On the representation of gravity waves in numerical models of the shallow-water equations

Gravity waves, or imbalanced motions, that develop during the evolution of vortical Rows in numerical models of the shallow water (SW) equations are examined in detail. The focus here is on nearly-balanced flows, with small but non-zero gravity-wave activity. For properly initialized Rows, it is reasonable to expect small GW activity when Froude numbers Fr < 1 and Rossby numbers Ro less than or similar to 1. The guiding principle in the present study is that an accurate representation of potential vorticity (PV) is the pre-requisite to a fair assessment of the generation of gravity waves. The contour-advective semi-Lagrangian (CASL) algorithm for the SW equations is applied, as it shows a remarkable improvement in the simulation of PV. However, it is shown that the standard CASL algorithm for SW leads to a noticeable numerical generation of gravity waves. The false generation of gravity waves can equivalently be thought of as the false, or numerical, breakdown of balance. In order to understand the maintenance of balance in the SW equations, a hierarchy of CASL algorithms is introduced. The main idea behind the new hierarchy is to implement PV inversion partially, balancing algorithms directly within the SW algorithm, while still permitting imbalanced motions. The results of the first three members of the hierarchy, CA(o) (standard CASL), CA(1), and CA(2), are described and are compared with the results of two other SW algorithms, a pseudo-spectral and a semi-Lagrangian one. The main body of results is obtained for a highly ageostrophic regime of flow, with \Ro\(max) similar to 1 and Fr-max similar to 0.5, where sub-index 'max' denotes maximum over the domain. Other Row regimes in the relevant parts of the Ro-Fr parameter space are also explored. It is found that, for a given resolution and Froude number, there is an optimal CASL algorithm, i.e. one which gives rise to the least numerical generation of gravity waves.