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 $\boldsymbol{E}\times\boldsymbol{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 $\boldsymbol{E}\times\boldsymbol{B}\ $ stress, as well as the density gradient drive term. Further, we identify the magnetic shear as a source of poloidal $\boldsymbol{E}\times\boldsymbol{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.