Changes between Version 34 and Version 35 of FluxLimitedDiffusion

Timestamp:

03/20/13 14:19:14 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -5 +5 @@
 = Physics of Radiation Transfer =
 
-|| [[latex(\tau = l \kappa=\frac{l}{\lambda_p})]] ||
-|| [[latex(\beta = \frac{u}{c})]] ||
+|| [[latex(\tau = l \kappa=\frac{l}{\lambda_p})]] || [[latex(\beta = \frac{u}{c})]] ||
 || [[latex(\tau << 1 )]] || streaming limit ||
 || [[latex(\tau >> 1 \mbox{, } \beta \tau << 1)]] || static diffusion limit ||
@@ -19 +18 @@
 ||  [[latex(\frac{1}{c^2} \frac{\partial \mathbf{F}}{\partial t} + \nabla \cdot \mathbf{P} = -\mathbf{G})]]  ||
 
-where 
+where the moments of the specific intensity are defined as
 
 ||  [[latex(cE=\int_0^\infty d \nu \int d \Omega I(\mathbf{n}, \nu))]]  ||
@@ -25 +24 @@
 ||  [[latex(c\mathbf{P}=\int_0^\infty d \nu \int d \Omega \mathbf{nn} I(\mathbf{n}, \nu))]]  ||
 
-and 
+and the radiation 4-momentum is given by
 
 ||   [[latex(cG^0 = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ])]]  ||
 ||   [[latex(c\mathbf{G} = \int_0^\infty d \nu \int d \Omega \left [ \kappa (\mathbf{n}, v)I(\mathbf{n}, \nu)-\eta(\mathbf{n},v) \right ] \mathbf{n})]]  ||
 
+If we had a closure relation for the radiation pressure then we could solve this system.  For gas particles, collisions tend to produce a Boltzmann Distribution which is isotropic and gives a pressure tensor that is a multiple of the identity tensor.  Photons do not "collide" with each other and they all have the same velocity 'c' but in various directions.  If the field were isotropic than [[latex(P^{ij}=\delta^{ij} 1/3 E)]] but in general [[latex(P^{ij}=f^{ij} E)]] where 'f' is the Eddington Tensor.
 
 == Simplifying assumptions ==