Tropical Cyclone Response to Time Depandant, Axially Symmetric Heating

Forecasting of Tropical Cyclone (TC) intensity remains a significant operational challenge. The fundamental process is conversion of latent heat extracted from the sea into available potential energy and kinetic energy of the wind. Progress on this key problem has been so slow that statistical techniques still provide the best forecast guidance.

As a first approximation, TC intensification can be modeled in terms of the response of an axially symmetric vortex to an imposed heat source. The analytical mean-vortex structures used in this study are realistic representations of actual TCs derived from a large sample of aircraft observations. The heating is represented as truncated Fourier series in time and azimuth. Only the axially symmetric, temporally steady component of the heat source supplies net heating. Its effect is readily computed with the classical Sawyer-Eliassen equation (SEE). The SEE is a Poisson-like equation for the mass flow stream function forced in the radius-height plane by heat and momentum sources. The computed radial and vertical motions are called the axially symmetric “secondary circulation”, in contrast with the “primary circulation”, the axially symmetric mean vortex. The SEE is derived by algebraic elimination of the time derivative of the swirling wind between the tangential momentum and the combined thermodynamic energy, thermal wind, and hydrostatic relations. Although the SEE is diagnostic, the solutions are not necessarily steady. Substitution of the computed secondary flow into the tangential momentum or thermodynamic energy equation allows computation of the gradual evolution of the primary vortex. Substitution of the secondary flow into the radial momentum equation introduces accelerations that act like excesses or deficits of pressure gradient, leading to super- or subgradient swirling wind. For the SEE to be valid, the forcing must change little during a rotation of the primary flow around the vortex and the induced secondary circulation must be small compared with the primary flow.

What happens when the heating is not steady in time? Relaxation of the steady forcing constraint leads to the next level of complexity in which the heat source is still axially symmetric but varies sinusoidally with time. This formulation uses the potentially nonhydrostatic and non-balanced Navier-Stokes equations linearized on a mean vortex in hydrostatic and gradient balance. Elimination of the swirling wind, buoyancy, and mass variable produces a single governing equation for the time dependant secondary-flow streamfunction. If the forcing frequency is low enough, the solutions here approach the solutions of the SEE. At higher frequencies, the solutions are different. Effects such as changes in local deformation radius due to transience or projection onto gravity-wave modes become factors. A still higher level of complexity considers a similarly derived equation for asymmetric heating with a given azimuthal wavenumber and frequency. Phenomena studied include changes in vortex response as a function of frequency, the roles buoyancy and dynamic pressure, forcing induced super- and sub-gradient winds, evolution of the mean vortex as a result of eddy fluxes of heat or momentum, and hydrodynamic stability of the realistic primary vortex.