General expression of the gyroviscous force

Assuming only small gyromotion periods and Larmor radii compared to any other time and length scales, and retaining the lowest significant order in δ=ρi∕L⪡1, the general expression of the ion gyroviscous stress tensor is presented. This expression covers both the “fast dynamics” (or “magnetohydrodynamic”) ordering, where the time derivative and ion gyroviscous stress are first order in δ relative to the ion gyrofrequency and scalar pressure, respectively, and the “slow dynamics” (or “drift”) ordering, where the time derivative and ion gyroviscous stress are, respectively, second order in δ. This general stress tensor applies to arbitrary collisionality and does not require the distribution function to be close to a Maxwellian. Its exact divergence (gyroviscous force) is written in a closed vector form, allowing for arbitrary magnetic geometry, parallel gradients, and flow velocities. Considering, in particular, the contribution from the velocity gradient (rate of strain) term, the final form of the momentum conservation equation after the “gyroviscous cancellation” and the “effective renormalization of the perpendicular pressure by the parallel vorticity” is precisely established.