1. Introduction
a. Vortex instability driven by inertia–gravity wave emission
The pertinent studies concern IG wave radiation from a monotonic cyclone (MC), whose potential vorticity (PV) consistently decays with distance from the central axis of rotation. The radial PV gradient allows Rossby-like waves to exist in the core region of the vortex (Kelvin 1880; McDonald 1968; Montgomery and Kallenbach 1997). The spectrum of vortex Rossby waves includes continuum and discrete modes. A continuum disturbance typically suffers spiral windup due to differential rotation of the mean flow. Here, we will focus on discrete vortex Rossby waves (DVRWs), which resist spiral windup and remain coherent over time. DVRWs often account for rotating tilts and elliptical (triangular, square, etc.) deformations of the vortex core. The angular phase velocity of a DVRW is of order Ω, but is less than the angular velocity of the core circulation.
Under condition (1), a DVRW will typically excite an outward propagating spiral IG wave in the peripheral region of the vortex. Basic perturbation theory establishes that the creation of any wave by another disturbance in a circular flow must occur in such a way that conserves net angular pseudomomentum (cf. McIntyre 1981; Haynes 1988; Guinn and Schubert 1993; Shepherd 2003). It has been shown for MCs that a DVRW and its IG wave emission have angular pseudomomenta of opposite sign, positive and negative, respectively (Schecter and Montgomery 2004, hereafter SM04; Schecter and Montgomery 2006, hereafter SM06). Consequently, a DVRW must grow in response to producing radiation. By this mechanism, the core of an MC will gradually deform.
Although an atmospheric vortex is in a continuously stratified fluid, the shallow-water model suffices to explain the essential physics of the radiation-driven instability. In the context of shallow-water theory, Ford (1994a, b) verified that IG wave emissions compel cyclone-scale DVRWs in MCs to grow exponentially with time, provided that the Rossby number Ro = Ω/f exceeds unity. Ford also showed that the growth rate vanishes algebraically as the Froude number, Fr = V/cg, tends to zero; here V is the azimuthal velocity of the vortex and cg is the ambient gravity wave speed. Specifically, the maximum growth rate of a DVRW is proportional to Fr4Ω in the regime where Fr ≪ 1. This result was later generalized by Plougonven and Zeitlin (2002) to “pancake” vortices in a continuously stratified fluid. It stands to reason that an atmospheric vortex at small Froude number would hardly feel the effects of radiation.
Nevertheless, mesoscale cyclones such as hurricanes and rotational thunderstorms (Fig. 1) can penetrate the “superspin” parameter regime, where both the Rossby and Froude numbers exceed unity. Recent numerical studies of a stratified MC indicate that superspin can shorten the e-folding time of a radiation-driven instability to less than four rotation periods (SM04). Chow and Chan (2003) further speculated that the spontaneous emission of spiral IG waves can remove up to one-tenth of the angular momentum of an intense hurricane in a single rotation period. A similar estimate was published earlier by Chimonas and Hauser (1997) for the negative radiation torque on a supercell mesocyclone under ideal conditions. In section 5, we will reconsider these estimates.
Of course, superspin does not guarantee that IG waves will greatly influence the evolution of a cyclone. An important mechanism for quenching the radiation-driven instability of an MC involves the resonant interaction between a DVRW and its critical layer. The critical layer of a DVRW is located beyond the core of an MC, precisely where the angular phase velocity of the wave equals the angular velocity of the mean flow. While creating radiation, a DVRW simultaneously stirs PV in its critical layer. Such stirring efficiently transfers angular pseudomomentum from the DVRW into the critical layer, and thereby acts to damp the wave. Damping will prevail over radiative pumping if the radial gradient of PV in the critical layer is sufficiently large (SM04; SM06). Figure 2 illustrates the two potential fates of a DVRW in an MC. Section 4 will demonstrate that a precise growth rate formula for the DVRW is readily extracted from an equation that expresses angular pseudomomentum conservation. For the first time, this formula will be derived for baroclinic MCs that possess active critical layers.2
b. Connection to the broader problem of spontaneous imbalance
Following the theme of the special issue for which this paper was prepared, let us now turn our attention to the general problem of spontaneous imbalance. In practical terms, balanced motions are theoretical approximations of synoptic or mesoscale flows that filter out IG waves. Quasigeostrophic (QG) flow, which applies only at small Rossby numbers, is the most familiar kind (Charney 1948; Hoskins et al. 1985; cf. Muraki et al. 1999). More general forms of balanced flow are contained in the semigeostrophic model (Hoskins 1975; Hoskins et al. 1985), the standard balance model (McWilliams 1985), and the asymmetric balance model (Shapiro and Montgomery 1993). Spontaneous imbalance (SI) is the divergence of an actual flow from the predictions of a balance model, which usually involves the production of IG waves. SI is evident in numerical simulations of baroclinic instability and frontogenesis (Snyder et al. 1993; O’Sullivan and Dunkerton 1995; Griffiths and Reeder 1996; Reeder and Griffiths 1996; Zhang 2004; Plougonven and Snyder 2005). At Ro ≪ 1, theoretical arguments suggest that balanced motions will produce exponentially weak IG wave emissions (Saujani and Shepherd 2002; Vanneste and Yavneh 2004). At high Rossby numbers, compact regions of unsteady balanced motions can resonate more directly with environmental IG waves and thereby exhibit stronger SI (Ford et al. 2000, 2002).
Typically, the intrinsic frequency of a DVRW will not exceed the characteristic inertial frequency of the vortex in which it resides, regardless of the Rossby number (cf. Montgomery and Lu 1997). From this local perspective, it is sensible to approximate DVRWs with a model that filters out IG waves. Section 2 will review the interaction of a DVRW with its critical layer in the context of the asymmetric balance (AB) model, which ostensibly applies to hurricanes as well as quasigeostrophic cyclones. However, at high Rossby number the balance approximation is unjustifiable in the far field where the inertial frequency reduces to f. As explained previously, this allows a DVRW to match onto an outward 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 feedback of IG wave radiation (see section 3). In this sense, the radiation-driven instability of an MC is a prime example of SI.
c. Outline
The remainder of this paper is organized as follows. Section 2 qualitatively reviews the balanced interaction of a DVRW and its critical layer in a shallow-water MC. Section 3 examines both linear and nonlinear conditions for the SI of a shallow-water MC. Section 4 derives a formula for the growth rate of a DVRW near marginal stability in a baroclinic MC. This formula accounts for the positive feedback of IG wave emission and the negative feedback of PV stirring in a 3D critical layer. Section 5 reexamines the torque that is applied by spontaneous radiation on a barotropic cyclone with parameters similar to those of a category 5 hurricane. Section 6 summarizes the main text and suggests future lines of investigation. For convenient reference, the appendices review the standard primitive equations that form the theoretical foundation of this paper.
2. The balanced evolution of a discrete vortex Rossby wave
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 theory on the f plane. Throughout, we will refer to a polar coordinate system whose origin is at the center of the vortex. As usual, the symbols r, φ, and t will denote radius, azimuth, and time. A generic flow field h(r, φ, t) will be separated into a basic state
a. The unperturbed cyclone
(i)
q is uniformly positive,(ii) d
q /dr is uniformly negative, and(iii) d
Ω /dr is uniformly negative.
Conditions (i)–(iii) define a monotonic cyclone (MC).
b. The asymmetric balance model for weak perturbations
Note that the AB model [Eqs. (9)–(10)] conveniently retains QG structure at high Rossby number. Furthermore, if the Rossby and Froude numbers satisfy the relation Fr2 ≪ Ro ≪ 1, the AB model properly reduces to the QG model. That is, (u′, υ′) becomes the geostrophic velocity perturbation, lD becomes the constant Rossby deformation radius lR, and q̂′ becomes l2R times the quasigeostrophic PV perturbation.
c. Basic kinematics of the DVRW–critical layer interaction
Through the extension of its velocity field, the DVRW can stir fluid everywhere beyond the core. Outside of the critical layer, fluid parcels moving with the mean flow experience rapid oscillations of positive and negative wave forcing, which will have little cumulative effect on the amplitude of q̂′. Within the critical layer, fluid parcels are essentially phase locked with the wave. It is therefore reasonable to assert that a DVRW will most profoundly disturb q̂′ in the neighborhood of r* and that we need only consider the interaction between a DVRW and its critical layer (cf. Lansky et al. 1997).
The reader may have noticed that a damped DVRW cannot possibly be an eigenmode of an MC. An eigenmode consists of a q̂′ distribution that decays exponentially with time everywhere in space. In contrast, a DVRW decays because of growing q̂′cl. For this reason, and others that are more technical, the DVRWs of MCs are more properly classified as “quasi-modes” (Briggs et al. 1970; Corngold 1995; Spencer and Rasband 1997). If the vortex were nonmonotonic, then d
3. The spontaneous imbalance of a shallow-water cyclone
a. The positive feedback of spiral radiation
At Rossby numbers greater than unity, any DVRW of a shallow-water MC can excite a frequency-matched outward propagating spiral IG wave in the environment. In other words, the geopotential (and velocity components) of the balanced DVRW can match onto a spiral IG wave at the periphery of the circular flow. As mentioned earlier, the spiral radiation has positive feedback on the DVRW.
Figure 5 contains a snapshot of an exponentially growing SI mode of a shallow-water cyclone at Fr2 = 0.74 and Ro = 53 (cf. SM06). For reference, Fig. 5a shows the basic state of the cyclone, which possesses a monotonic potential vorticity distribution. The angular velocity of the basic state is slightly nonmonotonic, but this feature is unimportant for our discussion. Figure 5b shows the complex radial velocity wavefunction U of the SI mode, which is defined by u′ = a(t)U(r)ei(nφ−ωt) + c.c. The inner part of U corresponds to a DVRW, and maintains an approximately constant phase far beyond the critical radius r*. The outer part of U corresponds to a spiral IG wave and has increasing phase with radius. Figure 5c verifies that the angular pseudomomenta of the DVRW and IG wave are positive and negative, respectively. Because the Froude number of the cyclone is less than unity, the IG wave emission is fairly weak. Consequently, the e-folding time of the SI mode is many (23) vortex rotation periods.
b. The growth rate of a DVRW and its coupled radiation field
It is important to emphasize that Eq. (23), in which γ is constant, corresponds to linearized dynamics. Henceforth, γ will refer exclusively to the growth rate of a DVRW in linear theory.
c. Nonlinear SI
Suppose that linear theory predicts the exponential decay of a DVRW, because the radial gradient of
d. An illustrative numerical simulation
Figures 6 –8 (adapted from SM06) illustrate the nonlinear SI of a numerically simulated shallow-water MC. The initial condition consists of a balanced elliptical deformation of the vortex core. Figure 6 shows the evolution of the potential vorticity field and the asymmetric component of the geopotential perturbation. The core perturbation consists of an n = 2 DVRW, which over time generates an outward propagating spiral IG wave in the far field. Early on, PV stirring in the critical layer (which appears as filamentation) tends to damp the DVRW and its radiation field. Later on, the DVRW and IG wave spontaneously amplify. The bottom panel of Fig. 6 provides a detailed picture of PV stirring in the critical layer. In time, trapped fluid parcels collectively coil the PV distribution into a pattern that is said to resemble a pair of cat’s eyes.
Figure 7 (top solid curve) shows a time series of the amplitude of the DVRW. For analytical purposes, the ratio |Ωb/γ|2 is substituted for a dimensional measure of the perturbation strength. Initially, the DVRW decays exponentially with time, exactly as predicted by linear theory (top dashed curve). Since the initial value of Ωb exceeds the magnitude of γ, the time series eventually diverges from linear theory. Specifically, the amplitude of the DVRW bounces after a time period of order 2π/Ωb. At this point, the critical layer starts to periodically return a fraction of the angular pseudomomentum that it absorbs. As a result, critical layer damping becomes inefficient. In the meantime, the positive feedback of IG wave radiation remains steady. Ultimately, radiative pumping prevails and SI ensues.
The bottom curves in Fig. 7 show the nonlinear (solid) and linear (dashed) evolution of the mode amplitude in a similar experiment in which the initial value of Ωb is less than the magnitude of γ. In contrast to the previous case, cat’s eyes do not fully develop in the critical layer of the DVRW. Moreover, there is no sign of nonlinear SI.
4. The spontaneous imbalance of a dry baroclinic cyclone
The theory of IG wave production by DVRWs, including the influence of critical layers, has been extended to 3D stratified cyclones with barotropic basic states and cloud coverage (SM04; Schecter and Montgomery 2007, hereafter SM07). Plougonven and Zeitlin (2002) had previously developed an analytical theory for IG wave emission by “pancake” vortices, but their analysis did not account for active critical layers. A general theory for baroclinic cyclones, whose tangential winds have vertical shear, was lacking before now. The following extension-by-analogy of shallow-water theory (SM06) to dry baroclinic cyclones has not been tested against numerical experiments, but is presented here for the purpose of motivating a deeper investigation.
As usual, we will assume that θ (the potential temperature) of the atmosphere increases monotonically with altitude and that the axisymmetric PV distribution
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. Accordingly, 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.
Note that we have departed from our previous convention, and have chosen to represent the rate of change of angular pseudomomentum in the environment by S rather than by an indefinite volume integral. This alternative approach will facilitate the derivation of a formula for the growth rate of a radiative DVRW.
b. The growth rate of a DVRW
As in shallow-water theory, a DVRW is here viewed as a nearly balanced vortex mode that matches onto a spiral IG wave in the environment. Presumably, the IG wave emission compels the DVRW to grow. In the outer region of the vortex, there exists a 3D critical layer in which the angular rotation frequency of the cyclone equals the angular phase velocity of the mode (Fig. 9). The PV disturbance in the critical layer is viewed (for the following analysis) as a relatively weak perturbation. Loosely speaking, the isentropic stirring of PV in the critical layer is slaved to the velocity fields of the DVRW and of the mean vortex. At “higher order,” the growth of q′ in the critical layer compels the DVRW to decay. The following formally examines the competition between the positive feedback of IG wave radiation and the negative feedback of PV stirring in the critical layer.
The second term of the integrand of dLcl/dt [Eq. (50)] is proportional to the time derivative of 
The preceding theory for γ relied on multiple assumptions that may have limited applicability. The least subtle is that the DVRW (should one even exist) is near marginal stability. A small value of γ is required for the presumed length scale separation between the thin critical layer and the bulk of the vortex. It is also required for the time scale separation between the e-folding and oscillation periods of the wave. Despite the imprecision of Eq. (56) in more general circumstances, it still demonstrates that wave–flow resonances can greatly hinder SI in MCs.
5. The potential impact of SI on tropical cyclones
The introduction referred to a theoretical study by Chow and Chan (2003) that estimated an appreciable rate of angular momentum loss by hurricanes due to IG wave radiation. Specifically, they proposed that a spiral IG wave can remove up to 10% of the core angular momentum in one rotation period. To the author’s knowledge, there are no numerical studies at this time that irrefutably support the 10% loss estimate.
On the other hand, there is numerical evidence that radiation-driven instabilities of hurricane-like vortices can occur relatively fast (Schecter and Montgomery 2003; SM04; D. Hodyss and D. Nolan 2005, personal communication). In the following, we will reexamine these instabilities with the goal of calculating the rate at which they remove angular momentum from a vortex. In addition, we will reexamine the magnitude of the spiral bands of vertical velocity that they create in the radiation zone. It has been speculated that such bands might encourage moist convection.
a. Radiation torque on a Rankine cyclone
b. Some computational results in the category 5 hurricane parameter regime
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.
Figure 11 (bottom) illustrates the variation of δ
According to Fig. 10, the first baroclinic SI modes of a category 5 “hurricane” do not remove angular momentum at an appreciable rate. For ε = 0.2, the rate is 0.006–0.028 core units per vortex rotation period. As such, it would take between a few days and one week for spiral radiation to deplete the core circulation (r ≤ Rυ) of the bulk of its original angular momentum, neglecting potential decay of the radiation torque and replenishment by radial inflow. In contrast, Chow and Chan (2003) estimated that δ
For comparison, the shaded region of Fig. 10 shows the estimated fractional loss of core angular momentum due to surface drag (δ
On another topic, the bottom curves of Fig. 10 show the maximum vertical velocities (w′) of the SI modes in the midtropospheric region (z = H/2) of the radiation zone (r > Rυ). For ε = 0.2, these velocities range from 0.002 to 0.015 times
c. Suppression of SI by skirts
Sections 5a and 5b focused on the radiation-driven instability of a Rankine cyclone, which might be exceptional 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
(i) increasing the density of cloud coverage,
(ii) decreasing the positive vertical potential temperature gradient,
(iii) increasing the intensity (inertial stability) of the vortex, or
(iv) decreasing the vertical length scale of the perturbation.
Typically, the radial PV gradient of a skirt sharply increases with decreasing r. Consequently, decreasing lD (to values below Rm) greatly increases the negative radial PV gradient in the critical layer of a DVRW and greatly accelerates resonant damping.10
Hurricane-like vortices have relatively small values of lD. For example, evaluating the inertial stability
The above conclusion seems more robust upon considering the effect of moisture. Figure 12, adapted from SM07, illustrates the suppression of a linear SI mode by cloud coverage in a barotropic cyclone. As usual, the SI mode consists of an inner DVRW and an outer spiral IG wave (not shown). The azimuthal wavenumber is n = 2, and the vertical wavelength is such that under dry conditions lD = 0.26 Rm in which lD is evaluated at r = Rm. Because the cyclone is nearly Rankine, the dry value of lD is not sufficiently small to quench SI. Nevertheless, lD is reducible by increasing cloudiness (decreasing Nm) in the vicinity of Rm. As shown here, reduction of lD by cloud coverage ultimately moves r* to a location where, in linear theory, critical layer damping prevails over radiative pumping. Interested readers may consult SM07 for further details.11
Of course, a steep skirt (or lD ≪ Rm) does not guarantee the suppression of SI in a real tropical cyclone; rather, it represents one inhibiting factor. As mentioned earlier, the angular pseudomomentum of a DVRW at sufficiently large amplitude can exceed the finite absorption capacity of its critical layer. In this case, nonlinear SI would occur. Furthermore, hurricanes typically have nonmonotonic cores that suffer shear-flow instabilities. Such instabilities can act as additional sources of spiral IG waves and are not sensitive to the gradient of the skirt. Finally, even if the cyclone has a monotonic potential vorticity distribution, it can exhibit a “nonmodal” shear-flow instability (cf. Nolan and Farrell 1999; Antkowiak and Brancher 2004) that temporarily amplifies the IG wave radiation field. Convective processes within the vortex might initiate such an event.
d. IG wave emission by sheared vortex Rossby waves
If a steep skirt is able to severely damp all DVRWs, then arbitrary forcing might prefer to excite sheared vortex Rossby waves (SVRWs). There is evidence that such waves exist in tropical cyclones and are capable of generating spiral cloud bands (McDonald 1968; Chen and Yau 2001; Wang 2002a, b; Chen et al. 2003). Balance theories suggest that SVRWs can intensify the storm by fluxing angular momentum to the radius of maximum wind (Montgomery and Kallenbach 1997; Möller and Montgomery 1999, 2000; Enagonio and Montgomery 2001). However, an SVRW may also create IG wave radiation that fluxes angular momentum away from the vortex. Because an SVRW loses coherence and attenuates, it is not likely to sustain radiation by itself. Nevertheless, persistent cycles of SVRW creation and dissipation may engender a notable time-averaged radiation torque.
6. Concluding remarks
This paper reviewed an important paradigm of spontaneous imbalance (SI) that operates at Rossby numbers greater than unity. The main text began by discussing the balanced dynamics of a discrete vortex Rossby wave (DVRW) in a monotonic cyclone (MC). In the context of balanced dynamics, the DVRW is a potential vorticity wave that acts at a distance on fluid beyond the core region of the MC. Section 2 explained that the external flow field of a DVRW most efficiently stirs PV in a critical layer where the angular phase velocity of the wave matches the angular velocity of the mean flow. Since total angular pseudomomentum is conserved, its production in the critical layer damps the DVRW. The damping rate is proportional to the mean radial gradient of critical layer PV.
Beyond the critical layer, balance conditions break down, and the external flow field of the DVRW excites an outward propagating spiral IG wave. Section 3 explained that the DVRW and IG wave have angular pseudomomenta of opposite sign, positive and negative, respectively. Therefore, producing IG wave radiation compels the DVRW to grow. The growth rate due to radiation alone increases algebraically from zero with the rotational Froude number of the vortex.
Nevertheless, SI will occur only if radiative pumping supersedes critical layer damping. The competition between these two processes has been quantified in linear theory, for both shallow-water cyclones [section 3; Eq. (24)] and continuously stratified cyclones [section 4; Eq. (56)]. It is sensible to suppose that increasing Fr would enable or intensify SI because it enhances the radiative pumping of a DVRW. However, increasing Fr at high Rossby number coincides with decreasing the vortex deformation radius lD. The reduction of lD moves the critical radius inward where the PV gradient can steepen by orders of magnitude (Fig. 12c). Hence, raising Fr can bolster critical layer damping and suppress SI.
Of course, linear damping does not guarantee suppression. Section 3d showed that nonlinear SI will occur if the angular pseudomomentum of the DVRW exceeds the finite absorption capacity of the critical layer. Such is the case when the bounce frequency Ωb of the DVRW is greater than the linear decay rate |γ|. When the wave amplitude is sufficiently large to satisfy this condition, the DVRW mixes the PV distribution in the critical layer and effectively levels the once stabilizing gradient.
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 barotropic cyclone at Rossby and Froude numbers characteristic of a category 5 hurricane. It was verified that IG wave radiation can triple the amplitude of a DVRW in just a few vortex rotation periods. However, the negative radiation torque appeared to be less significant than the expected influence of oceanic surface drag.
Despite recent progress, much remains to learn about the SI of intense mesoscale vortices. The extent to which DVRWs (as opposed to continuum perturbations) control IG wave emissions from baroclinic cyclones is unknown. Moreover, the effect of secondary circulation in the basic state is uncertain. The role of moisture is also relatively unexplored. Clearly, moisture cannot be ignored in the troposphere. At the most basic level of approximation, cloud coverage simply reduces the Brunt–Väisälä frequency of the system. As explained in section 5c, this can indirectly enhance the ability of critical layers to inhibit the SI of an MC (cf. SM07). Future research may better explain how a more realistic nonlinear coupling of vortex modes to cloud processes affects their ability to excite IG waves in the far field.
Acknowledgments
This paper was solicited for a special issue on the topics addressed at the SI workshop held in Seattle, Washington, 7–10 August 2006. The SI workshop was sponsored by the National Science Foundation and NorthWest Research Associates. The author thanks Dr. Paul D. Reasor for a helpful discussion regarding section 5. The author also thanks an anonymous referee for helpful suggestions on improving the style of this manuscript. Funding for this research was provided by NSF Grant ATM-0649944, channelled through the Department of Defense, Naval Postgraduate School Grant N00244-07-1-0002.
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APPENDIX A
The Shallow-Water Model
APPENDIX B
The Hydrostatic Primitive Equations in Isentropic Coordinates
APPENDIX C
The Hydrostatic Boussinesq Model in Pseudoheight Coordinates
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).
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
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 = 2 and angular phase velocity ω/n. The DVRW excites an outward propagating spiral IG wave in the environment, and stirs PV in its critical layer at r = r*. 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).
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
(a) Typical arrangement of the critical layers of the DVRWs of a shallow-water MC. The central radius r* of the critical layer decreases as the azimuthal wavenumber n increases. (b) Instantaneous streamlines in the critical layer of a DVRW, viewed in a reference frame that corotates with the wave. The symbol Ωb denotes the orbital angular frequency of fluid parcels near the center of the cat’s eye, and ltrap denotes the half-width of the trapping region that is enclosed by the separatrix. Both Ωb and ltrap are proportional to the square root of the wave amplitude, which can vary with time.
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
(top) Balanced dynamics of a DVRW. In a balance model, the core DVRW resonantly excites a PV perturbation at the critical radius r*. 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 γ represents the characteristic growth/decay rate of the DVRW. Note also that
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
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 
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
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 ϕ′. Note that ϕ′ decays and then grows. (bottom) Detailed evolution of the PV distribution in the critical layer. For all panels, time t is measured in units of
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
Conditional SI. The top curve (solid) shows the amplitude of the DVRW in Fig. 6. The horizontal bar (with double arrows) indicates the maximum bounce period of the wave. Since Ωb/|γ| initially exceeds unity, nonlinear SI occurs. The bottom curve (solid) is from a similar experiment, in which the initial wave amplitude is reduced by a factor of 0.25. Here, there is no sign of SI. The dashed curves are the exponential-decay predictions of linear theory (Ω2b ∝ eγt). Time t is measured in units of
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
The radial dependence of balance parameters in the numerical simulation of nonlinear SI (Fig. 6). Both
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
Illustration of a baroclinic MC and a cylinder that separates the vortex from the environment for analytical purposes. An arbitrary DVRW will emit spiral IG waves into the environment. Isentropic PV stirring in the critical layer, centered on the surface r = r*(θ), will potentially damp the DVRW and thereby suppress the IG wave radiation.
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
The SI modes (m = 1, 1 ≤ n ≤ 4) of a barotropic Rankine cyclone in the parameter regime of a category 5 hurricane. Each mode consists of an inner DVRW and an outer spiral IG wave. They are initially excited by deforming the isosurfaces of core potential vorticity, as sketched below the graph. The variables τγ and τω are, respectively, the e-folding time (1/γ) and oscillation period (2π/ω) of the mode, measured in vortex rotation periods (2πRm/
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
Snapshot of a radiating “hurricane.” The vertical wind (w′) and radiation torque (−δ
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
The suppression of a linear SI mode by increasing cloudiness in a barotropic cyclone at Ro ≡
Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2490.1
It is well known that the problems of IG wave radiation and acoustic radiation from a solitary vortex are analogous. Readers interested in the acoustic analog may consult Broadbent and Moore (1979), Kop’ev and Leont’ev (1983, 1985, 1988), Zeitlin (1991), Chan et al. (1993), and Howe (2003).
It is notable that the theory of modal growth in astrophysical disks must likewise treat the net influence of critical layer stirring and spiral density wave radiation. See, for example, Papaloizou and Pringle (1987) and Shukhman (1991).
The original derivation of the AB model (Shapiro and Montgomery 1993) followed a different path, which resulted in
The resonant damping of a DVRW is analogous to the “Landau damping” of a discrete plasma wave (Landau 1946; O’Neil 1965). Moreover, in nonneutral plasma physics the resonant damping of a DVRW is known as the Landau damping of a diocotron (slipping stream) mode (Briggs et al. 1970; cf. Davidson 1990, chapter 6).
Schecter et al. 2008 contains an explicit derivation and numerical verification of the acoustic analog of Eq. (24).
Sections 2 and 3 assumed that the angular pseudomomentum of the DVRW was negligible beyond the inner core of the vortex (i.e., for r > r* − δr*). Here, we have taken a more liberal view, and have allowed the angular pseudomomentum density of the DVRW to exist on both sides of the critical layer. In practice, the contribution to Ld from the inner core typically dominates.
Note that nonmodal (continuum) contributions to u′ are expected to dominate at very late times if the DVRW is damped. Nevertheless, we will assume that Eq. (51) stays valid over the time scale of interest.
Growth rate formulas that are similar to Eq. (56) have been numerically verified for DVRWs near marginal stability in moist barotropic MCs (SM07), dry barotropic MCs (SM04), and shallow-water MCs (SM06).
Using an extratropical value for the Coriolis parameter ( f = 10−4) instead of a tropical value does not matter, since Ro ≫ 1 in either case. Results for f = 5 × 10−5 are nearly indistinguishable from the results presented here.
Schecter et al. 2002 provides a rare counterexample.
