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. . , M. Parrot , J. Manninen
1 2 3

№ , І LPC2E/CNRS, Orleans, France 3 Sodankyla Geophysical Observatory, Finland

« » 14-18 2011 .,



. 2 7 [ ... . , 1984, . 87, 12, .. 18941904; Bespalov P.A. Effective saturation of absorption in a plasma magnetospheric maser. In book: Nonlinear Space Plasma Physics, R.Z. Sagdeev -Editor-in-Chief, American Institute of Physics, 1993, p.339-346]. . . . . . , , .


Magnetospheric plasma maser [Bespalov P.A, Trakhtengerts V.Yu., 1980]
Research in recent decades has shown that the regions of the radiation belts (RB) of the Earth and Jupiter, if cyclotron instability develops in them, are largely similar in their physical properties to laboratory masers and lasers.

In a magnetospheric plasma maser (MPM) the relatively dense magnetised plasma and the conjugate ends
of a magnetic trap form a quasioptical resonator for electromagnetic waves. An active substance consists of the plasma of the RB having a characteristic loss cone in velocity space. The nonequilibrium of the distribution function lead to a cyclotron instability development, the increment of which determines the balance between the processes of induced emission and absorption. The sources of energetic particles fill the pumping role. The working modes in MPM are whistler and Alfven waves, which are channelled by the magnetic field and, being reflected from the ends of the magnetic trap.


Saturable absorber
In laboratory lasers extensive use is made of saturating absorbers ­ specially chosen nonlinear elements in which, for a certain frequency range, absorbers ­ specially chosen nonlinear elements in which, for a certain frequency range, absorption decreases with an increase in the intensity of the electromagnetic radiation. In MPM in a number of cases with an increase in the energy density of electromagnetic waves, quasilinear relaxation occurs in such a way that the increment of cyclotron instability does not decrease but increase due to variation of the anisotropy of the distribution function [Bespalov P.A., 1982]. Passive synchronization of modes may electromagnetic signals is provided by the wave packet propagation between conjugate A study of the causes and consequences absorption permits a better understanding of SP emission in the whistler range, as well Alfven waves. develop, in which SP formation of a nonlinear regions of the ionosphere. of effective saturation of conditions of excitation of as pearls in the range of


Slow effective saturation of absorption [Bespalov P.A., 1984; 1993] Equation for the envelope of a pulse sequence
In the linear approximation a maser ( t Tgr Tb ) is characterised by a complex coefficient of transmission G = + i from input to input. We assume that a pulse E0 (t ) , short compared with period

of group propagation, with spectrum E0 ( ) is applied to the maser input. With allowance for the transmission coefficient ( ( ) < 0 )

Tgr

1 E (t ) = 2

+

To clarify the asymptotic laws, we can substitute the transmission coefficient in the form We have

-



E0 ( )e

- i t

1 + eG + e (

2G

1 + ... ) d = 2

+

-



E0 ( )e
0

- it

e - it d . G 1- e

G = g + i + iTgr - 2 , where = - i t expression for envelope E (t ) = E (t )e 0 : + 1 e - it E (t ) = E0 (0 + ) 1 - 1 + g - 2 ei( + 2 - ( )
i

.

2 Trg

)

d .

Therefore, the envelope of the individual pulse satisfies the equation

2 E i E + e + (1 + g - e 2 t t

)

E = 0,

where

= R + i

I

,

= Tp - Tgr ,

are constants.


Nonlinear interaction in an active medium
The distribution function of the energetic particles in the region of minimum of magnetic field varies in accordance with the equation

where Dij ( ) are known functions. Then the amplification of waves in a single passage through the resonator is determined by the expression

f 2 f = Dij | E | , t i j

Here Ki ( ) are known functions; g pulse. For a relatively low energy in the pulse, we can write:
t

( f - f0 ) 3 g eff = g + Ki d . i and f 0 ( ) correspond to the time long
t 2 2

before the arrival of the next

g

eff

= g + | E | dt - | E |2 - -

dt .

The explicit form of the coefficients is given in [6], where the possibility of the realisation of the inequalities g < 0, > 0 and > 0 is also shown. Then the problem of determining the shape of the envelope of a pulse sequence comes down to the solution of the nonlinear equation

2 E i E + e + 1 + g 2 t t

eff

- ei E = 0.


Soliton solution and its properties
It is easy to ascertain by a direct test that a soliton solution can be sought successfully in the form

where

E p , t p , a , b are constants which obtained as functions of I / R , g , / 2 . For example

t E = E p cosh tp

ia -1

exp

t ib tp

,
2 t

The mean frequency

(t )

1 2 ( -3 R + 9 R + 8 a= 2t
= 0 -

).

of the dynamic spectrum of the radiation varies within a pulse as

ba t - tanh , tp tp tp

where
I

whistlers, while for Alfven waves I <0 . The solution shows that the period between two successive pulses Tp Tgr . Without QL effects Whith QL effects

=- (1 2 ) ( Tgr ) > 0 for


Space and ground-based examples of short-period VLF emissions [Bespalov P.A., Parrot M., Manninen J., 2010]
Among many different types of natural VLF emissions there are SP emissions which have a spectral form repetition with periods of between 2 and 7 seconds. Such spectrograms are recorded by the low-altitude (710 km) satellite DEMETER. Figure shows a spectrogram of data recorded on June 3rd 2005 during 90 seconds between frequencies of 0 and 5000 Hz, when the satellite was in burst mode. Clear SP emissions can be observed between 1200 and 2500 Hz, with a period of 3.65 seconds. These occur at mid-latitude during the recovery phase of a magnetic storm. The regularity and shape of these elements, their location at mid-latitude, and their period indicate that these SP emissions cannot be mistaken with chorus. The cut-off frequency, observed to be at 228 Hz, is the bi-ion frequency which is close to the proton gyrofrequency (312 Hz at 09:04:30 UT). At higher frequencies the vertical lines are due to whistlers.


Nest Figure displays a spectrogram recorded on March 1st 2007 during a minute interval, between frequencies of 0 and 3000 Hz. The event occurred during a period of moderate magnetic activity. SP emissions with periods of 1.73 s are observed inside QP emissions between frequencies of 800 and 1500 Hz. Here also, they are very different from chorus emissions. As in the previous figure the cut-off frequency (570.3 Hz at 20:25:00 UT, 524.7 Hz at 20:26:00 UT) is the bi-ion frequency. At lower frequencies, electrostatic waves are seen. Furthermore, the spectrogram confirms theoretical conclusions about the general prerequisites to trigger SP and quasi-periodic (QP-2) VLF emissions. Many interesting examples of magnetospheric VLF emissions were obtained on the Earth's surface in the experiments performed at Sodankyla Geophysical Observatory (SGO). In particular, SP VLF emissions were registered. One example of a SP spectrogram from Kannuslehto at L=5.4 can be seen in Figure. Here, the period of these SP emissions is approximately 5 s. These emissions are in agreement with the examined theoretical formalism.



- . -, 2 7 . SP DEMETER SGO. , . , (QP 1, QP 2) . SP . SP . : * . * , . * .