INCORPORATING TOPOGRAPHY INTO THE MULTISCALE SYSTEMS FOR THE ATMOSPHERE AND OCEANS

Recently, new hyperbolic systems of equations that can be used to desc ribe smooth flows accurately in both the atmosphere and oceans have be en developed. These 'approximate systems' are derived by slowing down the speed of the fast waves instead of increasing their speed to infin ity as in the primitive equations. The approximate systems have a numb er of theoretical advantages over the traditional systems. The practic al implications of some of these advantages have already been demonstr ated for the oceanic case. There is another advantage of the new syste ms that has not been discussed extensively. A model based on either of the new systems can be used to describe different scales of motion, e .g. the large, medium, or small scale. In addition, a mechanism is pro vided for a smooth transition between these scales. The incorporation of topography into the approximate systems has also not been discussed . To demonstrate the multiscale nature of the transformed systems in t he presence of topography, numerical results from a model based on the approximate system for meteorology are compared with analytic solutio ns for three topographic scales. Removing the horizontal means of the density and pressure, which was necessary to obtain the proper scaling of the equations in the original papers, reduces the truncation error associated with a transformed system near steep mountains. For exampl e, in the atmospheric case a second-order method requires only approxi mately 10 points across the base of the mountain to achieve a 1% relat ive error for any of the three topographic solutions during the releva nt time scale of the associated motion.