Richardson Number Criterion for the Nonlinear Stability of Three-Dimensional Stratified Flow

With use of a method of Arnol'd, we derive the necessary and sufficient conditions for the formal stability of a parallel shear flow in a three-dimensional stratified fluid. When the local Richardson number defined with respect to density variations is everywhere greater than unity, the equilibrium is formally stable under nonlinear pertrubations. The essential physical content of the nonlinear stability result is that the total energy acts as a "potential well" for deformations of the fluid across constant density surfaces; this well is required to have definite curvature to assure stability under these deformations.