Changes between Version 15 and Version 16 of u/johannjc/scratchpad4

Timestamp:

10/01/15 07:57:16 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad4

@@ -118 +118 @@
 Now for the perpendicular term, we have
 
- $\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{-n \kappa_\perp}{B^2 \left (\Lambda+1\right )} \left ( \hat{b} \cdot \nabla T_*^{\Lambda+1} \right ) +  n \frac{n \kappa_\perp}{B^2 \left (\Lambda+1\right )}\nabla^2 T_*^{\Lambda+1} \right ]$
+ $\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{-n \kappa_\perp}{B^2 \left (\Lambda+1\right )} \left ( \hat{b} \cdot \nabla T_*^{\Lambda+1} \right ) +  n \frac{n \kappa_\perp}{B^2 \left (\Lambda+1\right )}\nabla T_*^{\Lambda+1} \right ]$
 
 And performing a Taylor expansion gives
 
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ -n \hat{b} \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) } \left ( \hat{b} \cdot \nabla \left ( -\Lambda T^{\Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda}T_{*} \right ) \right ) +  n \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) } \nabla^2 \left ( - \Lambda T^{ \Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda} T_{*} \right )  \right ] $
+$\frac{\partial T}{\partial t} = \nabla \cdot \left [ -n \hat{b} \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) } \left ( \hat{b} \cdot \nabla \left ( -\Lambda T^{\Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda}T_{*} \right ) \right ) +  n \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) } \nabla \left ( - \Lambda T^{ \Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda} T_{*} \right )  \right ] $
+
+And in Einstein notation gives
+
+$\partial_t T = - \partial_i n b_i \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) } \left ( b_j  \partial_j \left ( -\Lambda T^{\Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda}T_{*} \right ) \right ) +  \partial_i n \frac{n \kappa_\perp}{B^2 \left ( \Lambda+1 \right ) }   \partial_i \left ( - \Lambda T^{ \Lambda+1} + \left ( \Lambda + 1 \right ) T^{\Lambda} T_{*} \right )  $
+
+
+The first term is very similar to the parallel case...  
+
+$\phi' = \frac{\phi-\psi \Lambda}{\psi \left ( \Lambda + 1 \right )}$
+
+$B_j =  \phi' C_j $   
+
+$C_j =   -\kappa_\perp \psi \partial_i n^2 B^{-2} b_i b_j $
+
+$D_{ij}=\phi'E_{ij}$
+
+$E_{ij} =   -\kappa_\perp \psi  n^2 B^{-2} b_i b_j $