Integrals of the vorticity equation
for steady, 2D flow
The nonlinearity of the equations of motion makes them intractable from the viewpoint of analytic solutions : this we are accustomed to, and the use of computers to find approximate numerical solutions is accepted as the main solution strategy. In rare cases however, it is possible to make some progress towards an analytic solution for a restricted class of flows. An example is given in this section which has been particularly useful in the study of
orographic and deep convective flows.
Consider a steady ( in some non-accelerating reference frame), two-dimensional stratified flow in the plane where
If is to be zero everywhere then -- consistent with steadiness -- the Coriolis force must also be
considered to be zero. Now define the vorticity of the flow in the plane to be so that positive corresponds to anticlockwise rotation. Taking eq.(63) then gives:
or, using eq.(65):
Now and since the basic state entropy field does not generate horizontal vorticity, the above equation may be written as:
where . In this two-dimensional flow, the continuity equation (40) reduces to: