Rotation and momentum transport in tokamaks and helical systems

Poloidal and toroidal rotation has been recognized to play an important role in heat transport and magnetohydrodynamic (MHD) stability in tokamaks and helical systems. It is well known that the E × B shear due to poloidal and toroidal flow suppresses turbulence in the plasma and contributes to the improvement of heat and particle transport, while toroidal rotation helps one to stabilize MHD instabilities such as resistive wall modes and neoclassical tearing mode. Therefore, understanding the role of momentum transport in determining plasma rotation is crucial in toroidal discharges, both in tokamaks and helical systems. In this review paper, the driving and damping mechanisms of poloidal and toroidal rotation are outlined. Driving torque due to neutral beam injection and radio-frequency waves, and damping due to parallel viscosity and neoclassical toroidal viscosity (NTV) are described. Regarding momentum transport, the radial flux of momentum has diffusive and non-diffusive (ND) terms, and experimental investigations of these are discussed. The magnitude of the diffusive term of momentum transport is expressed as a coefficient of viscous diffusivity. The ratio of the viscous diffusivity to the thermal diffusivity (Prandtl number) is one of the interesting parameters in plasma physics. It is typically close to unity, but sometimes can deviate significantly depending on the turbulent state. The ND terms have two categories: one is the so-called momentum pinch, whose magnitude is proportional to (or at least depends on) the velocity itself, and the other is an off-diagonal term in which the magnitude is proportional to (or at least depends on) the temperature or/and pressure gradient, independent of the velocity or its gradient. The former has no sign dependence; rotation due to the momentum pinch does not depend on the sign of the rotation itself, whether it is parallel to the plasma current (co-direction) or anti-parallel to the plasma current (counter-direction). In contrast, the latter has a sign dependence; the rotation due to the off-diagonal residual term is either in the co- or counter-direction depending on the turbulence state, but not on the sign of the rotation itself. This residual term can also act as a momentum source for intrinsic rotation. The experimental results of investigations of these ND terms are described. Finally the current understanding of the mechanisms behind the ND terms in momentum transport, and predictions of intrinsic rotation driven by these terms are reviewed. (review article)