Changes between Version 3 and Version 4 of u/johannjc/scratchpad5

Timestamp:

10/01/15 10:07:39 (10 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad5

@@ -15 +15 @@
 or 
 
-$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} ]$
+$\frac{\partial T}{\partial t} = \nabla \cdot \left [ n \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$
+
+or in Einstein notation
+
+$\partial_t T = \partial_i \left [ n b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$
+
+or 
+
+$\partial_t T = \partial_i \left [ n b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right )  - \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j  T^{\lambda_\perp + 1} \right )  \right ) + \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$
+
+or 
+
+$\partial_t T = \partial_i n b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} +   \partial_i n \left ( \delta_{ij} - b_i b_j \right ) \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\parallel+1} $
+
+or 
+
+$\partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1}  \partial_i n b_i  b_j \partial_j T^{\lambda_\parallel+1} +   \frac{n \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_i n \left ( \delta_{ij} - b_i b_j \right )  \partial_j T^{\lambda_\parallel+1} $
+
 
 Let's first just consider the $\chi_\parallel$ term.