## 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

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 *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.

## 3. A method for analyzing perturbation growth

*q*whose unperturbed distribution

*r*. In the absence of forcing and diffusion, the linearized tracer equation is

*q*equal PV. In the hydrostatic Boussinesq approximation, the materially conserved PV is defined by

*ϕ*is the geopotential, and

*φ*yields

*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

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

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 *z* is then partitioned into components that are formally generated by the perturbations of vertical vorticity

*r*. The ODEs can be formally solved with a Green function technique. The result is

*r*and

*r*tends toward 0 or

*α*. For a generic disturbance, the partial velocity field ascribed to region

*α*amounts to the following sum over all azimuthal wavenumbers:

*a*is a complex amplitude, and c.c. denotes the complex conjugate required under the working assumption that

*n*or

*h*, the vertical wavefunction is given by

*k*is an integral multiple of

*q*by the velocity field ascribed to

*α*of the normal mode. The value of

*r*but not with

*z*, owing to the barotropic structure of the unperturbed vortex. Note that the value of

*q*is PV or an arbitrary passive tracer, since the relation

If the distributions of *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

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 *α* = vtx) and the radiation zone (*α* = rad). Such is evident by noting that inside the vortex, *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

## 4. Illustrative implementation of the method

*β*enhances the central vorticity deficit. Figure 1a shows the particular distribution with

*μ*,

*β*, Δ) and the following two dimensionless parameters:

*k*of the disturbance. The

*υ*subscripts on

*c*is taken to be 2 unless stated otherwise. The symbol

*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

*r*between 0 and

*R*.

*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):

*Q*and

*Q*(

*a*). It is seen that each vortex section defined above [Eq. (15)] contains a distinct peak of

Figure 3a shows the radial variation of the two components of

Figure 3b shows the radial variations of the four subcomponents of

The preceding growth-rate decomposition (Fig. 3) is found to exhibit only moderate sensitivity to variations of *R*. Clearly, variations of *R* to

It is worth remarking that we have conducted a simple test to gain confidence that

## 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.

*α*was defined by

*R*(M16).

Figure 4 presents the three diagnostics at issue for three selected instability modes of a cyclonic vortex with *n* eigenmode of the linearized dynamical system. The middle row corresponds to the dominant *N*, the Froude number may be viewed as a dimensionless vertical wavenumber or a dimensionless measure of vortex strength. Taking the former perspective with ^{−1}, the three depicted modes of instability would have vertical quarter-wavelengths *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.

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 ^{2} The

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

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

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

## 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 (

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

## Acknowledgments

The authors thank three anonymous reviewers for their constructive comments. This work was supported by the National Science Foundation under Grant AGS-1250533. Additional support was provided by the Natural Sciences and Engineering Research Council of Canada and Hydro-Quebec through the IRC program.

## REFERENCES

Antkowiak, A., and P. Brancher, 2004: Transient energy growth for the Lamb–Oseen vortex.

,*Phys. Fluids***16**, L1–L4, doi:10.1063/1.1626123.Bishop, C. H., 1996: Domain-independent attribution. Part I: Reconstructing the wind from estimates of vorticity and divergence using free space Green’s functions.

,*J. Atmos. Sci.***53**, 241–252, doi:10.1175/1520-0469(1996)053<0241:DIAPIR>2.0.CO;2.Briggs, R. J., J. D. Daugherty, and R. H. Levy, 1970: Role of Landau damping in crossed-field electron beams and inviscid shear flow.

,*Phys. Fluids***13**, 421–432, doi:10.1063/1.1692936.Bryan, G. H., 2012: Effects of surface exchange coefficients and turbulence length scales on the intensity and structure of numerically simulated hurricanes.

,*Mon. Wea. Rev.***140**, 1125–1143, doi:10.1175/MWR-D-11-00231.1.Corbosiero, K. L., J. Molinari, A. R. Aiyyer, and M. L. Black, 2006: The structure and evolution of Hurricane Elena (1985). Part II: Convective asymmetries and evidence for vortex Rossby waves.

,*Mon. Wea. Rev.***134**, 3073–3091, doi:10.1175/MWR3250.1.Ford, R., 1994: The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water.

,*J. Fluid Mech.***280**, 303–334, doi:10.1017/S0022112094002946.Hendricks, E. A., and W. H. Schubert, 2010: Adiabatic rearrangement of hollow PV towers.

,*J. Adv. Model. Earth Syst.***2**(4), doi:10.3894/JAMES.2010.2.8.Hendricks, E. A., B. D. McNoldy, and W. H. Schubert, 2012: Observed inner-core structural variability in Hurricane Dolly (2008).

,*Mon. Wea. Rev.***140**, 4066–4077, doi:10.1175/MWR-D-12-00018.1.Hodyss, D., and D. S. Nolan, 2008: The Rossby-inertia-buoyancy instability in baroclinic vortices.

,*Phys. Fluids***20**, 096602, doi:10.1063/1.2980354.Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric frontogenesis models: Mathematical formulation and solution.

,*J. Atmos. Sci.***29**, 11–37, doi:10.1175/1520-0469(1972)029<0011:AFMMFA>2.0.CO;2.Kossin, J. P., and M. D. Eastin, 2001: Two distinct regimes in the kinematic and thermodynamic structure of the hurricane eye and eyewall.

,*J. Atmos. Sci.***58**, 1079–1090, doi:10.1175/1520-0469(2001)058<1079:TDRITK>2.0.CO;2.Kossin, J. P., and W. H. Schubert, 2001: Mesovortices, polygonal flow patterns, and rapid pressure falls in hurricane-like vortices.

,*J. Atmos. Sci.***58**, 2196–2209, doi:10.1175/1520-0469(2001)058<2196:MPFPAR>2.0.CO;2.Kwon, Y. C., and W. M. Frank, 2005: Dynamic instabilities of simulated hurricane-like vortices and their impacts on the core structure of hurricanes. Part I: Dry experiments.

,*J. Atmos. Sci.***62**, 3955–3973, doi:10.1175/JAS3575.1.Levy, R. H., 1965: Diocotron instability in a cylindrical geometry.

,*Phys. Fluids***8**, 1288–1295, doi:10.1063/1.1761400.Menelaou, K., D. A. Schecter, and M. K. Yau, 2016: On the relative contribution of inertia–gravity wave radiation to asymmetric instabilities in tropical cyclone–like vortices.

,*J. Atmos. Sci.***73**, 3345–3370, doi:10.1175/JAS-D-15-0360.1.Michalke, A., and A. Timme, 1967: On the inviscid instability of certain two-dimensional vortex-type flows.

,*J. Fluid Mech.***29**, 647–666, doi:10.1017/S0022112067001090.Montgomery, M. T., and L. J. Shapiro, 1995: Generalized Charney–Stern and Fjortoft theorems for rapidly rotating vortices.

,*J. Atmos. Sci.***52**, 1829–1833, doi:10.1175/1520-0469(1995)052<1829:GCAFTF>2.0.CO;2.Montgomery, M. T., M. M. Bell, M. L. Black, and S. D. Aberson, 2006: Hurricane Isabel (2003): New insights into the physics of intense storms. Part I: Mean vortex structure and maximum intensity estimates.

,*Bull. Amer. Meteor. Soc.***87**, 1335–1347, doi:10.1175/BAMS-87-10-1335.Muramatsu, T., 1986: The structure of polygonal eye of a typhoon.

,*J. Meteor. Soc. Japan***64**, 913–921, doi:10.2151/jmsj1965.64.6_913.Naylor, J., and D. A. Schecter, 2014: Evaluation of the impact of moist convection on the development of asymmetric inner core instabilities in simulated tropical cyclones.

,*J. Adv. Model. Earth Syst.***6**, 1027–1048, doi:10.1002/2014MS000366.Nolan, D. S., and B. F. Farrell, 1999: Generalized stability analysis of asymmetric disturbances in one- and two-celled vortices maintained by radial inflow.

,*J. Atmos. Sci.***56**, 1282–1307, doi:10.1175/1520-0469(1999)056<1282:GSAOAD>2.0.CO;2.Park, J., and P. Billant, 2013: Instabilities and waves on a columnar vortex in a strongly stratified and rotating fluid.

,*Phys. Fluids***25**, 086601, doi:10.1063/1.4816512.Persing, J., M. T. Montgomery, J. C. McWilliams, and R. K. Smith, 2013: Asymmetric and axisymmetric dynamics of tropical cyclones.

,*Atmos. Chem. Phys.***13**, 12 299–12 341, doi:10.5194/acp-13-12299-2013.Plougonven, R., and V. Zeitlin, 2002: Internal gravity wave emission from a pancake vortex: An example of wave–vortex interaction in strongly stratified flows.

,*Phys. Fluids***14**, 1259–1268, doi:10.1063/1.1448297.Reasor, P. D., M. T. Montgomery, F. D. Marks Jr., and J. F. Gamache, 2000: Low-wavenumber structure and evolution of the hurricane inner core observed by airborne dual-Doppler radar.

,*Mon. Wea. Rev.***128**, 1653–1680, doi:10.1175/1520-0493(2000)128<1653:LWSAEO>2.0.CO;2.Renfrew, I. A., A. J. Thorpe, and C. H. Bishop, 1997: The role of the environmental flow in the development of secondary frontal cyclones.

,*Quart. J. Roy. Meteor. Soc.***123**, 1653–1675, doi:10.1002/qj.49712354210.Rogers, R., and Coauthors, 2013: NOAA’s Hurricane Intensity Forecasting Experiment: A progress report.

,*Bull. Amer. Meteor. Soc.***94**, 859–882, doi:10.1175/BAMS-D-12-00089.1.Rozoff, C. M., J. P. Kossin, W. H. Schubert, and P. J. Mulero, 2009: Internal control of hurricane intensity variability: The dual nature of potential vorticity mixing.

,*J. Atmos. Sci.***66**, 133–147, doi:10.1175/2008JAS2717.1.Saffman, P. G., 1992:

*Vortex Dynamics*. Cambridge University Press, 311 pp.Schecter, D. A., and M. T. Montgomery, 2004: Damping and pumping of a vortex Rossby wave in a monotonic cyclone: Critical layer stirring versus inertia–buoyancy wave emission.

,*Phys. Fluids***16**, 1334–1348, doi:10.1063/1.1651485.Schecter, D. A., D. H. E. Dubin, A. C. Cass, C. F. Driscoll, I. M. Lansky, and T. M. O’Neil, 2000: Inviscid damping of asymmetries on a two-dimensional vortex.

,*Phys. Fluids***12**, 2397–2412, doi:10.1063/1.1289505.Schubert, W. H., M. T. Montgomery, R. K. Taft, T. A. Guinn, S. R. Fulton, J. P. Kossin, and J. P. Edwards, 1999: Polygonal eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes.

,*J. Atmos. Sci.***56**, 1197–1223, doi:10.1175/1520-0469(1999)056<1197:PEAECA>2.0.CO;2.Shapiro, L. J., and M. T. Montgomery, 1993: A three-dimensional balance theory for rapidly rotating vortices.

,*J. Atmos. Sci.***50**, 3322–3335, doi:10.1175/1520-0469(1993)050<3322:ATDBTF>2.0.CO;2.Yang, B., Y. Wang, and B. Wang, 2007: The effect of internally generated inner-core asymmetries on tropical cyclone potential intensity.

,*J. Atmos. Sci.***64**, 1165–1188, doi:10.1175/JAS3971.1.

^{1}

It should be noted that regardless of vortical shear-flow instabilities, the enabling of asymmetric moist convection can alter the angular momentum fluxes and thermodynamics that regulate storm intensity, with various effects that may or may not be negative (e.g., Persing et al. 2013).