Thermal Diffusion
Castro incorporates explicit thermal diffusion into the energy equations. In terms of the specific internal energy, \(e\), this appears as:
where \(\kth\) is the thermal conductivity, with units \(\mathrm{erg~cm^{-1}~s^{-1}~K^{-1}}\).
Note
To enable diffusion, you need to compile with:
USE_DIFFUSION=TRUE
Thermal Diffusion related source codes are contained in the diffusion
directory.
Thermal Diffusion is treated explicitly, by constructing the contribution to the
evolution as a source term. This is time-centered to achieve second-order accuracy
in time.
Overall Procedure
Computing Thermal Conductivity
The main function that computes the diffusion term is getTempDiffusionTerm()
.
Within getTempDiffusionTerm()
, it first calculates the cell centered
thermal conductivity, \(\kth\) contained in the variable coeff_cc
using the function fill_temp_cond()
located in diffusion_util.cpp
.
fill_temp_cond()
fills an eos_state
using the
input conserved variables, which is used to calculate \(\kth\) via
conductivity(eos_state)
. conductivity()
routine is supplied via
the Microphysics
package. See Conductivities to see the
specific choices of conductivity routines available.
Note
The diffusion approximation breaks down at the surface of stars, where the density rapidly drops and the mean free path becomes large. In those instances, you should use the flux limited diffusion module in Castro to evolve a radiation field.
Now \(\kth\) is reset to 0 unless \(\rho \gt \mathrm{castro::diffuse\_cutoff\_density}\). And if \(\rho \lt \mathrm{castro::diffuse\_cutoff\_density\_hi}\), a linear scaling of \(\kth\) is done as:
Lastly, \(\kth\) is scaled with castro::diffuse_cond_scale_fac
,
a runtime parameter controlled by the user.
After obtaining cell-centered \(\kth\), we do an average along
i, j, and k depending on the direction to obtain face-centered MultiFabs.
This is stored in coeffs
, a vector of MultiFabs, and the number of
MultiFabs corresponds to geometry dimension, since a \(\nabla\) operator
will be applied to it later.
These Multifabs have 1 ghost cells due to the nature of MLMG solvers.
Computing Thermal Diffusion
We are now ready to compute \(\nabla \cdot \kth \nabla T\)
after obtaining \(\kth\). This is done in the applyop_mlmg()
function
in Diffusion.cpp
. It defines mlabec
an instance of class
MLABecLaplacian
which defines the Laplacian of the form:
where A and B are constant scalars, and \(\alpha\) and \(\beta\)
are scalar fields. In order to make it correctly represents our diffusion term,
we set A = 0 and B = -1, which is done via mlabec.setScalars(0.0, -1.0)
.
Now we recognize \(\beta = \kth\), which needs to be an array of MultiFab,
corresponding to dimension. This is done via mlabec.setBCoeffs()
.
One of the important flag that we need to pass in is to set setMetricTerm(true)
.
This enables modifications due to curvilinear coordinates.
Finally we create an instance of MLMG
using mlabec
, and call
mlmg.apply()
, which simply evaluates the LHS but do not solve it.
See more information in the amrex documentation:
https://amrex-codes.github.io/amrex/docs_html/LinearSolvers.html
Timestep Limiter
Castro integrates diffusion explicitly in time; this means that there is a diffusion timestep limiter.
To see the similarity to the thermal diffusion equation, consider the special case of constant conductivity, \(\kth\), and density, and assume an ideal gas, so \(e = c_v T\), where \(c_v\) is the specific heat at constant volume. Finally, ignore hydrodynamics, so \(\ub = 0\). This gives:
where \(D \equiv \kth/(\rho c_v)\).
The timestep limiter for this is:
This is implemented in estdt_temp_diffusion
.
Runtime Parameters
The following parameter affects diffusion:
castro.diffuse_temp
: enable thermal diffusion (0 or 1; default 0)A pure diffusion problem (with no hydrodynamics) can be run by setting:
castro.diffuse_temp = 1 castro.do_hydro = 0
castro.diffuse_cond_scale_fac
: a linear scaling to \(\kth\). (default 0).castro.diffuse_cutoff_density
: density under which \(\kth\) is set to 0. (Default: -1e200)castro.diffuse_cutoff_density_hi
: density under which a linear scaling is applied to \(\kth\), see section Computing Thermal Diffusion for details. (Default: -1e200)
Conductivities
To complete the setup, a thermal conductivity must be specified. These are supplied by Microphysics, and use an interface similar to the equation of state interface.
Note
The choice of conductivity must be specified at compile-time via
the CONDUCTIVITY_DIR
option.
The current choices of conductivity are:
constant
: A simple constant thermal conductivity. This can be selected by setting:CONDUCTIVITY_DIR := constant
in your
GNUmakefile
. To set the value of the conductivity (e.g., to \(100\)), you add to your input file:conductivity.const_conductivity = 100.0
constant_opacity
: A simple constant opacity. This is converted to an opacity as:\[\kth = \frac{16 \sigma_B T^3}{3 \kappa_\mathrm{const} \rho}\]where \(\kappa_\mathrm{const}\) is the opacity, with units \(\mathrm{cm^2~g^{-1}}\). This is selected by setting:
CONDUCTIVITY_DIR := constant_opacity
in your
GNUmakefile
. To set the value of the opacity, e.g., to 0.2 (for electron scattering), set:conductivity.const_opacity = 0.2
in the inputs file.
stellar
: This is the set of conductivities and radiative opacities appropriate for stellar interiors described in [68].
Unit Tests
A simple test problem that sets up a Gaussian temperature profile
and does pure diffusion is provided as diffusion_test
.