## 1. Introduction

### a. Background

*n*is a nonnegative integer that increases with the characteristic radial wavenumber,

*c*is the ambient sound speed, and

_{o}*a*is the radius of the vortex core. The derivation of Eq. (1) reasonably assumes that the squared Mach number of the vortex is much less than unity.

Substituting the typical values *c _{o}* = 330 m s

^{−1}and

*a*= 100 m into Eq. (1) yields

*f*

_{0}= 2.1 Hz. This value of

*f*

_{0}falls in the 0.5–10-Hz frequency band where tornadic thunderstorms are said to emit distinct infrasound (Bedard 2005; Bedard et al. 2004). It is therefore tempting to conclude that the infrasound comes from axisymmetric vortex oscillations (ibid.).

### b. Objection and objectives

Equation (1) predicts that the peak frequencies of tornado infrasound (in the neighborhood of 1 Hz) are inversely proportional to *a*, but have no dependence on wind speed. Both predictions are reportedly consistent with observations of infrasound emitted from the vicinity of a tornado.^{1} This does not mean that A66 provides the correct explanation. Appealing to basic intuition, it seems odd that the frequency spectrum for any class of modes attributable to the vortex does not change (provided *a* is fixed) as the wind speed goes to zero.

One purpose of this paper is to point out a subtle misstep in the derivation of Eq. (1) that casts doubt upon its fundamental credibility. In addition, this paper will show that the principal axisymmetric oscillations of a subsonic Rankine vortex (axisymmetric Kelvin modes) do not emit acoustic radiation. Finally, this paper will present numerical evidence that the axisymmetric radiation emitted after a generic disturbance of a Rankine vortex carries no significant vortex signature.

### c. Notational conventions

Insofar as possible, our notational conventions abide by contemporary norms. The variables *r*, *φ*, and *z* denote radius, azimuth, and height, respectively, in a cylindrical coordinate system centered on the vortex (see Fig. 1). The variable *t* denotes time. Overbars and primes are used to represent the basic state and perturbation of a generic fluid variable *G*, such that *u*, the azimuthal velocity *υ*, the axial (vertical) velocity *w*, the pressure *p*, and the potential temperature *θ*.^{2}

## 2. Reexamination of A66

### a. Matching conditions at the boundary of the vortex core

Following the derivation of Eq. (1) in A66, one arrives at a step where each component of the velocity perturbation is forced to be continuous at the core radius of the Rankine tornado [Eq. (31) of A66]. This continuity constraint may seem reasonable, but imposing it on the azimuthal velocity perturbation (*υ*′) is a mathematically subtle issue.

*φ*-derivatives, the prognostic equation for

*υ*′ in A66 takes the following linearized form:

*u*′ and

*υ*′. Because

*r*=

*a*, the consistency requirement results in perturbations that are characterized by the condition

*u*′ = 0 at

*r*=

*a*[see Eqs. (25) and (34) of A66]. Imposing such a node on every axisymmetric mode of oscillation seems unphysical; it suggests that the boundary of the vortex core suppresses radial flow.

The present author sees no valid reason to reject a *formal* discontinuity of *υ*′ across *r* = *a* (cf. Arendt et al. 1997). Physically, the discontinuity represents a rapid change across a thin transition layer. On the other hand, appendix A verifies that the radial eigenfunctions of *u*′ and *p*′ must be continuous at *r* = *a* for any normal mode of oscillation. Continuity of *u*′ and *p*′ at *r* = *a* is sufficient to derive an equation for the eigenfrequencies of all normal modes, after appropriate physical boundary conditions are applied at *r* = 0 and infinity (Kelvin 1880; Arendt et al. 1997). It is here claimed that imposing continuity of *υ*′ at *r* = *a* is not merely superfluous, but incorrect.

The reader may question how the extra continuity constraint at *r* = *a* did not lead Abdullah to an overdetermined problem. A solution was possible because the outer boundary condition was unspecified. The theory of A66 overlooks the requirement that acoustic waves created by the vortex must propagate outward as *r* tends toward infinity. Section 2b elaborates upon this oversight.

### b. Radiation condition

*J*and

_{ν}*Y*are Bessel functions of the first and second kinds,

_{ν}*ω*≡ 2

*πf*is the angular frequency of the mode,

*k*is (essentially) the vertical wavenumber of the mode, and

_{z}*Z*is the vertical structure function of the mode. The coefficients

*A*

_{1}and

*A*

_{2}are constants, and

*ω*is supposed to be sufficiently large to ensure that

*r*→ ∞, the Bessel functions in Eq. (5) become trigonometric functions (Abramowitz and Stegun 1972, hereafter AS72); specifically, it can be shown that

*A*

_{1}and

*A*

_{2}in Eq. (6) contain inward-propagating waves. This suggests that the A66 modes incorporate unphysical acoustic sources (or reflective walls) outside the vortex.

## 3. Can axisymmetric radiation reveal vortex features?

^{3}He showed that the vortex possesses a discrete set of axisymmetric eigenmodes of the following form:

*A*is an arbitrary constant,

*k*is positive by convention,

_{z}*r*

_{<}(

*r*

_{>}) is the lesser (greater) of the radial coordinate

*r*and the core radius

*a*. As usual,

*K*is a modified Bessel function of the second kind, and

_{ν}*c*.

*c*. denotes the complex conjugate of the term to its left. The eigenfrequencies are given by the following well-known dispersion relation:

*ω*}, in which

_{n}*n*∈ {1, 2, 3, … , ∞}. By construction, the value of

*ξ*increases with

*n*, so that

*n*may be viewed as a surrogate for the radial wavenumber. The solution curves for

*ω*(

_{n}*k*) fall on and below the

_{z}*n*= 1 branch that is shown in Fig. 2.

*M*≡

*Ωa*/

*c*is much less than unity. In this parameter regime, Eq. (9) should remain a valid approximation of the Kelvin mode dispersion relation. On physical grounds, a Kelvin mode can emit an acoustic wave into the environment only if

_{o}*k*.

_{z}*ak*≪ 1, the dispersion relation for the fastest (

_{z}*n*= 1) branch of the Kelvin modes reduces to

*j*

_{0,1}= 2.404 83 is the smallest zero of

*J*

_{0}. It follows that

*ω*

_{1}/

*ω*~ 2

_{a}*M*/

*j*

_{0,1}≪ 1 as

*ak*→ 0. The dispersion diagram further illustrates that

_{z}*ω*

_{1}asymptotes to the inertial frequency 2Ω as

*ak*→ ∞. More to the point,

_{z}*dω*

_{1}/

*dk*decays toward zero with increasing

_{z}*k*. Consequently, the axisymmetric Kelvin modes remain slower than acoustic waves for all

_{z}*k*, and cannot excite acoustic radiation.

_{z}It is reasonable to speculate that compressibility introduces faster oscillations of the vortex core that readily generate acoustic radiation (cf. A66). On the other hand, the pertinent spectrum of fast, axisymmetric eigenmodes could merely represent free-space sound waves,^{4} modified very slightly by the vortex. In the latter scenario, the character of the radiation would depend much more on the initial perturbation, or forcing, than on the structure of the vortex. Indeed, a brief numerical survey will provide evidence of the following:

The axisymmetric component of acoustic radiation emitted by a columnar vortex does not carry a robust vortex signature.

## 4. Numerical investigation of axisymmetric radiation

### a. Experimental setup

*θ*is the constant equilibrium potential temperature. By definition,

_{o}*p*is the ambient pressure,

_{o}*R*is the gas constant,

*c*is the isobaric specific heat, and

_{p}*ρ*is the local mass density.

^{5}The unperturbed radial velocity

*c*is the isovolumic specific heat, and (as before)

_{υ}*F*denotes a forcing function of the variable

_{G}The perturbation equations are solved numerically with low-order finite differencing in *r*, and fourth-order Runge–Kutta steps in *t*. The computational grid has 1-m increments and extends to *r* = 10 km. A linear sponge ring absorbs outward-propagating acoustic waves for *r* ≥ 8 km. The values of *θ _{o}* and

^{−1}, respectively.

*Ωa*is approximately the maximum tangential wind speed, and

*a*is approximately the radius of maximum wind. The forcing is either an extended thermal impulse, in which

*F*= 0 and

_{υ}*F*= 0 and

_{θ}*τ*(of

*F*) and radial length scale

_{θ}*b*are adjustable parameters. The dimensionless perturbation strength

*ϵ*does not change from one experiment to another. The specific value of

*ϵ*is arbitrary for the purpose of our discussion. The factor

*η*= 1 s

^{−1}is included in

*F*for dimensional consistency.

_{θ}Table 1 lists the key parameters of each numerical experiment. A few of the experiments involve perturbations of free space. These control experiments are labeled FS, followed by a lowercase letter. The remaining experiments involve perturbations of a modest vortex (*a* = 100 m, *M* = *Ωa*/*c _{o}* = 0.14) or a formidable vortex (

*a*= 300 m,

*M*= 0.43). The modest and formidable vortices are named V1 and V2, and the experiments are labeled accordingly.

Key parameters of the numerical experiments.

### b. Disturbances with moderate vertical wavelengths

Figure 3 shows the power spectra of *r* = 0) and in the acoustic radiation zone (*r* = 2.3 km) for experiments FSa, V1a, V1d, and V2a. In each case, the vertical wavelength of the perturbation (*λ _{z}* ≡ 2

*π*/

*k*) is 100 m. The power spectra are calculated after applying Hann windows to 12-min time series, starting from

_{z}*t*= 0. Similar results have been verified for 3-min time series.

The control experiment (FSa) involves a 1-s heat pulse applied to a resting atmosphere. Both power spectra in FSa have solitary peaks near *f* = *c _{o}*/

*λ*=

_{z}*ω*/2

_{a}*π*. This minimal acoustic frequency is characteristic of sound waves with

*λ*≪

_{z}*λ*, in which

_{h}*λ*is the horizontal wavelength. Such waves linger in the computational domain because of their infinite vertical extent and small horizontal group velocity,

_{h}Unlike the FSa experiment, the central perturbation of V1 consistently has strong spectral peaks for *f* ≪ *c _{o}*/

*λ*. The low-frequency peaks are prominent whether

_{z}*F*(in V1a) or

_{θ}*F*(in V1d) generates the disturbance. Figure 4 verifies that the low-frequency peaks correspond to the Kelvin modes of an ideal Rankine vortex with the same parameters as V1 [cf. Eq. (9)]. As explained previously, axisymmetric Kelvin modes do not radiate; consequently, their spectral signatures are not seen in the radiation zone. In fact, the power spectrum of the outer acoustic perturbation in experiment V1a has the same form as its counterpart in experiment FSa.

_{υ}The behavior of V2 is qualitatively similar to that of V1, as illustrated by the two experiments (V1a and V2a) in which a 1-s heat pulse generates the disturbance. In both cases, the heat pulse excites nonacoustic Kelvin modes and long-lived acoustic oscillations characterized by *λ _{z}* ≪

*λ*. Although V2 has thrice the radius of V1, the spectral peak of the acoustic perturbation in experiment V2a does not budge from

_{h}*f*=

*c*/

_{o}*λ*. In contrast to the prediction of A66, there is no clear evidence of a significant acoustic peak in which the central frequency varies as 1/

_{z}*a*.

There is one notable difference between experiments V1a and V2a. Comparison of Figs. 3b and 3d suggests that a 1-s heat pulse excites the Kelvin modes of V1 more efficiently than those of V2. However, the excitation of Kelvin modes depends on the parameters *ak _{z}* and

*a*/

*b*. Both of these parameters differ by a factor of 3 between the two experiments under consideration.

### c. Disturbances with infinite vertical wavelengths

Figure 5 displays time series of *Π*′ in the acoustic radiation zone, for disturbances with infinite vertical wavelengths. Figures 5a and 5b show infrasonic emissions generated by distinct thermal forcing functions. Both plots suggest that V2 emits slightly weaker infrasound than V1 in response to *F _{θ}*, but emissions from neither vortex differ significantly from the infrasound generated by

*F*in free space. Figure 5c shows infrasound generated by an angular impulse

_{θ}*F*. If the angular impulse scales in size and strength with the vortex, then the infrasonic emission reflects the size and strength of the vortex from which it comes. However, with the same angular impulse, V1 and V2 emit the same waves.

_{υ}^{6}

### d. Primary inference

The author concludes that the form of axisymmetric radiation is primarily determined by the forcing function, not the characteristics of the vortex. Therefore, in general, the axisymmetric component of vortex infrasound cannot be used to estimate vortex size or strength with a high degree of confidence.

## 5. Conclusions

In contrast to the speculations of A66, it has been shown that axisymmetric radiation does not carry a robust signature of the vortex core. The axisymmetric Kelvin modes are undamped, but nonradiative. Moreover, the infrasound of an axisymmetric disturbance is shaped by the forcing that creates it. These conclusions were derived from an investigation of Rankine vortices, but seem more general. The author has found similar results for other centrifugally stable vortices with either monotonic or nonmonotonic radial distributions of axial vorticity. One could hypothesize that growing axisymmetric perturbations of centrifugally unstable vortices are exceptional, and generate distinct infrasound. However, the considerations of appendix C suggest that if an axisymmetric vortex mode grows with time, it must be nonradiative.

There is no obvious reason (to the author) why the conclusions of this paper should not extend to helical vortices with nonzero *c _{o}k_{z}*, if the axial Mach number is much less than unity. In other words, a typical AKH mode does not satisfy the radiation condition given by Eq. (10).

On the other hand, it is well-known that *nonaxisymmetric* Kelvin modes and shear-flow instabilities can readily generate infrasound (Powell 1964; Broadbent 1984; Sozou 1987; Howe 2003; Roberts 2003). The frequency of such infrasound is proportional to Ω, and therefore carries some information about the vortex. For an ordinary tornado, Ω/2*π* is of order 0.1 Hz. Theoretical studies suggest that diabatic processes in cloud turbulence can be strong sources of infrasound in the very same 0.1-Hz frequency regime (Akhalkatsi and Gogoberidze 2009, 2011). Therefore, the infrasound of nonaxisymmetric Kelvin modes and shear-flow instabilities may not be discernible under usual circumstances. This conclusion is consistent with earlier field studies that found no clear relationship between severe weather infrasound and tornadoes at frequencies below 0.2 Hz (Bowman and Bedard 1971; Georges and Greene 1975). Nevertheless, the provisional simulations of Schecter et al. (2008) suggest that discernible signals generated by nonaxisymmetric Kelvin modes might be possible under exceptional circumstances, if the wind speed of the tornado greatly exceeds 50 m s^{−1}.

On empirical grounds, it seems reasonable to maintain that ordinary tornadoes cause abnormally strong levels of 0.5–10-Hz infrasound (Bedard 2005). That being said, a convincing theoretical explanation remains absent. This paper exposed critical deficiencies of the A66 theory, which has received considerable attention in the literature. A more successful theory of tornado infrasound may require consideration of noncolumnar structure, diabatic cloud processes (cf. Akhalkatsi and Gogoberidze 2009, 2011; Schecter and Nicholls 2010; Schecter 2011), or even electrodynamics (cf. Schmitter 2010). Suffice it to say, further investigation is necessary.

## Acknowledgments

This study was supported by NSF Grant AGS-0832320.

## APPENDIX A

### Eigenfunction Matching Conditions at the Boundary of the Vortex Core

*ω*are generally complex, but the vertical wavenumber

*k*is real.

_{z}*r*yields the following:

*r*. Therefore, the first matching condition at the radius

*a*of the vortex core is

*σ*. For the latter case, it is a jump formula for the radial derivative of Φ across a vanishingly small transition layer.

## APPENDIX B

### Asymptotic Solutions for the Largest Eigenfrequency of Axisymmetric Kelvin Modes

*ak*≪ 1 and

_{z}*ω*≪ 2Ω . In this parameter regime, the dispersion relation for axisymmetric Kelvin modes [Eq. (9)] reduces to

*K*(

_{ν}*ak*) has been replaced with the leading-order term in its asymptotic expansion for small

_{z}*ak*(AS72). Given that (

_{z}*ak*)

_{z}^{2}ln(

*ak*) ≪ 1 and

_{z}*y*/

*J*

_{1}(

*y*) is of order unity or greater, the smallest value of

*y*(corresponding to the largest value of

*ω*) that can satisfy Eq. (B1) is near the first root of

*J*

_{0}. In other words, min[

*y*] ≈

*j*

_{0,1}, which is an alternative expression of Eq. (11).

*ak*→ ∞. In approaching this limit, the Kelvin-mode dispersion relation [Eq. (9)] reduces to

_{z}*K*(

_{ν}*ak*) has been replaced with the leading-order term in its asymptotic expansion for large

_{z}*ak*(AS72). Given that (

_{z}*ak*)

_{z}^{−1}≪ 1, the smallest value of

*y*(corresponding to the largest value of

*ω*) that can satisfy Eq. (B3) is close to the first nonzero root of

*J*

_{1}. In other words, min[

*y*] →

*j*

_{1,1}= 3.831 71, or equivalently,

## APPENDIX C

### Nonexistence of Unstable, Axisymmetric, Radiative Vortex Modes

*r*= 0 to a distant radius

*r*, where the vortex velocity field is negligible, yields the following:

_{b}*independently*represent a physical perturbation. The imaginary part of Eq. (C1) can be written as

*ω*(

_{R}*ω*) is the real (imaginary) part of

_{I}*ω*,

*r*→ ∞.

_{b}Suppose that the eigenmode corresponds to an oscillation of the vortex core that emits outward-propagating acoustic radiation. If the vortex oscillation (the acoustic source) grows exponentially with time, causality requires that the radiation field (the outer part of *r*^{1/2}Φ) decays exponentially with increasing *r*. In other words, a radiative vortex mode must be bounded if *ω _{I}* > 0.

Setting *ω _{I}ω_{R}* = 0. It follows that bounded, axisymmetric eigenmodes must have purely real or purely imaginary eigenfrequencies. But a

*radiative*vortex mode must have nonzero

*ω*. One is led to conclude that unstable (

_{R}*ω*> 0), axisymmetric, radiative vortex modes cannot exist.

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^{1}

Supporting evidence for inverse proportionality between frequency and *a* can be found in Bedard (2005) and Bedard et al. (2004). Insensitivity of frequency (and amplitude) to wind speed was reported to the author by an anonymous reviewer for *Mon. Wea. Rev*.

^{2}

In contrast to our notation, A66 uses the letters *u* and *υ* to represent azimuthal and radial velocities, respectively.

^{3}

More recent discussions of perturbation theory for incompressible columnar vortices can be found in Saffman (1992), Arendt et al. (1997), and Fabre et al. (2006).

^{4}

In this paper, “free space” refers to an unbounded atmosphere without a vortex.

^{5}

In atmospheric modeling studies, *Π* and *θ* are traditional substitutes for pressure and entropy (e.g., Klemp and Wilhelmson 1978).

^{6}

Note that the wave amplitudes are the same, because the cores of V1 and V2 have equivalent angular rotation frequencies. In linear theory [Eqs. (15)], the amplitude of the acoustic emission created by any angular impulse *F _{υ}* decays to zero with Ω. By contrast, several tests have shown negligible sensitivity of the emitted wave form (frequency) to variation of Ω in the range 0.2–2 s

^{−1}, with

*a*and

*F*held unchanged.

_{υ}