An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics

Abstract

We describe a problem that can be tackled more-or-less routinely using the ideas of classical fluid mechanics, but it is a complex flow and even the linearised problem involves considerable algebraic complexity. The presentation here emphasises the approach that we adopt in order to formulate an accessible model of such a flow. The physical background is carefully described: the Equatorial Undercurrent, a particular phenomenon of the Pacific Ocean, the thermocline and the waves on both the free surface and on the thermocline. One of the results of this careful approach, coupled with a solution of the linearised problem for arbitrary wave numbers, is that we are able to provide well-grounded explanations for many of the fundamental processes that are relevant, and observed, in this region of the oceans; the result is a successful application of elementary principles. In the context of this system, we can describe the various types of wave dynamics, depending on wavelength, and also the differences between eastward and westward propagation. It is gratifying that the results of such a simple theory correspond closely to the observations reported in the literature. Of more interest, perhaps, from the theoretical-fluids viewpoint, is that the development leads directly to the prediction of critical layers and to a procedure for their analysis, which we outline here (and critical layers and their associated flows have been observed in the Pacific Ocean). Further, other possible rôles of nonlinearity are immediately accessible, such as wave evolution, for which we provide only an introduction, but this is sufficient to hint at tantalising prospects for further work.

Keywords:

• Ocean undercurrent
• nonlinearity
• waves
• asymptotic expansions

2000 Mathematics Subject Classification:

• 35R35
• 76B15
• 76B55