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2.4 Numerical Schemes for Barotropic Models

We note that the system of equations (2)--(4) has both parabolic and hyperbolic properties, the former associated with the diffusion terms and the latter with the pressure gradients and nonlinear terms. Diffusion of momentum will lead (by itself) to a parabolic partial differential equation. The coupling of the time derivatives (in all three equations) to the pressure gradients and the Coriolis terms (in equations (2) and (3)) will lead to a system that describes inertia-gravity waves and has hyperbolic characteristics. (Note: inertia-gravity waves here refer to gravity waves where the restoring buoyancy force is modified by the rotation of the Earth by the Coriolis force).

The basic techniques for the numerical solutions to these various classes of partial differential equations are given in the PDE chapter of this book. For geophysical fluid dynamic problems, an extensive survey of more specific finite-difference schemes has been given by Grammeltvedt [24] and Arakawa and Mesinger [45]. Another collection of numerical methods in oceanography can be found in O'Brien [51]. As the reader will see from both the outside literature and the various chapters of this book, the numerical schemes for solving time-dependent partial differential equations fall generally into two classes: explicit or implicit. To explain this terminology, let us assume that all variables are evaluated at discrete time and space intervals such that represents and is approximated as . Then the term ``explicit'' denotes a scheme where all terms on the right hand sides (r.h.s.) of system (2)--(4) are evaluated at time steps n, n-1, etc., i.e. at any given time the r.h.s. is known from previous time steps. On the other hand, ``implicit'' denotes a scheme where some of the terms on the r.h.s.\ are evaluated at time step and thus are not known at time . In order to proceed then, one needs to transfer these terms to the left hand sides (l.h.s) of the equations and invert the corresponding coefficient matrix for the unknown variables at .