Numerical simulation of shoaling internal solitary waves with a generalized vertical coordinate, nonhydrostatic ocean model

Internal gravity waves typically evolve into trains of weakly nonlinear internal solitary waves (ISWs) that shoal and break on continental slopes and provide a source of transport and mixing on continental shelves. Numerical simulation of shoaling ISWs is challenging owing to the multiscale nature of the shoaling process. In addition to the wavelength decrease upon shoaling, the waves undergo a transition from depression waves to elevation waves where the mixed-layer depth is less than half the total water depth. This transition and subsequent shoaling occur during ISW propagation over many wavelengths, thus placing significant numerical constraints on the underlying solver when considering real, field-scale problems. We present methods to accurately and efficiently simulate shoaling ISWs with an arbitrary Lagrangian-Eulerian (ALE) vertical coordinate, nonhydrostatic ocean model. A novel momentum advection scheme for horizontally-unstructured, staggered grids is described that is stable in the presence of arbitrarily small layer heights that arise where the density-following coordinate lines intersect the bottom topography. As the waves become increasingly nonlinear during the shoaling process, we present an adaptive technique that ensures a smoothly-varying vertical coordinate that closely follows the density lines while minimizing the spurious numerical diffusion of sharp, vertical density gradients. Results are presented that demonstrate the robust nature of the solver and its ability to accurately simulate shoaling ISWs in real, field-scale domains.