A spreading drop of shallow water

The theoretical solutions and corresponding numerical simulations of [Schär & Smolarkiewicz, J. Comput. Phys, 128 (1996) 101]) are revisited. The original abstract problem of a parabolic, slab-symmetric drop of shallow water spreading under gravity is extended to three spatial dimensions, with the initial drop defined over an elliptical compact support. An axisymmetric drop is considered as a special case. The elliptical drop exhibits enticing dynamics, which may appear surprising at the first glance. In contrast, the evolution of the axisymmetric drop is qualitatively akin to the evolution of the slab-symmetric drop and intuitively obvious. Besides being interesting per se, the derived theoretical results provide a simple means for testing numerical schemes concerned with wetting-drying areas in shallow water flows. Reported calculations use the libmpdata++, a recently released free/libre and open-source software library of solvers for generalized transport equations. The numerical results closely match theoretical predictions, demonstrating strengths of the nonoscillatory forward-in-time integrators comprising the libmpdata++.