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 re- view.) 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 re- sponse. 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- q1997 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) c, propagation. In the next section, a brief review of the regional Q 1 c 2 tan2cc 5 z , (1) oceanography and geography is provided. This is fol- xx zz 22 (1 2 v /N ) 12H 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 v 5 2 rz )1/2 is the buoyancy frequency with the density, model, while the baroclinic response is described in r0 r r0 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 (v . f ), (1) is hyperbolic with propa- gation along rays whose slope is given by 1/2 2. Regional oceanography v222f tanc 5 , (2) 22 A thorough review of the physical oceanography of 12N2v 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 a is supercritical if a/tanc . 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 a/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 fre- quency 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.