Interpreting solar wind turbulent spectra beyond Taylor's Hypothesis

In this paper we apply a methodology, recently proposed by Bourouaine and Perez (2019) (BP19), to interpret solar-wind turbulent power spectra beyond Taylor approximation (TA). The turbulent power spectra were measured using \emph{Helios} spacecraft data near 0.6 au. We use the model proposed in BP19 to reproduce the field-perpendicular power spectrum $E(k_\perp)$ of anti-sunward Alfv\'enic fluctuations in the plasma frame (where $k_\perp$ is the field-perpendicular wavenumber) from the corresponding measured frequency power spectrum $P_{\rm sc}(\omega,\theta_b)$ along the sampling angle $\theta_b$, which is the angle between the local magnetic field and the sampling direction. Here $\omega=2\pi f$ and $f$ is the frequency of the time signal. Interestingly enough, we found that for all corresponding measured frequency power spectrum $P_{\rm sc}(\omega,\theta_b)$ the reproduced field-perpendicular power spectrum $E(k_\perp)$ is the same and independent of the considered sampling angle $\theta_b$. This finding is consistent with the fact that the analyzed turbulence is strong and highly anisotropic with $k_\| \ll k_\perp$ (where $k_\|$ is the field-parallel wavenumber). Furthermore, for this specific time signal we found that the commonly used TA is still approximately valid with the important difference that a broadening in $k_\perp$ for each angular frequency $\omega$ is present. This broadening can be described in the context of the methodology proposed in BP19.