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On the Number of Conserved Quantities for the Two‐Layer Shallow‐Water Equations
Author(s) -
Montgomery P. J.,
Moodie T. B.
Publication year - 2001
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00166
Subject(s) - conservation law , inviscid flow , shallow water equations , mathematics , conserved quantity , flow (mathematics) , mathematical analysis , multinomial distribution , forcing (mathematics) , finite set , finite difference , physics , classical mechanics , geometry , mathematical physics , statistics
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.