Evaluation of Reduced Power Spectra from Three-Dimensional k-Space

We present a new tool to evaluate one dimensional reduced power spectral densities (PSD) from arbitrary energy distributions in $k$-space. This enables us to calculate the power spectra as they are measured in spacecraft frame for any given measurement geometry assuming Taylor's frozen-in approximation. It is possible to seperately calculate the diagonal elements of the spectral tensor and also to insert additional, non-turbulent energy in $k$-space (e.g.\ mirror mode waves). Given a critically balanced turbulent cascade with $k_\|\sim k_\perp^\alpha$, we explore the implications on the spectral form of the PSD and the functional dependence of the spectral index $\kappa$ on the field-to-flow angle $\theta$ between plasma flow and background magnetic field. We show that critically balanced turbulence develops a $\theta$-independent cascade with the spectral slope of the perpendicular cascade $\kappa(\theta{=}90^\circ)$. This happens at frequencies $f>f_\mathrm{max}$, where $f_\mathrm{max}(L,\alpha,\theta)$ is a function of outer scale $L$, critical balance exponent $\alpha$ and field-to-flow angle $\theta$. We also discuss potential damping terms acting on the $k$-space distribution of energy and their effect on the PSD. Further, we show that the functional dependence $\kappa(\theta)$ as found by \textit{Horbury et al.} (2008) and \textit{Chen et al.} (2010) can be explained with a damped critically balanced turbulence model.