The growth of a young frontal wave

In a two-layer model of the atmosphere with the upper warm air moving from the west relative to the cold air below the long waves are unstable and have a rather simple symmetric structure and behavior when the following approximations are used: (a) The latitude variation of the depth of the layers is ignored. (b) The motion is quasigeostrophic. (c) The planetary β-gradient of the earth's vorticity is ignored. (d) The tropopause is a rigid horizontal surface. When these approximations are removed one by one the wave structure and growth rate are modified somewhat, but the essential features of the wave mechanism remain unchanged.