Abstract:Rayleigh-Kuo theorem and Fjortoft theorem provide necessary conditions for barotropic instabilty of a geophysical fluid flow. Moreover, the barotropic instability of the geostrophic flow is of special importance for the geophysical flow. In this paper, two modified necessary conditions for the barotropic instability of the geostrophic flow are obtained. Based on energetics and potential vorticity dynamics theories, the implication of the necessary conditions for barotropic instability is explained. For the geostrophic flow, the Rayleigh-Kuo theorem can be revised as follws: A necessary condition for barotropic instability of the geostrophic flow is that the potential vorticity of the flow should have an extreme point. The Fjortoft theorem correspondingly becomes: A necessary condition for barotropic instability of the geostrophic flow is that Qy(U-Us)>0 in the field of the geostrophic flow, where ys is a point at which Qy=0 and Us=U(ys). That is, the absolute vorticity in the original theorems is modified to potential vorticity. These results are the consequence of geostrophic constraint and potential vorticity constraint. The energy relationship is the energetics requirement for the possibility of barotropic instability, the Rayleigh-Kuo theorem is the potential vorticity dynamics condition for the possibility of barotropic instability, and the Fjortoft theorem is the coordination of the two requirements of energetics and potential vorticity dynamics. For possible barotropic instability, all three requirements must be met.