Gerstner Edge Waves in a Stratified Fluid 

 Rotating about a Vertical Axis 



Erik Molo-Christensen 

 Massachusetts Institute of Technology 

 Cambridge, Massachusetts 



ABSTRACT 



An exact solution is obtained for edge waves along 

 one inclined planar boundary in a fluid rotating 

 about a vertical axis. The solution is based on a 

 modification of Gerstner 's rotational waves, and 

 includes the effect of mean drift. The solution re- 

 duces to Yih ' s edge wave solution for zero rotation 

 and to Pollard's rotational deep water Gerstner waves 

 in rotating flow. Satellite observations of sea sur- 

 face are shown which reveal patterns similar to those 

 which would be generated by Gerstner edge waves. 



1 . INTRODUCTION 



The early, exact solution by Gerstner (1802, see 

 1932, p. 419) was rediscovered by Rankine (1863) , 

 discussed by Lamb (1932) , found to be valid for free 

 surface waves in an arbitrarily stratified flow by 

 Dubreil-Jacotin (1932) , further modified to describe 

 edge waves by Yih (1966) , and free surface waves in 

 a rotating flow by Pollard (1970) . However, there 

 has been a tendency to dismiss Gerstner waves as of 

 limited applicability to phenomena in nature. As 

 Lamb (1932) has pointed out, the generation of 

 Gerstner, free surface waves by the application of 

 surface stresses requires a certain mean vorticity 

 distribution to exist in the fluid. It can be argued 

 that in a nonrotating fluid of uniform density it is 

 difficult to conceive how the required vorticity dis- 

 tribution can be established. However, in a strati- 

 fied and rotating fluid, there are mechanisms capable " 

 of generating vorticity without viscous diffusion. 

 In a stratified fluid, the baroclinic term, express- 

 ing the action of a pressure gradient normal to a 

 density gradient in generating vorticity will be 

 capable of establishing a horizontal vorticity field. 

 In a rotating fluid, the effects of vortex stretch- 

 ing and compression can establish distributed vertical 

 vorticity. 



There, in a rotating stratified flow, waves simi- 



lar to Gerstner waves are more likely to be encoun- 

 tered. In fact, the uniform flow, usually assumed 

 as the mean flow on which small perturbation waves 

 may ride, would be less likely to occur in a rotating 

 stratified fluid. But the small perturbation solu- 

 tions for waves, as well as exact, finite amplitude 

 solutions, are all useful as approximate descriptions 

 of real phenomena and actual observations. 



If such solutions do not fit the exact circum- 

 stances , they can possibly serve as starting points 

 for perturbation expansions. Furthermore, we may 

 learn about some of the special features of finite 

 amplitude exact wave solutions; there is a tendency 

 to forget some of these facts when preoccupied with 

 linear wave solutions. 



In the following, I shall present a Lagrangian 

 description of an edge wave field, point out where 

 it differs from previous solutions, and develop the 

 dispersion relation for the waves. 



2. COORDINATE SYSTEMS AND DISPLACEMENT FIELD 



Coordinate System 



The waves propagate in the x - direction, normal to 

 the plane of Figure 1. In the planes normal to the 

 x - direction we define the oyz- coordinates, with 

 oz vertical and the oyz-coordinates, with oy in the 

 plane of the inclined boundary, inclined at an angle 

 a with the vertical . The particle motion will be in 

 planes parallel to xy. 



While Yih (1966) could let the amplitude of 

 particle motion decay with negative y-distance, and 

 Pollard (1970) , for deep water waves away from a 

 side boundary, made the obvious and correct choice 

 of letting the particle motion decay with decreasing 

 vertical position; here I have to make a different 

 choice. The amplitude of particle motion will decay 

 along a direction - or, shown in Figure 1 as another 

 coordinate system, ors. 



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