Numerical Solution of The Equation System
The discretized equations are solved numerically on a staggered grid, i.e. the scalar quantities, and are defined at the cell centre while the velocity components, and are defined at the centre of the appropriate interfaces.
Temporal discretization of the prognostic equations is based on the explicit second order Adams-Bashforth scheme. There are two deviations from the Adams-Bashforth scheme: The first refers to the implicit treatment of the nonhydrostatic part of the mesoscale pressure perturbation . To ensure non-divergence of the flow field, an elliptic equation is solved. The elliptic equation is derived from the continuity equation wherein velocity components are expressed in terms of . Since the elliptic equation is derived from the discrete form of the continuity equation and the discrete form of the pressure gradient, conservativity is guaranteed (Flassak and Moussiopoulos, 1988). The discrete pressure equation is solved numerically with a fast elliptic solver in conjunction with a generalized conjugate gradient method. The fast elliptic solver is based on fast Fourier analysis in both horizontal directions and Gaussian elimination in the vertical direction (Moussiopoulos and Flassak, 1989).
The second deviation from the explicit treatment is related to the turbulent diffusion in vertical direction. In case of an explicit treatment of this term, the stability requirement may necessitate an unacceptable abridgement of the time increment. To avoid this, vertical turbulent diffusion is treated using the second order CrankâNicolson method.
On principle, advective terms can be computed using any suitable advection scheme. In the present version of MEMO, a 3-D second-order total-variation-diminishing (TVD) scheme is implemented which is based on the 1-D scheme proposed by Harten (1986). It achieves a fair (but not any) reduction of numerical diffusion, the solution being independent of the magnitude of the scalar (i.e., preserving transportivity).
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