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

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  • 1 Department of Earth Sciences, Florida International University, Miami, Florida
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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.

Corresponding author address: H. E. Willoughby, Department of Earth Sciences, Florida International University, University Park Campus, PC 344, Miami, FL 33199. Email: hugh.willoughby@fiu.edu

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.

Corresponding author address: H. E. Willoughby, Department of Earth Sciences, Florida International University, University Park Campus, PC 344, Miami, FL 33199. Email: hugh.willoughby@fiu.edu

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