1. Spontaneous rotation in stellarator and tokamak
K.Ida
National Institute for Fusion Science
2nd Coordinated Working Group Meeting (CWGM)
IPP Greifswald June 4-6, 2007
1 Spontaneous rotation in stellarator and tokamak
Non-diffusive term of momentum transport driving the spontaneous rotation
Parameter dependence of the non-diffusive term
Physics mechanism of spontaneous rotation and comparison with theory.
5 Summary

2. Spontaneous rotation in tokamak and stellarator plasmas
JFT-2M tokamak (Er<0)
Wendelstain 7AS (Er < 0)
spontaneous rotation in co-direction
spontaneous rotation in ctr-direction
For NBI plasma where the radial electric field is negative （Er<0)
Tokamak plasmas ： vf （co）＜ vf （ctr） [PRL74 (1995) 1990]
Helical plasmas ： vf（co）＞ vf （ctr） [EPS 18B I (1994)　p392]
The direction of spontaneous flow is opposite between tokamak and helical plasmas

3. Experimental study of non-diffusive momentum transport
Co-NBI
Parallel
to Ip
Ctr-NBI
Anti-parallel
Plasma current : Ip
JFT-2M
Non-diffusive momentum transport is studied by switching the NBI direction during the discharge in JFT-2M tokamak [PRL74 (1995) 1990]

4. Dynamic momentum transport analysis
Momentum transport analysis shows the non-diffusive term
(finite momentum flux at zero gradient)

5. Spontaneous rotation in the momentum transport
Momentum Flux
GM = mini[- cf dvf /dr + VinwardVf ]
GM = mini[- mD dvf /dr + mN (vth /Ti)(eEr) ]
eEr = dTi/dr
diffusive
(shear viscosity)
non-diffusive
(driving term)
K.Nagashima, Nucl Fusion
34 (1994) 449]
L=mode Plasma in JFT-2M
K.Ida
Phys Rev Lett 74 (1995 ) 1990
J.Phys.Soc.Jpn 67 (1998) 4089
Finite dVf/dr can be non-zero for GM = 0 because of the existence of non-diffusive term
Spontaneous rotation was observed as a non-diffusive term of momentum transport.

6. Magnitude of non-diffusive viscosity (q-dependence)
Spontaneous flow
B (small q) q
Ctr-NBI
Co-NBI f q
Spontaneous flow
B (large q)
Ctr-NBI
Co-NBI f
The non-diffusive term is proportinal to q value in JFT-2M [J. Phys. Soc. Jpn 67 (1998) 4089]
Similar q dependence is observed in Alcator C-mod
Dvf ~ (Wp/Ip) [J.E.Rice NF 41 (2001) 277]

7. Momentum transport in helical system
GM = mini[- m^D dvf /dr + mN (vth /Ti)(eEr) - m|| vf ]
Anomalous perpendicular viscosity
Non-diffusive term
Neoclassical parallel viscosity
Small ripple configuration
 Vmes << Vcal(NC)
Large ripple configuration
 Vmes ~ Vcal(NC)
In the configuration of small ripple the anomalous perpendicular viscosity is dominant in the plasma core ( r < 0.6) even in helical plasmas

8. Spontaneous toroidal rotation in CHS
Mod-B
Er > 0
ExB
force
<B>
Er < 0
Co-NBI + ECH
co-NBI B
The spontaneous toridal flow driven by ECH is proportional to the poloidal flow (radial electric field)

9. Sign of non-diffusive viscosity
● Helical (external current system) vq
helical effect
poloidal
Er <0 B
Er >0 vf
Helical symmetry
toroidal
●Tokamaks (internal current system) vq
toroidal effect
Er < 0
poloidal B
Toroidal symmetry
PPCF 44 (2002) 361
Er >0 vf
toroidal
Tokamak : negative Er  counter spontaneous flow V=1.3Er/Bq
Helical : positive Er  counter spontaneous flow V=0.16Er/Bq

10. Why the spontaneous rotation depends on Er in tokamak
Why the non-diffusivr terms has Er dependence?
GM = mini[- mD dvf /dr + mN (vth /Ti)(eEr) ] = - (1/r)∫ff r dr
Non-diffusive term can be expressed as toroidal force which proportional to Er shear
mN = csym(1/Bq )= csymqR/(rBf)
ffspon = (csymeqR/Bf)(vth/Ti)(1/r)(dEr/dr)
~ (csymeqRvth/Ti)(dwExB/dr)
In tokamak
Toroidal momentum is produced by symmetry breaking of non-zero <k||> ref PoP 14 (2007)
In stellarator
Because of the asymmetry of magnetic field, Er and spontaneous flow are produced by ripple loss too.

11. Summary
1 The spontaneous rotation as a result of off-diagonal term of transport has been observed in tokamak and stellarators.
2 The non-diffusive term has q(=1/i) dependence and Er(=dPi/dr) dependence as
csymqR/(rBf) (vth /Ti)(eEr)
This is equivalent to the
The toroidal force driven by the ExB velocity shear as
ffspon ~ (csymeqRvth/Ti)(dwExB/dr)
3 The sign of spontaneous flow is determined by Er and symmetry of B.
Tokamak (and core in helical?) :
toroidal symmetry (gB>gsym =0)  csym>0 ctr-rotation for Er < 0
Driving : Symmetry breaking + Er driven by turbulence  Reynolds stress
Drag : anomalous perpendicular viscosity
Helical: (additional mechanisms becomes important)
helical symmetry (gB< gsym)  csym< 0 co-rotation for Er < 0 near the plasma edge
Driving; Asymmetry of magnetic field + Er driven helical ripple  thermodynamic force
Drag : neoclassical parallel (toroidal) viscosity