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The square root depth wave equations

C. J. Cotter, D. D. Holm and J. R. Percival
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 466, No. 2124 (8 December 2010), pp. 3621-3633
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/40998102
Page Count: 13
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The square root depth wave equations
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Abstract

We introduce a set of coupled equations for multi-layer water waves that removes the ill-posedness of the multi-layer Green-Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear, they retain the same travelling wave solutions as MGN. We call the new model the square root depth $(\sqrt D )$ equations from the modified form of their kinetic energy of vertical motion. Our numerical results show how the $\sqrt D$ equations model the effects of multi-layer wave propagation and interaction, with and without shear.

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