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Breakdown of quasilinear approximation for incoherent 1-D Langmuir waves
Volume 80A, number 4
PHYSICS LETTERS
8 December 1980
BREAKDOWN OF QUASILINEAR APPROXIMATION FOR INCOHERENT 1-D LANGMUIR WAVES G. LAVAL and D. PESME Centre de Physique Th~orique de l'Ecole Polytechnique, 91128 Palaiseau C~dex, France Received 10 July 1980 Revised manuscript received 22 September 1980
It is shown that even for very small amplitudes, the mode coupling terms lead to a modification of the growth rate which cannot be neglected in the quasilinear evolution of 1-D Langmuir waves.
In this letter we discuss the validity of quasilinear theory of incoherent 1-D Langrnuir waves. In usual quasilinear theory [1], the 'frequency width of wave-particle resonances is the linear growth rate 7k for a mode with wave number k. Renormalized quasilinear theories [2] have shown that this frequency width has to be replaced by (k2D) 1/3 whenever (k2D)l/3 >>7k, D being the usual quasilinear diffusion coefficient in velocity space. However, renormalized theories do not lead to significant corrections for the growth rate and diffusion coefficient, even in the limit (k2D)ll 3 >>7k provided that the validity conditions of quasilinear are fulfilled. These conditions reduce to the assumption that the correlation time r c of the electric field seen by a resonant particle is small as compared with the evolution time of averaged quantities. In this letter we compute mode coupling terms which are neglected in renormalized theories, in the regime 7k>> (k2D)l/3 and we find that these terms lead to corrections of order [(k2D)l/317k] 6. Therefore our results indicate that these mode coupling terms have to be taken into account for 7k ~ (k2D) 1/3 and could lead to a zero order modification of the growth rate and diffusion coefficient in this regime. Let us first define our ordering. The usual validity conditions for quasilinear theory can be w r i t t e n "[k/6Ope 3,krc "~ 1 and (k2D)z 3 ~ 1. All over the quasilinear evolution we have 3,krc ~ 1 while (k2D)l/3/~tk , which can be initially very small, becomes much larger than unity at saturation. Consequently we shall restrict ourselves to the regime defined by
"Yk'rc "~ (k 2D)l13 [7k "~ 1 .
(1)
These conditions are fulfilled well before saturation during the evolution of the weak warm beam plasma instability. We compute the electric field spectrum (IEkl 2) as a power series in the parameter (k2D)l/3/7 k and we delete terms of order ~/kZc ;" "yk/W_e. In the regin3e defined by eq. (1), the relative modification of the averaged distribution function is of order ~k2D)[~/3] ~/2 z2 ~ 1 and thus will be neglected. We now expand the solution of the Vlasov equation as a power series in the fluctuating electric field E and we insert this series into the Poisson equation. We obtain
e(k, W)Ekt~ = PE(k, ~)EE +/a3(k , co)EEE + ....
(2)
where e(k, w) is the linear dielectric function at t = 0, and with
)ee =
266
k'
fd ' Vkto,k, ,eg,to,eg-g,
to
(3a)
Volume 80A, number 4
PHYSICSLETTERS
k~,k'co',k w
k to
k " t o " E k - k ' - k '' w - w ' - w "
8 December 1980
'
with (4a)
qe f g k w g k - k ' to-to'fo do, Vk w ,k, w, = ik60
qe
Vk~,k'~,',k"~" =~
f gk~ogk-k' ~o-~'gk-k'-k" ~,-~'-~"f0 do,
(4b)
where fo(v) is the space-averaged distribution function at time t = 0 and where gkw represents the operator gk~o = [(qe/m)/i(w - kv + in)] O/no in which a > 0 provides a prescription of contour for v integration. Higher order terms Pi can be put in the same standard form. Eq. (2) is a non-linear equation for the electric field which we solve iteratively by setting E = E(1) + E(2) + ... with eE (1) = 0 , eE (2) = P2E(1)E (1) , (5a,b) eE (3) =/~2E(1)E (2) + P2E(2)E (1) + P3E(1)E(1)E (1) .
(5c)
From eq. (5a) we deduce Ek(1)(t) =Ek(O)exp(--ioakt),
(6)
with 6Ok = g2k + iTk, ~2k and 7k being the linear frequency and growth rate for mode k. We now compute the field spectrum (IEkl 2) where the averaging is taken over initial conditions. We have
where E~13 is obtained from eq. (5) and where we have substituted Ek(1) as given by eq. (6). We so obtain (E~i)E(__'~)in terms of the initial conditions and the averaged quantities (E~i)E(__'~)are computed by assuming that the phases ofEk(0 ) are purely randomly distributed. By setting upper bounds to the resonance functions which appear in the mode coupling coefficients V, it can be easily shown that [{E~i)E(__'~)[ ~ (JEll)[ 2)(k 2D17 3 )(i*]) / 2 - 1 , when i +] is an even integer and is strictly zero otherwise. Thus this first result justifies the iterative procedure as given by eq. (5) that we have used when solving eq. (2). Furthermore we notice that E (2n) cannot oscillate at frequency Oape and consequently it is fohnd that these fields are at least of order 7 j W , e and have to be ignored. Thus the lowest order terms of the series are ~v~l)E(_3~))and ~L~3)E(I~)where E~3 k ) i[ given explicitly by
with £w = w k, + wk,, + OOk_k,_k,,. We find [(~'~1)E(3~)[ = O(k2Dr~pyk)(lE~l)[2), which we neglect. Thus the first significant terms correspond to i + ] = 6. They read (E(I~E~5)) + (Ek(1)E(_5)) + (E~3)E(_3k)). Detailed computations of these terms show that I(E~l)E(_Sk))l = O(k2Dr2 /Tk)2(IE~l )12> . On the other hand we fred that (E~3)E(_~)yields a contribution of the order of (k2D/s~)2(IE~:l)l 2)which can be 267
Volume 80A, number 4
PHYSICS LETTERS
8 December 1980
written explicitly -
,
,,
,12
(E~a)E(2k))= G G Irk z,~,~',~k,,~ ,~,k,, k' k"
(IE~l,)12)(lE~l,)12)([Efkl)_k,_k,,[2)
le(k, N~)I 2
3~r2 ( q ) 4 k 6 212 ~ ((IEkl2)) 3 , pe/k so that we have: (iEkl2) = (IE~1)12)( 1 +
(9)
[0.24(k2D)l/3/.yk ] 6}.
(10)
The quasilinear modification of the growth rate would have given: ([Ek 12) = (IE~1)12)[1 + O(k2Or2/~/k)]
,
(11)
which remains a negligible correction consistently with our ordering. The modification of the spectrum as given by eq. (10) shows that the mode-couping terms lead to an increase of the energy transfer from resonant particles towards waves. This result is strictly valid only for ~/b > (k2D) 1/3. However it shows that a relative modification of order unity for the growth rate could be expected in the regime 7k "~ (k2D) 1/3. In this regime we have (k2D)r2/Tk ~ 1, which implies that the distribution function has not yet been modified by quasilinear diffusion. We therefore conclude that the effect of mode coupling terms in quasilinear theory of 1-D Langmuir waves has not been correctly estimated in previous theories. In the regime ~/k ~ (k2D) 1/3, an approach from a renormalized theory of the correlation function [3,4] yields a zero order modification of the growth rate, corresponding also to an increase of the energy transfer between particles and waves. However, in this regime, no expansion in a power series of a small parameter is possible so that the result is questionable [5]. The main result of the present letter is therefore to provide a straightforward computation which clearly exhibits that mode coupling terms cannot be neglected in quasilinear theory for ~/k ~ (k2D) 1/3. It suggests that, in the regime 7k "~ (k2D) 1/3, where no quantitative result has been actually completely justified, one must expect a zero-order modification of the growth rate as found in refs. [3,4].
References [1] A.A. Vedenov, E.P. Velikhov and R.Z. Sagdeev, Nucl. Fusion, SuppL Pt. 2 (1962) 465;
W.E. Drummond and D. Pines, Nucl. Fusion, SuppL Pt. 3 (1962) 1049. [ 2] B.B. Kadomtsev, Plasma turbulence (AcademicPress, New York, 1965); T.H. Dupree, Phys. Fluids 9 (1966) 1773. [3] T.H. Dupree, Phys. Fluids 15 (1972) 334. [4] J.C. Adam, G. Laval and D. Pesme, Phys. Rev. Lett. 43 (1979) 1671. [5] D.F. Dubois and M. Espedal, Plasma Phys. 20 (1978) 1209.
268
PHYSICS LETTERS
8 December 1980
BREAKDOWN OF QUASILINEAR APPROXIMATION FOR INCOHERENT 1-D LANGMUIR WAVES G. LAVAL and D. PESME Centre de Physique Th~orique de l'Ecole Polytechnique, 91128 Palaiseau C~dex, France Received 10 July 1980 Revised manuscript received 22 September 1980
It is shown that even for very small amplitudes, the mode coupling terms lead to a modification of the growth rate which cannot be neglected in the quasilinear evolution of 1-D Langmuir waves.
In this letter we discuss the validity of quasilinear theory of incoherent 1-D Langrnuir waves. In usual quasilinear theory [1], the 'frequency width of wave-particle resonances is the linear growth rate 7k for a mode with wave number k. Renormalized quasilinear theories [2] have shown that this frequency width has to be replaced by (k2D) 1/3 whenever (k2D)l/3 >>7k, D being the usual quasilinear diffusion coefficient in velocity space. However, renormalized theories do not lead to significant corrections for the growth rate and diffusion coefficient, even in the limit (k2D)ll 3 >>7k provided that the validity conditions of quasilinear are fulfilled. These conditions reduce to the assumption that the correlation time r c of the electric field seen by a resonant particle is small as compared with the evolution time of averaged quantities. In this letter we compute mode coupling terms which are neglected in renormalized theories, in the regime 7k>> (k2D)l/3 and we find that these terms lead to corrections of order [(k2D)l/317k] 6. Therefore our results indicate that these mode coupling terms have to be taken into account for 7k ~ (k2D) 1/3 and could lead to a zero order modification of the growth rate and diffusion coefficient in this regime. Let us first define our ordering. The usual validity conditions for quasilinear theory can be w r i t t e n "[k/6Ope 3,krc "~ 1 and (k2D)z 3 ~ 1. All over the quasilinear evolution we have 3,krc ~ 1 while (k2D)l/3/~tk , which can be initially very small, becomes much larger than unity at saturation. Consequently we shall restrict ourselves to the regime defined by
"Yk'rc "~ (k 2D)l13 [7k "~ 1 .
(1)
These conditions are fulfilled well before saturation during the evolution of the weak warm beam plasma instability. We compute the electric field spectrum (IEkl 2) as a power series in the parameter (k2D)l/3/7 k and we delete terms of order ~/kZc ;" "yk/W_e. In the regin3e defined by eq. (1), the relative modification of the averaged distribution function is of order ~k2D)[~/3] ~/2 z2 ~ 1 and thus will be neglected. We now expand the solution of the Vlasov equation as a power series in the fluctuating electric field E and we insert this series into the Poisson equation. We obtain
e(k, W)Ekt~ = PE(k, ~)EE +/a3(k , co)EEE + ....
(2)
where e(k, w) is the linear dielectric function at t = 0, and with
)ee =
266
k'
fd ' Vkto,k, ,eg,to,eg-g,
to
(3a)
Volume 80A, number 4
PHYSICSLETTERS
k~,k'co',k w
k to
k " t o " E k - k ' - k '' w - w ' - w "
8 December 1980
'
with (4a)
qe f g k w g k - k ' to-to'fo do, Vk w ,k, w, = ik60
qe
Vk~,k'~,',k"~" =~
f gk~ogk-k' ~o-~'gk-k'-k" ~,-~'-~"f0 do,
(4b)
where fo(v) is the space-averaged distribution function at time t = 0 and where gkw represents the operator gk~o = [(qe/m)/i(w - kv + in)] O/no in which a > 0 provides a prescription of contour for v integration. Higher order terms Pi can be put in the same standard form. Eq. (2) is a non-linear equation for the electric field which we solve iteratively by setting E = E(1) + E(2) + ... with eE (1) = 0 , eE (2) = P2E(1)E (1) , (5a,b) eE (3) =/~2E(1)E (2) + P2E(2)E (1) + P3E(1)E(1)E (1) .
(5c)
From eq. (5a) we deduce Ek(1)(t) =Ek(O)exp(--ioakt),
(6)
with 6Ok = g2k + iTk, ~2k and 7k being the linear frequency and growth rate for mode k. We now compute the field spectrum (IEkl 2) where the averaging is taken over initial conditions. We have
where E~13 is obtained from eq. (5) and where we have substituted Ek(1) as given by eq. (6). We so obtain (E~i)E(__'~)in terms of the initial conditions and the averaged quantities (E~i)E(__'~)are computed by assuming that the phases ofEk(0 ) are purely randomly distributed. By setting upper bounds to the resonance functions which appear in the mode coupling coefficients V, it can be easily shown that [{E~i)E(__'~)[ ~ (JEll)[ 2)(k 2D17 3 )(i*]) / 2 - 1 , when i +] is an even integer and is strictly zero otherwise. Thus this first result justifies the iterative procedure as given by eq. (5) that we have used when solving eq. (2). Furthermore we notice that E (2n) cannot oscillate at frequency Oape and consequently it is fohnd that these fields are at least of order 7 j W , e and have to be ignored. Thus the lowest order terms of the series are ~v~l)E(_3~))and ~L~3)E(I~)where E~3 k ) i[ given explicitly by
with £w = w k, + wk,, + OOk_k,_k,,. We find [(~'~1)E(3~)[ = O(k2Dr~pyk)(lE~l)[2), which we neglect. Thus the first significant terms correspond to i + ] = 6. They read (E(I~E~5)) + (Ek(1)E(_5)) + (E~3)E(_3k)). Detailed computations of these terms show that I(E~l)E(_Sk))l = O(k2Dr2 /Tk)2(IE~l )12> . On the other hand we fred that (E~3)E(_~)yields a contribution of the order of (k2D/s~)2(IE~:l)l 2)which can be 267
Volume 80A, number 4
PHYSICS LETTERS
8 December 1980
written explicitly -
,
,,
,12
(E~a)E(2k))= G G Irk z,~,~',~k,,~ ,~,k,, k' k"
(IE~l,)12)(lE~l,)12)([Efkl)_k,_k,,[2)
le(k, N~)I 2
3~r2 ( q ) 4 k 6 212 ~ ((IEkl2)) 3 , pe/k so that we have: (iEkl2) = (IE~1)12)( 1 +
(9)
[0.24(k2D)l/3/.yk ] 6}.
(10)
The quasilinear modification of the growth rate would have given: ([Ek 12) = (IE~1)12)[1 + O(k2Or2/~/k)]
,
(11)
which remains a negligible correction consistently with our ordering. The modification of the spectrum as given by eq. (10) shows that the mode-couping terms lead to an increase of the energy transfer from resonant particles towards waves. This result is strictly valid only for ~/b > (k2D) 1/3. However it shows that a relative modification of order unity for the growth rate could be expected in the regime 7k "~ (k2D) 1/3. In this regime we have (k2D)r2/Tk ~ 1, which implies that the distribution function has not yet been modified by quasilinear diffusion. We therefore conclude that the effect of mode coupling terms in quasilinear theory of 1-D Langmuir waves has not been correctly estimated in previous theories. In the regime ~/k ~ (k2D) 1/3, an approach from a renormalized theory of the correlation function [3,4] yields a zero order modification of the growth rate, corresponding also to an increase of the energy transfer between particles and waves. However, in this regime, no expansion in a power series of a small parameter is possible so that the result is questionable [5]. The main result of the present letter is therefore to provide a straightforward computation which clearly exhibits that mode coupling terms cannot be neglected in quasilinear theory for ~/k ~ (k2D) 1/3. It suggests that, in the regime 7k "~ (k2D) 1/3, where no quantitative result has been actually completely justified, one must expect a zero-order modification of the growth rate as found in refs. [3,4].
References [1] A.A. Vedenov, E.P. Velikhov and R.Z. Sagdeev, Nucl. Fusion, SuppL Pt. 2 (1962) 465;
W.E. Drummond and D. Pines, Nucl. Fusion, SuppL Pt. 3 (1962) 1049. [ 2] B.B. Kadomtsev, Plasma turbulence (AcademicPress, New York, 1965); T.H. Dupree, Phys. Fluids 9 (1966) 1773. [3] T.H. Dupree, Phys. Fluids 15 (1972) 334. [4] J.C. Adam, G. Laval and D. Pesme, Phys. Rev. Lett. 43 (1979) 1671. [5] D.F. Dubois and M. Espedal, Plasma Phys. 20 (1978) 1209.
268