Simulation of Barotropic and Baroclinic Tides Off Northern British Columbia

762 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

Simulation of Barotropic and Baroclinic Tides off Northern British Columbia

PATRICK F. C UMMINS 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 ®nal 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 leading semidiurnal and diurnal constituents and experiments with strati®ed and homogeneous ¯uid are compared. The barotropic response shows good agreement with previously published studies of tides in the region. A comparison with tide gauge measurements indicates that average relative rms differences between observations and the model surface elevation ®eld are less than 5% for the largest constituents. An internal tide is generated in cases where the model is initialized with a vertical strati®cation. Diagnostic calculations of the baroclinic energy ¯ux are used to identify regions of generation and propagation of internal tidal energy. With a representative summer strati®cation, the integrated offshore ¯ux 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 signi®cant in¯uence of the internal tide.

1. Introduction ticularly M2 baroclinic energy ¯uxes, were highly sen- Motions at tidal frequency in the coastal ocean are sitive to speci®cation of the cross-shelf topographic sec- often composed of an astronomically driven surface tion. (barotropic) tide and an internal (baroclinic) tide, arising Tides along the northern coast of British Columbia from interaction of the barotropically forced, strati®ed have been the subject of a number of previous modeling ¯uid with bathymetry. In contrast to the barotropic tide, studies. Flather (1987, hereinafter F87) forced a ®nite the baroclinic component contributes relatively little to difference barotropic model of the northeast Paci®c with oscillations of the sea surface; its signature appears pri- the M2 and K1 tidal constituents. Foreman et al. (1993, marily in tidal currents and internal isopycnal motions. hereinafter FHWB93) presented results from a baro- Numerical models have been applied in several in- tropic ®nite element model with eight constituents for stances to study the response of a coastal region to spec- a somewhat smaller region. Extensive comparisons with i®ed tidal forcing. To date, however, models of open sea level data in these two papers showed generally coastlines have been concerned almost exclusively with excellent agreement between observations and the mod- the barotropic tide. (See Foreman et al. 1995 for a re- el surface elevation ®eld. This work was extended review.) While baroclinic models in three dimensions cently by Ballantyne et al. (1996, hereinafter BFCJ96) have been applied to embayments and channels of lim- who employ a three-dimensional diagnostic ®nite ele- ited extent (Matsuyama 1985; Wang 1989), to our ment model. The performance of the three-dimensional knowledge, baroclinic tidal studies of open coastlines model with respect to sea level is comparable to the have been limited to applications of two-dimensional, barotropic models. In addition, detailed comparisons of cross-shelf models (e.g., Craig 1988; Sherwin and Tay- tidal current ellipses with observations through the wa- lor 1990). Recently, Holloway (1996) applied an essen- ter column were presented for various locations. For the tially two-dimensional version of the Princeton Ocean M2 constituent, signi®cant discrepancies were found that Model to the Australian North West Shelf. Results, par- may be attributable to the presence of baroclinic tides. In each of the aforementioned studies, the nature of the model precluded the possibility of an internal tidal response. Corresponding author address: Dr. Patrick F. Cummins, Institute of Ocean Sciences, P.O. Box 6000, 9860 West Saanich Rd. Sidney, In the present paper, tidal simulations of the northern BC V8L 4B2, Canada. coastal waters of British Columbia with a fully prog- E-mail: [email protected] nostic three-dimensional model are discussed. The nu-

᭧1997 American Meteorological Society MAY 1997 CUMMINS AND OEY 763 merical formulation is based on the version of the in the North Paci®c. Amplitudes of both bands are sig- Princeton Ocean Model (G. Mellor 1993, personal com- ni®cant in the northeast Paci®c, and hence tides along munication) used by Oey and Chen (1992) to simulate the coast of British Columbia are classi®ed as mixed, ¯ows over the northeast Atlantic shelves. The model is predominantly semidiurnal. Mean tidal ranges are about driven at the boundaries with the principal diurnal and 2 m in deep water, increasing to about 3.5 m near the semidiurnal constituents. Numerical experiments with entrances of Queen Charlotte Sound and Dixon Entrance homogeneous and with vertically strati®ed ¯uid are and to about 5 m near Prince Rupert. The tides are compared. With homogeneous ¯uid, the solution is sim- subject to a spring±neap cycle, which arises through an ilar to that of BFCJ96. The barotropic tide is well rep- interference of M2 and S2, the dominant semidiurnal resented, but current ellipses in many locations bear constituents, and also K1 and O1, the largest diurnal little resemblance to observations. Inclusion of a vertical constituents. strati®cation allows an internal tide to be generated, In the presence of a sloping bottom, barotropic tides signi®cantly affecting current ellipses for the semidi- can force vertical oscillations of a stratifed ¯uid and urnal constituents. This leads to an improved represen- thus provide a source of baroclinic energy. In two di- tation of M2 currents in several regions over the shelf mensions (x, z), the dynamics can be expressed by an and slope. Calculations of the baroclinic energy ¯ux are equation governing the streamfunction (Huthnance used to identify regions of internal tide generation and 1989) ␺, propagation. In the next section, a brief review of the regional Q 1 ␺ Ϫ tan2c␺ ϭ z , (1) oceanography and geography is provided. This is fol- xx zz 22 (1 Ϫ ␻ /N ) ΂΃H xx lowed by a section containing a description of the model con®guration and outlining the numerical experiments. where Q is the barotropic cross-shelf volume ¯ux, H the bottom depth, and the tidal frequency; N ( g / Section 4 discusses the barotropic tidal response of the ␻ ϭ Ϫ ␳z )1/2 is the buoyancy frequency with the density, model, while the baroclinic response is described in ␳0 ␳ ␳0 a reference density, and g the gravitational constant. section 5. A summary and suggestions for future work 2 2 are given in the concluding section. Provided that (␻ Ͼ f ), (1) is hyperbolic with propagation along rays whose slope is given by

1/2 2. Regional oceanography ␻22Ϫf tanc ϭ , (2) 22 A thorough review of the physical oceanography of ΂΃NϪ␻ northern coast of British Columbia is available in Thom- where f is the local Coriolis parameter. The bottom slope son (1989). Our purpose here is to draw attention to the ␣ is supercritical if ␣/tanc Ͼ 1, and propagation toward features of the region that are pertinent to discussion of deep water is anticipated. Conversely, a subcritical slope the numerical simulations. The coastal geography and is associated with onshore propagation. bathymetry of the region are illustrated in Figs. 1a and Using typical temperature and salinity data from the 1b. The bathymetry is characterized by a broad, O(100 continental slope region to determine N (section 3, Fig. km wide) shelf, which is partially sheltered from the 3b), the ratio ␣/tanc is found to easily exceed unity over open ocean by the presence of Moresby and Graham the slope. Strati®cation in the upper 100±200 m of the Islands, known collectively as the Queen Charlotte Is- water column undergoes a seasonal modulation, asso- lands. The shelf is narrow on the west side of these ciated with coastal runoff and surface warming during islands with depths dropping to 2500 m over a distance summer months and vertical mixing during winter. Be- no greater than 30 km from the coast. Three connected low 200 m such variations are weak and internal tide seas separate the Queen Charlottes from the coastal generation with offshore propagation thus appears pos- mainland. Queen Charlotte Sound is open to the Paci®c sible throughout the year over the slope. Although the on its western side and marked by three relatively shal- barotropic forcing is constant, the internal tide may nev- low banks. Proceeding from south to north, these are ertheless be intermittent as physical processes affecting Cook Bank, Goose Island Bank and Middle Bank. To the strati®cation (e.g., mixing or wind-induced down- the north, Hecate Strait is a broad, shallow passage with welling) can modulate generation or propagation. Drak- an elongated submarine valley on its eastern side. Owing opoulos and Marsden (1993) examined evidence for in- to the presence of Dog®sh Banks, Hecate Strait is very ternal tides off the west coast of Vancouver Island, to shallow (less than 30-m depth) on its western side. The the southeast of our study area. They observed offshore third signi®cant sea is Dixon Entrance, a passage of O propagation and a seasonal modulation, with maximum (60 km) width connecting northern Hecate Strait to the intensity during summer months. open Paci®c. Learmonth Bank is a shallow barrier found near the opening of Dixon Entrance. 3. Numerical model Barotropic tides of the semidiurnal and diurnal frequency bands propagate approximately as large-scale The numerical model used in this study is based on Kelvin waves in a cyclonic sense around amphidromes the Princton Ocean Model (G. Mellor 1993, personal 764 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

FIG. 1a. Map of the northern British Columbia and southern Alaska coasts showing place names, the distribution of tide gauges sites (triangles), and locations of current meter moorings (circles). communication), and a detailed description is given in The numerical algorithm is described by Mellor in Oey and Chen (1992). Brie¯y, the model solves ®nite the user's guide (1993) for the Princeton Ocean Model. difference analogs of the primitive equations, with a The model employs a C grid in the horizontal and a nonlinear equation of state relating density to the tem- terrain following sigma coordinate system in the ver- perature and salinity. The level-2.5 turbulent kinetic en- tical. A mode-splitting technique is applied for integra- ergy closure scheme of Mellor and Yamada (1982) is tion of external and internal modes with different time used to determine vertical mixing coef®cients for mo- steps. The integration proceeds according to an explicit mentum and tracer (temperature and salinity) ®elds. leapfrog time stepping scheme, except for terms in- Horizontal mixing coef®cients are modeled with the volving vertical diffusion of momentum and that are Smagorinsky formulation for diffusion. Bottom fric- treated implicitly. A weak time ®lter is included to pre- tional stresses are modeled by a quadratic drag law with vent development of a computational mode with the a drag coef®cient formulated to yield a logarithmic bot- leapfrog scheme. tom boundary layer. Buoyancy and momentum ¯uxes Through a forced gravity wave radiation condition through the surface have been set to zero. (see Oey and Chen 1992; F87), the model is driven at MAY 1997 CUMMINS AND OEY 765

FIG. 1b. Bathymetry of the region with depth contours given in meters. open boundaries with the four leading tidal constituents. the horizontal (x, y) coordinates. The universal trans- Boundary elevation data for these constituents are ob- verse mercator map projection, used by Hannah et al. tained from the global tidal model of Schwiderski (1979, (1991), relates the (x, y) coordinate of each grid point 1981a±c). Depth-averaged tidal velocities required at to latitude±longitude coordinates. A rotation of 30Њ is open boundaries for the radiation condition are obtained applied to align the y coordinate of the model approx- from an iterative procedure similar to one outlined in imately with the shelf break, thus reducing the number F87. Boundary conditions differ from those of Oey and of land points. The Coriolis parameter is set to a constant Chen (1992) in certain respects. An Orlanski radiation appropriate to 53ЊN. In most cases, there are 21 sigma condition has been applied to the internal mode veloc- levels in the vertical with the following distribution: ␴k ities, temperature, and salinity at the open boundaries. ϭ (0, Ϫ0.003, Ϫ0.006, Ϫ0.015, Ϫ0.03, Ϫ0.05, Ϫ0.07, Barotropic tidal motions are driven entirely by os- Ϫ0.09, Ϫ0.11, Ϫ0.13, Ϫ0.165, Ϫ0.2, Ϫ0.3, Ϫ0.4, cillations at open boundaries; the earth tide, load tide, Ϫ0.5, Ϫ0.6, Ϫ0.7, Ϫ0.8, Ϫ0.9, Ϫ0.95, Ϫ1), where ␴k and tide-generating potential have been neglected. With ϭ 0 at the surface and ␴k ϭϪ1 at the bottom. Velocity a coastal model of similar extent, FHWB93 concluded and tracer ®elds are de®ned at midpoint between these that these forcings produce only slight modi®cations to levels. semidiurnal constituents. The model topography was constructed by averaging depth data within a 5 km radius at each grid point. The depth data for most of the domain were obtained from a. Model grid a high resolution (1 km) topographic database available The governing equations of the model are solved over for the British Columbia coast. It was necessary to sup- a Cartesian grid with a uniform resolution of 5 km in plement this with ETOPO5 topography for the northern 766 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

TABLE 1. Numerical experiments mentioned in this study. Strati®cation Experiment pro®le Comments 14v None Homogeneous ¯uid case 14u Summer 14s Winter 14p Summer Diagnostic case

RT1 Summer M2 only

RT2 Summer M2 only, 41 levels

RT3 Summer M2 only, 2.5-km grid

tally homogeneous vertically strati®ed ®elds representative of summer strati®cation. Figure 3a illustrates the vertical variation of temperature and salinity used for experiment 14u. The vertical pro®le is speci®ed with an analytic form composed of a linear region near the surface and an exponential form below. This pro®le, although idealized, provides a reasonable approximation to the density strati®cation typically observed over the continental slope and outer shelf during summer conditions (Fig. 3b). Additional experiments include a case with a ``winter'' strati®cation and a strati®ed diagnostic case. The initial strati®cation for the winter case, 14s, differs from the pro®le of Fig. 3a only in the upper 125 m where, to allow for a winter mixed layer, the ¯uid is homogeneous. Motivation for the diagnostic (®xed temperature and salinity) experiment, 14p, is to examine sensitivity to strati®cation in the absence of an internal tide. It is known that properties of tidal current ellipses can be FIG. 2. Model domain with contours of bottom topography. The horizontal arrow indicates the position of the cross-shelf sections sensitive to the value of the vertical viscosity (Prandle shown in Figs. 10 and 11. 1982). In the present case, this parameter is determined as part of the numerical solution through the turbulence closure submodel. Its value will depend on the several reaches of Dixon Entrance and for regions located well factors, in particular, the vertical strati®cation. In ex- offshore of the continental slope. No smoothing was periment 14p the ¯uid was strati®ed as in 14u, but an done to the resulting topographic ®eld except for an internal tidal response was precluded by relaxing the area adjacent to the offshore boundary (0 Ͻ X Ͻ 150 temperature and salinity ®elds to their initial values with km), where a smoother was applied, eliminating the a very short time constant. Except in the bottom bound- presence of Bowie and Union seamounts from the mod- ary layer where relatively minor differences were ob- el. The resulting model domain is shown in Fig. 2 with tained, results from this diagnostic case, in particular contours of bottom topography. The domain is open on tidal ellipse parameters, were nearly indistinguishable the two cross-shelf boundaries and on the offshore from those of 14v. The results of experiment 14p dem- boundary. Two narrow straits, Johnstone Strait to the onstrate that the signi®cant differences between 14u and south and Clarence Strait to the north (Fig. 1) have been 14v discussed below are not merely a result of strati- closed off. ®cation-dependent viscosity, but rather the presence of internal tides in 14u. b. Experiments For experiments 14v and 14u, the model was integrated for 34 days of simulated time. Hourly ®elds from The numerical experiments of this study are listed in the last 29 days were subject to an harmonic analysis Table 1. Most of the discussion is focused on a com- to separate the response at the four constituent fre- parison of the ®rst two cases, but mention is also made quencies. This analysis period is suf®cient for the har- of some additional runs. The ®rst case, denoted 14v, is monic analysis to separate the response within the di- a control run in which a homogeneous unstrati®ed ¯uid urnal and semidiurnal frequency bands, but also short is speci®ed. The baroclinic response of the model was enough that the mean strati®cation does not change ap- examined with a second experiment, denoted 14u. Here preciably. The initial spinup time is suf®cient to estab- the temperature and salinity are initialized as horizon- lish an equilibrium for the tide. MAY 1997 CUMMINS AND OEY 767

FIG. 3. (a) Vertical pro®les of temperature and salinity representing summer strati®cation used to initialize the model. (b) Comparison of the buoyancy frequency derived from the model pro®le shown in (a) with observations from two representative CTD stations at different locations over the continental slope. The observations were made on 13 and 14 July 1983. c. Resolution tests clinic currents of the model to variations in the horizontal or vertical resolution. The radiation pattern of the Simulations of the baroclinic tide require resolution internal tide was essentially unchanged in these cases of the scales of the topography and the wavelength as- and resembled the M response of the strati®ed case with sociated with the internal tide. It is also desirable to 2 four constituents (14u). Thus, the model does not appear minimize pressure gradient error and other numerical to be very sensitive to changes in resolution. It is likely truncation errors associated with sigma levels (Mellor that a model that resolved ®ner topographic scales et al. 1994). A length scale, H/tanc, based on (1) (Huth- would introduce new ®nestructure to the internal tidal nance 1989) yields a value of 25 km over the shelf, ®eld. However, as shown below, the large-scale structure assuming H ϭ 200 m and N ϭ 1 ϫ 10Ϫ2 rad sϪ1.In of the internal tide is determined by the larger scales of 2000 m of water with a typical deep value for N (3 ϫ the topography and the barotropic tide, which are both 10Ϫ3 rad sϪ1), the length scale is 75 km. The 5-km grid well resolved. resolution appears suf®cient for deep regions, but may be marginal over the shallower regions of the shelf. A limited convergence study was done to examine 4. Barotropic response sensitivity of the baroclinic response to variations in horizontal and vertical resolution. These cases are the The results in this section are drawn mainly from last three entries in Table 1 and include only forcing experiment 14v with homogeneous ¯uid. Strati®cation has little effect on the barotropic response of the model. with the M2 constituent. The runs were restricted to a duration of 5 or 6 days with harmonic analysis applied There are, however, subtle differences between 14v and to the last 3 or 4 days. In experiment RT1 the basic the strati®ed runs, the most notable of which is an ad- 5-km grid with 21 vertical levels is maintained. The ditional loss of barotropic tidal energy over the shelf vertical resolution is re®ned in experiment RT2 to 41 with strati®ed ¯uid. levels, while in RT3 the horizontal resolution is doubled to 2.5 km. The additional vertical levels in RT2 were a. Surface elevation placed at midpoint between those of the 21-level distribution. The topography in RT3 was obtained from The amplitude and phase of the surface elevation ®eld linear interpolation of the 5-km grid and does not in- at the M2 and K1 frequencies are presented in Figs. 4a troduce ®ner topographic scales. and 4b. The M2 phase progresses steadily northward An intercomparison of results from these three cases along the outer coast. As the tide propagates into Hecate showed only minor quantitative differences in the baro- Strait from the south, amplitudes increase due to shoal- 768 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

FIG. 4. Model co-amplitudes and co-phases (degrees UT) for the (a) M2 and (b) K1 constituents from experiment 14v. Contour intervals

for M2 (K1) are 10 (2) cm for amplitude and 5Њ (2Њ) for phase. ing effects on the shelf and reach almost 2 m on the with those of F87 than either FHWB93 or BFCJ96. Once east side of Graham Island. The shoaling also induces again, a likely reason for this is the tuning of boundary an east±west phase difference of about 10Њ across the data in these studies. (F87 made no such adjustments

Strait. The M2 amplitude and phase of Fig. 4a agree to his boundary forcing.) closely with comparable ®elds presented in earlier stud- Distributions for the S2 and O1 constituents (not ies (F87, FHWB93, BFCJ96). The most notable differ- shown) are similar in pattern to M2 and K1, respectively, ence with these last two studies is a small discrepancy and generally agree with unpublished results, provided in the phase over the northern part of the domain. In by M. Foreman, from the ®nite element models. Here

Fig. 4a, the 270Њ phase contour abuts on the outer coast S2 amplitudes are about 30% of M2, while O1 amplitudes of Alaska, whereas it passes through northern Dixon are about 60% of K1. Entrance in the ®nite element models. One factor that To provide a quantitative assessment of the elevation Ϫ1 may account for this difference is that FHWB93 and ®eld, averages of the absolute rms error, L ⌺L D, and Ϫ1 BFCJ96 tuned their boundary data (also taken from the the relative rms error, L ⌺L D/Ao, were computed for Schwiderski atlases) to obtain an improved agreement each constituent. The averaging is based on observations with sea level data over their domain. from 50 (ϭL) tide gauge sites, whose distribution is Figure 4b shows that the sea level amplitude ®eld for shown on Fig. 1a; D is the rms difference over a tidal

K1 is about 40% of M2 in the deep ocean with relatively cycle between model and observations, given by little ampli®cation in shallow regions. The amplitude 1/2 and phase distribution for K1 also agrees with earlier 1 Dϭ(A22ϩA)ϪAAcos(␾ Ϫ ␾ ) , (3) studies, even with respect to small-scale maxima and []2om omom minima over the shelf. Some discrepancies with respect to the distribution of the phase are evident over the deep where A and ␾ are amplitudes and phases for a given ocean regions where our results are in closer agreement constituent, and subscripts m and o refer to model and MAY 1997 CUMMINS AND OEY 769

TABLE 2. Average absolute and relative rms differences between observed and modeled sea surface elevation from experiment 14v for four tidal constituents. Corresponding values for experiment 14u are given in parentheses. Absolute error Relative error Constituent (cm) (%)

M2 4.1 (4.1) 3.1 (3.1)

K1 2.3 (1.8) 4.9 (3.9)

S2 1.7 (1.8) 4.3 (4.7)

O1 1.7 (1.5) 5.9 (5.0) observations, respectively. The 50 gauge sites are a sub- set of the 63 listed in Table 2 of FHWB93. As they are mainly distributed around the shelf region, where the greatest variations occur, these sites provide adequate coverage of elevation ®eld. For a given tide gauge, the closest corresponding model grid point was chosen for comparison. Of the 13 gauges omitted from consideration, 12 were situated at locations that were unresolved by the 5-km grid, usually because they were at the head of narrow inlets. One site (Alert Bay) was omitted because it was close to the closed-off entrance to Johnstone Strait (Fig. 1a), where realistic results are neither expected nor obtained. The average rms elevation errors from the comparison are given in Table 2 for the two basic cases, 14v and 14u. The average relative error is less than 5% for each constituent except O1, which is slightly higher. As expected, the two experiments have similar differences with observations. The largest absolute errors (e.g., D

ഠ 24 cm for M2) occur near the arti®cially closed entrance to Johnson Strait. However, the region of large error is locally con®ned and errors for the nearest available stations around Queen Charlotte Strait are generally smaller than the mean values of Table 2. Based on these results, we conclude that the barotropic tide is reason- ably well represented in the model. b. Tidal current ellipses FIG.5.M2 barotropic current ellipses from experiment 14v at every Tidal current ellipses for the depth-averaged velocity third grid point. Thin (thick) line ellipses indicate clockwise (coun- terclockwise) rotation and the line from the center gives the direction are presented in Fig. 5 for the M2 constituent. In deep regions, tidal currents are small, O(2 cm sϪ1), and it is of the current vector at the time of maximum tidal potential. only over the shelf that the ellipses assume a signi®cant magnitude. Large clockwise-rotating ellipses with rel- tably weaker in amplitude, especially in Dixon Entrance. atively large eccentricity (ratio of semiminor to semi- The largest K1 currents are found off Cape St. James, major axis) are found over the three shallow banks in over the shallow banks in Queen Charlotte Sound and Queen Charlotte Sound. Particularly large ellipses also on the Alaskan shelf. occur in the vicinity of Cape St. James. In Hecate Strait With homogeneous water, the tidal ellipses show little and especially in Dixon Entrance, the channel bound- depth dependence above the bottom Ekman layer. Figure aries constrain the motions to be nearly rectilinear, 6 compares the vertical variation of M2 and K1 tidal aligned along the axes of these channels. Near Rose ellipses from experiment 14v with observations at sta- Spit, the transition point between the two channels, there tion W02 located in central Hecate Strait (see Fig. 1a). is a rapid rotation in the orientation of the ellipses. This comparison, and others from Hecate Strait, show

FHWB93 show M2 tidal ellipses for northern Hecate that a homogeneous motel captures the essential aspects Strait, which are in good agreement with Fig. 5. In of M2 and K1 tidal currents in this region. Except in the comparison to M2, ellipses for K1 (not shown) are no- bottom boundary layer, tidal ellipses at W02 in the strat- 770 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

¯ux located south of Cape St. James and along the Alas- kan shelf. F87 suggested that similar features appearing at these locations in his solution were associated with a diurnal shelf wave component.

For the M2 constituent, barotropic energy leaks onto the shelf in a region of O (40 km) width south of Cape St. James. A fraction of this energy continues northward through Hecate Strait and Dixon Entrance, eventually rejoining the deep water ¯ux. A second branch recir- culates southward over the shelf, crossing over Goose Island Bank and Cook Bank, where it then rejoins the

deep water ¯ux. There are signi®cant M2 energy ¯uxes across the shelf break at three distinct locations: along the northern shelf region off Dixon Entrance, south of Cape St. James, and farther south off Cook Bank. A gap, where relatively little energy ¯ux crosses the shelf break, exists between the latter two locations.

The M2 energy ¯ux was integrated across various transects to infer the rate of energy loss in the barotropic tide over the shelf. The mean integrated ¯ux of barotropic energy into an enclosed region gives the net rate FIG. 6. Comparison of model M2 and K1 current ellipses from ex- of energy loss to all dissipation mechanisms, of which periment 14v with observations at the W02 site in central Hecate bottom friction is the most signi®cant. In strati®ed ex- Strait. To place model results and observations in a common coordinate system, a rotation of 30Њ has been applied, aligning the (x, y) periments, conversion to baroclinic energy represents components of model currents in the (north±south, east±west) direc- an additional potential sink for barotropic energy. Strat- tions. The arrow head on each ellipse indicates the sense of rotation i®cation effects may also modify bottom currents and of the current vector. hence the dissipation rate associated with bottom friction. Figure 8 illustrates three subregions of the model and i®ed experiments (not shown) are virtually identical to gives the net ¯uxes across the bounding transects for those of Fig. 6, suggesting that internal tidal motions experiments 14v and 14u. Barotropic losses within a are absent at this site. BFCJ96 compared model and subregion are the residual obtained from summing ¯ux- observed ellipses for four sites, including W02. At each es across the bounding transects. These values are given of these sites, their results are in close agreement with in Table 3 for the two experiments. In the unstrati®ed tidal ellipses computed from our experiment with ho- case, 14v, a net ¯ux of about 4 GW onto the southern mogeneous ¯uid. However, it is only at the W02 site shelf from the deep water is largely balanced by the that good agreement with observations is obtained. At ¯ux of 3.5 GW into Hecate Strait (1 GW ϭ 109 watts). the other sites, located in Queen Charlotte Sound, Dixon Hence there is a mean dissipation rate of about 0.5 GW Entrance, and on the west side of the Queen Charlotte over the Queen Charlotte Sound shelf. Most of the dis- Islands, observations show signi®cant vertical structure sipation, about 1.2 GW, occurs in Hecate Strait/Dixon that the model of BFCJ96 and our homogeneous ¯uid Entrance where relatively shallow water is found. model cannot capture. In the strati®ed case (14u), the northward ¯ux into Hecate Strait and out of Dixon Entrance is similar to the homogeneous case and the rate of energy loss in c. Energy ¯uxes this region is essentially unchanged. However, the net Propagation of the barotropic tide may be examined ¯ux onto the Queen Charlotte Sound increases to 4.9 from consideration of depth-integrated energy ¯uxes as- GW, implying an additional loss of about 1 GW over sociated with each tidal constituent. These ¯uxes are the southern shelf. Overall, losses in the M2 tide over determined from the harmonic analysis of the barotropic the shelf increase from 1.85 to 3.05 GW with intro- currents and the surface elevation ®eld according to a duction of the strati®cation. As discussed in the next method outlined in appendix A. The resulting energy section, baroclinic ¯ux calculations suggest that con- ¯ux vectors are shown in Fig. 7 for the principal semi- version to internal tidal energy accounts for a signi®cant diurnal and diurnal constituents. In both cases, the ®eld fraction of the additional barotropic losses. is dominated in the deep water by the northward propagation of energy associated with large-scale gyres of 5. Baroclinic response tidal energy in the Paci®c. Relatively little K1 energy enters Hecate Strait or Dixon Entrance. Near the shelf To examine the baroclinic response of the model, we break, the K1 constituent has circular gyres of energy consider results primarily from experiment 14u, the ba- MAY 1997 CUMMINS AND OEY 771

FIG. 7. Depth-integrated barotropic energy ¯ux vectors for the (a) M2 and (b) K1 constituents from experiment 14v. The dashed contour indicates the 1000-m isobath. Every third vector is shown for regions where the depth exceeds 500 m, while every second vector is shown for shallower regions. sic strati®ed case. As indicated above, the barotropic tidal in¯uence is greatest over the outer shelf of Queen response is nearly (but not completely) unchanged in Charlotte Sound and on either side of Learmonth Bank this run from the unstrati®ed case, 14v. However, a vig- in Dixon Entrance. There is also a signi®cant offshore orous internal tide is generated in 14u that affects tidal in¯uence emanating from the shelf break of these two currents over much of the domain. regions. In contrast, the internal tidal signal is relatively The regions of internal tidal in¯uence are delineated weak for the inner portions of the shelf and in Hecate in Fig. 9, which shows contours of the rms M2 baroclinic Strait. In addition, a region of comparatively weak in- current on a near-surface velocity sigma level. Here the ternal tides is indicated in deep water off the west coast baroclinic current is de®ned as the difference between of the Queen Charlotte Islands. the total and the depth-averaged current, as in Eq. (A5). In the deep water (H Ͼ 2500 m) beyond the shelf,

The intensity of the near-surface baroclinic component rms M2 currents from experiment 14u near the surface of the tidal current gives an indication of the strength (␴ ϭϪ0.0105) have an average value of 6.5 cm sϪ1 of the internal tide. (The comparable ®eld from 14v is with a standard deviation of 3.3 cm sϪ1. This compares close to zero everywhere; with homogeneous water, with an average of 2.1 cm sϪ1 (standard deviation 0.3 Ϫ1 near-surface and depth-averaged currents are nearly cm s ) for the rms depth-averaged M2 current. Near- identical.) The results of Fig. 9 indicate that the internal surface currents over deep regions are thus generally 772 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

TABLE 3. Inferred rates of energy loss of the M2 barotropic tide over subregions of the shelf for two experiments. The subregions are bounded by the transects indicated in Fig. 8. Values are in units of GW (109 watts). Subregion Expt 14v Expt 14u Queen Charlotte Sound 0.49 1.52 Hecate Strait/Dixon Entrance 1.21 1.20 Northern Shelf 0.15 0.33 Total 1.85 3.05

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 harmonics

of the form wncos(␻nt Ϫ ␭n) where wn and ␭n are the amplitude and phase at frequency ␻n. Since w ϭ d␨/dt, where ␨ is the vertical displacement, the amplitude and

phase of the latter is given by wn/␻n and (␭n ϩ 90Њ), respectively. Figure 10a presents contours of the vertical displace-

ment amplitude at M2 frequency for a representative x±

FIG. 8. Three subregions of the shelf nominally identi®ed as (1) Queen Charlotte Sound, (2) Hecate Strait/Dixon Entrance, and (3) northern shelf. Sectionally integrated barotropic M2 energy ¯uxes (in GW) for experiments 14v and 14u (in parentheses) across transects bounding these subregions are given. Arrows indicate the sense of the ¯uxes. dominated by the baroclinic component. These results are 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 (50ЊN, 145ЊW) and propagating eastward toward British Co- lumbia. Their average rms semidiurnal currents were practically the same for both 15-m and 120-m drogued drifters (5.9 cm sϪ1, with a standard deviation of 2.1 cm sϪ1). The minimum observed rms semidiurnal values (2.9 cm sϪ1) likely correspond to the barotropic currents.

a. Internal motions FIG.9.RmsM2 baroclinic current from experiment 14u on a near- surface velocity sigma level (␴ ϭϪ0.0105). The ®eld is shaded at The baroclinic response of the model is characterized the (8, 16, 24) cm sϪ1 levels. Dashed contours are for the 1000-m by internal vertical displacements and baroclinic cur- and 200-m isobaths. MAY 1997 CUMMINS AND OEY 773

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

z cross section that intersects the continental margin at K1 frequency, signi®cant (Ͼ15 m) displacements are Dixon Entrance. Semidiurnal M2 amplitudes in excess manifest over the continental slope. However, at the of 40 m are obtained over the steepest parts of the con- latitude of the B.C. coast, diurnal frequencies are smaller tinental slope. These large values result from the direct than the inertial frequency (␻ Ͻ f). In this frequency forcing of vertical motions over the slope by the baro- range Eq. (1) ceases to be hyperbolic, and free internal tropic tide and indicate generation regions for the in- wave propagation is no longer possible. As a result K1 ternal tide. Here the topographic slope is steeper than motions are trapped over the slope and there is no ra- the slope of characteristics (␣/tanc Ͼ 1) and propagation diation of energy offshore. A similar trapping occurs is directed toward deep water. In the deep regions, dis- with the O1 ®eld (not shown). placement amplitudes in the upper 500 m range from 1 A cross-sectional view of the rms M2 baroclinic cur- to 3 m near the surface to a maximum of about 10 m rent is given in Fig. 11. In deep water the vertical struc- at mid depth. These near-surface values are comparable ture resembles that of a baroclinic internal wave mode. to semidiurnal displacements of 1±4 observed in the There are maxima near the surface with rms values upper 200 m at station P (Thomson and Tabata 1982). reaching 5±10 cm sϪ1, minima at approximately mid- Ϫ1 Displacement amplitudes for S2 (not shown) are similar depth, and maxima of 1±2 cm s at abyssal depths. in pattern to the M2 ®eld but smaller by a factor of ϳ3 along this section. Contours of vertical displacement amplitude along the b. Energy ¯uxes cross section for K are presented in Fig. 10b. At the 1 Baroclinic energy ¯uxes can provide insight into generation and propagation of the internal tide. Three different views of this energy ¯ux are presented in Figs. 12a±c. Figure 12a shows, for the entire model domain, the depth-integrated, baroclinic energy ¯ux vectors at

the M2 frequency. (See appendix A for details.) Close-up views for northern and southern subregions of the shelf

are included as Figs. 12b and 12c. Energy ¯uxes for S2 (not shown) are similar in pattern to those for M2 but with amplitudes reduced by a factor of ϳ6. The baroclinic ¯uxes are generally much smaller than the cor-

responding barotropic ¯uxes. For example, the M2 energy ¯ux represented by the scale vector in Fig. 12a is 1% of that in Fig. 7a. The vector ®eld of Fig. 12a is dominated by an offshore ¯ux of baroclinic energy originating from the slope region near the 1000-m contour. The ®eld is highly inhomogeneous with offshore ¯uxes generated at three FIG. 11. M2 rms baroclinic current from a cross-shelf section along y ϭ 620 km (see Fig. 2). Contours are drawn at the (1, 2, 5, 10) cm principal locations. Along the shelf edge between Cape sϪ1 levels. The vertical arrow at the top indicates the position of St. James and Vancouver Island, two generation regions mooring QE6 along this section. are obtained, separated by a length of shelf where the 774 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

FIG. 12. (a) Depth-integrated M2 baroclinic energy ¯ux vectors over the entire model domain plotted at every second grid point. Similar ®elds are presented for (b) Queen Charlotte Sound and (c) Dixon Entrance, with vectors shown at each grid point where the depth is shallower than 300 m. The scale vector in (a) differs from those of (b) and (c). Dashed contours indicate the 1000-m isobath in (a), and the 300-m and 100-m isobaths in (b) and (c). Locations of several moorings mentioned in section 5c are indicated. MAY 1997 CUMMINS AND OEY 775

2 Ϫ2 TABLE 4. Net offshore baroclinic energy ¯ux from the shelf region TABLE 5. Partition of the M2 kinetic energy (cm s ) among the for M2 and S2 constituents in different experiments. Units are giga- barotropic (BT) and baroclinic (BC) modes from experiment 14u at watts (109 W). various moorings.

Experiment M2 S2 1st BC 2nd BC Higher Mooring BT mode mode mode BC modes 14u 0.56 0.08 14s 0.49 0.06 W02 222.2 7.5 2.7 6.9 RT1 0.61 Ð G04 67.3 59.1 9.6 3.0 RT2 0.61 Ð Q10 66.0 86.4 11.0 7.4 RT3 0.62 Ð QA3 35.0 63.9 6.1 2.9 QB2 11.8 8.3 0.2 1.9 QE6 8.6 7.1 0.2 2.2 QF1 139.6 24.4 2.4 4.0 offshore ¯ux is comparatively weak. The southernmost QF2 395.0 24.6 4.4 19.6 QF4 6.9 7.5 0.4 13.2 of these two source regions lies off Cook Bank. Farther B02 2.4 1.3 0.1 0.5 north, there is a stretch of the shelf break 150 km long B03 1.9 0.1 0.3 0.3 off Dixon Entrance and Alaska where a large offshore ¯ux of baroclinic energy is generated. Also evident is a region off the west side of the Queen Charlotte Islands The integrated baroclinic ¯ux is reduce somewhat from where the offshore ¯ux is relatively weak. It is this 14u in the winter strati®cation case (14s), but the basic ``shadow zone'' that accounts for the relatively weak pattern is essentially unchanged. Offshore ¯uxes in the near-surface signal obtained off the Queen Charlottes three resolution test cases (RT1, RT2, and RT3) show in Fig. 9. In the far ®eld, offshore ¯uxes spread geo- mutual agreement and differ only slightly from 14u, metrically from source regions at the slope. con®rming that energy ¯uxes are relatively insensitive Generation sites for baroclinic energy in the deep to variations in horizontal or vertical resolution. water correspond precisely to the three regions men- Of the net offshore baroclinic ¯ux of 0.56 GW in the tioned in section 4c where signi®cant ¯uxes of baro- basic case (14u), 0.17 GW emanates from the Dixon tropic energy cross the shelf break. In these regions the Entrance/Alaskan shelf region, while 0.34 GW origi- cross-slope component of the barotropic M2 tidal ellip- nates from Queen Charlotte Sound. The values represent ses (Fig. 7a) is largest and strati®ed ¯uid is forced most the baroclinic energy radiating offshore but do not in- vigorously up and down the continental slope. In terms clude ¯uxes radiating shoreward from the shelf break of the two-dimensional model [Eq. (1)], these are or local ¯uxes over the shelf or slope. It was noted in regions where the cross-slope volume ¯ux Q is maxi- section 4 that barotropic losses over the northern shelf mum. On the west side of the Queen Charlottes a dif- region were augmented by 0.18 GW and over the south- ferent situation prevails; cross-slope motions there are ern shelf region by 1.03 GW. Hence, baroclinic energy weak due to narrowness of the shelf and presence of conversion readily accounts for the additional barotropic the land mass. losses over the northern shelf region and a signi®cant fraction of the additional losses over Queen Charlotte An inhomogeneous pattern of baroclinic M2 energy generation and propagation over the shelf is evident in Sound. Bottom friction over shelves is the main dissi- Figs. 12b and 12c. South of Cape St. James, onshore pation mechanism for the barotropic tide and additional ¯uxes into Queen Charlotte Sound are obtained, which losses with strati®cation may be partly attributable to originate irregularly from near the 300-m depth contour enhancement of bottom dissipation. Rms currents above over a distance extending for about 100 km (Fig. 12b). the bottom are signi®cantly increased due to the internal Near Cape St. James, some of this baroclinic energy tide in some regions over shelf in Queen Charlotte leaks into the southwestern reaches of Hecate Strait. Sound and over the adjoining slope. There is also a broad ¯ux of baroclinic energy radiating The offshore M2 baroclinic energy ¯ux of about half off the southern ¯ank of Goose Island Bank, near the a gigawatt obtained here is signi®cantly larger than an 100-m contour, toward northern Vancouver Island. The earlier estimate made by Baines (1982) with a two- pattern of energy ¯uxes in the region of Dixon Entrance dimensional analytical model. Values for the northeast (Fig. 12c) is particularly inhomogeneous. It is possible Paci®c were not explicitly given but were implied to be to discern both onshore and offshore ¯uxes off the sides less than 0.1 GW. Reasons for the discrepancy are not clear. However, our results show localized sources and of Learmonth Bank. The results also show ¯uxes di- the inhomogeneous nature of baroclinic energy gener- rected in a cross-strait sense from the shoaling topog- ation, features that may be dif®cult to account for with raphy near the northern Queen Charlotte Islands. a two-dimensional model. Since a large fraction of the baroclinic energy is direct offshore, an estimate of the net baroclinic energy generation may be obtained by integrating the ¯ux normal c. Comparison of M2 current ellipses with to transects forming a box around the shelf region. The observations result of such a calculation for the M2 and S2 constituents As a means of verifying the model, comparisons be- is given in Table 4 for different strati®ed experiments. tween observed M2 tidal ellipses and results from ex- 776 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

periments 14v and 14u are presented. Current meter al. 1992) experiments. At QE6, the only existing de- moorings from regions where an internal tidal compo- ployment extends from October 1983 through June nent is indicated are considered. The discussion is lim- 1984. ited to M2, the leading semidiurnal constituent, however, For each model station the kinetic energy has been ellipses for S2 are similarly affected by the strati®cation. partitioned into contributions from barotropic and baro- Except for station QE6, the observations are drawn from clinic modes, according to a procedure outlined in ap- summer deployments (nominally mid-May to Sep) made pendix B. The modes are derived from a ¯at-bottom during the late 1970s and early 1980s as part of the theory, which is strictly inapplicable over variable to- CODE (Freeland et al. 1984) and NCODE (Huggett et pography. They are, however, still useful because they MAY 1997 CUMMINS AND OEY 777

FIG. 13. Comparison of model M2 tidal ellipses (rotated 30Њ,asin Fig. 6) with observations from moorings at various sites over the continental shelf and slope (see Figs. 1a and 12a±c). The ellipses are for the total (barotropic ϩ internal) current. Model results from experiment 14v (unstrati®ed) are on the left side of each panel, those of 14u (strati®ed) on the right, with observations in the center. The model results are sampled at the following velocity sigma levels (Ϫ0.0015, Ϫ0.1475, Ϫ0.25, Ϫ0.45, Ϫ0.65, Ϫ0.85, Ϫ0.975).

form an orthogonal set of basis functions that provides component. At other sites considered, the baroclinic an ef®cient decomposition of the model currents in sev- contribution is more signi®cant and usually dominated eral instances. by the ®rst mode.

Results of the modal partition applied to experiment M2 currents at two moorings situated in Queen Char- 14u are provided in Table 5, where contributions from lotte Sound, G04 and Q10, are shown in Figs. 13a and modes 3 and higher have been grouped together. At W02 13b. Station G04 is located on the southern ¯ank of in northern Hecate Strait, M2 currents from experiment Goose Island Bank, in a region subject to a southward 14u are similar to those of 14v (shown in Fig. 6) and, radiating ¯ux of baroclinic energy (see Fig 12b). Ellip- as expected, completely dominated by the barotropic ses from 14v are nearly identical to those presented by 778 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

BFCJ96 for this site and represent the barotropic re- both observations and our model. The authors noted that sponse. In the strati®ed case, the baroclinic and baro- complex local topography and the number of possible tropic modes have comparable kinetic energy levels. generation sites compound the dif®culties of applying Model ellipses capture the observed clockwise rotation ray tracing methods in this region. of phase with depth, a structure expected from super- position of barotropic and ®rst baroclinic mode com- 6. Conclusions ponents (Farmer and Freeland 1983, p. 171). Station Q10, located in the trough between Middle Bank and This study is concerned with simulation of tides over Goose Island Bank, is exposed to a baroclinic ¯ux ra- the northern coastal waters of British Columbia with a diating shoreward from the shelf break (Fig. 12b). In three-dimensional primitive equation model. Two basic the strati®ed case, the vertical structure, dominated by con®gurations of the model have been considered, one the ®rst baroclinic and barotropic modes, reproduces with homogeneous water and the second with a hori- relatively accurately the observed M2 ellipses. zontally homogeneous, vertically strati®ed ¯uid, rep- Figures 13c and 13d compare current ellipses at two resentative of summer conditions. Tidal forcing is in- sites in proximity of Cape St. James. Station QB2 is troduced along open ocean boundaries with four leading over the continental slope on the west side of the Queen constituents of the barotropic tide derived from a global Charlottes, lying in the path of baroclinic ¯uxes radi- tidal atlas. Sensitivity to variations in model resolution ating northwestward along the slope from a source re- and to a representative winter strati®cation have been gion off the Cape (Fig. 12a). Here the response of the explored. strati®ed model leads to near-surface ellipses that are The barotropic response of the model is similar to considerably larger than the unstrati®ed response, con- previous tidal simulations of the region, and compari- sistent with observations. At station QA3, near the shelf sons with tide gauge data indicate a relatively accurate break, observed ellipses have a clockwise rotation in representation of surface elevation ®elds. Tidal currents orientation and phase with depth, which is captured, in the unstrati®ed case also agree closely with the earlier albeit with some differences, in experiment 14u. studies, but capture little of the vertical structure ob- Results from a site near Dixon Entrance are presented served with the semidiurnal constituents in many in Fig. 13e. Station QF2 is located near the mouth of regions. the channel, at a point in the path of southward baro- With a strati®ed medium, an internal tide is estab- clinic ¯uxes emanating from Learmonth Bank (Fig. lished which has a signi®cant in¯uence on tidal currents.

12c). The model ellipses in both cases are dominated The rms magnitude of the M2 baroclinic current near by the nearly rectilinear, barotropic component. How- the surface delineates those regions most strongly af- ever, due to a stronger clockwise component, near-sur- fected by the internal tide. Most notably, these are face ellipses from 14u have an increased eccentricity, Queen Charlotte Sound and Dixon Entrance, where the consistent with observations. Comparisons with obser- shelf is broad and exposed to the open ocean. In Hecate vations at station QF1 (not shown), located on the op- Strait, which is sheltered from the open Paci®c by the posite side of Learmonth Bank (Fig. 12c), yield similar Queen Charlotte Islands, the internal tide has only lim- results. Generally, the eccentricity of near-surface ellip- ited in¯uence. Semidiurnal M2 tidal ellipses were com- ses throughout much of Dixon Entrance is increased by pared with observations drawn from a series of moor- the presence of the internal tide. ings situated in regions where an internal tide is indi- Finally, we show results in Figs. 13f and 13g for two cated. The vertical structure of the model ellipses is open ocean stations, QE6 located in 1100 m of water generally consistent with the observations from a num- off Dixon Entrance and B03, about 80 km southwest of ber of widely separated shelf and slope sites. These Brooks Peninsula (Fig. 1a), in 2300 m of water. Results comparisons suggest that, despite an idealized uniform from QF4 and B02 (not presented) are qualitatively sim- strati®cation, the essential characteristics of the internal ilar to QE6 and B03, respectively. At QE6 (which lies tide are retained by the model. Further improvements along the x±z section described in section 5a), the off- may require a more accurate initialization, taking ac- shore-directed internal tidal energy (Fig. 12a) produces count of inhomogeneities in the ambient strati®cation. a large clockwise-rotating ellipse at the depth of the Baroclinic energy ¯uxes computed from the model near-surface current meter. By comparison the unstrat- indicate that semidiurnal baroclinic tides radiate off- i®ed response is characterized by relatively weak coun- shore from localized sources along the continental slope. terclockwise motion. Similar comments apply to the At these generation sites the barotropic tidal ellipses B03 site, where the internal tide of the model affects have a pronounced cross-slope component. The inte- near-surface motions, consistent with observations. grated offshore ¯ux is about 0.5 GW, which gives an Drakopoulos and Marsden (1993) used a ray tracing average ¯ux of about 625 W mϪ1 with a model coastline approach to determine the pattern of characteristics for of 800 km. The analytical, two-dimensional model of the B02 and B03 stations. Their results indicated that a Baines (1982) provides (to our knowledge) the only beam of internal energy should be detected at a depth estimate of baroclinic energy generation available for of 1000 m for the B03 site, which is inconsistent with comparison. The magnitude of the offshore ¯ux ob- MAY 1997 CUMMINS AND OEY 779 tained here exceeds this earlier estimate by a factor of where the angle brackets denote a time-average over at least 5. The discrepancy may be due to dif®culties period T, inherent in application of a two-dimensional model to 1 T a region with localized sources of baroclinic energy. ͗(´)͘ϭ (´) dt. Satellite altimetry has permitted a precise determi- T ͵ 0 nation of the global rate of tidal energy dissipation for the largest constituents (Cartwright and Ray 1991). We consider the contributions to the mean horizontal However, the long standing problem of accounting for energy ¯ux density due to the rate of working by pres- these losses in the ocean remains open. In particular, sure forces associated with barotropic (subscript bt) and the signi®cance of losses to internal tides is uncertain baroclinic (subscript bc) tidal motions, and disparate estimates have been made. For example, M ϭ͗u p ͘ Baines (1982) calculated that internal tide generation bt bt bt along the continental margins accounts for a negligible Mbcϭ͗u bcp bc͘. (A3) fraction (0.3%) of the M2 dissipation rate. Applying the same analytic model to midocean ridges, Morozov Additional contributions to the ¯ux from advection and (1995) estimated that baroclinic conversion could rep- diffusion are neglected as these are expected to be of resent 25% of the global dissipation. The results from second-order for a small amplitude wave ®eld. For these this study suggest that incorporation of strati®cation ef- calculations, the horizontal velocity ®eld uh is decom- fects in three-dimensional models may be necessary to posed into a depth-averaged barotropic component simulate in a realistic fashion the dissipation of semi- 1 ␩ diurnal constituents over continental shelves. For our u ϭ u dz, (A4) bt H ͵ h coastal region we ®nd that strati®cation leads to sig- ϪH ni®cant additional barotropic losses as a consequence and an internal baroclinic component of the generation of internal tides.

ubc ϭ (uh Ϫ ubt). (A5) Acknowledgments. We thank Mike Foreman for help- ful discussions on tidal modelling. We have also ben- Here ubc is interpreted as current associated with the e®tted from advice provided by Bill Crawford, Howard baroclinic tide although it may be contaminated by vis- Freeland, and Rick Thomson. Contributions to software cous effects in the bottom boundary layer (see com- development were made by Jane Eert, Juijing Gu, and ments below). Dave Ramsden. This project was partially supported by The barotropic pressure ®eld pbt is associated with the Canadian Panel of Energy Research and Develop- displacements of the sea surface, ␩, ment, Project 67146. We acknowledge support from the p ϭ ␳ g(␩ Ϫ z). (A6) Pittsburgh Supercomputing Center. bt 0 A method similar to that of Holloway (1996) has been APPENDIX A adopted for calculation of the baroclinic ¯uxes. Pressure perturbations associated with internal tidal motions are zp ϭץ Calculation of Energy Fluxes assumed to be governed by hydrostatic balance ␳ ϭ ץ Ϫ␳g and a linearized adiabatic density equation The equation governing the rate of change of energy t ␳. The baroclinic pressure ®eld is then obtained ץϪw of an incompressible ¯uid is (Gill 1982; section 4.7) z from the relation Eץ ␳ץ p 0ץ (F ϭ G ϩ D, (A1´١ T ϩ (t bc ϭϪgwdz*, (A7ץ *zץ ͵ tץ z 2 where ET ϭ ␳[(1/2)ͦuͦ ϩ gz ϩ E] is the sum of kinetic, potential, and internal energy, with u the three-dimen- where w is the vertical velocity. A barotropic compo- ١H) with ␴ ϭ (z´sional velocity vector; G and D represent forcing and nent, wbt ϭ (1 ϩ ␴) D␩/Dt ϩ ␴ (ubt dissipative effects, respectively. The energy ¯ux density, Ϫ ␩)/(H ϩ ␩), may be removed from w, but this has been veri®ed to make little difference to the baroclinic F ϭ up ϩ uET ϩ ´´´ , ¯uxes. has contributions from the work done by pressure forces, To evaluate time averages it is useful to expand the advection of energy, and additional terms involving vis- velocity and pressure ®elds in a series of harmonics. For example, consider a rotation through an angle to cous ¯uxes and other nonconservative processes. Inte- ␪n gration of (A1) over a ®xed control volume and time de®ne coordinates (xЈ, yЈ), such that the xЈ axis is ori- averaging yields a balance between mean ¯uxes normal ented in the direction of the positive semimajor axis of to the volume surface and mean forcing and dissipation, the local tidal ellipse. Then barotropic and baroclinic velocities and the perturbation component of the pres- ∫∫ ͗F´n͘ dS ϭ ∫∫∫ ͗G ϩ D͘ dV, (A2) sure ®elds for each constituent are of the form 780 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

uЈϭuЈnnncos(␻ t Ϫ g ) current of the strati®ed case (14u). The recalculated depth-integrated ¯uxes were visually indistinguishable vЈϭ␷Јnnsin(␻ t Ϫ g n) from those given in Figs. 12a±c. The net integrated ¯ux pϭpcos(␻ t Ϫ ␾ ). (A8) from the shelf into the deep ocean was virtually un- nn n changed. It seems reasonable then to assume that there Here (uЈnn ,␷Ј ) are current amplitudes at frequency ␻n in is no signi®cant contamination of the energy ¯uxes due the (xЈ, yЈ) directions respectively; gn is the phase lag to viscous effects. of maximum current behind maximum tidal potential; pn and ␾n are the pressure amplitude and phase lag re- APPENDIX B spectively; and ␪n is the orientation of the semimajor axis with respect to (x, y) coordinates. Using the ex- Modal Decomposition pansion (A8), it is possible to express components of n The following procedure was applied to partition the the rotated mean energy ¯ux vector MxЈ ϭ͗uЈp͘n as kinetic energy into barotropic and baroclinic modes at 1 n the constituent frequencies of the tide. For each station M xЈ ϭ͗uЈp͘ϭnnnnnuЈpcos(␾ Ϫ g ) 2 the horizontal currents are expanded as

n 1 KϪ1 Mϭ͗␷Јp͘ϭ␷Јpsin(␾ Ϫ g ). (A9) (u, ␷) ϭ ⌺kϭ0 (⌸k(z)(uÃ k(t), ␷Ãk(t)), (B1) yЈ nnnnn2 where k is the mode index and K the number of vertical A rotation must be applied to recover the energy ¯uxes grid points. The ⌸k(z) are eigensolutions of the Sturm± in the (x, y) coordinate of the model: Liouville problem,

nn Ϫ2 Ϫ2 Mxnnxcos␪ Ϫsin␪ M Ј (N ⌸Јkk)Јϩ␭ ⌸k ϭ0, (B2) nnϭ . (A10) ΂΃Mynny ΂sin␪ cos␪ ΃΂΃M Ј with boundary conditions ⌸Јk ϭ 0atzϭ0, ϪH. Equa- Terms involving products of velocities and pressures at tion (B2) is appropriate to internal wave motions over 2 2 different frequencies have been omitted because these a ¯at bottom in the hydrostatic (␻ K N ), Boussinesq tend to zero as the averaging period T increases. Thus, limit (LeBlond and Mysak 1978); N is the buoyancy 2 provided T is large compared to the constituent periods frequency and␭k the eigenvalue. and that conditions are stationary, the total mean energy The modes are normalized according to the condition ¯ux is the sum of ¯uxes at each frequency M ϭ 1 0 n ⌺ M . Finally, the depth-integrated energy ¯ux vectors ⌸⌸kjdz ϭ ␦ k,j, (B3) n x H͵ presented in sections 4 and 5 are given by the following ϪH expressions, respectively: where ␦k,j is the Kronecker delta. Modal currents (uÃ k,

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