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

1
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

2
OUTLINE

• Introduction
• Non-diffusive (off-diagonal) term, internal
  (spontaneous) torque and spontaneous rotation
• 2 Pinch term and off-diagonal term in momentum
  transport
• Experimental results in LHD
• 3.1 radial electric field term
• 3.2 ion temperature gradient term
• 3.3 Causality between ?Ti and ?Vf
• 4 What is a driving mechanism of spontaneous
  rotation
• 5 Summary

3
Non-diffusive (off-diagonal ) term, internal
(spontaneous) torque and spontaneous rotation
Transport matrix
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
Pfr - M33 Vf - M31 n - M34 T - M32
Vq
Diagonal term (diffusive term)
Off-diagonal term (non-diffusive term)
Spontaneous rotation
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)
(1/r) ?r mini(-dVf/dt) Text dr mini- mD
dvf /dr off-diagonal term
(1/r) ?r mini(-dVf/dt) Text intrinsics
torque dr mini- mD dvf /dr
or
4
Diffusive and Non-diffusive terms in Momentum
Flux
Momentum flux is determined by the momentum input
and time derivative of Vf
GM (1/r) ?r mini(-dVf/dt) Text dr
Text external torque
Momentum flux has diffusive and non-diffusive
term
GM mini- mD dvf /dr VpinchVf mN (vth
/Ti)(eEr) mN (vth /Ti)(dTi/dr)
pinch
Er term
?Ti /?pi term
?Vf driven
K.Ida, PRL 86 (2001) 3040
K.Nagashima, NF 34 (1994) 449
K.Ida, PRL 74 (1995) 1990. M.Yoshida, PRL 100
(2008) 105002.
Diagonal term
off diagonal
diffusive (shear viscosity)
non-diffusive (driving terms)
It is not easy to distinguish Er driven ?Ti /?pi
driven, because they are coupled with each
others.
Is the pinch term really large enough to affect
the rotation profile?
5
Momentum pinch and off-diagonal term
Momentum pinch
second derivative becomes large at zero velocity
(not observed in experiment!)
m?Vf
VinwardVf
m?Vf
VinwardVf
momentum source at zero velocity is necessary
because of the conservation of momentum
Co-injection
Off diagonal term
m?Vf
mND?Ti
Artificial momentum source is NOT required at
zero velocity
Co-injection
m?Vf
Since the velocity shear affects the opn
transport, the causality between ?V and ?T is
important
mND?Ti
Ctr-injection
6
Toroidal effect on momentum transport
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
Vpinch 2mD(-e/R 1/Ln)
The pinch velocity can be evaluated as
er/R0
dI1 lt dI2
Inward
outward
V1 gtV2
The ratio of inward pinch term to diffusive term
is a order of 10-1 to 10-2
dI2
dI1
VpinchVf

(r/R) (LT/R0) ltlt 1
m dVf/dr
See O.D.Grucan PRL 100 135001 (2008) in details
7
Er Non-diffusive term
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 lt0 ) causes toroidal
rotation in co-direction (opposite to JT-60U)
Flux emimNni(vth /Ti)(Er)
Torque emimN (1/r) dr ni(vth /Ti)Er/dr
Er non-diffusive term is driven by the torque
with Er shear
8
Transition of spontaneous rotation
In TCV, a transition from ctr-rotation to
co-rotation is observed as 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??
9
Physics model of Er non-diffusive term
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.
emimN (1/r) dr ni(vth /Ti)Er/dr emi mN
ni(vth/Ti)(dEr/dr)
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).
See O.D.Gurcan Phys. Plasmas 14 (2007) 042306 in
details
Spontaneous velocity gradient
Internal toroidal torque
Spontaneous rotation
V 0 at the plasma edge
10
Torque scan experiment in LHD
1 co/ctr-NBI
2 balanced NBIs
2 balanced and 1 co/ctr-NBI
Near center (R lt 4.1m) ? NBI driven toroidal
rotation dominant Off center (R gt 4.1m) ?
spontaneous toroidal rotation dominant
The asymmetry of toroidal rotation is quite
significant at higher ion temperature. This
asymmetry is due to the Non-diffusive term in
momentum transport.
11
Spontaneous part of toroidal rotation velocity
Asymmetry part of the rotation (average of Vf
between co and ctr-NBI plasma) increases as the
?Ti is increased.
Near edge (R 4.6m ) ? spontaneous toroidal
rotation due to Er (gt 0). Core ? spontaneous
toroidal rotation due to ?Ti is dominant
12
?Ti Non-diffusive term
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-direction
• opposite 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.

13
?Ti and ?Vf causality
Since the toroidal rotation velocity shear affect
the ion transport, it is important to study the
causality between ?Ti and ?Vf at the transient
phase)
Early phase (t 2.09s) ? counter rotation is
driven direct effect of NBI Later phase (t
2.29s) ? co-rotation is driven secondary effect
of increase of ?Ti
Increase of velocity shear (in co-direction)
appears after the ?Ti is increased
14
What is physics mechanism of spontaneous rotation?
What we know
1 There is an non-diffusive term in momentum
transport 2 The non-diffusive terms are relating
to Er and ?Ti (or ?pi) 3 The direction of
spontaneous rotation observed is different
(even among tokamak experimets)
What we do not know
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?Ti,
?pi, etc
15
Summary

• Two Non-diffusive terms (off-diagonal term) of
  toroial momentum transport are observed
  separately in LHD one is Er terms and the other
  is ?Ti 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. ?Ti term is dominant at the half of plasma
  minor radius and causes a spontaneous rotation in
  the co-direction. (The causality is investigated
  (?Ti ? ?Vf)

16
(No Transcript)
17
Evidence of turbulence driven parallel Reynolds
stress
Cross correlation between parallel and radial
fluctuating velocities
Radial-parallel contribution to the production of
turbulent kinetic energy
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
18
Problem of concept of momentum pinch
Momentum pinch
Co-injection
second derivative (curvature) predicted
contradicts to that measured in experiment.
m?Vf
m?Vf
VinwardVf
VinwardVf
m?Vf
m?Vf
VinwardVf
VinwardVf
Ctr-injection
19
Velocity pinch due to turbulent equipartition
(TEP)
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.
Density profile
rotation profile
Skater makes a spin by reducing an angular
momentum inertia density, but he/she can not keep
the spin forever!
Particle pinch
velocity pinch
dI1 lt dI2
V1 gtV2
decay due to viscosity
sustained
dI2
dI1
See O.D.Grucan PRL 100 135001 (2008) in details
20
History of toroidal momentum transport studies
1980 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 plasmas CTR 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 Plasmas CTR
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
ltErxBqgt) PRL 86 (2001 ) 3040 CTR rotation
driven by ECH plasma in D-IIID PoP 11 (2004)
4323