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The Alfvénic reflection coefficient in the fast solar wind

The solar wind simulation software ZEPHYR (Cranmer 2007) has been developed to include an Alfénic module in order to acknowledge turbulent wave heating as a driving factor of the solar wind and coronal temperature.

The mathematical framework of the governing non-WKB wave equations has been described by (Barkhudarov 1991) and is explained in more detail in section \ref{sec:app}.

Alfvénic waves are modeled to come in two modes describing their direction of propagation: Inward and outward. In the equations to follow these are also denoted by the ’+’ and ’-’ signs, respectively.

The photosphere is assumed to be a force sphere emitting outward traveling Alfvénic waves which are partially reflected at gradients in the magnetic configuration of the fluxtube, coupling them to the inward traveling wave mode of that same frequency. Alfvénic wave dynamics is computed monochromatically for a constant number of bins. For ZEPHYR, 300 frequency bins were employed ranging from angular frequencies of \(\omega_{lo}=6.36\cdot 10^{-5}\frac{\mbox{rad}}{s}\) to \(\omega_{hi}=2.34\cdot 10^{-1}\frac{\mbox{rad}}{s}\), corresponding to frequencies of \(f_{lo}=1\cdot 10^{-5}\mbox{Hz}\) and \(f_{hi}=3.72\cdot 10^{-1}\mbox{Hz}\) or periods of \(T_{lo}=100000s\) and \(T_{hi}=26s\), respectively. Bins are spaced logarithmically between \(\omega_{lo}\) and \(\omega_{hi}\) and their contribution to solar wind heating is weighted by their width and their relative power densities. Power density spectra for several heights above the photosphere have been calculated by Cranmer and Ballegoijen (Cranmer 2005). For the computations of ZEPHYR the Alfvénic power spectrum has been assumed to be the one computed at the transition region (see figure 8 in (Cranmer 2005) or \ref{img:alfSpec} in this document) and to be constant throughout the entire fluxtube. The absolute power values of the different frequencies are not important for the computations, as only relative contributions are considered within ZEPHYR. Alfvénic contributions to this model are dependant on the shape of the spectrum, not its height.

Each frequency bin is treated sperately from the others making all of them good candidates for parallel computations. The goal is to obtain a numerical value of the monochromatic reflection coefficient throughout the fluxtube:

\begin{equation} R_{\omega}=\left|\frac{u+V_{A}}{u-V_{A}}\psi\right|\\ \end{equation}where \(\psi\) is the amplitude quotient between incoming(+) and outgoing(-) waves, \(u\) is the mean solar wind velocity and \(V_{A}=\frac{B_{0}}{\sqrt{\mu_{0}\rho}}\) is the Alvénic speed, \(\mu_{0}=4\pi\cdot 10^{-7}\frac{mkg}{s^{2}A^{2}}\) is the magnetic constant, \(B_{0}\) the unperturbed magnetic flux density and \(\rho\) the mass density of the plasma. All these values are evaluated on a non-equidistant, one-dimensional (radial) grid stretching from the photosphere to some height at about 1200 R_⊙ (solar radii). The computation requires a numeric integration starting from the Alfvén point \(r_{A}\) (where \(u=V_{A}\)) up and down the grid. Initial values of \(\psi(r_{A})\) and \(\gamma(r_{A})\) have to be computed separately, as the governing equations contain a singularity at that point. A fourth order Runge-Kutta scheme (RK4) has been employed to solve the following equations, which were derived in (Barkhudarov 1991), where one can also find instructions for finding the initial values \(\psi(r_{A})\) and \(\gamma(r_{A})\), which are not repeated here.

\begin{align}
\label{equ:psidash}\psi^{\prime} & =S\cos{(\gamma)}\big{(}\psi^{2}-1\big{)} \\
\label{equ:gammadash}\gamma^{\prime} & =S\frac{\psi^{2}+1}{\psi}\sin{(\gamma)}-\frac{2V_{A}}{u^{2}-V_{A}^{2}}\omega\\
\end{align}

The scale factor \(S=\frac{1}{2V_{A}}V_{A}^{\prime}\) incorporates the magnetic gradients of the fluxtube. A derivation of the equations above can be found in section \ref{sec:app}. The equations (see section 5 of (Barkhudarov 1991)) suggest a critical frequency value \(\omega_{0}\), which separates two different wave behavior domains: Waves of frequencies below \(\omega_{0}\) develop a standing wave pattern at high altitudes (\(\psi_{\infty}=1\)) whereas waves of a higher frequency (\(\omega>\omega_{0}\)) develop an outward propagating pattern (\(\psi_{\infty}=\alpha^{2}-\sqrt{\alpha^{2}-1}\)). \(\alpha=\frac{\omega}{\omega_{0}}\) is the ratio of Alfvénic wave frequency under consideration to critical frequency, which is determined by

\begin{equation}
\omega_{0}=\frac{1}{2}\frac{u_{\infty}^{3/2}}{r_{A}\sqrt{u_{A}}}\\
\end{equation}

with \(u_{\infty}\) the mean solar wind speed at high altitudes (or the right side of the computational grid) and \(u_{A}\) the mean solar wind speed at the Alfvén point \(r_{A}\). These relations are depicted by figure 2 of (Barkhudarov 1991), which is included in this document as figure \ref{img:asymp}.

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