## 1. Introduction

There have been many attempts to understand how wave motions within a tropical cyclone regulate its intensity (see below). Most of these efforts were based on wave theories that neglect moisture. Since clouds pervade tropical cyclones, such an approximation is not soundly justified. The warming or cooling of air parcels by condensation or evaporation of water content can dramatically affect the wave dynamics. In this paper, we will begin to see how.

In particular, we will examine waves in a nonprecipitating cloudy vortex whose basic state has no secondary circulation and is gravitationally stable. In general, the basic state has fluctuations of cloudy and unsaturated air, but those fluctuations are assumed to occur on smaller scales than the waves of interest. We will not consider the effects of boundary layer fluxes, radiative heat fluxes, nor eddy diffusion on the wave dynamics. Furthermore, we will suppose that perturbations are hydrostatic and governed by linearized equations.

In sections 2–5 we will formally develop this model. Although it applies to both hydrostatic inertia–gravity (IG) waves and vortex Rossby (VR) waves, we will focus our attention on the latter.

In section 6, we will briefly reexamine the eyewall instability of a baroclinic cyclone that resembles a category 3 hurricane. Nolan and Montgomery (2002) reported that this particular vortex destabilizes most rapidly by the interaction of counter propagating VR waves on opposite edges of the eyewall. We will show that increased fractional cloud coverage slows the instability, and thereby hinders asymmetric eyewall breakdown. The consequence on hurricane intensity is a topic of ongoing study. On the one hand, potential vorticity mixing after eyewall breakdown can directly reduce the maximum tangential wind speed of a hurricane (e.g., Schubert et al. 1999; Kossin and Schubert 2001). On the other hand, the thermo-fluid dynamics connected to eyewall breakdown is subtle, and may actually lead to a stronger hurricane (Emanuel 1997; Montgomery et al. 2002; Persing and Montgomery 2003; Montgomery et al. 2006).

In section 8, we will examine how moisture influences the discrete VR waves of a barotropic cyclone, whose vertical vorticity decreases monotonically with radius. These waves account for precessing tilts and elliptical (triangular, square, etc.) deformations of the vortex core. If the Rossby number of the cyclone exceeds unity, its baroclinic VR waves can efficiently excite frequency-matched spiral IG waves in the environment (Ford 1994a, b; Polvani et al. 1994; Plougonven and Zeitlin 2002; Schecter and Montgomery 2003, 2004, 2006). Surprisingly, producing IG wave radiation compels a VR wave to grow. Such growth is usually opposed by a critical layer that resonantly absorbs wave activity (Briggs et al. 1970; Pillai and Gould 1994; Schecter et al. 2000, 2002; Balmforth et al. 2001; Mallen et al. 2005). We will show that moisture tends to enhance damping by the critical layer and thereby suppress the radiative instability. Consequently, cloud coverage can inhibit the loss of angular momentum by IG wave radiation. Such loss might otherwise merit some consideration in the angular momentum budget of a hurricane (e.g., Chow and Chan 2003).

The theory of discrete VR waves, especially of the tilt mode, also provides a framework for understanding the response of a vortex to ambient vertical shear. Recent studies (e.g., Jones 1995; Reasor and Montgomery 2001; Schecter et al. 2002; Reasor et al. 2004) have shown that reducing the static stability of the atmosphere typically increases (i) the azimuthal phase velocity and (ii) the decay rate of the discrete tilt mode of a monotonic (or quasi-monotonic) cyclone. Both of these modifications increase the cyclone’s natural resistance to misalignment under the influence of ambient shear. As we will show in section 8, increasing the cloud coverage has similar effects on the tilt mode and thereby also improves vortex resilience (cf. Patra 2004). This result is quite reasonable, in light of the aforementioned studies, if one connects denser cloud coverage to lower static stability (e.g., Durran and Klemp 1982).

Another class of VR waves comprises continuum mode disturbances that are sheared by differential rotation of the mean flow. Sheared VR waves that are generated by localized bursts of convection can transport angular momentum toward the radius of maximum wind, and thereby intensify a cyclone (e.g., Montgomery and Kallenbach 1997; Montgomery and Enagonio 1998; Möller and Montgomery 1999, 2000; Enagonio and Montgomery 2001). They may also have some connection to moving spiral rainbands (e.g., Montgomery and Kallenbach 1997). Although we will not directly examine the influence of clouds on sheared VR waves, a theory based on our moist model would be a modest formal extension of the existing dry theory.

On a technical note, the moist wave equations of section 4 are nearly identical to the linearized dry hydrostatic primitive equations. Moisture simply reduces the Eulerian rate of cooling/warming by updrafts/downdrafts by a factor ϒ [see Eq. (31)]. The ϒ distribution is a property of the basic state of the vortex, and has only radial and vertical variation. Its value indicates the average reduction of the vertical buoyancy restoring force of air parcels along an azimuthal circuit (see appendix G). As the fraction of cloudy air in the circuit increases from zero, ϒ decreases from unity. In section 5, we will present a formula for ϒ [Eq. (37)] that makes a baroclinic vortex neutral to slantwise convection. In addition, we will examine the spatial distribution of ϒ for a realistically simulated hurricane. As expected, we will find that ϒ is smallest in the very moist eyewall.

The moist wave equations presented here are an improvement over the usual dry model, and their solutions provide insight into the most basic effects of cloud coverage. Nevertheless, they exclude several factors (see paragraph 2) that might significantly alter wave motions in actual or realistically simulated hurricanes (cf. Wang 2002a, b; Chen and Yau 2001; Chow et al. 2002; Chen et al. 2003; etc.). A more comprehensive theory should be the goal of future research.

The remainder of this paper is organized as follows: section 2 describes our parameterization of moist processes in a cloudy vortex. Section 3 presents the nonlinear hydrostatic fluid equations that are supposed to govern the vortex. Section 4 presents the linearized wave equations. Section 5 discusses some important ϒ distributions. Section 6 illustrates how cloud coverage can temper an asymmetric eyewall instability. Section 7 contains a formal discussion of the eigenmode problem for a cloudy vortex. Section 8 examines how cloud coverage can cause or accelerate the decay of discrete VR waves in a monotonic cyclone. Such decay inhibits the radiation of spiral IG waves and improves vortex resilience. Section 9 summarizes our main results. Appendices A–F provide some analytical and computational details for those who may wish to continue the analysis of waves in a cloudy vortex beyond the scope of this paper. Appendix G informally discusses the physical meaning of ϒ.

## 2. Moisture parameterization

### a. The water mass equations

*υ*) and liquid (

*l*). Further suppose that there is no precipitation, and that there is no diffusion of water mass. Then, the total water mixing ratio,

*q*

_{υ*}(

*T*,

*p*) is the saturation vapor mixing ratio,

*T*is temperature, and

*p*is total pressure.

### b. The heat equation

*c*and

_{pd}*c*are the specific heats of dry air (at constant pressure) and of liquid water;

_{l}*R*and

_{d}*R*are the gas constants of dry air and of water vapor;

_{υ}*p*is the pressure of dry air, whereas

_{d}*e*and

*e**(

*T*) are the actual and saturation values of the water vapor pressure;

*L*(

_{υ}*T*) is the latent heat of vaporization.

*ρ*is the total mass density of a fluid parcel and

*p*

_{0}is a reference pressure, taken to be 10

^{5}Pa.

*q*]. We may choose these variables to be

_{υ}*p*,

*s*and

_{m},*q*. In terms of these variables, the functional form of

_{t}*θ*differs for saturated (

_{ρ}*s*) and unsaturated (

*u*) air parcels; that is,

*H*(

*μ*) denote the unit step function. Then, we may rewrite Eq. (7) as follows:

*dH*/

*dμ*=

*δ*(

*μ*), in which

*δ*(

*μ*) is the Dirac distribution. Figure 1 is a contour plot of minus

*χ*for an air parcel with

*q*= 0.015. This value of

_{t}*q*is characteristic of, say, a parcel in the lower region of a tropical storm. Note that the negative value of

_{t}*χ*increases by orders of magnitude as a parcel moves from an unsaturated state (

*μ*< 0) to a saturated state (

*μ*> 0) along any path, such as a moist adiabat (dashed curve).

*χ*= 0 and

*θ*is conserved.

_{ρ}*χ*:

*ϵ*≡

*R*/

_{d}*R*= 0.622. Since

_{υ}*q*

_{υ}_{*}=

*q*

_{υ}_{*}(

*T*,

*p*) and

*L*=

_{υ}*L*(

_{υ}*T*), the right-hand sides of Eqs. (15) and (16) are functions of the measurable quantities

*T*,

*p*, and

*q*.

_{t}## 3. The moist primitive equations

For the remainder of this paper, we will assume that vertical accelerations are small and that hydrostatic balance applies. Consequently, we may use *p* as the vertical coordinate instead of height. In addition, we will assume that the vortex occupies a sufficiently small section of the globe where the *f*-plane approximation is valid.

**u**is the horizontal velocity vector,

*ω*≡

*dp*/

*dt*,

*f*is the constant Coriolis parameter, and

*ϕ*=

*gz*is the geopotential. Furthermore,

**ẑ**is the vertical unit vector, and

**∇**

*is the horizontal gradient operator, with spatial derivatives evaluated at constant*

_{p}*p*.

**∇**≡

**∇**

_{p}−

**ẑ**∂/∂

*p*, and

**∇**have different dimensions (inverse length and inverse pressure). We have used a similar convention for

**. Of course each term of**

*η***·**

*η***∇**(for example) has the same dimension.

## 4. The wave equations

This section presents a linear model for wave dynamics in a cloudy vortex whose basic state has no secondary circulation. We will use a vortex-centered cylindrical coordinate system in which *r* and *φ* are the radius and azimuth, and *u* and *υ* are the radial and azimuthal velocities. Overbars and primes will mark basic state and perturbation fields, respectively.

### a. Balanced basic state

*r*,

*p*). We may define

*υ*. For notational convenience, we introduce the following auxiliary fields:

*η*

*ω*

*= 0.*u

*r*,

*p*) and density potential temperature

*(*θ

_{ρ}*r*,

*p*) are defined to satisfy gradient balance,

The basic state also has a water mass profile * q _{t}*(

*r*,

*p*), which we take to be a tempero-azimuthal mean of

*q*. We further define

_{t}*χ*

*r*,

*p*) to be a tempero-azimuthal mean of

*χ*. Note that the difference between

*χ*

*r*,

*p*) and

*χ*(

*,*θ

_{ρ}*,*q

_{t}*p*) may be substantial if

*q*fluctuates between saturated and unsaturated values within the

_{t}*t*and

*φ*averaging intervals.

### b. Small perturbations

*q*′

_{t}, and is therefore closed.

*Note that the above wave equations are the same as those for a dry vortex, with* ∂* θ _{ρ}*/∂

*p*→ ϒ∂

*/∂*θ

_{ρ}*p*.

Note also that linearization of the heat equation is a bold approximation. Linear theory assumes that quadratic perturbation terms are subdominant to linear terms in the equations of motion. However, *ω*′*χ*′ is not necessarily negligible. Since *χ* varies discontinuously between unsaturated and cloudy air, *χ*′ can have fluctuations of order *χ*

## 5. The buoyancy reduction factor

For the remainder of this paper, we will treat the buoyancy reduction factor ϒ(*r*, *p*) as an adjustable parameter. We will attempt to choose ϒ distributions that have some meteorological significance.

### a. ϒ for symmetric stability

There is a common belief that intense atmospheric vortices, hurricanes in particular, have basic states that are nearly neutral to slantwise symmetric perturbations (e.g., Emanuel 1986). A necessary condition for the symmetric stability of a cloudy baroclinic vortex, with no secondary circulation, may be written as a local constraint on ϒ.

*θ*′

_{p}on the right-hand side of Eq. (33). The above substitutions are justifiable only if (i) the perturbation is roughly axisymmetric, (ii) the perturbation fields vary much more rapidly than the mean fields with

*r*and

*p*, and (iii) the radial wavelength of the perturbation is much less than

*r*. We will refer to the simplified equations as the local axisymmetric model.

*≡*M

*r*+

*fr*

^{2}/2. For global stability, this condition must hold at every point in the

*r*–

*p*plane (cf. Ooyama 1966).

We have not proven that condition (37) guarantees nongrowth of vortex-scale symmetric perturbations. Nevertheless, if ϒ = ϒ* _{nt}*, we will casually say that the vortex is neutral to slantwise convection. In section 6, we will see how the growth of an asymmetric perturbation can change as a vortex moistens and becomes neutral to slantwise convection.

It is worth mentioning that even complete cloud coverage can leave sections of a vortex symmetrically stable. Therefore, although setting ϒ = ϒ* _{nt}* is of theoretical interest, it is not always physical.

### b. ϒ for a simulated hurricane

Let us now consider the ϒ distribution of a mature hurricane in a “cloud-resolving” numerical simulation. This particular simulation was performed with the Regional Atmospheric Modeling System (RAMS), developed at Colorado State University (see Cotton et al. 2003). It was optionally set to include liquid cloud and rain, but not ice.

Figure 2 shows the instantaneous azimuthal means of the tangential and vertical velocity fields of the simulated hurricane. In addition, it shows the buoyancy reduction factor ϒ. To evaluate ϒ [the right-hand side of Eq. (32)], we used the instantaneous azimuthal means of *θ _{ρ}* and

*χ*.

^{1}To evaluate

*χ*at each grid point we used Eq. (15), Eq. (16), and the RAMS approximation for

*q*

_{υ}_{*}(

*T*,

*p*). As expected, ϒ is minimal in the main updraft region, where the air is very moist. This region also contains the strongest tangential winds. Low ϒ at high

Although tempting, we cannot use the theory of section 5a to properly evaluate the symmetric stability of the RAMS hurricane. Condition (37) strictly applies to a vortex that has no mean secondary circulation. In contrast, the RAMS hurricane is maintained by a strong radial inflow (not shown) that has a peak velocity of about 30 m s^{−1} near the surface. Still, the negative values of ϒ at the base of the main updraft might indicate that the vortex is gravitationally unstable in that region.

## 6. Eyewall instability

The circular eyewall of a mature hurricane can destabilize and evolve into a ring of mesovortices, which itself can destabilize and rapidly spread angular momentum into the eye. This process, the asymmetric instability of a cylindrical vortex sheet, has been studied extensively in laboratory experiments (e.g., Weske and Rankin 1963; Ward 1972; Vladimirov and Tarasov 1980; Montgomery et al. 2002). It has also been studied theoretically/computationally without much consideration of moisture (e.g., Michalke and Timme 1967; Rotunno 1978; Gall 1983; Davidson 1990, chap. 6; Schubert et al. 1999; Kossin and Schubert 2001; Nolan and Montgomery 2002; Terwey and Montgomery 2002). However, hurricane eyewalls are covered with relatively deep clouds (see Fig. 2). In this section, we will examine a case in which such cloud coverage gravely decelerates the linear stage of an asymmetric eyewall instability.

### a. Preliminary consideration: The quasigeostrophic shallow-water potential vorticity ring

The shallow-water potential vorticity ring is perhaps the simplest analogue of a mature hurricane. In this analogy, one relates cloud coverage to a reduction of the gravitational restoring force. As a preparatory exercise, we here examine the effect of lowering *g* (the gravitational acceleration) on the asymmetric breakdown of a shallow-water potential vorticity ring. For analytical simplicity, we confine our discussion to the quasigeostrophic (QG) parameter regime. Although hurricanes have basic states that are ageostrophic, we may expect their vortex-scale shear-flow instabilities to have QG-like properties (e.g., Montgomery and Shapiro 1995). We will return to this matter in section 6c.

*ψ*=

_{g}*gh*′/

*f*, in which

*h*′ is the surface-height anomaly of the shallow-water layer.

*l*that appears in the differential operator is the Rossby deformation radius. It is here defined by

_{R}*l*≡

_{R}*gh*

_{∞}

*f*, in which

*h*

_{∞}is the ambient surface height.

*r*

_{1}<

*r*

_{2},

*Z*

_{0}and

*Z*

_{1}are positive, and

*Z*

_{2}≡

*Z*

_{0}+

*Z*

_{1}. As

*r*increases, the PV jumps from

*Z*

_{0}to

*Z*

_{0}+

*Z*

_{1}at the radius

*r*

_{1}, and then drops to zero at

*r*

_{2}.

*n*= 0 member of the family that is defined by

*r*→ ∞, we have

*I*and

_{n}*K*are modified Bessel functions, and

_{n}*r*

_{>}(

*r*

_{<}) is the greater (lesser) of

*r*and

*r̃*. We note that if

*Z*

_{0}is zero,

*r*

_{1}and

*r*

_{2}, and then decays as

*r*tends to infinity.

_{1}and Ω

_{2}to denote the values of

*r*

_{1}and

*r*

_{2}, respectively. Using various analytical properties of modified Bessel functions that are, for example, listed in Abramowitz and Stegun (1972), one finds from Eq. (43) that

*n*. Both have the form

_{1}and Π

_{2}are complex constants, and c.c. denotes the complex conjugate of the preceding term. Evidently, both modes contain a pair of frequency-matched VR waves on opposite edges of the ring. The eigenfrequency is generally complex; that is,

*ν*=

*ν*+

_{R}*iν*, in which

_{I}*ν*and

_{R}*ν*are real.

_{I}*n*= 1, it can be shown that

*c*= 0 and the discrete modes are neutral. For

*n*≥ 2, there will exist a growing mode and a conjugate damped mode if the discriminant

*b*

^{2}− 4

*c*is negative.

^{2}we have found that

*l*→ 0. Therefore, any growing mode will neutralize as the deformation radius drops below a finite threshold. Such stabilization occurs in part because the VR waves on opposite edges of the ring become shielded from each other, and can no longer interact effectively. Of course, for more subtle reasons the growth rate may vary nonmonotonically with decreasing

_{R}*l*before shooting to zero.

_{R}Figure 3 illustrates the typical properties of an unstable mode of a shallow-water PV ring. The angular phase velocity of the mode is greater than Ω_{1} but less than Ω_{2}. Therefore, the inner VR wave is prograde, whereas the outer VR wave is retrograde. As predicted, the mode neutralizes as the gravitational restoring force (parameterized by *l _{R}*) drops below a finite threshold.

^{3}

### b. A category 3 hurricane

We now reexamine the eyewall instability of the category 3 “hurricane” in Nolan and Montgomery (2002, hereafter NM02). The reader may consult NM02 for an analytical description of the basic-state cyclonic vorticity *r*, *z*). The vorticity is easily integrated for azimuthal velocity *r*, *z*). The equilibrium profiles of pressure and density, * p*(

*r*,

*z*) and

*ρ*

*r*,

*z*), are found by an iterative algorithm that solves hydrostatic and gradient wind balance simultaneously. For this study, a Jordan mean sounding gives the thermal boundary condition at

*r*= 200 km (Jordan 1958). In addition, the Coriolis parameter is set to

*f*= 5 × 10

^{−5}s

^{−1}. Figure 4 portrays the basic state of the vortex on the

*r*–

*p*plane. Note that the eyewall (where

^{4}

*n*= 3. In order to understand how moisture influences the fastest growing mode, we here consider a set of numerical experiments with variable ϒ. Specifically, we let

*ε*increases from zero to one, the vortex goes from a dry state to a moist state that is neutral to slantwise convection. The lower right panel of Fig. 4 shows ϒ

*for the vortex under consideration.*

_{nt}*n*= 3 perturbation. Appendix C describes the numerical model that is used to evolve the disturbance forward in time. In all cases, a growing eigenmode prevails after a brief adjustment period. To be clear, the eigenmode is a disturbance of the form

*ν*≡

*ν*+

_{R}*iν*, in which

_{I}*ν*and

_{R}*ν*are real. Figure 5 illustrates the exponential growth of the mode for increasing levels of moisture. Evidently, cloud coverage tends to neutralize the mode.

_{I}^{5}

We note that the growth rate of the dry mode (3.3 × 10^{−4} s^{−1}) exceeds the value of 2.07 × 10^{−4} s^{−1} that appears in NM02. During the review process, D. Nolan (2006, personal communication) was able to repeat the computation of NM02 at a finer spatial resolution, which is now practical. With more than double the number of radial grid points, the growth rate of the dry mode increased to 2.7 × 10^{−4} s^{−1}. The remaining discrepancy between this value and our result is likely caused by (i) our different surface boundary condition (*ω*′ = 0 on a constant *p* surface as opposed to no stress on a constant *z* surface), (ii) our use of the hydrostatic approximation, and (iii) the absence of explicit eddy diffusion in our model.

### c. Comparison

^{6}To see this, let us examine its structure. The time-asymptotic PV perturbation approximately has the form

*ε*= 0 and for

*ε*= 0.8875. For

*ε*= 0, the dry case, the radial variation of |Π̂| has two distinct peaks. These peaks occur at the maximum positive and negative values of ∂

*r*. For

*ε*= 0.8875, |Π̂| has a similar structure, but the inner wave is fractured.

*σ*> 0, and below the arc

_{R}*σ*< 0. Whether dry or moist, the inner wave is above the arc, and is therefore prograde relative to the mean flow. In contrast, the outer wave is underneath the arc, and is therefore retrograde. These properties are consistent with the azimuthal propagation rule for VR waves in a cyclone: the propagation is prograde/retrograde if ∂

_{R}*r*is positive/negative (Montgomery and Kallenbach 1997).

*r*–

*p*variation of

^{2}for the dry and moist modes. In the eyewall, both modes appear to marginally satisfy condition (57).

^{7}

We note that if ^{2} ≪ 1, terms of order ^{2} and higher are subdominant in the perturbation equations. Neglecting these terms leads to the zero-order asymmetric balance model (Shapiro and Montgomery 1993; Montgomery and Shapiro 1995; Montgomery and Franklin 1998; Ren 1999; McWilliams et al. 2003). Like QG theory, the asymmetric balance model filters out inertia–gravity waves. Unlike QG theory, the basic state can have a Rossby number of order unity or greater.

In summary, the pertinent instabilities of a QG shallow-water ring and of the NM02 hurricane both consist of slow, counter propagating VR waves. By analogy to the QG shallow-water ring, it is no surprise that reduction of the gravitational/buoyancy restoring force by adding moisture suppresses the NM02 hurricane instability.

Of course, the analogy has limits. To begin with, the NM02 hurricane mode is not globally slow, as it is in QG theory. Figure 6 indicates severe violation of condition (57) in the periphery of the storm, in the eye of the storm, and in the upper troposphere. Previous studies have shown that such external violation can, in some cases, notably affect the growth rate of a vortex mode (Schecter and Montgomery 2003). Furthermore, moisture significantly alters the *r*–*p* structure of the mode. Figure 6 alone illustrates a vertical elevation and scale reduction of PV peaks. Figure 7 illustrates the changes of other perturbation fields. Note in particular that moisture causes mesoscale spatial oscillations to emerge in the eye. The emergence is most striking for *ω*′.^{8}

## 7. The eigenmode problem for a cloudy vortex

In section 8, we will examine the discrete eigenmodes of a cloudy vortex that resembles an incipient tropical cyclone. Before doing so, it is helpful to discuss the eigenmode problem more generally.

### a. Eigenmode problem for a generic vortex

*ν*that permit nontrivial solutions of Eq. (58), that is, the eigenfrequencies, are constrained by the radial and vertical (pressure) boundary conditions. We will specify these conditions when the need arises.

### b. Eigenmode problem for a barotropic vortex with separable ϒ

*can exist only if*

_{λ}*λ*is a constant. The vertical boundary conditions determine the permissible values of

*λ*. Higher values of

*λ*correspond to eigenfunctions with smaller vertical wavelengths. If, for example,

*ω*′ = 0 at

*p*=

*p*

_{min}and at

*p*=

*p*

_{max}, then the boundary conditions are

*d*Ψ

*/*

_{λ}*dp*= 0 at

*p*=

*p*

_{min}and at

*p*=

*p*

_{max}.

*yields*

_{λ}*l*here denotes the internal Rossby deformation radius, defined by

_{R}*l*decreases as the vertical wavenumber or the Coriolis parameter increases;

_{R}*l*also varies radially.

_{R}*cd*θ /

_{ρ}*dp*, in which

*c*is a constant. Then,

*l*

^{dry}

_{R}is the internal Rossby deformation radius in the absence of moisture. Since ϒ ≤ 1, it is clear that moisture reduces

*l*.

_{R}To complete the eigenmode problem, we must specify two radial boundary conditions. At the origin, we require that Φ* _{λ}* is zero (

*n*≥ 1) or finite (

*n*= 0). As

*r*→ ∞, we impose either a radiation or evanescence condition, depending on whether the mode is fast or slow relative to an ambient inertial oscillation. In appendix D, we give both the inner and outer asymptotic forms of Φ

*. From these, we can infer relations between Φ*

_{λ}*and*

_{λ}*d*Φ

*/*

_{λ}*dr*near the origin and at distant radii. We use these relations for mixed boundary conditions in our numerical eigenmode solver, which is based on the shooting method. The computational boundaries are typically at 0.01

*r*and 15

_{o}*r*, in which

_{o}*r*is the radius of the vortex core.

_{o}Interested readers may consult appendix E for a set of polarization equations that give the wavefunctions for *u*′, *υ*′, and *θ*′_{ρ} in terms of Φ* _{λ}*(

*r*) and Ψ

*(*

_{λ}*p*).

## 8. Suppression of spiral inertia–gravity wave radiation and improvement of vortex resilience by cloud coverage

During their early stages of development, and perhaps after eyewall breakdown, tropical cyclones are presently thought to have PV distributions that, roughly speaking, decrease monotonically away from the storm center. In this section, we examine how moisture affects the propagation and exponential growth of discrete VR waves in an ageostrophic monotonic cyclone. The most familiar of these waves include rotating tilts and elliptical deformations (e.g., Kelvin 1880; Schecter and Montgomery 2004). They are distinguished from continuum VR waves (e.g., Montgomery and Kallenbach 1997; Bassom and Gilbert 1998) in that their PV fields do not characteristically suffer spiral windup. Typically, spiral windup of PV leads to algebraic decay of the asymmetric geopotential perturbation, as opposed to exponential decay or growth.

Discrete VR waves have some role in regulating tropical cyclone intensity. This role is not completely understood, but some basic theory has been developed. In linear theory, discrete VR waves are compelled to decay by resonant transfer of wave activity into a critical layer (Briggs et al. 1970; Pillai and Gould 1994; Schecter et al. 2000, 2002; Balmforth et al. 2001; Schecter and Montgomery 2004, 2006; Mallen et al. 2005). During decay, the wave–mean flow interaction usually increases the maximum tangential wind speed in the vortex core (e.g., Montgomery and Enagonio 1998; Reasor and Montgomery 2001). On the other hand, if *ν _{R}* >

*f*, discrete VR waves are compelled to grow by exciting outward propagating spiral IG waves in the environment (Ford 1994a, b; Plougonven and Zeitlin 2002; Schecter and Montgomery 2004, 2006). This spontaneous radiation transports energy and angular momentum away from the vortex. The following demonstrates that core moisture increases the absorption rate of wave activity in the critical layer (indirectly) and thereby inhibits spontaneous IG wave radiation. In this sense, moisture can fortify the cyclone’s intensity.

The following demonstration also implies that cloud coverage improves the resilience of a monotonic cyclone that is exposed to ambient vertical shear. Suppose that the discrete VR wave under consideration is a tilt mode (*n* = 1). As mentioned previously, adding clouds to the cyclone either causes or accelerates decay of that mode. Therefore, a cloudy monotonic cyclone realigns more rapidly than its dry counterpart after it is tilted by transient vertical shear. As we will see, cloud coverage also boosts the azimuthal phase velocity of the mode. Intuition and basic modeling suggest that sustained vertical shear becomes relatively harmless if the phase velocity of the mode becomes greater than the shear rate (e.g., Jones 1995; Reasor et al. 2004).

### a. A barotropic cyclone

*r*increases from the origin,

*r*(see Fig. 8). The width of the transition layer is of order

_{o}*d*= 0.0375

*r*. The Rossby number of the cyclone, Ro ≡

_{o}*f*= 4, is characteristic of an incipient storm. Most importantly, it exceeds unity. The angular rotation frequency

*r*<

*r*, and decays nearly as

_{o}*r*

^{−2}for

*r*>

*r*.

_{o}*p*. Specifically, we let

*ε*)

^{2}at

*r*, and is unity for |

_{o}*r*−

*r*| ≫ 2

_{o}*d*(see Fig. 8). The

*ε*parameter can have any value between zero and one. Note that our choice of ϒ is minimal in the region of maximum azimuthal wind speed. This property was motivated by the simulated hurricane data in Fig. 2.

### b. The growth rate of a discrete vortex Rossby wave

Figure 9 shows a horizontal slice of the geopotential perturbation *ϕ*′ of a discrete VR wave in a cloudy tanh cyclone [Eqs. (71)–(72)]. The oscillation frequency *ν _{R}* of the wave exceeds the Coriolis parameter

*f*. Consequently, the VR wave resonantly excites an outward-propagating spiral IG wave in the environment. The coupled VR and IG waves constitute a single eigenmode.

*dry*barotropic vortex is given by the formula (cf. Schecter and Montgomery 2004)

*U*(

_{λ}*r*) and

*V*(

_{λ}*r*) are the radial wavefunctions of the radial and azimuthal velocities [see appendix E for their connection to Φ

*(*

_{λ}*r*)]. The superscript * is the complex conjugate operator, and ℜ[. . .] is the real part of the quantity in square brackets. The critical radius

*r*

_{*}is defined by

*r*defines the outer vortex boundary. Its exact value is somewhat arbitrary, but must exceed

_{υ}*r*

_{*}. The denominator

*M*is defined by

*r*

_{*}. Provided that our choice of

*r*is not orders of magnitude greater than

_{υ}*r*

_{*},

*M*is generally positive (Schecter and Montgomery 2004, 2006). Vertical boundaries do not appear explicitly in the equations above. However, the hidden vertical boundary conditions are

*ω*′ = 0 (and hence

*θ*′

_{ρ}= 0) at

*p*=

*p*

_{max}and at

*p*=

*p*

_{min}.

Equation (73) indicates that the growth rate of a discrete VR wave has two parts. The first term, *ν*_{rad}, is proportional to the outward angular momentum flux of the spiral IG wave that the discrete VR wave emits into the environment. This flux is generally positive; therefore, the IG wave emission destabilizes the discrete VR wave. The second term, −*ν _{cl}*, accounts for the resonant feedback of PV stirring in the critical layer. It is negative, since

*d*η /

*dr*=

*d*η /

*dr*< 0, and therefore acts to stabilize the discrete VR wave.

The preceding growth rate formula also applies to the discrete VR waves of a cloudy barotropic vortex in which ϒ is separable [see Eq. (65)]. Moisture implicitly changes the right-hand side of Eq. (73) by altering the wavefunctions, moving *r*_{*}, and giving *l _{R}* radial variation [see Eq. (70)]. Readers who would like to verify Eq. (73) in the moist context may follow the dry derivation in Schecter and Montgomery (2004). In doing so, one must replace the dry Boussinesq wave activity equation with its moist non-Boussinesq counterpart. Appendix F provides the pertinent equation.

### c. Computational results

Figures 10 and 11 illustrate the effects of moisture on the discrete VR waves of the tanh cyclone that is under consideration [Eqs. (71) and (72)]. These waves correspond to (nonsingular) numerical solutions of the eigenmode problem of section 7b. Figure 10 shows how the critical radii vary with the moisture parameter *ε*. The four plots cover azimuthal wavenumbers 1–4. The different curves in each plot correspond to different values of the dry Rossby deformation radius, *l*^{dry}_{R}. In general, adding clouds increases the azimuthal phase velocity of a wave and thereby decreases *r*_{*} toward *r*_{0}.^{9}

Figure 11 shows how the growth rates vary with *ε*. In general, adding clouds tends to decrease *ν _{I}* and stabilize a wave. Stabilization occurs because cloud coverage brings the critical radius to a region of relatively large negative PV gradient (

*d*η /

*dr*). This enables critical layer damping to prevail over radiative pumping (see below).

Figure 12 verifies the analytical growth rate formula of section 8b for a particular VR wave that is near marginal stability. Again, the growth rate is plotted as a function of cloudiness in the cyclone. The × symbols correspond to Eq. (73). To evaluate the right-hand side of this equation, we used the wavefunctions (*U _{λ}*,

*V*, Φ

_{λ}*) and critical radius (*

_{λ}*r*

_{*}) of the computed eigenmode. To evaluate

*M*, we removed the interval |

*r*−

*r*

_{*}| < 20|

*ν*/

_{I}*nd*

*dr*

_{*}| from the integral; that is, we removed an estimate of the linear critical layer (cf. Schecter et al. 2000; Schecter and Montgomery 2004). Evidently, Eq. (73) accurately reproduces the values of

*ν*(+ symbols) that are obtained directly from the eigenmode solutions.

_{I}Figure 12 also shows how the radiative pumping rate (*ν*_{rad}) and critical layer damping rate (−*ν _{cl}*) vary with the moisture parameter

*ε*. To obtain these rates,

*ν*

_{rad}and

*ν*were evaluated with

_{cl}*r*= 3

_{υ}*r*. As

_{o}*ε*increases from zero to one, the radiative pumping rate remains roughly constant. However, the negative PV gradient at

*r*

_{*}explosively grows, and the critical layer damping rate increases by orders of magnitude. Consequently,

*ν*becomes negative.

_{I}We note that as the growth rate becomes negative, the discrete VR wave transforms from an eigenmode into a quasimode. Quasimodes are not exact solutions to the eigenmode problem. For instance, the PV disturbance of a quasimode grows with time in the thin critical layer centered at *r*_{*}; in contrast, the PV disturbance of a damped eigenmode would decay everywhere. The numerical method for finding quasimodes is similar to the numerical method for finding eigenmodes, and has been described elsewhere (e.g., Spencer and Rasband 1997; Schecter and Montgomery 2004, 2006).

We must also emphasize that the above discussion is based on linear theory. Nonlinear processes can revive a linearly damped VR wave. For instance, dry studies have shown that radiative pumping ultimately overtakes critical layer damping if the initial VR wave amplitude exceeds a finite threshold (Schecter and Montgomery 2006).

## 9. Concluding remarks

In this paper, we presented a basic model for small-amplitude wave dynamics in a cloudy vortex. The wave equations were derived with the following set of approximations:

(i) an air parcel conserves its total water content (liquid plus vapor);

(ii) the mixing ratios of water vapor and liquid cloud in an air parcel instantaneously adjust to min[

*q*,_{t}*q*_{υ}_{*}] and max[0,*q*−_{t}*q*_{υ}_{*}], respectively;(iii) an air parcel conserves its total entropy;

(iv) the wave motions are in hydrostatic balance;

(v) the wave motions follow linearized equations;

(vi) the basic-state vortex has no secondary circulation;

Boundary layer drag and eddy viscosity were also ignored. Clearly, this model was not designed to precisely simulate atmospheric vortices. Rather, it was designed as a tool to easily explore the most basic effects of cloud coverage on vortex wave dynamics.

The moist wave equations that result from (i)–(vi) are identical to the linearized dry hydrostatic primitive equations, with reduced cooling/warming by updrafts/downdrafts. We note that a similar model was recently reported to have been used to numerically simulate the effect of moisture on vortices that are exposed to sustained vertical shear (Patra 2004). Here, we derived the model from “first principles,” in order to clarify the underlying assumptions. In addition, we provided an explicit formula for estimating the *r*–*p* variation of the buoyancy reduction factor ϒ from observations [Eq. (32)].

The latter sections of this paper briefly examined the influence of moisture on eyewall instability, spontaneous IG wave radiation and vortex resilience. We have by no means closed the book on these subjects. Rather, we have presented a few case studies to motivate future research. In one case study, we showed that cloud coverage can temper the asymmetric eyewall instability of a hurricane-like vortex. In another case study, we showed that clouds in the core of a monotonic cyclone can cause or accelerate the decay of discrete VR waves. This both inhibits the spontaneous radiation of IG waves and improves vortex resilience.

## Acknowledgments

The authors thank Prof. David S. Nolan and an anonymous referee for helpful reviews of this manuscript. This, work was supported by NSF Grants ATM-0528816, ATM-0132006, ATM-0349980, and ATM-0101781, and ONR Grant N00014-02-1-0474.

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## APPENDIX A

### Thermodynamic Formulas

In this appendix, we derive convenient formulas for the moist adiabatic pressure derivatives of density potential temperature.

#### Unsaturated air

*q*=

_{υ}*q*, we may write

_{t}*T*(

^{u}*p*,

*s*,

_{m}*q*) ≡

_{t}*T*(

*p*,

*s*,

_{m}*q*→

_{υ}*q*,

_{t}*q*). It follows that

_{t}*T*at constant

^{u}*s*and

_{m}*q*.

_{t}*z*(

*p*,

*s*,

_{m}*q*), defined such that

_{t}*z*derivative of

*T*is minus the adiabatic lapse rate of an unsaturated hydrostatic atmosphere, with constant

^{u}*q*. Standard texts [e.g., Emanuel 1994, Eq. (4.7.4)] derive the following formula for this lapse rate:

_{t}One may now execute the following chain of substitutions:

Here, primes indicate modified equations. The last substitution (iii) yields formula (15) of the main text.

#### Saturated air

*q*=

_{υ}*q*

_{υ}_{*}(

*p*,

*s*,

_{m}*q*), we may write

_{t}*T*(

^{s}*p*,

*s*,

_{m}*q*) ≡

_{t}*T*(

*p*,

*s*,

_{m}*q*→

_{υ}*q*

_{υ}_{*},

*q*). It follows that

_{t}*T*and

^{s}*q*

_{υ}_{*}at constant

*s*and

_{m}*q*.

_{t}*z*is defined by Eq. (A3), and

*e*

_{*}is the saturation vapor pressure. Since, to an excellent approximation,

*e*

_{*}is a function of temperature

*T*alone,

^{s}*q*

_{υ}_{*}is a function of only

*p*and

*T*. Accordingly, we may write

^{s}*p*, we obtain

*T*, and using the Clausius–Clapeyron equation,

^{s}One may now execute the following chain of substitutions:

Here, single or double primes indicate equations that have been modified once or twice, respectively. Simplifying Eq. (A7)″ yields formula (16) of the main text.

## APPENDIX B

### Symmetric Stability

*ae*

^{−i(νt−kpp−krr)}, in which

*a*is a complex constant. The dispersion relation is

*κ*≡

*k*/

_{r}*k*. Derivation of Eq. (B1) employs the thermal wind relation [Eq. (28)].

_{p}*ν*

^{2}≥ 0 for all

*κ*. Considering the limits |

*κ*| → ∞ and |

*κ*| → 0, we obtain the familiar conditions of gravitational and inertial stability:

*ν*

^{2}is minimized when

*ν*

^{2}

_{min}≥ 0 gives the condition of slantwise convective stability; that is, Eq. (37) of the main text.

## APPENDIX C

### Numerical Simulation

In this appendix, we briefly describe the numerical model that is used to integrate the linearized wave equations (29)–(34) forward in time.

#### Prognostic equations

*β*(

*r*,

*p*), to absorb inertia–gravity wave radiation at the top and outer radius of the simulation domain. The damping coefficient takes the form

*β*,

_{s}*r*,

_{s}*δr*,

_{s}*p*and

_{s},*δp*are adjustable constants. The simulation domain is

_{s}*p*

_{min}≤

*p*≤

*p*

_{max}and 0 <

*r*≤

*r*.

_{b}#### Diagnostic equations

*on the right-hand side of Eq. (C8). Since constraint (C11) must hold at all times, we require that*

_{n}*at*

_{n}*p*

_{max}is initially zero, and that

*r*for the function

*ϕ*(

_{n}*r*,

*p*

_{min},

*t*). To solve it, we must specify boundary conditions at

*r*= 0 and at

*r*=

*r*.

_{b}#### Radial boundary conditions

*r*, which is far from the vortex center, we impose an artificial wall, that is, we set

_{b}*r*at all times if we impose the condition

_{b}*υ*at

_{n}*r*.

_{b}*r*→ 0) we enforce the usual constraints on the Fourier components of the perturbation fields. For the scalar fields, we have

#### Discretization parameters

The computational grid has *N _{r}* radial grid points and

*N*pressure points. The radial grid spacing is Δ

_{p}*r*, and the pressure spacing is Δ

*p*. The first radial grid point is at

*r*= Δ

*r*and the last is at

*r*. The first pressure grid point is at

_{b}*p*=

*p*

_{min}, and the last is at

*p*

_{max}. The equations are marched forward in time with a fourth-order Runge–Kutta scheme, in steps of length Δ

*t*.

*β*to zero.

_{s}The numerical algorithm has passed a series of tests, in which simulated eigenmodes and quasimodes were shown to behave properly. Specifically, their complex frequencies matched those obtained from the shooting method.

## APPENDIX D

### Asymptotic Wavefunctions

In this appendix, we present the asymptotic (small and large *r*) solutions of the geopotential wavefunction of an eigenmode of a barotropic vortex that has a separable ϒ distribution. By assumption, the modal oscillation frequency (*ν _{R}*) is positive.

*n*is the azimuthal wavenumber.

*r*becomes infinite, there are two possibilities. Let

*k*

^{2}] > 0, then the modal oscillation can resonate with an environmental IG wave. Consequently, the mode is radiative, and

*y*=

*kr*, −

*π*/2 < arg(

*y*) ≤

*π*/2, and

*H*

^{(1)}

_{0}is a Hankel function of the first kind.

## APPENDIX E

### Polarization Equations for an Eigenmode of a Cloudy Barotropic Vortex with Separable ϒ

*(*

_{λ}*r*)Ψ

*(*

_{λ}*p*). The wavefunctions for the radial and azimuthal velocities are given by

*r*,

*φ*,

*p*,

*t*). Finally, the wavefunction of the pseudo PV perturbation, defined in appendix F, is

## APPENDIX F

### Wave Activity in a Cloudy Barotropic Vortex

*r*, bounded above and below at

_{υ}*p*

_{min}and

*p*

_{max}, yields

Suppose that *θ*′_{ρ} = 0 at *p*_{min} and *p*_{max}. Then, by the linearized heat equation (31) of a barotropic vortex, *ω*′ would also be zero on the vertical boundaries. As such, the second (radial) integral on the right-hand side of Eq. (F7) would vanish.

## APPENDIX G

### The Meaning of ϒϒ

**ẑ**) buoyancy restoring force that a parcel feels after a moist adiabatic vertical displacement, or pressure change (cf. Smith et al. 2005). The vertical buoyancy acceleration is defined by

*ρ*′ is the local perturbation from the basic-state density

*p*−

*δp*to

*p*. We may write the density change of the parcel as follows:

*c*is the specific heat of dry air at constant volume, and the coefficients of the infinitesimals are evaluated at the original parcel position.

_{υd}*r*fixed. Then, the density perturbation at the new parcel position is given by