Thermal Diffusion

Castro incorporates explicit thermal diffusion into the energy equations. In terms of the specific internal energy, $e$, this appears as:

$$\rho \frac{De}{Dt}+p\mathrm{\nabla }\cdot \mathbf{u}=\mathrm{\nabla }\cdot k_{\mathrm{th}}\mathrm{\nabla }T$$

where $k_{\mathrm{th}}$ is the thermal conductivity, with units $\mathrm{erg}{\mathrm{cm}}^{-1}{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$.

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, $k_{\mathrm{th}}$ 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 $k_{\mathrm{th}}$ 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 $k_{\mathrm{th}}$ is reset to 0 unless $\rho >\mathrm{castro}::\mathrm{diffuse}\mathrm{\_}\mathrm{cutoff}\mathrm{\_}\mathrm{density}$. And if $\rho <\mathrm{castro}::\mathrm{diffuse}\mathrm{\_}\mathrm{cutoff}\mathrm{\_}\mathrm{density}\mathrm{\_}\mathrm{hi}$, a linear scaling of $k_{\mathrm{th}}$ is done as:

$$k_{\mathrm{th}}=k_{\mathrm{th}}\cdot \frac{\rho -\mathtt{castro}\mathtt{.}\mathtt{diffuse}\mathtt{\_}\mathtt{cutoff}\mathtt{\_}\mathtt{density}}{\mathtt{castro}\mathtt{.}\mathtt{diffuse}\mathtt{\_}\mathtt{cutoff}\mathtt{\_}\mathtt{density}\mathtt{\_}\mathtt{hi}-\mathtt{castro}\mathtt{.}\mathtt{diffuse}\mathtt{\_}\mathtt{cutoff}\mathtt{\_}\mathtt{density}}$$

Lastly, $k_{\mathrm{th}}$ is scaled with castro::diffuse_cond_scale_fac, a runtime parameter controlled by the user.

After obtaining cell-centered $k_{\mathrm{th}}$, 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 $\mathrm{\nabla }$ 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 $\mathrm{\nabla }\cdot k_{\mathrm{th}}\mathrm{\nabla }T$ after obtaining $k_{\mathrm{th}}$. 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:

$$(A\alpha -B\mathrm{\nabla }\cdot \beta \mathrm{\nabla })\varphi =f$$

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 $\beta =k_{\mathrm{th}}$, 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, $k_{\mathrm{th}}$, 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 $\mathbf{u}=0$. This gives:

$$\frac{\partial T}{\partial t}=D{\mathrm{\nabla }}^{2}T$$

where $D\equiv k_{\mathrm{th}}/(\rho c_v)$.

The timestep limiter for this is:

$$\mathrm{\Delta }{t}_{\mathrm{diff}}\le \frac{1}{2}\frac{\mathrm{\Delta }{x}^2}{D}$$

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
  

• castro.diffuse_cond_scale_fac: a linear scaling to $k_{\mathrm{th}}$. (default 0).

• castro.diffuse_cutoff_density: density under which $k_{\mathrm{th}}$ is set to 0. (Default: -1e200)

• castro.diffuse_cutoff_density_hi: density under which a linear scaling is applied to $k_{\mathrm{th}}$, 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:

  $$k_{\mathrm{th}}=\frac{16{\sigma }_B{T}^3}{3{\kappa }_{\mathrm{const}}\rho }$$

  where ${\kappa }_{\mathrm{const}}$ is the opacity, with units ${\mathrm{cm}}^2{\mathrm{g}}^{-1}$. 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.