Electron-scale reduced fluid models with gyroviscous effects

Reduced fluid models for collisionless plasmas including electron inertia and finite Larmor radius corrections are derived for scales ranging from the ion to the electron gyroradii. Based either on pressure balance or on the incompressibility of the electron fluid, they respectively capture kinetic Alfv\'en waves (KAWs) or whistler waves (WWs), and can provide suitable tools for reconnection and turbulence studies. Both isothermal regimes and Landau fluid closures permitting anisotropic pressure fluctuations are considered. For small values of the electron beta parameter $\beta_e$, a perturbative computation of the gyroviscous force valid at scales comparable to the electron inertial length is performed at order $O(\beta_e)$, which requires second-order contributions in a scale expansion. Comparisons with kinetic theory are performed in the linear regime. The spectrum of transverse magnetic fluctuations for strong and weak turbulence energy cascades is also phenomenologically predicted for both types of waves. In the case of moderate ion to electron temperature ratio, a new regime of KAW turbulence at scales smaller than the electron inertial length is obtained, where the magnetic energy spectrum decays like $k_\perp^{-13/3}$, thus faster than the $k_\perp^{-11/3}$ spectrum of WW turbulence.