Abstract

Two topics will be covered in this lecture. The shallow water equations will be used as a test bed to introduce the ideas.

(i) For forced variational systems such as the potential flow shallow water wave equations, variational and symplectic time integrators will be extended using a new finite element approach. Here, a standard variational finite element discretization will be applied in space.

(ii) The shallow water equations formulated in terms of Clebsch variables will be discussed. The advantage of Clebsch variables is that they lead to canonical Hamilton’s equations for shallow water dynamics, in the Eulerian framework. A disadvantage is that the the system, is less compactly expressed in comparison to the usual formulation in terms of the velocity and fluid depth. I will make a link between a symmetry in the Hamiltonian and the associated mass weighted potential vorticity conservation law, also within the Eulerian framework. This will be done in two dimensions (2D) and in a quasi-2D symmetric form.