Mechanism of Balanced Flow and Frontogenesis

The final balanced state of an initial unbalanced flow is discussed with the same method as Vallis (1992). For the two-dimensional, inviscid, rotating and nonlinear model, the final state of the flow depends on the initial conditions. If the initial potential vortcity of the flow is non-uniform, the final state is not necessarily geostrophic. However, for the zero and uniform potential vorticity flow, the final state will satisfy the thermal wind relation when the length scale of the initial disturbance is large enough. Otherwise,discontinuity will occur in the geostrophic solution. In this case, the final balanced state will not be geostrophic any longer and an extended momentum coordinate is introduced to overcome the multi-value problem.