A Model Study of Internal Tides in Coastal Frontal Zonerainbow.ldeo.columbia.edu/~dchen/collection-dchen/jpo03a.pdfInternal tides near a midlatitude shelf–slope front are studied

170 VOLUME 33J 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

q 2003 American Meteorological Society

A Model Study of Internal Tides in Coastal Frontal Zone*

DAKE CHEN, HSIEN WANG OU, AND CHANGMING DONG

Lamont-Doherty Earth Observatory of Columbia University, Palisades, New York

(Manuscript received 18 December 2001, in final form 12 July 2002)

ABSTRACT

Internal tides near a midlatitude shelf–slope front are studied using an idealized numerical model, with emphasison their structure, energetics, and mixing effects. It is found that the properties of internal tides are highlydependent on frontal configuration and tidal frequency. At a winter front, energetic internal tides are generatedand arrested in the frontal zone; the cross-shelf flow tends to be surface (bottom) intensified by a large internalcirculation cell at the diurnal (semidiurnal) frequency. At a summer front, the diurnal internal tide is still trapped,but a semidiurnal internal tide propagates out of the frontal zone in the offshore direction while arrested at theinshore boundary. The presence of the shelf–slope front enhances the generation of internal tides, and it alsocauses an amplification of the semidiurnal internal tide by trapping its energy in the frontal zone. This ampli-fication is most prominent at the offshore boundary of the winter front and the inshore boundary of the summerfront, where strong tidal refraction takes place. Internal tides can cause significant mixing and dispersion in thefrontal zone, with the semidiurnal internal tide being most effective toward the frontal boundaries, and thediurnal internal tide more effective near the site of generation.

1. Introduction

Internal tides and coastal fronts are both frequentlydiscussed topics in oceanographic literature, but theyhave rarely been studied in the same context and littleattention has been paid to the interaction between them.Traditional theories of internal tides are based on linear,inviscid models with only vertical stratification (e.g.,Wunsch 1975; Baines 1982), and thus they are not read-ily applicable to coastal frontal regions where sharp hor-izontal density gradient and strong mixing exist. On theother hand, models of frontal dynamics usually do notexplicitly take into account the effect of internal tides,although it has been suggested that internal tides couldprovide a significant energy source for mixing in coastalfrontal zones (e.g., Huthnance 1989; Brickman and Lod-er 1993). In this study, we use a primitive-equationcoastal ocean model to examine the generation, prop-agation, and dissipation of internal tides in the coastalfrontal zone near the continental shelf break.

There is an increasing number of studies in whichprimitive-equation numerical models were applied toinvestigating tidal motions near the shelf break or bankedge. For example, Lamb (1994) used a nonhydrostatic

* Lamont-Doherty Earth Observatory Contribution Number 6326.

Corresponding author address: Dr. Dake Chen, Lamont-DohertyEarth Observatory, Columbia University, Palisades, NY 10964.E-mail: [email protected]

model to simulate the formation of internal wave boresby strong cross-shelf tidal flow, though his model wasinviscid and assumed a horizontally uniform stratifi-cation. Holloway (1996) studied the internal tides overa cross section of the Australian North West Shelf witha two-dimensional version of the Princeton Ocean Mod-el, but he also initialized his model with level isopyc-nals. Chen and Beardsley (1995, 1998), using the samePrinceton Ocean Model, examined the tidal rectificationand mixing over Georges Bank. They were able to pro-duce not only internal tides but also a tidal front at thebank edge, yet they did not try to explore the relationbetween the two. In their model, the net effect of internaltides was difficult to estimate because it could not beclearly separated from that of the surface tide.

The hydrography near the shelf break is often char-acterized by a seasonally varying density front. Duringwintertime, vertical stratification is largely destroyed bysurface cooling and wind stirring, and the fresher shelfwater is separated from the slope water by a front thatintersects both the bottom and the sea surface, a com-mon feature found along the mid-Atlantic seaboard ofthe North America. During summertime, a seasonalthermocline is formed in the upper water column byincreased insolation and diminished wind, but it maynot extend onto the shelf in regions of strong tidal cur-rents and shallow water depth. In such a case the tidallymixed shelf water is separated from the stratified slopewater by a double front: a lower branch that intersectsthe bottom and an upper branch that intersects the sea

JANUARY 2003 171C H E N E T A L .

surface, both of which merge to the seasonal thermo-cline offshore. The internal tides at a summer front maydiffer significantly from those at a winter front becauseinternal tidal energy can radiate away from the formerbut not from the latter.

The theoretical work of Ou and Maas (1988) is ofparticular relevance to the present study. They examinedthe internal tides at a winter shelf–slope front using atwo-layer analytical model in which the interface in-tersects both the bottom and the surface. According totheir model solution, internal tides should cause a sur-face (bottom) intensification of the cross-shelf flow atthe diurnal (semidiurnal) frequency, and the semidiurnaltide should be preferentially amplified due to the pres-ence of superinertial eigenmodes. Another study of rel-evance here is the earlier work of Chuang and Wang(1981). They developed a finite difference model of thelinear wave equations to evaluate the effects of a fixeddensity front on internal tides. Their front merges tolevel isopycnals away from the frontal zone, thus in away resembling the summer condition. They found thata shelf–slope front has strong effects on the generationand propagation of internal tides, and that these effectsare highly sensitive to the orientation of the front.

Here we evaluate the effects of both winter and sum-mer fronts on internal tides, as well as the roles of in-ternal tides in these two different frontal regimes. Inparticular, we address the following questions: What isthe structure of internal tides in the presence of a front?How does this structure vary from winter to summerfronts and from diurnal to semidiurnal tides? What con-trols the distribution and partition of internal tidal en-ergy in the frontal zone? And how do internal tidescontribute to the mixing and dispersion of passive trac-ers? The rest of this paper is organized as follows. Themodel configuration is described next in section 2. Thegeneral results for winter and summer fronts are pre-sented in sections 3 and 4, respectively. The energeticsof internal tides is considered in section 5 and the mixingeffects of them in section 6. Last, a summary and somediscussion are given in section 7.

2. Model configuration

The model used in this study is the primitive equation,free-surface, coastal ocean model originally developedby Wang (1982, 1985). It has been applied to variouscoastal studies, including simulations of the upwellingfront off the northern California coast (Chen and Wang1990) and the tidal front in the Celtic Sea (Wang et al.1990). In the present version of the model, we use theMellor and Yamada (1982) level 2.5 turbulence closurefor vertical mixing, Smagorinsky’s (1963) parameteri-zation for horizontal mixing, and the Smolarkiewicz(1984) antidiffusive scheme for density and tracer ad-vection. These features are essential for minimizingnonphysical mixing and maintaining a sharp front in themodel. Another modification is that here we directly

solve for density field instead of temperature and salinityso that the equation of state is not needed. Althoughthis model by design is fully three-dimensional, it isconfigured in this study on a two-dimensional cross-shelf section and uniformity is assumed in the alongshelfdirection. The effectiveness of this simplification hasbeen demonstrated by numerous previous studies, in-cluding those cited in the last section.

The model is placed at 418N so that the local inertialperiod is about 17 h. Since the model location is northof the critical latitude for the diurnal tide but south ofthat for the semidiurnal tide, the diurnal internal tide isexpected to be trapped in the region of generation whilethe semidiurnal internal tide is expected to travel away.This is representative of the situation in the midlatitudeoceans. The model domain is 135 km wide with a depthof 100 m on the shelf and 200 m offshore. For simplicity,the slope between the shelf and the offshore region hasa linear profile, with the corners at the shelf break andat the bottom of the slope being rounded over a fewgrid points. Figure 1 shows the middle portion (230 to30 km) of the model, within which the horizontal andvertical resolutions are 300 and 3.3 m, respectively. Theregions outside of this portion are sponge zones overwhich the horizontal grid spacing increases linearly byan order of magnitude. The time step of model inte-gration is about 3 min.

The model is forced by a barotropic tide through spec-ification of sea surface fluctuation at the offshore bound-ary. A radiation boundary condition is used for all othervariables at both offshore and inshore open boundaries.At the sea surface, momentum and buoyancy fluxes areset to zero. At the bottom, a slip boundary condition isused for most of the experiments described here. Thereis a good reason for neglecting bottom friction for themoment: In both observational data and comprehensivemodels, it is often difficult to cleanly separate baroclinicmotions from barotropic ones because of the presenceof bottom friction. A common practice to obtain internaltides is to subtract the vertical average from the totaltidal field. This of course cannot remove the verticalshears caused by the bottom drag, which would falselyappear as part of the internal tidal field. To prevent thisconfusion, we simply set the bottom friction to zero sothat barotropic tides do not have any shear and internaltides can be clearly determined. For comparison, we willlater show a case where bottom friction is retained.

The model experiments described in the followingsections all start with a preexisting density front. Twotypes of fronts, representing either winter or summerconditions, are shown in Fig. 1. These are quasi-equi-librium states obtained from free model runs of geo-strophic adjustment. Note that the density contrastacross the winter front is set to be the same as thatacross either the surface or the bottom branch of thesummer front. To ensure stability, a small backgroundstratification (about 0.2% of the vertical density gradient


FIG. 1. Initial density (shades) and alongshelf velocity (contour lines) in the middle of the modeldomain. Contour interval for velocity is 3 cm s21 and dotted lines are for values equal or lessthan zero (pointing out of paper). Density is in st units.

in the summer pycnocline) is uniformly applied in themodel.

3. Winter front

Two sets of experiments are discussed here, oneforced with a diurnal (K1) surface tide and the otherwith a semidiurnal (M2) surface tide. The amplitude ofthe barotropic cross-shelf current is about 16 cm s21 onthe shelf in both cases. The model is run for 20 tidalcycles in each experiment and a cyclic quasi-equilibriumstate is reached in about 10 cycles. In order to defineinternal tides and associated density variations, we needto know the barotropic tidal velocity and the tidallyadvected density field in the absence of the internaltides. These are obtained from two barotropic modelruns, which differ from the standard model experimentsonly in that the internal pressure gradient is set to zeroin the momentum equation, so that no internal wavesare generated and density is advected as a tracer. Theinternal tidal field is then simply the difference betweenthe standard and the barotropic model runs with thetidally averaged values subtracted.

Figure 2 shows the density and the cross-shelf flowfield during the 20th tidal cycle for the case of diurnal

tide. There are four snapshots, 1/4 tidal period apart,for the streamfunctions of both the total tidal flow andthe internal tide. The most prominent feature of theinternal tide is a large circulation cell confined to thefront. It rotates counterclockwise during ebb tide andclockwise during flood tide, causing a surface intensi-fication of the cross-shelf flow. This internal tidal cir-culation is nearly in phase with the surface tide, beingstrongest at the maximum ebb and flood and weakestat the slack times. A closer look at the sequence revealsthat this single large cell actually evolves from twosmaller cells of the same sign, which start to appeararound the slack tide. One of them is generated on thelower portion of the seaward side of the front, and theother on the upper portion of the shoreward side of thefront. As the tidal flow increases, these cells combinewith each other to form a single large cell that extendsthroughout the front and lasts till the tidal current re-verses direction.

This mode of diurnal internal tide is consistent withthe analytical model result of Ou and Maas (1988), whopredicted that near a winter front the cross-shelf flowwould be surface intensified at the diurnal frequency.The physical reason can be explained as follows. Duringthe ebb (flood) tide the front tends to be steepened (flat-


FIG. 2. Cross-shelf streamfunction (contour lines) and density (shades) at four different times, 1/4 tidal period apart, duringthe 20th tidal cycle for the case of diurnal tide and winter front. Contour interval is 1.5 m2 s21 for (left) total tidal flow and0.5 m2 s21 for (right) internal tide.

tened) by the barotropic tidal current. This is becausethe barotropic tide that acts on the upper (and fartherseaward) part of the front is weaker than that on thelower part of the front due to the sloping bottom to-pography. To restore the front to its geostrophically bal-anced position, a circulation cell that rotates counter-clockwise (clockwise) tends to be generated in responseto the steepening (flattening) of the front. Since the di-urnal tidal period is longer than the inertial period here,the geostrophic adjustment is almost instantaneous.Thus the cross-shelf component of the internal tide,which leads the alongshelf component by a quarter ofthe tidal period, has to be in phase with the forcing thattends to change the front steepness, namely the baro-tropic tide.

This simple explanation of course cannot count forall the details of the internal tidal field associated withthe front. For example, there is a weak cell at the inshoreboundary of the front that comes and goes with the largecell within the front but rotates in the opposite direction.It is actually induced by the main internal tidal cellrather than by the barotropic tide since the isopycnalsat this boundary are always on the shelf and the baro-tropic tidal velocity across them is uniform. Along theoffshore boundary of the front, there appears to be gen-eration and slow propagation of short internal waves.They escape out of the frontal region and decay withina few wavelengths. Strictly speaking, the wavy featuresin the weakly stratified offshore water are not internaltides. They are inertial gravity waves excited by the tidal


FIG. 3. As in Fig. 2 but for semidiurnal tide.

motions at the frontal boundary. These waves do notcarry much energy, but as we will see later, they cancause significant mixing in weakly stratified environ-ment.

The structure of the semidiurnal internal tide at thewinter front is quite different from that of the diurnalone. As shown in Fig. 3, the cross-shelf flow is nowbottom intensified at the front with a large clockwiseinternal tidal cell during the ebb tide and a counter-clockwise cell during the flood tide. This is again con-sistent with the prediction of Ou and Maas (1988). Thegeostrophic adjustment argument that we used abovefor the diurnal tide can be applied here as well. Thedifference is that the semidiurnal tidal period is shorterthan the inertial period, and thus the internal flow inthis case is no longer close to being geostrophic. The

time derivative of the internal cross-shelf velocity isproportional to the pressure gradient associated with thefront, which lags the barotropic tidal forcing by a quarterof the tidal period. Therefore the internal tidal cell atthe front lags the barotropic tide that causes it by halfa tidal cycle; the clockwise cell induced by the floodtide actually matures during the ebb tide, and vice versa.

Another big difference between the semidiurnal anddiurnal internal tides is that the former apparently prop-agates away from the site of generation while the latterseems to be dominated by a standing mode. Take theevolution of the counterclockwise cell as an example(see Fig. 3). It starts to appear in the upper slope regionat the peak ebb tide. As time proceeds, it grows upwardand propagates both seaward and shoreward, with theseaward propagation being more obvious. By the time


of the peak flood, the cell has split into two, a largestrong cell going offshore and a small weak cell goingonshore. The same sequence takes place for the clock-wise cell in the next half tidal cycle. As these internalcells move toward the boundaries of the front, theirwavelength and speed continue to decrease due to de-creasing stratification, forming a train of tightly packedalternating cells. As they finally move out of the front,they become strongly dissipated and excite a train ofinertial gravity waves, which are also eventually lost todissipation.

The preference of the internal tides to propagate sea-ward indicates a supercritical bottom slope, while in factthe slope is chosen here to be subcritical according tothe classic theory of internal tides (i.e., the bottom slopeis less than the slope of the internal wave characteristics,[(v2 2 f 2)/(N 2 2 v2)]1/2, where v is the wave fre-quency, f is the Coriolis parameter, and N is the buoy-ancy frequency). The contradiction is due to the pres-ence of the front. A front that slopes upward in theopposite direction of the bottom topography could sub-stantially decrease the slope of the internal wave char-acteristics, and such a front could also enhance the gen-eration of internal tides on the sloping bottom (Chuangand Wang 1981). Another noticeable asymmetry is thatthe counterclockwise internal tidal cells that move sea-ward are stronger but narrower than the correspondingclockwise ones, especially when approaching the offshoreboundary. This is due to the increasing nonlinear effectas the internal tide propagation gradually slows down.

One way to evaluate the impact of the winter fronton the generation and propagation of internal tides isto conduct parallel experiments with and without thefront. In the latter case, a vertical stratification corre-sponding to that in the middle of the front is uniformlyapplied throughout the model. Figure 4 compares thetime evolution of the internal cross-shelf velocity atdepth 20 m for different cases. Without the front, thesemidiurnal internal tide propagates away from the siteof generation in both onshore and offshore directionsat a phase speed close to that of the first-mode internalwave. The propagation is apparently modulated by thebarotropic tidal flow, especially on the shelf where tidalcurrent is stronger and wave speed is slower. The be-havior of the diurnal internal tide in the absence of thefront is somewhat unexpected from the linear theory.Although disturbances of diurnal frequency are trappednear the shelf break and the slope, there are waves ofsemidiurnal frequency traveling away. This secondarysemidiurnal internal tide is a result of the nonlinear in-teraction of the diurnal barotropic and baroclinic tides.

The presence of the winter front seems to have threemajor effects. First, it enhances the generation of in-ternal tides; second, it preferentially channels the prop-agating internal tidal energy to the offshore direction;and third, it confines most of the internal tide activitiesto the frontal region. As compared with the case withoutthe front, the amplification of the diurnal internal tide

is particularly large, and is characterized by a tidallyadvected standing mode that occupies the whole widthof the front. The semidiurnal internal tide, on the otherhand, is strongest at the boundaries of the front due tothe energy concentration associated with the refractionof the propagating internal tidal wave. There is no vis-ible reflection at the boundaries because of the strongdissipation there. In both diurnal and semidiurnal cases,there is a small amount of energy leaking out of thefront, carried by slowly moving waves. These wavesare quickly damped, and the ones that survive the lon-gest have a frequency close to the inertial frequencysince such a frequency relates to the longest possiblewavelength for inertial gravity waves.

4. Summer front

The model experiments and analyses performed forthe winter front are repeated for the summer front. Fig-ures 5, 6, and 7 discussed in this section are the sameplots as in Figs. 2, 3, and 4 in the previous section exceptfor a different type of front. The differences and sim-ilarities between the internal tides at the summer frontand those at the winter front should become obvious bycomparing the corresponding pairs of figures.

The structure of the diurnal internal tide at the sum-mer front (Fig. 5) is generally similar to that at the winterfront (Fig. 2). The dominant feature is a large internalcirculation cell in phase with the barotropic tide, rotatingcounterclockwise during the ebb tide and clockwise dur-ing the flood. The main difference from the winter frontcase is that the cell here is weaker and less developedin the upper water column so that there is no obvioussurface intensification. The upper branch of the front,due to its opposite pressure gradient, limits the growthof the circulation cell induced in the lower branch. Theinternal tide is still against the barotropic tide near thebottom, but it enhances the barotropic tide more in themiddle of the water column than near the surface. Ac-cording to the reasoning used for the winter front case,such a cell makes perfect sense because it tends to re-store both lower and upper branches of the front to theirgeostrophically balanced position. Another differencebetween the summer cell and the winter one is that theformer is broader and extends farther offshore. This isobviously because the existence of the pycnocline al-lows more energy to escape out of the front.

The situation is completely different for the semidi-urnal tide (Fig. 6). The internal tidal field is now char-acterized by energetic traveling circulation cells in bothoffshore and onshore directions. The cells that propagateseaward have much longer wavelength and faster phasespeed due to deeper water depth and stronger stratifi-cation. The first mode still dominates, but higher verticalmodes are evidently present since the cells change theirorientation as they travel along. The internal tidal energyactually propagates along the characteristics, going for-ward as well as up and down. As compared with the


FIG. 4. Time evolution of cross-shelf velocity at depth 20 m for cases (right) with or (left) without front. Contour interval for velocity is1 cm s21.

semidiurnal internal tide at the winter front (Fig. 3),there are two major differences here. First, the internaltidal cells are no longer trapped at the offshore boundaryof the front because the pycnocline enables them topropagate away. Second, the internal tide no longer pref-erentially propagates seaward because the upper branchof the front, which slants upward in the direction of thebottom slope, allows a substantial portion of the internaltidal energy to radiate onshore.

Although the same physical process is responsible forthe generation of the semidiurnal internal tide here asin the winter front case, there is no apparent bottomintensification of the cross-shelf flow at the front. Thereason is that the internal tidal cells move away fromthe site of generation so fast that, when they mature,they are already out of the frontal region. Let us again

follow the evolution of a counterclockwise cell. At thetime of the maximum ebb tide two counterclockwisecells are generated, one in each branch of the front. Theyquickly grow into a single large cell and start to prop-agate away. By the time of the maximum flood tide, ithas long split into two cells traveling in opposite di-rections. The large seaward going cell has moved wayout of the frontal region and the smaller shoreward goingcell has moved onto the shelf, leaving the center of thefront to the generation of the following clockwise cir-culation cell. It is interesting to note that the cells thatpropagate onshore are strongly refracted toward the in-shore boundary of the front, very much like what hap-pened to the semidiurnal internal tide at the offshoreboundary of the winter front.

The effects of the summer front are again measured


FIG. 5. As in Fig. 2 but for the summer front.

against the case with no front (Fig. 7). The horizontallyuniform vertical stratification in the case without thefront is chosen to be the same as that of the offshorewater in the case with the front. When the front is absent,internal tidal waves propagate away freely from the re-gion of generation in both directions. There is again anindication that part of the diurnal tidal energy is carriedaway by a nonlinearly produced semidiurnal wave. Inthe presence of the front, the shoreward propagation iscut short at the inshore boundary of the front. The am-plification of internal tides by the summer front is gen-erally not as strong as that by the winter front (see Fig.4) except at the inshore boundary where the amplitudeof the semidiurnal internal tide is largely increased bywave refraction. This, of course, is due to the fact thatthe summer front has an open end at the offshore pyc-

nocline that allows internal tidal energy to radiate awayfrom the front. Another reason for the remarkable lackof amplification of the diurnal internal tide is that thesummer double front does not permit the developmentof a strong circulation cell that extends from the bottomto the surface, as does the winter single front.

5. Energetics

The kinetic energy of internal tide is simply [(u 212

uT)2 1 (y 2 y T 2 yG)2 1 (w 2 wT)2], where u, y, andw are the cross-shelf, alongshelf, and vertical velocitiesof the total tidal flow (tidally mean is subtracted), uT,y T, and wT are the corresponding velocities of the bar-otropic tidal flow, and y G is the anomalous alongshelfgeostrophic velocity associated with the front excursion



by the barotropic tide. The available potential energyassociated with internal tide is estimated as (g2r92/1

2

r0N 2) (Gill 1982), where g is gravity, r0 is a referencedensity, N is the buoyancy frequency of the tidally ad-vected mean density field, and r9 is the perturbationfrom the mean. The sum of the kinetic energy and theavailable potential energy calculated this way definesthe total internal tidal energy here.

Figure 8 shows the mean energy density distributionsof the internal tides discussed in the previous two sec-tions. In the presence of the winter front, the energy ofthe diurnal internal tide is mostly confined within thefront with maxima near the bottom and the surface,clearly corresponding to the large circulation cell wehave seen in Fig. 2. The signature of the smaller cell atthe inshore boundary is evident too. The semidiurnal

internal tide also has most of its energy trapped to thewinter front, but the center of action is now at the off-shore boundary rather than the middle of the front. Theintense energy density maximum at the offshore corneris apparently due to the strong refraction of the internaltide there. In the case of the summer front, the energyof the diurnal internal tide is still mostly confined inthe frontal region though it has a tendency to spreadoffshore along the pycnocline. The maxima at the bot-tom and in the separation region of the two frontalbranches are associated with the circulation cell seen inFig. 5.

The only case in which significant internal tidal en-ergy is found outside of the frontal region is the caseof the semidiurnal tide with the summer front. The en-ergy radiates away in the offshore direction in a fashion



consistent with linear theory, that is, the propagation ofthe internal tidal energy tends to follow the rays ema-nating from the shelf break and the slope where theinternal tide is generated. Wave refraction and energyconcentration occur in places where these rays are re-flected from the sea surface. This explains why thereare localized maxima in the near-surface velocity, asevident in Fig. 7, for the cases with or without the front.Near the inshore boundary of the front, additional re-fraction of the internal tide occurs due to diminishingstratification, resulting in particularly strong energy con-centration. It is worth noting that in all cases the energydensity of the internal tide is relatively high in the bot-tom layer over the slope, the place where the internaltide is mostly generated, and that the energy there ishigher and more closely trapped to the bottom for thediurnal tide than for the semidiurnal tide.

The significance of the internal tide relative to thebarotropic tide can be measured by the ratio of theirkinetic energies (Fig. 9). The spatial distribution of thisratio is not much different from that of the internal tidalenergy shown in Fig. 8, except that the features in theoffshore region become more prominent than those onthe shelf since the energy density of the barotropic tidedecreases as the water depth increases. The kinetic en-ergy of the internal tide is generally much less than thatof the barotropic tide, but in some localized places itcan be as large as or even larger than the latter. Theseplaces include the surface layer in the middle of thewinter front for the diurnal tide, the surface layers onthe seaward side of the winter front and on the shore-ward side of the summer front for the semidiurnal tide,and the energy ray paths of the semidiurnal internal tidein and above the summer pycnocline. It should be point-


FIG. 8. Mean energy density (J m23) of internal tides for different cases (shades). Density fieldis shown as dashed contour lines.

ed out that internal tides may play a more significantrole in mixing than barotropic tides, even in placeswhere their energy level is much lower than that of thelatter, because they provide the main source of velocityshears away from the bottom frictional layer.

On the other hand, the places with a high level ofinternal tidal energy do not necessarily coincide withthe sites of strong mixing and dissipation because tur-bulence production depends not only on velocity shearbut also on density stratification. Figure 10 shows thespatial distribution of the turbulent kinetic energy(TKE), which is calculated here prognostically usingthe turbulence closure of Mellor and Yamada (1982).As compared with the distribution of the internal tidalenergy (Figs. 8 and 9), the most striking difference isthe lack of activity in the summer pycnocline. There theshears of the energetic internal tides are not large enough

to overcome the strong stratification. In all cases theTKE is intense in the bottom layer over the slope whereinternal tides are strong and stratification is weak. Forthe same reason, large TKE values are found at theoffshore boundary of the winter front and the inshoreboundary of the summer front for the case of semidi-urnal tide. While there is a moderate turbulence pro-duction throughout the winter front by either diurnal orsemidiurnal internal tide, the relatively strong stratifi-cation keeps the TKE level very low in the middle watercolumn of the summer front.

6. Mixing effects

We evaluate the mixing effects of internal tides byincluding a passive tracer in the model. In order to havean overall picture, the tracer is placed on a grid with a


FIG. 9. Ratio of mean kinetic energy of internal tide to that of surface tide for different cases(shades). Density field is shown as dashed contour lines.

horizontal spacing of 6 km and a vertical spacing ofapproximately 33 m. The horizontal grid lines are 10m thick and the vertical grid lines are 1.2 km wide. Thisgrid of a tracer is released at the slack before the ebbtide, after the model has reached a cyclic quasi-equilib-rium in each one of the model experiments describedin the previous sections. The model is then run for sev-eral tidal cycles more and the evolution of the tracerfield is recorded. To avoid the complication due to tidaladvection, we only check the tracer field at successiveslacks before the ebb. In such a sequence, the changesin the tracer distribution result mainly from the mixingby internal tides and the advection by tidally averagedresidue currents. We do not have a clean way to separatethese two here, but the latter should be less importantthan the former in the first few tidal cycles after thetracer release.

Figure 11 displays the evolution of the tracer fieldover six tidal cycles in the case of the semidiurnal tidefor both winter and summer fronts. Many features inthis figure are consistent with what one would expectfrom the TKE distribution (Fig. 10). The vertical mixingis very strong at the offshore boundary of the winterfront and the inshore boundary of the summer front.There is intense mixing in the bottom boundary layerfor both fronts, but only for the winter front the mixingtakes place in the whole water column throughout thefrontal region. The fact that the tracer is mixed hori-zontally as well as vertically cannot be directly attri-buted to the TKE distribution since in our model for-mulation only the vertical mixing term is related to theTKE. The horizontal mixing here is mostly caused byshear dispersion, a process in which vertical mixing actsupon an oscillatory shear flow to spread a tracer hori-


FIG. 10. Mean turbulent kinetic energy (J m23) for different cases (shades). Density field isshown as dashed contour lines.

zontally. The largest dispersion occurs in the bottomlayer over the slope where both the shear and the TKEare intense. The tracer distribution does not changemuch outside of the frontal region, except for the wig-gles associated with the inertial gravity waves seawardof the winter front. Some persistent distortions of thetracer grid are evidently caused by the residue flow.

The corresponding plots for the case of diurnal tideare shown in Fig. 12. As expected, there is little mixingin the summer front except near the bottom, while strongvertical mixing and horizontal dispersion take placethroughout the winter front. The main difference fromthe case of semidiurnal tide is that vertical mixing isnot nearly as strong at the offshore boundary of thewinter front and the inshore boundary of the summerfront, but the mixing in the middle of the winter frontis somewhat stronger. It is evident in Figs. 11 and 12

that density is much more resilient to mixing than thetracer. The density field does not change much from onetidal cycle to another while the tracer grid is being rap-idly mixed. This seems odd because both density andtracer are acted upon by the same diffusive and advec-tive processes. To understand this, we need to clarify acouple of points. First, density is not a passive tracer;it interacts with the dynamical processes to determinethe state of the system. Second, in our experiments thetracer is put into the model when a quasi-equilibriumstate has already been attained. In such a state, densitychanges little because its diffusion is balanced by ad-vection, but a newly introduced tracer field will bestrongly altered by the dynamical processes that seemto have no effect on the density, unless its spatial dis-tribution is exactly the same as that of the density.

In all the experiments so far, we used a slip bottom


FIG. 11. Tracer distribution at different times after release for the case of semidiurnal tide with either (left) winter or(right) summer front. Density field is shown as dashed contour lines.

boundary condition on purpose to prevent the internaltidal field from being contaminated by the shears of thesurface tide due to bottom friction. Now the questionis to what extent this simplification is valid or, in otherwords, how the internal tides obtained this way will beaffected by retaining the bottom friction. This is exactlywhat we show in Fig. 13, which is the same as Fig. 11except the corresponding experiments include bottomfriction. Away from the bottom boundary layer, the re-sults from the cases with bottom friction are almostidentical to those without. The main impact on the frontby retaining bottom friction is a strong shoreward in-trusion at the foot of the front. This is due to an Ekmantransport associated with the bottom stress on the along-shelf geostrophic flow. To a much less extent, such ashoreward bottom intrusion also exists in the case with-

out the bottom drag since the vertical shear in the geo-strophic current (Fig. 1) can produce an internal stress.The addition of the bottom drag greatly enhances thisprocess. In the weakly stratified regions outside of thefront, the effect of the bottom friction is to create a thickbottom mixed layer through largely increased verticalmixing.

It is worth noting that there is little horizontal dis-persion in places where internal tides are absent, evenwhen strong vertical mixing is generated there by thebottom-dragged surface tide. For example, on the innershelf shoreward of both the winter and summer fronts,there is no internal tidal activity (Fig. 11). In the pres-ence of bottom friction, the surface tide produces a 60-m-thick bottom mixed layer there and completely de-stroys the horizontal tracer grid lines in that layer, but


FIG. 12. As in Fig. 11 but for diurnal tide.

it does not have a significant effect on the vertical tracergrid lines (Fig. 13). This implies that, at least for theparameter range used in this study, the shears associatedwith internal tides are larger than those caused by bot-tom friction. Horizontal dispersion is not likely to beeffective without the internal tides even within the bot-tom mixed layer. Above this layer, the internal tidescertainly play a dominant role in mixing and dispersion,if atmospheric forcing is not considered.

7. Summary and discussion

We have studied the internal tides in the coastal fron-tal zone near the continental shelf break using an ide-alized numerical model, with emphases on their struc-ture, energetics and mixing effects. The model is set torepresent either winter or summer conditions at a mid-

latitude location, and it is forced with a surface tide ofeither diurnal or semidiurnal frequency. We found thatthe properties of internal tides are highly dependent onfrontal configuration and tidal frequency. At the winterfront, energetic internal tides are generated and arrestedin the frontal zone; the cross-shelf flow tends to besurface (bottom) intensified by a large internal circu-lation cell at the diurnal (semidiurnal) frequency. At thesummer front, the diurnal internal tide is still trapped,but the semidiurnal internal tide travels out of the frontalzone in the offshore direction while arrested at the in-shore boundary. The internal tides can cause significantmixing and dispersion in both winter and summer fronts;the semidiurnal internal tide has the strongest effectstoward the frontal boundaries while the diurnal internaltide is more effective close to the site of generation.

The presence of a shelf–slope front enhances the gen-


FIG. 13. As in Fig. 11 but for the case with bottom friction.

eration of internal tides, especially during winter time.This is attributable to larger disturbances of the densityfield by the surface tide because the isopycnals in thefront are more perpendicular to the barotropic tidal flowover the slope. The presence of the front also causes afurther amplification of the semidiurnal internal tide bytrapping its energy in the frontal zone. This amplifi-cation is most prominent at the offshore boundary ofthe winter front and at the inshore boundary of the sum-mer front, where strong refraction of the internal tidetakes place. The effect of the summer front on internaltides differs from that of the winter front in at least threeways: first, the existence of the summer pycnocline per-mits internal tidal energy to quickly radiate away to theopen ocean; second, the structure of the summer doublefront does not allow the development of a large bottom-to-surface internal circulation cell as does the winter

front; and third, the upper branch of the summer frontfacilitates a shoreward propagation of the semidiurnalinternal tide, while the winter front preferentially chan-nels it seaward.

The surface (bottom) intensification of the cross-shelfflow of the diurnal (semidiurnal) tide at the winter frontis consistent with the theoretical prediction of Ou andMaas (1988). Bottom intensified semidiurnal tide hasalso been identified in other observational and modelingstudies (e.g., Holloway 1984; Chen and Beardsley1995). However, another prediction of Ou and Maas(1988), the preferential amplification of the semidiurnaltide relative to the diurnal tide is not evident in thepresent model. Their theory is based on a linear, inviscidmodel, which allows the existence of standing modesin a front with closed ends. Since the eigenfrequenciesof these modes have a lower cutoff at the inertial fre-


FIG. 14. (left) Diurnal and (right) semidiurnal internal tidal fields similar to those shown in Figs. 2 and 3 except for awinter front that is only half as wide as the original one.

quency, the semidiurnal tide is preferred for resonancewith one of the superinertial modes. It seems that in thepresent model there are no standing modes for the pro-gressive semidiurnal internal tide. This is because of thelack of reflection from the ends of the front where therefracted semidiurnal tide is dissipated almost com-pletely. Therefore, the energetic semidiurnal tide foundin this model, which has maximum energy density to-ward the frontal boundaries, is a result of refractionrather than resonance.

Our model results are not particularly sensitive to thechoice of physical parameters such as the width of thefront and the slope of the bottom topography. For ex-ample, Fig. 14 shows the internal tidal fields at a winterfront that is 50% narrower than the original one. Thesimilarity between these fields and those shown in Figs.

2 and 3 is quite remarkable. All of the description anddiscussion presented in section 2 is still valid here,though the propagation of the semidiurnal internal tideis limited to a narrower frontal zone. We also did anexperiment with the front widened by 50% (not shown).The diurnal internal tidal cell again remains trapped tothe slope, while the semidiurnal tide propagates fartheroffshore and onshore to the frontal boundaries. The ef-fect of bottom slope on internal tide generation is wellknown from theory, and what we found in our sensitivitystudy is consistent with the theory. In short, a criticalslope is most favorable for internal tide generation; asupercritical slope prevents internal tides from propa-gating onshore; and a subcritical slope is less effectivein generating internal tides. As mentioned earlier, thebottom slope in the model is subcritical when only ver-


tical stratification is considered, but for the winter frontit becomes slightly supercritical. Increasing or decreas-ing the slope by 50% only changes the magnitude andthe onshore/offshore partition of internal tides, but nottheir general structure and behavior.

The model experiments discussed here are generallyin a weakly nonlinear regime. The tidal flow is sub-critical except when internal tides are highly refractednear the frontal boundaries, in which case nonlinearitybecomes important. This results in an asymmetry in thesemidiurnal internal tide, as evident in Figs. 3 and 4.What is not shown here is that the nonlinearity alsogenerates a residue flow field that is dominated by aseries of alternating circulation cells with the spatialscales of the internal tides. This kind of rectification hasalso been found in previous theoretical and modelingstudies (e.g., Maas and Zimmerman 1989; Chen andBeardsley 1995). Another instance in our experimentsthat points to the importance of nonlinearity is the oc-currence of a near-semidiurnal wave in the case of di-urnal forcing (Fig. 4). The implication is that, when thenonlinear interaction between surface and internal tidesbecomes more important, a significant portion of thediurnal tidal energy will no longer be trapped to the siteof generation, and any observed semidiurnal tidal fieldmay partly result from diurnal surface tide. We willexplore in a subsequent paper the interaction betweeninternal tides and coastal fronts in a strongly nonlinearregime.

Acknowledgments. This research is supported by theNational Science Foundation through Grant ATM9618260. We thank two anonymous reviewers for theirhelpful comments.

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Documents

A Model Study of Internal Tides in Coastal Frontal Zonerainbow.ldeo.columbia.edu/~dchen/collection-dchen/jpo03a.pdfInternal tides near a midlatitude shelf–slope front are studied