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H. E. Willoughby

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H. E. Willoughby

Abstract

This study is an attempt to model spiral rainbands in hurricanes as inertia-buoyancy waves. The model is based on the nonhydrostatic equations that describe, in cylindrical coordinates, linear perturbations on a barotropic vortex imbedded in a uniformly stratified atmosphere. Solutions are obtained by the assumption of imaginary exponential variation in all coordinates except radius algebraic elimination of all the dependent variables except geopotential, and numerical integration to obtain the radial structure. This system supports waves whose frequencies are confined to a passband between the local inertia frequency and the buoyancy frequency and which obtain energy at the expense of the mean flow's kinetic energy by the mechanism proposed by Kurihara (1976). All waves subject to this instability propagate wave energy from low values of their Doppler-shifted frequency toward high values and sustain an eddy flux of angular momentum out of the vortex center. Although the instability may double or triple the wave energy flux, it is not strong enough to explain the formation of outward propagating waves because of geometric spreading of the wave energy. In the eye wall the local inertia frequency is less than an hour, so the minimum possible frequency for the simulated waves excited there is higher than that observed for spiral rainbands in nature.

The differential equation governing the wave's radial structure becomes singular when the frequency is Doppler-shifted to the buoyancy frequency, resulting in the absorption of the wave by a process analogous to that occurring at critical levels for vertically propagating buoyancy waves.

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H. E. Willoughby

Abstract

Nonlinear motions of a shallow water barotropic vortex on a β plane differ substantially from the analogous linear motions. The nonlinear model described here, in which wavenumber 1–3 asymmetries interact with each other and the mean vortex, predicts that an initially completely cyclonic vortex will accelerate toward the NNW, reaching a speed of 2.5 m s−1 at 48 h. During the rest of the 240-h calculation, the speed varies by <0.5 m s−1 as the direction turns from NNW to NW. The vortex accelerations are in phase with temporal changes of vortex-relative angular momentum (LR). The turning of the track coincides with a transition of the wavenumber 1 asymmetry from a single dipole to a double dipole. The latter structure appears to be another orthogonal solution of the second-order radial structure equation for a neutral linear normal mode. The corresponding linear model, in which β forces only wavenumber 1, shows only the single dipole structure and straight NNW accelerating motion that reaches a speed of 9 m s−1 at 240 h. The slower motion in the nonlinear model stems from wave-induced changes in the axisymmetric vortex and vacillation between the orthogonal modal structures.

A nonlinear calculation with zero initial LR on a β plane follows a curving path dictated by a barotropically unstable linear mode for the first 144 h. Subsequently, the double dipole structure for that mode appears as the track turns toward the NW and the speed accelerates from 1 to 2 m s−1. A spatially uniform geostrophic environmental current on an f plane causes vortex motion by advection and by propagation. The potential vorticity (PV) gradient due to the current acts much as β does. Although the PV gradient is typically 0.1 of that due to β, the induced propagation toward high potential vorticity is ½–¼ of that on a β plane because super-position of the vortex on the geopotential gradient amplifies the PV gradient's effect. In a quiescent environment on an f plane, initial asymmetries that project onto the normal modes induce long-lasting motion that retains about half its speed to 240 h. If the initial speed is ≥2 m s−1, vacillation between orthogonal modal structures may cause dramatic turns and accelerations of the vortex track.

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H. E. Willoughby

Abstract

In hurricanes, linear, stationary, asymmetric, inertia-buoyancy oscillations can be resonantly forced at a radius where their tangential wavenumber times the orbital frequency of the mean swirling flow is equal to the local inertia frequency. If the hurricane is advected by an environmental geostrophic steering current without shear, the Coriolis force arising from the motion balances the environmental pressure gradient. However, if the motion of the storm differs from the geostrophic wind or if that wind has a horizontal shear, this balance is disrupted. For a uniform shear, resolution of the imbalance into radial and tangential components leads to a forcing that has a symmetric component and asymmetric components with tangential wavenumbers ±1 and ±2. The symmetric and wavenumber ±1 components have exponential horizontal structure, but the ±2 components have sinusoidal structure and are amplified by the wave action conservation mechanism of Willoughby (1978a,b). These waves resemble the spiral bands observed by research aircraft in Hurricane Anita of 1977.

As is generally the case for resonant forcing, the amplitude of the oscillations is sensitive to the dissipation rate, but for values of this quantity appropriate to cumulus friction and an environmental shear equal to 10% of the Coriolis parameter, the maximum horizontal velocity amplitude in the eye wall is several meters per second.

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H. E. Willoughby

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A barotropic model of tropical cyclone motion follows from calculation of linear wavenumber-1 perturbations on a moving axisymmetric, maintained vortex. The perturbations are Rossby waves that depend upon the radial gradient of axisymmetric relative vorticity. The vortex has normal modes at zero frequency and at the most anticyclonic orbital frequency; the latter mode is barotropically unstable. The structure of the perturbations is calculable for arbitrary motion of the vortex, but one can select the actual motion in a particular situation because that motion minimizes the Lagrangian of the system.

Motion of tropical cyclones may arise from environmental currents, convection, or the beta effect. In an environmental current that turns as time passes, the motion is nearly the same as the current, except when the frequency matches a normal mode. The effect of convection is simulated by an imposed, rotating mass source-sink pair, which excites both the normal modes and a perturbation that depends upon forcing at Rossby-wave critical radii. The latter response seems to correspond to the trochoidal motion of real tropical cyclones. It has the fastest vortex motion when its frequency is the same as the orbital frequency of the axisymmetric flow where the forcing is imposed. On a beta plane, the vortex motion is poleward with speed proportional to the total relative angular momentum of the vortex. Because of the normal mode at zero frequency, the poleward motion is much too fast when the vortex has cyclonic circulation throughout. This physically unreasonable result highlights the importance of nonlinear processes in tropical cyclone motion.

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H. E. Willoughby

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The model of hurricane rainbands developed by Willoughby (1977a, 1978) is here extended to simulate linear waves on a baroclinic mean vortex. Although the energetics are more complicated than in the case of barotropic mean flow, these results support the plausibility of Willoughby's model of rainbands as inward-propagating inertia-buoyancy waves. These waves are excited with small amplitude at the storm's periphery and amplify in consequence of the Eliassen-Palm theorem as they are Doppler-shifted to higher frequency during their propagation toward the storm's center. In a baroclinic mean flow, as in a barotropic one, the energy for the amplification comes primarily from the kinetic energy of the mean flow. The primary process which effects this energy exchange is associated with a horizontal export of angular momentum from the vortex center. A baroclinic mean vortex also supplies a lesser amount of energy to the perturbations at the expense of the mean available potential energy. In the upper troposphere, inward-propagating waves are absorbed as a quasi-horizontal critical layer; this leads to some loss of perturbation energy to the mean flow associated with the vertical momentum flux. The waves also experience critical radius absorption in the eye-wall region.

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H. E. Willoughby

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More than 900 radial profiles of in situ aircraft observations collected in 19 Atlantic hurricanes and tropical storms over 13 years confirm that the usual mechanism of tropical cyclone intensification involves contracting maxima of the axisymmetric swirling wind. Radar shows that annuli of convective echoes accompany the wind maxima. These features, called convective rings exist and move inward because latent heat released in the rings leads to descent, adiabatic warming, and rapid isobaric height falls in the area they enclose. The radial change in rate of isobaric height fall is concentrated at the inner edge of the wind maximum, causing the gradient wind to increase there and the maximum to contract. Vigorous convection organized in rings invariably causes well defined, inward moving wind maxima, but when convection is weak, the rings are also weak or even absent. In this case, the swirling wind may be nearly constant with radius and change slowly in time.

Hurricanes that have a single, vigorous, axisymmetric convective ring strengthen rapidly. Although a series of minor convective rings may support steady strengthening, development is more generally episodic. When asymmetric convection erupts near the center of tropical storms or weak hurricanes, it may cause intensification to falter and the cyclone tracks to become irregular. In intense hurricanes, outer convective rings may form around the preexistent eyewalls, contract, and strangle the original eyewalls, halting intensification or causing weakening.

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H. E. Willoughby

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This paper revisits calculation of motion for a shallow-water barotropic vortex with fixed mean axisymmetric structure. The algorithm marches the linear primitive equations for the wavenumber 1 asymmetry forward intime using a vortex motion extrapolated from previous calculations. Periodically, it examines the calculated asymmetry for the apparent asymmetry due to mispositioning of the vortex center, repositions the vortex to remove the apparent asymmetry, and passes the corrected vortex motion on to the next cycle.

This approach differs from the author's earlier variational determination of the steady-state motion after initial transients had died away. The steady-state approach demonstrated that the vortex had normal modes at zero frequency and, when an annulus of weak anticyclonic flow encircled the cyclonic inner vortex, at the most anticyclonic rotation frequency of the mean flow. Forcing of the former model led to too rapid steady-state poleward motion on a beta plane.

At least for the linear problem, the key to more realistic simulation of motion and structure is the normal modes' transient response to diverse forcing: environmental potential vorticity gradients, embedded sources and sinks of mass, and initial asymmetries. The beta effect and other environmental potential vorticity gradients excite the normal modes to induce an acceleration of the vortex center toward and to the left of the direction to maximum environmental vorticity. Times ~ 100 days would be required to reach the too fast motions predicted in the earlier work. A rotating mass source-sink pair drives the vortex along a cycloidal track, but does not force the normal modes. A nonrotating source-sink forces a motion from the source toward the sinkand excites the normal modes, leading to motion that persists after the forcing has ceased. Similarly, initial asymmetries that project onto the normal modes maintain themselves for times ≥ 10 days, leading to persistent vortex propagation that evolves as the complex normal-mode frequencies dictate. Understanding of these normal modes can contribute to better tropical cyclone motion forecasts through better initialization of numerical track prediction models.

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H. E. Willoughby

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Analysis of a large inventory of in situ observations from research aircraft shows that the gradient wind approximates the axisymmetric swirling flow in the free atmosphere within 150 km of the centers of Atlantic hurricanes and tropical storms. In the middle and lower troposphere, the rms difference between the azimuthal cream swirling and gradient winds is typically &lt 1.5 m s−1 with zero bias. This balance prevails only for the azimuthal mean, not locally, nor is balance to be expected in either the surface friction layer or the upper tropospheric outflow layer where the radial flow is comparable with the swirling flow.

It is theoretically possible that axisymmetric supergradient flow may occur in response to rapid radial acceleration where the radial flow slows in the friction layer beneath the eyewall or where it converges into intense diabatically forced updrafts. Nevertheless, the observations in the free lower and midtroposphere show that systematic departures of the azimuthal mean vortex from balance are too small to measure.

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H. E. Willoughby

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In gradient-balanced, cyclonic flow around low pressure systems, a golden radius exists where RG, the gradient-wind Rossby number, is ϕ −1 = 0.618 034, the inverse golden ratio. There, the geostrophic, cyclostrophic, and inertia-circle approximations to the wind all produce equal magnitudes. The ratio of the gradient wind to any of these approximations is ϕ −1. In anomalous (anticyclonic) flow around a low, the golden radius falls where RG = −ϕ = −1.618 034, and the magnitude of the ratio of the anomalous wind to any of the two-term approximations is ϕ. In normal flow, the golden radius marks the transition between more-nearly cyclostrophic and more-nearly geostrophic regimes. In anomalous flow, it marks the transition between more-nearly cyclostrophic (anticyclonic) and inertia-circle regimes. Over a large neighborhood surrounding the golden radius, averages of the geostrophic and cyclostrophic winds weighted as ϕ −2 and ϕ −3 are good approximations to the gradient wind. In high pressure systems Rg, the geostrophic Rossby number, must be in the range 0 > Rg ≥ −¼, and the pressure gradient cannot produce inward centripetal accelerations. An analogous radius where Rg = −ϕ −3 plays a role somewhat like that of the golden radius, but it is much less interesting.

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