Transcript Self-consistent mean field forces in turbulent plasmas

Self-consistent mean field forces in two-fluid
models of turbulent plasmas
C. C. Hegna
University of Wisconsin
Madison, WI
Hall Dynamo Get-together
PPPL via UW-Madison
June 11, 2004
Theses
• The properties of turbulent plasmas are described using the
two-fluid equations.
• Global constraints are derived for the fluctuation induced
mean field forces that act on the ion and electron fluids.
• Relationship between relaxation of parallel momentum
flows and parallel currents
C. C. Hegna, “Self-consistent mean-field forces in turbulent
plasmas: current and momentum relaxation,” Physics of
Plasmas 5, 2257 (1998); 3480 (1998). --- RFP physics
was largely the motivation
Outline
• Brief review of mean field resistive MHD theory relevant
to magnetized plasmas - applications to RFPs
• Two-fluid theory
– Constraints on the fluctuation induced mean-field forces
– Heuristic derivations of local forms for the mean-field forces
– A simple quasilinear theory
Subsequent work - Mirnov, et al ‘03 - is much more complete
– Relation to relaxation
Steinhauer, Ishida, ‘98-’03; Mahajan and co-workers ‘01
In resistive MHD dynamo theory, a mean field
force is identified
• Fluctuations affects mean field dynamics in resistive MHD
through a dynamo electric field
E  v  B  J
Write all quantities as mean field and fluctuations


˜
Q  Q  Q
The bracket <> notation denotes either an ensemble average or an
average over the “small” spatial scales or “fast” time scales of the
fluctuations
Mean field Ohm’s Law
 E    v    B  F    J 
˜
F  v˜  B 
Global conservation laws have motivated local
forms for the mean field force of resistive MHD
• In resistive MHD, fluctuations do not dissipate helicity, but do
dissipate energy. (Boozer, J. Plasma Physics, 1986; Bhattacharjee and
Hameiri, PRL 1986; Phys. Fluids 1987; Strauss, Phys. Fluids 1985).
d
d
3
x F   B   0,
3
x F   J   0.
These condititions are used to motivate a “local” form for the mean-field
force in toroidal confinement devices --- fluctuations generate an
additional
electron viscosity or hyper-resitivity, not a “dynamo.”

F|| 
Bo
Jo  Bo
2


(K

)
Bo2
Bo2
K2 is a profile dependent positive function satisfying boundary conditions.
Consistent with the Taylor state, F gets large, ---> J||/B = constant

Two-fluid equations can be written in a concise
form
• The exact two-fluid momentum balance equations
vs
 vs  vs )  nsqs (E  vs  B)  ps    s  Rs
t
– These equations can be written more concisely with the
identification of the canonical momentum.
msns (

m s v s
(
 v s  v s )  E  v s  B
qs t
ms
msv s2
As  A 
v s,  s   
qs
2qs
Momentum balance equations
A
ps
R
  s
 s   s  v s  Bs 
 s 
t
nsqs nsqs
nsqs
Plasma flow for each species
vs  u 
J
m e mi
nsqsms me  mi


ms 
m v2
m
A
v s   s s  v s  (  s v s ) 
   v s  B
qs t
2qs
qs
t


m
m v2
m
(A  s v s )  (  s s )  v s    (A  s v s )
t
qs
2qs
qs

As
  s  v s  Bs
t
A pressure equation is also used for each species
• The pressure evolution equations

ps
 vs  ps  ps  vs  ( 1)(Qs    qs  s : vs )
t
– Q = collision energy transfer and Ohmic heating, last term
represents viscous heating.
– In general, compressibility is allowed. This modifies the usual
definition of the mean field force and allows for anomalous
particle transport.
– In what follows, the effects of heat flux q are simplified.
A weakness in the theory and a potential new area of investigation.
Fluctuations induce mean field forces on both the
ion and electron species.
• For simplicity, we consider a cylindrical plasmas with all
the usual boundary conditions. Quantities are split into
equilibrium and fluctuating quantities, Q  Qo  Q˜
Nonlinearities produce fluctuation induced mean field forces (actually
forces per unit charge)
T˜s ln(1 n˜ s /nso )
msv˜ s2
˜
˜
Fs  v s  Bs   



qs
2qs
Note, the first term contains both the MHD and Hall dynamo terms.
For the electrons, ve=u - J/ne + O(me/mi).

˜
J
˜
˜
Fe  u˜  B    ( )  B  ...
ne
Three global properties of the mean-field forces
can be shown
• Mean field momentum balance equations


Aso
pso
R
  s
  so  v so  Bso  Fs 
 ( s )o  (
)o
t
nso qs
nsqs
nsqs
• Three global constraints to be shown
 dV Bo  Fe  0 to O()
 dV B  F  0 to O()
 dV(n ev  F  nev  F )  0
o
i
o
io
i
eo
e
The last condition can also be written using F||M = F||i -F||e, F||O =
(miF|||e+meF||i)/(mi+me)

 dV(n eu  F
o
M
 Jo  FO )  0
A number of assumptions are used to prove the
three global constraints
• Simplifying assumptions used in the constraint derivations:
– Fluctuation amplitudes are small compared to the mean magnetic
field, typically valid in all MFE devices
2
–
1
1 ˜2
B
msn sv˜ s2, p˜ ,
B  o
2
2o
2o
The equilibrium quantities evolve on a slow diffusive time scale


 Q ~
Q
t
o a 2

Viscosities
and radial mean flow are ordered with resistivity. Parallel
heat flux is ordered small to be consistent with the neglect of heat

flux, (again, probably a weak point)
B  T

~ O(
)
B
oVA a2

A number of assumptions are used to prove the
three global constraints
• Assumptions (continued)
– The viscous force is dissipative for both species
: v s  0
– All other equilibrium flows are ordered small - probably not a
crucial assumption,may be generalized to equilibrium with flow
Jo  Bo  po
– Ion and electron skin depths are small. With b~ 1, r ~ d
s
s
c
ds 
 ps
 a
 velocity and magnetic field fluctuations are small, gradients of
While
fluctuating quantities may be large, in general

J˜ au˜
~
~ O(1)
Jo
VA
The first two conditions can be shown from the
generalized helicity evolution equations
• Two separate ways to generate the evolution of the mean
generalized helicity
– The first from the total
momentum balance
– The last from the mean
momentum balance
• Subtracting the average of
the first equation from the last
equation.
2Fs  Bso    C1 
 ˜ ˜
˜ ˜
˜   s
 As  Bs  2  J  B  2  Bs 

t
n sqs
2
2
 (1 ln n so )Bso  Tso   (1 ln n s )Bs  Ts 
q
qs
With the assumed orderings and appropriate
boundary conditions, the first two conditions are
derived
• The previously derived condition
2Fs  Bso    C1 
 ˜ ˜
˜ ˜
˜   s
 As  Bs  2  J  B  2  Bs 

t
n sqs
2
2
 (1 ln n so )Bso  Tso   (1 ln n s )Bs  Ts 
q
qs
– All the terms on the right hand side are smaller than O()


 dV 2F  B
s
so

 dS  C  0
1
to O()
– C1 is the fluctuation induced generalized helicity flux.
– In the me = 0 limit, the electron condition corresponds to the same
as that derived for resistive MHD. In two-fluid theory, there two
constraints, one for each fluid.
Energy balance relations are used to prove the
third condition
• Total energy conservation
 B2
nsmsv s2
p
nsmsv s2
p v
(


)    [
v s  E  B  ( s s  v s  s )]  0
t 2o s
2
 1
2
 1
s
s
•

Construct mean magnetic energy evolution from
 pressureequations   nsqsvso   Momentumbalances  0
s
s
– Subtract this from O() average of the top equation


 p˜ sv˜ s 
˜ : v˜ 
noev io  Fi  noev eo  Fe  
 pso    C2    J˜ 2   
s
s

p
os
s
s
– C2 is the leading-order energy flux caused by the fluctuations
– Third term denotes anomalous cross-field transport --- similar bits show
up in resistive MHD - Hameiri and Bhattacharjee, ‘87.
By accounting for the cross-field diffusion in our
definition of F, the fluctuations are shown to
dissipate energy
• One can redefine the mean field force to account for turbulence
induced cross field heat and particle transport
– This redefinition doesn’t affect
The first two conditions
– The final condition is derived
 dV(n ev
o
io
F  F  F ,
f  p˜ ev˜ e  (1 f )  p˜ iv˜i 
F  Bo [

]
peo
poi
˜ : v˜ )  0
 Fi  noev eo  Fe )    dV(  J˜ 2   
s
s

s
- This condition can also be written using FM = Fi - Fe, FO = (miFe +
meFi)/(me + mi)
 dV(n eu  F

o
M
 Jo  FO )  0
• Fluctuations dissipate energy. Energy delivered to electrons and ions
via Ohmic and viscous heating.

The global constraints imply a particular choice
for the local form of the mean field forces
• In analogy with the hyper-resistivity form implied by the
resistive MHD global constraints, local forms are implied
by the two-fluid global constraints.
– The first two conditions imply
 dV B  F
o
s||
 0  Bo  Fs||   xs
where xs vanishes on the boundary
– The third condition can then be written

 dV qsnsvso  Fs||    dV xs  (
s
s
– A solution that guarantees the above is
n v B
xs   f st ( to to2 o )

Bo
t
with conditions on the coefficients fst.
qsnsv so  Bo
)0
2
Bo
The implied local forms suggest a coupling
between current and flow evolution
• The parallel components of the turbulent mean field force

F|| e 
Bo
n v B
n v B
  [ke2( o eo2 o )  Le ( o io 2 o )],
2
Bo
Bo
Bo
F|| i 
Bo
n ov io  Bo
n ov eo  Bo
2


[k
(
)

L
(
)],
i
ei
Bo2
Bo2
Bo2
– Can rewrite these equations as
Those that appear in Ohm’s law
And the total momentum balance
F||O 
Bo
Jo  Bo
eno uo  Bo
2


[

(
)


(
)],
e
e
Bo2
Bo2
Bo2
F|| M 
Bo
eno uo  Bo
Jo  Bo
2


[

(
)


(
)],
i
i
Bo2
Bo2
Bo2
Coefficients are spatially
dependent functions that
vanish on the boundary
and satisfy ke2 > 0, ki2 > 0,
(Le + Li)2 < ke2ki2/4
e2 > 0, i2 > 0,
(e + i)2 <
e2i2/4
The local forms are also derivable from quasilinear tearing mode theory
• Simplified, incompressible, low-b, me = 0, s = 0 limit
– Mean field forces are derived from quasilinear expression for the
tearing mode resonant on some surface. < > = average over layer
width
˜
Fs B o    v˜ s (Bo  As ) ,
˜
˜ ˜ J
FO  Bo    (Bo  A)(u  ) 
ne
m
1 ˜
˜
FM  Bo    i u˜ (Bo  u˜ )  J (Bo  A) 
e
ne

• In the Ohm’s law, the resistive MHD and Hall dynamos
• In the total momentum balance, the Reynolds and Maxwell
stresses (to within a factor of ne).
A simple model is used in the linear layer analysis
• A four field model
B  bˆ B  bˆ  ,
u  bˆ V  bˆ  
• Equilibrium near the
 rational surface
x2
Bo  Bo (ro ),o  o '' ,
2
J o  J o (ro )  J o '(ro )x
Vo  Vo (ro )  Vo '(ro )x
The mean field forces derived from quasilinear
theory exhibit the same structure as that implied
from the global constraints
• After linear theory
dJo
dV
 n oeN e o ),
dx
dx
dV
dJ
FM  Bo    (n oeMi2 o  N i o ),
dx
dx
FO  Bo    (M e2

– In the resistive MHD limit, Me2 is the largest coefficient - hyperresistivity dominates and FM is small.
– In the two-fluid limit, the dominant drive comes in the
combination dVe/dx=dVo/dx - dJo/dx (ne)-1
• Highly simplified theory --- A much more complete job
calculation of two-fluid tearing mode growth rates has
been calculated by Mirnov, et al.
Relationship to relaxation theory
• Constraints imply a relaxation theory via the minimization
of W - eKe - iKi where Ks =  dV As.Bs
– To lowest order in c/(pia), the minimizing solutions yield
J o  1Bo
n oeuo  2 Bo
A state discussed by many - Sudan, ‘79; Finn and Antonsen, ‘83;
Avinash and Taylor ‘91; Steinhauer and Ishida, ‘98; Mahajan et al
‘01.

Summary
• Using a modest number of assumptions, three global constraints are
derived for turbulence induced mean field forces in two-fluid models
of plasmas.
 dV Bo  Fe  0 to O()
 dV B  F  0 to O()
 dV(n ev  F  nev  F )  0
o
i
o
io
i
eo
e
• These constrains imply functional forms for the parallel mean-field
forces in the Ohm’s law and the total momentum balance equations
suggesting
the fluctuations relax the plasma to states with field aligned

current and bulk plasma momentum.
• Applications to flow profile evolution during discrete dynamo events
on MST?