James, Middle Bank, Goose Island Bank, and the Scott. Islands. ... an idealized Cape St. James. ..... only hope to represent such features if their scale is larger.
JOURNAL
OF GEOPHYSICAL
A Finite
RESEARCH,
Element
VOL. 98, NO. C2, PAGES 2509-2531, FEBRUARY
Model
15, 1993
for Tides and Resonance
Along the North Coast of British Columbia M. G. G. FOREMAN
AND R. F. HENRY
Institute of Ocean Sciences, Sidney, British Columbia, Canada
R. A. WALTERS U.S. Geological Survey, Tacoma, Washington
V. A. BALLANTYNE Institute of Ocean Sciences, Sidney, British Columbia, Canada A finite element, barotropic, tidal model is developed for the north coast of British Columbia.
The modelis run with eight tidal constituentsand the resultsare comparedwith the Flather (1987) finite difference model, and with extensive tide gauge and current meter observations. Although the tidal potential, Earth tide, and loading tide are included in the forcing, their inclusion is shown
to changethe largestM2 amplitudesby only 2.5% and the largest K1 amplitudes by lessthan 1%. Root mean square differencesbetween observed and calculated sea level amplitudes and phases are within 1.9 cm and 2.9ø for all but one constituent, but the model currents do not in general, compare as favourably. The barotropic currents observedin Hecate Strait are reproduced well, but elsewhere
evidence
is shown
that
model
inaccuracies
are due to baroclinic
effects.
Tidal
residual
currents calculated by the model suggest the existence of eddies off the tip of Cape St. James, Cape Chacon, and around Goose Island and Learmonth Banks. The shallow water constituents in Hecate Strait are shown to have significant contributions from the constructive interference of signals propagating into Dixon Entrance and Queen Charlotte Sound. Using the model, the longest resonant period of the system is estimated to be 7.6 hours with an energy dissipation parameter, Q, of 9.5.
1.
INTRODUCTION
and 2). These islandsare separatedfrom the mainland
This paper describesthe developmentand validation of a barotropic tidal model for the northern coastal waters of British Columbia. This work is part of a project
by three bodies of water: Dixon Entrance, Hecate Strait, and Queen Charlotte Sound. Whereas Dixon Entrance is approximately 150 km in length, 55-65 km in width and
has depthsas large as 400 m, the other two inland waters (henceforthPERD) to study surfacecurrentsin Queen are not so regular. Hecate Strait is very shallow (less Charlotte Sound,Hecate Strait, and Dixon Entrance (see than 30 m) on its western half and has only a narrow Figures1 and 2). Exploratoryoffshoredrillinghasindicated north-south trench with depths greater than 100 m on the possibilityof oil reservesin the region and this project its eastern side. Queen Charlotte Sound is characterized
funded by the Panel of Energy Researchand Development
aims to establish a basic understandingof surface flows in the event of an oil spill. In addition to the tidal model to be describedshortly, a wind-driven model has also been
developedfor the sameregion [Hannahet al., 1991], and
by three relatively deep troughs that separate Cape St. James, Middle Bank, Goose Island Bank, and the Scott Islands. Other notable features of the region are the highly irregular mainland coastline comprising numerous narrow passesand fiords, the narrow continental shelf off the west coast of the Queen Charlotte Islands, and the
a buoyancy-drivenmodel will follow. The P ERD project also comprisesseveral years of observationalwork which includes current meter and tide gauge deployments, ship- presenceof seamounts(e.g., Union and Bowie) beyond mounted acoustic Doppler current profiler measurements, the continental slope. Detailed descriptionsof the physical of the regionhavebeengivenby Crean[1967], CTD surveys,and the deploymentand trackingof drifters. oceanography Dodimead [1980], Thomson[1981],Freelandet al. [1984], Only the current meter and tide gauge observationswill Crawfordet al. [1988],and Crawfordand Thomson[1991]. be discussedhere as they are used in the validation of the There have been severalprevioustidal models for this retidal model. gion. Bell and Boston[1962,1963]developeda non-rotating The Queen Charlotte Islands are a prominent feature hydraulic model for Hecate Strait, Dixon Entrance, and of the north coast of British Columbia (see Figures i Queen Charlotte Sound that gave a reasonably successful This paper is not subject to U.S. copyright. Published in 1993 representationof the diurnal and semi-diurnaltidal heights. However, their M2 tidal currents had large phase shifts by the American Geophysical Union. relative
Paper number 92JC02470.
to observations
in Dixon
Entrance
and inaccurate
residual currents near the forcing boundaries. The first nu2509
FOaEMAN ET AL.' TIDESANDRESONANCE, NORTHCOAST OFBRITISH COLUMBIA
2510
126
130
138
57
Depth contours inmetres
• Learmonth Bank
oo
•)
50
Union --•
49 49
138
130
126
Fig.1. Geography anddepth contours (m)forthenorthcoast ofBritish Columbia andthesouthern coast of Alaska.
modelof tidally-rectifiedcurrentsaround mericalmodel,developed by Kinneyet al. [1976Jto predict three-dimensional an idealized Cape St. James. Their resultscompared oil spill movementsfor the Kitimat PipelineProject, was a M2 barotropicmodel with 9.3 km resolution. Perry et well with both current meter observations and satellite
Finally,Bowmanet al. [1992]useda model al. [1983]usedcurrentspredicted by this modelto study photographs. that simplified the geometry of DixonEntranceandHecate Strait in order to study tidally-rectified currents(the Rose Flather[1987](henceforth F87) developed a finite dif-
tidal frontsand the plankton distributionin HecateStrait.
ference model with 18 km resolution for the region north Spit Eddy) in Dixon Entrance. All the foregoing modelsusedfinitedifference techniques of 45ø and east of 136ø. He included only constituents
of the coastline M:• and Ki and obtained generally excellentagreement and a regulargrid. Giventhe complexity with the elevationamplitudesand phasesmeasuredat 63 and bathymetryand the scalesof variationin the tidal
to followthe example of tide gaugelocations.However,no verification of the tidal flows(e.g.,seeF87), we decided FW) and employ currentswaspresented.Thomsonand Wilson[1987]used Foremanand Walters[1990](henceforth for this region. As explainedin the results of Flather's model to force a finer resolution finite elementtechniques
FOREMANET AL.' TIDES AND RESONANCE, NORTH COASTOF BRITISH COLUMBIA 130
136
,
56
2511 126
,
I
i
56
ß Current Meter
ß TideGauge
ß DIXON ,
ß
ß
"•
ßß
ß
ß
ß
-•
"d•/-
ß
Rose
I[ sp.
•
) ß
HECATE
BRITISH
ß,',
COLUMBIA
•• ßQUEEN ß
CHARLOTTE
CapeSt. Jamesß
ß ß SOUND i
i
Scottoo Islands ß ß
ß
VANCOUVE 1 ISLAND
50--
5O
ß
49
'
'
'
'
'
I
136
I
•
i
130
49 126
Fig. 2. Place names,and tide gauge (square),and current meter (circle) observationsites for evaluatingthe performanceof the tidal model. Tide gaugesitesmarkedI (QC14) and 2 (Cape Ball) are referencedin the text. FW, a triangular grid permits more flexibility in reducing numerical errors and providing a better fit of an irregular coastline and bathymetry with elements of an arbitrary
size, shape, and orientation (within certain regularity constraints). In particular, this means that smaller
observations. In section 4, two shallow water constituents are examined
and the model is modified
results are summarized
elements can be placed in regions where the elevation or velocity field is changing rapidly and higher accuracy is desired.
This paper is organized as follows. In section 2, we briefly describe the numerical technique and the triangular grid. In section 3, tidal height and current results are presentedfor eight astronomicalconstituentsand compared with both F87, and extensive tide gauge and current meter
to find the resonant
frequency of the Queen Charlotte Sound, Hecate Strait, and Dixon Entrance system. Finally, in section 5, the
2.
and future
work is outlined.
NUMERICAL TECHNIQUE
Tidal heights and currents were calculated with a finite element
model
that
solves
the
two-dimensional
shallow
water equations:
oq--• + •7-[(H+ r/)u]-- 0
(1)
2512
FOREMAN ET AL.' TIDESANDRESONANCE, NOR• COASTov BRITISHCOLUMBIA culatedby Wahr [1981]and summarizedin Table 1. These ulul (2) values are recommended by the TOPEX/Poseidon Satellite +. _ AhV2U Tidal Committee(P. Woodworth,personalcommunication,
where (x,y) are eastwardand northwardspatialcoordi- 1992),and showthat the P1 and K1 h valuesare affected nates,t is time, r/(x,y, t) is surfaceelevation,u(x, y, t) is by the narrow "freecorenutation"(or "nearly-diurnalfree horizontalvelocity,H(x,y) is depth, f(y) is the Coriolis wobble")resonance at approximately 0.0419cycles/hour. parameter,g is gravity,• is the bottomfrictioncoefficient, The equilibrium tide for constituent n, namely •n, is Ah is the horizontalviscosityparameter,{(x,y,t) is the expressed as [Cartwright, 1978;Schwiderski, 1978] equilibriumtide, and a, •3 are parametersto accoun•for ½.= co?cos(.t + 2x+ v.) (4) the tidal potential,Earth tide, and loadingtide. As in FW, we set •- 0.003 thoughout this study. (An experiment with • = 0.004 scarcelychangedthe results from those reported herein.) The valuesfor a, •3, and Ah will be
for semi-diurnals
- Hn sin2•bcos(wnt+ X -t-Vn)
(5)
discussedshortly. for diurnals, where the amplitudesHn are given in Table The numerical technique is virtually the same harmonic 1; (•b,X) are the latitude and east longitude;•vn is the
wave equation method [Walters, 1986, 1987] as was frequency;and Vn is the astronomicalargument. In used by FW for their model of the southwestcoast of particular, the Vn values are the same as those in the Vancouver Island. The only differencesare the inclusion boundary forcing and are calculatedfor the arbitrary time of tidal potential forcing and sea bed deformationdue 0000 UT, August 1, 1984. to the Earth and loading tides, and a true variation of f with latitude.
Although the governing equations are
not expressedin sphericalcoordinates,the grid units are measuredwith respectto a Universal TransverseMercator
TABLE 1. Love Numbersand Tidal Potential Amplitudes
(UTM) projection[McDonnell,1979]. This projectionis widely usedfor mappingand military purposes.Although the longitudinalrange of our grid (from approximately
127øEto 139øE)is largerthan the 6ø width of standard UTM zones,choosingthe centralmeridianto be 133ø producesonly the smallrangeof 0.9996to 1.002074in the scalingfactor. Thus our grid distancesand elementareas are very closeto their true values. The eight tidal constituentsM2, S2, N2, K2, K1, O1,
P1, and Q1 are includedin the boundaryand astronomical forcing, howeverthe presenceof nonlinearterms in the governingequationsmeansthat a residualtide and several compoundtidesand overtideswill alsobe generatedwithin the model domain. At Prince Rupert (seeFigure 2), the eight forcingconstituentsaccountfor 89% of the tidal heightrange in the daily and half-daily frequencybands. Boundaryconditionsfor the tidal model are zero flow normal to the coast and specified elevations along the
Constituent
k
h
Hn
Q1
0.298
0.603
0.01940
O1 P1 K1 N2 M2 S2 K2
0.298 0.287 0.256 0.302 0.302 0.302 0.302
0.603 0.581 0.520 0.609 0.609 0.609 0.609
0.10128 0.04713 0.14246 0.04674 0.24408 0.11355 0.03090
In accordance with Pekerisand Accad[1969],the ocean loading tide (the deformationof the elastic earth due to the redistribution of mass in the ocean tide) is assumed to be in phase with the ocean tide and a fixed percentage of that value. Following the recommendationof Ray and Sanchez[1989] and the TOPEX/Poseidon Satellite Tidal Committee(P. Woodworth,personalcommunication, 1992), we set a- 0.940,0.953 for the diurnal and semidiurnal constituents, respectively. As the load tide is
open sea. Theseelevationswere computedpredominantly expected to have a relatively minor influence, it was from observations with the tidal harmonics from Bowie not felt that more accuratecalculations[e.g., Francisand and Union Seamounts(Figure 1) playing a critical role Mazzega,1990; Hendershott,1972]werenecessary. along the westernboundary. Values for the northern and The triangular grid for the l•nite element model was northwestern boundaries, where there are no observations, created using the Henry and Walters [1992] packageof wereinitiallyestimatedfromF87 andthe Schwiderski [1978] interactive computer programs and is shown in Figure 3. world tidal model, and then refined(via trial and error) It has 4893 nodes, 8004 elements, and triangle sides that in accordance with the ensuing model results in Dixon vary from approximately 35 km in the deep ocean to about Entrance. This boundary condition tuning may be viewed 1.2 km off Cape St. James. The grid generally representsa as a crude form of the inverse methods approach used by compromiseamongthe followingdesignrequirements:(1)
Bennettand Mcintosh[1982]. closefitting of the coastline,(2) near-equilateralelement In addition to boundary forcing, the effectsof the tide shape;(3) an elementsizethat decreases with rapid changes generatingpotential, the Earth tide, and ocean loading in the velocity and elevation fields.
tide are also included in the model. Assuming the Earth
In addition to a generally finer resolution, the model
to be elastic,the first two forcingterms are usually(e.g., used in this study has other features that should lead to F87, Schwiderski[1978])expressedas f]•, where more accurate results than were described in F87. They
f] -- 1 -t-k- h
(3)
are as follows.
1.
More
constituents
are included
in the
simulation.
This will not only give a more completepicture of the total (andassociated perturbations) to the potential.Although tide but will also improve the nonlinear representation the values k- 0.3 and h- 0.61 are often assumed for all of bottom friction and advection, as more constituent
and k, h are Lovenumbersrelatingthe body Earth tide
constituents, we usedthe frequencydependent valuescal-
interactions
will be included.
FOREMANET AL.: TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
2513
Fig. 3. Triangular grid for the finite element model. Numbered open boundaries are referenced in the text.
2. More accurateLove numbers(particularlyfor K1)
3.
MODEL
EVALUATIONS
FOR THE MAJOR
CONSTITUENTS
are used in the tidal potential and Earth tide calculations.
Also an oceanloading tide is now included as an additional forcing. 3. The specified boundary conditions have been im-
The model was evaluated by comparing its results with observations at the 63 tide gauge and 67 current meter sites shown in Figure 2. Actually, observations exist at proved. ObservationssinceF87 and a careful re-analysis more sites, but for reliability, only those of at least 50 of the Bowie Seamount record that corrected for a timdays duration were chosen. The set of tide gauge sites is ing problem have enabled more accurate values to be about double the number used for the same region in F87. specified along the offshore boundaries. Whereas Flather For tidal heights, the amplitudes and phases of the eight
obtainedhis boundaryconditionsfrom $chwiderski[1978], major constituentswere calculatedusing Foreman's[1977] the boundary conditionsused in this model were calculated mainly from observations.
harmonic analysis programs and compared at each site.
For tidal currents,the ellipseparameters(major semi-axis,
FOREMAN ET AL.: TIDESANDRESONANCE, NORTHCOASTOFBRITISHCOLUMBIA
2514
minor semi-axis, angle of inclination, and Greenwich phase
lag) were calculated[Foreman,1978] and comparedat
AlaskanPanhandlein Figure 4b arisebecausethe northern boundaryelevationsare not perfectly compatiblewith the
each site. Observedbarotropic currents were calculated by
northwardpropagatingdiurnal shelfwave. Flather (F87)
weightedaveragesat a minimumof two depths(unlessthe water depth was lessthan 100 m), with the weightsbeing
had similar problemsin the sameregion. Though co-amplitudeand co-phaseplots for the remaining six constituentsare not shown, constituents$2, N2, and K2 havecharacteristics similarto M2; whereasO1, P1, and Q1 are similar to K1. In particular, $2, N2, and K2 have amplitudesthat are approximately30, 20, and 8% of
proportional to the vertical separation between the current
meters. For both tidal elevations and currents, model values were calculated by interpolating from the nodes to the exact observation
location.
As discussedand illustrated in FW, it shouldbe pointed
the M2 amplitudes; and phase lags that are approximately
out that the observed harmonics are only estimates of the true tides. Thus discrepancies between the model
30ø, -25 ø, and 25ø greater. Analogously, the Ox, P•, and
results and the observations
are due to both uncertainties
in the observed harmonics, and inaccuracies in the model. In addition to measurement errors, uncertainities in the observed harmonics can arise from both real influences, such as seasonal changes due to the nonlinear interaction between tides and wind- or buoyancy-driven currents, and from statistical factors such as the length of the time series and the number
of constituents
included
in the harmonic
analysis. The harmonic analysis provides estimates of the latter
uncertainties
in the form of standard
deviations
for all
the cosineand sine (constituent)coefficients.For example,
Qx amplitude ratios with respect to K1 are approximately
60, 31, and 11%, and the phaselags are 16ø, 4ø, and 23ø smaller.
It is interesting to also compare our M2 and K• results with those in F87. Since Flather compared observations
with the nearestgrid points (rather than interpolatingto the precisetide gaugesites),we haveattemptedto provide a fair comparison by interpolating our model results to those same Flather locations. Employing our set of
observedharmonics(which in somecasesare more recent than Flather's), rms errors for the 34 sites common to both
models
are shown in Table
3b.
The
results
from
our
the one year analysis of elevations at Beauchemin Channel model are seen to be consistently more accurate for both and the four month analysis at Cape Ball respectively constituents. estimated these standard deviations to be about 0.16 cm It is also interesting to compare the effects of the tidal and 0.43 cm for all constituents. As these estimates are potential, Earth tide, and loading tide with those noted in virtually identical for all constituents, they are not the F87. This was done by repeating all model calculationswith best measures of variability. A better indication, given a---- 1 and/• = 0 in equation(1). WhereasFlather found a sufficiently long time series, is obtained by performing that the tidal potential, Earth and loading tides increased a seriesof short (e.g., monthly) analyses(with inference) the maximum M2 elevation amplitude in Hecate Strait by and calculating averages and standard deviations. This about 5% and increasedthe phase by approximately2ø, technique was used with three time series in FW, but it we found the inclusion of these terms only increased the
has not been appliedhere. See Foremanand Henry [1989] amplitudes by about 2.5% and increasedthe phasesby for further
discussion
of the
confidence
limits
associated
with the harmonic tidal analysis of time series. Table 2 shows observed and calculated M2 and K1 elevation amplitudes and phases for the 63 sites shown in Figure 2. Differences between the two sets of harmonic constants are calculated as distances in the complex plane; that is,
about 0.5 ø.
However, as in F87, we found the effects of these extra
forcingterms on Kx (and all diurnal constituents)to be very small. When the tidal potential, and Earth and loading tides were removed from the model forcing, the Kx amplitudes decreased by a maximum of 0.4 cm in Hecate Strait, and the phases decreased by a maximum of about
D ---{(Aocosgo-Am cosgm)2•-(Aosingo-Amsingm)2} 1/2, 1ø, both alongthe AlaskanPanhandlejust north of Dixon Entrance and in eastern Hecate Strait. As concluded by where Ao, Am, go, and gm are the observedand modelled amplitudes and phases. The largest M2 differencesoccur at Bella Coola, Kitimat, Kemano Bay and Queen Charlotte City; sites at the end of fiords or channels where, presumably, the model resolution is inadequate and the
Flather (F87), the tide-generatingpotential and the body and Earth tides should be included to obtain high accuracy for the semi-diurnal constituents, but their presenceis not essential
for the diurnals.
Figures 5a and 5b show the major semi-axis magnitudes
true dissipation(via bottom and side-wall friction, and for M2 and Kx. Theseplots havethe samefeatures(but in the generationof internal waves over sills) has not been greaterdetail) as Figure 4 in F87. In particular,noticethe accurately represented by the model. Table 3a shows root
meansquare(rms) elevationdifferences betweenmodeland observationfor each of the eight major constituents. Notice that with the exception of M2, all the rms amplitude differences are within 1.9 cm, and with the exception of
Qx, all the rms phasedifferencesare within 2.9ø. The M2 and K• model results are also displayed in Figures 4a and 4b. They are seen to have the same features as the analogous plots in F87. Most notable are the large M2 amplitude increases eastward in Dixon Entrance and northward in Hecate Strait. K1 is also seen to have an amplitude increase in the same region, but to a lesserdegree than M2. The irregular patterns along the
large M2 values off Rose Spit, Cape St. James, Cape Scott, the mid-western shoreline of Hecate Strait, and around Goose Island and Learmonth Banks. Observed vertically averaged major semi-axes are also shown in Figure 5 and are generally seen to be in good agreement with the model results. Notable exceptions for M2 are the large observed currents off Cape Chacon and on the continental slope to the west of Dixon Entrance; and the large model currents at the most
western
site in the central
Hecate
Strait
line
and the two sites nearest Rose Spit. Agreement for K1 is not as good; however, to some degree this may be a problem with the observations. The Kx confidence limits are proportionately much larger than those for M2 and
FOREMANET AL.: TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
2515
TABLE 2. Observed and CalculatedM2 and K• ElevationAmplitudes(Amp.) and Phasesfor the 63 SitesShownin Figure2 M2
Site Dixon Entrance Brooks Pen. S. Brooks Pen. N. Kyuquot Hunt Islets Q05 QC10 QC12 QC14 QC15 Hakai Passage Beauchemin Ch. Hecate Sandspit Welcome Hat. Foggy Bay McLean Arm Cape Chacon Cape Muzon Security Cove Houston Stewart Dawson Harbour Nesto Inlet Heater Harbour Atli Inlet W05 Cape Ball HEC7 Dixon Rose Spit Egeria Bay Langara Island Union Seamout
Alert Bay Port Hardy Cape Scott Egg Island Wadhams Bella Coola Bella Bella Milne Island McKenney Is. Gillen Harbour Kitimat Kemano Bay Qlawdzeet Brundige Inlet Prince Rupert Port Simpson Tasu Sound Copper Island Q.C. City Wiah Point Langara Island Ketchikan Davis River Bowie Seamt. QC13 Nakat Harbour Kelp Island Nichols Bay View Cove G01 G05 Q06
Latitude
Longitude
54.562 49.837 50.267 50.217 50.473 51.367 51.847 51.005 51.053 51.120 51.690 52.782 53.275 54.023 54.935 54.793 54.667 54.640 54.747 52.077 53.163 53.557 52.120 52.713 53.173 53.717 54.122 54.260 54.225 54.242 49.583 50.583 50.717 50.783 51.250 51.517 52.383 52.167 52.617 52.650 52.967 53.983 53.517 54.217 54.617 54.317 54.567 52.750 52.350 53.250 54.117 54.250 55.333 55.767 53.317 51.680 54.820 54.877 54.717 55.083 51-.600 51.363 51.483
132.937 127.943 128.352 127.353 128.317 130.017 129.578 128.698 129.438 127.945 128.107 129.298 130.982 130.613 130.960 131.968 132.025 132.688 132.845 131.123 132.458 132.925 131.033 131.577 131.275 131.733 131.024 131.950 132.965 133.163 132.783 126.933 127.483 128.417 127.833 127.517 126.800 128.133 128.767 129.483 129.600 128.700 128.117 130.767 130.850 130.333 130.433 132.033 131.167 132.067 132.317 133.050 131.633 130.167 135.633 130.565 130.702 131.300 132.133 133.017 128.883 128.915 130.483
K1
Observed Amp. Phase
Model D, Amp. Phase cm
128.2 98.5 98.3 99.0 101.1 107.9 120.1 113.2 106.8 123.1 126.0 145.3 167.8 190.7 182.9 170.9 162.9 137.3 125.7 103.4 103.1 105.2 119.4 151.4 167.3 191.5 186.0 160.0 125.3 119.1 91.2
268.7 242.0 244.0 241.4 243.6 250.5 251.9 245.9 248.1 247.1 248.0 253.9 263.0 264.8 268.1 269.9 270.3 270.2 270.9 252.8 259.8 262.1 264.3 267.6 266.6 265.2 264.1 263.6 263.0 260.9 251.5
128.7 98.2 100.5 100.0 101.5 106.5 119.4 111.1 105.6 120.0 124.2 145.3 168.6 192.6 183.6 168.6 161.3 136.8 125.4 102.1 104.2 105.7 117.3 151.6 168.1 194.5 187.6 158.7 130.9 119.8 89.5
127.1 133.0 108.0 118.1 124.7 142.6 129.7 128.2 137.8 144.6
259.5 251.9 244.2 249.0 249.0 252.6 249.8 250.7 255.1 255.8
131.6 132.9 106.4 120.0 123.7 147.4 133.0 138.5 140.5 148.1
165.0 172.5 183.8 179.4 195.1 186.2 101.9 131.7 197.4 146.5 130.6 187.4 203.8 100.9 114.4 181.3 177.6 163.6 152.2 119.0 115.0 111.7
257.8 258.5 267.0 270.0 267.4 266.9 257.9 267.5 271.5 263.0 262.8 270.3 268.4 265.9 257.7 271.7 270.6 269.8 272.1 249.0 247.4 256.8
160.9 162.2 189.7 184.8 197.5 190.4 102.3 132.0 206.5 150.2 125.0 180.3 206.0 99.5 106.9 186.5 178.0 156.8 138.7 118.5 113.8 111.5
268.3 241.9 244.4 240.0 244.7 251.4 251.9 246.1 248.6 246.6 246.9 252.5 263.4 264.5 270.7 271.4 270.7 270.9 271.4 256.9 260.2 261.6 264.9 268.6 266.7 266.8 264.8 264.3 261.6 262.3 251.8 261.1 252.7 243.3 246.6 246.2 247.3 248.4 250.6 253.6 254.8 253.6 253.3 265.7 267.9 266.2 268.0 258.7 266.9 274.2 261.8 261.6 271.8 268.7 266.2 254.8 269.2 270.3 271.4 272.9 249.1 248.2 256.2
1.1 0.3 2.3 2.6 1.9 2.2 0.6 2.1 1.6 3.3 2.9 3.6 1.5 2.3 8.3 5.0 2.0 1.8 1.2 7.4 1.3 1.0 2.4 2.7 0.8 6.1 2.7 2.4 6.5 3.0 1.8 5.8 1.9 2.3 5.4 6.1 14.3 4.6 10.3 4.6 4.4 12.8 18.4 7.2 8.6 4.7 5.7 1.5 1.4 13.2 4.7 6.2 8.5 2.5 1.5 9.3 9.4 1.0 8.1 13.7 0.6 2.1 1.3
Observed Amp. Phase 48.1 44.8 45.2 47.9 43.5 44.5 46.9 45.5 45.2 46.5 45.9 47.4 49.7 51.7 50.5 50.2 49.6 47.8 46.4 43.6 44.9 48.0 44.4 47.9 48.3 49.7 50.8 48.9 45.3 46.5 43.3 51.6 49.9 44.3 44.6 46.7 48.2 45.8 43.6 44.5 45.6 48.5 48.8 49.4 48.7 50.7 50.5 43.7 44.1 51.0 45.9 49.1 50.7 51.2 44.6 49.6 48.6 48.9 47.2 47.3 44.9 44.5 45.8
260.6 243.7 244.8 245.8 247.0 249.2 250.0 246.8 246.9 248.2 250.1 252.8 258.8 258.4 260.2 260.8 261.4 261.5 261.1 247.6 252.7 253.5 257.7 259.6 261.0 260.3 258.9 258.6 257.1 257.2 248.9 256.6 249.8 246.4 251.8 251.1 253.0 251.7 253.7 253.9 256.1 254.9 255.4 259.4 260.6 259.4 259.3 256.0 259.8 261.9 258.1 258.0 261.6 260.2 256.9 248.6 262.3 261.1 257.0 262.5 251.5 250.5 253.4
Amplitudes arein centimeters, andphases arein degrees, UT. D isdefined byequation (6).
Model D, Amp. Phase cm 47.1 44.5 44.9 44.9 45.2 47.6 47.7 45.9 45.9 46.4 45.5 47.4 49.2 52.8 52.2 49.4 49.0 47.5 46.8 46.5 46.5 46.9 44.7 47.4 48.4 50.4 51.3 49.3 48.0 47.2 42.0 51.2 50.3 45.7 45.9 46.1 48.0 46.1 46.6 47.0 47.1 48.9 48.6 52.3 51.3 53.4 51.9 46.4 45.7 51.4 49.0 47.8 49.6 53.1 42.9 47.2 52.0 50.1 48.7 48.4 45.7 45.4 46.9
260.6 244.9 245.6 244.1 245.6 248.0 250.1 247.1 246.4 249.0 249.4 249.4 256.6 257.3 261.7 262.2 262.1 261.9 261.8 251.7 255.2 256.5 255.5 258.8 258.7 260.8 258.5 259.8 258.3 258.1 249.8 256.7 250.9 245.7 249.7 249.9 250.2 250.2 250.8 251.2 251.5 252.5 251.4 259.1 260.9 259.0 260.8 253.5 257.1 264.1 258.7 257.7 263.1 261.3 256.7 250.9 262.0 261.7 262.3 263.6 249.2 247.8 251.7
1.0 1.0 0.7 3.3 2.0 3.2 0.8 0.4 0.8 0.7 0.7 2.8 1.9 1.5 2.2 1.4 0.8 0.4 0.7 4.3 2.6 2.7 1.7 0.9 2.0 0.8 0.6 1.1 2.9 1.0 1.5 0.4 1.1 1.5 2.1 1.2 2.4 1.2 3.7 3.3 4.0 2.1 3.4 2.9 2.6 2.7 1.9 3.3 2.6 2.0 3.1 1.4 1.8 2.2 1.7 3.1 3.4 1.4 4.7 1.4 2.0 2.3 1.8
FOREMAN ET AL.' TIV•.s ANyRESONANCE, NORTHCOASTOFBRITISHCOLUMBIA
2516
TABLE 3a.
RMS Elevation Differencesat All the Sites
M2
Amp.,cm 4.0 Phase,deg 1.7
K1
$2
O1
N2
P1
some of the observed values do not seem to be consistent
with their neighbours.
Listed in Table 2 and Shown in Figure 2
K2
Q1
1.6 1.9 0.9 1.3 0.7 0.7 0.7 1.9 2.1 1.7 2.2 2.0 2.9 6.0
Table 4 summarizes the rms differences between the
model and observedM2 and K1 currents. As was found with the FW model for the southwestcoast of Vancouver
Island,thesecurrent differences are muchlargerthan the elevation differences. Again this is probably due to a combination of the same three reasons discussed
TABLE 3b. I•MS ElevationDifferences at 34 SitesCommon
to the Flather[1987]modelandThisModel,and Interpolationto the Flather Locations This Model
Amp.,cm Phase,deg
in FW (pages277-278),namely,(1) crudeaveraging of the observationsat specificdepths in order to obtain a
Flather's Model
M2
K1
M2
K1
2.5 1.5
1.6 2.0
3.4 2.9
2.2 2.9
barotropiccurrent;(2) baroclinicprocesses presentin the observations but not represented in the model; and (3) largervariabilityin the observedtidal currents. Grid resolutionmay be an additional reason that was not discussed in FW.
Whereas the current observations are
pointmeasurements thatcanreflect verylocalfeatures in
140
M2 co-ampl]tudes[cm)
and co-phases[OMT)'
Fi•. •. •ode•½o-•mp•i•.d• (½m)(•o•id½..•) •-d •o-ph• (d.•r•. UT) (d•h•d •.r•)
•or(•) •= •.d (•)
FOREMANET AL.' TIDES ANDRESONANCE, NORWHCOASTOF BRITISHCOLUMBIA
2517
b co-ampltudes[cm]
and
co-phases[¸P1T]"
Fig.4 (continued)
the bathymetryand/or coastline,the modelsimulationcan only hope to represent such features if their scale is larger than the grid resolution. Figure 6 servesto illustrate the degree of baroclinicity in four representative regions of the model domain. It shows model and observed M2 and K1 current ellipses versus depth. At site D04 in the middle of Dixon Entrance south of Cape Chacon, the M2 ellipsesare seento rotate clockwise with depth while the K1 ellipsesrotate counterclockwise. Although the model K1 ellipse is a reasonably good representation of the averaged observation, the fact that the shallowest M2 ellipse is fatter than the deeper ellipses
suggeststhe presenceof a larger cross-straitcomponent.As there can be significant stratification in Dixon Entrance in
At site W02 in the center of easternHecate Strait, both constituentsare seento be predominantly barotropic and in good agreement with the model. At site G04 in the trench south of GooseIsland Bank and QC2 off the west coast of the Queen Charlotte Islands, M2 internal tides also seem to be presentas variationsin the ellipseangle of inclination with depth suggesta componentthat is perpendicular to both the bathymetry contours and the inclination of the
model (barotropic)ellipse. The fact that the K1 currents are comparable in magnitude to the M2 currents at QC2 (though the K1 elevation amplitudes are less than half
the M2 amplitudes)is similar to the situation(FW) off Vancouver Island and suggeststhat diurnal shelf waves may exist along the narrow continental shelf to the west
the summermonths(whentheseobservations were made) of the Queen Charlotte Islands. An extensionof Figure due to freshwater input from the Skeena and Nass l•ivers, 5b northwardsuggests(as in F87) that diurnal shelfwaves it likely that this component is an internal tide.
are also present along the Alaskan shelf north of Dixon
\
FOREMAN ET AL.' Tress ANy RESONANCE, NORTI-ICOASTOF BRITISHCOLUMBIA
2518
'2
H2 major
semi-axis
[cm/s]
Fig. 5. Major semi-axismagnitudes(cm/s) for (a) M2 and (b) K1. Asterisk and number denoteobservationsite and observed depth-averaged value.
Entrance; however, there are no current meter observations Figure 4a). As M2 is dominantin this region,this means for confirmation.
that at floodtide, the currents(and thuspassiveparticles)
Figure 7 showsM2 current ellipsesfor northeast Hecate Strait. It is interestingto note that the anglesof inclination
will converge along the northeast coast of the Queen Charlotte Islands, while at ebb tide they will diverge from
convergeto the region of maximal tidal amplitude (see
that
location.
FOREMANET AL.' TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
2519
•t.O
o/
0 ß
;
;, /o
ß
1.4
,•,
.,-
'...
.-
•
,'
.-.
_-. '-"
,
.,
:..• '.•,. ,..•
,"'- ,,.
R ! maj oc sem •- ax • s ( cm/s ) Fig. 5. (continued) TABLE 4. RMS Differences for Ellipse Parameters at All Current Meter Sites Shown in Figure 2
Figure 8 shows the barotropic residual tidal velocities calculated by the model. For this computation, the sea level was set to zero along all eight open boundaries(i.e., ßonly generationwithin the modeldomainis assumed),and
Major semi axis,
Minor semi axis,
inclination,
cm/s
cm/s
deg
deg
M2
7.5
5.3
23.1
26.4
a smalllateralviscosity (Ah= 10m2/s)wasaddedto the
K1
3.2
1.7
24.8
47.5
momentum equations in order to assist convergencein some
S2 O1
3.0 2.1
2.1 0.9
22.1 30.5
31.3 61.1
of the narrow passagesand fiords and in the region around Cape St. James. Clockwise eddies are evident around
phase lag,
M2
K1 Mode 1 I>
Observed :
2
Observed
Mode ]
cm/s
:
b
20
cm/s
(• (•
cI
M2 Mode l
13bserved
Mode ]
Observed
c)
•
-
2
cm/s
-
20
cm/s
c)
Fig.6. •1•2andK1current ellipses at sites (a)D04inthemiddle ofDixon Entrance south ofCapeChacon, (5) •V02in center ofeastern Hecate Strait,(c)G04in thetrench south ofGoose IslandBank,and(•) (•C2midway alongthecontinental slopewestof the(•ueenCharlotte Islands.Thelinewithineachellipse denotes O, the positionof the currentvectorat the timeof maximumtidal potential,andthe arrowdenotes the sense of rotation.
FOl•EMAN ETAL.' TIDESANDRESONANCE, NOl•THCOAST OFBI•ITISHCOLUMBIA
K1 Mode I
•2 Observed
Mode I
Observed
c
K1 Mode ]
I)
M2 Observed
-
2
cm/s
d Fig.6 (continued)
Mode 1
•
Observed
-
2
cm/s
2521
2522
FOREMANET AL.: TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
M2 current
ellipses
Fig.7. M2 modelcurrent ellipses fornortheast HecateStrait.Thelargest majorsemi-axis is71cm/s.Theline
within eachellipsedenotesG, the currentvectorpositionat the time of maximumtidal potential.Solidline ellipses denoteclockwise rotationof the currentvector,dashed ellipses denotecounterclockwise rotation.
GooseIsland and Learmonth Banks, while counterclockwise the cape and a setup along the coast to north and west eddies are seen off Cape Chacon and south of Learmonth of the cape. The former feature is consistent with the
eddy of Figure 8, while the latter feature Bank. (The complicatedflows near Learmonth Bank counterclockise suggest the needfor furthergrid refinement in this region.) would be consistent with a clockwise eddy were it not Strong clockwiseflows are also evident to the northwest landfast. Although it is possible that better resolution of Rose Spit, but they do not closeto form an eddy. In of the coastline and the narrow continental shelf might addition, there is no evidenceof the large, counterclockwise, produce a clockwise eddy consistent with Thomson and RoseSpit Eddy studiedby Bowmanet al. [1992].A strong Wilson[1987],both stratificationand a three-dimensional
counterclockwise eddy (the maximumspeedis 69 cm/s)
model are probably necessaryto accurately reproduce both the Cape St. James Eddy, and the Rose Spit Eddy. The final interesting feature of Figure 8 is a southward there is no evidence of the clockwise eddy west of the cape that was calculatedand observedby Thomsonand residual flow along the western side of Hecate Strait that Wilson[1987]. However,the elevationcontoursassociated is accompanied by a maximum setup along the coast of with these residual flows show both a depressioneast of approximately 2 cm.
is also predicted to the east of Cape St.
James but
FOREMANET AL.' TIDES ANDR•so•^•c•,
NORTHCOASTOF BRITISHCOLUMBIA
-•
-
2523
10.0
.
ß
o
.
ß
ß
o
,
, .
..
.
. o
ß
Fig. 8. Barotropic residual tidal velocities.
.
cm/s
2524 4.
FOREMANET AL.: TIDESANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA SHALLOW-WATER CONSTITUENTS AND RESONANCE
Figures 4a and 4b show that the amplitude increase northward
in Hecate
Strait
and eastward
in Dixon
Entrance
is greater for M2 than K1. These results lead to the natural question: What is the resonant frequency of the system?
The
tidal
model
confirms
these
conclusions.
When
run with radiation (normal flow) conditionson all eight sea boundaries(see Figure 3) except number I where the elevation is set to zero (it is necessaryto specify at least one boundary elevation in order to avoid an
ill-conditioned problem), the M4 phases indicated the
ChoosingQC14 (becauseit was the longesttime seriesof
expected outward radiation from northwest Hecate Strait all the tide gauges at the entrance to Queen Charlotte and the M4 amplitudes were reasonably accurate. For Sound), and Cape Ball (becauseit was situated closeto example, the amplitudes at Cape Ball and QC14 were 4.3 the largest M2 amplitude in Hecate Strait) as reference and 0.3 cm, respectively. However, the $K3 model solution sites, Table 5 shows the observed amplitude ratios and did not agree with the observations as it calculated a value phasedifferencesfor severaltidal constituents.(Both these of only 0.6 cm for Cape Ball and phases that increase sites are shown in Figure 2.) A resonantfrequencyin away from that general region. In order to confirm that the terdiurnal or quarter-diurnal frequency band is clearly reasonably accurate $K3 values could be obtained, the suggested. model simulation was repeated with specified elevations However the situation is complicated by the fact that alongboundaries3 and 4 (seeFigure3) that are consistent these so-called shallow water constituents also have conwith the observations from Bowie and Union Seamounts, tributions that arise from their local nonlinear generation. and radiation conditions on the other six outer boundaries. For example, the relatively large signals for SK3 and M4 As shown in Figure 9c, the $K3 amplitudes and phasesat at Cape Ball will have contributions from the nonlinear both QC14 and Cape Ball now agree reasonably well with interactions(primarily via the secondterms in equations the observed values, and the phases indicate a net energy (1) and (2)) of $2 and K1, and M2 and M2, respectively. propagation into Hecate Strait. Figures 9a and 9b show the observed amplitudes and Having demonstrated that the values in Table 5 should phases at selected sites for these two constituents. As the not be used to estimate the resonant frequencies of the smaller amplitudes are comparable to the background noise region, accurate estimates were then calculated by making level, it is not surprising that the harmonic analysis has the following modifications to the finite element model. produced some values that appear to be inconsistent with The tidal potential and Earth and loading tide forcing neighbouring values and trends. However, general patterns were turned off, and periodic forcing with frequency c•0 are apparent. For both constituents, the amplitudes was specifiedon boundaries3 and 4 (see Figure 3). The increase uniformly to maxima in the northwest region of amplitude and phase of this forcing varied linearly from Hecate Strait. However, whereas the M4 phases increase 10 cm and 90 ø in the southwest to 18 cm and 180 ø in away from that region, suggestingoutward radiation, the the northwest. The associated amplitude ratios and phase $K3 phases increase toward that region and suggestinward differences are close to those chosen for the $K3 simulation radiation. The M4 signal at Cape Ball would therefore illustrated in Figure 9c and represent propagation to the appear to arise largely from local nonlinear generation northwest. All the other open boundaries were given a
(consistentwith both the elevationand velocity gradients normal flow condition (i.e., the computed velocity was apparentin Figures4a and 5a), while the $K3 signalwould restrictedto be normalto the boundary)in order to permit appear to have a significant contribution from •he resonant growth of energy propagating into Hecate Strait. TABLE 5.
Freq.
Constituent
(cph)
the radiation of incident waves.
Following a procedure
similarto that employedby Walters[1988]in SanFrancisco
Tidal Harmonics at Cape Ball and QC14
QC 14
Amp., cm
Cape Ball
Phase
Amp., cm
Phase
Amp.
Phase
Ratio
Diff.
Q1
0.0372
5.30
230.9
6.01
238.5
1.13
7.6
O1 P• K• N2 M2 $2 K2 M03 M3 MK3 $K3 MN4 M4 M$4 2M K5 2MN6
0.0387 0.0416 0.0418 0.0790 0.0805 0.0833 0.0836 0.1192 0.1208 0.1223 0.1251 0.1595 0.1610 0.1638 0.2013 0.2400
27.90 14.02 45.22 22.16 106.80 31.87 8.66 0.12 0.23 0.12 0.22 0.11 0.34 0.22 0.10 0.11
232.7 242.4 246.9 223.7 248.2 274.6 265.7 134.7 309.0 239.3 90.0 100.2 110.4 127.1 130.6 116.4
30.91 15.41 49.72 39.26 191.50 59.60 15.72 1.09 2.20 1.37 2.58 1.91 4.92 2.45 0.39 0.18
242.5 257.2 260.3 240.6 265.2 297.6 292.2 6.3 326.7 19.3 147.3 16.5 40.4 93.9 41.0 19.6
1.11 1.10 1.10 1.77 1.79 1.87 1.95 9.08 9.59 11.42 11.72 17.36 14.47 11.14 3.90 1.64
9.8
Phases are degrees UT.
14.8
13.4 16.9 17.0 23.0
26.5 231.6 17.7
140.1 57.2 276.3 290.0 326.8 270.4 263.2
FOREMANET AL..' TIDES AND RESONANCE, NORTH COASTOF BRITISH COLUMBIA
I .0
2525
149
0.4•114
"Q
)•147 0.!
1
! .9)• .9•154
!32
I .6•1
0.5•E123
0.4•120
0.2•E99
0.8 •98
0.7)•97 0.2•90
0.6•9%
0.2)•52 ß
0.!•67
(]
sk3
observed
amplitudes
and
phases
Fig. 9. Observed amplitudes(cm) andphases(degrees, UT) at selecttide gaugesites(asterisk)for (a) SK3 and (b) M4. Amplitudes are shownto the left of the siteand phasesto the right. (c) Modelco-amplitudes (cm) (solid)andco-phases (degrees, UT) (dashed)for $K3 with nonlineargeneration andspecified elevations alongthe western and northwestern outer boundaries. Asterisk and associated numbers denote select tide gauge sites and observed amplitudes and phases.
Bay, coowas then steppedthrough a wide range of values geometry is not that of a classicalharbor. In this case, and the amplitude ratio and phase differencesbetween resonanceoccurs when the wave travelling northward into Cape Ball and QC14 were calculated. (It should be QueenCharlotte Soundand the wavetravellingeastwardin mentioned that this response-versus-frequency procedure is much easier to explore with a harmonic model than a
time-steppingmodel.) The resultsare shownin Figure 10. Although a classical quarter-wavelength resonance is evident in Figure 10 when coo= 0.132 cycles/hour,the
Dixon Entrance, meet in phase along the western shoreline of Hecate Strait. Figure 11 showsthe co-amplitudesand co-phasesfor this resonant frequency. Notice the phase shifts of 90 ø •om
the entrances of Hecate Strait
and
Dixon Entrance to Cape Ball, and a factor of 14 increase
2526
FOREMANET AL.: TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
265
0.6•226
0.6 •1•4
1.0•3
•
1.3
41
t.5•3
O.
ß2 •(268
2.6•(76
I .8•9
0.5 •(108
0.7•(113
I .5•94
0.7•(74 ••'•
0.5•(147 0.3•110
0.2•213 b
m4 observed
amp ] i tudes
and
phases
Fig.9 (continued)
in amplitude from QC14 to Cape Ball.
Figure 10 also
The
Table
5 observed amplitude
ratios and phase
shows3/4-wavelengthresonance at coo= 0.255cycles/hour, differences for individual tidal constituents are also shown and an interestingpeak near 0.310 cycles/hour. A closer in Figure 10. Considering the simple boundary forcing, inspection of the latter reveals amphidromes east of both Rose Spit and Cape St. James, and amplitude growth both near Cape Ball and to the northeast of Vancouver Island. Cape Ball amplitudes at these frequencies are about 120 cm and 130 cm, respectively. Were Figure 10 continued for larger frequencies,shorter wavelengthresonanceswould also appear.
the diurnal and semi-diurnal constituents Q1, O1, P1, K1, N2, M2, 52, and K2 are seen to agree reasonably well with the model curves. Amplitude ratios for the terdiurnal constituents also agree reasonably well but the observed quarter-diurnal amplitude ratios are seen to be much larger than predicted by the model. As was demonstrated with $K3 and M4, this is because the
FOREMANET AL.' TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
2527
ß ,
i
ß
ß
ß
,
ß ß
,
.½
--. /0.2
,'•90
..' ,"-
'
ß
SK3 co-amplitudes[cm)
and
co-phases[O-MT]
Fig.9 (continued) energy associated with these shallow water constituents does not arise solely through inward radiation from the deep ocean. In particular, the extent to which the large observed amplitude ratios for MN4, M4, and M$4 are not reproduced by the model is an indication of the local nonlinear generation of these constituents near Cape Ball.
thatthe$K3 signal is largerthantheMK3 signal.As the M2 elevations and velocities are approximately three times larger than their $2 counterparts, we would expect approximately the same proportionality for MK3 and $K3 if nonlinear generationwere the only energy source. It should be noted that although all eight major tidal
A furtherindicationthat the terdiurnalsignalat Cape
constituents
Ball is not primarily due to local nonlinear generation is
that
were included
in the bottom
friction
the forced waves in the resonance calculation
term
so
had an
2528
FOREMANET AL.' TIDESANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
Tidal
Harmonics'
(Z) observed
ampl•_tude
•
phase
observed
Cape
Ball
rat•_os
d:•fferences
vs.
QC14
model
amplltude
model
phase
ratlos
dlfferences
Or)
/ /
H(
/
I
/
................. .......... / ........................ '.................. ;' I I
/
I
/ /
" /
/ • '••,// \\/ /
/
I
I
I
I l
/
\/'
!I
I
I
I
_• ¸ e-
I
o '•-
I I I
o¸
I
o•
I I
! I I !
o('0
/ /
/
..... • .....
! ...............
ii
iI
I
ß
'
/
/ / /
ß
.oo
I
0.05
I
O.lO
I
0.15
I
I
I
I
0.20
0.25
0,30
0.35
0
frequency
Fig.10. Amplitude ratios andphase differences (degrees) between CapeBallandQC14asa function offrequency(cycles/hour). Modelamplitude ratios (solid), model phase differences (dashed), observed amplitude ratios (circles), observed phasedifferences (asterisks).
Defining2•r/Q as the fractionof the energydissipated appropriatedrag, their inclusiononly affectedthe results slightly. Also increasingthe magnitudeof the forcing in onecycleand usingthe sameformulaeas Garrett[1972] producedessentially the samecurvesas thoseshownin for determiningresonancein the Bay of Fundy and Gulf of Figure 10. So, at least for the purposeof resonance,the Maine, we calculatedQ for this system. Using the M3 and system is essentiallylinear.
SK3 observationsof Table 5, Q was computedto be 11.3,
whereasusing the numericalmodel resultsat frequencies for-medwith different values of the amplitudes and phases 0.12 and 0.13 cycles/hour,Q was foundto be 9.5. Unlike the Bay of Fundy and Gulf of Maine region alongthe westernboundariesin order to mimic radiation from another source. For example, when our calculations where the primary resonantfrequencyis very closeto the wererepeatedwith constantvaluesof amplitudeand phase N2 frequency[Garrett, 1972],resonancein Hecate Strait (in order to representradiation from the southwest),the has not been previously noticed because there is little Obviously other resonancecalculationscould be per-
results differed slightly from those shown in Figures 10 and oceanicenergy at the peak frequenciesshownin Figure 10. 11. Thus the direction of wave propagation does influence However,this is not to say that resonancein Hecate Strait resonance.
is not important. With the imminent possibility of an
FOREMAN ET AL.' TIDES ANDRESONANCE, NORTHCOASTOF BRITISHCOLUMBIA
2529
IO0
70•
Resonance
co-ampltudes{cm)
and co-phase
Fig. 11. Co-amplitudes (cm) (solid)andco-phases (degrees) (dashed) for the resonant frequency cv0-- 0.132 cycles/h.SitesQC14 and Cape Ball are denotedby asterisks.
of Vancouver Island and other intermittent earthquakes around the North Pacific Ocean, there is a constant threat
Further investigationsshould therefore be carried out to determine the shorter resonant periods for Hecate Strait and the possibilitythat they might enhancethe danger of
of tsunamis along the west coast of British Columbia.
future
earthquakealongthe Juan de Fuca Plate off the west coast
tsunamis.
2530
FOREMAN ET AL.: TIDESANDRESONANCE, NORTHCOASTOFBRITISHCOLUMBIA 5.
SUMMARY AND DISCUSSION
Bowman,M. J., A. W. Visser,and W. R. Crawford,The Rose
The preceding presentation has detailed the verification of a barotropic tidal model for the northern coastal waters
Spit Eddy in Dixon Entrance: Evidence for its existenceand
underlyingdynamics,Atmos.Ocean,30(1), 70-93, 1992.
Cartwright,D. E., Oceantides. Int. Hydrogr.Rev. Monaco,
60(œ),35-84, 1978. of British Columbia. Although the model has on average, W. R. and R. E. Thomson, Physical oceanography reproduced the elevation amplitudes and phases of the Crawford,
eight largest constituents to within 4.0 cm and 6.0ø when compared with observations at 63 sites, it has not been
as successfulwith the tidal currents. This is primarily because in most regions of the model, the true currents contain significant baroclinic effects that the barotropic model cannot reproduce. The
shallow
water
constituents
in Hecate
Strait
were
shown to have significant contributions from the constructive interferenceof signalspropagating into Dixon Entrance and Queen Charlotte Sound. By forcing the model from
the western boundarieswith a wide range of frequencies, the longestresonantperiod of the systemwas calculatedto be 7.6 hours. This estimate was confirmedwith analyses of observationsfrom tide gaugesin the middle of Queen Charlotte Sound and near the peak responsepoint in Hecate Strait. The energy dissipation parameter Q was estimated to be 9.5 for the primary resonantfrequency. The effects of tidal potential, Earth tide, and loading tide forcing were not found to be significantfor this model. In particular, the inclusion of these forcingsincreasedthe largest M2 and K1 amplitudesin Hecate Strait by 2.5%, and lessthan 1%, respectively. With referenceto the P ERD project goal of providing a better understanding of surfacecurrents in northern British Columbia waters, it is clear that a descriptionof the tidal currentswith a barotropicmodel is not sufficient.Although
of the westernCanadiancontinental shelf,Continental Shelf Res., 11, 669-683, 1991.
Crawford,W. R., W. S. Huggert,and M. J. Woodward, Water transportthroughHecateStrait. Atmos. Ocean,26(3), 301320, 1988.
Crean,P. B., Physicaloceanography of DixonEntrance,British Columbia,Bull. Fish. Res. BoardCan., 156,66 pp., 1967. Dodimead,A. J., A generalreviewof the oceanography of the QueenCharlotteSound-Hecate Strait-DixonEntrance region, Can. Manuscr.Rep. Fish. Aquat.Sci. 157•, 248pp., Dep. of Fish. andOceans, Resour.Serv.Branch,Pac. Biol. Sta., Nanaimo, B.C., 1980.
.Flather,R. A., A tidal modelof the northeastPacific,Atmos. Ocean,25(1), 22-45, 1987.
Foreman, M. G. G., Manualfortidal heightsanalysis andprediction. Pacific Mar. Sci. Rep. 77-10, 101 pp., Inst. of Ocean Sci., Patricia Bay, Sidney,B.C., 1977.
Foreman,M. G. G., Manualfor tidal currentsanalysis andpre-
diction. PacificMar. Sci. Rep. 78-6, 70 pp., Inst. of Ocean Sci., Patricia Bay, Sidney,B.C., 1978.
Foreman, M. G. G., andR. F. Henry,The harmonic analysis of tidal modeltime series,Adv. WaterResour.,12, 109-120, 1989.
Foreman,M. G. G., and R. A. Walters, A finite-elementtidal model for the southwestcoast of Vancouver Island. A tmos. Ocean,28(3), 261-287, 1990. Francis, O., and P. Mazzega, Global charts of the ocean tide
loading effects, J. Geophys. Res.,95(C7),11,411-11,424, 1990. Freeland,H. J., W. R. Crawford,and R. E. Thomson,Currents alongthe Pacificcoastof Canada,Atmos.Ocean,22(2), 151172, 1984.
Garrett, J. G. R., Tidal resonance in the Bay of Fundyand Gulf
of Maine, Nature, 238, 441-443, 1972. observedtidal currentsare predominantlybarotropicin HecateStrait wheremost of the oil leasesare located, Hannah, C. G., P. H. LeBlond, W. R. Crawford, and W. P. Budgell,Wind-drivendepth-averaged circulationin QueenCharevidenceof vertical variationsin other regionsof the
model necessitates a full three-dimensional model that includesdensity variations and mean flow currents. Such a
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V. A. Ballantyne, M. G. G. Foreman, and R. F. Henry, Department of Fisheries and Oceans, Institute of Ocean Sciences, P.O. Box 6000, Sidney, British Columbia, VSL 4B2, Canada. R. A. Walters, U.S. GeologicalSurvey, 1201 Pacific Ave., Suite 901, Tacoma, WA 98402. Received March 9, 1992; revised August 31, 1992; accentedSeptember29, 1992.)
