Cascade mechanism for developing hydrodynamical model of nonlinear plasma turbulenc

Tchen, C. M.

English

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CASCADE MECHANISM OF NONLINEAR INTERACTIONS BETWEEN 
MODES I N  A TURBULENT PLASMA 
C. M. Tchen 
T h e  C i t y  C o l l e g e  of The C i t y  U n i v e r s i t y  of New York 
O c t o b e r  1969 
https://ntrs.nasa.gov/search.jsp?R=19700002649 2020-03-12T04:48:12+00:00Z
SHORT ABSTRACT 
(For Information Retrieval) 
A hydrodynamic model of plasma turbulence is developed. The cascade 
concept is utilized for the problem of closure. 
electrostatic energies are investigated for both the collisional and collision- 
less cases. The anomalous diffslsiw, called Bohm diffusion, is also derived, 
and found to play an important role in the nonlinear transport properties of 
the problem. 
The spectra of kinetic and 
1. INTRODUCTION 
There are several areas of marked similarity between the nonlinear 
wave interactions in electron-ion gas plasmas and in electrQn-hole solid plasmas. 
Such a similarity in turbulence has been demonstrated by Betsy Ancker-Johnson 
in solid plasmas by an experiment equivalent to that in gas plasmas, and has 
been stressed by Hoh2 on theoretical cowiderations . 
in gas and solid plasmas have been very sketchy, and up to dace, only some 
phenomenological and quasilinear considerations have been treated, valid for a 
weak turbulence. 
turbulence in the aim o f  analyzing the nonlinear interactions #cross the indivi- 
dual spectra and between the spectra of kinetic and electrostatic energies. In 
view of the similarity in turbulence between the solid and gas pla$miis, as 
mentioned above, we shall retain the notations of gas pJasmas in our theoretical 
development, and shall not attempt to discriminate the singular distinctions of 
solid plasmas. 
1 
3 Theories on turbulence 
In the following, we attempt; a hydrodynarpicsl model of strong 
2. CASCME SYSTEM 
We consider a quasineutral plasms consisting of ions and electrons. 
The ions have a velocity u, a charge e and, a mass M. A magnetic field of 
constant strength O,O,B, has a cyclotron frequency dc = eB/M, and is assumed 
large compared to the pressure force. The electrons are hot, and asswe a 
Boltzmann distribution 
ry 
r z L =  
for the number density n for electrons or ioos, n is the average dengity, a 
is the phase velocity 
0 
with an electron temperature Te and the Boltzmann constant K. 
product of the self-consistent electric potential by e/M. Write the Navier- 
Stokes equation of motion and the equation of continuity for the fluctuacions 
Further ? is the 
-2- 
transverse to the magnetic field; 
Here 3 is the kinematic viscosity, .E is an antisymmetric unit tepsor, and 
b' = 0,0,1. 
: Using:.the cascade deeampositim 
k 
Similarly for 3, f '  4 ', we find the following cascade 
s y s t em : 
t k O * V  
D a 
Dt a t  - - = -  
-3- 
where 
g o =  - 6x9 a d N  E ) =  - d V y  
represent the self-consistent electric fields mwltiplied by e/M. 
The variables go 4 y o  are called the large scale flucfua- 
tions, and the variables MI Cd- y'are called the small scale fluctuations. 
N 
The system o f  equations (3) and (4) has been derived from (l), upon 
substituting for (2) and Qn the basis o f  che following assumptions: 
(i) The variables %'and yovary slowly and the variables 2' and p' 
vary rapidly, so that a variable transition scale k can be used to distinguish 
between the parts by a local average, denoted by (. r >  . A Le- 
average over an infinite length will be denoted by a bar. Tbis is the assump- 
tion known as the quasi-stationarity assumption. 
(ii) The turbulent motions are isotropic 
in the plane perpendicular to the magn9t;ic field. 
and 
homogeneous, i.e., within an interval g-', as their averages have been so de- 
fined. 
The quantities U, y, &4* 
%' y '  are Qnly locally 
cc 
y o  are homogeneous generally, but the qvantities 
The equations of kinetic and potential energies are derived by 
multiplying (3a) and (3b) by Lc"; and y o  respectively, by averaging and using 
where 
or identically 
as shown from averaging (3b). We have denQted 
-iy = R " ( k = q ,  J = jo(k=D0) 
-4- 
The express ions  (6) and (7) a r e  t r a n s p w t  f enc t iops ;  Fore spec i -  
gk i s  a product ion func t ion ,  t ransforming t h e  k i n e t i c  energy i n t o  f i c a l l y  
t h e  p o t e n t i a l  energy i n  t h e  form o f  e l e c t r o s t a t i c  f l u c t u a t i o n s .  and 
),, J "  
of energy ac ross  t h e  k i n e t i c  and p o t e n t i a l  spec t r a  r e spec t ive ly .  
a r e  k i n e t i c  and p o t e n t i a l  t r a n s f e r  func t ions ,  desc r ib ing  t h e  t r a n s f e r  
The equat ions  (5) r e p r e s q t  the  evo lu t ion  of t h e  k i n e t i c  and poten- 
t i a l  energy spec t r a .  I n  t h e  un ive r sa l  range, i.e., i n  t h e  range of not  t oo  
small  4 , t h e  r a t e  of change of energy i n  t h e  po r t ion  (0,k) of the spectrum 
i s  not: very much d i f f e r e n t  from t h a t  i n  t h e  whole spectrum (0, DO ) which i s  
a cons tan t  i n  equi l ibr ium. Under such circumstances,  we s impl i fy  (5) t o  
no t ing  t h a t  
as,  by d e f i n i t i o n ,  t h e r e  i s  no net: energy t r a n s f e r  i n  t;he whole g g e c t r m .  
3.  TRANSPORT FUNCTIONS 
The t r a n s f e r  func t ions  (7) involve the  s t a t i s t i c a l  e f f e c t s  from 
t h e  f l u c t u a t i o n s  5' and J/' which a r e  c a l c u l a t e d  by incegra t ing  (4a) .  
During t h i s  procedure,  it i s  important t o  note  t h a t  t h e  func t ions  (7) con ta in  
a coupling wi th  uo o r  Po t o  i t s  f i r s t  power, and t h e r e f o r e  t h e  i n t e g r a t i o n  
of (4a) should s e l e c t  terms of such a coupl ing only,  f o r  t h e  genera l  average 
t o  make a non zero  con t r ibu t ion ,  a l l  o t h e r  terms having no con t r ibu t loe .  
Hence, w e  f i n d  t h e  stresses i n  (3) and (7) :  
Z Q  cyk au/L/ax > = -9, Q w L  
d 
(5'. V P J )  = ?A6 u y  
y ie ld ing ,  by a genera l  average, 
-5- 
u: (l4; a&& > - - %  "2 V Z U O  
k 
L'k R" - 
yo  (Lc'-VyJ'> - = -A/+ y0V')D0 
- = AkJe 
The terms V2LLQr N and V 2 P L  have been dropped on account of the homo- 
and 
genei ty  assumption (ii). From the  above ope ra t ions ,  we f i n d  
P O  
I n  t h e  d e r i v a t i o n  of ( loa)  from (4a) ,  we have dropped fhe  viscous a f f e c t .  
(b) The product ion func t ion  (6b), on the Other hand, does not  i n -  
volve a coupl ing with U* and y o  . Therefore ,  t he  de te rmina t ion  of  %' from 
(4a),  f o r  t h e  purpose of eva lua t ing  (6b),  should s e l e c t  terne without  a 
coupl ing,  a s  any t e r m  wi th  a coupl ing wi th  e Q o r  t o  i t s  f i r a t  power 
would vanish by a genera l  average. 
N 
Hence we f i n d  t h e  product ipn func t ion  
(6b) t o  be 
aga in  we have dropped t h e  viscous e f f e c t .  
4. TRANSPORT COEFFICIENTS 
The time integrationl of t h e  au to -co r re l a t ions  of v e l o c i t y  apd 
f i e l d  f l u c t u a t i o n s  de f ine  t h e  t r a n s p o r t  c o e f f i c i e n t s  
i n  (10). They a r e  very s i m i l a r .  
Using t h e  Four i e r  t ransform of a func t ion  t runca ted  wi th in  a t i m e  
i n t e r v a l  7 and a length  i n t e r v a l  w i th  
3~ = 72;+/rx3 
w e  can wri te  
where t h e  ba r  with s u p e r s c r i p t s  ( t ,x )  denotes  an average over  t i m e  and space 
wi th in  t h e  above mentioned t r u n c a t i o n  i n t e r v a l g .  Fu r the r  T, and 
Y-XL k k t  and t h e  Lagangian time i n t e g r a t i o n  g ives  
C V -  r y -  I 
- 6 -  
Here we have introduced the approximation 
Similarly we conclude that 
?% L G ~  4& 
- I  
Y 
'Ug 
Hence (loa) and (1Oc) become 
'lk = F q - '  ($'> 
9 = 32 wq-' q'=> 
k . 4  
Introduce the spectral functions F(k) and G(k) for the kinetic and poten- 
- tial energies respectively. We cae write - 
L- yot = /"..A 4 k 
/ 2 
Uo2 = I 2 , "  
- 
0 0 
Ro = L J ~ ~ ~ F ,  J " =  2 & P C ;  
a 0 
and reduce (lla) to 
Since 
and 
we can rewrite (llb) in the form 
with 
the diffusion coefficient of Bob. It is foung to be proportional to 
Wc-: in contrast t o  the classical diffusion acrosg the magnetic field whiah 
is proportional to 
without demonstration. 
coefficient smaller than in (12c). 
";' . The formula (12~) has been proposed by Bohm 
His proposed formula contained an ewpirical numerical 
Formula (12b) determines 
$ = *  r U Q  
which substitution into (8b) produces 
The evlluation of according t o  (lob) requires equally g 
k 
Lagrangian time integration,with the paramgter Wc 
variable relaxation frequency Qy and we find 
in ( l l a )  replaced by a 
c! "h)  
( '2 9 
c.3 
J 4.c ( u l c ~ ) .  2 7  f 2 k  F / d *  0 3") 
k * T c l o  
The relaxation frequency @is to be determined by the very 
. We write quch a process for 
"k 
cascade process (5b) which introduced 
the energy band Gdk, according to (5b) 
or simply 
as 
in the spectral region dominated by 
We rewrite the system (13 ) ,  
follows : 
with the consideration of (lob), as 
yielding the solution 
-8- 
The formulas (12) and (14) aye the important expressions of 
transport coefficients needed in the determination of the spectra. 
5. ENERGY SPECTRA 
With the use of (12b), (12d) and (12e), we can reduce (8) to the 
sys tem 
t = v R  
determining the flow o f  energy across and between the spectra. Each spec- 
trum F or G possesses a source, a sink and a nonlinear cransfer across the 
spectrum. The source maintains the high Reynolds number of tyrbulence, 
in such a way that its structure does not appear directly in the universal 
range of the spectrum (range of large k), but its amwnt has to balance the 
total rate of energy dissipation. which was written on the right hand sides 
of respective equations (15a) and (15b). The sink is a dissipation of 
thermal origin: it is Yf?" proportional to the molecular viscosity due to 
collisions in the F-spectrum, and is 
diffusion in the G-spectrum. It: is to be remarked that the Bohm diffusion 
can also be considered as Qf thermal nature,since 
to a 2 = k G . / M  , according to (12c). Hente the qrnallest scale of 
the G-spectrum is the Larmor radius a c or A B / &  
dual spectrum, there exist6 an energy transfer characteristic of the non- 
linear interactions. This transfer is t R o  in the F-spectrum and &To 
in the G-spectrum. This description completes the energy flow in the G- 
spectrum, as indicated by (15b). Hpwever, the transfeF pf electrohtatic 
energy has to be driven by the kinetic energy, requiring a coupling term of 
equal value 
7" proportional to the pohm 
3 
A& is proportional 
. Across each indivi- 
= X,J" 
see (12e). This coupling is an additional loss to the F-spectrum. In this 
way, we have completed the flow of energy in the F-spectrum, as indicated by 
(15a). 
-9- 
6. SPECTRUM IN A COLLISIONLFSS PLASMA 
The equations (15) represent a system of rather complicated and 
comprehensive dynamical processes. Sivplifications are necessary for their 
solutions. To this end, consider first the case wlpere the collision is 
negligible. The F-spectrum transfers its energy across the $Recfrum by an 
amount L' R" , to be converted into electrostatic energy, 
perpetuates its full pattern (15b). In order to assure this flow effect- 
ively, the F-spectrum,'khich plays the role of a driving force, phould have 
a high Reynolds number, and therefore should be in the inertial range, 
while the G-spectrum, having a low Reynolds,number, is in the dissipative 
range, in view of the dominant Bohm diffusion causing an early drop of the 
G-spectrum. 
The latter k 
The interactions between tan inertial F-spectrum and a dissi- 
pative G-spectrum can be simply described by a differential form of (15), 
d J "  - , \  - = o  written as follows: dR" ?4 &k B dR 
and obtained by the approximations 
R" Z 0, 7"s 7, 
An addition of the two equations (16) gives 
d R 0  
'k 
/\A A. 
yielding the solution 
in the inertial subrange. It entails frov ( $ 4 )  and (15b) 
in the dissipative subrange. 
-5 
The spectral law k , as predicted by (47b), has been verified 
Dimensional, arguments based on the parameters a and WC 
4 
by experiments. 
have proposed earlier a spectral law, forprovisional experiment31 usage, 
- 10- 
Since a spectrum should have Lts magnitude dependent on the rate 
of dissipation, the proposed formula (17c) cannot be fully valid. 
7. SPECTRUM IN A COLLISIONAL PLASMA 
The collision is assumed to dominate the coupling term 
decoupling the two energy equations (15a) and (15b). The F-spectrum in 
the inertial subrange is determined by (15a), reduced to 
giving the solution 
Under the inertial regime (18) of the F-spqctrum, the G-spectrum 
is determined by (15b), rewritten in the form 
or, after integration and use of (14). 
where 
% 'A (3 = (Z7/&) , 
The formula (19) for the G-spectrum reduces to its jnerfial law for $<< /$ 
and to its dissipative law, for R 5* kB , 
The transitions from the inertial law (20a) to the dissipative law (20b) 
occurs at the critical wave-number j3'L'34B . 
8. CONCLUSIONS 
By means of a hydrodynamic model, valid f o r  a plaspa of hot 
electrons and cold ions, we have established the development of spectra of 
kinetic and electrostatic energies. The nonlinear interactions have been 
described by two transfer functions valid for the nonlinear transfer of 
modes across each spectrum, and by a productian function serving the coupling 
between the two spectra. The closure problem, as characteristic to any 
nonlinear system, has of course to be solved, and in the present theory, the 
approximation is resorted to the cascade concept. The cqncept helped in 
the derivation of the equations for the energy spectre, and in the formula- 
tion of a "turbulent relaxation frequency". 
important in determining the structure o f  turbulent transport coefficients 
entering in all the above nonlinear transport phenomena. Finally the 
equations of energy spectra are solved for the collisonal and collisionless 
cases. 
in gas plasmas. It seems that experiments in solid plasmas, which often 
are simpler, should soon be able to check the theoretical predictions 
more fully. 
The latter frequency is very 
Some of the theoretical results have been confirmed by experiments 
- 12- 
ACKNOWZEDGME NT 
This research was supported by the Natipnal Aeronautics and 
Space Administration, under contrac6 NGR 33-013-034.  
-.4. 
- 13- 
1 
Betsy Ancker-Johnson, "Some Nonlinear Properties of Electron-Bole Plasmas 
Sustaining the Helical Instability", Princeton Universify Plasma Physics 
Laboratory Report D1-82-0339, pp. 1-35, April 1964. 
'F. C. Hoh, "Anomalous Diffusion and Turbulence in the Positive Column", 
Proc. International SympQsium on Diffusion of Plasma Across B Magnetic - Field, Feldafing, Germany, June-July 1964, pp. 183-189.' 
B. B. Kadomtsev, Pld.qma Turbulence, London and New York: Academic Press, 
1965, pp. 65-123. 
3 
J. R. Apel, "Harmonic Generation and Turbulence-like Spectrum in a 
Beam-Plasma Interaction", Phys. Rev. Letters, V Q ~ .  19, pp. 744-746, 
September 1967. 
4 
- 14- 


Cascade mechanism of nonlinear interactions between modes in a turbulent plasma

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