  Considered herein is a class of Boussinesq systems of Bona-Smith type that
describe water waves in bounded two-dimensional domains with slip-wall boundary
conditions and variable bottom topography. Such boundary conditions are
necessary in situations involving water waves in channels, ports, and generally
in basins with solid boundaries. We prove that, given appropriate initial
conditions, the corresponding initial-boundary value problems have unique
solutions locally in time, which is a fundamental property of deterministic
mathematical modeling. Moreover, we demonstrate that the systems under
consideration adhere to three basic conservation laws for water waves: mass,
vorticity, and energy conservation.
  The theoretical analysis of these specific Boussinesq systems leads to a
conservative mixed finite element formulation. Using explicit, relaxation
Runge-Kutta methods for the discretization in time, we devise a fully discrete
scheme for the numerical solution of initial-boundary value problems with
slip-wall conditions, preserving mass, vorticity, and energy. Finally, we
present a series of challenging numerical experiments to assess the
applicability of the new numerical model.
