## 1. Introduction

Recent observations and longstanding theoretical considerations suggest that a developing tornado has a detectable signature in the infrasound^{1} of a severe storm (Bedard 2005; Bedard et al. 2004; Georges 1976; Passner and Noble 2006; Szoke et al. 2004). To reliably distinguish the vortex signal from extraneous noise, we must improve our current understanding of the various mechanisms that produce infrasound in atmospheric convection. Without detailed observations of the acoustic sources within a storm, numerical modeling may be the best method of investigation.

On the other hand, standard cloud models were not designed for the study of atmospheric infrasound. Their capabilities and limitations in this area of research are not fully understood. Here, we consider a special version of the Regional Atmospheric Modeling System (RAMS) that has been customized to simulate acoustics (Nicholls and Pielke 2000). Cotton et al. 2003 provide an overview of standard RAMS, whereas details of the microphysics parameterizations can be found in a number of additional papers (Walko et al. 1995, 2000; Meyers et al. 1997; Saleeby and Cotton 2004). Appendix A briefly describes the customized version of the model, which is called c-RAMS hereinafter.

The literature contains some evidence that c-RAMS has an adequate foundation for modeling the infrasound of convective storms. Schecter et al. (2008) recently showed that c-RAMS can simulate the adiabatic generation of infrasound by the Rossby-like waves of substorm-scale vortices. Nicholls and Pielke (1994a,b, 2000) previously showed that c-RAMS can simulate the creation of low-frequency compression waves (30-min Lamb waves) by storm-scale heating. However, no prior study has verified that c-RAMS accurately simulates the infrasound that is generated by phase transitions of moisture in the 0.1–10-Hz frequency band. This critical frequency band is where severe storms are observed to produce abnormally strong and distinct signals (Bedard 2005). In theory, adiabatic vortex motions could account for the observations (Bedard 2005; Schecter et al. 2008). However, entropy and mass fluctuations associated with phase transitions may also contribute. A recent analytical study concluded that such fluctuations in the moist turbulence of a severe storm are likely to dominate other sources of 0.1-Hz infrasound (Akhalkatsi and Gogoberidze 2009).

In this paper, we illustrate the fundamental mechanism by which phase transitions produce infrasound in c-RAMS. We focus on a conceptually simple paradigm—the evaporation of an isolated, homogeneous cloud of water droplets. Analytical expressions are derived for the acoustic emissions in dry and humid environments. Under sufficiently dry conditions, the evaporation may occur in a few seconds or less. Successful comparison of the analytical results to numerical experiments verifies that the practical output of c-RAMS agrees with the theoretical thermo–acoustics of the model. Such verification helps justify using c-RAMS for future numerical studies of infrasound generated by convective storms.

The remainder of this paper is organized as follows: section 2 presents a relatively simple theory for the acoustic emissions of an evaporating cloud in a dry environment, using two different single-moment microphysics parameterizations. Section 3 compares the analytical results of section 2 with numerical simulations. Section 4 presents a theory for the acoustic emissions of an evaporating cloud in a humid environment, and compares the results with numerical simulations. Section 5 contains a summary and concluding remarks. The appendixes discuss our customization of RAMS, the infrasound of icy hydrometeors, and theoretical subtleties regarding evaporation under humid conditions.

## 2. Theory of infrasound generated by evaporation in a dry environment

This section derives analytical formulas for the infrasound of an evaporating cloud of water droplets in a dry environment. Different results are obtained for different constraints on the evolution of the droplet size distribution, which are commonly imposed by c-RAMS and other cloud models. The analytical results provide benchmarks for evaluating the ability of c-RAMS to simulate infrasound consistent with its theoretical foundation.

### a. The pressure equation

**x**is the position vector relative to the center of the cloud,

*t*is time,

*ϵ*is the cloud radius, and Θ is the Heaviside step function. The value of Θ is zero or unity if its argument is negative or positive, respectively.

If the air is subsaturated, the water droplets evaporate. Evaporation cools the air and increases the vapor mixing ratio *r _{υ}*(

**x**,

*t*) within the cloud. The combination of local cooling and gaseous mass production generates an outward propagating acoustic pulse. The following derives a formal expression for the pressure perturbation

*p*′(

**x**,

*t*) associated with the pulse. The derivation is based on the thermo-mechanical core of c-RAMS. For simplicity, the ambient pressure

*p*

_{0}, mass density

*ρ*

_{0}, absolute temperature

*T*

_{0}, and sound speed

*c*

_{0}≡ (

*c*

_{p}p_{0}/

*c*

_{υ}ρ_{0})

^{1/2}are treated as constants. The symbols

*c*,

_{p}*c*, and

_{υ}*R*represent the specific heat at constant pressure, the specific heat at constant volume, and the gas constant of dry air. In general, a prime or zero subscript denotes a perturbation or basic-state variable, respectively.

*ρ*is the mass density of the gaseous component of moist air and

**u**is the velocity field. The linearized momentum equation is

*L*

_{lv}is the latent heat of vaporization (per unit mass),

*L*

_{iv}is the latent heat of sublimation,

*r*is the mixing ratio of ice, and

_{i}*T*is the absolute temperature of the air. Taking into consideration that

*T*> 253 K,

*r*= 0,

_{i}*r*≪ 1,

_{υ}*L*

_{lv}

*r*/

_{l}*c*≪ 1, and

_{p}T*r*′

*= −*

_{l}*r*′

*, the linearized heat equation reduces to*

_{υ}**u**,

*ρ*′, and

*θ*′

*with Eqs. (3), (5), and (7) yields*

_{υ}*t̂*≡

*t*− |

**x**−

**y**|/

*c*

_{0}is the retarded time. The reader may consult appendix B for a generalization of Eq. (8) that accounts for ice.

### b. Evaporation equations for two single-moment schemes

*r*

_{*c}is the saturation mixing ratio at the surface of a cloud droplet. The bulk rate of evaporation is largely determined by the value of

*ψ*is the vapor diffusivity,

*N*is the number concentration of cloud droplets,

*D*is the mean droplet diameter, and

_{m}*S*is the droplet shape parameter. Here, we have neglected a small correction to

*μ*associated with the characteristic Reynolds number of the droplet motion. To close the evaporation equation [Eq. (10)],

*μ*and

*r*

_{*c}must be expressed as functions of

*r*.

_{υ}*μ*closure is to consider the relationship between

*r*,

_{l}*N*, and

*D*In RAMS, the probability distribution of the droplet diameter

_{m}.*D*is given by

*ν*is an adjustable parameter and Γ is the standard gamma function. The mass density of cloud droplets is given by

*m*is the mass of an individual droplet. By assumption,

*m*=

*αD*, in which

^{β}*α*and

*β*are empirical parameters. The mixing ratio of cloud water is thus

*N*or

*D*is specified as a fixed parameter of the hydrometeor distribution. An expression for the other parameter as a function of

_{m}*r*in the evaporating cloud is obtained from Eq. (13) and the conservation of water mass (

_{υ}*r*′

_{l}= −

*r*′

_{υ}). Substituting the result into Eq. (11) yields

*r*≡

_{t}*r*

_{υ0}+

*r*

_{l0}and

*r*

_{υ0}is the initial vapor mixing ratio. Although the constant-

*N*model (top) may seem more physical for the problem under consideration, the constant-

*D*model (bottom) is often used for precipitating hydrometeors.

_{m}*r*

_{*c}, we suppose that the net heat flux (conductive plus latent) at the surface of each cloud droplet is approximately zero. This leads to an equation of the form

*T*is the temperature at the surface of a droplet and

_{c}*κ*is the thermal conductivity of air (e.g. Pruppacher and Klett 1997). In this section, we simplify Eq. (15) by assuming

*r*equal to zero on the left-hand side of Eq. (15). We further assume that the liquid water mass is sufficiently small for all other variables in the equation to stay nearly fixed over the course of evaporation. A standard analytical formula for

_{υ}*T*(

_{c}*r*

_{*c},

*p*

_{0}) (e.g., Emanuel 1994; Walko et al. 1995) may then be used to convert Eq. (15) into a time-invariant equation for

*r*

_{*c}.

*r*

_{*c}into Eq. (10) yields an autonomous, first-order, ordinary differential equation (ODE) for

*r*. Assuming again that the air remains sufficiently dry throughout the evaporation process [Eq. (16)], we replace

_{υ}*r*

_{*c}−

*r*with

_{υ}*r*

_{*c}on the right-hand side. The solution of the ODE for the constant-

*N*model is then

*r*

_{υ0}= 0, which corresponds to the numerical experiments of section 3. Note that evaporation ends (

*r*=

_{υ}*r*

_{l0}) at

*t*=

*γ*

^{−1}. The solution of the ODE for the constant-

*D*model is given by

_{m}### c. Solutions of the pressure equation

*r*, the integral equation for the pressure perturbation [Eq. (9)] is readily evaluated. Consider a spherical coordinate system whose origin is at the observation point, which is a distance

_{υ}*x*from the center of the cloud. Let ϖ,

*λ*, and

*φ*denote the radius, azimuth, and polar angle (measured from the axis connecting the origin and the center of the cloud). In this coordinate system, we may write ∂

*≡*

_{tt}r_{υ}*P*(

*t*)Θ(

*ϵ*−

*y*), in which

*y*≡ (

*x*

^{2}+

*ϖ*

^{2}− 2

*xϖ*cos

*φ*)

^{1/2}. The specific form of

*P*depends on the single-moment model that is used for the microphysics. Both the constant-

*N*(top) and constant-

*D*(bottom) forms are given below:

_{m}*δ*is the Dirac distribution. The top and bottom definitions of

*γ*are given by Eqs. (18) and (20), respectively.

*B*≡

*ρ*

_{0}(1.61 −

*L*

_{lv}/

*c*

_{p}T_{0})/4

*π*. Then the integral equation [Eq. (9)] may be written

*t̂*=

*t*− ϖ/

*c*

_{0}. Upon further reduction, we obtain

*x*>

*ϵ*. What remains is to substitute (21) into (23) with

*t*→

*t̂*, and to perform the integration.

*N*we have

*F*

_{1}for any

*τ*is obtained by numerical quadrature. For the case of evaporation with constant

*D*we have

_{m}## 3. C-RAMS simulations of infrasound generated by evaporation in a dry environment

The analytical pressure perturbations of section 2 are now compared with the infrasound simulated by c-RAMS when a cloud evaporates in a dry environment. We consider six numerical experiments with variable microphysics. The common experimental settings (of interest) are listed in Table 1.

A subset of the parameters in Table 1 pertains to the discretization. The model is configured to contain a fine inner grid (*i*) and a coarse outer grid (*o*). The inner horizontal grid spacing (*dx _{i}* =

*dx*/3) is one-tenth of the cloud radius

_{o}*ϵ*. The vertical grid spacing (

*dz*=

*dx*) is uniform. Both the inner and outer grids have sufficient spatial resolutions to simulate 1-s (and longer) thermo-acoustic processes; the temporal resolutions (

_{i}*dt*=

_{i}*dt*/2 = 0.02 s) are also adequate. The horizontal widths of the inner and outer grids (

_{o}*X*and

_{i}*X*) are roughly 10 and 40 times

_{o}*ϵ*, respectively. The lateral boundary conditions of the outer grid are set to allow free passage of acoustic waves (cf. Klemp and Wilhelmson 1978). The vertical boundaries at the ground and height

*Z*are purely reflective; however, they are far enough removed from the cloud to prevent echoes from returning over the time period under consideration.

The basic state of the atmosphere is dry and isentropic. The vertical density and pressure gradients that would ordinarily appear, because of enforcement of hydrostatic balance, are eliminated by setting the gravitational acceleration *g* to a very small value (10^{−4} m s^{−2}). At *t* = 0, a cloud is introduced by adding a uniform spherical distribution of *r _{l}* [Eq. (1)] in the center of the domain, and adjusting the ice–liquid potential temperature [Eq. (6)] accordingly. In principle, either “cloud water” or “rain” (two distinct hydrometeor categories in RAMS) may be used to form the cloud mass. Here, we use a customized rain category, which has zero fall speed and an initial temperature of

*T*= 281.2 K, which corresponds to a surface saturation mixing ratio of

_{c}*r*

_{*c}= 6.8 × 10

^{−3}.

All six experiments use Marshall–Palmer distributions (*ν* = 1) for the droplet size. The distributions have three distinct initial conditions, summarized in rows 1–3 of Table 2. For each initial condition, the cloud is allowed to evaporate keeping either *N* or *D _{m}* fixed. Figure 1 shows the acoustic emission (diamonds) for each case at

*x*= 424.3 m. The pressure perturbation is multiplied by

*x̃*to compensate for “geometric decay” with distance from the edge of the cloud.

The top row of Fig. 1 shows the variation of the acoustic emission with the number density *N*, when *N* is held fixed during evaporation. Decreasing *N* by two orders of magnitude severely damps the peak wave amplitude. Although reducing *N* has little influence on the width of the leading (negative) pulse, it dramatically widens the trailing (positive) wave. The bottom row shows the variation of the acoustic emission with the mean droplet diameter *D _{m}*, when

*D*is held fixed during evaporation. Each plot is directly underneath the constant-

_{m}*N*experiment with the same microscopic initial conditions (Table 2). Evidently, changing the single-moment scheme significantly modifies the shape of the trailing wave, but not the characteristic amplitude or time scale of the emission.

The basic properties of the acoustic emissions are readily explained. To begin with, the leading pulse is created by the initial shock of the evaporation rate over the entire cloud. Because of the finite propagation speed of acoustic signals, an observer senses this shock over a time period of 2*ϵ*/*c*_{0}. The duration of the trailing wave is the evaporation time scale, which dilates with decreasing *N* or increasing *D _{m}*. Finally, since the amplitude of the acoustic source decreases with the evaporation rate, so must the emission attenuate with decreasing

*N*or increasing

*D*, for a given

_{m}*r*

_{l0}[see Eq. (14)].

The solid curves in each figure correspond to the integral on the right-hand side of Eq. (22), with *P*(*t̂*) given by the simulated time series of *r _{υ}*. All solid curves very closely match the pressure waves that are generated by c-RAMS. In this sense, c-RAMS correctly simulates the infrasound of evaporation.

^{2}

The dotted curves in each figure correspond to the analytical approximations of section 2 [Eqs. (25) and (28)]. When *D _{m}* is held constant during evaporation, the approximations are excellent. When

*N*is held constant during evaporation, a quantitative discrepancy is noticeable. The error appears primarily because our calculated value of

*T*, based on Eq. (15) with

_{c}*r*= 0, is roughly 1 K less than the time-averaged hydrometeor temperature (

_{υ}*T*

_{c}) in c-RAMS. The associated 5% discrepancy in

*T*−

*T*is plausible, since the vapor mixing ratio grows with time, and the microphysics algorithm in c-RAMS does not strictly enforce Eq. (15) (Walko et al. 2000). The dash–dotted curves are the analytical approximations using

_{c}*T*

_{c}. For fixed

*N*, this revision clearly improves the quantitative agreement between the analytical waves and the infrasound that is generated by c-RAMS. For fixed

*D*, the revised curves are nearly indistinguishable from the data.

_{m}## 4. Infrasound generated by evaporation in a humid environment

This section examines the infrasound of an evaporating cloud in a humid environment. For analytical convenience, we restrict our analysis to the single-moment parameterization in which *D _{m}* is constant.

### a. Basic theory

*μ*into two parts, we may rewrite the evaporation equation as follows:

*r*

_{*c}−

*r*on the right-hand side of Eq. (30) can change over time by a large fraction of its initial value. We may express this term as a function of the variable

_{υ}*r*alone, by substituting the following approximations into Eq. (15):

_{υ}*r*

_{*0}and (∂

*r*

_{*}/∂

*T*)

_{0}are the saturation mixing ratio and its temperature derivative at constant pressure, evaluated at

*T*

_{0}and

*p*

_{0}. The new version of Eq. (15) is given by

*γ*can be positive or negative. When

*γ*is positive (

*ζ*

_{0}< 1), the cloud fully evaporates over time. When

*γ*is negative (

*ζ*

_{0}> 1), saturation occurs before the condensate is fully removed. In this case, the final value of the liquid mixing ratio is

*r*=

_{lf}*r*

_{l0}−

*ξ*

_{0}/(1 +

*b*). Note that

*ζ*

_{0}is the ratio of the initial liquid water mass (

*r*

_{l0}) to the maximum that can evaporate (

*r*

_{l0}−

*r*).

_{lf}*x̃*,

*τ*,

*γ̃*, and

*τ*= 0, whereas a positive maximum occurs at

*τ*= 1. If

*γ̃*≪ 1, one may derive the following expressions for the minimum (−) and maximum (+) values of the pressure perturbation:

*p*

_{+}depends on the cloud size (

*ϵ*) only through

*M*

_{c0}, and is proportional to the square of

*Ê*

_{0}. Because a cloud-size shock creates the leading wave,

*p*

_{−}varies with

*ϵ*through both

*M*

_{c0}and

*c*

_{0}/

*ϵ*, which is the regularized shock frequency. In contrast to

*p*

_{+}, the variation of

*p*

_{−}with

*Ê*

_{0}is linear.

*r*

_{l∞}is the time-asymptotic liquid mixing ratio. If the air saturates without consuming the entire cloud, then

*r*

_{l∞}is nonzero and

*E*

_{0}>

*Ê*

_{0}. Figure 2 illustrates how

*E*

_{0}varies with relative humidity, liquid water mass, droplet size, and air temperature.

^{3}In general, increasing the relatively humidity (

*r*

_{υ0}/

*r*

_{*0}) reduces the evaporation rate until

*ζ*

_{0}exceeds unity, after which

*E*

_{0}remains constant. For fixed relative humidity, the evaporation rate decays with the liquid water mass until

*ζ*

_{0}drops below unity. The evaporation rate also decays with increasing droplet size, as

*D*

_{m}^{−2}if

*β*= 3. As a final remark on the matter, lowering the air temperature (with an adiabatic pressure drop) tends to decelerate evaporation for a given relative humidity, droplet size, and liquid water mass.

Figure 3 illustrates how the peak amplitude (*p*_{+}) of the trailing wave of the infrasonic emission varies with relative humidity, liquid water mass, droplet size, and air temperature. In general, the numerical values correspond to pressure perturbations at *x* = 10 km from the center of a cloud of radius *ϵ* = 100 m. Sensibly, increasing the relative humidity to 100% damps the wave amplitude to zero. Decreasing the liquid water mass also leads to a weaker emission. Increasing the droplet size decreases the trailing wave amplitude asymptotically as *D _{m}*

^{−4}, assuming that

*β*= 3. Lowering the air temperature (with an adiabatic pressure drop) also attenuates the wave, for a given relative humidity, droplet size, and liquid water mass.

*p*

_{+}/

*p*

_{−}tends toward a constant value as the relative humidity tends toward 100%. From Eq. (41), we obtain the following approximation for this value:

*p*

_{+}/

*p*

_{−}with increasing droplet size (as

*D*

_{m}^{−2}if

*β*= 3).

The paradigm of infrasound generated by the evaporation of an isolated cloud, suddenly introduced into a subsaturated environment, was conceived for the purpose of testing c-RAMS. It does not adequately represent evaporation in a realistic storm, which may result from turbulent mixing or falling hydrometeors.^{4} Nevertheless, the results shown here provide some basis for estimating the influence of evaporation (or condensation) on the infrasound in a storm simulation. We are primarily interested in the 0.1–10-Hz component of the infrasound, where severe storm signals are relatively strong and may contain detectable signatures of tornadoes (Bedard 2005). At 10 km from the source, the observed signals have wave amplitudes of the order 0.1 Pa (Bedard 2005; Schecter et al. 2008). The plots shown here loosely suggest that evaporation over 100-m-scale regions of high relative humidity (>90%) may noticeably affect these signals only if the cloud water has a mean droplet size of about 10 *μ*m or less, or a liquid mixing ratio greater than 0.1 g kg^{−1}. Otherwise, phase transitions are too slow and the acoustic emissions are too weak.

### b. Comparison with c-RAMS

Figure 5 compares the analytical pressure perturbations of section 4a with the infrasound simulated by c-RAMS due to evaporation under humid conditions. For proper comparison, all three numerical experiments use the single-moment parameterization in which *D _{m}* is held fixed. Table 1 lists the common parameters of all experiments (note that Table 1 neglects very small corrections to the ambient mass density and sound speed, associated with finite relative humidity), whereas Table 2 (rows 4–6) lists the distinguishing parameters. The left and middle plots correspond to simulations with the same relative humidity and liquid mixing ratio, but different values of

*D*. The middle and right plots correspond to simulations with the same

_{m}*D*, but different relative humidities and liquid mixing ratios.

_{m}Regardless of the specific conditions, the infrasonic emissions simulated by c-RAMS (solid curves) compare favorably to the analytical predictions (dotted curves) that are given by Eq. (39). We conclude that c-RAMS properly simulates the infrasound of parameterized evaporation in a humid environment.

## 5. Conclusions

In this paper, we derived analytical formulas for the infrasound generated by evaporating clouds in dry and humid environments. The derivations were based on the dynamical core of c-RAMS and standard single-moment microphysics parameterizations. The theoretical development elucidated the fundamental mechanism by which (liquid–vapor) phase transitions produce infrasound in the model. Furthermore, it explained the potential variation of 0.1–10-Hz emissions with the selected microphysics parameterization.

More important is that the analytical solutions gave us a benchmark for evaluating the ability of c-RAMS to simulate infrasound consistent with theory. Successful comparisons between the analytical results and computational output verified the adequacy of the numerics, under both dry and humid conditions. A more general theoretical benchmark covering the infrasound of ice and mixed-phase hydrometeors was left for another study. Nevertheless, appendix B shows that c-RAMS correctly generates the infrasound of an equilibrating cloud of mixed-phase hydrometeors, under the assumption that the microphysics algorithm properly models the ice–liquid–vapor phase conversions.

In summary, we have demonstrated that c-RAMS is built upon a solid foundation for simulating the infrasound associated with phase transitions of moisture. Fine details of the acoustic power spectrum (between 0.1 and 10 Hz) may vary with the microphysics parameterization, because phase transition rates are influenced by the imposed constraints on hydrometeor distributions. In our view, the subtle variation of spectral details should not discourage researchers from using c-RAMS (or any other cloud model) to investigate the conditions under which tornado infrasound may dominate the infrasound of generic moist processes within a storm. However, microphysics sensitivity tests may be necessary to establish definitive conclusions.

## Acknowledgments

The authors thank Stephen Saleeby for a helpful discussion regarding the microphysics parameterizations in RAMS. The authors also thank three anonymous reviewers for their constructive comments on the original manuscript. This work was supported by NSF Grant ATM-0832320.

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## APPENDIX A

### Customization of RAMS

*c*(

_{p}*p*/

*p*

_{0})

^{R/cp}. The complete form of this equation (including a term connected to mass exchange between air and hydrometeors) is given by

*c*

^{2}≡

*R*Π

*θ*/

_{υ}*c*. All terms in Eq. (A1) are kept in the customized version of RAMS that is used here to study atmospheric acoustics (cf. Nicholls and Pielke 2000). The standard version of RAMS neglects the entire right-hand side.

_{υ}Note that while perhaps unnecessary, the sedimentation routine was commented out of the code for this study. Furthermore, the macroscopic diffusion coefficients (connected to subgrid eddies) were given negligible values.

## APPENDIX B

### Linearized Pressure Equation with Ice

**u**,

*ρ*′,

*θ*′

*, and*

_{υ}*r*′

*with Eqs. (3), (5), (B1), and (B2) yields*

_{l}*r*and

_{υ}*r*, the solution of (B3) outside of a uniform cloud reduces to an integral similar to the right-hand side of Eq. (23).

_{i}Figure B1 verifies that the integral for *p*′ agrees with the infrasound generated by c-RAMS, when a uniform sphere of hail (*ϵ* = 200 m) is suddenly introduced into a warm (*T*_{0} = 300 K) and saturated (*r*_{υ0} = *r*_{*0}) environment. The experiment employs a single-moment parameterization in which the mean diameter (*D _{m}*) of the hail distribution is held constant at 1 cm, and

*ν*= 2. The initial value of the hail mixing ratio is 1.14 g kg

^{−1}, and the initial liquid mass fraction is zero. Over time, the ice melts under the actions of heat and vapor diffusion, as explained in standard textbooks (e.g., Pruppacher and Klett 1997). Because the mean surface-to-volume ratio of hail is relatively small, phase transitions occur slowly. Consequently, the only significant infrasound is produced by the initial shock. The dotted curves show the individual contributions to the “shock wave” computed from the output vapor and ice (top and bottom) terms on the right-hand side of (B3). The solid curve is the sum of both contributions, which matches the pressure perturbation generated by c-RAMS (diamonds).

Note that the time step (*dt _{i}*) for the hail simulation is 2 ms, and the standard value of

*L*

_{iv}= 2.834 × 10

^{6}J kg

^{−1}is used to calculate the theoretical infrasound.

## APPENDIX C

### Approximations in the Evaporation Equation under Humid Conditions

*T*that is derived from the following reformulated heat equation [Eq. (7)]:

*E*is sufficiently small for the acoustic wavelength

*c*

_{0}/

*E*to greatly exceed the length scale

*l*of the source region. Under this condition, Eq. (8) suggests that

*p*′ ∼

*l*

^{2}

*E*

^{2}

*ρ*

_{0}

*L*

_{lv}

*r*′

_{υ}/

*c*

_{p}T_{0}. Accordingly, the ratio of the

*p*′ term over the

*r*′

*term in Eq. (C1) is of the order (*

_{υ}*l*

^{2}

*E*

^{2}/

*c*

_{0}

^{2})(

*R*/

*c*) ≪ 1. For this reason, Eq. (32) neglects the partial variation of

_{υ}*T*′ with

*p*′.

*T*. This formula assumes that

_{c}*dT*≡

_{c}*T*−

_{c}*T*

_{0}is very small. From the relations

*R*(

_{υ}*R*) is the gas constant of water vapor (dry air) and

*e*

_{*}(

*T*) ≪

*p*denotes the saturation vapor pressure (e.g., Emanuel 1994). Using Eq. (33) to evaluate

*dr*

_{*c}≡

*r*

_{*c}−

*r*

_{*0}, and using our previous estimate of

*dp*≡

*p*′, one may show that neglecting the term proportional to

*dp*in Eq. (C3) is typically consistent.

Common experimental parameters, defined in the text.

Distinct initial conditions for experiments in dry (rows 1–3) and humid (rows 4–6) environments. All parameters are defined in sections 2 and 4.

^{1}

The term “infrasound” refers to sound waves at frequencies less than the lower limit of unimpaired human hearing, which is roughly 20 Hz.

^{2}

A similar test has verified the simulated infrasound of evaporation governed by the two-moment microphysics option of c-RAMS (Meyers et al. 1997). In this case, the trailing wave exhibits gradual decay at late times, qualitatively similar to the infrasound generated with fixed *D _{m}*.

^{3}

Table 1 gives the unspecified parameters that are required to evaluate most curves in Figs. 2 –4, for which *T*_{0} = 300 K. For the exceptional cases where *T*_{0} = 275 K, the pressure is reduced (dry adiabatically) to *p*_{0} = 10^{5} (275/300)^{cp/R} = 73 757 Pa; furthermore, the values of *c*_{0}, *ρ*_{0}, *ψ*, *κ*, *r*_{*0}, and (∂*r*_{*}/∂*T*)_{0} are adjusted according to standard formulas (e.g., Pruppacher and Klett 1997; Emanuel 1994).

^{4}

In a practical storm simulation, the subgrid turbulence parameterization may have a switch that turns on when the local Richardson number is sufficiently small. Once activated, the model could rapidly mix finescale inhomogeneities of the entropy and moisture variables. Conceivably such mixing could (artificially) trigger the type of sudden phase transition considered here.