Dynamics of Atmospheres and Oceans,9 (1985) 29'l -374Elsevier Science Publishers 8.V.. Amsterdam - Printed in The Netherlands

297

INTERNAL TIDE GENERATION AT A CONTINENTALSHELF/SLOPE JUNCTION: A COMPARISON BETWEENTHEORY AND A LABORATORY EXPERIMENT

PETER G. BAINES

CSIRO Diuision of Atmospheric Research, Aspendale, Vic. (Australia)

FANG XIN-HUA

Shandong College of Oceanography, Qingdao (People's Republic of China)

(Received August 20, 1984; revised March 4, 1985; accepted April 29,1985)

ABSTRACT

Baines, P.G. and Fang, X.-H., 1985. Internal tide generation at a continental shelf/slopejunction: a comparison between theory and a laboratory experiment. Dyn. Atmos. Oceans,9: 297-374.

Observations of internal tide generation over continental slopes in a laboratory experimenthave been carrried out, with the objectives of making comparisons with linear generationtheory and investigating its limitations. Both continuous and layered stratification have beenconsidered. A measure of the amplitude of the barotropic tidal forcing (and hence ofnon-linearity) is given by the Froude Number F:uso/c*, where u,6 is the maximumbarotropic tidal velocity at the shelf break, and c* is the long-wave speed of the lowestinternal mode.

For continuous stratification, good agreement was obtained for "steep" slopes (a/c>1,where a is the slope at the continental slope and c is the slope of the internal wave rays oftidal frequency), even for quite large amplitude motions (f < 1.6), and the upper limit of itsquantitative usefulness was not reached. For "flat" slopes (a/c<1) reasonable agreementwas also obtained, even up to quite large amplitudes (F < 3.1), although some departure fromlinear theory was apparent.

For two-layer flows the applicability of linear theory was much more restricted. ForF-0.5 there was only qualitative agreement and for larger F (>1) significant differenceswere observed. The latter were due to the substantial advection and associated hydraulicjumps which occurred seaward of the shelf-break during the ebb-phase of the barotropic tide.Shelf-break values of F>t arc common in the ocean.

l.INTRODUCTION

There has been a considerable amount of theoretical work on the genera-tion of internal tides near the junction of continental shelves and slopes (e.g.,

0377-0265/85/$03.30 o 1985 Elsevier Science Publishers B.V.

298

Rattray et a1.,1969; Baines, 1973,1974,1982-see references in the last, fora complete list). The problem has been tackled in several (essentiallyequivalent) ways, all of which are linear. This paper describes some experi-mental observations which have been carried out to provide a test of thelinear theory, and where possible to establish the limits of its applicabilityand the nature of its breakdown. Both continuous and layered stratificationhave been used.

Some previous experiments on internal wave generation by tidal mecha-nisms have been described by Maxworthy (1979) and by Farmer andFreeland (1983). Both of these studies used a steep-sided "sill" rather than ashelf-break, used two-layer stratification (approximately), and gave a quali-tative description of the formation of non-linear wave trains by this process.The present paper is generally more quantitative and is directed to the mostcommon oceanic situation, namely continental shelf/slope geometry. Itindicates the degree of usefulness of linear theory for the continuouslystratified and layered situations, but it is not exhaustive.

The experiment does not model the effects of the Earth's rotation. In thelinear theoretical models of internal wave generation with two-dimensionaltopography, rotation does not change the character of the motion if thefrequency <,0 lies in the range f < <.r ( N, where / is the Coriolis frequencyand N is the Brunt-Vbisblb frequency. It simply alters the ray slope c, where

| ) r ) \ \ / z

I Q - - r - t 'c : l - |

I N z - t . t z I\ r '

(1 .1 )

and introduces along shore velocities so that the particle paths are elongatedinto ellipses in the along shore direction. On the other hand, lf u: < f ( N, cis complex and there is no internal wave propagation in this two-dimensionalsystem; this occurs in the ocean for the diurnal tide poleward of 30o latitude,for example. Consequently these experiments model internal tide generationin the ocean when /< o < N.

The plan of the paper is as follows: the nature of the experiment isdescribed in detail in section 2, and the equations for linear-theory motionsare given in section 3. The observations are described in sections 4 and 5,together with a comparison with the predictions of linear theory; where theseare significantly different (notably, the two-layer case at large amplitudeforcing) a physical interpretation of the process is given. The conclusions aresummarized in section 6.

2.THE EXPERIMENT

The experiments were carried out inside a tank of width 23 cm as shownin Fig. 1. At the right-hand end of the tank a solid triangular piston (face

lntefnqt wove

Absorber

299

lnlerDol Wove

Absofbel

( inser ied o f te r f i l l i ng)

Fig. 1. Schematic side view of experimental arrangement.

angle 45o) was positioned. This piston could be oscillated vertically with arange of amplitudes and frequencies; at the lowest point of the stroke thevertex of the piston just reached the bottom of the tank. The other end of theworking region of the tank was blocked by a barrier which was also inclinedat an angle of 45o, parallel to the face of the piston. A fixed horizontalcontinental shelf with an upper surface situated 22.3 cm above the floor ofthe tank was inserted, contiguous with this angled barrier. The tank was thenfilled either with fresh water, or with salt-stratified water using the familiartwo-tank process, to a depth of (typically) 26(+ 3) cm; the fact that theright-hand end barrier and the piston face had the same angle (45') ensuredthat the cross-sectional volume of the tank was uniform with height, so thatthe two-tank process yielded a stratification with constant density gradient(to within

'J,Vo accvr&cy). When the tank was filled, the continental slope was

slid into place at an angle of 20.8' to the horizontal, as shown in Fig. 1; thejunctions at the shelf break and tank floor were made smooth and water-tightwith the use of tape. A relatively thin layer of immiscible kerosene orkerosene mixed with cooking oil, was then placed on top of the water insome experiments; in others a thin layer of fresh water was placed thereinstead. This thin homogeneous upper layer represented a possible mixedlayer in the ocean.

We therefore have a firute tank with continental shelf/slope geometry,filled with water which may be homogeneous or salt-stratified and possiblysurmounted by a homogeneous lighter layer. If the amplitude of the verticaloscillation of the piston is sufficiently small (a few cm in practical terms), itdirectly forces barotropic motion; baroclinic motion in the tank is thenproduced by the interaction of this motion with the continental shelf/slope.

In general, baroclinic motions (internal waves) were generated near theshelf-break and propagated away in both directions; care was taken toensure that this wave energy was absorbed atboth the deep and the shallowends. For the deep end, a sheet of wire mesh spanning the width of the tankwas inserted between the piston and the topography, inclined at an angle of- 35" to the vertical and covering the total depth (see Fig. 1); this did not

300

affect the barotropic motion but acted as a partial barrier to the internalwaves (both interfacial and continuous), and appeared to result in a reflec-tion coefficient of practically zero for these waves from the deep-water endin most cases. On the shallow shelf side only the interfacial waves weresignificant; it was found that the reflection coefficient from the beach couldbe made very small by inserting a vertical V-shaped barrier on the shelf,pointing toward the piston end, which blocked the middle two-thirds of thewidth of the tank.

For the stratified fluid, three means of flow visualization were used: (1)dye lines (blue vegetable dye) were inserted whilst the tank was filling; thesegave a good description of vertical displacements; (2) neutrally buoyantbeads, which were used to obtain velocity profiles; and (3) shadowgraph,which was used to inspect the character of mixing near the shelf-break,where this occurred. With suitable changes in lighting all three techniquescould be used for the one experiment. Al1 data were recorded photographi-cally using stills and movie film, and quantitative information was extractedfrom the prints. The kerosene-water interface also showed up clearly. Forstreak photographs with the neutrally buoyant beads, a flash at the begin-ning of the exposure was used to remove ambiguity in the direction ofmotion.

3. THE EQUATIONS FOR LINEAR THEORY

We use the equations and models of Baines (1973, 1982) which aredescribed here in summary form only. For situations where the stratificationconsists of a surface mixed Iayer, a strong thin thermocline and deepstratified region, the tidal velocity field may be written (Baines, 1982)

u : u 1 + u ' u ; : u 7 f u 3 (3 .1 )

Here u, denotes the barotropic tidal motion (i.e., the tidal motion whichwould occur if the fluid were homogeneous), u, the baroclinic motion, ur thetidal motion associated with the interface (thermocline) displacements, andu" the additional baroclinic flow due to the deep stratification. In thequantitative results described here, either the interface or the lower stratifica-tion is not present, so that the only one of ul or u, is non-zeto.

The determination of u, : (ut, wt) in a finite tank such as this requiressome care. For sufficiently low frequencies c.r the horizontal mass flux

Io ,uru': QQ) cos .ur (3.2)

varies linearly from the piston to the "beach", as in the experiments ofBaines (1983). However, a finite tank has resonances at higher frequencies,

301

and the lowest of these frequencies corresponds to the quarter-wavelengthresonance on the shelf. Hence the longer the shelf the lower the resonancefrequency. The shelf in the present experiments is much longer (L50 cm) thanin the previous experiments, and the linear form of QQ) cannot be assumed.This resonance occurs on some oceanic continental shelves, and adds a touchof realism to these experiments.

In the ocean where bottom slopes are small, rzt is independent of depth.In the present experiments with larger bottom slopes of - 20o some verticalvariation in u, occurs but it is still only about 4Vo (Baines, 1983). Thisimplies that the barotropic flow is approximately hydrostatic, and we makethis assumption here. With the additional assumption that the flow is linear(which is valid for frequencies sufficiently below the lowest resonance), thespatial variation of QG) is given by

, . 2O - - + - : = = O : 0

sh\x ) - (3.3)

with Q: hnilp at the piston (taken at the mean position where the pistonface intersects the true surface), where ft* is the fluid depth and uo themaximum vertical piston velocity, and Q:0 at the beach. This equation wassolved to obtain Q(x) (over the slope its representation involves Besselfunctions) and in particular its value at the shelf break, Q.o.

Baroclinic wave generation occurs over the slope. In this region there istypically a *l5Vo variation in QQ) in the present experiments. For theinterfacial motion and for the continuous stratification case when a/c>I,however, the generation is concentrated near the shelf break (here a is theslope of the continental slope and the slope of the internal wave raysc: a/(N2 - ,n2)7/2, where N is the Brunt-Vhishlb frequency). This suggeststhat one may make the approximation that Q is independent of x in thegeneration region for baroclinic wave generation purposes. The approxima-tion is less justifiable when o/c< 1, but the baroclinic motion is generallysmall here in any case (see next section). We also apply the radiationcondition to the wave motion on each side of the generation region.

For the purposes of calculation the baroclinic velocity field may beexpressed asu , : u2 * u3 (3 .4 )

whereu2 : -u1 (3 .5 )

so that u, denotes the total motion. Further, the stream function rfi3 for u3,which is defined by

' - t : (uz r"): (-gb o-t: ) .- ' " ' (3.6)\ " 5 " t t \ d z 0 x I '

302

satisfies a homogeneous equation

( l r ' - " ' ) t r , ,

- 12*2"" : o

with the boundary conditions

. p r : O z : - h ( x ) \

*r: - QQ) on the free surface /If the variation in Q is small over the generation region and the free surfacedisplacement is negligible (the two are related), eq. 3.8 is equivalent to

\ J . t )

(3 .8 )

(3.10)

(3.r2)

(3.e)

These equations are independent of the form of the stratification. However,the results quoted below for two-layer stratification from Baines (1982) werecalculated using a slightly different procedure.

For two-layer stratification the interfacial displacement fo on the con-tinental shelf (x < 0) is given by

, l r , : 0 z : 0 \: a z : - n ( * ) |

ro: - H; tgry(R,r )ewhere the real part is taken and

Ap

R:+ t ' : t i r : ! ; k :

- i (kx+ at)

L(7 - R)" '( 3 .11 )

d is the depth of the upper (mixed) Iayer, h, the total depth on the shelf andLp/p is the fractional change in density across the interface (assumed small).The origin (x:0) is taken at the shelf break. y(R, 7) is complex and isgiven in fig. 7 of Baines (1982).

For the case of continuous stratification with constant N" the solution ofeqs.3.7-3.9 for the velocity field u, gives, for d/c> 1, for the cartesiancomponents

I Q"au t : - 4 h ,

v(() - v(-n/2t")

- Lnh- I ) - v ( : ! \p ' \ ' p J ' \ z v r )

r > o \

,=o,lwj : - cu3

wherez z

e : - + x + y , n : - - x - y ,' c

' c

' L

0 t - c( : t /2u f r . : hr /c p:a . + c

(3 .13)

- ftsq[ pqpf

-- Imoginofy port

303

\ v (E)

o = : - = 0 . 1

Vetocity prof i te V (( )

Fig. 2. (a) The function Z(f) representing the velocity profile on the f characteristics forp :0.1,1.0. The solid curve (real part) is in phase with the on-off.shore barotropic tide, p :1implies a vertical continental slope, whereas for p : 0.1 the continental slope is only slightlysteeper than a characteristic. (b) The velocity field in the stratified fluid for steep topography(p:0.1) with a rigid upper surface. Solid lines correspond to the real part of Z, dashed linesto the complex part.

Z(l) is given in Fig. 2a and the associated velocity field for I :0.1 isshown in Fig. 2b. The calculation assumes infinite depth on the deep(right-hand) side.

When a/c< f- it is necessary to solve the relevant integral equation toobtain u3e Ers described in Baines (1973), and this has been done to obtainthe theoretical velocity profiles shown in Fig. 4.

4. OBSERVATION WITH CONTINUOUS STRATIFICATION

We anticipate that the linear theory will give a satisfactory description ofthe observed flow at small amplitudes, but that differences may becomeapparent as the amplitude increases. A suitable measure of the amplitude isthe Froude number

F: u"o/cn

where ar6 is the maximum barotropic velocity at the shelf break and c* isthe long wave speed of the lowest internal mode on the shelf. For continuousstratification we have

c*: Nhr/n

If ,F > 1, then internal waves will not be able to propagate onto the shelf forat least some portion of the tidal cycle.

When u/c> 1 the linear theoretical flow pattern consists of a o'beam"

(4.1)

(4.2)

304

KEYH L 1

> - - - - { 1 2

H L 1

F - - - { 1 2

a-------------4 L1& - - - { 1 2

Fig. 3. Velocity profiles for three runs with a/c>7 (see Table I) measured in the characteris-tic direction at two locations, lines L1 and L2 (see inset). The solid curves denote the profile

derived from linear theory. Run 12, F : 0.66; run 13, F:1.10; run 14, F :1.62.

over the continental slope which varies very little with d, and changes in ccause changes in angle but no changes in structure (see Fig. 2). Velocityprofiles across such a beam at four phases of the tide are shown in Fig. 3, forthree values of F with u/c : L44. The experimental parameters are given inTable I. Theoretical profiles obtained from the procedure of section 3 arealso shown. Given the method of velocity measurement (measuring streaklengths in time-exposure photographs) the differences between the variousvelocity profiles were not considered to be significant and the general patternis in reasonable agreement with linear theory, even for F values up to L.62.A notable difference is the absence of the "rabbit ears" at the flood and ebbphases from the observed profiles. This is not surprising, because these"spikes" are afiif.acts of the inviscid theory and may be regarded as due to

1?

1 l

305

TABLE I

Details for runs with a/c >l

Runnumber

oL/c ht/h^ usb Distance from shelfbreak along a lineperpendicular to the2 axts

Line 1 Line2(cm) (cm)

^ TUsb-

Nft.u

7213l.+

0.380.380.38

444653

0.264 7.44 0.200.264 1.44 0.200.264 1..44 0.20

1.01 0.661.70 1.102.57 1.62

3023.53l

For these runs there was a slight "sill" at the shelf break with a depth ft,6 of 5.0 cm; on theshelf the depth ha was 5.7 cm.

very small-scale waves generated at the shelf break. One would expectviscosity to severely attenuate these waves as they propagate away from theshelf break. Further, the group velocity for such small scale disturbances isvery low so that they would have difficulty escaping frbm the generationregion, and advective effects should hamper their generation anyway. Thesame remarks apply to the somewhat larger spikes at the high and low tidephases. Also, at high and low tide phases the lower peak (near the rayemanating downward from the shelf break) is larger and narrower than theupper; this is presumably because the latter incurs some energy loss onreflection from the upper surface. The observed flood and ebb profiles arevery similar (apart from a change in sign) and the same applies to the highand low-tide profiles; consequently there is no asymmetry in the motiondetectable in these experiments. No velocity profiles were measured on theshallow (5.0 cm) continental shelf, but no baroclinic motion was evidentthere either during the experiments or on the photographs. This is consistentwith linear theory which predicts that the motion on the shelf should bemostly barotropic, with "spiky" baroclinic motion the latter would berapidly damped in practice, if generated at all, as discussed above. Also, forthe experiments described in this section, no mixed layer was observed toform on the continental shelf.

For a/c < 1 the linear internal wave motion generated from the theoreti-cal model is generally small unless a/c lies in the range

f f i q g < 1 (4.3)

so that the topography may be spanned by one ray (or characteristic) whichreflects once from the upper surface (Baineg, 7973); for a/c smaller than this

306

KEY

RUN 60

RUN 63

- 0 2 0 0 2 0 4 0 6 u / u . 6

(o )0 2 0.1+ 0.6 08 ulu,6 0 0 2 u/ur6

Fig. 4a.

range the internal tide generating forces tend to cancel each other. Theresults for four runs with d/c<1 are shown in Fig.4, where observedhofizontal velocity profiles over the slope are compared with theoretical onescalculated from section 3. The relevant parameters are given in Table ILNote that inequality (4.3) is clearly satisfied for runs 60 and 63, but is notsatisfied for runs 61 and 62. As expected, the sharp peaks of the inviscidtheoretical model are not observed, for the reasons given above. Apart fromthis there is general agreement between the observed and theoretical profiles,even for quite large values of F. The high and low-tide profiles in Fig. 4c,dare exceptions; the differences between theory and experiment are consistentand systematic, and cannot be attributed to a timing error. However, thevelocities induced are small (note scale) and the discrepancy is probably dueto the along-tank variation of the mass flux QQ) over the slope region,which is neglected in the theoretical model. In all four cases the barotropic

H

- 0 . 8 - 0 6 - 0 4 - 0 2 0

30'7

- l

-10

-15

- L V

-0 'B -06 -04-02 0 u /u r6

KEY

IH RUN 60

H RUN 63

-0 6 -04 -02 0 02 u /u .60 0 '2 0/* 06 08 u/ur6(b)

Fig. 4b.

motion is generally dominated by the barotropic motion, particularly forruns 61 and 62. Further, no conspicuous baroclinic motion could be ob-served on the shelf.

308

H

H

KEY

THEORY

RUN 61

RUN 62

U/usb

0 0 0 2 0 4 0 6 u / u . 6(c)

Fig.4c.

5. OBSERVATIONS WITH TWO-LAYER STRATIFICATION

A number of runs with two-layer stratification were made. Of these, 17cases with various amplitudes, frequencies and ratio of layer depths werestudied in detail and compared with linear predictions. For two-layer stratifi-cation, linear theory predicts significant shoreward interfacial wave propa-gation on the shelf, with larger amplitude propagation seaward over theslope. Observations of the interface displacements were made, mainly on theshelf. The Froude number F is again given by eq. 4.I but with

.-: ( +('- ( '- +Aen(r - D)"')) (5 .1 )

where p is the density of the lower layer. To make the interface displace-ments observable, most experiments were conducted with F> 0.5.

-2

309

H

|H

- 2- 4- b

- 8

- 1 0

-12

- u z

(d)

Fig.4. Velocity profiles for four runs with a/c>\(see Table II) measured horizontally attwo locations, lines A and B (see inset). (a) Runs 60 and 63 with u/c:0.90,line A; the solidline denotes the theoretical profile; (b) same as (a) but with line B; (c) runs 61 and 62 witha/c :0.745,Iine A; (d) same as (c) but with line B. Depths are in cm.

The results of these experiments, when compared with the predictions oflinear theory, were generally disappointing. The general character of thepropagating wave motion on the shelf was as expected but there was scatterin the quantitative details, both for amplitudes and phases. Observed waveamplitudes were all less than or equal to the corresponding predictions of

KEY

THEORY

RUN 62

RUN 61

0 z- 0 2-0 6 - 04 -02 0

- 2- 1- 6

-10

-16

t o

0 0 2 0 r * 0 6

Run s. cnumber

d /c h t / hn 7 - hL /hR i l sb1 + h L / h R ( c m s - 1 ;

._TU"tt Horizontal' - Nh t distance from

shelf-break

Line A Line B(cm) (cm)

310

TABLE II

Details for runs with a/c <7

46.646.646.646.6

2.22.62.43 .1

60636762

0.38 0.42 0.90 0.1490.38 0.42 0.90 0.1490.38 0.51 0.745 0.7490.38 0.51 0.745 0.149

0.740.740 ;740.74

2.7J . Z

2.93.75

75.71.5.71.5.7t5.7

linear theory (within the errors of measurement) and the ratio (observed/theoretical amplitude) decreased with increasing -F. It was inferred from thisthat non-linear effects become important for F> 0.5. The number of runsmade was not sufficient to describe the quantitative behaviour pattern,partly because of the number of variable parameters, and we give only aqualitative description of some typical results here.

Figure 5 shows the interface displacements during eight phases of a tidalcycle for three runs which differ only in the amplitude of the barotropicforcing; the parameters are given in Table III. As F increases, the flowpattern changes from one describable by linear theory to one in which asubstantial hydraulic jump forms just seaward of the shelf break, during theebb phase of the tidal cycle. The calculation of finite amplitude effects onthe characteristic propagation speed of long internal waves shows that, forthe amplitudes observed in these experiments, the onshore wave propagation

TABLE III

Parameters for the two-layer runs described in the text

Runnumber

h L

(cm)d(cm)

o Ap /p usb, . _ 1 . , _ 1 .(rad s ') (cm s ')

F: u",o/c*

4l42445556

2.72.72.73.451 1I . I

4.74.7A 1

5.353.0

0.364 0.4060.364 0.4080.364 0.4100.364 0.4320.364 0.542

0.020.020.020.020.06

2.403.816.483.464t.4

0.500.801..360.745.04

Values of 2,6 (and hence F) are based on linear hydrostatic calculations, rather thanmeasurement. For run 56 this value is probably over estimated because of the proximity toresonance, where non-linear factors will reduce the ampttude.

311

(0

-b r '5 Fo *

. L

U i

! ! 9 *q 7 _ "

o ob € . fT ; . )3 9 3

= 8 6F B :

: ( € ^ /> , , . t ^Q H o

{ F c'.lr .. I

s . 9 oo o . 4 )3 9 Eb 9 . gE A P

g r * e6 F X> o f- 9 i -r Y q Ze . ^ d

; tor^b b F f ii i t "

o

cl

372

Fig. 6. As for Fig. 5, but with a relatively thicker upper layer (Run 55 in Table III).

Fig. 7. As for Fig. 5, but with a relatively tlunner upper layer (Run 56 in Table III).

speed is not sensitive to the shear and is given approximately by

Co,rho," : c* (1 - .F cos c.rr) (5 .2)

Hence for F > 1 the wave cannot propagate onshore during part of the ebbphase. This results in a distorted waveform on the shelf, and a surge of upperlayer water onto the shelf at approximate$ the "low" phase of the tide. Thisdistortion will still be present in a weaker form for F <I.

/

\

\

\

3r3

From the viewpoint of linear theory, significant advection by the baro-tropic tide will reduce the effectiveness of the generation mechanism byspreading it over a wavelength. The horizontal displacement 2Lx due toadvection by the barotropic tide a,o is given by

(s.3)2A,x :2u"o/a

and hence

2Ax^

: ' / ' (5 .4)

where tr is the linear internal wavelength. F is therefore a good measure ofthe magnitude of advection, and this ratio is appreciable for F > 0.5.

Two other runs show typical phenomena when the lower (Fig. 6) or upper(Fig. 7) layer is shallow. In these cases the waves show non-linear propa-gation characteristics with their fronts propagating in the form of internalbores (denoted by arrows), particularly in Fig. 6, although the shallowdepths in these experiments probably limit their development. We can,therefore, identify two distinct non-linear phenomena in these experiments,(1) the formation of a hydraulic jump seaward of the shelf-break during theebb phase when F > 1, due to the flow on the shelf becoming supercritical,and (2) the formation of internal bores when one layer is significantly thickerthan the other.

For F > L the flow is effectively turbulent seaward of the hydraulic jump.This turbulence introduces random fluctuations in the interface displace-ment which are apparent in Figs. 5-7, and these fluctuations are advectedonshore with the turning tidal flow. These fluctuations, though small,contribute to the difficulties of measurement.

6. SUMMARY

We have made quantitative comparisons between linear generation theoryand laboratory experimental observations of internal tides, for both continu-ous and two-layer stratifications. For continuous stratification, where baro-clinic tides on the shelf are small, linear theory is a satisfactory model for themotion over the continental slope for values of the Froude number F (takenat the shelf break) substantially greater than unity, and the limit of its

'. usefulness was not reached in these experiments. On the continental shelf thebaroclinic velocities predicted by linear theory are generally sharply peaked;

r such motion has low propagation speed and is rapidly dissipated, if it isgenerated at all. The fact that the obserued shelf flow is relatively featurelesspartly explains why linear theory describes the flow over the slope so well at

314

large amplitudes-water on the shelf appears in the "beam" when it isadvected past the shelf break over the slope, so that large displacementamplitudes have little effect.on the flow structure.

For two-layer stratification, on the other hand, where baroclinic motionon the shelf is significant, non-linear effects causing deviation from lineartheory were apparent for F > 0.5, and possibly smaller. In this case, forexperimental reasons, the region of accurate predictions from linear theorywas not obtained, although qualitatively the observed motion had the samecharacter.

Values of F > 1 are easily realized in the ocean, and this seems to be alikely explanation for the commonly observed non-linear character of inter-nal waves of tidal origin on continental shelves, frequently observed bysatellite. Non-linear generation models are required to describe these phe-nomena adequately.

ACKNOWLEDGEMENT

The able assistance of David Murray with the experiments-and computa-tions is much appreciated.

REFERENCES

Baines, P.G., 1973. The generation of internal tides by flat-bump topography. Deep Sea Res.,20:7'19-205.

Baines, P.G., L974. The generation of internal tides over steep continental slopes. Philos.Trans. R. Soc. London, A227:27-58.

Baines, P.G., 1982. On internal tide generation models. Deep Sea Res., 29: 307-338.Baines, P.G., 1983. Tidal motion in submarine canyons-a laboratory experiment. J. Phys.

Oceanogr., 13: 310 -328.Farmer, D.M. and Freeland, H.J., 1983. The physical oceanography of Fjords. Prog. Oc-

eanogr., t2:147-219.Maxworthy, T., 1,979. A note on the internal solitary waves produced by tidal flow over a

three-dimensional ridge. J. Geophys. Res., 84: 338-346.Rattray, M., Jr., Dworski, J.G. and Kovala, P.E., 1969. Generation of long internal waves at a

continental slope. Deep Sea Res., 76:'779-795 (suppl.).

if