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 , and rewrite
(, ) in the form
In this case, the Hamiltonian is the sum of the potential and kinetic energy, without qualifications.