-ditions (Fig. 3b). Additional experiments include a casewith a ‘‘winter’’ stratification and a stratified diagnosticcase. The initial stratification for the winter case, 14s,differs from the profile of Fig. 3a only in the upper 125m where, to allow for a winter mixed layer, the fluid ishomogeneous.

Motivation for the diagnostic (fixed temperature andsalinity) experiment, 14p, is to examine sensitivity tostratification in the absence of an internal tide. It isknown that properties of tidal current ellipses can besensitive to the value of the vertical viscosity (Prandle1982). In the present case, this parameter is determinedas part of the numerical solution through the turbulenceclosure submodel. Its value will depend on the severalfactors, in particular, the vertical stratification. In ex-periment 14p the fluid was stratified as in 14u, but aninternal tidal response was precluded by relaxing thetemperature and salinity fields to their initial values witha very short time constant. Except in the bottom bound-ary layer where relatively minor differences were ob-tained, results from this diagnostic case, in particulartidal ellipse parameters, were nearly indistinguishablefrom those of 14v. The results of experiment 14p dem-onstrate that the significant differences between 14u and14v discussed below are not merely a result of strati-fication-dependent viscosity, but rather the presence ofinternal tides in 14u.

For experiments 14v and 14u, the model was inte-grated for 34 days of simulated time. Hourly fields fromthe last 29 days were subject to an harmonic analysisto separate the response at the four constituent fre-quencies. This analysis period is sufficient for the har-monic analysis to separate the response within the di-urnal and semidiurnal frequency bands, but also shortenough that the mean stratification does not change ap-preciably. The initial spinup time is sufficient to estab-lish an equilibrium for the tide.


FIG. 3. (a) Vertical profiles of temperature and salinity representing summer stratification used to initialize the model. (b) Comparison ofthe buoyancy frequency derived from the model profile shown in (a) with observations from two representative CTD stations at differentlocations over the continental slope. The observations were made on 13 and 14 July 1983.

c. Resolution tests

Simulations of the baroclinic tide require resolutionof the scales of the topography and the wavelength as-sociated with the internal tide. It is also desirable tominimize pressure gradient error and other numericaltruncation errors associated with sigma levels (Melloret al. 1994). A length scale, H/tanc, based on (1) (Huth-nance 1989) yields a value of 25 km over the shelf,assuming H 5 200 m and N 5 1 3 1022 rad s21. In2000 m of water with a typical deep value for N (3 31023 rad s21), the length scale is 75 km. The 5-km gridresolution appears sufficient for deep regions, but maybe marginal over the shallower regions of the shelf.

A limited convergence study was done to examinesensitivity of the baroclinic response to variations inhorizontal and vertical resolution. These cases are thelast three entries in Table 1 and include only forcingwith the M2 constituent. The runs were restricted to aduration of 5 or 6 days with harmonic analysis appliedto the last 3 or 4 days. In experiment RT1 the basic5-km grid with 21 vertical levels is maintained. Thevertical resolution is refined in experiment RT2 to 41levels, while in RT3 the horizontal resolution is doubledto 2.5 km. The additional vertical levels in RT2 wereplaced at midpoint between those of the 21-level dis-tribution. The topography in RT3 was obtained fromlinear interpolation of the 5-km grid and does not in-troduce finer topographic scales.

An intercomparison of results from these three casesshowed only minor quantitative differences in the baro-

clinic currents of the model to variations in the hori-zontal or vertical resolution. The radiation pattern of theinternal tide was essentially unchanged in these casesand resembled the M2 response of the stratified case withfour constituents (14u). Thus, the model does not appearto be very sensitive to changes in resolution. It is likelythat a model that resolved finer topographic scaleswould introduce new finestructure to the internal tidalfield. However, as shown below, the large-scale structureof the internal tide is determined by the larger scales ofthe topography and the barotropic tide, which are bothwell resolved.

4. Barotropic response

The results in this section are drawn mainly fromexperiment 14v with homogeneous fluid. Stratificationhas little effect on the barotropic response of the model.There are, however, subtle differences between 14v andthe stratified runs, the most notable of which is an ad-ditional loss of barotropic tidal energy over the shelfwith stratified fluid.

a. Surface elevation

The amplitude and phase of the surface elevation fieldat the M2 and K1 frequencies are presented in Figs. 4aand 4b. The M2 phase progresses steadily northwardalong the outer coast. As the tide propagates into HecateStrait from the south, amplitudes increase due to shoal-


FIG. 4. Model co-amplitudes and co-phases (degrees UT) for the (a) M2 and (b) K1 constituents from experiment 14v. Contour intervalsfor M2 (K1) are 10 (2) cm for amplitude and 58 (28) for phase.

ing effects on the shelf and reach almost 2 m on theeast side of Graham Island. The shoaling also inducesan east–west phase difference of about 108 across theStrait. The M2 amplitude and phase of Fig. 4a agreeclosely with comparable fields presented in earlier stud-ies (F87, FHWB93, BFCJ96). The most notable differ-ence with these last two studies is a small discrepancyin the phase over the northern part of the domain. InFig. 4a, the 2708 phase contour abuts on the outer coastof Alaska, whereas it passes through northern DixonEntrance in the finite element models. One factor thatmay account for this difference is that FHWB93 andBFCJ96 tuned their boundary data (also taken from theSchwiderski atlases) to obtain an improved agreementwith sea level data over their domain.

Figure 4b shows that the sea level amplitude field forK1 is about 40% of M2 in the deep ocean with relativelylittle amplification in shallow regions. The amplitudeand phase distribution for K1 also agrees with earlierstudies, even with respect to small-scale maxima andminima over the shelf. Some discrepancies with respectto the distribution of the phase are evident over the deepocean regions where our results are in closer agreement

with those of F87 than either FHWB93 or BFCJ96. Onceagain, a likely reason for this is the tuning of boundarydata in these studies. (F87 made no such adjustmentsto his boundary forcing.)

Distributions for the S2 and O1 constituents (notshown) are similar in pattern to M2 and K1, respectively,and generally agree with unpublished results, providedby M. Foreman, from the finite element models. HereS2 amplitudes are about 30% of M2, while O1 amplitudesare about 60% of K1.

To provide a quantitative assessment of the elevationfield, averages of the absolute rms error, L21 SL D, andthe relative rms error, L21 SL D/Ao, were computed foreach constituent. The averaging is based on observationsfrom 50 (5L) tide gauge sites, whose distribution isshown on Fig. 1a; D is the rms difference over a tidalcycle between model and observations, given by

1/2

1 2 2D 5 (A 1 A ) 2 A A cos(f 2 f ) , (3)o m o m o m2[ ]where A and f are amplitudes and phases for a givenconstituent, and subscripts m and o refer to model and


TABLE 2. Average absolute and relative rms differences betweenobserved and modeled sea surface elevation from experiment 14v forfour tidal constituents. Corresponding values for experiment 14u aregiven in parentheses.

ConstituentAbsolute error

(cm)Relative error

(%)

M2

K1

S2

O1

4.1 (4.1)2.3 (1.8)1.7 (1.8)1.7 (1.5)

3.1 (3.1)4.9 (3.9)4.3 (4.7)5.9 (5.0)

FIG. 5. M2 barotropic current ellipses from experiment 14v at everythird grid point. Thin (thick) line ellipses indicate clockwise (coun-terclockwise) rotation and the line from the center gives the directionof the current vector at the time of maximum tidal potential.

observations, respectively. The 50 gauge sites are a sub-set of the 63 listed in Table 2 of FHWB93. As they aremainly distributed around the shelf region, where thegreatest variations occur, these sites provide adequatecoverage of elevation field. For a given tide gauge, theclosest corresponding model grid point was chosen forcomparison. Of the 13 gauges omitted from consider-ation, 12 were situated at locations that were unresolvedby the 5-km grid, usually because they were at the headof narrow inlets. One site (Alert Bay) was omitted be-cause it was close to the closed-off entrance to JohnstoneStrait (Fig. 1a), where realistic results are neither ex-pected nor obtained.

The average rms elevation errors from the comparisonare given in Table 2 for the two basic cases, 14v and14u. The average relative error is less than 5% for eachconstituent except O1, which is slightly higher. As ex-pected, the two experiments have similar differenceswith observations. The largest absolute errors (e.g., Dø 24 cm for M2) occur near the artificially closed en-trance to Johnson Strait. However, the region of largeerror is locally confined and errors for the nearest avail-able stations around Queen Charlotte Strait are generallysmaller than the mean values of Table 2. Based on theseresults, we conclude that the barotropic tide is reason-ably well represented in the model.

b. Tidal current ellipses

Tidal current ellipses for the depth-averaged velocityare presented in Fig. 5 for the M2 constituent. In deepregions, tidal currents are small, O(2 cm s21), and it isonly over the shelf that the ellipses assume a significantmagnitude. Large clockwise-rotating ellipses with rel-atively large eccentricity (ratio of semiminor to semi-major axis) are found over the three shallow banks inQueen Charlotte Sound. Particularly large ellipses alsooccur in the vicinity of Cape St. James. In Hecate Straitand especially in Dixon Entrance, the channel bound-aries constrain the motions to be nearly rectilinear,aligned along the axes of these channels. Near RoseSpit, the transition point between the two channels, thereis a rapid rotation in the orientation of the ellipses.FHWB93 show M2 tidal ellipses for northern HecateStrait, which are in good agreement with Fig. 5. Incomparison to M2, ellipses for K1 (not shown) are no-

tably weaker in amplitude, especially in Dixon Entrance.The largest K1 currents are found off Cape St. James,over the shallow banks in Queen Charlotte Sound andon the Alaskan shelf.

With homogeneous water, the tidal ellipses show littledepth dependence above the bottom Ekman layer. Figure6 compares the vertical variation of M2 and K1 tidalellipses from experiment 14v with observations at sta-tion W02 located in central Hecate Strait (see Fig. 1a).This comparison, and others from Hecate Strait, showthat a homogeneous motel captures the essential aspectsof M2 and K1 tidal currents in this region. Except in thebottom boundary layer, tidal ellipses at W02 in the strat-


FIG. 6. Comparison of model M2 and K1 current ellipses from ex-periment 14v with observations at the W02 site in central HecateStrait. To place model results and observations in a common coor-dinate system, a rotation of 308 has been applied, aligning the (x, y)components of model currents in the (north–south, east–west) direc-tions. The arrow head on each ellipse indicates the sense of rotationof the current vector.

ified experiments (not shown) are virtually identical tothose of Fig. 6, suggesting that internal tidal motionsare absent at this site. BFCJ96 compared model andobserved ellipses for four sites, including W02. At eachof these sites, their results are in close agreement withtidal ellipses computed from our experiment with ho-mogeneous fluid. However, it is only at the W02 sitethat good agreement with observations is obtained. Atthe other sites, located in Queen Charlotte Sound, DixonEntrance, and on the west side of the Queen CharlotteIslands, observations show significant vertical structurethat the model of BFCJ96 and our homogeneous fluidmodel cannot capture.

c. Energy fluxes

Propagation of the barotropic tide may be examinedfrom consideration of depth-integrated energy fluxes as-sociated with each tidal constituent. These fluxes aredetermined from the harmonic analysis of the barotropiccurrents and the surface elevation field according to amethod outlined in appendix A. The resulting energyflux vectors are shown in Fig. 7 for the principal semi-diurnal and diurnal constituents. In both cases, the fieldis dominated in the deep water by the northward prop-agation of energy associated with large-scale gyres oftidal energy in the Pacific. Relatively little K1 energyenters Hecate Strait or Dixon Entrance. Near the shelfbreak, the K1 constituent has circular gyres of energy

flux located south of Cape St. James and along the Alas-kan shelf. F87 suggested that similar features appearingat these locations in his solution were associated witha diurnal shelf wave component.

For the M2 constituent, barotropic energy leaks ontothe shelf in a region of O (40 km) width south of CapeSt. James. A fraction of this energy continues northwardthrough Hecate Strait and Dixon Entrance, eventuallyrejoining the deep water flux. A second branch recir-culates southward over the shelf, crossing over GooseIsland Bank and Cook Bank, where it then rejoins thedeep water flux. There are significant M2 energy fluxesacross the shelf break at three distinct locations: alongthe northern shelf region off Dixon Entrance, south ofCape St. James, and farther south off Cook Bank. Agap, where relatively little energy flux crosses the shelfbreak, exists between the latter two locations.

The M2 energy flux was integrated across varioustransects to infer the rate of energy loss in the barotropictide over the shelf. The mean integrated flux of baro-tropic energy into an enclosed region gives the net rateof energy loss to all dissipation mechanisms, of whichbottom friction is the most significant. In stratified ex-periments, conversion to baroclinic energy representsan additional potential sink for barotropic energy. Strat-ification effects may also modify bottom currents andhence the dissipation rate associated with bottom fric-tion.

Figure 8 illustrates three subregions of the model andgives the net fluxes across the bounding transects forexperiments 14v and 14u. Barotropic losses within asubregion are the residual obtained from summing flux-es across the bounding transects. These values are givenin Table 3 for the two experiments. In the unstratifiedcase, 14v, a net flux of about 4 GW onto the southernshelf from the deep water is largely balanced by theflux of 3.5 GW into Hecate Strait (1 GW 5 109 watts).Hence there is a mean dissipation rate of about 0.5 GWover the Queen Charlotte Sound shelf. Most of the dis-sipation, about 1.2 GW, occurs in Hecate Strait/DixonEntrance where relatively shallow water is found.

In the stratified case (14u), the northward flux intoHecate Strait and out of Dixon Entrance is similar tothe homogeneous case and the rate of energy loss inthis region is essentially unchanged. However, the netflux onto the Queen Charlotte Sound increases to 4.9GW, implying an additional loss of about 1 GW overthe southern shelf. Overall, losses in the M2 tide overthe shelf increase from 1.85 to 3.05 GW with intro-duction of the stratification. As discussed in the nextsection, baroclinic flux calculations suggest that con-version to internal tidal energy accounts for a significantfraction of the additional barotropic losses.

5. Baroclinic response

To examine the baroclinic response of the model, weconsider results primarily from experiment 14u, the ba-


FIG. 7. Depth-integrated barotropic energy flux vectors for the (a) M2 and (b) K1 constituents from experiment 14v. The dashed contourindicates the 1000-m isobath. Every third vector is shown for regions where the depth exceeds 500 m, while every second vector is shownfor shallower regions.

sic stratified case. As indicated above, the barotropicresponse is nearly (but not completely) unchanged inthis run from the unstratified case, 14v. However, a vig-orous internal tide is generated in 14u that affects tidalcurrents over much of the domain.

The regions of internal tidal influence are delineatedin Fig. 9, which shows contours of the rms M2 barocliniccurrent on a near-surface velocity sigma level. Here thebaroclinic current is defined as the difference betweenthe total and the depth-averaged current, as in Eq. (A5).The intensity of the near-surface baroclinic componentof the tidal current gives an indication of the strengthof the internal tide. (The comparable field from 14v isclose to zero everywhere; with homogeneous water,near-surface and depth-averaged currents are nearlyidentical.) The results of Fig. 9 indicate that the internal

tidal influence is greatest over the outer shelf of QueenCharlotte Sound and on either side of Learmonth Bankin Dixon Entrance. There is also a significant offshoreinfluence emanating from the shelf break of these tworegions. In contrast, the internal tidal signal is relativelyweak for the inner portions of the shelf and in HecateStrait. In addition, a region of comparatively weak in-ternal tides is indicated in deep water off the west coastof the Queen Charlotte Islands.

In the deep water (H . 2500 m) beyond the shelf,rms M2 currents from experiment 14u near the surface(s 5 20.0105) have an average value of 6.5 cm s21

with a standard deviation of 3.3 cm s21. This compareswith an average of 2.1 cm s21 (standard deviation 0.3cm s21) for the rms depth-averaged M2 current. Near-surface currents over deep regions are thus generally


FIG. 8. Three subregions of the shelf nominally identified as (1)Queen Charlotte Sound, (2) Hecate Strait/Dixon Entrance, and (3)northern shelf. Sectionally integrated barotropic M2 energy fluxes (inGW) for experiments 14v and 14u (in parentheses) across transectsbounding these subregions are given. Arrows indicate the sense ofthe fluxes.

TABLE 3. Inferred rates of energy loss of the M2 barotropic tideover subregions of the shelf for two experiments. The subregions arebounded by the transects indicated in Fig. 8. Values are in units ofGW (109 watts).

Subregion Expt 14v Expt 14u

Queen Charlotte SoundHecate Strait/Dixon EntranceNorthern Shelf

0.491.210.15

1.521.200.33

Total 1.85 3.05

FIG. 9. Rms M2 baroclinic current from experiment 14u on a near-surface velocity sigma level (s 5 20.0105). The field is shaded atthe (8, 16, 24) cm s21 levels. Dashed contours are for the 1000-mand 200-m isobaths.

dominated by the baroclinic component. These resultsare consistent with values that Thomson et al. (1997)obtained from an analysis of near-surface drifters de-ployed in the region of Ocean Weather Station P (508N,1458W) and propagating eastward toward British Co-lumbia. Their average rms semidiurnal currents werepractically the same for both 15-m and 120-m drogueddrifters (5.9 cm s21, with a standard deviation of 2.1 cms21). The minimum observed rms semidiurnal values(2.9 cm s21) likely correspond to the barotropic currents.

a. Internal motions

The baroclinic response of the model is characterizedby internal vertical displacements and baroclinic cur-

rents at the constituent frequencies of the barotropic tide.The internal displacements can be examined by decom-posing the vertical velocity into a series of harmonicsof the form wncos(vnt 2 ln) where wn and ln are theamplitude and phase at frequency vn. Since w 5 dz/dt,where z is the vertical displacement, the amplitude andphase of the latter is given by wn/vn and (ln 1 908),respectively.

Figure 10a presents contours of the vertical displace-ment amplitude at M2 frequency for a representative x–


FIG. 10. Vertical displacement amplitude for the (a) M2 and (b) K1 constituents for a cross-shelf section along y 5 620 km. Contours aredrawn at the (1, 3, 6, 10, 20, 40) m levels.

FIG. 11. M2 rms baroclinic current from a cross-shelf section alongy 5 620 km (see Fig. 2). Contours are drawn at the (1, 2, 5, 10) cms21 levels. The vertical arrow at the top indicates the position ofmooring QE6 along this section.

z cross section that intersects the continental margin atDixon Entrance. Semidiurnal M2 amplitudes in excessof 40 m are obtained over the steepest parts of the con-tinental slope. These large values result from the directforcing of vertical motions over the slope by the baro-tropic tide and indicate generation regions for the in-ternal tide. Here the topographic slope is steeper thanthe slope of characteristics (a/tanc . 1) and propagationis directed toward deep water. In the deep regions, dis-placement amplitudes in the upper 500 m range from 1to 3 m near the surface to a maximum of about 10 mat mid depth. These near-surface values are comparableto semidiurnal displacements of 1–4 observed in theupper 200 m at station P (Thomson and Tabata 1982).Displacement amplitudes for S2 (not shown) are similarin pattern to the M2 field but smaller by a factor of ;3along this section.

Contours of vertical displacement amplitude along thecross section for K1 are presented in Fig. 10b. At the

K1 frequency, significant (.15 m) displacements aremanifest over the continental slope. However, at thelatitude of the B.C. coast, diurnal frequencies are smallerthan the inertial frequency (v , f). In this frequencyrange Eq. (1) ceases to be hyperbolic, and free internalwave propagation is no longer possible. As a result K1

motions are trapped over the slope and there is no ra-diation of energy offshore. A similar trapping occurswith the O1 field (not shown).

A cross-sectional view of the rms M2 baroclinic cur-rent is given in Fig. 11. In deep water the vertical struc-ture resembles that of a baroclinic internal wave mode.There are maxima near the surface with rms valuesreaching 5–10 cm s21, minima at approximately mid-depth, and maxima of 1–2 cm s21 at abyssal depths.

b. Energy fluxes

Baroclinic energy fluxes can provide insight into gen-eration and propagation of the internal tide. Three dif-ferent views of this energy flux are presented in Figs.12a–c. Figure 12a shows, for the entire model domain,the depth-integrated, baroclinic energy flux vectors atthe M2 frequency. (See appendix A for details.) Close-upviews for northern and southern subregions of the shelfare included as Figs. 12b and 12c. Energy fluxes for S2

(not shown) are similar in pattern to those for M2 butwith amplitudes reduced by a factor of ;6. The baro-clinic fluxes are generally much smaller than the cor-responding barotropic fluxes. For example, the M2 en-ergy flux represented by the scale vector in Fig. 12a is1% of that in Fig. 7a.

The vector field of Fig. 12a is dominated by an off-shore flux of baroclinic energy originating from theslope region near the 1000-m contour. The field is highlyinhomogeneous with offshore fluxes generated at threeprincipal locations. Along the shelf edge between CapeSt. James and Vancouver Island, two generation regionsare obtained, separated by a length of shelf where the


FIG. 12. (a) Depth-integrated M2 baroclinic energy flux vectors overthe entire model domain plotted at every second grid point. Similarfields are presented for (b) Queen Charlotte Sound and (c) DixonEntrance, with vectors shown at each grid point where the depth isshallower than 300 m. The scale vector in (a) differs from those of(b) and (c). Dashed contours indicate the 1000-m isobath in (a), andthe 300-m and 100-m isobaths in (b) and (c). Locations of severalmoorings mentioned in section 5c are indicated.


TABLE 4. Net offshore baroclinic energy flux from the shelf regionfor M2 and S2 constituents in different experiments. Units are giga-watts (109 W).

Experiment M2 S2

14u14sRT1RT2RT3

0.560.490.610.610.62

0.080.06———

TABLE 5. Partition of the M2 kinetic energy (cm2 s22) among thebarotropic (BT) and baroclinic (BC) modes from experiment 14u atvarious moorings.

Mooring BT mode1st BCmode

2nd BCmode

HigherBC modes

W02G04Q10QA3QB2QE6QF1QF2QF4B02B03

222.267.366.035.011.8

8.6139.6395.0

6.92.41.9

7.559.186.463.9

8.37.1

24.424.6

7.51.30.1

2.79.6

11.06.10.20.22.44.40.40.10.3

6.93.07.42.91.92.24.0

19.613.2

0.50.3

offshore flux is comparatively weak. The southernmostof these two source regions lies off Cook Bank. Farthernorth, there is a stretch of the shelf break 150 km longoff Dixon Entrance and Alaska where a large offshoreflux of baroclinic energy is generated. Also evident isa region off the west side of the Queen Charlotte Islandswhere the offshore flux is relatively weak. It is this‘‘shadow zone’’ that accounts for the relatively weaknear-surface signal obtained off the Queen Charlottesin Fig. 9. In the far field, offshore fluxes spread geo-metrically from source regions at the slope.

Generation sites for baroclinic energy in the deepwater correspond precisely to the three regions men-tioned in section 4c where significant fluxes of baro-tropic energy cross the shelf break. In these regions thecross-slope component of the barotropic M2 tidal ellip-ses (Fig. 7a) is largest and stratified fluid is forced mostvigorously up and down the continental slope. In termsof the two-dimensional model [Eq. (1)], these areregions where the cross-slope volume flux Q is maxi-mum. On the west side of the Queen Charlottes a dif-ferent situation prevails; cross-slope motions there areweak due to narrowness of the shelf and presence ofthe land mass.

An inhomogeneous pattern of baroclinic M2 energygeneration and propagation over the shelf is evident inFigs. 12b and 12c. South of Cape St. James, onshorefluxes into Queen Charlotte Sound are obtained, whichoriginate irregularly from near the 300-m depth contourover a distance extending for about 100 km (Fig. 12b).Near Cape St. James, some of this baroclinic energyleaks into the southwestern reaches of Hecate Strait.There is also a broad flux of baroclinic energy radiatingoff the southern flank of Goose Island Bank, near the100-m contour, toward northern Vancouver Island. Thepattern of energy fluxes in the region of Dixon Entrance(Fig. 12c) is particularly inhomogeneous. It is possibleto discern both onshore and offshore fluxes off the sidesof Learmonth Bank. The results also show fluxes di-rected in a cross-strait sense from the shoaling topog-raphy near the northern Queen Charlotte Islands.

Since a large fraction of the baroclinic energy is directoffshore, an estimate of the net baroclinic energy gen-eration may be obtained by integrating the flux normalto transects forming a box around the shelf region. Theresult of such a calculation for the M2 and S2 constituentsis given in Table 4 for different stratified experiments.

The integrated baroclinic flux is reduce somewhat from14u in the winter stratification case (14s), but the basicpattern is essentially unchanged. Offshore fluxes in thethree resolution test cases (RT1, RT2, and RT3) showmutual agreement and differ only slightly from 14u,confirming that energy fluxes are relatively insensitiveto variations in horizontal or vertical resolution.

Of the net offshore baroclinic flux of 0.56 GW in thebasic case (14u), 0.17 GW emanates from the DixonEntrance/Alaskan shelf region, while 0.34 GW origi-nates from Queen Charlotte Sound. The values representthe baroclinic energy radiating offshore but do not in-clude fluxes radiating shoreward from the shelf breakor local fluxes over the shelf or slope. It was noted insection 4 that barotropic losses over the northern shelfregion were augmented by 0.18 GW and over the south-ern shelf region by 1.03 GW. Hence, baroclinic energyconversion readily accounts for the additional barotropiclosses over the northern shelf region and a significantfraction of the additional losses over Queen CharlotteSound. Bottom friction over shelves is the main dissi-pation mechanism for the barotropic tide and additionallosses with stratification may be partly attributable toenhancement of bottom dissipation. Rms currents abovethe bottom are significantly increased due to the internaltide in some regions over shelf in Queen CharlotteSound and over the adjoining slope.

The offshore M2 baroclinic energy flux of about halfa gigawatt obtained here is significantly larger than anearlier estimate made by Baines (1982) with a two-dimensional analytical model. Values for the northeastPacific were not explicitly given but were implied to beless than 0.1 GW. Reasons for the discrepancy are notclear. However, our results show localized sources andthe inhomogeneous nature of baroclinic energy gener-ation, features that may be difficult to account for witha two-dimensional model.

c. Comparison of M2 current ellipses withobservations

As a means of verifying the model, comparisons be-tween observed M2 tidal ellipses and results from ex-


periments 14v and 14u are presented. Current metermoorings from regions where an internal tidal compo-nent is indicated are considered. The discussion is lim-ited to M2, the leading semidiurnal constituent, however,ellipses for S2 are similarly affected by the stratification.Except for station QE6, the observations are drawn fromsummer deployments (nominally mid-May to Sep) madeduring the late 1970s and early 1980s as part of theCODE (Freeland et al. 1984) and NCODE (Huggett et

al. 1992) experiments. At QE6, the only existing de-ployment extends from October 1983 through June1984.

For each model station the kinetic energy has beenpartitioned into contributions from barotropic and baro-clinic modes, according to a procedure outlined in ap-pendix B. The modes are derived from a flat-bottomtheory, which is strictly inapplicable over variable to-pography. They are, however, still useful because they


FIG. 13. Comparison of model M2 tidal ellipses (rotated 308, as inFig. 6) with observations from moorings at various sites over thecontinental shelf and slope (see Figs. 1a and 12a–c). The ellipses arefor the total (barotropic 1 internal) current. Model results from ex-periment 14v (unstratified) are on the left side of each panel, thoseof 14u (stratified) on the right, with observations in the center. Themodel results are sampled at the following velocity sigma levels(20.0015, 20.1475, 20.25, 20.45, 20.65, 20.85, 20.975).

form an orthogonal set of basis functions that providesan efficient decomposition of the model currents in sev-eral instances.

Results of the modal partition applied to experiment14u are provided in Table 5, where contributions frommodes 3 and higher have been grouped together. At W02in northern Hecate Strait, M2 currents from experiment14u are similar to those of 14v (shown in Fig. 6) and,as expected, completely dominated by the barotropic

component. At other sites considered, the barocliniccontribution is more significant and usually dominatedby the first mode.

M2 currents at two moorings situated in Queen Char-lotte Sound, G04 and Q10, are shown in Figs. 13a and13b. Station G04 is located on the southern flank ofGoose Island Bank, in a region subject to a southwardradiating flux of baroclinic energy (see Fig 12b). Ellip-ses from 14v are nearly identical to those presented by


BFCJ96 for this site and represent the barotropic re-sponse. In the stratified case, the baroclinic and baro-tropic modes have comparable kinetic energy levels.Model ellipses capture the observed clockwise rotationof phase with depth, a structure expected from super-position of barotropic and first baroclinic mode com-ponents (Farmer and Freeland 1983, p. 171). StationQ10, located in the trough between Middle Bank andGoose Island Bank, is exposed to a baroclinic flux ra-diating shoreward from the shelf break (Fig. 12b). Inthe stratified case, the vertical structure, dominated bythe first baroclinic and barotropic modes, reproducesrelatively accurately the observed M2 ellipses.

Figures 13c and 13d compare current ellipses at twosites in proximity of Cape St. James. Station QB2 isover the continental slope on the west side of the QueenCharlottes, lying in the path of baroclinic fluxes radi-ating northwestward along the slope from a source re-gion off the Cape (Fig. 12a). Here the response of thestratified model leads to near-surface ellipses that areconsiderably larger than the unstratified response, con-sistent with observations. At station QA3, near the shelfbreak, observed ellipses have a clockwise rotation inorientation and phase with depth, which is captured,albeit with some differences, in experiment 14u.

Results from a site near Dixon Entrance are presentedin Fig. 13e. Station QF2 is located near the mouth ofthe channel, at a point in the path of southward baro-clinic fluxes emanating from Learmonth Bank (Fig.12c). The model ellipses in both cases are dominatedby the nearly rectilinear, barotropic component. How-ever, due to a stronger clockwise component, near-sur-face ellipses from 14u have an increased eccentricity,consistent with observations. Comparisons with obser-vations at station QF1 (not shown), located on the op-posite side of Learmonth Bank (Fig. 12c), yield similarresults. Generally, the eccentricity of near-surface ellip-ses throughout much of Dixon Entrance is increased bythe presence of the internal tide.

Finally, we show results in Figs. 13f and 13g for twoopen ocean stations, QE6 located in 1100 m of wateroff Dixon Entrance and B03, about 80 km southwest ofBrooks Peninsula (Fig. 1a), in 2300 m of water. Resultsfrom QF4 and B02 (not presented) are qualitatively sim-ilar to QE6 and B03, respectively. At QE6 (which liesalong the x–z section described in section 5a), the off-shore-directed internal tidal energy (Fig. 12a) producesa large clockwise-rotating ellipse at the depth of thenear-surface current meter. By comparison the unstrat-ified response is characterized by relatively weak coun-terclockwise motion. Similar comments apply to theB03 site, where the internal tide of the model affectsnear-surface motions, consistent with observations.

Drakopoulos and Marsden (1993) used a ray tracingapproach to determine the pattern of characteristics forthe B02 and B03 stations. Their results indicated that abeam of internal energy should be detected at a depthof 1000 m for the B03 site, which is inconsistent with

both observations and our model. The authors noted thatcomplex local topography and the number of possiblegeneration sites compound the difficulties of applyingray tracing methods in this region.

6. Conclusions

This study is concerned with simulation of tides overthe northern coastal waters of British Columbia with athree-dimensional primitive equation model. Two basicconfigurations of the model have been considered, onewith homogeneous water and the second with a hori-zontally homogeneous, vertically stratified fluid, rep-resentative of summer conditions. Tidal forcing is in-troduced along open ocean boundaries with four leadingconstituents of the barotropic tide derived from a globaltidal atlas. Sensitivity to variations in model resolutionand to a representative winter stratification have beenexplored.

The barotropic response of the model is similar toprevious tidal simulations of the region, and compari-sons with tide gauge data indicate a relatively accuraterepresentation of surface elevation fields. Tidal currentsin the unstratified case also agree closely with the earlierstudies, but capture little of the vertical structure ob-served with the semidiurnal constituents in manyregions.

With a stratified medium, an internal tide is estab-lished which has a significant influence on tidal currents.The rms magnitude of the M2 baroclinic current nearthe surface delineates those regions most strongly af-fected by the internal tide. Most notably, these areQueen Charlotte Sound and Dixon Entrance, where theshelf is broad and exposed to the open ocean. In HecateStrait, which is sheltered from the open Pacific by theQueen Charlotte Islands, the internal tide has only lim-ited influence. Semidiurnal M2 tidal ellipses were com-pared with observations drawn from a series of moor-ings situated in regions where an internal tide is indi-cated. The vertical structure of the model ellipses isgenerally consistent with the observations from a num-ber of widely separated shelf and slope sites. Thesecomparisons suggest that, despite an idealized uniformstratification, the essential characteristics of the internaltide are retained by the model. Further improvementsmay require a more accurate initialization, taking ac-count of inhomogeneities in the ambient stratification.

Baroclinic energy fluxes computed from the modelindicate that semidiurnal baroclinic tides radiate off-shore from localized sources along the continental slope.At these generation sites the barotropic tidal ellipseshave a pronounced cross-slope component. The inte-grated offshore flux is about 0.5 GW, which gives anaverage flux of about 625 W m21 with a model coastlineof 800 km. The analytical, two-dimensional model ofBaines (1982) provides (to our knowledge) the onlyestimate of baroclinic energy generation available forcomparison. The magnitude of the offshore flux ob-


tained here exceeds this earlier estimate by a factor ofat least 5. The discrepancy may be due to difficultiesinherent in application of a two-dimensional model toa region with localized sources of baroclinic energy.

Satellite altimetry has permitted a precise determi-nation of the global rate of tidal energy dissipation forthe largest constituents (Cartwright and Ray 1991).However, the long standing problem of accounting forthese losses in the ocean remains open. In particular,the significance of losses to internal tides is uncertainand disparate estimates have been made. For example,Baines (1982) calculated that internal tide generationalong the continental margins accounts for a negligiblefraction (0.3%) of the M2 dissipation rate. Applying thesame analytic model to midocean ridges, Morozov(1995) estimated that baroclinic conversion could rep-resent 25% of the global dissipation. The results fromthis study suggest that incorporation of stratification ef-fects in three-dimensional models may be necessary tosimulate in a realistic fashion the dissipation of semi-diurnal constituents over continental shelves. For ourcoastal region we find that stratification leads to sig-nificant additional barotropic losses as a consequenceof the generation of internal tides.

Acknowledgments. We thank Mike Foreman for help-ful discussions on tidal modelling. We have also ben-efitted from advice provided by Bill Crawford, HowardFreeland, and Rick Thomson. Contributions to softwaredevelopment were made by Jane Eert, Juijing Gu, andDave Ramsden. This project was partially supported bythe Canadian Panel of Energy Research and Develop-ment, Project 67146. We acknowledge support from thePittsburgh Supercomputing Center.

APPENDIX A

Calculation of Energy Fluxes

The equation governing the rate of change of energyof an incompressible fluid is (Gill 1982; section 4.7)

]ET 1 =·F 5 G 1 D, (A1)]t

where ET 5 r[(1/2)zuz2 1 gz 1 E] is the sum of kinetic,potential, and internal energy, with u the three-dimen-sional velocity vector; G and D represent forcing anddissipative effects, respectively. The energy flux density,

F 5 up 1 uET 1 ··· ,

has contributions from the work done by pressure forces,advection of energy, and additional terms involving vis-cous fluxes and other nonconservative processes. Inte-gration of (A1) over a fixed control volume and timeaveraging yields a balance between mean fluxes normalto the volume surface and mean forcing and dissipation,

∫∫ ^F·n& dS 5 ∫∫∫ ^G 1 D& dV, (A2)

where the angle brackets denote a time-average overperiod T,

T1^(·)& 5 (·) dt.ET 0

We consider the contributions to the mean horizontalenergy flux density due to the rate of working by pres-sure forces associated with barotropic (subscript bt) andbaroclinic (subscript bc) tidal motions,

M 5 ^u p &bt bt bt

M 5 ^u p &. (A3)bc bc bc

Additional contributions to the flux from advection anddiffusion are neglected as these are expected to be ofsecond-order for a small amplitude wave field. For thesecalculations, the horizontal velocity field uh is decom-posed into a depth-averaged barotropic component

h1u 5 u dz, (A4)bt E hH

2H

and an internal baroclinic component

ubc 5 (uh 2 ubt). (A5)

Here ubc is interpreted as current associated with thebaroclinic tide although it may be contaminated by vis-cous effects in the bottom boundary layer (see com-ments below).

The barotropic pressure field pbt is associated withdisplacements of the sea surface, h,

pbt 5 r0g(h 2 z). (A6)

A method similar to that of Holloway (1996) has beenadopted for calculation of the baroclinic fluxes. Pressureperturbations associated with internal tidal motions areassumed to be governed by hydrostatic balance ]zp 52rg and a linearized adiabatic density equation ]tr 52w]zr. The baroclinic pressure field is then obtainedfrom the relation

0]p ]rbc 5 2g w dz*, (A7)E]t ]z*z

where w is the vertical velocity. A barotropic compo-nent, wbt 5 (1 1 s) Dh/Dt 1 s (ubt·=H) with s 5 (z2 h)/(H 1 h), may be removed from w, but this hasbeen verified to make little difference to the baroclinicfluxes.

To evaluate time averages it is useful to expand thevelocity and pressure fields in a series of harmonics.For example, consider a rotation through an angle un todefine coordinates (x9, y9), such that the x9 axis is ori-ented in the direction of the positive semimajor axis ofthe local tidal ellipse. Then barotropic and baroclinicvelocities and the perturbation component of the pres-sure fields for each constituent are of the form


u9 5 u9cos(v t 2 g )n n n

v9 5 y9sin(v t 2 g )n n n

p 5 p cos(v t 2 f ). (A8)n n n

Here ( , ) are current amplitudes at frequency vn inu9 y9n n

the (x9, y9) directions respectively; gn is the phase lagof maximum current behind maximum tidal potential;pn and fn are the pressure amplitude and phase lag re-spectively; and un is the orientation of the semimajoraxis with respect to (x, y) coordinates. Using the ex-pansion (A8), it is possible to express components ofthe rotated mean energy flux vector 5 ^u9p&n asnMx9

1nM 5 ^u9p& 5 u9p cos(f 2 g )x9 n n n n n2

1nM 5 ^y9p& 5 y9p sin(f 2 g ). (A9)y9 n n n n n2

A rotation must be applied to recover the energy fluxesin the (x, y) coordinate of the model:

n nM cosu 2sinu Mx n n x95 . (A10)n n1 2 1 21 2M sinu cosu My n n y9

Terms involving products of velocities and pressures atdifferent frequencies have been omitted because thesetend to zero as the averaging period T increases. Thus,provided T is large compared to the constituent periodsand that conditions are stationary, the total mean energyflux is the sum of fluxes at each frequency M 5

. Finally, the depth-integrated energy flux vectorsnS Mn x

presented in sections 4 and 5 are given by the followingexpressions, respectively:

0

n n nJ 5 M dz 5 HM ,bt E bt bt

2H

0

n nJ 5 M dz. (A11)bc E bc

2H

The harmonic amplitudes and phases of (A8) and theun were extracted for each tidal constituent using theleast squares fit method of Foreman (1977, 1978). Forthe barotropic energy flux, these parameters are two-dimensional fields, defined at each horizontal grid pointof the model. For the baroclinic fluxes, they are three-dimensional, additionally dependent on the vertical co-ordinate.

As noted above, two processes contribute to ubc: thebaroclinic tide, which we wish to isolate, and unwantedviscous effects in the bottom Ekman layer, which shouldbe most significant over shallow shelf regions. To ex-amine whether viscous effects significantly contaminatethe energy fluxes, these were recalculated with the baro-clinic current in (A3) replaced by (uh 2 ubt) z14u 2 (uh

2 ubt) z14v. That is, the ‘‘internal’’ component of thecurrent from the unstratified experiment (14v), which isdue to bottom friction, is subtracted from the baroclinic

current of the stratified case (14u). The recalculateddepth-integrated fluxes were visually indistinguishablefrom those given in Figs. 12a–c. The net integrated fluxfrom the shelf into the deep ocean was virtually un-changed. It seems reasonable then to assume that thereis no significant contamination of the energy fluxes dueto viscous effects.

APPENDIX B

Modal Decomposition

The following procedure was applied to partition thekinetic energy into barotropic and baroclinic modes atthe constituent frequencies of the tide. For each stationthe horizontal currents are expanded as

(u, y) 5 (Pk(z)(uk(t),K21S y (t)),k50 k (B1)

where k is the mode index and K the number of verticalgrid points. The Pk(z) are eigensolutions of the Sturm–Liouville problem,

(N22 )9 1 Pk 5 0,22P9 lk k (B2)

with boundary conditions 5 0 at z 5 0, 2H. Equa-P9ktion (B2) is appropriate to internal wave motions overa flat bottom in the hydrostatic (v2 K N2), Boussinesqlimit (LeBlond and Mysak 1978); N is the buoyancyfrequency and the eigenvalue.2lk

The modes are normalized according to the condition01

P P dz 5 d , (B3)E k j k,jH2H

where dk,j is the Kronecker delta. Modal currents (uk,are thus given byy )k

01(u , v ) 5 P (u, y) dz. (B4)k k E kH

2H

The (uk, vk) are harmonically analyzed in the usual man-ner and mean kinetic energy in each mode for constit-uent n is obtained as ( 1 )/4, where An,k and Bn,k

2 2A Bn,k n,k

are the semimajor and semiminor axes lengths of themodal current ellipse, respectively.

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simulation of barotropic and baroclinic tides off …...tanc 5 , (2) 12 n222v where f is the local...

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