Thermal Diffusion

Castro incorporates explicit thermal diffusion into the energy equation. In terms of the specific internal energy, \(e\), this appears as:

\[\rho \frac{De}{Dt} + p \nabla \cdot \ub = \nabla \cdot \kth \nabla T\]

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

It 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.

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:

\[\frac{\partial T}{\partial t} = D \nabla^2 T\]

where \(D \equiv \kth/(\rho c_v)\).

The timestep limiter for this is:

\[\Delta t_\mathrm{diff} \le \frac{1}{2} \frac{\Delta x^2}{D}\]

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
```

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. However, if your interest is only on the diffusion in the interior, you can use the parameters:

castro.diffuse_cutoff_density

castro.diffuse_cutoff_density_hi

to specify a density, below which, diffusion is not modeled. This is implemented in the code by linearly scaling the conductivity to zero between these limits, e.g.,

\[\kth = \kth \cdot \frac{\rho - \mathtt{castro.diffuse\_cutoff\_density}}{\mathtt{castro.diffuse\_cutoff\_density\_hi} - \mathtt{castro.diffuse\_cutoff\_density}}\]

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 [66].

Unit Tests

A simple test problem that sets up a Gaussian temperature profile and does pure diffusion is provided as diffusion_test.