Changes between Version 13 and Version 14 of u/johannjc/scratchpad4

Timestamp:

10/01/15 07:32:01 (9 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad4

@@ -118 +118 @@
 Now for the perpendicular term, we have
 
- $\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{-n \kappa_\perp}{B^2 \left (\Lambda+1\right )} \left ( \hat{b} \cdot \nabla T_*^{\Lambda+1} \right ) -  \frac{-n \kappa_\perp}{B^2 \left (\Lambda+1\right )}\nabla^2 T_*^{\Lambda+1} \right ]$
+ $\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \frac{-n \kappa_\perp}{B^2 \left (\Lambda+1\right )} \left ( \hat{b} \cdot \nabla T_*^{\Lambda+1} \right ) +  \frac{n \kappa_\perp}{B^2 \left (\Lambda+1\right )}\nabla^2 T_*^{\Lambda+1} \right ]$
+
+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 ] $