Inter-Species Collisionless Energy transfer by drift wave-zonal flow Turbulence

L. Zhao and P. H. Diamond

1.CMTFO and CASS; UCSD, La Jolla, CA 920932. WCI, NFRI, Korea

1

1 1,2

We reconsider the classic problem of “ turbulent heating” and the inter-species transfer of energy in drift wave turbulence

Motivation: Transfer vs Transport Roles in energy budget Consider

Net volumetric heating Does turbulence heat a given volume of plasma?Electron ion collisonless energy transfer channels

Calculate and Estimate Energy Transfer Channels Electron cooling : quasilinear Ion heating : quasilinear, nonlinear, Ion Pol & Dia

Implication for ITER/Result and discussionTurbulent vs collisonal transferTurbulent transport vs Turbulent transfer

Special Topics ongoing:

Nonlinear electron cooling in CTEM

Origin of temperature profile "stiffness"

Conclusion

Outline

•

Background

2 3 4 11 1 2 0D T He n+ → +

ECRH electronelectron transport

ion

electron

ion

electron transport

3.5 Mev 14.1 Mev

• Two stages in energy flow

External energy

collisional collisionless

Heat flux

2 3 4 11 1 2 0D T He n+ → +

( ),3 ...2

e ee e e i e i

i

T mn Q E J n T Tt m

ν∂

+∇ ⋅ = ⋅ − − +∂

• ITER: collisionless, electron heated plasma

• What is ultimate fate of the energy? Collisionless energy transfer mechanisms?

Motivation• Transfer vs Transport

( )3 ...2

ee i

i

T mn Q E J n T Tt mα

α α ν∂

+∇ ⋅ = ⋅ − +∂

∓ heat balance;

Transport collsionlesstransfer Collisional transfer

Q heat flux, energy loss by turbulent transport

electron and ion collisionless energy transfer, local energy "soure" or " sink"

E Jα⋅

ITER: low collisionality, electron heated plasma

,e iα=

( )ee i

i

mn T Tm

ν −∓ : electron and ion heat transfer by collisions

Collisionless energy transfer likely dominant!!•

turbulent heating for single species: electron cooling,ion heating

,e iE J E Jα

α=

⋅ = ⋅∑

0E J⋅ =

• Issues with turbulent heating:

• Is the net heating zero?( Manheimer 77)

• If periodic boundary condition , no boundary contributions to , so

But 2

1

( )r

rr

dr E J J dr Jϕ ϕ⋅ = − + ∇⋅∫ ∫ 0≠

Surface term survives! Net heating

Boundary effect in a finite annular region

Classic Question:

E J⋅

• Another perspective: wave energy theorem

0W S E Jt

∂+∇⋅ + ⋅ =

∂

W≡ Wave energy density S≡wave energy density flux

• At steady state 2 2

11

r r

rrr

dr E J S⋅ = −∫

wave energy flux differential net heating

We need reconsider both the total turbulent heating and energy transfer chanels in an annular region!

( )2

*, 22 2

21s r

r gr r

s

k k VS Vkθ ω ω

ωρ ε ε

ερ⊥

= = −+

2r rV V k kθ θ ϕ= ∑−

Point : 0E J⋅ ≠

Related to Reynolds work

• Collisionless, inter-species energy transfer– Where does the net energy transfer go?– How is energy transferred from electrons to ions

(turbulent transfer channelsturbulent transfer channelsturbulent transfer channelsturbulent transfer channels)? – How relate to saturation mechanisms?– Role of ZF in heating?

– ZF is important to saturation, so must enter energy ZF is important to saturation, so must enter energy ZF is important to saturation, so must enter energy ZF is important to saturation, so must enter energy transfer as well!?transfer as well!?transfer as well!?transfer as well!? Zonal flow damping by friction is an energy sink Zonal flow damping by friction is an energy sink Zonal flow damping by friction is an energy sink Zonal flow damping by friction is an energy sink Nonlinear wave-particle interaction ( considered in Nonlinear wave-particle interaction ( considered in Nonlinear wave-particle interaction ( considered in Nonlinear wave-particle interaction ( considered in

future) is another possibilityfuture) is another possibilityfuture) is another possibilityfuture) is another possibility

Turbulent Energy flow Channels

8

ST & ZF>0ST & ZF>0

,n T∇ ∇ E J⋅

( )20iE J >

( )40iE J >

( )2ipolE J⊥ ⊥⋅

Ion LandauDamping

Beat ModeResonance

Collisional Zonal Flow Damping

( )2i

diaE J⊥ ⊥⋅

STb: Bounce averageST: Surface Term

( )20eE J <

Primary mode resonance

Nonliear Landau damping

Necessary Correspondence: Nonlinear Saturation and Energy Transfer Nonlinear saturation in turbulent state implies energy transfer

from source to sink Schematically, saturation implies some balance condition

must be satisfied i.e.

Channels for electron ion energy transfer mustmustmustmust be consistent with saturation balance

In particularIn particularIn particularIn particular:• If zonal flows control saturation, they mustmustmustmust contribute to energy

transfer and dissipation.• As zonal flows are nonlinearly generated (Reynolds stress), we

should consider other nonlinear heating channels.9

( ),eT n∇ ∇

0 ...Linear Linear Zonal NLLDelectron ion Flow

γ γ γ γ γ= = + + + +

>0 <0<0 <0

Quasilinear Turbulent Heating in Electron Drift Wave

Calculate in quasilinear theory DKE for electron Take non-adiabatic electron distribution function •

the electrons cool via inverse electron the electrons cool via inverse electron the electrons cool via inverse electron the electrons cool via inverse electron Landau damping Landau damping Landau damping Landau damping

Similarly, calculate for ion•

the ions gain energy via ion Landau damping the ions gain energy via ion Landau damping the ions gain energy via ion Landau damping the ions gain energy via ion Landau damping 10

( )2

eE J

*( ) ,kk e

z z e

eg fk v T

ϕω ωω

−=

− * ,y s s

en

k cLρ

ω =ef

( )2

e z z kE J e dvv E g= ∫ ( )

2

*z

ke e e

kk e z the

enT fT k V

ωϕ ω

π ω ω= −∑

*2 2 ,

1e

skω

ωρ⊥

=+

( )20eE J <

( )2

iEJ

( )2

2

*z

k ii i e i

kk i z the e

e TE J nT fT k V T

ωϕ ωπ ω ω

⎛ ⎞= +⎜ ⎟

⎝ ⎠∑

( )20,iE J >

(Prototype; CTEM- specific later)

is Maxwellian

Perpendicular Current Induced Turbulent Heating• The turbulent heating induced by ion polarization current

– –

( )pol pol poli i iE J J Jϕ ϕ⊥ ⊥ ⊥ ⊥ ⊥ ⊥⋅ = − ∇ ⋅ + ∇ ⋅

( )0

0

2 2

0 0

... ...rR

r

dz rd drπ π

θ+∆

−∆

= ∫ ∫ ∫

rpol ri i r r rr

E J nm A V V V dr V V Vrθ θ θ θ

+∆+∆⊥ ⊥ −∆ −∆

∂⎛ ⎞⋅ = −⎜ ⎟∂⎝ ⎠∫

• Directly linked to zonal flow drive

• At steady state 2 0,

rpol

i colr

E J dr Vθυ+∆

⊥ ⊥−∆

⋅ = >∫ Zonal flow frictional damping is the final fate of net electron-ion energy transfer

• Diamagnetic current induced turbulent heating

2Dia r

i rB pE J nc

Bϕ +∆

⊥ ⊥ −∆

×∇⋅ =−

Heat flux differential

Defining a annular region

• wave energy flux

•Perpendicular heating Reynolds work on zonal flow

• other damping possible

Turbulent Turbulent Turbulent Turbulent heating heating heating heating analytical analytical analytical analytical

Using mixing length Using mixing length Using mixing length Using mixing length approximationapproximationapproximationapproximationfor fluctuation levelsfor fluctuation levelsfor fluctuation levelsfor fluctuation levels

( )2

eE J⋅

( )2

iE J⋅

( )2

polE J⋅

( )2*e

ee z the

enTT k V

ω ω ωϕπ

−∑

( )

22 ** 1 s

the

Rq F kVω

ρ ρ⊥

( )2 2* expi

ii z thi z thi

enTT k V kV

ω ω ωϕ ωπ

− ⎛ ⎞−⎜ ⎟⎝ ⎠

∑

( )2

2 ** 2 s

thi

Rq F kVω

ρ ρ⊥

2i colnm Vθν 2 3/ 2 2

* *thi

i sVm CRq

ρ ν ε

*e

eTϕ

ρ∼

Overview of ResultsEstimation of the turbulent heating contributions:

Short wavelength

ITER Parameters R=6.2m, ITER Parameters R=6.2m, ITER Parameters R=6.2m, ITER Parameters R=6.2m, a=2m,q=2a=2m,q=2a=2m,q=2a=2m,q=2

1sk ρ⊥ ∼

21.56 10−×

*0.8ν

( )2

iE J⋅

( )2

polE J⋅

Basic comparision of channels

Ion Landau Damping Zonal flow friction

2 * 3* 10 , 10ν ρ− −= =

i

e

E JRatio

E J

⋅=

⋅

3 * 3* 10 , 10ν ρ− −= =

Ratios of energy dissipation channels at differernt collisionality

Zonal flow frictional damping can be a significant dissipation channel"Collisionless drift wave"

* 0ω ν >≫

Turbulent Heating NL wave-particle Interaction • The ion diffusion equation in electron drift wave turbulence:

• ZF coupling to , so need calculate parallel heating to 4e

Tϕ

ο⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

4eTϕ

ο⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

ivv i vx i xv i xx i

fD f D f D f D f

t v v v x x v x x∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(2) (4) ...vv vv vvD D D= + + (2) (4) ...vx xv vx vxD D D D= = + + (2) (4) ...xx xx xxD D D= + +Diffusion coefficient in quasilinear and nonlinear order

The resonant ion kinetic energy evolution : 21

2res

iE dv mv ft t

∂ ∂=

∂ ∂∫

( ) ( ) ( ) ( ) ( ) ( )2 4 2 4 2 4( ...) ( ...) ( ...)vv vv vx vx

E J E J E J E J q q= ⋅ + ⋅ + + ⋅ + ⋅ + + ∇⋅ +∇⋅ +

• The nonlinear ion turbulent heating up to 4e

Tϕ

ο⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

( ) ( ) ( )4 4 4

i vv vxE J E J E J⋅ = ⋅ + ⋅

21 ( )2vv i vx i xv i xx idvmvD f dvmvD f dv mv D f D f

v x x v x∂ ∂ ∂ ∂ ∂

=− − + +∂ ∂ ∂ ∂ ∂∫ ∫ ∫

q is energy flux 4

eTϕ

ο⎛ ⎞⎜ ⎟⎝ ⎠

• The forth order nonlinear diffusion coefficient

( ) ( ) ( )2 24

2vv

F t F tD d

ττ

∞

−∞

+= ∫

perturbation theory (Dupree 68)

• The particle orbit is expanded as : ( ) 20 0 1 2( ) ( ) ...,x t x v t x t x tε ε= + + + +

• The force in perturbative electric field

( ) ( ) ( ) ( )4 4 4 4zx zx zy zy zz zzvv v v v v v vD D D D= + +

'' '

'' '

'' '

22 ' ' ' '22' ' '' ''

, , ,

22 ' ' ' '22' ' '' ''

, , ,

42 2

, , ,

Re

Re

x z x zk k

k k k k z z z z z

x z x zk k

k k k k z z z z z

zz z

k k k k

k k k k k kec imB k v k v k v

k k k k k kec imB k v k v k v

ke E Em

θ θ

θ θ

ϕ ϕω ω ω

ϕ ϕω ω ω

− −

− −

− −

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟ − − −⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟ − − −⎝ ⎠ ⎝ ⎠

−⎛ ⎞+ ⎜ ⎟⎝ ⎠

∑

∑

∑

2'

' ' '' ''Re( )( )

z

z z z z z

k ik v k v k vω ω ω

⎛ ⎞⎜ ⎟− − −⎝ ⎠

( ) ( ) ( )4 4 4 2*1/

zx zx zy zy zz zzv v v v v vD D D ρ≈ ≈≫

Nonlinear diffusion correction by E×B scattering is bigger than the one by parallel acceration

Nonlinear diffuion in velocity space corrected by E×B drift motion

Nonlinear diffuion in Vlasov plasma, corrected by scattering (Dupree 68)

v

v

• Similarly, the cross term of nonlinear diffusion coefficient

( ) ( ) ( )2 24

2vx

F t V tD d

ττ

∞

−∞

+= ∫

( ) ( ) ( )4 4 4 2*1/

zz xx zz xy zz xzv x v x v xD D D ρ≈ ≈≫and

• The nonlinear turbulent heating for ions ( ) ( ) ( )

'

' '

4 4 4

2 2 23 ' '''

2 ' ''' '', , ,

i vv vx

thi k k x xzi

k k k k ci i i z z z zz z

E J E J E J

eV e k k k kknTT T k v k vk k

θ θϕϕ ω

πω ω ω− −

⋅ = ⋅ + ⋅

⎛ ⎞≈ +⎜ ⎟− −⎝ ⎠∑

''

''

''

*'' 0iz e i

z e k

Tk fk T ω

ωω

⎛ ⎞+ >⎜ ⎟

⎝ ⎠

• Estimate using the mixing length approximation

Beat mode resonance : '' ' '' ', k k kω ω ω= ± = ±

due to Ion Compton Scattering

Turbulent energy transfer : Electron wave IonNLLD

( )4

iE J⋅

( ) ( )4 2

i iE J E J⋅ > ⋅ Mixing length approximation overestimates overestimates overestimates overestimates the turbulence

intensity in the weak turbulence theory. (T.S. Hahm, 1991)

2 0L NL ZFN N Nγ γ γ+ + = N has to be self-consistentself-consistentself-consistentself-consistent (Diamond, 2005)

electronsenergy transport loss

ions

eQ

heat transport,T n∇ ∇

collisional collisionless

( )32

e ee e e i

i

T mn Q E J n T Tt m

ν∂

+ ∇ ⋅ = ⋅ − −∂

Comparison of the collisonal and collisionless energy transfer.

Comparison of energy transfer in collisonlesstransfertransfertransfertransfer and energy transport transport transport transport by heat flux Q

Implication Bottom Line • Electon turbulent energy transport

Can be of the same order!

Electron heat balance

Collisionless transfer can dominate at low collisionaltiy

heat transfer

Collisionality • Collisonality in ITER

– dimensionless

• Collisionality at crossover of collisional and collisionless coupling– Energy transfer in collision :

– Quasilinear trapped electron cooling in CTEM

– At crossover At crossover At crossover At crossover : The collisonless turbulent energy transfer then beats

collisional inter-species coupling process!

*ν3

* 10ν −∼

e ei e

i

nmQ Tmν

≈

( )3/2(2)

1/2 1/2 2* *4

2 e

s s

e e e RTEb k C

RE J nT fa

θ

ωρ

π ε ρ ω ω=

⎛ ⎞⋅ −⎜ ⎟⎝ ⎠

≃ ( )2

ib

Q E J≈ ⋅ 2

* 10ν −∼

3/2

*e

the

RqV

ε νν

−

=

Collisionless process will control electron-ion transfer in ITER

Transfer vs Transport• The and energy loss in CTEM

– Compare the volume integral of the electron cooling to the surface integrated of the electron heat flux

– The heat flux for electrons :

– The pressure fluctuation

3e boundaryAQ d r E J= ⋅∫

( )1Ime r e ek

cQ v P k PB θ ϕ= =− ∑

( )*

5 21 1 222 2

*4de e

n

the ee n e RTE

n e e L

V k nTR eQ fL T

θωω

ϕπ ε ω ω

=

⎛ ⎞= −⎜ ⎟ Ω⎝ ⎠∑

( )2 1e boundary

r E J aRQ

ο∆ ⋅

≈

∼

The ratio

transfer transport

( )1 3 212e bp d v mv g= ∫

The rate of electron energy lost by collisonless energy transfer is comparable to turbulent transport by CTEM !

Result and Discussion• Net heating

• Quasilinear turbulent energy transfer in drift wave• Nonlinear ion heating by beat wave resonance

•Energy transfer channels• Identify important energy transfer channels• Zonal flow frictional damping can be comparable to LD

• For low collisionality ITER plasma, collisionless energy transfer is a critical element of transport model.

• it is same order as transport

In the future, an important task for ITER transport modelling

• Energy flux differential gives rise to the net heating Zonal flow

• Develope a tractable representation of the turbulent heating and collisionless energy transfer

Realistic application: details of turbulent energy transfer in CTEM

•

Toroidal procession of trapped particles

Trapped electron precession frequency:s s

de

k c ER T

θ ρω ≈

wave-trapped particle resonance: dω ω=

Some details for CTEM

• Bounce everaged kinetic equation for CTEM*

,k ee d

eg fT iω

ω ωω ω ν

−= −

− + ,

inqk b

e θωϕ−

• turbulent heating for quasilinear trapped electron(2)

3,e d k

bE J e d vV Eg ω⋅ = ∫ ( )

3/21/2 1/2 2

* *42 e

s s

e e RTEk C

RnT fa

θ

ωρ

π ε ρ ω ω=

⎛ ⎞= −⎜ ⎟⎝ ⎠

∑

(2)*2 2 0

1 ebs

E Jkω

ωρ⊥

= ⇒ ⋅ <+

Trapped electron cooling via trapped Trapped electron cooling via trapped Trapped electron cooling via trapped Trapped electron cooling via trapped electron precession resonanceelectron precession resonanceelectron precession resonanceelectron precession resonance

Nonlinear trapped electron coolingNonlinear trapped electron coolingNonlinear trapped electron coolingNonlinear trapped electron coolingThe beat mode resonance effect for trapped electrons

The 3rd order nonlinear trapped electron response function

the nonlinear trapped electron-wave interaction

The nonlinear turbulent heating for trapped electrons

22

2 2(3)

2ˆe e inq e inqn n rr nb b

nqh d e d er r

θ θθθ ϕ ϕ− −∂⎛ ⎞= − +⎜ ⎟ ∂⎝ ⎠

'

''

2 2 '

'' '' '

1 1e in q e een bn de e

c ed e fB r T

θθθ

ω ωϕ

ω ω ω ω ω

∗ ∗− ⎛ ⎞∂⎛ ⎞= −⎜ ⎟⎜ ⎟ − ∂⎝ ⎠ ⎝ ⎠

∑

'

''

22 '' 2

'' '' '

1 1e in q e err en bn de e

c n q ed e fB r T

θ ω ωϕ

ω ω ω ω ω

∗ ∗− ⎛ ⎞⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎜ ⎟ −⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑

'' ''1/ deω ω−

( )43 (3)ˆ e

e d d nb

E J e d vV E h⋅ = ∫

( ) ( )' '

''' ''

31 22 4 4 '' 22 2, ,2 2 '2 2 '2 '2 20 ''2 ''

,

02 e

s s

n n the s ee r r e RTEn n e e s se k c

ee V RTn T k k k k k k fT T k ck

ω ωθ θ θ θ ω

θ ρ

ϕϕ ρ ωεπ

ρ =

⎛ ⎞⎛ ⎞→ − − + <⎜ ⎟⎜ ⎟ Ω⎝ ⎠ ⎝ ⎠∑

In CTEM, trapped electron can have quasilinear and nonlinear In CTEM, trapped electron can have quasilinear and nonlinear In CTEM, trapped electron can have quasilinear and nonlinear In CTEM, trapped electron can have quasilinear and nonlinear cooling. Nonlinear cooling can be significant.cooling. Nonlinear cooling can be significant.cooling. Nonlinear cooling can be significant.cooling. Nonlinear cooling can be significant.

(Gang'90)(L.Chen'77)

Related experimental phenomenon • Electron temperature profile "stiffness"

Experimental electron temperature profile measured in DIII-D tokamak.

C.C . Petty 94'

• temperature profile react weakly to changes of auxiliary heating deposition

Heat pinch : inward flow Nondiffusive term in heat flux Electron- ion energy transfer in

the core: " Sink"

• They are two different and independent effects . which one is more efficient? Both must be examined..

e e e e e e eQ n T V n Tχ= − ∇ +

ee e

T Q E Jt

∂+ ∇ ⋅ = ⋅

∂

Possibledynamicalcause

heat pinch (Weiland ' 89), ( L. Wang '2011)

Conclusion• ITER plasma is low collisionality, electron heated plasma • Turbulent heating and Turbulent energy transfer channels

– Net heating occurs!– Zonal flow (& frictional damping) can be significant energy

transfer channel (and sink) for CTEM turbulence• Collisonless energy transfer likely dominant in energy

coupling and a critial element in transport analysis– Does this process excite ITG?

• Future work extend to :– Realistic model: CTEM– Representation for energy transfer in ITER modelling– Future experiment :electron temperature profile "stiffness"–

4eTϕ

ο⎛ ⎞⎜ ⎟⎝ ⎠

contributions to Residual stress, in momentum flux