Hamiltonian Aspects of Three-Layer Stratified Fluids

Abstract

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

Analysis of the Oscillations of Stratified Liquid with Elastic Ice

“…take place, then problem (6) has a unique strong solution on ] [0;T . The rest of the proof is based on the reverse transition from the Cauchy problem (6) to the initial boundary value problem (5) and then to a problem (2). Thus, the Cauchy problem (2) has a unique solution, strong with respect to t, under the initial conditions of Theorem 1 of the work [13].…”

“…To date, various analytical and numerical approaches have been developed to study the behaviour of various structures interacting with a liquid [1][2][3][4][5][6][7]. At the same time, a number of methods based on functional analysis and the theory of operators in abstract Hilbert spaces are known in mathematical physics.…”