Kinetic-Alfven and whistler waves in astrophysical plasma turbulence at subproton scales

The analytical study of sub-proton electromagnetic fluctuations in a collisionless plasma with plasma beta of order unity is presented. In the linear limit, a rigorous derivation from the kinetic equation is conducted focusing on the role and physical properties of kinetic-Alfven and whistler waves. Then, nonlinear fluid-like equations for kinetic-Alfven waves and whistler modes are derived, with the special emphasis on the similarities and differences in the corresponding plasma dynamics. The kinetic-Alfven modes exist in the lower-frequency region of the phase space, $\omega \ll$ k$_{perp}$ v$_{Ti}$, where they are described by the kinetic-Alfven system. These modes exist both below and above the ion cyclotron frequency. The whistler modes, which are qualitatively different from the kinetic-Alfv\'en ones, occupy a different region of the phase space, k$_{perp}$ v$_{Ti} \ll \omega \ll$ k$_{zvTe}$, and they are described by the electron MHD system or the reduced electron MHD system if the propagation is oblique. Here k$_z$ and k$_{perp}$ are the wave numbers along and transverse to the background magnetic field, and v$_{Ti}$ and v$_{Te}$ are ion and electron thermal velocities. The models of sub-proton plasma turbulence are discussed and the results of numerical simulations are presented. Possible implications for solar-wind observations are pointed out.