Non-Diffusive Part -- Leapfrog Scheme

The time stepping used for processes other than diffusion is the well-known leapfrog scheme [Mesinger and Arakawa, 1976]. This scheme is widely used for advection processes in low-viscosity fluids. It is a time centred scheme, $i.e.$ the RHS in (3.1) is evaluated at time step $t$, the now time step. It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, but not for diffusion terms. It is an efficient method that achieves second-order accuracy with just one right hand side evaluation per time step. Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by amplifying the oscillations. These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme, and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes. However, the scheme allows the coexistence of a numerical and a physical mode due to its leading third order dispersive error. In other words a divergence of odd and even time steps may occur. To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter (hereafter the LF-RA scheme). This filter, first designed by Robert [1966] and more comprehensively studied by Asselin [1972], is a kind of laplacian diffusion in time that mixes odd and even time steps:

$$x_F^t = x^t + \gamma \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right]$$ (3.2)

where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is initialized as rn_atfp (namelist parameter). Its default value is rn_atfp=$10^{-3}$ (see § 3.5), causing only a weak dissipation of high frequency motions ([Farge, 1987]). The addition of a time filter degrades the accuracy of the calculation from second to first order. However, the second order truncation error is proportional to $\gamma$, which is small compared to 1. Therefore, the LF-RA is a quasi second order accurate scheme. The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes, because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme. When used with the 2nd order space centred discretisation of the advection terms in the momentum and tracer equations, LF-RA avoids implicit numerical diffusion: diffusion is set explicitly by the user through the Robert-Asselin filter parameter and the viscosity and diffusion coefficients.