numerical-conduction.tex

\subsection{Thermal conduction}
\label{sec.num.conductions}

\enzo\ implements the equations of isotropic heat conduction in a
manner similar to that of \citet{2007ApJ...664..135P}.  The isotropic
flux of heat is given by equation~(\ref{eq:conduction}) and we use a
value for the Spitzer conduction coefficient, $\kappa_{\rm sp} = 4.6
\times 10^{-7}$~T$^{5/2}$ erg s$^{-1}$~cm$^{-1}$~K$^{-1}$
\citep{1962pfig.book.....S}.  In this situation we are using a value
for the Coulomb logarithm, $\log \Lambda = 37.8$, that is appropriate
for the intracluster medium \citep{1988xrec.book.....S} -- in
astrophysically-relevant, fully ionized plasmas this value varies by
no more than 50\% (see, e.g., Smith et al. 2013, submitted).  It is
quite possible that the local heat flux computed in this way can
become unphysically large in the high-temperature, low-density cluster
regime when using this formulation; therefore, we take into account
the saturation of the heat flux \citep{1977ApJ...211..135C} at a
maximum level of

\begin{equation}
F_{sat} \simeq 0.4 n_e k_b T \left( \frac{2 k_b T}{\pi m_e} \right)^{1/2}.
\end{equation}

To ensure a smooth transition between the Spitzer and saturated
regimes, we define an effective conductivity using the formalism of
\citet{1988xrec.book.....S}

\begin{equation}
\kappa_{eff} = \frac{\kappa_{\rm cond}}{1 + 4.2 \lambda_e / \ell_T} \; ,
\end{equation}

where $\lambda_e$ is the electron mean free path and $\ell_T \equiv T
/ |\grad T|$ is the characteristic length scale of the local
temperature gradient.  We also assume that the conductivity of the
plasma can be described in terms of an effective conductivity, which
can be expressed as a fraction f$_{\rm sp}$ of the Spitzer
conductivity (where f$_{\rm sp} \leq 1.0$ are considered physically
realistic values).  This takes into account physical processes below
the resolution limit of the simulation, such as tangled magnetic
fields, that can suppress heat transport.

Thermal conduction in a plasma can be strongly affected by the
presence of magnetic field lines, which may suppress heat flow
perpendicular to the magnetic field.  In that case, we allow for heat
transport only parallel to the magnetic field lines in
magnetohydrodynamic simulations.  Mathematically, this is given by
equation~(\ref{eq:anisotropic_conduction}.  As with the isotropic
thermal conduction, we allow a multiplicative factor f$_{\rm sp}$ to
take into account the possible suppression of magnetic fields below
the resolution limit of the simulation.

Both isotropic and anisotropic thermal conduction in \enzo\ are
treated in an operator-split manner.  Furthermore, within the heat
transport module, transport along the x, y, and z directions are
computed in a directionally-split fashion, with heat flux along each
direction calculated at the + and - faces of the cell using the
arithmetic mean of the cell-centered temperature in cells $n$ and
$n+1$ or $n-1$ and $n$, respectively (empirically, this is more stable
than taking the geometric mean of the cell-centered temperatures).
The addition of transport along magnetic field lines requires the
calculation of cross-terms in the temperature derivatives at cell
faces, which can result in spurious oscillations in the temperature
field in regions where the temperature gradient is strong in more than
one spatial direction.  Controlling these oscillations requires the
addition of a flux limiter for calculations of the temperature field.
In this case, we choose the monotonized central difference flux
limiter \citep{1977JCoPh..23..263V}, which serves to maintain
numerical stability without sacrificing substantial speed or accuracy.