The interfacial conditions for a cylindrical and an axial vortex sheet or thin fluid layer are obtained for a general class of vortex flows in a radius and gravity-stratified environment. The flow is assumed to be inviscid and incompressible. No Boussinesq approximation is required. In addition to the kinematic and dynamic conditions that the flow has to satisfy in the centrifugal and gravitational directions, a third condition, which restrains the interaction of the centrifugal and gravitational force fields, has to be imposed on those vortex sheets. This is consistent with the previous derived criteria for this type of vortex motions, in which a third condition based on pressure and force balance must be satisfied. Nonaxisymmetric instability for a special flow profile is examined and the stability boundary is obtained to show the behavior of this type of stratified vertical vortex. The results provide us with some information on the instability mechanism for the generation of the horizontal vortices in the ocean and for the spiral type of vortex breakdown in tornadoes and waterspouts in the atmosphere.

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