Spectral Anisotropy of Elsässer Variables in Two Dimensional Wave-vector Space as Observed in the Fast Solar Wind Turbulence

Abstract: 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.