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OCR for page 46
4
Acoustic Radiations from Lightning
ARTHUR A. FEW, JR.
Rice University
ACOUSTIC SOURCES IN THUNDERSTORMS
Electric storms produce a variety of acoustic emis-
sions. 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 elec-
tric processes are thunder, which is produced by the rap-
idly 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
46
shock wave is roughly 3 x 103 m/see and decreases rap-
idly as the shock wave expands; in comparison the mea-
sured speeds for various lightning-breakdown processes
range from 104 to 108 m/see (U~man, 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 or-
ganized. Other electric processes occur over longer pe-
riods (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 dis-
charge. Hill's (1971) computer simulation indicated
that approximately 95 percent of the total channel en-
ergy is deposited within the first 20 ,usec with the peak
electric power dissipation occurring at 2 ,usec; during
the 20-,usec period of electric energy input, the shock
wave can only move approximately 5 cm. This simula-
tion may actually be slower than real lightning because
Hill used a slower current rise time than indicated by
OCR for page 47
ACOUSTIC RADlATlONS FROM LIGHTNING
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 ,usec and the
pressure of the luminous channel dropping to atmo-
spheric 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 pe-
riods exceeding 100 ,usec, the processes that are impor-
tant to the generation of thunder occur very quickly
~ < 10 ,usec) and in a very confined volume (radius ~ 5
cm). The strong shock wave propagates outward be-
yond the luminous channel, which returns to atmo-
sphere pressure within 40 ,usec. 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 en-
ergy is in the form of internal heat energy behind the
shock wave.
As the strong shock wave expands it must do thermo-
dynamic work (PdV) on the surrounding fluid. The ex-
pected 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, Rs' and cylindrical, Rc' relaxation radii are
Rs= (3Et14~Po)~/3
and
Rc= (EiI~Po)i'2,
(4.1)
(4.2)
where E' is the total energy for the spherical shock wave,
Ei is the energy per unit length for the cylindrical shock
wave, and P0 is the environmental atmospheric pres-
sure. Table 4.1 gives RC over a range of values that have
been suggested in the literature for En. Nondimensional
distances denoted by X may be defined for spherical
problems by dividing by Rs and for cylindrical problems
by dividing by RC.
Figure 4.1 shows the propagation of the strong shock
47
TABLE 4.1 Relaxation Radii (R..) (in meters) for Different
Energies per Unit Length (El) of Cylindrical Shock Waves
P(,= lOOkPa
( ~ surface)
0.18
0.25
0.40
0.56
0.80
1.26
1.78
Pi, = 60 kPa Pi, = 30 kPa
( ~ 4-km height) ( ~ 9-km height)
0.23
0.32
0.52
0.72
1.03
1.63
2.30
104
2x 104
5x 104
105
2x 105
5x 105
lO'5
0.33
0.46
0.73
1.02
1.46
2.30
3.25
into the transition region (X ~ 1) and beyond into the
weak-shock region. As the shock front passes the relaxa-
tion radius (X = 1) the central pressure falls below am-
bient pressure as postulated in the definition of the re-
laxation radius. The momentum gained by the gas
during the expansion carries it beyond X = 1 and forces
the central pressure to go momentarily below atmo-
spheric. At this point the now weak-shock pulse decou-
ples from the hot-channel remnant and propagates out-
ward. 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
50
20
It
Con 5 _
14
ly
,!
//, /
1 ~
l ~
_ _ I
l
_ ~ .
~ 1
/ 1
/ ~
l
1
1
SPHERICAL SHOCK WAVE
CYLINDRICAL SHOCK WAVE
,1'}
~1
l
id .
1
1
X
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 nondimensiona-
lized 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).
OCR for page 48
48
p/po
. ,
.03
.02
.01
.00
.99
.98 ~
7 8 9
X
10 11
FIGURE 4.2 The weak shock wave formed from the spherical strong
shock wave. This is the final pressure profile computed by Brode
(1956). For an energy input of 105 Jim (R. = 0.56 m for P.`, = 105 Pa)
this weak shock wave would be approximately 6 m from the lightning
channel.
in greater detail later; for now we note that owing to
tortuosity we cannot expect the shock wave to continue
to perform as a cylindrical wave once it has propagated
beyond a distance equal to the effective straight section
of the channel that generated it. If the transition from
cylindrical to spherical occurs near X = 1 as suggested
by Few (1969), then the spherical weak-shock solutions
of Brode provide a good means of estimating the wave
shapes of lightning-caused acoustic pulses.
Figure 4.4 presents a graphical summary of the vari-
ous transitions that are thought to take place. The initial
strong shock will behave cylindrically following the
dashed line based on Plooster's (1968) computations;
this must be the case for the line source regardless of the
tortuosity because the high-speed internal waves (3 x
103 m/see) will hydrodynamically adjust the shape of the
channel during this phase. The transition from strong
shock to weak shock occurs near X = 1, and the transi-
tions from cylindrical divergence to spherical diver-
gence will occur somewhere beyond X = 0.3 and proba-
bly beyond X = 1 depending on the particular geometry
of the channel at this point. The family of lines labeled x
in Figure 4.4 represent transitions occurring at different
points. x is the effective length, L, of the cylindrical
source divided by Rig (x = LIRC); it is approximately
equal to the value of X at which the transition to spheri-
cal divergence takes place.
Comparisons with Numerical Simulations and
Experiments
In the numerical solutions of Plooster (1971a, b) and
Hill (1971) the energy inputs to the cylindrical problem
ARTHUR A. FEW, JR.
were computed as a function of time for specified cur-
rent wave shapes and channel resistance obtained from
the computations in the numerical model. These model
results predicted that the energy input to the lightning
channel was an order of magnitude or more below the
values obtained from electrostatic estimates or from
other indirect measurements of lightning energy (Few,
1982~. The major differences might be due to the as-
sumed current wave forms used in the models. The re-
cent data obtained with fast-response-time equipment
yields current rise times for natural cloud-to-ground
lightning in the 35-50 kA/,usec range (Weidman and
Krider, 1978~. These values are considered as represen-
tative of normal strokes; extraordinary strokes have
been measured with current rise times in the 100-200
kA/,usec range. By way of comparison, Hill's (1971) cur-
rent rise time was 2.5 kA/,usec.
Laboratory simulations of lightning have been suc-
cessfully performed in a series of experiments conducted
at Westinghouse Research Laboratories; these results
provide us with our best quantitative information on
thunder generation. In these tests a 6.4 x 106 V impulse
generator was used to produce 4-m spark discharges in
air (Uman et al., 1970~. Circuit instrumentation al-
lowed the measurement of the spark-gap voltage and
current from which the power deposition can be com-
puted. Calibrated microphones were used to measure
the shock wave from the spark as a function of distance.
The results of the research (Uman et al., 1970) have been
compared with the theory of Few (1969) and with other
2.2 t
_ _
o
CL 1.8
0
-
~ 1.4
a)
.=
In
In
~ 1.0
-Spherical Pressure Wave /:
~~Cylindrical Pressure Wave /,
.4
~ {'
ll
.8
I.C
1~2
FIGURE 4.3 Comparison of spherical and cylindrical shock-wave
shapes near X = 1. These profiles are for the point-source, ideal-gas
solutions of Brode (1955) and Plooster (1968). In the transition region
of strong shock to `` eak shock, these w eve shapes are nearly identical.
From Fee (1969) faith permission of the American Geophysical
Union.
OCR for page 49
ACOUSTIC RADIATIONS FROM LIGHTNING
possible interpretations (Plooster, 1971a). The data
were found to be consistent with the theory developed
by Few.
Figure 4.5 compares a measured spark-pressure pulse
with the profile that is predicted from the theory; both
represent conditions in the plane perpendicular to the
spark channel. Figures 4.6 and 4.7 summarize the exten-
sive series of spark measurements. Figure 4.6 is in the
same format as Figure 4.4. The center line passing _.Ol
through the scattered points and labeled L = 0.5 m cor
responds (using the measured energy input of 5 x 103
him, which gives RC = 0.126 m) to x = 4 in Figure 4.4.
The two boundary lines L = 6.25 cm and L = 4.0 m
would correspond to x values 0.5 and 32. The lower
bound is very close to the lower limit value of one third
indicated in Figure 4.4. The upper bound of Figure 4.6
(x ~ 32) is too large to be depicted in Figure 4.4, where
x = 4 is the last line shown.
The data points of Figure 4.6 corresponding to the
larger x or L values could represent situations where the
shock-wave expansion was following the cylindrical be-
havior over a long distance, hence large x. However, if
the expansions were truly cylindrical to that extent, then
to 100 _
~ to
= ~.
-
o
10
o
in
~ of\ -Spherical Divergence
- \ --Cylindrical Divergence
Lower bound for ^~
.01 ~ ~ ~ ~ ~ ~ ~ ~ 1
.1 .2 .5 1
X
FIGURE 4.4 Line-source shock-wave expansion. The overpressure
of the shock front is given for spherical (Brode, 1956) and cylindrical
(Plooster, 1968) shock waves. Line sources must initially follow cylin-
drical behavior, but on expanding to distances of the same size as line
irregularities they change to spherical expansion following curves simi-
lar to the depicted curves. From Few (1969) with permission of the
American Geophysical Union.
49
.02- |t Pressure Profile
Co .ol- ~ \: Spark (5 X 103 j/m) at 3 meters
at: ~\\Predicted profile
> ~ ~_. TIME
.! .2 .3 .4 .5 .6 .7 .S .9
m see
FIGURE 4.5 Comparison of theory with a pressure wave from a
long spark. The measured pressure wave from a long spark (Uman et
al., 1970) is compared with the predicted pressure from a section of a
mesotortuous channel having the same energy per unit length. x is
assumed to be 4/3. From Few (1969) with permission of the American
Geophysical Union.
the length of the pulse would be longer, as required by
the cylindrical-wave predictions. The data of Figure 4.7
indicate that this cannot be the case. The lengths of the
positive-pressure pulses shown in Figure 4.7 are clearly
not in the cylindrical regime; if anything, they tend to
be even shorter than predicted by the spherical expan-
sion. (See also Figure 4.5.)
It is obvious from both the spark photographs and
wave forms in Uman et al. (1970) that the spark is tortu-
ous and produces multiple pulses. They found that the
wave shapes, more distant from the spark where pulse-
transit times were most similar, showed evidence of an
in-phase superposition of pulses; at closer range the
pulses exhibited greater relative phase shifts and more
multiplicity aspects. The in-phase superposition of
spherical waves would reproduce the distributions
shown in Figures 4.6 and 4.7. The pressure amplitude
would be increased relative to a single pulse, but the
wavelength would not be substantially affected.
The measured spark wave forms (Uman et al., 1970)
were systematically shorter than predicted by the the-
ory. As shown in Figure 4.5, the tail of the wave was
compressed, and the data of Figure 4.6 indicate that the
positive pulse was similarly shortened. This shortening
could be due simply to an inadequacy in the numerical
shock-wave model; we think, instead, that the differ-
ence results from the energy input being instantaneous
in the one case (Brode, 1956) and of longer duration for
the spark case. If energy, even in small quantities, con-
tinues to be input into the low-density channel core after
the shock front has moved outward then the core will be
kept at temperatures much higher than predicted by the
theories, having an instantaneous energy input followed
by expansion. Owing to the elevated sound speed associ-
ated with the higher core temperature the part of the
OCR for page 50
so
1 n
.010
nn
_ \ 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 _
~ \\
_\ \N
\ \ ~ Plooster, 3
- \ <~Cylinder, W=5.0xlO J/m
\ i\
_ \- .~. N~ _
I; . a\ ~` Brode, Sphere,
- / N;:# IN . \ WL =2.0x 10 J -
B rode, \, ~ . < ` L = 4. 0 m
Sphere, V ~ +\ ~ /
WL=3.1X102J \~. i+ +V~`
L =6.25 cm >by If \ ~
B rode, \t it\+ \ ~ ``
Sphere, +: ~ '\ \ `~
WL=2.5x103J I \ \ ~
L=0.5m +~ \:
\ +N
I ~
1 1 1 1 1 11 1 \
10 30
1 1 1 1 1 1
1 111 1 1
.
0.1 1.0
Distance, meters
FIGURE 4.6 Shock-front overpressure as a function of distance from
the spark. The dots represent data obtained with a piezoelectric micro-
phone; the crosses data obtained with a capacitor microphone. The
total electric energy per unit length computed from measurement of
the spark voltage and current is 5 x 103 J/m. Also shown are theoreti-
eal values for cylindrical and spherical shock waves. From Uman et al.
(1970).
0.7 ~ ~ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1
0.6 _
=. 0.4
-
-
c 0.3
to
to
-
`= 0. 2
1
C
PlOoster. ~~/
Cylinder /
W =5.0xlO3J/m / /
/ /
/~ Brode. 4
a,' Sphere, WL=2.0xlO J
L =4.0m
4 _
1 _
O .......
0.1 1.0
Distance, meters
~T
/ / Brode, +
/// Sphere, WL=2.5xlO J +
L=0.5 ~
1 ~ + +
+
+
B rode, 2
Sphere, WL = 3.1 x 10 J
L=6.25cm
1 1 1 1 ~1 1 1 1 ~
+
+
I ~ 1 1 1 1 ,
10 ' 30
FIGURE 4.7 Duration of positive part of the shock wave from the
long.spark. For the same data of Figure 4.6, we see here the length of
the positive pressure pulse for the 5 x 103 J/m sparks at various dis-
tances. From Uman et al. ( 1970) .
ARTHUR A . FEW, JR.
wave following the shock front will form and propagate
outward faster than predicted by theory. We expect,
therefore, that the elevated core temperature associated
with sparks and lightning can reasonably produce the
shortened wave forms.
The wave shape produced by the shock wave is re-
lated to the energy per unit length of the lightning flash;
thunder is superposition of many such pulses from the
lightning channel; hence, the power spectrum of the
thunder, with simplifying assumptions, can be related
to the energy per unit length of the channel (Few, 1969~.
Other properties (tortuosity and attenuation) that influ-
ence the spectrum of thunder are discussed later.
The assumptions in this theory all affect the thunder
spectrum in the same sense; the peak of the theoretical
spectrum will occur at higher frequencies than the peak
of the real thunder spectrum (Few, 1982~. The light-
ning-channel energy that one estimates from the peak
will therefore be an overestimate of the actual lightning-
channel energy. Holmes et al. (1971) provided the most
complete published thunder spectra to date; these spec-
tra show a lot of variation. Most of the spectra are con-
sistent with the qualitative expectations of thunder pro-
duced by multiple-stroke lightning, but a few of them
exhibit very-low-frequency (< 1 Hz) components that
are dominant during portions of the record and appear
to be totally inconsistent with the thunder-generation
theory from the hot explosive channel. Dessler (1973),
Bohannon et al. (1978), and Balachandran (1979) sug-
gested that these lower-frequency components might be
electrostatic in origin; Holmes et al. (1971) also consid-
ered that this was a possible explanation.
Tortuosity and the Thunder Signature
With respect to the effects of lightning-channel tortu-
osity on the thunder signal there is almost unanimous
agreement among researchers. Lightning channels are
undeniably tortuous and are tortuous apparently on all
scales (Few et al., 1970~. For convenience in discussing
channel tortuosity Few (1969) employed the terms mi-
crotortuosity, mesotortuosity, and macrotortuosity rel-
ative to the relaxtion radius of the lightning shock wave.
For a lightning channel having an internal energy of
105 l/m (see Table 4.1), Rc ~ 1/2 m. The microtortuous'
features smaller than Rc, although optically resolvable,
are probably not important to the shock wave as mea-
sured at a distance because the high-speed internal
waves (3 x 103 m/see) are capable of rearranging the
distribution of internal energy along the channel while
the shock remains in the strong-shock regime. At the me-
sotortuous scale ~ ~ Rc) the outward propagating shock
wave decouples from the irregular line source because
OCR for page 51
ACOUSTIC RADIATIONS FROM LIGHTNING
the acoustic waves from the extended line source can no
longer catch up with the shock wave. Somewhere in this
mesotortuous range the divergence of the shock waves
makes the transition from cylindrical to spherical.
Whereas the mesotortuous channel segments are im-
portant in the formation and shaping of the individual
pulses being emitted by the channel the macrotortuous
segments are fundamental to the overall organization of
the pulses and the amplitude modulation of the resulting
thunder signature. Few (1974a) computed that 80 per-
cent of the acoustic energy from a short spark was con-
fined to within + 30° of the plane perpendicular to the
short line source. A macrotortuous segment of a light-
ning channel will direct the acoustic radiations from its
constituent mesotortuous, pulse-emitting segments into
a limited annular zone. An observer located in this zone
(near the perpendicular plane bisecting the macrotortu-
ous segment) will perceive the group of pulses as a loud
clap of thunder, whereas another observer outside the
zone will perceive this same source as a lower-amplitude
rumbling thunder. This relationship between claps,
rumbles, and channel macrotortuosity has been con-
firmed by experiment (Few, 1970) and in computer sim-
ulations (Ribner and Roy, 1982~.
Loud claps of thunder are produced, as mentioned
above, near the perpendicular plane of macrotortuous
channel segments; there are three contributory effects
(Few, 1974a, 1975) to the formation of the thunder
claps. The directed acoustic radiation pattern described
above is one of the contributing factors, and this effect is
distributed roughly between + 30° of the plane.
A second effect, which occurs only very close to the
plane, is the juxtaposition of several pulses in phase,
which increases the pulse amplitude to a greater extent
than would a random arrival of the same pulses.
The third effect contributing to thunder clap forma-
tion is simply the bunching in time of the pulses. In a
given period of time more pulses will be received from a
nearly perpendicular macrotortuous segment of chan-
nel than from an equally long segment that is perceived
at a greater angle owing to the overall difference in the
travel times of the composite pulses.
In this section we have examined the complex nature
of the formation of individual pulses from hot lightning
channels and how a tortuous line source arranges and
directs the pulses to form a thunder signature. The re-
sulting thunder signature depends on (l) the number
and energy of each rapid channel heating event (leaders
and return strokes); (2) the tortuous and branched con-
figuration of the individual lightning channel; and (3)
the relative position of the observer with respect to the
lightning channel.
Perhaps the most convincing discussion of thunder
51
generation as described above comes not from analytical
evidence but from research using sophisticated com-
puter models of thunder. Ribner and Roy (1982) synthe-
sized thunderlike acoustic signals utilizing computer-
generated waves formed by the superposition of N
waveless from tortuous geometric sources. The resulting
"thunder" is highly similar to natural thunder (see Fig-
ures 4.8 and 4.9~. Where the computer models are used
to simulate laboratory experiments, there is also close
agreement.
PROPAGATION EFFECTS
Once generated, the acoustic pulses from the light-
ning channel must propagate for long distances through
the atmosphere, which is a nonhomogeneous, aniso-
tropic, turbulent medium. Some of the propagation ef-
fects can be estimated by modeling the propagation us-
ing appropriate simplifying assumptions; however,
other effects are too unpredictable to be reasonably
modeled and must be considered in individual situa-
tions.
Three of the largest propagation effects finite-am-
plitude propagation, attenuation by air, and thermal
refraction can be treated with appropriate models to
account for average atmosphere effects. Reflections
from the flat ground can also be easily treated. Once the
horizontal wind structure between the source and the
receiver are measured, the refractive effects of wind
shear and improved transient times may also be calcu-
lated. Beyond these effects, elements such as vertical
,
O - a. .L~/ u~ ~ _ ~ 4d~
~ 7b9~ \~-
in
~ .
> L ~ ~ ~
p'~
-ale
_~.0 .lIIl.~lll~lllillIlllllllllllllllllllllllll-~-lJlllllllllllllllllllli
0 ~ ~ taco 4n .~ ~ m . - .o
TIME, s
FIGURE 4.8 Schematic depiction of the synthetic generation of
thunder by computer by the superimposition (upper trace) of N wave-
lets from a tortuous line source (Ribber and Roy, 1982); the summed
signal is shown on the lower trace.
OCR for page 52
52
I.6 ~
a Ill.lll .
O ~ 41115111
Ct
G1 1.:
1 ~ I ~I I ~
1.4 1.6 1.8 ~0
TIME, s
FIGURE 4.9 Comparison of synthetic (upper trace) and real (lower
trace) thunder signals (Ribner and Roy, 1982).
winds, nonsteady storm-related horizontal winds, tur-
bulence, aerosol effects, and reflections from irregular
terrain produce complications that must be either ig-
nored or examined on a case-by-case basis.
Finite-Amplitude Propagation
As large-amplitude acoustic waves propagate
through air, theory predicts that the shape of the wave
must evolve with time. A single pulse will evolve to the
shape of an N wave (see, for example, the spark wave in
Figure 4.5~; further propagation of the wave produces a
lengthening of this N wave. The best theoretical treat-
ment of this process for application to the thunder prob-
lem is the one developed by Otterman (1959~. His for-
mulation addressed the lengthening of a Brode-type
pulse, such as Figure 4.2, from an initial length (Lo) at
an initial altitude (Ho) down to the surface; his treat-
ment differs from many others that do not include the
change of ambient pressure (P0) with altitude. Few
ARTHUR A . FEW, JR.
(1982) used the Otterman theory to develop an expres-
sion for the lengthening of acoustic pulses generated by
mesotortuous lightning-channel elements. The result
for the length of the positive-pressure pulse at the
ground, Lg. is given by
2 (`L 3/2 _ Lo3/2) =
+ 1 RoLoi/2Ho
4y
[ ( Rocose ) 2 H ] (4 3)
Ro is the distance from the channel to the front of the
pulse at the initial state where the fractional overpres-
sure at the pulse front is IIo = bPo/Po. The angle ~ is
measured between the acoustic ray path and vertical; By
is the ratio of specific heats; and Hg is the atmosphere
scale height.
Equation (4.3) provides the finite-amplitude stretch-
ing that should be applied to the waves predicted by
strong-shock theory. Uman et al. (1970) demonstrated
that pulse stretching occurred beyond Brode's final pres-
sure profile shown in Figure 4.2; we see this clearly in
Figure 4.7. Few (1969) used linear propagation beyond
the profile of Figure 4.2 to estimate the power spectrum
of thunder but commented that nonlinear effects may
be important. The need for application of nonlinear or
finite-amplitude theory to the thunder signal has been
voiced in a number of papers in addition to these men-
tioned above (e.g., Holmes et al., 1971; Few, 1975,
1982; Hill, 1977; Bass, 1980~.
If the Brode pressure pulse (shown in Figure 4.4) is
used as the initial condition for the finite-amplitude
propagation effect, the following values for input to Eq.
(4.3) are R0 = 10.46RC, Lo = 0.53RC, andII0 = 0.03. In
addition, if By = 1.4 and Hg = 8 x 103 m are used in Eq.
(4.3), the following equation is obtained:
L. = RCLO.386 + 0.147~1n: °
g ~ 10.46RC cos
16 X 103 ]3
Equation (4.4) has been used to generate the values in
Table 4.2. The relaxation radii (Rc) cover the entire
range of values for Rc in Table 4.1. Three values for ~ are
represented, as are three heights for the source. In gen-
eral, the finite-amplitude propagation causes a dou-
bling in the length of the positive pulse within the first
kilometer, but beyond this range the wavelength re-
mains approximately constant. The theory developed
by Otterman did not include attenuation of the signal;
because attenuation reduces wave energy, which in turn
OCR for page 53
ACOUSTIC RADIATIONS FROM LIGHTNING
TABLE 4.2 Finite-Amplitude Stretching of a Positive Pulse (Length, Lo) for a Range of Cylindrical Relaxation Radii (Rc),
Source Heights (Ho), and Angles (id, See Eq. (4.4)
53
Rc (m) 0.20 0.40 0.60 0.80 1.00 1.50 2.00 2.50 3.00 3.50
L,, (m) 0.11 0.21 0.32 0.42 0.53 0.80 1.06 1.33 1.59 1.86
= Do
Lg(m),H(,= lkm 0.24 0.45 0.65 0.84 1.03 1.49 1.93 2.35 2.77 3.17
L~(m),HO = 4km 0.26 0.49 0.71 0.93 1.14 1.66 2.16 2.65 3.12 3.59
L~(m),H`, = 8km 0.26 0.51 0.74 0.96 1.18 1.72 2.24 2.75 3.25 3.74
= 45°
L~(m),H,,= lkm 0.24 0.46 0.67 0.87 1.06 1.54 1.99 2.44 2.87 3.30
L(m), H., = 4km 0.26 0.50 0.73 0.96 1.18 1.71 2.22 2.73 3.22 3.71
L~(m),H,' = 8km 0.27 0 52 0.76 0.99 1.22 1.77 2.31 2.83 3.35 3.85
= 60°
Lg(m),H,,= lkm 0.25 0.47 0.69 0.89 1.10 1.59 2.06 2.52 2.98 3.42
L~(m),H,, = 4km 0.27 0.51 0.75 0.98 1.21 1.76 2.29 2.81 3.32 3.82
Lg(m),H,,= 8km 0.28 0.53 0.77 1.01 1.25 1.81 2.37 2.91 3.44 3.97
reduces the wave stretching, this theory should be
viewed as a maximum estimator of the pulse length.
The finite-amplitude propagation effect does, how-
ever, help to resolve the overestimate of lightning-chan-
nel energy made by acoustic power-spectra measure-
ments. Few (1969) noted that the thunder-spectrum
method yielded a value for E' that was an order of mag-
nitude greater than an optical measurement by Krider
et al. (1968~. By assuming a doubling in wavelength by
the finite-amplitude propagation, the energy estimate is
reduced by a factor of 4, bringing the two measurements
into a range of natural variations and measurement pre
. .
clslon.
Attenuation
There are three processes on the molecular scale that
attenuate the signal by actual energy dissipation; the
wave energy is transferred to heat. Viscosity and heat
conduction, called classical attenuation, represent the
molecular diffusion of wave momentum and wave in-
ternal energy from the condensation to the rarifaction
parts of the wave. The so-called molecular attenuation
results from the transfer of part of the wave energy from
the translational motion of molecules to their internal
molecular rotational and vibrational energy during the
condensation part of the wave and back out during the
rarifaction part of the wave. The phase lag of the energy
transfer relative to the wave causes some of the internal
energy being retrieved from the molecules to appear at
an inappropriate phase; thus it goes into heat rather
than the wave. These three processes can be treated the-
oretically within a common framework (Kinsler and
Frey, 1962; Pierce, 1981~. The amplitude of a plane
wave, ~ P. as a function of the distance, x, from the coor-
dinate origin is given by
hP= bPOe~~X,
(4.5)
where TYPO is the wave amplitude at the origin. The coef-
ficient of attenuation, cat, can be shown in the low-fre-
quency regime to be
<,,2
2e
(4.6)
In Eq. (4.6), ~ is the angular frequency and T iS the re-
laxation time (or e-folding time) for the molecular pro-
cess being considered; c is the speed of sound. The low-
frequency condition above assumes that cur < 1. The
expressions for depend on the particular molecular pro-
cesses under consideration; it is important to note, how-
ever, that cat is proportional to w2 for the assumed condi-
tions; hence, attenuation alters the spectral shape of the
propagating signals.
For thunder at frequencies below 100 Hz it can be
shown (Few, 1982) that the total attenuation is insignifi-
cant. However, for the many small branches having
much lower energy than the main channel, the frequen-
cies will be much higher and attenuation is important.
Because of lower initial acoustic energies, spherical di-
vergence, and attenuation it is unlikely that acoustics
emitted by the smaller branches and channels can be
easily detected over longer distances (see also Bass, 1980;
Arnold, 1982~.
Scattering andAerosol Effects
The scattering of acoustic waves from the cloud parti-
cles is similar to the scattering of radar waves from the
particles; both are strongly dependent on wavelength.
The intensity of the scattered sound waves from a plane
acoustic wave of wavelength, )\' incident on a hard sta
OCR for page 54
54
tionarv.snhere of rarli'~.s ra
, ~ is proportional to (brat (alkyd;
this is the same relationship that appears in the radar
cross-section expression for these parameters. For thun-
der wavelengths ~ ~ 1 m3 and cloud particles ~ ~ 10 - 3 m)
the ratio (alkyd is 10- ]2. The cloud is, therefore, trans-
parent to low-frequency thunder just as it is to meter-
wavelength electromagnetic radiation, although insig-
nificant fractions of the radiation do get scattered.
There are, however, eddies in the same size range as
low-frequency thunder wavelengths, and these fea-
tures, 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 de-
scribed with geometric acoustics. For the low-frequency
thunder, turbulent scattering will attenuate the high-
amplitude beamed parts of the thunder signal; this in-
creases 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 parti-
cles; this changes the local vapor-to-liquid or vapor-to-
solid conversion rates. For example, during the com-
pressional part of the wave the air temperature is
increased and the relative humidity is decreased relative
to equilibrium; the droplets partially evaporate in re-
sponse and withdraw some energy from the wave to ac-
complish it. The opposite situation occurs during the ex-
pansion 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 vis-
cosity" 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 conduc-
tion 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 ampli-
tude of the fluid displacement, (, produced by an acous
ARTHUR A. FEW, JR.
tic wave of pressure amplitude lip and angular fre-
quency ~ is (Kinsler and Frey, 1962)
it=
hip
.
po c cd
(4.7)
Using 50 Pa as a representative value of lip for thunder
inside a cloud we find for a 100-Hz frequency that ~ =
100 ,um. The part of the cloud particle population whose
diameter is much smaller than this, say 10 ,um, should,
owing to viscous drag, come into dynamic equilibrium
with the wave flow. [Dessler (1973) computed the re-
sponse time for a ~ 10-,um droplet to re-establish dy-
namic equilibrium with drag forces; only 10 - 3 see is re-
quired. ~ 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 impe-
dence of the medium. For higher-frequency waves,
fewer cloud particles participate, so the effect is re-
duced; whereas lower-frequency waves include greater
percentages of the population and are more strongly af-
fected. 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. As-
suming a total water content of order 5 g/m3, we esti-
mate that the order of magnitude of the effect on sound
speed and impedence is 10 ~ 3; 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" altering the wave-prop-
agation 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 ef-
fectively attenuate higher-frequency components of
thunder; this is in support of the conclusions of the pre-
vious section. We have, in addition, found three pro-
cesses 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 inter-
act with the cloud population dynamically; as a result,
cloud boundaries may act as partial reflectors and in-
cloud propagation may be dispersive.
Refraction
There is a wide range of refractive effects in the envi-
ronment of thunderstorms. In the preceding section we
OCR for page 55
ACOUS TIC RADIA TIONS FROM LIGHTNING
found that turbulence on the scale of the acoustic wave-
length and smaller could be treated with scattering the-
ory. Turbulence larger than acoustic wavelengths, up to
and including storm-scale motions, should be describ-
able by geometric acoustics or ray theory. To actually do
this is impractical because it requires detailed informa-
tion (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 acous-
tic 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 adia-
batic, we do not expect that the thermal perturbations
due to turbulence will be systematic. In fact, the turbu-
lent 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 gra-
dients 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/see through 2 km to the cloud
boundary where the vertical velocity is assumed to be
zero; we also assume a linear decrease in vertical veloc-
ity between the center and boundary. The ray will be
"adverted" by 90 m upward during this transit, which
requires approximately 6 see, while the direction of
propagation of the ray will be rotated through 5 down-
ward (maximum angle ~ tan- ~ /\ VIC). Owing to this
rotation, which is a maximum computation, the "ap-
parent" source by straight-ray path would be 180 m
above the real source. These two effects have been esti-
mated 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
55
km then this error represents 5 percent of the range; over
the length in which it occurs, 2 km, it represents 13 per-
cent error.
Now we turn our attention to the large-scale refrac-
tion effects that can be incorporated in an atmospheric
model that employs horizontal stratification. The two
strongest refractive effects of the atmosphere the ver-
tical thermal gradient and boundary-layer wind
shears fall into this category along with other winds
and wind shears of less importance.
The nearly adiabatic thermal structure of the atmo-
sphere during thunderstorm conditions has been recog-
nized for a long time as a strong influence on thunder
propagation (Fleagle, 1949~. This thermal gradient is
effective because it is spatially persistent and unavoid-
able. Even though the temperature in updrafts and
downdrafts inside and outside the cloud may 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 struc-
ture. If we assume no wind, a constant lapse rate (F =
- ~ Tl~z), and is Tl To << 1 (A T is the change in temper-
ature and To is the maximum temperature along the
path), then the ray path may be described as a segment
of a parabola
2 = 4To h.
(4.8,
In Eq. (4.8), To also corresponds to the vertex of the
parabola where the ray slope passes through zero and
starts climbing. h and I 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 be-
cause 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 To 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
So, 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
OCR for page 56
56
FIGURE 4.10 Parabolic acoustic ray from
sources So, S. or S3 tangent to the surface at P.
This ray was generated utilizing Eq. (4.8)
with T.`, = 30°C and ~ = 9.8 K/km. Observ-
ers on the surface to the right of P cannot de-
tect sound from sources So, S. S3, or S.'; an
observer at P can only detect sound originat-
ing on or above the parabolic ray shown.
ARTHUR A . FEW, JR.
l
-
o Ss
SO - ~
-
~. . . . . . , , . , , , , ,,77717
S 0 5 10
40 3S 30
gent point. The shaded zone in Figure 4.10 corresponds
to a shadow zone that receives no sound from any point
source on the tangent ray beyond the tangent point.
Point sources below the tangent ray, such as source S4 in
Figure 4.10, have their tangent ray shifted to the left in
this representation and similarly cannot be detected in
the shadow zone. However, sources above the tangent
ray, S5 for example, can be detected in some parts of the
shadow zone.
For each observation point on the ground one can de-
fine a paraboloid of revolution about the vertical gener-
ated by the tangent ray through the observation point;
the observer can only detect sounds originating above
this parabolic surface. For this reason we usually hear
only the thunder from the higher parts of the lightning
channel unless we are close to the point of a ground
strike. For evening storms, which can often be seen at
long distances, it is common to observe copious li~ht-
ning activity but hear no thunder at all; thermal refrac-
tion is the probable cause of this phenomenon. For To =
30°C, 1~ = 9.8K/km,andh = 5kmwefindthatl= 25
km; as noted by Fleagle (1949) thunder is seldom heard
beyond 25 km. (See also the discussion in Ribner and
Roy, 1982.)
Winds and wind shears also produce curved-ray
paths but are more difficult to describe because they af-
fect the rays in a vectorial manner, whereas the temper-
ature was a scalar effect. If you are downwind of a
source and the wind has positive vertical shear (bu/3z >
0), the rays will be curved downward by the shear; on
the upwind side, the rays are curved upward. Wind
shears are very strong close to the surface and can effec-
tively bend the acoustic rays that propagate nearly par-
allel to the surface. The combined effects of tempera-
ture gradients, winds, and wind shears can best be
handled with a ray-tracing program on a computer.
With such a program one can accurately trace ray paths
through a multilevel atmosphere with many variations
in the parameters; it is usually necessary in these pro-
grams to assume horizontal stratification of the atmo-
sphere. The accuracy of the ray tracing by these tech
2S 20 t 5 tO
Distance in km
A
,0;
i_
s ~
._
-
niques can be very high, usually exceeding the accuracy
with which temperature and wind profiles can be deter-
mined.
MEASUREMENTS AND APPLICATIONS
A number of the experimental and theoretical re-
search papers dealing with thunder generation have
been discussed in earlier sections and will not be re-
peated here. In this section we describe additional
results, techniques, and papers that deal with thunder
measurements.
Propagation Effects Evaluation
The reader should have, at this point, an appreciation
for the difficulty in quantitatively dealing with the
propagation effects on both the spectral distribution of
thunder and the amplitude of the signal. If, however,
we are willing to forfeit the information content in the
higher-frequency ~ > 100 lIz) portion of the thunder sig-
nal, which is most strongly affected by propagation, we
can recover some of the original acoustic properties
from the low-frequency thunder signal.
If the peak in the original power spectrum of thunder
is assumed to be below 100 Hz, then the "2-attenuation
effects deplete the higher frequencies without shifting
the position of the peak. Most spectral peaks of thunder
tend to be around or below 50 Hz; therefore, this as-
sumption appears to be safe even with finite-amplitude
stretching effects considered. Further assume that the
spectra are not substantially altered by turbulent scat-
tering and cloud aerosols. To the extent that these as-
sumptions are valid, the finite-amplitude stretching can
be removed from the thunder signal and its peak fre-
quency at the source can be estimated. This technique
enables a rough estimate of the energy per unit length of
the stroke to be made; the result is corrected for first-
order propagation effects. Holmes et al. (1971) found
that the spectral peak overestimated the channel energy
using Few's (1969) method; if corrected for stretching
OCR for page 57
A CO US TIC RADIA TIONS FR OM LIGH TNING
these measurements are in closer agreement, with the
exception of those events containing other lower-fre-
quency acoustic sources.
There are a number of experiments that could and
should be done to evaluate the propagation effects. Us-
ing thunder as the acoustic source, several widely sepa-
rated arrays of microphones could compare signals from
the same source at several distances. If carefully exe-
cuted this experiment could quantify some of the propa-
gation effects. Another approach would be to employ a
combination of active and passive experiments such as
point-source explosions inside clouds from either bal-
loons or rockets. This experiment provides an additional
controllable factor that can yield more precise data; it
also involves greater cost and hazard.
Acoustic Reconstruction of Lightning Channels
In the section on refraction we mentioned the utility
of ray-tracing computer programs that could accurately
calculate the curved path of an acoustic ray from its
source to a receiver; the accuracy is limited to the preci-
sion with which we are able to define the atmosphere.
An obvious application of thunder measurements is to
invert this process; one measures thunder then traces it
backward from the point of observation along the ap-
propriate ray to its position at the time of the flash. Few
(1970) showed that by performing this reverse-ray prop-
agation for many sources in a thunder record it was pos-
sible to reconstruct in three dimensions the lightning
channel producing the thunder signal. The sources in
this case were defined by dividing the thunder record
into short ~ ~ 1/2 see) intervals and associating the acous-
tics in a given time interval with a source on the channel.
Within each time interval the direction of propaga-
tion of the acoustic rays are found by cross correlating
the signals recorded by an array of microphones. The
position of the peak in the cross-correlation fraction
gives the difference in time of arrival of the wave fronts
at the microphones; from this and the geometry of the
array, one calculates the direction of propagation. At
least three noncollinear microphones are required.
Close spacing of the microphones produces higher corre-
lations and shorter intervals thus more sources; how-
ever, the pointing accuracy of a small array is less than
that of a large array. Based on experiences with several
array shapes and sizes, 50 m2 has been adopted as the
optimum by the Rice University Group (see Few,
1974a).
The reconstruction of lightning channels by ray trac-
ing was described by Few (1970) and Nakano (1973~. A
discussion of the accuracy and problems of the tech-
nique is given in Few and Teer (1974) in which acousti
57
cally reconstructed channels were found to agree closely
with photographs of the channels below the clouds. The
point is dramatically made in these comparisons that the
visual part of the lightning channel is merely the "tip of
the iceberg."
Nakano (1973) reconstructed, with only a few points
per channel, 14 events from a single storm. Teer and
Few (1974) reconstructed all events during an active pe-
riod of a thunderstorm cell. MacGorman et al. (1981)
similarly performed whole-storm analyses by acoustic
channel reconstruction and compared statistics from
several different storm systems. Reconstructed lightning
channels by ray tracing have been used to support other
electric observations of thunderstorms at the Langmuir
Laboratory by Weber et al. (1982) and Winn et al.
(1978~.
A second technique for reconstructing lightning
channels has been developed that is called thunder rang-
ing. This technique was developed to provide a quick
coarse view of channels (within minutes after lightning
if necessary) as opposed to the ray-tracing technique,
which is slow and time consuming. Thunder ranging
requires thunder data from at least three noncollinear
microphones separated distances on the order of kilome-
ters. Experience with cross-correlation analysis of thun-
der signals has shown that the signals become spatially
incoherent at separations greater than 100 m owing to
differences in perspective and propagation path. How-
ever, the envelope of the thunder signals and the gross
features such as claps remain coherent for distances on
the order of kilometers. As discussed earlier these gross
features are produced by the large-scale tortuous sec-
tions of the lightning channel. Thunder ranging works
as follows: (1) The investigator identifies features in the
signals (such as claps) that are common to three thunder
signatures on an oscillograph. (2) The time lags between
the flash and the arrival of each thunder feature at each
measurement point are determined. (3) The ranges to
the lightning channel segments producing each thunder
feature are computed. (4) The three ranges from the
three separated observation points for each thunder fea-
ture define three spheres, which should have a unique
point in space that is common to all of them. (5) The set
of points gives the locations of the channel segments pro-
ducing the thunder features (see Few, 1974b; Uman et
al., 1978~.
The basic criticism of the thunder-ranging technique
is that the selection of thunder features is the subjective
judgment of the researcher; for many features the selec-
tion is unambiguous; other features, which are close to-
gether, may appear separated at one location and
merged at another. The program developed by Bohan-
non (1978) included these uncertainties in the estima
OCR for page 58
58
lion of errors associated with such points. Most of the
recent thunder research has used a combination of rang-
ing and ray tracing.
The whole-storm studies in which an extended series
of channels are reconstructed have proven to be the most
valuable use of thunder data to date. They define the
volume of the cloud actually producing lightning, the
evolution of the lightning-producing volume with time,
and the relationship of individual channels with other
cloud observations such as radar reflectivity and envi-
ronmental winds (Nakano, 1973; Few, 1974b; Teer and
Few, 1974; Few et al., 1977, 1978; MacGorman and
Few, 1978; MacGorman et al., 1981~.
ELECTROSTATICALLY PRODUCED
ACOUSTIC EMISSIONS
The concept of electrostatically produced acoustic
waves from thunderclouds goes all the way back to the
ARTHUR A . FEW, JR.
writings of Benjamin Franklin in the eighteenth cen-
tury; Wilson (1920) provided a rough quantitative esti-
mate of the magnitude of the electrostatically produced
pressure wave. McGehee (1964) and Dessler (1973) de-
veloped quantitative models for this phenomenon
McGehee for spherical symmetry and Dessler for spheri-
cal, cylindrical, and disk symmetries. The theory
developed by Dessler is of particular importance be-
cause it made specific predictions regarding the direc-
tivity and shape of the wave. The predictions were sub-
sequently verified in part by Bohannon et al. (1977) and
Balachandran (1979, 1983~.
The charge in a thundercloud resides principally on
the cloud drops and droplets. In a region of the cloud
where the charge is concentrated producing an electric
field E, the charged particles will experience an electric
force, which is directed outward with respect to the
charge center, in addition to the other forces expressed
on them. These particles quickly (on the order of milli
,._. , _ ~, ~'a ~ ~7 ~ ~' T i ! ~ ~ S !. ~t' ~' j j ~ ' ~
~' ~ ~ T ~' i i ' ~ ! ~ · i ; ~,
__ ~,! r ' i ~ ~ ~' ~~~~ ~ ~ ~
1 ~
__ - ~ = {_ ~
FIGURE 4.11 Low-frequency acoustic pulse thought to have been generated by an electrostatic pressure change inside the cloud during a light-
ning flash. The higher-frequency signals from thunder have been removed from this record. From Balachandran (1979) with permission of the
American Geophysical Union.
OCR for page 59
ACOUSTIC RADIATIONS FROM LIGHTNING
seconds) come into dynamic equilibrium where the hy-
drodynamic drag force associated with their motion is
balanced by the sum of all the externally expressed
forces. When the electric field is quickly reduced by a
lightning flash the cloud particles readjust to a new dy-
namic equilibrium. The change in the hydrodynamic
drag force requires a change in the pressure distributions
surrounding all the charged cloud particles, hence, the
pressure in the volume continuing the cloud particles is
altered by the sudden reduction of the electric field.
Since the electric force from a charge concentration is
outward, the pressure inside the charged volume will be
slightly lower than the surrounding air. When E is re-
duced by the lightning flash the charged volume pro-
duces a slight implosion; this radiates a negative wave.
Few (1982) derived a general expression for the internal
pressure gradient produced by the electrostatic force;
when integrated the result is
p p ~ 1y co(Eo - E2)
· (49)
In Eq. (4.9) the parameter n takes the value 0 for
plane geometry, 1 for cylindrical geometry, and 2 for
spherical geometry; P0 and En are the values at the edge
of the charged volume.
The amplitude of this pressure signal is related to the
electric field, the wavelength to the thickness of the
charged region, and the directivity of the wave to
the geometry to the source (Dessler, 1973~. If the theory
can be quantitatively verified, the signal can be used to
determine remotely internal cloud electric parameters.
The experimental search for electrostatic pressure
waves has been difficult. The wave is low frequency
(~ 1 Hz), small amplitude (~ 1 Pa), and buried in large
background pressure variations produced by wind, tur-
bulence, and thunder. Prior to Dessler's prediction of
beaming, one wondered why the signal was not more
frequently seen in thunder measurements. Holmes et al.
(1971) measured a low-frequency component in a few of
their power spectra of thunder but found these compo-
nents completely missing in others. Dessler showed that
that signal would be beamed for cylindrical and disk
geometry; the disk case would require that the detectors
be placed directly underneath the charged volume for
observation. This relationship has been observed by Bo-
hannon et al. (1977) and by Balachandran (1979, 1983~.
The electrostatic pressure wave predicted by the the-
orv discussed above is a negative pulse. The measured
acoustic signature thought to be the verification of the
prediction actually exhibits a positive pulse followed by
a negative pulse (see Figure 4.11~. The negative pulse
appears to fit the theory, but the theory is deficient in
59
that the positive component of the wave is not de
scribed. Recently, Few (1984) suggested that the dia-
batic heating of the air in the charged volume by posi-
tive streamers may be the source of the positive pulse.
Colgate and McKee (1969) described theoretically an
electrostatic pressure pulse using this same mechanism
but applied to a volume of charged air surrounding a
stepped leader. This particular signature has not been
experimentally verified because it has the regular thun-
der signal, which is 300 times more energetic, superim-
posed on it.
ACKNOWLEDGMENT
The author's research into the acoustic radiations
from lightning has been supported under various grants
and contracts from the Meteorology Program, Division
of Atmospheric Sciences, National Science Foundation,
and the Atmospheric Sciences Program, Office of Naval
Research; their support is gratefully acknowledged.
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Representative terms from entire chapter:
lightning channels
