Inverse cascade and magnetic vortices in kinetic Alfv\'en-wave turbulence

A Hamiltonian two-field gyrofluid model for kinetic Alfv\'en waves (KAWs) in a magnetized electron-proton plasma, retaining ion finite-Larmor-radius corrections and parallel magnetic field fluctuations, is used to study the inverse cascades that develop when turbulence is randomly driven at sub-ion scales. In the directions perpendicular to the ambient field, the dynamics of the cascade turns out to be nonlocal and sensitive to the ratio $\chi_f$ of the wave period to the characteristic nonlinear time at the driving scale. When $\chi_f$ is not too large, decay instability can develop, enhancing for a while inverse transfers. The balanced state, obtained at early time when the two counter-propagating waves are equally driven, also becomes unstable for small $\chi_f$, leading to an inverse cascade restricted to a limited spectral range. For $\beta_e$ smaller than a few units, the cascade slows down when reaching the low-dispersion spectral range. For higher $\beta_e$, the ratio of the KAW to the Alfv\'en frequencies displays a local minimum. At this transverse wavenumber, a condensate is formed, associated with the development of ion-scale magnetic vortices. The cascade towards larger scales is then inhibited. In the parallel direction, depending on the parameters, a local inverse cascade can develop, leading to elongated vortices.