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Supersonic langmuir collapse in the semiclassical limit
Volume 131, number 4,5
PHYSICSLETTERSA
22 August 1988
SUPERSONIC L A N G M U I R COLLAPSE IN THE SEMICLASSICAL L I M I T S.V. BULANOV and S.G. SHASHARINA Plasma Physics Department. General Physics Institute, Academy qf Sciences o['the USSR. l/avilovst. 38. Moscow. USSR Received 28 December 1987;revised manuscript received 10 June 1988;accepted for publication 10 June 1988 Communicatedby R.C. Davidson
Exact selfsimilar solutions of the nonlinear scalar equation which describes the evolutionof the nonlinear waveenvelope in the semiclassical limit are found. The supersonic regimewith the ion nonlinearity is investigated. The possibilityof fast ion production is demonstrated. The formation of the pancake structures of the collapson is found to be typical.
The problem of the wave collapse occupies one of the central places in modern investigations of nonlinear waves [1-5]. In some particular cases this phenomenon can be described by the scalar equation [1-4]
iO,~,+ ½ O , ~ , - a N ~ u = O
.
(1)
For example, the evolution of the Lanqmuir wave envelope can be reduced to ( 1 ). Here, 0, and 0i stand for derivatives with respect to t and x~, ~, is the electric field envelope, a N = N - 1 is the ion density variation. Here and below the dimensionless variables introduced in ref. [1] are used (x in the units (9mi/8me)l/22D, t in 3m~/2mewo, N in 4meno/3mi, I~ul2 in 16rcmenoTe/3mi). To solve the problem we are to add to ( 1 ) the ion motion equation. In accordance with it one can distinguish the supersonic and subsonic regimes [ 1-5 ]. Previously, only one type of nonlinearity (connected with the HF-pressure dependence on I~tl 2) was investigated. In the present paper the nonlinear effects of the ion motion in the supersonic regime is taken into account. So, we Use the following equations:
O,N+O,(NV,) = 0 ,
(2)
o , v , + ~oj v, = - 0 , I~,12 .
(3)
Here, ~ is the plasma velocity of quasineutral motion, N i s the ion density. In eqs. (2), (3) the thermal pressure is neglected in comparison with the HFpressure. 298
Representing ~,=A exp(iqb) where A is the amplitude and • is the phase of ~, and neglecting the higher derivatives in the semiclassical limit (like in refs. [ 3 - 5 ] ) one obtains the equations for the squared amplitude and the phase of the Langmuir oscillations: O, I A I 2 + O i ( I A I 2 0 i ~ ) = O ,
(4)
0,q)+ ½0,q~O,q~= - a N ,
(5)
which together with (2), (3) form the coupled set of equations. This set permits the selfsimilar ansatz I A I : = n( t) + ½n,~ ( t)x~xk ,
(6)
@= ~0(t) + ½~Oik(t)x~xk,
(7)
N = N ( t ) + ½N,k(t)XiX~ ,
(8)
V~ = W,~(t)xk.
(9)
The solution analogous to (6), (7) in the subsonic limit was analyzed in ref, [4]. These solutions represent the local approximation of the variables near the extremal points. It also means that the collapson has the form of a 3-D ellipsoid. After substituting ( 6 ) - ( 9 ) into (4), (5) one obtains a set of ordinary differential equations. In order to solve it it is convenient to introduce the Lagrange variables for (4), (5) related to the Euler coordinates by
xi=Mij(t)x ° ,
~oij(t)=~I~(t)M~(t),
(10)
x,=K~j(t)y~,
Wo(t)=k,k(t)K~l(t).
(11)
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 131, number4,5
PHYSICS LETTERSA
22 August 1988
A similar approach was used in ref. [6]. The matrices K Uand M~j determine rotations and deformations in the course of time of Lagrange surfaces frozen into the plasma and into of the effective fluid described by (4), ( 5 ). The solutions of (2), (4) in Lagrange variables yield to
n ( x °, t ) = [n(O)+½n,k(O)x°x°]/D(t) ,
(12)
N ( y °, t ) = [N(O)+½N, k(O)y°y°]/Q(t) ,
(13)
where D(t) = d e t ( M a ( t ) ) and Q(t) = d e t ( K a ( t ) ) . Substituting ( 6 ) - (9) into ( 3 ) - ( 5 ) with ( 10)- ( 13 ) taken into account one obtains the system of ordinary differential equations for M a and Ka:
I
×,-- Xs I
A
=
o
~ a = - ~_~QNkt(O)(KGIKff, +KkmK. - t - , )Mmj, (14) 1
Ka= - ~) nkt(O)MGtM~K..j.
(15)
The initial conditions for (14), ( 15 ) are Ma(0)=d a , ~/a(0)=~a(0),
(16)
Ka(O)=Sa,
(17)
[(a(O)=Wa(O).
Let us study the case, when the intial state is decribed by the diagonal matrices: N a (0) = diag{ xl, x2, x3}, n~j(0)--diag{a~, a2, a3}, ~a(0!=diag{.h~(0), ~i2(0), d3(0)} and Wa(0)=diag{Xt(0), X2(0), Jr3 (0) }. From eqs. ( 14 ), ( 15 ) one can easily see that in this case the matrices Mo(t), Ka(t) remain diagonal for t>0: Ma(t)=diag{al(t), a2(t), a3(t)}, Ka(t) =diag{Xl (t), X2(t), Xa(t)}. This means that in this case the axes of the 3-D ellipsoidal collapson conserve their orientation. It follows from (14), ( 15 ) that the functions X~(t) and a~(t) yield
X~=-a,X~/a3aja~., ?i, = - x, a~/X 3XjXk ,
i#j~k,
0.5
1.
"/:
Fig. 1. Evolution of ai(t) and Xi(t) for the initial conditions: aA0) =XR0) = 1, hi(0) = J(i(0) =0, cti = - 1, x l = 1 . 5 , X2 = X 3 = 1. In finite time a~(t) turns to zero and Xt (t) goesto infinity. being finite is reached in finite time. This results in formation of the pan-cake collapson as in ref. [2]. Let us investigate the behaviour of the solutions of (18), (19) near the singularity in the cases of the spherical ( d = 3 ) , cylindrical ( d = 2 ) and plane ( d = 1 ) symmetries. From (18) and (19) one can find
X= - a X / a 2+a ,
(20)
ii= - x a / X 2+a .
(21)
If x > 0 (plasma density minimum), a < 0 (maximum of the amplitude of the Langmuir oscillations) and the initial values J( and ~ are small, then the function a(t) decreases and X ( t ) increases. This leads to the falling down of the r.h.s, of (21) and to the following behavior of a(t) near the singularity
(18)
a(t)=al(1 -t/to),
( 19 )
where a~ and to are some constants. The solutions (20) by means of (22) can be written for t-,to as
where the summation over the repeated indices is not implied. The solution of the set ( 18 ), ( 19 ) obtained numerically for the initially formed local minimum of the plasma density and maximum of the amplitude of Langmuir oscillations is shown in fig. 1. One can see from it that the singularity where one of the functions X~(t) goes to infinity and one of the functions a~(t) turns to zero with the other functions
(22)
X ( t ) =XI ( 1 - t / t o ) (2-a)/4 ×exp[2d(at~/a2+d) t/2( 1 --t/to) -a/2 ] .
(23)
From (23) we have a(t)--.O, X ( t ) ~ o o for t~to. This means that near the singularity the cavity contracts: X=xoa(t), the plasma concentration turns to zero, the amplitude of the Langmuir oscillations goes to 299
Volume 131, number 4,5
PHYSICS LETTERS A
infinity: N~ N(O)/X a(t). The p l a s m a velocity tends to zero at the center o f cavity: V= (J(/X)x a n d tends to infinity near its edge: x=a(t)lo, while t-,to. The m a x i m u m velocity can be e s t i m a t e d for instance from the the c o n d i t i o n for the self-intersection o f electron trajectories to occur at the final stage o f the collapse. The a m p l i t u d e o f the electron oscillations in the cavity increases according to the law r=eE/moj2~ (t-to)-a/2; it becomes equal to the cavity length a(t)lo or to the wave length o f the plasma oscillations 2~/k at a (t) = max{(ro/lo) l/~2+d~, (kro)2/d}, here ro=eEo/m09~. F r o m this: Vmax=(lo/to)Xmin{(ro/lo) -d/2(d+2), (kro)-~}. The relations contain large p a r a m e t e r s lo/ro, (kro)- '. This also means that during the t i m e t << oJ ~- ' / (kro) 2 one can neglect the n o n l i n e a r electron effects. The energy spectrum o f the fast ions averaged over the cavity has the exponential form N ( g ) ~ exp ( - ~x/~oo ), where go = mil~/2t2. Thus, the o b t a i n e d solutions describe the behav-
300
22 August 1988
iour o f the L a n g m u i r oscillations which differs from the m o v e m e n t o f the plasma. That is, the m o v e m e n t o f ions remains forced near the singularity when the L a n g m u i r oscillations b e c o m e free. We also pay attention to the possibility o f fast ion p r o d u c t i o n at the final stage o f the supersonic collapse. The authors are grateful to A.S. Sakharov for the useful discussions.
References [ 1] V.E. Zakharov, Sov. Phys. JETP 35 ( 1972 ) 908. [2] P.A. Robinson, D.L. Newman and M.V. Goldman, Preprint No. 1109, University of Colorado (February 1988). [ 3 ] V.E. Zakharov, E.A. Kuznetsov and S.M. Musher, Pis'ma Zh. Eks. Teor. Fiz. 41 (1985) 125. [4] V.E. Zakharov and E.A. Kuznetsov, Zh. Eksp. Teor. Fiz. 91 (1986) 1310. [5] G. Pelletier, Physica D 27 (1987) 187. [6] S.V. Bulanov and M.A. Ol'shatetskij, Phys. Lett. A 100 (1985) 35.
PHYSICSLETTERSA
22 August 1988
SUPERSONIC L A N G M U I R COLLAPSE IN THE SEMICLASSICAL L I M I T S.V. BULANOV and S.G. SHASHARINA Plasma Physics Department. General Physics Institute, Academy qf Sciences o['the USSR. l/avilovst. 38. Moscow. USSR Received 28 December 1987;revised manuscript received 10 June 1988;accepted for publication 10 June 1988 Communicatedby R.C. Davidson
Exact selfsimilar solutions of the nonlinear scalar equation which describes the evolutionof the nonlinear waveenvelope in the semiclassical limit are found. The supersonic regimewith the ion nonlinearity is investigated. The possibilityof fast ion production is demonstrated. The formation of the pancake structures of the collapson is found to be typical.
The problem of the wave collapse occupies one of the central places in modern investigations of nonlinear waves [1-5]. In some particular cases this phenomenon can be described by the scalar equation [1-4]
iO,~,+ ½ O , ~ , - a N ~ u = O
.
(1)
For example, the evolution of the Lanqmuir wave envelope can be reduced to ( 1 ). Here, 0, and 0i stand for derivatives with respect to t and x~, ~, is the electric field envelope, a N = N - 1 is the ion density variation. Here and below the dimensionless variables introduced in ref. [1] are used (x in the units (9mi/8me)l/22D, t in 3m~/2mewo, N in 4meno/3mi, I~ul2 in 16rcmenoTe/3mi). To solve the problem we are to add to ( 1 ) the ion motion equation. In accordance with it one can distinguish the supersonic and subsonic regimes [ 1-5 ]. Previously, only one type of nonlinearity (connected with the HF-pressure dependence on I~tl 2) was investigated. In the present paper the nonlinear effects of the ion motion in the supersonic regime is taken into account. So, we Use the following equations:
O,N+O,(NV,) = 0 ,
(2)
o , v , + ~oj v, = - 0 , I~,12 .
(3)
Here, ~ is the plasma velocity of quasineutral motion, N i s the ion density. In eqs. (2), (3) the thermal pressure is neglected in comparison with the HFpressure. 298
Representing ~,=A exp(iqb) where A is the amplitude and • is the phase of ~, and neglecting the higher derivatives in the semiclassical limit (like in refs. [ 3 - 5 ] ) one obtains the equations for the squared amplitude and the phase of the Langmuir oscillations: O, I A I 2 + O i ( I A I 2 0 i ~ ) = O ,
(4)
0,q)+ ½0,q~O,q~= - a N ,
(5)
which together with (2), (3) form the coupled set of equations. This set permits the selfsimilar ansatz I A I : = n( t) + ½n,~ ( t)x~xk ,
(6)
@= ~0(t) + ½~Oik(t)x~xk,
(7)
N = N ( t ) + ½N,k(t)XiX~ ,
(8)
V~ = W,~(t)xk.
(9)
The solution analogous to (6), (7) in the subsonic limit was analyzed in ref, [4]. These solutions represent the local approximation of the variables near the extremal points. It also means that the collapson has the form of a 3-D ellipsoid. After substituting ( 6 ) - ( 9 ) into (4), (5) one obtains a set of ordinary differential equations. In order to solve it it is convenient to introduce the Lagrange variables for (4), (5) related to the Euler coordinates by
xi=Mij(t)x ° ,
~oij(t)=~I~(t)M~(t),
(10)
x,=K~j(t)y~,
Wo(t)=k,k(t)K~l(t).
(11)
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 131, number4,5
PHYSICS LETTERSA
22 August 1988
A similar approach was used in ref. [6]. The matrices K Uand M~j determine rotations and deformations in the course of time of Lagrange surfaces frozen into the plasma and into of the effective fluid described by (4), ( 5 ). The solutions of (2), (4) in Lagrange variables yield to
n ( x °, t ) = [n(O)+½n,k(O)x°x°]/D(t) ,
(12)
N ( y °, t ) = [N(O)+½N, k(O)y°y°]/Q(t) ,
(13)
where D(t) = d e t ( M a ( t ) ) and Q(t) = d e t ( K a ( t ) ) . Substituting ( 6 ) - (9) into ( 3 ) - ( 5 ) with ( 10)- ( 13 ) taken into account one obtains the system of ordinary differential equations for M a and Ka:
I
×,-- Xs I
A
=
o
~ a = - ~_~QNkt(O)(KGIKff, +KkmK. - t - , )Mmj, (14) 1
Ka= - ~) nkt(O)MGtM~K..j.
(15)
The initial conditions for (14), ( 15 ) are Ma(0)=d a , ~/a(0)=~a(0),
(16)
Ka(O)=Sa,
(17)
[(a(O)=Wa(O).
Let us study the case, when the intial state is decribed by the diagonal matrices: N a (0) = diag{ xl, x2, x3}, n~j(0)--diag{a~, a2, a3}, ~a(0!=diag{.h~(0), ~i2(0), d3(0)} and Wa(0)=diag{Xt(0), X2(0), Jr3 (0) }. From eqs. ( 14 ), ( 15 ) one can easily see that in this case the matrices Mo(t), Ka(t) remain diagonal for t>0: Ma(t)=diag{al(t), a2(t), a3(t)}, Ka(t) =diag{Xl (t), X2(t), Xa(t)}. This means that in this case the axes of the 3-D ellipsoidal collapson conserve their orientation. It follows from (14), ( 15 ) that the functions X~(t) and a~(t) yield
X~=-a,X~/a3aja~., ?i, = - x, a~/X 3XjXk ,
i#j~k,
0.5
1.
"/:
Fig. 1. Evolution of ai(t) and Xi(t) for the initial conditions: aA0) =XR0) = 1, hi(0) = J(i(0) =0, cti = - 1, x l = 1 . 5 , X2 = X 3 = 1. In finite time a~(t) turns to zero and Xt (t) goesto infinity. being finite is reached in finite time. This results in formation of the pan-cake collapson as in ref. [2]. Let us investigate the behaviour of the solutions of (18), (19) near the singularity in the cases of the spherical ( d = 3 ) , cylindrical ( d = 2 ) and plane ( d = 1 ) symmetries. From (18) and (19) one can find
X= - a X / a 2+a ,
(20)
ii= - x a / X 2+a .
(21)
If x > 0 (plasma density minimum), a < 0 (maximum of the amplitude of the Langmuir oscillations) and the initial values J( and ~ are small, then the function a(t) decreases and X ( t ) increases. This leads to the falling down of the r.h.s, of (21) and to the following behavior of a(t) near the singularity
(18)
a(t)=al(1 -t/to),
( 19 )
where a~ and to are some constants. The solutions (20) by means of (22) can be written for t-,to as
where the summation over the repeated indices is not implied. The solution of the set ( 18 ), ( 19 ) obtained numerically for the initially formed local minimum of the plasma density and maximum of the amplitude of Langmuir oscillations is shown in fig. 1. One can see from it that the singularity where one of the functions X~(t) goes to infinity and one of the functions a~(t) turns to zero with the other functions
(22)
X ( t ) =XI ( 1 - t / t o ) (2-a)/4 ×exp[2d(at~/a2+d) t/2( 1 --t/to) -a/2 ] .
(23)
From (23) we have a(t)--.O, X ( t ) ~ o o for t~to. This means that near the singularity the cavity contracts: X=xoa(t), the plasma concentration turns to zero, the amplitude of the Langmuir oscillations goes to 299
Volume 131, number 4,5
PHYSICS LETTERS A
infinity: N~ N(O)/X a(t). The p l a s m a velocity tends to zero at the center o f cavity: V= (J(/X)x a n d tends to infinity near its edge: x=a(t)lo, while t-,to. The m a x i m u m velocity can be e s t i m a t e d for instance from the the c o n d i t i o n for the self-intersection o f electron trajectories to occur at the final stage o f the collapse. The a m p l i t u d e o f the electron oscillations in the cavity increases according to the law r=eE/moj2~ (t-to)-a/2; it becomes equal to the cavity length a(t)lo or to the wave length o f the plasma oscillations 2~/k at a (t) = max{(ro/lo) l/~2+d~, (kro)2/d}, here ro=eEo/m09~. F r o m this: Vmax=(lo/to)Xmin{(ro/lo) -d/2(d+2), (kro)-~}. The relations contain large p a r a m e t e r s lo/ro, (kro)- '. This also means that during the t i m e t << oJ ~- ' / (kro) 2 one can neglect the n o n l i n e a r electron effects. The energy spectrum o f the fast ions averaged over the cavity has the exponential form N ( g ) ~ exp ( - ~x/~oo ), where go = mil~/2t2. Thus, the o b t a i n e d solutions describe the behav-
300
22 August 1988
iour o f the L a n g m u i r oscillations which differs from the m o v e m e n t o f the plasma. That is, the m o v e m e n t o f ions remains forced near the singularity when the L a n g m u i r oscillations b e c o m e free. We also pay attention to the possibility o f fast ion p r o d u c t i o n at the final stage o f the supersonic collapse. The authors are grateful to A.S. Sakharov for the useful discussions.
References [ 1] V.E. Zakharov, Sov. Phys. JETP 35 ( 1972 ) 908. [2] P.A. Robinson, D.L. Newman and M.V. Goldman, Preprint No. 1109, University of Colorado (February 1988). [ 3 ] V.E. Zakharov, E.A. Kuznetsov and S.M. Musher, Pis'ma Zh. Eks. Teor. Fiz. 41 (1985) 125. [4] V.E. Zakharov and E.A. Kuznetsov, Zh. Eksp. Teor. Fiz. 91 (1986) 1310. [5] G. Pelletier, Physica D 27 (1987) 187. [6] S.V. Bulanov and M.A. Ol'shatetskij, Phys. Lett. A 100 (1985) 35.