A Problem Related to Nonlinear Stability Criteria for Multi-layer Quasi-geostrophic Flow

The second author studied the nonlinear stability of N-layer quasi－geostrophic flow subject to perturbations of parameters and initial data, and established the stability criteria for the flow in question, which involve finding out the lowest eigenvalue of an elliptic boundary value problem.In this paper when the domain is a periodic zonal channel, a formula of the lowest eigenvalue is established, which is useful for further studies and practical applications.