Nonlinear gyrokinetic theory of toroidal momentum pinch

Abstract

The turbulent convective flux of the toroidal angular momentum density is derived using the nonlinear toroidal gyrokinetic equation which conserves phase space density and energy ͓T. S. Hahm, Phys. Fluids, 31, 2670 ͑1988͔͒. A novel pinch mechanism is identified which originates from the symmetry breaking due to the magnetic field curvature. A net parallel momentum transfer from the waves to the ion guiding centers is possible when the fluctuation intensity varies on the flux surface, resulting in imperfect cancellation of the curvature drift contribution to the parallel acceleration. This mechanism is inherently a toroidal effect, and complements the k ʈ symmetry breaking mechanism due to the mean E ϫ B shear ͓O. Gurcan et al., Phys. Plasmas 14, 042306 ͑2007͔͒ which exists in a simpler geometry. In the absence of ion thermal effects, this pinch velocity of the angular momentum density can also be understood as a manifestation of a tendency to homogenize the profile of "magnetically weighted angular momentum density," nm i R 2 ʈ / B 2. This part of the pinch flux is mode-independent ͑whether it is trapped electron mode or ion temperature gradient mode driven͒, and radially inward for fluctuations peaked at the low-B-field side, with a pinch velocity typically, V Ang TEP ϳ −2 / R 0. Ion thermal effects introduce an additional radial pinch flux from the coupling with the curvature and grad-B drifts. This curvature driven thermal pinch can be inward or outward, depending on the mode-propagation direction. Explicit formulas in general toroidal geometry are presented.

Poloidal rotation and its relation to the potential vorticity flux

“…͑18a͒ and ͑18b͒. The perpendicular stress as well as the first contribution to the parallel stress can be recognized as E ϫ B convection of perpendicular 11,31 and parallel momentum, [32][33][34][35] respectively. The second contribution to Eq.…”

Residual parallel Reynolds stress due to turbulence intensity gradient in tokamak plasmas

“…24 On the theoretical front, there has been a multitude of attempts to explain this phenomenon. Note that an inward convective flux of momentum [25][26][27] ͑i.e., a "momentum pinch"͒ that transports scrape-off layer ͑SOL͒ flows into the core, cannot explain the L-H spin-up, since the direction of the SOL flows can be changed by changing the location of the x-point ͑or where the plasma touches the wall͒, yet the direction of the H-mode rotation in the core remains unaltered. 28 A theory that has some possibility of explaining this phenomenon is based on "residual" Reynolds stress 29,30 ͑or more generally, a residual component of the toroidal stress tensor͒.…”

“…Parallel flow shear itself, 36 magnetic curvature ͑curvature from B ʈ ‫ء‬ in laboratory frame͒, 26 or the effect of Coriolis drift in rotating frame, 25 can also lead to k ʈ symmetry breaking but give diffusive and pinchlike contributions to the Reynolds stress. Here we only consider the truly off-diagonal ͑i.e., residual͒ terms, which do not contain the transported field itself, since only these can explain the formation of a nonvanishing field from an initial value of zero, or the anomalous residual "torque" that acts on the plasma when the field and its gradients are set to vanish.…”

On the efficiency of intrinsic rotation generation in tokamaks

“…The first term is diffusive part, the second term is pinch, [11][12][13] and the last term is the residual stress. Intrinsic torque in toroidal plasmas, which is related to the residual stress via =−ٌ · ⌸ rʈ res , was observed for a plasma with no flow and unbalanced NBI injection ͑1 co+ 2 counter͒.…”