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Diffusion due to parametric instability turbulence
Volume 45 A, number 2
10 September
PHYSICS LETTERS
1973
DIFFUSION DUE TO PARAMETRIC INSTABILITY TURBULENCE * J.J. THOMSON University of California, Lawrence Livermore Laboratory,
USA
Received 5 June 1973 We derive the diffusion coefficient for a weakly turbulent plasma in the presence of a driver field, E. cos wet and find low velocity components due to diffusion at the beat phase VelOcitY, (w-nwo)/k.
We consider the non-linear state of a plasma driven by a strong external field E,,(r) = E, cos w,f, where CA+,5 wp, the plasma frequency. This problem is of interest to laser fusion, since the non-linear state determines average reflection coefficients [ 11, heated particle distribution [2,3], and other aspects of light absorption. We assume here that a stationary turbulent state has been reached, without considering particular saturation mechanisms. We assume that particle trapping is not important. Refs. [2] and [3] have shown that evolution of the particle distribution function and the resultant heating can be well described by a diffusion equation. Here we derive the self-consistent [4] diffusion coefficient due to the turbulent spectrum. The Vlasov equation is
af -+v.&f+~[E(t)+E’]
(1)
$f=O
at
where E(t) is the coherent external field and E’ is the stochastic internal field. Ensemble averaging eq. (1) subtracting the average (denote by ( ) from eq. (1) we obtain equations for f = (f>, and f’ = f -f, which may be solva ed, correct to order )E’21, in terms of a diffusive Green function [4] ,g(Rut/R’u’t’), describing the statistically perturbed orbits. The Green function equation is
~+v~&+~E(~)-~-;~-D-i
g(Rvt/R,v,t,)
= 0, g(Rvt,/R,v,t,)
= 6 (R-R,)
6 (v-u,)
1
(2)
where D(u) = (q2,m2)~E(R.t)jdfl~dRIdvlg(Rvt,R~qfi)E(R1,tl)L
(3)
In the following, we assume D(u) to be a weak function of velocity. This is probably not good in general. In ref. [3], it was shown that for w, = ape, the important part of the heating is done in the region where D(u) - u. However, in the interests of simplicity, we follow the articles of ref. [4] in assuming D(u) = constant. We make the coordinate
~=t-t~,q=v-v~-
transformation
t ;~E(t’)dr’+&D,
t E(t’-
e=R-R,-vr+zJ
to
t,)dr’-
12a Y r au * D.
(4)
t0
Eq. (2) becomes
(5)
’ Work performed under the auspices of the U.S. Atomic Energy Commission.
111
10 September 1973
PHYSICS LETTERS
Volume 45A, number 2
Eq. (5) has been solved forg(g,q, 7) in ref. [S]. We use this solution in eq. (3). Using the random phase approximation and defining n = (qk *E,/mw~), we have
(6) X exp [i(w - nw,
-k*v)7-iiT2k
l
Silin [6] derived a diffusion coefficient with a background coherent field starting from the first two equations of the BBGKY hierarchy. However, he expressed the background fields in terms of the linear dielectric function and the driver field assuming the internal fieldtnegligible. As shown in refs. [2,3], and many other works on the saturated parametric instability, the background fields may become as large as the driver field. We have thus expressed D(v) in terms of the stationary turbulent fields. In the two limits /.I + 0 and w, + 0, eq. (6) collapses to the ordinary expression for the diffusion coefficient. - ku) = 0. Since we have assumed w. e w, the For I_(,w, # 0, D(v) is the sum of terms appreciable at (o-no, only appreciable terms will be n = 0, + 1. Since we generally have /._L~ & 1, we may write eq. (6) correctly to terms of order n2 as q2 dkdw D(u) z--- s -lEk”,l tn2 (2794
~d7(1-fP2(l+~l+w~T2)) 0
+$ p2J dr(1 +iwor) ex p[‘(1 w - kv-wO)r+(..)] 0
(7)
exp[i(w-kv)rt(..)]
+;~2~dr(I-i~or)exp[iw-ku+wo)r+(..)] 0
where (. .) are the diffusive terms. We may evaluate the resonant part of the beat terms in eq. (7) in the following way. In the integral, we may replace ioor by w. alaw. Integrating by parts, the integrand becomes (1 -w. a/aw)lEkil 6 (w - w. - ku), neglectI follows a power law. ing the diffusive terms. For electron waves, i3/aw = (k/3k2 h2 ) (a/ak). Let us assume IElI. The resonance i.e. lE;l - ken. Then the beat term has the factor (l+ rzw,fl k2Xt) IEl{ = diffusion coefficient is thus approximately 6(w-ku)t$$-
no0
3k2h2D
6(w-w,-ku).
1
The diffusive terms in the exponentials of eq. (7) broaden the resonance, but do not appreciably magnitude of D(u), since the resoannce width is much less than the frequency.
References [l] W.L. Kruer, KG. Estabrook and K.H. Sinz, UCRL-74676; to be published; D.W. Forslund, J.M. Kindel and E.L. Lindman, Phys. Rev. Lett. 30 (1973) 739. [2] J. Katz et al., UCRL-74334, to be published. [3] J.J. Thomson, R.J. Thomson, R.J. Faehl and W.L. Kruer, UCRL-74686, to be published. [4] T.H. Dupree, Phys. Fluids 9 (1466) 1773; J.J. Thomson and G. Benford, Phys. Rev. Lett. 28 (1972) 590. [5] F.L. Hinton and C. Oberman, Phys. Fluids 11 (1968) 1982. [6] V.P. Silin, JETP 30 (1970) 105.
112
(8) affect the
10 September
PHYSICS LETTERS
1973
DIFFUSION DUE TO PARAMETRIC INSTABILITY TURBULENCE * J.J. THOMSON University of California, Lawrence Livermore Laboratory,
USA
Received 5 June 1973 We derive the diffusion coefficient for a weakly turbulent plasma in the presence of a driver field, E. cos wet and find low velocity components due to diffusion at the beat phase VelOcitY, (w-nwo)/k.
We consider the non-linear state of a plasma driven by a strong external field E,,(r) = E, cos w,f, where CA+,5 wp, the plasma frequency. This problem is of interest to laser fusion, since the non-linear state determines average reflection coefficients [ 11, heated particle distribution [2,3], and other aspects of light absorption. We assume here that a stationary turbulent state has been reached, without considering particular saturation mechanisms. We assume that particle trapping is not important. Refs. [2] and [3] have shown that evolution of the particle distribution function and the resultant heating can be well described by a diffusion equation. Here we derive the self-consistent [4] diffusion coefficient due to the turbulent spectrum. The Vlasov equation is
af -+v.&f+~[E(t)+E’]
(1)
$f=O
at
where E(t) is the coherent external field and E’ is the stochastic internal field. Ensemble averaging eq. (1) subtracting the average (denote by ( ) from eq. (1) we obtain equations for f = (f>, and f’ = f -f, which may be solva ed, correct to order )E’21, in terms of a diffusive Green function [4] ,g(Rut/R’u’t’), describing the statistically perturbed orbits. The Green function equation is
~+v~&+~E(~)-~-;~-D-i
g(Rvt/R,v,t,)
= 0, g(Rvt,/R,v,t,)
= 6 (R-R,)
6 (v-u,)
1
(2)
where D(u) = (q2,m2)~E(R.t)jdfl~dRIdvlg(Rvt,R~qfi)E(R1,tl)L
(3)
In the following, we assume D(u) to be a weak function of velocity. This is probably not good in general. In ref. [3], it was shown that for w, = ape, the important part of the heating is done in the region where D(u) - u. However, in the interests of simplicity, we follow the articles of ref. [4] in assuming D(u) = constant. We make the coordinate
~=t-t~,q=v-v~-
transformation
t ;~E(t’)dr’+&D,
t E(t’-
e=R-R,-vr+zJ
to
t,)dr’-
12a Y r au * D.
(4)
t0
Eq. (2) becomes
(5)
’ Work performed under the auspices of the U.S. Atomic Energy Commission.
111
10 September 1973
PHYSICS LETTERS
Volume 45A, number 2
Eq. (5) has been solved forg(g,q, 7) in ref. [S]. We use this solution in eq. (3). Using the random phase approximation and defining n = (qk *E,/mw~), we have
(6) X exp [i(w - nw,
-k*v)7-iiT2k
l
Silin [6] derived a diffusion coefficient with a background coherent field starting from the first two equations of the BBGKY hierarchy. However, he expressed the background fields in terms of the linear dielectric function and the driver field assuming the internal fieldtnegligible. As shown in refs. [2,3], and many other works on the saturated parametric instability, the background fields may become as large as the driver field. We have thus expressed D(v) in terms of the stationary turbulent fields. In the two limits /.I + 0 and w, + 0, eq. (6) collapses to the ordinary expression for the diffusion coefficient. - ku) = 0. Since we have assumed w. e w, the For I_(,w, # 0, D(v) is the sum of terms appreciable at (o-no, only appreciable terms will be n = 0, + 1. Since we generally have /._L~ & 1, we may write eq. (6) correctly to terms of order n2 as q2 dkdw D(u) z--- s -lEk”,l tn2 (2794
~d7(1-fP2(l+~l+w~T2)) 0
+$ p2J dr(1 +iwor) ex p[‘(1 w - kv-wO)r+(..)] 0
(7)
exp[i(w-kv)rt(..)]
+;~2~dr(I-i~or)exp[iw-ku+wo)r+(..)] 0
where (. .) are the diffusive terms. We may evaluate the resonant part of the beat terms in eq. (7) in the following way. In the integral, we may replace ioor by w. alaw. Integrating by parts, the integrand becomes (1 -w. a/aw)lEkil 6 (w - w. - ku), neglectI follows a power law. ing the diffusive terms. For electron waves, i3/aw = (k/3k2 h2 ) (a/ak). Let us assume IElI. The resonance i.e. lE;l - ken. Then the beat term has the factor (l+ rzw,fl k2Xt) IEl{ = diffusion coefficient is thus approximately 6(w-ku)t$$-
no0
3k2h2D
6(w-w,-ku).
1
The diffusive terms in the exponentials of eq. (7) broaden the resonance, but do not appreciably magnitude of D(u), since the resoannce width is much less than the frequency.
References [l] W.L. Kruer, KG. Estabrook and K.H. Sinz, UCRL-74676; to be published; D.W. Forslund, J.M. Kindel and E.L. Lindman, Phys. Rev. Lett. 30 (1973) 739. [2] J. Katz et al., UCRL-74334, to be published. [3] J.J. Thomson, R.J. Thomson, R.J. Faehl and W.L. Kruer, UCRL-74686, to be published. [4] T.H. Dupree, Phys. Fluids 9 (1466) 1773; J.J. Thomson and G. Benford, Phys. Rev. Lett. 28 (1972) 590. [5] F.L. Hinton and C. Oberman, Phys. Fluids 11 (1968) 1982. [6] V.P. Silin, JETP 30 (1970) 105.
112
(8) affect the