**A.R. Mohebalhojeh & D.G. Dritschel**

*Quarterly Journal of the Royal Meteorological Society*
**126 , 669-688 (2000)**

We examine in detail the gravity waves, or imbalanced motions, that develop during the evolution of vortical flows in numerical models of the shallow water (SW) equations. The focus here is on nearly-balanced flows, with small but non-zero gravity wave (GW) activity. For properly initialised flows, it is reasonable to expect small GW activity when Froude numbers Fr < 1 and Rossby numbers Ro = O(1) or less.

The guiding principle in the present study is that an accurate
representation of potential vorticity (PV) is the prerequisite to a
fair assessment of the generation of gravity waves. The
Contour-Advective Semi-Lagrangian (CASL) algorithm for the shallow
water equations (Dritschel *et al* 1999) 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 GWs 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 partially implement PV-inversion, balancing algorithms directly within the SW algorithm, while still permitting imbalanced motions. The results of the first three members of the hierarchy, CA0 (standard CASL), CA1, and CA2 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 max(Ro) approximately 1 and max(Fr) approximately 0.5, where 'max' denotes maximum over the domain. We also explore other flow regimes in the relevant parts of the Ro-Fr parameter space. We find 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.