STOCHASTIC STABILIZATION OF THE ExPLoSIVE

INSTABILITY IN A NONEQUILIBRIUM PLASMA

S. Ya. Vyshkind, E. A. Dubinina, S. L. Sakovskaya, and S. M. Fainshtein UDC 533.9:530

The explosive instability of longitudinal waves in a beanr-plasma system is examined. It is found that under certain conditions in the system indicated, stochastic stabilization of the "explosion" is possible, and in this case complex attracting regions with random characteristics are formed in phase space. Estimates for laboratory plasma are presented.

It is well known (see, e.g., [1-4]) that an explosive instability, characterized by a sharp increase in the amplitude of the interacting waves, is possible in a nonequilibrium plasma. Possible mechanisms for stabilizing this instability by a nonlinear shift in fre- quency or nonlinear damping of the oscillations were analyzed in [5-7]. We note that in all of the papers cited a dynamic regime of saturation of the instability appears, i.e., the amplitudes and phases of the oscillations are strictly determined and depend on the non- linear detuning from synchronism or nonlinear damping. However, stochastic regimes of parametric instabilities, whose characteristic property is the tangling of phase trajectories of the system and the appearance of small mode turbulence, are being intensively discussed in �9 the literature (see, e.g., [8-10]).

In this paper, we discuss a fundamentally new mechanism for stabilizing the "explosion" related to the appearance of complex motions, which correspond to attracting sets in the phase space of the system, similar to a strange attractor. Although the presence of a strange attractor is not rigorously proved, it is shown that the characteristics of these motions, in particular the time period of motion along the loops of the phase trajectories, are a random function with a normal distribution. The results obtained are illustrated for the nonequilibrium beam-plasma system. Conditions are presented for the appearance of stochastic motion in a plasma and the amplitudes of the excited oscillations are estimated. These results are of general physical interest and are also useful for plasma applications, since they show the conditions for generating noise fluctuations in beamlike plasma systems.

i. We shall examine a nonisothermal plasma into which a charged particle beam has penetrated. The perturbations of the longitudinal electric field are given at the plasma boundary (x = 0), i.e., in what follows we find the stationary distribution of field ampli- tudes in an infinite plasma. The starting one-dimensional system of equations has the form*

aE -- ---- 4~e (p~ - - Pt + ~), Ox

Ov~, t e xT~, ~ ap~, dOx OqJe, I -Jr- "0,, I --" - - E v elf Ve i ,

Ot ' Ox m~. l m~, l (N~. i + p~, ~) '

Opt, ~ Ova, t 0 (p~,, + v,,~), Ot . + N ~ , ~ + v~,z = - O"-x

Ovs Ov~ e O~ s o-7 + v o , s , - - , O X m e " - - - - - v s O X

*We assume that T e >> T i (Te, i are the electron and ion temperatures). In what follows, for simplicity, T i ~ 0.

Gorki Polytechnic Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 25, No. 7, pp. 773-778, July, 1982. Original article submitted May 27, 1981.

554 0033-8443/82/2507-0554507.50 �9 1983 Plenum Publishing Corporation

Ov~ 00~ O o ~ 1~ N~ 4- V,, " ~ . . . . . . (n~'~'~), (1 ) Ot - ~ x Ox Ox

w h e r e ~ e , i , V e , i , s a r e t h e d e v i a t i o n s o f t h e e l e c t r o n a n d i o n c o n c e n t r a t i o n s o f t h e p l a s m a a n d t h e beam p a r t i c l e s a n d t h e c o r r e s p o n d i n g v e l o c i t i e s f r o m t h e i r e q u i l i b r i u m v a l u e s . (The l a t t e r a r e i n d i c a t e d a s N e = N i , Ns , Ve0 = 0 , v i 0 = O, Vs0 = V0, r e s p e c t i v e l y . ) F u r t h e r ,

Vef f is the effective collision frequency.

The system of equations (i) is described in the linear approximation by the dispersion equation*

(.~)g t%2 i to ~ 1 - - 0, -- 0; (2)

(~ - i,) - ~ #~ ~ (~ - i,) ~- (<0 - ~ vD ~.

u)~ 4=Ne ~ , ~O~o1_ 4~N~e._....._~'- , o~s = -~,4~Nse~ v~re == "~r Te (3 ) t i t l i t I l i t e 17"l e

Under the conditions

Z - < < I ~ ~ .T< , N.~<< t

Eq. (2) describes HF beam waves and ion-sound waves in a nonisothermal plasma:

~, --~ k~c~ (1 + ~ ~2i~2 ~-l,'~ (4)

~,3--kz3Vo~•177 y ~ % ~ j (5)

y, ~ are small increments and decrements of the corresponding modes, whose values were computed on a computer.

The waves (4) and (5) can satisfy the following matching conditions:

~3 = ~2 + ~ l , k~ = k2 + k~ + 8 ( 6 )

(6 is the detuning from synchronism). It is easy to see that

i ~ ~'>,,sc: V s 1 ( 7 )

(we r e c a l l t h a t V 0 >> Cs , s o t h a t t h e u s u a l i o n - s o u n d i n s t a b i l i t y c a n b e n e g l e c t e d ) s w h i l e

Using the standard procedure [ii, 12], we obtain the stationary equations for the dimensionless complex wave amplitudes:#

Oa,, ~/Ox~ = a;, lag e -is.,._ vl, 2a,,~ + i [m (] a~ ]2 4_ [ a, r) + l i a3 [2] al, ~,

OaUOx~ = a i a z e -i~-" + "i3a~ - - i [m ( l ag I' + [a , I s) + l t as [-~] a3,

(9 )

where

[ a , ] = [ ls ~j2, i J= r r n,

':'2, 3 ~ e ( 4 m y ~ ) - I N, N71 (,,,~, 3,,,J~ (~%,~oD-2,

*We assume that the waves propagate along the particle beam. #We note that the weak increment y appears due to the interaction of negative energy waves with the dissipative medium.

555

01 1 e

8 mics

k~ e2 toa ~ N e ~O ot

4 efioi N s C s

We transform to new variables:

X = l aa I cos (arg aa - arg a2 - arg al + ~x),

Y = l a3 [ sin (arg aa - - arg a~ - - arg at + ~x),

Z = l a , l la~ [, V = l a , l/la=l.

(io)

Then system (9) is rewritten in the form

,K = Z + ~X - - ~r -}- Y~ (V + V . x) + 3mZY (V + V -x) -}- 31Y (X 2 + Y=),

? = .~v + ~ x - x v ( v + v - , ) - 37nzx(v + v = , ) - 3tx(x~ + y~),

2 = z x (v + v - o - - z (~ + ~,),

~7 - - X ( 1 - - V z) + V(~2 - - ~,).

(ii)

The dot indicates differentation d/dx6.

System (ii), aside from the zero equilibrium, has two more equilibrium states:

X,~ = V ~, "~2, Y~ 2 = + h~2 -st- -- J

6zV~,~--~ [ 36t w ~ s l j (12) ~-0 v o

Z l , 2 ~ 0, l , 2 = V312/'r �9

It is easy to determine that for X ~ 0 the zero equilibrium state is always unstable. The nonzero equilibrium states (12) are also unstable.

Figure i shows the trajectories of the roots of the characteristic equation, calculated on a computer, as a function of the nonlinear detuning parameter 1 for different y. It is evident that for some value of I two real roots coalescing form a pair of complex-conjugate roots. The factor 7 qualitatively has no effect on the behavior of the roots of the charac- teristic equation, and only shifts the of the roots indicates the oscillatory of the trajectories in the phase space account the instability, gives a basis subsequent analysis was performed near impossible to study (ii) analytically,

bifurcation point. It is evident that the complexity nature of the process, i.e., it assumes "returnability" of the system, and this circumstance, taking into for assuming the appearance of complex motions. The the bifurcation points and, in addition, since it is (ii) was modeled on an AVK-2 analog computing complex.

2. Numerical integration of the system showed that for small linear detuning ~, the instability becomes bounded only for large values of the nonlinear detuning and either an equilibrium stationary state (stable equilibrium state in phase space) or a dynamic stabili- zation state (limit cycle in phase space) is established. For large linear detuning, the "explosion" is bounded even for small nonlinear detuning. Thus, a decrease in the linear detuning is analogous to an increase in the nonlinear detuning and vice versa. As 1 in- creases, a limit cycle appears in phase space which at first increases in size, and then a double cycle is formed from the single cycle. With further increase in 1 a fourfold cycle appears, which in its turn becomes unstable and a region with complex behavior arises in phase space. We note that the values of the bifurcation parameter 1 obtained in the numerical experiment satisfy the universality law [13]:

l = - - l n ~ " ~ ~ Const, (13)

556

"h s

- t

-2

P

! ! 1

0,45 0,@6 0,9

Fig. i. Dependence of the "trajectories"

of the roots of the characteristic equa- tion P on the coefficient 1 with YI = 10-5, Y3 = 3"i0-5, Y2 = 2"I0-5,Y4 = 4"10 -5 ,

~i ~ ~2 ~ 10-3, ~ = 0, n = 0.

t = 0.13 t = 0,65

t = 0,'78 C=0,8

Fig. 2. Projection of the four-dimensional phase space on three-dimensional space (XYZ).

where I n is the value of the parameter for which a cycle with period 2 n is is created; 60 =

4~ universal Feigenbaum constant; and /cr, value of the parameter after which chaos' appears in the system.

Figure 2 shows several successive bifurcations leading, subsequently, with continuous variation of the nonlinear detuning parameter, to the appearance of a stochastic attractor.

Thus, in the case y = 0.4.10 -4 , 6 = I, v I = 0.0012, ~2 = 0.0015, the singlefold cycle (n = i) is created for l I = 0.13, the twofold cycle (n = 2) is created for 22 = 0.65, and the four- fold (n = 4) cycle is created for l~ = 0.78. Using relation (13), it is easy to calculate the value of the parameter /cr for which chaos should appear: /cr ~ 0.79 (here const = 3.06). In the numerical experiment, the attractor was observed at 1 = 0.8. Analogous bifurcations were also obtained for other combinations of parameters; in particular, when the linear de-

tuning decreases (6 = 0.9) and for the same values of the decrements and increments as above, a singlefold cycle appears for 11 = 0.3, a twofold cycle appears for 12 = 0.67~ and a four- fold cycle appears for 14 = 0.75. Now, already, const = 2.01, while /cr = 0.75. In the

557

experiment, in this case, an attractor was obtained for 1 = 0.76. Thus, the state with stochastic stabilization of the explosive instability arises for a value of the nonlinear detuning parameter /cr = 0.75-0.79 and exists up to values ~cr = 1.05. As the nonlinear detuning increases further, stabilization is impossible in the system -- the waves are no longer synchronized and the slow beam wave in this model grows without bound. The effect of the parameter m on the dynamics of the process is analogous to 1.

Thus, together with the well-known states in which the explosion is stabilized, under certain conditions a stochastic bound on the instability is also observed. An analysis of such statesshowed that the spectrum of interacting waves is broadened.

We note, however, that although in this work we have not proved the property of hyper- bolicity, the correlations do not completely decouple (the correlation function does not reduce to zero), and we can confidently Speak of the presence of a stable attracting set, whose time of existence (according to computer calculations) is at least not less than 103m~, and on which the behavior of the phase trajectories is random. The regions of complex stabilization states are not an exotic exception in the parameter space. A con- tinuous variation (which we performed by both increasing and decreasing their values) revealed the sequence in which the doubling bifurcations, corresponding to qualitatively different states, were passed, and it was possible to follow how one changed into another. In so doing, we observed the coupling-hysteresis effect of the states: Appearance of one regime of stabilization and its loss of stability occurred for different (for the forward and reverse path) values of the parameter, i.e., there is a rigid bound on the "explosion."

The regions of attraction of different stabilization regimes for a number of combina- tions of values of the system parameters were determined on the surface of initial condi- tions (IC), i.e., transient processes were examined for IC taken within the region bounded by the attractor and outside it. For small initial perturbations the stabilization has a dynamic character (limit cycle in phase space); if, on the other hand, the initial wave intensities exceed some threshold, then a stochastic stabilization regime develops. In particular, for 6 = i, X = 0.4"i0-4, Vl = 0.0012, Y2 =0.0015, a stochastic regime arises

for X.in =--0.05, Y.ln = 0.335, Zin = 0.01, Vin = 0.99.

3. In conclusion, we shall estimate the characteristic parameters of the exciting waves in the electron beam-plasma system with the following parameters:

N e ~ 1 0 1 t c m "3 , coo ~ 3 �9 1 0 1 ~ ' N o s / N o ~ 10 -4,

V o ~ 3 �9 1 0 % m / s e c , Veff/r ~ 10 -3, 7 ~ 1 0 - 4 % t ,

• ~ 10 ev.

Under the initial conditions a01 ~ 0.36 V/cm, a02 ~ 0.28 V/cm, a03 ~ 0.27 V/cm, over the time T ~ 10 -8 sec a complex motion is established with wave amplitudes a I ~ 8.5 V/cm, a 2 % 9.2 V/cm, a 3 ~ 3.2 V/cm and Am/m ~ 0.2.

Thus, under certain conditions stochastic stabilization of the explosive instability is possible in a nonequilibrium plasma. This stabilization is characterized by broadening of the spectrum of interacting waves and has properties of random processes, whose forms in phase space are a result of attracting sets of the strange attractor type, existing stably for a long time.

We thank O. A. Delektorskaya for help in the computer calculations.

2.

.

.

5. 6. 7.

LITERATURE CITED

H. Wilhelmsson, J. Plasma Phys., ~, 215 (1969). S. M. Fainshtein, Zh. Tekh. Fiz., 4_~5, 1334 (1975); Izv. Vyssh. Uchehn. Zaved., Radio- fiz., i_88, 1059 (1975) E. E. Plotkin and S. M. Fainshtein, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 17, 62 (1974). V. V. Tamoikin and S. M. Fainshtein, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 18, 1134 (1975). J. Fukai, S. Krishnan, and E. Harris, Phys. Rev. Lett., 23, 910 (1969). M. I. Rabinovich and V. P. Reutov, Phys. Rev. Lett., 23, 910 (1969). S. M. Fainshtein, Zh. Eksp. Teor. Fiz., 71, 1021 (1969).

558

8. 9.

i0.

I!. 12.

13.

M. I. Rabinovich, Usp. Fiz. Nauk, 125, 123 (1978). S. Ya. Vyshkind and M. I. Rabinovich, Zh. Eksp. Teor. Fiz., 72, 557 (1976). S. Ya. Vyshkind, M. I. Rabinovich, and A. L. Fabrikant, Izv, Vyssh. Uchebn. Zaved., Radiofiz., 20, No. 2, 318 (1977). V. N. Tsytovich, Nonlinear Effects in Plasma [in Russian], Nauka, Moscow (1967). A. V. Gaponov, L. A. Ostrovskii, and M. I. Rabinovich, Izv. Vysh. Uchebn. Zaved., Radiofiz., 13, No. 2, 163 (1970); V. P. Reutov, Dissertation, Gorki State University, Gorki (1976). Ya. G. Sinai, Nonlinear Waves [in Russian], IPF Akad. Nauk SSSR, Gorki (1980), Pt. II, p. 24.

MARKOVIAN FLUCTUATION-DISSIPATION THEORY OF OPEN

RADIOPHYSiCAL SYSTEMS

R. L. Stratonovich UDC 538.56:519.25

An equality expressing the time reversibility of processes in Markovian open systems is derived. The reciprocity relations for processes involving relaxation to a nonequilibrium stationary state are analyzed. An equation is derived that describes the nonequilibrium stationary distribution for mixed open systems.

Nonequilibrium processes in open systems have been studied in various papers (see, e.g., [i]). Bochkov and Kuzovlev analyzed open systems from a different point of view, in particular in [2], using the generating function of the non-Markovian fluctuatiorr-dissipa- tion theory. In this paper we start from the relations of the Markovian theory, which ex- presses the condition for time reversibility of the processes in the system being examined. From the equations obtained for closed systems, by passing to the limit, we derive the basic equalities that describe open systems.*

In this paper, we are concerned primarily with mixed open systems, i.e., systems that are open with respect to some parameters and closed with respect to others. In such sys- tems it is possible to have a stationary probability distribution with respect to part of the variables and relaxation to stationary values of the variables. In what follows, we introduce two variants of the kinetic potential which differ from the variant of the potential introduced in [4]. We study the linearized equations that are valid near a non- equilibrium stationary state; for these equations, we discuss the satisfaction of the reci- procity relations.

In the conclusion we examine the kinetic equation for nondiffusively varying variables. We introduce an equation that determines their nonequilibrium stationary distribution or, which is equivalent, the corresponding quasifree energy.

i. EQUALITIES EXPRESSING TIME REVERSIBILITY

Assume that the starting closed system is described by the variables B = (BI, ..., Br), forming, as a set, a Markov process. We write the corresponding kinetic equation in the form

~(B, t) = (kT)-' M(--kT(O/OB), B) w (B, t), (1) where

7V!(u,B)~ l__(kT)1_ s ~ u ..... u%K ..... %(B). s,=I Sl

*The limiting transition from closed systems to open systems was examined in [3].

(2)

Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radio- fizika, Vol. 25, No. 7, pp. 779-790, July, 1982. Original article submitted March 27, 1081.

0033-8443/82/2507-0559507.50 �9 1983 Plenum Publishing Corporation 559