Tide models and finite elements
Barotropic tides can be modeled by linearized rotating shallow water equations plus possibly nonlinear bottom friction terms. We present energy estimates that describe the effective damping rates of these friction terms. In the linear case, one obtains exponential energy damping rates, but without smoothing effects. In the nonlinear case, careful analysis gives sub-exponential growth rates under fairly general structural assumptions. These estimates demonstrate long-time stability of forced systems and a global attracting solution. This justifies the practice of “spinning up” tide models used in applications. Continuous dependence results that bypass Gronwall-type techniques also allow us to provide rigorous uniform-in-time finite element error estimates for spatially-discrete models. Extensions to a multi-layered tide model will also be discussed. This is joint work with Colin Cotter (Imperial College, London) and Jameson Graber and Alan Mullenix (Baylor).