An axisymmetric model of a mature tropical cyclone incorporating azimuthal vorticity

Pearce, R. P.

doi:10.1256/qj.02.86

Cited by 8 publications

(12 citation statements)

References 41 publications

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“…The present analysis is also background for a companion paper (Willoughby 2009) that reformulates the SEQ as a circumferential vorticity equation (e.g., Smith et al 2004;Pearce 2004), which describes responses to time-varying forcing. We cast the problem in height coordinates, where the horizontal ocean surface is also a coordinate surface.…”

Section: Introductionmentioning

confidence: 99%

Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing

Pendergrass

Willoughby

2009

Monthly Weather Review

141

121

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The Sawyer-Eliassen Equation (SEQ) is here rederived in height coordinates such that the sea surface is also a coordinate surface. Compared with the conventional derivation in mass field coordinates, this formulation adds some complexity, but arguably less than is inherent in terrain-following coordinates or interpolation to the lower physical boundary. Spatial variations of static stability change the vertical structure of the mass flow streamfunction. This effect leads to significant changes in both secondary-circulation structure and intensification of the primary circulation. The SEQ is solved on a piecewise continuous, balanced mean vortex where the shapes of the wind profiles inside and outside the eye and the tilt of the specified heat source can be adjusted independently. A series of sensitivity studies shows that the efficiency with which imposed heating intensifies the vortex is most sensitive to intensity itself as measured by maximum wind and to vortex size as measured by radius of maximum wind. Vortex shape and forcing tilt have impacts 20%-25% as great as intensity and size, suggesting that the aspects of tropical cyclones that predispose them to rapid intensification are environmental or thermodynamic rather than kinematic.

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Section: Introductionmentioning

confidence: 99%

Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing

Pendergrass

Willoughby

2009

Monthly Weather Review

141

121

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“…As shown in the next section, algebraic elimination of all variables except the radial and vertical velocities leaves an equation for the circumferential component of the vorticity (e.g., Pearce 2004Pearce , 2005Smith 2005) in which the perturbation vortex tubes encircle the 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 hydrostatic balance.…”

Section: Introductionmentioning

confidence: 99%

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

Willoughby

2009

Monthly Weather Review

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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.

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“…In contrast, if C > 1, corresponding with the regime of relatively weak stability and strong rotation, the * The same limitation holds for alternative profile v(r) = V R (R/r) α (α a constant < 1) in the extended theory described by Pearce (2004). † The profile has the desirable property also, that that the radially-integrated kinetic energy and angular momentum are bounded as r increases.…”

Section: An Alternative Calculationmentioning

confidence: 99%

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mentioning

confidence: 99%

“…In this case h(0) > 0 for C < 1, which conforms with our intuition that for sufficiently strong stability (g' relatively large) and sufficiently weak rotation (v m relatively small) the interface is not depressed as far as the surface at the axis. In contrast, if C > 1, corresponding with the regime of relatively weak stability and strong rotation, the * The same limitation holds for alternative profile v(r) = V R (R/r) α (α a constant < 1) in the extended theory described by Pearce (2004). † The profile has the desirable property also, that that the radially-integrated kinetic energy and angular momentum are bounded as r increases.…”

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"Why must hurricanes have eyes?" ? revisited

Smith¹

2005

wea

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In a recent thought-provoking article, Pearce (2005) poses the question "Why must hurricanes have eyes"? In the article he explains aspects of the inner-core dynamics of a mature hurricane in terms of a simple axisymmetric model in which the eye is non-rotating and less dense than the vortex surrounding it. He describes a calculation in which the eye always has a finite radius at the surface and the inference appears to be that a similar dynamical constraint applies to a mature hurricane. My aim here is to review some of the important issues raised by Prof. Pearce and to present a slightly different view of the inner-core dynamics of a hurricane and an alternative explanation for the hurricane eye. In particular, I will argue that the foregoing calculation is unrealistic in one important respect and show how it can be repaired. Even so, its relevance to the hurricane eye remains unclear.Prof. Pearce makes a commendable attempt to simplify the concepts required to understand the dynamics involved by considering parts of the problem separately. For example he explains q how azimuthal vorticity is produced in a rising thermal; q why the azimuthal wind speed must decrease with height above the surface friction layer; and q how this decrease leads to an azimuthal vorticity tendency that balances the tendency associated with the negative radial temperature gradient observed in a hurricane.He argues that subsidence must occur in the eye so that the radial temperature gradient in the eye can balance the production of azimuthal vorticity by vortex tilting in the eyewall. However there are some misleading aspects of the discussion that I will try to explain below. An analysis of Pearce's simple modelFirst let me examine the simple model that always predicts the existence of an eye of finite radius at the surface. The model considers two layers of immiscible incompressible fluid with densities ρ -∆ρ and ρ, where ∆ρ > 0. The lighter fluid that represents the eye is at rest while the heavier one representing the surrounding vortex core is rotating. In the model, an expression is derived for the shape of the surface, h(r), separating the eye from the vortex outside it. The assumption is made that Coriolis forces can be neglected, which is not unreasonable for the rapidly rotating inner core of a hurricane. It is shown that with these assumptions, h(r) satisfies the ordinary differential, where v is the azimuthal (tangential) wind speed, r is the radius, g' = g∆ρ/ρ is the reduced gravity and g is the acceleration due to gravity. It is then assumed, for simplicity, that the vortex outside the eye has uniform angular momentum, i.e. the azimuthal wind profile v(r) satisfies rv(r) = V R R, where V R is the wind speed at some radius R. Then the equation for h can be integrated to give], where H = h(∞) is the value of h at large r and r e = V R R/ √(2g'H). It follows that h is zero when r = r e , which is always finite. In other words, the eye region has a finite width at the surface irrespective of the prescribed strength of the an...

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An axisymmetric model of a mature tropical cyclone incorporating azimuthal vorticity

Cited by 8 publications

References 41 publications

Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing

Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing

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

"Why must hurricanes have eyes?" ? revisited

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