Some effects of horizontal discretization on linear baroclinic and symmetric instabilities

The effects of horizontal discretization on linear baroclinic and symmetric instabilities are investigated by analyzing the behavior of the hydrostatic Eady problem in ocean models on the B and C grids. On the C grid a spurious baroclinic instability appears at small wavelengths. This instability does not disappear as the grid scale decreases; instead, it simply moves to smaller horizontal scales. The peak growth rate of the spurious instability is independent of the grid scale as the latter decreases. It is equal to cf/Ri where Ri is the balanced Richardson number, f is the Coriolis parameter, and c is a nondimensional constant that depends on the Richardson number. As the Richardson number increases c increases towards an upper bound of approximately 1/2; for large Richardson numbers the spurious instability is faster than the Eady instability. To suppress the spurious instability it is recommended to use fourth-order centered tracer advection along with biharmonic viscosity and diffusion with coefficients (Δ_x)⁴f/(32Ri) or larger where Δ_x is the grid scale. On the B grid, the growth rates of baroclinic and symmetric instabilities are too small, and converge upwards towards the correct values as the grid scale decreases; no spurious instabilities are observed. In B grid models at eddy-permitting resolution, the reduced growth rate of baroclinic instability may contribute to partially-resolved eddies being too weak. On the C grid the growth rate of symmetric instability is better (larger) than on the B grid, and converges upwards towards the correct value as the grid scale decreases.