Deep-Sea Research, 1971. Vol. 18, pp. 477 to 483. Petlgamon Pr~s. Printed in Great Britain.

The ray paths of topographic Rossby waves

RONALD SMITH*

(Received 20 March 1970; in revised forra 29 December 1970; accepted 30 December 1970)

Abstract--The geometry of the ray (or group-velocity) paths of low frequency topography Rossby waves is found to depend in a simple way on the ' geostrophic vector'

= bY(f ib) .

Simple models are used to show that the energy of topographic Rossby waves tends to be refracted away from oceanic ridges or the equator, absorbed by shore-liues, and trapped along escarpments.

1. INTRODUCTION

CLASSICAL Rossby waves in an ocean of constant depth exist because of the variation in planetary vorticity with latitude. When the water is not of constant depth there is the additional effect of vortex stretching as a column of fluid is displiieed, i nto a deeper region. The local interplay between these two effects is well understood, a~i~t~ addi- t ion a few idealized problems involving topographic Rossby waves have been solved (PARSONS, 1968; RHINES, 1969).

The aim of this paper is to obtain a more global understanding of topographic Rossby waves. The property which enables us to achieve this is that the wavelength of Rossby waves decreases with the frequency. Thus for low frequency Rossby waves we can employ the powerful short wave asymptotic solutions of geometrical optics.

Subject to several implicit assumptions (see ~4), the equations which we must solve are either the linearised "r igid l id" equation (RHINES, 1969)

v . (h-lye,) - ~ov¢,. (k x V(f/h)) = 0, (1)

or the linearised shallow water-wave equation, one form of which is

h i [1 co 2

---o • - t - ( 7 1 t"

+ f ~ ( ~ _- ~o2) v ~ . k x V + i Vf -- O. (2)

In these equations, co and f denote the wave frequency and Coriolis parameter re- spectively (both non-dimensionalised with respect to the frequency fl of the earth's rotation), k the unit vertical vector, h the equilibrium depth, $ a depth integrated stream function, and t/ the surface elevation above the equilibrium level.

As the frequency approaches zero, equation (1) has a wave-like solution of the form

~ A exp(/q,/co)

• Fluid Mechanics Research Institute, University of Essex.

477

478 RONALD SMISH

provided that A and tp satisfy the " e ikona l " and " t r a n s p o r t " equations of geometrical optics:

V~ . (V~ - k × hV(y/h)) = 0 (3) and

h r . (h-aAV~) + VA. V¢ - VA. (k x hV(f/h)) = 0. (4)

To the same level of approximation, equation (2) has the closely related solution

q ",~ fhexp(i~p/~o)

in the derivation of which it is implicitly assumed t h a t f 2 > ~o z. S incefequa ls zero at the equator, this second solution may not be accurate near the equator.

We note that equation (3) is equivalent to the dispersion relation for shortwaves,

~o = (k x hV(f /h)) . 1/]I[ 2

where I is the local real wavenumber.

2. RAY P A T H S

In order to solve equation (3), we shall use the method of characteristics or rays (COLmANT and HILn~T, 1962, Chapter 2). The ray path is in the direction of the vector

½k × hV( . f /h) - V~,

which, in the limit to - , 0 with the angle of wave crests fixed, is also the direction of the energy propagation. In this limit, the local phase-velocity is in the direction of Vcp. The relationship between these directions and the "geostrophic vector"

G = hV(f /h) = V f - f V h / h

is shown in Fig. 1 which is also the wavenumber locus for fixed o~. (The above limiting process should not be confused with the limit co ~ 0 with the length scale fixed, for which the paths of propagation are forced to be f ib contours i.e. geotrophic flow.)

Eet t denote the unit tangent vector and s the arc-length along a ray path. The derivative of a function b with respect to s is defined by

Ob 1 - ( t . V)b = ([k x G - 2Vrp] V)b. (5)

0s ~ - "

Using this definition we can recast equation (3) into the form

3-~ = ½(t. (k x G) - [G[) (6)

which is simply an ordinary differential equation along a ray path. In order to extend a ray path we must know both its direction t and its geodesic-

curvature ~¢. The geodesic-curvature measures how the ray path deviates from a great circle on the earth's surface and is defined by

8t 1 (n . V [G[ + W. G) (7) ~-n.8--)G1

The ray paths of topographic Rossby waves 479

/ ~kxG / k x G ;~ \

2 / ~ . . i - "~

Fig. 1. Wavenumber locus for fixed small to.

where the unit vector n is chosen so that t, n, k form a right-handed triad C~EATHER-

BURN, 1955, Chapter 6). Equation (7) shows that the bending of ray paths is associated with two features

of the geostrophic vector field: firstly that rays tend to go into regions in which I G [ is large, and secondly that rays are bent towards the left or right according to whether G is divergent or convergent. The interaction of these two effects gives rise to many possibilities, as shown below. For convenience great circles are regarded as straight lines, cartesian axes x, y, z are used where the z axis points along k, and t is written in component form (tl, t2,0).

Example 1

I G I has a pole: G = (x- 1 0, 0).

It follows that = (t 2 + 1)x - I .

Thus the ray paths are circles tangential to the line x = 0, as shown in Fig. 2. The geotrophic vector has a pole of this kind at a shoreline. Thus we deduce that

the energy of any free waves inshore of the continental rise is refracted towards the shore. Since nonlinear and friction effects are enhanced as the depth tends to zero, we are led to expect that the shoreline will act as a partial energy sink and only reflect a portion of the energy into further topographic Rossby waves.

Example 2

[ G [ has a simple zero: G = ( - x, O, 0).

It follows that x = - ( 1 + t2 sign x ) [x [ -1 .

480 RONALD SMITH

Thus, as in example l, the ray paths are tangential to the line x = 0, though the energy flow is reversed.

Although it is extremely rare for the geostrophic vector to vanish, along the crest of an oceanic ridge the usually large topographic contribution to G vanishes while the small Vf contribution remains. Thus the general features shown in Fig. 1 are found near oceanic ridges. Consequently we deduce that the energy of topographic Rossby waves tends to be refracted away from oceanic ridges (See RHINES, 1969).

/ i /

Fig. 2. Energy propagation near a shoreline.

T

x JW --

F i g . 3. Energy propagation along an escarpment with a > 0.

The ray paths of topographic Rossby waves 481

Example 3

[G[ has a turning point: G = (1 + ½ax 2, b, 0).

It follows that for small x

g ~ a x t 2

We observe that the curvature of the rays is strongest when t 2 is negative and, as sketched in Fig. 3, that there exist trapped rays which propagate in the direction of increasing ay.

The geostrophic vector has a turning point of this kind on an escarpment. Thus we deduce that the energy of topographic Rossby waves can be trapped along escarp- ments (See RmNES, 1969, LONGUET-HIC, GINS, 1968).

Example 4

G has a shear: G = (ay, 1, 0).

It follows that r = tla2y[(1 + a2y).

Thus, as sketched in Fig. 4, the curvature tends to refract the rays away from the line y - - 0 .

This form for G can be found near the equator if the ocean depth varies from east to west. Thus in agreement with the results of PARSONS (1968) we find that the net effect of variation in bot tom topography near the equator is to refract the energy of topographic Rossby waves away from the equator.

Nay p a t ~ Equator

Fig. 4. Energy propagation near the equator with east to west depth variation.

3 . T R A N S P O R T E Q U A T I O N

Utilising the definition (5) of differentiation along a ray path we find that the trans- port equation (4) can be written

1 aA 1 ~---~ = IGI (v2~p - h- lVh" Vcp),

482 RONALD Slvrrrn

which, like equation (6), is an ordinary differential equation along a ray path. We can obtain an explicit solution for A by introducing the Jacobian

• x O s ] '

where x is the vector position of a ray point and p is a parameter indexing the different ray paths. After a straightforward but lengthy calculation it can be shown that

1 0 J 1 1 J o ~ = IGI ((k x G). h - W h - 2V2~0) - ] - ~ (k x G - 2Vtp). V IGI.

Using this result together with more direct applications of the definition (5) we can rewrite the transport equation as

I OA l (~ Oh l OJ 1 0 , G l ~ A Os - 2 Os JOs [GI ~S ]"

Finally, integrating this equation we obtain

• [Jo[ G [ o h ' '/2 ( ~ ]V(f/h) '°] '/z (8) A = A o t - ) - I ~ Z o / =Ao ~ ! ,

where Ao, Jo etc. are evaluated at any convenient point along the ray path. Physically the expression (8) for A can be derived from the conservation of the

energy propagated between two adjacent group-velocity paths in the limit o~-~ 0, the separation of the paths being proportional to d. Thus, along a ray path we have

½P l ut 2h [Co I J = constant,

where eg is the group-velocity. Now, to the lowest order

lul2= tlf202 and

I%1 = D.h IV(f/h)p/lll 2,

where ! is the local real wave-number. Hence, to the lowest order

02J I v ( f /h) l = constant,

which is a variant of (8). From equation (8) we observe that A becomes infinite, and the geometrical optics

solution invalid, when either J or I GI is zero i.e. at a caustic of a family of ray paths or at a null point of the geostrophic vector.

4. APPLICABILITY

Two necessary conditions for the use of the approximate equations (1) and (2) are that the wavelength of the topographic Rossby waves be greater than the local water depth and that the advection be small. Thus the two measures

h IG[/o) and UtGI/V 2 f~

The ray paths of topographic Rossby waves 483

must both be small, where Uis a typical current velocity. These measures work against the geometric optics requirement that c0 be small. I f h = l0 s cm, [G[ = 10 -7 cm -1, Q = 10 -4 sec -1, then the two measures would be

10-2/O9 and 10- 3 U/co 2

(U being measured in cm sec-1) and, except in strong currents, we would have a non-zero range of o~ in which none of the three requirements are violated. Other impli- cit assumptions in the usage of equations (1) and (2) are that the density stratification and all friction effects may be neglected.

The primary contribution of the geometrical optics solution is the global picture which we gain of the energy propagation of topographic Rossby waves. We can expect this to be qualitatively accurate even when we cannot justify the quantitative results. For example, the ray solution predicts that waves can be trapped along escarp- ments yet a family of ray paths corresponding to such a trapped wave has caustics at both sides of the escarpment and the geometrical optics solution is not valid near caustics. Nevertheless waves can indeed be trapped along escarpments (RHINOS, 1969; LONGUET-HIOGINS, 1968), and for small co an asymptotic solution for these trapped waves can be obtained by the technique described by SMrrH (1970).

Finally, it must be emphasised that the four simple examples given in this paper by no means exhaust the variety of features and phenomena associated with topographic Rossby waves.

Acknowledgement--I wish to thank the referees for their exceptionally helpful comments.

R E F E R E N C E S

COUgANT R. and D. HILBERT (1962) Methods of mathematical physics, Vol. 2. Interscience, New York, 830 pp.

LoNotnyr-I-h(u3rNs M. S. (1968) Double Kelvin waves with continuous depth profiles. J. fluid Mech., 34, 49-80.

PARSONS A. T. (1968) Two Problems in Oceanography. Thesis, Bristol University. RmN~ P. B. (1969) Slow oscillations in an ocean of varying depth. Part 1: Abrupt topo-

graphy. J. fluid Mech., 37, 161-189. SMrrH R. (1970) Asymptotic solutions for high frequency trapped wave propagation. Phil.

Trans., A268, 289-324. WEATHVatnUgN C. E. (1955) Differentialgeometry, Vol. 1, C.U.P., Cambridge, 268 pp.