To get to the Kuo criterion, the imaginary part is integrated over the domain () . The stream function at the boundaries of the domain is zero, , as already stated in the assumptions. The zonal flow must vanish at the boundaries of the domain. This leads to a constant stream function which is set to zero for convenience.
The first integral can be solved:
So the first integral is equal to zero. This means that the second integral should also be zero, making it possible to solve this integral numerically.
When  is zero, we are dealing with a stable amplitude of the solution, this means that the solution is stable. We are looking for un unstable situation, so then  should be zero. Since the fraction in front of  is non-zero and positive, this leads to the conclusion that  should be zero. This leads to the final formulation, the Kuo criterion:
Here,  is the mean zonal flow and  is the derivative of the planetary vorticity  with respect to .