Mathematics of Stability, Breaking and Mixing for Waves in Shallow Fluids

Abstract: Shallow-fluid models are often the first step in modeling many geophysical flows. These models apply when the horizontal scales of motion are much larger than the vertical scales. Examples range from tsunamis to internal waves in the ocean to large scale atmospheric waves. For a single fluid layer with a free-surface the shallow water approximation results in a hyperbolic system of partial differential equations which is mathematically well understood and broadly applied. In shallow water theories waves generically steepen and break creating hydraulic jumps (discontinuous solutions also called shocks or bores). These are understood mathematically as satisfying the PDEs in a weak sense, and, physically, that one must ensure that certain conserved quantities - usually mass and momentum - are preserved across shocks. For the single layer case this closes the problem and results in a prediction of the small scale energy dissipation at the shock. In layered shallow water models the situation is complicated by at least two issues: the flow may be unstable to the formation of billows (the Kelvin-Helmholtz instability) and that breaking waves may mix the fluids. We discuss some physically motivated mathematical results on these issues.

Refreshments will be served at 2:45 p.m. (MATX 1115, Math Lounge).