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. 1Introduction Non-diffusive (off-diagonal) term, internal (spontaneous) torque and spontaneous rotation 2 Pinch term and off-diagonal term in momentum transport 3Experimental 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

3. 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TnVVT rPrPrqrrPrPrqr Transport matrix P  r = - M 33 V  - M 31 n - M 34 T  - M 32 V  Diagonal term (diffusive term) Off-diagonal term (non-diffusive term) V  = 0 even for P  r = 0 Spontaneous rotation (1/r) ∫r[ m i n i (-dV  /dt) + T ext ] dr = m i n i [-  D dv  /dr + off-diagonal term] (1/r) ∫r[ m i n i (-dV  /dt) + T ext + intrinsics torque] dr = m i n i [-  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 )

4. 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    = m i n i [-  D dv  /dr + V pinch V  +  N (v th  /T i )(eE r )  +  N (v th  /T i )(dT i /dr)  ] pinchE r term ∇ T i / ∇ p i term ∇ V  driven off diagonal Diagonal term Is the pinch term really large enough to affect the rotation profile?   =(1/r) ∫r[ m i n i (-dV  /dt) + T ext ] dr T ext : external torque Momentum flux has diffusive and non-diffusive term It is not easy to distinguish E r 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

5. Momentum pinch and off-diagonal term Momentum pinch V inward V  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 ∇ T i ∇V∇V Off diagonal term  ND ∇ T i ∇V∇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 V inward V  ∇V∇V ∇V∇V Co-injection Ctr-injection

6. 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  I 1 <  I 2 I2I2 I1I1 V 1 >V 2 V pinch = 2  D (-  /R + 1/L n ) The pinch velocity can be evaluated as Inwardoutward V pinch V   dV  /dr ~ (r/R) (L T /R 0 ) << 1 The ratio of inward pinch term to diffusive term is a order of 10 -1 to 10 -2  =r/R 0

7. E r Non-diffusive term Flux : em i  N n i (v th  /T i )(E r ) Torque : em i  N (1/r) d[r n i (v th  /T i )E r ]/dr E r non-diffusive term is driven by the torque with E r 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 E r change its sign from positive to negative and the tnegative E r (or dE r /dr <0 ) causes toroidal rotation in co-direction (opposite to JT-60U)

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 E r non-diffusive term See O.D.Gurcan Phys. Plasmas 14 (2007) 042306 in details em i  N (1/r) d[r n i (v th  /T i )E r ]/dr ~ em i  N n i (v th /T i )(dE r /dr) The E r non-diffusive term is nearly equivalent to the spontaneous torque due to E r 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

10. Torque scan experiment in LHD Near center (R < 4.1m)  NBI driven toroidal rotation dominant Off center (R > 4.1m)  spontaneous toroidal rotation dominant 1 co/ctr-NBI2 balanced NBIs2 balanced and 1 co/ctr-NBI 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 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

12. ∇ T i 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. ∇ T i and ∇ V  causality 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 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

14. What is physics mechanism of spontaneous rotation? What we know What we do not know 1 There is an non-diffusive term in momentum transport 2 The non-diffusive terms are relating to E r 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 E r and ∇ T i, ∇ p i, etc……

15. Summary 1.Two Non-diffusive terms (off-diagonal term) of toroial momentum transport are observed separately in LHD : one is E r terms and the other is ∇ T i term. (Their coupling is too strong in tokamk) 2. E r term is dominant near the plasma edge and positeive E r 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  )

17. 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

18. Problem of concept of momentum pinch Momentum pinch V inward V  second derivative (curvature) predicted contradicts to that measured in experiment. V inward V  ∇V∇V ∇V∇V Co-injection V inward V  ∇V∇V Ctr-injection V inward V  ∇V∇V

19. 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!  I 1 <  I 2 I2I2 I1I1 V 1 >V 2

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 ) [PRL 86 (2001 ) 3040] CTR rotation driven by ECH plasma in D-IIID [PoP 11 (2004) 4323]