Initialization of the Primitive Equations by the Bounded Derivative Method

Large-amplitude high-frequency motions can appear in the solution of a hyperbolic system containing multiple time scales unless the initial conditions are suitably adjusted through a process called initialization. We observe that a solution of such a system which varies slowly with respect to time must have a number of time derivatives on the order of the slow time scale. Given a variable which is characteristic of low-frequency motions (e.g., vorticity), we can apply this observation at the initial time to find constraints which determine the rest of the initial data so that the amplitudes of the ensuing high-frequency motions remain small. Boundary conditions of the system must be taken into account in the derivation of the constraints. This procedure is referred to as the bounded derivative method. For a general linear version of the shallow-water equations, we prove that if the initial kth order time derivative is of the order of the slow time scale, then it will remain so for a fixed time interval. For the corresponding constant coefficient system, we compare the present initialization procedure with the normal mode approach. We then apply the new procedure to initialize the nonlinear shallow-water equations including the effect of orography for both the midlatitude and equatorial beta plane cases. In the midlatitude case, the initialization scheme based on quasi-geostrophic theory can be obtained from the bounded derivative method by certain simplifying assumptions. In the equatorial case, the bounded derivative method provides an effective initialization scheme and new insight into the nature of equatorial flows.