A Nodal Spectral Element Method for the Simulation of Internal Solitary Waves Over an Actual Bathymetry.

Internal Solitary Waves (ISW) are large amplitude internal waves which travel with relatively unchanging form, playing an important role in near-shelf energetics and transporting salt, heat, and nutrients into shallower coastal waters. Much remains unknown about ISWs and accurate numerical modelling of such waves using realistic bathymetry can provide insight. The existence of an ISW is a precarious balance between dispersive and nonlinear effects, therefore the use of a high-order numerical methods is imperative. The Nodal Spectral Element Method has been used extensively for the solution of the Incompressible Navier-Stokes equations in complex geometries. Realistic ocean bathymetry has a high-aspect ratio, causing great difficulty in the solution of elliptic problems arising in many time-splitting. One approach to mitigate this difficulty is static condensation. The degrees of freedom on each spectral element are reduced to a smaller system on the interfaces between elements with improved conditioning. Subsequently, independent solves are performed for the degrees of freedom on the spectral element interiors, leading to a highly parallelizable algorithm. The efficacy of this method will be demonstrated by solving numerically the Helmholtz problem.