| Abstract: | Recently, new hyperbolic systems of equations that can be used to describe smooth flows accurately in both the atmosphere and oceans have been developed. These ‘approximate systems’ are derived by slowing down the speed of the fast waves instead of increasing their speed to infinity as in the primitive equations. The approximate systems have a number of theoretical advantages over the traditional systems. The practical implications of some of these advantages have already been demonstrated for the oceanic case. There is another advantage of the new systems 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 provided 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 the presence of topography, numerical results from a model based on the approximate system for meteorology are compared with analytic solutions 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 example, in the atmospheric case a second-order method requires only approximately 10 points across the base of the mountain to achieve a 1% relative error for any of the three topographic solutions during the relevant time scale of the associated motion. |
