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Induced wave scattering in ionospheric plasma
Planet. Space Sci. 1974, Vol. 22, pp. 95 to 98. Pergamon Press. Printed in Northern Ireland
I N D U C E D WAVE S C A T T E R I N G IN I O N O S P H E R I C PLASMA N. A. MITYAKOV, V. O. RAPOPORT and V. YU. TRAKHTENGERTS
Scientific Research Radiophysical Institute (NIRFI), Lyadov Street 25/14, Gorki, U.S.S.R.
(Received 28 August 1972) Abstract--Parametric excitation of plasma oscillationsin the ionosphere by an electromagnetic wave near the reflection level has been considered. The spectrum of plasma waves forms as a result of action of the source (a pump wave), non-linear transfer towards large scales and damping. The results of the theory are in satisfactory agreement with the experiment. Powerful radio emission experiments (Wong and Taylor, 1971) on the ionosphere show that the F-layer is of interest for plasma turbulence investigations. In reality, steady-state conditions, the absence of boundaries and the small number of collisions are the main features ofF-layer plasma in contrast to a laboratory plasma. On the other hand, the highly developed methods for ionospheric diagnostic measurements enable detailed quantitative data on all main parameters of plasma turbulence to be obtained. This paper is concerned with the quantitative analysis of non-linear effects (the induced scattering of a transverse wave into the plasma one, and the non-linear interaction of plasma waves) in ionospheric plasma disturbed by powerful radio emission under the experimental conditions of Wong and Taylor (1971). The strong electromagnetic wave incident on the inhomogeneous plasma layer generates the plasma waves in the vicinity of the reflection level. This non-linear effect is associated with the induced scattering of an incident wave on ions* (Litvak and Trakhtengerts, 1971 ; Tsitovich, 1967) in quasi-equilibrium plasma (T, ~ Ti, T,.i are the temperatures of electrons and ions). The effective excitation of plasma waves in the region near the reflection level is explained by a small difference between the frequencies of the transverse and plasma waves (Tsitovich, 1967) and by the effect of the transverse wave field growth (Ginzburg, 1967). The excitation of plasma waves begins with some amplitude threshold of the transverse wave when the non-linear growth rate 7N exceeds the full damping of plasma waves 7- At a sufficiently high level of plasma waves the saturation of plasma oscillations occurs due to their energy redistribution over the spectrum in the phase space of wave numbers k (a non-linear transfer into the other scales). If the duration of pumping pulse exceeds the growth time of plasma waves this non-linear interaction may be described in terms of stationary solutions. Under experimental conditions (Wong and Taylor, 1971) the nonlinear absorption of an incident wave is small and it is possible to use the undisturbed solution for the transverse wave field in the form of the Airy function (Ginzburg, 1967). Assuming this, the initial equation for the intensity of plasma waves has the form (Tsitovich, 1967)
a In rrk Ot
:-
- e + f w(k, ki)Ze dkl = 0
(i)
where W ~ = Eo2/81r, f Wk dk are the energy density of the transverse and plasma waves, y, 7~. = ~W t are the linear damping and the non-linear growth rate of plasma waves. The * In a linear approximation the process considered corresponds to parametric instabilities in plasma, discussed in Fejer and Leer (1972); Perkins and Kaw (1971) and Silin (1965). 95
96
N, A. M i T Y A K O V . ~. (~ R A P O P O t t / | ' and ~,. YU. T R A K H T E N G I R ] 5
interaction coefficients of waves ~. and w for ~i~c case )'N "'- kvT, depend on the t:emperatv~c ratio T,/T~ and on the relative difference frequency of interacting waves.v := 5,)flk - k~{ with VTe = ( 2 T I M ) 1/" lhe thermal ion velocity. 5~, is the absolute difference of frequencies; of interacting waves
w(k,
k,)
....
~,., .
,,,o,,~ cos~ (e, eO . . . F(.v, ~,~ neT~
.
T,/TJ,
~:?i~
where e, el are the wave polarization vectors, (-')o~-- (4rre~n~,/m)1/2 is the plasma frequency of electrons. At T,/T~ ~,~ 1 the function F(x) has the maximum near x,~ ~ 1 with the width Ax ,~ 1. The maximum value of the function F(x~, TdT~) = F,,~is given in Table 1 tbr the different T~/T~. TABLE 1
TJT~ F,~ X,~
1 0.3 1.2
1.5 0-43 1.4
2 0-57 1-6
The character of non-linear energy transfer over the spectrum of plasma waves depends on two values: the maximum frequency shift in one act of scattering [k -- kll VT~and on the spectrum width of the initial signal &o. At Om < Ik -- kl[ vT~ the energy transfer over the spectrum is of the integral character. If &o in this case is substantially less than the frequency interval, influenced by non-linear pumping, the process of non-linear interaction is step by step interaction between discrete satellites. The spectrum width of each satellite ~oo, < kvT~ and the distance between the satellites ~kv~.~. Kingsep and Rudakov (1970) give some reasons largely responsible for this kind of non-linear interaction. The case may arise at the incidence of a monochromatic electromagnetic wave on the plasma layer. It should be pointed out that the energy transfer over the spectrum connected with the variation of the wave number is accompanied by the intensive transformation of the angular spectrum of waves which leads to isotropy distribution in k space. Isotropy is always of an integral character (it occurs without frequency shift). These facts allow the plasma turbulence to be considered as isotropic and the use, instead of Equation (1), of the equivalent system of ordinary differential equations, describing the evolution of different satellites. The stationary solution of Equation (1) may then be written in the form (see some details, Mitjakov, Rapoport and Trakhtengerts, 1973)
(O~o/Ok ( W'
k ~k k- ~v ) , k > k > k -- W w' (3)
outside the interval where ~k = 2x,~ Ok]Oookvri, k = (z/3L)I/2VT~/V/2 0%~,VT, = (2T,/m) 1/~, L = n,(an,/Oz)-l, is the scale of the linear layer; the coordinate z is read off from the reflection level of a transverse wave. From Equation (3) Fig. 1 gives the calculated intensity distribution of plasma waves as a function of frequency f , wave number k and the distance from the reflection level z. The initial data (the frequency of pumping wave 5.62 Mc/sec, the intensity of pumping wave 30 #W/m 2, T, = 2T~ ---- 2 × 10a °K, L = 50 kM, 7 = 1.6 × 10~ sec-t, Y~[7 = 6) correspond to the experiment made in Arecibo (Wong and Taylor, 1971). As the
INDUCED WAVE SCATTERING IN IONOSPHERIC PLASMA
97
0
0.5
bO
.2-5
/ 5 52
5 "47 f,
557
5 62
m c/see
F I G . 1. T H E PLASMA WAVE INTENSITY DISTRIBUTION AS A FUNCTION OF THE FREQUENCY f , THE WAVE NUMBER k AND THE DISTANCE TO THE REFLECTION POINT 2.
registration of plasma waves in Arecibo was carried out by radar at the frequency fr = 430 Mc/sec, the back scattering cross-section at the combination frequency 435"62 Mc/sec is determined by the intensity of plasma waves with the wave number k0 _ 4~fr _ 0-18 cm -1. C
It follows from Equation (3) that at fixed/} = ko the spectrum of plasma waves is of triangular form and only its height is determined by the power of the incident wave W ~ (see Fig. 1). That is why the spectrum of the scattered signal at the combination frequency of radar 435"62 Mc/sec must also have the triangular form and the total intensity is described by square law (W*)2. Just this law was observed in the experiment by Wong and Taylor (1971). At the parameters mentioned above the calculated width of the triangular spectrum is 60 kHz and in the experiment A / ' ~ 5 0 kHz (Wong and Taylor, 1971). The total intensity of the scattered signal must exceed, according to the calculations, 7 × 103 times the intensity of the scattered signal for the non-perturbed level of plasma waves determined by photoelectrons. The corresponding ratio obtained in the experiment is equal to (I -- 3) 10a (Wong and Taylor, 1971; Evans, 1967). The value of plasma waves damping 7 = 1.6 × 103, measured in the experiment (Wong and Taylor, 1971) is used in the calculations. These values exceed the collision frequency of electrons and ions ~,~ by 3-4 times. It shows the determining role of Landau damping for plasma waves. When noting that Landau damping in cold plasma becomes essential only in the region z > 4 km below the reflection level, the existence of fast electrons (photoelectrons or the electrons generated as the result of non-linear interaction) must be assumed in quantitative explanation of the experimental value 7. It is sufficient that at the temperature T~ = 50T~ the concentration of fast electrons is equal to n~ := 5 × 10-She. 7
98
N . A . MITYAKOV~ vo O. RAPOPORT and V. YU. TRAKHTENGERT}_,
The satisfactory agreement of theory and experiment allows one to hope ti~at at pumping, powers, which do not substantially exceed the threshold power, the approximations used at,::. realistic, that the induced wa~.e scattering in plasma is the main non-linear cl~'ect, and ~!~ the saturation level o f plasma waves occurs as the result o f non-linear energy trans~'e~~ ~ the other scales with the subsequent damping. The ordinary c o m p o n e n t used a~ i i~, p u m p i n g wave, taking account of the influence o f the Earth's magnetic field, has not ;c:~,A to any essential difficulties in the calculations. The problem is reduced to ~he k n o w n ot~e with some corrections in the coefficienls of non-linear interactions and the dispers~x.c equations. The qualitatively new effects occur with the increase o f the p u m p i n g wave power, when the approximation of the undisturbed transverse wave cannot be used and the condition ?'~, <: kr:r~ is violaled. REFERENCES EvaNs, J. V. (1966). Proc. IEEE 57, 496. FrJER, J. A. and LEER,E. (1972). Radio Sei. 7, 481. GINZBURG,V. L. (1967). Electromagnetic Wave Propagation in Plasma. Nauka. KINGStP, A. S. and RUDAKOV,L. I. (1970). JETP 58, 582. LITVAK,A. G. and TRAKHTENGERTS,V. Yu. (1971). JETP 60, 170. MrrJAKOV, N. A., RAPOeORT,V. O. and T~rd~TENGERTS, V. Yu. (1973). Geomag. Aeron. In press. P~RKINS, F. W. and KAw, P. K. (1971) J. geophys. Res. 76, 282. SIt[N, V. P. (1965). JETP 48, 1679. TSITOWCH,V. N. (1967). Non-linear Effects in Plasma. Nauka. WONG, A. Y. and TAYLOR,R. J. (1971). Phys. Rev. Lett. 27, 644.
I N D U C E D WAVE S C A T T E R I N G IN I O N O S P H E R I C PLASMA N. A. MITYAKOV, V. O. RAPOPORT and V. YU. TRAKHTENGERTS
Scientific Research Radiophysical Institute (NIRFI), Lyadov Street 25/14, Gorki, U.S.S.R.
(Received 28 August 1972) Abstract--Parametric excitation of plasma oscillationsin the ionosphere by an electromagnetic wave near the reflection level has been considered. The spectrum of plasma waves forms as a result of action of the source (a pump wave), non-linear transfer towards large scales and damping. The results of the theory are in satisfactory agreement with the experiment. Powerful radio emission experiments (Wong and Taylor, 1971) on the ionosphere show that the F-layer is of interest for plasma turbulence investigations. In reality, steady-state conditions, the absence of boundaries and the small number of collisions are the main features ofF-layer plasma in contrast to a laboratory plasma. On the other hand, the highly developed methods for ionospheric diagnostic measurements enable detailed quantitative data on all main parameters of plasma turbulence to be obtained. This paper is concerned with the quantitative analysis of non-linear effects (the induced scattering of a transverse wave into the plasma one, and the non-linear interaction of plasma waves) in ionospheric plasma disturbed by powerful radio emission under the experimental conditions of Wong and Taylor (1971). The strong electromagnetic wave incident on the inhomogeneous plasma layer generates the plasma waves in the vicinity of the reflection level. This non-linear effect is associated with the induced scattering of an incident wave on ions* (Litvak and Trakhtengerts, 1971 ; Tsitovich, 1967) in quasi-equilibrium plasma (T, ~ Ti, T,.i are the temperatures of electrons and ions). The effective excitation of plasma waves in the region near the reflection level is explained by a small difference between the frequencies of the transverse and plasma waves (Tsitovich, 1967) and by the effect of the transverse wave field growth (Ginzburg, 1967). The excitation of plasma waves begins with some amplitude threshold of the transverse wave when the non-linear growth rate 7N exceeds the full damping of plasma waves 7- At a sufficiently high level of plasma waves the saturation of plasma oscillations occurs due to their energy redistribution over the spectrum in the phase space of wave numbers k (a non-linear transfer into the other scales). If the duration of pumping pulse exceeds the growth time of plasma waves this non-linear interaction may be described in terms of stationary solutions. Under experimental conditions (Wong and Taylor, 1971) the nonlinear absorption of an incident wave is small and it is possible to use the undisturbed solution for the transverse wave field in the form of the Airy function (Ginzburg, 1967). Assuming this, the initial equation for the intensity of plasma waves has the form (Tsitovich, 1967)
a In rrk Ot
:-
- e + f w(k, ki)Ze dkl = 0
(i)
where W ~ = Eo2/81r, f Wk dk are the energy density of the transverse and plasma waves, y, 7~. = ~W t are the linear damping and the non-linear growth rate of plasma waves. The * In a linear approximation the process considered corresponds to parametric instabilities in plasma, discussed in Fejer and Leer (1972); Perkins and Kaw (1971) and Silin (1965). 95
96
N, A. M i T Y A K O V . ~. (~ R A P O P O t t / | ' and ~,. YU. T R A K H T E N G I R ] 5
interaction coefficients of waves ~. and w for ~i~c case )'N "'- kvT, depend on the t:emperatv~c ratio T,/T~ and on the relative difference frequency of interacting waves.v := 5,)flk - k~{ with VTe = ( 2 T I M ) 1/" lhe thermal ion velocity. 5~, is the absolute difference of frequencies; of interacting waves
w(k,
k,)
....
~,., .
,,,o,,~ cos~ (e, eO . . . F(.v, ~,~ neT~
.
T,/TJ,
~:?i~
where e, el are the wave polarization vectors, (-')o~-- (4rre~n~,/m)1/2 is the plasma frequency of electrons. At T,/T~ ~,~ 1 the function F(x) has the maximum near x,~ ~ 1 with the width Ax ,~ 1. The maximum value of the function F(x~, TdT~) = F,,~is given in Table 1 tbr the different T~/T~. TABLE 1
TJT~ F,~ X,~
1 0.3 1.2
1.5 0-43 1.4
2 0-57 1-6
The character of non-linear energy transfer over the spectrum of plasma waves depends on two values: the maximum frequency shift in one act of scattering [k -- kll VT~and on the spectrum width of the initial signal &o. At Om < Ik -- kl[ vT~ the energy transfer over the spectrum is of the integral character. If &o in this case is substantially less than the frequency interval, influenced by non-linear pumping, the process of non-linear interaction is step by step interaction between discrete satellites. The spectrum width of each satellite ~oo, < kvT~ and the distance between the satellites ~kv~.~. Kingsep and Rudakov (1970) give some reasons largely responsible for this kind of non-linear interaction. The case may arise at the incidence of a monochromatic electromagnetic wave on the plasma layer. It should be pointed out that the energy transfer over the spectrum connected with the variation of the wave number is accompanied by the intensive transformation of the angular spectrum of waves which leads to isotropy distribution in k space. Isotropy is always of an integral character (it occurs without frequency shift). These facts allow the plasma turbulence to be considered as isotropic and the use, instead of Equation (1), of the equivalent system of ordinary differential equations, describing the evolution of different satellites. The stationary solution of Equation (1) may then be written in the form (see some details, Mitjakov, Rapoport and Trakhtengerts, 1973)
(O~o/Ok ( W'
k ~k k- ~v ) , k > k > k -- W w' (3)
outside the interval where ~k = 2x,~ Ok]Oookvri, k = (z/3L)I/2VT~/V/2 0%~,VT, = (2T,/m) 1/~, L = n,(an,/Oz)-l, is the scale of the linear layer; the coordinate z is read off from the reflection level of a transverse wave. From Equation (3) Fig. 1 gives the calculated intensity distribution of plasma waves as a function of frequency f , wave number k and the distance from the reflection level z. The initial data (the frequency of pumping wave 5.62 Mc/sec, the intensity of pumping wave 30 #W/m 2, T, = 2T~ ---- 2 × 10a °K, L = 50 kM, 7 = 1.6 × 10~ sec-t, Y~[7 = 6) correspond to the experiment made in Arecibo (Wong and Taylor, 1971). As the
INDUCED WAVE SCATTERING IN IONOSPHERIC PLASMA
97
0
0.5
bO
.2-5
/ 5 52
5 "47 f,
557
5 62
m c/see
F I G . 1. T H E PLASMA WAVE INTENSITY DISTRIBUTION AS A FUNCTION OF THE FREQUENCY f , THE WAVE NUMBER k AND THE DISTANCE TO THE REFLECTION POINT 2.
registration of plasma waves in Arecibo was carried out by radar at the frequency fr = 430 Mc/sec, the back scattering cross-section at the combination frequency 435"62 Mc/sec is determined by the intensity of plasma waves with the wave number k0 _ 4~fr _ 0-18 cm -1. C
It follows from Equation (3) that at fixed/} = ko the spectrum of plasma waves is of triangular form and only its height is determined by the power of the incident wave W ~ (see Fig. 1). That is why the spectrum of the scattered signal at the combination frequency of radar 435"62 Mc/sec must also have the triangular form and the total intensity is described by square law (W*)2. Just this law was observed in the experiment by Wong and Taylor (1971). At the parameters mentioned above the calculated width of the triangular spectrum is 60 kHz and in the experiment A / ' ~ 5 0 kHz (Wong and Taylor, 1971). The total intensity of the scattered signal must exceed, according to the calculations, 7 × 103 times the intensity of the scattered signal for the non-perturbed level of plasma waves determined by photoelectrons. The corresponding ratio obtained in the experiment is equal to (I -- 3) 10a (Wong and Taylor, 1971; Evans, 1967). The value of plasma waves damping 7 = 1.6 × 103, measured in the experiment (Wong and Taylor, 1971) is used in the calculations. These values exceed the collision frequency of electrons and ions ~,~ by 3-4 times. It shows the determining role of Landau damping for plasma waves. When noting that Landau damping in cold plasma becomes essential only in the region z > 4 km below the reflection level, the existence of fast electrons (photoelectrons or the electrons generated as the result of non-linear interaction) must be assumed in quantitative explanation of the experimental value 7. It is sufficient that at the temperature T~ = 50T~ the concentration of fast electrons is equal to n~ := 5 × 10-She. 7
98
N . A . MITYAKOV~ vo O. RAPOPORT and V. YU. TRAKHTENGERT}_,
The satisfactory agreement of theory and experiment allows one to hope ti~at at pumping, powers, which do not substantially exceed the threshold power, the approximations used at,::. realistic, that the induced wa~.e scattering in plasma is the main non-linear cl~'ect, and ~!~ the saturation level o f plasma waves occurs as the result o f non-linear energy trans~'e~~ ~ the other scales with the subsequent damping. The ordinary c o m p o n e n t used a~ i i~, p u m p i n g wave, taking account of the influence o f the Earth's magnetic field, has not ;c:~,A to any essential difficulties in the calculations. The problem is reduced to ~he k n o w n ot~e with some corrections in the coefficienls of non-linear interactions and the dispers~x.c equations. The qualitatively new effects occur with the increase o f the p u m p i n g wave power, when the approximation of the undisturbed transverse wave cannot be used and the condition ?'~, <: kr:r~ is violaled. REFERENCES EvaNs, J. V. (1966). Proc. IEEE 57, 496. FrJER, J. A. and LEER,E. (1972). Radio Sei. 7, 481. GINZBURG,V. L. (1967). Electromagnetic Wave Propagation in Plasma. Nauka. KINGStP, A. S. and RUDAKOV,L. I. (1970). JETP 58, 582. LITVAK,A. G. and TRAKHTENGERTS,V. Yu. (1971). JETP 60, 170. MrrJAKOV, N. A., RAPOeORT,V. O. and T~rd~TENGERTS, V. Yu. (1973). Geomag. Aeron. In press. P~RKINS, F. W. and KAw, P. K. (1971) J. geophys. Res. 76, 282. SIt[N, V. P. (1965). JETP 48, 1679. TSITOWCH,V. N. (1967). Non-linear Effects in Plasma. Nauka. WONG, A. Y. and TAYLOR,R. J. (1971). Phys. Rev. Lett. 27, 644.