Generation of Moving Spiral Bands in Tropical Cyclones

K. C. Chow Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

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Kwing L. Chan Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

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Alexis K. H. Lau Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

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Abstract

A theory based on the shallow water equation is developed to explain the generation of moving spiral-banded structures in tropical cyclones. The theory proposes that fluctuation of the vorticity distribution in the compact core region can act as a source, generating gravity waves that produce the banded structures. This mechanism is usually called the Lighthill radiation mechanism. A numerical experiment using the Fifth Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) code is used to test this theory. The moving bands in the simulated tropical cyclone show good agreement with the theoretically calculated gravity waves that emerge from a core region with an elliptical distribution of vorticity. Results of the simulated cyclone also illustrate that spiral cloud bands are triggered by the propagating gravity waves.

Corresponding author address: K. C. Chow, Dept. of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. Email: mackc@ust.hk

Abstract

A theory based on the shallow water equation is developed to explain the generation of moving spiral-banded structures in tropical cyclones. The theory proposes that fluctuation of the vorticity distribution in the compact core region can act as a source, generating gravity waves that produce the banded structures. This mechanism is usually called the Lighthill radiation mechanism. A numerical experiment using the Fifth Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) code is used to test this theory. The moving bands in the simulated tropical cyclone show good agreement with the theoretically calculated gravity waves that emerge from a core region with an elliptical distribution of vorticity. Results of the simulated cyclone also illustrate that spiral cloud bands are triggered by the propagating gravity waves.

Corresponding author address: K. C. Chow, Dept. of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. Email: mackc@ust.hk

1. Introduction

The spiral-banded structure is a prominent feature of tropical cyclones. Apart from its distinctive appearance, the spiral pattern provides information on the cyclone intensity. Dvorak (1984) has shown that the cloud pattern of a tropical cyclone (from satellite pictures) can be used as an indicator of the cyclone intensity. This technique is now widely used in many meteorological centers, and considered a basic tool for cyclone intensity analysis. The relationship between the spiral pattern and the intensity suggests a close dynamical link. To investigate this linkage, one important step is to find out how the spiral bands are generated and maintained. The objective of the present paper is to explore such a mechanism.

Spiral bands are commonly classified in two different ways. One classification is based on the motion. There are stationary bands and moving bands (Willoughby et al. 1984). Stationary bands normally locate at fixed positions relative to the cyclone center, for an order of one day. Moving bands rotate around the center with a speed close to the maximum wind speed of the cyclone. The second classification is based on location and dimension; there are inner and outer bands (Guinn and Schubert 1993). Inner bands are located less than 500 km from the vortex center; they are usually not visible on satellite pictures because of cirrus overcast, but are evident on radar images. Typical spiral bands are 20–40 km in width beyond about 40 km from the center (Atkinson 1981). However, there are some small-scale spiral bands, 5–10 km in width and within 100 km from the center, as identified by Gall et al. (1998) and Atkinson (1981). The focus of the present investigation is on the large-scale moving bands. Depending on the means of obtaining the image—such as radar or satellite pictures with visible, infrared, and microwave sensors—spiral bands may have different manifestations. In the present study, we refer to spiral bands as the banded structure of strong convection that can be observed from the images of radar or microwave sensors, and the regions are normally associated with strong vertical motions.

Current theories for the spiral bands can be roughly divided into two groups. In the first group, spiral bands are considered to be manifestation of inertia–gravity waves (Diercks and Anthes 1976; Kurihara 1976; Willoughby 1978). In the second group, spiral bands are generated by vortex Rossby waves (Guinn and Schubert 1993; Montgomery and Kallenbach 1997; Chen and Yau 2001). In the work of Guinn and Schubert, the inner bands were explained as vortex Rossby wave breaking, and the outer bands as results of the nonlinear breakdown of the intertropical convergence zone through barotropic instability. Such interpretation for the outer bands, however, cannot explain the observed moving spirals, which are moving with speeds larger than the speed of the local mean wind (Willoughby et al. 1984). Willoughby et al. had proposed that the moving spiral bands are inertia–gravity waves that could be formed by oscillatory motions of the vortex center. Stationary bands, on the other hand, were hypothesized to be formed by the motion of the vortex through an environmental flow with westerly vertical shear. An alternative theory for the stationary bands has recently been proposed by Weber (1999). In a nondivergent barotropic model, Weber found that motion of a symmetric vortex in beta plane could induce slowly evolving Rossby waves, and form cyclonically curved spiral bands of vorticity with positive and negative anomalies in the rear-right quadrant of the moving vortex.

In summary, it is fair to say that the origin of the spiral banded structures has not been clearly identified. Spiral bands may be of more than one type, and there may be different generation mechanisms. In this paper, we focus on the large-scale moving spirals that extend far beyond the vortical core region. We adopt the interpretation of Willoughby et al. (1984) for this kind of spiral bands and proceed to work out a connection between the outgoing waves and the central vortex.

The theory of vortex sound by Lighthill (1952) and Howe (1975), showed that time fluctuation of a compact region of vorticity may act as a source of acoustic waves. When a similar principle is applied to the shallow water equations (as shown in section 4), the waves generated by the vorticity fluctuation are gravity waves. The term Lighthill radiation is now usually used to describe this kind of spontaneous generation of waves. Using the shallow water equation in an f plane, Ford (1994a,b) and Ford et al. 2000 have analyzed gravity wave radiation from compact vortical flow regions. They found that the radiation is generally weak (the wave amplitudes are second order in the Froude number and the back reactions are fourth order in the Froude number), and depending on the size of the Coriolis parameter, the rotation can have significant effect on the wave radiation characteristics. Here, we apply this radiation mechanism to tropical cyclones. The propagating waves are in spiral form, and the vorticity distribution can be considered as compact since it concentrates in the core region.

In section 2, a numerical model [Fifth Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5)] of an idealized tropical cyclone is described. Development of the structure of the cyclone and the moving spiral bands are discussed in section 3, so that one can obtain guidance for developing the analytical model (discussed in section 4), and comparisons can be made. The relation between the triggered cloud bands and the propagating gravity waves is discussed in section 5. Relevance to realistic examples is discussed in section 6. A summarizing conclusion is given in section 7.

2. The numerical model

The mesoscale numerical model PSU–NCAR MM5 version 2.9 is adopted as the basic model to simulate a tropical cyclone in an idealized environment. The simulation is idealized in the sense that no real sounding is used. The simulated cyclone evolves from a weak, balanced, axisymmetric vortex to a super typhoon. The model is designed to run over a flat sea surface and in a quiescent environment with no environmental wind. A radiation lateral boundary condition is adopted in the outermost mesh. Only explicit moisture scheme (Goddard graupel) but no parameterization scheme for cumulus is used. Processes at the boundary layer are parametrized by the Bulk Aerodynamic Planetary Boundary Layer Scheme. Also, effects such as atmospheric radiation, topography, variation of sea surface temperature, and latitude dependence of the Coriolis parameter are omitted. Further details on the idealization and modification of the MM5 can be found in Chow (2001).

The current MM5 model uses three nested domains as shown in Table 1. The dimension of the largest domain is chosen to be 3150 km in order to minimize the effects of the lateral boundaries on the results. The finest domain of horizontal spacing 5 km is used to resolve the cloud bands. For all domains, 20 sigma levels are used.

Sea surface temperature at 28°C and Coriolis parameter of 5 × 10−5 s−1 (about 20° latitude) are set constant throughout the experiment. To set up the initial condition of the atmosphere, the background temperature is specified as T = Ts + 50 ln(P/Ps), where Ts = 303 K is the ground temperature, and Ps = 1010 mb is the sea level pressure. The initial background relative humidity (RH) decreases linearly with the geopotential height (H), as RH = RHs − (RHs − RHt) × (H/Ht), where RHs = 85% is the surface RH, and RHt = 10% is the RH at Ht, the geopotential height at P = 100 mb.

With an artificial initial sounding and radiating lateral boundary condition at the outermost mesh, the resulting numerical model retains only essential parameterization of physical processes such as heat and mass transfer at the surface and the process of precipitation. It is anticipated that the numerical model may produce clearer illustration of the key features in cyclone intensification and spiral band formation.

An axisymmetric vortex is used as the initial flow field. The horizontal wind is given by
i1520-0469-59-20-2930-eq1
where r is the distance from the center of the vortex and the vertical profile is
i1520-0469-59-20-2930-eq2
The subscript max denotes quantities associated with the maximum wind. The size of the initial vortex is 1800 km in diameter; that is, V = 0 beyond a radius of 900 km. The maximum velocity Vmax is 11 m s−1 and the radius at maximum wind rmax is 99 km. The flow field having been specified, the atmosphere is balanced again by using the thermal wind equation and the hydrostatic equation. The resulting atmosphere is in a state of conditional instability. It is favorable for developing a tropical cyclone.

The center of the vortex is defined here as the location with minimum surface pressure. The axisymmetric and asymmetric values for each variable are calculated based on this center. To study two-dimensional features, depth-averaged values are obtained by averaging the required variables at all sigma levels at each horizontal grid location. From the definition of the sigma levels, the depth averaging process is approximately equal to mass averaging in the vertical direction.

The evolution of the axisymmetric vortex is summarized in the diagrams of Fig. 1. Figures 1a and 1b show that the simulated tropical cyclone approximately attains a steady state after 100 h, with a minimum sea level pressure of about 912 mb and a maximum surface wind of about 72 m s−1. It is consistent with the theoretical maximum intensity given by Holland (1997) for a sea surface temperature of 28°C. Figure 1c shows that the radius of maximum wind is about 30 km at the time of 131 h.

3. Moving spiral structures of the simulated tropical cyclone

As the simulated tropical cyclone intensifies, asymmetries in the flow field start to develop. Before about 50 h, convection in the central region is not well organized, several spots of convection with radius about 10 km appear, unevenly distributed around the central region. Outside the central region, no band structures are recognizable. After 50 h, the spots of convection disappear and are replaced by one or two larger convection regions. Spiral bands with positive vertical velocity start to form but are transient in time. The bands usually have a wavenumber-1 or -2 pattern, with wavenumber-1 bands appearing more frequently. However, after about 85 h, only spiral bands with wavenumber 2 can be observed.

The depth-averaged values of the vertical velocity perturbation from 130.5 h (7830 min) to 131.8 h (7908 min) are shown in Fig. 2. It is clear in the left column of Fig. 2 that the pattern resembles a spiral structure with two arms (wavenumber 2) with positive and negative anomalies spiraling together. The spiral structure rotates like a solid body with a period of about 75 min, which is comparable to the period of the mean wind at the radius of maximum wind. As shown in Fig. 1, at 130.5 h the depth-averaged maximum wind at 30-km radius is 53 m s−1, and the corresponding period is about 59.3 min. Beyond a radius of 35 km, the rotational speed of the spiral pattern is faster than that of the depth-averaged mean wind.

From the diagrams in the middle column of Fig. 2, we can see that the distribution of the vertical velocity perturbation in the central region has a quadrupole form. Positive anomalies are separated by negative anomalies, with maximum values located near the major axis of the elliptical vortex patch. Spiral patterns corresponding to positive and negative values of the contours appear to be originated from the corresponding extremities, and rotate together with the elliptical vortex patch.

The distribution of the depth-averaged absolute vorticity in the central region is shown in the right column of Fig. 2. The distribution is compact, and its shape is close to an ellipse. The magnitude of the vorticity decreases outward; in other words, it forms a monopolar vortex. Apparently, the elliptical vortex rotates steadily with the same speed as the spiral structure. The formation of a elliptical region of vorticity in the eye of a tropical cyclone has been discussed by Kossin et al. (2000); the theory is based on the barotropic instability of an axisymmetric vortex.

Although the depth-averaged absolute vorticity is elliptical, the horizontal distribution of the absolute vorticity takes different shapes at different altitudes (as shown in Fig. 3). Elliptical shapes can only be found below 4 km. The color contours of Fig. 3 at 1 and 2 km show two filaments of vorticity emanating from the tips of the ellipse, indicating the presence of axisymmetrization at low altitudes. The magnitude of the absolute vorticity drops in the central region above a height of 7 km, but remains positive up to a height of 13 km.

The spiral structures shown in Fig. 2 are in depth-averaged quantities. To illustrate the propagation of the perturbations at various height levels, azimuth–height profiles of the vertical velocity perturbation at different radii and time instances are shown in Fig. 4, while Fig. 5 shows the corresponding distributions for the depth-averaged values. Both figures show wavenumber-2 features in the azimuthal direction. Although the contours in Fig. 4 show the shapes of the patterns at the azimuthal–height cross section are not uniform or slightly distorted during their azimuthal propagations, their depth-averaged values in Fig. 5 show clear evidence of wave propagation through different radii. Notice the temporal change of the perturbation at the same radius; that is also a feature of the spiral wave propagation. Figure 6 shows radius–height profiles of the vertical velocity perturbation for the same time instances as Figs. 4 and 5. Figure 7 shows the corresponding distributions of the depth-averaged values. From the movement of the wave crests, the radial propagation speed can be estimated to be approximately 60 m s−1. In Figs. 6 and 7, examples of strong updraft (at 7830 min) and downdraft (at 7848 min) in the eye region can also be found.

4. Theory

In the present study, the theory of Lighthill radiation for spiral gravity waves is developed based on the two-dimensional shallow water equation. The original theory of aerodynamic sound by Lighthill (1952) was developed using the Euler equation for compressible gas. It is well known that the shallow water equation is equivalent to the two-dimensional Euler equation with the ratio of specific heats of the gas chosen to be 2 and the density replaced by the height. The shallow water model cannot describe the full physical situation of a tropical cyclone since the vertical structure, the diabatic, and frictional exchanges are all omitted. However, the model allows for analytical treatment of some fundamental hydrodynamical processes that are not vitally dependent on the ignored physics. The shallow water equation is widely used in geophysical fluid study; it is the simplest governing equation that admits vortices, gravity waves, and vortex Rossby waves; it also allows for the nonlinear interactions among these modes of motions. In an f plane, the equation takes the form
i1520-0469-59-20-2930-e1
where d/dt = ∂/∂t + (V·∇), V and ∇ are in horizontal dimensions only, k is a unit vector in the vertical direction, and V and h are the velocity and height, respectively.
Through the vector identities (V·∇)V = ζ × V + ∇(V2/2) and V·(fk × V) = 0, the shallow water equations can be written in the form
i1520-0469-59-20-2930-e2
where ϕ = gh + V2/2 is the total energy per unit mass, and ζa = fk + ζ = (f + ζ)k = ζak is the absolute vorticity. The two equations in (2) can be combined to yield the convected wave equation
i1520-0469-59-20-2930-eq3

This equation is very similar to the equation obtained by Howe (1975) using the Euler equation. The main difference is that Howe's equation uses the “stagnation enthalpy” ∫ dp/ρ + ∫ T dS + (V2/2), S being the entropy, instead of ϕ defined above. Also, the term gh is replaced by the square of the sound speed c2, and h is replaced by the density ρ.

The two equations in (2) can also be combined to obtain the following equation:
i1520-0469-59-20-2930-e3
where h = h0 + h′ and h0 is the constant background equivalent depth. Analogous to the argument of Lighthill (1952), in an uniform outer region (outside the wave generation region) with steady and irrotational mean flow, the right-hand side of Eq. (3) is small since it is of second order of velocity perturbations. Therefore, in the outer region, neglecting the terms at the right-hand side, Eq. (3) represents a free wave equation, and the fluctuations in the total energy ϕ within the inner region are propagated as free shallow water waves.

Equation (3) is exactly derived from the shallow water equation; its application to the wave radiation phenomenon requires some simplifications. The first assumption is compactness of the source. That is, the length scale of the eddy region l that produces the waves is small compared to the corresponding wavelength λ. The compactness assumption is basically equivalent to the low Froude number approximation. The Froude number Fr is analogous to the Mach number in gas dynamics and is defined as Fr = υ/gh0, where υ is a velocity scale, in the present analysis chosen to be the magnitude of velocity fluctuation in the eddy region. Therefore, the frequency can be scaled as υ/l, and hence the wavelength is scaled as λc0/(υ/l), where c0 = gh0 is the shallow water wave speed. For compactness, l/λ ≪ 1, which requires υ/c0 = Fr ≪ 1.

Under the assumption that the Froude number and the perturbations are small (i.e., h′ ≪ h0), Eq. (3) can be reduced to a linear wave equation with a source
i1520-0469-59-20-2930-e4
From the identity ∇·(ζa × V) = ∇·(ζ × V) − fζ, we can see that the source term in (4) is compact, because, for the case of tropical cyclone, relative vorticity is generally concentrated in the core region. In this region, the magnitude of the relative vorticity is actually much greater than that of the earth (typically more than 50 times). Therefore, we may neglect the effects of the Coriolis force in the source in Eq. (4). There are two other physical reasons for neglecting the effects of Coriolis force in the case of tropical cyclones. One is that the timescale of the pattern (∼2 h) is much shorter that of the earth (1 day). The other reason is that the domain of interest for the waves is well within the Rossby deformation radius. Putting f = 5 × 10−5 s−1 and c0 = 60 m s−1 in the formula of Rossby deformation radius for the shallow water situation r = c0/(f/2), one gets r ≃ 2400 km. With this further simplification, Eq. (4) becomes
i1520-0469-59-20-2930-e5
For a compact source emitting to free space, the far field solution of this equation can be written as (see appendix A)
i1520-0469-59-20-2930-e6
where ϕ′(x, t) = ϕϕ0, and ϕ0 is the value of ϕ without perturbations, x = (x1, x2) is the position vector of the observing point, and yh = (y1, y2) is the position vector of the source function. The domain of integration for the inner integral in Eq. (6) is over the area of the source, and z is a dummy variable for the improper integral.
It has been shown by Powell (1995) that, for a compact source, the integral involving the term of vortex force can be reduced to an integral with time derivative of the vorticity alone. Then Eq. (6) can be written as,
i1520-0469-59-20-2930-e7

The derivation of this formulation, based on Powell (1995), is included in appendix B. Note that the time derivative in Eq. (7) should be evaluated at the retarded time tx2 + z2/c0. We can see from this formulation that the fluctuation of vorticity in the inner region can generate gravity waves [linear solutions of the homogeneous part of Eq. (5)] that propagate to the far field. Therefore, if the flow field in this compact inner region is known, characteristics of the propagating waves can be calculated. In a real tropical cyclone, vorticity is mainly generated in the core region and usually fluctuates in time. It makes a good candidate to apply Eq. (7).

a. Solution for a rotating elliptical vortex

The far-field solution for acoustic waves generated by a two-dimensional elliptical vortex with constant vorticity has been obtained by Howe (1975), who derived Eq. (5) from Euler's equation. The differences between Eq. (7) and Howe's solution are as follows.

  1. Equation (7) is in terms of the time variation of the vorticity only, so that it is not necessary to evaluate the velocity for the vortex force term of Eq. (5).

  2. The wave speed here is the gravity wave speed, rather than the sound speed.

  3. The perturbed variable here is mainly h′ instead of p′.

Furthermore, we are going to get a solution for a slightly more general vorticity distribution. The vortex region is still elliptical but the vorticity is allowed to decrease continuously through a layer in which contours of vorticity are similar ellipses. Equation (7) can then be evaluated as following.

The two-dimensional elliptical region is assumed to be only a slight deviation of the circle with radius a; the semimajor and semiminor axes are, respectively, a(1 ± ϵ), where 0 < ϵ < 1. The whole patch rotates with an angular velocity ω/2. In cylindrical coordinates r, λ, z, the equation of its outer boundary is represented by r(t) = a[1 + ϵ cos(2λωt)]. The vorticity ζ is a constant ζ0 from r = 0 to r = a0[1 + ϵ cos(2λωt)], and decreases linearly to a minimum value ζm from a0 to a. That is, the value of vorticity ζ at r = k[1 + ϵ cos(2λωt)], k > a0, is given by ζ = [1/(a0a)][k(ζ0ζm) − aζ0 + a0ζm]. So, the distribution of vorticity at r > a0[1 + ϵ cos(2λωt)] is, ζ = [1/(a0a)]{r (ζ0ζm)/[1 + ϵ cos(2λωt)] − aζ0 + a0ζm}. In terms of the cylindrical coordinates, x1 = R cosθ, x2 = R sinθ, y1 = r cosλ, and y2 = r sinλ, where R and θ are the radial and azimuthal coordinates of the observing point. The area integral I in (7) can then be evaluated as
i1520-0469-59-20-2930-eq4
where
i1520-0469-59-20-2930-eq5
It can be seen that for a0 = a, A = 1, it reduces to Howe's solution for a constant vortex. For small ellipticity
i1520-0469-59-20-2930-eq6
Therefore, Eq. (7) can be written as
i1520-0469-59-20-2930-e8
It is the same integral obtained by Howe (1975) who used the vortex force formulation [Eq. (5)]. Here, the third-order time derivative of vorticity is used.
The integral in Eq. (8) has been evaluated by Howe using the method of stationary phase. The result is
i1520-0469-59-20-2930-e9

As aω and ζ0a are both scaled by the velocity scale υ of the the eddy region, it can be seen from Eq. (9) that the amplitude of the wave is on the order of υ2Fr2(Λ/R), where Λ = 2πc0/ω is the wavelength. Since the ϕ associated with the driver vortex in the inner region has a scale υ2, the amplitude of the wave has a relative size O(Fr2), as found in Ford (1994a) and Ford et al. (2000). Ford (1994a) obtained the solution for a rotating ellipse (Kirchoff's vortex) in shallow water by the method of matched asymptotic expansion, using the small Froude number as the expansion parameter. His solution shows that the leading-order solution is the classical Kirchoff's vortex solution without gravity wave generation. To second order, the far field solution with the Coriolis parameter omitted is of the form Kω2H(1)2(ωR)ei(2θωt), where H(1)2 represents the second-order Hankel function of the first kind, and K is a constant depending on the geometry of the ellipse; this corresponds to the gravity wave generation. It can be seen that Eq. (9) is consistent with this solution, even though it can be applied to a more general situation of a rotating elliptical vortex other than that of the Kirchoff's vortex.

In the far field where the velocity scale of the mean flow is small compared with the gravity wave speed, ϕ′ ≃ gh′. Also, for shallow water wave propagation in a barotropic atmosphere, the vertical velocity perturbation can be found as w′ = −(1/g)(∂ϕ′/∂t). So from Eq. (9) we have
i1520-0469-59-20-2930-e10
This equation shows that the vertical velocity perturbation is a quadrupole with wavenumber 2, and the pattern rotates with the same angular speed as the central elliptical vortex. The propagation speed of the wave in the radial direction is c0 = gh0, the same as the speed of free gravity waves in shallow water. Equation (10) also shows that the amplitude of the wave decreases with distance from the center as 1/R. The amplitude contains the maximum vorticity ζ0 and the fourth power of the rotational speed of the vortex as factors. The two quantities are closely related. For example, a Kirchoff vortex with constant vorticity ζ0 rotates with a speed ζ0/4. Therefore, the amplitude of the wave generated by the vortex is highly sensitive to the central vorticity.
It can be seen from the phase angle in Eq. (10), 2θω(tR/c0) + π/4, that the spiral is trailing since the azimuthal wavenumber m has the same sign as the radial wavenumber k = ω/c0. At a fixed instance, the contour for the same phase of the wave is represented by the equation mθ + kr = constant; that is, Δθ = −(k/mr. A trailing spiral is depicted in Fig. 8. The shape of the spiral is determined by the pitch angle δ, given by
i1520-0469-59-20-2930-e11
where Ωp = ω/m is the rotational speed of the pattern. For approximately constant pattern and wave speeds, the pitch angle decreases with radius r. The shape of the spiral gets closer to a circle in the far field. The spiral pattern can also be interpreted as a result of the bending of wave path. The wave propagation is a superposition of the radial component (c0) and the azimuthal component (Ωpr). As the rotational speed of the pattern Ωp is generally proportional to the magnitude of the vortex vorticity while the gravity wave speed is probably less sensitive, it is anticipated that a stronger tropical cyclone can have a tighter spiral pattern than that of a weaker one. This is in agreement with the observation of Dvorak (1984).

b. Evaluation and comparison

In order to assess the applicability of the analytical model, we apply it to the MM5 model described in section 2 and compare the results. The theoretical vertical velocity perturbation given by Eq. (10) is evaluated for the time 130.5 h of the MM5 model. The core region parameters required in Eq. (10) are estimated from information on the depth-averaged vorticity shown in Fig. 2. The maximum and minimum values of relative vorticity are taken to be 55 × 10−4 s−1 and 15 × 10−4 s−1, respectively. The period of rotation is 75 min. The shape parameters for the elliptical vortex are a = 32.75 km, ϵ = 0.145, and a0 = 13.45 km, so that A in (10) has the value 0.517. The gravity wave speed c0 is estimated from the radial propagation speed of the depth-averaged vertical velocity perturbation in the numerical simulation and is found to be 62 m s−1. The origin of the time parameter t in Eq. (10) is calibrated by the azimuthal positions of the elliptical axes at the specified moment (130.5 h). At t = 0, the major and minor axes of the ellipse are to be aligned with the x and y axes, respectively. In other words, t is shifted to match the position of the ellipse in the theory with that observed from the numerical results at the specified instance.

Contour maps of the depth-averaged vertical velocity perturbation from MM5 mesh C (450 km × 450 km) and from Eq. (10) are shown in Fig. 9. The length scales of the spirals are similar, but the magnitudes given by the MM5 model are generally larger. The patterns also show differences in the central region. This, however, is to be expected as Eq. (10) is only a far field approximation. Nevertheless, we can still see the coincidence of the two maxima spots in the eye region, and the agreement of phases in the far field. In the larger domain (mesh B: 1080 km × 1080 km) the MM5 spirals become tightly wound in the far-field region, in agreement with the predication of Eq. (11).

Comparison of the MM5 spiral pattern with Eq. (10) can be better illustrated in radial and azimuthal cuts of the depth-averaged vertical velocity field. Figure 11 shows the radial profiles of the field along a cut from the SW corner to the NE corner of the mesh at 130.5 h (7830 min) and 130.8 h (7848 min). Figure 12 shows the azimuthal profiles of the field at the radii of 100 and 200 km from the vortex center, for the same instances. From these two figures, we can see that agreements between the phases and the amplitudes of the waves both get better in the far field.

The amplitude of the analytical wave can substantially deviate from that of the MM5 model for a number of reasons. First, Eq. (10) is a solution to a simple shallow water model with many approximations. Second, the amplitude of the analytical solution depends on the dimension a of the vortex, the square of the gravity wave speed c0, as well as the fourth power of the rotational speed ω. Small changes in the estimates of these parameters can induce large change in the amplitude of the wave. Third, the presence of deep cloud bands in the central region may significantly enhance the vertical motions in that region. Concerning this aspect, one might raise the question of whether the moist processes are critical to the local maintenance of the spiral bands. For this reason, we have performed a test experiment with the MM5 model. In this experiment, a mesh with dimensions of 120 km × 120 km is introduced at 4710 min so that the core region of the tropical cyclone can just be fitted into this mesh. Moist processes are then turned off in all meshes except in this innermost mesh. Even without the moist processes, spiral bands are still observed outside this mesh. In addition, the shapes and phases of the spiral patterns remain the same as those generated in the original run for at least 30 min after the small mesh is added. This indicates that the spirals are not locally generated by the moist processes.

Another comparison of Eq. (10) with the MM5 results is made at an earlier time (74.2 h or 4452 min) when the cyclone is less mature. The maximum and minimum values of the relative vorticity are estimated to be 48 × 10−4 s−1 and 10 × 10−4 s−1, respectively. The vortex rotates with a period of 96 min, longer than that at 130.5 h, when the maximum vorticity reaches 55 × 10−4 s−1. The shape parameters for the elliptical vortex are a0 = 7 km, a = 31.25 km, and ϵ = 0.2, so that the value of A in Eq. (10) is 0.412. The gravity wave speed is estimated to be 35 m s−1. Comparison of the analytical and numerical patterns over the 450 km × 450 km scale is shown in Fig. 13. Although the MM5 field still shows a wavenumber-2 feature, the spirals are not as clear and symmetric as those at 130.5 h. Nevertheless, the scale and phase of the analytical pattern still provide a good description for the MM5 pattern.

5. Formation of cloud bands

In the MM5 model, the clouds in the eyewall region are deep; rain water is mainly deposited in this region. The eyewall, however, is not in the form of a complete ring. Clouds concentrate at two spots of strong vertical motions (see Fig. 2). Outside the core region, as the vertically velocity perturbation spirals outward, the clouds form bands as the perturbation passes. But they generally form at high altitudes, and there is little associated rainwater.

Evolution of depth-averaged cloud water is shown in Fig. 14. The diagrams show the life cycles of several cloud bands. Each cloud band can only be identified for a short time period, indicating a lifetime of about 18–25 min. This is in agreement with some observational studies (e.g., Anthes 1982). A cloud band labeled A starts to form and becomes identifiable at time 130.8 h (7848 min). It becomes mature at 130.9 h (7854 min), then diminishes (7866 min) and vanished at 131.1 h (7866 min). The band labeled B can be identified from 131 h (7860 min) to 131.3 h (7878 min). If we follow the subsequent evolution of band A and band B, we can see that they apparently rotate around the center, but appear discontinuously since it actually involved different life cycles of cloud formation and dissipation. The labeling of bands A and B from 7884 min to 7914 min is based on the consideration that labels follow the bands that originate from the same tips of the elliptical eyewall. If one follows the evolution of band B, one can see that it almost completed a circular trip in 60 min, which is close to the period of rotation of the vertical velocity perturbation. The radial propagation speeds of the bands vary. The speed estimate of band B from Fig. 14 is about 9.5 m s−1 from 131 h (7860 min) to 131.4 hr (7884 min), and it is about 25 m s−1 from 131.4 h (7884 min) to 131.7 h (7902 min).

To illustrate how the formation of cloud bands is related to the vertical velocity perturbation, the azimuthal profiles of the depth-averaged cloud distribution and vertical velocity perturbation at a radius of 160 km are plotted from 130.8 h (7848 min) to 131.9 h (7914 min) in Fig. 15 (same corresponding period as Fig. 14). It can be clearly seen that the vertical velocity perturbation has a wavenumber-2 feature, and the phase is ahead that of the clouds. Clouds start to form at a position where a maximum of positive vertical velocity perturbation (updraft) just passes, and diminish rapidly when a maximum of negative vertical velocity (downdraft) passes through that position. The formation of cloud bands behind passing gravity waves agrees with the hypotheses that inertia–gravity waves can trigger convection within the convective lines of a tropical cyclone (May 1996).

6. Discussion

a. Observations

Both the elliptical core and spiral bands are key elements in the theory being investigated here. However, observational records of moving spiral bands are rare, though there are more for elliptical eyes. From the radar observations described in Willoughby et al. (1984), one can see that the velocity field in the core region of hurricane David 1979 was in the form of an ellipse. Moving spiral bands were also recorded for the same hurricane. For typhoon Herb 1996, Kuo et al. (1999) reported that an elliptical eye was observed with a rotation period of about 144 min. The elliptical eye had a maximum radar reflectivity at the vertices of the major axis, in a form similar to those shown in Fig. 2 for the MM5 model. However, spiral bands outside the eye region were not clearly shown in the paper.

Further observational examples of spiral bands and elliptical eyes are shown in Fig. 16. These are satellite images obtained by microwave sensors, so that the bands observed correspond to regions with strong convection. From the image of hurricane Floyd 1999 we can see that it contained an elliptical eye. The eyewall was dominated by two spots, and it had two spiral bands. The images of typhoon Bilis in 2000 show that it also contained an elliptical eye. One information that may be obtained from these images is that the large-scale spiral-banded structures with quasi-wavenumber-two features are usually connected with the presence of elliptical eyewalls in the core region. Unfortunately, no more images other than those of Willoughby et al. (1984) show the evolution of spiral structures.

b. Dvorak's intensity analysis

The present theory of spiral band generation may help to provide some information about the strength of a tropical cyclone that is consistent with the guide for intensity forecasting developed by Dvorak (1984). The background of this guide is mainly based on two facts. First, it is the pattern formed by the clouds, not the amount of clouds, that is correlated with the tropical cyclone's intensity. Second, an increase in the distance the overcast cloud bands coil around the center of the system indicates an increase of the cyclone's intensity, and the cyclone's intensity at any given time is related to the distance the curved bands wrap around the center. For the first point, it has been discussed in section 4 that the spiral pattern of a stronger tropical cyclone is tightly wound compared with that of a weaker one. For the second point, it can be seen in Eq. (10) that the amplitude of the perturbation increases with the magnitude of vorticity in the eye region, and decreases as the square root of distance from the center. Therefore, an elliptical vortex with higher vorticity can deliver a sufficiently strong perturbation to a farther distance. In general, a tropical cyclone of high intensity means that it has high maximum wind and thus high maximum vorticity.

c. Speculations

It can be inferred from the theoretical results here that the formation of an incomplete eyewall, as well as some unusually strong downdrafts at the eyewall (Black et al. 1994) may be due to the generation of gravity waves in this region. If it is the case, then the horizontal dimension of vorticity distribution in the eye region may determine the scale of the eyewall. The vertical motions induced may also act as a complement to the process of conditional instability of the second kind (CISK) for cyclone intensification. The occurrence of stationary spiral bands, which are more frequently observed, were not explained by the present theory. Weber (1999) has shown that a moving axisymmetric, barotropic vortex can induce a wavenumber-1 anomaly of vorticity that remains almost stationary relative to the vortex center. However, he used a nondivergent barotropic numerical model where gravity waves were eliminated. Since the movement of the axisymmetric vortex is equivalent to a fluctuation of vorticity relative to a fixed location, it is speculated that the movement may induce gravity waves by the same principle of the present theory. Other than the well-organized moving spiral bands discussed in the present paper, the formation of some irregular rain bands that usually exist in the outer region of a tropical cyclone may also be due to radially propagating gravity waves. The waves are intermittently generated by vorticity fluctuation in the core region during the intensification process.

7. Concluding remarks

A theory of Lighthill radiation has been developed for the shallow water equation, and is proposed as a generation mechanism of moving spiral bands in tropical cyclones. The theory shows that fluctuation of vorticity distribution in a compact vortical region can act as a source to generate gravity waves. The principle of spontaneous generation of gravity waves by this theory may be considered as one form of adjustment. For the potential vorticity in shallow water to be conserved, fluctuation of vorticity in a region should be accompanied by a fluctuation of height, leading to the radiation of gravity waves if this fluctuation is confined to a small region. The intrinsic connection can be understood through the fact that both the Lighthill radiation Eq. (3) and the potential vorticity equation are exact consequences of the shallow water equation. Another interpretation is to view the waves as the difference in adjustment response between the shallow water dynamics and the two-dimensional nondivergent barotropic dynamics. In dimensionless form, the only difference between the two sets of equations is in the continuity equation, which takes the form ∇·V = 0 in the barotropic case but ∇·V = O(Fr2) in the shallow water case (Ford et al. 2000). This probably explains the O(Fr2) amplitudes of the radiated waves.

Another interpretation for this principle can be as follows. If the irrotational flow in the far field is induced by the vorticity in the core region, variation of vorticity distribution in this region can also induce a corresponding variation in the far irrotational flow field, which, in turn, produces the perturbed height field.

The moving bands in the simulated tropical cyclone show good agreement with the analytical calculation for an elliptical core of vorticity in the central region. This suggests that as far as the spirals are concerned, the shallow water approach provides a reasonable approximation to the more complex atmosphere. Although the amplitudes of the gravity waves are low (the depth-averaged value of the vertical velocity perturbation is only a few cm s−1), the MM5 results suggest that the organized motion can trigger cloud formation in the conditionally unstable atmosphere, and thus moving spiral bands are generated. The theory provides a possible framework to address traditional questions about cyclone-scale spiral bands such as the number of arms and the connection with cyclone intensity. However, it is necessary to emphasize that spiral band generation by the principle of Lighthill radiation does not exclude other possibilities for generation of spiral bands, such as vortex Rossby waves. In fact, the present theory can only deal with large-scale spiral structures that have wavelengths larger than the scale of the vortical core.

Acknowledgments

This work was partially supported by the research Grant WITI93/94.RC01 (R5326) of the Center of Coastal and Atmospheric Research (CCAR) at HKUST and the research Grant HKUST6/18/00P provided by the Hong Kong Research Grant Council.

REFERENCES

  • Anthes, R. A., 1982: Tropical Cyclones—Their Evolution, Structure and Effects. Meteor. Monogr., No. 41, Amer. Meteor Soc., 208 pp.

  • Atkinson, B. W., 1981: Mesoscale Atmospheric Circulations. Academic Press, 495 pp.

  • Black, R. A., H. B. Bluestein, and M. L. Black, 1994: Unusually strong vertical motions in Caribbean hurricane. Mon. Wea. Rev., 122 , 27222739.

    • Search Google Scholar
    • Export Citation
  • Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification. J. Atmos. Sci., 58 , 21282145.

    • Search Google Scholar
    • Export Citation
  • Chow, K. C., 2001: Moving spiral structure in tropical cyclones. Ph.D. thesis, Hong Kong University of Science and Technology, 124 pp.

  • Diercks, J. W., and R. Anthes, 1976: Diagnostic studies of spiral rainbands in a nonlinear hurricane model. J. Atmos. Sci., 33 , 959975.

    • Search Google Scholar
    • Export Citation
  • Dvorak, V. F., 1984: Tropical cyclone intensity analysis using satellite data. NOAA Tech. Rep. NESDIS-11, 47 pp.

  • Ford, R., 1994a: The response of a rotating ellipse of uniform potential vorticity to gravity wave radiation. Phys. Fluids, 6 ((11),) 36943704.

    • Search Google Scholar
    • Export Citation
  • Ford, R., . 1994b: Gravity wave radiation from vortex trains in rotating shallow water. J. Fluid Mech., 281 , 81118.

  • Ford, R., M. E. McIntyre, and W. A. Norton, 2000: Balance and the slow quasimanifold: Some explicit results. J. Atmos. Sci., 57 , 12361254.

    • Search Google Scholar
    • Export Citation
  • Gall, R., J. Tuttle, and P. Hildebrand, 1998: Small-scale spiral bands observed in Hurricanes Andrew, Hugo, and Erin. Mon. Wea. Rev., 126 , 17491766.

    • Search Google Scholar
    • Export Citation
  • Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50 , 33803403.

  • Holland, G. J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54 , 25192541.

  • Howe, M. S., 1975: Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech., 71 , 625673.

    • Search Google Scholar
    • Export Citation
  • Kossin, J. P., W. H. Schubert, and M. T. Montgomery, 2000: Unstable interactions between a hurricane's primary eyewall and a secondary ring of enhanced vorticity. J. Atmos. Sci., 57 , 38933917.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., 1976: On the development of spiral bands in tropical cyclone. J. Atmos. Sci., 33 , 940958.

  • Kuo, H. C., R. T. Williams, and J. H. Chen, 1999: A possible mechanism for the eye rotation of Typhoon Herb. J. Atmos. Sci., 56 , 16591673.

    • Search Google Scholar
    • Export Citation
  • Lighthill, M. J., 1952: On sound generated aerodynamically-I. General theory. Proc. Roy Soc. London, A211 , 564587.

  • May, P. T., 1996: The organization of convection in the rainbands of Tropical Cyclone Laurence. Mon. Wea. Rev., 124 , 807815.

  • Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., 123 , 435465.

    • Search Google Scholar
    • Export Citation
  • Powell, A., 1995: Vortex sound theory: Direct proof of equivalence of ‘vortex force’ and ‘vorticity alone’ formulations. J. Acoust. Soc. Amer., 97 , 15341537.

    • Search Google Scholar
    • Export Citation
  • Weber, H. C., 1999: A numerical study of tropical-cyclone structure: Quasi-stationary spiral bands. Quart. J. Roy. Meteor. Soc., 125 , 811836.

    • Search Google Scholar
    • Export Citation
  • Willoughby, H. E., 1978: A possible mechanism for the formation of hurricane rainbands. J. Atmos. Sci., 35 , 838848.

  • Willoughby, H. E., F. D. Marks Jr.,, and R. J. Feinberg, 1984: Stationary and moving convective bands in hurricanes. J. Atmos. Sci., 41 , 31893211.

    • Search Google Scholar
    • Export Citation

APPENDIX A

Far-Field Solution—Eq. (6)

The solution of (5) in free space can be represented by a convolution integral of the free space Green's function:
i1520-0469-59-20-2930-eq7
where G(x, t|y, τ) = δ(c0(tτ) − |xy|)/(4π|xy|) is the free space Green's function in three dimensions, and c0 = gh0 is the wave speed; x = (x1, x2) is the position vector of the observing point (on the x3 = 0 plane), and y = (y1, y2, y3) is the position vector of the source function.
Integration by parts gives
i1520-0469-59-20-2930-ea1
Since the source is compact in the horizontal dimension, in the far field |x| ≫ |yh|, where yh = (y1, y2) is the horizontal position vector of the source. Thus, |xy|−1 ≃ (|x|2 + y23)−1/2 is small, and we have
i1520-0469-59-20-2930-eq8
So,
i1520-0469-59-20-2930-ea2
Now |xy| = (|x|2 − 2x·yh + |yh|2 + y23)1/2|x|2 + y23 − (x·yh/|x|2 + y23), since |x| ≫ |yh|. Variation of the source in retarded time t − |xy|/c0 can be expanded in Taylor series as
i1520-0469-59-20-2930-ea3
Since the rate of change of the total vortex force ∂/∂tζ × V(y, t) dy2 ≃ 0 at any time t, to a first-order approximation retaining only the term with first time derivative in (A3) and put into (A2), the far-field solution given by (A1) is then
i1520-0469-59-20-2930-eq9

APPENDIX B

Vorticity Alone Formulation—Powell (1995)

We wish to prove
i1520-0469-59-20-2930-eq10
where L(yh, t) = ζ × V(yh, t), x is a unit vector, and ′ denotes the time derivative
From the vector identity
i1520-0469-59-20-2930-eq11
and putting a = yh and b = L, we have
i1520-0469-59-20-2930-eb1
From Eq. (2), we have the Helmholtz's vorticity equation
ζLfkV
In our application (tropical cyclones), the domain of interest is less than the Rossby radius, and the relative vorticity is generally much larger than the vorticity contributed by the earth. We can neglect the Coriolis force, and consider the source to be in the compact core region. Therefore, we have
yhLyhζ
Now, since
i1520-0469-59-20-2930-eq14
(B1) becomes ∇(yh;th·;thL) = −yh × ζ′ − L + ∂(yiLj)/∂yi, which implies
i1520-0469-59-20-2930-eb2

For any scalar a, x·∇a = ∇·(xa) since the gradient operator is acting on the source coordinate y only.

So, (yh;th·;thx)(x;th·;th∇a) = yh;th·;thx∇;th·;th(xa) = ∇;th·;th(xayh;th·;thx) − ∇(yh;th·;thx);th·;thxa = ∇;th·;th(xayh;th·;thx) − a.

Letting a = yh;th·;thL′, we have (yh;th·;thx)x;th·;th∇(yh;th·;thL′) = ∇;th·;th(x(yh;th·;thL′)yh;th·;thx) − yh;th·;thL′. Also,
i1520-0469-59-20-2930-eq145
Multiplying (B2) by (yh·x), we then have
i1520-0469-59-20-2930-eq15
Next, take the two-dimensional “volume” integral of the above equation. Since L′ = 0 at the boundary of the source, the divergence terms vanish, and we have the integral equation
i1520-0469-59-20-2930-eb3
Taking the time derivative of this equation, we have
i1520-0469-59-20-2930-eb4
The term ∫ yh;th·;thLdy1 dy2 should be equal to 0. It is based on the fact that this integral is independent of the position x. If x1 and x2 are two positions perpendicular to each other, then for a quadrupole, we have ϕ(x1) + ϕ(x2) = 0. From (B4), it required that ∫ yh;th·;thLdy1 dy2 = 0. Therefore, we have the required relation
i1520-0469-59-20-2930-eq16

Fig. 1.
Fig. 1.

Evolution of the axisymmetric vortex. (left) Evolution of minimum sea level pressure. (middle) Evolution of maximum tangential wind. (right) Profile of axisymmetric depth-averaged flow field at time of 131 h

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 2.
Fig. 2.

Evolution of the spiral structure for a period of 78 min from time = 130.5 h (7830 min). Dotted lines denote negative values. (left) Vertical velocity perturbations with positive values in bright color (contour = 0.4 m s−1, domain size = 450 km × 450 km). (middle) Enlarged view of vertical velocity perturbation contour (contour = 0.3 m s−1) with color contour of absolute vorticity. (right) Absolute vorticity with contour = 5 × 10−4 s−1. Domain sizes = 180 km × 180 km for middle and right columns

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 2.
Fig. 2.

(Continued)

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 3.
Fig. 3.

Absolute vorticity at various heights at time = 131 h. (Domain shown = 180 km × 180 km)

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 4.
Fig. 4.

Azimuthal–height cross sections of vertical velocity perturbations from the MM5 results of the 450 km × 450 km mesh at different radii and time periods. (top) Radius = 100 km. (bottom) Radius = 200 km. The horizontal axis shows the azimuthal direction and spatial scale along the perimeter. Here, A and B are bands with positive phases that can be identified in these time periods

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 5.
Fig. 5.

Azimuthal propagation of depth-averaged vertical velocity perturbations at different radii from the MM5 results of the 450 km × 450 km mesh. (left) Radius = 100 km. (right) Radius = 200 km

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 6.
Fig. 6.

Radial–height cross sections of vertical velocity perturbations from the MM5 results of the 450 km × 450 km mesh at different time periods. (left) Time = 130.5 h (7830 min). (right) Time = 130.8 h (7848 min). The cross sections are cut from the SW corner to the NE corner of the mesh through the vortex center. Here, A and B are bands with positive phases that can be identified in these time periods

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 7.
Fig. 7.

Radial propagation of depth-averaged vertical velocity perturbations with details corresponding to Fig. 6. The wave crests labelled A and B are the same bands in these two time periods corresponding to Fig. 6.

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 8.
Fig. 8.

The pitch angle of a spiral

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 9.
Fig. 9.

Depth-averaged vertical velocity perturbations at time = 130.5 h (7830 min) of the 450 km × 450 km mesh. Negative values are represented by dotted lines. (left) MM5 results with contour = 0.04 m s−1. (right) From Eq. (10) with contour = 0.02 m s−1

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 10.
Fig. 10.

Depth-averaged vertical velocity perturbation at time = 130.5 h (7830 min) of the 1080 km × 1080 km mesh. Dotted lines representing negative values. (left) MM5 output with contour = 0.02 m s−1. (right) From Eq. (10) with contour = 0.02 m s−1

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 11.
Fig. 11.

Radial profiles of depth-averaged vertical velocity perturbations from results of MM5 and theory [Eq. (10)] at different time periods. (left) Time = 130.5 h (7830 min). (right) Time = 130.8 h (7848 min). The wave crests labeled A and B are the same bands in these two time periods corresponding to Figs. 6 and 7

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 12.
Fig. 12.

Azimuthal profiles of depth-averaged vertical velocity perturbations from results of MM5 and theory [Eq. (10)] at different radii and time periods. (top) Radius = 100 km. (bottom) Radius = 200 km

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 13.
Fig. 13.

Same as Fig. 9, but for time = 74.2 h (4452 min).

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 14.
Fig. 14.

Evolution of cloud bands (depth-averaged cloud water mixing ratio) from time = 130.8 h (7848 min) to 131.9 h (7914 min) at 6-min intervals. Contour = 0.0005 kg kg−1; domain = 450 km × 450 km. The labeling of bands A and B is based on the consideration that the bands with the same labels are originating from the same tip of the elliptical eyewall

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 15.
Fig. 15.

Azimuthal distributions of depth-averaged vertical velocity perturbations and clouds at radius of 160 km from the vortex center. Time period: from 130.8 h (7848 min) to 131.9 h (7914 min) at 6-min intervals. Labels A and B are the cloud bands corresponding to Fig. 14

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Fig. 16.
Fig. 16.

Satellite images of hurricanes with elliptical eyewalls and approximately wavenumber-2 spiral bands. (top left) Hurricane Floyd by special sensor microwave imager (SSM/I) 85H sensor. (top right; bottom left and right) Hurricane Bilis by SSM/I–Tropical Rainfall Measuring Mission 85-GHz sensor at time 0919, 1123, and 1214 UTC, respectively. Images from the Naval Research Laboratory Monterey Marine Meteorology Tropical Cyclone page.>

Citation: Journal of the Atmospheric Sciences 59, 20; 10.1175/1520-0469(2002)059<2930:GOMSBI>2.0.CO;2

Table 1. 

Meshes of the numerical model

Table 1. 
Save
  • Anthes, R. A., 1982: Tropical Cyclones—Their Evolution, Structure and Effects. Meteor. Monogr., No. 41, Amer. Meteor Soc., 208 pp.

  • Atkinson, B. W., 1981: Mesoscale Atmospheric Circulations. Academic Press, 495 pp.

  • Black, R. A., H. B. Bluestein, and M. L. Black, 1994: Unusually strong vertical motions in Caribbean hurricane. Mon. Wea. Rev., 122 , 27222739.

    • Search Google Scholar
    • Export Citation
  • Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hurricane. Part I: Vortex Rossby wave verification. J. Atmos. Sci., 58 , 21282145.

    • Search Google Scholar
    • Export Citation
  • Chow, K. C., 2001: Moving spiral structure in tropical cyclones. Ph.D. thesis, Hong Kong University of Science and Technology, 124 pp.

  • Diercks, J. W., and R. Anthes, 1976: Diagnostic studies of spiral rainbands in a nonlinear hurricane model. J. Atmos. Sci., 33 , 959975.

    • Search Google Scholar
    • Export Citation
  • Dvorak, V. F., 1984: Tropical cyclone intensity analysis using satellite data. NOAA Tech. Rep. NESDIS-11, 47 pp.

  • Ford, R., 1994a: The response of a rotating ellipse of uniform potential vorticity to gravity wave radiation. Phys. Fluids, 6 ((11),) 36943704.

    • Search Google Scholar
    • Export Citation
  • Ford, R., . 1994b: Gravity wave radiation from vortex trains in rotating shallow water. J. Fluid Mech., 281 , 81118.

  • Ford, R., M. E. McIntyre, and W. A. Norton, 2000: Balance and the slow quasimanifold: Some explicit results. J. Atmos. Sci., 57 , 12361254.

    • Search Google Scholar
    • Export Citation
  • Gall, R., J. Tuttle, and P. Hildebrand, 1998: Small-scale spiral bands observed in Hurricanes Andrew, Hugo, and Erin. Mon. Wea. Rev., 126 , 17491766.

    • Search Google Scholar
    • Export Citation
  • Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci., 50 , 33803403.

  • Holland, G. J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54 , 25192541.

  • Howe, M. S., 1975: Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech., 71 , 625673.

    • Search Google Scholar
    • Export Citation
  • Kossin, J. P., W. H. Schubert, and M. T. Montgomery, 2000: Unstable interactions between a hurricane's primary eyewall and a secondary ring of enhanced vorticity. J. Atmos. Sci., 57 , 38933917.

    • Search Google Scholar
    • Export Citation
  • Kurihara, Y., 1976: On the development of spiral bands in tropical cyclone. J. Atmos. Sci., 33 , 940958.

  • Kuo, H. C., R. T. Williams, and J. H. Chen, 1999: A possible mechanism for the eye rotation of Typhoon Herb. J. Atmos. Sci., 56 , 16591673.

    • Search Google Scholar
    • Export Citation
  • Lighthill, M. J., 1952: On sound generated aerodynamically-I. General theory. Proc. Roy Soc. London, A211 , 564587.

  • May, P. T., 1996: The organization of convection in the rainbands of Tropical Cyclone Laurence. Mon. Wea. Rev., 124 , 807815.

  • Montgomery, M. T., and R. J. Kallenbach, 1997: A theory for vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., 123 , 435465.

    • Search Google Scholar
    • Export Citation
  • Powell, A., 1995: Vortex sound theory: Direct proof of equivalence of ‘vortex force’ and ‘vorticity alone’ formulations. J. Acoust. Soc. Amer., 97 , 15341537.

    • Search Google Scholar
    • Export Citation
  • Weber, H. C., 1999: A numerical study of tropical-cyclone structure: Quasi-stationary spiral bands. Quart. J. Roy. Meteor. Soc., 125 , 811836.

    • Search Google Scholar
    • Export Citation
  • Willoughby, H. E., 1978: A possible mechanism for the formation of hurricane rainbands. J. Atmos. Sci., 35 , 838848.

  • Willoughby, H. E., F. D. Marks Jr.,, and R. J. Feinberg, 1984: Stationary and moving convective bands in hurricanes. J. Atmos. Sci., 41 , 31893211.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Evolution of the axisymmetric vortex. (left) Evolution of minimum sea level pressure. (middle) Evolution of maximum tangential wind. (right) Profile of axisymmetric depth-averaged flow field at time of 131 h

  • Fig. 2.

    Evolution of the spiral structure for a period of 78 min from time = 130.5 h (7830 min). Dotted lines denote negative values. (left) Vertical velocity perturbations with positive values in bright color (contour = 0.4 m s−1, domain size = 450 km × 450 km). (middle) Enlarged view of vertical velocity perturbation contour (contour = 0.3 m s−1) with color contour of absolute vorticity. (right) Absolute vorticity with contour = 5 × 10−4 s−1. Domain sizes = 180 km × 180 km for middle and right columns

  • Fig. 2.

    (Continued)

  • Fig. 3.

    Absolute vorticity at various heights at time = 131 h. (Domain shown = 180 km × 180 km)

  • Fig. 4.

    Azimuthal–height cross sections of vertical velocity perturbations from the MM5 results of the 450 km × 450 km mesh at different radii and time periods. (top) Radius = 100 km. (bottom) Radius = 200 km. The horizontal axis shows the azimuthal direction and spatial scale along the perimeter. Here, A and B are bands with positive phases that can be identified in these time periods

  • Fig. 5.

    Azimuthal propagation of depth-averaged vertical velocity perturbations at different radii from the MM5 results of the 450 km × 450 km mesh. (left) Radius = 100 km. (right) Radius = 200 km

  • Fig. 6.

    Radial–height cross sections of vertical velocity perturbations from the MM5 results of the 450 km × 450 km mesh at different time periods. (left) Time = 130.5 h (7830 min). (right) Time = 130.8 h (7848 min). The cross sections are cut from the SW corner to the NE corner of the mesh through the vortex center. Here, A and B are bands with positive phases that can be identified in these time periods

  • Fig. 7.

    Radial propagation of depth-averaged vertical velocity perturbations with details corresponding to Fig. 6. The wave crests labelled A and B are the same bands in these two time periods corresponding to Fig. 6.

  • Fig. 8.

    The pitch angle of a spiral

  • Fig. 9.

    Depth-averaged vertical velocity perturbations at time = 130.5 h (7830 min) of the 450 km × 450 km mesh. Negative values are represented by dotted lines. (left) MM5 results with contour = 0.04 m s−1. (right) From Eq. (10) with contour = 0.02 m s−1

  • Fig. 10.

    Depth-averaged vertical velocity perturbation at time = 130.5 h (7830 min) of the 1080 km × 1080 km mesh. Dotted lines representing negative values. (left) MM5 output with contour = 0.02 m s−1. (right) From Eq. (10) with contour = 0.02 m s−1

  • Fig. 11.

    Radial profiles of depth-averaged vertical velocity perturbations from results of MM5 and theory [Eq. (10)] at different time periods. (left) Time = 130.5 h (7830 min). (right) Time = 130.8 h (7848 min). The wave crests labeled A and B are the same bands in these two time periods corresponding to Figs. 6 and 7

  • Fig. 12.

    Azimuthal profiles of depth-averaged vertical velocity perturbations from results of MM5 and theory [Eq. (10)] at different radii and time periods. (top) Radius = 100 km. (bottom) Radius = 200 km

  • Fig. 13.

    Same as Fig. 9, but for time = 74.2 h (4452 min).

  • Fig. 14.

    Evolution of cloud bands (depth-averaged cloud water mixing ratio) from time = 130.8 h (7848 min) to 131.9 h (7914 min) at 6-min intervals. Contour = 0.0005 kg kg−1; domain = 450 km × 450 km. The labeling of bands A and B is based on the consideration that the bands with the same labels are originating from the same tip of the elliptical eyewall

  • Fig. 15.

    Azimuthal distributions of depth-averaged vertical velocity perturbations and clouds at radius of 160 km from the vortex center. Time period: from 130.8 h (7848 min) to 131.9 h (7914 min) at 6-min intervals. Labels A and B are the cloud bands corresponding to Fig. 14

  • Fig. 16.

    Satellite images of hurricanes with elliptical eyewalls and approximately wavenumber-2 spiral bands. (top left) Hurricane Floyd by special sensor microwave imager (SSM/I) 85H sensor. (top right; bottom left and right) Hurricane Bilis by SSM/I–Tropical Rainfall Measuring Mission 85-GHz sensor at time 0919, 1123, and 1214 UTC, respectively. Images from the Naval Research Laboratory Monterey Marine Meteorology Tropical Cyclone page.>

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