A comparison of methods to balance geophysical flows

We compare a higher-order asymptotic construction for balance in geophysical flows with the method of 'optimal balance', a purely numerical approach to separating inertia-gravity waves from vortical modes. Both methods augment the linear geostrophic mode with dependent inertia-gravity wave mode contributions, the so-called slaved modes, such that the resulting approximately balanced states are characterized by very small residual wave emission during subsequent time evolution. In our benchmark setting - the single-layer rotating shallow water equations in the quasi-geostrophic regime - the performance of both methods is comparable across a range of Rossby numbers and for different initial conditions. Cross-balancing, i.e. balancing the model with one method and diagnosing the imbalance with the other, suggests that both methods find approximately the same balanced state. Our results also reinforce results from previous studies suggesting that spontaneous wave emission from balanced flow is very small. We further compare two numerical implementations of each of the methods: one pseudospectral, and the other a finite difference scheme on the standard C-grid. We find that a state that is balanced relative to one numerical scheme is poorly balanced for the other, independent of the method that was used for balancing. This shows that the notion of balance in the discrete case is fundamentally tied to a particular scheme.