The shallow-water equations for two-layer inviscid flow with a free surface overlying a rigid horizontal bottom subject to gravitational forcing only are examined to determine the possible forms of conservation laws that the equations permit. In the case of a single layer with flow in only one horizontal direction, it is known that there are an infinite number of associated equations in conservation form, where the conserved quantity is a multinomial in the layer variables. The method used to determine this result is generalized to show that in the two-layer case, the result does not generalize, and it is discovered that only a finite number of conservation equations exist when the density difference between the layers is nonzero. The subsequent conservation equations are given explicitly, and a systematic method for deriving conservation laws from an arbitrary first-order system is described. For the case when the flow is in both horizontal dimensions, the method of analysis is straightforward in the one-layer case, and the finite number of conservation equations are derived. The two-layer case is similar, and the finite number of generalized conserved quantities are stated, although the question of whether or not there are only a finite number is posed as an open question.

On the number of conserved quantities for the two-layer shallow-water equations

The steady-state propagation of cold domes of dense fluid lying beneath a layer of less dense fluid and overlying a sloping bottom have been studied previously by Nof [I]. Such research is important in improving the understanding of heat and mass transport in the oceans, and much of the presentday understanding relies on the assumption that the flow is in geostrophic balance. Such an assumption is not valid either when inertial effects in the flow dominate, or when the flow is near the equator. The research presented concerns the initial release problem of a finite volume of dense fluid underlying a deeper layer of less dense fluid. The flow is modelled by the shallow-water equations in three spatial dimensions which are expressed as a system of nonlinear hyperbolic conservation laws. Numerical solutions are achieved through the use of a finite-difference relaxation method developed previously by Jin and Xin [2], and modified for use in this specific situation by Montgomery [3]. Results are presented showing the resulting time-dependent flow for the initial release problem of a cylinder of dense fluid, with varying rotation rates and bottom slope to help investigate the times required for the inertial spreading to become primarily geostrophic.

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P.J. Montgomery

Papers

The Transition From Rest To GeostrophicBalance Of A Dense Volume Of Fluid