Double criticality and the two-way Boussinesq equation in stratified shallow water hydrodynamics

Bridges, Thomas J. and Ratliff, Daniel (2016) Double criticality and the two-way Boussinesq equation in stratified shallow water hydrodynamics. Physics of Fluids, 28 (6). 062103. ISSN 1070-6631

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Abstract

Double criticality and its nonlinear implications are considered for stratified N–layer shallow water flows with N = 1,  2,  3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector and one generalized eigenvector) and semi-simple (two independent eigenvectors). Using a multiple scales argument, dictated by the type of singularity, it is shown that the weakly nonlinear problem near double criticality is governed by a two-way Boussinesq equation (non-semisimple case) and a coupled Korteweg-de Vries equation (semisimple case). Parameter values and reduced equations are constructed for the examples of two-layer and three-layer stratified shallow water hydrodynamics.

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