# -8/3 spectrum in kinetic Alfv\'en wave turbulence: implications for the solar wind

## TL;DR

This paper demonstrates that a nonlinear diffusion model of weak kinetic Alfvén wave turbulence can produce an $-8/3$ magnetic energy spectrum, explaining observed solar wind turbulence features at sub-proton scales.

## Contribution

It introduces a self-similar, non-stationary turbulence model that accounts for the observed $-8/3$ spectral index in solar wind at sub-proton scales.

## Key findings

- The $-8/3$ spectrum can emerge from a non-stationary turbulence model.
- The model predicts a front propagation law $k_f \,\sim\ (t_*-t)^{-3/4}$.
- Observations support the non-stationarity of solar wind turbulence at small scales.

## Abstract

The nature of solar wind turbulence at large scale is rather well understood in the theoretical framework of magnetohydrodynamics. The situation is quite different at sub-proton scales where the magnetic energy spectrum measured by different spacecrafts does not fit with the classical turbulence predictions: a power law index close to $-8/3$ is generally reported which is far from the predictions of strong and wave turbulence, $-7/3$ and $-5/2$ respectively. This discrepancy is considered as a major problem for solar wind turbulence. Here, we show with a nonlinear diffusion model of weak kinetic Alfv\'en wave turbulence where the cascade is driven by local triadic interactions (Passot and Sulem, 2019), that a magnetic spectrum with a power law index of $-8/3$ can emerge. This scaling corresponds to a self-similar solution of the second kind with a front propagation following the law $k_f \sim (t_*-t)^{-3/4}$, with $t<t_*$. This solution appears when we relax the implicit assumption of stationarity generally made in turbulence. The agreement between the theory and observations can be interpreted as an evidence of the non-stationarity of solar wind turbulence at sub-proton scales.