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Three wave interaction in turbulent plasmas
Volume 43A, number 1
PHYSICS LETTERS
THREE WAVE INTERACTION
IN TURBULENT
12 February 1973
PLASMAS
V.N. TSYTOVICH Lebedev Physical Institute, Moscow, USSR
and L. STENFLO
Institute of Physics,90187 Ume~,Sweden Received 18 December 1972 A new possibility for enhancement of three wave interaction processes is pointed out. This could be of interest for plasma heating by means of high frequency radio waves as well as by laser radiation.
Although much work [e.g., 1 - 3 ] recently has been devoted to the coupling of waves in plasmas there was no general theory for three wave interactions in turbulent plasmas. Such a theory should be of interest as there are regions where one can observe enhancement of the wave-wave interactions even if the turbulence is weak. This is not surprising as three wave interactions in cold plasmas, which are not turbulent, in general are quite weak as the coupling is due to the spatial inhomogeneities, which are produced by the waves. New nonlinear terms should thus be present in a turbulent plasma as the turbulence can introduce spatial gradients, connected with the turbulence itself, which can enhance the wave-wave interactions. Another reason for nonlinearities is the fact that the turbulent collision frequency can be very large if the turbulence level is high enough. The collision term might then be responsible for wave coupling [4]. This paper considers wave-wave interactions in a turbulent plasma and verifies that the normal interaction processes can be significantly enhanced if the turbulence exceeds a critical limit. In connexion with plasma heating experiments it is of interest to study the possibility to enhance the absorption o f a high frequency electromagnetic wave, which penetrates into the plasma. The nonlinear process, which we then want to consider, is the decay of this wave into two longitudinal plasma waves. This process might be enhanced due to the presence of ion sound turbulence. It should then be mentioned that anomalous heating occurs when the incident wave propagates towards an increasing plasma gradient. Ion sound waves are produced by the decay of the transverse wave [5, 6] and ion sound turbulence is thus almost always present when a wave interacts with an inhomogeneous plasma in the region where its frequency is twice the electron plasma frequency (6Op). Three-wave interaction can be described by [2]
/ k = .~. f Si/(k' kl' k2) Eikl ~k28(k-k1-k2) d4kl d4k2 l,]
where k = { k, co}, d4k = dkdoa, j is the second order current density and E is the electric field of the waves, which are to be studied. We will only treat the simplest case when Sii can be expanded in powers of the square of the turbulent electric fields, i.e., IEKI2
f s,,.,7
=¢, + St.
From the Vlasov and Maxwell equations, the final result of a very lengthy calculation is
Volume 43A, number 1
+ r ' ~ -a~ s k - K
PHYSICS LETTERS
-- ap/1 ~s
a
a
a~o
k - l q A i-~,k2 +K.-~Sk+~ap] - - 1 s k2aPi2
12 February 1973
Ai A/ A/ ) k2_r,k 1 k2_K,k I k2_K
, ( ~~i l s,~:-~ A ,-~,~: i + ~ ' ~a' , ~ a~;i ~ , ~a ~o' + A ~:-~,~, 'i A /~:-9 \dp' - c~_~(k-~),A'.~_~ f~,._~ ~k: - A~,+~(k •
.
where ap]l
I
+~)'~ a s~_~(A£~,~:
.
.
w1 ]
.
+
601 d aP t'
a%\l Ag_A~2-~).~I ] aPil
s,,K'~p+K'~ s k ,aPil - %,
k2-g
=Ck~_,,fs~_,,A_,,, k
Ck = 47rq2/ik2ek , s k = - i ( w - k ' o ) -1 , e is the dielectric constant, ~bo is the velocity distribution function, q is the electron charge and a prime means that o in a symbol is replaced by o'. Let us finally estimate when ion sound turbulence enhances the process where a transverse wave decays into two plasma waves. We found that
I St S ~ "~
C;W~IKId4K cu nT(.~p
ue ~p
where I~K is the spectral energy density of ion sound turbulence, nT the thermal energy density, oe the thermal electron velocity, c the light velocity and u the effective turbulent collision frequency [2]. The most important r vectors have a length o f the order o f Wp/Oe. Thus we obtain strong enhancement of the nonlinear interaction when W / n T > Oe/C, where W = f WKd4~ is the energy density o f the ion sound turbulence. Ion damping must be considered in the domain where [gl is large. One must also take the hot ions, which usually are created by ion sound turbulence, into account. This leads to a complex value of S t, whereas it was real in the absence of turbulence. It is therefore possible [3] that the above mentioned creation o f plasma waves is explosive. This could be of great importance for plasma heating by lasers or high frequency radio waves.
References [1] [2] [3] [4] [5 ] [6]
8
V.N. Tsytovich, Nonlinear effects in plasma, ed. S.M. Hamberger (Plenum, New York, 1970). V.N. Tsytovich, An introduction to the theory of plasma turbulence (Pergamon Press, Oxford, 1972). H. Wilhelmsson, L. Stenflo and F. Engelmann, J. Math. Phys. 11 (1970) 1738. L. Stenflo, Plasma Phys. 14 (1972) 713. V.N. Tsytovich, Soviet Phys.-Techn.Phys. 14 (1970) 1321 ; Eight Intern. Conf. on Phen. in Ionized Gases, 1AEA 1967, p.408. V.G. Makhankov and V.N. Tsytovich, Joint Institute for Nuclear Research (preprint), Dubna 1972.
PHYSICS LETTERS
THREE WAVE INTERACTION
IN TURBULENT
12 February 1973
PLASMAS
V.N. TSYTOVICH Lebedev Physical Institute, Moscow, USSR
and L. STENFLO
Institute of Physics,90187 Ume~,Sweden Received 18 December 1972 A new possibility for enhancement of three wave interaction processes is pointed out. This could be of interest for plasma heating by means of high frequency radio waves as well as by laser radiation.
Although much work [e.g., 1 - 3 ] recently has been devoted to the coupling of waves in plasmas there was no general theory for three wave interactions in turbulent plasmas. Such a theory should be of interest as there are regions where one can observe enhancement of the wave-wave interactions even if the turbulence is weak. This is not surprising as three wave interactions in cold plasmas, which are not turbulent, in general are quite weak as the coupling is due to the spatial inhomogeneities, which are produced by the waves. New nonlinear terms should thus be present in a turbulent plasma as the turbulence can introduce spatial gradients, connected with the turbulence itself, which can enhance the wave-wave interactions. Another reason for nonlinearities is the fact that the turbulent collision frequency can be very large if the turbulence level is high enough. The collision term might then be responsible for wave coupling [4]. This paper considers wave-wave interactions in a turbulent plasma and verifies that the normal interaction processes can be significantly enhanced if the turbulence exceeds a critical limit. In connexion with plasma heating experiments it is of interest to study the possibility to enhance the absorption o f a high frequency electromagnetic wave, which penetrates into the plasma. The nonlinear process, which we then want to consider, is the decay of this wave into two longitudinal plasma waves. This process might be enhanced due to the presence of ion sound turbulence. It should then be mentioned that anomalous heating occurs when the incident wave propagates towards an increasing plasma gradient. Ion sound waves are produced by the decay of the transverse wave [5, 6] and ion sound turbulence is thus almost always present when a wave interacts with an inhomogeneous plasma in the region where its frequency is twice the electron plasma frequency (6Op). Three-wave interaction can be described by [2]
/ k = .~. f Si/(k' kl' k2) Eikl ~k28(k-k1-k2) d4kl d4k2 l,]
where k = { k, co}, d4k = dkdoa, j is the second order current density and E is the electric field of the waves, which are to be studied. We will only treat the simplest case when Sii can be expanded in powers of the square of the turbulent electric fields, i.e., IEKI2
f s,,.,7
=¢, + St.
From the Vlasov and Maxwell equations, the final result of a very lengthy calculation is
Volume 43A, number 1
+ r ' ~ -a~ s k - K
PHYSICS LETTERS
-- ap/1 ~s
a
a
a~o
k - l q A i-~,k2 +K.-~Sk+~ap] - - 1 s k2aPi2
12 February 1973
Ai A/ A/ ) k2_r,k 1 k2_K,k I k2_K
, ( ~~i l s,~:-~ A ,-~,~: i + ~ ' ~a' , ~ a~;i ~ , ~a ~o' + A ~:-~,~, 'i A /~:-9 \dp' - c~_~(k-~),A'.~_~ f~,._~ ~k: - A~,+~(k •
.
where ap]l
I
+~)'~ a s~_~(A£~,~:
.
.
w1 ]
.
+
601 d aP t'
a%\l Ag_A~2-~).~I ] aPil
s,,K'~p+K'~ s k ,aPil - %,
k2-g
=Ck~_,,fs~_,,A_,,, k
Ck = 47rq2/ik2ek , s k = - i ( w - k ' o ) -1 , e is the dielectric constant, ~bo is the velocity distribution function, q is the electron charge and a prime means that o in a symbol is replaced by o'. Let us finally estimate when ion sound turbulence enhances the process where a transverse wave decays into two plasma waves. We found that
I St S ~ "~
C;W~IKId4K cu nT(.~p
ue ~p
where I~K is the spectral energy density of ion sound turbulence, nT the thermal energy density, oe the thermal electron velocity, c the light velocity and u the effective turbulent collision frequency [2]. The most important r vectors have a length o f the order o f Wp/Oe. Thus we obtain strong enhancement of the nonlinear interaction when W / n T > Oe/C, where W = f WKd4~ is the energy density o f the ion sound turbulence. Ion damping must be considered in the domain where [gl is large. One must also take the hot ions, which usually are created by ion sound turbulence, into account. This leads to a complex value of S t, whereas it was real in the absence of turbulence. It is therefore possible [3] that the above mentioned creation o f plasma waves is explosive. This could be of great importance for plasma heating by lasers or high frequency radio waves.
References [1] [2] [3] [4] [5 ] [6]
8
V.N. Tsytovich, Nonlinear effects in plasma, ed. S.M. Hamberger (Plenum, New York, 1970). V.N. Tsytovich, An introduction to the theory of plasma turbulence (Pergamon Press, Oxford, 1972). H. Wilhelmsson, L. Stenflo and F. Engelmann, J. Math. Phys. 11 (1970) 1738. L. Stenflo, Plasma Phys. 14 (1972) 713. V.N. Tsytovich, Soviet Phys.-Techn.Phys. 14 (1970) 1321 ; Eight Intern. Conf. on Phen. in Ionized Gases, 1AEA 1967, p.408. V.G. Makhankov and V.N. Tsytovich, Joint Institute for Nuclear Research (preprint), Dubna 1972.