Diabatically Induced Secondary Flows in Tropical Cyclones. Part II: Periodic Forcing

Abstract

The linearized equation for the time-varying, axially symmetric circumferential component of the vorticity in a hurricane-like vortex closely resembles the classical Sawyer-Eliassen equation for the quasi-steady, diabatically induced secondary-flow streamfunction. The salient difference lies in the coefficients of the second partial derivatives with respect to radius and height. In the Sawyer-Eliassen equation, they are the squares of the buoyancy and isobaric local inertia frequencies; in the circumferential vorticity equation they are the differences between these quantities and the square of the frequency with which the imposed forcing varies. The coefficient of the mixed partial derivative with respect to radius and height is the same in both equations. Thus, for low frequencies the response to periodic forcing is a slowly varying analog to steady Sawyer-Eliassen solutions. For high frequencies, the solutions are radially propagating inertia-buoyancy waves. Since the local inertia frequency, which approximately defines the boundary between quasi-steady and propagating solutions, decreases with radius, quasi-steady solutions in the vortex core transform into radiating ones far from the center. Periodic forcing will always lead to some wave radiation to the storm environment unless the period of the forcing is longer than a half-pendulum day.

Evolution of Multiscale Vortices in the Development of Hurricane Dolly (2008)

“…2e-f, we notice that the square root of the ratio of the divergence power to the vorticity power (scaled as the Rossby number) exhibits an obvious reduction at the cluster and system scales; this ratio is much smaller than 0.5 at the system scale during most of the spinup period of Dolly, suggesting that the simulated development of Dolly becomes increasingly more balanced in the system-scale sense. This is in agreement with the well-held argument in TC dynamics that, to a first order of approximation, the primary circulations of TCs are vortices in near gradient balance (Shapiro and Willoughby 1982;Willoughby 1990). …”

“…Since the system scale is a well-organized vortex (Fig. 1b), the adjustment processes can be further explored by diagnosing the axisymmetric secondary circulation forced by diabatic heating through the Sawyer-Eliassen equation (e.g., Schubert et al 1980;Shapiro and Willoughby 1982;Hack and Schubert 1986;Hendricks et al 2004;Pendergrass and Willoughby 2009;Willoughby 2009). …”

Application of the Concepts of Rossby Length and Rossby Depth to Tropical Cyclone Dynamics

“…In the present paper we consider idealized vortex structures and idealized vertical structures of Q that allow the transverse circulation equation to be solved analytically via three different methods. These simple theoretical arguments illustrate the usefulness of the complementary concepts of Rossby length (also sometimes referred to as the “Rossby radius of deformation”) and Rossby depth, and thereby elaborate on ideas discussed in the extensive literature on applications of the balanced vortex model to tropical cyclones (e.g., Ooyama , 1969, Sundqvist , 1970a, b, Smith , 1981, Shapiro and Willoughby , 1982, Schubert and Hack , 1982, Hack and Schubert , 1986, Nolan et al , 2007, Vigh and Schubert , 2009, Pendergrass and Willoughby , 2009, and Willoughby , 2009).…”

Sensitivity of tropical cyclone intensification to inner-core structure

“…Under the constraint of both the hydrostatic and gradient wind balance, the axisymmetric secondary circulation forced by the diabatic heating can be obtained through the Sawyer-Eliassen (SE) balance equation (Hendricks et al, 2004;Willoughby, 2009). This method is extensively used to diagnose the adjustment to reach a quasi-balanced state.…”