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Basic-state experiment I: (a) tangential wind (m s 21 ), (b) vorticity (s 21 ), (c) angular velocity (s 21 ), and (d) radial gradient of vorticity (km s 21 ).
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Despite the fact that asymmetries in hurricanes, such as spiral rainbands, polygonal eyewalls, and mesovortices, have long been observed in radar and satellite imagery, many aspects of their origin, space–time structure, and dynamics still remain unsolved, particularly their role on the vortex intensification. The underlying inner-core dynamics nee...
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... first simulation in experiment I is initialized with the basic-state tangential wind y 0 (r), angular velocity V 0 (r), vorticity j 0 (r), and its radial gradient g 0 (r) de- picted in Fig. 1. In particular, the equilibrium vorticity profile is given ...
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... z 0 5 0.0009 s 21 , r denotes radius or distance in kilometers from the central axis, and r 0 5 35 km. This particular equilibrium profile describes a weak storm with a radius of maximum wind (RMW) located at about 70 km and a maximum tangential wind of about 20 m s 21 (Fig. 1a). Because the mean vorticity (Fig. 1b) is a mono- tonic function of radius, the vortex satisfies Rayleigh's sufficient condition for linear stability (Gent and McWilliams 1986). The angular velocity (Fig. 1c) decreases monotonically with radius, and g 0 (r) is negative over the entire domain with two local minima, one situated at about ...
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... z 0 5 0.0009 s 21 , r denotes radius or distance in kilometers from the central axis, and r 0 5 35 km. This particular equilibrium profile describes a weak storm with a radius of maximum wind (RMW) located at about 70 km and a maximum tangential wind of about 20 m s 21 (Fig. 1a). Because the mean vorticity (Fig. 1b) is a mono- tonic function of radius, the vortex satisfies Rayleigh's sufficient condition for linear stability (Gent and McWilliams 1986). The angular velocity (Fig. 1c) decreases monotonically with radius, and g 0 (r) is negative over the entire domain with two local minima, one situated at about 21 km and the other at about 75 km ...
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... profile describes a weak storm with a radius of maximum wind (RMW) located at about 70 km and a maximum tangential wind of about 20 m s 21 (Fig. 1a). Because the mean vorticity (Fig. 1b) is a mono- tonic function of radius, the vortex satisfies Rayleigh's sufficient condition for linear stability (Gent and McWilliams 1986). The angular velocity (Fig. 1c) decreases monotonically with radius, and g 0 (r) is negative over the entire domain with two local minima, one situated at about 21 km and the other at about 75 km (Fig. 1d). The equilibrium profile (2) is perturbed according to Schecter and Montgomery (2006) to generate an ini- tially (t 5 0) elliptical vortex with a total vorticity ...
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... (Fig. 1b) is a mono- tonic function of radius, the vortex satisfies Rayleigh's sufficient condition for linear stability (Gent and McWilliams 1986). The angular velocity (Fig. 1c) decreases monotonically with radius, and g 0 (r) is negative over the entire domain with two local minima, one situated at about 21 km and the other at about 75 km (Fig. 1d). The equilibrium profile (2) is perturbed according to Schecter and Montgomery (2006) to generate an ini- tially (t 5 0) elliptical vortex with a total vorticity field j(r, l, 0) given ...
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... term of (2) is used in the basic state, the decay process occurs via sheared VRWs. The first term in (2) describes a Gaussian vortex basic state; when the second term in (2) is included, however, the profile of vorticity is slightly flattened around 48 km, creating the two local extrema on the radial gradient of vorticity profile observed in Fig. 1d. The profile in Fig. 1b is commonly found in hurricanes ( Mallen et al. 2005); furthermore, as will be shown next, it supports asymmetric disturbances that have a large impact on the parent ...
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... in the basic state, the decay process occurs via sheared VRWs. The first term in (2) describes a Gaussian vortex basic state; when the second term in (2) is included, however, the profile of vorticity is slightly flattened around 48 km, creating the two local extrema on the radial gradient of vorticity profile observed in Fig. 1d. The profile in Fig. 1b is commonly found in hurricanes ( Mallen et al. 2005); furthermore, as will be shown next, it supports asymmetric disturbances that have a large impact on the parent ...
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... theoretical periods of these modes are 6.2 h for mode 1 and 6.3 h for mode 2. The period predicted by the Landau pole is about 5.8 h (v R q ' 0.0003 s 21 ), which is in good agree- ment with the periods from the numerical experiment. Figure 10 shows the vorticity ENM spatial patterns corresponding to the first two modes of wavenumber-2 anomalies. The cosine and sine (real and imaginary) components for mode 1 and mode 2 in (13) are shown in Figs. ...
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... by the Landau pole is about 5.8 h (v R q ' 0.0003 s 21 ), which is in good agree- ment with the periods from the numerical experiment. Figure 10 shows the vorticity ENM spatial patterns corresponding to the first two modes of wavenumber-2 anomalies. The cosine and sine (real and imaginary) components for mode 1 and mode 2 in (13) are shown in Figs. 10a,b and Figs. 10c,d, respectively. The ENM spatial patterns are smooth functions in the bulk of the domain and they have two local extrema, one at about 23 km and the other at about 73 km. Note that the ENM spatial patterns satisfy the relation j (2) n2 (r) } rg 0 (r) for n 5 1, 2 (Figs. 10b,d), which usually is a good approxi- mation for the spatial ...
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... components for mode 1 and mode 2 in (13) are shown in Figs. 10a,b and Figs. 10c,d, respectively. The ENM spatial patterns are smooth functions in the bulk of the domain and they have two local extrema, one at about 23 km and the other at about 73 km. Note that the ENM spatial patterns satisfy the relation j (2) n2 (r) } rg 0 (r) for n 5 1, 2 (Figs. 10b,d), which usually is a good approxi- mation for the spatial patterns of a wavenumber-2 quasi mode ( Schecter et al. 2000). Note also that the spatial structures of the leading ENMs (Fig. 10a) resemble the eigenfunction of the discrete-like mode (n 5 141) de- picted in Fig. 6b (bottom). The cross correlation be- tween the pairs of diagonal ...
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... local extrema, one at about 23 km and the other at about 73 km. Note that the ENM spatial patterns satisfy the relation j (2) n2 (r) } rg 0 (r) for n 5 1, 2 (Figs. 10b,d), which usually is a good approxi- mation for the spatial patterns of a wavenumber-2 quasi mode ( Schecter et al. 2000). Note also that the spatial structures of the leading ENMs (Fig. 10a) resemble the eigenfunction of the discrete-like mode (n 5 141) de- picted in Fig. 6b (bottom). The cross correlation be- tween the pairs of diagonal patterns in Figs. 8a,d is 299.75% and between Figs. 8b,c is 99.9%. This large cross similarity between the spatial patterns together with the results from the wave activity spectra and the ...
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... we investigate the effects of radially propagating VRWs on the mean vortex using the small-amplitude approach of the EP flux theory. Figures 11a,b show the contribution of the wavenumber-2 mode 112 anomalies to the numerator of the rhs of (18) Fig. 11a that a dipole pattern exists in the EP flux divergence map. The general picture is maximum acceleration occurring slightly outside the RMW (r 5 70 km) at 80 km, and maximum deceleration occurring farther outside at 130 km. ...
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... we investigate the effects of radially propagating VRWs on the mean vortex using the small-amplitude approach of the EP flux theory. Figures 11a,b show the contribution of the wavenumber-2 mode 112 anomalies to the numerator of the rhs of (18) Fig. 11a that a dipole pattern exists in the EP flux divergence map. The general picture is maximum acceleration occurring slightly outside the RMW (r 5 70 km) at 80 km, and maximum deceleration occurring farther outside at 130 km. Using (18), we con- clude that the total effect on the mean tangential wind is net spinup slightly outside the ...
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... the EP fluxes, Fig. 11b shows that fluxes are positive throughout the domain and a maximum is lo- cated slightly outside the critical radius. This indicates that the inward flux of cyclonic eddy angular momentum starts to decrease after reaching the critical ...
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... inspection of the wave activity spectra (Fig. 12) indicates that wave modes populate both regions of the spectra, implying that both prograde and retrograde VRWs are of importance. The change in sign in the pseudomomentum spectrum suggests that barotropi- cally unstable modes may be excited (see Held ...
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... 13a,b depict the time series for the first pair of ENMs of wavenumber-4 anomalies. Figure 14 is as in Fig. 13, but for the last pair of ENMs (modes 144 and 145). The power spectra for modes 1 and 2 (Fig. 13c) have their maxima at 0.55 and 0.50 h, respectively, and for the modes 144 and 145 (Fig. 14c) at 0.53 h. ...
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... 13a,b depict the time series for the first pair of ENMs of wavenumber-4 anomalies. Figure 14 is as in Fig. 13, but for the last pair of ENMs (modes 144 and 145). The power spectra for modes 1 and 2 (Fig. 13c) have their maxima at 0.55 and 0.50 h, respectively, and for the modes 144 and 145 (Fig. 14c) at 0.53 h. The correlation between the two pairs of diagonal panels of vorticity ENM space patterns of modes 1 and 2 in Figs. 15a,d and Figs. ...
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... 13a,b depict the time series for the first pair of ENMs of wavenumber-4 anomalies. Figure 14 is as in Fig. 13, but for the last pair of ENMs (modes 144 and 145). The power spectra for modes 1 and 2 (Fig. 13c) have their maxima at 0.55 and 0.50 h, respectively, and for the modes 144 and 145 (Fig. 14c) at 0.53 h. The correlation between the two pairs of diagonal panels of vorticity ENM space patterns of modes 1 and 2 in Figs. 15a,d and Figs. 15b,c are 99.55% and 299.60%, respectively; be- tween the two pairs of diagonal panels of vorticity, ...
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... 13a,b depict the time series for the first pair of ENMs of wavenumber-4 anomalies. Figure 14 is as in Fig. 13, but for the last pair of ENMs (modes 144 and 145). The power spectra for modes 1 and 2 (Fig. 13c) have their maxima at 0.55 and 0.50 h, respectively, and for the modes 144 and 145 (Fig. 14c) at 0.53 h. The correlation between the two pairs of diagonal panels of vorticity ENM space patterns of modes 1 and 2 in Figs. 15a,d and Figs. 15b,c are 99.55% and 299.60%, respectively; be- tween the two pairs of diagonal panels of vorticity, ENM space patterns of modes 144 and 145 in Figs. 16a,d and Figs. 16b,c are 99.64% and ...
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... Figure 14 is as in Fig. 13, but for the last pair of ENMs (modes 144 and 145). The power spectra for modes 1 and 2 (Fig. 13c) have their maxima at 0.55 and 0.50 h, respectively, and for the modes 144 and 145 (Fig. 14c) at 0.53 h. The correlation between the two pairs of diagonal panels of vorticity ENM space patterns of modes 1 and 2 in Figs. 15a,d and Figs. 15b,c are 99.55% and 299.60%, respectively; be- tween the two pairs of diagonal panels of vorticity, ENM space patterns of modes 144 and 145 in Figs. 16a,d and Figs. 16b,c are 99.64% and 299.52%, respectively. The excellent match between the observed periods of the first and last pair of modes together with the large values ...
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... 0.50 h, respectively, and for the modes 144 and 145 (Fig. 14c) at 0.53 h. The correlation between the two pairs of diagonal panels of vorticity ENM space patterns of modes 1 and 2 in Figs. 15a,d and Figs. 15b,c are 99.55% and 299.60%, respectively; be- tween the two pairs of diagonal panels of vorticity, ENM space patterns of modes 144 and 145 in Figs. 16a,d and Figs. 16b,c are 99.64% and 299.52%, respectively. The excellent match between the observed periods of the first and last pair of modes together with the large values of cross correlations among the complex spatial patterns suggests a phase locking between counterpropagating VRWs formed by the modes on the extrema of the wave ...
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... phase locking could even- tually result in barotropic instability. Figure 17 depicts the contribution from modes 1, 2, 144, and 145 to the EP flux divergence (Fig. 17a) and EP flux (Fig. 17b) of the wavenumber-4 anomalies. Similar to experiment I, a dipole structure is observed in the EP flux divergence map, but in this case the whole pattern is shifted toward the vortex center with maximum ac- celeration located inside the RMW (r 5 60 km) and maximum deceleration at/outside that radius. ...
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... values of cross correlations among the complex spatial patterns suggests a phase locking between counterpropagating VRWs formed by the modes on the extrema of the wave activity spectra. This phase locking could even- tually result in barotropic instability. Figure 17 depicts the contribution from modes 1, 2, 144, and 145 to the EP flux divergence (Fig. 17a) and EP flux (Fig. 17b) of the wavenumber-4 anomalies. Similar to experiment I, a dipole structure is observed in the EP flux divergence map, but in this case the whole pattern is shifted toward the vortex center with maximum ac- celeration located inside the RMW (r 5 60 km) and maximum deceleration at/outside that radius. The result ...
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... among the complex spatial patterns suggests a phase locking between counterpropagating VRWs formed by the modes on the extrema of the wave activity spectra. This phase locking could even- tually result in barotropic instability. Figure 17 depicts the contribution from modes 1, 2, 144, and 145 to the EP flux divergence (Fig. 17a) and EP flux (Fig. 17b) of the wavenumber-4 anomalies. Similar to experiment I, a dipole structure is observed in the EP flux divergence map, but in this case the whole pattern is shifted toward the vortex center with maximum ac- celeration located inside the RMW (r 5 60 km) and maximum deceleration at/outside that radius. The result is a ring that contracts. ...
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... tant in our datasets. The time series of the wavenumber-4 leading (prograde and retrograde) modes exhibits an exponential growing behavior during the first few hours of the experiment. These modes form a discrete spec- trum of unstable VRWs that counterpropagate and phase lock as reflected from the match in frequencies between FIG. 17. As in Fig. 11, but for experiment II for the sum of contributions from modes 1, 2, 144, and 145 of wavenumber-4 disturbances. TABLE 1. Summary of the ENM diagnostic results for the two experiments. Table shows mode number, wavenumber, variance explained var (%), theoretical periods Tth (h), observed periods To (h), and the correlation among the ...
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An analysis of a high-resolution dataset from a realistic simulation of Hurricane Wilma (2005) was performed to understand the mechanism for the formation of a prominent polygonal eyewall and mesovortices during the rapid intensifying stage of the hurricane. The impact of these asymmetries on the intensity change of the hurricane vortex was assessed using the empirical normal mode (ENM) method and Eliassen–Palm (EP) flux calculations.
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The present results complement previous findings of theoretical and highly idealized numerical studies that a polygonal eyewall and mesovortices are the result of barotropic instability, thereby furnishing a bridge between idealized studies and observations. The work also provides new insight on the role of asymmetries and VRWs in the intensification of hurricanes.
... The emerging paradigm of tropical cyclone spin-up as articulated above recognizes the intrinsic fluid dynamics and thermodynamics of rotating deep convection as well as its collective effects in producing system-scale inflow. It recognizes also the potential role of vortex Rossby waves (VRWs 2 ) and their wave-mean and wave-wave interaction as well as their coupling to the boundary layer and convection (Chen and Yau, 2001;Wang, 2002a, b;Chen et al., 2003;Martinez, 2008;Martinez et al., 2010Martinez et al., , 2011. In other words, the tropical cyclone intensification process generally comprises a turbulent system of rotating, deep moist convection and vortex Rossby waves. ...
We present the results of idealized numerical experiments to examine the
difference between tropical cyclone evolution in three-dimensional (3-D)
and axisymmetric (AX) model configurations. We focus on the prototype
problem for intensification, which considers the evolution of an
initially unsaturated AX vortex in gradient-wind balance on an f-plane.
Consistent with findings of previous work, the mature intensity in the
3-D model is reduced relative to that in the AX model. In contrast with
previous interpretations invoking barotropic instability and related
horizontal mixing processes as a mechanism detrimental to the spin-up
process, the results indicate that 3-D eddy processes associated with
vortical plume structures can assist the intensification process by
contributing to a radial contraction of the maximum tangential velocity
and to a vertical extension of tangential winds through the depth of the
troposphere. These plumes contribute significantly also to the
azimuthally-averaged heating rate and the corresponding azimuthal-mean
overturning circulation. The comparisons show that the
resolved 3-D eddy momentum fluxes above the boundary layer exhibit
counter-gradient characteristics and are generally not represented
properly by the subgrid-scale parameterizations in the AX configuration.
The resolved eddy fluxes act to support the contraction and
intensification of the maximum tangential winds. The comparisons
indicate fundamental differences between convective organization in the
3-D and AX configurations for meteorologically relevant forecast time
scales. While the radial and vertical gradients of the system-scale
angular rotation provide a hostile environment for deep convection in
the 3-D model, with a corresponding tendency to strain the convective
elements in the tangential direction, deep convection in the AX model
does not suffer this tendency. Also, since during the 3-D
intensification process the convection has not yet organized into
annular rings, the azimuthally-averaged heating rate and radial gradient
thereof is considerably less than that in the AX model. This lack of
organization results broadly in a slower intensification rate in the 3-D
model and leads ultimately to a weaker mature vortex after 12 days of
model integration. While axisymmetric heating rates in the 3-D model are
weaker than those in the AX model, local heating rates in the 3-D model
exceed those in the AX model and at times the vortex in the 3-D model
intensifies more rapidly than AX. Analyses of the 3-D model output do
not support a recent hypothesis concerning the key role of small-scale
vertical mixing processes in the upper-tropospheric outflow in
controlling the intensification process. In the 3-D model,
surface drag plays a particularly important role in the intensification
process for the prototype intensification problem on meteorologically
relevant time scales by helping foster the organization of convection in
azimuth. There is a radical difference in the behaviour of the 3-D and
AX simulations when the surface drag is reduced or increased from
realistic values. Borrowing from ideas developed in a recent paper, we
give a partial explanation for this difference in behaviour. Our results provide new qualitative and quantitative insight into the
differences between the asymmetric and symmetric dynamics of tropical
cyclones and would appear to have important consequences for the
formulation of a fluid dynamical theory of tropical cyclone
intensification and mature intensity. In particular, the results point
to some fundamental limitations of strict axisymmetric theory and
modelling for representing the azimuthally-averaged behaviour of
tropical cyclones in three dimensions.
Observational analyses of hurricanes in the tropical atmosphere indicate the existence of spiral rainbands which propagate outwards from the eye and affect the structure and intensity of the hurricane. These disturbances may be described as vortex Rossby waves. It has been suggested that vortex Rossby waves may play a role in the eyewall replacement cycle observed in tropical cyclones in which concentric rings of high-intensity wind develop and propagate in towards the centre of the cyclone. In previous work with Nikitina, we investigated the dynamics of vortex Rossby waves in a cyclonic vortex in a two-dimensional configuration on a beta-plane and derived analytical solutions. In the current work, some results of numerical simulations are presented.
Spiral cloud bands dominate tropical cyclones’ appearance in satellite and radar images. It is generally accepted that at least some of them are vortex Rossby waves that propagate on the radial gradient of mean-flow-relative vorticity. This study models these features in Fourier and time domains as linear, barotropic, nondivergent waves on a maintained mean vortex scaled to resemble tropical cyclones. This formulation is the simplest one imaginable that encompasses the essential rotational dynamics.
The modeled waves are episodically forced by a rotating annular train of sinusoidal vorticity sources and sinks that crudely represents eyewall convection. Substantial quiescent time intervals separate forced intervals. The waves propagate wave energy predominantly outward and converge angular momentum inward. Waves’ energy is absorbed as their perturbation vorticity becomes filamented near the outer critical radii where their Doppler-shifted frequencies and radial group velocities approach zero. The waves can propagate spatially only in narrow annular waveguides because of their slow tangential phase velocity and the restricted Rossby wave frequency domain. Although radial shear of the mean flow distorts their velocity field into tightly wound spirals, their streamfunction and geopotential fields assume the form of elliptical gyres or broad trailing spirals that do not resemble observed hurricane rainbands.
