Non-diffusive terms of momentum transport as a driving force for spontaneous rotation in toroidal plasmas

K.Ida, M.Yoshinuma, LHD experimental group

National Institute for Fusion Science

1 April 2009

Transport & Confinement ITPA Meeting

JAEA, Naka Japan

1 Introduction Non-diffusive (off-diagonal) term, internal (spontaneous) torque and

spontaneous rotation

2 Pinch term and off-diagonal term in momentum transport

3 Experimental results in LHD 3.1 radial electric field term 3.2 ion temperature gradient term 3.3 Causality between T∇ i and V∇

4 What is a driving mechanism of spontaneous rotation

5 Summary

OUTLINE

Non-diffusive (off-diagonal ) term, internal (spontaneous) torque and spontaneous rotation

Toroidal momentum transport has a diagonal and an off-diagonal term

K.Itoh, S-I Itoh and A.Fukuyama “Transport and structural formation in plasmas” IOP publishing 1999

= - M

nV

V

T

r

Pr

Pr

qr

Transport matrix

Pr = - M33 V - M31 n - M34 T - M32 V

Diagonal term(diffusive term)

Off-diagonal term(non-diffusive term)

V = 0 even for Pr = 0

Spontaneous rotation

(1/r) ∫r[ mini(-dV/dt) + Text] dr = mini[- D dv/dr + off-diagonal term]

(1/r) ∫r[ mini(-dV/dt) + Text+ intrinsics torque] dr = mini[- D dv/dr ]or

off-diagonal term is equivalent to intrinsic torque (Residual stress, Reynolds stress etc. O.D.Gurcan PoP 14 (2007) 042306, B.Concalves, PRL 96 (2006) 145001)

Diffusive and Non-diffusive terms in Momentum Flux

diffusive (shear viscosity) non-diffusive (driving terms)

Momentum flux is determined by the momentum input and time derivative of V

= mini[- D dv/dr + VpinchV+ N (vth/Ti)(eEr)+ N (vth/Ti)(dTi/dr)]pinch Er term ∇Ti / p∇ i term∇V driven

off diagonalDiagonal term

Is the pinch term really large enough to affect the rotation profile?

=(1/r) ∫r[ mini(-dV/dt) + Text] dr Text : external torque

Momentum flux has diffusive and non-diffusive term

It is not easy to distinguish Er driven T∇ i / p∇ i driven, because they are coupled with each others.

K.Ida, PRL 74 (1995) 1990.

M.Yoshida, PRL 100 (2008) 105002.

K.Nagashima, NF 34 (1994)

449

K.Ida, PRL 86 (2001) 3040

Momentum pinch and off-diagonal termMomentum pinch

VinwardV momentum source at zero velocity is necessary because of the conservation of momentum

second derivative becomes large at zero velocity (not observed in experiment!)

ND∇Ti∇V

Off diagonal term

ND∇Ti

∇V

Artificial momentum source is NOT required at zero velocity

Since the velocity shear affects the opn transport, the causality between V ∇and T is important∇

VinwardV∇V

∇V

Co-injection

Co-injection

Ctr-injection

-100

-50

0

50

3.6 3.7 3.8( )R m

See O.D.Grucan PRL 100 135001 (2008) in details

Because of the toroidal effect moment of inertia density, the conservation of the toroidal angular momentum causes an “apparent” momentum pinch in the linear momentum in the toroidal direction

Toroidal effect on momentum transport

I1 < I2

I2 I1

V1 >V2

Vpinch = 2D(-/R + 1/Ln) The pinch velocity can be evaluated as

Inward outward

VpinchV

dV/dr ~ (r/R) (LT/R0) << 1

The ratio of inward pinch term to diffusive term is a order of 10-1 to 10-2

=r/R0

Er Non-diffusive term

Flux : emiNni(vth/Ti)(Er) Torque : emiN (1/r) d[r ni(vth/Ti)Er]/dr

Er non-diffusive term is driven by the torque with Er shear

In LHD radial electric field can be controlled by changing the electron density slightly by taking advantage of the ion-root electro root transition

As the electron density is increased the Er change its sign from positive to negative and the tnegative Er (or dEr/dr <0 ) causes toroidal rotation in co-direction (opposite to JT-60U)

-10

-5

0

5

10

4.2 4.3 4.4 4.5R (m)

ne=0.4x10

19m

-3

1x1019

m-3

(a)

0.45x1019

m-3

-15

-10

-5

0

5

4.2 4.3 4.4 4.5( )R m

ne=0.4 10x

19m

-3

1 10x19

m-3

h/

t=4.17( )b

0.45 10x19

m-3

-10

-5

0

5

0 2 4 6 8 10E

r( / )kV m

=4.4R m

Electron Root

weak positive Er

( )c

Transition of spontaneous rotation

In TCV, a transition from ctr-rotation to co-rotation is observedas the electron density is increased. (ref : A.Bortolon, PRL 97 (2006) 235003)The sign of spontaneous rotation is same as that in LHD.But the same physics??

Physics model of Er non-diffusive term

See O.D.Gurcan Phys. Plasmas 14 (2007) 042306 in details

emiN (1/r) d[r ni(vth/Ti)Er]/dr ~ emi N ni(vth/Ti)(dEr/dr)

The Er non-diffusive term is nearly equivalent to the spontaneous torque due to Er shear if the derivative radial electric field much rather than that of non-diffusivity coefficient and temperature.

The symmetry breaking of turbulence and existence of radial electric field shear can produce the internal toroidal torque and results in the spontaneous velocity gradient).

Internal toroidal torque

V = 0 at the plasma edge

Spontaneous rotation

Spontaneous velocity gradient

Torque scan experiment in LHD

Near center (R < 4.1m) NBI driven toroidal rotation dominantOff center (R > 4.1m) spontaneous toroidal rotation dominant

3.8 4.0 4.2 4.4 4.6R(m)

43210

100

50

0

-50

balanced injection

balanced injection

( )b

3.8 4.0 4.2 4.4 4.6( )R m

- co injection

- ctr injection

- co injection

- ctr injection

100

50

0

-5043210

( )c

1 co/ctr-NBI 2 balanced NBIs 2 balanced and 1 co/ctr-NBI

3.8 4.0 4.2 4.4 4.6( )R m

43210

100

50

0

-50

- co injection

- ctr injection

- co injection

- ctr injection

( )a

The asymmetry of toroidal rotation is quite significant at higher ion temperature. This asymmetry is due to the Non-diffusive term in momentum transport.

Spontaneous part of toroidal rotation velocityAsymmetry part of the rotation (average of V between co and ctr-NBI plasma) increases as the T∇ i is increased.

Near edge (R ~ 4.6m ) spontaneous toroidal rotation due to Er (> 0).Core spontaneous toroidal rotation due to T∇ i is dominant

-30

-20

-10

0

10

20

3.8 4.0 4.2 4.4 4.6-10

-5

0

5

10

15

20

R(m)

-30

-20

-10

0

10

20

-10

-5

0

5

10

15

20

3.8 4.0 4.2 4.4 4.6

R(m)

∇Ti Non-diffusive term

-4

-2

0

2

4

6

8

10

0.0 0.5 1.0 1.5 2.0 2.5 3.0-grad Ti (keV/m)

R=4.35m

ΔVT

Δ grad Ti~4.3 /km s

/keV m

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

4.0 4.1 4.2 4.3 4.4 4.5 4.6R (m)

NBI1+NBI2+NBI3+ECH

NBI1+NBI2+NBI3

NBI2+NBI3

NBI3

(a)

increasing of grad Ti

-10

-5

0

5

10

15

20

4.0 4.1 4.2 4.3 4.4 4.5 4.6R (m)

NBI1+NBI2+NBI3+ECHNBI1+NBI2+NBI3

NBI2+NBI3

NBI3

(b)

There is a clear relation between the ion temperature gradient and change in toroidal rotation in the power scan experiment in LHD.

Ion temperature gradient causes spontaneous toroidal rotation in co-directionopposite to that observed in JT-60U [Y.Koide, et. al., PRL 72 (1994) 3662, Y.Sakamoto, NF 41 (2001) 865] same as that observed in JET [G.Eriksson PPCF 34 (1992) 863] and Alcator C-mod [J.Rice et al., NF 38 (1998) 75].

∇Ti and V∇ causality

-150

-100

-50

0

50

100

3.6 3.8 4.0 4.2 4.4 4.6( )R m

2.29s

2.09s

beam momentum

gradTiterm

-35

-30

-25

-20

-15

-10

-5

0

3.6 3.8 4.0 4.2 4.4 4.6( )R m

2.04s

2.14s2.24s

2.34s

-50

-40

-30

-20

-10

0

10

2.5 3.0 3.5 4.0 4.5 5.0 5.5dT

i/ ( / )dr keV m

=4.261R m

Early phase (t = 2.09s) counter rotation is driven : direct effect of NBILater phase (t = 2.29s) co-rotation is driven : secondary effect of increase of Ti ∇

Since the toroidal rotation velocity shear affect the ion transport, it is important to study the causality between T∇ i and V∇ at the transient phase)

Increase of velocity shear (in co-direction) appears after the T∇ i is increased

What is physics mechanism of spontaneous rotation?

What we know

What we do not know

1 There is an non-diffusive term in momentum transport2 The non-diffusive terms are relating to Er and T∇ i (or p∇ i)3 The direction of spontaneous rotation observed is different (even among tokamak experimets)

1 How the direction of spontaneous rotation is determined?2 How the magnitude of the non-diffusive term (or magnitude of spontaneous torque) is determined? 3 Does the multi non-diffusive terms suggests multi physics mechanism in the plasma or just expansion of complicated term, which include to Er and T∇ i, p∇ i, etc……

Summary

1. Two Non-diffusive terms (off-diagonal term) of toroial momentum transport are observed separately in LHD : one is Er terms and the other is T∇ i term. (Their coupling is too strong in tokamk)

2. Er term is dominant near the plasma edge and positeive Er causes a spontaneous rotation in the counter-direction.

3. T∇ i term is dominant at the half of plasma minor radius and causes a spontaneous rotation in the co-direction. (The causality is investigated ( T∇ i V∇ )

Evidence of turbulence driven parallel Reynolds stress

In TJ-II stellarator, significant radial-parallel component of the Reynolds stress, which drives spontaneous parallel flow is observed

See B.Concalves, Phys. Rev. Lett. 96 (2006) 145001 in details

Cross correlation between parallel and radial fluctuating velocities

Radial-parallel contribution to the production of turbulent kinetic energy

Problem of concept of momentum pinchMomentum pinch

VinwardV

second derivative (curvature) predicted contradicts to that measured in experiment.VinwardV

∇V ∇V

Co-injection

VinwardV∇V

Ctr-injection

VinwardV

∇V

-100

-50

0

50

3.6 3.7 3.8( )R m

See O.D.Grucan PRL 100 135001 (2008) in details

Velocity pinch is possible under the condition of conservation of angular momentum during the transition phase when density profile changes from flat to peaked ones bur not in the steady-state.

Velocity pinch due to turbulent equipartition (TEP)

Density profile rotation profile

Particle pinch velocity pinch

sustained decay due to viscosity

Skater makes a spin by reducing an angular momentum inertia density, but he/she can not keep the spin forever! I1 < I2

I2 I1

V1 >V2

History of toroidal momentum transport studies1980’ Toroidal rotation of Ohmic plasma

CTR rotation of Ohmic plasma in PLT [NF 21 (1981) 1301] PDX [NF 23 (1983) 1643] and Alcator C-mod [NF 37 (1997) 421]

Early 90’ Toroidal rotation of ICRF plasmasCTR rotation in JIPP-TIIU [NF 31 (1991) 943]Co rotation in JET [PPCF 34 (1992) 863] in Alcator C-mod [NF 38 (1998) 75]

Mid 90’ Non-Diffusive term of momentum transport in NBI heated PlasmasCTR rotation in JT-60U [NF 34 (1994) 449] in JFT-2M [PRL 74 (1995 ) 1990]CTR Spontaneous toroidal flow in helical plasma in LHD [2005]

Spontaneous toroidal flow in the plasma with ITB CTR rotation in JT-60U [PRL 72 (1994) 3662, PoP 3 (1996) 1943, NF 41 (2001) 865] CTR rotation in TFTR [PoP 5 (1998) 665] CTR rotation in Alcator C-mod [ NF 41 (2001) 277]

Early 2000’ Spontaneous toroidal flow driven by ECH

CTR rotation in CHS (anti-parallel to <ErxB>) [PRL 86 (2001 ) 3040]CTR rotation driven by ECH plasma in D-IIID [PoP 11 (2004) 4323]