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Hugh E. Willoughby

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.

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Angeline G. Pendergrass and Hugh E. Willoughby

Abstract

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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John L. Mcbride and Hugh E. Willoughby

Abstract

This comment presents a detailed examination of the published model results of Kurihara and Kawase in an attempt to clarify the role of wave-CISK in the development of tropical cyclones. Kurihara and Kawase's model simulates the development of a tropical depression, although the vertical structure differs significantly from observations. The physical roles of vertical shear and nonlinear dynamics in the development in this model are unclear. The authors propose that the nonlinear terms in the equations promote rapid growth by increasing the “inertial stiffness”. A major concern, however, is that the enhanced development may occur because the nonlinear terms excite modes with high horizontal wavenumbers. These modes grow rapidly through wave-CISK. From considerations of the climatological importance of horizontal shear to tropical-cyclone development in nature, this model may be less relevant to tropical cyclogenesis than one that allows horizontal sheer of the environmental flow. The authors discuss the model's response to changes in the vertical shear of the basic state, which appears to have the opposite effect in the model from what it has in nature.

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Michael L. Black and Hugh E. Willoughby

Abstract

Hurricane Gilbert of 1988 formed an outer eyewall as it intensified rapidly toward a record minimum pressure of 888 hPa in the western Caribbean. The outer eyewall strengthened and contracted, while the inner eyewall showed some signs of weakening before landfall on the Yucatan Peninsula. Remarkably, both eyewalls survived passage over land, but the storm was much weaker when it entered the Gulf of Mexico. Although the primary cause of weakening was passage over land, the effect of the contracting outer eyewall may have contributed. Later, the outer eyewall completely replaced the inner eyewall. Subsequently, it contracted steadily but slowly as Gilbert maintained nearly constant intensity over the cooler waters of the Gulf before final landfall on the mainland of Mexico.

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Thomas P. Carsey and Hugh E. Willoughby

Abstract

Measurements of ozone (O3) concentrations obtained during aircraft eyewall crossings of Tropical Cyclones Floyd (September 1999) and Georges (September 1998) by NOAA P-3 hurricane research aircraft showed marked changes between the intensifying and weakening stages of the storms’ life cycles. Renewed deepening appeared to be underway near landfall of both storms. During intensification, ozone levels indicated that air either descended from an altitude <1 km above flight level or was strongly diluted with low-O3 eyewall air. During weakening, ozone concentrations were low throughout the eye and eyewall, consistent with the eye’s being filled with boundary layer air.

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Hugh E. Willoughby and Marcy B. Chelmow

Abstract

An algorithm for location of hurricane centers by least squares using aircraft data has been developed. As the aircraft traverses the eye, lines of position normal to the wind are constructed each 100 m along its track. An additional line of position is constructed normal to the track at the closest point of approach to the center. The center coordinates are then chosen such that the sum of the squares of the normal distances from the center to the lines of position is minimized. A cubic spline storm track is first constructed using centers based on winds in a coordinate system fixed to the earth. A track based upon winds in moving, storm-centered coordinates may be obtained by transformation of the winds into such a coordinate system and iterative redetermination of the centers.

For intense hurricanes, the centers can be located with an accuracy of 3 km and the mean motion over a period of four to six hours determined to within 4° of direction and 0.5 m s−1 of speed. The details of the track oscillations analyzed by this techique agree with those observed simultaneously by land-based radar. The technique is used routinely to prepare storm-centered composites and has potential for real-time operational application.

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Stephen J. Lord, Hugh E. Willoughby, and Jacqueline M. Piotrowicz

Abstract

Results of an axisymmetric, nonhydrostatic hurricane model are analyzed with emphasis on the role of a parameterized ice-phase microphysics Inclusion of ice processes produces dramatic differences in the structure and evolution of the simulated hurricane vortex. Mesoscale convective features are wore plentiful with ice, and the simulated vortex grows more slowly.

Time and space-averaged budgets of key model varibles show that cooling due to melting ice particles can initiate and maintain model downdrafts on a horizontal scale of tens of kilometers. This scale depends critically on both the horizontal advection of the parameterized snow particles detrained from the tops of convective updrafts and the mean fall speed of the particles toward the melting level. In situ0 production of snow particles results from a wide variety of parameterized microphysical processes and is significant factor in maintaining upper-level snow concentration These processes are strongly height-dependent.

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Robert W. Jones, Hugh E. Willoughby, and Michael T. Montgomery

Abstract

A nonlinear, two-layer, vortex-tracking semispectral model (i.e., Fourier transformed in azimuth only) is used to study the evolution of dry, but otherwise hurricane-like, initially tilted vortices in quiescent surroundings on f and β planes. The tilt projects onto vorticity asymmetries that are dynamically vortex Rossby waves.

Since the swirling wind in the principal mean vortex used here decays exponentially outside the eyewall, it has an initial potential vorticity (PV) minimum. The resulting reversal of PV gradient meets the necessary condition for inflectional (i.e., barotropic or baroclinic) instability. Thus, the vortex may be inflectionally stable or unstable. On an f plane, the tilt precesses relatively slowly because the critical radius, where the phase speeds of the waves match the mean swirling flow, is far from the center. An alternative Gaussian-like PV monopole that has a monotonic outward decrease of PV is stable to inflectional instability. It has a smaller critical radius and rapid tilt precession. Generally, vortices with fast tilt precession are more stable, as are stronger vortices in higher latitudes.

On a β plane, the interaction between the symmetric vortex and the planetary PV gradient induces β gyres that push the vortex poleward and westward. The interaction between the β gyres and the planetary PV gradient may either create a PV minimum or intensify a minimum inherited from the initial condition. Thus, the nonlinear β effect reduces the ability of the vortex to recover from initial tilt, relative to the same vortex on an f plane. This result contrasts with previous studies of barotropic vortices on f planes, where the linear and nonlinear solutions were nearly identical.

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Erika L. Navarro, Gregory J. Hakim, and Hugh E. Willoughby

Abstract

A modified version of the Sawyer–Eliassen equation is applied to determine the impact of periodic diurnal heating on a balanced vortex. The TC diurnal cycle is a coherent signal that arises in the cirrus canopy. However, despite thorough documentation in the literature, the dynamical mechanism remains unknown. Recent work demonstrates that periodic radiative heating in the TC outflow layer is linked with an anomalous upper-level circulation; this heating is also associated with a cycle of latent heating in the lower troposphere that corresponds to a cycle in storm intensity. Using a method that is analogous to the Sawyer–Eliassen equation, but for solutions having the same time scale as time-periodic forcing, these distributions are analyzed to determine the effect of periodic diurnal heating on an axisymmetric vortex.

Results for periodic heating in the lower troposphere show an overturning circulation that resembles the Sawyer–Eliassen solution. The model simulates positive perturbations in the azimuthal wind field of 2.5 m s−1 near the radius of maximum wind. Periodic heating near the top of the vortex produces a local overturning response in the region of heating and an inertia–buoyancy wave response in the storm environment. Comparison of the results from the modified Sawyer–Eliassen equation to those of an idealized axisymmetric solution for both heating distributions shows similarities in the structure of the perturbation wind fields, suggesting that the axisymmetric TC diurnal cycle is primarily a balanced response driven by periodic heating.

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Hugh E. Willoughby, Frank D. Marks Jr., and Robert J. Feinberg

Abstract

Aircraft observations in hurricanes indicate that the hurricane vortex may be subdivided into an inner gyre where the air trajectories form closed paths and an outer envelope where they do not. In the closed gyre, a core of air moves with the vortex; in the envelope, environmental air passes through the vortex and around the core. A system of spiral bands, termed the stationary band complex (SBC), forms near the boundary between the core and the envelope where the Rossby number is of order unity. The SBC differs dynamically both from convective rings because it is asymmetric and from propagating gravity-wave bands because its Doppler-shifted frequency is below the local inertia frequency. In more intense systems with stronger convective instability, the SBC may evolve into a convective ring and move into the vortex core. Outward propagating gravity-wave bands have also been observed. Such bands are often associated with track oscillations as the storm makes landfall or recurves.

Spiral-shaped entities within the SBC tend to lie across the streamlines when the convective instability is small and along them when it is large. Storms moving through an environmental flow with westerly vertical shear exhibit an east-to-west drift across the vortex. This phenomenon is expressed in the asymmetric streamfunction as an anticyclonic eddy northeast of the center and a cyclonic eddy south of the center. The velocity potential has a divergent cell west of the center and convergent cell that extends along the inside of the SBC east of the center. This pattern is apparently forced by potential vorticity conservation along the trajectories of the rotational flow and by heating in the SBC. The irrotational flow between the two cells substantially cancels the rotational drift within the vortex core.

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