Three-layer water flows: Dirichlet–Neumann operators and approximations

The object of investigation in this paper is the nonlinear equations of motion for two-dimensional inviscid water flows with piecewise constant density stratification in a three-layer fluid with a flat bottom, a free surface and two interfaces. We establish a Hamiltonian formulation for the nonlinear governing equations in this set-up. The Hamiltonian of the system and the equations of motion of the surface and of the interfaces are expressed with the help of the Dirichlet–Neumann (DN) operators, which are introduced for each of the layers. Then the linear equations for small amplitudes of the elevation of the surface and of the interfaces in the leading order are derived, from which a bi-cubic equation for the dispersion relation is obtained, whose solutions are analysed. The six real solutions for the possible propagation speeds (three positive, related to right-moving waves, and three negative, related to left-moving waves) have magnitudes of different order. Upper and lower bounds for the previously mentioned roots are also given in terms of the coefficients of the equation. Subsequently, approximate formulas for the propagation speeds are derived. The importance of the DN operators is further illustrated in a separate analysis of the three-layer model with flat surface (rigid lid). The full nonlinear evolution equations are expressed again in terms of the DN operators, and the equations in the linear regime and the weakly nonlinear propagation regime (the Boussinesq approximation) are derived by a proper expansion of the DN operators. Limits to the two-layer free surface model are obtained as well. The obtained results are applicable to internal waves in lakes and in the ocean as well as to laboratory experiments with three superimposed fluid layers.