tides in the sea of okhotsk - uaf school of fisheries and ocean

JULY 1998 1389K O W A L I K A N D P O L Y A K O V

q 1998 American Meteorological Society

Tides in the Sea of Okhotsk

ZYGMUNT KOWALIK AND IGOR POLYAKOV

Institute of Marine Science, University of Alaska, Fairbanks, Alaska

(Manuscript received 8 October 1996, in final form 25 August 1997)

ABSTRACT

Eight major tidal constituents in the Sea of Okhotsk have been investigated using a numerical solution oftidal equations on a 59 space grid. The tides are dominated by the diurnal constituents. Diurnal tidal currentsare enhanced in Shelikhov Bay and Penzhinskaya Guba, at Kashevarov Bank, in proximity to the Kuril Islandsand at a few smaller locations. The major energy sink for diurnal tides (over 60% of the total energy) is ShelikhovBay and Penzhinskaya Guba. The major portion of semidiurnal tide energy is dissipated in the northwesternregion of the Sea of Okhotsk and in Shelikhov Bay and Penzhinskaya Guba. Nonlinear interactions of diurnalcurrents are investigated through K1 and O1 constituent behavior over Kashevarov Bank. These interactionsgenerate residual circulation of the order of 10 cm s21, major oscillations at semidiurnal and fortnightly periods(13.66 days), and higher harmonics of basic tidal periods. The M2 tidal current, caused by the nonlinear interactionof the diurnal constituents over Kashevarov Bank, constitutes approximately a half of the total M2 tide currentthere. The fortnightly current, through nonlinear interactions, also influences basic diurnal tidal currents byinducing fortnightly variations in the amplitude of these currents.

1. Introduction

The Sea of Okhotsk (SO) is a region of large tidalsea-level oscillations and strong tidal currents. In shal-low Penzhinskaya Guba, total tidal sea level oscillationsreach 13 m. Because of the strong currents and sea levelchanges, the tides significantly influence water mass for-mation in the SO. Total tidal currents of up to 4 knotsoccur in the Kuril Straits (Leonov 1960). These largecurrents cause mixing of the upper-ocean layer aroundthe Kuril Islands, generating a front between the Sea ofOkhotsk and the Pacific Ocean (Gladyshev 1995). Kitaniand Shimazaki (1971) found an almost homogeneousvertical structure of the temperature, salinity, and dis-solved oxygen over Kashevarov Bank and on the shelfin the mouth of Penzhinskaya Guba. During winter,strong vertical mixing sustains a polynya over Kash-evarov Bank. According to Alfultis and Martin (1987),to maintain this polynya a vertical heat flux of 50–100W m22 is necessary. One source of this vertical mixingis a strong tidal current.

Enhanced tidal currents in both the semidiurnal anddiurnal bands usually occur in shallow areas due to to-pographic amplification. In the diurnal band of oscil-lations the maximum current can also be associated withthe occurrence of shelf waves. Enhanced velocity alongthe shelf break and over isolated seamounts (see sum-

Corresponding author address: Dr. Zygmunt Kowalik, Institute ofMarine Science, University of Alaska, Fairbanks, AK 99775-7220.E-mail: [email protected]

mary by Foreman et al. 1995) is caused by near-resonantamplification of diurnal currents by topography. Inves-tigations of the resonance band of frequencies over sea-mounts by Chapman (1983, 1989), Brink (1989), Hun-kins (1986), and Haidvogel et al. (1993) delineated thedependence of trapped waves on the range of topo-graphic sizes and stratification. From their results onecan conclude that regions of local resonance are morelikely to be found in the polar oceans, where large valuesof the Coriolis parameter occur.

There are at least two regions of well-defined tidallygenerated topographic waves in the SO: off HokkaidoIsland and along Sakhalin Island. Investigations of tidalcurrent and sea level at the northern coast off HokkaidoIsland and in Soya Strait by Aota and Matsuyama (1987)and Odamaki (1994) reveal that the differences in tidalphase and amplitude between the Sea of Okhotsk andthe Sea of Japan result in strong diurnal currents at SoyaStrait. Odamaki (1994) demonstrated that these strongcurrents generate a shelf wave along the coast of Hok-kaido. Aota and Matsuyama (1987) carried out an anal-ysis of a 32-month current data series from the mooringstation near Soya Strait and observed a high temporalvariability of the tidal current harmonic constants.

Strong diurnal currents were also observed by Ra-binovich and Zhukov (1984) along the coast of Sakha-lin. Their data were compared against an analytical mod-el that describes the barotropic shelf wave and Kelvinwave (Yefimov et al. 1985). Computations showed thattidal currents are controlled by the first mode of thebarotropic shelf wave, whereas sea level is related to

1390 VOLUME 28J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

both the shelf and Kelvin waves. Superposition of theshelf wave and Kelvin wave results in a set of amphid-romic points along the coast of Sakhalin Island.

Suzuki and Kanari (1986) developed a tidal model ofthe Okhotsk Sea with a resolution of 18.4 km. Ratherhigh spatial resolution allowed the determination oftrapped tidal energy in the diurnal band of oscillationsnot only off Sakhalin but also above Kashevarov Bankand in the Kuril Islands region. In the latter region shelfwave dynamics was demonstrated by Yefimov et al.(1985) through direct measurements and theory.

In this study we shall compute eight dominant tidalconstituents using a high spatial resolution. Analysis ofcomputed sea level and currents is aimed primarily atdescribing regions of enhanced currents and patterns ofenergy flow in the entire SO. The focus of this study isinvestigation of mechanisms of tidal current enhance-ment over Kashevarov Bank, an area where two strongdiurnal components, O1 and K1, interact nonlinearly,resulting in semidiurnal and fortnightly periods andhigher harmonics of the basic tidal periods.

2. Tidal equations and parameters

To obtain the distribution of tides in the SO we shalluse the vertically averaged equations of motion and con-tinuity in a spherical coordinate system (Gill 1982):

]u u ]u y ]u uy sinf1 1 2 f y 2

]t R cosf ]l R ]f R cosfbg ] tl5 2 (az 2 bz ) 2 1 Au (1)0R cosf ]l rH

]y u ]y y ]y uu sinf1 1 1 fu 1

]t R cosf ]l R ]f R cosfbg ] t f5 2 (az 2 bz ) 2 1 Ay (2)0R ]f rH

]z 1 ](Hu) 1 ]1 1 (Hy cosf) 5 0. (3)

]t R cosf ]l R cosf ]f

The operator A in the horizontal friction term in (1) and(2) is

21 ] 1 ] ]A 5 N 1 cosf . (4)h 2 2 2 2 1 2[ ]R cos f ]l R cosf ]f ]f

The bottom stress components are taken as

5 rru u2 1 y 2; 5 rry u2 1 y 2.b bt tÏ Ïl f (5)

In (5) r denotes the bottom drag coefficient; it will betaken as r 5 2.6 3 1023.

The following notation has been used in the aboveequations: l and f denote longitude and latitude, t istime, z is free surface elevation, u and y are velocitycomponents along longitude and latitude respectively,r is water density, Nh is horizontal eddy viscosity (52.53 106 cm2 s21), H denotes depth (it does not include

sea level), f is the Coriolis parameter, g denotes gravityacceleration, R is the radius of the earth, z0 denotes theequilibrium tide, and a and b are parameters accountingfor tidal potential perturbations. The above value of thehorizontal eddy viscosity Nh is required to preserve nu-merical stability.

Tidal forcing is described in (1) and (2) through theterms that are multiplied by coefficients a and b. Theseterms include the tide-generating potential, but they alsocontain various corrections due to earth tide and oceanloading (Schwiderski 1979, 1981a–g). Coefficient a de-fines ocean loading; its value ranges from 0.940 to 0.953according to Ray and Sanchez (1989). A higher-ordercorrection for the loading effect can be implemented aswell (e.g., Francis and Mazzega 1990). The term bz0

includes both the tide-generating potential and correc-tion due to the earth tide. It is usually expressed as(Hendershott 1977)

bz0 5 (1 1 k 2 h)z0. (6)

Here k and h denote Love numbers, which are equal to0.302 and 0.602, respectively. These numbers are av-eraged over all tidal constituents. Expressions for theequilibrium tides z0 are given by Schwiderski (1979,1981a–g).

In the ensuing computations one additional simpli-fication is introduced. Because the SO is only partlycovered by pack ice in winter and its effect on tides isusually quite small (e.g., Kowalik 1981), the influenceof pack ice is neglected.

3. Results of computation

The model domain is shown in Fig. 1. The locationof the open boundary is marked by the double line. Thedomain includes the Sea of Okhotsk, the northern partof the Sea of Japan, and a small portion of the PacificOcean.

The SO is bounded by Hokkaido, the Kuril Islands,the Kamchatka Peninsula, Siberia, and Sakhalin Island(Fig. 1). The two major domains of the sea are a broadshelf area along the Siberian coast and a relatively flatcentral basin with depths of approximately 1000–1500m. The Kuril Basin is the deepest region with depthsof 3000 to 3200 m. For numerical solution of the tidalequations a spatial grid of 59 was applied. Boundaryconditions at the open boundaries are specified by thesea level oscillations. The tidal constants for the eightmajor constituents at the Pacific Ocean open boundaryare taken from Schwiderski’s computations (Schwid-erski 1979, 1981a–g). The spatial resolution of thesedata is 18. The missing values for the 59 grid resolutionwere obtained by linear interpolation of Schwiderski’sresults. The Schwiderski data compare well with sat-ellite data (Cartwright et al. 1991) and with recent worldtide models by LeProvost et al. (1994) and Kantha(1995).

The mixed tide (eight constituents) was computed for


FIG. 1. Bathymetry of the Sea of Okhotsk. The double line denotes the open boundaries.

a two-month period. During the second month, whenthe total energy of the system became stationary, sealevel and velocity were recorded hourly at each pointof the domain. The standard for tidal harmonic analysisis a 29-day-long series with one-hour sampling (Fore-man et al. 1995). The same approach was applied toinvestigate the tides over Kashevarov Bank, but for sim-plicity of analysis the mixed tide was represented bytwo major tidal constituents, K1 and O1. In the latterexperiment, the tide was computed for a four-monthperiod, and sea level and currents were recorded hourlyat each point of the domain during the last three monthsof the computation. The P1 and K2 harmonic constantswere extracted from the record using an inference meth-od described by Foreman et al. (1995). Additionally,results obtained for the P1 constituent from the com-putation of eight constituents were compared against P1

computed alone without interaction with the remainingconstituents.

Cotidal charts for four major constituents, K1, O1, M2,and S2, are given in Figs. 2–5. Coamplitudes (cm) andcophases (deg) referred to Greenwich are shown by solidand dashed lines, respectively. The S2, N2, and K2 con-stituent charts (the latter two not shown) generally re-peat the pattern of the dominant M2 tidal constituent,

whereas the diurnal O1, P1, and Q1 constituents (thelatter two not shown as well) repeat the pattern of K1,the dominant diurnal constituent.

Amplitudes of the diurnal constituents are large inthe northeastern part of the SO. In the narrow Penzhin-skaya Guba the K1 and O1 constituents have maximumamplitudes of approximately 2.5 and 1.5 m, respective-ly. The amplitude of the semidiurnal constituent M2 isonly 1.3 m there. Along the southern and central Kam-chatka coast the amplitudes of the diurnal constituentsare slightly larger than those of semidiurnal constituents.

The largest calculated amplitudes for the semidiurnalband occur in Udskaya Guba. For example, amplitudeof the M2 constituent is 1.81 m there. Diurnal tides arerelatively weak in this bay (up to 0.6 m for the K1

constituent). Semidiurnal tides prevail along the Sibe-rian coast, with amplitudes up to 1.3 and 0.52 m for theM2 and S2 constituents, respectively. However, the larg-est amplitudes of the diurnal constituents are in the rangeof 0.4–0.6 m along this coast.

Model results were verified by sea level observationsfrom 108 stations of the SO and northern part of theSea of Japan. Unfortunately, along the Siberian coastand Kamchatka Peninsula only 10 tide stations are avail-able. The remaining 98 stations are located along the


FIG

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TABLE 1. Comparison of observed and computed tidal amplitudes of the Sea of Okhotsk. In the table NN is the number of stations usedin the analysis; Mean and SD are the mean value and standard deviation of the computed and observed time series; Corr and SD1 are thecorrelation coefficient and standard deviation between the computed and observed data.

Tidalconstitu-

ent

Amplitude

NN

Mean (cm)

Com Obs

SD (cm)

Com Obs CorrSD1(cm)

Phase

Mean (deg)

Com Obs

SD (deg)

Com Obs CorrSD1(deg)

K1

O1

P1

Q1

M2

S2

N2

K2

108108

2119

108107

1921

32.921.618.1

7.526.912.0

8.65.3

27.822.115.8

7.429.711.010.9

5.0

34.317.222.3

6.526.3

8.25.63.7

32.819.821.9

8.229.2

9.510.8

4.4

0.9650.9370.9420.9200.8900.8240.7270.806

6.76.94.05.66.34.74.42.1

111191123148181213192202

127182132148205240207218

109146

72130

6974

10199

113132

82104

909097

115

0.9230.9420.7760.8110.8690.8740.8420.887

1820332126261741

TABLE 2. Comparison of computed (Comp) and observed (Obs)M2/K1 amplitude h and phase g at coastal tide gauges.

Gauge

Latitude(N)

Longi-tude (E)

h (cm)

Comp Obs

g (deg)

Comp Obs

Kuril Islands 4484391478219

28.432.0

28.033.0

169.37.5

187.27.0

4880891538169

22.030.8

18.029.0

169.29.1

185.312.1

5085091558399

39.859.4

34.053.0

198.25.0

229.6352.7

Sakhalin Island 4781491438029

18.821.3

17.019.0

186.649.0

185.845.3

4981491438089

29.320.2

25.017.0

182.542.4

182.642.2

Kamchatka 5084991568309

28.147.0

22.045.0

161.1347.1

191.9354.9

5581991558339

71.498.9

88.094.0

253.526.1

305.065.0

Penzhinskaya Guba 6282391648309

105.0246.5

132.9252.1

274.6202.6

339.8183.0

Shelikhov Bay 5981391558099

21.6119.7

26.5123.4

237.1113.9

222.5101.3

Northwestern coast 5682791388099

96.256.0

86.645.1

85.283.9

109.665.2

Udskaya Guba 5485491368469

149.468.3

181.461.0

204.3131.2

198.396.6

Kuril Islands, Sakhalin, and Hokkaido. At the majorityof Russian tidal stations information is available onlyon K1, O1, M2, and S2 constituents. The main source ofinformation on the minor tidal constituents are 21 sta-tions along Hokkaido and the Kuril Islands.

Statistical analysis of observed and computed sea lev-el data for the eight constituents is shown in Table 1.It includes the number of stations (NN), mean values(Mean) and standard deviations (SD) of the computedand observed series, correlation coefficients (Corr) be-tween computed (Com) and observed (Obs) amplitudesand phases, and standard deviation (SD1) between com-puted and observed amplitudes and phases. High cor-

relation coefficients attest to satisfactory agreement be-tween the observed and computed sea level. The rep-resentative magnitude of errors between measured andcomputed data for the major constituents (K1, O1, andM2) is approximately 6.5 cm in amplitude and 1.0 h inphase. These errors are not distributed uniformly overthe coastal line of SO. Due to presence of only 10 sta-tions along northern and eastern shorelines the com-parison is skewed toward the Kuril Islands, Hokkaido,and Sakhalin.

To identify the source of these errors the comparisonbetween computed and observed amplitude and phasefor a few locations is given in Table 2. Major sourcesof errors are shallow Penzhinskaya Guba and UdskayaGuba. Whereas in Penzhinskaya Guba, the computedamplitude of 246.5 cm for the K1 constituent is closeto the observed amplitude of 252.1 cm, for the M2 am-plitude the difference between observation and com-putation is 27.9 cm. A similar situation occurs in Uds-kaya Guba. The set of experiments carried out in Pen-zhinskaya Guba indicates that the major cause of theseerrors is related to the bathymetry. A 4-m increase ofdepth in Penzhinskaya Guba resulted in a 25% increaseof M2 amplitude and small change in the K1 amplitude.The K1 amplitude is less sensitive to small changes inbathymetry, rather, it is influenced by the large-scaleresonance conditions.

Generally, the semidiurnal constituents are the leastaccurate. The higher frequency and shorter wavelengthof the semidiurnal waves are more sensitive to smallvariations of the bathymetry and coastline. The 59 gridsmooths local variations of the shallow water bathym-etry and the detailed structure of the coastline and nar-row straits.

Comparison of the computed cotidal charts for thedominant constituents of the diurnal (K1: Fig. 2) andsemidiurnal (M2: Fig. 4) bands of tidal oscillationsshows that at the eastern coast of Sakhalin, KashevarovBank, the Kuril Islands, and the southwestern part ofKamchatka, local maxima occur in the K1 pattern,whereas there is no local amplification of the M2 tidalamplitudes in the same regions. In Figs. 6 and 7 tidal


current ellipses for the K1 and M2 constituents areshown. General patterns for the other calculated con-stituents of the diurnal and semidiurnal bands are closeto the K1 and M2 waves, respectively. Both diurnal andsemidiurnal currents increase in shallow areas andstraits. Udskaya Guba, Penzhinskaya Guba, Kuril, andSoya Straits are areas of M2 and K1 current enhancement.Amplification of the diurnal currents is not, however,limited to these regions. Additional regions occur inproximity to the Kuril Islands, Sakhalin Island, off Hok-kaido, over Kashevarov Bank, over escarpments locatedbetween Kashevarov Bank and Kamchatka (Fig. 1), andin proximity to Kamchatka. Diurnal band amplificationis different from semidiurnal band amplification. Themanner in which the amplification takes place overKashevarov Bank sheds some light on the differentphysics in these bands. While M2 currents close to thebank are 10 cm s21, over a tiny portion of the bank thetopographic amplification increases currents to 20 cms21, thus doubling them. The K1 constituent generatesenhanced currents over the whole domain of the bank,amplifying currents approximately 10 times, from 5–10cm s21 to 85 cm s21. Thus, the different amplificationimplies that different mechanisms are at work in diurnaland semidiurnal bands of oscillations. In comparing M2

and K1 currents near the Kuril Islands, one also notesmuch stronger enhancement of the diurnal currents.

The energy flux for the K1 constituent, calculated ac-cordingly to Kowalik and Proshutinsky (1993), is de-picted in Fig. 8. The flux of tidal energy is directed fromthe open boundary in the Pacific through the KurilStraits toward the region of high frictional dissipationin Shelikhov Bay and Penzhinskaya Guba (big arrowsshow energy fluxes crossing several transects includingopen boundaries). The general pattern of energy flowis broken by larger and smaller domains of a circularor semicircular flux of energy. These are regions oftrapped tidal energy and enhanced flux. The areasaround the Kuril Islands (especially on the Pacific side),Kashevarov Bank, and the entrance to Shelikhov Bayare major domains of trapped energy. Lesser domainsare located at escarpments between Kashevarov Bankand the entrance to Shelikhov Bay, along Kamchatkaand Sakhalin. The energy flux of the M2 constituent isgiven in Fig. 9. The magnitude of the energy flux di-rected into Shelikhov Bay and Penzhinskaya Guba andinto the northwestern region of the SO is similar; there-fore, these domains play approximately the same rolein dissipation of the M2 tide. It is difficult to discernthe local domains of the circular or semicircular trappingof energy in the energy flux pattern; therefore, one canconclude that semidiurnal waves do not behave like di-urnal waves.

Figures 10 and 11 depict the rate of energy dissipationper unit surface in the SO due to K1 and M2 constituents,respectively. Again, these figures demonstrate the dif-ference in the dissipation pattern of diurnal and semi-diurnal tides in the SO. The trapping of the energy flux

over local bathymetry depicted in Fig. 8 for the K1 con-stituent resulted in the local maxima for the rate ofenergy dissipation (Fig. 10). Patterns of dissipated en-ergy in Figs. 10 and 11 are in close agreement withcomputations made by Suzuki and Kanari (1986). Thebalance of tidal energy is presented in Table 3. Termsin the energy conservation equation were calculated ac-cording to Marchuk and Kagan (1989). The total fluxof energy across open boundaries and the overall rateof energy generation due to astronomical tidal forcingis balanced by the total rate of energy dissipation. Thenet energy flux through the open boundaries is the prin-cipal source of energy. The overall rate of energy gen-erated by astronomical forces is relatively small. Forthe K1 tide astronomical forces generate only 6% of thetotal energy. Slight discrepancies between the sourcesand sinks of energy in Table 3 are due to errors asso-ciated with the averaging of the velocity components inthe staggered C-grid used in the computations.

The total rate of energy dissipation due to M2 tide,averaged over one tidal period, is estimated to be 49.23 1016 erg s21 in the entire SO. This magnitude is closeto 40.0 3 1016 erg s21 given by Jeffreys (1920), but issmaller than 73.0 3 1016 erg s21 derived by Lyard andLeProvost (1997) and is much smaller than 210.0 31016 erg s21 estimated by Miller (1966). We did not findany reference to the overall rate of energy dissipated inthe diurnal band of oscillation for comparison againstdata given in Table 3. According to our results, morethan 60% of the K1 energy is dissipated in ShelikhovBay and Penzhinskaya Guba (Fig. 9); therefore, thisrather small basin has a very important role in the bal-ance of tidal energy.

Due to variable bathymetry, strong oscillating diurnaland semidiurnal currents transfer vorticity to the meanmotion, generating residual currents through the non-linear interaction. In Fig. 12 residual currents due to theeight major constituents are depicted. The maximumvelocity is close to 16 cm s21 in a numerical lattice ofapproximately 6.5-km resolution. To extract residualmotion, tidal currents were averaged over a 29-day pe-riod. The residual circulation shows well-developedtrapped eddies over Kashevarov Bank, in proximity toKuril Islands and in the Shelikhov Bay–PenzhinskayaGuba area. Stationary clockwise eddies in the proximityof the Kuril Islands were described by Yefimov et al.(1985), but only recently Rogachev et al. (1996) dem-onstrated, with the help of Argos buoys and in situmeasurements, that the diurnal tide is the source of en-ergy for these eddies.

To investigate behavior of the numerical model in theshelf wave regions we first compare the model to mea-surements off the coast of Hokkaido. This region is welldescribed through observations of tidal amplitude andcurrent made by Aota and Matsuyama (1987) and Oda-maki (1994). Odamaki carried out current meter obser-vations in many locations at the 10-m level for onesummer month, while Aota and Matsuyama measured


FIG. 6. Computed K1 tidal current ellipses. Ellipses are depicted at every sixth point of numerical grid. Thearrow at each ellipse shows the direction of rotation.

current from February 1980 to September 1982 at asingle location. This current meter was deployed at the20-m level in a water depth of 35 m. The long timerecord was divided into 150-day segments by Aota andMatsuyama to study time variability of the currents.Observed and calculated sea level amplitudes andphases for major tidal constituents are given in Table4. The calculated elevations are in close agreement withthose obtained from observations.

Table 5 contains calculated and observed harmonicconstants for the tidal currents (orientation of the majoraxis is relative to the north). This table shows that themodel produced realistic diurnal tidal currents along thestraight part of the coast [from station 1 to station 9 inthe Aota and Matsuyama (1987) notation]. The calcu-lated and observed diurnal tidal velocities are in agree-ment at Soya Strait northern station S1, whereas at thesouthern station S2 the observed current is almost twiceas large as the calculated current. The reason for thisunderestimation seems to be proximity to the nodal

point in Soya Strait where a small change in the nodalpoint location may cause a large change in current. Thestrong spatial variability of diurnal currents in the vi-cinity of the nodal point is corroborated by Odamaki(1994). According to his measurements the major axisof the tidal ellipse at two stations in Soya Strait differsby 29 cm s21. Observed tidal current amplitudes havenoticeable time variability. The nature of this variabilityremains uncertain (Aota and Matsuyama 1987). Becausedensity stratification is weak and tidal temperature fluc-tuations do not occur in the observed temperature rec-ords, the internal tides must be rejected as a source ofthis variability. Due to low phase velocity of the shelfwave the occurrence of variability in the diurnal bandmay be related to the interaction of the Soya Currentwith the shelf wave. Some variability may be causedby the nonlinear interaction of the various tidal con-stituents. As demonstrated in Table 5, the length of themajor axis of the K1 tidal current ellipses varied in timefrom 19.6 to 33.5 cm s21 at station 2 in 1980–82 (Aota


FIG. 7. Computed M2 tidal current ellipses. Ellipses are depicted at every sixth point of numerical grid.The arrow at each ellipse shows the direction of rotation.

and Matsuyama 1987). The phase and orientation of thetidal ellipses are very stable. According to Odamaki(1994), the major axis at the same station is only 12.3cm s21. The computed value falls in this range andequals 22.5 cm s21.

Odamaki (1994) suggested that the difference be-tween the tidal amplitude and phase of the SO and theSea of Japan generates a shelf wave, which propagatesalong the coast of Hokkaido from Soya Strait. The ob-served phase velocity of the K1 tidal current is equal to5.5 m s21. This almost coincides with the estimate ofthe phase velocity of 6.6 m s21 for the first mode of theshelf wave obtained by using a simple analytical model(Clarke 1991). Odamaki calculated the travel time(phase lag) of the shelf wave from Soya Strait, for allstations given in Table 5, using a phase speed of 5.5 ms21 and distances between the stations and Soya Strait.In Fig. 13 the relation between the phase lags of the K1

tidal currents and station distances from Soya Strait ac-

cording to Odamaki and to our computation is shown.The computed phase velocity of the K1 tidal current is7.1 m s21.

4. Tides over Kashevarov Bank

The selective interaction of diurnal and semidiurnaltides with a seamount is exhibited over KashevarovBank. There are no large peculiarities in the cotidalcharts of the semidiurnal tides there (Figs. 4 and 5).Increase in the semidiurnal tidal currents is relativelysmall (Fig. 7) when compared to the diurnal currents(Fig. 6).

A special behavior of the diurnal tides around Kash-evarov Bank is easily detected in energy flux and dis-sipation charts (Figs. 8 and 10). Tidal energy is trappedaround the bank and the rate of energy dissipation de-picts a local maximum there. A noticeable transfor-mation of the diurnal amplitudes and currents is also


FIG. 8. Tidal energy flux for K1 wave. Large arrows show the net energy flux through transects. The valuesinside arrows should be multiplied by 1016 erg s21.

evident. Results are discussed for the dominant K1 con-stituent. The cotidal chart of the K1 constituent for theKashevarov Bank region is shown in Fig. 14. The am-plitude at the top of bank increases, reaching a maximumof 65 cm. A similar pattern in the O1 tidal level oscil-lations occurs as well. The maximum current for K1 (Fig.15) and O1 (not shown) is 85 and 75 cm s21, respec-tively, relative to off-bank values of 5–10 cm s21. Thecircular shape of the tidal current ellipses above the bank(Fig. 6) changes to rectilinear oscillations at the steepestslopes of the bank, located south and southwest fromthe top of the bank. This tidal flow behavior is typicalfor trapping or partial trapping of tidal energy by bottomirregularity (Kowalik 1994).

Local behavior of tidal oscillations often depends onthe resonance phenomena in local water bodies (e.g.,Platzman 1972). To demonstrate the possibility of res-onance we depict the distribution of natural oscillationsin the SO and especially at Kashevarov Bank over therange of diurnal and semidiurnal tides. The investigation

is done with the help of the basic set of equations (1)–(3), but the nonlinear and frictional effects are neglected.Sea level is set to zero at the open boundaries and atthe solid boundaries the normal derivative of velocityvanishes. To force oscillation through this system ofequations one can apply initial forcing at the openboundary or over the entire domain (Marchuk and Ka-gan 1989). We utilize the latter approach and a randomsea level distribution is prescribed initially. With suchinitial and boundary conditions the set of equations (1)–(3), without nonlinear and frictional terms, is integratedin time. Because this set of equations is conservative,the system is allowed to oscillate for a long enough timeso that the initial oscillations are redistributed to variousportions of the free wave spectra. After an initial periodof 100 h the stationary regime occurs and the model isrun for 1024 h. The series of hourly data is used for thepower spectra analysis of amplitude and velocity of os-cillations over the whole SO. The natural period of 26.3h occurs in the entire SO both in the sea level and in


FIG. 9. Tidal energy flux for M2 wave. Large arrows show the net energy flux through transects. The valuesinside arrows should be multiplied by 1016 erg s21.

velocity spectra. This period is very close to the O1

period (25.82 h), therefore enhancement of the O1 tideoccurs, but the 26.3 h resonant peak is broad and someenhancement of oscillations takes place at the K1 period(23.93) as well. Observations taken near Sakhalin Island(Rabinovich and Zhukov 1984), off Hokkaido (Odamaki1994), and at the Kuril Islands (Yefimov et al. 1985)show that the O1 constituent is much more amplifiedthan the K1 constituent, corroborating the possibility ofresonance enhancement in the entire SO through a 26.3-h oscillation.

At the given spectral band, as Marchuk and Kagan(1989) showed, the enhancement of oscillations of sealevel and velocity caused by resonance can be inves-tigated through the spatial distribution of sea level andvelocity of the free oscillations over the same spectralband. For the SO, the velocity and sea level of the freeoscillations obtained above are filtered using the tech-nique described by Marchuk and Kagan. After filtration,the velocity at the given period is weighted by its stan-

dard deviation taken over the entire SO. Therefore, thespatial distribution of the normalized velocity representscoefficient of amplification. A similar technique is ap-plied to the sea level amplitude. This approach gives apossibility for comparison of enhancement over the var-ious domains of the SO. For example, the largest am-plification of the elevation amplitude of diurnal con-stituents occurs in Shelikhov Bay and PenzhinskayaGuba. The amplification coefficient for the diurnal tideover these regions is approximately 10 times greaterthan over Kashevarov Bank.

In Fig. 16 spatial distributions of the amplificationcoefficient for the sea level (left side of Fig. 16) andfor the maximum velocity (right side of Fig. 16) aregiven over Kashevarov Bank at three periods in thediurnal range of oscillations: the natural oscillation pe-riod of 26.3 h (top), the O1 constituent (center), and theK1 constituent (bottom). The spatial location of the sealevel and velocity maxima differs for the different pe-riods, and for the K1 and O1 periods compares well with


FIG

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TABLE 3. Energy balance for the major tidal constituents.

Source (3 1016 erg s21)

K1 O1 M2 S2

Sink (31016 erg s21)

K1 O1 M2 S2

Net energy flux throughopen boundary:

Overall rate of energydissipation due to:

Japan Seasouthern boundaryeastern boundary

0.3227.8100.5

0.3211.4

34.4

20.05215.8

56.3

0.0021.3

5.4

bottom frictionhorizontal friction

58.120.2

17.57.8

39.99.3

3.41.1

Overall rate of energy generation dueto astronomical forcing 4.7 1.3 8.1 0.5Total 77.7 24.6 48.6 4.6 78.2 25.2 49.2 4.5

FIG. 12. Residual tidal currents due to eight tidal constituents. Vectors are shown at every fifth grid point.

the location of the computed maxima for the tides (seeFigs. 14 and 15). In Fig. 16 the coefficient of amplifi-cation for the amplitude decreases for the shorter pe-riods, whereas in some locations the amplification ofcurrent is stronger at the K1 period than at the O1 period.It follows from the above results that the resonant en-hancement of velocity and sea level at the O1 period iscaused by proximity of the natural oscillation period

and O1 period. The strong amplification coefficient ofvelocity at the K1 period shows that the resonant am-plification of velocity is substantial for a wide range ofperiods.

The nonlinear interaction of tidal constituents abovea seamount is an important element of tidal dynamics.Recent studies, based on current observations over Fie-berling Guyot in the North Pacific (Brink 1995) and on


TABLE 4. Comparison of observed (upper row) and computed (lower row) tidal amplitudes (A) and phases (Ph) along Hokkaido Island.

Station Lat (N) Long (E)

K1

A (cm) Ph (deg)

O1

A (cm) Ph (deg)

M2

A (cm) Ph (deg)

S2

A (cm) Ph (deg)

1 458259 1418419 74.0

238226

74.1

215216

31.4

199193

23.8

233250

2 458319 1418579 56.0

35625

43.5

3448

611.5

193191

46.0

247249

3 448569 1428369 1719.9

3835

1716.0

1722

1724.6

174194

88.1

221238

4 448359 1428589 2021.5

4242

2018.6

1829

1725.5

178196

88.3

221239

5 448219 1438229 2221.8

4845

2319.4

2433

1825.5

176197

88.3

218240

6 448019 1448169 2121.1

7248

2118.1

3539

1824.3

186199

88.1

240241

7 438449 1458279 2725.7

2720

2217.1

35037

2931.2

169201

1312.5

223244

8 448029 1458519 2721.2

1110

1816.7

3226

3038.5

152196

1313.0

205222

9 438209 1458359 2325.8

2821

1816.7

35020

3131.5

166184

1416.5

219235

investigations by Butman et al. (1983), show that thenonlinear interaction of diurnal constituents K1 and O1

results in new oscillations with periods at the semidi-urnal M2 tidal frequency (sum of K1 and O1 frequencies)and a fortnightly tide (difference of K1 and O1 frequen-cies) with a 13.66-day period. The rectification of thestrong diurnal currents produces a mean (residual)clockwise circulation around the seamount.

We applied this mechanism to study nonlinear tidalinteraction over Kashevarov Bank due to the two majorconstituents K1 and O1. A series of experiments wascarried out to assess the effects of nonlinear interactionsof these constituents and to demonstrate the balancebetween a linear and nonlinear tendency in the tidalvelocity.

The sum of K1 and O1 tides in the SO was calculatedby the same equations as those used for the eight tidalwaves computation. The boundary conditions at theopen boundaries and astronomic forcing included onlythe K1 and O1 sea level oscillations. The energy of thesystem became stationary after one month, but we con-tinued the calculation to obtain a three-month time se-ries. Two sets of experiments were carried out: 1) allnonlinear terms are included and 2) nonlinear advectiveterms are rejected and bottom friction terms are linear-ized. Comparison of results with and without advectiveterms and nonlinear bottom friction in the equations ofmotion shows that their omission removes the residualcirculation. To investigate the differences between linearand nonlinear tidal dynamics, we considered the tem-poral variability of the tidal currents. The time series ofthe tidal currents at point P (see Fig. 14) over Kash-evarov Bank after 30 days of simulation is shown inFig. 17 for both nonlinear and linear cases.

The temporal variability of the tidal currents resultingfrom the interaction of K1 and O1 constituents exhibits

a fortnightly (13.66 day) oscillation that constitutes theupper and lower envelope of the diurnal signals (Fig.17). In the linear case (Fig. 17, top) the fortnightly en-velopes of the positive and negative values are sym-metrical and no residual currents are generated. In thenonlinear case the upper and lower envelopes of thetidal currents are asymmetrical (Fig. 17, bottom). Thevalues of the upper envelope of the north–south (V)velocity component are greater than those of the lowerenvelope, resulting in a 13–14 cm s21 residual current.Particularly striking is the difference of 45 cm s21 be-tween the upper and lower envelopes of the east–west(U) velocity component.

For additional analysis we used a digital filter tunedto the narrow frequency band around K1 and O1 con-stituents. The digital filter was constructed by IDL andit allows us to apply a variable number of the filterweights (IDL Reference Guide 1995, 1–235). To achieveproper resolution we used a three-month time series ofhourly values and a filter with 501 weights. This filterloses 501 h (close to 21 days) from each end of theoriginal time series. Some results of filtration for theeast–west component at point P over Kashevarov Bankfor the fully nonlinear solution are given in Fig. 18. Thetop portion depicts results of filtration at the K1 periodand the bottom shows a 13.66-day period and meanresidual current derived by low-pass filter. The K1 tidalcurrent over Kashevarov Bank is modulated by a fort-nightly period. The energy alternates between the K1

period and fortnightly period when the K1 constituentoscillates with the maximum velocity the minimum ofthe fortnightly velocity occurs and vice versa. The fort-nightly modulation is different at various points of thedomain due to different intensity of the fortnightly cur-rents. In Fig. 19 the fortnightly current ellipses areshown. The maximum fortnightly current is close to 7


TA

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tude

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LS

Dir

O1

LS

Dir

M2

LS

Dir

S 2

LS

Dir

S1

4584

8914

2803

962

.260

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.9 4.6

138

114

80.2

46.6

11.3 4.5

130

112

27.3

26.2

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165 80

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6.7

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160 77

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4583

8914

1859

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64.

714

311

498

.351

.110

.8 3.3

141 84

35.5

19.6

23.7 0.6

174

105

15.4

12.2

10.3 0.8

228

100

145

8289

1428

279

21.1

17.4

6.7

1.7

135

117

21.1

18.9

3.1

1.4

131

120

5.7

5.3

3.6

0.0

99 103

5.1

1.9

1.5

0.1

82 104

2 2a 2b 2c 2d 2e

4581

8914

2819

912

.329

.719

.627

.933

.530

.622

.5

0.5

4.7

1.7

1.9

5.3

5.1

1.2

137

152

142

160

153

151

135

11.3

29.4

25.4

21.9

35.2

37.7

17.4

0.5

3.5

1.9

1.9

3.7

3.5

3.8

140

141

148

154

150

152

107

3.6

9.7

8.8

8.5

10.9

14.1 6.7

0.5

0.8

0.5

1.1

0.8

1.1

0.4

135

145

144

147

141

144

129

2.1

3.3

2.9

2.5

5.1

4.6

2.5

0.5

0.5

0.2

0.8

0.8

1.3

0.0

130

146

134

149

148

145

126

345

8119

1428

409

11.3 9.9

3.6

0.6

131

136

10.8 9.8

5.1

1.2

107

121

3.6

1.9

1.0

0.1

22 146

4.1

0.6

1.5

0.0

203

168

445

8119

1428

309

16.5

13.1

5.1

0.5

144

140

15.9

11.4

3.1

0.9

154

120

3.6

2.8

1.0

0.1

231

143

3.1

0.9

0.5

0.1

75 157

544

8569

1428

569

8.2

7.3

0.5

0.5

141

140

6.7

8.7

0.5

1.0

142

136

2.1

1.2

0.0

0.0

144

194

3.6

0.4

1.5

0.2

163

187

644

8159

1428

449

8.7

8.0

0.0

0.2

146

147

7.2

8.1

0.0

0.5

147

137

1.0

1.2

0.5

0.1

158

191

1.0

0.4

0.5

0.0

118

191

744

8419

1438

129

6.2

4.7

0.0

0.3

127

139

7.2

6.7

3.1

0.3

166

135

1.5

1.1

0.5

0.0

74 225

3.6

0.4

1.5

0.1

19

844

8369

1438

059

7.2

4.8

1.0

0.0

140

143

6.7

6.9

0.5

0.3

135

135

0.5

0.9

0.5

0.1

173

228

0.5

0.3

0.0

0.0

15 219

944

8349

1438

349

5.1

3.8

0.5

0.6

133

132

8.7

6.5

0.5

0.9

133

131

1.5

1.4

0.0

0.1

159

237

3.1

0.5

1.0

0.1

55 236


FIG. 13. (top) Computed amplitudes (solid lines, cm) and phases(dashed lines, degrees) of surface elevation for the K1 tide at thenorthern coast of Hokkaido. Stars and numbers show the locationand numbering of current stations taken by Aota and Matsuyama(1987). Dots and numbers show the location and numbering of thecoastal sea level gauges. (bottom) Phase lags (travel time) from SoyaStrait to measuring stations. The diamonds denote Odamaki’s (1994)observations, crosses are results of computations, and the solid lineshows a least squares fit to computations.

cm s21. The distribution of tidal ellipses depicts oppositerotation of the tidal currents at the northern and southernslopes of Kashevarov Bank.

To investigate further the nonlinear interaction of thetwo diurnal constituents on tide formation, we describea few additional properties. The fortnightly oscillationsrepresent a rather strong variation in velocity. However,this variation is not reflected as strongly in the sea leveloscillations because the maximum fortnightly sea levelchange is 10 cm. The nonlinear interaction of the diurnaltides influences the pattern of the M2 constituent. TheK1 and O1 tides generate a rather strong current at theM2 frequency (Fig. 20). The current shows a maximumof approximately 9 cm s21 at the top of KashevarovBank where the nonlinear interactions are greatest andrapidly diminishes to 1 cm s21 off the bank where thenonlinear terms are small. The 9 cm s21 current is ap-proximately a half of the total M2 tidal current above

the bank, as computed by the model which incorporatedthe eight tidal constituents (Fig. 7). The nonlinear in-teraction of the diurnal tides has no significant effecton sea level at the M2 tidal frequency and the maximumof amplitude is approximately 1 cm at the bank top.

To identify additional important aspects of the inter-action of K1 and O1 constituents, power spectra basedon FFT with the Hanning spectral window were em-ployed. The purpose of these experiments was to learnhow energy is redistributed from the major tidal con-stituents, K1 and O1, to various parts of the tidal spectrafor the linear and nonlinear interactions. Magnitudes ofthe energy spectra at O1 and K1 periods for the nonlinearand linear experiments are governed by different phys-ics. These magnitudes are controlled by bottom friction,and only nonlinear bottom friction [Eq. (5)] reproducestidal amplitude well. The linearized formula is based onmean values that do not take local conditions into ac-count.

The power spectra of the east–west component of thetidal current at point P is given in Fig. 21. The linearresult is shown in the top panel, the full nonlinear in-teraction is given in the center panel, and the nonlinearinteraction without advective terms (only nonlinear bot-tom friction remains) in the equation of motion is de-picted in the bottom panel.

The power spectrum for the linear problem (Fig. 21,top) shows only one major maximum with energy at K1

and O1 wave periods. In the case of nonlinear interac-tion, the existence of several major and minor maximain the power spectra (Fig. 21, center) is revealed. Themajor maxima occur at the semidiurnal, diurnal, andfortnightly periods. Minor maxima are located close to8 and 6 h. The major maxima can be explained bynonlinear interaction of two original tidal constituents(the so-called compound tides). Moreover, each basicconstituent (K1 or O1) produces, through the nonlinearterms, overtides represented in the power spectra as bothmajor and minor maxima. Assuming the K1 period tobe , this constituent through the advective terms,TK1

generates overtide oscillations whose periods are 5TKi

(Parker 1991). Overtides due to nonlinear bottomT /2iK1

friction are 5 . Here i 5 1, 2, 3, · · · .T T /(2i 1 1)K Ki 1

Thus, for the K1 tide, the major overtides are located at11.96 and 5.98 h (due to advective terms) and 7.98 h(due to bottom friction) and, for the O1, tide the majorovertides are at 12.91 and 6.45 h (due to advectiveterms) and 8.61 h (due to bottom friction). In the semi-diurnal band, the power spectra depict a large maximumof energy caused by several oscillations: a compoundtide at the M2 tide period, an overtide due to the K1 tideat 11.96 h, and an overtide due to the O1 tide at 12.91h. In the bottom plot, the nonlinear interaction is causedby the bottom friction only (i.e., advective terms areneglected), the dominant oscillation is located at thediurnal band, and the first higher harmonic is, as onewould expect, close to 8 h. The latter maximum is ofsecondary magnitude and one can conclude that the bot-


FIG. 14. Computed amplitudes (solid lines, cm) and phases (dashed lines, deg) of surfaceelevation for the K1 constituent above Kashevarov Bank. Bathymetry in meters is given by dottedlines. Point P denotes location for the time series analysis.

FIG. 15. Contours of the maximum K1 tidal currents (solid lines, cm s21) and bathymetry(dotted lines, m) above Kashevarov Bank.


FIG. 16. Computed distribution of the amplification coefficients for sea level (left panels) and currents(right panels) for 26.3 h (top), O1 period (center), and K1 period (bottom) over Kashevarov Bank. Dot–dashlines depict bottom contours.

tom frictional terms do not transfer energy as effectivelyas advective terms. In summary, the nonlinear termstransfer energy from the K1 and O1 constituents towardlonger and shorter periods. It can be deduced from Fig.21 (center panel) that the energy from these waves main-ly sustains oscillations in the fortnightly and semidiurnalbands. At the shorter periods, the magnitudes of energymaxima are very small.

The power spectra shed some light on the nature ofthe fortnightly periodicity, linking it to the advectiveaccelerations. The time variations of the diurnal andfortnightly oscillations are reciprocally connected,which can be inferred through comparison of the topand the bottom panels in Fig. 18. There, larger K1 cur-rents correspond to smaller fortnightly currents. The roleof the bottom friction term in the fortnightly modulationof the K1 constituent depicted in Fig. 18 (top) cannot

be established explicitly. From Fig. 21 (bottom panel)it follows that nonlinear bottom friction does not gen-erate 14-day oscillations. Therefore, bottom friction canonly be an additional factor in K1 modulation if fort-nightly oscillations are already present in the systemdue to advective acceleration.

5. Discussion and conclusions

A tidal model with 59 spatial resolution was appliedto simulate tides in the Sea of Okhotsk. This rather highspatial resolution demonstrated peculiarities that werenot previously disclosed by measurements or modelingstudies. The results of the simulation show enhancementof the semidiurnal tidal amplitudes and currents in theshallow bays (Shelikhov Bay, Penzhinskaya, and Uds-kaya Guba of the Sea of Okhotsk and Tartar Bay of the


FIG. 17. Time series of the U (east–west) and V (south–north) components of tidal current at point P overKashevarov Bank (see Fig. 14) for the linear (top) and nonlinear (bottom) simulations. Forcing is due to K1 1 O1.

FIG. 19. Computed fortnightly tidal current ellipses. The arrow ateach ellipse shows the direction of the current vector rotation. Dashedlines depict bottom contours.

FIG. 18. Time series of the U (east–west) component of tidal currentafter filtration (at point P over Kashevarov Bank): (top) K1 waveperiod, (bottom) 13.66-day period and mean residual current. Forcingis due to K1 1 O1.

Sea of Japan). Amplification of the diurnal amplitudesand currents is associated with the trapping of tidal en-ergy by bottom irregularities like banks and steep slopes.The computation delineated strong diurnal currents atthe eastern coast of Sakhalin, the northern coast of Hok-kaido, the Kuril Islands, Kashevarov Bank, and the cen-tral Kamchatka coast.


FIG. 21. Power spectra of the U (east–west) component of the tidalcurrent at point P in Fig. 14. (top) Linear simulation, (center) fullynonlinear simulation, and (bottom) partly nonlinear simulation, dueto the bottom friction. Forcing is due to K1 1 O1.

FIG. 20. Contours of the maximum M2 tidal current (solid lines,cm s21) due to nonlinear interaction of the K1 and O1 constituentsand depth contours (dotted lines, m) above Kashevarov Bank.

Analysis of the energy budget in the Sea of Okhotskshowed that rather small areas may play an extremelyimportant role in the dissipation of tidal energy. Forexample, in Shelikhov Bay and Penzhinskaya Guba therate of energy dissipation of the diurnal tides is over60% of the overall rate of energy dissipation of thediurnal tidal constituents in the entire Sea of Okhotsk.The total tidal energy dissipated in the Sea of Okhotskis greater than the energy dissipated in the Arctic Ocean(Kowalik and Proshutinsky 1993). Thus, small waterbodies can be significant sinks of tidal energy globally.

Investigations over Kashevarov Bank revealed en-hanced diurnal tidal currents, a clockwise flux of tidalenergy, and a local maximum in the magnitude of energydissipation. The computed diurnal currents above thebank are approximately 10 times greater than the far-field values. Enhancement of diurnal tides over Kash-evarov Bank is related to the near-resonant trapping ofthe tidal energy. This is due to the 26.3-h period thatoccurs in the free oscillation spectra both over Kash-evarov Bank and in the entire SO. Detailed study of thenatural oscillation spectra over Kashevarov Bank showsthat the resonant amplification of velocity takes placeover a wide range of periods.

Diurnal K1 and O1 currents, through the nonlinearinteraction, generate new oscillations at semidiurnal M2

and fortnightly periods, and a residual current. An anal-ogous phenomenon was observed at the Fieberling Guy-ot in the North Pacific (Brink 1995). The nonlinear in-teractions are studied by application of the filter. Thisline of research shows that the amplitude of the basicconstituents (K1 and O1) is modulated by the fortnightlyoscillations. The application of power spectra for in-vestigation of K1 and O1 interactions over KashevarovBank indicates that the nonlinear terms transfer energyfrom the K1 and O1 tides toward longer and shorterperiods. It can be deduced that the energy from these constituents mainly sustains oscillations in fortnightly

and semidiurnal bands. At the shorter periods the mag-nitude of energy maxima is very small.

Acknowledgments. We would like to express our grat-itude to A. Yu. Proshutinsky from the Institute of MarineScience, University of Alaska, Fairbanks, for his helpthroughout the work. We are indebted to anonymousreferees for the thoroughness of their comments andmany improvements that stengthened the paper. Supportof the Office of Naval Research under Grant N00014-95-1-0929 is gratefully acknowledged.

REFERENCES

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Aota, M., and M. Matsuyama, 1987: Tidal current fluctuations in theSoya Current. J. Oceanogr. Soc. Japan, 43, 276–282.

Brink, K. H., 1989: The effect of stratification on seamount-trappedwaves. Deep-Sea Res., 36, 825–844.


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Cartwright, D. E., R. D. Ray, and B. V. Sanchez, 1991: Oceanic tidesmaps and spherical harmonic coefficients from Geosat altimetry.NASA Tech. Memo. 104544, 75 pp.

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