Dynamics of Atmospheres and Oceans, g (1985) 85*120 85Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

THE REFLECTION AND DIFFRACTION OF INTERNAL WAVESFROM THE JUNCTION OF A SLIT AND A HALF-SPACE, WITHAPPLICATION TO SUBMARINE CANYONS

R.H.J. GRIMSHAW

Department of Mathematics, Uniuersity of Melbourne, Parkuille, Vic. 3052 (Austalia)

P.c. BAINES and R.C. BELL

CSIRO Diuision of Atmospheric Research, Aspendale, Vic. 3195 (Australia)

(Received June 25, 1984; revision accepted February 15, 1985)

ABSTRACT

Grimshaw, R.H.J., Baines, P.G. and Bell, R.C.,internal waves from thejunction of a slit and acanyons. Dyn. Atmos. Oceans, 9: 85-120.

1985. The reflection and diffraction ofhalf-space, with application to submarine

We consider the three-dimensional reflection and diffraction properties of internal wavesin a continuously stratified rotating fluid which are incident on thejunction of a vertical slitand a half-space. This geometry is a model for submarine canyons on continental slopes inthe ocean, where various physical phenomena embodying reflection and diffraction effectshave been observed. Three types of incident wave are considered: (1) Kelvin waves in the slit(canyon); (2) Kelvin waves on the slope; and (3) plane internal waves incident from thehalf-space (ocean). These aie scattered into Kelvin and Poincar6 waves in the slit, a Kelvinwave on the slope and Poincard waves in the half-space. Most of the discussion is centredaround case (1). Various properties of the wave field are calculated for ranges of theparameters c/cot 0, ya and f /u where cot d is the topographic slope, c is the internal waveray slope, a is the canyon half-width, 1 is the down-slope wave-number, / is the Coriolisparameter and o is the wave frequency. Analytical results are obtained for small ya andsome approximate results for larger values of ya. The results show that significant wavetrapping may occur in oceanic situations, and that submarine canyons may act as sourceregions for internal Kelvin waves on the continental slope.

1. INTRODUCTION

Field observations of currents in submarine canyons have shown that tidalvelocities and internal wave energies in general are substantially higher therethan on the neighbouring continental slope (Wansch and Webb, 1979;Shepard et a1.,1979; Hotchkiss and Wunsch, 1982). These have significance

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

86

for a range of physical and biological processes, because of the enhancedmixing and enhanced fluid transport between continental shelf and slopewater, and also for geological processes because of the increased bottomstress which may cause sediment movement. Baines (1983) showed that, inlaboratory experiments, this enhanced inbound wave activity is dependenton the ratio cot 0/c, where cot 0 is the bottom slope and c is the slope(relative to the horizontal) of the internal wave rays, and that it is partlyassociated with a phenomenon of trapping of internal waves in canyons, in amanner similar to sound-wave containment in organ pipes. To obtain a basisfor the interpretation of these phenomena in oceanographic field observa-tions it is necessary to obtain quantitative estimates of the associated wavereflection, diffraction and internal wave trapping which can occur forrealistic canyon geometries, and this is the primary motivation for the workdescribed in this paper.

Since many oceanic internal waves (and in particular the internal tides)have frequencies <,.r which are comparable with / (the Coriolis frequency),the inclusion of the Earth's rotation is a necessary factor. Some theoreticalresults for a non-rotating system were presented in Baines (1983) and werebased on the analysis given in detail here. The addition of rotation intro-duces new phenomena, mostly associated with Kelvin waves, and also causesa substantial increase in mathematical complexity.

We consider the reflection and diffraction properties of internal (inertial)waves in a rotating stratified fluid which are incident on the junction of avertical slit and a half-space, which models the canyon mouth. The geometryis shown in Fig. 1, and the vertical slit (width 2a) is normal to the planebounding the half-space. The waves may be incident from within the slit orfrom the half-space. We believe this to be the first detailed study of a fullythree-dimensional diffraction problem of this type.

The idealized geometry of this paper does not consider the effects of theupper, free (reflecting) surface of the ocean, or the finite length and depth ofactual canyons. These considerations will impose upper limits to the lengthscales of the motion for which our results are applicable to actual canyons.The analysis presented here can be regarded as a local analysis, which, forapplication to actual canyons, requires matching with a global analysiswhich takes account of the global features of the canyon and oceanicenvironment. For "non-rotating" canyons this matching process has beendiscussed by Baines (1983).

The generation of internal tides in "non-rotating" submarine canyons hasbeen considered experimentally and theoretically by Baines (1983). Thecorresponding problem with rotation has yet to be studied, but we maypresume that the internal tides generated in the canyon will have thecharacter of Kelvin waves, and will have similar wavelengths to the non-

87

'-'it-

-,/CONTINE NTAt

S L O P Et,- 1 / -----t

{ c }

Fig. 1. Geometry of the idealized canyon on a continental slope. (a) perspective, (b) side-view,

(c) plan-view.

7 -

88

rotating case. The theoretical results described in this paper can be regardedas a preliminary local analysis necessary for the satisfactory solution of thegeneration problem.

Internal wave propagation in two dimensions for given frequency o where

f < us < N (N is the Brunt-Vbisblb frequency) is fairly well understood. In ahorizontal (x-y) plane (with given vertical structure) the wave properties aregoverned by a Helmholtz equation, whilst in a vertical (x-z) plane they aredescribed by a hyperbolic equation; the latter permits certain simplifications,and the procedure for solution is quite different from the elliptic Helmholtzcase (e.g., Baines, I911a,b). The governing equation for the present problemmay be reduced to an equation in two variables which may be either ellipticor hyperbolic, depending on geometrical factors. This difference has implica-tions for the character of the wave field, but the same solution proceduresare used for both types, and are typical of those used for elliptic problems.

The plan of the paper is as follows. The basic equations and formalism forthe solution are set out in section 2. For the case when a Kelvin wave withinthe canyon (sliQ is incident on the canyon mouth, the diffraction problem isreduced to an integro-differential equation across the gap, with a kernelformed from the Green's function for the half-space. The diffracted fieldconsists of a reflected Kelvin wave in the canyon, possibly some reflectedPoincar6 waves in the canyon, a radiated Poincar6 wave field in the half-space(i.e., the ocean), and possibly a Kelvin wave on the continental slope (seeFig. 7). Some relevant properties of the Green's function are also set outhere. Then in section 3 we describe some analytical approximations for thereflection coefficient for the reflected Kelvin wave in the canyon. For narrowgaps analytical expressions are obtained. For wider gaps a plane-waveapproximation is used and evaluated numerically for various values of therelevant parameters. In section 4 we describe an alternative approximationprocedure based on conformal mapping, which can be used for the casewhen the governing equation is elliptic. In section 5 we consider the case of aKelvin wave on the continental slope incident on the canyon mouth, and thesimilar problem when a plane Poincar6 wave in the ocean is incident on thecanyon mouth. Finally we summarize the results in section 6. In theAppendix we describe a Galerkin approximation which was also used toevaluate the reflection coefficient in the non-rotating case (/:0); theseresults were used in Baines (1983).

2. FORMULATION

The linearized equations of motion for an incompressible, Boussinesqfluid, rotating about a vertical axis with angular velocity if , are

u, - fu * p , :0 (2.1a)

u , * f u + p r : 0

w , * p # p , : 0

p , - N 2 w : Q

u r * u r * w r : 0

Here u, u, and w are the velocity conlponents in the x, y and z directions,respectively, p is the reducdd presSure, p is the buoyancy and N is theBrunt-Vbisblh frequency which we assume henceforth is a constant. Thetotal pressure is po(z) -f pop and the total density is po(z)(1 * g-tp), wherepoQ) and ps(z) are the basic state pressure and density fields, respectively,

Po": -EPo, and TPo,: -poN'. If all variables have the time factorexp(-lc,;r), then the velocity components can be expressed in terms of thepressure

89

(2.1b)

(2.1c)

(2.1d)

(2.1e)

( r ' - f ' )u : - iap,* fp ,

( r ' - f ' ) u : - f p , - i r p ,

(N'- r lz)w: iap,

Then,'eliminating the velocity components through (2.7e), we obtain

Pr' + Pyy - czPrr: 0

where

t 0 ) 2 - f 2 .f - : --

N 2 - , , 2 ' " " i '

(z.za)

(2.2b)

(2.2c)

(2.3a)

(2.3b)

: , . J ' - , ' : ' j ! : j , - : j ' i

l : :

We propose to.model the continental s1op9 6y the plane x * z tan 0 :0,where d(0 < 0 < 1T /2) is the angle between the, slop.g,ppd,the vertical (see Fig.1). The canyon we shall rnodel by a. semi-inf,lnite cha,ryrgl of infinite depthand along-slope horizontal width of 2a. To describg this.geometry it isconvenient to introduce new co-ordinates in the z-x plane, r and s, givenb y :

r : i r c o s 0 + z s i n d \s : -x s in 0 + z cos 0 l

(2.4)

Thus r is a co-ordinate normal to the continental slope, which is given byr:0, = oo <r< co and lyl>a (see Fig. 1a,b). The channel is then givenby r<0, -@ <,s< oo and ly l<a.Theboundarycondi t ionstobesat is f iedoa the slope and channel boundades are that the velocity c-omponentsnormal to the boundary should vadsh. At the junction of the slope and the

90

channel we shall require that the pressure and normal velocity be continu-ous. Hence

u : O f o r r 1 0 , - c o < s < o o a n d l y l : a ( Z . S a )

u c o s 0 * w s i n 0 : 0 f o r r : 0 , - o o < . r < o o a n d l y l > a ( 2 . 5 b )

p, ucos 0 + w s in dcont inuous at r :0 , - m <.s < co andly l<a \2 .5c)

Equation (2.3a) and the boundary conditions (2.5a, b and c) form a problemwhich has a close analogy with a tidal diffraction problem discussed byBuchwald (797I); it 0 :0, the analogy is exact. We pursue a similar methodof solution to that used by Buchwald (1971). The boundary conditions(2.5a,b and c) suggest that it is appropriate to seek a solution which has aprescribed wavenumber y in the s-direction (i.e., in the direction parallel tothe continental slope). Hence we put

(2.6a)

(2.6b)

Here y, like c".r, is specified a priori. Substituting (2.6a) into (2.3a) it followsthat

er, * (cos20 - c2 sirf 0) q,, +y2c2

q : 0 (2.7)(cos2d - c2 sin2d)

An important distinction emerges at this stage. For a "steep" slope, whencot20> c2, the governing equation is elliptic; but for a "flat" slope, whencot20 < c2, the governing equation is hyperbolic. One consequence, whichwill emerge as we develop the analysis, is that for a "steep" slope waves canbe radiated in all directions, but for a "flat" slope waves are confined to thewedge 12 > y2(c2 sir'20 - cos2d;. The boundary conditions (2.5a,b and c) canalso be expressed in terms of q

if fyc2 sin 0- a c o s 0 q , * 4 y * f f i Q : 0 ,

/ ( 0 , l y l : a

(cos20 - c2 si*O)q, + {"o, lqr:0, r :0, I yl > a

q, (cos20 - c2 sirtL)q" + {cos lqrcontinuous at t :0, lyl< a

The boundary conditions at infinity are that all incoming waves should bespecified a priori, and then eq.2.7, with the boundary conditions (2.8a,b and

(2.8a)

(2.8b)

(2.8c)

q : p , ( y ) e x p ( l a r )

where

po(il- "*o( - X4

k2 , ct2- : f : : -

m 2 " u

N 2 - a 2

and

p,(y): kf sinfij + o) - ry:""'#0 + o)- . I n r \ 2u ' * \ n ) , , ' - f '

^ : -m 2

- N 2 _ a 2

91

c), determines only outgoing waves. In the channel (r < 0), it is readilyestablished that eq. 2.7 and the boundary condition (2.8a) can be satisfied by

(2.ea)

(2.eb)

(2.ec)

(2.ed)

(2.ee)

Here ft and m are wavenumbers in the xfrom (2.6a) and (2.9a)

p : p,(y) exp(tkx * imz)

and

and z directions, defined so that

(2.I}a)

(2.10b)a * M : / c c o s 0 + m s i n d \

k s i n 0 - l m c o s 0 |

The n: 0 mode is a Kelvin wave, and the remaining modes n: I ,2, 3, . . .we shall designate as Poincar6 waves. The corresponding dispersion relations(2.9c) or (2.9e) are to be interpreted as equations for a when y and @ arespecified.

For the Kelvin wave mode, if. c( > 0 both values of a determined from(2.9c) are real; one represents an incoming wave, and the other, which wedesignate henceforth as a' (and similarly k', m') is an outgoing wave. Theyare distinguished by computing the component of group velocity in ther-direction from (2.9c)

0, : a3m^l

0 a N 2 k 3(2.r1)

It may then be shown that if c2o < cot20, the incoming wave has c,rk > 0 andthe outgoing wave has ak' <0, and for both waves my (or m'y)>O. lfc2o> cot20 then the incoming wave has am10 and the outgoing wave hasrim' > 0, and for both waves ky @r k''D < 0. On the other hand if .o' ( 0,

92

both values of c are complex-valued, and we must choose that for whichIm a < 0, so that the corresponding mode is evanescent.

For each Poincar6 wave mode there are again two values of a, determinednow from (2.9e). They are given by

(2.12)

lf cot20<c2, then both these values of a are real; one represents anincoming wave and the other, designated by o', represents an outgoing wave.They are distinguished by considering the sign of the r-component of groupvelocitv

o, : (" '- f ') '(cos2d - c2sirto)a0c (u , - f , )ac4m2

Thus the incoming wave has aa(Nz - f').0, while the outgoing wave hasaa'(N2 - f

') r 0. If cot2d > c2, but .l > ^ln where

a(cos2 0 - c2sin2d) : + {t 'r '

-(#)'{"or'a - r ' r in' i l}/

Y.:( #\ry -'i'fo)"'

(2.13)

(2.74)

so that both values of a are again real, then the incoming wave hasacr(Nz - f')r 0 while the outgoing wave has oa'(N2 - f').0. Since 7n --

@ as n -- m (2.14) it follows that only a finite number of Poincare wavescan propagate, and for sufficiently small values of ya (y < yr) there are nopropagating modes. When a is complex-valued, we choose that value forwhich Im a < 0, so that the corresponding mode is evanescent.

We propose that in the channel the solution of (2.7) and the boundarycondition (2.8a) is

Q : Q t * { n , f o r r < 0 , l y l < a

where

qt: A^p^(y) exp(tu^r)

and@

4n: I A',pi,(y) exp(ta!,r)n : o

(2.Isa)

(2.15b)

(2.1sc)

Here we have indexed the r-wavenumber a according to the mode number,and the direction of propagation of the mode. The incident w;ave q, has aspecified amplitude A^,and its modenumber rn is such that a^ is real. Insubsequent applications we shall be concerned only'with'the case tn:0, so

93

that the incident wave is a Kelvin wave. The reflected wave q* contains allmodes, and their amplitudes are to be determined by matching with theocean solution from.the boundary condition (2.5c).

In the ocean (r > 0) it is readily verified that a solution of (2.3a) whichdescribes a propagating internal wave is

p: exp(ikx + ily + imz)

where

k 2 + 1 2 : c ' * '

Comparing (2.16b) with (2.9c) we see that this is thechannel Poincar6 wave modes, and we shall likewisePciincar6 wave. The corresponding expression for 4 is

q: exp(iur + ily)

where

a(cos2a - cz si*o)- + { y'c'- 12(cos20 - r' ,irfl)\t/'

(z.I6a)

(2.16b)

counterpart of therefer to this as a

(2.r7a)

(2.r7b)

(2.18a)

(2.18b)

(2.18c)

If cot20 < c2 then both values of a are real; the direction of propagation isagain determined from the sign of the r-component of group velocity (2.13).Thus an outgoing wave has aa(N2 -f').0, while lf. cot20 > c2, and a isagain real, then an outgoing wave has aa(N2 - f')r 0. In both cases, wemust choose the square root in (2.17b) to have the same sign as ,n(N2 - ft).If a is complex-valued, then we must choose fm a> 0 so that the corre-sponding mode is evanescent.

The boundary condition to be satisfied on r:0 for I yl> a is (2.8b). It isimmediately apparent that no single Poincare wave can satisfy this boundarycondition, although a combination of an incoming and outgoing wave withsuitable amplitude ratio can. However, if cot20>cl>0, then it may beshown that there is a Kelvin wave mode which satisfies (2.7) and (2.8b). Thisis given by

q = exp(- dr + ilsy)

where

&.(cos2o - c2 si,,:e): - fu"o, o

and

tl(cos2o - cl sirfo): ctyz

To ensure that Re A > 0, we must choose /o from (2.18c) so that flo/r':<0.The group velocity in the y-direction is oy27to(y'+ til and so the Kelvin

94

wave propagates in the negative (positive) y-direction when / is positive(negative).

We now propose as a solution of (2.7) in the ocean region a Fouriersuperposition of outgoing Poincar6 waves

(2.1e)

Here a is given by (2.77b); if the square root is real, it has the same sign as@(N2 - f'); otherwise Im a > 0. Next, it is useful to define

u ( y ) : ( c o s 2 0 - c z s i r j : 0 ) q , + { " o , 0 q , a t r : 0 , - o o < y < @ ( 2 . 2 0 )

Apart from the constant factor -iu(az -f')-', U(y) exp(iys) is thevelocity component normal to r :0. The boundary condition (2.8b) shows

that U(y) is zero for ly l>a. Next we determine B(/) in terms of U(y).Using the Fourier inversion theorem, the result is

q: + I l* t |>exp(lar + i ty)d/ for /) 0, -co < y < @

r @B( t )D( t ) : J_* r r r ) . "p ( - i t y )dy

where

D(t): i a(cos2d - c2 sin2o) - f l"o, o

On substituting (2.27a) into (2.19) it follows thatr @

q : J _ _ o ( r ,

y - i ) U ( i ) d i f o r r > 0 , - @ < ) , < o o

where

I r@ exp(iar + i ly) dlG( , , y ) : ; J_ * 'T

(2.21a)

(2.27b)

12.22a)

(2.22b)

Note that since U(y) is zero for lyl> a,the integral in(2.22a) is only over

I j,l<a. Here G(r, y) is the Green's function for (2.7) with the boundarycondition that U(y) have a D-function singularity at y:0. If it is desired toinclude a forcing term qo in r>- 0, then go must be specified a priori andadded to (2.22a). Also qo should satisfy (2.1) and the boundary conditionthat U(y):o for all y. For instance qo could consist of a Kelvin wavepropagating along r:0, or it could consist of incident and reflected Poin-card waves. These possibilities are taken up in section 5.

The boundary condition (2.8c) shows that at r :0, p and U can beevaluated from the channel solution (2.15a). Thus

"aq t + q R : I G ( 0 , y - r ) U ( r ) d i t , a t r : 0 , l y l < a ( 2 . 2 3 a ){ _ a

95

where

u ( y ) : ( c o s 2 d - c 2 s i n 2 | ) ( q r , + q R ) + { " o s 0 ( q r r * q * r ) ,

a t r : 0 , (2.23b)

Equation (2.23a) is an integro-differential equation for U(y) where q, isspecified. From the series representation (2.15c) it is also an infinite set ofalgebraic equations for the reflected wave amplitudes A', with the incidentwave amplittde A^ specified. In general (2.23a) must be solved numerically,but we shall not develop this approach here. Instead, in section 3, wedescribe various approximations which are generally most useful for narrowgaps ( ftal < < 1). For "steep" slopes (cot20 > c2) and narrow gaps, confor-mal mapping methods can be used to obtain approximate solutions; this isdescribed in section 4.

We close this section with a discussion of the Green's function G(r, y)(2.22b), which is similar to a Green's function used by Buchwald (1971) in atidal diffraction problem. Hence we simply quote the relevant results here.In the far field, where 12 + y2 + oo, the integral in (2.22b) may be evaluatedby the method of steepest descent. If cot20 > c(> 0, D(l) has a zero on thereal /-axis at l: /o; this zero corresponds to the Kelvin wave mode describedby (2.18a,b and c), and so the contour of integration is indented around thepole in a manner to ensure that the Kelvin wave is outgoing. We find that

G - Gr* G* as r2 + y2 -, ss (2.24)

where Gn is the contribution from the critical point in the method ofsteepest descent, and represents radiating Poincare waves, while G* is theKelvin wave. Here

" p( i ry * iur *

(2.25a)

x {t2c2 + 12 (c2sirf e - "or 'o)}t/ '

(2.zsb)

and we recall that a is defined as a function of / by (2.17b). The sign of thesquare root in (2.25b) is that of o(N2 - f'); also t in the exponent is thesign of -,,r(N2 - f'). For a given value of y/r (2.25b) determines a uniquevalue of I; btrt this value of / can be used in (2.25a) only if it is real. Forcot20 > c2 > O this is the case for all y /r. and Poincar6 waves are radiated in

< a

G P

i n \4 )

ora(2,,1#l)"'where

y : - d or d l

l nrl[r' + y2 (coszo - c' sin'o)f'/'x:

\ (""" 'd - r 'si , fd) -

and

, - sien(c,,(-,^r 'z-l1)) \r*rrrrr($(cos2,

- c2 sirf 0)* r) '")-" ' 1r.rr",(t- fi*' e) \ \ r-

Here H is the Heaviside function. The Kelvin wave is the contribution fromthe pole on the real axis, occurs only if cot20> c3>0, and is given by

- i u l f l c o s d /G "

: f f iexp(

- dr + i t sy) H (- fy)

where & and /o are defined by (2.18b and c).

all directions. However.for cot20 <c2. a real value of / can be found onlvfor

, 'u y ' (r ' s i f i? - cos2d) (2.26)

and the Poincar6 waves are confined to the wedge described by (2.26);outside this wedge G" is exponentially small. lf c2 < 0, there are no Poincar6waves and G" is exponentially small. Evaluating / from (2.25b) we find that

| ,2 " l cos20 \ \Gp:Aexp( i1 ) ,n l - +y ' l + -s in 'zd l l (2 .27a)

\ c ' \ c ' l lwhere

- f')) (2.27b)

(2.28)

The behaviour of the Green's function in the near field is given inAppendix A. The structure here is not of particular interest in itself, but it isused in the approximations described in the next section.

3. ANALYTICAL APPROXIMATIONS

Henceforth we shall assume that c2o > 0 and that the incident wave is aKelvin wave (i.e., m : 0).The main aim of the analysis is the determinationof the reflected wave amplitvde A'o from the integro-differential eqs.2.23a,b.For this purpose it is convenient to recast eqs. 2.23a,b into a form fromwhich A'o can be directly determined. The procedure is similar to that usedby Miles (L967) in an analysis of surface wave scattering.

With q given by (2.15a,b,c) we find f,rom (2.20) that

u (y ) : Aouo( i l + d 'ou 'oQ) * i n ' ^u ; ( y ) , l y l . o

i ) 'o't '(N'

(3 .1a)

97

(3 .1d)

Here k'^ are determined from the dispersion relation (2.9e) with y and c,rspecified, while ko and k'o are determined from (2.9c).In particular, we notethat

where

uo( i l : * !p ,01c6mo

u'o( i l : **pa(v)c6mo

and

u',(y): t fczm'i l sinff i | + o) *,r4(#)A;cos d - cf;m', sin 0)

* "otffi1 + o)

ko k 'o / \ ( to - t6 ) (cos2o-c ls inzo)- - ) : L 0 S r g l - r \ w i ) : -

mo m'g -u --(

2y cos 0

and

mo(* ') : y(cos 0 T co sign(c.ry) s in 0)-t

[" "t

oi,u: + pluL] d.v : o if n * s

* I:.1:"o(0, y - i,)u(i,)uo(il di dy

(3.1b)

(3.1c)

(z.za)

Q.aa)

(3.2b)

Here the upper (lower) sign refers to ms(m'J. The next step is to obtain anexpression for A'" in terms of U(y). However, this is difficult to obtaindirectly from (3.1a) as the set {u',(y); n:0, I, 2,... } is not orthogonal.Instead we note the biorthogonality relation

(3 .3 )

Remarkably, the relation (3.3) also holds between incident and reflectedmodes. It is readily established from the definitions (2.9cd) and (3.1c,d). Itsphysical significance arises from the constancy of the energy flux down thechannel, and this aspect is developed at the end of this section. To use (3.3),we multiply (3.1a) by p!, and (2.15a) by u',, and integrate both equationsover the range - a < y < c. Using (2.23a) we obtain

l ' . a \ r a

1 2 l p o u o d ! l A o : I U \ y ) p o U ) d y\ . / _ a J r - a

(z l ' 76", or)na : I _,rtrl p'00) dy

* I:"1:,o(0, y - r)u(r)u'o(il di,dy

(z | _ y:"; o r) o; : I _"uf rl pi,(y) d y

* I:,1:,o@, y - i lu(i,)u',(y) di,dy

-v

_v

d

d

i

i

d

d

)

)

v

_v

(

(

vo

Po

, )

v )

(

(

U

U

, )

, )

v

-v

(o

(o

G

G

.l:""aJ"oJ

I

I+

v

v

d

d

v )

v )

(

(

p'o

Po

v )

v )

(

(

U

Ua

a

? a

JJ

(3.4b)

(3.ac)

Equations (3.4a,b,c) express As, A's and A'n in terms of U(y) alone. Inparticular, substitution of (3.4b,c) into the series (2.15c) converts theintegro-differential eq. 2.23a into an integral equation for U(y) alone.Alternatively, substitution of the series (3.1a) into the right-hand sides of(3.4b,c) yields an infinite set of equations A',, which can form the basis of aGalerkin approximation. This approach was pursued in the non-rotatingcase (/:0) and is described in Appendix B. However, in the rotating casean attempt to use a Galerkin approximation showed that a very high orderof truncation in the series (3.1a) is required due to the very slow convergenceof the series. This is because t(y) has a singularity of 0( lyT a'-tz:; asy -+ +a, with the consequence that A',uL(y) is 0(n-2/s1as /? --) oo.

As a preliminary to obtaining approximations for A'o we divide (3.ab) by

Q.aQ to obtain the following expression for the reflection coefficient

A t I H \ 1 / 2R : - ; Y l ; I

A o \ D I

where

(3 .5a)

(3.sb)b(b,): l_73@ dr(l_"0'3t i l dy)

and

^ c z k ot : t

4 m oHere we have also used (3.1b,c). The normalizing factor (b'7bytz has beenincluded in (3.5a) so that lR |

2 measures the ratio of reflected to incidentenergy flux (see (3.19a) below). The expression (3.5a) is scale-invariant inU(y) and hence lends itself to approximate procedures.

(3.sc)

99

3. L Narrow-gap approximation

Firstly, we consider approximations for narrow gaps when lyal< <1.From (3.4c) it can be shown that relativeto Ao, A', is0((ya)2; for n27.Thus the Poincare wave modes (n>- 1) make a contribution of 0(ra) to thepressure field (see (2.75a,b,c)), but an 0(1) contribution to the velocity fieldU(y) since in (3.1a) each A'nu', is O(1).Note that this is the case even whencot20 > c2 and the Poincard waves are evanescent down the channel forsmall ya. However, it can be shown that the contribution of the Poincardwave modes to (3.5a) is 0(ya), since to within this error poj) atd p'oQ) areconstant, and the integral of A',u',(y) over the range - a < y < a is O((ya)')for n> 1. Thus to leading order in ya, U(y) in (3.5a) can be approximatedby the first two terms in (3.1a). Further, poj) and p'oQ) in (3.5a) can beapproximated by constants. The result is

^ : - { ' -

Here e "

is an 0(1) term, and €pyd represents the total contribution of thePoincar6 wave modes. To evaluate the integral in (3.6) we use the approxi-mate expressions (A.5a), or (A.6a), for G(0, y). Thus, for cot20 > c2 > O

G(0, y) : . ' " (

*y)* r rs ign y + c3+o(ro) (3.7)

1 L, l_.o(0, y - i l d iur)( t +o((r ,) ' ) ) *epya (3.6)

a n d F : t { 2 , + h " - 1 }

o r F : " { 7 , + b " - } }

for cot20 > c2 >0

fot cot20 < c2

(3 .8 )

(3.ea)

(3.eb)

(3.ec)

(3.ed)

For cot20 1c2, Ct is replaced by -iC, sign(or(N2 -f')) and C, by Co,while for c2 <0, C, is replaced by Co lf cot20 < cl and by G if cot20 > c20.When (3.7) is substituted into (3.6) we find that

R : - r * Ea^(+) * r 'o t e ,ya+ o((vr ) ' )\ r I " \

Here, the constants E and F are given by

E:4(ko- k ' r ) ( t - , , tur f 01' / t for cot20 > c2

or E : - ] l o r - k ' o l k ' t a f i | - t 1 t / ' f o r co t20 <c2

If c2 < 0, F is again given by (3.9d) if cot20 < co2, but Ca is replaced by C, ifcot20 > cf. Note that we have retained the term Fa in (3.8) even though it is

100

of the same order as epla, which at this stage is not known explicitly.However, for cot20 > c2 > 0, it can be shown that the leading order term ine, is pure imaginary, and hence only contributes a phase shift to R.. Insection 4 we present an alternative method of deriving the narrow-gapapproximation which allows e

" to be evaluated (see (4.7)). For this "steep"

slope case it then follows from (3.8) that

I R | : 1 - olko - k 'ol( l - r ' tan20)t /2 + 0((7a ln ya)2)

i f cot20 > ,3> , '

or

lRl : 7 - a lko- k 'of { t r - r ' tun' | ) ' / '' ' t '

i f c26> cot20 > c2

(3.10a)

Note that when cot20 > cB a Kelvin wave can propagate along the continen-tal slope, and the reflection coefficient for this case (3.10a) is smaller than inthe case (3.10b) when no Kelvin wave can propagate. In the "flat" slopecase, cot20 1c2, E is real and dominates any contribution of the Poincar6wave term e , to lR l. We find that

) | a o l ^ \l R | : 1 +

; l k o - k ' o l a t n \ :

) + o ( 1 a )

if cot20 < c2 (3 .11 )

The reflection coefficient is smaller by an order of magnitude than its valuein the "steep" slope case. lf c2 < 0, then lRl is given by (3.10a) ifcot20 > cf,, and is 1 with an error of 0((ya ln ya)2) If cot20 < c?0.

Although A',(n>- 1) and hence U(y) have not been found explicitly, wecan find an expression for q in the far field, where r'+y' --+ oo. From(2.22a)

(3.I2a)

(3.12b)

Here the Poincard wave modes contribute only to the error term in (3.12b).From the far field expression for G(r, y) (2.24) the amplitudes of theradiated Poincar6 waves and the Kelvin wave, if this latter exists, can now bedetermined. Note that (3.12a) is not valid in the regions (?' - y'):0(o')when cot20 < c2.

q - G(r, i l f" "u(i,1 ai,

where

I_,rtr l d,i , : iy4aA,(, - +u'"(+) * of wl)

- l* l) + o((Ydtn ya)2)

(3.10b)

101

P lane -w au e approximation

To extend the approximate expression (3.8) to larger valuesreturn to the exact expression (3.5a) and exploit the fact that itvariant. Thus, in (3.5a) we put

U(y): (constant)po(y)

of yc, weis scale-in-

(3.13)

Q.Iaa)

(3.14b)

(3.1ac)

This can be called the plane-wave approximation (Miles, 1967) and isequivalent to a one-term Galerkin approximation. Making the substitution

ft=-(+)"'\*#)where

b'so : I-,or(v) p(,(v) dv (3.14d)

b'st: t, : I:,f .G (0, y - i,) poj) p'00) d i d y

bs2: J2: I_,1_,o(0, y - i l poj,) poj) di , dy

and

The approximation (3.1aa) ignores any contribution from the Poincar6 wavemodes in the channel, and in this respect is similar to the narrow-gapapproximation (3.8) with e" omitted. However, unlike the narrow-gap ap-proximation, there is no restriction on fa and, in particular, the fullcontribution of the Poincaft and Kelvin waves in the ocean is retained.Further, it is readily shown that (3.14a) agrees with the narrow-gap ap-proximation (3.6) (when e

" is omitted) as ya --> 0, and hence agrees with

(3.8) in this limit. Thus (3.14a) can be regarded as an extension of (3.8) tolarger values of ya. There is, of course, no indication of the range of validityof (3.I4a), but experience with similar approxtmations elsewhere suggeststhat it may be useful (see Buchwald, 1971). The integrals Jt., (3.I4b,c) in(3.I4a) must be evaluated numerically. Firstly, however, we substitute theexpression (2.22b) for G into (3.14b,c) and then interchange the order ofintegration; this can be justified here even though the integral defining G isonly conditionally convergent. {hen cot20 > cl tne integral (2.22b) has apole on the real axis at /0, representing the Kelvin wave contribution. This istreated by indenting the contour around the pole in the sense required togive an outgoing Kelvin wave, extracting the residue, and evaluating the

t02

remaining integral as a principal value. Thus we find that, f.or cot20 > c!

t,: + I:*L( l )L ' ( t ) * d t i " l f l cos ?L ( l )L ' ( l ) *D ( t ) z("' - f')(cos2a - cl sir:20)

i c ^ r l l l c o s 0 l L ( t ) 1 2r _ 7 f * l L ( t ) | ' z d lr 2 -

2 , I _ * D A

(3.15a)

(3.15b)2(r ' - f ' ) (cos'a - c l s in2o)

where

l k " f \ - r l I k " f a \ | k " f a \ \L U ) : l : + i / l \ e x p l i t a + _ - _ _ y j _ l _ e * p l

_ i t a _ " " l ) ( 3 . 1 5 c )\ o / \

- \ ( , ) I '

\ ( . ) l )

while t'(/) is given by (3.15c) with fto replaced by k'o.For cot20 <coz theresidue term in (3.15a,b) is omitted. The remaining integral in (3.15a,b) wasevaluated numerically with the exception of a small interval about theprincipal value point /s, which was evaluated analytically.

The reflection coefficient R is a function of the dimensionless parameters

f/a, c/cot 0,0 and ya.The results from evaluating (3.14a) are shown inFigs. 2-6 as graphs of lRland phase (R) as functions of ya for selectedvalues of f /a, c/cot 0 and d. For simplicity only results for positive valuesof ya (i.e., up-slope propagation) are shown; the results for negative valuesof ya are similar. Indeed, for the narrow-gap approximation (3.8) R is afunction of lya l. In general we find that the results for negative values of yadiffer significantly from those for the corresponding positive values of yaonly for lyal> 0.5 with the difference increasing as c/cot 0 --I. For theresults shown in Figs. 2-4, the difference in lR I between positive andnegative values of ya was never >207o, and usually <107o. For thenon-rotating case (and also for d :0") the reflection coefficient is a functiono f l y a l .

In Figs. 2 and 4 we put f /" :0.709 which is a typical value for M, tid,alforcing in mid-latitudes. The results shown are for a fixed value of N/a:35.5 and a range of values for c/cot 0 varying from 0.1 to 1.9; correspond-ingly, d varied over the range 79-89", which are typical of continentalslopes. We also did some calculations with d fixed at 84o and for the samerange of values for c/cot 0; the results differed very little from those shown.For this range of 0, R is insensitive to 0 but shows significant variation withc/cot 0.ln all cases I R I decreases from one as ya increases, while phase (R)increases from -180o. In Fig.2 we show the cases c/cot 0:0.1and 0.5,corresponding to a 'osteep" slope; also, for both these cases, cl < cot20 andhence a Kelvin wave can propagate along the continental slope. In addition,ko and k'o have opposite signs (see (3.2a,b) or the discussion following(2.11)) and hence the incident and reflected Kelvin waves in the channel are

103

0.0.0

000

Phose(R)

- 60'

- 8 0

- 100'

- 120"

-'l/*0

- 180

f =oror

quodroiic opprox

ccoi 0 = 0 '5

,2' c

c o t O = 0 1

1001 010-11 0 2'10-l

T t

Fig.2.The reflection coefficient jR (3.5a), which is the ratio of the reflected to incident energyflux, for the case when an internal Kelvin wave is incident from inside the canyon (Fig. 7a).The plots show -R as a function of ya for various cases when cot 0 > ca. The solid curves(-) denote the "plane-wave" approximation (3.14a), and the quadratic approximation isalso shown for the case c/cot d:O.l.rThe dashed curves (---) denote the analyticsmall-gap approximation (3.8). Herc f/a:0.709. (a) shows the magnitude of R, lRl, and(b) shows the argument of R, phase (R).

104

| = o r o e

-l-- = o.scol u

-ro o'

- 1

Phose( R ) - rL0

- 1 60

- 18010-3

T o

Fig. 3. As for Fig. 2, but for a case when c < cot d < cs.

confined to opposite channel walls (see (2.9b)). In both cases lR I decreasesto zero; for large values of. ya phase (R) changes rapidly and lR I isoscillatory, but the oscillations are too small to be apparent in the Figure. InFig. 3 we show the case c/cot 0:0.9; this is still a "steep" slope but nowc2o> cot20 and there is no Kelvin wave on the continental slope. Moresignificantly ko and ki now have the same sign, and the incident andreflected Kelvin waves in the channel are now confined to the same channelwall. As 7a increases lRldecreases from one and phase (R) increases from-180o, but as ye--) @, R tends to a constant. This can be shown analyti-cally from (3.14a). The explanation is that with both Kelvin waves confinedto the same wall there can be significant transfer of energy even for verywide channels. In Fig.4 we show the cases c/cot 0:7.L and 1.5, corre-

10-110-2

105

T s

Fig. 4. As for Fig. 2, but for cases when 0 < cot 0 < c.-fhe analytic small-gap approximation(- - -) is shown only for the case c/cot 0:7.1; for c/cot" 0:I.5 it is indistinguishable fromthe "plane-wave" approximation for ya op to 10-l. For phase (R), the analytic small-gapapproximation is equal to -180o, and is not shown.

sponding to a "flat" slope; also for all these cases c3 > cot20. The results aregenerally similar to those shown in Fig. 3, and the same comments apply. Itappears from these,results that the transition cf, S cot20 is just as significantas the ,transition between o'steep" and "flat" slopes. In Figs. 2-4 we alsoshow the results from the narrow-gap approximate formula (3.8) where the

10r10{10-l

I-^ =r s qucidroiic opproxcot u

106

10-l 10-2'!0{ 1 0 100

T o

Fig. 5. As for Fig. 2, but with / :0 (the non-rotat ing case), with 0 < c<cot d. (x) denotethe results of the Galerkin approximation (Appendix B), which is shown only for f a >-70-2,as there is complete agreement for smaller values of ya.

Poincard wave contribution e o is evaluated from (4.7) when co(0 > c2 and isignored when cot2d < c2. The narrow-gap approximation is reasonably goodfor ya < 0.1.

r07

lRl

1 .0

0 5

\\\

".\..\ \ r. , \ . . \ - c , = 1 s

, .#\<k"t', . *

= 1 1 - \ r \ l

Fig. 6. As for Fig. 2, but with / :0, and 0 <cot d<c. For both the "plane-wave"approximation and the analytic small-gap approximation, the argument of R equals -180',

and is not shown.

1 0 - 1

%

rc-21 0 - '

A possible improvement to the plane-wave approximation is

, r I h \ t / 2 , lu ( y ) : ( cons tan t ) ( po (y ) - \ ; , )

np t t y l 1 (3.16)

which, although it again ignores the Poincar6 wave modes in the channel,does give the correct expression for both the incident and reflected waves.When (3.16) is substituted into (3.5a), we obtain a quadratic equation for Rwith coefficients involving integrals similar to Jr., (3.14b,c). This was testedin a few cases. For ya < 1.0, the difference between this quadratic approxi-mation and (3.14a) was insignificant, and for larger values of ya was alwaysless than a few percent.

In Figs. 5 and 6 we put /:0 to obtain some results pertinent tolaboratory experiments. The remaining parameters are chosen to model theexperimental configuration of Baines (1983). In Fig. 5 we put cot 0: 0.615and show results for a range of values of c/cot 0 :0.7, 0.5 and 0.9,corresponding to the "steep" slope case. The results are generally similar tothose shown in Fig. 2; lRl decreases from one to zero as "ya increases, withsmall-scale oscillations as ya-+ co. Also shown in Fig. 5 are the results of aGalerkin approximation, which is described in Appendix B. We can see thatthere is reasonable agreement between the Galerkin approximation, and thepresent plane-wave approximation. In Fig.6 we put N/r:1.8 and showresults for c/cot 0:1.1 and 1.5, corresponding to the "flat" slope case.Again lR I decreases from one to zero as ya increases although, in contrastto all other cases, lRl increases with c/cot d. Also note that phase (R) is

108

now constant, and equal to -180o. In both Figs.5 and 6 we also show theresults for the narrow-gap approximation (3.8), which is again reasonablygood for ya <0.L

3.3. Energt balance

To conclude this section we examine the consequences of conservation ofenergy. It is readily established from the eqs. 2.1a-e that

( u ? ) p ) , + ( u p ) r : 0

where

and

(3.17a)

T ("'"r )dr (3 .17b)

u ( ' ) : u c o s 0 - t w s i n 0 (3.17c)

Here uu) is the velocity component in the r-direction (along the channel),and so (uQ)p) is the time-averaged energy flux along the channel. Integrat-ing (3.17a) across the channel, we find that, in the channel for r < 0,

g: I"

"@(')p) dy: constant (3 .18 )

(3.1ea)

(3.1eb)

Evaluating ,%' from (2.15a) and (3.1a), and using the biorthogonality relation(3.3), we find that

. q : \ t A ^ 3 Y k o b - + l A ; t ' Y k o b '@tl lo QtTlo

* + i 1t',3ek€!-!E!)Re\a,)bin : o l r ' - f ' )

where^a

b ' , : I l p i , l " d v

Since ai is real only when the Poincard wave mode can propagate down thechannel the sum in (3.19a) contains only the finite number of propagatingmodes for which .{ > yn (see (2.14)). For y < yt there are.no propagatingPoincard wave modes and I is given entirely by the first two termsrepresenting the incident and reflected Kelvin wave modes. Next, applyingGreen's theorem in the plane to (3.17a) for a region bounded by the channelwalls ( l y l : a for r < 0), the continental slope ( r : 0 for l y l > a), a cross-

109

channel strip (lyl<a for r<0), and a large semi-circle in the ocean(rt + y'- @), it is readily shown that

.q: f im ["" ((u"' cos Q * u sin Q)p) d{ (3.20), 2 + v 2 - a J - n / 2

Here y : r Lan Q, and the integral in (3.20) represents the outgoing energyflux due to radiating Poincare and Kelvin waves in the ocean. This integralcan be evaluated using the far field expression (3.12a), and the asymptoticexpressions (2.24), (2.25a) and (2.27a) for G(r, y) as 12 * yt - co. We findthat, for the "steep" slope case when cot20 > c2

s: +t [ ' u(y) ayl l , ] '1." . f l rygl +["or,a - cz sin2| l tn-t \ - t' r - a \ ( , , r r - / r ) \ l @ |

' ' l

(3.27)

Here the first term represents the contribution from the radiating Poincarewaves and the second term is the contribution from the outgoing Kelvinwave. For cot20>cl the ratio of the contribution from the Kelvin wave tothe contribution from the Poincar6 wave rs

I f /"1 {l tt"l + (1 - , ' tan'o)'/ '}(3.22)

(t - f '/, ') (t - ,3 ta*o)For fixed values of cs, f this increases without,limit as lco/cot dl increasesfrom zero to unity. The Kelvin wave also dominates as o approaches /. Forthe "flat" slope case when cot20 < cz the integral in (3.20) cannot beevaluated from the asymptotic expression (2.25a) because there is a non-inte-grable singularity along the characteristic directions r':y2(c2 sin20 -

cos20;. Along these directions the asymptotic expression (2.25a) must bereplaced by an expression involving Airy functions and these also contributeto (3.20). However, the resulting integrals cannot be evaluated explicitly andso will not be displayed here. The equality between (3.19a) and (3.21)provides a useful check on the approximations discussed earlier in thissection, and it can be shown that the narrow-gap approximations (3.10a,b)and (3.12b) satisfy this. Further, in the plane-wave approximation (3.20) canbe estimated from (3.4a) and (3.13) in terms of Ao. For the case y < yr (i.e.,no propagating Poincar6 wave modes in the channel) the required equalitybetween (3.19a) and (3.20) yields

7 r k o ( 1 - l R l r ) = | I t 2 2 a h ( W ) t r G . 2 3 )z ' ) m o ' ' = l i l l 6 l o J ' u n \ , l ' )

Here the term [ ...] is precisely that occurring in (3.27). The expression

110

(3.23) provides a useful check on the validity of the plane-wave approxima-tion, where R is estimated from (3.1aa). In all cases we find that thepercentage error in satisfying (3.23) increases wrth ya, but was very smallwhen 7 was significantly less than 7r; the percentage error is <I7o fory < 0.1Tr, and < 70Vo for y < 0.5yr.

4. APPROXIMATIONS OBTAINED BY CONFORMAL MAPPING

When cot20> c2, ag.2.7 is elliptic, and for narrow channels the derivateterms dominate. The equation is then approximately Laplace's equation in yand ? (see (2.30b)) to within a relative error of 0((ya)z).This suggests theuse of conformal mapping methods to obtain an approximate solution.Buchwald (1971) used this technique in a tidal diffraction problem. Ouranalysis is very similar and we shall only give a brief description. In the nearfield (i.e., the region which is a distance 0(a) from the junction of thechannel with the ocean) we propose that

- -- l. I if cos o? \ /yco2 sin d lQ : a b o \ r - l y I r /

\ \ ar(cos2d - c2 sin20)"' I ,(cos2d - c! sin'z?) J

(4.1)

where Bo and B, are constants to be determined. Here 4 and r/, are the realand imaginary parts of the complex function w(? + ry). The boundarycondition (2.8b) on r:0, lyl> a is satisfied by choosing ry' equal to aconstant there. Similarly, the boundary condition (2.8a) on r:0, I yl: a isalso satisfied to within an error of 0((ya)2 ln ya) by choosing rf equal to aconstant there.

In the far field in the ocean (r>0) it follows from(2.22a) that

q - aDG(r , y ) (4 .2 )

where D is also a constant to be determined. To match (4.1) with (4.2) thenear field approximation (A.5a) for G(r, y) is used. It may then be shownthat an appropriate choice for w(? + ry) is (see Buchwald (1971))

w : Q + i r ! , : 2 l n r - i r r + T

where

i t , * iy) : zln r - 2 tn{(r2 - r)" ' + t }

- 2i(r2 - r)1/2

(a3a)

(4.3b)

111

Then matching with (4.2) shows that

I l o l - a \ \Bo: D | , . , * cr lnl ;

))B r : l C r D

@.aa)

(4.4b)

$.ac)z to l f I 81 2f yrt sin 0Bo

B z : -or(1 - ,'tarf o)t/' oa(coszo - cl sir'o)

To match with the channel solution (2.1'5a,b,c) we note that when i -+ - oo(see Buchwald (1971))

w - yQ + iy)+ 2(ln 2 - r) * to, "*p(+(r + ,/))

. '#?'* '(( "- +)irr. wt)Here c, and d, are real coefficients obtained from the mapping (4.3a,b); inpart icular h:2e-1 and dt: -2e-2. Matching with (2.15a) shows that

#: -r * Ea{^(? .

3 t tn2- r) . o((to) ' rn ya)

* : 4i i + o(ya tn ya)

Here E is the same constant defined in (3.9a).

Comparing @.6a) with (3.8) we see that the only difference is the Poincard

wave contribution erya in (3.8). Thus we can infer that

/ 1 a \. , : E{ i

- r " i }+uQatn ya) (4 .7)

Also note that (4.6b) agrees with (3.12b). The amplitudes of the Poincarewaves are given by, to leading order,

. , (k r - k ' ) (1 - c ' tan ' l ) t " I z \A, , : i t #1 , * ) ( - t ) * - ' d * r f n : 2m i s even

(4 .8a)

of

.L ' , : o( ( ta) ' tn 7a) i f n :2m - l is odd

Note that here we have been able to obtain an explicit expression for A'nwhen n is even, and for all n (4.8a,b) agree with our previous estimation of

the order of magnitude of A',.

(4.s)

(a.6a)

(4.6b)

(4.8b)

772

5. FORCING FROM THE OCEAN

In the previous sections we have considered the effect of a Kelvin wave inthe channel incident on the boundary r: 0 between the channel and theocean, as shown schematically in Fig. 7a. ln this section we give a briefdiscussion of the opposite case when there is forcing from the ocean. As wementioned in section 2 (see the discussion following (2.22a,b)) this may berepresented by a term qo in r ) 0 where qo satisfies (2.1) and the boundarycondition (2.8b) on the whole of r: 0. In particular we consider two cases.Case (a) is a Kelvin wave (2.18a) travelling along the continental slope. Thus

eo: Ax exp( - d.r + i loy)

where d and lo arc defined by (2.18b and c), respectively, and A*incident wave amplitude. For this case to occur, cot20> co2>0. Casethe combination of an incident and reflected Poincard wave, and so

eo: Aw exp(iar + ilry) + A* exp(iar * itry)

Fig. 7. Schematic diagrams of the three cases discussed in the text. The double arrow ( : > )denotes incident energy flux, and the other arrows (-+) denote wave energy emanating fromthe junction. (a) Shows incident Kelvin waves in the canyon, scattered Kelvin waves in thecanyon and on the slope, and Poincare waves in the ocean. Possible Poincar6 waves in thecanyon are not shown. (b) Shows an incident Kelvin wave on the slope, with a scattered wavefield similar to (a). (c) Shows an incident (and reflected) Poincare wave from the ocean, with ascattered wave field similar to (a).

(s.1)is the(b) is

(s.2a)

113

where

- ia (cos2o - c2 s in2 | ) (A* -A ' * ) - L f ! (A*+ A ' * ) :o

and

u(cos2l - c2 s\*0): (s ign ,){y ' r ' - t l (cos2l - , ' , i r }01}t / '

boundary condition (2.8c) now gives, in place of (2.23a)

.a

Qp. : ao+ | G(0 , y - i )U ( i ) d , a t r : 0 , I y l < ar - a

where

U(y) : (cos20 - c2 s i *0 )q^ ,+ Wf a^ , a t r :0 , l y l < a

(s.2b)

(s.2c)

Here the sign of a has been chosen so that A* is the amplitude of theincident Poincare wave (see the discussion following (2.I7a,b)). To obtainpropagating waves, /, must be chosen so that a is real. These two cases (aand b) are shown schematically in Fig. 7b and c, respectively.

In the channel, the solution is given by (2.15a,b,c) with 1,:0. Thus

e: Q,^ : L A ' ,p i ,Q) exp( ia ! " r ) for r < 0, ly l < an : 0

In the ocean

r @

Q : Q o I I G ( r , y - y ) U j ) a y f o r r > 0 , - @ < / < o o. , _ 6

(s.3)

(s.+)

Herc G(r, y) is the Green's function (2.22b) afi U(y) is again defined by(2.20), where it is importanl to note that 4, does not contribute to U(y).The

t).)a.,

(5 .5b)

This integro-differential equation can be reformulated in a manner similar tothat described in section 3. Thus, in place of (3.5a) we obtain

(5.6)

Here the right-hand side is preciselytlfie same expression which occurs insidethe brackets in (3.5a), and hence provides a scale invariant expression forA'0, the amplitude of the scattered Kelvin wave in the channel.

1r4

For the narrow-gap approximation, when lyal< < 1 (and l loal, l l { l< < 1) we can adapt the procedure described in section 3. Thus, we find that

A'O - l a o l \: 1 - ina tnl+) - lro - epra+ o((r,) ')4o (0, o)

(5 .7 )

(s.eb)

where E, F and e, have precisely the same meanings as in section 3 (see(3.9a-d)). The reflected Kelvin wave is directly forced by the pressure fieldof the incident wave qo at the entrance of the channel. For case (a), qo(O, 0)ts A* and the reflected wave amplitude is also approximately Au; for case(b), qo(0,0) is (A.+A'r) which is also the approximate reflected waveamplitude. The plane-wave approximation is now

U(y): (constant)p[(y) (5 .8)

which is substituted into the righrhand side of (5.6). This leads to anexpression similar to (3.14a); however, we shall not pursue this approach anyfurther here. In the far field where 12 + y2 --+ a (3.I2a,b) is replaced by

. , f a

Q - Q o + G ( r , y ) | U ( y ) d y. , - u

(s.la)

where

I_,rrr l dy:2aq,(0, o)(r - ino^(+) *ott, l )

Thus for narrow channels, the far field is only an O(ya) perturbation of theforcing field 40.

6. SUMMARY AND DISCUSSION

The main purpose of this work has been to obtain some quantitativeestimates of the extent to which internal wave energy can be trapped insubmarine canyons. From this point of view the main results for oceanicapplications are contained in Figs.2a-4a. These show the reflection coeffi-cient lR I when a Kelvin wave in the canyon is incident on the canyonmouth. Although based on an approximation, the results agree with thenarrow-gap approximation over a wide range of. ya(< 0.1), and are physi-cally plausible. For values of 0 typical of continental slopes, the graphs of

lRl as a function of ya are relatively insensitive to 0, but show significantvariation with c/cot 0.

For c2 <cot20 and also c2o<cot20, we expect wave trapping to occur forya < 0.I but significant leakage of energy for larger values of ya, and a totalloss of energy as Ia -+ oo. The reflection coefficient decreases markedly asc/cot d is increased. The simplest physical explanation is that in this case

115

the incident and reflected Kelvin waves propagate on opposite walls of thecanyon, while a Kelvin wave can also propagate along the continental slope.Thus, as ya increases there is an increasing mismatch across the canyonmouth between the incident and reflected waves" which is enhanced asc / cot d is increased. Note that the across-channel scale of the Kelvin wavesis proportional to ko I which from (3.2a,b) decreases (relative to the canyonwith 2a) as'fa, or c/cot d, increases. However, the across-channel scale ofthe oceanic Kelvin wave is /o I (see (2.18a)) and from (2.18c) this remainscomparable with ko 1. Thus as ya, ot c/cot d, increases there is an enhance-ment of energy leakage, primarily into the oceanic Kelvin wave. Note thatfrom (3.22) the ratio of energy flux which goes into the oceanic Kelvin waveto that which goes into radiating Poincare waves, is independent of ya, andincreases as c/cot d increases.

For c2 < cot20 but cfr > cot20 there is again significant wave trapping forya<0.I, and leakage of energy for larger values of ya. However, there isnow substantial reflection as ya - @, and some wave trapping for all valuesof ya.In this case the incident and reflected Kelvin waves propagate on thesame canyon wall, and hence can remain well-matched across the canyonmouth as ya increases. Also in this case there is no oceanic Kelvin wave, andhence energy leakage into the ocean is diminished. For c2 > cot20 similarcomments apply. Generally the reflection coefficient diminishes slightly asc/cot d is increased. From these results it follows that for moderate or largevalues of. ya the transition co'S cot2d is just as significant as the transitionbetween "steep" and "flat" slopes.

The results shown in Figs. 2-4 are for a fixed value of f/u:. However, weexpect similar results for other values of f/a. Indeed, the narrow-gapapproximation (3.10a,b) or (3.11) is valid for all values of f/o: and showsthat the major sensitivity in the graph of lR I as a function of yc is to thevalue of c/cot 0. In Figs. 5 and 6 we show the results for the non-rotatingcase, f :0. The major differences between this and the oceanic case are theabsence of an oceanic Kelvin wave, and that the channel Kelvin waves areuniform across the channel. Nevertheless, with this proviso, the graphs aregenerally similar to those in the oceanic case, the most notable differencebeing the lack of any major distinction between the cases clS cot20.

The main emphasis in this paper has been on the case when the incidentenergy is due to a Kelvin wave in the channel. For the opposite case, whenthe incident energy arrives from the ocean, either a Kelvin wave on thecontinental slope or a Poincard lvave, we have given a brief discussion insection 5. In this case the incident wave has zero energy flux in thealong-channel direction at the channel mouth, and the forcing of thereflected wave in the channel is due to the pressure field of the incident waveat the channel mouth. Further, the energy flux of the waves generated in the

116

channel is exactly equal to the back-scattered energy flux in the ocean. Fromthe narrow-gap approximation (5.7) we see that the amplitude of thescattered Kelvin wave in the canyon is comparable with the magnitude of thepressure forcing term at the channel mouth. Although we have not obtainedany quantitative results for larger values of ya we can anticipate thequalitative nature of the results, since a plane-wave approximation to (5.6)leads to an expression similar to the plane-wave approximation of section 3.Thus as yc increases, we expect a decrease in the normalized amplitude (i.e.,b'1/2A'o/2a 4o(0, 0)) of the reflected Kelvin wave. The simplest physicalexplanation is that the incident wave is now oscillatory across the channelmouth (see (5.1) and (5.2a)) and we would then expect a significant excita-tion of Poincar6 waves, both in the channel and in the ocean, as Iaincreases.

We conclude with some realistic numbers. With a:3 km, and a typicalinternal wavelength of 20 km (along the continental slope), we have ya:0.9.The corresponding horizontal and vertical wavelengths of the incident Kelvinwave can be deduced from (3.2a,b). For instance, with N/a:35.5 and0 :84o, the horizontal wavelength is about 54 km and the vertical wave-length is about 1.5 km. In the ocean, c/cot 0 may take a range of values oneither side of unity, and hence the effects described in this paper can berealized in oceanic situations.

ACKNOWLEDGEMENT

We gratefully acknowledge the assistance of Janette Kaval who carriedout the numerical evaluation of the reflection coefficients in section 3.

APPENDIX A

Near-field Green's function

To examine the behaviour of the Green's function (2.22b) in the nearfield, where 12 + _r-2 - 0, we follow Buchwald (1971) and put G in the form

G ( r , y ) :("' - f')("os2g - cfi si#0) { , , ,

, ,a2

. ry Il *t,{r, }) sign(y - y) exp(;/ ol y -i lsign f ) di

i/o(sign f) cos20 (ct - r')

2(cos2o - cz sir,2o)clI ( r , j ' )

x exp( - i to ly - j l s ign i l o t | (A.1a)

t7'l

where

1 16 exp(iar - l i ly) ̂ ,I ( r , v ) : = I - - s t2 7 i l J _ 6 d

(A.1b)

Here lois defined by (2.18c); if /o is real then we choose fto/r< 0, but if /ois complex-valued, we choose Im(lof) <0. From (A.1b) I(t, y) can beidentified as a Bessel function (see, for instance, Abramowitz and Stegun(1964), Ch. 9). We find that, if cot20 > c2 > 0 and or(N2 - f')r 0, then

I( r , y) : - ; (cos2g - c2 s in2 l ) r / 'n [ ' t ( t " ( t ' * y ' ) " ' )

where

r : ? lcosz? - c2 sitf 0lrtz

and

l,: l^tcl lcos2d - c2 sin2?;ttz (A.2c)

Here Hj1) is the Hankel function of the first kind; if a(N2 - f').0, then itis replaced by - u[2t . For cot2d < cz and 6(N' - f

') r 0, then

I(r, y) : trk'sin2d - cos' l)t / 'H["(t,(rt - y')" ') for ?2 > y2 (A.3a)

or

I ( r , y ) :i(cz sirf l-"or ' l) ' / t

Ko( t " (y ' - r ' ) " ' ) s ign( , (w ' - f ' ) )

for ?2 <y2 (A.3b)

Here K6 is a modified Bessel function. If o(N2 - f').0, then in (2.31a),HJ') ir replaced by ajtl. Finally if cz < 0, then

, (cos2 o - c2 sin'o\'/ ' -- | . r ̂ . t .ttt/2\I (r , y) Ko\I"(?') + y')"

') (A.4)

The near field approximations for G(r, y) now follow on using the smallargument approximations for the Bessel functions. We find that, if cot20 >c 2 > o . t h e n a s ? 2 + v 2 - - o

(A.2a)

(A.2b)

G ( , , y): ct^(+O' * r')"')

*. r ( Ztun- '4 - i tor ignf ( v -

y'))

if cos 07

+ c3+ o((n'z+ y') tn(t' +

co(cos2o - ,' tirf o)'/'

118

where

but in (A.5d),real, where

i a-

. Sl$n <,)L

(A.5b)

(A.6a)

(A.6b)

is replaced

,21cos2o - c2 sirf o l t tzn(r' - f2)(cosze - cfr stnzo)

c t :

c z : (A.sc)

(A.5d)

("( l r ' - f ' ))

ru(rt - v')

, 2 -

, l ( 1

d ,

- c

2 "

o i s

+

2 ( o

l 0 )

l n r

v ) ,

c3CT

I 1 2 l 1 / 2 \*13 - ' l Il ' , I l

cot20 < cz

/,\ + c" nlY + il" '\ .I

' l y - ' r l / t ' s o

, i f cos9- r -

,(r' sirf0 - "ostl)t/'

iaf cos 0

f2)(coszo - cl sir,2o)

t f t _L , l , | ! l- c' tanzo)"' l l

'" I

Euler's constant. For

( t o l ^i { c r r n l i t , - y ' t '

Icz{s ien yn(y ' - ? ' )

Here

G ( , ,

-; lo sien /(

+ c 4 + o ( l ? 2

c3by (A.5a)C, if /o is

1 2 l 1 / 2 \

T + t l Il z I I- c I l

If c2 < 0, then G(r, y) is again givenby Co if /o is complex-valued and by

, , l l+l . l' I or l(r - c' tan20)'/' l, | '. | |

(A.7)

119

APPENDIX B

A Galerkin approximation when f : Q

When f :0, both p'" (2.9bd) and u', (3.1c,d) are proportional tocos nn/Za(y * a), n : 0, 1,...,which form a complete set for the represen-tation of qr* qn in the integro-differential equation (2.23a). Also, when

f : 0, the orthogonality relation (3.3) reduces to the well-known orthogonal-ity relation for cos nn/2a(y * c). Thus an appropriate Galerkin approxima-tion for the solution of (2.23a) is the substitution of (2.75a) into the left-handside of (2.23a), (3.1a) into the right-hand side of (2.23a), followed by theprojection of (2.23a) onto p'n. This yields an infinite set of equations for A'n.Equivalently, we substitute (3.1a) into (3.4b,c) with the result

( i + r d r a \A o \ 1 - ; _ l / c ( 0 , y - i ) d y d i , l

\ z u J _ a J _ a )

+ a'o{t . + I _.1 _,G(0, y - i ay ar}

* E, o;{- * t:,t:.G(o, y - y)p^(y)or or) : om : I

o"{ , , l _" ! _"G(0, y - i l p ; f i l a y a r }

* n,,(- ,i I: "l: "G (0, y - il pi,@ a y d rl

* F_,o;(

6^,1_,0'1dy+ iK^1" , l_,o(0, y- i)

xphj)pi , ( j , ) dydt) :

(B.1a)

(B.1b)

(n.tc)

(B.1d)

where

K', : (k ' ,cos d - c lm, sin 0)

and

G ( 0 , y ) : 1 ( 0 , y )

Here 1(0, y) is a Bessel function defined by (A.2a) lf cot20 > c2 > 0, and by(A.3a) or (A.4) otherwise. The Galerkin approximation now consists oftruncating the infinite series in (B.1a,b) to M terms, and simultaneouslyrestricting the infinite set of equations in (B.lb) to the first M equations.This gives (M + l) algebraic equations for (M * 1) unknowns ,4'0, . . ., A'r.

120

These algebraic equations were solved numerically. The integrals in the

coefficients were evaluated numerically by replacing the Bessel function

I(0, y) with a finite power series representation. The results, using a low

order of truncation (M <70) are shown in Figs. 5 and 6, and were also

reproduced in Baines (1983). It can be seen that there is reasonable agree-

ment with the plane-wave approximation. Indeed, the M: 0 truncation is

exactly equivalent to the plane-wave approximation.

REFERENCES

Abramowitz, M. and Stegun, I., 1964. Handbook of Mathematical Functions. Nat. Bur.

Stand. Appl. Math. Ser., 55.Baines, P.G., 1971a. The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid

Mech.,46: 2'73-29\.Baines. P.G.. 1971b. The reflexion of internal/inertial waves from bumpy surfaces. Patt 2.

Split reflexion and diffraction. J. Fluid Mech., 49: 113-131.Baines, P.G., 1983. Tidal motion in submarine canyons-a laboratory experiment. J. Phys.

Ocean, 13: 310-328.Buchwald, V.T., 1971. The diffraction of tides by a narrow channel. J. Fluid Mech., 46:

501-511.Hotchkiss, F.L. and Wunsch, C., 1982. Internal waves in Hudson canyon with possible

geological implications. Deep-Sea Res., 29: 415-442-Miles, J.W., 1967. Surface-wave scattering matrix for a shelf. J. Fluid Mech., 28:'755--16'7.Shepard, F.P., Marshall, N.F., Mcloughlin, P.A. and Sullivan, G.G., 1979. Currents in

submarine canyons and other sea valleys. Am. Assoc. Petrol. Geol., Tulsa, 173 pp.

Wunsch, C. and Webb, S., 1979. The climatology of deep ocean internal waves. J. Phys.

Ocean.. 9:235-243.