A pseudo-spectral algorithm and test cases for the numerical solution of the two-dimensional rotating Green–Naghdi shallow water equations

A pseudo-spectral algorithm is presented for the solution of the rotating Green–Naghdi shallow water equations in two spatial dimensions. The equations are first written in vorticity–divergence form, in order to exploit the fact that time-derivatives then appear implicitly in the divergence equation only. A nonlinear equation must then be solved at each time-step in order to determine the divergence tendency. The nonlinear equation is solved by means of a simultaneous iteration in spectral space to determine each Fourier component. The key to the rapid convergence of the iteration is the use of a good initial guess for the divergence tendency, which is obtained from polynomial extrapolation of the solution obtained at previous time-levels. The algorithm is therefore best suited to be used with a standard multi-step time-stepping scheme (e.g. leap-frog).Two test cases are presented to validate the algorithm for initial value problems on a square periodic domain. The first test is to verify cnoidal wave speeds in one-dimension against analytical results. The second test is to ensure that the Miles–Salmon potential vorticity is advected as a parcel-wise conserved tracer throughout the nonlinear evolution of a perturbed jet subject to shear instability. The algorithm is demonstrated to perform well in each test. The resulting numerical model is expected to be of use in identifying paradigmatic behavior in mesoscale flows in the atmosphere and ocean in which both vortical, nonlinear and dispersive effects are important.