National Academies Press: OpenBook

The Earth's Electrical Environment (1986)

Chapter: Shock-Wave Formation and Expansion

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Suggested Citation:"Shock-Wave Formation and Expansion." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
Page 46
Suggested Citation:"Shock-Wave Formation and Expansion." National Research Council. 1986. The Earth's Electrical Environment. Washington, DC: The National Academies Press. doi: 10.17226/898.
Page 47

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ACOUSTIC RADIATIONS FROM LIGHTNING 46 4 Acoustic Radiations from Lightning Arthur A. Few, JR. Rice University ACOUSTIC SOURCES IN THUNDERSTORMS Electric storms produce a variety of acoustic emissions. The acoustic emissions can be broadly divided into two categories—those that are related to electric processes (i.e., they correlate with lightning) and those that either do not depend on cloud electricity or for which no correlations with electric changes have been observed. Only the first group will be discussed here (see Few, 1982; Georges, 1982, for reviews of nonelectrical acoustics). Two types of acoustic emissions correlated with electric processes are thunder, which is produced by the rapidly heated lightning-discharge channel, and infrasonic emissions produced by electrostatic fields throughout the charged regions of the cloud. Thunder is probably the most common of all loud natural sounds, while other acoustic emissions are not ordinarily observed without special devices. THUNDER—THE RADIATION FROM HOT CHANNELS Spectrographic studies of lightning return strokes (Orville, 1968) show that this electric-discharge process heats the channel gases to a temperature in the 24,000 K range. At high temperatures the expansion speed of the shock wave is roughly 3 × 103 m/sec and decreases rapidly as the shock wave expands; in comparison the measured speeds for various lightning-breakdown processes range from 104 to 108 m/sec (Uman, 1969; Weber et al., 1982). Therefore, the electric breakdown process in a discharge event is completed in a given length of the channel before the hydrodynamic responses are fully organized. Other electric processes occur over longer periods (e.g., continuing currents), but the energy input to the hot channel is strongly weighted toward the early breakdown processes when channel resistance is higher (Hill, 1971). Shock-Wave Formation and Expansion The starting point for developing a theory for a shock-wave expansion to form thunder is the hot ( ~ 24,000 K), high-pressure (> 106 Pa) channel left by the electric discharge. Hill's (1971) computer simulation indicated that approximately 95 percent of the total channel energy is deposited within the first 20 µsec with the peak electric power dissipation occurring at 2 µsec; during the 20-µsec period of electric energy input, the shock wave can only move approximately 5 cm. This simulation may actually be slower than real lightning because Hill used a slower current rise time than indicated by

ACOUSTIC RADIATIONS FROM LIGHTNING 47 more recent measurements (Weidman and Krider, 1978). The time-resolved spectra of return strokes (Orville, 1968) show the effective temperature dropping from ~ 30,000 to ~ 10,000 K in a period of 40 µsec and the pressure of the luminous channel dropping to atmospheric in this same time frame. During this period the shock wave can expand roughly 0.1 m. Even though channel luminosities and currents can continue for periods exceeding 100 µsec, the processes that are important to the generation of thunder occur very quickly (<10 µsec) and in a very confined volume (radius ~ 5 cm). The strong shock wave propagates outward beyond the luminous channel, which returns to atmosphere pressure within 40 µsec. The channel remnant cools slowly by conduction and radiation and becomes nonconducting at temperatures between 2000 and 4000 K perhaps 100 msec later (Uman and Voshall, 1968). Turning our attention now to the shock wave itself we can divide its history into three periods—strong shock, weak shock, and acoustic. The division between strong and weak can be related physically to the energy input to the channel; the weak-shock transition to acoustic is somewhat arbitrary. Calculations and measurements have shown that the radiated energy is on the order of 1 percent of the total channel energy (e.g., Uman, 1969; Krider and Guo, 1983), hence most of the available energy is in the form of internal heat energy behind the shock wave. As the strong shock wave expands it must do thermodynamic work (Pd V) on the surrounding fluid. The expected distance though which the strong shock wave can expand will be the distance at which all the internal thermal energy has been expended in doing the work of expansion. Few (1969) proposed that this distance, which he called the ''relaxation radius," would be the appropriate scaling parameter for comparing different sources and different geometries. The expressions for the spherical, R s , and cylindrical, R c , relaxation radii are where E t is the total energy for the spherical shock wave, El is the energy per unit length for the cylindrical shock wave, and Po is the environmental atmospheric pressure. Table 4.1 gives R c over a range of values that have been suggested in the literature for E l . Nondimensional distances denoted by X may be defined for spherical problems by dividing by Rs and for cylindrical problems by dividing by R c. Figure 4.1 shows the propagation of the strong shock into the transition region (X ~ 1) and beyond into the weak-shock region. As the shock front passes the relaxation radius (X = 1) the central pressure falls below ambient pressure as postulated in the definition of the relaxation radius. The momentum gained by the gas during the expansion carries it beyond X = 1 and forces the central pressure to go momentarily below atmospheric. At this point the now weak-shock pulse decouples from the hot- channel remnant and propagates outward. Figure 4.2 shows on a linear coordinate system the final output from Brode's (1955) numerical solution, the weak-shock pulse at a radius of X = 10.5 Figure 4.3 shows Plooster's (1968) cylindrical shock wave near X = 1 with Brode's (1955) spherical shock wave. The effects of channel tortuosity will be discussed Figure 4.1 The expansion of spherical and cylindrical shock waves from the strong-shock region into the weak- shock region. The radii of both spherical and cylindrical geometries have been nondimensionalized using the relaxation radii defined in Eqs. (4.1) and (4.2). The spherical shock wave is that of Brode (1956), and the cylindrical shock wave is from a similarity solution by Sakurai (1954).

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This latest addition to the Studies in Geophysics series explores in scientific detail the phenomenon of lightning, cloud, and thunderstorm electricity, and global and regional electrical processes. Consisting of 16 papers by outstanding experts in a number of fields, this volume compiles and reviews many recent advances in such research areas as meteorology, chemistry, electrical engineering, and physics and projects how new knowledge could be applied to benefit mankind.

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