Inertial Stability and Tropical Cyclone Development

We consider the frictionless, axisymmetric, balanced flow occurring in a thermally forced vortex on an f-plane. Following Eliassen (1952) we derive the diagnostic equation for the forced secondary circulation. This equation contains the spatially varying coefficients A (static stability), B (baroclinity), C (inertial stability), and the thermal forcing Q. Assuming that A is a constant, B = 0, and that C and Q are piecewise constant functions of radius, we obtain analytical solutions for the forced secondary circulation. The solutions illustrate the following points. 1) For a given Q an increase in inertial stability leads to a decrease in the forced secondary circulation and a change in the radial distribution of local temperature change, with enhanced ∂θ/∂t; in the region of high inertial stability. 2) Lower tropospheric tangential wind accelerations are larger inside the radius of maximum wind, which leads to a collapse of the radius of maximum wind. 3) The fraction of Q which ends up as ∂θ/∂t; increases during the tropical cyclone development, particularly if the horizontal extent of Q is small and close to the region of high inertial stability. 4) One can regard the formation of an eye as a process which tends to stabilize the vortex since it removes Q from the protected, highly stable inner region.