Shallow Moment Equations—Comparing Legendre and Chebyshev Basis Functions

Depth-averaged free surface flow models are widely used in geophysical applications. The classical shallow water system assumes a vertical plug flow profile and can be generalized to account for quasi-steady vertical velocity profiles. In contrast to this, shallow moment equations constitute a hierarchical set of depth-averaged equations, able to describe transient changes in the vertical profile, for instance, induced by a change of bottom topography. Recent work leverages a Legendre expansion of the vertical velocity, generating a cascaded set of equations with increasing information about the flow structure as the number of basis functions is increased. In this work, we shall first generalize the shallow moment approach to arbitrary basis functions, and continue by specifically analyzing the feasibility of Chebyshev basis functions. We compare low-level systems and evaluate the errors of a numerical simulation in an idealized setting.