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 theorySubsequent work - Mirnov, et al ‘03 - is much more complete

– Relation to relaxationSteinhauer, 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

Write all quantities as mean field and fluctuations

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

€

rE +

r v ×

r B = η

r J

€

Q =< Q > + ˜ Q

€

< r

E > + <r v > × <

r B > +

r F = η <

r J >

r F ≡<

r ̃ v ×r ̃ 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).

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.”

K2 is a profile dependent positive function satisfying boundary conditions.

Consistent with the Taylor state, F gets large, ---> J||/B = constant

€

d3x∫r F ⋅<

r B > = 0,

d3∫ xr F ⋅<

r J > < 0.

€

rF || =

r B oBo

2∇ ⋅(K 2∇

r J o ⋅

r B o

Bo2

)

Two-fluid equations can be written in a concise form

• The exact two-fluid momentum balance equations

– These equations can be written more concisely with the identification of the canonical momentum.

Momentum balance equations

Plasma flow for each species

€

msns(∂

r v s

∂t+

r v s ⋅∇

r v s) = nsqs(

r E +

r v s ×

r B ) −∇ps −∇ ⋅

t π s −

r R s

€

rA s =

r A +

ms

qs

r v s, ϕ s = ϕ +

msvs2

2qs

€

ms

qs

(∂

r v s

∂t+

r v s ⋅∇

r v s) −

r E −

r v s ×

r B

=ms

qs

∂

∂t

r v s +∇

msvs2

2qs

−r v s × (∇ ×

ms

qs

r v s) +

∂r A

∂t+∇ϕ −

r v s ×

r B

=∂

∂t(

r A +

ms

qs

r v s) +∇(ϕ +

msvs2

2qs

) −r v s ×∇ × (

r A +

ms

qs

r v s)

=∂

r A s∂t

+∇ϕ s −r v s ×

r B s

€

−∂

rA s∂t

−∇ϕ s +r v s ×

r B s =

∇ps

nsqs

+

r R s

nsqs

+∇ ⋅

t π s

nsqs

€

rv s =

r u +

r J

nsqsms

memi

me + mi

A pressure equation is also used for each species

• The pressure evolution equations

– 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.

€

∂ps

∂t+

r v s ⋅∇ps + γps∇ ⋅

r v s = (γ −1)(Qs −∇ ⋅

r q s −

t π s :∇

r v s)

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,

Nonlinearities produce fluctuation induced mean field forces (actually forces per unit charge)

Note, the first term contains both the MHD and Hall dynamo terms. For the electrons, ve=u - J/ne + O(me/mi).

€

Q = Qo + ˜ Q

€

rF s =<

r ̃ v s ×r ̃ B s > − <

˜ T s ln(1+ ˜ n s /nso)

qs

> − <∇ms ˜ v s

2

2qs

>

€

rF e =<

r ̃ u ×r ̃ B > − < (

r ̃ J

ne) ×

r ̃ B > +...

Three global properties of the mean-field forces can be shown

• Mean field momentum balance equations

• Three global constraints to be shown

The last condition can also be written using F||M = F||i -F||e, F||O = (miF|||

e+meF||i)/(mi+me)

€

−∂

rA so

∂t−∇ϕ so +

r v so ×

r B so +

r F s =

∇pso

nsoqs

+ (

r R s

nsqs

)o + (∇ ⋅

t π s

nsqs

)o

€

dV∫r B o ⋅

r F e = 0 to O(η )

dVr B o ⋅

r F i = 0 to O(η )∫

dV (noer v io ⋅

r F i − ne

r v eo ⋅

r F e ) < 0∫

€

dV∫ (noer u ⋅

r F M +

r J o ⋅

r F O ) < 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

– The equilibrium quantities evolve on a slow diffusive time scale

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)

€

1

2msns ˜ v s

2, ˜ p ,1

2μo

˜ B 2 <<Bo

2

2μo

€

rB ⋅∇T

B~ O(

η

μoVA a2)

€

∂∂t

< Q >~η

μoa2

< Q >

A number of assumptions are used to prove the three global constraints

• Assumptions (continued)– The viscous force is dissipative for both species

– All other equilibrium flows are ordered small - probably not a crucial assumption,may be generalized to equilibrium with flow

– Ion and electron skin depths are small. With ~ 1, s ~ s

While velocity and magnetic field fluctuations are small, gradients of fluctuating quantities may be large, in general

€

tπ :∇

r v s < 0

€

rJ o ×

r B o =∇po

€

δs =c

ωps

<< a

€

˜ J

Jo

~a∇ ˜ u

VA

~ O(1)

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.

€

−2r F s ⋅Bso +∇ ⋅

r C 1 =

∂

∂t<

r ̃ A s ⋅r ̃ B s > +2η <

r ̃ J ⋅r ̃ B > +2 <

r ̃ B s ⋅∇ ⋅

t π s

nsqs

>

+2

q(1− ln nso)

r B so ⋅∇Tso −

2

qs

< (1− lnns)r B s ⋅∇Ts >

With the assumed orderings and appropriate boundary conditions, the first two conditions are

derived

• The previously derived condition

– All the terms on the right hand side are smaller than 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.

€

−2r F s ⋅Bso +∇ ⋅

r C 1 =

∂

∂t<

r ̃ A s ⋅r ̃ B s > +2η <

r ̃ J ⋅r ̃ B > +2 <

r ̃ B s ⋅∇ ⋅

t π s

nsqs

>

+2

q(1− ln nso)

r B so ⋅∇Tso −

2

qs

< (1− lnns)r B s ⋅∇Ts >

€

dV 2r F s ⋅

r B ∫

so= d

r S ⋅

r C 1∫ = 0 to O(η )

Energy balance relations are used to prove the third condition

• Total energy conservation

• Construct mean magnetic energy evolution from

– Subtract this from O() average of the top equation

– 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.

€

∂∂t

(B2

2μo

+nsmsvs

2

2s

∑ +p

γ −1) +∇ ⋅[

nsmsvs2

2s

∑ r v s +

r E ×

r B + (

γps

r v s

γ −1+

r v s ⋅

t π s

s

∑ )] = 0

€

< pressure equations >s

∑ + nsqs

r v so⋅< Momentum balances >= 0

s

∑

€

noer v io ⋅

r F i − noe

r v eo ⋅

r F e +

< ˜ p sr ̃ v s >

λpos

⋅∇pso +∇ ⋅r C 2 = −η < ˜ J 2 > + <

t ̃ π s :∇r ̃ v s >

s

∑s

∑

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

- This condition can also be written using FM = Fi - Fe, FO = (miFe + meFi)/(me + mi)

• Fluctuations dissipate energy. Energy delivered to electrons and ions via Ohmic and viscous heating.

€

rF ⇒

r F +

r F ⊥,

r F ⊥ =

r B o ×[

f < ˜ p er ̃ v e >

γpeo

+(1− f ) < ˜ p i

r ̃ v i >

γpoi

]

€

dV∫ (noer v io ⋅

r F i − noe

r v eo ⋅

r F e ) = − dV (η < ˜ J 2 > − <

t ̃ π s :∇r ̃ v s

s

∑∫ >) < 0

€

dV∫ (noer u ⋅

r F M +

r J o ⋅

r F O ) < 0

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

where s vanishes on the boundary– The third condition can then be written

– A solution that guarantees the above is

with conditions on the coefficients fst.

€

dVr B o∫ ⋅

r F s|| = 0 ⇒

r B o ⋅

r F s|| =∇ ⋅

r ξ s

€

dV qs

s

∑∫ ns

r v so ⋅

r F s|| = − dV∫

r ξ s ⋅∇(

qsns

r v so ⋅

r B o

Bo2

s

∑ ) < 0

€

r s = f st

t

∑ ∇(nto

r v to ⋅

r B o

Bo2

)

The implied local forms suggest a coupling between current and flow evolution

• The parallel components of the turbulent mean field force

– Can rewrite these equations as

Those that appear in Ohm’s law

And the total momentum balance

€

rF ||e =

r B oBo

2∇ ⋅[−ke

2∇(no

r v eo ⋅

r B o

Bo2

) + Le∇(no

r v io ⋅

r B o

Bo2

)],

r F ||i =

r B oBo

2∇ ⋅[ki

2∇(no

r v io ⋅

r B o

Bo2

) − Lei∇(no

r v eo ⋅

r B o

Bo2

)],

Coefficients are spatially dependent functions that vanish on the boundary and satisfy ke

2 > 0, ki2 > 0,

(Le + Li)2 < ke2ki

2/4

€

rF ||O =

r B oBo

2∇ ⋅[κ e

2∇(

r J o ⋅

r B o

Bo2

) + Λe∇(eno

r u o ⋅

r B o

Bo2

)],

r F ||M =

r B oBo

2∇ ⋅[κ i

2∇(eno

r u o ⋅

r B o

Bo2

) + Λi∇(

r J o ⋅

r B o

Bo2

)],

e2 > 0, i

2 > 0, ( e + i)2 <

e2 i

2/4

The local forms are also derivable from quasi-linear tearing mode theory

• Simplified, incompressible, low-, me = 0, πs = 0 limit– Mean field forces are derived from quasilinear expression for the

tearing mode resonant on some surface. < > = average over layer width

• 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).

€

rF s ⋅

r B o= −∇⋅<

r ̃ v s(r B o ⋅

r ̃ A s) >,

r F O ⋅

r B o = −∇⋅< (

r B o ⋅

r ̃ A )(r ̃ u −

r ̃ J

ne) >

FM ⋅r B o = −∇⋅<

mi

e

r ̃ u (r B o ⋅

r ̃ u ) +1

ne

r ̃ J (r B o ⋅

r ̃ A ) >

A simple model is used in the linear layer analysis

• A four field model

• Equilibrium near the

rational surface

€

rB = ˆ b B + ˆ b ×∇Ψ,r u = ˆ b V + ˆ b ×∇Φ

€

Bo = Bo(ro),Ψo = Ψo ' 'x 2

2,

Jo = Jo(ro) + Jo '(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

– In the resistive MHD limit, Me2 is the largest coefficient - hyper-

resistivity 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.

€

rF O ⋅

r B o =∇ ⋅(Me

2 dJo

dx+ noeNe

dVo

dx),

r F M ⋅

r B o =∇ ⋅(noeM i

2 dVo

dx+ N i

dJo

dx),

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

A state discussed by many - Sudan, ‘79; Finn and Antonsen, ‘83; Avinash and Taylor ‘91; Steinhauer and Ishida, ‘98; Mahajan et al ‘01.

€

rJ o = λ1

r B o

noer u o = λ 2

r B o

Summary

• Using a modest number of assumptions, three global constraints are derived for turbulence induced mean field forces in two-fluid models of plasmas.

• 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?

€

dV∫r B o ⋅

r F e = 0 to O(η )

dVr B o ⋅

r F i = 0 to O(η )∫

dV (noer v io ⋅

r F i − ne

r v eo ⋅

r F e ) < 0∫