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Renormalized theory of magnetic field generation in resonance absorption
Volume 66A, number 1
PhYSICS LETTERS
17 April 1978
RENORMALIZED THEORY OF MAGNETIC FIELD GENERATION IN RESONANCE ABSORPTION Patrick MORA Commissariat ~ I’Energie Atomique, Centre d’Etudes de Limeil, Villeneuve-Saint-Georges, France
and René PELLAT Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau Cédex, France Received 16 June 1977 Revised manuscript received 23 November 1977
DC magnetic field generation in resonance absorption is studied, in a non-linear regime, when it becomes of sufficient order of magnitude to affect wave propagation, as well as electron—ion collisions or thermal dispersion do. It is shown that a simple expression obtained in the linear regime is still valid in the non-linear theory. Scaling laws are set up.
In the theory for resonance absorption, wave propagation in the resonance layer is governed either by electron—ion collisions, by thermal dispersion or by non-linear effects (wave breaking, etc.) [1—4]. However if the plasma is even weakly magnetized, the resonance properties will be modified as soon as the electronic cyclotron frequency wb = eB/me will be of the same order as the electron—ion collision frequency Vei, or the equivalent collision frequency v X (XDIL)2/3 describing the thermal dispersion [1] (w is the incident wave frequency, XD the Debye length and
fe(rut)
=
F(r,w, i)
(1)
,
with W = U Uh eA Ime; uh(r,t) is the time-dependent quiver velocity: aUh/at= —(e/me)Eh;Eh and Bh are the high-frequency radiation fields;A and ~ are the low-frequency vector and scalar potentials: (B) = V X A; (E) = V~ ~A/~ t (we use K ) to denote time averaging over a light wave period). We then isolate the time averaged part of F in this new velocity frame by writing: F(r, w, t) = F 1(r, w, t) + Fh(r, w, t), Fl = (F) (2) —
—
—
—
.
L the density gradient length). DC magnetic field generation has been studied until now in a linear regime [5] (Wb ~ ~ r’~).However, in laser plasma experiments, the expected magnetic field is of sufficient order of magnitude to affect wave propagation [6] (wb ~ In this letter, we derive a non-linear theory of magnetic field generation in resonance absorption, letting be of the same order of magnitude as Vei, v~.In most laser and microwave plasma experiments [7,81, the resonance is limited by thermal dispersion (r’eq ~ ~ei)’ which allows to neglect collisions in the theory. Let us introduce the following velocity-space transformation of the electron distribution function ~ 1: Wb
28
If we introduce this transformation in the Vlasov equation, we obtain two coupled equations for F1 and Fh in the steady state:
(
w+
aFh
eA) e VF+~V(4~o_AW)~=(iiIFh>, (3) me 1 me ~
+1W ~
at
aFh me/ VFh +LV(~0 Aw)~
=
t
The procedure is the same as in ref. [5] , but the lowfrequency vector potentialA was not included.
(4)
Volume 66A, number 1
PHYSICS LETTERS
where ‘h is the operator
where (u’)
Ih=V((W+eAIm()uh)a/aWUhV,
((uh• v)sh),
=
(14)
(5)
and
~
2). (6) (m~/2e)((u~) + (eAlme) We may obviously solve eq. (4) to lowest order in the smallness parameters wb/W, v~/w.The limits of validity in the resonance region are not changed as compared with the standard theory [5J We obtain: =
17Apr11 1978
—
me
~
(15)
The low-frequency current is given by: (J)
—
e(fd3 vfe(r,u,t))
—
e((nhuh)
(16) =
+
(n) Ku’)),
.
where
aF me Fh = f~Fldt= V((w+~) where sh reads: (IhFh)
=
=
3v’F~,
1
~
— ~•
V~, (7)
I uhdt. The right-hand side of eq. (3) then
((Uh
V)s~VFl
~h
(n)=fd3v’F.
(17)
=fd
Using eqs. (12) and (14), we obtain: (J)=e(uhV((n)sh)+(n)(VV)uh),
(18)
so that we may write:
_V((uh.V)((w+eA/me)sh))aFj/aw,
(8)
and substitution of eq. (8) in eq. (3) gives:
(J)~eVX(n)(uhXsh).
(19)
Finally, using Ampere’s law:
(w+—~A).VFl+-~ V(~ me
1—A 1 w)~—~=0, (9) aw
1
m ~ ((uh v)sh)
~i
~o
+((hhi~)(u1
•~)).(10)
It should be emphasized that eq. (9) takes the same form as eq. (3) without the high-frequency fields (i.e. with the right-hand side set equal to zero), with a new definition thesimple potentials (10)). solved: Due for to its form(formulas eq. (9)is easily Fl~’m(w2 _(2e/m~)(~ — 1 A1.w)), (II) where Fm is a maxwel}ian distribution. It is convenient to write the distribution function in the velocity frame defined by: F(r, w,
(20)
~pOe(n)(uh Xsh).
We can remark here that in this formula Wb does not appear explicitly. The only explicit effect of the
where A1 A
(B)
t) =
F’(r,u’, t),
where u’ = w + eA/me = U Uh. The high-frequency and time-averaged part ofF’ then reads: —
F~= V(u’.sh).aF/au’ _ShVFj F’
2 1~”Fm ((i)’
—
—
,
(2e/me)t~’),
(12)
magnetic field in this calculation is a modification of the potential energy of particles (see eq. (15)). However ~b may be taken into account if necessary in the wave propagation and in the calculation of Uh. To finish this letter, we give an analytic expression of the self-generated magnetic field in 213). the linear re-inTaking gime for propagation (wb/w <(XD/L) to account the solution for the electrostatic part of the incident field [3]: E
=
—
.~) C sin OB~(0)i D2/3
exp (~ir3 + T~)dr, 0
(L where ~ = x/(LX~)113,B~(0)is the high-frequency magnetic field at the resonance point and 0 the angle of incidence (7(n) is parallel to Ox and the plane of incidence is xOy). We then find:
(13)
(u’)) 29
Volume 66A, number 1
PHYSICS LETTERS
17 April 1978
C’)
(X) C’)
—
b1(A~(~) fA1(~’)d~’
with
=
v 2 113sin0~ (k0L)h/3r~o2(r)(_.~) —~-; eq i=(k~L)
References [1] V. L. Ginzburg, Propagation of electromagnetic waves in plasmas (Pergamon, New York, 1970) S. 20. [2] N. G. Denisov, Zh. Eksp. Teor. Fiz. 31(1956)609; Engi. transi. Soy. Phys. JETP 4 (1957) 544. [3] A. D. Piliya, Zh. Tekh. Fiz. 36 (1966) 818; EngI. transl. Soy. Phys. Tech. Phys. 11(1966)609. [4] P. Koch and J. Albritton, Phys. Rev. Left. 32 (1974) 1420; J. Aibritton and P. Koch, Phys. Fluids 18(1975)1136. [5] B. Bezzerides, D. F. Dubois, D. W. Forslund and E. L. Lindman, Phys. Rev. Lett. 38 (1977) 495.
p(r) is the function quoted by Ginzburg [1], k 0 the incident wave number, v0 the oscillating electron yelocity in the incident field,A~the Airy function, and G1 is related to the Airy function [101. One expects from our results that numerical simulations at high fluxes could verify formula (20) for the B field generation.
30
[6 J. J. Thomson, C. F. Max and K. Estabrook, Phys. Rev. Lett. 35 (1975) 663. Divergiio, A. Y. Wong, H. C. Kim and Y. C. Lee, Phys. Rev. Lett. 38 (1977) 541. [8] A. Raven and D. T. Rumsby, Phys. Lett. 60A (1977) 42. 19] B. Bezzerides and D.F. Dubois, Phys. Rev. Lctt. 34 (1975) 1381. [10] Handbook of mathematical functions, eds. M. Abramowitz and I. Stegun (Dover Publications, New York, 1970) pp. 346 ff.
[71 W. F.
PhYSICS LETTERS
17 April 1978
RENORMALIZED THEORY OF MAGNETIC FIELD GENERATION IN RESONANCE ABSORPTION Patrick MORA Commissariat ~ I’Energie Atomique, Centre d’Etudes de Limeil, Villeneuve-Saint-Georges, France
and René PELLAT Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau Cédex, France Received 16 June 1977 Revised manuscript received 23 November 1977
DC magnetic field generation in resonance absorption is studied, in a non-linear regime, when it becomes of sufficient order of magnitude to affect wave propagation, as well as electron—ion collisions or thermal dispersion do. It is shown that a simple expression obtained in the linear regime is still valid in the non-linear theory. Scaling laws are set up.
In the theory for resonance absorption, wave propagation in the resonance layer is governed either by electron—ion collisions, by thermal dispersion or by non-linear effects (wave breaking, etc.) [1—4]. However if the plasma is even weakly magnetized, the resonance properties will be modified as soon as the electronic cyclotron frequency wb = eB/me will be of the same order as the electron—ion collision frequency Vei, or the equivalent collision frequency v X (XDIL)2/3 describing the thermal dispersion [1] (w is the incident wave frequency, XD the Debye length and
fe(rut)
=
F(r,w, i)
(1)
,
with W = U Uh eA Ime; uh(r,t) is the time-dependent quiver velocity: aUh/at= —(e/me)Eh;Eh and Bh are the high-frequency radiation fields;A and ~ are the low-frequency vector and scalar potentials: (B) = V X A; (E) = V~ ~A/~ t (we use K ) to denote time averaging over a light wave period). We then isolate the time averaged part of F in this new velocity frame by writing: F(r, w, t) = F 1(r, w, t) + Fh(r, w, t), Fl = (F) (2) —
—
—
—
.
L the density gradient length). DC magnetic field generation has been studied until now in a linear regime [5] (Wb ~ ~ r’~).However, in laser plasma experiments, the expected magnetic field is of sufficient order of magnitude to affect wave propagation [6] (wb ~ In this letter, we derive a non-linear theory of magnetic field generation in resonance absorption, letting be of the same order of magnitude as Vei, v~.In most laser and microwave plasma experiments [7,81, the resonance is limited by thermal dispersion (r’eq ~ ~ei)’ which allows to neglect collisions in the theory. Let us introduce the following velocity-space transformation of the electron distribution function ~ 1: Wb
28
If we introduce this transformation in the Vlasov equation, we obtain two coupled equations for F1 and Fh in the steady state:
(
w+
aFh
eA) e VF+~V(4~o_AW)~=(iiIFh>, (3) me 1 me ~
+1W ~
at
aFh me/ VFh +LV(~0 Aw)~
=
t
The procedure is the same as in ref. [5] , but the lowfrequency vector potentialA was not included.
(4)
Volume 66A, number 1
PHYSICS LETTERS
where ‘h is the operator
where (u’)
Ih=V((W+eAIm()uh)a/aWUhV,
((uh• v)sh),
=
(14)
(5)
and
~
2). (6) (m~/2e)((u~) + (eAlme) We may obviously solve eq. (4) to lowest order in the smallness parameters wb/W, v~/w.The limits of validity in the resonance region are not changed as compared with the standard theory [5J We obtain: =
17Apr11 1978
—
me
~
(15)
The low-frequency current is given by: (J)
—
e(fd3 vfe(r,u,t))
—
e((nhuh)
(16) =
+
(n) Ku’)),
.
where
aF me Fh = f~Fldt= V((w+~) where sh reads: (IhFh)
=
=
3v’F~,
1
~
— ~•
V~, (7)
I uhdt. The right-hand side of eq. (3) then
((Uh
V)s~VFl
~h
(n)=fd3v’F.
(17)
=fd
Using eqs. (12) and (14), we obtain: (J)=e(uhV((n)sh)+(n)(VV)uh),
(18)
so that we may write:
_V((uh.V)((w+eA/me)sh))aFj/aw,
(8)
and substitution of eq. (8) in eq. (3) gives:
(J)~eVX(n)(uhXsh).
(19)
Finally, using Ampere’s law:
(w+—~A).VFl+-~ V(~ me
1—A 1 w)~—~=0, (9) aw
1
m ~ ((uh v)sh)
~i
~o
+((hhi~)(u1
•~)).(10)
It should be emphasized that eq. (9) takes the same form as eq. (3) without the high-frequency fields (i.e. with the right-hand side set equal to zero), with a new definition thesimple potentials (10)). solved: Due for to its form(formulas eq. (9)is easily Fl~’m(w2 _(2e/m~)(~ — 1 A1.w)), (II) where Fm is a maxwel}ian distribution. It is convenient to write the distribution function in the velocity frame defined by: F(r, w,
(20)
~pOe(n)(uh Xsh).
We can remark here that in this formula Wb does not appear explicitly. The only explicit effect of the
where A1 A
(B)
t) =
F’(r,u’, t),
where u’ = w + eA/me = U Uh. The high-frequency and time-averaged part ofF’ then reads: —
F~= V(u’.sh).aF/au’ _ShVFj F’
2 1~”Fm ((i)’
—
—
,
(2e/me)t~’),
(12)
magnetic field in this calculation is a modification of the potential energy of particles (see eq. (15)). However ~b may be taken into account if necessary in the wave propagation and in the calculation of Uh. To finish this letter, we give an analytic expression of the self-generated magnetic field in 213). the linear re-inTaking gime for propagation (wb/w <(XD/L) to account the solution for the electrostatic part of the incident field [3]: E
=
—
.~) C sin OB~(0)i D2/3
exp (~ir3 + T~)dr, 0
(L where ~ = x/(LX~)113,B~(0)is the high-frequency magnetic field at the resonance point and 0 the angle of incidence (7(n) is parallel to Ox and the plane of incidence is xOy). We then find:
(13)
(u’)) 29
Volume 66A, number 1
PHYSICS LETTERS
17 April 1978
C’)
(X) C’)
—
b1(A~(~) fA1(~’)d~’
with
=
v 2 113sin0~ (k0L)h/3r~o2(r)(_.~) —~-; eq i=(k~L)
References [1] V. L. Ginzburg, Propagation of electromagnetic waves in plasmas (Pergamon, New York, 1970) S. 20. [2] N. G. Denisov, Zh. Eksp. Teor. Fiz. 31(1956)609; Engi. transi. Soy. Phys. JETP 4 (1957) 544. [3] A. D. Piliya, Zh. Tekh. Fiz. 36 (1966) 818; EngI. transl. Soy. Phys. Tech. Phys. 11(1966)609. [4] P. Koch and J. Albritton, Phys. Rev. Left. 32 (1974) 1420; J. Aibritton and P. Koch, Phys. Fluids 18(1975)1136. [5] B. Bezzerides, D. F. Dubois, D. W. Forslund and E. L. Lindman, Phys. Rev. Lett. 38 (1977) 495.
p(r) is the function quoted by Ginzburg [1], k 0 the incident wave number, v0 the oscillating electron yelocity in the incident field,A~the Airy function, and G1 is related to the Airy function [101. One expects from our results that numerical simulations at high fluxes could verify formula (20) for the B field generation.
30
[6 J. J. Thomson, C. F. Max and K. Estabrook, Phys. Rev. Lett. 35 (1975) 663. Divergiio, A. Y. Wong, H. C. Kim and Y. C. Lee, Phys. Rev. Lett. 38 (1977) 541. [8] A. Raven and D. T. Rumsby, Phys. Lett. 60A (1977) 42. 19] B. Bezzerides and D.F. Dubois, Phys. Rev. Lctt. 34 (1975) 1381. [10] Handbook of mathematical functions, eds. M. Abramowitz and I. Stegun (Dover Publications, New York, 1970) pp. 346 ff.
[71 W. F.