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On finite amplitude planetary waves Clarke, Richard Allyn


Finite amplitude planetary waves are studied on a homogeneous fluid on both the rotating sphere and on a mid-latitude β-plane. The integrated equations of motion are rederived both on the rotating sphere, in a spherical polar co-ordinate system whose axis is tilted relative to the rotation axis, and on a mid-latitude β-plane. The linear solutions are re-examined and the errors associated with the non-divergent and the β-plane approximations are each shown to be about 10 to 15% for waves of a few thousand kilometers wavelength. Using the integrated equations of motion both on the sphere and on the β-plane, the linear non-divergent Rossby wave solutions are shown to be exact finite amplitude solutions. An exact topographic wave solution is also given for the case of an exponential depth profile. Such behaviour is not found for the divergent waves. Using a Stokes-type expansion in terms of an amplitude parameter, the second order solution for divergent Rossby waves is obtained, and it is found that, as in surface gravity wave theory, the first order correction to the phase velocity is zero. It is also shown that the linear non-divergent Rossby wave solution on a uniformly sheared zonal current is not a finite amplitude solution, and the second order solution is then calculated. Once again, the phase speed is correct to the first order. A class of long waves of permanent form analogous to the solitary and cnoidal waves of surface wave theory is obtained for a β-plane channel of either constant or exponentially varying depth. Such waves are found to exist in the divergent case in the absence of any zonal current; however, if the divergence is weak, or if the non-divergent approximation is made , then it is found, as it was by Larsen (1965), that these waves will exist only in the presence of a weakly sheared zonal current. On the exponential depth profile, such waves exist in the absence of a sheared zonal current, even if the non-divergent approximation is made. It is suggested that such waves may also exist trapped along long ocean ridges or scarps.

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