### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. General formalism
- 3. Spectral decomposition
- 4. Infrasound correlation with tornadoes
- 5. Conclusions
- Acknowledgements
- References

We consider the generation of acoustic waves by turbulent convection and perform spectral analysis of a monopole source of sound related to the heat production by condensation of moisture. A quantitative explanation of the correlation between intensity of infrasound generated by supercell storms and later tornado formation is given. It is shown that low lifting condensation level (LCL) and high values of convective available potential energy (CAPE), which are known to favour significant tornadoes, also lead to a strong enhancement of a supercell's low-frequency acoustic radiation. Copyright © 2011 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. General formalism
- 3. Spectral decomposition
- 4. Infrasound correlation with tornadoes
- 5. Conclusions
- Acknowledgements
- References

Recent observations of infrasound originating from regions of severe weather show that infrasound dominant frequency occurs in a passband from 0.5 to 2.5 Hz, with peak frequencies between 0.5 and 1 Hz (e.g. Bedard, 2005; Bedard *et al.*, 2004a). Analysing acoustic radiation from severe thunderstorms, Bedard (2004a) concluded that radiation of infrasound in this passband is not a natural consequence of all severe weather. Infrasound of a tornadic thunderstorm is much stronger than infrasound of a non-severe weather system and, therefore, mesocyclones or tornadoes may be a primary mechanism for infrasound production in this frequency range (Szoke *et al.*, 2004).

Detection of infrasound appears to have significant potential for improving tornado forecasting. The acoustic power radiated by a strong convective storm system could be as high as 10^{7} W (Georges, 1988) and infrasound below 1 Hz can travel for distances of thousands of kilometres from a source without significant absorption.

Over the years, several potential sound generation mechanisms were compared with measured characteristics of infrasound (Georges and Greene, 1975; Georges, 1988; Bedard and Georges, 2000; Beasley *et al.*, 1976). Such mechanisms include release of latent heat, dipole radiators, boundary layer turbulence, lightning, electrostatic sources and vortex sound (radial vibrations and the co-rotation of suction vortices). Georges (1976) eliminated many sources as likely candidates and concluded that vortex sound was the most likely model. Bedard (2005) also found that the radial vibration model (Abdullah, 1966) is most consistent with infrasonic data. This model predicts that the fundamental frequency of radial vibration will be inversely proportional to the core radius. A radius of about 200 m will produce a frequency of 1 Hz. Schecter *et al.* (2008) performed a numerical simulation of the adiabatic generation of infrasound by tornadoes and also simulated the infrasound radiated from a single-cell non-tornadic thunderstorm in a shear-free environment, using the Regional Atmospheric Modeling System (RAMS). In the latter simulation, dominant infrasound in the 0.1 to 10 Hz frequency band (so-called ‘thunderstorm noise’) appeared to radiate from the vicinity of the melting level, where diabatic processes involving hail were active. They found that 3D Rossby waves of tornado-like vortices can generate stronger 0.1 to 10 Hz infrasound at or above the simulated non-tornadic thunderstorm noise if the maximum wind speed of the vortex exceeds a modest threshold.

Georges and Green (1975) noted that infrasound often precedes an observed tornado by upto an hour. Many other case-studies of infrasound associated with tornadoes and tornadic storms (Bedard *et al.*, 2004a; Bedard *et al.*, 2004b) are consistent with this result. Bedard *et al.* (2004b) analysed all significant infrasonic signals data collected in 2003 during the continuous Infrasonic Network (ISNeT) operation. Summarizing differences between predicted acoustic signal arrival times from reported tornadoes and start times of infrasonic detection, they concluded that infrasound is usually produced substantially before (0.5 to 1 h) reports of tornadoes and that there could be other sound generation processes active not related to tornadic vortices. Bedard *et al.* (2004b) also indicated the difference between the infrasonic bearing sectors and the vortex location and concluded that, although the storm environment wind and temperature gradient could be responsible for bearing deviations, there remains the possibility that another storm feature could radiate infrasound from another location within the storm (Bedard, 2004).

Broad and smooth spectra of observed infrasound radiation indicates that turbulence is a promising source of the radiation. Acoustic radiation from turbulent convection was studied by Akhalkatsi and Gogoberidze (2009) taking into account effects of stratification, temperature fluctuations and moisture of air, using Lighthill's acoustic analogy. It was shown, for typical parameters of strong convective storms, that infrasound radiation should be dominated by a monopole acoustic source related to the moisture of the air. The total power of this source is two orders higher than thermo-acoustic and Lighthill's quadrupole radiation power, being of order 10^{7} W, in qualitative agreement with observations of strong convective storms (Bowman and Bedard, 1971; Georges, 1973; Georges and Greene, 1975; Bedard and Georges, 2000).

This paper represents an extension of the study performed by Akhalkatsi and Gogoberidze (2009), developing the earlier study in two directions. Firstly, we perform detailed spectral analysis of a monopole source related to heat production during the condensation of moisture. Assuming homogeneous and stationary turbulence we calculate the spectrum of acoustic radiation. Secondly, we perform detailed analysis of the sound generated by a monopole source as well as infrasound observations and present a qualitative explanation of the observed high correlation between intensity of low-frequency infrasound signals from supercell storms and the probability of later tornado formation. We show that acoustic power of a monopole source related to the moisture of the air strongly depends on the same parameters that are the most promising in discriminating between non-tornadic and tornadic supercells according to a recent study of tornadogenesis (Markowsky and Richardson, 2009). In particular, low lifting condensation level (LCL) is known to favour significant tornadoes (Rasmussen and Blanchard, 1998; Thompson *et al.*, 2003). On the other hand, low LCL height means that air in the updraught is saturated at lower heights and consequently has a higher temperature at the level of saturation. We show that the increase of temperature causes rapid enhancement of monopole acoustic power related to heat production during condensation of moisture.

Another widely used supercell and tornado forecasting parameter is supercell environmental convective available potential energy (CAPE). It is known that high values of CAPE (especially occurring closer to the surface) assist tornado formation (Weisman and Klemp, 1982; Rotunno and Klemp, 1982; Rasmussen and Blanchard, 1998; Rasmussen, 2003; Thompson *et al.*, 2003). Furthermore, high values of CAPE means high updraught velocity and therefore the increased r.m.s. of turbulent velocities, which results in a strong enhancement of total acoustic power.

The paper is organized as follows. General formalism is presented in section 2. Spectral analysis of acoustic radiation is performed in section 3. Correlation between intensity of infrasound radiation by supercell storm and probability of tornado formation is discussed in section 4. Conclusions are given in section 5.

### 2. General formalism

- Top of page
- Abstract
- 1. Introduction
- 2. General formalism
- 3. Spectral decomposition
- 4. Infrasound correlation with tornadoes
- 5. Conclusions
- Acknowledgements
- References

The dynamics of convective motion of moist air is governed by the continuity, Euler, heat, humidity and ideal gas state equations:

- (1)

- (2)

- (3)

- (4)

where **v***,ρ*, *s* and *p* are velocity, density, entropy and pressure, respectively; *D/Dt* ≡ *∂/∂t* + **v** · **∇** is the Lagrangian time derivative; *L*_{v} is the latent heat of condensation and *q* is the humidity mixing ratio

- (5)

Here *ρ*_{v} is the mass of water vapour in a unit volume, Φ is the gravitational potential energy per unit mass, **∇**Φ ≡ −**g**, is the ratio of molecular masses of water and air, *a* = 0.608 and *R* is the universal gas constant.

In the set of Eqs. (1)–(4), diffusion and viscosity effects are neglected due to the fact that they have a negligible influence on low-frequency acoustic wave generation and propagation.

Using the standard procedure of Lighthill's acoustic analogy (Goldstein, 2002), manipulation of Eqs. (1)–(4) yields the equation governing sound generation by turbulence for a moist atmosphere in terms of the total enthalpy (Akhalkatsi and Gogoberidze, 2009)

- (6)

where *γ* ≡ *c*_{p}*/c*_{v} is the ratio of specific heats,

- (7)

is the sound velocity and enthalpy *B* is determined by the generalized Bernoulli theorem

- (8)

where *E* is internal energy. *B* is constant in the steady irrotational flow and, at large distances from acoustic sources, perturbations of *B* represent acoustic waves (Howe, 2001).

To make further mathematical analysis more tractable, we will make standard simplifying assumptions: for low-Mach-number flow, all convective derivatives in Eq. (6) can be replaced by time derivatives *∂/∂t* (Goldstein, 1976); for the acoustic pressure in the far field, we can use

- (14)

we neglect nonlinear effects of acoustic wave propagation and scattering of sound by vorticity; we also neglect the influence of stratification on the acoustic wave generation process and consider background thermodynamic parameters in Eq. (6) as constants. The later assumption is valid if the wavelength of generated acoustic waves do not exceed the stratification length-scale (Stein, 1967):

- (15)

The first three terms on the right-hand side of Eq. (16) represent well-known sources of sound: the first term represents Lighthill's quadrupole source (Lighthill, 1952); the second term is a dipole source related to temperature fluctuations (Goldstein, 1976); *S*_{γ} is a monopole source related to variability of adiabatic index and usually has negligible acoustic output (Howe, 2001). As was shown in Akhalkatsi and Gogoberidze (2009), in the case of saturated moist air turbulence there exist two additional monopole sources of sound, *S*_{q} and *S*_{m}, related to non-stationary heat and mass production during condensation of moisture, respectively.

*S*_{q} and *S*_{m} produce monopole radiation and physically have the following nature: suppose there exist two saturated air parcels of unit mass with different temperatures *T*_{1} and *T*_{2} and water masses *m*_{v}(*T*_{1}) and *m*_{v}(*T*_{2}). Mixing of these parcels leads to the condensation of water due to the fact that

- (22)

Condensation of water leads to two effects important for sound generation: production of heat and decrease of gas mass. Both of these effects are known to produce monopole radiation (Goldstein, 1976; Howe, 2001). Consequently, turbulent mixing of saturated air with different temperatures will lead not only to dipole thermo-acoustical radiation, but also to monopole radiation.

As was shown by Akhalkatsi and Gogoberidze (2009), for typical parameters of supercell storms, acoustic radiation is dominated by a monopole source related to the heat production *S*_{q}. The total acoustic power can be estimated as

- (23)

where Δ*q* is the r.m.s. of humidity mixing ratio perturbations; *l* is the length-scale of energy-containing turbulent eddies; is the turbulent Mach number and *F*_{1} is the volume occupied by saturated moist air turbulence. For typical parameters, *S*_{q} is much greater than other sources of sound and two orders of magnitude stronger than Lighthill's quadrupole source.

### 3. Spectral decomposition

- Top of page
- Abstract
- 1. Introduction
- 2. General formalism
- 3. Spectral decomposition
- 4. Infrasound correlation with tornadoes
- 5. Conclusions
- Acknowledgements
- References

In this section we study the spectrum of acoustic radiation related to *S*_{q}. Dropping all other source terms, Eq. (16) reduces to the inhomogeneous wave equation

- (24)

Using standard methods for spectral analysis of the inhomogeneous wave equation (Goldstein, 1976; Howe, 2001; Gogoberidze *et al*., 2007), after a long but straightforward calculation, for the spectrum of temperature fluctuations *I*(**x***,ω*) we obtain

- (25)

where *H*(**x**′,**k***,ω*) is a spectral tensor of temperature fluctuations and represents a Fourier transform of a two-point time-delayed fourth-order correlation function of temperature fluctuation and *q*_{s} is the humidity mixing ratio of saturated air.

Equation (25) allows us to calculate the spectrum of sound radiated by a monopole source related to the moisture, if statistical properties of the turbulent source can be determined. For our calculations we consider the von Kármán model of stationary and homogeneous turbulence, which is given by (Hinze, 1975)

- (26)

where 2*π/k*_{d} is the dissipation length-scale, *C*_{K} is the Kolmogorov constant, *ε* is energy dissipation rate and *k*_{0} is characteristic wave number of energy containing eddies.

The von Kármán spectrum reduces to the Kolmogorov spectrum in the inertial interval (*k*), but also satisfactorily describes the energy spectrum in the energy-containing interval, which is a dominant contributor to acoustic radiation.

As is known (Monin and Yaglom, 1975), temperature fluctuations of homogeneous isotropic and stationary turbulence behave like a passive scalar and therefore have the same spectrum as velocity fluctuations. Therefore for the spectral function of temperature fluctuation *F*(**k***,τ*), which is the spatial Fourier transform of the temperature correlation function

we assume

- (27)

where

- (28)

In Eq. (27), Δ*T* is the r.m.s. of the temperature fluctuation, *η*_{k} is the autocorrelation function defined as

- (29)

and *f*(*η*_{k}*,τ*) characterizes the temporal decorrelation of turbulent fluctuations, such that it becomes negligibly small for *τ* ≫ 1*/η*_{k}.

For *f*(*η*_{k}*,τ*) we use a square exponential time dependence (Kraichnan, 1964):

- (30)

For homogeneous turbulence, the two-point time-delayed fourth-order correlation function of temperature fluctuations *R*(**x**′,**x**′ + **r***,τ*) is related to the temperature correlation function by means of the following relation (Monin and Yaglom, 1975):

- (31)

Using Eqs. (27),(29),(30) and the convolution theorem, we find

- (32)

where

- (33)

For low-Mach-number turbulence, one can use the so-called aeroacoustic approximation (Goldstein, 1976), which for the Fourier spectrum implies that in Eq. (25) instead of *H*(**x**′*,ω***x**/|**x**|*,ω*) one can set *H*(**x**′,0*,ω*).

Performing the integration with respect to frequency in Eq. (32), as well as angular variables in wave number space, we obtain

- (34)

The aeroacoustic approximation simplifies finding asymptotic limits for the spectrum. In the low-frequency regime, taking the limit *ω* 0 we obtain

- (35)

For the spectrum this means . Physically, these frequencies are lower than the lowest characteristic frequency in the problem, corresponding to the eddy turnover time on the energy-containing scale.

At high frequencies (*ω* ≫ *k*_{0}*MR*^{1/2}), the integral in Eq. (34) is dominated by the contribution from its upper limit and we get

- (36)

and

- (37)

The functional form of the high-frequency suppression is determined by the specific form of the time autocorrelation function of turbulence, Eq. (30) (Kraichnan, 1964), but for any autocorrelation the amplitude of emitted waves should be very small in this band. Physically, this limit corresponds to radiation frequencies which are larger than any frequencies of turbulent motions; consequently, no scale of turbulent fluctuations generates these radiation frequencies directly, and the resulting small radiation amplitude is due to the sum of small contributions from many lower-frequency source modes. Since the integral is dominated by the upper integration limit, the highest-frequency source fluctuations (which contain very little of the total turbulent energy) contribute most to this high-frequency radiation tail.

In the intermediate frequency regime, *k*_{0}*Mc*_{s}*< ω < k*_{0}*Mc*_{s}*R*^{1/2}, the integral in Eq. (34) is dominated by the contribution around *k*_{1}, where . The width of the dominant interval is Δ*k*_{1} ∼ *k*_{1}. Physically, this implies that radiation emission at some frequency in this range is dominated by turbulent vortices of the same frequency. Consequently, for the inertial interval we obtain the following estimates:

- (38)

and

- (39)

We performed numerical integration of Eq. (34) for the von Kármán model of turbulence and determined the normalized spectrum of acoustic radiation to be

- (40)

The normalized spectrum for a characteristic length-scale of energy-containing eddies *l* = 2*π/k*_{0} = 15 m and a characteristic r.m.s. velocity *v*_{0} = 5m s^{−1} is presented in Figure 1. As can be seen for these typical parameters, the peak frequency of infrasound radiation is *ν*_{peak} ≈ 0.8 Hz.

As shown by Akhalkatsi and Gogoberidze (2009), the peak frequency of acoustic radiation is inversely proportional to the turnover time of energy containing turbulent eddies *ν*_{peak} ∼ *v*_{0}*/l*, whereas total acoustic power is proportional to , Δ*T*^{4} and inversely proportional to *l*.

### 4. Infrasound correlation with tornadoes

- Top of page
- Abstract
- 1. Introduction
- 2. General formalism
- 3. Spectral decomposition
- 4. Infrasound correlation with tornadoes
- 5. Conclusions
- Acknowledgements
- References

Severe storm forecasting operations are based on several large-scale environmental, storm-scale, mesobeta-scale kinematic and thermodynamic parameters (Lemon and Doswell, 1979; Davies-Jones, 1993; Markowski *et al.*, 1998, Markowski *et al.*, 2002) used to study the potential of severe weather, thunderstorm structure and organization to produce tornadoes. Recent climatological studies of thunderstorms using real-time radar data combined with observations of near-storm environment have been focused on the utility of various supercell and tornado forecast parameters (CAPE, storm relative helicity –SRH, bulk Richardson number –BRN, and other parameters). Two parameters have been established to be the most promising in discriminating between non-tornadic and tornadic supercells: boundary layer water vapour concentration (LCL height) and low-level vertical wind shear (Thompson and Edwards, 2000; Rasmussen, 2003; Markowski and Richardson, 2009).

Examining a baseline climatology of parameters commonly used in supercell thunderstorm forecasting and research, Rasmussen and Blanchard (1998) established that the parameter that shows the most utility for discriminating between soundings of supercells with significant tornadoes and supercells without significant tornadoes is LCL height. The height of the LCL appeared to be generally lower for supercells with significant tornadoes than those without. Rasmussen and Blanchard (1998) also found that for storms producing large (at least 5 cm) hail only, without at least F2 strength tornadoes, the LCL heights were significantly higher than for ordinary thunderstorms. In their study, half of the tornadic supercell soundings had LCLs below 800 m, while LCL heights above about 1200 m were associated with decreasing likelihood of significant tornadoes. They concluded that stronger evaporational cooling of the moist downdraught leads to greater outflow dominance of storms in high LCL settings, and low heights increase the likelihood of supercells being tornadic.

The work of Thompson and Edwards (2000) on assessing utility of various supercell and tornado forecasting parameters supports the finding that supercells above deeply mixed convective boundary layers, with relatively large dew-point depressions and high LCL, often do not produce tornadoes even in environments of large CAPE and/or vertical shear. They found the LCL to be markedly lower for supercells producing significant tornadoes than for those producing weak tornadoes, which were in turn lower than for non-tornadic supercells. Particularly, no strong and violent tornadoes occurred for supercells with LCL > 1500 m.

Studying the relationship between rear flank downdraught (RFD) thermodynamic characteristics and tornado likelihood, Markowski *et al.* (2002) found that low LCL favours the formation of significant tornadoes because the boundary layer relative humidity (RH) somehow alters the RFD and outflow character of supercells. Relatively warm and buoyant RFDs, which are supposed to be necessary for the genesis of significant tornadoes, were more likely in moist low-level environments than in dry low-level environments (Markowski *et al.*, 2002). It appeared that relatively dry boundary layers, characterized by higher LCLs, support more low-level cooling through the evaporation of rain, leading to stronger outflow, which could have decreased the likelihood of significant tornadoes in supercells. These are possible explanations for finding that the LCL height is generally lower in soundings associated with tornadic versus non-tornadic supercells (Rasmussen and Blanchard, 1998; Thompson and Edwards, 2000).

Thompson *et al.* (2003) reinforced the findings of previous studies related to LCL height as an important discriminator between significantly tornadic and non-tornadic supercells and concluded that the differences in LCL heights across all storm groups studied were statistically significant, though the differences appeared to be operationally useful only when comparing significantly tornadic and non-tornadic supercells. The lower LCL heights of the significantly tornadic storms supported the hypothesis of Markowski *et al.* (2002) that increased low-level RH may contribute to increased buoyancy in the rear flank downdraught and an increased probability of tornadoes.

In idealized numerical simulations Markowski *et al.* (2003) investigated the effects of ambient LCL and the precipitation character of a rain curtain on the thermodynamic properties of the downdraught, and ultimately on tornado intensity and longevity. The simulations were consistent with the observation that high boundary layer RH values (i.e. low LCL height and small surface dew-point depression) are associated with relatively warmer RFDs and more significant tornadogenesis than environments of relatively low boundary layer RH.

These findings of low LCL favouring significant tornadoes could explain observed high correlation between low-frequency infrasound signals from supercell storm and later tornado formation.

The acoustic power of a monopole source related to the heat production during condensation of moisture can been estimated as (Akhalkatsi and Gogoberidze, 2009):

- (41)

where

- (42)

and *T*_{C} = *T* − 273.15 is the temperature (in °C).

Due to the numerator in the exponent, *f*(*T*_{C}) strongly depends on temperature, e.g. *f*(*T*_{C} = 10°)*/f*(*T*_{C} = 0°) ≈ 2, and according to multiplier *f*^{2}(*T*_{C}) in Eq. (41), increase of saturated air temperature causes rapid enhancement of total acoustic power radiated by a monopole source.

On the other hand, low LCL height means low-level air in the updraught motion being saturated and consequently, a higher temperature of saturated moist air. Therefore, the lower LCL heights contribute to increased total acoustic power radiated by a monopole source related to the heat production during the condensation of moisture. As a result, enhanced low-frequency infrasound signals from a supercell storm appear to be in strong correlation with an increased probability of tornadoes.

It is also known (Weisman and Klemp, 1982; Rotunno and Klemp, 1982; Rasmussen and Blanchard, 1998; Rasmussen, 2003; Thompson *et al.*, 2003) that high values of supercell CAPE assist tornado formation. Indeed, high values of CAPE lead to an increase of the updraught persistence and thunderstorm activity and therefore increase the probability of tornado formation. According to recent studies, relatively larger CAPE occurs closer to the surface, which could produce more intense low-level stretching of vertical vorticity required for low-level mesocyclone intensification and perhaps tornadogenesis (McCaul, 1991; McCaul and Weisman, 1996; Rasmussen, 2003) is associated with low LCLs. Rasmussen (2003) found the 0–3 km above ground level CAPE to be possibly important in discriminating between environments supportive of significant tornadoes and those that are not.

High values of CAPE mean high updraught velocity caused by large low-level accelerations and increased r.m.s. of turbulent velocities. According to Eq. (41), . Therefore increasing r.m.s. of turbulent velocities results in strong enhancement of total acoustic power.

### 5. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. General formalism
- 3. Spectral decomposition
- 4. Infrasound correlation with tornadoes
- 5. Conclusions
- Acknowledgements
- References

In this paper we have performed detailed spectral analysis of a monopole source of sound related to heat production during condensation of moisture, which is supposed to be dominant in the infrasound radiation observed from strong convective storms. We have also discussed the relationship between the acoustic power of this source and certain significant tornado forecast parameters. In particular, low LCL, which is known to favour significant tornadoes (Rasmussen and Blanchard, 1998; Thompson *et al.*, 2003), implies warmer air at the level of saturation. We have shown that the increase of temperature causes rapid enhancement of acoustic power. High values of CAPE (especially occurring closer to the surface), which assist tornado formation (Rotunno and Klemp, 1982; Weisman and Klemp, 1982; Rasmussen and Blanchard, 1998; Rasmussen, 2003; Thompson *et al.*, 2003), mean high updraught velocity and therefore increased r.m.s. of turbulent velocities, which results in strong enhancement of total acoustic power.

ISNeT data combined with the information from Doppler radar may help to improve tornado forecasts and reduce false alarms from non-tornadic supercells. Recent studies comparing ISNeT output with occurrences of tornadoes (Bedard *et al.*, 2004a) and correlating ISNeT signals with detailed radar output (Szoke *et al.*, 2004) show that the infrasound of a tornadic thunderstorm is much stronger than the infrasound of a non-severe weather system. Correlation between the intensity of infrasound signals and probability of later tornado formation indicates the potential for discriminating between supercells that produce tornadoes and those that do not. Therefore, information from an infrasound-detecting system may help to determine potentially tornadic storms.