% $Id: notes.tex,v 1.65 2007/09/22 20:31:53 brandenb Exp $
\documentclass{article}
\usepackage{graphicx}
\usepackage{bm}
\graphicspath{{./fig/}{./png/}}
\input apj
\title{Test field procedure}
\author{}
\date{\today,~ $ $Revision: 1.65 $ $}
\begin{document}
\maketitle

\section*{Equations}

The MHD equation for the magnetic vector potential $\AAA$ and
the velocity $\UU$ for an isothermal equation of state
with constant speed of sound, $c_{\rm s}$,
and external forcing functions $\FFF_{\rm ext}$ and $\EEE_{\rm ext}$
can be written in the form
\EQ
{\partial\AAA\over\partial t}=\UU\times\BB-\eta\mu_0\JJ+\EEE_{\rm ext}-\nab\Phi,
\EN
\EQ
{\DD\UU\over\DD t}=\JJ\times\BB/\rho-\nu\QQ+\FFF_{\rm ext}-\nab h,
\EN
\EQ
{\DD h\over\DD t}=-c_{\rm s}^2\nab\cdot\UU,
\EN
where
$\DD/\DD t=\partial/\partial t+\UU\cdot\nab\UU$ is the advective derivative,
$h=c_{\rm s}^2\ln\rho$ is the pseudo enthalpy, $\rho$ is the density,
$\phi$ is the electric potential,
$\BB=\nab\times\AAA$ is the magnetic field,
$\JJ=\nab\times\BB/\mu_0$ is the current density, and
$\QQ=\nab\times\WW-{4\over3}\nab\nab\cdot\UU+\SSSS\cdot\nab\ln\rho$ is a
modified curl of the vorticity $\WW=\nab\times\UU$, where
${\sf S}_{ij}={1\over2}(U_{i,j}+U_{j,i})-{1\over3}\delta_{ij}\nab\cdot\UU$
is the traceless rate of strain tensor.

In addition to the equation for $\AAA$ we consider the equations for the
mean field, $\meanAA$, and for the fluctuation, $\aaa=\AAA=\meanAA$, i.e.\
\EQ
{\partial\meanAA\over\partial t}=
\meanUU\times\meanBB+\overline{\uu\times\bb}
-\eta\meanJJ-\nab\meanPhi
\EN
\EQ
{\partial\aaa\over\partial t}=\meanUU\times\bb+\uu\times\meanBB
+\uu\times\bb-\overline{\uu\times\bb}-\eta\jj+\EEE_{\rm ext}-\nab\phi.
\EN
In the following we consider the evolution of the fluctuating field,
$\bb^{pq}=\nab\times\aaa^{pq}$, that results from arbitrary test fields,
$\meanBB^{pq}$,
\EQ
{\partial\aaa^{pq}\over\partial t}=\meanUU\times\bb^{pq}+\uu\times\meanBB^{pq}
+\uu\times\bb^{pq}-\overline{\uu\times\bb^{pq}}
-\eta\jj^{pq}+\EEE_{\rm ext}-\nab\phi^{pq}.
\EN
Under the assumption of SOCA the terms that are nonlinear in the
fluctuations are neglected, i.e.\
\EQ
{\partial\aaa^{pq}\over\partial t}=\meanUU\times\bb^{pq}+\uu\times\meanBB^{pq}
-\eta\jj^{pq}+\EEE_{\rm ext}-\nab\phi^{pq}\quad\mbox{(SOCA)}.
\EN
In either of the two cases the turbulent transport coefficients would
be calculated from $\meanEMF^{pq}=\overline{\uu\times\bb^{pq}}$.

If the mean flow contains linear shear, it is advantageous to rewrite
$\meanUU=Sx\yyy+\meanUU^1$ so that\footnote{Note that
$(\meanUU^0\times\nab\times\aaa)_i=\meanU^0_j(a_{j,i}-a_{i,j})$
and $\meanU^0_j a_{j,i}=-\meanU^0_{j,i} a_j$ + gradient term.
}
\EQ
{\DDD\aaa^{pq}\over\DDD t}=
-Sa_y^{pq}\xxx
+\meanUU^1\times\bb^{pq}+\uu\times\meanBB^{pq}
+\uu\times\bb^{pq}-\overline{\uu\times\bb^{pq}}
-\eta\jj^{pq}+\EEE_{\rm ext}-...
\EN
where $\DDD/\DDD t=\partial/\partial t+Sx\partial/\partial y$
is the advective derivative with respect to the linear shear flow
and the dots indicate additional gradient terms that can be
removed by a gauge transformation.

We now assume two-dimensional (horizontal) averages, i.e.\
\EQ
\meanBB(z,t)={1\over L_x L_y}\int\BB(x,y,z,t)\;\dd x\,\dd y.
\EN
We assume that the emf is given by
\EQ
\meanemf_i^{pq}=\meanemf_i^{0}+
\alpha_{ij}\meanB_j^{pq}+\eta_{ij3}\meanB_{j,3}^{pq}.
\EN
In the absence of an imposed mean field there can still be
an emf, $\meanemf_i^{0}$, due to large scale dynamo action.

Define four test fields as follows:
\EQ
\meanBB^{11}=\pmatrix{\cos k z\cr0\cr0},\quad
\meanBB^{21}=\pmatrix{\sin k z\cr0\cr0},
\EN
\EQ
\meanBB^{12}=\pmatrix{0\cr\cos k z\cr0},\quad
\meanBB^{22}=\pmatrix{0\cr\sin k z\cr0}.
\EN
with
\EQ
\meanJJ^{11}=\pmatrix{0\cr-k\sin k z\cr0},\quad
\meanJJ^{21}=\pmatrix{0\cr k\cos k z\cr0},
\EN
\EQ
\meanJJ^{12}=\pmatrix{k\sin k z\cr0\cr0},\quad
\meanJJ^{22}=\pmatrix{-k\cos k z\cr0\cr0}.
\EN
So, for each of the four combinations of $i$ and $j$ ($=p$) the set of two
coefficient, $\alpha_{ij}$ and $\eta_{ij3}$, is obtained as
\EQ
\pmatrix{\alpha_{ij}\cr\eta_{ij3}k}=\MMMM^{-1}
\pmatrix{\meanemf_i^{~1j}\cr\meanemf_i^{~2j}},
\EN
where the matrices
\EQ
\MMMM=\pmatrix{
\cos k z & -\sin k z\cr
\sin k z & ~~\cos k z}
\quad\MMMM^{-1}=\pmatrix{
~~\cos k z & \sin k z\cr
-\sin k z & \cos k z}
\EN
are the same
for each value of $p$ and each of the two components $i=1,2$
of $\meanemf_i^{pq}$.
Finally, $\tilde\eta$ is calculated using \Eq{tildeETArel}.
Note that $\det\MMMM=1$.
The $\eta_{ij3}$ are related to $\eta^*_{ij}$ in the formulation
\EQ
\meanemf_i^{pq}=\alpha_{ij}\meanB_j^{pq}-\eta^*_{ij}\meanJ_j^{pq}
\EN
via
\EQ
\pmatrix{
\eta^*_{11} & \eta^*_{12} \cr
\eta^*_{21} & \eta^*_{22} }
=\pmatrix{
\eta_{123} & -\eta_{113} \cr
\eta_{223} & -\eta_{213} }.
\EN
At first instance, we might only need the test fields
$\meanBB^{11}$ and $\meanBB^{21}$, so we get
\EQ
\pmatrix{\alpha_{i1}\cr\eta_{i13}k}=\MMMM^{-1}
\pmatrix{\meanemf_i^{~11}\cr\meanemf_i^{~21}},
\EN
so we get only $\alpha_{11}$ and $\alpha_{11}$, as well as
$\eta_{113}=-\eta^*_{12}$ and $\eta_{213}=-\eta^*_{22}$.
This can be useful in simple cases (e.g.\ isotropy).

\section*{Roberts flow}

Use the Roberts flow as a test flow.
Defined here as
\EQ
\UU(x,y)=k_{\rm f}^{-1}\nab\times(\varphi\zz)
+k_{\rm f}^{-2}\nab\times\nab\times(\varphi\zz),
\label{UUrobflow}
\EN
with the stream function $\varphi=\sqrt{2}\,U_0\cos k_x x\cos k_y y$,
where $k_x=k_y=\pi/a$.
This flow is fully helical with $\WW=k_{\rm f}\UU$, were
$k_{\rm f}^2=k_x^2+k_y^2$ and $\WW=\nab\times\UU$.
The flow is normalized such that $\bra{\UU^2}=U_0^2$.
In components form, the velocity is given by
\EQ
\UU=U_0\pmatrix{
-\cos kx\,\sin ky\cr
+\sin kx\,\cos ky\cr
\sqrt{2}\,\cos kx\,\cos ky}
\EN

\begin{figure}[h!]\begin{center}
\includegraphics[width=\textwidth]{palp_robflow.ps}
\end{center}\caption[]{
Roberts flow
}\label{p1}\end{figure}

\section*{Test field procedure}

We calculate
\EQ
{\partial\bb^{(p,q)}\over\partial t}=\nab\times\left(
\uu\times\meanBB^{(p,q)}\right)+\eta\nabla^2\bb^{(p,q)}+\GG
\EN
and $\meanemf^{(p,q)}\equiv\overline{\uu\times\bb^{(p,q)}}$
for each of the 9 test field $\meanBB^{(p,q)}$,
\EQ
\meanBB^{(1,0)}=\pmatrix{\sin z\cr0\cr0},\quad
\meanBB^{(2,0)}=\pmatrix{0\cr\sin z\cr0},\quad
\meanBB^{(3,0)}=\pmatrix{0\cr0\cr\sin z},
\EN
\EQ
\meanBB^{(1,1)}=\pmatrix{\cos x\cr0\cr0},\quad
\meanBB^{(2,1)}=\pmatrix{0\cr\cos x\cr0},\quad
\meanBB^{(3,1)}=\pmatrix{0\cr0\cr\cos x},
\EN
\EQ
\meanBB^{(1,3)}=\pmatrix{\cos z\cr0\cr0},\quad
\meanBB^{(2,3)}=\pmatrix{0\cr\cos z\cr0},\quad
\meanBB^{(3,3)}=\pmatrix{0\cr0\cr\cos z},
\EN
where we have assumed that $\meanBB$ is an average over the $y$ direction,
so there is no dependence on $y$.
For each of these test fields there is only a single nonvanishing
(and constant) component of the derivative matrix $\meanB_{j,k}$.
This gives the components of
\EQ
\left.\eta_{ijk}=\left(\overline{\cal E}_i^{(p,q)}
-\alpha_{ij}\meanB^{(p,q)}_j\right)\right/\meanB_{j,k}^{(p,q)}
\quad\mbox{(second step, $q=4,5,...,9$)}.
\EN

\section*{Results for Roberts flow}

Consider dimensional quantities using $2a=2\pi$, so $k_1=1$,
$k_{\rm f}=\sqrt{2}$, $U_0=1$, and $\eta=0.05$,
so $R_{\rm m}=U_0/(\eta k_{\rm f})=14.14$.
We first adopt horizontal $xy$ averages, and find in that case
\EQ
\alpha_\perp=-4.714,\quad
\eta_\perp=3.333,
\EN
independent of $z$.

\begin{table}[htb]\caption{
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
            & \multicolumn{2}{c}{SOCA} & \multicolumn{2}{c}{non-SOCA} \\
$R_{\rm m}$ & $\alpha_\perp$ & $\eta_\perp$ & $\alpha_\perp$ & $\eta_\perp$ \\
\hline
14.14 & $-4.708$ & 3.329 & $-0.190$ & 0.005 \\
 7.07 & $-2.354$ & 1.664 & $-0.241$ & 0.026 \\
 3.54 & $-1.177$ & 0.832 & $-0.281$ & 0.106 \\
 1.41 & $-0.471$ & 0.333 & $-0.298$ & 0.188 \\
 0.71 & $-0.235$ & 0.166 & $-0.193$ & 0.133 \\
\label{Ttimescale}\end{tabular}}\end{table}

Next, we consider only $y$ averages, and consider $\alpha_{ij}$
and $\eta_{ijk}$ as functions of $x$ and $z$.
\EQ
\alpha_{ij}(x,z)=\pmatrix{
-4.714 & 0 & 0\cr
0 & -4.714(1+\cos2x) & 0\cr
0 & -3.333\sin2x & 0}
\EN
\EQ
\eta_{ijz}(x,z)=\pmatrix{
0 & 3.333(1+\cos2x) & 2.357\sin2x\cr
-3.333(1+\cos2x) & 0 & 0 \cr
2.357\sin2x & 0 & 0}
\EN
For $x$-derivatives, we need $\sin x$ and $\cos x$ test fields.
\EQ
\alpha_{ij}(x,z)=\pmatrix{
-2.828 & 0 & 0\cr
0 & -8.5(1+\cos2x) & 2\sin 2x\cr
0 & 4\sin2x & 0}
\EN
\EQ
\eta_{ijz}(x,z)=\pmatrix{
0 & 0 & 0 \cr
0 & 5.65\sin2x & 3(1+\cos2x) \cr
0 & 3(1+\cos2x) & 0}
\EN

\section*{Isolating $\ddelta$ effect, etc.}

Split
\EQ
\eta_{ijk}\meanB_{j,k}=
\eta_{ijk}^{\rm(A)}\meanB_{j,k}^{\rm(A)}+
\kappa_{ijk}\meanB_{j,k}^{\rm(S)}.
\EN
where
\EQ
\meanB_{j,k}^{\rm(A)}=-\half\epsilon_{jkl}\meanJ_l
\EN
so
\EQ
\eta_{ijk}^{\rm(A)}\meanB_{j,k}^{\rm(A)}=
-\half\eta_{ijk}^{\rm(A)}\epsilon_{jkl}\meanJ_l
\equiv-\tilde{\eta}_{il}\meanJ_l
\EN
where
\EQ
\tilde{\eta}_{il}=\half\eta_{ijk}^{\rm(A)}\epsilon_{jkl}
\EN
\EQ
\tilde{\eta}_{il}=\eta_{il}+\epsilon_{ilm}\delta_m
\EN
where $\meanemf=...+\ddelta\times\meanJJ$ gives the contribution to the
$\ddelta$ effect, representing the antisymmetric part of $\tilde{\eta}_{il}$.
To calculate $\ddelta$ we write
\EQ
\delta_m=-\half\epsilon_{mil}\tilde{\eta}_{il}
\EN
so
\EQ
\delta_m
=\quarter\epsilon_{mil}\eta_{ijk}\epsilon_{jkl}
=\quarter(\eta_{kmk}-\eta_{jjm})
\EN
\EQ
\delta_1
=\quarter(\eta_{k1k}-\eta_{jj1})
=\quarter(\eta_{111}+\eta_{313}-\eta_{111}-\eta_{331})
=\quarter(\eta_{313}-\eta_{221}-\eta_{331})
\EN
\EQ
\delta_2
=\quarter(\eta_{k2k}-\eta_{jj2})
=\quarter(\eta_{121}+\eta_{323}-\eta_{112}-\eta_{332})
=\quarter(\eta_{121}+\eta_{323})
\EN
\EQ
\delta_3
=\quarter(\eta_{k3k}-\eta_{jj3})
=\quarter(\eta_{131}+\eta_{333}-\eta_{113}-\eta_{333})
=\quarter(\eta_{131}-\eta_{113}-\eta_{223})
\EN
Symmetric part of $\eta_{il}$ as in $\meanemf_i=...-\eta_{ij}\meanJ_j$ is
\EQ
\eta_{il}=\quarter(\eta_{ijk}\epsilon_{jkl}+\eta_{ljk}\epsilon_{jki})
\EN
\EQ
\eta_{11}=\half\eta_{1jk}\epsilon_{jk1}=\half(\eta_{123}-\eta_{132})=+\half\eta_{123}
\EN
\EQ
\eta_{22}=\half\eta_{2jk}\epsilon_{jk2}=\half(\eta_{231}-\eta_{213})
\EN
\EQ
\eta_{33}=\half\eta_{3jk}\epsilon_{jk3}=\half(\eta_{312}-\eta_{321})=-\half\eta_{321}
\EN
\EQ
\eta_{12}=\quarter(\eta_{131}+\eta_{113}-\eta_{223})
\EN
\EQ
\eta_{23}=-\quarter(\eta_{313}+\eta_{221}-\eta_{331})
\EN
\EQ
\eta_{13}=\quarter(\eta_{323}-\eta_{121})
\EN
Antisymmetric part of $\alpha$ tensor
\EQ
\gamma_j=\half\epsilon_{ijk}\alpha_{ik}
\EN
\EQ
\gamma_1=\half(\alpha_{32}-\alpha_{23})
\EN
\EQ
\gamma_2=\half(\alpha_{13}-\alpha_{31})
\EN
\EQ
\gamma_3=\half(\alpha_{21}-\alpha_{12})
\EN

\subsection*{If there is only $z$-dependence}

If there is only $z$-dependence, only $2\times2\times2$ components will
be relevant, i.e.\ $\alpha_{ij}$ and $\eta_{ij3}$ for $i,j=1,2$.
Furthermore, because there are only $z$ derivatives,
\EQ
\meanB_{j,k}^{\rm(A)}=\meanB_{j,k}^{\rm(S)},
\EN
so the information contained in $\kappa_{ijk}$ must be the same
as in $\tilde{il}$, and so we can write
\EQ
\meanemf_i=\alpha_{ij}\meanB_j-\tilde{\eta}_{ij}\meanJ_j,
\quad i,j=1,2,
\EN
where $\meanJJ=\nab\times\meanBB$ is the mean current density, and
\EQ
\tilde{\eta}_{il}=\eta_{ijk}\epsilon_{jkl}
\label{tildeETArel}
\EN
is the resistivity tensor operating only on the mean current density.
Express these as
\EQ
\delta_3=-\half(\eta_{113}+\eta_{223})
\EN
\EQ
\eta_{11}=+\eta_{123}
\EN
\EQ
\eta_{22}=-\eta_{213}
\EN
\EQ
\eta^{\rm S}_{12}=\eta^{\rm S}_{21}=\half(\eta_{113}-\eta_{223})
\EN
and for the $\alpha$ tensor
\EQ
\gamma_3=\half(\alpha_{21}-\alpha_{12})
\EN
\EQ
\alpha^{\rm S}_{11}=\alpha_{11}
\EN
\EQ
\alpha^{\rm S}_{22}=\alpha_{22}
\EN
\EQ
\alpha^{\rm S}_{12}=\alpha^{\rm S}_{21}=\half(\alpha_{12}+\alpha_{21})
\EN

In the 1-D case it may be advantageous to stick to the original
$\alpha_{ij}$ and $\eta_{ij3}$ tensors, so we note therefore
\EQ
\tilde\eta_{11}=+\half\eta_{123}
\EN
\EQ
\tilde\eta_{22}=-\half\eta_{213}
\EN
\EQ
\tilde\eta_{12}=-\half\eta_{113}
\EN
\EQ
\tilde\eta_{21}=+\half\eta_{223}
\EN

\section*{Isotropic turbulence}

$\alpha_{ij}(k=1)$
\begin{verbatim}
    -0.150    -0.001     0.001
    -0.001    -0.151     0.000
     0.001     0.000    -0.147
\end{verbatim}
$\eta_{ijz}(k=1)$
\begin{verbatim}
     0.000    -0.032     0.000
     0.032     0.000     0.000
     0.000     0.000     0.000
\end{verbatim}

$\alpha_{ij}(k=2)$
\begin{verbatim}
    -0.144    -0.001     0.001
    -0.001    -0.145     0.000
     0.001     0.000    -0.131
\end{verbatim}
$\eta_{ijz}(k=2)$
\begin{verbatim}
    -0.000    -0.061     0.000
     0.060    -0.000     0.000
     0.000     0.001     0.000
\end{verbatim}

$\alpha_{ij}(k=3)$
\begin{verbatim}
    -0.136     0.000     0.001
     0.001    -0.133     0.000
     0.001    -0.000    -0.108
\end{verbatim}
$\eta_{ijz}(k=3)$
\begin{verbatim}
     0.001    -0.083     0.000
     0.083     0.001    -0.000
     0.001     0.000     0.001
\end{verbatim}

nosoca32c31 (SOCA):
$\alpha_{ij}$
\begin{verbatim}
    -4.708    -0.000
     0.000    -4.708
\end{verbatim}
$\eta_{ij}$
\begin{verbatim}
     3.329     0.000
    -0.000     3.329
\end{verbatim}

$\alpha_{ij}$
\begin{verbatim}
    -0.190    -0.000
     0.000    -0.190
(nosoca32c32_4proc, non-SOCA):
\end{verbatim}
$\eta_{ij}$
\begin{verbatim}
    -0.005     0.000
    -0.000    -0.005
\end{verbatim}

\section*{Helical homogeneous turbulence}

\begin{verbatim}
etat    alpha   alp0     eta   urms   brms   tmax  run
0.0108 -0.0283 -0.0157   5e-3  0.119  0.128   337  impNOSOCA32a2 (started)
diverges.....  -0.0128   1e-3  0.141  0.151   601  impNOSOCA64a2
0.0078 -0.0261 -0.0091   1e-3  0.153  0.166   186  impNOSOCA128a2 (giga)
0.0088 -0.0410 -0.0157   5e-3  0.119  0.128 17514  imp32c
0.0145 -0.0647 -0.0128   2e-3  0.141  0.152  1235  imp64b
0.0156 -0.0711 -0.0089   1e-3  0.151  0.166   418  imp128b
0.0089 -0.0411 -0.0157   2e-3  0.119  0.128  8520  imp32b
0.0155 -0.0699 -0.0091   1e-3  0.150  0.165   487  imp64a (restarted)
0.0149 -0.0697 -0.00550  5e-4  0.155  0.175   525  imp128a
\end{verbatim}

\section*{Helical homogeneous turbulence}

Use $\eta=\nu=0.1$

\begin{verbatim}
etat     alpha   urms      force   tmax  run
7.52e-5  3.83e-4 3.88e-02  5.0e-2  527   kin16a
7.52e-7  3.83e-6 3.89e-03  5.0e-3  588   kin16a
\end{verbatim}

\section*{Nonhelical homogeneous turbulence with rotation}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\textwidth]{prot32}
\end{center}\caption[]{
Dependence of $\eta_{\rm t}$ and $\delta-\kappa$ on $\Omega$.
Solid line is for kinetic forcing, and dashed line is
for magnetic forcing.
}\label{p1}\end{figure}

Note nonlinearity in Omega dependence of delta and etat.

\begin{verbatim}
A negative value of force0 indicates magnetic forcing.
etat   2del-kap Omega urms   brms   force0 eta   tmax  run
0.00622 0.00003 0.0   0.0663 0.131  -0.01  5e-3  4960  brot32o
0.00595 0.00027 0.01  0.0663 0.131  -0.01  5e-3  7027  brot32h
0.00615 0.00046 0.02  0.0663 0.131  -0.01  5e-3  5052  brot32g
0.00578 0.00063 0.05  0.0662 0.131  -0.01  5e-3  5218  brot32f
0.00539 0.00069 0.1   0.066  0.131  -0.01  5e-3  4526  brot32a
0.0049  0.0006  0.2   0.065  0.130  -0.01  5e-3  1690  brot32b (giga2)
0.00408 0.00039 0.5   0.060  0.134  -0.01  5e-3  4843  brot32c
0.00383 0.00018 1.0   0.0524 0.139  -0.01  5e-3  4909  brot32d
0.00363 0.00006 2.0   0.0432 0.145  -0.01  5e-3  4967  brot32e
                0.01                -0.01  5e-3        brotNOSOCA32h
                0.02                -0.01  5e-3        brotNOSOCA32g
                0.05                -0.01  5e-3        brotNOSOCA32f
0.00489 0.00090 0.1   0.0659 0.131  -0.01  5e-3  3411  brotNOSOCA32b
0.00417 0.00092 0.2   0.0695 0.131  -0.01  5e-3  8063  brotNOSOCA32a (stopped)
0.00381 0.00053 0.5   0.0600 0.134  -0.01  5e-3  3563  brotNOSOCA32c
0.00354 0.00024 1.0   0.0524 0.139  -0.01  5e-3  3541  brotNOSOCA32d
0.00340 0.00012 2.0   0.0432 0.145  -0.01  5e-3  3723  brotNOSOCA32e
0.0229  0.00078 0.01  0.149  0.000  +0.05  5e-3  3553  rot32f
0.0228  0.00133 0.02  0.149  0.000  +0.05  5e-3  3407  rot32g
0.0220  0.00245 0.05  0.149  0.000  +0.05  5e-3  3542  rot32h
0.0214  0.00354 0.1   0.149  0.000  +0.05  5e-3 10279 rot32c
0.02157 0.00470 0.2   0.150  0.000  +0.05  5e-3 12722  rot32a
0.0175  0.0040  0.5   0.150  0.000  +0.05  5e-3 13006  rot32b
0.01416 0.00177 1.0   0.152  0.000  +0.05  5e-3  4056  rot32d
0.00964 0.00046 2.0   0.151  0.000  +0.05  5e-3  4203  rot32e
0.01839 -.00005 0.0   0.149  0.000  +0.05  5e-3 12236  rotNOSOCA32o
                0.01  0.149  0.000  +0.05  5e-3        rotNOSOCA32g
                0.02  0.149  0.000  +0.05  5e-3        rotNOSOCA32g
                0.05  0.149  0.000  +0.05  5e-3        rotNOSOCA32f
0.01605 0.00174 0.1   0.149  0.000  +0.05  5e-3  3039  rotNOSOCA32c
0.01668 0.00371 0.2   0.149  0.000  +0.05  5e-3  3106  rotNOSOCA32a
0.0147  0.0077  0.5   0.150  0.000  +0.05  5e-3  4310  rotNOSOCA32b
0.0137  0.00401 1.0   0.152  0.000  +0.05  5e-3  3010  rotNOSOCA32d
0.0097  0.00101 2.0   0.150  0.000  +0.05  5e-3  3008  rotNOSOCA32e
0.0122  0.00048 0.1   0.100  0.159  -0.01  2e-3   538  brot64a
0.0094  0.00070 0.2   0.099  0.161  -0.01  2e-3   338  brot64b (lost data)
0.01003 0.00028 0.0   0.100  0.159  -0.01  2e-3  2433  brotNOSOCA64o (OK)
0.00866 0.00149 0.2   0.098  0.160  -0.01  2e-3  2508  brotNOSOCA64b
0.00529 0.00064 2.0   0.071  0.192  -0.01  2e-3  2976  brotNOSOCA64e
0.0257  0.00046 0.0   0.193  0.000  +0.05  2e-3  1724  rotNOSOCA64o (STOPPED)
0.0304  0.00325 0.5   0.202  0.000  +0.05  2e-3   805  rot64a (STOPPED)
0.0270  0.00045 0.2   0.195  0.000  +0.05  2e-3  2434  rotNOSOCA64b (ANALYZE)
0.0327  0.01212 0.5   0.202  0.000  +0.05  2e-3   748  rotNOSOCA64c
0.02221 0.00159 1.0   0.255  0.000  +0.05  2e-3  1891  rotNOSOCA64d (stopped)
0.02244 0.00130 2.0   0.240  0.000  +0.05  2e-3  2248  rotNOSOCA64e
0.01542 0.00004 0.0   0.119  0.179  -0.01  1e-3   213  brot128o (workq)
0.0088  0.0003  0.5   0.112  0.187  -0.01  1e-3   198  brot128a (cant read)
0.0110  0.00126 0.0   0.116  0.177  -0.01  1e-3   308  brotNOSOCA128o (giga2, XX)
0.00781 0.00160 0.5   0.112  0.187  -0.01  1e-3   259  brotNOSOCA128a3 giga2)
0.0121  0.00035 0.2   0.11   0.18   -0.01  1e-3   400  brot128b (restarted XX)
0.03324 -.03000 0.0   0.215  0.000  +0.05  1e-3   276  rotNOSOCA128o (giga2)
0.0394  0.00076 0.5   0.229  0.000  +0.05  1e-3   406  rotNOSOCA128a2 (blew up!?)
                0.5          0.000  +0.05  1e-3        rotNOSOCA128a3 (giga2)
0.04529 0.00011 0.0   0.216  0.000  +0.05  1e-3   209  rot128o2 (workq, ANALYSE)
0.0354  0.00171 0.5   0.229  0.000  +0.05  1e-3   318  rot128c
\end{verbatim}

New simulations, SOCA.

\begin{verbatim}
Omega nu   eta   etatm     etaxym    etayxm   urms   force tmax run
5e-2  5e-1 5e-1  1.17e-05  2.36e-08 -1.65e-08 0.037  1e-1  462  SOCArot16c
2e-1  5e-1 5e-1  1.17e-05  4.94e-08 -6.54e-08 0.037  1e-1  270  SOCArot16d
1e0   5e-1 5e-1  1.17e-05  3.31e-07 -3.19e-07 0.037  1e-1  442  SOCArot16a
2e0   5e-1 5e-1  1.17e-05  5.09e-07 -6.26e-07 0.037  1e-1  121  SOCArot16b
\end{verbatim}
In all cases $\tau_c=0.045$ and $St=0.0083$.

\section*{Nonhelical homogeneous turbulence with shear}

There is no rotation here (except in 2 cases).

\begin{verbatim}
etat   etaxy  etayx  Shear urms   brms  force0 eta   tmax SOCA run
0.0134 0.0090 -.0028 -0.20 0.141  0.000 0.05   5e-3   53   N  kinshear32a Om=.5
0.0142 0.0139 0.0014 -0.20 0.150  0.000 0.05   5e-3  380   Y  kinshear32b
0.0209 0.0098 -.0040 -0.20 0.200  0.000 0.05   1e-3   12   N  kinshear64a Om=.5
0.00008 .0098 -.0040 -0.20 0.200  0.000 0.05   1e-3  887   Y  kinshear16a
\end{verbatim}

Runs with $S=-0.2$, $f_0=0.05$.
\begin{verbatim}
nu   eta   etat      etaxy     etayx    urms   force tmax SOCA run
1e-1 1e-1  7.98e-05  2.50e-06  3.19e-06 0.038  5e-2  330   Y  kinshear16a
5e-2 5e-2  3.42e-04  2.80e-05  2.55e-05 0.052  5e-2  567   Y  kinshear16d
2e-2 2e-2  2.54e-03  6.54e-04  3.80e-04 0.083  5e-2  887   Y  kinshear16b
\end{verbatim}

Runs with $S=-0.2$, $f_0=0.05$.
\begin{verbatim}
nu   eta   etat      etaxy     etayx    urms   force tmax SOCA run
1e-1 2e-1  7.98e-05  2.50e-06  3.19e-06 0.038  5e-2  330   Y  kinshear16a
\end{verbatim}

Runs with $S=-0.2$, $f_0=0.05$.
\begin{verbatim}
nu   eta   etat      etaxy     etayx    urms   force tmax SOCA run
1e-1 2e-1  5.29e-05  8.66e-07  1.60e-06 0.038  5e-2  458   N  NOSOCkinshear16f
1e-1 1e-1  7.98e-05  2.74e-06  3.29e-06 0.038  5e-2  632   N  NOSOCkinshear16a
1e-1 5e-2  1.07e-04  5.43e-06  3.74e-06 0.038  5e-2  458   N  NOSOCkinshear16g
1e-1 2e-2  1.33e-04  1.04e-05  3.14e-06 0.038  5e-2  445   N  NOSOCkinshear16e
1e-1 1e-2  1.46e-04  1.37e-05  2.23e-06 0.038  5e-2  917   N  NOSOCkinshear16b
1e-1 5e-3  1.52e-04  1.52e-05  1.43e-06 0.038  5e-2  458   N  NOSOCkinshear16c
1e-1 2e-3  1.57e-04  1.91e-05  1.11e-06 0.038  5e-2  458   N  NOSOCkinshear16d
1e-1 1e-3  1.69e-04  1.83e-05 -3.05e-06 0.038  5e-2  110   N  NOSOCkinshear16h
\end{verbatim}

Magnetic forcing, runs with $S=-0.2$, $f_0=0.05$.
\begin{verbatim}
nu   eta   etat      etaxy     etayx    urms   force tmax SOCA run
1e-1 2e-1  1.09e-06  1.21e-07  4.87e-08 0.007  5e-2  369   N  NOSOCmagshear16a
1e-1 1e-1  1.49e-05  2.75e-06  7.39e-07 0.016  5e-2  464   N  NOSOCmagshear16a
1e-1 5e-2  1.35e-04  3.77e-05  6.20e-06 0.035  5e-2  449   N  NOSOCmagshear16a
\end{verbatim}

Magnetic forcing, runs with $S=-0.2$, $f_0=0.02$.
\begin{verbatim}
nu   eta   etat      etaxy     etayx    urms   force tmax SOCA run
1e-1 2e-2  6.07e-08  3.97e-09  9.51e-10 0.001  2e-2  289   N  NOSOCmagShear32f
1e-1 1e-2  5.41e-04  2.50e-04  2.80e-06 0.031  2e-2  448   N  NOSOCmagShear32b
1e-1 5e-3  1.49e-03  8.88e-04 -2.51e-05 0.048  2e-2  409   N  NOSOCmagShear32c
\end{verbatim}

Magnetic forcing, runs with $S=-0.2$, no SOCA
\begin{verbatim}
 nu     eta     etat       etaxy     etayx      urms      brms    force  tmax      run
1e-01  1e-01  2.59e-05   3.48e-06   1.12e-06  1.50e-02  1.80e-01   0.1   92.73  magnohel32A
1e-01  1e-01  3.36e-04   3.40e-05   1.08e-05  5.60e-02  3.60e-01   0.1   30.27  magnohel32B
1e-01  1e-01  2.98e-03   1.63e-04   1.81e-05  1.95e-01  6.86e-01   0.2   11.07  magnohel32C
1e-01  1e-01  1.75e-02   5.59e-04   1.73e-04  5.80e-01  1.26e+00   0.4    4.19  magnohel64D
1e-01  1e-01  3.94e-02   2.07e-03   1.67e-03  1.25e+00  2.01e+00   0.8    0.61  magnohel64E
1e-01  1e-02  2.49e-02   4.12e-04  -6.39e-04  5.00e-01  1.22e+00   0.2    3.88  magnohel64H
1e-01  1e-03  2.95e-02  -1.35e-04  -2.40e-04  5.44e-01  1.44e+00   0.2    3.63  magnohel64F
1e-01  1e-04  2.95e-02  -2.30e-04   6.14e-05  5.48e-01  1.51e+00   0.2    3.62  magnohel64G
\end{verbatim}

Kinetic forcing, runs with $S=-0.2$, 
\begin{verbatim}
  nu      eta     etat      etaxy     etayx    urms   force    tmax   SOCA      run
1.0e-01 1.0e-01 8.53e-05  2.83e-06  3.02e-06 3.60e-02  0.05   143.0     N    kinohel32D
\end{verbatim}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\textwidth]{kinshear16c}
\end{center}\caption[]{
Dependence of $\eta_{\rm t}$ on $\eta$. Kinetic forcing.
}\label{kinshear16c}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\textwidth]{magshear16c}
\end{center}\caption[]{
Dependence of $\eta_{\rm t}$ on $\eta$. Magnetic forcing.
}\label{magshear16c}\end{figure}

\section*{New runs (September 2007)}

Kinetic forcing, runs with $S=-0.1$.

Define $q_{ij}=\tilde\eta_{ij}/\tilde\eta_{\rm t}$
and $C_S=S/(\eta_{\rm t}k_1^2)$, so the growth rate becomes
\EQ
\lambda=\eta_{\rm t}k_1^2\left(-1+\sqrt{q_{21}(q_{12}+C_S)}\right)
\EN
with $q_{21}\approx-0.1$, $q_{12}\approx+0.5$, $C_S\approx30$,
so that $\lambda/(\eta_{\rm t}k_1^2)\approx2$.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\textwidth]{kinshear_Re14}
\end{center}\caption[]{
Roberts flow
}\label{kinshear_Re14}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\textwidth]{kinshear_Re14_crit}
\end{center}\caption[]{
Roberts flow
}\label{kinshear_Re14_crit}\end{figure}


\newpage
\section*{Galloway--Proctor flow}

%----------------------
\EQ
\eta=0.01\quad %128^3
\alpha_{ij}=\pmatrix{
 -6.66 &  -7.63 \cr
  8.44 &   0.80 }
\quad
\eta_{ij}=\pmatrix{
 -1.26 & -11.40 \cr
  4.98 &  -1.04 }
\EN
%----------------------
\EQ
\eta=0.02\quad %128^3
\alpha_{ij}=\pmatrix{
  1.58 &  -0.53 \cr
  0.44 &   0.01 }
\quad
\eta_{ij}=\pmatrix{
  1.27 &  -0.35 \cr
  0.48 &   1.24 }
\EN
%----------------------
\EQ
\eta=0.05\quad %64^3
\alpha_{ij}=\pmatrix{
  0.83 &  -0.57 \cr
  0.56 &   0.11 }
\quad
\eta_{ij}=\pmatrix{
  1.02 &  -0.74 \cr
  0.34 &   1.02 }
\EN
%----------------------


\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/tex/mhd/alphaeta/notes.tex,v 1.65 2007/09/22 20:31:53 brandenb Exp $
\end{verbatim}





\end{document}
