Note on Analyzing Perturbation Growth in a Tropical1

Cyclone-Like Vortex Radiating Inertia-Gravity Waves2

David A. Schecter1∗ and Konstantinos Menelaou2

1NorthWest Research Associates, Boulder, Colorado, USA

2Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

To appear in Journal of the Atmospheric Sciences.

3

Submitted October 13, 2016; revised January 12, 2017;4

updated with minor edits on March 22, 2017.5

∗Corresponding author address: NorthWest Research Associates, 3380 Mitchell Lane, Boulder, CO, USA,80301. E-mail: schecter@nwra.com

1

6

Abstract7

A method is outlined for quantitatively assessing the impact of inertia-gravity8

wave radiation on the multimechanistic instability modes of a columnar strati-9

fied vortex that resembles an intense tropical cyclone. The method begins by10

decomposing the velocity field into one part that is formally associated with11

sources inside the vortex and another part that is attributed to radiation. The12

relative importance of radiation is assessed by comparing the rates at which the13

two partial velocity fields act to amplify the perturbation of an arbitrary tracer14

field– such as potential vorticity –inside the vortex. Further insight is gained15

by decomposing the formal vortex contribution to the amplification rate into16

subparts that are primarily associated with distinct vortex Rossby waves and17

critical layer perturbations.18

2

1. Introduction19

20

Tropical cyclones may exhibit various asymmetric instabilities as their basic states freely21

evolve or adjust to changing environmental conditions. Such instabilities can give rise22

to commonly seen elliptical cores, polygonal eyewalls and mesovortices [Muramatsu 1986;23

Reasor et al. 2000; Kossin and Schubert 2001; Corbosiero et al. 2006; Montgomery et24

al. 2006; Hendricks et al. 2012]. They may also induce horizontal mixing processes that25

efficiently redistribute angular momentum and equivalent potential temperature [Schubert26

et al. 1999; Kossin and Eastin 2001; Hendricks and Schubert 2010]. The immediate conse-27

quence of asymmetric instability and mixing can be the slowdown of intensification or a28

reduction of maximum wind speed in the primary circulation of the vortex [Schubert et29

al. 1999; Naylor and Schecter 2014; cf. Rozoff et al. 2009]. The possible negative influence of30

asymmetric instabilities may factor into why three-dimensional (3D) cloud-resolving tropi-31

cal cyclone models often yield moderately or slightly weaker storms than their axisymmetric32

counterparts [Yang et al. 2007; Bryan 2012; Persing et al. 2013; Naylor and Schecter 2014].133

In short, there is reason to believe that the theory of tropical cyclone intensity cannot be34

fully detached from the theory of vortex instability.35

There are several well-known mechanisms of asymmetric vortex instability that are36

potentially relevant to the behavior of intense tropical cyclones. Classical barotropic insta-37

bility mechanisms include (1) the mutual amplification of phase-locked counter-propagating38

vortex Rossby waves in the vicinity of the eyewall [Levy 1965; Michalke and Timme 1967;39

Schubert et al. 1999], and (2) the mutual amplification of a vortex Rossby wave and the poten-40

tial vorticity (PV) anomaly that it generates in a suitably conditioned critical layer [Briggs41

et al. 1970]. Another viable mechanism of asymmetric perturbation growth is the positive42

feedback of inertia gravity wave radiation on the vortex Rossby wave that is responsible for43

1It should be noted that regardless of vortical shear-flow instabilities, the enabling of asymmetric moistconvection can alter the angular momentum fluxes and thermodynamics that regulate storm intensity, withvarious effects that may or may not be negative [e.g., Persing et al. 2013].

3

its excitation [e.g., Ford 1994; Plougonven and Zeitlin 2002; Schecter and Montgomery 200444

(SM04); Hodyss and Nolan 2008 (HN08); Park and Billant 2013]. Instabilities related to45

baroclinic vortex structure [Kwon and Frank 2005] and the transient growth of nonmodal46

perturbations [Nolan and Farrell 1999; Antkowiak and Brancher 2004] are also pertinent,47

but will not be considered explicitly in this note.48

The dominant modes of instability can involve multiple mechanisms operating simulta-49

neously [Menelaou et al. 2016 (M16)]. Under these circumstances, the role of each mecha-50

nism in destabilizing the vortex is difficult to assess without the right diagnostic. The main51

purpose of this note is to briefly present an alternative method to quantitatively compare52

the importance of inertia-gravity wave radiation to that of other processes in driving the53

growth of vortex perturbations. The method amounts to comparing the rates at which the54

perturbation to a tracer field inside the vortex– such as PV –is amplified by velocity fields55

attributed to radiation and to sources within the vortex itself. The seeds for such an analy-56

sis were planted in a qualitative discussion of vortex instability in section 1 of HN08. The57

following broadens the discussion and explicates our procedure for quantitatively assessing58

the nature of an instability.59

60

2. A Simple Model Suitable for a Study of Complex Instabilities61

62

Consider a barotropic vortex in gradient-wind and hydrostatic balance. Herein, we shall63

assume that 3D perturbations of the balanced state obey linearized hydrostatic primitive64

equations, simplified with a Boussinesq approximation. The Coriolis parameter f and the65

static stability N2 are given constant values, and viscosity is entirely neglected. The reader is66

referred to section 2 of M16 for a detailed description of the applicable perturbation equations67

and the method used for this study to computationally find the dominant modes of vortex68

instability. Additional discussion of the linear model can be found in SM04.69

Analysis of the perturbation dynamics is facilitated by introducing a cylindrical coordi-70

4

nate system that is coaligned with the central axis of the vortex. As usual, r and ϕ represent71

the radial and azimuthal coordinates. To simplify various equations, the vertical coordinate72

z is chosen to be the pressure based pseudo-height of Hoskins and Bretherton [1972]. The73

variables u, v and w denote (in order) the radial, azimuthal and pseudo-vertical components74

of the vector velocity field v. Field variables dressed with overbars and primes respectively75

represent equilibrium and time-dependent perturbation fields. Each fluid variable is the sum76

of its equilibrium and perturbation components, as exemplified by v = v(r) + v′(r, ϕ, z, t), in77

which t is time.78

Henceforth, we will assume that the unperturbed vortex features an off-center relative79

vorticity peak [Fig. 1a] similar to that found in the eyewall region of a strong tropical80

cyclone [Rogers et al. 2013]. We will further assume that the angular velocity of the vortex81

greatly exceeds f , and the azimuthal velocity is comparable to the characteristic speed of an82

internal gravity wave. Under the preceding conditions, the dominant modes of asymmetric83

instability may involve a pair of vortex Rossby waves on opposite sides of the relative vortic-84

ity peak, two critical layer perturbations with distinct signatures in the PV field, and an85

outward propagating spiral inertia-gravity wave. Figures 1b and 1c illustrate the horizontal86

structure of such a growing perturbation. Further discussion of this figure is deferred to87

section 4.88

89

3. A Method for Analyzing Perturbation Growth90

91

Consider a fluid tracer q whose unperturbed distribution q depends only on radius r. In92

the absence of forcing and diffusion, the linearized tracer equation is93

∂q′

∂t+ Ω

∂q′

∂ϕ= −u′dq

dr, (1)94

in which Ω(r) ≡ v/r is the equilibrium angular rotation frequency. It is suitable for the95

5

present study to let q equal PV. In the hydrostatic Boussinesq approximation, the materially96

conserved PV is defined by q ≡ (ζ + f z) · ∇(∂φ/∂z), in which ζ ≡ ∇× u, u ≡ v− wz, φ is97

the geopotential, and ∇ is the 3D gradient operator [SM04]. The linearized PV perturbation98

is q′ = N2ζ ′ + η∂2φ′/∂z2, in which ζ ≡ z · ζ, η ≡ ζ + f , and N2 ≡ ∂2φ/∂z2.99

Multiplying both sides of Eq. (1) by q′ and averaging over the azimuth ϕ yields100

1

2

∂ 〈(q′)2〉∂t

= −〈u′q′〉 dqdr, (2a)101

in which 〈. . .〉 is the averaging operator. In theory, the radial velocity perturbation u′ that102

advects q inward and outward can be viewed as a sum of contributions from each identifiable103

component (α) of the growing mode and fluid boundaries should they exist. In other words,104

the small-amplitude tendency equation for 〈(q′)2〉 can be decomposed as follows:105

1

2

∂ 〈(q′)2〉∂t

= −∑α

〈u′αq′〉dq

dr. (2b)106

Assuming that there exists a u′α in the vortex core attributable to the outer radiation field,107

the relative magnitude of its anti-correlation with q′dq/dr would quantify the importance of108

radiation in driving the local growth of q′.109

The method for partitioning the velocity perturbation is neither straightforward nor110

unique. One conceivable approach for a Boussinesq fluid might start by expressing the111

3D-nondivergent velocity perturbation v′ as a Biot-Savart-like volume integral of the vector112

vorticity perturbation and its image located beyond physical boundaries [e.g., Saffman 1992].113

One could then separate the volume integral into several parts associated with different114

regions of the fluid and a part associated with the boundary conditions. Another approach115

might begin by separating v′ into balanced and unbalanced components. Further decompo-116

sition of the initial separation could involve partial velocities attributable to different flow117

structures, such as Rossby-like waves, critical layer perturbations and inertia-gravity waves.118

A preliminary concern would be the best choice for the mesoscale balance model.119

6

The following explores the usefulness of a simpler partitioning scheme that is deemed120

reasonable for analyzing the 3D instability of a barotropic vortex. Both binary and multi-121

component decompositions of the velocity perturbation are considered. The former begins122

by separating the flow-domain into a cylindrical region containing the vortex core, and an123

exterior radiation zone [Fig. 2a]. The boundary radius (R) corresponds to the outermost124

turning point where the modal perturbation starts to locally exhibit the characteristics of an125

inertia-gravity wave [M16, appendix B]. The multicomponent decomposition begins similarly,126

but further divides the vortex region into annular subregions that contain distinct peaks of127

wave activity associated with either a vortex Rossby wave or a critical layer disturbance128

[Fig. 2b]. The horizontal velocity perturbation u′ at arbitrary z is then partitioned into129

components that are formally generated by the perturbations of vertical vorticity (ζ ′) and130

horizontal divergence (σ′) in each section of the fluid [cf. Bishop 1996; Renfrew et al. 1997].131

Importantly, each component of u′ has a (2D) irrotational and nondivergent extension beyond132

the section containing its vortical and divergent sources [Fig. 2c], and therefore contributes133

to the stirring of external tracers such as PV. Further details are forthcoming.134

Constructing the partial velocity field associated with each section of the fluid is a135

relatively simple matter. The procedure begins by expressing the horizontal velocity pertur-136

bation as the gradient of a scalar potential (χ′) added to the cross-gradient of a streamfunc-137

tion (ψ′). In other words, let138

u′

v′

=

−1

r

∂ψ′

∂ϕ+

∂χ′

∂r∂ψ′

∂r+

1

r

∂χ′

∂ϕ

. (3)139

For compact flows in unbounded domains, ψ′ and χ′ are unique up to arbitrary constants.140

The compactness condition essentially applies to the problem at hand, because the radiation141

field of a temporally growing normal mode decays exponentially with increasing radius.142

7

Taking the 2D curl and divergence of Eq. (3) yields143

(1

r

∂

∂rr∂

∂r+

1

r2

∂2

∂ϕ2

) ψ′

χ′

=

ζ ′

σ′

, (4)144

in which ζ ′ ≡ [∂(rv′)/∂r − ∂u′/∂ϕ]/r and σ′ ≡ [∂(ru′)/∂r + ∂v′/∂ϕ]/r.145

The Poisson equations for ψ′ and χ′ are readily solved upon expanding each perturbation146

field (g′) into an azimuthal Fourier series of the form147

g′ ≡∞∑

n=−∞

gn(r, z, t)einϕ.148

Substituting the appropriate Fourier series into Eq. (4) leads to a set of independent second-149

order ordinary differential equations (ODEs) for ψn and χn in the variable r. The ODEs can150

be formally solved with a Green function technique. The result is151

ψn(r, z, t)

χn(r, z, t)

=

∫ ∞0

drrGn(r, r)

ζn(r, z, t)

σn(r, z, t)

, (5)152

in which153

Gn(r, r) ≡

− 1

2|n|

(r<r>

)|n|n 6= 0,

ln(r/r)Θ(r − r) n = 0.

(6)154

The notation r< (r>) is used above to denote the lesser (greater) of r and r. The Heaviside155

step function is defined such that Θ(r − r) = 1 for r < r and 0 for r > r. The Green156

function Gn defined by Eq. (6) enforces appropriate boundary conditions in which the veloc-157

ities corresponding to ψn and χn are non-infinite as r tends toward 0 or ∞.158

8

Taking the Fourier transform of Eq. (3) and using Eq. (5) yields159

un

vn

=

∫ ∞0

drr

γun

γvn

, (7)160

in which161 γun

γvn

≡ −inr Gn(r, r)ζn(r, z, t) +

∂

∂rGn(r, r)σn(r, z, t)

∂

∂rGn(r, r)ζn(r, z, t) +

in

rGn(r, r)σn(r, z, t)

. (8)162

Decomposing the integral in Eq. (7) into segments associated with the various regions of the163

fluid depicted in Fig. 2 yields164

un

vn

=∑α

unα

vnα

≡∑α

∫α

drr

γun

γvn

, (9)165

in which∫α

denotes integration over region α. For a generic disturbance, the partial velocity166

field ascribed to region α amounts to the following sum over all azimuthal wavenumbers:167

(u′α, v′α) =

∑n(unα, vnα)einϕ, in which (unα, vnα) is given by the integral-summand on the168

far-right hand side of Eq. (9).169

The normal modes of a barotropic vortex are single-wavenumber perturbations whose170

pertinent fields have the form171

u′

v′

φ′

ζ ′

σ′

q′

= a

U(r)

V (r)

Φ(r)

Z(r)

D(r)

Q(r)

Υ(z)ei(nϕ−ωt) + c.c., (10)172

in which ω ≡ ωR+ iωI is a complex frequency, a is a complex amplitude, and c.c. denotes the173

9

complex conjugate required under the working assumption that n or ωR is nonzero [SM04;174

M16]. Taking the vertical boundaries to be isothermal (∂φ′/∂z = 0) at z = 0 and h, the175

vertical wavefunction is given by Υ = cos(kz), in which k is an integral multiple of π/h [ibid].176

The subcomponents of (u′, v′) are expressible as177

u′α

v′α

= a

Uα(r)

Vα(r)

Υ(z)ei(nϕ−ωt) + c.c., (11a)178

in which179 Uα(r)

Vα(r)

=

∫α

drr

−inr Gn(r, r)Z(r) +∂

∂rGn(r, r)D(r)

∂

∂rGn(r, r)Z(r) +

in

rGn(r, r)D(r)

(11b)180

by virtue of Eq. (9).181

Substituting Eq. (10) for q′ and Eq. (11a) for u′α into Eq. (2b) yields the following modal182

growth rate formula:183

ωI =∑α

−<[UαQ∗]

|Q|2dq

dr≡∑α

ωIα(r). (12)184

Each partial growth rate ωIα corresponds to one-half the local rate of change of 〈(q′)2〉185

resulting from the radial advection of the tracer q by the velocity field ascribed to ζ ′ and186

σ′ in region α of the normal mode. The value of ωIα varies with r but not with z, owing187

to the barotropic structure of the unperturbed vortex. Note that the value of ωIα is the188

same regardless of whether q is PV or an arbitrary passive tracer, since the relation Q =189

−iU(dq/dr)/(ω − nΩ) is general.190

If the distributions of ζ ′ and σ′ in region α fully and exclusively constituted those of a191

particular dynamical element of the modal perturbation— such as a vortex Rossby wave,192

critical layer disturbance or inertia-gravity wave —one might reasonably connect ωIα to the193

destabilizing (or stabilizing) influence of that element. On the other hand, one should bear194

in mind that the foregoing condition can be satisfied under normal circumstances only in195

some approximate sense, regardless of how carefully the vortex is partitioned. For example,196

10

the velocity fields of a vortex Rossby wave are traditionally obtained by inverting a localized197

PV or pseudo-PV perturbation according to specific balance conditions. As such, the wave198

distributions of ζ ′ and σ′ generally extend beyond the localized PV or pseudo-PV anomaly,199

into regions that are formally ascribed to other perturbation elements. While the far-200

reaching extensions of vorticity and divergence may be weak, they could be relevant to201

slow instabilities.202

One might also worry about the appropriateness of instantaneous attribution. Although203

mathematically valid, the idea of attributing part of the velocity field within the vortex to204

simultaneous sources in the outer radiation field may seem physically questionable, owing to205

the finite propagation speed of inertia-gravity waves. That being said, the actual information206

contained in this partial velocity field amounts to the normal component of u′ − u′vtx at the207

boundary between the vortex (α = vtx) and the radiation zone (α = rad). Such is evident208

by noting that inside the vortex, u′rad = ∇hϑ, in which ∇2hϑ = 0 subject to ∂ϑ/∂r =209

u′ − u′vtx at r = R. Here we have let ∇h denote the horizontal gradient operator. Based210

on the preceding consideration, one might view u′rad within the vortex as a flow-adjustment211

connected to inertia-gravity wave emission at the boundary, without envisioning external212

sources and sinks. In practice, u′rad may be readily obtained from the difference u′ − u′vtx213

without any additional computation. For this study, the preceding expression is cross-checked214

by calculating u′rad with the appropriate Green function integral between R and a sufficiently215

large radius that ensures convergence within a very small fractional error.216

As a final remark, over the bulk of the vortex region, the dominant modes of instability217

considered herein are intrinsically slow relative to inertial oscillations [M16]; that is, the218

magnitude of ωR−nΩ is appreciably less than [(2Ω + f)(ζ + f)]1/2. The preceding condition219

suggests that the perturbation dynamics within the vortex is quasi-balanced [Shapiro and220

Montgomery 1993]. As such, attributing local partial velocity fields to instantaneous nonlo-221

cal sources within the vortex or on its boundary seems consistent (in a general sense) with222

normal practice and thinking.223

11

224

4. Illustrative Implementation of the Method225

226

For illustrative purposes, we consider the asymmetric normal modes of cyclonic vortices227

whose unperturbed relative vorticity distributions have the form228

ζ ≡ ζ0

1

1 + (r/rv)∆− β

1 + [r/(µrv)]∆

, (13)229

in which 0 < µ < 1, 0 < β < 1, ∆ 1, rv approximates the radius of maximum wind speed,230

and ζ0 is a positive scaling factor. The vorticity distribution defined by Eq. (13) possesses231

an off-center peak between µrv and rv, whose edges become square as ∆ → ∞. Increasing232

the dimensionless parameter β enhances the central vorticity deficit. Figure 1a shows the233

particular distribution with µ = 0.6, β = 0.8 and ∆ = 25, along with the corresponding234

angular velocity field Ω. The modal instabilities are completely controlled by the variables235

shaping ζ (µ, β, ∆) and the following two dimensionless parameters:236

Ro ≡ 2Ωv

fand Fr ≡ vv

Nk−1, (14)237

in which Ro is the Rossby number and Fr is a rotational Froude number based on the vertical238

wavenumber k of the disturbance. The v-subscripts on Ω and v indicate that the variables239

are evaluated at r = rv. The perturbation depicted in Figs. 1b and 1c is the fastest growing240

n = 2 normal mode of the vortex in Fig. 1a, with Ro = 100 and Fr = 2.6. It is equivalent to241

the normal mode appearing in Fig. 2 of M16.242

12

The fluid partitioning sketched in Fig. 2b can be summarized as follows:243

α ∈

iw : 0 ≤ r ≤ rc excluding icl;

icl : r−∗i ≤ r ≤ r+∗i;

ow : rc < r ≤ R excluding ocl;

ocl : r−∗o ≤ r ≤ r+∗o;

rad : r > R.

(15)244

Here we have introduced notations for the inner critical radius (r∗i) and the outer critical245

radius (r∗o) of the instability mode, which represent the two solutions of246

nΩ(r∗) = ωR. (16)247

Moreover, we have let r±∗ = r∗ ± δr∗, in which δr∗ ≡ c∣∣ωI/(ndΩ/dr)

∣∣r∗

is the nominal half-248

width of the linear critical layer [Schecter et al. 2000]. The constant c is taken to be 2 unless249

stated otherwise. The symbol rc denotes the nonzero finite radius at which dζ/dr = 0, or r+∗i250

if the latter is larger. The inner wave section (iw) is the central circle of radius rc, excluding251

the inner annular critical layer (icl). The outer wave section (ow) is the annulus between252

rc and the inner boundary radius R of the radiation zone (rad), excluding the outer critical253

layer (ocl). Although the inner and outer wave sections (iw and ow) may each contain two254

disconnected regions separated by a critical layer, the former reduces to a single disc of255

radius r−∗i when rc = r+∗i. The vortex region (vtx) comprises all sections of the fluid but the256

radiation zone, and therefore covers the entire interval of r between 0 and R.257

Separating the vortex region from the radiation zone at the outermost turning point R258

of the instability mode seems relatively uncontroversial. The rationale for further decom-259

position of the vortex region requires additional discussion. To begin with, each subsection260

of the vortex region contains a distinct extremum of the angular pseudomomentum density261

of the instability mode. The angular pseudomomentum density is a standard measure of262

13

local wave activity in systems with cylindrical geometry. Averaging over ϕ and z, the263

modal angular pseudomomentum density at any given time is proportional to the following264

function [SM04, M16]:265

L(r) ≡ LPV + Lvφ, (17a)266

in which267

LPV ≡ −|a|2r2|Q|2

2dq/drand Lvφ ≡ −|a|2k2r2< [V Φ∗] . (17b)268

Here, Q and q are the perturbation wavefunction and basic state distribution of PV, as269

opposed to a generic tracer. For all of the instability modes under consideration, LPV tends270

to dominate Lvφ for r < r+∗o. Figure 1b is essentially a plot of

∣∣LPV ∣∣1/2 cos(nϕ+ϕq +ϕa), in271

which ϕq (ϕa) is the phase of Q (a). It is seen that each vortex section defined above [Eq. (15)]272

contains a distinct peak of∣∣LPV ∣∣. It has been verified with various diagnostics that the273

peaks within the inner and outer wave sections of the vortex (near µrv and rv) correspond274

to counter-propagating vortex Rossby waves [M16, section 3b therein]. The peaks within275

the inner and outer critical layers (near r∗i and r∗o) are obviously generated by resonant276

stirring of PV. Similar structure is found in all of the instability modes examined in this277

note and in the more comprehensive study of M16. A caveat is that one or more of the278

modal elements (such as the outer critical layer disturbance) may be negligible.279

Figure 3a shows the radial variation of the two components of ωI attributed to radiation280

and internal vortex dynamics, for the instability mode appearing in Figs. 1b and 1c. The281

top graph shows ωIrad and ωIvtx, while the bottom graph shows the aforementioned partial282

growth rates multiplied by∣∣LPV ∣∣. Also shown are ωI and ωI

∣∣LPV ∣∣; whereas the former is a283

constant, peaks in the latter correspond to regions of maximal vortex Rossby wave activity284

and critical layer stirring. It is seen that ωIrad considerably exceeds ωIvtx in regions of peak285

vortex Rossby wave activity, whereas the opposite holds in the critical layers. In other words,286

the horizontal velocity field attributed to radiation is primarily responsible for the growth of287

q′ associated with the vortex Rossby waves, whereas the horizontal velocity field generated288

14

by ζ ′ and σ′ within the vortex primarily controls the growth of q′ in the critical layers.289

Figure 3b shows the radial variations of the four subcomponents of ωIvtx. The subcom-290

ponents paint a more complex picture of the instability mode. Cancellations between the291

subcomponents account for the smallness of ωIvtx where the vortex Rossby wave activity292

is concentrated. The inner and outer wave regions are seen to generate velocity fields that293

act to amplify q′ in each other but mostly damp q′ locally. The velocity field produced by294

sources in the inner critical layer hinders the growth of q′ in both vortex Rossby waves. The295

velocity field produced by sources in the outer critical layer adds slightly to the damping296

effort in the outer wave. The positive vortex contribution to the growth of q′ in the inner297

critical layer is due to the positive influence of u′iw exceeding the negative influence of u′ow.298

The growth of q′ in the outer critical layer is mostly due to the stirring induced by u′ow.299

Additional information on the nature of the instability can be obtained by splitting each300

partial growth rate ωIα into subparts associated with ζ ′ and σ′ individually. That is, let301

ωIα = ωIαζ + ωIασ, in which302

ωIαs ≡−<[UαsQ

∗]

|Q|2dq

dr(18a)303

and304

Uαs =

∫α

drr−inrGn(r, r)Z(r) s = ζ,

∫α

drr∂

∂rGn(r, r)D(r) s = σ.

(18b)305

Figure 3c shows the radial variations of ωIαζ and ωIασ for the instability mode at hand, with306

α ∈ vtx, rad. Unsurprisingly, it is found that ωIradσ ωIradζ . Less anticipated, one can307

see that ωIvtxσ has values comparable and opposite to those of ωIvtxζ in the regions of peak308

vortex Rossby wave activity.309

The preceding growth rate decomposition [Fig. 3] is found to exhibit only moderate310

sensitivity to variations of δr∗, rc and R. Clearly, variations of δr∗ and rc that are constrained311

to prevent regional overlap have no bearing on the values of ωIvtx or ωIrad. Variation of312

δr∗ from one-half to twice its standard value most notably coincides with a proportional313

15

amplification of the ratio of ωIicl to ωIiw in the neighborhood of µrv. Reduction of rc to r+∗i314

decreases the positive magnitude of ωIow by 37% (39%) at µrv (rv). The corresponding local315

changes to ωIiw are equal in absolute value but opposite in sign. Reduction of R to r+∗o most316

notably increases the positive value of ωIrad by 26% at rv, and commensurately intensifies317

the local negative value of ωIvtx.318

It is worth remarking that we have conducted a simple test to gain confidence that319

ωIrad ωIvtx implies the importance of radiation in driving the local growth of the PV320

perturbation in a tropical cyclone-like vortex. It is well known that a monotonic cyclone (β =321

0) would be stable in the absence of inertia-gravity waves [Montgomery and Shapiro 1995].322

Moreover, it is reasonably well established that the dominant mode of instability of a323

monotonic cyclone involves the positive feedback between a vortex Rossby wave at the edge324

of the potential vorticity core (r = rv) and inertia-gravity wave radiation [Ford 1994; SM04].325

We have verified that when β = 0, the condition ωIrad ωIvtx holds very well in the vicinity326

of rv for a number of vortices with Ro 1 and Fr <∼ 1. In each case considered, ∆ was made327

sufficiently large to prevent significant opposition to modal growth by PV stirring in the328

outer critical layer [SM04]. Note that the extremely opposite condition, ωIrad = 0, agreeably329

holds for all nondivergent barotropic (Fr, k = 0) instabilities. In this limit, ζ ′ and σ′ vanish330

outside the vortex, and the integral expression for Urad [Eq. (11b)] in the definition of ωIrad331

[Eq. (12)] is clearly zero.332

333

5. Comparison to Alternative Diagnostics334

335

The tracer based instability analysis offers a perspective on the importance of inertia-gravity336

wave radiation that may not fully agree with tentative assessments gleaned from simpler337

diagnostics. Discrepancies primarily occur when more than one mechanism has substantial338

impact on the amplification of a perturbation.339

Recently, Menelaou and coauthors [M16] provisionally assessed the importance of radia-340

16

tion by examining its contribution to the wave activity budget of a growing mode. The wave341

activity of region α was defined by342

Wα ≡∫α

drLe2ωI t. (19)343

For modes in which |Wow| >∼ |Wiw|, conservation of total wave activity was expressed in the344

form345

dWow

dt= −

∑α 6=ow

dWα

dt. (20a)346

Substituting Eq. (19) into the left-hand side of Eq. (20a) and dividing through by 2Wow347

yields348

ωI =∑α6=ow

ωIα, (20b)349

in which ωIα ≡ −(dWα/dt)/(2Wow). A large relative magnitude of ωIrad on the right-350

hand side of Eq. (20b) simply implies that amplification of the radiation field (possessing351

negative wave activity) has an important role in balancing the growth of positive outer vortex352

Rossby wave activity. One might tentatively infer from such a result that radiation has an353

important role in driving the instability, but rigorous justification of this conclusion generally354

requires supplemental analysis and reasoning. We note that in practice, the calculation of355

ωIrad is simplified by reducing the integral dWrad/dt to an equivalent algebraic expression356

proportional to the angular momentum flux at R [M16].357

Hodyss and Nolan [HN08] examined the radial distribution of the contribution358

Sr ≡ −r 〈u′v′〉 dΩ/dr (21)359

to the growth rate of kinetic energy in the asymmetric perturbation. They showed that360

Sr is concentrated beyond the edge radius (rv) of the vorticity distribution if the instabil-361

ity primarily involves the positive feedback of inertia-gravity wave radiation on an outer362

vortex Rossby wave that is responsible for its emission. By contrast, they found that Sr363

17

is concentrated inward of rv if the instability primarily involves the interaction of counter-364

propagating vortex Rossby waves. One might therefore speculate that the importance of365

radiation in driving the instability of a tropical cyclone-like vortex could be assessed simply366

by comparing the magnitudes of Sr inward and outward of rv.367

Figure 4 presents the three diagnostics at issue for three selected instability modes of a368

cyclonic vortex with µ = 0.8, β = 0.9, ∆ = 40 and Ro = 100. The top row corresponds to the369

dominant n = 2 instability when the Froude number Fr has a subcritical value of 0.8. As in370

M16, the term “subcritical” refers to the small-Fr parameter regime in which inertia-gravity371

wave radiation has minimal influence on the fastest growing wavenumber-n eigenmode of372

the linearized dynamical system. The middle row corresponds to the dominant n = 2 insta-373

bility when Fr has a strongly supercritical value of 6. The bottom row corresponds to the374

dominant n = 2 instability when Fr has a transitional value of 3. Assuming constant N , the375

Froude number may be viewed as a dimensionless vertical wavenumber or a dimensionless376

measure of vortex strength. Taking the former perspective with vv having a severe tropi-377

cal cyclone value of 65 m s−1, the three depicted modes of instability would have vertical378

quarter-wavelengths (π/2k) of (top) 12.8 km, (middle) 1.7 km and (bottom) 3.4 km. Here it is379

assumed that N is adequately approximated by a dry tropospheric value of 0.01 s−1; a reduc-380

tion of N due to moisture would increase the vertical lengthscale associated with each mode.381

The diagnostics under consideration offer a consistent picture of the subcritical instability382

mode. The binary growth rate partitioning advocated herein [Fig. 4a] suggests that inertia-383

gravity wave radiation is much less relevant to the amplification of q′ than sources of the384

velocity perturbation (ζ ′ and σ′) inside the vortex. The wave activity based growth rate385

partitioning of M16 [Fig. 4b] consistently suggests that radiation has minimal impact.2 The386

Sr profile [Fig. 4c] indicates that kinetic energy is transferred from the mean shear flow to387

the asymmetric perturbation primarily in the “eyewall” (µ < r/rv < 1). There is little388

2The negative values of ωIocl and ωIicl in Fig. 4b do not imply that both critical layer perturbationshinder the growth of q′ in the vicinity of the outer vortex Rossby wave; the value of ωIicl is found to bepositive at rv (not shown).

18

evidence of such transfer in the outer core (r/rv > 1), where under different circumstances389

enhancement of Sr might have reflected appreciable positive feedback from radiation.390

The opposite picture is found for the strongly supercritical instability mode. The tracer391

based instability analysis [Fig. 4d] reveals a dominant partial growth rate attributable to392

inertia-gravity wave radiation in all pertinent regions of the vortex, except the outer critical393

layer. The wave activity based growth rate decomposition [Fig. 4e] yields ωIrad ωIα for all394

α 6= rad. The distribution of Sr [Fig. 4f] is concentrated in the outer core, as in the principal395

radiation-driven instabilities of monotonic vortices.396

The transitional instability mode exemplifies how the three diagnostics under consider-397

ation can leave different impressions. The tracer based instability analysis [Fig. 4g] suggests398

that radiation is equally or more responsible for the amplification of q′ than sources of the399

velocity perturbation inside the vortex. A notable exception is in the inner critical layer,400

where the velocity field generated by vortex sources prevails. The wave activity based growth401

rate decomposition [Fig. 4h] consistently suggests that radiation is relevant to the instability.402

However, the relation ωIrad < ωIiw+ωIicl+ωIocl leaves the inconsistent overall impression that403

radiation is less important than internal vortex dynamics. The Sr profile [Fig. 4i] indicates404

that kinetic energy is transferred from the mean shear flow to the asymmetric perturbation405

in both the eyewall and the outer core of the vortex, with no obvious discrimination. It is406

unclear to the authors how one might confidently assess the relative importance of radiation407

to the instability from the information contained in Sr. In contrast to the tracer based408

analysis, an assessment based solely on the location of where Sr is peaked (r < rv) might409

encourage one to believe that internal vortex dynamics has the leading role in driving the410

instability. Alternatively, comparing the inner (r ≤ rv) and outer (r > rv) radial integrals of411

rSr yields an ambiguous inner-to-outer ratio of 1.2.412

Note that of the three diagnostics under consideration, only the tracer based analysis413

was expressly designed to isolate and quantify the relative importance of radiation in forcing414

the growth of a perturbation field within the vortex. There may be no rigorous justification415

19

for having presumed that one could find distinct patterns in the wave activity budget or the416

Sr-distribution to reliably convey the same information. A limited search for such patterns417

was deemed worthwhile, because variants of the aforementioned diagnostics are commonly418

examined and simpler to calculate. However, the preceding analysis of the transitional insta-419

bility mode suggests that relatively simple diagnostics may be inescapably ambiguous when420

more than one mechanism has appreciable influence on the growth of a perturbation.421

422

6. Summary423

424

This note has expounded a previously underdeveloped method for evaluating the relative425

importance of inertia-gravity wave radiation in driving the instability of a columnar vortex426

resembling a tropical cyclone. The procedure begins by dividing the fluid volume into vortex427

and radiation zones. The velocity perturbation is then decomposed into one part that is428

formally associated with sources (ζ ′ and σ′) inside the vortex and another part that is429

attributed to radiation. The importance of radiation is assessed by comparing the rates at430

which the two partial velocity fields act to amplify a tracer perturbation, denoted by the431

variable q′ and exemplified by the PV perturbation in the vortex core.432

As illustrated in section 4, the foregoing instability analysis can be readily extended to433

see how different sources of the velocity perturbation residing within the vortex individually434

contribute to the amplification of q′. Sources deemed relevant include those found in distinct435

critical layers and regions of enhanced vortex Rossby wave activity. In principle, an extended436

analysis can be beneficial for elucidating the true intricacy of a multimechanistic instability.437

438

Acknowledgments: The authors thank three anonymous reviewers for their constructive439

comments. This work was supported by the National Science Foundation under grant AGS-440

1250533. Additional support was provided by the Natural Sciences and Engineering Research441

Council of Canada and Hydro-Quebec through the IRC program.442

20

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24

-1 -0.5 0 0.5 1 x

1

0.5

0

-0.5

-1-1

-0.5

0

0.5

1

y

rvmrv

r*i

r*o

-3 -2 -1 0 1 2 x

-2

-1

0

1

2

y

-3

rc

(b) (c)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2r/rv

z

V

m = 0.6b = 0.8D = 25

(a)z

Figure 1: (a) Basic state relative vorticity (ζ) and angular velocity (Ω) distributions of a tropicalcyclone-like vortex whose shape parameters [Eq. (13)] are printed on the upper-right corner of thegraph. Both distributions are normalized to Ωv ≡ Ω(rv). Note that the unshown PV of the basicstate [q = (ζ+f)N2] closely resembles ζ. (b) The scaled PV perturbation q′r/|dq/dr|1/2 at arbitraryz of the fastest-growing n = 2 eigenmode, when the Rossby and Froude numbers [Eq. (14)] arerespectively given by Ro = 100 and Fr = 2.6. The color scale is normalized to the peak value ofthe plotted field. In the same units, the contour values are ±[0.04, 0.4, 0.9]. (c) The geopoten-tial perturbation φ′. Solid/dashed contours correspond to the following positive/negative values:±[0.06, 0.13, 0.21, 0.5, 0.75, 0.95] times the peak magnitude of the plotted field. The boundary ofthe yellow circle of radius R = 1.83 separates the vortex region from the radiation zone. All lengthsin all parts of this figure are in units of the core radius rv. The eigenfrequency of the mode depictedin (b) and (c) is ω = (1.16 + 0.06i)Ωv.

25

rad

vtx

vtx

a

z+,

s-(a) (b) (c)

R

z

,

Figure 2: (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 associatedwith the inner vortex Rossby wave (iw, dark gray), the inner critical layer (icl, white), the outervortex Rossby wave (ow, black) and the outer critical layer (ocl, light gray). (c) Sketch of thevelocity fields associated with localized positive relative vorticity (ζ ′+) and negative divergence (σ′−)perturbations in an arbitrary region (α) of the fluid.

26

-0.4

0

0.4

0.8

0.4 0.6 0.8 1 1.2 1.4r/rv

ωα |L

PV|

α = rad-ζvtx-ζ

rad+vtxrad-σ

vtx-σ

-0.5

0

0.5

1

0.4 0.6 0.8 1 1.2 1.4

α = iwowicloclvtx

ωα |L

PV|

r/rv

-0.04

0

0.04

0.08

0

0.2

0.4

0.6

0.8

1

0.4 0.6 0.8 1 1.2 1.4

ωIα

|LP

V|

r/rv

r*i

r*i

- + rc

r*o

r*o

- +

α = radvtx

rad+vtx

ωIα

/ Ω(r

v)

(a)

(b)

(c)ω

Iα |L

PV|

ωIα

|LP

V|

µ

µ

µ

Figure 3: Partial growth rates of the instability mode shown in Figs. 1b and 1c. (a) Top panel:binary decomposition of the total growth rate (dotted black) into one part associated with vorticityand divergence anomalies inside the vortex (blue) and another part attributed to radiation (red).Bottom panel: similar to top panel, but with the growth rates multiplied by

∣∣LPV ∣∣ and normalizedto the peak value of ωI

∣∣LPV ∣∣. (b) Similar to the bottom panel of (a) but for partial growth ratesattributed 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) Similar to the bottom panel of (a), but withthe partial growth rates attained from velocity sources in the vortex (blue) and in the radiationzone (red) each split into those attained from vorticity (ζ ′, solid) and divergence (σ′, dashed). Theirsum (dotted black) is shown for reference.

27

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

-r<

u'v

'>dΩ

/dr

r/rv

vtx rad

0

0.02

0.04

0.06

iw

sum

icl

ocl0

2

4

6

10

2 x

ω

Iα /

Ω(r

v)

^

-0.5

0

0.5

1

1.5

0.6 0.8 1 1.2 1.4

r*i

r*i

- + rc r

*or*o

- +,

r/rv

ωIα

|LP

V|

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4

ωIα

|LP

V|

r/rv

r*i

r*i

- + rc r

*or*o

- +

α = radvtxrad+vtx

,

(g) (h) (i)

0

0.02

0.04

0.06

vtx rad

0

6

iw

sum

iw + icl

ocl

4

^1

02 x

ω

Iα /

Ω(r

v)

2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

-r<

u'v

'>dΩ

/dr

r/rv

α = radvtxrad+vtx

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4r/rv

r*i

r*i

- + rc r

*or*o

- +,

ωIα

|LP

V|

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

vtx rad

iw

sum

ocl

icl-2

2

4

6

10

2 x

ω

Iα /

Ω(r

v)

^

0

8

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

-r<

u'v

'>dΩ

/dr

r/rv

α = radvtxrad+vtx

(d) (e) (f)

(a) (b) (c)

Figure 4: (a-c) Instability diagnostics for the fastest-growing n = 2 eigenmode when µ = 0.8,β = 0.9, ∆ = 40, Ro = 100 and Fr = 0.8. (a) Partial growth rates ωIα attributed to vorticity anddivergence anomalies inside the vortex (vtx, blue) and to radiation (rad, red). Each is multipliedby∣∣LPV ∣∣ and then normalized to the maximum of ωI

∣∣LPV ∣∣. Their scaled sum (dotted-black) isshown for reference. (b) The alternative partial growth rates ωIα of M16. Those associated withthe inner vortex Rossby wave (iw), inner critical layer (icl) and outer critical layer (ocl) are stackedin the blue column. The red column shows the formal contribution to ωI from radiation. (c) Theproduction rate of kinetic energy in the n = 2 perturbation associated with the radial shear ofΩ, normalized to its maximum value. (d-f) As in (a-c) but when Fr = 6.0. (g-i) As in (a-c) butwhen Fr = 3.0. 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 rv. The eigenfrequencies of the top, middleand bottom modes are respectively ω = 1.59 + 0.06i, 1.35 + 0.07i and 1.15 + 0.10i in units of Ωv.

28