Angular momentum and rotational energy of mean flows in toroidal magnetic fields

We derive the balance equation for the Favre averaged angular momentum in toroidal not necessarily axisymmetric magnetic field equilibria. We find that the components of angular momentum are given by the covariant poloidal and toroidal components of E x B and parallel flow velocities and we separately identify all relevant stress tensors, torques and source terms for each of these components. Our results feature the Favre stress generalisations of previously found Reynolds stresses like the diamagnetic or parallel E x B stress, as well as the density gradient drive term. Further, we identify the magnetic shear as a source of poloidal E x B angular momentum and discuss the mirror and the Lorentz force. Here, we find that the geodesic transfer term, the Stringer-Winsor spin-up term and the ion-orbit loss term are all part of the Lorentz force and are in fact one and the same term. Discussing the relation to angular velocity we build the inertia tensor with the help of the first fundamental form of a flux-surface. In turn, the inertia tensor is used to construct a flux-surface averaged rotational energy for E x B surface flows of the plasma. The evolution of this rotational energy features a correction of previous results due to the inertia tensor. In particular, this correction suggests that density sources on the high-field side contribute much more to zonal flow energy generation than on the low field side. Our derivation is based on a full-F, electromagnetic, gyro-kinetic model in a long-wavelength limit. The results can be applied to gyro-kinetic as well as gyro-fluid theories and can also be compared to drift-kinetic and drift-fluid models. Simplified cases for the magnetic field geometry including the axisymmetric purely toroidal and purely poloidal magnetic fields are discussed, as are the angular momentum balance of the electromagnetic fields, the ion-orbit loss mechanism and the parallel acceleration.