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Strong turbulence of short-wavelength electrostatic perturbations of a magnetized plasma
Volume 145, number 5
PHYSICS LETTERS A
16 April 1990
S T R O N G TURBULENCE OF S H O R T - W A V E L E N G T H E L E C T R O S T A T I C P E R T U R B A T I O N S OF A M A G N E T I Z E D P L A S M A A.B. M I K H A I L O V S K I I , S.V. N O V A K O V S K I I 1. v. Kurchatov Institute of Atomic Energy, Moscow 123182, USSR and O.G. O N I S H C H E N K O Space Research Institute, Moscow 117810, USSR Received 31 May 1989; revised manuscript received 15 December 1989; accepted for publication 12 February 1990 Communicated by R.C. Davidson
Steady-state spectra of strong Kolmogorov turbulence of short-wavelength (in comparison with the ion Larmor radius) electrostatic perturbations of a magnetized plasma are found. Locality of these spectra is shown and directions of spectral fluxes are elucidated.
1. Introduction According to ref. [ 1 ], the investigation of strong turbulence of short-wavelength perturbations (k±pi >> 1, where k± is the transverse wavenumber, pi is the ion L a r m o r radius) is of interest for the problem of anomalous thermal conductivity of a plasma in a strong magnetic field. Nonlinear equations for these perturbations were obtained in ref. [ 1 ]. One spectrum of the K o l m o g o r o v turbulence of these perturbations was found in ref. [2]. In the present paper we show that the electrostatic shortwavelength turbulence is characterized by two spectra of Kolmogorov type. We also investigate the question of the turbulence locality and show that both spectra are local. Besides that, we elucidate directions of spectral fluxes and show that, as in the case of two-dimensional hydrodynamic turbulence [ 3 ], these directions are opposite.
2. Starting equations We study two-dimensional electrostatic perturbations of a homogeneous plasma, supposing their 272
electric field E = - V . 0 , where 0 is the electrostatic potential, V± = (O/Ox, O/Oy). The equilibrium magnetic field Bo is assumed to be directed along the z axis. Let us consider the wavelength of perturbations to be small in comparison with the ion Larmor radius, k±pi >> 1, and taking into account that the ion nonlinearity is small in comparison with the electron nonlinearity (cf. ref. [ 1 ] ), we take the ion perturbed density fi depending in the linear approximation on 0. Then using, for example, the method of integration along particle trajectories [4], we obtain the well-known formula fik
=
--
enoO____k[ 1 -- (2~zi) - 1/2 ] •
Ti
( 1)
Here fik and Ok are spatial Fourier-components of fi and 0, zi= (k±pi) 2, no and Ti are the equilibrium plasma density and ion temperature, e is the ion charge, k is the wave vector. Besides that, we use the electron continuity equation Ofe/Ot+ VE" V I f e = 0 ,
(2)
where hie is the perturbed electron density, VE=c(ez×VO)/Bo is the drift velocity in crossed (E, Bo) fields. Using the transition to the Fourier trans-
Volume 145, number 5
PHYSICS LETTERS A
form in k in (2) and taking the quasineutrality condition ni= ne into account, we obtain with use of ( 1 ) the nonlinear equation for ¢~k:
16 April 1990
Here
Gq=(og+qq) -l ,
(10)
and qq is determined by
i ~ o z f V(k,k,,k2)
qq= -4 ~ G"~,lq2V(k, kl, k2) V*(k~, k, k2)
Xq~10~2~(k-l-kl -t-k2) dk I dk 2 .
Here
(3)
V(k, kl, k2) is the interaction matrix element,
V(k, kl,k2)=i(k '
~-~)(k,×k2)z,
(4)
X6(q+ql +q2) dq~ dq2.
Taking flq do9ocWk (see (7)) and the fact that (cf. ref. [5]) Im
kl and k2 are modulae of vectors k~ and k2, the asterisk means the complex conjugation. The subscript Z on wavenumbers is omitted for simplicity. We note that
V(k, kx,kE)+V(kl,kE,k)+V(k2, k, kl)=O, k-IV(k, kl, k2) + k i -1V(kl, k2, k) +ky I V(k2, k, k, ) = 0 .
(11 )
f oglqdogocOW,/Ot
(12)
into account, we obtain from ( 9 ) - ( 11 ) that
0wk
Ot oc
g(k,k~,k2)[F*(k, kl,k2)aflq, I~
(5) -'}-V* (kl,
k, k2 )tTqlIqlcz -t- I/'*(k2, k, kl )ff q2IqIql ]
X~(q+q~ +q2) dq~ dq2 dog, (6)
Eq. (3) has two integrals of motion connected with (5) and (6): energy W and generalized enstrophy H defined by
w~c f 10,12~- f W.dk,
(7)
Hoc f K-'l~kl2dk- f H, dk.
(8)
where aq=-ImGq. Using (10) and the standard prescription for the calculation o f I m Gq (o9--,o9+iA, A> 0), we conclude that aq > O.
4. Steady-state spectra We consider (cf. ref. [6] )
Iq=k-tV+s)f(o9/ks), eq=k-s#(o9/ks), Therefore, by analogy with the hydrodynamic turbulence [ 3 ], one can expect that a steady-state spectrum corresponds to each integral of motion. We shall show this below.
We represent ~k in the form of the Fourier integral in o9-space, introducing by this way ¢q, where q= (k, to). We also introduce Iq defined by (O~,~q)= Iq5(q-q' ), where ( ) means averaging over the statistical ensemble. In accordance with ref. [ 5 ] we find from (3) that Iq satisfies the equation
Iq=21Gql2 ~ Ig(k,k~,k2)l 2 XIq~Iq2~(q"{-ql -t-q2) dqt dq2.
(9)
(14)
where p and s are numbers, f and # are functions. Changing in (13) to positive frequencies o9, o9~ and o92, and using the factorization method of ref. [ 6 ], we obtain
OWk/Otoc f dk~ dk2 ~ ( k + k l
3. Equations of turbulence
(13)
+kE)K
X ~ Rd(og- ogl -o92) do9 do91 do92.
( 15 )
Here
K= V(k, kl, k2) + (k/k~)~V(kl, k2, k) + (k/kE)~V(k2, kl, k), (16) R= V*(k, kl, k2)aqlq,Iq2+ V*(k~, k, k2)trqllqlq2 + V*(k2, k, ki )trq,IqIq., (17) where a = 6 - s - 2p. Taking (5) and (6) into ac273
Volume 145, number 5
PHYSICS LETTERS A
count, we find that K = 0, i.e. the right-hand side o f (15) vanishes at a=ot u~, i = 1 or 2, where a ( ~ ) = 0 a n d ct(2) = 1. We conclude from this that steady-state spectra, for which s + 2 p = 6 - a (o, can be realized. Using (11 ) a n d considering that Gq~ck-~g(og/k~), where g ( x ) is a function connected w i t h / t ( x ) , we find one m o r e correlation between s a n d p: 2s + p = 4. Thus we obtain two pairs o f n u m b e r s p=p(O and s = s ~°, i = 1 or 2, c o r r e s p o n d i n g to steady-state spectra.p~,)__8 • - 3 and s ( ~ )-_~ 2, o r p (2)=2 and s(2)= 1. Since at Iq o f the form ( 1 4 ) Wkock -p, we obtain that energy densities o f these spectra
W(ki)oc(k-S/3, k - 2 ) ,
i=1,2.
(18)
Respectively, for E k = k W k we have
16 April 1990
K ( ° ( u , v)=u-~'"'( 1 - l /v) In u -v-'~"'(1-1/u)
lnv,
(22)
2°)(u, v) = [.i dxdxl dx2 ,~(x-xl uS"'-x2v ~"') × [ ( 1/u- 1/v)/~(x)f(x, )f(x2 ) + (l/v-- l
)~L~(Xl)f(x)f(x2)
+ ( 1- 1 / u ) p ( x 2 ) f ( x ) f ( x l ) ] ,
(23)
A2(u, v) is the dimensionless area o f the triangle m a d e by the vectors k, k~ and k2,
A2 = [ (u+ v) 2 - 1 ] 1 / 2 [ 1 - ( u - v ) 2] 1/2
The spectrum with i = 1 was o b t a i n e d in ref. [2]• Both spectra c o r r e s p o n d to the turbulence o f Kolm o g o r o v type. Eq. ( 1 5 ) has also steady-state solutions when R = 0 , in this case p = (0, 1) a n d s = ( 1 , ½). These solutions correspond to the Rayleigh-Jeans turbulence (i.e. t h e r m o d y n a m i c e q u i l i b r i u m solutions, see ref. [ 9 ] ).
The integration in (21 ) is carried out over a range in which A22> 0. Using (21), we find that both spectral fluxes are finite. This means that both spectra are local. F o r elucidation o f the signs o f spectral fluxes, we consider by analogy with ref. [ 9 ] that a ( x ) ozf(x). Then it can be shown that P ( 1 ) < 0 a n d P ( 2 ) > 0 , i.e. the spectral flux in the spectrum with i = 1 is directed on the side o f small k, while in the spectrum with i = 2 the flux is directed on the side o f large k.
5. Spectral fluxes and turbulence locality
References
We assume in ( 1 5 ) that a = a ( i ) + 6 ~ , where 6 , are small a d d i t i o n a l terms. Then, by analogy with ref. [ 8 ], we carry out the limiting transition to 6 , ~ 0 a n d obtain the transfer equation (cf. ref. [8] )
[ 1] B.B. Kadomtsev and O.P. Pogutse, JETP Lett. 39 (1984) 269. [2] A.V. Gruzinov and O.P. Pogutse, Dokl. Akad. Nauk SSSR 290 (1986) 322. [3] R.H. Kraichnan, Phys. Fluids 10 (1967) 1417. [4] A.B. Mikhailovskii, Theory of plasma instabilities, Vol. 2 (Consultants Bureau, New York, 1974). [ 5 ] B.B. Kadomtsev, Plasma turbulence (Academic Press, New York, 1965 ). [6] V.E. Zakharov and V.S. Lvov, Radiofizika 18 ( 1975 ) 1470. [ 7 ] V.E. Zakharov, in: Handbook of plasma physics, Vol. 2, eds. A.A. Galeev and R.N. Sudan (North-Holland, Amsterdam, 1984) p. 3. [8] A.B. Mikhailovskii, S.V. Novakovskii, V.P. Lakhin, S.V. Makurin, E.A. Novakovskaya and O.G. Onishchenko, Zh. Eksp. Teor. Fiz. 94 ( 1988 ) 159. [9] R.H. Kraichnan, J. Fluid Mech. 47 ( 1971 ) 525.
E~i)oc(k-5/3, k - l ) ,
i=i,2.
OD~° 1 OP (° o----7-+ k o~- _ o .
(19)
(20)
Here D~ s) = ( Wk, Hk) a n d p u ) are spectral fluxes defined by
- f (uv)-P"'A2(u, v)K(i)(u, v) X2i(u, v)u du vdv ,
274
(21)
PHYSICS LETTERS A
16 April 1990
S T R O N G TURBULENCE OF S H O R T - W A V E L E N G T H E L E C T R O S T A T I C P E R T U R B A T I O N S OF A M A G N E T I Z E D P L A S M A A.B. M I K H A I L O V S K I I , S.V. N O V A K O V S K I I 1. v. Kurchatov Institute of Atomic Energy, Moscow 123182, USSR and O.G. O N I S H C H E N K O Space Research Institute, Moscow 117810, USSR Received 31 May 1989; revised manuscript received 15 December 1989; accepted for publication 12 February 1990 Communicated by R.C. Davidson
Steady-state spectra of strong Kolmogorov turbulence of short-wavelength (in comparison with the ion Larmor radius) electrostatic perturbations of a magnetized plasma are found. Locality of these spectra is shown and directions of spectral fluxes are elucidated.
1. Introduction According to ref. [ 1 ], the investigation of strong turbulence of short-wavelength perturbations (k±pi >> 1, where k± is the transverse wavenumber, pi is the ion L a r m o r radius) is of interest for the problem of anomalous thermal conductivity of a plasma in a strong magnetic field. Nonlinear equations for these perturbations were obtained in ref. [ 1 ]. One spectrum of the K o l m o g o r o v turbulence of these perturbations was found in ref. [2]. In the present paper we show that the electrostatic shortwavelength turbulence is characterized by two spectra of Kolmogorov type. We also investigate the question of the turbulence locality and show that both spectra are local. Besides that, we elucidate directions of spectral fluxes and show that, as in the case of two-dimensional hydrodynamic turbulence [ 3 ], these directions are opposite.
2. Starting equations We study two-dimensional electrostatic perturbations of a homogeneous plasma, supposing their 272
electric field E = - V . 0 , where 0 is the electrostatic potential, V± = (O/Ox, O/Oy). The equilibrium magnetic field Bo is assumed to be directed along the z axis. Let us consider the wavelength of perturbations to be small in comparison with the ion Larmor radius, k±pi >> 1, and taking into account that the ion nonlinearity is small in comparison with the electron nonlinearity (cf. ref. [ 1 ] ), we take the ion perturbed density fi depending in the linear approximation on 0. Then using, for example, the method of integration along particle trajectories [4], we obtain the well-known formula fik
=
--
enoO____k[ 1 -- (2~zi) - 1/2 ] •
Ti
( 1)
Here fik and Ok are spatial Fourier-components of fi and 0, zi= (k±pi) 2, no and Ti are the equilibrium plasma density and ion temperature, e is the ion charge, k is the wave vector. Besides that, we use the electron continuity equation Ofe/Ot+ VE" V I f e = 0 ,
(2)
where hie is the perturbed electron density, VE=c(ez×VO)/Bo is the drift velocity in crossed (E, Bo) fields. Using the transition to the Fourier trans-
Volume 145, number 5
PHYSICS LETTERS A
form in k in (2) and taking the quasineutrality condition ni= ne into account, we obtain with use of ( 1 ) the nonlinear equation for ¢~k:
16 April 1990
Here
Gq=(og+qq) -l ,
(10)
and qq is determined by
i ~ o z f V(k,k,,k2)
qq= -4 ~ G"~,lq2V(k, kl, k2) V*(k~, k, k2)
Xq~10~2~(k-l-kl -t-k2) dk I dk 2 .
Here
(3)
V(k, kl, k2) is the interaction matrix element,
V(k, kl,k2)=i(k '
~-~)(k,×k2)z,
(4)
X6(q+ql +q2) dq~ dq2.
Taking flq do9ocWk (see (7)) and the fact that (cf. ref. [5]) Im
kl and k2 are modulae of vectors k~ and k2, the asterisk means the complex conjugation. The subscript Z on wavenumbers is omitted for simplicity. We note that
V(k, kx,kE)+V(kl,kE,k)+V(k2, k, kl)=O, k-IV(k, kl, k2) + k i -1V(kl, k2, k) +ky I V(k2, k, k, ) = 0 .
(11 )
f oglqdogocOW,/Ot
(12)
into account, we obtain from ( 9 ) - ( 11 ) that
0wk
Ot oc
g(k,k~,k2)[F*(k, kl,k2)aflq, I~
(5) -'}-V* (kl,
k, k2 )tTqlIqlcz -t- I/'*(k2, k, kl )ff q2IqIql ]
X~(q+q~ +q2) dq~ dq2 dog, (6)
Eq. (3) has two integrals of motion connected with (5) and (6): energy W and generalized enstrophy H defined by
w~c f 10,12~- f W.dk,
(7)
Hoc f K-'l~kl2dk- f H, dk.
(8)
where aq=-ImGq. Using (10) and the standard prescription for the calculation o f I m Gq (o9--,o9+iA, A> 0), we conclude that aq > O.
4. Steady-state spectra We consider (cf. ref. [6] )
Iq=k-tV+s)f(o9/ks), eq=k-s#(o9/ks), Therefore, by analogy with the hydrodynamic turbulence [ 3 ], one can expect that a steady-state spectrum corresponds to each integral of motion. We shall show this below.
We represent ~k in the form of the Fourier integral in o9-space, introducing by this way ¢q, where q= (k, to). We also introduce Iq defined by (O~,~q)= Iq5(q-q' ), where ( ) means averaging over the statistical ensemble. In accordance with ref. [ 5 ] we find from (3) that Iq satisfies the equation
Iq=21Gql2 ~ Ig(k,k~,k2)l 2 XIq~Iq2~(q"{-ql -t-q2) dqt dq2.
(9)
(14)
where p and s are numbers, f and # are functions. Changing in (13) to positive frequencies o9, o9~ and o92, and using the factorization method of ref. [ 6 ], we obtain
OWk/Otoc f dk~ dk2 ~ ( k + k l
3. Equations of turbulence
(13)
+kE)K
X ~ Rd(og- ogl -o92) do9 do91 do92.
( 15 )
Here
K= V(k, kl, k2) + (k/k~)~V(kl, k2, k) + (k/kE)~V(k2, kl, k), (16) R= V*(k, kl, k2)aqlq,Iq2+ V*(k~, k, k2)trqllqlq2 + V*(k2, k, ki )trq,IqIq., (17) where a = 6 - s - 2p. Taking (5) and (6) into ac273
Volume 145, number 5
PHYSICS LETTERS A
count, we find that K = 0, i.e. the right-hand side o f (15) vanishes at a=ot u~, i = 1 or 2, where a ( ~ ) = 0 a n d ct(2) = 1. We conclude from this that steady-state spectra, for which s + 2 p = 6 - a (o, can be realized. Using (11 ) a n d considering that Gq~ck-~g(og/k~), where g ( x ) is a function connected w i t h / t ( x ) , we find one m o r e correlation between s a n d p: 2s + p = 4. Thus we obtain two pairs o f n u m b e r s p=p(O and s = s ~°, i = 1 or 2, c o r r e s p o n d i n g to steady-state spectra.p~,)__8 • - 3 and s ( ~ )-_~ 2, o r p (2)=2 and s(2)= 1. Since at Iq o f the form ( 1 4 ) Wkock -p, we obtain that energy densities o f these spectra
W(ki)oc(k-S/3, k - 2 ) ,
i=1,2.
(18)
Respectively, for E k = k W k we have
16 April 1990
K ( ° ( u , v)=u-~'"'( 1 - l /v) In u -v-'~"'(1-1/u)
lnv,
(22)
2°)(u, v) = [.i dxdxl dx2 ,~(x-xl uS"'-x2v ~"') × [ ( 1/u- 1/v)/~(x)f(x, )f(x2 ) + (l/v-- l
)~L~(Xl)f(x)f(x2)
+ ( 1- 1 / u ) p ( x 2 ) f ( x ) f ( x l ) ] ,
(23)
A2(u, v) is the dimensionless area o f the triangle m a d e by the vectors k, k~ and k2,
A2 = [ (u+ v) 2 - 1 ] 1 / 2 [ 1 - ( u - v ) 2] 1/2
The spectrum with i = 1 was o b t a i n e d in ref. [2]• Both spectra c o r r e s p o n d to the turbulence o f Kolm o g o r o v type. Eq. ( 1 5 ) has also steady-state solutions when R = 0 , in this case p = (0, 1) a n d s = ( 1 , ½). These solutions correspond to the Rayleigh-Jeans turbulence (i.e. t h e r m o d y n a m i c e q u i l i b r i u m solutions, see ref. [ 9 ] ).
The integration in (21 ) is carried out over a range in which A22> 0. Using (21), we find that both spectral fluxes are finite. This means that both spectra are local. F o r elucidation o f the signs o f spectral fluxes, we consider by analogy with ref. [ 9 ] that a ( x ) ozf(x). Then it can be shown that P ( 1 ) < 0 a n d P ( 2 ) > 0 , i.e. the spectral flux in the spectrum with i = 1 is directed on the side o f small k, while in the spectrum with i = 2 the flux is directed on the side o f large k.
5. Spectral fluxes and turbulence locality
References
We assume in ( 1 5 ) that a = a ( i ) + 6 ~ , where 6 , are small a d d i t i o n a l terms. Then, by analogy with ref. [ 8 ], we carry out the limiting transition to 6 , ~ 0 a n d obtain the transfer equation (cf. ref. [8] )
[ 1] B.B. Kadomtsev and O.P. Pogutse, JETP Lett. 39 (1984) 269. [2] A.V. Gruzinov and O.P. Pogutse, Dokl. Akad. Nauk SSSR 290 (1986) 322. [3] R.H. Kraichnan, Phys. Fluids 10 (1967) 1417. [4] A.B. Mikhailovskii, Theory of plasma instabilities, Vol. 2 (Consultants Bureau, New York, 1974). [ 5 ] B.B. Kadomtsev, Plasma turbulence (Academic Press, New York, 1965 ). [6] V.E. Zakharov and V.S. Lvov, Radiofizika 18 ( 1975 ) 1470. [ 7 ] V.E. Zakharov, in: Handbook of plasma physics, Vol. 2, eds. A.A. Galeev and R.N. Sudan (North-Holland, Amsterdam, 1984) p. 3. [8] A.B. Mikhailovskii, S.V. Novakovskii, V.P. Lakhin, S.V. Makurin, E.A. Novakovskaya and O.G. Onishchenko, Zh. Eksp. Teor. Fiz. 94 ( 1988 ) 159. [9] R.H. Kraichnan, J. Fluid Mech. 47 ( 1971 ) 525.
E~i)oc(k-5/3, k - l ) ,
i=i,2.
OD~° 1 OP (° o----7-+ k o~- _ o .
(19)
(20)
Here D~ s) = ( Wk, Hk) a n d p u ) are spectral fluxes defined by
- f (uv)-P"'A2(u, v)K(i)(u, v) X2i(u, v)u du vdv ,
274
(21)