% $Id: imposed.tex,v 1.4 2006/01/29 14:32:25 brandenb Exp $
\documentclass{article}
\usepackage{graphicx}
\usepackage{bm}
\input apj
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\title{Test field plus imposed field}
\author{}
\date{\today,~ $ $Revision: 1.4 $ $}
\begin{document}
\maketitle

Ignore diffusion term for now.
\EQ
{\partial\over\partial t}\BB=\nab\times\left
[\UU\times\left(\BB_0+\BB\right)\right]
\EN
Same for test field:
\EQ
{\partial\over\partial t}\BB^{\rm(test)}
=\nab\times\left[\UU\times\left(\BB_0+\BB^{\rm(test)}\right)\right]
\EN
The test field consists of an imposed test field,
$\BB_0^{\rm(test)}$, and the deviations from it,
which are split into a mean and fluctuating component, i.e.\
\EQ
\BB^{\rm(test)}
=\BB_0^{\rm(test)}+\meanBB^{\rm(test)}+\bb^{\rm(test)},
\EN
where $\BB_0^{\rm(test)}$ is independent of time and
the other components are not.
The evolution equation is thus
\EQ
{\partial\over\partial t}
\left(\meanBB^{\rm(test)}+\bb^{\rm(test)}\right)
=\nab\times\left[\UU\times
\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}+\bb^{\rm(test)}\right)
\right]
\label{fullTestFieldNoImposed}
\EN
Average this equation
\EQ
{\partial\over\partial t}\meanBB^{\rm(test)}
=\nab\times\left[\overline{\UU\times
\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}+\bb^{\rm(test)}\right)}
\right]
\EN
\EQ
{\partial\over\partial t}\meanBB^{\rm(test)}
=\nab\times\left[
\meanUU\times\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}\right)
+\overline{\uu\times\bb^{\rm(test)}}\right]
\EN
Subtract this from \Eq{fullTestFieldNoImposed}
\EQA
{\partial\over\partial t}\bb^{\rm(test)}
=\nab\times\left[\left(\meanUU+\uu\right)\times
\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}+\bb^{\rm(test)}\right)\right]
\nonumber
\\
-\nab\times\left[
\meanUU\times\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}\right)
+\overline{\uu\times\bb^{\rm(test)}}\right]
\ENA
or
\EQ
{\partial\over\partial t}\bb^{\rm(test)}
=\nab\times\left[
\meanUU\times\bb^{\rm(test)}
+\uu\times\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}+\bb^{\rm(test)}\right)
-\overline{\uu\times\bb^{\rm(test)}}\right]
\EN
or
\EQ
{\partial\over\partial t}\bb^{\rm(test)}
=\nab\times\left[
\meanUU\times\bb^{\rm(test)}
+\uu\times\left(\BB_0+\BB_0^{\rm(test)}+\meanBB^{\rm(test)}\right)\right]+\GG
\label{flucTestFieldNoImposed}
\EN
where
\EQ
\GG=\nab\times\left(\uu\times\bb^{\rm(test)}
-\overline{\uu\times\bb^{\rm(test)}}\right)
\EN
is the nonlinear term.

\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/tex/mhd/alphaeta/imposed.tex,v 1.4 2006/01/29 14:32:25 brandenb Exp $
\end{verbatim}

\end{document}
