Nonlinear, Non-Rotating Shallow Waters

For the fully nonlinear shallow-water equations in (, ), waves and vorticity no longer decouple (in fact, the nonlinear interaction of waves and vorticity is among the main theoretical obstacles to a full description of turbulence.) However, it is still true that a flow which starts irrotational stays so forever. Hence we may restrict ourselves to considering this scenario, introduce again the scalar potential $\phi$, and rewrite (, ) in the form

$$h_t + \nabla \cdot (h \nabla \phi) = 0\, ,$$

$$\phi_t + \frac{1}{2} \vert\phi\vert^2 + h = 0\, .$$

This system is also Hamiltonian, with

$${\cal H}= \frac{1}{2}\int \left( h^2 + h \vert\nabla\phi\vert^2 \right) \, dx \, ,$$ (8)

and canonical equations

$$h_t = \frac{\delta {\cal H}}{\delta \phi} \, ,$$

$$\phi_t = - \frac{\delta {\cal H}}{\delta h} \, .$$

In this case, the Hamiltonian is the sum of the potential and kinetic energy, without qualifications.