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According to some observational analyses, it is reasonable to give the order of ten days in size as a representative time scale for the zonal motion of ultra-long wave disturbances with a long scale in meridional direction in the troposphere and the lower stratosphere. In this paper, we examine sets of equations applicable to this kind of disturbances by minimizing fundamental scales to be provided, that is, by providing each magnitude for space and time scales first. As a result, it is derived that the magnitude of meridional motion must be one order of magnitude less than that of zonal motion in the disturbances examined here. The first-approximate set of equations becomes a type of the quasi-geostrophic system in the troposphere and the barotropic equation in the stratosphere, and a type of linearized balance system is derived as the second-approximate one in the troposphere and the stratosphere.

Scale Analysis of Ultra-Long Wave Motions with a Long Scale in Meridional Direction