Spectral Anisotropy of Els\"asser Variables in Two Dimensional Wave-vector Space as Observed in the Fast Solar Wind Turbulence

Intensive studies have been conducted to understand the anisotropy of solar wind turbulence. However, the anisotropy of Els\"asser variables ($\textbf{Z}^\pm$) in 2D wave-vector space has yet to be investigated. Here we first verify the transformation based on the projection-slice theorem between the power spectral density PSD$_{2D}(k_\parallel,k_\perp )$ and the spatial correlation function CF$_{2D} (r_\parallel,r_\perp )$. Based on the application of the transformation to the magnetic field and the particle measurements from the WIND spacecraft, we investigate the spectral anisotropy of Els\"asser variables ($\textbf{Z}^\pm$), and the distribution of residual energy E$_{R}$, Alfv\'en ratio R$_{A}$ and Els\"asser ratio R$_{E}$ in the $(k_\parallel,k_\perp)$ space. The spectra PSD$_{2D}(k_\parallel,k_\perp )$ of $\textbf{B}$, $\textbf{V}$, and $\textbf{Z}_{major}$ (the larger of $\textbf{Z}^\pm$) show a similar pattern that PSD$_{2D}(k_\parallel,k_\perp )$ is mainly distributed along a ridge inclined toward the $k_\perp$ axis. This is probably the signature of the oblique Alfv\'enic fluctuations propagating outwardly. Unlike those of $\textbf{B}$, $\textbf{V}$, and $\textbf{Z}_{major}$, the spectrum PSD$_{2D}(k_\parallel,k_\perp )$ of $\textbf{Z}_{minor}$ is distributed mainly along the $k_\perp$ axis. Close to the $k_\perp$ axis, $\left| {E}_{R}\right|$ becomes larger while R$_{A}$ becomes smaller, suggesting that the dominance of magnetic energy over kinetic energy becomes more significant at small $k_\parallel$. R$_{E}$ is larger at small $k_\parallel$, implying that PSD$_{2D}(k_\parallel,k_\perp )$ of $\textbf{Z}_{minor}$ is more concentrated along the $k_\perp$ direction as compared to that of $\textbf{Z}_{major}$. The residual energy condensate at small $k_\parallel$ is consistent with simulation results in which E$_{R}$ is spontaneously generated by Alfv\'en wave interaction.