Theory of the residual stress and the origins of intrinsic

We present recent results in the theory of turbulent momentum transport pertinent to the description of intrinsic rotation. A minimal model of intrinsic rotation is developed for H-mode plasmas based on an extended L-H mode bifurcation model [1]. The primary novel components introduced into this model are a residual stress [2,3] (i.e. the portion of the momentum flux not proportional to $v_\phi$ or $v^\prime_\phi$) and a turbulent equipartition pinch [4]. This reduced model reproduces salient features of the ITPA H-mode database [5]. Furthermore, a quasilinear analysis of a phase space conserving gyrokinetic equation is utilized to identify a novel mechanism by which microturbulence may spin-up a plasma from rest, in addition to the $\mathbf{E}\times\mathbf{B}$ shear driven mechanism discussed in Refs. [2,3]. This mechanism, which appears in the gyrokinetic formulation through the parallel nonlinearity, emerges due to charge separation induced by the polarization drift, and is not tied to $\mathbf{E} \times\mathbf{B}$ shear. Thus, this mechanism is likely relevant in regimes of weak $\mathbf{E}\times\mathbf{B}$ shear. [1] F. L. Hinton, Phys. Fluids B 3, 696 (1991), [2] O. D. Gurcan et al. Phys. Plasmas 14, 042306 (2007), [3] Dominguez et al. Phys. Fluids B 5, 3876 (1993), [4] T.S. Hahm et al. Phys. Plasmas 14, 072302 (2007), [5] J. E. Rice et al. Nucl. Fusion 47, 1618 (2007).