On the Boussinesq model for two-dimensional wave motions in heterogeneous fluids

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Abstract
In the Boussinesq model, which is a standard frame for the analysis of internal-wave phenomena, the fluid has variable density but is incompressible, inviscid and non-diffusive. Without further approximations, which will not be made here, the dynamical equations are nonlinear and the evolutionary problem posed cannot be solved explicitly except by numerical means; but various interesting properties are accessible. In §1, where a previous summary account is recalled (Benjamin 1984), the model is reformulated as a system of integro-differential equations in which the dependent variables are density ρ and density-weighted vorticity σ. The aim subsequently is to survey the model's mathematical consequences in general rather than to examine particular solutions. Very few exact solutions are yet known although approximate solutions are on record describing solitary and periodic waves of permanent form.In § 2 the Hamiltonian representation of the two-component system is noted, being the key to much of the analysis that follows. The complete symmetry group for this system is given in § 3. It is composed of nine one-parameter subgroups which are listed first in Theorem 1; then their collective significance in relation to Hamiltonian structure is discussed. In § 4 two theorems are given specifying necessary and sufficient conditions for a scalar function to be a conserved density for solutions of the Boussinesq system. There are found to be basically eight such conserved densities which are listed in Theorem 4; and the corresponding conservation laws in integral form for motions between horizontal planes are stated in Theorem 5.The meaning of impulse according to the Boussinesq model is examined in § 5. The two linear components of impulse density and the density of impulsive couple are revealed by the preceding examination of symmetries and local conservation laws; but care is needed to identify physical interpretations of the integral conservation laws that involve impulse. Two laws relating impulse to kinematic properties of the density distribution are particularly strange. Separate treatments are needed for the cases where the fluid-filled domain D is the whole of ℝ2, where D is a half-space with rigid horizontal boundary (which case is in several respects the most delicate) and where D is a horizontal infinite strip. Finally, in § 6, a variational characterization of steady wave motions is explained as a concomitant of Hamiltonian structure, and its implications concerning the stability properties of such motions are reviewed.Appendix A notes a semi-Lagrangian formulation which has a simpler Hamiltonian structure but a narrower range of application. Appendix B outlines an alternative confirmation of Hamiltonian properties by use of a lemma due to Olver (1980b).

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