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Temperature anisotropy estimation

The simplification of the method proposed in GS and implemented here lies in the reduction to a 1d integration, made possible through the use of the scalar approximation and an isotropic distribution. However we can attempt to estimate the anisotropy - i.e. the random velocities parallel and perpendicular to the magnetic field - and interpret the result as two temperature components $T_{\parallel}$ and $T_{\perp}$. Here we describe the estimation.

The $ij^{{\rm th}}$ element of the measured pressure tensor ${\bf p}_m$ can be expressed

\begin{displaymath}p_{m_{ij}} = P_{m_{ij}} - m_e(NV)_{mi}V_{mj}\end{displaymath} (4.23)

which can be decomposed in terms of the unit magnetic field vector ${\bf b}$ in the case of a gyrotropic distribution:
\begin{displaymath}p_{m_{ij}} = p_{m\parallel}b_ib_j + p_{m\perp}(\delta_{ij}-b_ib_j).\end{displaymath} (4.24)

The trace of ${\bf p}_m$ is then
\begin{displaymath}{\rm Tr}({\bf p}_m) = p_{m\parallel} + 2p_{m\perp}.\end{displaymath} (4.25)

Finally the measured anisotropy $A_m$ is found as
\begin{displaymath}
A_m = \frac{p_{m\perp}}{p_{m\parallel}} = \frac{{\rm Tr}({\bf p}_m) - p_{m\parallel}}{2p_{m\parallel}}.
\end{displaymath} (4.26)

We make the assumption that this measured anisotropy is an approximation to the underlying anisotropy and so we can partition the corrected temperature $T_f$ according to:
\begin{displaymath}
3T_f = T_{f\parallel} + 2T_{f\perp}
\end{displaymath} (4.27)

with
\begin{displaymath}
\frac{T_{f\perp}}{T_{f\parallel}} \equiv A = A_m
\end{displaymath} (4.28)

As a check we rotate ${\bf p}_m$ into a field-aligned frame such that:

\begin{displaymath}
\left[\begin{array}{ccc}
p_{m00} & p_{m01} & p_{m02}\\
p_{m...
...{m\perp}_1} & 0 \\
0 & 0 & p_{{m\perp}_2}
\end{array} \right]
\end{displaymath}

In practice the off-diagonal terms are not identically zero and we use the ratio of these to the diagonal terms as an error check. Additionally $p_{m\perp 1} \neq p_{m\perp 2}$ provides a second check.


next up previous contents
Next: Estimation parameters Up: The algorithm Previous: Adding high energy moments   Contents
Steve Schwartz 2005-03-26