An initial-value problem for testing numerical models of the global shallow-water equations

We present an initial-value problem for testing numerical models of the global shallow-water equations. This new test case is designed to address some of the difficulties that have recently been uncovered in the canonical test case suite of Williamson et al. The new test case is simple to set up, yet able to generate a complex and realistic flow. The initial condition consists of an analytically specified, balanced, barotropically unstable, mid-latitude jet, to which a simple perturbation is added to initiate the instability. The evolution is comprised of an early adjustment phase dominated by fast, gravity wave dynamics, and a later development characterized by the slow, nearly balanced roll-up of the vorticity field associated with the initial jet. We compute solutions to this problem with a spectral transform model to numerical convergence, in the sense that we refine the spatial and temporal resolution until no changes can be visually detected in global contour plots of the solution fields. We also quantify the convergence with standard norms. We validate these solutions by recomputing them with a different model, and show that the solutions thus obtained converge to those of the original model. This new test is intended to serve as a complement to the Williamson et al. suite, and should be of particular interest in that it involves the formation of complicated dynamical features similar to those that arise in numerical weather prediction and climate models.