Characteristics of Internal Tides in the Upper Layer of ... · PDF filetides observed in the bay can be explained as internal Kelvin waves. 1. Introduction Internal tides are mostly

Journal of Oceanography, Vol. 53, pp. 143 to 159. 1997

Keywords:⋅ Internal tide,⋅ internal inertialgravity wave,

⋅ internal Kelvinwave,

⋅ reflection andinterference ofinternal tide,

⋅Poincaré mode,⋅ amphidromic point,⋅Sagami Bay,⋅ Izu Ridge.

143Copyright The Oceanographic Society of Japan.

Characteristics of Internal Tides in the Upper Layerof Sagami Bay

YUJIRO KITADE and MASAJI MATSUYAMA

Department of Ocean Sciences, Tokyo University of Fisheries,4-5-7, Konan, Minato-ku, Tokyo 108, Japan

(Received 18 March 1996; in revised form 26 August 1996; accepted 28 August 1996)

Internal tides in the upper layer of Sagami Bay were investigated by long-term temperatureobservations at seven stations during the period from summer to fall in 1991. Statisticalanalysis of the data shows the following features of internal tides. The temperaturefluctuations due to the semidiurnal internal tides are predominant in the bay throughoutthe observational period, but the amplitudes and phases are considerably different amongthe stations. The phases at each station shift with the variations of the basic density fieldas well. Amplitudes of the diurnal internal tides are nearly the same along the bay coast,and phases are mostly independent of the variation of the basic density field. In order toexplain these features of internal tides in the bay, numerical experiments using a two-layer model were performed. The results showed that the semidiurnal and diurnalinternal tides are generated at the northern part of the Izu Ridge and south of the BosoPeninsula. The semidiurnal internal tides generated are propagated into Sagami Bay withcharacteristics of Poincaré and Kelvin modes, and they are reflected at the bay head andinterfere with the incident internal tides. Phase relations of the semidiurnal internal tidesobserved in the bay can be well explained as an interference of the internal tide. Thediurnal internal tide in Sagami Bay is generated south of the Boso Peninsula, andpropagated into the bay trapping along the coast. Characteristics of the diurnal internaltides observed in the bay can be explained as internal Kelvin waves.

1. IntroductionInternal tides are mostly generated by coupling with

barotropic tides at steep slope regions such as a continentalshelf break, sea mounts, and ridges (Rattray, 1960; Baines,1973, 1982; Hibiya, 1986). Significant internal tides are notexpected to be generated at the continental shelf break alongthe Japan coast because the shelf region is narrow, except inthe East China Sea (Kuroda and Mitsudera, 1995). However,the Izu Ridge is anticipated as a generation region of theinternal tides observed along the southern coast of Japan(Maeda, 1971; Matsuyama, 1985; Ohwaki et al., 1994; Kitadeand Matsuyama, 1996), because the water depth around theIzu Ridge abruptly changes and the co-tidal lines of thesemidiurnal and diurnal barotropic tides are nearly parallelto the bottom contours (Ogura, 1933; Nishida, 1980). Thus,the internal tides observed around Sagami and Suruga Baysare likely to be generated at the northern part of the Izu Ridge(Matsuyama, 1985). Recently, we found the existence ofstrong barotropic tidal currents and evidence of the generationof internal tides from the repeated ADCP and XBT mea-surements across the Izu Ridge (Kitade and Matsuyama,1996).

In general, internal tides over a continental shelf aredissipated by bottom friction and turbulent or viscous pro-cess (e.g. Sherwin, 1988; Brink, 1988; Holloway, 1991). Aninternal tide on wide continental shelves usually propagatesshoreward across the shelf in most places (Baines, 1986),and does not set up a cross-shelf standing mode due todissipation. On the other hand, the internal tide is reflectedat the coast with little dissipation on a narrow shelf (Winantand Bratkovich, 1981). Indeed, Matsuyama (1985, 1991)indicated an evidence of reflection of semidiurnal internaltides from the temperature and current measurements inUchiura Bay, located at the head of Suruga Bay in Japan.Further, the semidiurnal internal tide is resonant with thelongitudinal internal seiches of Uchiura Bay under thestratification in summer and fall. By using a two-layernumerical model, Ohwaki et al. (1994) demonstrated thatthe internal tides generated north of the Izu Ridge propagateinto Suruga and Sagami Bays and are reflected at the coastin a complicated manner, especially in Sagami Bay.

Sagami Bay is a very deep bay, having a maximumdepth of more than 1500 m and a narrow continental shelf.The internal tides propagating into the bay are expected to

144 Y. Kitade and M. Matsuyama

be reflected at the coast with slight dissipation. The horizon-tal scale of Sagami Bay (about 40 km) is larger than theinternal radius of deformation (about 10 km), so the Coriolisforce plays an important role in the propagation of internaltides. When the barotropic tidal waves are reflected at thecoast of a rectangular bay, incident waves interfere withreflected waves in the bay, and have the characteristics ofthe Poincaré mode (Taylor, 1921; Defant, 1961; Brown,1973). Internal tidal waves can also have the same charac-teristics as the Poincaré mode in a bay when they behave asstanding waves. From the analysis of temperature and cur-rent data obtained at three mooring stations in Sagami Bay,Kitade et al. (1993) indicated the possibility of reflection ofthe semidiurnal internal tide at the bay coast from thedistributions of amplitude and phase. However, the currentand temperature measurements were performed at onlythree moored stations, which are not enough to clarify thecomplicated characteristics of the semidiurnal internal tide.

The purpose of this study is to clarify propagationcharacteristics of the internal tide in Sagami Bay throughintensive field measurements. Attention is concentrated onthe temporal variations and horizontal distributions of am-plitudes and phases of the semidiurnal and diurnal internaltides. In addition, we discuss dynamic features of internaltides observed in the upper layer of the bay, i.e., generation,propagation, and reflection, by using a numerical model.

2. ObservationLong-term temperature observations were made at

several depths of the upper layer at seven stations in SagamiBay (Fig. 1) during the period from summer to fall in 1991.Thermometer arrays were moored from the bottom withsubsurface floats and a weight at JO and OK, while theywere suspended down from surface floats at the otherstations. In order to record temperature variations in theseasonal thermocline, most of the thermometers were locatedfrom 10 m to 50 m depth. The sampling interval was 20minutes at all sites. The mooring periods and sensor depthsfor each station are presented in Table 1. Temperaturerecords at all stations were obtained simultaneously fromJuly 20 to September 25, 1991. Each thermometer wascalibrated before and after the observations.

Vertical profiles of temperature and density obtainedmonthly at thirteen stations by Kanagawa PrefecturalFisheries Station were also used in this study. Sea level dataobtained at AB (Aburatsubo) were offered from the Geo-graphical Survey Institute.

3. Temperature FluctuationsFigure 2 shows the time series of three-hour running-

averaged temperature obtained at four stations, KS, EN, IT,and PY (see Fig. 1) during the period from July 12 toSeptember 24, 1991. The first three stations were arrangedalong the bay coast and the last station was near center of the

Station Period Sensor depths (m)

OK 1991 Jun. 12–1991 Sep. 25 45, 60, 75JO 1991 Jun. 12–1991 Sep. 25 30, 45, 60, 75KS 1991 Jul. 12–1991 Nov. 6 5, 10, 15, 20, 30, 40, 50EN 1991 Jul. 20–1991 Nov. 16 5, 10, 15, 20, 30IT 1991 Jul. 6–1991 Dec. 13 5, 10, 20, 30, 40, 50KM 1991 May 15–1991 Oct. 15 10, 45PY 1991 Jul. 1–1991 Oct. 30 10, 20, 30, 40

Table 1. The mooring periods and thermometer depths at each station.

Fig. 1. Locations of the mooring stations and bottom topographyin and around Sagami Bay. Numerals in the figure are inmeters. Inset shows the location of Sagami Bay. Symbol �indicates CTD stations.

Characteristics of Internal Tides in the Upper Layer of Sagami Bay 145

bay. Periodic fluctuations with semidiurnal and diurnalperiods are predominant at all four stations throughout mostof the observation period. These fluctuations are intermit-tently intensified at the same time. But the magnitudes of thetemperature fluctuations are different among the four sta-tions, that is, they are larger at KS and IT than at EN and PY.Also, they change with depth at a station. In addition, themodulations of temperature fluctuations are not directlyrelated to the fortnight period, i.e., a spring-neap cycle ofbarotropic tides.

In order to indicate the above periodic fluctuations oftemperature as being due to the internal tides, the time seriesof vertical displacements of isotherms, constructed from thetemperature records at various depths at KS, EN, IT, and PY,are shown in Fig. 3, together with sea level variations at AB.The vertical displacement of the isotherms with thesemidiurnal and/or diurnal periods is over 10 m at allstations, while the sea level displacement is only about 1.6m. The observed temperature fluctuations at the mooredstations are mostly due to the internal tides for the followingreasons: (a) the temperature fluctuations of the semidiurnaland diurnal periods are predominant; (b) the fluctuations aresimultaneously intensified at all stations intermittently; (c)the vertical displacements of the isotherms mostly exceed10 m; (d) the amplitude modulations are not directly relatedto the fortnight period of spring-neap cycle.

The temperature fluctuations at the four stations areamplified in the three periods (see Fig. 2), i.e., the middle ofJuly, the first and middle parts of August, and the first part

Fig. 2. Time variations of three-hours running-averaged temperature obtained at KS, EN, IT, and PY during the period from July 12to September 24, 1991. The spring tides are indicated with open and solid circles at the uppermost part of the figure.

Fig. 3. Time variations of one-hour running-averaged tempera-ture contours with 1.5°C spacing at KS, EN, IT, and PY (lowerfour panels), and predicted sea level at Stn. AB (upper panel)during the period from August 10 to 14, 1991.


of September. Then, the amplification of internal tides isinvestigated to see how it occurred under the basic tempera-ture field. Figure 4(a) shows the time series of 25-hourrunning-averaged temperature obtained at every 10 m depthat KS. The temporal variations of stratification in the upper50 m are shown in this figure. The seasonal thermocline wasseen at the bottom of this surface layer from the beginningof the observations to the first part of October. A remarkable

warming of surface water occurred at the end of July. Lowfrequency variations often occurred through the water col-umn of several tens of meters. In order to compare horizontalvariations of the basic temperature field in the bay, the timeseries of 25-hour running-averaged temperatures at thedepth of 30 m at five stations are shown in Fig. 4(b). Thetemperature variations for all the cases are similar in theoverall feature. Therefore, it is found that the basic tem-perature stratification for the internal tides in Sagami Bay isvariable in time but uniform horizontally. A large increaseof temperature occurred on July 29, August 15, and September9. To understand the temporal variation of internal tides, weseparated the long-term temperature records into three 15-day periods, which have relatively large temperature fluc-tuations and strong stratification (Period 1: July 14 to July29; Period 2: August 2 to August 16; Period 3: August 31 toSeptember 14).

We obtained long-term temperature records from thesea surface to 75 m depth (Table 1). In addition, CTD dataobtained at 13 stations in the bay provided vertical profilesof average temperature and density for each period. Figure5 shows vertical profiles of the temperature and σt averagedover all the CTD stations for each month, namely on July 4,August 2, and September 3 in 1991. These profiles can beregarded as typical stratification corresponding to Periods 1,2, and 3, respectively. The July 4 profile served as a typicalone for Period 1, because little variation of temperatureexisted throughout the month of July as shown in Fig. 4(b).The thermal stratification was much weaker in July than inAugust or September.

4. Characteristics of Internal Tides

4.1 Statistical propertiesFigure 6 shows the power spectra of temperature

fluctuations at each station for Period 1. The figure indicates

Fig. 5. Spatial mean vertical-profiles of temperature (left panel) and σt (right panel) observed in Sagami Bay on July 4 (dashed line),August 2 (dotted lines), and September 3 (solid lines) in 1991.

Fig. 4. (a) Time series of 25-hours running-averaged temperatureobtained at 10 m interval from 10 to 50 m depths at KS duringthe period from July 12 to October 29, 1991. (b) Time seriesof 25-hours running-averaged temperature obtained at a depthof 30 m at each station during the period from July 1 to October29, 1991.


the predominance of the semidiurnal and diurnal compo-nents at most stations, as expected. Other peaks also existedat the biharmonic periods at all stations. The energy level ofthe semidiurnal component changed from station to stationat a given depth, i.e., the highest at IT and the lowest at KM,located only 12 km from IT. The energy level for the diurnalcomponent was also different among the stations.

As the basic temperature field was horizontally uniform(Fig. 4(b)), energy spectra of temperature fluctuations ob-tained at a given depth are proportional to the squaredamplitude of internal tides, i.e., the potential energy. Thepeak heights of energy spectra of the semidiurnal anddiurnal components for each period are shown in Fig. 7. Thepeaks at 12.6 and 24.4 hours with high energy levels wereselected as representative of the semidiurnal and diurnalcomponents, respectively. The spectra showed mostly highenergy levels at 10 m depth for Period 1, and at 30 m depthfor Periods 2 and 3, so that the energy densities at 10 m depthwere displayed for Period 1 and those at 30 m depth forPeriods 2 and 3.

The energy levels for both semidiurnal and diurnalcomponents changed with the period at all the stations. Theenergy densities for the semidiurnal component in Periods 2and 3 are considerably higher than those in Period 1, whilethose for the diurnal component in Periods 2 and 3 are lowerthan those in Period 1.

The energy density for the semidiurnal componentobtained at a given depth is clearly different among thestations. For example, the energy density for the semidiurnalcomponent in Period 1 was the largest at IT (west of the bayhead) and the smallest at KM (west side of the bay). Further,it was considerably larger at IT or KS (east side of the bay)than at the other stations in Periods 2 and 3. Following tothese results, a remarkable feature of the semidiurnal com-ponent is that energy density is variable in space. On theother hand, energy densities for the diurnal component ateach station along the coast of the bay were nearly constantin Period 1. In Period 2, energy density was maximum at EN(center of the bay head). In Period 3, energy densities at theedges of the bay were larger than those at the center of the bay.

Fig. 6. Power spectra of the temperature fluctuations observed at each station in Period 1 (from July 14 to 29).


Table 2 presents the coherence and phase of crossspectra between temperature fluctuations for the semidiurnalcomponent at different stations. Relatively high coherencewas found in the 30 m depth records at every station, so thevalues at the 30 m depth were indicated except OK and KM.The 99% confidence limit of the coherence squared is about0.4 for twenty degrees of freedom. High coherence exists inthe records between different stations, except OK45 inPeriod 1, JO30 in Period 2, and PY30 in Period 3. Thehorizontal distribution of the phase is very complicated inevery period. For Period 1, the phase difference relative toJO30 ranges from 135° to 283° with high coherence, whileOK45 has no correlation with any other records. For Period2, OK45 has a high correlation with the other records andshows a remarkable phase difference from them, whilecoherence between JO30 and other stations is low. ForPeriod 3, OK45 also shows a high correlation with thestations except JO30. A large phase difference exists be-tween OK45 or JO30 and the remaining record. Thesesuggest the complicated distributions of co-phase lines inthe bay. Furthermore, the phase difference between any twoof the records has complicated variations in every period,even for high coherence. For example, the phase differencebetween KS30 and IT30 is 37° in Period 1, while it is 311°in Period 2. The phase difference between OK45 and KM30or IT30 in Period 3 was approximately 180°. This phaserelation is in good agreement with the result observed inSeptember 1986 (Kitade et al., 1993). Although the tem-

perature fluctuations for the semidiurnal component areclosely correlated with the other stations in the bay through-out the three periods, phase difference for the correspondingcombination varies with period, namely with variation ofthe stratification.

Table 3 presents the coherence and phase of the diurnalcomponent obtained for each period. For the diurnal com-ponent, the 10 m data at KS, IT, and KM, with relatively highenergy level, were combined for Period 1. All combinationsof KS10, IT10, and KM10, along the coast of the bay, havehigh coherence. The coherence between JO30 and KS30 hasover a 99% confidence level throughout the three periods,and the phase differences are nearly constant among theperiods. The phase difference between KS and IT is 198° inPeriod 1, which supports that the diurnal period is not due tosurface heating or cooling but the propagation of the diurnalinternal tides along the coast of the bay. The coherence of thediurnal component is smaller than that for the semidiurnalcomponent, except for Period 1. Furthermore, the diurnalcomponent has a lower energy density than the semidiurnalcomponent, so the propagation of the diurnal internal tidescan be detected only along the east coast of the bay in thisobservation. The diurnal component has high coherencealong part of the east coast and the phase difference betweenthe stations JO and OK remains constant with time.

4.2 Vertical structureIn order to investigate the vertical structure of internal

Fig. 7. Distributions of the spectral peak for temperature fluctuations of semidiurnal (left three panels) and diurnal (right three panels)components in each period. The values obtained at 10 m depth for Period 1, and the values at 30 m depth for Periods 2 and 3, areindicated. The values obtained near 10 or 30 m depth are also indicated with a parenthesis for reference.


tides in detail, the temperature fluctuations at each depth areconverted into vertical displacements due to the internaltides. First, the temperature data sets are interpolated to 5 mintervals. The data set for each period is demodulated byFourier Transformation, and then the amplitude and phaseof the temperature fluctuations at each depth for semidiurnaland diurnal components are estimated. Finally, verticaldisplacements of a water particle are calculated from theamplitude and vertical gradient of temperature (e.g. Halpern,1971).

Figure 8 shows the amplitude and phase of verticaldisplacement due to internal tides for the M2 and K1 con-stituents in each period. Since the temperatures at KS, EN,and IT were obtained throughout the water column, vertical

structures of internal tides can be investigated. The distancefrom the origin and angle from the A-axis indicate ampli-tude and phase at each depth, respectively (Webb and Pond,1986). For the M2 constituent, the amplitude at middle depthis the largest and phases are nearly constant throughout thewater column at almost all stations in Periods 1 and 2. Theamplitude for the M2 constituent at IT reaches 15 m in Period2. In Period 3, the amplitudes for the M2 constituent in thelower layer are relatively large, and phase difference be-tween the upper and lower layers is about 45° at KS and EN.On the other hand, the vertical displacements for the K1

constituents are larger in the upper and lower layers than inthe middle layer and their phases in the former are out ofphase to the latter in Period 1. In Periods 2 and 3, theamplitudes for the K1 constituent are smaller than those inPeriod 1 and the vertical variations of the phase becomelarge. Since the vertical displacements are different in each

Table 2. Values of coherence squares (upper) and phase differ-ence (lower) of semidiurnal components. Numerals attachedafter the station name indicate the record depth. Coherencesquared of over 0.4 is shaded. (a) Period 1; (b) Period 2; (c)Period 3.

Table 3. As for Table 2, but for diurnal components. (a) Period 1;(b) Period 2; (c) Period 3.


period, modal structures of the internal tides which dependon stratification are investigated at these three stations next.

The vertical structure of internal long waves over a flatbottom can be expressed as the sum of a series of verticalmodes, which are the eigensolutions of the equation (e.g.LeBlond and Mysak, 1978)

d2φn z( )dz2 + N z( )2 − ω2

cn2 φn z( ) = 0, 1( )

where cn2 = ghn (g is the gravitational acceleration and hn is

the equivalent depth for mode n) is the separation constant.The φn(z) is the vertical displacement at depth z (negativedownward) for mode n, N(z) the buoyancy frequency(N2(z) = –g/ρ·∂ρ/∂z, where ρ is the density), and ω the fre-quency. Given the boundary condition φ(z) = 0 at z = 0 and–H, Eq. (1) constitutes an eigenvalue problem. Verticalmodes are computed from Eq. (1) by using a numericalscheme under the mean density profile at each period.Figure 9 shows the typical vertical profile of the first threemodes at KS computed from the density field observed onSeptember 3, 1991. The modal structures at other stationswere essentially similar to those at KS.

In general, the variations of vertical displacement due

Fig. 8. Amplitude and phase of vertical displacement at 5 m interval for the M2 and K1 constituents at KS, EN, and IT. The deepestdepth is marked by a larger symbol.

Fig. 9. Vertical profiles of vertical displacement for the lowestthree internal modes at the depth of 55 m calculated from thedensity field observed on September 3, 1991.


Mode KS EN IT

P. Energy Phase P. Energy Phase P. Energy Phase

×10 –2 (J m –3) (%) (Deg.) ×10 –2 (J m –3 ) (%) (Deg.) ×10–2 (J m –3) (%) (Deg.)

Period 1 1 3.17 64.4 –113 — 28.70 86.8 –1262 0.16 3.3 0 — 0.53 1.6 1023 0.26 5.3 36 — 2.13 6.4 28

Total 4.92 — 33.08

Period 2 1 36.97 69.2 –125 4.32 39.8 167 35.41 39.3 –1522 14.98 28.0 69 2.87 26.3 –1 46.47 51.8 133 0.35 0.6 –27 2.16 19.9 145 3.67 4.1 168

Total 53.42 10.88 90.10

Period 3 1 34.58 69.5 75 5.56 44.4 62 41.83 78.6 922 7.33 14.7 84 4.74 37.8 –123 0.16 0.3 773 5.57 11.2 170 1.94 15.5 54 6.80 12.8 114

Total 49.76 12.54 53.24

(b) S2 constituent

Mode KS EN IT


×10 –2 (J m –3 ) (%) (Deg.) ×10 –2 (J m –3 ) (%) (Deg.) ×10 –2 (J m –3 ) (%) (Deg.)

Period 1 1 36.84 88.3 –175 — 177.64 92.4 –1132 0.79 1.9 52 — 11.23 5.8 –1093 0.13 0.3 –52 — 1.37 0.7 –83

Total 41.70 — 192.21

Period 2 1 121.63 74.0 –61 59.22 64.6 –120 381.48 60.1 –1152 39.61 24.1 114 7.58 8.3 69 234.35 35.9 613 0.24 0.2 95 16.13 17.6 –125 4.04 0.6 –165

Total 164.41 91.73 634.61

Period 3 1 255.35 76.0 –148 41.22 67.9 –118 173.17 86.6 1702 37.84 11.3 –26 16.56 27.3 42 6.68 3.3 –583 0.23 6.9 –155 1.76 2.9 –123 9.04 4.5 144

Total 336.18 60.72 200.04

(a) M2 constituent

Table 4. Available potential energy density.

to the internal long waves can be represented by asuperposition of a large number of vertical modes M (e.g. Gill,1982), namely,

A z( ) = anφn z( )n=1

M

∑ , B z( ) = bnφn z( )n=1

M

∑ , 2( )

where A(z) and B(z) are observed displacement at the depthof z as shown in Fig. 8, and an and bn are the amplitude formode n. To estimate these amplitudes, the vertical modeswere fitted by the least squares method to the verticaldisplacement at each station of each period. As a result, 95percent of the total variance was in the lowest seven modesat all stations, i.e., M = 7. The potential energy density of


Mode KS EN IT


×10–2 (J m –3) (%) (Deg.) ×10–2 (J m –3) (%) (Deg.) ×10 –2 (J m –3) (%) (Deg.)

Period 1 1 3.63 10.6 152 — 55.64 74.6 132 25.07 73.5 173 — 6.23 8.4 –593 0.91 2.7 103 — 9.03 12.1 16

Total 34.12 — 70.45

Period 2 1 9.00 78.2 140 8.40 66.6 70 8.63 46.9 –332 1.04 9.0 –126 1.42 11.2 105 7.23 39.2 1283 0.22 1.9 39 1.56 12.3 –36 0.11 0.6 40

Total 11.51 12.62 18.42

Period 3 1 10.77 67.9 –138 6.42 55.9 –172 13.31 63.0 –802 0.17 10.8 –169 3.21 28.0 131 6.22 29.5 1183 2.19 13.8 –116 1.00 8.7 –160 0.12 0.6 0

Total 15.86 11.48 21.12

(b) K1 constituent

Mode KS EN IT


×10 –2 (J m –3) (%) (Deg.) ×10 –2 (J m –3 ) (%) (Deg.) ×10–2 (J m –3) (%) (Deg.)

Period 1 1 8.41 62.7 –174 — 1.73 24.9 32 3.23 24.1 –95 — 2.41 34.8 –913 1.19 8.9 –157 — 1.59 23.0 26

Total 13.42 — 6.92

Period 2 1 1.87 31.6 101 2.00 19.8 171 3.19 56.7 332 3.60 60.9 45 3.19 31.5 –10 0.25 4.5 –1733 0.05 0.8 –166 3.36 33.2 179 0.92 16.3 –179

Total 5.92 10.11 5.63

Period 3 1 10.53 28.5 –97 8.38 58.9 –45 3.66 37.9 –1482 19.58 53.0 1 2.65 18.6 –80 4.19 43.4 –1663 0.95 2.6 –113 2.81 19.7 –21 0.54 5.6 –120

Total 36.91 14.24 9.66

(a) O1 constituent

Table 5. Available potential energy density.

internal tides averaged over the entire depth is estimated as(e.g. Gill, 1982)

P. E. = 14H

ρ0 N 2 z( )ηn2 z( )dz

−H

0

∫ , 3( )

where ρ0 is density in the undisturbed state, and ηn(z) is the

calculated amplitude of the vertical displacement for then-th vertical mode at z.

Table 4 indicates potential energy densities per unitvolume,

P. E. , of the first three modes for the M2 and S2

constituents, together with the phase of each mode obtainedby fitting. The first mode is predominant except for the S2

constituent of Period 2 at IT. The sum of the potential energy


densities of the first mode for the M2 and S2 constituents isthe largest at IT in Periods 1 and 2, but at KS in Period 3. Thepotential energy density for the M2 constituent is four to tentimes larger than that for the S2 constituent at all stations.The phase difference of the first mode between KS and ITfor the M2 constituent is 62° in Period 1, 306° in Period 2,and 318° in Period 3. These phase differences agree roughlywith those for the semidiurnal component estimated by thespectra analysis of temperature fluctuations (Table 2).Therefore, the phase differences between the temperaturefluctuations presented in Table 2 can be considered to bemainly due to the first mode internal tides.

Tables 5 presents the potential energy densities of thefirst three modes for the O1 and K1 constituents. In manycases, the second mode is more predominant than the firstmode for the O1 constituent, but the first mode is predomi-nant for the K1 constituent. Total potential energy density ateach station is smaller in Periods 2 and 3 than in Period 1.However, the sum of energy densities for the O1 and K1

constituents at each station is almost the same in eachperiod. Although the energy density of the temperaturefluctuations at 10 m depth is higher for the diurnal componentthan for the semidiurnal component at KS for Period 1, thesum of potential energy density for the M2 and S2 constituentsis comparable to that for the K1 and O1 constituents.

4.3 Summary of characteristics of observed internal tidesThe characteristics of internal tides obtained from the

temperature measurements at seven stations in Sagami Bayare summarized as follows: (1) the semidiurnal internal tidewas nearly comparable to the diurnal one in Period 1, but theformer dominates the latter in Periods 2 and 3; (2) amplitudeof the semidiurnal internal tides was larger at IT (west of thebay head) or KS (east side of the bay) than the other stations;(3) the phase of the semidiurnal internal tide, even for highcoherence, showed complicated distributions as well; (4)amplitudes of the diurnal internal tides were nearly equalalong the coast and were larger than that at center of the bay;(5) coherences for diurnal internal tides between two sta-tions were not very high, but a high coherence region existedalong part of the eastern coast with almost the same phasedifference; (6) most of the vertical structure of internal tideswere in the first seven modes, and the first mode wasparticularly predominant.

5. Modeling of the Internal Tides in Sagami Bay

5.1 ModelingIn order to understand deeply the behavior of the

internal tides observed in Sagami Bay, numerical experi-ments were performed. The internal tides observed in theseasonal thermocline have the properties of an interfacialwave (Baines, 1982; Largier, 1994). Further, the dominanceof the first internal mode indicates that the internal tides

observed in the upper layer of the bay behave as an interfa-cial wave. Thus, we can use a two-layer model to investigatethe generation and propagation process of the observedinternal tides.

Figure 10 shows the computational domain of 155 km(from north to south) and 126 km (from east to west). Thebottom topography used in the model was obtained from abathymetric chart with 1 km × 1 km grid point resolution,and depth in the areas shallower than 50 m were set to 50 m.As Sagami Bay is very deep (see Fig. 1), the assumption ofthe shallowest depth of 50 m is expected to have little effecton the results. In this model, only barotropic tides wereforced from the open boundaries, as shown by dotted linesin Fig. 10.

Under hydrostatic and Boussinesq approximations,linearized basic equations for the upper layer are given by

∂U1

∂t+ f×U1

= −g h1 + ζ1 − ζ 2( )∇ hζ1 + Ah∇ h2U1 − γ i

2u′ u′ , 4( )

Fig. 10. Model ocean with realistic bottom topography. Dashedlines indicate open boundaries. The mooring stations of fieldobservations and tidal stations are indicated.


∂ζ1

∂t= −∇ h ⋅U1 − ∇ h ⋅U 2 , 5( )

and for the lower layer

∂U 2

∂t+ f×U 2 = − ρ1

ρ2

g h2 + ζ 2( )∇ hζ1

− ∆ρρ2

g h2 + ζ 2( )∇ hζ 2 + Ah∇ h2U 2

+γ i2u′ u′ − γb

2 U 2 U 2

h2 + ζ 2( )2 , 6( )

∂ζ 2

∂t= −∇ h ⋅U 2 , 7( )

and ∇ h = i∂/∂x + j∂/∂y, where i and j indicate unit vectorsalong x (eastward) and y (northward) axes, respectively. Heresubscripts 1 and 2 indicate the upper and lower layers,respectively. U is the volume transport vector, u′ the dif-ference of velocity vector between upper and lower layers,and and ζ1 and ζ2 are the vertical displacement of surfaceand interface, respectively. The h is the mean thickness ofthe upper layer, Ah the coefficient of horizontal eddy vis-cosity, and γi and γb the coefficients of interface and bottomfriction, respectively. The f is the vector of Coriolis’ pa-rameter (|f| = 8.36 × 10–5 s–1), g the gravitational acceleration(9.8 m s–2), and ρ and ∆ρ the density and the density dif-ference between the upper and lower layers, respectively.The coefficient of bottom friction is put into γb

2 = 0.0026everywhere in the model (e.g. Imasato, 1983). The coeffi-cients of horizontal eddy viscosity and interface frictionwere assumed constant at Ah = 50 m2s–1 (Imasato, 1983) andγi

2 = 0.0013 (Ohwaki et al., 1994), respectively, except near

Station Semidiurnal (M 2 constituent) Diurnal (K1 constituent)

Amplitude difference Phase difference Amplitude difference Phase difference

(cm) (%) (Deg.) (cm) (%) (Deg.)

MERA 1.16 3.2 2.72 0.10 0.4 –0.81TATEYAMA –1.30 –3.5 –1.17 0.20 0.8 1.88ABURATUBO 0.30 0.8 1.65 –0.69 –3.0 –1.42YOKOSUKA –1.44 –3.5 –0.14 –0.25 –1.0 0.02MANAZURU 0.04 0.1 –2.57 0.04 0.2 –0.82ITO 1.33 3.8 –1.91 –0.31 –1.3 0.30MINAMIIZU –0.42 –1.1 –0.16 –1.11 –4.6 2.31OKADA –0.44 1.3 1.26 0.72 3.1 –1.47

Table 6. Difference between harmonic constants for surface tides from observed and calculated.

the open boundary. In order to suppress the reflection ofinternal tides at the open boundary, coefficients of hori-zontal eddy viscosity and interface friction are set to becomelarger exponentially toward the outer ten grid points at theboundary. The non-slip boundary condition is applied at thecoast. Surface elevations due to barotropic tides at the openboundary are given by

ζ1 = A x, y( )sin ωt − θ x, y( )( ), 8( )

and

ζ 2 = h2

hζ1, 9( )

where A(x, y) and θ(x, y) are the amplitude and phase at theboundary, respectively. In this model, the M2 and K1 con-stituents, having amplitude greater than other constituentsin the bay, are chosen for the semidiurnal and diurnalcomponents, respectively. The amplitude and phase alongthe open boundary are following to Ohwaki et al. (1991). Inorder to make the difference between the calculated andobserved surface elevations as small as possible at the eightmonitoring stations, open boundary conditions were adjustedin amplitude and phase. Table 6 indicates the difference ofharmonic constants for the M2 and K1 constituents betweenthe calculated and observed surface elevations at each tidalstation. The amplitude and phase differences between theobserved and calculated values were less than 5% and ±3°,respectively. Thus, barotropic tides in and around SagamiBay were considered to be well simulated by applying sucha boundary condition.

5.2 Generation and propagation of the internal tidesIn the two layer model, the forcing term for the gen-

eration of the internal tides is expressed by |–u·∇ H/H|, where

12345678


Table 7. Conditions of density stratification in the numerical experiment.

u is the velocity of the barotropic tide, and H the water depth(Serpette and Maze, 1989; Ohwaki et al., 1991). Figure 11shows the distribution of magnitude of this term for the M2

and K1 constituents. Large values are distributed over thenorthern part of the Izu Ridge and off the Boso and IzuPeninsulas, and the M2 and K1 have a similar distribution.Therefore, internal tides can be mostly generated over thenorthern part of the Izu Ridge and in the south of the Bosoand Izu Peninsulas around Sagami Bay.

For the semidiurnal and diurnal components, threenumerical experiments were performed using the same openboundary condition under different stratification. Threestratification conditions shown in Table 7 were given byreferring to the averaged density profiles for July, August,and September, 1991 (see Fig. 5).

Figure 12 shows the co-range and co-tidal charts of theinterface displacement for the semidiurnal component. Thesecharts were constructed from the data obtained during theone tidal period from 132 to 144 lunar hours. The amplitudeover the entire bay is larger for both Cases 2 and 3 than forCase 1. The amplitude has a very complicated distribution inall three cases, especially near the coast. Some amphidromic

points for the semidiurnal internal tide are formed in the bay.The existence of the amphidromic points leads to the com-plicated amplitude distribution. The locations of theseamphidromic points are slightly different in each case de-pending on the slightly different stratification. The ampli-tude distributions of semidiurnal internal tides calculated bythe numerical experiments agree with those of the observedtemperature fluctuations shown in Fig. 7. The amplitudesare larger at the east and west of the bay head than near thecenter of the bay.

Table 8(a) indicates the phase at the seven monitoringstations for semidiurnal internal tides obtained by the nu-merical experiments with the observed phase. The sevenobserved stations for the numerical results were chosen asthe monitoring stations for comparing the observed andnumerical results. The phase at each station for the observedresults is expressed by the difference from IT or KS tocompare with the results of numerical experiment. Thephases at most stations in the numerical experiment agreewith those observed within ±30° except at JO. The ampli-tude at JO is the smallest for both the observed and ex-perimental results, so there are large phase differences. On

Fig. 11. Distributions of the maximum value of the forcing term |–u·∇ H/H| for M2 (left panel) and K1 (right panel) constituents. Unitsare the 1.0 × 10–5 s–1.

Experiment Thickness of upper layer ∆ρ/ρ2

Case 1 (Period 1) 25 m 0.0025Case 2 (Period 2) 30 m 0.0033Case 3 (Period 3) 35 m 0.0030


Fig. 12. Co-range (left) and co-tidal (right) charts for the M2

constituent of internal tides calculated by numerical experi-ment. (a) Case 1; (b) Case 2; (c) Case 3.

the contrary, phases at KS, IT, and KM agree well with theobserved results in all cases. Especially, phases in Case 3agree with those observed in Period 3, in such a way that thephases at KS and EN are almost in phase, but OK is out ofphase to IT (or KM).

Figure 13 shows the co-range and co-tidal charts of theinterface displacement for the diurnal component constructedfrom the data obtained during the period from 120 to 144hours. The amplitudes of the diurnal internal tide in SagamiBay are relatively small except around Oshima Island andalong the western coast of Boso Peninsula. The amphidromicpoint is formed near the center of the bay, and the phase linesalong the coast rotate cyclonically around it. In the bay, theco-amplitude lines of the diurnal internal tide are large alongthe coast of the bay in Cases 1 and 2. The phase andamplitude distribution in the bay imply that the diurnalinternal tide behaves like a free internal Kelvin wavepropagating along the bay coast.

Table 8(b) indicates the phases obtained by the nu-merical experiments and observations. The calculated phasesagree well with the observed ones. In addition, the phase ateach monitoring station is similar among the three cases,

Table 8. The phases of internal tides obtained by the numericalexperiments and observations. The phases with parenthesishave coherence squared less than 0.4. (a) Semidiurnal compo-nent; (b) diurnal component.

*Standardized phase.

Fig. 13. As in Fig. 12 but for the K1 constituent of internal tides.(a) Case 1; (b) Case 2; (c) Case 3.


i.e., the phase differences between JO and KS calculated bythe numerical experiments are 48°, 45°, and 28° for Cases 1,2, and 3, respectively. Thus, the calculated phase for thediurnal internal tides is little dependent on the variation ofthe basic density fields. These characteristics of amplitudeand phase obtained by the numerical experiments agree wellwith those of the observed semidiurnal and diurnal internaltides.

Consequently, the characteristics of internal tides cal-culated by the numerical experiments are as follows: (1) theamplitude of the semidiurnal internal tide in Sagami Bay isapproximately comparable to that of the diurnal internal tidein Case 1, while the former is larger than the latter in Cases2 and 3; (2) the amplitude of vertical displacement for thesemidiurnal internal tide shows a complicated distributionin the bay, that is, amplitudes are larger at KS and IT than atEN, JO, and PY; (3) the phase differences between thevertical displacements for the semidiurnal internal tide alsohave complicated variations among the cases; (4) the ampli-tude of the vertical displacement for the diurnal internal tideis relatively larger along the coast than at center of the bay;(5) phase distributions for the diurnal component are similaramong the three cases. Although the amplitudes of internaltides calculated by the numerical experiments are generallysmaller than those by the observations, the distributions ofamplitude and phase of the internal tides are in goodagreement between the numerical experiments and obser-vations.

6. DiscussionThe amplitudes of internal tides estimated by the nu-

merical experiments are smaller than those observed in theoverall feature. For example, the amplitude of the internaltide observed at IT for the M2 constituent in Period 2 exceeds10.0 m in the middle layer, whereas the amplitude calculatedby the experiment is only 3.4 m. Then we compare thepotential energy between the model and observation. Themean potential energy density per unit area for the two-layermodel is given by

P. E.= 14

δρgζ 22 , 10( )

where δρ is the density difference between the upper andlower layers, and ζ2 the amplitude of interface displacement(Phillips, 1977). Therefore, the calculated potential energydensity per unit area at IT in Case 2 is 93.5 J m–2. Whenaveraged over the water depth of 55 m, the potential energydensity per unit volume is estimated as 1.70 J m–3. Then, thepotential energy density calculated by the numerical experi-ment is the same order as the observed one (3.81 J m–3) asshown in Table 4. The small amplitude of the verticaldisplacement in the numerical experiments can be inter-preted as due to the concentration of potential energy in the

interface for a two-layer model. The phase of the internal tidesobserved in the upper layer at the seven stations in the bayis satisfactorily simulated by the two-layer numerical model.Thus it is concluded that most of the observational resultscan be explained by the two-layer numerical model.

Next we shall discuss the propagation process of thesemidiurnal and diurnal internal tides in Sagami Bay. Forsimplicity, the internal-tide propagation process is consid-ered in a rectangular bay with uniform depth. By using theboundary condition of no normal flow at both sides and headof the bay, the dispersion relation of internal waves in arectangular bay is given by

K 2 = ω2 − f 2

ghn

− m2π2

L2 , 11( )

where K is the along-bay component of wave number, m thecross-bay mode number, L the width of the bay, and ω, f, g,and hn are given in the previous section (e.g. LeBlond andMysak, 1978). The conditions for propagation of internalwaves having the characteristics of Poincaré modes in arectangular bay can be obtained as

ω2 > f 2 + m2π2ghn

L2 ≡ ωc2 12( )

from the dispersion relation. Thus, internal waves can bepropagated with the Poincaré mode when the angular fre-quency of incident waves, ω, is larger than ωc. When ap-plying this condition to Sagami Bay, the width of the bay isabout 40 km, f = 8.36 × 10–5 s–1, and m = 1. The condition(12) becomes

ghn <L ω2 − f 2

π≈ 1.44 m s−1( ). 13( )

This condition is satisfied for the M2 constituent when thephase velocity of the internal gravity wave is smaller than1.44 m s–1. Under the two-layer approximation, the phasevelocity of the internal gravity wave is smaller than 1.44m s–1 for every case. Consequently, the semidiurnal internaltide can propagate with the characteristics of the Poincarémode in Sagami Bay. Then semidiurnal internal tides can bereflected at the bay head and interfere with incident internaltides in the bay. The interference is considered to be importantto produce the amphidromic points along the coast of thebay. On the other hand, since the inertial period at SagamiBay is about 20.9 hours, the diurnal internal tide can propagateonly with the characteristics of Kelvin modes (Ohwaki et al.,1994). Thus, the diurnal internal tide is propagated along thecoast, and its amplitude is maximum at the coast of the bay,i.e., as an internal Kelvin wave.


7. SummaryThe internal tides in the upper layer of Sagami Bay are

investigated by using the temperature observations fromsummer and fall in 1991. The observed results are attemptedto be explained by numerical experiments. Then, the char-acteristics of internal tides can be summarized as follows.

The semidiurnal internal tide observed in Sagami Bayis generated at the northern part of the Izu Ridge and thesouth of the Boso Peninsula, and then is propagated into thebay having the characteristics of Poincaré and Kelvin waves.The internal tidal waves are reflected at the bay coast withless dissipation and the reflected internal waves interferewith incident ones. Then, the internal tidal wave behaves asa standing wave, so the vertical displacement for thesemidiurnal internal tide has a specific distribution, such aslarge amplitudes east and west of the bay head and smallamplitude regions (i.e., the amphidromic points) at thecenter of the bay, at the center of the bay head, and at the westcoast. The locations of internal amphidromic points shiftwith variations of the wave length of the internal tide,namely the variations of the density stratification. Since theco-phase lines are concentrated at the amphidromic points,the phase distributions are changed largely together with thevariations of density stratification. The phase relations ofthe semidiurnal internal tides observed in the bay are suf-ficiently explained as the interference between the incidentand reflected internal waves.

The diurnal internal tide is also generated at the northernpart of the Izu Ridge and south of the Boso Peninsula. Thediurnal internal waves can propagate only as internal Kelvinwaves in and around Sagami Bay because the diurnal periodis longer than the inertia period. Thus, the diurnal internaltide generated at the northern part of the Izu Ridge cannotpropagate into Sagami Bay (Ohwaki et al., 1994), while thepart of the diurnal internal tide generated south of the BosoPeninsula propagates into the Bay. The diurnal internalwaves in Sagami Bay propagate along the bay coast asinternal Kelvin waves.

AcknowledgementsWe wish to thank Drs. H. Sudo and H. Nagashima for

their useful discussions and valuable comments. We alsowish to thank Dr. S. Iwata for his help in carrying out theobservations and Drs. A. Kamatani, A. Ohtsuki, J. Yoshida,and A. Ohwaki for their useful comments. Thanks areextended to the captain and crew of the R/V Ushio for theirhelp in the observations. Numerical experiments wereconducted on the FACOM M760/6 at Tokyo University ofFisheries.

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Characteristics of Internal Tides in the Upper Layer of ... · PDF filetides observed in the bay can be explained as internal Kelvin waves. 1. Introduction Internal tides are mostly