Asymptotic Behaviour of Solutions to Laplace's Tidal Equations At Low Frequencies

Abstract

S U M M A R YThe asymptotic behaviour of solutions to Laplace's tidal equations at low frequencies is considered. The method used is based on perturbation in small parameters, these being the ratios of tidal frequency and the coefficient of bottom friction to the angular frequency of the Earth's rotation. It is shown that the resulting solutions are unstable in that the functions involved in the zero-order approximation are not uniquely determined by the zero-order equations, but depend on first-order terms as well. Because of this instability, direct methods of numerical integration are inefficient. We propose a different procedure, replacing the original set of equations in partial derivatives by ordinary differential equations that have a stable solution. The equations are examined qualitatively. It is shown, in particular, that for the case of an ocean of uniform depth over the whole Earth, they coincide with the well-known Lamb's equations. The asymptotic behaviour of the solutions is examined as modified by basin shape, bottom topography and bottom friction.

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“…The asymtotic behaviour of solutions to Laplace's tidal equations (L.t.e.) at low (for example, Chandlerian) frequencies was considered in recent years in may papers (see, for example, Dickman , 1986O'Connor, 1986;Carton, Wahr, 1986;Molodensky, 1989;Groten, Lenhardt, Molodensky, 1990). It was shown in the last two papers, that for the limiting case a/u -»0 (where a is the tidal frequency and w is the angular velocity of the Earth's diurnal rotation)) these solutions are unstable in that the functions involved in the zero-order approximation are not uniquely determined by the zero-order equations, but depend on first-order terms (terms of the order of CT/W) as well.…”

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