Unsteady analytical solutions of the spherical shallow water equations

Abstract

A new class of unsteady analytical solutions of the spherical shallow water equations (SSWE) is presented. Analytical solutions of the SSWE are fundamental for the validation of barotropic atmospheric models. To date, only steady state analytical solutions are known from the literature. The unsteady analytical solutions of the SSWE are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system. Fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles. With the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system. An application to the atmospheric model PLASMA shows the benefit of unsteady analytical solutions for the quantification of convergence properties.

On the use of exponential time integration methods in atmospheric models

“…We first consider one of the time-dependent flow cases of La¨uter et al (2005). This unsteady case is a useful test because it has a known analytic solution, allowing us to measure the numerical accuracy.…”

“…2, we plot the errors for a 10-d simulation of the La¨uter et al (2005) unsteady case. Here, for interest, we also include forecasts with the exponential methods at Dt0900s, matching the semi-implicit TÁABT.…”

“…Table 2. Average execution time (minutes) per simulated day, for the cases of the unsteady flow (La¨uter et al, 2005) and the mountain and RossbyÁHaurwitz wave of Williamson et al (1992) Table 3. Average time taken, in seconds, for the calculation of the Jacobi matrix, the forcing terms and the execution of the phipm routine at each step over the 14- …”

A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals

“…In a second, time dependent test, the analytic solution of the shallow water equations on the sphere derived in [37] has been employed to assess the performance of the proposed discretization. More specifically, the analytic solution defined in formula (23) of [37] was used.…”

“…More specifically, the analytic solution defined in formula (23) of [37] was used. Since the exact solution is periodic, the initial profiles also correspond to the exact solution an integer number of days later.…”