Tropical Cyclone Response to Periodic Forcing

Tropical cyclones (TCs) intensify because convective updrafts around the eye release latent heat extracted from the sea. The dominant factor in intensification is strengthening of the maximum axially symmetric balanced wind. In order to remain nearly balanced, the forcing must be steady, weak enough to keep the induced secondary flow small relative to the primary flow, and predominantly symmetric about the vortex axis of rotation.

“Steady” means that rate of intensity change is slow compared with the local inertial frequency I² in a neighborhood around the locus of heating. I² is the square root of the radial gradient of angular momentum divided by the cube of the radius. The local inertia period at the RMW can vary from a half hour to several hours, dependent upon TC size and strength. Far from the center, it decreases asymptotically to a half pendulum day. When the steadiness, weakness, and symmetry conditions are met, the Sawyer-Eliassen (SEQ) describes the heating-induced, quasi-steady mass-flow streamfunction. Advection of primary-flow angular momentum and mass by this secondary flow accounts for the intensification process. In a TC with convective heating concentrated in the eyewall, the secondary flow that penetrates a short distance into the high-angular-momentum vortex core, increasing the wind at the radius of maximum wind and inward from it, so that the eye contracts as the maximum wind strengthens. This “convective ring” process was studied extensively during the 1980s.

Periodic forcing requires a different formulation in which the governing equations are linearized about a mean balanced vortex. Algebraic elimination of all variables except the radial and vertical velocities leaves an equation for the azimuthal component of the vorticity, that is, for the component in which the perturbation vortex tubes encircle the mean vortex. Like the SEQ, this equation is a second order partial differential equation for the mass-flow streamfunction, but one that relaxes the requirements for gradient and hyrdrostatic balance. In the SEQ, the coefficients of the second partial derivatives with respect to radius and height are the squares of the buoyancy and inertia frequencies. In the azimuthal-vorticity equation, these coefficients are the differences between the squares of the buoyancy and inertia frequencies and the frequency of periodic forcing. In both equations, the coefficient of the mixed partial derivative is proportional to the radial mean buoyancy gradient. For frequencies within a passband defined by the leading coefficients, the equation is hyperbolic and its solutions are radiating inertia-buoyancy waves. Below the passband, it elliptical with quasi-steady secondary circulation solutions, as described by the SEQ. Since I² decreases with distance from the center, the solution's character changes radially. Unless the forcing oscillates very rapidly in time, as might happen in a convective burst, the solution in the core is quasi steady, but within 100-200 km of the center it transforms into low-frequency radiating waves. For fixed intensity and vortex profile shape, the quasi-steady regime should generally be smaller in low latitudes where the half pendulum day is longer.