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ACOUSTIC RADIATIONS FROM LIGHTNING 54 tionary sphere of radius Î± is proportional to (Ï Î± 2) (Î±/Î»)4; this is the same relationship that appears in the radar cross- section expression for these parameters. For thunder wavelengths (~ 1 m) and cloud particles (~ 10â3 m) the ratio (Î±/Î»)4 is 10â12. The cloud is, therefore, transparent to low-frequency thunder just as it is to meterwavelength electromagnetic radiation, although insignificant fractions of the radiation do get scattered. There are, however, eddies in the same size range as low-frequency thunder wavelengths, and these features, owing to small thermal changes and flow shears, produce a distortion of wave fronts and scattering-type effects. For the part of the turbulent spectrum having wavelengths smaller than the acoustic wavelengths of interest, the turbulence can be treated statistically by scattering theory. Larger-scale turbulence must be described with geometric acoustics. For the low-frequency thunder, turbulent scattering will attenuate the high-amplitude beamed parts of the thunder signal; this increases the rumbles at the expense of claps. In the first part of this subsection we discussed the cloud particles as sources of acoustic scattering; there are other and probably more important ways in which these aerosol components interact with the acoustic waves. First, the surface area of the cloud particles within a volume provides preferred sites for enhanced viscosity and heat conduction; hence, the presence of particles increases the classical attenuation coefficient. Another totally different process produces attenuation by changing the thermodynamic parameters associated with the acoustic wave over the surfaces of cloud particles; this changes the local vapor-to-liquid or vapor-to-solid conversion rates. For example, during the compressional part of the wave the air temperature is increased and the relative humidity is decreased relative to equilibrium; the droplets partially evaporate in response and withdraw some energy from the wave to accomplish it. The opposite situation occurs during the expansion part of the wave. Because the phase-change energy is ideally 180 out of phase with the acoustic-wave energy this process produces attenuation. Landau and Lifshitz (1959) included this effect in their "second viscosity" term. This attenuation process differs from the other microscopic processes in that it can be effective at the lower frequencies. The magnitude of this effect plus the enhanced attenuation by viscous and heat conduction at the surface exceed that of particle-free air by a factor of 10 or greater depending on the type, size, and concentration of the cloud particles (Kinsler and Frey, 1962). Finally, there is a mass-loading effect with respect to the cloud particles that must be considered. The amplitude of the fluid displacement, Î¶, produced by an acoustic wave of pressure amplitude Î´p and angular frequency Ï is (Kinsler and Frey, 1962) Using 50 Pa a representative value of Î´p for thunder inside a cloud we find for a 100-Hz frequency that Î¶ = 100 Âµm. The part of the cloud particle population whose diameter is much smaller than this, say 10 Âµm, should, owing to viscous drag, come into dynamic equilibrium with the wave flow. [Dessler (1973) computed the response time for a ~ 10-Âµm droplet to re-establish dynamic equilibrium with drag forces; only 10â3 sec is required.] These cloud particles, which participate in the wave motion, add their mass to the effective mass of the air; this effects both the speed of sound and the impedence of the medium. For higher-frequency waves, fewer cloud particles participate, so the effect is reduced; whereas lower-frequency waves include greater percentages of the population and are more strongly affected. Clouds are, therefore, dispersive with respect to low-frequency waves. Also, the cloud boundary acts as a partial reflector of the low-frequency acoustic signals because of the impedence change at the boundary. Assuming a total water content of order 5 g/m3, we estimate that the order of magnitude of the effect on sound speed and impedence is 10â3; this is not large, but it may be detectable. The cloud aerosols interact with the acoustic waves in three different ways depending on their size relative to the amplitude of air motion of the sound. The smallest fraction "ride with the wave"âaltering the wave-propagation parameters. The largest particles are stationary and act as scatterers of the acoustic waves. The particles in the middle range provide a transition scale for the above effects but are primarily responsible for enhanced viscous attenuation. In summary, there are several processes that can effectively attenuate higher-frequency components of thunder; this is in support of the conclusions of the previous section. We have, in addition, found three processes that affect the low-frequency components. Low frequencies can be attenuated by turbulent scattering and, in the cloud, by coupling wave energy to phase changes. We have also found that low frequencies interact with the cloud population dynamically; as a result, cloud boundaries may act as partial reflectors and incloud propagation may be dispersive. Refraction There is a wide range of refractive effects in the environment of thunderstorms. In the preceding section we
ACOUSTIC RADIATIONS FROM LIGHTNING 55 found that turbulence on the scale of the acoustic wavelength and smaller could be treated with scattering theory. Turbulence larger than acoustic wavelengths, up to and including storm-scale motions, should be describable by geometric acoustics or ray theory. To actually do this is impractical because it requires detailed information (down to the turbulent scale) of temperature and velocity of the air everywhere along the path between the source and the observer. since the thunder sources are widely distributed we would require complete knowledge of the storm environment down to the meter scale to trace accurately the path of an individual acoustic ray. These requirements can be relieved if we relax somewhat our expectations regarding the accuracy of our ray path. The three fluid properties that cause an acoustic ray to change its direction of propagation are the components of thermal gradient, velocity gradient, and velocity that are perpendicular to the direction of propagation. Beyond the overall thermal structure of the environment, which will be approximately adiabatic, we do not expect that the thermal perturbations due to turbulence will be systematic. In fact, the turbulent thermal perturbations should be random with a zero average value; hence, an acoustic ray propagating through turbulence should not deviate markedly owing to thermal gradients associated with the turbulence from the path predicted by the overall thermal structure of the environment. Similarly, velocity and velocity gradients should produce a zero net effect on the acoustic ray propagating through the turbulence. This argument of compensating effects is not valid for large eddies whose dimensions are equal to or greater than the path length of the ray because the ray path is over a region containing a systematic component of the gradients associated with the large eddy. We can obtain a worst-case estimate of these effects by examining a horizontal ray propagating from a source at the center of an updraft of 30 m/sec through 2 km to the cloud boundary where the vertical velocity is assumed to be zero; we also assume a linear decrease in vertical velocity between the center and boundary. The ray will be "advected" by 90 m upward during this transit, which requires approximately 6 sec, while the direction of propagation of the ray will be rotated through 5 downward (maximum angle tanâ1 â V/C). Owing to this rotation, which is a maximum computation, the "apparent" source by straight-ray path would be 180 m above the real source. These two effects have been estimated independently when, in fact, they are coupled and are to some extent compensatory; when we merely add them the result is an overestimate of the apparent source shift, which in this example is 270 m. If this worst case is the total error in propagation to the receiver at 5 km then this error represents 5 percent of the range; over the length in which it occurs, 2 km, it represents 13 percent error. Now we turn our attention to the large-scale refraction effects that can be incorporated in an atmospheric model that employs horizontal stratification. The two strongest refractive effects of the atmosphereâthe vertical thermal gradient and boundary-layer wind shearsâfall into this category along with other winds and wind shears of less importance. The nearly adiabatic thermal structure of the atmosphere during thunderstorm conditions has been recognized for a long time as a strong influence on thunder propagation (Fleagle, 1949). This thermal gradient is effective because it is spatially persistent and unavoidable. Even though the temperature in updrafts and downdraftsâinside and outside the cloudâmay differ (sometimes significantly), the thermal gradients in all parts of the system will be near the adiabatic limit (or pseudoadiabatic in some cases) because of the vertical motion. Hence, the acoustic rays propagate in this strong thermal gradient throughout its existence. We can employ a simplified version of ray theory to illustrate some of the consequences of this thermal structure. If we assume no wind, a constant lapse rate (Î = â âT/âz), and â T/T0 << 1 (â T is the change in temperature and T0 is the maximum temperature along the path), then the ray path may be described as a segment of a parabola In Eq. (4.8), T0 also corresponds to the vertex of the parabola where the ray slope passes through zero and starts climbing. h and l are respectively the height above the vertex and the horizontal displacement from the vertex. To apply Eq. (4.8) to all rays it is necessary to ignore (mathematically) the presence of the ground because the vertices of rays reflecting from the ground are mathematically below ground. In addition, we must in other cases visualize rays extending backward beyond the source to locate their mathematical vertices. If T 0 is set equal to the surface temperature, a special acoustic ray that is tangent to the surface when it reaches the surface is defined; this is depicted in Figure 4.10. This same ray is applicable to any source, such as S1, S2, or S3, that lies on this ray path. For the conditions assumed in this approximation it is not possible for rays from a point source to cross one another (except those that reflect from the surface). The other acoustic rays emanating from S2 must pass over the point on the ground where the tangent ray makes contact; this is also true for rays reflecting from the surface inside the tan