## 1. Introduction

Intense tropical cyclones (TCs) often develop polygonal eyewalls, double eyewalls, and spiral rainbands in the inner-core regions (e.g., Willoughby et al. 1982; Marks and Houze 1987; Liu et al. 1997). These features, superimposed on the mean rotational flow, play an important role in determining the structural and intensity changes of TCs (Black and Willoughby 1992; Lee and Bell 2007; Chen et al. 2011). Earlier views on the dynamics of these perturbation structures were based on the theory of internal inertio-gravity waves (IGWs) (e.g., Kurihara 1976; Willoughby 1978; Elsberry et al. 1987). One caveat with this theory is that these observed structures often propagate much slower than pure IGWs. Thus, more attention later has been shifted to the vortex Rossby wave (VRW) theory of Macdonald (1968) who drew an analogy between spiral rainbands and Rossby waves around a rotating sphere (e.g., Guinn and Schubert 1993; Montgomery and Kallenbach 1997, hereafter MK; Wang 2002). Recent observational and modeling studies have shown the presence of intense divergence and cyclonic vorticity in the eyewall and spiral rainbands (Jorgensen 1984; Liu et al. 1999; Hogsett and Zhang 2009). Obviously, the IGW and VRW theories, describing the respective divergent and rotational flows, cannot provide a complete description of wave dynamics in TCs.

*f*-plane, shallow-water equations and justifying the radial variation of the Bessel parameter based on a real-data cloud-resolving TC simulation, ZZL were able to obtain a form of the Bessel equation for an azimuthal harmonic oscillator in the cylindrical coordinates with the following cubic wave-frequency equation exhibiting the coexistence of IGWs, VRWs, and mixed VRIGWs:where

*ω*is the intrinsic frequency; the Froude number

*F*

_{r}is the squared ratio of the rotational speed to the phase speed of surface gravity waves; the vortex Rossby number

*R*

_{0}is the ratio of the mean rotation to its radial gradient of vertical absolute vorticity

*η*;

*T*

_{r}is the curvature of azimuthal flows; Ω is the angular velocity; and

*m*(

*n*) is the percentage distance of each node, corresponding to the positive roots of

*n*th-order Bessel function of the first kind, between the TC center and the boundary radius for a given azimuthal wavenumber (WN)

*n.*Note that all the variables in Eq. (1a) are nondimensional (see ZZL for more details). An important result from ZZL is that the normal-mode characteristics of the three different waves are determined by the following discriminant parameter associated with Eq. (1a); that is,Using a real-data model-simulated TC vortex as the basic state, ZZL found that (i) when

*Q*> 0, there exist two high-frequency oppositely propagating IGWs and a low-frequency VRW propagating against the mean flow; (ii) when

*Q*= 0, there are only two intermediate-frequency waves exhibiting the characteristics of mixed VRIGW; and (iii) when

*Q*< 0, there is a single real solution corresponding to a cyclonic-propagating low-frequency VRIGW and two complex conjugate solutions associated with dynamically unstable VRIGWs. Thus, it is clear from Eq. (1b) that the smaller

*m*is, the more likely it is that mixed VRIGWs will occur.

Because the analytical solution in Eq. (1a) was obtained after applying the above-mentioned two simplifying procedures, it is desirable to examine to what extent the results of ZZL can be generalized. Furthermore, little is understood about the spectral characteristics of mixed VRIGWs in relation to IGWs and VRWs as described by the linearized shallow-water equations. Thus, in this study, we will use the same dynamical framework as in ZZL, but without making any approximation, to study various classes of waves propagating in TC-like vortices, with more attention given to mixed VRIGWs. This will be done by invoking numerical solutions of the linearized shallow-water equations. Numerical solutions are also desirable because ZZL assumed nonzero Doppler-shifted frequency like other theoretical wave-motion studies in order to derive the analytical normal-mode solutions.

Montgomery and Lu (1997, hereafter ML) have also invoked numerical solutions of the linearized shallow-water equations to study the spectrum and structures of VRWs and IGWs propagating in TC-like vortices. In addition, they examined the nature of “balanced” and “unbalanced” flows dominated by the respective rotational and divergent components of barotropic vortices. The divergent flows account for the adjustment of the mass and wind fields, depending upon the time scale of the motions. Recent studies have revealed the presence of quasi-balanced flows in intense TCs and other mesoscale convective systems (Davis and Weisman 1994; Wang and Zhang 2003; Zhang and Kieu 2006; Zhong et al. 2008). This implies that both divergence and rotation are important in these weather systems. Zeng et al. (1990) studied the wave spectrum and eigenfunction of two-dimensional shallow-water equations under the influences of westerly wind shears and found that the wave spectrum can be easily separated into two classes in the presence of weak flows: low-frequency Rossby waves with a continuous spectrum and high-frequency IGWs with a discrete spectrum; continuous versus discrete spectra will be discussed in section 3a. However, the spectrum of IGWs could extend into the continuous spectrum and overlaps with Rossby wave spectrum in the presence of strong westerly flows. As a result, low-frequency IGWs would resemble in many aspects those of Rossby waves, including those typical characteristics of low-frequency waves (e.g., quasi-balanced features).

The objectives of this study are (i) to examine the spectral and propagation characteristics of mixed VRIGWs in relation to VRWs and IGWs in rapidly rotating TC-like vortices and (ii) to provide a more complete understanding of wave spectrum using a linearized, finite-differenced shallow-water equations model, the so-called shallow-water vortex perturbation analysis and simulation (SWVPAS), developed by Nolan et al. (2001, hereafter NMG). They are achieved by numerically solving the shallow-water normal-mode equations and then examining their wave spectrum distribution for given TC-like vortices. A third objective is to compare the wave propagation characteristics from the numerical solutions to those from the mixed-wave theory of ZZL.

The next section shows discretized normal-mode shallow-water equations (i.e., SWVPAS), discusses two different vortices used for VRIGWs studies, and then presents their frequency–WN relations for different signs of *Q*. Section 3 performs an eigenfrequency analysis of mixed VRIGWs, VRWs, and IGWs in SWVPAS, following ML and Zeng et al. (1990). Section 4 shows the structures and evolution of each class of the waves on a monopolar vortex with SWVPAS, and then compares the numerical solutions of propagating waves on a TC-like vortex to the analytical solutions of ZZL.

## 2. Numerical model and basic states

*f*-plane, shallow-water equations in polar (

*r*,

*λ*) coordinates (ML; ZZL), which is also the basic framework of SWVPAS, given bywhere

*u*′ and

*υ*′ are the radial and azimuthal perturbation velocity, respectively;

*h*′ and

*r*) is the mean azimuthal wind, and

*r*) =

*r*)/

*r*is the mean angular velocity;

*f*) varying with radius;

*κ*, set to either 1 or 0, is used to indicate the effect of the radial advection of

*κ*was set to null in order to obtain a second-order ordinary differential equation with constant coefficient, but

*κ*= 1 is used herein.

*u*

_{n},

*υ*

_{n}, and

*h*

_{n}denote the perturbation amplitudes in terms of complex functions of radius and time, as in NMG, and

*n*denotes the azimuthal WN; and

^{−1}), we obtain the azimuthal Fourier spatial version of the linearized perturbation equations [Eqs. (2a)–(2c)], as in SWVPAS; that is,To ensure the finite amplitudes of perturbation quantities, we require that

*r*= 0 and as

*r*→∞ for

*n*≥ 1.

*r*

_{b}] equally into 2

*N*pieces, giving the grid distance of

*δr*=

*r*

_{b}/2

*N*, and then define the horizontal winds at even grid points and the mass variables at odd grid points; namely,where

*k*= 1, 2, …,

*N*. The radial domain [0,

*r*

_{b}] is truncated at an outer radius of

*r*

_{b}= 2000 km where

*r*

_{b}is greater than the radius of Rossby deformation. Since we are only interested in the cases of

*n*> 0, the boundary conditions are simply

*r*= 0 and

*r*=

*r*

_{b}. Then, the discretized system [Eqs. (6a)–(6c)] can be considered as a standard matrix eigenvalue problem; that is,where

*N*− 1) × (3

*N*− 1) matrix representing discretization of the differential operator of Eqs. (6a)–(6c) with the boundary conditions included. The matrix eigenvalue problem is numerically solved using SWVPAS, whose functions have been improved herein to include all resolvable eigenvalues.

It is evident from Eq. (7) that each *N*, we can minimize the distortion of frequency spectrum caused by numerical discretization. We find that the use of *N* = 2000 with *δr* = 1 km is satisfactory for this purpose.

Next, we need to consider the impact of a basic-state or mean vortex on wave frequencies and structures. Two types of vortex profiles—that is, monopolar and hollow in terms of *r*_{m}). Figure 1c shows the radial distribution of azimuthal flow, ^{−1} is the maximum azimuthal flow at the RMW and *R* = *r*/*r*_{m} is the nondimensional radius. Figure 1d shows the corresponding height field that is in gradient balance with the azimuthal flow. This type of vortex profile has been shown by MK and ML to favor the generation and propagation of VRWs, and it will be shown herein to also allow for the generation and propagation of mixed VRIGWs.

In contrast, a hollow vortex is characterized by a minimum in *ζ*) that is located slightly inside the RMW. Hollow vortex profiles have been used to examine the local frequency relation of IGWs, VRWs, and VRIGWs by ZZL and the dynamics of barotropic or algebraic instability in TCs by Schubert et al. (1999), NMG, Nolan and Montgomery (2002), and Zhong et al. (2010). Figure 1 shows an example of a hollow vortex, with *r*_{m} = 57.5 km, that is based on the simulation of Hurricane Andrew (1992) by Liu et al. (1997). Note that its peak

Note the two distinct *r* = 0 and then decreases with radius, whereas for the hollow vortex, it manifests an outward increasing tendency at *r* = 0 and attains a peak value before reaching the RMW. Although both

Figure 1d compares the height fields between the monopolar and hollow vortices. Because of the larger amplitude of

At this point, one may wonder if the monopolar vortex would also allow the development of VRIGWs. For this purpose, we have repeated all the calculations as those in ZZL, including the radial distribution of the Bessel parameter and several nondimensional parameters given in Eq. (1) and found little qualitative differences in these parameters between the monopolar and hollow vortices (not shown), except that the former has no singularity near *R*_{0} (see Fig. 2b in ZZL for an example at the RM*ζ* associated with a hollow vortex). Figure 2 shows that the existence of such a singularity does affect the radial distribution and magnitude of *Q* between the two types of vortices but affects little the theoretical implication of mixed-wave motions. That is, the condition of *Q* ≤ 0 takes place in the core region of the monopolar vortex, albeit with much smaller magnitudes, as compared to the eyewall region near the RMW of the hollow vortex. Note that the value of *Q* in Fig. 2, calculated with *m* = 0.8, is one order of magnitude smaller than that calculated with *m* = 1.5 in Fig. 7 of ZZL. This difference could be attributed to the use of large *Q* ≤ 0 is met, depending upon the combination of their basic-state parameters and *m* [see Eq. (1b)].

To see further the generation of VRIGWs in relation to the other two classes of pure waves in the monopolar vortex, Fig. 3 shows a frequency–WN diagram for three different values of *Q*, based on Eq. (1), in which *ω* denotes the nondimensional intrinsic wave frequency; see Fig. 6 in ZZL for a similar diagram associated with a hollow vortex. When *Q* > 0, there are a pair of high-frequency IGWs propagating in opposite directions and a low-frequency VRW propagating against the mean flow. The two classes of waves have fundamentally different frequency distributions with WN *n*. That is, the frequencies of IGWs increase rapidly from the value of *f* at WN 0, whereas the VRW frequency increases slowly from the origin and reaches a peak at WN 1 and then decreases slowly with WN *n*. In the case of *Q* = 0, we see two allowable waves with frequencies increasing slowly at higher WNs but faster at lower WNs, which are similar in physical characteristics to those of VRWs and IGWs, respectively; they are so-called mixed VRIGWs. The mixed waves occur in the vicinity of the peak *Q* < 0, there are a mixed VRIGW propagating in the same direction as the mean flow (dashed–dotted lines in Fig. 3) and a pair of growing (decaying) mixed waves (not shown). All of these properties are similar to those of a hollow vortex shown in Fig. 6 of ZZL.

## 3. Spectrum and eigenmode structures of free waves

In this section, we examine the wave spectrum and eigenmode structures in the discretized system [Eqs. (6)] with the monopolar and hollow vortices as the basic state, respectively.

### a. Wave spectral analysis

Wave spectrum represents the aggregation of eigenvalues for

By definition (Zeng et al. 1990; Eidelman et al. 2004), when the resolvent (^{−1}, where ^{−5} s^{−1} corresponding to a pair of pure inertial waves. In addition, the frequencies of IGWs increase with WN *n*, which is consistent with the theory of IGWs (Pedlosky 2003) and the eigenmode analysis of ML.

However, the wave spectral structures become complicated in the presence of nonvanishing mean flows. If *n**n**n**n**n**n**n**n**r* = *r*_{c}, at which *r*_{c}) = 0, implying that a singularity occurs at *r*_{c}—the so-called critical radius (ML). In this case, only an integration of its eigenfunction over the spectrum range can constitute the particular solution of Eqs. (6) (Zeng et al. 1990). Thus, the pertinent eigenvalues are distributed in the continuous spectrum of

*r*=

*r*

_{0}from the nondivergent barotropic vorticity equation,where

*r*=

*r*

_{0}as assumed in MK and ZZL; and

*l*is the radial WN. Clearly, VRWs owe their existence to the radial gradient in

^{−1}, which corresponds to a frequency range of 10

^{−3}–10

^{−4}s

^{−1}. It is apparent from Figs. 1 and 4b that VRWs occur mostly within

*r*=100 km where the radial gradient in

*n*

*r*

_{c}. In addition, Schecter and Montgomery (2004) indicated that VRWs tend to be damped at

*r*=

*r*

_{c}, so little perturbation structures of VRWs could be seen beyond

*r*=

*r*

_{c}. We can also see from the above scale analysis that

^{−4}–10

^{−5}s

^{−1}, and

*n*

*n*

*n*

*n*

*m*/

*R*

_{m}is the dimensional form of

*m*(

*n*), and

Evidently, the IGW frequencies are closely associated with vortex intensity. It can be shown that when the rotational flow is weak, *n**c*_{0} ~ 10^{2} m s^{−1}, *m* ~ 10^{−1}, *m*/*R*_{m} ~ 10^{−6}, and *n* ~ 10^{0}, the threshold value of ^{−4}–10^{−5} s^{−1}, and it decreases with increasing WN *n*. Thus, the IGW frequencies tend to move toward the lower-frequency range for strong TCs and enter the range of continuous spectrum when the basic-state rotation exceeds

Based on the above analysis, we may see the following three scenarios. First, for a monopolar vortex, when its intensity satisfies *n**n**Q* ≤ 0 (ZZL). This is the main reason for the generation of mixed VRIGWs in both monopolar and hollow vortices. Third, the sign change of *ζ* in a hollow vortex could generate two oppositely propagating VRWs on each side, making *n*^{−5} s^{−1}) for WN *n* = 1–3: the continuous spectrum between 0 and *n*

However, Fig. 5a shows that the wave spectral structures change substantially when the vortex intensity exceeds a threshold. Although the wave spectrum still exhibits continuous and discrete distributions inside and outside the range [0, *n**n*, shorter VRWs tend to have more mixed VRIGWs properties.

### b. Eigenmode analysis

Since both observational and modeling studies show the slower-than-mean-flow propagation of disturbances in the eyewall (i.e., with positive Doppler-shifted frequencies), we may simply examine the eigenstructures of the three classes of waves as being characterized by their amplitudes: high, intermediate, and low frequencies, respectively. High-frequency waves include discrete spectral points that are greater than *n**n**n*^{−3} and 7.13 × 10^{−4} s^{−1} as two representative waves and plot their radial structures in Figs. 6a and 6b and in Figs. 6c and 6d, respectively.

It is apparent that the high-frequency WN-1 wave exhibits radial wavelike structures for the perturbation variables ^{−7} to 10^{−8} s^{−1} inside *r* = 100 km and then to 10^{−9} s^{−1} outside *r* = 100 km). Although divergence is slightly smaller than *r* = 30 km, it is one to two orders of magnitude greater than

In contrast, the low-frequency wave has little radial wavelike structures (Figs. 6c,d), which confirms the earlier analysis of the dispersion relation of VRWs. As mentioned before, each VRW mode has a critical radius *r*_{c}, at which the Doppler-shifted frequency vanishes. In fact, *r*_{c} is determined by wave frequency, and it is smaller for higher frequency. At *r*_{c}, *r*_{c}, suggesting that VRWs can develop only in the monopolar core region. Unlike IGWs, *r*_{c} and is about two orders of magnitude greater than that of divergence. In addition, *r*_{c}, indicating that the low-frequency wave is balanced and rotationally dominated. All of these are the typical characteristics of VRWs.

A critical radius also occurs for the intermediate-frequency WN-1 wave, whose Doppler-shifted frequency is close to *n**r*_{c}. Outside it, however, the wavelike structures are similar to those of IGWs, with much smaller

However, as the azimuthal WN increases, the decreased frequency of VRWs and the extended frequency range of IGWs make some of their radial wave structures differ from those of WN-1 waves. Figure 7 shows WN-2 wave structures associated with a high-frequency wave at ^{−3} s^{−1} (Figs. 7a,b), an intermediate-frequency wave at ^{−3} s^{−1} (Figs. 7c,d), and a low-frequency wave at ^{−4} s^{−1} (Figs. 7e,f). Although the radial structures of the high-frequency WN-2 wave are similar to its corresponding WN-1 structures (cf. Figs. 7a,b and 6a,b), its *r* = 30 km, indicating an important gravitational impact at higher WNs.

Figures 7c and 7d show an example of intermediate-frequency wave with *r*_{c} = 50 km. Its perturbation structures within and near *r*_{c} are similar to those shown in Figs. 6c and 6d except for the half- versus one-quarter wavelength oscillation of *r* = 0. The presence of very high amplitude of *r*_{c}.

The low-frequency WN-2 wave also has a critical radius, at which *r*_{c} appears in the outer region, near *r* = 180 km, rather than in the inner-core region. Just like the VRW in Fig. 7c, the discontinuity of *r*_{c}. Despite the outward shift of *r*_{c}, the inner-core region is dominated by convergence with its magnitude decreasing rapidly toward *r*_{c}. Although the magnitude of *r*_{c}. Moreover, *r*_{c}, indicating that the cyclonic flows corresponds to a low pressure region, like VRWs, and to a high pressure region, like IGWs, respectively, inside and outside *r* = 60 km. Thus, this mode should be regarded as a mixed VRIGW in the core region. Similar features can also be found for higher-WN waves (not shown). It follows that at higher WN, low-frequency waves tend to exhibit more mixed-wave characteristics. ML also examined the low-frequency waves at WN 2 and higher, and found this “unexpected” mixed-wave structure in their Rossby shear modes. However, the theoretical framework ML used to understand VRWs, which is based on nondivergent absolute vorticity equation, cannot describe the fundamental properties of mixed VRIGWs. So ML could only speculate the existence of some commonalities with equatorial mixed Rossby–gravity waves found by Matsuno (1966).

Figure 8 shows the WN-2 eigenmodes of intermediate- and low-frequency waves on the hollow vortex for the purpose of comparing them to those on the monopolar vortex given in Figs. 7c–f. Only their propagation solutions are selected because of the presence of mixed-wave instability near the RM*ζ* where *Q* < 0 (see Fig. 5 in ZZL). In this regard, use of monopolar vortices may be more suitable for theoretical studies of wave motions. We see that the general wave structures, the larger magnitude of *r*_{c}. This indicates that these intermediate- (Figs. 8a,b) and low- (Figs. 8c,d) frequency waves correspond to VRWs and mixed VRIGWs, respectively.

This mixed-wave mode could play an important role in geostrophic adjustment as discussed by ML, but in different ways over different regions of a hollow vortex. Here, let us consider a rapidly rotating vortex by defining *S*^{2} = *S*^{2} < 1, the wave motion is more balanced, like VRWs, with positive (negative) *S*^{2} > 1, the mass and wind fields are unbalanced, and IGWs should exert more influences on the adjustment. To help visualize the above points, Fig. 9 shows the radial distribution of *S*^{2} associated with the low-frequency modes for the monopolar (as in Figs. 7e,f) and hollow (as in Figs. 8c,d) vortices. Both contours exhibit wavelike radial distributions of *S*^{2}. Of particular relevance is that *S*^{2} is less than unity in the core region (i.e., 0–50 km) and near *r*_{c} (i.e., 100–200 km), whereas it is greater than unity elsewhere. Clearly, these features reflect the balanced and unbalanced characteristics of the mixed-wave mode, corresponding to the respective roles of VRWs and IGWs at different radii of the vortices. Thus, this mode should not be classified as a VRW mode as ML.

## 4. Propagation of WN-2 waves on the monopolar and hollow vortices

*N*wind grid points (

*k*= 1, 3, 5, …, 2

*N*− 1), andat the

*N*− 1 height grid points (

*k*= 2, 4, 6, …, 2

*N*− 2). Here,

*a*is the wave index, whose value of 1–3 represents IGWs, VRWs, and mixed VRIGWs, respectively;

*b*is the eigenfrequency index ranging from 1 to

*B*

_{1}, from 1 to

*B*

_{2}, and from 1 to

*B*

_{3};

*B*

_{1},

*B*

_{2}, and

*B*

_{3}are the total eigenmodes of IGWs, VRWs, and mixed VRIGWs, respectively, as classified by eigenfrequencies in Fig. 5 and wave characteristics in Fig. 7 or Figs. 8c and 8d, and they satisfy

*B*

_{1}+

*B*

_{2}+

*B*

_{3}

*=*3

*N*− 1;

*; k*is the radial index; and the coefficient

*C*

_{b}should be typically inverted from Eq. (10) for the given initial perturbations

In contrast, numerical integrations of the SWVPAS model from the initial composite perturbations cannot guarantee the propagation of a pure class of waves, even started from a pure one, because the model contains the dynamical mechanisms for the generation of all the three classes of waves, as indicated by Eq. (2). Thus, this model property has some limitations on the above-mentioned comparative analyses.

### a. Propagation of WN-2 waves on the monopolar vortex

Figure 10 compares the propagation characteristics of the three classes of waves, based on the linear superimposition and numerical integration approaches. Eight eigenmodes from each wave class are selected with their eigenfrequency indices *b* and wave characteristics determined from the knowledge of Figs. 5 and 7, respectively; their summations for each wave class at *t* = 0 are then defined as the initial (composite) perturbations. Note that for each wave class the coefficient *C*_{b} should be set to be either 1 for selected eigenmodes or 0 for unselected eigenmodes when the summations in Eq. (10) are executed at *t* = 0 or any subsequent time using the linear superimposition method.

We see that the initial composite perturbations represent well the WN-2 characteristics of IGWs (Figs. 10a,j), VRWs (Figs. 10b,k), and VRIGWs (Figs. 10c,l) in terms of rotation and divergence, and the mass–wind relation; their radial structures and wave amplitudes [as well as vorticity and divergence (not shown)] are similar to those associated with single waves shown in Fig. 7. Moreover, we can still see a wide radial range of the IGWs and VRIGWs activity (i.e., greater than 200 km) but a limited radial range of the VRWs activity (i.e., less than 50 km), as determined by their *r*_{c} (cf. Figs. 10 and 7).

Because of its fast azimuthal propagation, the IGWs from both approaches develop lengthy spiral bands after 60 min (Figs. 10d,g). By comparison, the VRWs exhibit slow clockwise propagation within *r*_{c}, only having slight shape changes even after 120 min (Figs. 10e,h). Large cross-isobaric (divergent) and nearly isobaric (rotational) flows can be clearly seen from the IGWs and VRWs, respectively. However, numerical integrations tend to inevitably produce slightly more rotational and divergent components associated with the respective IGWs and VRWs than those from the linear superimposition, as expected from the earlier discussion. The modeled flows are also stronger (weaker) in the central area for the IGWs (VRWs) than those superimposed (cf. Figs. 10d,e and 10g,h) because of the influences of the other waves, similarly for the faster clockwise movements of the VRWs. So, strictly speaking, the waves shown in Figs. 10g and 10h are not pure ones.

The right column of Fig. 10 displays the mixed-wave characteristics of the IGWs and VRWs with complicated mass and wind relations. That is, the initial perturbations (Fig. 10c) exhibit rotational (divergent) characteristics of VRWs (IGWs) within *r* = 50 km (outside roughly *r*_{c} = 130 km), with their structures similar to those shown in Fig. 10b (Fig. 10a). Between the two radii, we see the coexistence of both rotation and divergence with 45° phase shift, as indicated by local flow vectors with respect to isobars, which are the exact characteristics of mixed VRIGWs. At *t* = 120 min, Fig. 10f shows well-structured spiral bands within *r* = 200 km of the linear superimposed mixed VRIGWs, like the IGWs, but with the major highs and lows trapped within *r*_{c} = 130 km, like the VRWs (Fig. 10f). Of importance is that like the equatorial mixed Rossby–gravity waves, the mixed VRIGWs are characterized by near-isobaric circulations associated with the highs and lows, but cross-isobaric flows outside *r*_{c}. Of further importance is that their height and wind perturbations attenuate at rates (Figs. 10c,f) that are much slower than those of the IGWs (Figs. 10a,d) and VRWs (Figs. 10b,e), implying that such quasi-balanced wave properties may have more significant impact than the other waves on the maintenance of rainbands in TCs.

By comparison, numerical integrations produce more pronounced distortions of the mixed VRIGWs than the VRWs (Figs. 10h,i). They can be seen from the height and wind perturbations that are completely out of phase outside *r* = 100 km, with more cross-isobaric components over wide regions. As will be shown next, these features are transient during this adjustment period in which all the three classes of waves interact.

### b. Propagation of WN-2 waves on the hollow vortex

A different set of eight eigenmodes is also selected for each wave class propagating on the hollow vortex in order to see individually different WN-2 wave characteristics from those associated with the monopolar vortex. It is necessary to select such a different set of eigenmodes, although it is still close to the one used in Fig. 10, because the distribution of the wave classes differ under different basic states, as shown in Fig. 5. Indeed, Figs. 11a–c exhibit similar composite perturbation structures to those shown in Figs. 10a–c at *t* = 0, if viewed within *r* = 70 km, except for nearly vanishing flows in the central region and the elongated *h*′ pattern associated with the VRWs (cf. Figs. 11b and 10b). Moreover, to facilitate comparisons to the analytical solutions of ZZL in which the critical radius is absent after assuming nonzero intrinsic frequency (i.e., *r*_{c} (50 km for the VRWs, and 300 km for the VRIGWs, as shown in Figs. 8b and 8d) by performing nine-point smoother. As a result, the initial composite-wave structures resemble quite well the analytical solutions shown in Fig. 9 of ZZL, albeit with some differences in detailed structures.

Like in the monopolar vortex, the linear superimposition also produces well-structured spiral bands in the hollow vortex for the IGWs at *t* = 60 min, and the other two waves *t* = 120 min. However, because of the reduced vortex intensity, their azimuthal propagations are slower than those in ZZL (cf. Figs. 9b,e,h in ZZL and Figs. 11d–f herein). Of significance is that after 120 min, the mixed waves develop more rotational characteristics inside the RMW, like the VRWs, because of the presence of strong inertial stability, but more divergent characteristics outside, like the IGWs (Fig. 11f). This can also be seen from the distribution of *S*^{2} in Fig. 9 showing a balanced regime within *r* = 50 km (i.e., *S*^{2} < 1), and unbalanced regimes beyond the RMW (i.e., *S*^{2} > 1). This inseparable property is unique for mixed VRIGWs compared to IGWs. Despite the presence of the IGW characteristics, the peak *h*′ amplitude of the mixed waves only attenuates 40% after 120 min, as compared to the 50% reduction of the VRWs.

Because of the unstable basic state of the hollow vortex, as also indicated by their *h*′ amplitudes (cf. Figs. 11g–i and 11d–f), numerical integrations produce more significant distortions of all the three waves than those associated with the monopolar vortex (cf. Figs. 11g–i and 10g–i), such as the radially coupled *h*′ within *r* = 40 km for the IGWs, and the more circular motions with an opposite phase relationship between the wind and height perturbations for both the VRWs and VRIGWs. In particular, the numerically integrated waves exhibit little spiral structures for the lower-frequency waves (Figs. 11h,i), as compared to the results of ZZL and linear superimposition. These waves also show sharp intensification within the RMW (Figs. 11h,i) and outward dispersion of the VRIGWs outside the RMW (Fig. 11i). In this regard, NMG indicated the growth of VRWs on a hollow vortex from numerical integrations using SWVPAS. Nevertheless, the long period of integrations shows little unbalanced flow, but balanced circulation of VRWs within the RMW (Figs. 11j–l); this is clearly determined by the SWVPAS dynamics, since the influences of the initial conditions decrease with time.

## 5. Summary and conclusions

In this study, the SWVPAS numerical model is used as a tool to study the matrix eigenvalue problems associated with the propagations of IGWs, VRWs, and mixed VRIGWs on the monopolar and hollow vortices. An eigenfrequency analysis indicates that as long as *n*

An eigenfunctional analysis reveals three distinct radial wave structures associated with three classes of waves in the SWVPAS model: (i) IGWs exhibit more wavelike structures with dominant divergent flows, and their

Eight eigenmodes for each wave class are selected and linearly superimposed in time to gain insight into different propagation characteristics of the three wave classes on the monopolar and hollow vortices. Results show that cross-isobaric flows and isobaric flows associated with the IGW and VRWs, respectively, as expected, whereas the VRIGWs exhibit more balanced flows, like the VRWs, inside the RMW, but more unbalanced characteristics outside, like the IGWs. Moreover, the IGWs propagate rapidly outward as spiral bands, in significant contrast to the slow propagation of the VRWs and VRIGWs. The latter two waves tend to be trapped within their critical radii. On the other hand, the IGWs weaken rapidly with time, whereas both the VRWs and VRIGWs show slow reductions in amplitude. This indicates the potential importance of the VRWs and VRIGWs in maintaining spiral rainbands and organizing deep convection in the eyewall.

It is shown that numerical solutions of pure classes of the composite waves on the hollow vortex are generally similar to the analytical results of ZZL, except for some differences in small-scale details owing to the use of Doppler-shifted frequencies and the existence of *r*_{c}. Thus, we may conclude that VRIGWs, at least at WN 2, containing both intense vortical and divergent flows, should be common in TCs, especially in intense hurricanes. In a forthcoming study, we will examine the structural evolution of these waves using real-data-simulated hurricane cases and investigate their roles in the formation of spiral rainbands and the eyewall replacement cycle.

## Acknowledgments

We are grateful to Prof. David Nolan and three anonymous reviewers for their constructive comments that have helped significantly improve the presentation of this article, and Prof. Hancheng Lu for his kind support. This work was funded by the Major State Basic Research Development Program of China (2013CB430103), Natural Science Foundation of China (41275002 and 41175054), the US NSF Grant ATM-0758609, Natural Science Key Foundation of China (41230421), and China Postdoctoral Science Foundation (2013M531321).

## APPENDIX

### Frequency Separation between IGWs and VRWs

*n*

By definition, we may assume

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