Rapid development of the tropical cyclone warm core

This study seeks to isolate the conditions that control the pace of development of a tropical cyclone's warm-core thermal structure. The theoretical argument is based on the balanced vortex model and, in particular, on the associated geopotential tendency equation, a second order partial differential equation. This equation contains a diabatic forcing term and the spatially-varying coefficients A (static stability), B (baroclinity), and C (inertial stability). Thus, the temperature tendency in a tropical vortex depends not only on the diabatic forcing, but also on the spatial distribution of A, B, and C. Experience shows that a tropical cyclone's large radial variations of inertial stability C typically dominate the vortex response. Under certain simplifying assumptions on the vertical structure of the diabatic forcing and on the spatial variability of A, B, and C, the geopotential tendency equation can be solved via separation of variables. The resulting radial structure equation, a second order ordinary differential equation, retains the dynamically-important radial variation of the Rossby length, which is related to the radial variation of C. This radial structure equation can then be solved using the Green function G(r,r_h) to yield the temperature tendency at radius r resulting from a Dirac delta function in the diabatic heating at radius r_h. The differential equation for the Green function can be solved analytically only if the radial variation of the Rossby length takes some simple form. We present the simple case of a Rankine-like vortex which possesses piecewise constant Rossby length, having a small value in the vortex core and a large value in the far-field. From these Green function solutions it is clear that the temperature tendency is very sensitive to the location of the diabatic heating relative to the radius of maximum wind. Rapid intensification tends to occur when at least some of the eyewall diabatic heating is located inside the radius of maximum wind.