Downloaded 09/30/21 05:50 AM UTC

2498 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

The Spontaneous Imbalance of an Atmospheric Vortex at High Rossby Number

DAVID A. SCHECTER NorthWest Research Associates, Redmond, Washington

(Manuscript received 2 April 2007, in final form 12 October 2007)

ABSTRACT

This paper discusses recent progress toward understanding the instability of a monotonic vortex at high Rossby number, due to the radiation of spiral inertia–gravity (IG) waves. The outward-propagating IG waves are excited by inner undulations of potential vorticity that consist of one or more vortex Rossby waves. An individual vortex Rossby wave and its IG wave emission have angular pseudomomenta of opposite sign, positive and negative, respectively. The Rossby wave therefore grows in response to producing radiation. Such growth is potentially suppressed by the resonant absorption of angular pseudomomentum in a critical layer, where the angular phase velocity of the Rossby wave matches the angular velocity of the mean flow. Suppression requires a sufficiently steep radial gradient of potential vorticity in the critical layer. Both linear and nonlinear steepness requirements are reviewed. The formal theory of radiation-driven instability, or “spontaneous imbalance,” is generalized in isentropic coordinates to baroclinic vortices that possess active critical layers. Furthermore, the rate of angular momentum loss by IG wave radiation is reexamined in the hurricane parameter regime. Numerical results suggest that the negative radiation torque on a hurricane has a smaller impact than surface drag, despite recent estimates of its large magnitude.

1. Introduction on the vortex. This paper will review and generalize some notable results.1 a. Vortex instability driven by inertia–gravity wave The pertinent studies concern IG wave radiation emission from a monotonic cyclone (MC), whose potential vor- Mesoscale vortices are among the most intriguing co- ticity (PV) consistently decays with distance from the herent structures in the troposphere. Figure 1 illustrates central axis of rotation. The radial PV gradient allows their many forms, which range from hurricanes to ro- Rossby-like waves to exist in the core region of the tational thunderstorms. The angular velocity ⍀ of a me- vortex (Kelvin 1880; McDonald 1968; Montgomery and soscale vortex typically satisfies the condition Kallenbach 1997). The spectrum of vortex Rossby waves includes continuum and discrete modes. A con- f Ͻ⍀ϽN, ͑1͒ tinuum disturbance typically suffers spiral windup due to differential rotation of the mean flow. Here, we will focus on discrete vortex Rossby waves (DVRWs), in which f is the Coriolis parameter and N is the ambi- which resist spiral windup and remain coherent over ent Brunt–Väisälä frequency. This condition allows time. DVRWs often account for rotating tilts and ellip- vortex-scale disturbances to resonate with ambient in- tical (triangular, square, etc.) deformations of the vor- ertia–gravity (IG) waves, and thereby efficiently pro- tex core. The angular phase velocity of a DVRW is of duce IG wave radiation. Recent studies have examined the magnitude of such radiation and its consequences 1 It is well known that the problems of IG wave radiation and acoustic radiation from a solitary vortex are analogous. Readers Corresponding author address: David A. Schecter, NorthWest interested in the acoustic analog may consult Broadbent and Research Associates, 4118 148th Ave. NE, Redmond, WA 98052. Moore (1979), Kop’ev and Leont’ev (1983, 1985, 1988), Zeitlin E-mail: [email protected] (1991), Chan et al. (1993), and Howe (2003).

DOI: 10.1175/2007JAS2490.1

JAS2490 AUGUST 2008 S CHECTER 2499

FIG. 1. It has been speculated that Rossby-like waves in vortices like these (at high Rossby and moderate-to-high Froude numbers) might produce relatively strong levels of spiral IG wave radiation. (top left) Hurricane Katrina at 0820 EDT 29 Aug 2005 (National Aeronautics and Space Administration). (top right) A mesocyclone in Duncan, OK, on 17 Mar 2003 (courtesy of Dr. Robert J. Conzemius). (bottom left) A polar low over the Barents Sea on 27 Feb 1987 (NOAA-9). (bottom right) Wake vortices behind Selkirk Island on 15 Sep 1999 (United States Geological Survey, Earth Resources Observation and Science; NASA). order ⍀, but is less than the angular velocity of the core lishes that the creation of any wave by another distur- circulation. bance in a circular flow must occur in such a way that Under condition (1), a DVRW will typically excite an conserves net angular pseudomomentum (cf. McIntyre outward propagating spiral IG wave in the peripheral 1981; Haynes 1988; Guinn and Schubert 1993; Shep- region of the vortex. Basic perturbation theory estab- herd 2003). It has been shown for MCs that a DVRW

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2500 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 and its IG wave emission have angular pseudomomenta dial gradient of PV in the critical layer is sufficiently of opposite sign, positive and negative, respectively large (SM04; SM06). Figure 2 illustrates the two poten- (Schecter and Montgomery 2004, hereafter SM04; tial fates of a DVRW in an MC. Section 4 will demon- Schecter and Montgomery 2006, hereafter SM06). Con- strate that a precise growth rate formula for the DVRW sequently, a DVRW must grow in response to produc- is readily extracted from an equation that expresses ing radiation. By this mechanism, the core of an MC angular pseudomomentum conservation. For the first will gradually deform. time, this formula will be derived for baroclinic MCs Although an atmospheric vortex is in a continuously that possess active critical layers.2 stratified fluid, the shallow-water model suffices to explain the essential physics of the radiation-driven insta- b. Connection to the broader problem of bility. In the context of shallow-water theory, Ford spontaneous imbalance (1994a,b) verified that IG wave emissions compel cy- Following the theme of the special issue for which clone-scale DVRWs in MCs to grow exponentially with this paper was prepared, let us now turn our attention time, provided that the Rossby number Ro ϭ⍀/f ex- to the general problem of spontaneous imbalance. In ceeds unity. Ford also showed that the growth rate van- practical terms, balanced motions are theoretical ap- ϭ ishes algebraically as the Froude number, Fr V/cg, proximations of synoptic or mesoscale flows that filter tends to zero; here V is the azimuthal velocity of the out IG waves. Quasigeostrophic (QG) flow, which ap- vortex and cg is the ambient gravity wave speed. Spe- plies only at small Rossby numbers, is the most familiar cifically, the maximum growth rate of a DVRW is pro- kind (Charney 1948; Hoskins et al. 1985; cf. Muraki et portional to Fr4⍀ in the regime where Fr K 1. This al. 1999). More general forms of balanced flow are con- result was later generalized by Plougonven and Zeitlin tained in the semigeostrophic model (Hoskins 1975; (2002) to “pancake” vortices in a continuously stratified Hoskins et al. 1985), the standard balance model fluid. It stands to reason that an atmospheric vortex at (McWilliams 1985), and the asymmetric balance model small Froude number would hardly feel the effects of (Shapiro and Montgomery 1993). Spontaneous imbal- radiation. ance (SI) is the divergence of an actual flow from the Nevertheless, mesoscale cyclones such as hurricanes predictions of a balance model, which usually involves and rotational thunderstorms (Fig. 1) can penetrate the the production of IG waves. SI is evident in numerical “superspin” parameter regime, where both the Rossby simulations of baroclinic instability and frontogenesis and Froude numbers exceed unity. Recent numerical (Snyder et al. 1993; O’Sullivan and Dunkerton 1995; studies of a stratified MC indicate that superspin can Griffiths and Reeder 1996; Reeder and Griffiths 1996; shorten the e-folding time of a radiation-driven insta- Zhang 2004; Plougonven and Snyder 2005). At Ro K 1, bility to less than four rotation periods (SM04). Chow theoretical arguments suggest that balanced motions and Chan (2003) further speculated that the spontane- will produce exponentially weak IG wave emissions ous emission of spiral IG waves can remove up to one- (Saujani and Shepherd 2002; Vanneste and Yavneh tenth of the angular momentum of an intense hurricane 2004). At high Rossby numbers, compact regions of in a single rotation period. A similar estimate was pub- unsteady balanced motions can resonate more directly lished earlier by Chimonas and Hauser (1997) for the with environmental IG waves and thereby exhibit negative radiation torque on a supercell mesocyclone stronger SI (Ford et al. 2000, 2002). under ideal conditions. In section 5, we will reconsider Typically, the intrinsic frequency of a DVRW will these estimates. not exceed the characteristic inertial frequency of the Of course, superspin does not guarantee that IG vortex in which it resides, regardless of the Rossby waves will greatly influence the evolution of a cyclone. number (cf. Montgomery and Lu 1997). From this local An important mechanism for quenching the radiation- perspective, it is sensible to approximate DVRWs with driven instability of an MC involves the resonant inter- a model that filters out IG waves. Section 2 will review action between a DVRW and its critical layer. The criti- the interaction of a DVRW with its critical layer in the cal layer of a DVRW is located beyond the core of an context of the asymmetric balance (AB) model, which MC, precisely where the angular phase velocity of the ostensibly applies to hurricanes as well as quasigeo- wave equals the angular velocity of the mean flow. strophic cyclones. However, at high Rossby number the While creating radiation, a DVRW simultaneously stirs PV in its critical layer. Such stirring efficiently transfers 2 It is notable that the theory of modal growth in astrophysical angular pseudomomentum from the DVRW into the disks must likewise treat the net influence of critical layer stirring critical layer, and thereby acts to damp the wave. and spiral density wave radiation. See, for example, Papaloizou Damping will prevail over radiative pumping if the ra- and Pringle (1987) and Shukhman (1991).

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2501

FIG. 2. The two potential fates of a DVRW in the core of a stratified MC, bounded above and below by rigid surfaces (cf. Schecter and Montgomery 2003; SM04). Evolution (top) of a PV isosurface deformed by the DVRW and (bottom) of the horizontal flow at fixed height z in a reference frame that corotates with the DVRW. An initial perturbation of the balanced cyclone (middle) excites a baroclinic DVRW with azimuthal wavenumber n ϭ 2 and angular phase velocity ␻/n. The DVRW excites an outward propagating spiral IG wave in the environment, and r ϭ r stirs PV in its critical layer at *. If the positive feedback of IG wave radiation prevails, the DVRW will grow (left to right). If the negative feedback of PV stirring prevails, the DVRW will decay (right to left). balance approximation is unjustifiable in the far field lines of investigation. For convenient reference, the ap- where the inertial frequency reduces to f. As explained pendices review the standard primitive equations that previously, this allows a DVRW to match onto an out- form the theoretical foundation of this paper. ward propagating spiral IG wave. The balance approximation will qualitatively fail for the DVRWs of MCs if critical layer damping is unable to restrain the positive 2. The balanced evolution of a discrete vortex feedback of IG wave radiation (see section 3). In this Rossby wave sense, the radiation-driven instability of an MC is a prime example of SI. This section qualitatively reviews the balanced interaction of a DVRW with its critical layer. For simplicity, we will limit our discussion to linear shallow-water c. Outline theory on the f plane. Throughout, we will refer to a The remainder of this paper is organized as follows. polar coordinate system whose origin is at the center of Section 2 qualitatively reviews the balanced interaction the vortex. As usual, the symbols r, ␸, and t will denote of a DVRW and its critical layer in a shallow-water MC. radius, azimuth, and time. A generic flow field h(r, ␸, t) Section 3 examines both linear and nonlinear condi- will be separated into a basic state h(r) and a pertur- tions for the SI of a shallow-water MC. Section 4 de- bation hЈ(r, ␸, t). rives a formula for the growth rate of a DVRW near marginal stability in a baroclinic MC. This formula ac- a. The unperturbed cyclone counts for the positive feedback of IG wave emission and the negative feedback of PV stirring in a 3D critical By definition, the basic state of the vortex satisfies layer. Section 5 reexamines the torque that is applied gradient balance; that is, by spontaneous radiation on a barotropic cyclone with 2 parameters similar to those of a category 5 hurricane. d␾ ␷ u ϭ 0 and ϭ ϩ f␷, ͑2͒ Section 6 summarizes the main text and suggests future dr r

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2502 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

in which u is the radial velocity, ␷ is the azimuthal ve- Condition (6) is the cornerstone of the AB model, locity, ␾ is the geopotential, and f is the constant Co- which filters out IG waves and provides a reduced set of riolis parameter. For notational convenience, we intro- equations for vortex Rossby wave dynamics in ageo- duce the following auxiliary variables: strophic vortices (Shapiro and Montgomery 1993; Montgomery and Franklin 1998; Ren 1999; Möller and ⍀ ϵ ␷րr, ␨ ϵ rϪ1d͑r␷͒րdr, Shapiro 2002; McWilliams et al. 2003). It is straightforward to derive the lowest-order AB ␩ ϵ ␨ ϩ f, ␰ ϵ 2⍀ ϩ f, ͑3͒ model from the linearized primitive equations. To sim- plify notation, let in which ⍀ is the angular rotation frequency, ␨ is the ␩ Ѩ Ѩ relative vertical vorticity, is the absolute vertical vor- DV ␰ ϵ ϩ ⍀ . ͑7͒ ticity, and is the modified Coriolis parameter. The Dt Ѩt Ѩ␸ basic-state potential vorticity is defined by Operating on both sides of the linearized momentum

q ϵ ␩ր␾. ͑4͒ equations with DV/Dt yields 1 1 Ѩ␾Ј 1 D Ѩ␾Ј As mentioned earlier, the radial gradient of q supports Ј ϭϪ ϩ V Ϫ 2 Ј u ␩ ͩ Ѩ␸ Ѩ ͪ D u , the existence of vortex Rossby waves. Unless stated r ␰ Dt r otherwise, we will assume that the basic state has the Ѩ␾Ј Ѩ␾Ј 1 1 DV following properties: ␷Ј ϭ ͩ Ϫ ͪ Ϫ D 2␷Ј ͑8͒ ␰ Ѩr r␩ Dt Ѩ␸ (i) q is uniformly positive, Ј ␷Ј (ii) dq/dr is uniformly negative, and in which u is the radial velocity perturbation, is the ␾Ј (iii) d⍀/dr is uniformly negative. azimuthal velocity perturbation, and is the geopotential perturbation. Substituting the right-hand sides Conditions (i)–(iii) define a monotonic cyclone (MC). of Eqs. (8) into the linearized potential vorticity equation (McWilliams et al. 2003), and neglecting terms of 2 b. The asymmetric balance model for weak order D and higher yields perturbations Ј Ѩ␾Ј DV qˆ 1 dq 1 ϭ . ͑9͒ Disturbances to the basic state are exactly governed Dt r dr q 2 Ѩ␸ by the primitive equations (appendix A). Standard balance approximations of the primitive equations rely Here we have introduced the quasi–potential vorticity upon the smallness of the Rossby number, the Froude perturbation, number, or both (cf. McWilliams 1985; Polvani et al. Ѩ Ѩ␾Ј 2 ␯ Ѩ2␾Ј 1 l D 1994). Here, we consider an alternative that has be- qˆЈ ϵ ͩrl 2 ͪ ϩ Ϫ ␾Ј, ͑10͒ r Ѩr D Ѩr 2 Ѩ␸2 come somewhat popular in hurricane meteorology. To r begin with, let us classify perturbations according to the in which magnitude of ␾ Ѩ Ѩ 2 l ͑r͒ ϵ ͱ ͑11͒ 2 D ␩ ␰ D ϵ ͩ ϩ ⍀ ͪ Ͳ␩ ␰. ͑5͒ Ѩt Ѩ␸ is a local deformation radius, and ␯(r) ϭ (2␰ Ϫ ␩)/␩.3 The value of D 2 is the squared rate of change of the Equation (9) is the only prognostic equation in the AB disturbance in a reference frame that follows the basic model. By Eq. (10), it is essentially an equation for the flow, divided by the regional inertial stability. Loosely temporal evolution of ␾Ј. Equations (8) reduce to di- speaking, vortex Rossby waves satisfy D 2 Ͻ 1, whereas agnostic formulas for uЈ and ␷Ј by neglecting the far IG waves satisfy D 2 Ͼ 1 (cf. Montgomery and Lu 1997). right terms that are proportional to D 2. In other words, vortex Rossby waves are intrinsically slow, whereas IG waves are intrinsically fast. For low azimuthal wavenumbers, and Rossby numbers of order 3 The original derivation of the AB model (Shapiro and Mont- ␯ ϭ unity or less, the vortex Rossby waves of a cyclone gomery 1993) followed a different path, which resulted in 1. This discrepancy has no bearing on the present discussion, and typically satisfy the more restrictive condition that becomes negligible for small but finite Rossby numbers. Different AB models are possible because their derivation paths tacitly re- D 2 K 1. ͑6͒ quire different secondary conditions of validity.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2503

Note that the AB model [Eqs. (9)–(10)] conveniently retains QG structure at high Rossby number. Further- more, if the Rossby and Froude numbers satisfy the relation Fr2 K Ro K 1, the AB model properly reduces to the QG model. That is, (uЈ, ␷Ј) becomes the geostrophic velocity perturbation, lD becomes the constant Ј 2 Rossby deformation radius lR, and qˆ becomes l R times the quasigeostrophic PV perturbation. c. Basic kinematics of the DVRW–critical layer interaction A DVRW has a quasi-PV distribution of the form

Ј ϭ ͑ ͒ ˆ ͑ ͒ i͑n␸Ϫ␻t͒ ϩ ͑ ͒ qˆ d a t Q r e c.c., 12 in which a is the complex wave amplitude, n is the azimuthal wavenumber, and c.c. denotes the complex conjugate of the preceding term. We will assume that the complex radial wavefunction Qˆ vanishes beyond some core radius. We will further assume that the phase of a varies at a much slower rate than the real wave frequency ␻. Note that a DVRW has fixed radial structure and therefore differs from a generic continuum disturbance that is sheared by the differential rotation FIG. 3. (a) Typical arrangement of the critical layers of the DVRWs of a shallow-water MC. The central radius r* of the of the mean flow (e.g., Montgomery and Kallenbach critical layer decreases as the azimuthal wavenumber n increases. 1997; Bassom and Gilbert 1998; Brunet and Montgom- (b) Instantaneous streamlines in the critical layer of a DVRW, ery 2002; McWilliams et al. 2003). viewed in a reference frame that corotates with the wave. The ⍀ In the outer region of the vortex, there is a critical symbol b denotes the orbital angular frequency of fluid parcels near the center of the cat’s eye, and l denotes the half-width of layer where the phase velocity of the DVRW equals the trap the trapping region that is enclosed by the separatrix. Both ⍀ and angular rotation frequency of the mean flow. The cen- b ltrap are proportional to the square root of the wave amplitude, tral radius r of the critical layer is precisely defined by * which can vary with time. the resonance condition

⍀͑r ͒ ϵ ␻րn. ͑13͒ r͑qqˆЈ͒2 * L͑AB͒ ϵ ͵ dx2 , ͑14͒ Ϫ 2dqրdr Figure 3a illustrates a typical set of critical layers for the DVRWs of a shallow-water MC. in which the integral is over all space (Ren 1999). Note (AB) Through the extension of its velocity field, the that L is positive definite because we have assumed DVRW can stir fluid everywhere beyond the core. Out- that dq/dr is uniformly negative. If only the DVRW and (AB) side of the critical layer, fluid parcels moving with the critical layer contribute significantly to L , its con- mean flow experience rapid oscillations of positive and servation law reduces to negative wave forcing, which will have little cumulative d r͑qqˆЈ͒2 d r͑qqˆЈ ͒2 effect on the amplitude of qˆЈ. Within the critical layer, ͵ 2 d ϭϪ ͵ 2 cl dx Ϫ ր dx Ϫ ր , fluid parcels are essentially phase locked with the wave. dt core 2dq dr dt cl 2dq dr It is therefore reasonable to assert that a DVRW will ͑15͒ most profoundly disturb qˆЈ in the neighborhood of r * and that we need only consider the interaction between in which the left integral is over the area of the vortex a DVRW and its critical layer (cf. Lansky et al. 1997). core and the right integral is over the area of the critical Any interaction between different components of the layer. For convenience, we have introduced the nota- Ј Ј perturbation must conserve total angular pseudomo- tion qˆ cl to represent qˆ in the critical layer. Suppose that Ј mentum. In AB theory, the angular pseudomomentum qˆ cl is initially zero and then grows under the influence is given by of the DVRW. According to Eq. (15), the DVRW must

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2504 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

FIG. 4. (top) Balanced dynamics of a DVRW. In a balance model, the core DVRW resonantly excites a PV perturbation r at the critical radius *. To conserve total angular pseudomomentum, the DVRW must decay. (bottom) Spontaneous imbalance. If the Rossby number exceeds unity, the DVRW will also emit a frequency-matched spiral IG wave into the environment. Since the IG wave has negative angular pseudomomentum, its creation compels the DVRW to grow. In this cartoon, radiative pumping dominates critical layer damping of the DVRW and causes an instability that is a form of spontaneous imbalance. In both panels, the symbol ␥ represents the characteristic growth/decay rate of the DVRW. Note also that q is sketched for spatial reference only, and is not drawn to scale relative to qЈ in the vertical.

decay in response. Figure 4 (top) illustrates the kine- 3. The spontaneous imbalance of a shallow-water matics.4 cyclone The reader may have noticed that a damped DVRW cannot possibly be an eigenmode of an MC. An eigen- a. The positive feedback of spiral radiation mode consists of a qˆЈ distribution that decays exponen- At Rossby numbers greater than unity, any DVRW tially with time everywhere in space. In contrast, a of a shallow-water MC can excite a frequency-matched Ј DVRW decays because of growing qˆ cl. For this reason, outward propagating spiral IG wave in the environ- and others that are more technical, the DVRWs of MCs ment. In other words, the geopotential (and velocity are more properly classified as “quasi-modes” (Briggs components) of the balanced DVRW can match onto a et al. 1970; Corngold 1995; Spencer and Rasband 1997). spiral IG wave at the periphery of the circular flow. As If the vortex were nonmonotonic, then dq/dr could mentioned earlier, the spiral radiation has positive have opposite signs in the core and at r . According to * feedback on the DVRW. Eq. (15), the DVRW would then grow in response to its We may understand the positive feedback by recon- action on the critical layer (Briggs et al. 1970; Schecter sidering conservation of angular pseudomomentum. To et al. 2000, 2002; Benilov 2005; Mallen et al. 2005). lowest order in the perturbation fields, the angular Ј Simultaneous growth of a DVRW and qˆ cl is consistent pseudomomentum of an unfiltered perturbation in a with the behavior of a genuine eigenmode. shallow-water cyclone is given by (cf. Guinn and Schu- bert 1993)

L͑SW͒ ϵ ͵ d 2 ͑SW͒ ͑ ͒ 4 The resonant damping of a DVRW is analogous to the “Lan- x J , 16 dau damping” of a discrete plasma wave (Landau 1946; O’Neil 1965). Moreover, in nonneutral plasma physics the resonant in which the area integral is over all space, and damping of a DVRW is known as the Landau damping of a dio- ␾ 2͑ Ј͒2 ͑ ͒ r q cotron (slipping stream) mode (Briggs et al. 1970; cf. Davidson J SW ϵ Ϫ r␷Ј␾Ј. ͑17͒ 1990, chapter 6). Ϫ 2dqրdr

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2505

FIG. 5. The radial structure of an SI mode. (a) The basic state of the shallow-water cyclone that suffers SI. The radial coordinate r ⍀ ␾ ⍀ ␾ is normalized to the characteristic vortex scale ro. The variables and are in units of max and the ambient geopotential a. The ϵ⍀ ϭ potential vorticity q is normalized to one-half of its peak value. The Rossby number of the cycloneisRo max/f 53, and the squared 2 ϵ ⍀ 2 ␾ ϭ Froude number is Fr ( maxro) / a 0.74. (b) The magnitude (solid curve) and phase (dotted curve) of the radial velocity ϭ wavefunction U of the n 2 SI mode.␸ The magnitude is normalized to have a peak value of 1. (c) The azimuthally averaged angular pseudomomentum density rJ (SW) of the mode, normalized to its peak value at the instant under consideration. The DVRW and IG r wave components have positive and negative angular pseudomomenta, respectively. The bump at * accounts for the absorption of wave activity by the critical layer. In this case, radiative pumping prevails over critical layer damping, but is fairly weak. The growth ␥ ⍀ rate of the mode is merely 0.007 max.

The first component of J (SW) is proportional to the of an IG wave that has negative angular pseudomo- square of the exact PV perturbation qЈ and corresponds mentum. Such growth is in sharp contrast to balanced to the angular pseudomomentum density that appears dynamics, where the DVRW is either neutral or in most balance models. The new term, proportional to damped. ␷Ј␾Ј, has emerged by the removal of balance con- Figure 5 contains a snapshot of an exponentially straints. growing SI mode of a shallow-water cyclone at Fr2 ϭ With the addition of radiation, conservation of angu- 0.74 and Ro ϭ 53 (cf. SM06). For reference, Fig. 5a lar pseudomomentum now takes the form shows the basic state of the cyclone, which possesses a monotonic potential vorticity distribution. The angular d d velocity of the basic state is slightly nonmonotonic, but ͵ 2 ͑SW͒ ϭϪ ͵ 2 ͑SW͒ dx J d dx J rad dt core dt env this feature is unimportant for our discussion. Figure 5b shows the complex radial velocity wavefunction U of the d Јϭ i(n␸Ϫ␻t) ϩ Ϫ ͵ 2 ͑SW͒ ͑ ͒ SI mode, which is defined by u a(t)U(r)e dx J cl . 18 dt cl c.c. The inner part of U corresponds to a DVRW, and maintains an approximately constant phase far beyond The integral on the left-hand side of Eq. (18) covers the the critical radius r . The outer part of U corresponds to * core DVRW. The top integral on the right-hand side a spiral IG wave and has increasing phase with radius. covers the environmental radiation field, whereas the Figure 5c verifies that the angular pseudomomenta of bottom integral covers the critical layer disturbance. It the DVRW and IG wave are positive and negative, (SW) has been shown that the PV component of J d domi- respectively. Because the Froude number of the cy- nates the angular pseudomomentum of the DVRW, clone is less than unity, the IG wave emission is fairly and yields a positive integral (SM06). On the other weak. Consequently, the e-folding time of the SI mode ␷Ј␾Ј (SW) hand, the component of J rad dominates the an- is many (23) vortex rotation periods. gular pseudomomentum of the spiral IG wave radia- In the parameter regime where the vortex motion is tion, and yields a negative integral (SM06). Hence, by approximately governed by 2D Euler flow, say f ϭ 0 exciting a spiral IG wave, the DVRW creates angular and Fr2 K 1, the radiation-driven instability is readily pseudomomentum of opposite sign. Neglecting the explained in more familiar terms. In the 2D Euler critical layer, the DVRW must grow in response. In model, the conserved potential vorticity reduces to q ϭ ␨ ␾ ␾ other words, the top term on the right-hand side of / a in which a is the constant ambient geopotential. Eq. (18) is positive. Figure 4 (bottom) illustrates the Furthermore, the mean torque per unit mass on the spontaneous growth of a DVRW due to the emission cyclone at the radius r is given to lowest order by

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2506 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

␸ 5 Ѩ Ѩ ␾ ͑ Ј͒2 in which ␸ r a q r␷ ␸ ϭϪr␨u ϭ , ͑19͒ Ѩ Ѩ ր n t t 2dq dr ␲n͑n Ϫ 1͒2 ␥ ϭ 2n⍀ Fr o ␸ ͑n!͒2 2 2n in which h is the azimuthal average of h. Equation (19) indicates that an MC (in which dq/dr Ͻ 0) will lose ␲ n Ϫ 1 nϪ3ր2 dq ϩ ͩ ͪ ␾ ͩ ͪ ͑ ͒ angular momentum as a DVRW [(qЈ)2] grows. It is aro 24 4n n dr rϭr therefore sensible that the outward angular momentum * flux of IG wave radiation acts to amplify the DVRW. for n Ն 2, and ␥ ϭ 0 for the n ϭ 1 pseudomode that corresponds to a static displacement. As in the more b. The growth rate of a DVRW and its coupled general case of section 4, the first (top) term of the ␥ radiation field growth rate accounts for the positive feedback of IG wave radiation (cf. Ford 1994a). Its magnitude de- Of course, an instability will occur only if the positive creases algebraically with the Froude number, Fr ϭ feedback of IG wave radiation prevails over the nega- ⍀ ͌␾ oro / a. The second (bottom) term accounts for the tive feedback of PV stirring in the critical layer. In sec- negative feedback of the critical layer disturbance (cf. tion 4, we will derive a fairly general solution for the Briggs et al. 1970). It is directly proportional to the growth rate of a DVRW in a baroclinic cyclone that negative radial gradient of basic-state PV at r . This is * accounts for both feedbacks. Here we cite a closed- because a steeper gradient at r permits stirring (the * form result (based on linear perturbation theory) for radial redistribution of fluid elements) to create a larger the growth rate of a DVRW in a special shallow-water PV perturbation in the critical layer. Equivalently, a cyclone whose relative PV distribution consists of a steeper gradient at r increases the capacity of the criti- * constant core, surrounded by a low amplitude skirt. cal layer to absorb angular pseudomomentum. Specifically, let It is important to emphasize that Eq. (23), in which ␥ is constant, corresponds to linearized dynamics. Hence- ⍀ Ͻ f 2 o 1, r ro, forth, ␥ will refer exclusively to the growth rate of a q ϭ ϩ ͭ ͑ ͒ ␾ ␾ ␣͑ ͒ Ͼ 20 a a r , r ro, DVRW in linear theory.

⍀ ␣ K ␣ Ͻ in which o is a positive constant, 1, and d /dr c. Nonlinear SI ϭ 0. For simplicity, we will again suppose that f 0 and Suppose that linear theory predicts the exponential 2 K Fr 1. In this case, a Rankine circulation is an excel- decay of a DVRW, because the radial gradient of q is lent approximation of the basic flow out to very large steep at the critical radius r . Nonlinear dynamics al- * distances; that is, lows q, interpreted as the azimuthal mean of q,to change with time (cf. Killworth and McIntyre 1985; r ␷͑ ͒ Ӎ ⍀ Ͻ ͑ ͒ Maslowe 1986). In nonlinear balance models, dq/dr at r oro 21 rϾ r tends to oscillate and become negligible, provided * that the initial DVRW amplitude exceeds a threshold. in which rϾ (rϽ) is the greater (lesser) of r and r . Fur- Equation (24) suggests that leveling of q at r would o * thermore, the frequency and critical radius of the nth enable radiative pumping to prevail over critical-layer DVRW are given by damping. The amplitude threshold that is required to “flatten” n q in the critical layer has been examined theoretically ␻ ϭ ⍀ ͑n Ϫ 1͒ and r ϭ r ͱ ͑22͒ (Briggs et al. 1970; Balmforth et al. 2001; cf. Le Dizes o * o n Ϫ 1 2001), numerically (Bachman 1998; SM06; cf. Rossi et al. 1997), and experimentally (Pillai and Gould 1994; to zero order in Fr2 and ␣ (Kelvin 1880; Briggs et al. Cass 1998; Schecter et al. 2000). A good estimate for 1970; Ford 1994a). this threshold is obtained by considering basic critical After a brief transition period, the amplitude of a layer dynamics. Figure 3b shows that the critical layer DVRW in the cyclone under consideration obeys an contains a region where fluid parcels (within one azi- equation of the form

d|a| 5 ϭ ␥|a|, ͑23͒ Schecter et al. 2008 contains an explicit derivation and nu- dt merical verification of the acoustic analog of Eq. (24).

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2507

Ј muthal wavelength) are bound to orbit a central fixed The above estimate for qcl follows from material con- point. The region of trapped fluid is enclosed by a sepa- servation of PV. In addition, we may suppose that ratrix. The radial width of the separatrix is given by ␦r ϳ max͑l , l␥͒. ͑30͒ 2aU 1ր2 * trap l ϵ 2ͯ ͯ , ͑25͒ trap ⍀ր nd dr rϭr * The above estimate for ␦r considers two possibilities. * in which |2aU| is the peak radial velocity of the DVRW The first candidate ltrap is the initial half-width of the at a given radius (cf. section 3a). The orbital angular separatrix [Eq. (25)]. The second candidate l␥ is derived frequency of a trapped fluid parcel has a characteristic from linear theory; it is the radial length scale of the Ј → ϱ value of order peak that develops in qcl as t . Schecter et al. (2000) demonstrates that, to lowest order in ␥, ⍀ ϵ | ⍀ր | 1ր2 ͑ ͒ b 2naUd dr rϭr , 26 * ␥ ϭ ͯ ͯ ͑ ͒ l␥ . 31 which is commonly called the “bounce frequency.” In ⍀ր nd dr rϭr ⍀ * linear theory, both ltrap and b decrease exponentially with time, in proportion to the square root of a ϭ a e␥t. o If ␥ K ␻ and the correction to balanced dynamics is Linear theory tacitly assumes that trapped fluid par- small, then we may infer from SM06 (cf. section 4) that cels move outside of the collapsing separatrix before making too much progress along their orbital cycles. 2 2 2 The escape of trapped fluid prevents thorough mixing ␲ r |U| ␾ dqրdr ␥ Ϸ ͩ ͪ . r 2 2 2 | ⍀ր | of PV. However, if the initial bounce frequency exceeds c r ␾ |Q| nd dr rϭr ͵ * the linear decay rate of the wave; that is, if dr Ϫ ր 0 dq dr ⍀ ͑a ͒ տ |␥|, ͑27͒ b o ͑32͒ then trapped fluid parcels would complete their orbital Ј ϭ in␸ ϩ ϭ ␥ cycles and collectively coil PV inside the separatrix. Using qd aoQ(r)e c.c. at t 0, Eq. (32) for , and Such coiling tends to level q in the vicinity of r and either l or l␥ for ␦r , condition (28) amounts to (27) * trap * thereby arrest critical layer damping. In other words, up to an undetermined constant of proportionality on, condition (27) is sufficient grounds for a nonlinear ra- say, the right-hand side. diation-driven instability. An alternative line of reasoning leads to the same d. An illustrative numerical simulation conclusion. Let us first posit that a nonlinear radiation- driven instability will occur if the initial angular pseudo- Figures 6–8 (adapted from SM06) illustrate the non- momentum of the DVRW exceeds the finite absorption linear SI of a numerically simulated shallow-water MC. capacity of the critical layer. Assuming that the PV The initial condition consists of a balanced elliptical component dominates the angular pseudomomentum, deformation of the vortex core. Figure 6 shows the evo- this condition can be written lution of the potential vorticity field and the asymmetric component of the geopotential perturbation. The ␲ 2 2 rc r 2 ␾ ͑qЈ͒2 core perturbation consists of an n ϭ 2 DVRW, which ͵ ͵ ␸ d ͯ d dr Ϫ ր over time generates an outward propagating spiral IG 2dq dr tϭ 0 0 0 wave in the far field. Early on, PV stirring in the critical 2␲ r ϩ␦r 2 ␾ 2͑ Ј ͒2 layer (which appears as filamentation) tends to damp * * r qcl Ն maxͩ͵ ͵ d␸ dr ͪ, ͑28͒ the DVRW and its radiation field. Later on, the Ϫ ր 0 r Ϫ␦r 2dq dr * * DVRW and IG wave spontaneously amplify. The bot- in which r Ͻ (r Ϫ ␦r ) is the radius of the vortex core, tom panel of Fig. 6 provides a detailed picture of PV c * * and ␦r is the half-width of the critical layer. As usual, stirring in the critical layer. In time, trapped fluid par- * the subscripts d and cl refer to the DVRW and critical- cels collectively coil the PV distribution into a pattern layer perturbation, respectively. that is said to resemble a pair of cat’s eyes. To estimate the right-hand side of the above inequal- Figure 7 (top solid curve) shows a time series of the ity, we may first suppose that amplitude of the DVRW. For analytical purposes, the ⍀ ␥ 2 ratio | b / | is substituted for a dimensional measure dq 2 ͓͑ Ј ͒2͔ ϳ ͩ␦ ͪ ͑ ͒ of the perturbation strength. Initially, the DVRW de- max qcl r . 29 * dr rϭr cays exponentially with time, exactly as predicted by *

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2508 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

FIG. 6. Numerical simulation of the nonlinear SI of a shallow-water MC on the f plane. (top left) Evolution of the PV distribution q. Note that core ellipticity (the DVRW) decays and then grows. (top right) Evolution of the asymmetric component of the geopotential perturbation ␾Ј. Note that ␾Ј decays and then grows. (bottom) Detailed ⍀Ϫ1 evolution of the PV distribution in the critical layer. For all panels, time t is measured in units of max, and lengths ϵ⍀ ϭ are normalized to a characteristic vortex-scale ro. The Rossby number of the cyclone is Ro max/f 40, and the ϵ⍀ ͌␾ ϭ Froude number is Fr maxro / a 0.7. The data shown here correspond to simulation G1 of SM06.

linear theory (top dashed curve). Since the initial value (solid) and linear (dashed) evolution of the mode am- ⍀ ␥ of b exceeds the magnitude of , the time series even- plitude in a similar experiment in which the initial value ⍀ ␥ tually diverges from linear theory. Specifically, the am- of b is less than the magnitude of . In contrast to the plitude of the DVRW bounces after a time period of previous case, cat’s eyes do not fully develop in the ␲ ⍀ order 2 / b. At this point, the critical layer starts to critical layer of the DVRW. Moreover, there is no sign periodically return a fraction of the angular pseudomo- of nonlinear SI. mentum that it absorbs. As a result, critical layer damp- To conclude this section, let us briefly address an ing becomes inefficient. In the meantime, the positive important technicality. The preceding discussion as- feedback of IG wave radiation remains steady. Ulti- serted without proof that the inner part of the excited mately, radiative pumping prevails and SI ensues. mode was dominated by a vortex Rossby wave. A vor- The bottom curves in Fig. 7 show the nonlinear tex Rossby wave should possess the basic characteris-

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2509

FIG. 7. Conditional SI. The top curve (solid) shows the amplitude of the DVRW in Fig. 6. The horizontal bar (with double FIG. 8. The radial dependence of balance parameters in the arrows) indicates the maximum bounce period of the wave. Since numerical simulation of nonlinear SI (Fig. 6). Both D 2 and ⌬ (at ⍀ /|␥| initially exceeds unity, nonlinear SI occurs. The bottom b t ϭ 429.18) are less than unity to a radius beyond r , suggesting curve (solid) is from a similar experiment, in which the initial * that the DVRW and its critical layer are approximately balanced. wave amplitude is reduced by a factor of 0.25. Here, there is no In contrast, D 2 and ⌬ are much greater than unity in the IG wave sign of SI. The dashed curves are the exponential-decay predic- ⍀2 ϰ ␥t radiation zone. Radius r is measured in units of the characteristic tions of linear theory ( b e ). Time t is measured in units of ⍀Ϫ1 vortex scale, as were all lengths in Fig. 6. max. The data shown here correspond to simulations G1 (top curve) and G3 (bottom curve) of SM06. The reader may consult ⍀ this reference for details on the measurement of b and the cal- ␥ culation of . 4. The spontaneous imbalance of a dry baroclinic cyclone tics of a balanced perturbation. The formal theory of asymmetric balance requires that The theory of IG wave production by DVRWs, in- cluding the influence of critical layers, has been ex- ͑␻ Ϫ n⍀͒2 tended to 3D stratified cyclones with barotropic basic 2 ϭ K 1, ͑33͒ D ␩ ␰ states and cloud coverage (SM04; Schecter and Mont- gomery 2007, hereafter SM07). Plougonven and Zeitlin in which, as usual, ␻ is the oscillation frequency of the (2002) had previously developed an analytical theory mode. The weaker condition, D 2 Ͻ 1, suffices to ensure for IG wave emission by “pancake” vortices, but their that the intrinsic frequency of the mode is less than the analysis did not account for active critical layers. A local inertial frequency. Figure 8 (solid curve) verifies general theory for baroclinic cyclones, whose tangential that the asymmetric balance condition weakly holds in winds have vertical shear, was lacking before now. the vortex core and strongly holds in the critical layer. The following extension-by-analogy of shallow-water For standard balance (McWilliams 1985; Polvani et al. theory (SM06) to dry baroclinic cyclones has not been 1994; Montgomery and Franklin 1998), the central per- tested against numerical experiments, but is presented turbation would normally satisfy here for the purpose of motivating a deeper investigation. ␦ As usual, we will assume that ␪ (the potential tem- ⌬ ϵ n K ͑ ͒ ␨ 1, 34 perature) of the atmosphere increases monotonically n with altitude and that the axisymmetric PV distribution ␦ ␨ ␪ in which n and n are the peak values (at a given ra- q(r, ) of the unperturbed cyclone decreases monotoni- dius) of the divergence and vorticity components of the cally with radius on a surface of constant ␪. With suit- mode. Figure 8 (broken curve) indicates that the inner able boundary conditions, such a vortex is stable in the part of the mode satisfies condition (34). On the other context of balanced dynamics (Montgomery and Sha- hand, neither D 2 nor ⌬ is less than unity far beyond the piro 1995; Ren 1999). On the other hand, stability is not critical layer, where the DVRW transitions to an IG guaranteed when IG waves are allowed to interact with wave. DVRWs—in which case SI can occur.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2510 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

a. PV and angular pseudomomentum equations It is most convenient to analyze the dynamics of a baroclinic cyclone using the hydrostatic primitive equations in isentropic coordinates (see Appendix B). The assumption of hydrostatic balance does not permit accurate modeling of high-frequency IG waves. Accord- ingly, we will confine our attention to IG wave radiation that has a characteristic frequency greater than f but much less than the Brunt–Väisälä frequency N. There are two prominent equations that we must ex- plicitly consider in order to discuss the SI of a baroclinic MC. The first is the linearized PV equation: Ј Ѩ DVq q ϭϪuЈ . ͑35͒ Dt Ѩr

The second is the lowest-order flux conservative equa- FIG. 9. Illustration of a baroclinic MC and a cylinder that sepa- rates the vortex from the environment for analytical purposes. An tion for the angular pseudomomentum density J.In arbitrary DVRW will emit spiral IG waves into the environment. isentropic coordinates, Isentropic PV stirring in the critical layer, centered on the surface r ϭ r ␪), will potentially damp the DVRW and thereby suppress *( r␴2͑qЈ͒2 the IG wave radiation. J ϵϪ Ϫ r␴Ј␷Ј ͑36͒ 2͑ѨqրѨr͒

and The surface integrals that define S are associated with IG wave radiation, and possibly waves that are re- ѨJ 1 Ѩ Ѩ pЈ Ѩ␺Ј flected back into the vortex from distant objects should ϭ ͑r 2␴uЈ␷Ј͒ Ϫ ͩ ͪ Ѩt r Ѩr Ѩ␪ g Ѩ␸ they exist. Specifically, the top integral is the outward angular momentum flux on the lateral boundary of in ր V Ѩ ␷ ␴ R͑pЈ͒2 p R cp ␷ ϭ Ϫ ͫ ϩ ͑ Ј2 Ϫ ␷Ј2͒ Ϫ ͩ ͪ ͬ the absence of ambient rotation [assuming that 0at Ѩ␸ J u , r 2 2gp pa r ϭ R␷; cf. Eq. (B8)]. The bottom integral is the net outward angular momentum flux on the vertical bound- ͑37͒ aries of V for any value of f. in which ␴ is the isentropic mass density, ␺ is the Mont- Note that we have departed from our previous con- gomery streamfunction, g is gravitational acceleration, vention, and have chosen to represent the rate of change of angular pseudomomentum in the environ- R is the gas constant of dry air, cp is the specific heat of ment by S rather than by an indefinite volume integral. dry air at constant pressure p, and pa is the constant ambient surface pressure (cf. Chen et al. 2003). This alternative approach will facilitate the derivation The total angular pseudomomentum L in a fixed vol- of a formula for the growth rate of a radiative DVRW. ume V, containing a compact vortex, is the integral b. The growth rate of a DVRW

L ϵ ͵ d␸ d␪ drrJ. ͑38͒ As in shallow-water theory, a DVRW is here viewed V as a nearly balanced vortex mode that matches onto a spiral IG wave in the environment. Presumably, the IG Let V be a cylinder of radius R␷, with top and bottom wave emission compels the DVRW to grow. In the surfaces at ␪ and ␪ (Fig. 9). Integrating Eq. (37) max min outer region of the vortex, there exists a 3D critical over V, we obtain layer in which the angular rotation frequency of the ␲ ␪ dL max cyclone equals the angular phase velocity of the mode ϭ ͵ ͵ ␸ ␪͓ 2␴ Ј␷Ј͔ d d r u rϭR␷ (Fig. 9). The PV disturbance in the critical layer is dt Ϫ␲ ␪ min viewed (for the following analysis) as a relatively weak ␲ ␪ R␷ pЈ Ѩ␺Ј max perturbation. Loosely speaking, the isentropic stirring Ϫ ͵ ͵ d␸ drr ͯ of PV in the critical layer is slaved to the velocity fields Ϫ␲ g Ѩ␸ ␪ 0 min of the DVRW and of the mean vortex. At “higher or- ϵ S. ͑39͒ der,” the growth of qЈ in the critical layer compels the

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2511

␲ ␪ DVRW to decay. The following formally examines the max R␷ ϵ ͵ ͵ ͵ ␸ ␪ competition between the positive feedback of IG wave Ld – d d drrJ and Ϫ␲ ␪ 0 radiation and the negative feedback of PV stirring in min ␲ ␪ r ϩ␦r the critical layer. max * * L ϵ ͵ ͵ ͵ d␸ d␪ drr ͑ ͒ In a baroclinic cyclone, the perturbation fields of an cl J. 44 Ϫ␲ ␪min r −␦r ideal DVRW have the form * * Here ͐– is a radial integral that excludes the critical uЈ U͑r,␪͒ d layer, and the value of R␷ corresponds to the inner edge ␷Ј ͑ ␪͒ 6 d V r, of the radiation zone. With the above decomposition, ␴Ј ⌺͑ ␪͒ conservation of total L [Eq. (39)] becomes d r, ϭ a͑t͒ei͑n␸Ϫ␻t͒ ϩ ͑ ͒ Ј ͑ ␪͒ c.c., 40 pd P r, dLd dLcl ΄ ΅ ΄ ΅ ϭ S Ϫ . ͑45͒ ␺Ј ⌿͑ ␪͒ d r, dt dt Ј ͑ ␪͒ qd Q r, The restructured conservation law [Eq. (45)] is readily transformed into an equation for the amplitude in which the phase of a varies at a much slower rate |a| of the DVRW. Inserting Eq. (40) into the integral than ␻. Far from the vortex core, the above wavefunc- expression for L [Eq. (44)] and taking the time deriva- tions match onto those of a spiral IG wave. d tive yields The critical layer is centered on the surface r ϭ r (␪) * in which r satisfies the following resonance condition: | | 2 * dLd d a ϭ 2␲M , ͑46͒ dt dt ⍀͑r , ␪͒ ϵ ␻րn. ͑41͒ * in which ␦ ␪ In linear theory, the radial length scale r ( )ofthe ␪ * max R␷ r 2␴2 |Q| 2 critical layer is proportional to M ϵ ͵ ͵– d␪ drͫ Ϫ r 2͑⌺V* ϩ c.c.͒ͬ Ϫ ѨqրѨr ␪min 0 ␥ ͑␪͒ ϵ ͯ ͯ ͑ ͒ ͑47͒ l␥ , 42 Ѩ⍀րѨ n r rϭr * is the angular pseudomomentum of the DVRW divided ␲ 2 in which ␥ ϵ d ln|a|/dt. In principle, the constant of by 2 |a| . Likewise, inserting Eq. (40) into the surface proportionality is of order unity (cf. Schecter et al. integrals that define S yields 2000), but a more conservative choice of order 10 is S ϭ 4␲⑀ |a| 2, ͑48͒ frequently used (SM04; SM06; SM07). If at some ver- rad tical level r happens to be near the origin, where the * in which radial derivative of ⍀ vanishes, then a basic exten- ␪ sion of the analysis in Schecter et al. (2000) suggests 1 max ⑀ ϵ ͵ ␪ ͑ 2␴ ϩ ͒ ␦ ϰ ͌ ␥ rad d r UV* c.c. rϭR␷ that r | |. 2 ␪ * min Note that Eq. (40) cannot precisely describe a quasi- R␷ mode that is damped by the growth of qЈ in the region 1 ␪ Ϫ ͵ ͑Ϫ ⌿ ϩ ͒| max ͑ ͒ dr inrP * c.c ␪ . 49 Јϭ Ј min of resonance. In particular, q qd would be accurate 2g 0 only outside the critical layer. Inside the critical layer, let It is consistent with the barotropic problem (SM04; ⑀ SM07) to expect positive values for M and rad. Ј ϭ ͑ ␪ ͒ in␸ ϩ | Ϫ ͑␪͒| Ͻ ␦ ͑␪͒ A different path is required to evaluate the time de- q q˜ cl r, , t e c.c., r r r , * * rivative of angular pseudomomentum in the critical ͑43͒

6 Sections 2 and 3 assumed that the angular pseudomomentum in which q˜ cl is to be determined. To find the growth rate of a DVRW, let us first divide of the DVRW was negligible beyond the inner core of the vortex r Ͼ r Ϫ ␦r ). Here, we have taken a more liberal view, (i.e., for * * the total angular pseudomomentum into DVRW and and have allowed the angular pseudomomentum density of the ϭ ϩ critical-layer components; that is, let L Ld Lcl in DVRW to exist on both sides of the critical layer. In practice, the

which contribution to Ld from the inner core typically dominates.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2512 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 layer where qЈ is potentially deviant. First, we may in which write ␪ max r 2 |U| 2␴ 2ѨqրѨr ⑀ ϵϪ␲͵ d␪ͩ ͪ ͑ ͒ ␪ ϩ␦ cl . 54 max r r ␪ |nѨ⍀րѨr| dLcl * * min rϭr ϭ 2␲͵ ͵ d␪ dr * dt ␪ r Ϫ␦r min * * ⑀ Note that the positive value of cl increases from zero r in the criticalץ/qץ with the “average” negative value of 2 2 r ␴ Ѩ Ѩ ␸ ϫ ͫ ͑q˜ q˜*͒ Ϫ r 2 ␴Ј␷Ј ͬ, ͑50͒ layer. Ϫ ѨqրѨr Ѩt cl cl Ѩt Upon substituting Eqs. (46), (48), and (53) into Eq. ␸ (45) we obtain in which h is the azimuthal average of h, as in section 3. The integrand contains two terms that we will exam- d|a| ϭ ␥|a|, ͑55͒ ine separately. dt To begin with, consider the linearized PV equation 8 [Eq. (35)] in the critical layer, in which ⑀ Ϫ ⑀ Ѩ Ѩ ␥ ϵ rad cl ͑ ͒ q˜ cl q . 56 ϩ in⍀q˜ ϭϪaUeϪi␻t . ͑51͒ M Ѩt cl Ѩr ⑀ The first term of the growth rate ( rad /M ) accounts for Јϭ Ј Here we have used the modal approximation u ud on the positive feedback of IG wave radiation. The second Ϫ⑀ the right-hand side. Presumably, nonmodal contribu- term ( cl /M ) accounts for the negative feedback of .r is sufficiently weak in PV stirring in the critical layerץ/qץ tions to uЈ are negligible if the critical layer.7 For simplicity, suppose that the initial The preceding theory for ␥ relied on multiple as- value of q˜ cl equals zero. Furthermore, suppose that the sumptions that may have limited applicability. The least DVRW is near marginal stability so that a remains ap- subtle is that the DVRW (should one even exist) is near proximately constant over many oscillation periods. marginal stability. A small value of ␥ is required for the Then, integration of Eq. (51) yields presumed length scale separation between the thin critical layer and the bulk of the vortex. It is also re- Ѩ 2␲|U| 2͑ѨqրѨr͒2 quired for the time scale separation between the e- ͑q˜ q˜*͒ → ␦͑r Ϫ r ͒|a| 2 ͑52͒ folding and oscillation periods of the wave. Despite the Ѩt cl cl |nѨ⍀րѨr| * imprecision of Eq. (56) in more general circumstances, it still demonstrates that wave–flow resonances can for t k ␻Ϫ1 (cf. SM04). The delta-function response is greatly hinder SI in MCs. a signature of resonance. The second term of the integrand of dL /dt [Eq. cl ␸ (50)] is proportional to the time derivative of ␴Ј␷Ј . For 5. The potential impact of SI on tropical cyclones the special case of a barotropic cyclone, it has been argued that the integral of this term is negligible The introduction referred to a theoretical study by (SM04). The argument was built upon a Frobenius Chow and Chan (2003) that estimated an appreciable analysis of the critical layer disturbance and on numeri- rate of angular momentum loss by hurricanes due to IG cal evidence. Let us suppose that the second term of wave radiation. Specifically, they proposed that a spiral IG wave can remove up to 10% of the core angular dLcl /dt is also negligible in the more general baroclinic problem, and leave rigorous proof for future research. momentum in one rotation period. To the author’s Inserting Eq. (52) into the right-hand side of Eq. (50) knowledge, there are no numerical studies at this time and neglecting the second term yields that irrefutably support the 10% loss estimate. On the other hand, there is numerical evidence that radiation-driven instabilities of hurricane-like vortices dLcl ϭ 4␲⑀ |a| 2, ͑53͒ can occur relatively fast (Schecter and Montgomery dt cl 2003; SM04; D. Hodyss and D. Nolan 2005, personal

7 Note that nonmodal (continuum) contributions to uЈ are ex- 8 Growth rate formulas that are similar to Eq. (56) have been pected to dominate at very late times if the DVRW is damped. numerically verified for DVRWs near marginal stability in moist Nevertheless, we will assume that Eq. (51) stays valid over the barotropic MCs (SM07), dry barotropic MCs (SM04), and shal- time scale of interest. low-water MCs (SM06).

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2513

communication). In the following, we will reexamine Suppose that the perturbation is dominated by a these instabilities with the goal of calculating the rate at single mode. Furthermore, suppose that which they remove angular momentum from a vortex. ␧ ␷ In addition, we will reexamine the magnitude of the ϭ m ͑ ͒ ao , 61 2 |U| ϭ spiral bands of vertical velocity that they create in the r Rm radiation zone. It has been speculated that such bands such that the initial amplitude of the radial velocity might encourage moist convection. ϭ ϭ ␧ perturbation at r Rm and z 0is times the maximum tangential wind speed. Then it is readily shown a. Radiation torque on a Rankine cyclone that the initial rate of change of L satisfies the following The study that we shall revisit is SM04, which is based equation: on the hydrostatic Boussinesq primitive equations (see ␲␧2ℜ͓ ͔ 1 dL UV* rϭR␷ appendix C). In the Boussinesq model, the relative ϭϪ , ͑62͒ L d␶ 1 R 2 angular momentum of a cyclone (per unit mass) is de- ͫ Ϫ ͩ mͪ ͬ| | 2 1 U rϭR fined by 2 R␷ m ℜ 1 in which [. . .] is the real part of the quantity in square L ϵ ͵ dx3r␷, ͑57͒ brackets and ␶ is time normalized to the vortex rotation ␲ 2 R␷ H V ␲ ␷ period, 2 Rm / m. The rhs of Eq. (62) is expected to increase monotonically from zero as the rotational in which the volume integral is over a cylinder V of Froude number of the cyclone increases from zero to height H and radius R␷. For simplicity, we assume rigid order one (cf. section 3b; Ford 1994a,b; Plougonven and vertical boundaries at z ϭ 0 and at z ϭ H. Then, the Zeitlin 2002). In other words, at fixed N, the dimen- rate of change of angular momentum due to spiral IG sionless radiation torque is expected to increase mono- wave radiation is given by ␷ tonically with m into the superspin parameter regime. For comparison, surface drag would remove angular dL ␸,z ϭϪ2uЈ␷Ј ͑58͒ momentum (per unit mass) from the unperturbed cy- dt clone at the following rate: ␸,z in which h is the average of h over the lateral bound- dL 2C R␷ ͩ ͪ ϭϪ D ͵ 2␷ 2 ͑ ͒ ary of the cylinder V [cf. Eq. (C10) with condition (C7)]. 2 drr , 63 ␷ dt s R␷ H 0 Consider a barotropic Rankine cyclone in which m is the maximum tangential velocity and Rm is the radius of in which CD is the average dimensionless drag coeffi- maximum wind. The unperturbed angular momentum cient. For Rankine cyclones, Eq. (63) implies that (per unit mass) of this cyclone is R␷ 4 ␲ 3 ͩ Ϫ ͪ 2 4 CDRm 1 Rm 1 dL R 5 L ϭ ␷ R ͫ1 Ϫ ͩ ͪ ͬ, ͑59͒ ͩ ͪ ϭϪ m ͑ ͒ m m , 64 2 R␷ L d␶ 1 R 2 s 2 ͫ Ϫ ͩ mͪ ͬ R␷ H 1 Ͼ 2 R␷ assuming that R␷ Rm. If the Brunt–Väisälä frequency Ͼ N is constant, the linear modes of any barotropic cy- again assuming that R␷ Rm. The absolute value of the clone have the form rhs of Eq. (64) provides a threshold on the dimensionless radiation torque, above which IG wave emission Ј ␲ ր u U͑r͒ cos͑m z H͒ supersedes surface drag as an angular momentum sink. ␷ ␷Ј V͑r͒ cos͑m␲zրH͒ Notably, this threshold does not vary with m.Ifan ϭ a e␥te i͑n␸Ϫ␻t͒ ϩ c.c., ΄wЈ΅ o ΄W͑r͒ sin͑m␲zրH͒΅ intense cyclone cannot generate radiation torque above this threshold, neither can a weak cyclone. ␾Ј ⌽͑r͒ cos͑m␲zրH͒ ͑60͒ b. Some computational results in the category 5 hurricane parameter regime in which uЈ, ␷Ј, wЈ, and ␾Ј are the radial velocity, azimuthal velocity, vertical velocity, and geopotential per- Figure 10 presents key features of the first baroclinic turbations, respectively. The reader may consult appen- (m ϭ 1) SI modes of a barotropic Rankine cyclone. The dix C for formulas that relate the velocity wavefunc- parameters of the cyclone resemble those of a category ⌽ ␷ ϭ Ϫ1 ϭ ϭ tions to . 5 hurricane: m 75ms ; Rm 50 km; H 10 km;

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2514 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

eigenmode solver approximated the Rankine cyclone with the following vorticity distribution:

␷ ͑ Ϫ ͒ m r Rm ␨͑r͒ ϭ ͭ1 Ϫ tanhͫ ͬͮ, ͑66͒ Rm bRm

in which b ϭ 0.025. Two of the four solutions (n ϭ 2, 4) were independently verified with numerical integra- tions of the initial value problem. The top curves of Fig. 10 show the e-folding times and wave periods of the DVRWs and their connected radiation fields. Notably, the minimum e-folding time is 3.5 rotation periods, which is just over 4 h. It is important to emphasize that this instability is driven entirely by IG wave radiation. It does not require millions, thousands, hundreds, or even tens of rotation periods to be seen, as would be the case at small Froude number. FIG. 10. The SI modes (m ϭ 1, 1 Յ n Յ 4) of a barotropic The middle curves of Fig. 10 show the fractional loss Rankine cyclone in the parameter regime of a category 5 hurri- of core angular momentum (␦L) in one vortex rotation cane. Each mode consists of an inner DVRW and an outer spiral period [minus the rhs of Eq. (62)]. This quantity in- IG wave. They are initially excited by deforming the isosurfaces of ␧ core potential vorticity, as sketched below the graph. The vari- creases quadratically with the perturbation strength , ␧ϭ ␦ ables ␶␥ and ␶␻ are, respectively, the e-folding time (1/␥) and and is shown for 0.2. The two curves for L use oscillation period (2␲/␻) of the mode, measured in vortex rotation different values of R␷. For the top curve, R␷ is chosen to ␲ ␷ ␦ periods (2 Rm / m). The variable L is the fraction of core relative maximize ␦L. All maxima occur in the domain 1.05 Ͻ angular momentum that is lost by radiation in one vortex rotation R␷ /R Ͻ 1.07. For the bottom curve, R␷ ϭ r ϩ 10l␥. period. The variable wЈ is the maximum vertical velocity of the IG m * ␷ ␦ Ј With this formula, R␷ /Rm decreases from 2.1 to 1.2 as n wave radiation, normalized to m. Both L and w depend on the instantaneous mode amplitude ␧, which is the peak value of uЈ at increases from 1 to 4. For all cases considered, ␷ ␧ϭ the radius of maximum wind, normalized to m. Here, 0.2. The two curves for 4␦L correspond to two reasonable definitions of the NH K ͑ ͒ core radius R␷ (see section 5b). The shaded region covers the R␷ ␲ ␥ , 67 ␦ m estimated range of 4 L s, which is (four times) the fraction of core relative angular momentum that would be lost by surface drag over the ocean in one vortex rotation period. in which the rhs is an estimate (valid for ␻ k f )ofthe radial distance that the spiral IG wave travels in one Ϫ1 e-folding time (␶␥ ϭ ␥ ) of the perturbation. Condition Ϫ4 Ϫ1 Ϫ2 Ϫ1 9 ℜ f ϭ 10 s ; and N ϭ 10 s . The Rossby and (67) guarantees that [UV*]rϭR␷ in Eq. (62) approxi- Froude numbers are thus mately corresponds to the radiation flux created by the core DVRW at the instant under consideration. ␷ ␲␷ Figure 11 (bottom) illustrates the variation of ␦L with ϵ m ϭ ϵ m ϭ ͑ ͒ Ro 15 and Fr 2.36, 65 R␷. The limit as r → ϱ is irrelevant since it is zero. In fRm NH part, ␦L vanishes at infinity because the relative angular in which NH/␲ is the characteristic horizontal phase momentum of a Rankine cyclone diverges with increas- speed of an ambient gravity wave with vertical wave- ing radius. To avoid this problem, we could regularize number m ϭ 1. Each SI mode under consideration, the vortex. However, the instantaneous radiation field which consists of an inner DVRW and an outer spiral would still decay exponentially with increasing r. Such IG wave, corresponds to a numerical solution of the decay follows from causality; the outer waves were cre- linear eigenmode problem (cf. SM04; appendix C). The ated when the amplitude of the source (the DVRW) was exponentially smaller. According to Fig. 10, the first baroclinic SI modes of a category 5 “hurricane” do not remove angular mo- 9 ϭ Using an extratropical value for the Coriolis parameter (f ␧ϭ 10Ϫ4) instead of a tropical value does not matter, since Ro k 1in mentum at an appreciable rate. For 0.2, the rate is either case. Results for f ϭ 5 ϫ 10Ϫ5 are nearly indistinguishable 0.006–0.028 core units per vortex rotation period. As from the results presented here. such, it would take between a few days and one week

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2515

the midtropospheric region (z ϭ H/2) of the radiation zone (r Ͼ R␷). For ␧ϭ0.2, these velocities range from ␷ Ϫ1 0.002 to 0.015 times m, or from 0.1 to 1.1 m s . For perspective, Fig. 11 (top) shows the wЈ field of the n ϭ 2 mode. To the author’s knowledge, there is no indis- putable evidence from realistic hurricane simulations that the spiral wЈ bands of SI modes engender deep convection. The case study by Chow et al. (2002) suggests that cloud bands producing little rainwater are more likely. Perhaps future investigations will reveal circumstances in which the IG wave component of an SI mode produces heavy rainbands.

c. Suppression of SI by skirts Sections 5a and 5b focused on the radiation-driven instability of a Rankine cyclone, which might be excep- tional in its propensity for SI. Comprehensive observations (e.g., Mallen et al. 2005) and basic theory (e.g., Emanuel 1986) suggest that, in contrast to the Rankine model, a typical hurricane will have a significant skirt of ␨ cyclonic vorticity that extends far beyond Rm and de- FIG. 11. Snapshot of a radiating “hurricane.” The vertical wind (wЈ) and radiation torque (Ϫ␦L) are generated by an SI mode with cays with increasing r. The linear theories of sections 3 (m, n) ϭ (1, 2) and ␧ϭ0.2. The solid wЈ curve is along a fixed and 4 suggest that such skirts enhance the damping of azimuth at z ϭ H/2. The dashed curve shows the maximum of wЈ DVRWs owing to wave–flow resonances, and thereby Ј at the same vertical level. Note that w is negligible in the critical hinder SI. layer [cf. Eq. (C13)]. The dark triangle and diamonds indicate the measurements that were used in Fig. 10. Studies of barotropic MCs in a continuously stratified fluid indicate that the damping power of a skirt is largely controlled by the deformation radius [cf. Eq. for spiral radiation to deplete the core circulation (r Յ (11)]: R␷) of the bulk of its original angular momentum, neglecting potential decay of the radiation torque and re- N l plenishment by radial inflow. In contrast, Chow and ϵ m z ͑ ͒ lD . 68 Chan (2003) estimated that ␦L is of order 0.1. Their ͌␩ ␰ large estimate may have resulted from using a small Froude-number formula to approximate the radiation Here N is the characteristic Brunt–Väisälä frequency Ͼ m flux at Fr 1. of moist air within the cyclone, ␩␰ is the characteristic For comparison, the shaded region of Fig. 10 shows inertial stability of the cyclone, and lz is the vertical the estimated fractional loss of core angular momentum length scale of the DVRW. The literature generally due to surface drag (␦ ) in one vortex rotation period L s indicates that the critical layers of the DVRWs of MCs [minus the rhs of Eq. (64)]. The upper and lower move radially inward with decreasing l (Jones 1995; bounds correspond to C ϭ 0.001 and C ϭ 0.003, as D D D Reasor and Montgomery 2001; Schecter et al. 2002; might be expected for a hurricane over the ocean (cf. Schecter and Montgomery 2003; SM04; SM07; Reasor Black et al. 2007). The value of R␷ was set equal to et al. 2004). Equation (68) indicates that decreasing l 1.17R where ␦L is maximized. The upper bound of D m s occurs by the radiation torque barely matches the lower bound of the surface torque. So, it is sensible to assume that ra- (i) increasing the density of cloud coverage, diation torque is typically subdominant. This assump- (ii) decreasing the positive vertical potential tempera- tion is a tacit component of current theories for the ture gradient, maximum potential intensity of a hurricane (e.g., (iii) increasing the intensity (inertial stability) of the Emanuel 1986, 1995). vortex, or On another topic, the bottom curves of Fig. 10 show (iv) decreasing the vertical length scale of the pertur- the maximum vertical velocities (wЈ) of the SI modes in bation.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2516 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

ϵ⍀ ϭ FIG. 12. The suppression of a linear SI mode by increasing cloudiness in a barotropic cyclone at Ro max/f 2. All lengths are normalized to the radius of maximum wind Rm, and all frequencies are normalized to the peak relative vorticity. (a) Relative vorticity ␨ ⍀ ( ) and angular velocity ( ) of the basic state. The basic state is cloudy in the vicinity of Rm, but the environmental air is dry. (b) The ␥ ϭ ϭ growth rate of an n 2 DVRW whose vertical length scale yields lD 0.26Rm under dry conditions. As cloudiness increases, lD decreases and ␥ ultimately becomes negative. The ϩ symbols correspond to numerical solutions of the eigenmode/quasi-mode problem, and are connected by the solid curve. The dotted curves are theoretical values of the (top) radiative pumping rate and (bottom) critical ϫ layer damping rate. Each corresponds to the sum of the dotted curves at a specific value of lD. (c) As lD decreases, the critical radius r of the DVRW moves inward, where the vorticity (PV) gradient is relatively steep. Each diamond marks the location of r for a data * * point in (b), with progressively smaller lD from right to left. Note: the data in (b) and (c) are from Figs. 12 and 10 of SM07.

Typically, the radial PV gradient of a skirt sharply in- a location where, in linear theory, critical layer damping

creases with decreasing r. Consequently, decreasing lD prevails over radiative pumping. Interested readers 11 (to values below Rm) greatly increases the negative ra- may consult SM07 for further details. K dial PV gradient in the critical layer of a DVRW and Of course, a steep skirt (or lD Rm) does not guar- greatly accelerates resonant damping.10 antee the suppression of SI in a real tropical cyclone; Hurricane-like vortices have relatively small values rather, it represents one inhibiting factor. As men- ␩␰ of lD. For example, evaluating the inertial stability at tioned earlier, the angular pseudomomentum of a ϭ ␲ ϭ Rm and letting lz H/ , we obtain lD 0.3 Rm for the DVRW at sufficiently large amplitude can exceed the model hurricane of section 5b. This small value of lD finite absorption capacity of its critical layer. In this ensured that the critical layers of all baroclinic DVRWs case, nonlinear SI would occur. Furthermore, hurri- were centered close to the core; in particular, r Յ 1.6 canes typically have nonmonotonic cores that suffer * Rm for each mode. Consequently, in linear theory, a shear-flow instabilities. Such instabilities can act as ad- skirt of very modest extension would have sufficed to ditional sources of spiral IG waves and are not sensitive quench all DVRWs and thereby suppress radiation (cf. to the gradient of the skirt. Finally, even if the cyclone SM04). has a monotonic potential vorticity distribution, it can The above conclusion seems more robust upon con- exhibit a “nonmodal” shear-flow instability (cf. Nolan sidering the effect of moisture. Figure 12, adapted from and Farrell 1999; Antkowiak and Brancher 2004) that SM07, illustrates the suppression of a linear SI mode by temporarily amplifies the IG wave radiation field. Con- cloud coverage in a barotropic cyclone. As usual, the SI vective processes within the vortex might initiate such mode consists of an inner DVRW and an outer spiral an event. IG wave (not shown). The azimuthal wavenumber is n ϭ 2, and the vertical wavelength is such that under d. IG wave emission by sheared vortex Rossby ϭ waves dry conditions lD 0.26 Rm in which lD is evaluated at ϭ r Rm. Because the cyclone is nearly Rankine, the dry If a steep skirt is able to severely damp all DVRWs, value of lD is not sufficiently small to quench SI. Nev- then arbitrary forcing might prefer to excite sheared ertheless, lD is reducible by increasing cloudiness (de- vortex Rossby waves (SVRWs). There is evidence that

creasing Nm) in the vicinity of Rm. As shown here, reduction of l by cloud coverage ultimately moves r to D * 11 The results in SM07 (section 8) are expressed in terms of a cloudiness parameter ⑀ and the dry Rossby deformation radius dry ϭ Ϫ ⑀ dry l R . Here we have used the relation lD (1 )l D in which 10 dry ϭ dry ͌␩␰ Schecter et al. 2002 provides a rare counterexample. l D fl R / .

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2517 such waves exist in tropical cyclones and are capable of can steepen by orders of magnitude (Fig. 12c). Hence, generating spiral cloud bands (McDonald 1968; Chen raising Fr can bolster critical layer damping and sup- and Yau 2001; Wang 2002a,b; Chen et al. 2003). Bal- press SI. ance theories suggest that SVRWs can intensify the Of course, linear damping does not guarantee sup- storm by fluxing angular momentum to the radius of pression. Section 3d showed that nonlinear SI will occur maximum wind (Montgomery and Kallenbach 1997; if the angular pseudomomentum of the DVRW exceeds Möller and Montgomery 1999, 2000; Enagonio and the finite absorption capacity of the critical layer. Such ⍀ Montgomery 2001). However, an SVRW may also cre- is the case when the bounce frequency b of the ate IG wave radiation that fluxes angular momentum DVRW is greater than the linear decay rate |␥|. When away from the vortex. Because an SVRW loses coher- the wave amplitude is sufficiently large to satisfy this ence and attenuates, it is not likely to sustain radiation condition, the DVRW mixes the PV distribution in the by itself. Nevertheless, persistent cycles of SVRW cre- critical layer and effectively levels the once stabilizing ation and dissipation may engender a notable time- gradient. averaged radiation torque. Section 5 reexamined SI in the superspin parameter regime where both Ro and Fr are above unity. In particular, it considered the first baroclinic SI modes of a 6. Concluding remarks barotropic cyclone at Rossby and Froude numbers characteristic of a category 5 hurricane. It was verified This paper reviewed an important paradigm of spon- that IG wave radiation can triple the amplitude of a taneous imbalance (SI) that operates at Rossby num- DVRW in just a few vortex rotation periods. However, bers greater than unity. The main text began by dis- the negative radiation torque appeared to be less sig- cussing the balanced dynamics of a discrete vortex nificant than the expected influence of oceanic surface Rossby wave (DVRW) in a monotonic cyclone (MC). drag. In the context of balanced dynamics, the DVRW is a Despite recent progress, much remains to learn potential vorticity wave that acts at a distance on fluid about the SI of intense mesoscale vortices. The extent beyond the core region of the MC. Section 2 explained to which DVRWs (as opposed to continuum perturba- that the external flow field of a DVRW most efficiently tions) control IG wave emissions from baroclinic cy- stirs PV in a critical layer where the angular phase ve- clones is unknown. Moreover, the effect of secondary locity of the wave matches the angular velocity of the circulation in the basic state is uncertain. The role of mean flow. Since total angular pseudomomentum is conserved, its production in the critical layer damps the moisture is also relatively unexplored. Clearly, mois- DVRW. The damping rate is proportional to the mean ture cannot be ignored in the troposphere. At the most radial gradient of critical layer PV. basic level of approximation, cloud coverage simply re- Beyond the critical layer, balance conditions break duces the Brunt–Väisälä frequency of the system. As down, and the external flow field of the DVRW excites explained in section 5c, this can indirectly enhance the an outward propagating spiral IG wave. Section 3 ex- ability of critical layers to inhibit the SI of an MC (cf. plained that the DVRW and IG wave have angular SM07). Future research may better explain how a more pseudomomenta of opposite sign, positive and nega- realistic nonlinear coupling of vortex modes to cloud tive, respectively. Therefore, producing IG wave radia- processes affects their ability to excite IG waves in the tion compels the DVRW to grow. The growth rate due far field. to radiation alone increases algebraically from zero with the rotational Froude number of the vortex. Acknowledgments. This paper was solicited for a spe- Nevertheless, SI will occur only if radiative pumping cial issue on the topics addressed at the SI workshop supersedes critical layer damping. The competition be- held in Seattle, Washington, 7–10 August 2006. The SI tween these two processes has been quantified in linear workshop was sponsored by the National Science Foun- theory, for both shallow-water cyclones [section 3; Eq. dation and NorthWest Research Associates. The au- (24)] and continuously stratified cyclones [section 4; thor thanks Dr. Paul D. Reasor for a helpful discussion Eq. (56)]. It is sensible to suppose that increasing Fr regarding section 5. The author also thanks an anony- would enable or intensify SI because it enhances the mous referee for helpful suggestions on improving the radiative pumping of a DVRW. However, increasing Fr style of this manuscript. Funding for this research was at high Rossby number coincides with decreasing the provided by NSF Grant ATM-0649944, channelled vortex deformation radius lD. The reduction of lD through the Department of Defense, Naval Postgradu- moves the critical radius inward where the PV gradient ate School Grant N00244-07-1-0002.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2518 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

APPENDIX A 1976) replaces the vertical Cartesian coordinate z with the potential temperature The Shallow-Water Model Rրcp pa ␪ ϵ Tͩ ͪ ͑B1͒ The shallow-water equations constitute the simplest p model of geophysical flow that accounts for the interaction of vortical motion with IG waves. They form the in which T is absolute temperature, R is the gas con- basis of the theoretical and numerical results of section stant of dry air, cp is the specific heat of dry air at 3. We may write the shallow-water equations in the constant pressure p, and pa is the ambient surface pres- following compact form: sure. In isentropic coordinates, the momentum equation Ѩu u2 ١ͩ␾ ϩ ͪ ϭ 0, ͑A1͒ takes the form ϩ ␩ ϫ u ϩ Ѩt 2 Ѩu u2 ͒ ͑ ϩ ␺ͪ ϭ ͩ ١ Ѩ␾ ϩ ␩ ϫ ϩ zˆ u ␪ 0, B2 ␾u ϭ 0. ͑A2͒ Ѩt 2 · ١ ϩ Ѩt ١ Above, u is the horizontal velocity field, ␾ is the geo- in which u is the horizontal velocity field, ␪ is the isen- ϫ ϩ ١ ␩ ϵ potential (gravitational acceleration g times the free tropic horizontal gradient operator, zˆ · ␪ u f ␺ ϵ ϩ is the horizontal gradi- is the absolute vertical vorticity, and cpT gz is the ١ surface height), t is time, and ent operator. The absolute vorticity vector is defined by Montgomery streamfunction. The hydrostatic relation is given by ϫ u ϩ f zˆ, ͑A3͒ ١ ␩ ϵ ր Ѩ␺ p R cp in which zˆ is the vertical unit vector. ϭ c ͩ ͪ . ͑B3͒ Ѩ␪ p p It is well known that Eqs. (A1) and (A2) conserve a PV (q) along material trajectories. That is, Finally, mass continuity takes the form Ѩq Ѩ␴ ١q ϭ 0, ͑A4͒ · ϩ u ١␪ · ␴u ϭ 0, ͑B4͒ Ѩ ϩ t Ѩt in which in which zˆ · ␩ q ϵ . ͑A5͒ 1 Ѩp ␾ ␴ ϭϪ . ͑B5͒ g Ѩ␪ In addition, manipulation of Eqs. (A1) and (A2) yields the following flux-conservation law for absolute angu- With appropriate boundary conditions, Eqs. (B2)–(B5) lar momentum density: form a closed system. In the isentropic coordinate system, potential vortic- 2 2 2 Ѩ fr fr r␾ ity is given by ,ͬ ˆu␾ͩr␷ ϩ ͪ ϩ ␸ͫ · ␾ͩr␷ ϩ ͪͬ ϭϪ١ͫ Ѩt 2 2 2 q ϵ ␩ր␴ ͑B6͒ ͑A6͒ and is conserved along material trajectories; that is, in which r is radius from a central vertical axis, ␷ is the azimuthal velocity field, and ␸ˆ is the azimuthal unit Ѩq ١␪q ϭ 0. ͑B7͒ · ϩ u vector. Conservation of mass, PV, and absolute angular Ѩt momentum form the basis of an exact angular pseudomomentum conservation law. The reader may consult In addition, the absolute angular momentum density is Guinn and Schubert (1993) for a detailed derivation. governed by Ѩ fr 2 fr 2 ͪ ␴ͩ ␷ ϩ ͫ APPENDIX B ͫ␴ͩ ␷ ϩ ͪͬ ϭϪ١ r ␪ · u r Ѩt 2 2

The Hydrostatic Primitive Equations in Isentropic ր c rp p R cp Coordinates ϩ p ͩ ͪ ␸ͬ ͑ ϩ ր ͒ ˆ g 1 cp R pa The theoretical results of section 4 are based on the hydrostatic primitive equations in isentropic coordi- Ѩ p Ѩ␺ ϩ ͩ ͪ , ͑B8͒ nates. The isentropic coordinate system (e.g., Dutton Ѩ␪ g Ѩ␸

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2519 v ϭ 0. ͑C7͒ · ١ .in which ␷ again represents the azimuthal velocity field As in shallow-water theory, conservation of mass, PV, With appropriate boundary conditions, Eqs. (C2)–(C4) and absolute angular momentum form the basis of an and (C7) form a closed system. exact angular pseudomomentum conservation law. It can be shown that the Boussinesq equations conserve the following potential vorticity, APPENDIX C ١␪, ͑C8͒ · ͒ˆϫ u ϩ fz ١͑ q͑x,t͒ ϵ The Hydrostatic Boussinesq Model in along material trajectories. That is, Pseudoheight Coordinates Ѩq ١q ϭ 0. ͑C9͒ · ϩ v The results of section 5 are based on a hydrostatic Ѩt Boussinesq model that uses the following pseudoheight as the vertical coordinate: In addition, the relative angular momentum per unit mass is governed by ր p R cp c p ͑ ͒ ϵ Ϫ p a ͑ ͒ Ѩ ␷ 2 z p ͫ1 ͩ ͪ ͬ ␳ , C1 ͑r ͒ fr vͩr␷ ϩ ͪ ϩ r␾␸ˆ ͬ ϭ 0 ͑C10͒ͫ · ١ pa R ag ϩ Ѩt 2 in which p and ␳ are ambient surface values of pres- a a ␷ sure and density (Hoskins and Bretherton 1972). Note in which (as usual) is the azimuthal velocity field. ϭ ␪ ␪ SM04 derives an exact angular pseudomomentum con- that dz dz˜ a / , in which z˜ is the genuine height co- ␪ servation law that stems from Eqs. (C9) and (C10). ordinate and a is the ambient surface value of potential temperature. The difference between z and z˜ is gener- Equation (58) of the main text follows from Eq. (C10), ally small in the lower 10 km of the atmosphere and Eq. (C7), and rigid, frictionless vertical boundary con- vanishes if the stratification is dry adiabatic. Therefore, ditions. the pseudo vertical velocity (w ϵ dz/dt) of a fluid parcel It is straightforward to derive linearized perturbation approximately equals the genuine vertical velocity in equations from the above model. Suppose that the ba- the region of interest. sic state is a barotropic cyclone in gradient balance, and In pseudoheight coordinates, the horizontal momen- that the vertical boundaries are rigid. Furthermore, sup- 2 ץ 2␾ץ͌ ϭ tum equation takes the form pose that the Brunt–Väisälä frequency N / z is constant. Then, the eigenmodes are formally given by Ѩu Eq. (60) of the main text. Furthermore, it has been ␾ ϭ 0, ͑C2͒ ١ ١u ϩ fzˆ ϫ u ϩ · ϩ v Ѩt z shown (SM04) that the geopotential eigenfunction must satisfy the following equation: in which u is horizontal velocity, v ϵ u ϩ zˆw, ␾ ϵ gz˜ is d r d n d ␰ 1 ١ the geopotential, z is the horizontal gradient operator, ͫ ⌽ͬ Ϫ ͫ ͬ⌽ ␻ 2 2 ץ ץ ϩ ١ ϵ ١ and z zˆ / z. Furthermore, the adiabatic heat r dr ␩ ␰ Ϫ ␻ˆ dr ˆ r dr ␩ ␰ Ϫ ␻ˆ equation is given by n2 m␲ 2 Ѩ␪ Ϫ ͫ ϩ ͩ ͪ ͬ⌽ ϭ 0, ͑C11͒ ١␪ ϭ 0. ͑C3͒ r 2͑␩ ␰ Ϫ ␻ˆ 2͒ NH · ϩ v Ѩt in which The hydrostatic relation and mass continuity equations ␻͑ ͒ ϵ ␻ Ϫ ⍀ ͑ ͒ take the forms ˆ r c n C12 ␻ ϵ ␻ ϩ ␥ ␪ Ѩ␾ and c i is the complex eigenfrequency. The a ␪ ϭ ͑C4͒ polarization equations are g Ѩz and i d⌽ n␰ U͑r͒ ϭ ͩ␻ˆ Ϫ ⌽ͪ, ␳ ϭ ͑ ͒ ␩ ␰ Ϫ ␻ˆ 2 dr r ١ · pv 0, C5 ⌽ ␻ respectively. Above, we have introduced a pseudoden- 1 d n ˆ V͑r͒ ϭ ͩ␩ Ϫ ⌽ͪ, sity, defined by ␩ ␰ Ϫ ␻ˆ 2 dr r

ր p c␷ cp im␲␻ˆ ␳ ͑ ͒ ϵ ␳ ͩ ͪ ͑ ͒ ͑ ͒ ϭ ⌽ ͑ ͒ p z a . C6 W r 2 C13 pa HN ␳ In the Boussinesq approximation, p is treated as a con- for the radial, azimuthal, and vertical velocities, respec- stant in (C5) and divided through on both sides to obtain tively. The boundary conditions that determine the pos-

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC 2520 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

␻ sible values of c are regularity at the origin and out- Emanuel, K. A., 1986: An air-sea interaction theory for tropical ward IG wave propagation as r tends toward infinity cyclones. Part I: Steady state maintenance. J. Atmos. Sci., 43, (SM04). The eigenmodes of section 5 were obtained 585–604. ϭ ——, 1995: Sensitivity of tropical cyclones to surface exchange numerically with a radiation condition imposed at r coefficients and a revised steady state model incorporating 9.5Rm. eye dynamics. J. Atmos. Sci., 52, 3969–3976. Enagonio, J., and M. T. Montgomery, 2001: Tropical cyclogenesis REFERENCES via convectively forced vortex Rossby waves in a shallow- water primitive equation model. J. Atmos. Sci., 58, 685–706. Antkowiak, A., and P. Brancher, 2004: Transient energy growth Ford, R., 1994a: The instability of an axisymmetric vortex with for the Lamb-Oseen vortex. Phys. Fluids, 16, L1–L4 monotonic potential vorticity in rotating shallow water. J. doi:10.1063/1.1626123. Fluid Mech., 280, 303–334. Bachman, D. A., 1998: Nonlinear phenomena in a pure electron ——, 1994b: The response of a rotating ellipse of uniform poten- plasma studied with a 2D fluid code. Ph.D. dissertation, Cali- tial vorticity to gravity wave radiation. Phys. Fluids, 6, 3694– fornia Institute of Technology, 143 pp. 3704. Balmforth, N. J., S. G. Llewellyn Smith, and W. R. Young, 2001: ——, M. E. McIntyre, and W. A. Norton, 2000: Balance and the Disturbing vortices. J. Fluid Mech., 426, 95–133. slow quasi-manifold: Some explicit results. J. Atmos. Sci., 57, Bassom, A. P., and A. D. Gilbert, 1998: The spiral wind-up of 1236–1254. vorticity in an inviscid planar vortex. J. Fluid Mech., 371, ——, ——, and ——, 2002: Reply. J. Atmos. Sci., 59, 2878–2882. 109–140. Griffiths, M., and M. J. Reeder, 1996: Stratospheric inertia-gravity Benilov, E. S., 2005: The effect of ageostrophy on the stability of waves generated in a numerical model of frontogenesis. I: thin oceanic vortices. Dyn. Atmos. Oceans, 39, 211–226. Model solutions. Quart. J. Roy. Meteor. Soc., 122, 1153–1174. Black, P. G., and Coauthors, 2007: Air–sea exchange in hurri- Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. canes: Synthesis of observations from the Coupled Boundary Atmos. Sci., 50, 3380–3403. Layer Air–Sea Transfer Experiment. Bull. Amer. Meteor. Haynes, P. H., 1988: Forced, dissipative generalizations of finite Soc., 88, 357–374. amplitude wave-activity conservation relations for zonal and Briggs, R. J., J. D. Daugherty, and R. H. Levy, 1970: Role of Lan- nonzonal basic flows. J. Atmos. Sci., 45, 2352. dau damping in crossed-field electron beams and inviscid Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric fronto- shear flow. Phys. Fluids, 13, 421–432. genesis models: Mathematical formulation and solution. J. Broadbent, E. G., and D. W. Moore, 1979: Acoustic destabiliza- Atmos. Sci., 29, 11–37. tion of vortices. Philos. Trans. Roy. Soc. London, A290, 353– ——, 1975: The geostrophic momentum approximation and the 371. semigeostrophic equations. J. Atmos. Sci., 32, 233–242. Brunet, G., and M. T. Montgomery, 2002: Vortex Rossby waves ——, M. E. McIntyre, and A. W. Robertson, 1985: On the use and on smooth circular vortices: I. Theory. Dyn. Atmos. Oceans, significance of isentropic potential vorticity maps. Quart. J. 35, 153–177. Roy. Meteor. Soc., 111, 877–946. Cass, A. C., 1998: Experiments on vortex symmetrization in mag- Howe, M. S., 2003: The Theory of Vortex Sound. Cambridge Uni- netized electron columns. Ph.D. dissertation, University of versity Press, 216 pp. California, San Diego, 64 pp. Jones, S. C., 1995: The evolution of vortices in vertical shear. I: Chan, W. M., K. Shariff, and T. H. Pulliam, 1993: Instabilities of Initially barotropic vortices. Quart. J. Roy. Meteor. Soc., 121, two-dimensional inviscid compressible vortices. J. Fluid 821–851. Mech., 253, 173–209. Kelvin, Lord, 1880: On the vibrations of a columnar vortex. Phi- Charney, J. G., 1948: On the scale of atmospheric motions. Geo- los. Mag., 10, 155–168. phys. Publ., 17, 1–17. Killworth, P. D., and M. E. McIntyre, 1985: Do Rossby-wave criti- Chen, Y., and M. K. Yau, 2001: Spiral bands in a simulated hur- cal layers absorb, reflect, or over-reflect? J. Fluid Mech., 161, ricane. Part I: Vortex Rossby wave verification. J. Atmos. 449–492. Sci., 58, 2128–2145. Kop’ev, V. F., and E. A. Leont’ev, 1983: Acoustic instability of an ——, G. Brunet, and M. K. Yau, 2003: Spiral bands in a simulated axial vortex. Sov. Phys. Acoust., 29, 111–115. hurricane. Part II: Wave activity diagnostics. J. Atmos. Sci., ——, and ——, 1985: Energy aspect of the acoustic instability of 60, 1239–1256. certain steady-state vortices. Sov. Phys. Acoust., 31, 205–207. Chimonas, G., and H. M. Hauser, 1997: The transfer of angular ——, and ——, 1988: Acoustic instability of planar vortex flows momentum from vortices to gravity swirl waves. J. Atmos. with circular streamlines. Sov. Phys. Acoust., 34, 276–278. Sci., 54, 1701–1711. Landau, L., 1946: On the vibration of the electronic plasma. J. Chow, K. C., and K. L. Chan, 2003: Angular momentum trans- Phys. U.S.S.R., 10, 25. ports by moving spiral waves. J. Atmos. Sci., 60, 2004–2009. Lansky, I. M., T. M. O’Neil, and D. A. Schecter, 1997: A theory of ——, ——, and A. K. H. Lau, 2002: Generation of moving spiral vortex merger. Phys. Rev. Lett., 79, 1479–1482. bands in tropical cyclones. J. Atmos. Sci., 59, 2930–2950. Le Dizes, S., 2000: Non-axisymmetric vortices in two-dimensional Corngold, N. R., 1995: Linear response of the two-dimensional flows. J. Fluid Mech., 406, 175–198. pure electron plasma: Quasi-modes for some model profiles. Mallen, K. J., M. T. Montgomery, and B. Wang, 2005: Reexamin- Phys. Plasmas, 2, 620–628. ing the near-core radial structure of the tropical cyclone pri- Davidson, R. C., 1990: Physics of Nonneutral Plasmas. Addison- mary circulation: Implications for vortex resiliency. J. Atmos. Wesley, 733 pp. Sci., 62, 408–425. Dutton, J. A., 1976: The Ceaseless Wind: An Introduction to the Maslowe, S. A., 1986: Critical layers in shear flow. Annu. Rev. Theory of Atmospheric Motion. McGraw-Hill, 579 pp. Fluid. Mech., 18, 405–432.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC AUGUST 2008 S CHECTER 2521

McDonald, N. J., 1968: The evidence for the existence of Rossby- Reeder, M. J., and M. Griffiths, 1996: Stratospheric inertia-gravity like waves in the hurricane vortex. Tellus, 20, 138–150. waves generated in a numerical model of frontogenesis. II: McIntyre, M. E., 1981: On the ‘wave-momentum’ myth. J. Fluid Wave sources, generation mechanisms and momentum Mech., 106, 331–347. fluxes. Quart. J. Roy. Meteor. Soc., 122, 1175–1195. McWilliams, J. C., 1985: A uniformly valid model spanning the Ren, S., 1999: Further results on the stability of rapidly rotating regimes of geostrophic and isotropic, stratified turbulence: vortices in the asymmetric balance formulation. J. Atmos. Balanced turbulence. J. Atmos. Sci., 42, 1773–1774. Sci., 56, 475–482. ——, L. P. Graves, and M. T. Montgomery, 2003: A formal theory Rossi, L. F., J. F. Lingevitch, and A. J. Bernoff, 1997: Quasi- for vortex Rossby waves and vortex evolution. Geophys. As- steady monopole and tripole attractors for relaxing vortices. trophys. Fluid Dyn., 97, 275–309. Phys. Fluids, 9, 2329–2338. Möller, J. D., and M. T. Montgomery, 1999: Vortex Rossby waves Saujani, S., and T. G. Shepherd, 2002: Comments on “Balance and and hurricane intensification in a barotropic model. J. Atmos. the slow quasimanifold: Some explicit results.” J. Atmos. Sci., Sci., 56, 1674–1687. 59, 2874–2877. ——, and ——, 2000: Tropical cyclone evolution via potential Schecter, D. A., D. H. E. Dubin, A. C. Cass, C. F. Driscoll, I. M. vorticity anomalies in a three-dimensional balance model. J. Lansky, and T. M. O’Neil, 2000: Inviscid damping of asym- Atmos. Sci., 57, 3366–3387. metries on a two-dimensional vortex. Phys. Fluids, 12, 2397– ——, and L. J. Shapiro, 2002: Balanced contributions to the in- 2412. tensification of Hurricane Opal as diagnosed from a GFDL ——, and M. T. Montgomery, 2003: On the symmetrization rate of model forecast. Mon. Wea. Rev., 130, 1866–1881. an intense geophysical vortex. Dyn. Atmos. Oceans, 37, 55–87. Montgomery, M. T., and R. J. Kallenbach, 1997: A theory of vor- ——, and ——, 2004: Damping and pumping of a vortex Rossby tex Rossby-waves and its application to spiral bands and in- wave in a monotonic cyclone: Critical layer stirring versus tensity changes in hurricanes. Quart. J. Roy. Meteor. Soc., inertia-buoyancy wave emission. Phys. Fluids, 16, 1334–1348. 123, 435–465. ——, and ——, 2006: Conditions that inhibit the spontaneous ——, and C. Lu, 1997: Free waves in barotropic vortices. Part I: radiation of spiral inertia–gravity waves from an intense me- Eigenmode structure. J. Atmos. Sci., 54, 1868–1885. soscale cyclone. J. Atmos. Sci., 63, 435–456. ——, and J. L. Franklin, 1998: An assessment of the balance ap- ——, and ——, 2007: Waves in a cloudy vortex. J. Atmos. Sci., 64, proximation in hurricanes. J. Atmos. Sci., 55, 2193–2200. 314–337. ——, and L. J. Shapiro, 1995: Generalized Charney–Stern and ——, ——, and P. D. Reasor, 2002: A theory for the vertical align- Fjortoft theorems for rapidly rotating vortices. J. Atmos. Sci., ment of a quasigeostrophic vortex. J. Atmos. Sci., 59, 150– 52, 1829–1833. 168. Muraki, D. J., C. Snyder, and R. Rotunno, 1999: The next-order ——, M. E. Nicholls, J. Persing, A. J. Bedard Jr., and R. A. Pielke corrections to quasigeostrophic theory. J. Atmos. Sci., 56, Sr., 2008: Infrasound emitted by tornado-like vortices: Basic 1547–1560. theory and a numerical comparison to the acoustic radiation Nolan, D. S., and B. F. Farrell, 1999: Generalized stability analy- of a single-cell thunderstorm. J. Atmos. Sci., 65, 685–713. ses of asymmetric disturbances in one- and two-celled vorti- Shapiro, L. J., and M. T. Montgomery, 1993: A three-dimensional ces maintained by radial inflow. J. Atmos. Sci., 56, 1282–1307. balance theory for rapidly rotating vortices. J. Atmos. Sci., 50, O’Neil, T. M., 1965: Collisionless damping of nonlinear plasma 3322–3335. oscillations. Phys. Fluids, 8, 2255–2262. Shepherd, T. G., 2003: Hamiltonian dynamics. Encyclopedia of O’Sullivan, D., and T. J. Dunkerton, 1995: Generation of inertia– Atmospheric Sciences, J. R. Holton, J. A. Curry, and J. A. gravity waves in a simulated life cycle of baroclinic instability. Pyle, Eds., Academic Press, 929–938. J. Atmos. Sci., 52, 3695–3705. Shukhman, I. G., 1991: Nonlinear evolution of spiral density Papaloizou, J. C. B., and J. E. Pringle, 1987: The dynamic stability waves generated by the instability of the shear layer in a of differentially rotating discs—III. Mon. Not. Roy. Astron. rotating compressible fluid. J. Fluid Mech., 233, 587–612. Soc., 225, 267–283. Snyder, C., W. C. Skamarock, and R. Rotunno, 1993: Frontal dy- Pillai, S., and R. W. Gould, 1994: Damping and trapping in 2D namics near and following frontal collapse. J. Atmos. Sci., 50, inviscid fluids. Phys. Rev. Lett., 73, 2849–2852. 3149–3212. Plougonven, R., and V. Zeitlin, 2002: Internal gravity wave emis- Spencer, R. L., and S. N. Rasband, 1997: Damped diocotron sion from a pancake vortex: An example of wave-vortex in- quasi-modes of nonneutral plasmas and inviscid fluids. Phys. teraction in strongly stratified flows. Phys. Fluids, 14, 1259– Plasmas, 4, 53–60. 1268. Vanneste, J., and I. Yavneh, 2004: Exponentially small inertia– ——, and C. Snyder, 2005: Gravity waves excited by jets: Propa- gravity waves and the breakdown of quasigeostrophic bal- gation versus generation. Geophys. Res. Lett., 32, L18802, ance. J. Atmos. Sci., 61, 211–223. doi:10.1029/2005GL023730. Wang, Y., 2002a: Vortex Rossby waves in a numerically simulated Polvani, L. M., J. C. McWilliams, M. A. Spall, and R. Ford, 1994: tropical cyclone. Part I: Overall structure, potential vorticity, The coherent structures of shallow-water turbulence: Defor- and kinetic energy budgets. J. Atmos. Sci., 59, 1213–1238. mation-radius effects, cyclone/anticyclone asymmetry and ——, 2002b: Vortex Rossby waves in a numerically simulated gravity-wave generation. Chaos, 4, 177–186. tropical cyclone. Part II: The role in tropical cyclone structure Reasor, P. D., and M. T. Montgomery, 2001: Three-dimensional and intensity changes. J. Atmos. Sci., 59, 1239–1262. alignment and co-rotation of weak, TC-like vortices via linear Zeitlin, V., 1991: On the backreaction of acoustic radiation for vortex-Rossby-waves. J. Atmos. Sci., 58, 2306–2330. distributed two-dimensional vortex structures. Phys. Fluids ——, ——, and L. D. Grasso, 2004: A new look at the problem of A, 3, 1677–1680. tropical cyclones in shear flow: Vortex resiliency. J. Atmos. Zhang, F., 2004: Generation of mesoscale gravity waves in upper- Sci., 61, 3–22. tropospheric jet-front systems. J. Atmos. Sci., 61, 440–457.

Unauthenticated | Downloaded 09/30/21 05:50 AM UTC