Abstract
In this paper we derive an approximate multi‐dimensional model of dispersive waves propagating in a two‐layer fluid with free surface. This model is a “two‐layer” generalization of the Green–Naghdi model. Our derivation is based on Hamilton's principle. From the Lagrangian for the full‐water problem we obtain an approximate Lagrangian with accuracy O(ɛ2), where ɛ is the small parameter representing the ratio of a typical vertical scale to a typical horizontal scale. This approach allows us to derive governing equations in a compact and symmetric form. Important properties of the model are revealed. In particular, we introduce the notion of generalized vorticity and derive analogues of integrals of motion, such as Bernoulli integrals, which are well known in ideal Fluid Mechanics. Conservation laws for the total momentum and total energy are also obtained.
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