An aerodynamic-acoustic theory of high-altitude fluctuation phenomena

r. Sound vib. (1967) 6 (2), 199-208

AN AERODYNAMIC-ACOUSTIC FLUCTUATION

THEORY OF HIGH-ALTITUDE PHENOMENA-t

R. S. IYENGAR Mount Allison University,

Sackville, New Brunswick, Canada

(Received 24 November 1966)

A dynamical coupling exists between the troposphere and the lower ionosphere via the acoustic waves generated in the turbulence of the tropospheric regions. The propagation of such waves through the ionospheric regions causes density fluctuations of geophysical interest. Several types of high-altitude pulsation effects can be linked by this mechanism of modulation, inclusiveof infrasonicwaves associatedwith aurora1regions during periods of disturbances on the sun. I. INTRODUCTION Atmospheric turbulence can radiate sound. This sound forms an important background for infrasonic ground measurements. The propagation to the ionosphere of these acoustic waves is of geophysical interest due to the resulting density fluctuations there. These two aspects of aerodynamic sound are discussed here with special reference to some geophysical phenomena observed in the aurora1 zone. 2. THEORY

Aerodynamic sound is that generated by a distribution of convected quadrupoles [I]. The variously sized eddies radiate sound of different frequencies. Radiated frequency is determined by the characteristic time of the eddy concerned while the power radiated at that frequency is determined by the velocity amplitude of the eddy [z]. The energy of turbulent motion will reside in the large-scale eddies, and most of the acoustic radiation comes from the large-scale eddies. The number and size of these depend on the manner in which turbulence is produced. The large-scale eddies produce the low-frequency sound [3]. The high frequency sound (that is, wsl/cs
R. S. IYENGAR

200

ordinary law of acoustics. given by

The instantaneous

equivalent

applied stress at a point [4] is

Here p is the mass density, pvi, the momentum density, vi, the gas velocity vector (~i,va,vs), and c,, the sound velocity. The first term on the right-hand side gives the fluctuating Reynolds stresses, i.e. the direct convection of momentum component pv by the velocity component vi. The second term Pij gives the real stresses, i.e. the stress between adjacent elements of the fluid. The third term gives the acoustic density fluctuation outside the field. The conversion from kinetic energy of fluctuating shearing motion into acoustic energy of fluctuating longitudinal motions proceeds via mass fluctuations in a fixed region of space, or via the rates of mass flux or momentum flux across fixed surfaces. 3.

3. I.

ACOUSTIC

OBSERVATIONS

PROPAGATION

THROUGH

AND

DISCUSSION

THE IONOSPHERE

In the present discussion we can confine our attention to the experimentally observed acoustic waves (from the troposphere) propagating upwards [5]. Acoustic background at high altitudes (near 30 km) has been monitored with balloon-borne, battery-powered, infrasonic microphones [5,6,7]. Measurements have been carried out at frequencies from 0.2 to 200 c/s. The pressure amplitudes from 0.03 to I dyne cme2 at I c/s have been observed. With ascending sensors, that is, in the far-field of pressure fluctuations, a non-Gaussian signal has been observed, but a Gaussian signal in the near field. These results are in accordance with Lighthill’s theory. The cancellation effects considered in the rapid pressure fluctuations in the near field by small-scale eddies are mitigated for observation in the far-field. This is due to the fact that the signals from the eddies (radiating sound) do not all originate at the same time. The acoustic waves mentioned earlier propagate to the ionosphere. Finally they get quenched where the sound wavelength becomes comparable to the mean free path. This propagation is of great geophysical interest due to the resulting fluctuations of electron density in the region 90 to 200 km or higher. Consequent on the latter, various types of concurrently observed pulsation phenomena will be explained. 3.2.

QUENCHING

ALTITUDES

The mean free path around 200 km is of the order 10~ cm [8]. Acoustic waves quenched around this height would have a frequency of about 4 c/s ; those quenched around I 20 km (mean free path about 10~ cm), about 120 c/s. At higher altitudes around 220 to 250 km, the frequency would be of the order 0.4 to 0.04 c/s (or periods of the order 2.5 to 25 set). The sound pressure P, at a given height can be obtained by equating the integrated intensity

for two levels, using the known pressure P, for one level ; c, is given by (~P/P)“~, where y is the specific heat ratio, p, the density and P, the ambient pressure. From the measured P*(NI dyne cmm2) at 20 km one can get a value P,= 3.75 x IO-~ dyne cme2 at 120 km. The ambient pressure at this level is 3.4 x IO- 2 dyne cme2 [9]. The ratio of the two pressures is IO-~, thus giving a fluctuation of I o/o ; this would be about 5% for the IOO km level for P, = 5 to 6 dyne cme2 at 20 km. Electron density fluctuations of this order follow those of the neutral particles. Now it becomes easy to explain the geophysical consequences.

HIGH-ALTITUDE

3.3.

RELATED FLUCTUATION

PHENOMENA

FLUCTUATIONS

201

OF THE AURORAL ZONE

Electron density fluctuations during disturbed conditions can cause a number of types of events known in the aurora1 zone. For example, a 5% fluctuation in the electron density will cause a 3 gamma fluctuation in a total field of 60 gamma [IO]. Luminosity effects in aurorae are governed by the rate of ionization and probability of simultaneous ionization and excitation, and thus fluctuations in aurora1 luminosity could be expected. Absorption effects depend on the electron density fluctuations. Fluctuations in Joule heating due to currents are of interest as a source of infrasonic waves observed concurrently with other events. The fractional change in current density (J) would be directly proportional to the fractional change in conductivity (u). The latter can be obtained from the fractional change in the electron density (n,). Thus dJ/J=dn,/ n,. The fractional variation in the field is thus connected with that of the current density. One can also consider the fluctuation effects in the far-field of aurora1 turbulence of the IOO km region due to the acoustic waves generated in the latter. When the sound pressure is comparable to the ambient pressure at higher levels (for example, above 220 km the pressure ratio is of the order 1.3 to 2) shock waves can be caused. Shock waves are in themselves of great geophysical interest in their propagation whether in a neutral gas or in a plasma. 3.4.

V.H.F.

SIGNAL FLUCTUATIONS

It is shown elsewhere [II] that V.H.F. events are not always correlated with other events of the aurora1 zone. During normal periods this correlation is poor; and the V.H.F. events can be thought to arise at a higher level ; for example, in the F-region by the acoustic propagation from the tropospheric regions [s]. However, during intense aurora1 conditions, V.H.F. events do show a good correlation with other concurrent events (see Figure I). .Ilong with some of the other events, they can also originate near IOO km region.

Figure I (top to bottom): Aurora1 x 5577 scatter signal fluctuations, riometer absorption 14

fluctuations, V.H.F. (41 MC/S) forward (30 MC/S), all for the same hours of universal time.

A intensity

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202

Due to intensified electron bombardment at this height, the ionization rate will increase. It is proportional to (nJ2, where n, is the electron density per cme3, when the two types of ionizations are considered equal (in number per cm”). For the known values of 71, during aurora1 conditions [IZ], the peak ionization can occur around IOO km region. The inhomogeneities causing signal fluctuations can be assumed to be electron density fluctuations, and magnetic field effects may be neglected. For large frequencies o of the propagating signal (that is w > wP, where wp is the plasma angular frequency corresponding to which in the ionosphere the frequency f, is around 3 MC/S) and signal propagation velocity c, the propagation constant tz ZCU/C. If the mean square of amplitude of fluctuation is denoted by A2, the correlation distance by a, and the path length traversed by the signal by L (in the inhomogeneous medium) then Meecham [s] has obtained the following results, namely : A2

=

w_1 ‘ <@;)‘a,

(1 w

(1)

n0

2 _ ve’n0 WI - ---.

where

(2)

m

Here, w is the signal frequency, e, the electronic charge, m, the electronic mass, no, the average density of electrons and dn, the fluctuation in the average electron density, and ( ) denotes average. The energy ratio of noise to signal is given by

N2 -. = $k2A2nL. S2 From

(I),

This result holds when small. Using k zw/c and the result

(2)

N2 ?F=-

in (4) we get dna (477’e2)2 I 4 c2 (mo)2 w2

(pq2)L.

It follows from (5) that the energy ratio of noise to signal is inversely square of the frequency and directly proportional to

proportional

to the

((h~)~) L = j(dn)2dn 0

integrated over the path traversed. tion is involved in the discussion. N2 p=

As is seen from (s), the actual electron (5) or (4) can also be written as *k2e)l((@)')aL. 4

density

fluctua-

(6)

Now let us consider the numerical parameters involved in the foregoing theory. The correlation distance a is of the order of an eddy size (I km). The thickness of the region traversed (L) by the acoustic wave is about IO km. The ratio (OJ~/CO)x10-~ when the signal frequency is around 40 MC/S and the frequency equivalent corresponding to CO*is of the order of a few MC/S. The propagation constant k ~2~r/750 for a signal frequency of 40 MC/S. The fractional change in electron density as deduced earlier is about 5% or (IO-~ x 5). The noise will turn out to be roughly of the same order as signal. Electron

Figure 2. (top) Magnetic (n-s) fluctuations; hours of universal time.

(bottom) V.H.F.

backscatter

(41

MC/S) signal fluctuations, both for the same

204

R. S. IYENGAR

densities during aurora1 conditions can be around (10~ to 10~) cme3 and fluctuations can be caused by the propagation of acoustic waves mentioned earlier. Signal periods can be obtained by the autocorrelation methods and discrete frequencies, by the power spectrum which is the Fourier transform of the autocorrelation function. A closely related signal pattern in the simultaneously recorded magnetic field variations and V.H.F. backscatter

Time delay (x 0.47

set)

Figure 3. (a) Autocorrelation of the signal due to optical fluctuations trum of the signal in (a) (reference 14).

in aurora.

(b) Power

spec-

activity is shown in Figure 2. Several types of events such as aurora1 brightness fluctuations, magnetic fluctuations, fluctuations in the signal due to absorption of cosmic noise and infrasonic signal activity have been found to be well related with one another during disturbed conditions in the upper atmosphere [IO]. In several of these events (including Bremsstrahlung X-rays at balloon heights), signal periods of a few seconds are known by rapid recording techniques [IO, 121. Rapid-run and small angular field records of optical signal fluctuations from aurorae, as manifestations of fluctuating electron densities, have also been obtained [13, 14, IS]. Periods of a few seconds [14] [see, for example, Figure 3(a, b)], and occasionally fractions of a second in active forms of aurorae [I 31 (see Figure 4) are observed.

HIGH-ALTITUDE

FLUCTUATIONS

205

Figure 4. Power spectrum showing a few c/s for a signal from an actively pulsating aurora1 form CI31.

4. TURBULENCE-ACOUSTICS

DURING

AURORAL

CONDITIONS

The foregoing theory can be extended to the turbulence-generated sound during aurora1 conditions in the lower ionosphere with the following remarks. Normally, the turbulence energy in the lower ionosphere is too small compared with that in the troposphere. For a turbulent gust of IO m/s in the troposphere with an eddy size of I km, the energy density is r erg cmW3 set- *. In the lower ionosphere, for parameters of the same order, energy density is less by a factor IO- 5 due to decrease in density at higher altitudes. But energy contribution from aurora improves this energy density considerably. Thermal energy dissipation in aurorae of the order IOO erg cmP2 see-’ is easily possible [ 161. During intense aurora1 periods one can expect three to four times this order of flux. For a linear size of I km for the eddy, the energy density would be about IO-~ erg cmm3 set-‘. Or, if one would consider wind shear of 50 to IOO m/s known in the lower ionosphere, the energy density given by pv3/1 (where v is the fluctuating velocity, 1, the eddy scale size and p, the density) will be of the order (IO-~ to IO-~) erg cme3 set-‘. For much larger wind shear it can be increased by another order of magnitude. Contributions from tidal and atmospheric gravity waves in the range IO-~ or a little more would be small in comparison. Joule heating by currents in the lower ionosphere would be much smaller still. During disturbed conditions solar radiation could be considered as the major contribution of energy. Thus, to an order of magnitude approximation, the energy density in the lower ionosphere competes with the magnitude in the tropospheric activity, as also the pressure fluctuation in the lower ionosphere during aurora1 periods. Actually, the observed infrasonic pressure amplitudes up to 20 dyne cmm2 [IO] associated with aurora1 regions could be in evidence of the fact that the pressure fluctuations are at times much greater than the modest estimates made earlier. Moreover, collaborating and conflicting types of description of audible effects from aurorae are not rare in literature ; such as, for example, “ hissing “, “ crackling “, “ burning of leaves ” and other types of description. A great challenge yet to any aurora1 observatory is in the detection of sound definitely ascribable to aurorae. Tropospheric acoustics would have to be watched to some extent in

R. S. IYENGAR

206

any ground-based observation. Detection from above the regions of aurora1 displays using suitably designed telemetering systems may not perhaps prove ineffective. However, at higher altitudes the rarefied atmosphere makes audibility feeble, but an encouraging sign is in the observed aurora1 infrasonics observed on the ground [IO]. Some of the authors have attributed the infrasonics so observed to heating fluctuations by currents in the lower ionosphere [IO] or in the heating fluctuations during ionization and excitation caused by the bombarding particles [I 71. It is also possible to explain some of the observed infrasonic activity as originating in the pressure fluctuations of the intensified turbulence of the aurora1 conditions.

2-

l-

o-

-l_ 0

I 2

I 4

I 6

log E (erg gm-’ set’ -‘1

Figure 5. Time scales of eddies in turbulence versus energy dissipation rates.

The energy dissipation through turbulence in the lower ionosphere which is normally low, of the order (IO-~ to IO-‘) erg cmF3 set-‘, can go up to (IO-~ to IO-~) during disturbed conditions assuming such dissipation of the available thermal contributions from aurorae. The eddy time scales given by (Z/s) for the larger eddy sizes ranging from an average 0.5 km to I km [IS], and for the known wind speeds of about 50 m/s, can easily be of the order of a few seconds. If one were to obtain the time scales from the smallest inhomogeneities, these are given by 7s= (v/E)l12 where v is the kinematic viscosity (cm2 set-i) and E, the energy rate of dissipation (erg gm-* set-*). As the dissipation rate decreases the time scale gets longer (Figure 5). We can consider a thickness of IO km for the turbulent layer around IOO km region in accordance with experimental results of Blamont [IS]. Corresponding to dissipation rates of the order IO-~ erg cmm3 set- i, the energy flux would be about IOO erg cme2 see-I. This is roughly the order of flux required by Maeda and Watanabe [17] for a surface pressure amplitude of I dyne cmP2 in their treatment of infrasonic wave sources. Corresponding to values of the order 20 dyne cmm2[IO] for ground-based observations, the sound pressure at zoo km will be about IO- 2 dyne cme2. The ambient pressure at this level is

HIGH-ALTITUDE

FLUCTUATIONS

207

1.71 x I0 -I dyne cme2 [g]. Thus the fractional variation of pressure is (0.5 x IO-~), or about 5%. At go km where the pressure is (g-5 x 10-l) dyne cmm2the pressure fluctuations will be about 3 o/o; smaller values for smaller infrasonic amplitudes observed. Next, we can theoretically obtain the pressure fluctuations in the turbulence of the aurora1 conditions. The order of this fluctuation will be denoted by Sp. The only quantity having the dimension of pressure and formed from the quantities such as the velocity fluctuation (u), the density of the medium (p) and the eddy scale size (I) is given by Sp = Pi. For wind shear magnitudes of the order 50 to 60 m/s the numerical magnitude of the right-hand term will be around I to 5%. Or, if z, is taken to represent the average deviation from the surrounding mean molecular (thermal) velocity (zI,,,)which is roughly of the order of the speed of sound in the undisturbed medium, by Bernoulli’s result we have

With v= 40 to IOO m/s, and v,= 4 x 10~ cm set-*, one can get Splp as approximately I to 5%. This order of fractional change in the pressure is in accordance with the observed surface pressure amplitudes. 5. CONCLUSIONS

The propagation to the ionosphere of turbulence-generated acoustic waves from the tropospheric regions provides a modulation mechanism with which to link several types of concurrently observed fluctuation phenomena in the aurora1 zone. A special feature of this mechanism is that fluctuations caused in the entire range go to 200 km or higher can be explained. V.H.F. events can be related, through this mechanism, to the other events during aurora1 conditions. The significance of a dynamical coupling between the lower and upper atmospheres is suggested through this mechanism. Electron density fluctuations and the consequent Joule current fluctuations being supported through this mechanism, possibility of the pressure fluctuations in the aurora1 turbulence as a source of infrasonic waves is indicated. ACKNOWLEDGMENTS

I am thankful to Drs. W. C. Meecham, M. J. Lighthill and J. M. Burgers for reading the more detailed work and their kind encouragement expressed. I am grateful to Dr. B. W. Currie of the University of Saskatchewan, Canada, for all the help given. Some of the experimental work and analysis of data were done while at the Physics Department of the University of Saskatchewan, and it is a pleasure to express my thanks.

REFERENCES I. M. J. LIGHTHILL 1954 PYOC. R. Sot. A, 221, I. On sound generated aerodynamically, II (turbulence as a source of sound). 2. W. C. MEECHAMand J. W. WESCOTT1965 Priwate communication. High-altitude noise background. 3. W. C. MEECHAMand G. W. FORD 19583: acoust. Sot. Am. 30, 318. Acoustic radiation from isotropic turbulence. 4. M. J. LIGHTHILL 1952 Proc. R. Sot. A, 211, 564. On sound generated aerodynamically, I (general theory). 5. W. C. MEECHAM1964.7. geophys. Res. 69, 3175. Satellite signal fluctuations caused by ionospheric irregularity.

R. S.

208

IYENGAR

6. W. L. WEBB, J. W. COFFMANand G. Q. CLARK1959 Spec. Rept, 28, U.S. Army Signal Missile Support Agency, AD-z30726L. A high-altitude acoustic sensing system. I, 182. Acoustic 7. J. W. WESCOTT 1961 Proc. Symp. Atmos. Acoust. Propagation, AD-408716, background at high altitudes. 8. ROCKETPANEL 1952 Phys. Rev. 88, 1027. Pressures, densities, and temperatures in the upper atmosphere. 9. F. S. JOHNSON1961 Satellite Environment Handbook. Stanford University Press. IO. W. H. CAMPBELL 1964 Central Radio Propagation Lab. Rept. A review of seven studies of geomagnetic pulsation associated with aurora1 zone disturbance phenomena. II. R. S. IYENGAR1964 Nature, Lond. 202, 372. Auroral-zone magnetic events. 12. W. H. CAMPBELLand M. H. REES 1961 J. geophys. Res. 66,41. Aurora1 coruscations. ‘3. R. S. IYENGAR1961 Thesis, University of Saskatchewan. Rapid fluctuations in aurora1 intensity. 14. R. S. IYENGARand G. G. SHEPHERD1961 Can. J. Phys. 39, 1911. Observations of aurora1 luminosity fluctuations. 15. K. V. PAULSONand G. G. SHEPHERD1966 Can.J. Phys. 44,837. Fluctuation in brightness from quiet form auroras. 16. J. W. CHAMBERLAIN 1961 Physics ofthe Aurora and Airglow. New York: Academic Press Inc. 17. K. MAEDAand T. WATANABE1964J. Met. 21, 15. Pulsating aurorae and infrasonic waves in the polar atmosphere. 18. J. E. BLAMONT1963 PEanet. Space Sci. IO, 89. Upper atmospheric turbulence near the IOOkm level. APPENDIX-NOTATION eddy scale size speed of sound acoustic angular frequency mass density momentum density velocity vector with components or, v2, v3 stress between adjacent fluid elements instantaneous applied stress at a point sound pressure ambient pressure ratio of specific heats current density fluctuation in J conductivity electron density fluctuation in n, angular frequency of the propagating signal plasma angular frequency propagation constant speed of signal propagation mean square of the amplitude of fluctuation electronic charge electronic mass average electron density noise to signal energy ratio average correlation distance kinematic viscosity energy rate of dissipation in turbulence density fluctuation pressure fluctuation velocity fluctuation (from the surrounding velocity) mean molecular velocity (thermal) time scale of small eddies