Physical vs. Numerical Dispersion of Internal Waves

Sean Vitousek, Stanford University

Many large-scale simulations of internal waves are computed with ocean models solving the primitive (hydrostatic) equations. Internal waves, however, are a delicate dynamical balance of nonlinearity and nonhydrostasy (dispersion), and thus may require computationally expensive nonhydrostatic simulations to be well resolved. Many ocean models solving the primitive equations are second order accurate, and thus from odd-order terms in the truncation error (3rd order and higher), induce numerical dispersion which mimics physical dispersion due to nonhydrostasy. In numerical simulations of internal waves, problems with numerical dispersion arise due to lack of horizontal resolution and the fundamental limitations of the hydrostatic assumption.
Through the use of the modified equivalent partial differential equation analysis (Hirt analysis), we determine the numerical dispersion coefficient associated with the discretization of the primitive equations. Comparing this coefficient with the physical dispersion coefficient from the Boussinesq equations we determine a condition on the horizontal grid spacing required for proper resolution of nonhydrostatic effects. This condition can be equivalently obtained in determining the analytical dispersion relation of the discrete equations. The condition for convergence to the correctly dispersive solution constrains the grid leptic ratio (the ratio of the grid spacing to wave depth). When this condition is not satisfied the numerical dispersion overwhelms the physical dispersion, and modeled internal waves exist with a dynamical balance between nonlinearity and numerical dispersion. The fulfillment of this condition is a significant additional resolution (computational cost) requirement relative to the current state of ocean modeling.

Abstract Author(s): Sean Vitousek, Oliver Fringer.