35 | | $\rho c_v \frac{\partial T}{\partial t} = \nabla \cdot \left [ \hat{b} \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( \hat{b} \cdot \nabla T^{\lambda_\parallel+1} \right ) - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( \hat{b} \cdot \nabla T^{\lambda_\perp + 1} \right ) \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \nabla T^{\lambda_\perp+1} \right ]$ |
36 | | |
37 | | == Einstein simplification == |
38 | | |
39 | | or in Einstein notation |
40 | | |
41 | | $\rho c_v \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right ) - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j T^{\lambda_\perp + 1} \right ) \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \partial_i T^{\lambda_\perp+1} \right ]$ |
42 | | |
43 | | or |
44 | | |
45 | | $\rho c_v \partial_t T = \partial_i \left [ b_i \left ( \frac{\kappa_\parallel}{\lambda_\parallel+1} \left ( b_j \partial_j T^{\lambda_\parallel+1} \right ) - \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp+1 \right )} \left ( b_j \partial_j T^{\lambda_\perp + 1} \right ) \right ) + \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp +1 \right )} \delta_{ij} \partial_j T^{\lambda_\perp+1} \right ]$ |
46 | | |
47 | | or |
48 | | |
49 | | $\rho c_v \partial_t T = \partial_i b_i \frac{\kappa_\parallel}{\lambda_\parallel+1} b_j \partial_j T^{\lambda_\parallel+1} + \partial_i \left ( \delta_{ij} - b_i b_j \right ) \frac{n^2 \kappa_\perp}{B^2 \left ( \lambda_\perp + 1\right )} \partial_j T^{\lambda_\perp+1} $ |
50 | | |
51 | | or |
52 | | |
53 | | $\rho c_v \partial_t T = \frac{\kappa_\parallel}{\lambda_\parallel+1} \partial_i b_i b_j \partial_j T^{\lambda_\parallel+1} + \frac{\kappa_\perp}{\left ( \lambda_\perp + 1\right )} \partial_i n^2 B^{-2} \left ( \delta_{ij} - b_i b_j \right ) \partial_j T^{\lambda_\perp +1} $ |
| 46 | $\rho c_v \partial_t T = \partial_i \kappa_\parallel b_i b_j \partial_j T + \partial_i \kappa_\perp \left ( \delta_{ij} - b_i b_j \right ) \partial_j T$ |
78 | | $ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i B_{ij} \partial_j T_*^{\lambda+1}$ |
79 | | |
80 | | It is simpler to linearize the equation in terms of $T_*$ and then subsitute then vice-versa - though both give the same answer |
81 | | |
82 | | $T_*^{\lambda + 1} \approx T^{\lambda + 1} + \left ( \lambda + 1 \right ) T^{\lambda} \left ( T_{*} - T \right ) = T^{\lambda + 1} + \left ( \lambda + 1 \right ) T^{\lambda} \phi \left ( T' - T \right )$ |
83 | | $= \left(1 - \phi \left ( \lambda + 1\right ) \right ) T^{\lambda + 1}+ \phi \left ( \lambda + 1 \right ) T^{\lambda} T' $ |
84 | | |
85 | | |
86 | | So we have |
87 | | |
88 | | $\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i B_{ij} \partial_j \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) T^{\lambda} T' \right ]}$ |
89 | | |
90 | | $\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i B_{ij} \left [ \left ( 1 - \phi \left ( \lambda + 1 \right ) \right ) \partial_j T^{\lambda + 1} + \phi \left ( \lambda + 1 \right ) \partial_j T^{\lambda} T' \right ]}$ |
91 | | |
92 | | $\partial_t T = \displaystyle \sum_{\parallel, \perp}{A \partial_i B_{ij} \left [ C \partial_j T^{\lambda + 1} + D \partial_j T^{\lambda} T' \right ]}$ |
93 | | |
94 | | where $ C= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D = \phi \left ( \lambda + 1 \right )$ |
| 71 | $ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i B_{ij} \partial_j T_*$ |
| 72 | |
| 73 | $ \partial_t T = \displaystyle \sum_{\parallel, \perp} A \partial_i B_{ij} \left [ C \partial_j T + D \partial_j T' \right]$ |
| 74 | |
| 75 | |
| 76 | where $ C= \left ( 1 - \phi \right )$ and $ D = \phi $ |
104 | | $ \partial_i B_{ij} \partial_j T^{\lambda+1} = \left [ \frac{B^{ij}_{\hat{i}} T^{\lambda+1}_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda+1}_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda+1}_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda+1}_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ] \left ( 1-\delta_{ij} \right )$ |
105 | | $ + \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda+1}_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda+1}_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda+1}_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda+1}_{\hat{-i}}}{\Delta x^2} \right ] \delta_{ij} $ |
106 | | |
107 | | $ \partial_i B_{ij} \partial_j T^{\lambda}T' = \left [ \frac{B^{ij}_{\hat{i}} T^{\lambda}_{\hat{i}+\hat{j}}T'_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T^{\lambda}_{\hat{i}-\hat{j}}T'_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T^{\lambda}_{-\hat{i}+\hat{j}}T'_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T^{\lambda}_{-\hat{i}-\hat{j}}T'_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ] \left ( 1-\delta_{ij} \right ) $ |
108 | | $+ \left [ \frac{B^{ij}_{\hat{i/2}} T^{\lambda}_{\hat{i}}T'_{\hat{i}} - B^{ij}_{\hat{i/2}}T^{\lambda}_{0}T'_{0}-B^{ij}_{-\hat{i/2}} T^{\lambda}_{0}T'_{0} + B^{ij}_{-\hat{i/2}}T^{\lambda}_{-\hat{i}}T'_{-\hat{i}}}{\Delta x^2} \right ] \delta_{ij}$ |
109 | | |
110 | | |
111 | | $ \partial_i \epsilon_{ijk} b_j \partial_k T= \frac{\epsilon_{ijk} \left ( B^j_{\hat{i}} \left (T_{\hat{i}+\hat{k}} - T_{\hat{i}-\hat{k}} \right ) - B^j_{-\hat{i}} \left (T_{-\hat{i}+\hat{k}} - T_{-\hat{i}-\hat{k}} \right ) \right )}{\Delta x}$ |
| 86 | $ \partial_i B_{ij} \partial_j T = \left [ \frac{B^{ij}_{\hat{i}} T_{\hat{i}+\hat{j}} - B^{ij}_{\hat{i}}T_{\hat{i}-\hat{j}}-B^{ij}_{-\hat{i}} T_{-\hat{i}+\hat{j}} + B^{ij}_{-\hat{i}}T_{-\hat{i}-\hat{j}}}{4 \Delta x^2} \right ] \left ( 1-\delta_{ij} \right )$ |
| 87 | $ + \left [ \frac{B^{ij}_{\hat{i/2}} T_{\hat{i}} - B^{ij}_{\hat{i/2}}T_{0}-B^{ij}_{-\hat{i/2}} T_{0} + B^{ij}_{-\hat{i/2}}T_{\hat{-i}}}{\Delta x^2} \right ] \delta_{ij} $ |
117 | | $T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} AC \left [ \alpha_0 T^{\lambda+1}_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $ |
118 | | $T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [ \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$ |
| 93 | $T_0 + \Delta t \displaystyle \sum_{\parallel, \perp} AC \left [ \alpha_0 T_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T_{\pm \hat{i} \pm \hat{j}} \right ] = $ |
| 94 | $T'_0 - \Delta t \displaystyle \sum_{\parallel, \perp} AD \left [ \alpha_0 T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,i \ne j} \alpha_{\pm i, \pm j} T'_{\pm \hat{i} \pm \hat{j}} \right ]$ |
142 | | || $A$ || $\frac{\kappa_\parallel}{\rho c_v \left ( \lambda_\parallel + 1 \right ) }$ || $\frac{\kappa_\perp}{\rho c_v \left (\lambda_\perp + 1\right )}$ || |
143 | | || $B_{ij} $ || $b_i b_j$ || $n^2 B^{-2} \left (\delta_{ij} - b_i b_j \right )$ || |
144 | | || $C$ || $\left ( 1 - \phi \left ( \lambda_\parallel + 1 \right ) \right )$ || $ \left ( 1 - \phi \left ( \lambda_\perp + 1 \right ) \right )$ || |
145 | | || $D$ || $\phi \left ( \lambda_\parallel + 1 \right )$ || $ \phi \left ( \lambda_\perp + 1 \right ) $|| |
| 118 | || $A$ || $\frac{c_\parallel}{\rho c_v }$ || $\frac{c_\perp}{\rho c_v }$ || |
| 119 | || $B_{ij} $ || $b_i b_j T^{\lambda_{\parallel}}$ || $n^2 B^{-2} T^{\lambda_{\perp}} \left (\delta_{ij} - b_i b_j \right )$ || |
| 120 | || $C$ || $\left ( 1 - \phi \right )$ || $ \left ( 1 - \phi \right )$ || |
| 121 | || $D$ || $\phi $ || $ \phi $|| |