1. Introduction

Tropical cyclones may exhibit various asymmetric instabilities as their basic states freely evolve or adjust to changing environmental conditions. Such instabilities can give rise to commonly seen elliptical cores, polygonal eyewalls, and mesovortices (Muramatsu 1986; Reasor et al. 2000; Kossin and Schubert 2001; Corbosiero et al. 2006; Montgomery et al. 2006; Hendricks et al. 2012). They may also induce horizontal mixing processes that efficiently redistribute angular momentum and equivalent potential temperature (Schubert et al. 1999; Kossin and Eastin 2001; Hendricks and Schubert 2010). The immediate consequence of asymmetric instability and mixing can be the slowdown of intensification or a reduction of maximum wind speed in the primary circulation of the vortex (Schubert et al. 1999; Naylor and Schecter 2014; cf. Rozoff et al. 2009). The possible negative influence of asymmetric instabilities may factor into why three-dimensional (3D) cloud-resolving tropical cyclone models often yield moderately or slightly weaker storms than their axisymmetric counterparts (Yang et al. 2007; Bryan 2012; Persing et al. 2013; Naylor and Schecter 2014).¹ In short, there is reason to believe that the theory of tropical cyclone intensity cannot be fully detached from the theory of vortex instability.

There are several well-known mechanisms of asymmetric vortex instability that are potentially relevant to the behavior of intense tropical cyclones. Classical barotropic instability mechanisms include 1) the mutual amplification of phase-locked counterpropagating vortex Rossby waves in the vicinity of the eyewall (Levy 1965; Michalke and Timme 1967; Schubert et al. 1999) and 2) the mutual amplification of a vortex Rossby wave and the potential vorticity (PV) anomaly that it generates in a suitably conditioned critical layer (Briggs et al. 1970). Another viable mechanism of asymmetric perturbation growth is the positive feedback of inertia–gravity wave radiation on the vortex Rossby wave that is responsible for its excitation (e.g., Ford 1994; Plougonven and Zeitlin 2002; Schecter and Montgomery 2004, hereafter SM04; Hodyss and Nolan 2008, hereafter HN08; Park and Billant 2013). Instabilities related to baroclinic vortex structure (Kwon and Frank 2005) and the transient growth of nonmodal perturbations (Nolan and Farrell 1999; Antkowiak and Brancher 2004) are also pertinent but will not be considered explicitly in this note.

The dominant modes of instability can involve multiple mechanisms operating simultaneously (Menelaou et al. 2016, hereafter M16). Under these circumstances, the role of each mechanism in destabilizing the vortex is difficult to assess without the right diagnostic. The main purpose of this note is to briefly present an alternative method to quantitatively compare the importance of inertia–gravity wave radiation to that of other processes in driving the growth of vortex perturbations. The method amounts to comparing the rates at which the perturbation to a tracer field inside the vortex—such as PV—is amplified by velocity fields attributed to radiation and to sources within the vortex itself. The seeds for such an analysis were planted in a qualitative discussion of vortex instability in section 1 of HN08. The following broadens the discussion and explicates our procedure for quantitatively assessing the nature of an instability.

2. A simple model suitable for a study of complex instabilities

Consider a barotropic vortex in gradient wind and hydrostatic balance. Herein, we shall assume that 3D perturbations of the balanced state obey linearized hydrostatic primitive equations, simplified with a Boussinesq approximation. The Coriolis parameter f and the static stability are given constant values, and viscosity is entirely neglected. The reader is referred to section 2 of M16 for a detailed description of the applicable perturbation equations and the method used for this study to computationally find the dominant modes of vortex instability. Additional discussion of the linear model can be found in SM04.

Analysis of the perturbation dynamics is facilitated by introducing a cylindrical coordinate system that is coaligned with the central axis of the vortex. As usual, r and φ represent the radial and azimuthal coordinates. To simplify various equations, the vertical coordinate z is chosen to be the pressure-based pseudoheight of Hoskins and Bretherton (1972). The variables u, υ, and w denote (in order) the radial, azimuthal, and pseudovertical components of the vector velocity field . Field variables dressed with overbars and primes respectively represent equilibrium and time-dependent perturbation fields. Each fluid variable is the sum of its equilibrium and perturbation components, as exemplified by , in which t is time.

Henceforth, we will assume that the unperturbed vortex features an off-center relative vorticity peak (Fig. 1a) similar to that found in the eyewall region of a strong tropical cyclone (Rogers et al. 2013). We will further assume that the angular velocity of the vortex greatly exceeds f and the azimuthal velocity is comparable to the characteristic speed of an internal gravity wave. Under the preceding conditions, the dominant modes of asymmetric instability may involve a pair of vortex Rossby waves on opposite sides of the relative vorticity peak, two critical-layer perturbations with distinct signatures in the PV field, and an outward propagating spiral inertia–gravity wave. Figures 1b and 1c illustrate the horizontal structure of such a growing perturbation. Further discussion of this figure is deferred to section 4.

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(a) Basic-state relative vorticity and angular velocity distributions of a tropical cyclone–like vortex whose shape parameters [Eq. (13)] are printed on the upper-right corner of the graph. Both distributions are normalized to . Note that the unshown PV of the basic state closely resembles . (b) The scaled PV perturbation at arbitrary z of the fastest-growing eigenmode, when the Rossby and Froude numbers [Eq. (14)] are respectively given by and . The color scale is normalized to the peak value of the plotted field. In the same units, the contour values are . (c) The geopotential perturbation . Solid (dashed) contours correspond to the following positive (negative) values: times the peak magnitude of the plotted field. The boundary of the yellow circle of radius separates the vortex region from the radiation zone. All lengths in all parts of this figure are in units of the core radius . The eigenfrequency of the mode depicted in (b) and (c) is .

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0289.1

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3. A method for analyzing perturbation growth

Consider a fluid tracer q whose unperturbed distribution depends only on radius r. In the absence of forcing and diffusion, the linearized tracer equation is

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in which is the equilibrium angular rotation frequency. It is suitable for the present study to let q equal PV. In the hydrostatic Boussinesq approximation, the materially conserved PV is defined by , in which , , ϕ is the geopotential, and is the 3D gradient operator (SM04). The linearized PV perturbation is , in which , , and .

Multiplying both sides of Eq. (1) by and averaging over the azimuth φ yields

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in which is the averaging operator. In theory, the radial velocity perturbation that advects q inward and outward can be viewed as a sum of contributions from each identifiable component α of the growing mode and fluid boundaries should they exist. In other words, the small-amplitude tendency equation for can be decomposed as follows:

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Assuming that there exists a in the vortex core attributable to the outer radiation field, the relative magnitude of its anticorrelation with would quantify the importance of radiation in driving the local growth of .

The method for partitioning the velocity perturbation is neither straightforward nor unique. One conceivable approach for a Boussinesq fluid might start by expressing the 3D-nondivergent velocity perturbation as a Biot–Savart-like volume integral of the vector vorticity perturbation and its image located beyond physical boundaries (e.g., Saffman 1992). One could then separate the volume integral into several parts associated with different regions of the fluid and a part associated with the boundary conditions. Another approach might begin by separating into balanced and unbalanced components. Further decomposition of the initial separation could involve partial velocities attributable to different flow structures, such as Rossby-like waves, critical-layer perturbations, and inertia–gravity waves. A preliminary concern would be the best choice for the mesoscale balance model.

The following explores the usefulness of a simpler partitioning scheme that is deemed reasonable for analyzing the 3D instability of a barotropic vortex. Both binary and multicomponent decompositions of the velocity perturbation are considered. The former begins by separating the flow domain into a cylindrical region containing the vortex core and an exterior radiation zone (Fig. 2a). The boundary radius R corresponds to the outermost turning point where the modal perturbation starts to locally exhibit the characteristics of an inertia–gravity wave (M16, their appendix B). The multicomponent decomposition begins similarly but further divides the vortex region into annular subregions that contain distinct peaks of wave activity associated with either a vortex Rossby wave or a critical-layer disturbance (Fig. 2b). The horizontal velocity perturbation at arbitrary z is then partitioned into components that are formally generated by the perturbations of vertical vorticity and horizontal divergence in each section of the fluid (cf. Bishop 1996; Renfrew et al. 1997). Importantly, each component of has a (2D) irrotational and nondivergent extension beyond the section containing its vortical and divergent sources (Fig. 2c) and therefore contributes to the stirring of external tracers such as PV. Further details are forthcoming.

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(a) Binary decomposition of the fluid into regions associated with the vortex (vtx, black) and radiation (rad, white). (b) Further decomposition of the vortex region into parts associated with the inner vortex Rossby wave (iw, dark gray), the inner critical layer (icl, white), the outer vortex Rossby wave (ow, black), and the outer critical layer (ocl, light gray). (c) Sketch of the velocity fields associated with localized positive relative vorticity and negative divergence perturbations in an arbitrary region α of the fluid.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0289.1

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Constructing the partial velocity field associated with each section of the fluid is a relatively simple matter. The procedure begins by expressing the horizontal velocity perturbation as the gradient of a scalar potential added to the cross-gradient of a streamfunction . In other words, let

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For compact flows in unbounded domains, and are unique up to arbitrary constants. The compactness condition applies to the problem at hand, because the radiation field of a temporally growing normal mode decays exponentially with increasing radius. Taking the 2D curl and divergence of Eq. (3) yields

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in which and .

The Poisson equations for and are readily solved upon expanding each perturbation field into an azimuthal Fourier series of the form

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Substituting the appropriate Fourier series into Eq. (4) leads to a set of independent second-order ordinary differential equations (ODEs) for and in the variable r. The ODEs can be formally solved with a Green function technique. The result is

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in which

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The notation is used above to denote the lesser (greater) of r and . The Heaviside step function is defined such that for and 0 for . The Green function defined by Eq. (6) enforces appropriate boundary conditions in which the velocities corresponding to and are noninfinite as r tends toward 0 or .

Taking the Fourier transform of Eq. (3) and using Eq. (5) yields

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in which

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Decomposing the integral in Eq. (7) into segments associated with the various regions of the fluid depicted in Fig. 2 yields

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in which denotes integration over region α. For a generic disturbance, the partial velocity field ascribed to region α amounts to the following sum over all azimuthal wavenumbers: , in which is given by the integral summand on the far-right-hand side of Eq. (9).

The normal modes of a barotropic vortex are single-wavenumber perturbations whose pertinent fields have the form

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in which is a complex frequency, a is a complex amplitude, and c.c. denotes the complex conjugate required under the working assumption that n or is nonzero (SM04; M16). Taking the vertical boundaries to be isothermal at and h, the vertical wavefunction is given by , in which k is an integral multiple of (SM04; M16). The subcomponents of are expressible as

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in which

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by virtue of Eq. (9).

Substituting Eq. (10) for and Eq. (11a) for into Eq. (2b) yields the following modal growth-rate formula:

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Each partial growth rate corresponds to one-half the local rate of change of resulting from the radial advection of the tracer q by the velocity field ascribed to and in region α of the normal mode. The value of varies with r but not with z, owing to the barotropic structure of the unperturbed vortex. Note that the value of is the same regardless of whether q is PV or an arbitrary passive tracer, since the relation is general.

If the distributions of and in region α fully and exclusively constituted those of a particular dynamical element of the modal perturbation—such as a vortex Rossby wave, critical-layer disturbance, or inertia–gravity wave—one might reasonably connect to the destabilizing (or stabilizing) influence of that element. On the other hand, one should bear in mind that the foregoing condition can be satisfied under normal circumstances only in some approximate sense, regardless of how carefully the vortex is partitioned. For example, the velocity fields of a vortex Rossby wave are traditionally obtained by inverting a localized PV or pseudo-PV perturbation according to specific balance conditions. As such, the wave distributions of and generally extend beyond the localized PV or pseudo-PV anomaly, into regions that are formally ascribed to other perturbation elements. While the far-reaching extensions of vorticity and divergence may be weak, they could be relevant to slow instabilities.

One might also worry about the appropriateness of instantaneous attribution. Although mathematically valid, the idea of attributing part of the velocity field within the vortex to simultaneous sources in the outer radiation field may seem physically questionable, owing to the finite propagation speed of inertia–gravity waves. That being said, the actual information contained in this partial velocity field amounts to the normal component of at the boundary between the vortex (α = vtx) and the radiation zone (α = rad). Such is evident by noting that inside the vortex, , in which subject to at . Here we have let denote the horizontal gradient operator. Based on the preceding consideration, one might view within the vortex as a flow adjustment connected to inertia–gravity wave emission at the boundary, without envisioning external sources and sinks. In practice, may be readily obtained from the difference without any additional computation. For this study, the preceding expression is cross-checked by calculating with the appropriate Green function integral between R and a sufficiently large radius that ensures convergence within a very small fractional error.

As a final remark, over the bulk of the vortex region, the dominant modes of instability considered herein are intrinsically slow relative to inertial oscillations (M16); that is, the magnitude of is appreciably less than . The preceding condition suggests that the perturbation dynamics within the vortex is quasi balanced (Shapiro and Montgomery 1993). As such, attributing local partial velocity fields to instantaneous nonlocal sources within the vortex or on its boundary seems consistent (in a general sense) with normal practice and thinking.

4. Illustrative implementation of the method

For illustrative purposes, we consider the asymmetric normal modes of cyclonic vortices whose unperturbed relative vorticity distributions have the form

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in which , , , approximates the radius of maximum wind speed, and is a positive scaling factor. The vorticity distribution defined by Eq. (13) possesses an off-center peak between and , whose edges become square as . Increasing the dimensionless parameter β enhances the central vorticity deficit. Figure 1a shows the particular distribution with , , and , along with the corresponding angular velocity field . The modal instabilities are completely controlled by the variables shaping (μ, β, Δ) and the following two dimensionless parameters:

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in which Ro is the Rossby number and Fr is a rotational Froude number based on the vertical wavenumber k of the disturbance. The υ subscripts on and indicate that the variables are evaluated at . The perturbation depicted in Figs. 1b and 1c is the fastest-growing normal mode of the vortex in Fig. 1a, with and . It is equivalent to the normal mode appearing in Fig. 2 of M16.

The fluid partitioning sketched in Fig. 2b can be summarized as follows:

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Here we have introduced notations for the inner critical radius and the outer critical radius of the instability mode, which represent the two solutions of

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Moreover, we have let , in which is the nominal half-width of the linear critical layer (Schecter et al. 2000). The constant c is taken to be 2 unless stated otherwise. The symbol denotes the nonzero finite radius at which or if the latter is larger. The inner wave section (iw) is the central circle of radius , excluding the inner annular critical layer (icl). The outer wave section (ow) is the annulus between and the inner boundary radius R of the radiation zone (rad), excluding the outer critical layer (ocl). Although the inner and outer wave sections (iw and ow) may each contain two disconnected regions separated by a critical layer, the former reduces to a single disc of radius when . The vortex region (vtx) comprises all sections of the fluid but the radiation zone, and therefore covers the entire interval of r between 0 and R.

Separating the vortex region from the radiation zone at the outermost turning point R of the instability mode seems relatively uncontroversial. The rationale for further decomposition of the vortex region requires additional discussion. To begin with, each subsection of the vortex region contains a distinct extremum of the angular pseudomomentum density of the instability mode. The angular pseudomomentum density is a standard measure of local wave activity in systems with cylindrical geometry. Averaging over φ and z, the modal angular pseudomomentum density at any given time is proportional to the following function (SM04; M16):

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in which

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Here, Q and are the perturbation wavefunction and basic-state distribution of PV, as opposed to a generic tracer. For all of the instability modes under consideration, tends to dominate for . Figure 1b is essentially a plot of , in which is the phase of Q (a). It is seen that each vortex section defined above [Eq. (15)] contains a distinct peak of . It has been verified with various diagnostics that the peaks within the inner and outer wave sections of the vortex (near and ) correspond to counterpropagating vortex Rossby waves (M16, section 3b therein). The peaks within the inner and outer critical layers (near and ) are obviously generated by resonant stirring of PV. Similar structure is found in all of the instability modes examined in this note and in the more comprehensive study of M16. A caveat is that one or more of the modal elements (such as the outer critical-layer disturbance) may be negligible.

Figure 3a shows the radial variation of the two components of attributed to radiation and internal vortex dynamics for the instability mode appearing in Figs. 1b and 1c. The top graph shows and , while the bottom graph shows the aforementioned partial growth rates multiplied by . Also shown are and ; whereas the former is a constant, peaks in the latter correspond to regions of maximal vortex Rossby wave activity and critical-layer stirring. It is seen that considerably exceeds in regions of peak vortex Rossby wave activity, whereas the opposite holds in the critical layers. In other words, the horizontal velocity field attributed to radiation is primarily responsible for the growth of associated with the vortex Rossby waves, whereas the horizontal velocity field generated by and within the vortex primarily controls the growth of in the critical layers.

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Partial growth rates of the instability mode shown in Figs. 1b and 1c. (a) (top) Binary decomposition of the total growth rate (dotted black) into one part associated with vorticity and divergence anomalies inside the vortex (blue) and another part attributed to radiation (red). (bottom) As in (top), but with the growth rates multiplied by and normalized to the peak value of . (b) As in the bottom panel of (a), but for partial growth rates attributed to velocity sources in the inner wave region (iw, solid orange), outer wave region (ow, solid black), inner critical layer (icl, dashed orange), and outer critical layer (ocl, dashed black). Their sum (vtx, light blue) is shown for reference. (c) As in the bottom panel of (a), but with the partial growth rates attained from velocity sources in the vortex (blue) and in the radiation zone (red) each split into those attained from vorticity (solid) and divergence (dashed). Their sum (dotted black) is shown for reference.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0289.1

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Figure 3b shows the radial variations of the four subcomponents of . The subcomponents paint a more complex picture of the instability mode. Cancellations between the subcomponents account for the smallness of where the vortex Rossby wave activity is concentrated. The inner and outer wave regions are seen to generate velocity fields that act to amplify in each other but mostly damp locally. The velocity field produced by sources in the inner critical layer hinders the growth of in both vortex Rossby waves. The velocity field produced by sources in the outer critical layer adds slightly to the damping effort in the outer wave. The positive vortex contribution to the growth of in the inner critical layer is due to the positive influence of exceeding the negative influence of . The growth of in the outer critical layer is mostly due to the stirring induced by .

Additional information on the nature of the instability can be obtained by splitting each partial growth rate into subparts associated with and individually. That is, let , in which

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and

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Figure 3c shows the radial variations of and for the instability mode at hand, with . Unsurprisingly, it is found that . Less anticipated, one can see that has values comparable and opposite to those of in the regions of peak vortex Rossby wave activity.

The preceding growth-rate decomposition (Fig. 3) is found to exhibit only moderate sensitivity to variations of , , and R. Clearly, variations of and that are constrained to prevent regional overlap have no bearing on the values of or . Variation of from one-half to twice its standard value most notably coincides with a proportional amplification of the ratio of to in the neighborhood of . Reduction of to decreases the positive magnitude of by 37% (39%) at . The corresponding local changes to are equal in absolute value but opposite in sign. Reduction of R to most notably increases the positive value of by 26% at and commensurately intensifies the local negative value of .

It is worth remarking that we have conducted a simple test to gain confidence that implies the importance of radiation in driving the local growth of the PV perturbation in a tropical cyclone–like vortex. It is well known that a monotonic cyclone would be stable in the absence of inertia–gravity waves (Montgomery and Shapiro 1995). Moreover, it is reasonably well established that the dominant mode of instability of a monotonic cyclone involves the positive feedback between a vortex Rossby wave at the edge of the potential vorticity core and inertia–gravity wave radiation (Ford 1994; SM04). We have verified that when , the condition holds very well in the vicinity of for a number of vortices with and . In each case considered, was made sufficiently large to prevent significant opposition to modal growth by PV stirring in the outer critical layer (SM04). Note that the extremely opposite condition, , agreeably holds for all nondivergent barotropic instabilities. In this limit, and vanish outside the vortex, and the integral expression for [Eq. (11b)] in the definition of [Eq. (12)] is clearly zero.

5. Comparison to alternative diagnostics

The tracer-based instability analysis offers a perspective on the importance of inertia–gravity wave radiation that may not fully agree with tentative assessments gleaned from simpler diagnostics. Discrepancies primarily occur when more than one mechanism has substantial impact on the amplification of a perturbation.

Recently, M16 provisionally assessed the importance of radiation by examining its contribution to the wave activity budget of a growing mode. The wave activity of region α was defined by

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For modes in which , conservation of total wave activity was expressed in the form

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Substituting Eq. (19) into the left-hand side of Eq. (20a) and dividing through by yields

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in which . A large relative magnitude of on the right-hand side of Eq. (20b) simply implies that amplification of the radiation field (possessing negative wave activity) has an important role in balancing the growth of positive outer vortex Rossby wave activity. One might tentatively infer from such a result that radiation has an important role in driving the instability, but rigorous justification of this conclusion generally requires supplemental analysis and reasoning. We note that in practice, the calculation of is simplified by reducing the integral to an equivalent algebraic expression proportional to the angular momentum flux at R (M16).

HN08 examined the radial distribution of the contribution

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to the growth rate of kinetic energy in the asymmetric perturbation. They showed that is concentrated beyond the edge radius of the vorticity distribution if the instability primarily involves the positive feedback of inertia–gravity wave radiation on an outer vortex Rossby wave that is responsible for its emission. By contrast, they found that is concentrated inward of if the instability primarily involves the interaction of counterpropagating vortex Rossby waves. One might therefore speculate that the importance of radiation in driving the instability of a tropical cyclone–like vortex could be assessed simply by comparing the magnitudes of inward and outward of .

Figure 4 presents the three diagnostics at issue for three selected instability modes of a cyclonic vortex with , , , and . The top row corresponds to the dominant instability when the Froude number Fr has a subcritical value of 0.8. As in M16, the term “subcritical” refers to the small- Fr parameter regime in which inertia–gravity wave radiation has minimal influence on the fastest-growing wavenumber-n eigenmode of the linearized dynamical system. The middle row corresponds to the dominant instability when Fr has a strongly supercritical value of 6. The bottom row corresponds to the dominant instability when Fr has a transitional value of 3. Assuming constant N, the Froude number may be viewed as a dimensionless vertical wavenumber or a dimensionless measure of vortex strength. Taking the former perspective with having a severe tropical cyclone value of 65 m s⁻¹, the three depicted modes of instability would have vertical quarter-wavelengths of (top) 12.8 km, (middle) 1.7 km, and (bottom) 3.4 km. Here it is assumed that N is adequately approximated by a dry tropospheric value of 0.01 s⁻¹; a reduction of N due to moisture would increase the vertical length scale associated with each mode.

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(a)–(c) Instability diagnostics for the fastest-growing eigenmode when , , , , and . (a) Partial growth rates attributed to vorticity and divergence anomalies inside the vortex (vtx, blue) and to radiation (rad, red). Each is multiplied by and then normalized to the maximum of . Their scaled sum (dotted black) is shown for reference. (b) The alternative partial growth rates of M16. Those associated with the inner vortex Rossby wave (iw), inner critical layer (icl), and outer critical layer (ocl) are stacked in the blue column. The red column shows the formal contribution to from radiation. (c) The production rate of kinetic energy in the perturbation associated with the radial shear of , normalized to its maximum value. (d)–(f) As in (a)–(c), respectively, but with . (g)–(i) As in (a)–(c), respectively, but with . The turning points not shown alongside other important radii in (a),(d), and (g) respectively occur at R = 2.17, 1.54, and 1.80 in units of . The eigenfrequencies of the top, middle, and bottom modes are respectively , , and in units of .

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0289.1

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The diagnostics under consideration offer a consistent picture of the subcritical instability mode. The binary growth rate partitioning advocated herein (Fig. 4a) suggests that inertia–gravity wave radiation is much less relevant to the amplification of than sources of the velocity perturbation ( and ) inside the vortex. The wave-activity-based growth rate partitioning of M16 (Fig. 4b) consistently suggests that radiation has minimal impact.² The profile (Fig. 4c) indicates that kinetic energy is transferred from the mean shear flow to the asymmetric perturbation primarily in the “eyewall” . There is little evidence of such transfer in the outer core , where under different circumstances enhancement of might have reflected appreciable positive feedback from radiation.

The opposite picture is found for the strongly supercritical instability mode. The tracer-based instability analysis (Fig. 4d) reveals a dominant partial growth rate attributable to inertia–gravity wave radiation in all pertinent regions of the vortex except the outer critical layer. The wave-activity-based growth rate decomposition (Fig. 4e) yields for all . The distribution of (Fig. 4f) is concentrated in the outer core, as in the principal radiation-driven instabilities of monotonic vortices.

The transitional instability mode exemplifies how the three diagnostics under consideration can leave different impressions. The tracer-based instability analysis (Fig. 4g) suggests that radiation is equally or more responsible for the amplification of than sources of the velocity perturbation inside the vortex. A notable exception is in the inner critical layer, where the velocity field generated by vortex sources prevails. The wave-activity-based growth-rate decomposition (Fig. 4h) consistently suggests that radiation is relevant to the instability. However, the relation leaves the inconsistent overall impression that radiation is less important than internal vortex dynamics. The profile (Fig. 4i) indicates that kinetic energy is transferred from the mean shear flow to the asymmetric perturbation in both the eyewall and the outer core of the vortex, with no obvious discrimination. It is unclear to the authors how one might confidently assess the relative importance of radiation to the instability from the information contained in . In contrast to the tracer-based analysis, an assessment based solely on the location of where is peaked might encourage one to believe that internal vortex dynamics has the leading role in driving the instability. Alternatively, comparing the inner and outer radial integrals of yields an ambiguous inner-to-outer ratio of 1.2.

Note that of the three diagnostics under consideration, only the tracer-based analysis was expressly designed to isolate and quantify the relative importance of radiation in forcing the growth of a perturbation field within the vortex. There may be no rigorous justification for having presumed that one could find distinct patterns in the wave activity budget or the distribution to reliably convey the same information. A limited search for such patterns was deemed worthwhile because variants of the aforementioned diagnostics are commonly examined and simpler to calculate. However, the preceding analysis of the transitional instability mode suggests that relatively simple diagnostics may be inescapably ambiguous when more than one mechanism has appreciable influence on the growth of a perturbation.

6. Summary

This note has expounded a previously underdeveloped method for evaluating the relative importance of inertia–gravity wave radiation in driving the instability of a columnar vortex resembling a tropical cyclone. The procedure begins by dividing the fluid volume into vortex and radiation zones. The velocity perturbation is then decomposed into one part that is formally associated with sources ( and ) inside the vortex and another part that is attributed to radiation. The importance of radiation is assessed by comparing the rates at which the two partial velocity fields act to amplify a tracer perturbation, denoted by the variable and exemplified by the PV perturbation in the vortex core.

As illustrated in section 4, the foregoing instability analysis can be readily extended to see how different sources of the velocity perturbation residing within the vortex individually contribute to the amplification of . Sources deemed relevant include those found in distinct critical layers and regions of enhanced vortex Rossby wave activity. In principle, an extended analysis can be beneficial for elucidating the true intricacy of a multimechanistic instability.