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New mechanism for electromagnetic field generation by acoustic waves in partially ionized plasmas
17 February
1997
PHYSICS
ELSESIER
LETTERS
A
Physics Letters A 226 f 1997) 199-204
New mechanism for electromagnetic field generation by acoustic waves in partially ionized plasmas George D. Aburjania I, George Z. Machabeli IN.V&u
Institute ofAppiied ~uthe~ut~c~. Tbilisi State University. Received
13 Novem~r
1996; accepted for publication Communicated
Uniue~~~j~ 2,38#43
20 November
Tbilisi, Georgia
19%
by M. Porkolab
Abstract
A new mechanism for the transformation of long-wavelength acoustic waves into electromagnetic ones in partially ionized plasmas is suggested. It is shown that an acoustic wave involves in collective motion charged particles of media by means of collisions. Relative motion of charged particles excites the alternating current with arbitrary phase and, consequently, there occurs parametric generation of electromagnetic fields. The growth rates, excitement tresholds, spectra and polarization of electromagnetic disturbances are determined. PACS: 53.35.G; 52.35.Bj; 52.35.Dm; Keywnrcfs:
Ionospheric
52.35.M~ plasma; Parametric instability;
Growth
rate; Spectra; Arbitrary
1. The interest in the study of wave phenomena connected with seismic and anthropogenic activity has recently increased. The results of numerous land and satellite observations show that before an earthquake or at the time of influencing artificial sources, acoustic wave excitation in the atmosphere and electromagnetic field amplification in the ionosphere take place (see, for example, Refs. [l-4]). The acoustic waves in question may come from any natural and artificial source, including earthquakes, when propagating upwards, and they reach the ionospheric layers [5]. On the satellite “Oreol-3”, for instance, the effective energy transformation of anthropogenic explosions - the acoustic perturbation of the infrasonic band - into energy of electromagnetic perturbations is detected [2]. So, quite a large
’ E-mail: d~d~pha.~heta.ge. 0375~9601/97/$17.~ Copyright SSDI 0375-9601~96)0089i-2
Q 1997 Pubiish~
phase
amount of experimental data has been acumulated, which point to the important role of the impact of the interaction of the neutral component on the dynamics of the upper atmospheric electromagnetic perturbations. In this context, the problem of the investigation of the mechanisms of the acoustic wave energy transfer - in particular, that of the waves of seismic origin to elec~omagnetic disturbances becomes urgent. The observed data fit in the following physical picture: an acoustic wave propagating upwards in the atmosphere with an exponentially decreasing density increases its amplitude and modulates the parameters of the medium at ionospheric heights. This fact directly leads us to the concept that a natural or artificial acoustic wave may interact parametrically with the atmospheric-ionospheric medium. This was ignored by previous authors that considered analogous ionospheric problems (see, for example Refs.
by Elsevier Science B.V. All rights reserved.
200
G.D. Ahurjunia.
G.Z. Muchaheli/
16-81). These papers were restricted to the consideration of the linear problem. In the equation for the velocities they did not take into account the nonlinear term of (u - V)v. In doing so they neglected the only available possibility of transformation of the waves in a quasihomogeneous medium: nonlinear interaction of waves. Therefore the autors had to assume the existence of a sharp border line between a neutral gas and a fully ionized plasma, which is not the case, and so had to reduce the problem, in fact, to a slightly modified mechanism of the linear transformation of waves on inhomogeneities of the medium. As a result, the coefficient of the transformation for waves in the ionosphere has been found to be unimportant [6]. The nature of our case essentially differs from the ordinary parametric instability: the pump-wave is an acoustic one, i.e. oscillations of neutral particles. So a direct interaction of the pump-wave with electromagnetic modes is impossible. An intense acoustic wave reaching ionospheric altitudes draws charged particles into the collective motion by friction with the neutral particles. It is important that the friction coefficients of the electrons and the ions with the neutral particles differ from each other, which causes a relative motion of the charged particles. The alternating currents induced by this motion can generate electromagnetic fields due to the parametric pumping. It is also important that such alternating currents have a chaotic phase that must be taken into account. The specificity of the case in question requires a new technique to be developed, somewhat different from that for the ordinary parametric instability [9]. This technique we will propose below. 2. The medium in question consists of neutral particles, ions and electrons. Supposing the collisional ionospheric medium to be quasihomogeneous, quasineutral (n, = noi = n,), and neglecting the inertia of the electrons (which is true for AlfvCn and magneto-acoustic disturbances), the dynamics of wave disturbances is described by the equations [IO]
avi Mn, -$f(Vi'V)Ui i
1
= -V(P,
+P,) + &[rot
+ Mn, Vi”(Un- Vi),
B, B] (1)
Physics Letters A 226 (1997) 199-204
Mn,
l>+(v..v)v,I
= - VP” - Mn, Vi”( 0”
2
+ div( n,v,)
-
Vi) )
= 0,
VP,= ynTaVn,,
a=e,
(2) (3)
i,n.
(4)
Here the following was taken into account, v
Mui,
‘in
me ‘en
e” -N-w
M (
4
“* 1
>> 1.
(5)
In Eqs. (l)-(4) P, = P,, + Fa, n, = n,, + ii,, the pressure and the density of the ionospheric plasma, B = B, + B and E are the magnetic and electric fields (the index “0” means an equilibrium value, and a tilde above the average quantities means their disturbed values), vap is the average frequency of collisions for a particle of a kind “(Y ” with those of a kind “B”, c is the speed of light, M is the mass of an ion or a molecule (neutral), e and m, are the charge and the mass of the electron, ‘y, is the adiabatic index, T, is the temperature. In the following we shall omit, as a rule, the tilde for disturbed quantities. We describe the magnetic and electric field by Maxwell equations. When the collisions are scarce enough (w,, =B Y,,) and the processes are slow (o e oCi, W~is k2c2; here k and w are the wave number and the characteristic frequency of the generated disturbances; 6~;~= 4re2noi/M, wcn = eB,/m, c, a = i, e) [IO], the induction equation takes the form of aB
-
at
= rot[viB].
In Eq. (1) the term proportional to VP, is decisive, because it describes the motion of electrons relative to ions. To define VP,<= y,TCVn,) correctly, one must have the equation connecting the velocities of electrons and ions. In our case (ionospheric medium) this connection is achieved by collisions of electrons and ions with neutral particles. To find this connection we will use the calculation scheme described in Ref. [I I]. Let the homogeneous geomagnetic field B, be directed along the z axis, and the electric field
G.D. Ahurjuniu,
G.Z. ~ucbfl~~~i/Pby.si~s
have components along the x and z axes, E = (E,, 0, E,). In such a model one can easily find the solution of the equation of motion’of charged particles. Rewriting these expressions separately for electrons and ions (a! = e, i> and excluding E and B, one can easily find the connection between velocity components (u~)~ and (~~1, ( j = x, y. z or 1, 2, 3),
Here
:‘;;;; ,
p,, = _ ;y;y I”
c1
en
ce
d vi +w,“l P22= 7w,i vb”, f co,25 ’ Pn=
mi
vin
----g-y-‘ e
(8)
e”
The self-consistent set of equations (l)-(4), (6), (7) describes the generation of an elec~omagnetic field when an acoustic wave propagates in the ionosphere.
dently, and the velocity v, can be presented in the form of oscillations with an arbitrary phase $. v,= v, cos(Wor+$).
(9)
Here V, is a time-invariant amplitude of the neutral particle velocity, o0 is the acoustic (pump) wave frequency, $ is the initial phase of the neutral particles or ions. We describe a large set of particles. So it is clear that $ has a random value. Therefore the qu~i-homogeneous atmosphere is supplied by the external constant source of perturbation (9) (for instance, by the oscillation of the terrestrial surface which, because of superficial Rayleigh wave excitation, generates sound propagating up to ionospheric heights). Let us investigate temporal evolution of the ionospheric perturbation created by the external source (9) in the frequency range vniK o -=KwCi. As was mentioned above, one should pass to the rest frame of neutral particles. and for convenience let us introduce new variables, V= vi_vn=
3. Hereafter we will consider the problem in the frame of the oscillating neutral particles, in order to take into account the influence of the acoustic wave on the charged particles. In our case the influence of the acoustic pump wave on the charged particles is achieved only by collisions. It is obvious that collisions with different particles occur at different moment of time. After each collision of a neutral with charged particles their momenta change - the particles act as if they begin to move with a new phase. Hence, for an adequate description of the statistical state of many particles one has to introduce the arbitrary phase. In the case of two fluid hydrodynamics one has to introduce the arbitrary phases cp and I/J for the plasma electrons and ions, respectively, with a consequent averaging over these phases. Let us consider the case when the frequency of the generated perturbation w is much higher than the frequency of the collision of neutral particles with ions ZJ,,~, w B vni (vni - ~~i/~~~ vin -SCvi,>. In this frequency range the motion of the charged particles does not influence the motion of neutral particles. So the solution of the equation of the neutral particle dynamics (2)-(4) can be accomplished indepen-
201
Letters A 226 (19971 199-204
U(t)e-iUSin([u,lt+~f,
k - V, a=-------
(10)
WO
where k is the wave vector of the generated perturbation. For these new variables we get from Eqs. (1). (6). with an accuracy of the terms of the second order, au(t) at
Xc-i”
- z$“U( t)
sin(w,,t’+J,)
eio
dt’
sin(o,,t+
li,)
(11)
i
It should be noted that the ions and the electrons do
not interact with the neutral particles in a synchronous way. So the initial oscillation phases for the electrons cp and the ions I,!I are not the same. Taking this (as well as (7)) into account, the solution
202
G.D. Ahurjaniu,
G.Z. Muchoheli/Physics
Letters A 226 (1997) 199-204
of the continuity equation for electrons and ions (3) can be presented in the form n,( t> = _n,e-ih
sin(w,,r+p) = AijVj( w) = 0.
X
X
Here
i Jn( b) J,( u)ei(“‘p-“+) n,s= -7z
y,TJM;
z k, pjjuj( ~‘)ei[“‘+(“-s)w~~l’ do, I -cc
9
w’+(n-s)o,
(12) ni( t) = __inoe-io sin(w,r+llr) ‘k * U(t) /
dr’,
(‘3)
where b_
PjjkjVOj
,
--
j=x,
y, ~(1,233).
(14)
WO
sin(w,r++)
=
5 n=
sion equation,
-A,3A22A3,
Jn(a)ein(w++)
2
,= --r
J,(a)V(
1
I+,,~
+A,3A2,A32
-A,,A,,A,,-A,,A,,A33=O~ (16)
o-
0 l+i--l[l. “w V(w) -k,(k, -k(k.
V,: = orthogo-
nal and parallel (to the outer magnetic field B,) components of the wave vector, respectively. Vector equation (15) or the corresponding system of three scalar equations define the relation between the acoustic and electromagnetic waves in the ionosphere. The nontrivial solution of Eq. (15) exists if the determinant constructed from the coefficients of the system is zero. This expression is the desired disper-
-5.z
fwo)ei’$,
*V*(4);
where +i%-~(V~+V$+p,,AoVs:).
A,,=1 A,,=
-z
kv k, w (vA’ +
A,3 = - F(VGi
which is the Fourier transform of Eq. (10). If here the phase 4 is fixed as it occurs in the case of usual parametrics [9], the infinite sum of various harmonics is obtained. To describe such a system one has to cut off the infinite system of the overlapping equation, using the additional resonance conditions. As for our case, at the averaging over arbitrary phases cp and I,!J,the system of equations for V is automatically simplified. Indeed, as the result of such an averaging over arbitrary phases Eq. (11) eventually acquires the form
(
V;i = yiTi/M, k, , k,, are the
=Al,A22A33+A,2A23A31
was used in Eq. (11). Here U(w) is the Fourier component of U(r). We substitute expressions (12) and (13) into ( 11). Then we integrate (11) over f’, perform a Fourier transformation and pass back from the variable U(w) to V(w) using the expression U( 0) =
Vi = Bi/4rn,M, k = (kt +ki)‘/2;
IIAij II
Summation is over j. The expansion eio
(15)
A,, = -
+
P22
AOVse2)v
+Pj3AOK;),
kykx ~(vk2+vT?i+P,,AOK~)~
A,,=1+iz-
A,,=
‘T’i
$(
-- ‘f;(V:i
VA’
+
VT’i
+
+@,,AoVsz),
A32= - $(V;i
+ p22 AOVsz)T
A,,=
A,=
AoK:),
+P33AOVsz),
A,, = - %(V:i
v,’
P22
1 +i~-$(V$+/333AoVs~). Cc c n=-r
oJ,2(b) b=Pjj---, w---w,
kjVoj (17) WO
G.D. hbur~nia~ G.Z. Mu~~beli/
which determines the spectra of generated perturbations 0 = w(k). We note that in (17) the arguments of Bessel’s functions comprise the velocity of neutral particles V, (b = j3,,kjVoj). Thanks to this (16) defines the nonlinear dependence of the spectra of generated perturbations of the amplitude of the pumping wave. In this sense the obtained dispersion equation describes the nonlinear effect of the impact of the acoustic signal (pumping) on the ionospheric plasma. 4. It is well-known from the theory of parametric instabilities that in a conducting medium, in an outer electric field, when the frequency of the outer field w0 is close to that of the self-oscillation frequencies of the medium w(k), increasing disturbances may possibly appear. In our case the motion of neutral particles (acoustic pumping wave> in the ionosphere causes separation of charges through collisions and, subsequently, induction of an electric field. The induced field can change the parameters of the ground state of the ionospheric medium and can lead to parametric instabilities, i.e. to the generation of an electromagnetic field. To demonstrate this, let us consider the solution of (16) in an extreme case - transversal propagation of generated perturbations. Let the perturbation propagate transversally to the geomagnetic field, k _LB,(k = (k,, k,, 0)). Assuming k _= 0 in ( 171, from (16) one can obtain ~2+i~i”~-k~(V~~V~i) - ( Pnk;
- I P,r 1kt)K:
Now let us consider the frequencies of medium oscillations, close to the pumping wave harmonics, i.e. the sonic wave, w(k) = wk + iy,
(19) where wk=ZuO (t= 1,2 ,... > and yew,. Then, substituting (19) into (18) and exp~ding over the degrees of y/ok we will obtain for the oscillation spectrum w; = k: ( VA2+ V;i) (20)
Physics Letters A 226 (1997)199-21w
203
Here, the effect of the outer acoustic (pump) wave manifests itself in the correction terms and specifies the dependence of the frequency on the wave vector (because b = pjjkjVoj/wo). For wavelengths greatly exceeding the characteristic pump-wavelength V,/w, (b -c I), the oscillation frequency (20) equals the frequency of a fast magneto-acoustic wave. The growth rate of the oscillations (20) is equal to y=
1 --zvin
(21) It follows that the generation of electromagnetic perturbations (y > 0) is possible in an ionospheric plasma caused by external acoustic waves, in the area of the following wave numbers,
(22) Accordingly from (21) one can obtain the relation for the threshold values of the pumping wave amplitude where the instability becomes possible,
(23) When V, + 0, expression (21) gives the known result y = - vi”,/2 and we have strong damping of elec~omagnetic (magneto-acoustic) disturbances in a collisional medium. The set of equations (IS) also allows the polarization of the generated waves under consideration to be determined. If we take the quantity P = b’JV, as a wave polarization characteristic, then, from (151, with allowance for (181, we have p+;_ Y
Y
It is clear that the polarization factor P is a real number and the generated perturbations (20) have a linear pol~ization in the plane X, y. It should be noted that the form of the external source (9) presumes that tiOt * k, - r, where k, ( IIV,> is the wave vector of the pump-wave, i.e. the area of the generation of ionospheric perturbation is less than the pump-wavelength A, = 2r/k,. We
204
G.D. Aburjuniu.
G.Z. Muchubeli/Physics
ignore also the inhomogeneity of the pump-wave and the ionospheric medium. ‘Ihis imposes restrictions on the wavelengths of the disturbances under study. Also the wavelengths A are to be considered small as compared with the characteristic dimension of the inhomogeneity of the outer field L, and the medium L Ani mint h,, L,, L,) .
(25)
In the above discussion we neglected by the atmosphere viscosity too. During the upward propagation of acoustic waves the higher frequencies are absorbed due to the viscosity of the atmosphere. The viscosity effect is small for frequencies that satisfy the condition o0 c VJL (here V, is the sound speed, 1 is the mean free path for the molecules) [ 12J. Therefore, the upper atmospheric layers are only reached by low-frequency, long acoustic waves (for instance, the waves of frequencies under low2 SC’ propagate up to an altitude of 400 km without any absorption). 5. The possibility of nonlinear transformation of natural and artificial acoustic waves (e.g., those of seismic origin) into electromagnetic ones in the upper layers of a quasi-homogeneous atmosphere is examined. The range of frequencies covering geomagnetic pulsations vni K o =S oCi has been studied. The phase of the outer acoustic wave (pumping) is arbitrary. This is the main difference of our consideration with the usual parametrics 191.As a result of averaging over the arbitrary phases we will have one vector equation instead of the infinite system of connected equations for particle velocities (which is intrinsic to the parametric& It is important that the equation contains the contribution of electron motion relative to ions (the last term in Eq. (15)), which causes excitation of nonstable modes. The theory stated here makes possible a qualitative interpretation of the initiation of the acoustic and
Letters A 226 (1997)
199-204
electromagnetic disturbances in the atmosphere that have been observed in several experiments: lowfrequency acoustic waves (infrasound) generated at the earth-air boundary (for instance, due to Rayleigh waves>, after reaching the ionospheric layer, bring about disturbances of the current-carrying medium parameters and transform their energy into that of electromagnetic disturbances with high efficiency. Knowing the polarization of the generated field, one can evaluate the direction towards the perturbation source center. The suggested mech~ism not only has a general-physical mewing, but also explains the observed data and predicts the character of atmospheric wave-processes before an earthquake.
References [l] E. Blance, Ann. Geophys. 3 (1985) 673. [2] Jul. Galperin et al., Ann. Ceophys. 10 (1984) 823. [3] M.B. Gokhberg, V.A. Pilipenko and O.A. Pokhotelov, Dokl. Akad. Nauk SSSR 268 (1982) 56. [4] Z.S. Sharadze et al., Izv. Akad. Nauk SSSR, Fizika Zemli 11 (1991) 106. I51 V.K. Petukbov and N.N. Romanova, Izv. Akad. Nauk SSSR, Fizika atmospheri i okeana 7 (1971) 219. [6] S.L. Kohalas and D.A. McNeil& Phys. Fluids 7 (1964) 1321. [7] L.S. Alperovich, M.B. Gokhberg and V.M. Sorokin, Izv. Akad. Nauk SSSR, Fizika Zemli 3 (1979) 58. [S] V.L. Saveliev and E.V. Zholeznjakov, Planet. Space Sci. 40 ( 1992) 509. [9] V.P. Silin, Parametric influence high-intensy radiation upon plasma (Nat&a, Moskow, 1973) [in Russian]. [lo] S.L. Braginskii, The phenomenon of transfer in plasma, in: Voprosy teorii plasmy (Atomizdat, Moscow, 1963) [in Russian]. [i I] R. Bostrom, 2 Electrodynamics of the ionosphere, in: Cosmical geophysics UJniversitetsforlaget, Oslo-Bergen-Tromso, 1973). [12] E.E. Gossard and W.H. Hooke, Waves in the atmosphere (Elsevier, Amsterdam, 197.5).
1997
PHYSICS
ELSESIER
LETTERS
A
Physics Letters A 226 f 1997) 199-204
New mechanism for electromagnetic field generation by acoustic waves in partially ionized plasmas George D. Aburjania I, George Z. Machabeli IN.V&u
Institute ofAppiied ~uthe~ut~c~. Tbilisi State University. Received
13 Novem~r
1996; accepted for publication Communicated
Uniue~~~j~ 2,38#43
20 November
Tbilisi, Georgia
19%
by M. Porkolab
Abstract
A new mechanism for the transformation of long-wavelength acoustic waves into electromagnetic ones in partially ionized plasmas is suggested. It is shown that an acoustic wave involves in collective motion charged particles of media by means of collisions. Relative motion of charged particles excites the alternating current with arbitrary phase and, consequently, there occurs parametric generation of electromagnetic fields. The growth rates, excitement tresholds, spectra and polarization of electromagnetic disturbances are determined. PACS: 53.35.G; 52.35.Bj; 52.35.Dm; Keywnrcfs:
Ionospheric
52.35.M~ plasma; Parametric instability;
Growth
rate; Spectra; Arbitrary
1. The interest in the study of wave phenomena connected with seismic and anthropogenic activity has recently increased. The results of numerous land and satellite observations show that before an earthquake or at the time of influencing artificial sources, acoustic wave excitation in the atmosphere and electromagnetic field amplification in the ionosphere take place (see, for example, Refs. [l-4]). The acoustic waves in question may come from any natural and artificial source, including earthquakes, when propagating upwards, and they reach the ionospheric layers [5]. On the satellite “Oreol-3”, for instance, the effective energy transformation of anthropogenic explosions - the acoustic perturbation of the infrasonic band - into energy of electromagnetic perturbations is detected [2]. So, quite a large
’ E-mail: d~d~pha.~heta.ge. 0375~9601/97/$17.~ Copyright SSDI 0375-9601~96)0089i-2
Q 1997 Pubiish~
phase
amount of experimental data has been acumulated, which point to the important role of the impact of the interaction of the neutral component on the dynamics of the upper atmospheric electromagnetic perturbations. In this context, the problem of the investigation of the mechanisms of the acoustic wave energy transfer - in particular, that of the waves of seismic origin to elec~omagnetic disturbances becomes urgent. The observed data fit in the following physical picture: an acoustic wave propagating upwards in the atmosphere with an exponentially decreasing density increases its amplitude and modulates the parameters of the medium at ionospheric heights. This fact directly leads us to the concept that a natural or artificial acoustic wave may interact parametrically with the atmospheric-ionospheric medium. This was ignored by previous authors that considered analogous ionospheric problems (see, for example Refs.
by Elsevier Science B.V. All rights reserved.
200
G.D. Ahurjunia.
G.Z. Muchaheli/
16-81). These papers were restricted to the consideration of the linear problem. In the equation for the velocities they did not take into account the nonlinear term of (u - V)v. In doing so they neglected the only available possibility of transformation of the waves in a quasihomogeneous medium: nonlinear interaction of waves. Therefore the autors had to assume the existence of a sharp border line between a neutral gas and a fully ionized plasma, which is not the case, and so had to reduce the problem, in fact, to a slightly modified mechanism of the linear transformation of waves on inhomogeneities of the medium. As a result, the coefficient of the transformation for waves in the ionosphere has been found to be unimportant [6]. The nature of our case essentially differs from the ordinary parametric instability: the pump-wave is an acoustic one, i.e. oscillations of neutral particles. So a direct interaction of the pump-wave with electromagnetic modes is impossible. An intense acoustic wave reaching ionospheric altitudes draws charged particles into the collective motion by friction with the neutral particles. It is important that the friction coefficients of the electrons and the ions with the neutral particles differ from each other, which causes a relative motion of the charged particles. The alternating currents induced by this motion can generate electromagnetic fields due to the parametric pumping. It is also important that such alternating currents have a chaotic phase that must be taken into account. The specificity of the case in question requires a new technique to be developed, somewhat different from that for the ordinary parametric instability [9]. This technique we will propose below. 2. The medium in question consists of neutral particles, ions and electrons. Supposing the collisional ionospheric medium to be quasihomogeneous, quasineutral (n, = noi = n,), and neglecting the inertia of the electrons (which is true for AlfvCn and magneto-acoustic disturbances), the dynamics of wave disturbances is described by the equations [IO]
avi Mn, -$f(Vi'V)Ui i
1
= -V(P,
+P,) + &[rot
+ Mn, Vi”(Un- Vi),
B, B] (1)
Physics Letters A 226 (1997) 199-204
Mn,
l>+(v..v)v,I
= - VP” - Mn, Vi”( 0”
2
+ div( n,v,)
-
Vi) )
= 0,
VP,= ynTaVn,,
a=e,
(2) (3)
i,n.
(4)
Here the following was taken into account, v
Mui,
‘in
me ‘en
e” -N-w
M (
4
“* 1
>> 1.
(5)
In Eqs. (l)-(4) P, = P,, + Fa, n, = n,, + ii,, the pressure and the density of the ionospheric plasma, B = B, + B and E are the magnetic and electric fields (the index “0” means an equilibrium value, and a tilde above the average quantities means their disturbed values), vap is the average frequency of collisions for a particle of a kind “(Y ” with those of a kind “B”, c is the speed of light, M is the mass of an ion or a molecule (neutral), e and m, are the charge and the mass of the electron, ‘y, is the adiabatic index, T, is the temperature. In the following we shall omit, as a rule, the tilde for disturbed quantities. We describe the magnetic and electric field by Maxwell equations. When the collisions are scarce enough (w,, =B Y,,) and the processes are slow (o e oCi, W~is k2c2; here k and w are the wave number and the characteristic frequency of the generated disturbances; 6~;~= 4re2noi/M, wcn = eB,/m, c, a = i, e) [IO], the induction equation takes the form of aB
-
at
= rot[viB].
In Eq. (1) the term proportional to VP, is decisive, because it describes the motion of electrons relative to ions. To define VP,<= y,TCVn,) correctly, one must have the equation connecting the velocities of electrons and ions. In our case (ionospheric medium) this connection is achieved by collisions of electrons and ions with neutral particles. To find this connection we will use the calculation scheme described in Ref. [I I]. Let the homogeneous geomagnetic field B, be directed along the z axis, and the electric field
G.D. Ahurjuniu,
G.Z. ~ucbfl~~~i/Pby.si~s
have components along the x and z axes, E = (E,, 0, E,). In such a model one can easily find the solution of the equation of motion’of charged particles. Rewriting these expressions separately for electrons and ions (a! = e, i> and excluding E and B, one can easily find the connection between velocity components (u~)~ and (~~1, ( j = x, y. z or 1, 2, 3),
Here
:‘;;;; ,
p,, = _ ;y;y I”
c1
en
ce
d vi +w,“l P22= 7w,i vb”, f co,25 ’ Pn=
mi
vin
----g-y-‘ e
(8)
e”
The self-consistent set of equations (l)-(4), (6), (7) describes the generation of an elec~omagnetic field when an acoustic wave propagates in the ionosphere.
dently, and the velocity v, can be presented in the form of oscillations with an arbitrary phase $. v,= v, cos(Wor+$).
(9)
Here V, is a time-invariant amplitude of the neutral particle velocity, o0 is the acoustic (pump) wave frequency, $ is the initial phase of the neutral particles or ions. We describe a large set of particles. So it is clear that $ has a random value. Therefore the qu~i-homogeneous atmosphere is supplied by the external constant source of perturbation (9) (for instance, by the oscillation of the terrestrial surface which, because of superficial Rayleigh wave excitation, generates sound propagating up to ionospheric heights). Let us investigate temporal evolution of the ionospheric perturbation created by the external source (9) in the frequency range vniK o -=KwCi. As was mentioned above, one should pass to the rest frame of neutral particles. and for convenience let us introduce new variables, V= vi_vn=
3. Hereafter we will consider the problem in the frame of the oscillating neutral particles, in order to take into account the influence of the acoustic wave on the charged particles. In our case the influence of the acoustic pump wave on the charged particles is achieved only by collisions. It is obvious that collisions with different particles occur at different moment of time. After each collision of a neutral with charged particles their momenta change - the particles act as if they begin to move with a new phase. Hence, for an adequate description of the statistical state of many particles one has to introduce the arbitrary phase. In the case of two fluid hydrodynamics one has to introduce the arbitrary phases cp and I/J for the plasma electrons and ions, respectively, with a consequent averaging over these phases. Let us consider the case when the frequency of the generated perturbation w is much higher than the frequency of the collision of neutral particles with ions ZJ,,~, w B vni (vni - ~~i/~~~ vin -SCvi,>. In this frequency range the motion of the charged particles does not influence the motion of neutral particles. So the solution of the equation of the neutral particle dynamics (2)-(4) can be accomplished indepen-
201
Letters A 226 (19971 199-204
U(t)e-iUSin([u,lt+~f,
k - V, a=-------
(10)
WO
where k is the wave vector of the generated perturbation. For these new variables we get from Eqs. (1). (6). with an accuracy of the terms of the second order, au(t) at
Xc-i”
- z$“U( t)
sin(w,,t’+J,)
eio
dt’
sin(o,,t+
li,)
(11)
i
It should be noted that the ions and the electrons do
not interact with the neutral particles in a synchronous way. So the initial oscillation phases for the electrons cp and the ions I,!I are not the same. Taking this (as well as (7)) into account, the solution
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Letters A 226 (1997) 199-204
of the continuity equation for electrons and ions (3) can be presented in the form n,( t> = _n,e-ih
sin(w,,r+p) = AijVj( w) = 0.
X
X
Here
i Jn( b) J,( u)ei(“‘p-“+) n,s= -7z
y,TJM;
z k, pjjuj( ~‘)ei[“‘+(“-s)w~~l’ do, I -cc
9
w’+(n-s)o,
(12) ni( t) = __inoe-io sin(w,r+llr) ‘k * U(t) /
dr’,
(‘3)
where b_
PjjkjVOj
,
--
j=x,
y, ~(1,233).
(14)
WO
sin(w,r++)
=
5 n=
sion equation,
-A,3A22A3,
Jn(a)ein(w++)
2
,= --r
J,(a)V(
1
I+,,~
+A,3A2,A32
-A,,A,,A,,-A,,A,,A33=O~ (16)
o-
0 l+i--l[l. “w V(w) -k,(k, -k(k.
V,: = orthogo-
nal and parallel (to the outer magnetic field B,) components of the wave vector, respectively. Vector equation (15) or the corresponding system of three scalar equations define the relation between the acoustic and electromagnetic waves in the ionosphere. The nontrivial solution of Eq. (15) exists if the determinant constructed from the coefficients of the system is zero. This expression is the desired disper-
-5.z
fwo)ei’$,
*V*(4);
where +i%-~(V~+V$+p,,AoVs:).
A,,=1 A,,=
-z
kv k, w (vA’ +
A,3 = - F(VGi
which is the Fourier transform of Eq. (10). If here the phase 4 is fixed as it occurs in the case of usual parametrics [9], the infinite sum of various harmonics is obtained. To describe such a system one has to cut off the infinite system of the overlapping equation, using the additional resonance conditions. As for our case, at the averaging over arbitrary phases cp and I,!J,the system of equations for V is automatically simplified. Indeed, as the result of such an averaging over arbitrary phases Eq. (11) eventually acquires the form
(
V;i = yiTi/M, k, , k,, are the
=Al,A22A33+A,2A23A31
was used in Eq. (11). Here U(w) is the Fourier component of U(r). We substitute expressions (12) and (13) into ( 11). Then we integrate (11) over f’, perform a Fourier transformation and pass back from the variable U(w) to V(w) using the expression U( 0) =
Vi = Bi/4rn,M, k = (kt +ki)‘/2;
IIAij II
Summation is over j. The expansion eio
(15)
A,, = -
+
P22
AOVse2)v
+Pj3AOK;),
kykx ~(vk2+vT?i+P,,AOK~)~
A,,=1+iz-
A,,=
‘T’i
$(
-- ‘f;(V:i
VA’
+
VT’i
+
+@,,AoVsz),
A32= - $(V;i
+ p22 AOVsz)T
A,,=
A,=
AoK:),
+P33AOVsz),
A,, = - %(V:i
v,’
P22
1 +i~-$(V$+/333AoVs~). Cc c n=-r
oJ,2(b) b=Pjj---, w---w,
kjVoj (17) WO
G.D. hbur~nia~ G.Z. Mu~~beli/
which determines the spectra of generated perturbations 0 = w(k). We note that in (17) the arguments of Bessel’s functions comprise the velocity of neutral particles V, (b = j3,,kjVoj). Thanks to this (16) defines the nonlinear dependence of the spectra of generated perturbations of the amplitude of the pumping wave. In this sense the obtained dispersion equation describes the nonlinear effect of the impact of the acoustic signal (pumping) on the ionospheric plasma. 4. It is well-known from the theory of parametric instabilities that in a conducting medium, in an outer electric field, when the frequency of the outer field w0 is close to that of the self-oscillation frequencies of the medium w(k), increasing disturbances may possibly appear. In our case the motion of neutral particles (acoustic pumping wave> in the ionosphere causes separation of charges through collisions and, subsequently, induction of an electric field. The induced field can change the parameters of the ground state of the ionospheric medium and can lead to parametric instabilities, i.e. to the generation of an electromagnetic field. To demonstrate this, let us consider the solution of (16) in an extreme case - transversal propagation of generated perturbations. Let the perturbation propagate transversally to the geomagnetic field, k _LB,(k = (k,, k,, 0)). Assuming k _= 0 in ( 171, from (16) one can obtain ~2+i~i”~-k~(V~~V~i) - ( Pnk;
- I P,r 1kt)K:
Now let us consider the frequencies of medium oscillations, close to the pumping wave harmonics, i.e. the sonic wave, w(k) = wk + iy,
(19) where wk=ZuO (t= 1,2 ,... > and yew,. Then, substituting (19) into (18) and exp~ding over the degrees of y/ok we will obtain for the oscillation spectrum w; = k: ( VA2+ V;i) (20)
Physics Letters A 226 (1997)199-21w
203
Here, the effect of the outer acoustic (pump) wave manifests itself in the correction terms and specifies the dependence of the frequency on the wave vector (because b = pjjkjVoj/wo). For wavelengths greatly exceeding the characteristic pump-wavelength V,/w, (b -c I), the oscillation frequency (20) equals the frequency of a fast magneto-acoustic wave. The growth rate of the oscillations (20) is equal to y=
1 --zvin
(21) It follows that the generation of electromagnetic perturbations (y > 0) is possible in an ionospheric plasma caused by external acoustic waves, in the area of the following wave numbers,
(22) Accordingly from (21) one can obtain the relation for the threshold values of the pumping wave amplitude where the instability becomes possible,
(23) When V, + 0, expression (21) gives the known result y = - vi”,/2 and we have strong damping of elec~omagnetic (magneto-acoustic) disturbances in a collisional medium. The set of equations (IS) also allows the polarization of the generated waves under consideration to be determined. If we take the quantity P = b’JV, as a wave polarization characteristic, then, from (151, with allowance for (181, we have p+;_ Y
Y
It is clear that the polarization factor P is a real number and the generated perturbations (20) have a linear pol~ization in the plane X, y. It should be noted that the form of the external source (9) presumes that tiOt * k, - r, where k, ( IIV,> is the wave vector of the pump-wave, i.e. the area of the generation of ionospheric perturbation is less than the pump-wavelength A, = 2r/k,. We
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G.D. Aburjuniu.
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ignore also the inhomogeneity of the pump-wave and the ionospheric medium. ‘Ihis imposes restrictions on the wavelengths of the disturbances under study. Also the wavelengths A are to be considered small as compared with the characteristic dimension of the inhomogeneity of the outer field L, and the medium L Ani mint h,, L,, L,) .
(25)
In the above discussion we neglected by the atmosphere viscosity too. During the upward propagation of acoustic waves the higher frequencies are absorbed due to the viscosity of the atmosphere. The viscosity effect is small for frequencies that satisfy the condition o0 c VJL (here V, is the sound speed, 1 is the mean free path for the molecules) [ 12J. Therefore, the upper atmospheric layers are only reached by low-frequency, long acoustic waves (for instance, the waves of frequencies under low2 SC’ propagate up to an altitude of 400 km without any absorption). 5. The possibility of nonlinear transformation of natural and artificial acoustic waves (e.g., those of seismic origin) into electromagnetic ones in the upper layers of a quasi-homogeneous atmosphere is examined. The range of frequencies covering geomagnetic pulsations vni K o =S oCi has been studied. The phase of the outer acoustic wave (pumping) is arbitrary. This is the main difference of our consideration with the usual parametrics 191.As a result of averaging over the arbitrary phases we will have one vector equation instead of the infinite system of connected equations for particle velocities (which is intrinsic to the parametric& It is important that the equation contains the contribution of electron motion relative to ions (the last term in Eq. (15)), which causes excitation of nonstable modes. The theory stated here makes possible a qualitative interpretation of the initiation of the acoustic and
Letters A 226 (1997)
199-204
electromagnetic disturbances in the atmosphere that have been observed in several experiments: lowfrequency acoustic waves (infrasound) generated at the earth-air boundary (for instance, due to Rayleigh waves>, after reaching the ionospheric layer, bring about disturbances of the current-carrying medium parameters and transform their energy into that of electromagnetic disturbances with high efficiency. Knowing the polarization of the generated field, one can evaluate the direction towards the perturbation source center. The suggested mech~ism not only has a general-physical mewing, but also explains the observed data and predicts the character of atmospheric wave-processes before an earthquake.
References [l] E. Blance, Ann. Geophys. 3 (1985) 673. [2] Jul. Galperin et al., Ann. Ceophys. 10 (1984) 823. [3] M.B. Gokhberg, V.A. Pilipenko and O.A. Pokhotelov, Dokl. Akad. Nauk SSSR 268 (1982) 56. [4] Z.S. Sharadze et al., Izv. Akad. Nauk SSSR, Fizika Zemli 11 (1991) 106. I51 V.K. Petukbov and N.N. Romanova, Izv. Akad. Nauk SSSR, Fizika atmospheri i okeana 7 (1971) 219. [6] S.L. Kohalas and D.A. McNeil& Phys. Fluids 7 (1964) 1321. [7] L.S. Alperovich, M.B. Gokhberg and V.M. Sorokin, Izv. Akad. Nauk SSSR, Fizika Zemli 3 (1979) 58. [S] V.L. Saveliev and E.V. Zholeznjakov, Planet. Space Sci. 40 ( 1992) 509. [9] V.P. Silin, Parametric influence high-intensy radiation upon plasma (Nat&a, Moskow, 1973) [in Russian]. [lo] S.L. Braginskii, The phenomenon of transfer in plasma, in: Voprosy teorii plasmy (Atomizdat, Moscow, 1963) [in Russian]. [i I] R. Bostrom, 2 Electrodynamics of the ionosphere, in: Cosmical geophysics UJniversitetsforlaget, Oslo-Bergen-Tromso, 1973). [12] E.E. Gossard and W.H. Hooke, Waves in the atmosphere (Elsevier, Amsterdam, 197.5).