A closer look at the 'Flux Limiter'

\lambda is the 'flux limiter'. It comes into the equations in the diffusion term:

\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)

This is because instead of solving the following conservation equation,

\frac{\partial E}{\partial t} =- \nabla \cdot F

(which would require an additional equation for F), we make the 'flux-limited' approximation that,

F=-\frac{c \lambda}{\kappa_R \rho}\nabla E

This then turns the conservation equation into a diffusion equation:

\frac{\partial E}{\partial t} = \nabla \cdot (\frac{c \lambda}{\kappa_R \rho} \nabla E)

The value of \lambda controls whether the radiation is diffusing in the free-streaming limit (\lambda \rightarrow 0), i.e. at the speed of light, or is diffusing as it would in the optically thick limit (\lambda \rightarrow \frac {1}{3}). The FLD approximation does well at these two limits, but not in between. Let's examine these two limits in more detail and then consider why it doesn't perform as well in between. Here is the functional form of \lambda,

\lambda = \frac{1}{R}(\coth{R}-\frac{1}{R})

where,

R=|\frac{\nabla E}{\kappa_R \rho E}|

Graphically, we have:

For what follows, it will be helpful to quickly recall the following. Opacity (\kappa) is related to the absorption coefficient (\alpha) by,

\kappa \rho=\alpha (cm^{-1})

(where the absorption coefficient reduces the intensity of the ray by dI_\nu=-\alpha_\nu I_\nu ds)

and the optical depth is defined by,

\tau_\nu (s)=\int^s_{s_0} \alpha_\nu(s')ds'

When \tau>1 (integrated along a typical path through the medium), the material is optically thick, and when \tau<1, optically thin.

The mean optical depth of an absorbing material can be shown to =1, and so in terms of the mean free path (l_\nu) we have:

\bar{\tau_\nu}=\alpha_\nu l_\nu = 1

or

l_\nu=\frac{1}{\rho \kappa}

Thus, given the opacity and the density of the material, one can compute the mean free path.

Optically thick limit

From the graph above, we see that as R\rightarrow 0, \lambda\rightarrow \frac{1}{3}. How to interpret this? R is essentially the ratio of the mean free path l=1/\kappa \rho, to the radiation's scale height, h^{-1}=\frac{\nabla E}{E} (shown below). Thus, as R\rightarrow 0, we have:

\boxed{R=\frac{l}{h}\rightarrow 0 ~,~ \lambda\rightarrow \frac{1}{3}}

(note there is no frequency dependence now in the mean free path, as K_R integrated over frequency space to give an 'average' opacity).

That is, the radiation travels infinitesmally small distances before it is absorbed or scattered - and thus, we are in the optically thick regime.

The rad diffusion equation then becomes,

\frac{\partial E}{\partial t} = \nabla \cdot (\frac{1}{3}\frac{c}{\kappa_R \rho} \nabla E)

which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer.

Free streaming limit

In the other limit, R\rightarrow \infty, we have:

\boxed{R=\frac{l}{L}\rightarrow \infty~,~\lambda\rightarrow 0}

That is, a photon travels an infinite distance before it interacts with another particle — i.e. we are in the free-streaming limit.

In this case, the rad diffusion equation becomes:

\frac{\partial E}{\partial t} = 0

That is, the radiation diffuses instantly through the grid. Recall, how this radiative energy couples to the gas is given by the coupling term in the radiation equation, not shown here.

Estimating the diffusion time

Estimating \lambda…

Depending on the value of R, we will get different values of \lambda. So, we start by approximating R.

Note that,

\frac{\nabla E}{E}=\frac{d(\ln E)}{dx}

so if we make the approximation,

\frac{\nabla E}{E}\approx -\frac{1}{h}

(by dimensional arguments and assuming the gradient is negative), we have:

\frac{d(\ln E)}{dx}=-\frac{1}{h}

which integrates to:

\ln E = \ln E_0 - \frac{x}{h}

or,

E = E_0 e^{ - \frac{x}{h}}

From this equation, it is clear that h is the scale-height. Thus, by writing R as:

R \approx |-\frac{1}{h \kappa_R\rho}| \approx \frac{1}{h \kappa_R\rho}

we can set h to be the distance between the sink and box side (h=L), and imagine it as the scale height for the radiation.

Now, Offner et al. 2009 state that the Rosseland specific opacity is best fit by

\kappa_R=0.23~\frac{cm^2}{g}

for 10 K gas.