## 1. Introduction

During its development, an atmospheric or oceanic vortex may experience episodes of external vertical shear. This shear can destroy the vortex before it fully matures. However, many vortices survive, because they have a dominant tendency to stand upright. In this article, we will present a new theory for what drives a vortex to a state of vertical alignment.

Past studies of vertical alignment (Polvani 1991; Viera 1995; Sutyrin et al. 1998; Reasor and Montgomery 2001) typically assume that the Rossby number is much less than unity, so that the dynamics is quasigeostrophic. For simplicity, we will also use the quasigeostrophic approximation. Furthermore, we will neglect diabatic and frictional processes. These effects can be added later.

Figure 1 shows a numerical simulation of the vertical alignment of a quasigeostrophic vortex under conservative dynamics, taken directly from Reasor and Montgomery (2001). The shaded object is an isosurface of potential vorticity (PV). The PV distribution extends radially far beyond this surface. At *t* = 0, the vortex is tilted by an episode of external vertical shear, and then the shear is turned off. In time, the orientation of the tilt rotates, while the amplitude of the tilt decays. Eventually the vortex relaxes to an upright position. We will show that this vertical alignment occurs by the resonant damping of a mode.

The core of a quasigeostrophic vortex can have discrete modes of oscillation (vortex Rossby modes). A misalignment, such as the tilt in Fig. 1, corresponds to the excitation of a discrete core mode. The angular phase velocity of this mode is resonant with the fluid rotation frequency at a critical radius *r*∗ in the outer skirt of the vortex. We will explain how this resonance damps the core mode exponentially with time, provided that the radial derivative of PV at *r*∗ is negative. An analytical expression for the decay rate will be derived and then compared to numerical simulations: the two are in excellent agreement.

Our analysis will parallel the theory of plasma waves, which can decay exponentially by a “wave-particle” resonant interaction (Landau 1946; O'Neil 1965; Briggs et al. 1970; Timofeev 1992). The analogous “wave–fluid” resonant interaction, central to our theory of vertical alignment, has been studied previously to understand the exponential decay of two-dimensional vortex modes (Briggs et al. 1970; Pillai and Gould 1994; Corngold 1995; Spencer and Rasband 1997; Bachman 1997; Schecter et al. 2000; Balmforth et al. 2001), and to understand asymmetric vortex merger (Lansky et al. 1997).

*l*

_{R}, defined as

*h*is the height of the vortex,

*f*is the Coriolis parameter at the latitudinal position of the vortex,

*N*is the ambient Brunt–Väisälä frequency, and

*m*is the number of vertical nodes in the core mode that dominates the misalignment. For a simple tilt,

*m*= 1. The Brunt–Väisälä frequency measures the vertical (

*ẑ*) stratification of the atmosphere or ocean. For atmospheres,

*N*

^{2}≡

*g*

*θ*

^{−1}

_{o}

*dθ*

_{o}/

*dz,*where

*g*is gravitational acceleration and

*θ*

_{o}(

*z*) is the ambient potential temperature. For oceans,

*N*

^{2}≡ −

*g*

*ρ*

^{−1}

_{o}

*dρ*

_{o}/

*dz,*where

*ρ*

_{o}is the ambient mass density.

The alignment rate versus *l*_{R} is not a universal curve. To illustrate this point, we will examine how the alignment rate depends on *l*_{R} for a “Rankine-with-skirt” vortex (section 5), and a “Gaussian” vortex (section 6). Rankine-with-skirt (RWS) vortices are prevalent in turbulence simulations that employ contour dynamics (Dritschel 1998). An RWS vortex can also be used to model the stratospheric polar vortex (Dritschel and Saravanan 1994). On the other hand, Gaussian vortices more closely resemble incipient tropical cyclones (for data, see Willoughby 1990). We will focus on the regime where *l*_{R} is within a factor of 10 of the radius of the vortex core. In this regime, we will show that, for the case of an RWS vortex, the alignment rate increases with increasing *l*_{R}. On the other hand, for a Gaussian vortex, the alignment rate decreases with increasing *l*_{R}.

In section 6, we will show that the core modes of a vortex disappear if *l*_{R} is reduced below a vortex-dependent threshold. Clearly, if *l*_{R} is below this threshold, the vortex can not align by the resonant damping of a mode. Rather, the vortex will align by another mechanism, such as spiral wind-up of PV (Montgomery and Kallenbach 1997; Bassom and Gilbert 1998). In this case, the misalignment generally decays nonexponentially with time.

We note that Reasor and Montgomery (2001) were the first to show that vertical alignment can occur by the decay of a core mode. This article presents a new way to understand the damping, and provides an analytical expression for the decay rate. Reasor and Montgomery (2001) analyzed the damped core mode as a packet of *freely* propagating continuum eigenmodes, with a sharply peaked frequency spectrum. From this point of view, the core mode decays by destructive interference, as its constituent continuum modes disperse. Here, we view the core mode as a single wave. In this picture, the core mode decays by its *interaction* with fluid in a critical layer.

We now give an outline of the remaining text. Section 2 summarizes the quasigeostrophic fluid equations that are used to model the vortex dynamics. Section 3 reviews the properties of a Rankine vortex (uniform vortex column), which has no tendency to align (Reasor and Montgomery 2001). Section 4 explains how vortices with broader PV distributions align by the resonant damping of modes. In addition, section 4 contains a simple formula for the alignment rate. Section 5 examines the alignment of an RWS vortex. Section 6 examines the alignment of a Gaussian vortex. Section 7 recapitulates our results and discusses their relevance to the survival of incipient tropical cyclones.

## 2. Basic equations

### a. Conservative quasigeostrophic model

The following summarizes the conservative quasigeostrophic (QG) fluid equations (e.g., Pedlosky 1987) that are used here to model the vortex dynamics. These equations are formally valid only if the Rossby number *R*_{o} is much less than unity, that is, only if the vortex rotation frequency is much less than the local Coriolis parameter *f*. In addition, the QG model is valid only if *h* ≪ *L* ≪ *r*_{e}, where *h* is the height of the vortex, *L* is the horizontal length scale of the vortex, and *r*_{e} is the radius of the earth. Such scale relations are satisfied by the stratospheric polar vortex, extratropical cyclones, and midlatitude oceanic eddies, but not by tropical cyclones, which have *R*_{o} ≳ 1. Quasigeostrophic theory is only a starting point for understanding tropical cyclone dynamics, as discussed in section 7.

**υ**

_{g}

*ẑ*

_{h}

*ψ.*

_{h}is the horizontal gradient operator, and

*ψ*(

**r**,

*t*) is the “geostrophic streamfunction.” The geostrophic streamfunction is related to the pressure anomaly

*p*(

**r**,

*t*) in the vortex, and the ambient atmospheric/oceanic density

*ρ*

_{o}, by the equation

*ψ*≡

*p*/

*fρ*

_{o}.

*q*(

**r**,

*t*) following the geostrophic flow:

*N*vary over length scales greater than the vertical extent

*h*of the vortex. In this “Boussinesq” regime, the invertibility relation is given by

*N*≃ constant.

*ψ,*Eqs. (2)–(4) completely describe the evolution of the horizontal flow. If the vertical velocity

*υ*

_{z}(

**r**,

*t*) is desired, it may be obtained from the derivatives of the geostrophic streamfunction by the following:

### b. Perturbations

We now derive the conservative evolution equations for small-amplitude perturbations on a QG vortex. To begin with, we introduce a cylindrical coordinate system (*r,* *φ*, *z*). The PV distribution *q*_{o} and the streamfunction *ψ*_{o} of the unperturbed vortex are assumed to be independent of the vertical coordinate *z* and the polar angle *φ*.

*q*and Δ

*ψ*are perturbations, perhaps associated with a tilt. For analysis, each perturbation field is decomposed into an azimuthally symmetric part (S), and an asymmetric part (A); that is,

*δq*

^{(S)}≡ (2

*π*)

^{−1}

^{2π}

_{o}

*dφ*Δ

*q,*and

*δψ*

^{(S)}≡ (2

*π*)

^{−1}

^{2π}

_{o}

*dφ*Δ

*ψ.*The asymmetric perturbation fields satisfy

^{2π}

_{o}

*dφ*

*δq*

^{(A)}=

^{2π}

_{o}

*dφ*

*δψ*

^{(A)}= 0.

*δq*

^{(A)}and

*δψ*

^{(A)}are expanded into Fourier series in

*φ*and

*z.*Motivated by tropical cyclone observations (Hawkins and Rubsam 1968) and numerical simulations (Rotunno and Emanuel 1987), we will assume that the potential temperature is constant along the top and bottom surfaces of the vortex. This implies that ∂

_{z}Δ

*ψ*vanishes at

*z*= 0 and at

*z*=

*h.*We can then write

*n*= 0, because such terms constitute the azimuthally symmetric parts of the perturbation fields. The cosine vertical eigenfunctions in Eq. (8) are special to the Boussinesq approximation. In the non-Boussinesq regime, similar, but less familiar, eigenfunctions replace the cosines.

*asymmetric*perturbation fields. If we neglect terms in Eq. (3) that are of second order in the perturbation amplitude, we obtain the following linear equation for each Fourier component of

*δq*

^{(A)}:

_{o}(

*r*) ≡

*r*

^{−1}

*ψ*

^{′}

_{o}

*r*) is the angular rotation frequency of the unperturbed vortex, and the prime denotes the radial derivative. The Fourier components of the streamfunction are obtained from Eq. (4):

*l*

_{R}≡

*Nh*/

*mπf*is the internal Rossby deformation radius. The boundary conditions are

*δψ*

^{(m,n)}→ 0 as

*r*approaches zero and infinity.

*symmetric*part of the PV perturbation can be derived by substituting Eqs. (2) and (6)–(8) into Eq. (3), and averaging over

*φ*. The result is

*δq*

^{(m′,n)*}is the complex conjugate of

*δq*

^{(m′,n)}. From Eq. (11), it is clear that the evolution of the symmetric part of the PV perturbation is coupled to the asymmetric part. This is why the asymmetric part of the PV perturbation (e.g., a tilt) can decay in the absence of viscosity while, for example, total bulk kinetic energy is conserved: the energy lost by the asymmetric perturbation is absorbed by the symmetric perturbation.

## 3. Perpetual tilt of a compact vortex

*Q*

_{o}is a positive constant. For

*r*<

*r*

_{υ}the angular rotation frequency of a Rankine vortex has the constant value Ω

_{o}(

*r*) =

*Q*

_{o}/2. For

*r*>

*r*

_{υ}, the angular rotation frequency decays monotonically with

*r.*Specifically,

^{1}

*A*is a dimensionless complex amplitude,

*ω*is the eigenfrequency, and

*δ*(

*r*−

*r*

_{υ}) is the Dirac delta function centered at

*r*

_{υ}. The Green function is given by

*G*

_{mn}

*r,*

*r*

*I*

_{n}

*r*

_{<}

*l*

_{R}

*K*

_{n}

*r*

_{>}

*l*

_{R}

*I*

_{n}and

*K*

_{n}are modified Bessel functions of the

*n*

^{th}order, and

*r*

_{>}(

*r*

_{<}) is the larger (smaller) of

*r*and

*r*′.

*G*

_{mn}depends implicitly on

*m*through

*l*

_{R}[Eq. (1)].

*r*∗ of a mode is defined by the following resonance condition:

_{o}

*r*

*ω*

*n.*

*r*∗ is where the unperturbed fluid rotation frequency equals the angular phase velocity of the mode. Using Eqs. (13) and (16), we obtain

*r*

*r*

_{υ}

*G*

_{mn}

*r*

_{υ}

*r*

_{υ}

^{−1/2}

The modes of Eqs. (14)–(18) are examples of discrete “vortex Rossby modes.” Like planetary Rossby waves, they are supported by a PV gradient, *q*^{′}_{o}*q*^{′}_{o}*ω*/*n* of a mode is less than the angular fluid velocity Ω_{o} at the radius where that mode is peaked (*r*_{υ} for a Rankine vortex).

In linear theory, a generic deformation of a Rankine vortex evolves as a superpositon of its discrete modes. Vertical alignment requires the decay of discrete modes with *n* = 1. Since the discrete modes of a Rankine vortex are undamped, a Rankine vortex will not align.

In fact, many other vortices stay misaligned because they too possess undamped discrete modes. To begin with, all vortices have undamped *n* = 1 “pseudomodes” in the limit of infinite *l*_{R}.^{2} A pseudomode is a discrete mode with *ω* = 0 and radial PV eigenfunction *ξ*(*r*) ∝ *q*^{′}_{o}*r*). The excitation of a pseudomode can be viewed as a change in the equilibrium state of the vortex; for example, a change from an upright to a tilted equilibrium.

Even if *l*_{R} is finite, there is an infinite set of “compact” vortices that support undamped discrete modes, with *ω* ≠ 0. By compact we mean simply that *q*_{o} is zero beyond some core radius *r*_{υ}, as is the case for the Rankine vortex. Because compact vortices can support undamped discrete modes, they need not align.

Undamped discrete modes and pseudomodes share a common trait: *q*^{′}_{o}*r*∗. One can easily check that *r*∗ > *r*_{υ}, and therefore that *q*^{′}_{o}*r*∗) = 0 for the discrete modes of a Rankine vortex. For a pseudomode, *r*∗ is at infinity, where the PV and its derivatives are zero. Section 4 will explain why zero PV gradient at *r*∗ neutralizes the decay of a discrete mode.

## 4. Alignment by the resonant damping of a mode

The following contains a new theory of vertical alignment. Section 4a establishes a framework for analysis, by dividing the vortex into an inner core and an outer skirt. A misalignment (e.g., a tilt) is viewed as the excitation of a discrete core mode. In general, this mode is resonant with the fluid rotation frequency at a critical radius *r*∗ in the outer skirt. Section 4b invokes conservation of canonical angular momentum [Eq. (21)] to explain why this resonance damps the mode with time (and causes the vortex to align), provided that *q*^{′}_{o}*r*∗) < 0. In section 4.c, an analytical expression for the decay rate is derived. In section 4d, we show how conservation of “wave activity” can also explain the decay of a core mode. Readers familiar with wave activity may prefer to read section 4d before reading section 4b.

### a. Vortex core and its modes

*r*

_{υ}and an outer skirt. The unperturbed PV distribution can be written

*q*

_{o}

*r*

*q*

_{o,c}

*r*

*q*

_{o,s}

*r*

*c*” and “

*s*” denote core and skirt, respectively. We will only consider cases in which

*q*

_{o,c}decreases monotonically with radius. The PV perturbation can also be written as a sum of core and skirt contributions: Δ

*q*(

*r,*

*φ*,

*z,*

*t*) = Δ

*q*

_{c}(

*r,*

*φ*,

*z,*

*t*) + Δ

*q*

_{s}(

*r,*

*φ*,

*z,*

*t*). By definition,

*q*

_{o,c}= Δ

*q*

_{c}= 0 for

*r*≳

*r*

_{υ}, and

*q*

_{o,s}= Δ

*q*

_{s}= 0 for

*r*≲

*r*

_{υ}. Note that the value of the core radius

*r*

_{υ}is somewhat arbitrary. Precise definitions are given in sections 5 and 6 for two different vortices. For now, let it suffice to say that the bulk of the PV is contained within the core, that is, within a cylinder of radius

*r*

_{υ}.

*ω*is real. The streamfunction produced by such a mode is given by

*r*) and

*ξ*(

*r*) are related to each other by Eq. (10), with

*δψ*

^{(m,n)}→ Ψ and

*δq*

^{(m,n)}→

*ξ.*By analogy to a Rankine vortex (section 3), we may assume that the core modes are retrograde. Provided that Ω

^{′}

_{o}

*r*> 0, this implies that the critical radius

*r*∗ [Eq. (17)] of each mode is in the skirt (

*r*∗ >

*r*

_{υ}).

Unlike the discrete modes of a Rankine vortex, the core modes considered here can decay by their interactions with the skirt. Furthermore, in general, the core modes are not exact solutions to the linear eigenmode equation. For this reason, they are sometimes referred to as “quasi modes” (Reasor and Montgomery 2001, and references therein). As we will see, the amplitude |*a*| of a quasi mode exhibits an early stage of exponential decay. However, in contrast to the behavior of a pure eigenmode, the amplitude of a quasi mode decays nonexponentially at late times.

### b. Resonant damping

*P*(Sutyrin et al. 1998); that is,

*dV*≡

*rdrdφdz,*and the integral is over the entire fluid. The canonical angular momentum consists of contributions from the core and skirt of the vortex:

*q*

_{c}≡

*q*

_{o,c}+ Δ

*q*

_{c}, and

*q*

_{s}≡

*q*

_{o,s}+ Δ

*q*

_{s}. By conservation of total

*P,*any gain of

*P*

_{s}is compensated by a loss of

*P*

_{c}. This has a simple geometrical interpretation: any increase in the (PV weighted) mean square radius of the skirt requires a decrease in the mean square radius of the core.

Suppose that at *t* = 0 a single core mode is excited. The wavenumbers of this mode are arbitrary, but for illustrative purposes we consider the specific case in which (*m,* *n*) = (1, 1). This initial condition is depicted in Fig. 3a and is similar to the initial condition in Fig. 1. It is created by horizontally displacing the PV distribution at all vertical levels, such that the displacement amplitude varies with *z* like cos(*πz*/*h*). Provided that *q*^{′}_{o,c}*r* > 0, this initial perturbation increases the PV-weighted mean square radius of the vortex core; that is, the perturbation increases *P*_{c}.

Figure 3a also illustrates our expectation that the mode is damped with time, due to its interaction with the skirt. We now explain why the alignment in Fig. 3a follows from conservation of total *P* [Eq. (22)].

To begin with, consider the flow perturbation, in the skirt, that is created by the mode. Figure 3b is a sketch of the horizontal flow in the skirt, at an arbitrary vertical level. This flow is shown in a reference frame that corotates with the mode, so that the streamlines form a “cat's-eye” about the critical radius *r*∗ [Eq. (17)].

By inspection of the streamlines, one can imagine that the PV is effectively redistributed about *r*∗. As illustrated in Fig. 3(c), this redistribution tends to create a radial plateau about *r*∗ in the PV distribution, averaged over *φ*. If *q*^{′}_{o,s}*r*∗) < 0, the formation of a plateau increases the mean square radius of the skirt and thereby increases *P*_{s}. By conservation of total *P* [Eq. (22)], the mean square radius of the vortex core (i.e., *P*_{c}) must decrease. This is accomplished by the decay of the core mode, that is, the decay of the tilt, as illustrated in Fig. 3a.

Note that if *q*^{′}_{o,s}*r*∗) = 0, then the redistribution of resonant PV elements (PV elements near *r*∗) produces no change in *P*_{s}. Consequently, there is no change in *P*_{c}, and the core mode does not decay. This is why compact vortices, which have no skirts, generally do not align. Furthermore, if *q*^{′}_{o,s}*r*∗) > 0, then the redistribution of resonant PV elements decreases *P*_{s}. In this case, the core mode grows with time; that is, the wave–fluid resonant interaction causes an instability.

*a*(

*t*)| of a core mode [Eqs. (20)] decays or grows exponentially with time, due to the resonance at

*r*∗. Specifically, we will show that there is an early period during which |

*a*| ∝

*e*

^{γt}, where

*ξ,*

*ξ*〉 is the angular pseudomomentum (total wave activity) of the core mode, defined by

*q*

^{′}

_{o,c}

*ξ,*

*ξ*〉 is positive. So, in accordance with our discussion of conservation of

*P,*

*γ*has the same sign as

*q*

^{′}

_{o,s}

*r*∗).

We have explained the decay of a core mode as the “equal and opposite” reaction to the formation of a PV plateau about a critical radius (Fig. 3c). This plateau is a nonlinear (second order) perturbation; however, the early stage of its formation, which coincides with the early exponential decay of the mode, is described by linear theory [Eqs. (9)–(10)]. This result is well known from previous studies of the resonant damping of modes in fluids and plasmas (O'Neil 1965; Briggs et al. 1970; Pillai and Gould 1994; Bachman 1997; Schecter et al. 2000; Balmforth et al. 2001).

*π*/

*ω*

_{b}, where

*δψ*

^{(m,n)}(

*r*∗)| is the initial amplitude of the streamfunction perturbation, evaluated at the critical radius

*r*∗. We expect that for times greater than 2

*π*/

*ω*

_{b}, the amplitude of the core mode will “bounce” (begin to increase) and then equilibrate at some finite value. Experiments have demonstrated such behavior during the evolution of two-dimensional vortex modes (e.g., Schecter et al. 2000).

^{3}However, the nonlinear numerical simulations in this article (sections 5 and 6) run for times less than 2

*π*/

*ω*

_{b}, and bouncing is not observed.

Of course, if the decay rate *γ* of the mode is much greater than *ω*_{b}, the closed streamlines in Fig. 3b will open before the PV elements complete their orbits. That is, if *γ*/*ω*_{b} ≫ 1, we expect that linear theory remains accurate forever, barring other nonlinear processes such as mode–mode interaction.

### c. Weak damping

Suppose that a transient episode of external vertical shear perturbs a vortex, exciting the (*m,* *n*) discrete mode of its core. We now calculate the rate at which the core mode decays, due to a *weak* PV gradient at its critical radius *r*∗. We are concerned primarily with the damping of an *n* = 1 mode, since this causes vertical alignment. Nevertheless, our analysis covers all azimuthal wavenumbers. The decay of a core mode with *n* ≥ 2 is associated with horizontal axisymmetrization (e.g., Schecter et al. 2000).

*d*/

*dt*

*P*

_{c/s}is referred to as the “torque” on the core/skirt. The torques on the core and on the skirt are given by the following integrals:

*δ*

*q*

^{(S)}

_{c/s}

*r,*

*z,*

*t*) is the azimuthally symmetric part of the PV distribution in the core/skirt.

*m*′,

*n*′) different from the wavenumbers (

*m,*

*n*) of the core mode. Such neglect is valid to second order in the initial perturbation amplitude. To the same order of accuracy, it is valid to use the

*linear*solutions for

*δψ*

^{(m,n)}and

*δ*

*q*

^{(m,n)}

_{c/s}

*r*≲

*r*

_{υ}) consists entirely of a core mode [Eq. (20a)]; that is,

*δq*

^{(S)}

_{c}

*ξ,*

*ξ*〉 was previously defined by Eq. (24).

*r*≳

*r*

_{υ}). From the linearized PV equation [Eq. (9)], we have

*q*

^{′}

_{o,s}

*δψ*

^{(m,n)}in the skirt. Substituting Eq. (34) into Eq. (33) and integrating, we obtain

*δ*

*q*

^{(S)}

_{s}

*t*≲ |

*a*(

*da*/

*dt*)

^{−1}|. Then

*a**(

*t*′) is approximately

*a**(

*t*) and can be pulled out of the time integral. Furthermore, suppose that

*t*≫

*ω*

^{−1}. Then

*a*| ∝

*e*

^{γt}, where

*γ*is given by Eq. (23). For a discussion of this result, we refer the reader back to section 4b. Note that Eq. (40) can hold over times much longer than |

*γ*|

^{−1}, even though our simplified derivation employed the condition that

*t*≲ |

*γ*|

^{−1}. This fact is later illustrated in Figs. 6 and 8.

### d. Wave activity

Sections 4b and 4c explained the damping of a core mode (i.e., vertical alignment) using conservation of canonical angular momentum *P.* Alternatively, one can explain the damping of a core mode using conservation of wave activity (e.g., Held 1985).

*q*is the PV perturbation,

*q*

^{′}

_{o}

*r*) is the radial derivative of the unperturbed PV distribution, and the integral is over the entire fluid. For our purpose, it is convenient to express conservation of total wave activity as a balance between the rates of change of wave activity in the core and in the skirt:

A discrete mode in the vortex core corresponds to a perturbation Δ*q*_{c}. In time, the core mode creates a PV perturbation Δ*q*_{s} in the skirt. After a few rotation periods, Δ*q*_{s} is effectively concentrated about the critical radius *r*∗ (Fig. 3c). Suppose that *q*^{′}_{o,c}*r*) and *q*^{′}_{o,s}*r*∗) are both negative. Then, by conservation of total wave activity [Eq. (42)], the perturbation Δ*q*_{s} created at *r*∗ necessitates a decrease in the magnitude of Δ*q*_{c}, that is, a decay of the mode amplitude.

Conservation of wave activity, Eq. (42), can also replace conservation of canonical angular momentum, Eq. (26), as a starting point for a quantitative analysis of the decay rate *γ.* That is, after substituting the linear solutions for Δ*q*_{c} and Δ*q*_{s} into Eq. (42), one can derive Eq. (23) for *γ.*

## 5. Alignment of an RWS vortex

### a. Analysis

*ξ*(

*r*) is the PV eigenfunction, Ψ(

*r*) is the streamfunction eigenfunction, and

*δ*(

*r*−

*r*

_{υ}) is the Dirac delta function, which has units of one over length. The oscillation frequency

*ω*and the critical radius

*r*∗ of each core mode are given approximately by Eqs. (16) and (18), respectively.

*γ*is obtained by substituting Eq. (44) into Eq. (23). This yields

*n*and with the internal Rossby deformation radius

*l*

_{R}. Although

*l*

_{R}does not appear explicitly in Eq. (45), it is implicit in the Green function

*G*

_{mn}and in the critical radius

*r*∗. The

*alignment rate*of the vortex is the decay rate

*γ*of the

*n*= 1 core mode that dominates the initial perturbation.

*γ*on

*l*

_{R}, we consider the limits where

*l*

_{R}approaches zero and infinity. The first step is to calculate the asymptotic forms of the critical radius. For small

*l*

_{R}, Eq. (18) reduces to

*r*∗ approaches the radius of the uniform core as

*l*

_{R}approaches zero. Equation (46) indicates that

*r*∗ does not depend on the azimuthal wavenumber

*n,*to first order in the asymptotic expansion about

*l*

_{R}= 0. On the other hand, for large

*l*

_{R}, Eq. (18) reduces to

*r*∗ diverges with

*l*

_{R}for all modes with

*n*= 1. On the other hand, if

*n*≥ 2,

*r*∗ asymptotes to a finite limit that is different for each

*n*but generally greater than

*r*

_{υ}. Figure 4a shows the full dependence of the critical radius

*r*∗ on

*l*

_{R}for

*n*= 1, 2, 3, and 4.

*r*∗, and Eq. (13) for Ω

_{o}(

*r*∗), we can obtain asymptotic expressions for the decay rate of each mode, for small and large

*l*

_{R}. For small

*l*

_{R}, Eq. (45) reduces to

*n*and bounded

*q*

^{′}

_{o,s}

*r*

_{υ}+

*l*

_{R}/2), the decay rate vanishes as

*l*

_{R}approaches zero. This result is physically reasonable, since the skirt becomes shielded from the core mode as

*l*

_{R}becomes zero.

*l*

_{R}, Eq. (45) reduces to

*n*≥ 2, we have omitted a small correction due to finite

*l*

_{R}, since it is unimportant. The pertinent result for

*n*≥ 2 is that

*γ*asymptotes to a finite value as

*l*

_{R}becomes infinite. This value of

*γ*is equivalent to the decay rate that was derived in Briggs et al. (1970) and Schecter et al. (2000) for the core modes of an RWS vortex in strictly two-dimensional dynamics.

Figure 4b offers a more complete picture of how the decay rate *γ* depends on *l*_{R}, for *n* = 1, 2, 3, and 4. For this figure, *γ* was obtained from Eq. (45), using Eq. (18) for *r*∗, and Eq. (13) for Ω_{o}(*r*∗). The asymptotic behaviors discussed previously are evident. Note that the *n* = 1 curve does not imply that *γ* diverges with *l*_{R}. It appears this way, because the decay rates in Fig. 4b are divided by *q*^{′}_{o,s}*r*∗). For *n* = 1, *γ* actually vanishes as *l*_{R} becomes infinite, that is, as *q*^{′}_{o,s}*r*∗) → *q*^{′}_{o,s}*l*_{R}, all modes with *n* = 1 are undamped pseudomodes (Michalke and Timme 1967; Levy 1968).

### b. Simulations

*l*

_{R}. Figure 5 shows the equilibrium PV distribution and angular rotation frequency of the smoothed RWS vortex that is used for our numerical simulations. Note that the core has approximately uniform PV, and that the skirt has relatively small PV. Although the exact functional form of the PV distribution is unimportant, it is given below for the sake of reproducibility. The PV distribution of the core is given by

*a*= 1.01,

*b*= 0.025,

*δr*=

*r*

_{υ}/30,

*r*

_{z}= 1.09

*r*

_{υ}, and

*Q*

_{o}is a constant with units of frequency (

*Q*

_{o}≪

*f*for QG theory to apply). The parameter

*δr*measures the width of the transition layer, centered at

*r*

_{υ}, where the PV distribution drops rapidly to zero. The PV distribution of the skirt is given by

*q*

_{o,s}is negligible in the core. The total equilibrium PV of the vortex is

*q*

_{o}=

*q*

_{o,c}+

*q*

_{o,s}.

*t*= 0, we set

*C*is a real constant and

*q*

^{′}

_{o}

_{max}is the maximum equilibrium PV gradient. All other Fourier components of the perturbation are initialized to zero. Figures 1 and 3a show examples of vortices that have simple tilts. A simple tilt is dominated by the (1, 1) discrete mode of the vortex core. This follows intuitively from the fact that

*δq*

^{(1,1)}(

*r,*

*t*= 0) [given by Eq. (52)] is exactly the radial eigenfunction

*ξ*(

*r*) of the (1, 1) pseudomode, that is, the core mode in the limit of infinite

*l*

_{R}. The radial eigenfunction of the (1, 1) pseudomode is plotted below the equilibrium profile in Fig. 5.

Figure 6a shows the linear decay of the tilt, for cases in which *l*_{R} = 0.5*r*_{υ}, *l*_{R} = *r*_{υ}, *l*_{R} = 1.5*r*_{υ}, and *l*_{R} = 3*r*_{υ}. Here, the tilt is measured by the amplitude of the (1, 1) component of the perturbation streamfunction, evaluated at the core radius *r*_{υ}. The solid curves are complete solutions of Eqs. (9) and (10), obtained from a numerical technique that is described in Schecter et al. (2000) and Sutyrin (1989). The dashed curves in Fig. 6a correspond to pure resonant damping; that is, |*δψ*(*r*_{υ}, *t*)| ∝ *e*^{γt}, where *γ* is given by Eq. (45). Early on, there is excellent agreement between pure resonant damping and the linear simulations; however, at late times, when *t* ≫ |*γ*|^{−1}, the streamfunction tends toward power-law decay. Note that the damping rate (alignment rate) is zero for *l*_{R} = 3*r*_{υ}. This is because the critical radius, *r*∗ = 3.44*r*_{υ}, is in the extreme outer region where *q*_{o,s} = *q*^{′}_{o,s}

Figure 6b shows that the linear results accurately describe the vertical alignment that is observed in nonlinear simulations (diamonds). In these nonlinear simulations, the initial tilt amplitude is *C* = 0.3. The nonlinear simulations are based on a semispectral method that is described in appendix A of Reasor and Montgomery (2001). The nonlinear simulations include viscosity; however, they are terminated before viscous effects are noticeable.

We have focused on the decay of a simple tilt, in which case the vertical wavenumber *m* is equal to one. However, in linear theory, the decay rate depends on *m* only through *l*_{R} = *Nh*/*mπf*. So, the variation with *l*_{R} in Fig. 6 can also be viewed as a variation with *m,* keeping *Nh*/*f* fixed.

## 6. Alignment of a Gaussian vortex

In section 5, we examined the alignment rate of a tilted RWS vortex. We showed that a simple tilt [Eq. (52)] decays exponentially with time through the resonant damping of the (1, 1) core mode, and that the decay rate vanishes as the Rossby radius *l*_{R} approaches zero. In contrast, we now show that the alignment rate of a Gaussian vortex increases as *l*_{R} approaches zero. In addition, we show that the alignment of a Gaussian vortex occurs by the resonant damping of a core mode *only if* *l*_{R} is less than roughly one-fourth of the radius of the vortex core.

*r*

*q*

^{′}

_{o}

*r*)|). We define the vortex core as that part of a Gaussian vortex with

*r*≲ 3

*r*

*r*

_{υ}. In addition, we define the skirt as that part of a Gaussian vortex with

*r*≳

*r*

_{υ}. Figure 7 shows the equilibrium PV and angular rotation frequency of a Gaussian vortex. Figure 7 also shows the PV radial eigenfunction

*ξ*(

*r*) ∝

*q*

^{′}

_{o}

*r*) of a Gaussian's

*n*= 1 pseudomodes, which exist in the limit of infinite

*l*

_{R}. The amplitude of

*ξ*(

*r*) is very small for

*r*≳

*r*

_{υ}; thus, we have defined the vortex core so that it contains the bulk of a pseudomode. Note that a Gaussian vortex differs from an RWS vortex simply by a more gradual transition from the level of PV in the core to the level of PV in the skirt.

Suppose that at *t* = 0, a Gaussian vortex is given a simple tilt, as defined by Eq. (52). In the limit of infinite *l*_{R}, the simple tilt exclusively excites the (1, 1) pseudomode. Because this mode is undamped, the vortex remains tilted forever.

If *l*_{R} is large but finite, we expect that a simple tilt excites a (1, 1) core mode with a small frequency *ω,* a large critical radius *r*∗ > *r*_{υ}, and a spatial structure similar to that of the pseudomode (Reasor and Montgomery 2001). Because the critical radius *r*∗ is finite, this mode can interact with the skirt effectively, that is, resonantly. Since *q*^{′}_{o,s}

Figure 8 shows the evolution of the perturbation streamfunction at *r* = *r**l*_{R} = 0.28*r*_{υ}, *l*_{R} = 0.75*r*_{υ}, and *l*_{R} = 1.51*r*_{υ}. The solid curves are from numerical integrations of the linearized equations [Eqs. (9), (10)], and the diamonds are from nonlinear simulations, taken directly from Reasor and Montgomery (2001). In the nonlinear simulations, the initial tilt amplitude is *C* = 0.3.

Figure 8 clearly demonstrates that the streamfunction perturbation exhibits an early stage of exponential decay. This suggests that, even for a Gaussian vortex, the alignment process occurs through the resonant damping of a core mode. However, in contrast to the alignment rate of an RWS vortex (Fig. 5b), here the alignment rate appears to *increase* as *l*_{R} decreases toward zero.

To verify that the alignment of a tilted Gaussian vortex occurs through a wave–fluid resonant interaction, we will compare the observed decay rates in Fig. 8 to those predicted by the theory of resonant damping. The theoretical decay rates are given by Eq. (23), provided that the damping is weak, that is, |*γ*/*ω*| ≪ 1. However, to evaluate Eq. (23), we must know the eigenfunction *ξ*(*r*) and the critical radius *r*∗ of a Gaussian's core mode. We do not have an exact analytical solution to this problem. Furthermore, the damping may be strong, as for *l*_{R} = 0.28*r*_{υ} in Fig. 8, in which case Eq. (23) is inaccurate. So, we will use a numerical technique to find the decay rate, due to resonant damping.

A precise numerical solution is obtained by a procedure that is described in Briggs et al. (1970), Corngold (1995), Spencer and Rasband (1997), and Schecter et al. (2000). These articles discuss the resonant damping of modes on a two-dimensional vortex; however, their conclusions are readily generalized to the damping of modes on a three-dimensional vortex. In the following, we briefly describe the numerical method, without explanation. In the appendix, we explain why this method gives the decay rate of a core mode due to resonant damping.

*ω*of the following eigenmode equation (Briggs et al. 1970; Corngold 1995; Spencer and Rasband 1997; Schecter et al. 2000):

*r*is defined along a contour in the complex plane that arcs above the point where

*n*Ω

_{o}(

*r*) =

*ω,*as sketched in Fig. 9. If the arc is underneath this singular point, then the eigenmode problem will have no solution. The eigenfrequency is sometimes called a “Landau pole,” after L. Landau, who used a similar technique to calculate the damping rate of plasma waves (Landau 1946). We will denote the real and imaginary parts of the Landau pole by

*ω*

_{L}and

*γ*

_{L}, respectively.

Figure 10 illustrates the *l*_{R} dependence of the *n* = 1 Landau pole of a Gaussian vortex. This Landau pole was obtained from a numerical “shooting” solution to Eq. (54). Figure 10a shows that the critical radius *r*∗, defined by *n*Ω_{o}(*r*∗) ≡ *ω*_{L}, increases monotonically with *l*_{R}. This implies that the eigenfrequency *ω*_{L} decreases monotonically with *l*_{R}. On the other hand, Fig. 10b shows that the predicted decay rate *γ*_{L} is a nonmonotonic function of *l*_{R}, peaked at about 0.13*r*_{υ}.

The imaginary part (*γ*_{L}) of the Landau pole should be consistent with Eq. (23) in the regime of weak damping. To evaluate Eq. (23), we use the critical radius *r*∗ obtained from *ω*_{L}. In addition, we use *r*_{υ}*q*^{′}_{o}*r*) as a good approximation for the radial eigenfunction *ξ*(*r*). Finally, we use Ψ(*r*∗) = ^{∞}_{o}*drrG*_{mn}(*r*∗, *r*)*ξ*(*r*), where *G*_{mn} is defined by Eq. (15). The result is shown as a dashed curve in Fig. 10b. For *l*_{R} greater than about *r*_{υ}/2, in which case *γ*_{L}/*ω*_{L} ≤ 0.1, there is excellent agreement between *γ*_{L} and Eq. (23).

We now return to Fig. 8. The dashed curves in Fig. 8 show exponential decay, given solely by the Landau pole; that is, |*δψ*(*r**t*)| ∝ *e*^{γLt}*γ*_{L} is obtained from Fig. 10b. There is excellent agreement between this pure resonant damping and the simulations. Of course, at later times (not shown), the amplitude exhibits power-law decay, in accord with time-asymptotic linear theory.

For all cases in Fig. 8, the critical radius is outside or at the edge of the vortex core; that is, *r*∗ ≳ *r*_{υ}. So for all cases in Fig. 8, there is a spatial separation between the core mode and the critical layer. We now consider the alignment of a Gaussian vortex, in an atmosphere or ocean with *l*_{R} ≲ *r*_{υ}/4. In this regime the critical radius *r*∗ is well within the core. So the picture of a core mode interacting with a skirt breaks down and, as we will see, *γ*_{L} does not accurately give the alignment rate.

Figure 11 shows the evolution of the perturbation streamfunction at *r* = *r**l*_{R} = 0.1*r*_{υ} and *r*∗ = 0.24*r*_{υ}. The solid curve is from a numerical integration of the linearized equations [Eqs. (9)–(10)], and the diamonds are from a numerical integration of the nonlinear equations. The dashed curve shows the exponential decay that is given solely by the Landau pole; that is, |*δψ*(*r**t*)| ∝ *e*^{γLt}*γ*_{L} is obtained from Fig. 10b. It is evident that the simulations do not agree with the dashed curve; that is, the alignment observed in the simulations is not explained by the resonant damping of a core mode.

Furthermore, when *l*_{R} ≲ *r*_{υ}/4, the PV perturbation in a Gaussian's core does not resemble a damped mode in any way. Figure 12a shows the linear evolution of the PV perturbation for the case in which *l*_{R} = 0.1*r*_{υ}. The solid curves give the amplitude of the perturbation, whereas the dash–dot curves give the phase. Figure 12a indicates that the peak amplitude of the PV perturbation propagates radially outward, from *r* = 0.33*r*_{υ} to *r* ≃ 0.4*r*_{υ}. In addition, the PV perturbation undergoes “spiral wind-up;” that is, it develops rapid phase oscillations in the radial direction. In Reasor and Montgomery (2001), a PV perturbation with such features is referred to as a *sheared* vortex Rossby wave. Detailed discussions of sheared vortex Rossby waves, in the analogous two-dimensional problem, can be found in Montgomery and Kallenbach (1997) and Bassom and Gilbert (1998).

In contrast, Fig. 12b shows the evolution of the PV perturbation for the case in which *l*_{R} = 0.75*r*_{υ}, and the critical radius, *r*∗ = 1.24*r*_{υ}, is in the skirt. As before, the solid curves give the amplitude of the perturbation, whereas the dash–dot curves give the phase. As expected, the PV perturbation behaves like a damped mode in the vortex core. The short-dashed curves correspond to the theoretical behavior of this mode: *δq*^{(1,1)}(*r,* *t*) ≃ *a*_{o}*q*^{′}_{o}*r*)*e*^{γLt−iωLt}*a*_{o} is determined by a fit at *t* = 4 central rotation periods, at which point *δψ*^{(1,1)}(*r**t*) (Fig. 8) begins to decay at an exponential rate. In addition, we have used *ξ*(*r*) ≃ *r*_{υ}*q*^{′}_{o}*r*). Figure 12b validates our assertion that a vortex aligns by the resonant damping of a core mode, provided that *r*∗ is in the skirt.

We have demonstrated that a Gaussian vortex supports a (1, 1) discrete core mode only if *l*_{R} ≳ *r*_{υ}/4, or equivalently if 1 ≲ 4*Nh*/*πfr*_{υ}. More generally, an *n* = 1 core mode with vertical wavenumber *m* exists only if *m* ≲ 4*Nh*/*πfr*_{υ}. Therefore, resonant damping of a core mode can explain the alignment of a Gaussian vortex only if the dominant vertical wavenumber of the misalignment satisfies *m* ≲ 4*Nh*/*πfr*_{υ}.

Note also that for an ideal RWS vortex, where *q*^{′}_{o,c}*δ*(*r* − *r*_{υ}), the critical radius *r*∗ (for any *n*) is greater than the radius of the vortex core *r*_{υ} for all values of *l*_{R}. This suggests, in contrast to the Gaussian vortex, that an ideal RWS vortex can align at an exponential rate by the resonant damping of a core mode at any level of atmospheric/oceanic stratification, and for any vertical wavenumber *m* that characterizes the misalignment.

## 7. Discussion

### a. Recap

We have presented an important conservative mechanism for the vertical alignment of a QG vortex—the resonant damping of a vortex Rossby mode. In this picture, a misalignment corresponds to the weak excitation of an *n* = 1 discrete mode of the vortex core. If there is no PV gradient at the critical radius *r*∗ [Eq. (17)], then the mode amplitude remains constant; that is, the vortex stays misaligned forever. However, even a slight negative PV gradient at the critical radius causes the mode to decay exponentially, and the vortex to align, by a wave–fluid resonant interaction (section 4).

In sections 5 and 6, we examined the alignment of an RWS vortex and a Gaussian vortex, respectively. In both cases, we showed that the theory of resonant damping accurately predicts the alignment observed in numerical simulations (Figs. 6, 8), provided that *l*_{R} is sufficiently large. At the end of section 6, we argued that, for an ideal RWS vortex, resonant damping can explain the alignment for all *l*_{R}. However, for a Gaussian vortex, we showed explicitly that the theory of resonant damping fails to explain alignment when *l*_{R} ≲ *r*_{υ}/4. This is because the core of a Gaussian vortex does not support an *n* = 1 discrete mode when *l*_{R} ≲ *r*_{υ}/4.

### b. Resonant damping during tropical cyclogenesis?

To survive its early stages of development, a tropical cyclone (TC) must counter the destructive influence of external vertical shear. Moist convection likely assists in keeping TCs vertically aligned. However, they may also benefit from conservative alignment mechanisms, such as the resonant damping of a core vortex Rossby mode.

Suppose that a transient episode of external vertical shear misaligns a weak TC. Presumably, this vortex can better resist the next episode if it rapidly realigns. In QG theory, if the perturbation is dominated by a discrete vortex Rossby mode of the core, then the alignment rate (due to resonant damping) increases with the magnitude of the PV gradient at the critical radius *r*∗ of that mode. So, with a larger (negative) PV gradient at *r*∗, the vortex has a better chance to survive episodes of external vertical shear and develop into a violent storm.

Of course, alignment by resonant damping can occur only if the vortex core supports a discrete *n* = 1 mode. This depends on the form of the vortex, and on the internal Rossby deformation radius *l*_{R}. Figure 13 (Willoughby 1990) shows the azimuthal velocity profiles of four weak TCs. In each case, the azimuthal velocity *V*(*r*) decays more slowly than 1/*r* past the radius of maximum wind, *r*_{mw}. Therefore, in each case there exists a significant amount of cyclonic vorticity beyond *r*_{mw}. In this sense, the TCs resemble Gaussian vortices more than RWS vortices. In analogy to the QG theory of Gaussian vortices, we expect that TCs support *n* = 1 core modes if *l*_{R} ≳ *r*_{υ}/4.

*R*

_{o}, we use

*f*= 2Ω

_{e}sin(

*λ*), where Ω

_{e}= 7.29 × 10

^{−5}s

^{−1}is the angular rotation frequency of the earth, and

*λ*is the latitudinal position of the vortex. Table 1 indicates that

*R*

_{o}(

*r*

_{mw}) is actually greater than unity for all of the TCs in Fig. 13. Therefore, using QG theory to understand the behavior of TCs, even weak TCs, is debatable. Nevertheless, TCs may still align in part by the resonant damping of modes.

*R*

_{o}. The AB equations are similar to the QG equations, but the constant Rossby radius of QG theory is replaced by the following radially dependent Rossby radius:

*η*(

*r*) is the local absolute vorticity of the unperturbed vortex, and Ξ(

*r*) is the average absolute vorticity within the radius

*r*; that is,

*l*

^{(QG)}

_{R}

*f*over the geometric mean of

*η*and Ξ.

At present, we anticipate that AB dynamics has the same qualitative dependence on the average value of *l*^{(AB)}_{R}*l*^{(QG)}_{R}*l*^{(AB)}_{R}*r*_{υ}/4, the cores of the Gaussian-like TCs in Fig. 13 support *n* = 1 discrete vortex Rossby modes, which decay by resonant damping. Using *m* = 1, *h* ≃ 8 km, *N* ≃ ^{−2} s^{−1}, and *r*_{υ} ≃ 2*r*_{mw}, we obtain values of *l*^{(AB)}_{R}*r*_{υ} between 0.3 and 0.5 (see Table 1). So, the TCs in Fig. 13 are candidates to support *n* = 1 core modes, and align by resonant damping. Future work will clarify the conditions for discrete core modes to exist (if at all) for vortices with *R*_{o} ≳ 1.

Future work will also examine how the alignment rate changes when the initial perturbation is more complicated than the simple tilt that is defined by Eq. (52). For example, convection in the core of a TC can excite single cluster PV anomalies, as described by Montgomery and Enagonio (1998), and Reasor and Montgomery (2001). These spatially concentrated PV anomalies can have a negligible overlap with discrete modes of the vortex core and behave like a superposition of sheared, radially propagating vortex Rossby waves (Montgomery and Kallenbach 1997; Bassom and Gilbert 1998). Accordingly, the perturbation streamfunction would quickly exhibit power-law decay with time, in contrast to the exponential decay that is associated with resonant damping of a core mode.

Finally, we have considered an ideal scenario in which the vortex is free of environmental stress during its alignment. In reality, the vortex may experience sustained forcing (as in Jones 1995). Future work will address the effect of such forcing on the resonant damping of core modes.

## Acknowledgments

D. A. Schecter thanks Dr. R. Rotunno and Dr. R. Saravanan for their helpful comments. This work was funded in part by NSF Grants ATM-9732678 and ATM-0101781.

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## APPENDIX

### The Landau Pole

In this appendix we will describe the Laplace transform solution of the linearized perturbation equations [Eqs. (9)–(10)]. In doing so, we will explain why the Landau pole (section 6) gives the frequency and decay rate of a core mode.

Briggs et al. (1970; hereafter BDL) described the Laplace transform solution for the particular case of infinite *l*_{R} (the two-dimensional limit). We will not repeat their detailed analysis; rather, we will summarize their solution in a form that is suitable for the more general case of finite *l*_{R}. For an extended discussion, we recommend that the interested reader consult BDL and several follow-up articles (Corngold 1995; Spencer and Rasband 1997; Schecter et al. 2000). In BDL, subtleties are discussed that for the sake of brevity, we omit from our presentation.

*m,*

*n*) to simplify notation. In addition,

*α*is a positive real number, so that the inverse transform is over a straight line, from right to left, in the upper-half of the complex-

*ω*plane. Note that the inverse transform in BDL is in the lower half of the complex-

*ω*plane: the present convention is chosen to be consistent with the articles that have followed BDL (Corngold 1995; Spencer and Rasband 1997; Schecter et al. 2000).

*δψ*:

*δq*(

*r,*

*t*= 0) is the initial PV perturbation, and

*χ*is given by

*r*

_{>}(

*r*

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

*r*and

*r*′. In addition, Ψ

_{1}and Ψ

_{2}are solutions to the homogeneous equation

_{ω}Ψ

_{i}= 0, with the end-point conditions Ψ

_{1}(0) = 0 and Ψ

_{2}(∞) = 0. The function

*W*(

*r*′,

*ω*) is the Wronskian determinant, defined by

*W*≡ Ψ

_{1}

^{′}

_{2}

^{′}

_{1}

_{2}.

*ω*for which Ψ

_{1}= Ψ

_{2}. That is,

*W*= 0 at an eigenfrequency of the following eigenmode problem:

*q*

^{′}

_{o}

*r*> 0). Then the eigenmode problem [Eqs. (A7)], defined along the real-

*r*axis, generally has no solution. On the other hand, if

*r*defines a contour (e.g., Fig. 9) that arcs sufficiently far above the real-

*r*axis, an eigenfrequency will emerge (Briggs et al. 1970; Corngold 1995; Spencer and Rasband 1997; Schecter et al. 2000). This eigenfrequency is a pole of

*δψ*(

*r,*

*ω*), provided that the contour of integration in (A5) is deformed appropriately into the upper half of the complex-

*r*′ plane.

As in the main text (section 6), we will denote the real and imaginary parts of this Landau pole by *ω*_{L} and *γ*_{L}, respectively. The inversion contour in Eq. (A1) can be made to wrap around the Landau pole, producing a component of *δψ*(*r,* *t*) that varies like Ψ_{L}(*r*)*e*^{γLt−iωLt}*γ*_{L} is negative, and the Landau pole contribution behaves like an exponentially damped mode. We emphasize that the location of the Landau pole is a property of the equilibrium profile, as opposed to the initial PV perturbation.

The “eigenfunction” Ψ_{L}(*r*) generally has a discontinuity at the critical radius *r*∗, defined by *n*Ω_{o}(*r*∗) = *ω*_{L} (Briggs et al. 1970). This discontinuity becomes negligible as *γ*_{L} approaches zero. Of course, the complete solution to *δψ*(*r,* *t*) is smooth. The smoothness is restored by an additional and essential contribution to *δψ,* referred to as the “continuum,” or “branch-cut” contribution. The continuum contribution decays as a power law at late times and therefore eventually dominates the exponentially damped contribution from the Landau pole.

We now calculate the Landau pole of a vortex that consists of a core of radius *r*_{υ}, and a skirt of relatively small PV [Fig. 2]. By assumption, the vortex core would support an undamped discrete mode in the absence of the skirt. This “root mode” satisfies the same eigenvalue problem [Eq. (A7)] that is used to calculate the Landau pole of the entire vortex, with *q*^{′}_{o}*q*^{′}_{o,c}*q*^{′}_{o,s}

*q*

^{′}

_{o,s}

*γ*

_{L}. To obtain an expression for

*γ*

_{L}, we first multiply the eigenvalue equation by

*r*Ψ*(

*r*) and then integrate over

*r*:

*L*denotes a “Landau contour” that extends along the real-

*r*axis from zero to infinity, except near

*r*∗, where it must arc just above the point at which

*n*Ω

_{o}(

*r*) =

*ω*

_{L}+

*iγ*

_{L}. Since

*q*

^{′}

_{o,s}

*γ*

_{L}are small, we may approximate Eq. (A8) with the following:

*r*),

*ω*

_{L}, and

*r*∗ are the eigenfunction, eigenfrequency, and critical radius of the root mode, respectively. Furthermore,

*P*denotes the principal part of the integral. In deriving Eq. (A9), we used the Plemelj formula (Muskhelishvili 1953) and the boundary conditions Ψ(0) = Ψ(∞) = 0.

*r*

_{υ}, since the skirt contribution (multiplied by

*γ*

_{L}) is negligible. In section 4, we obtained an expression [Eq. (23)] for the decay rate that involved the vorticity eigenfunction

*ξ*(

*r*) of the root mode of the vortex core. This vorticity eigenfunction is related to Ψ(

*r*) by the equation

*r*within the core. For

*r*beyond the core,

*ξ*(

*r*) = 0. Substituting this result into Eq. (A10), we obtain

*ξ,*

*ξ*〉 ≡ −

^{rυ}

_{o}

*dr*

*r*

^{2}

*ξ**

*ξ*/

*q*

^{′}

_{o,c}

Estimated tropical cyclone parameters

^{*}

The National Center for Atmospheric Research is funded by the National Science Foundation.

^{1}

The undamped discrete modes derived in QG theory for a vertically stratified Rankine vortex are similar to the discrete modes of a uniform density Rankine vortex, discussed by Kelvin (1880) and Lamb (1945).

^{2}

If *l*_{R} is infinite, there is no vertical motion, and the horizontal flow is governed by 2D Euler dynamics. The general existence of *n* = 1 pseudomodes in 2D Euler flow is pointed out in section 3 of Michalke in Timme (1967), and in Levy (1968).

^{3}

Balmforth et al. (2001) address the bouncing and equilibration of two-dimensional vortex modes in greater detail. They also demonstrate that such effects are not observed if the initial mode amplitude is below a critical value.