1. Introduction

Recent observations and longstanding theoretical considerations suggest that a developing tornado has a detectable signature in the infrasound¹ 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

Consider an atmosphere at rest that contains a spherical cloud of water droplets. The cloud is characterized by the liquid mixing ratio

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in which 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 p₀, mass density ρ₀, absolute temperature T₀, and sound speed c₀ ≡ (c_pp₀/c_υρ₀)^1/2 are treated as constants. The symbols c_p, c_υ, and 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.

We assume that all perturbations are sufficiently weak to justify linearizing the compressible gas dynamics. The linearized mass continuity equation, including a source term due to evaporation, is given by

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in which ρ is the mass density of the gaseous component of moist air and u is the velocity field. The linearized momentum equation is

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assuming that gravitational effects and viscosity are unimportant over the time scale of interest.

The definition of virtual potential temperature,

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here serves as the equation of state. Linearizing Eq. (4) yields

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Neglecting diffusion by subgrid eddies and sedimentation, the heat equation in RAMS is approximated by the material conservation of ice–liquid potential temperature, defined by (Tripoli and Cotton 1981)

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Here, L_lv is the latent heat of vaporization (per unit mass), L_iv is the latent heat of sublimation, r_i is the mixing ratio of ice, and T is the absolute temperature of the air. Taking into consideration that T > 253 K, r_i = 0, r_υ ≪ 1, L_lvr_l/c_pT ≪ 1, and r′_l = −r′_υ, the linearized heat equation reduces to

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Taking the time derivative of Eq. (2), and eliminating u, ρ′, and θ′_υ with Eqs. (3), (5), and (7) yields

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Note that the factor in parentheses is typically negative and much greater than unity, meaning that evaporative cooling dominates the mass source of the acoustic emission. A formal solution to Eq. (8) in an infinite domain is given by

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in which t̂ ≡ t − |x − y|/c₀ 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

Neglecting the convective component of the material derivative, which is second order in the velocity and mixing ratio, the rate of change of water vapor is given by

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in which 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

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in which ψ is the vapor diffusivity, N is the number concentration of cloud droplets, D_m is the mean droplet diameter, and 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.

The first step of the μ closure is to consider the relationship between r_l, N, and D_m. In RAMS, the probability distribution of the droplet diameter D is given by

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in which ν is an adjustable parameter and Γ is the standard gamma function. The mass density of cloud droplets is given by , in which m is the mass of an individual droplet. By assumption, m = αD^β, in which α and β are empirical parameters. The mixing ratio of cloud water is thus

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In a single-moment model, either N or D_m is specified as a fixed parameter of the hydrometeor distribution. An expression for the other parameter as a function of r_υ in the evaporating cloud is obtained from Eq. (13) and the conservation of water mass (r′_l = −r′_υ). Substituting the result into Eq. (11) yields

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in which 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_m model (bottom) is often used for precipitating hydrometeors.

To determine 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

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in which T_c is the temperature at the surface of a droplet and κ is the thermal conductivity of air (e.g. Pruppacher and Klett 1997). In this section, we simplify Eq. (15) by assuming

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which makes sense only for low-density clouds in dry environments. The condition in Eq. (16) permits setting r_υ 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 T_c(r_*c, p₀) (e.g., Emanuel 1994; Walko et al. 1995) may then be used to convert Eq. (15) into a time-invariant equation for r_*c.

Substituting Eq. (14) and a constant value of r_*c into Eq. (10) yields an autonomous, first-order, ordinary differential equation (ODE) for r_υ. Assuming again that the air remains sufficiently dry throughout the evaporation process [Eq. (16)], we replace r_*c − r_υ with r_*c on the right-hand side. The solution of the ODE for the constant-N model is then

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in which

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Here (and below), we have assumed an initial vapor mixing ratio of 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_m model is given by

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in which

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In this case, evaporation continues forever.

c. Solutions of the pressure equation

Given either of the above formulas for r_υ, 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 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 ∂_ttr_υ ≡ P(t)Θ(ϵ − y), in which y ≡ (x² + ϖ² − 2xϖ cosφ)^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_m (bottom) forms are given below:

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in which δ is the Dirac distribution. The top and bottom definitions of γ are given by Eqs. (18) and (20), respectively.

Continuing, let B ≡ ρ₀(1.61 − L_lv/c_pT₀)/4π. Then the integral equation [Eq. (9)] may be written

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in which t̂ = t − ϖ/c₀. Upon further reduction, we obtain

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under the assumption that x > ϵ. What remains is to substitute (21) into (23) with t → t̂, and to perform the integration.

The solutions are conveniently expressed in terms of the following dimensionless variables:

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For the case of evaporation with constant N we have

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in which

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and

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The value of F₁ for any τ is obtained by numerical quadrature. For the case of evaporation with constant D_m we have

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in which

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Section 3 discusses the basic properties of solutions (25) and (28), and the extent to which they agree with the infrasound that is simulated by c-RAMS.

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_o/3) is one-tenth of the cloud radius ϵ. The vertical grid spacing (dz = dx_i) is uniform. Both the inner and outer grids have sufficient spatial resolutions to simulate 1-s (and longer) thermo-acoustic processes; the temporal resolutions (dt_i = dt_o/2 = 0.02 s) are also adequate. The horizontal widths of the inner and outer grids (X_i and X_o) are roughly 10 and 40 times 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_c = 281.2 K, which corresponds to a surface saturation mixing ratio of 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_m is held fixed during evaporation. Each plot is directly underneath the constant-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 2ϵ/c₀. 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_m, for a given 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_c, based on Eq. (15) with r_υ = 0, is roughly 1 K less than the time-averaged hydrometeor temperature (T_c) in c-RAMS. The associated 5% discrepancy in T − T_c 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 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_m, the revised curves are nearly indistinguishable from the data.

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

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

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in which

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If the air is nearly saturated, the term r_*c − r_υ 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 r_υ alone, by substituting the following approximations into Eq. (15):

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in which r_*0 and (∂r_*/∂T)₀ are the saturation mixing ratio and its temperature derivative at constant pressure, evaluated at T₀ and p₀. The new version of Eq. (15) is given by

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in which

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The reader may consult appendix C for further discussion of the key approximations used in deriving this result.

The solution of Eq. (30) is conveniently expressed in terms of the following two variables, related to the initial degree of subsaturation and the initial liquid mixing ratio:

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Specifically, we have

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in which

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and

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The value of γ can be positive or negative. When γ is positive (ζ₀ < 1), the cloud fully evaporates over time. When γ is negative (ζ₀ > 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 − ξ₀/(1 + b). Note that ζ₀ is the ratio of the initial liquid water mass (r_l0) to the maximum that can evaporate (r_l0 − r_lf).

As before, we may use Eq. (23) to calculate the pressure perturbation outside of the cloud. Solving the integral yields

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in which

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The dimensionless variables x̃, τ, γ̃, and were defined earlier in Eq. (24).

The wave form is similar to the acoustic pulse generated by evaporation in a dry environment. A negative minimum occurs at about 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:

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Here, we have introduced the (approximate) initial cloud mass,

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and the initial decay rate of the cloud mass,

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Note that p₊ depends on the cloud size (ϵ) only through M_c0, and is proportional to the square of Ê₀. Because a cloud-size shock creates the leading wave, p₋ varies with ϵ through both M_c0 and c₀/ϵ, which is the regularized shock frequency. In contrast to p₊, the variation of p₋ with Ê₀ is linear.

The decay rate (characteristic frequency) of the trailing wave corresponds to the “proper” evaporation rate, defined by

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in which r_l∞ is the time-asymptotic liquid mixing ratio. If the air saturates without consuming the entire cloud, then r_l∞ is nonzero and E₀ > Ê₀. Figure 2 illustrates how E₀ 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 ζ₀ exceeds unity, after which E₀ remains constant. For fixed relative humidity, the evaporation rate decays with the liquid water mass until ζ₀ 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.

Figure 4 illustrates how the trailing wave amplitude varies relative to the leading wave amplitude as the ambient water vapor and droplet size increase. Evidently, the magnitude of 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:

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Equation (41) also predicts the readily seen decay of 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 μm 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.

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.