27.2.1 damp_inertial_oscillation

Option damp_inertial_oscillation damps inertial oscillations by treating the Coriolis term semi-implicitly. Why treat the Coriolis terms semi-implicitly? It only makes sense in coarse resolution $(\Delta_x \ge 5^\circ)$ global models where the time step allowed by the CFL condition does not resolve the inertial period which is 1/2 day at the poles. Treating the Coriolis term semi-implicitly damps the inertial oscillation and allows a longer time step. For global models with $\Delta_x \le 2^\circ$, the time step allowed by the CFL condition is typically small enough (less than 2 hours) to resolve the inertial period at the poles and so semi-implicit treatment is not needed. Consider the simple system for inertial oscillations

u_t - fv = 0 (27.31)

v_t + fu = 0 (27.32)

where $f=2\Omega\sin\phi$, u is zonal velocity, and v is meridional velocity. When discretizing and solving the Coriolis term explicitly, the solution is given by

$$u^{\tau+1} u^{\tau-1} + 2\Delta\tau\cdot fv^{\tau}$$ = (27.33)

$$v^{\tau+1} v^{\tau-1} - 2\Delta\tau\cdot fu^{\tau}$$ = (27.34)

When treating the Coriolis term semi-implicitly, an implicit Coriolis factor $0.5 \le acor < 1$ is used and the system becomes

$$u^{\tau+1} - 2\Delta\tau\cdot acor\cdot fv^{\tau+1} u^{\tau-1} + 2\Delta\tau\cdot(1-acor)\cdot fv^{\tau-1}$$ = (27.35)

$$v^{\tau+1} + 2\Delta\tau\cdot acor\cdot fu^{\tau+1} v^{\tau-1} - 2\Delta\tau\cdot(1-acor)\cdot fu^{\tau-1}$$ = (27.36)

The solution, after a little algebra, is given by

$$u^{\tau+1} = u^{\tau-1} + 2\Delta\tau\cdot \frac{fv^{\tau-1} - (2... ...a\tau\cdot acor\cdot f)\cdot fu^{\tau-1}}{1 + (2\Delta\tau\cdot acor\cdot f)^2}$$ (27.37)

$$v^{\tau+1} = v^{\tau-1} - 2\Delta\tau\cdot \frac{fu^{\tau-1} + (2... ...a\tau\cdot acor\cdot f)\cdot fv^{\tau-1}}{1 + (2\Delta\tau\cdot acor\cdot f)^2}$$ (27.38)

Note that option damp_inertial_oscillation will also damp external Rossby waves.