The Second Quadratic Invariant of Astrophysical Gyrokinetics

We show that gyrokinetics in a spatially uniform equilibrium (or `astrophysical gyrokinetics') possesses a previously unknown quadratic invariant, which we call the gyrokinetic helicity. We derive the limiting forms of the gyrokinetic helicity in five subsidiary expansions of astrophysical gyrokinetics: the isothermal electron fluid (ITEF) approximation, kinetic reduced magnetohydrodynamics (KRMHD), electron reduced magnetohydrodynamics (ERMHD), finite-Larmor-radius magnetohydrodynamics (FLR-MHD), and the kinetic reduced electron heating model~(KREHM). We subdivide the real part of the gyrokinetic helicity into `magnetofluid' and `phase-space' components, which are separately conserved in KRMHD, ERMHD, and FLR-MHD (in which there is no phase-space helicity). The magnetofluid helicity is equivalent to the Alfv\'enic part of the cross helicity in KRMHD, the magnetic helicity in~ERMHD, and the generalized helicity in FLR-MHD and~KREHM. We derive an analytic expression for the rate at which magnetofluid helicity is converted into phase-space helicity at length scales comparable to the proton gyroradius in the ITEF approximation. We also explain how this conversion circumvents the helicity barrier and enhances turbulent heating in coronal holes and the near-Sun solar wind.