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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: [email protected]
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].
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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