84 | | [[latex($\partial (B^2)/\partial t + \nabla \cdot S = - \textbf{J}\cdot\textbf{E}$)]][[BR]][[BR]] |
85 | | Since we know that:[[BR]][[BR]] |
86 | | [[latex($\textbf{J}=\eta\textbf{E}$)]][[BR]] |
87 | | and also:[[BR]][[BR]] |
88 | | [[latex($\textbf{E}=\nabla \times \textbf{B}$)]][[BR]][[BR]] |
89 | | This leads to an extra energy source term:[[BR]][[BR]] |
90 | | [[latex($\rho\frac{\partial e}{\partial t}=\eta (\nabla \times \textbf{B})^2$)]][[BR]][[BR]] |
91 | | This term can be large in the fast-varying field region, and can be a major mechanism of the dissipation of energy released by the magnetic reconnection. [[BR]][[BR]][[BR]] |
| 84 | [[latex($\partial (B^2)/\partial t + \nabla \cdot \textbf{S} = - \textbf{J}\cdot\textbf{E} = -j^2/\eta$)]][[BR]][[BR]] |
| 85 | where [[latex($\textbf{S}=\textbf{J} \times \textbf{B}$)]] is the magnetic energy flux caused by resistive diffusion and [[latex($j=|\textbf{J}|$)]] is the magnitude of the diffusive current.[[BR]][[BR]] |
| 86 | In this equation, the [[latex($\textbf{S}$)]] term accounts for the redistribution of magnetic energy (and thus the redistribution of total energy), and the [[latex($j^2/\eta$)]] term accounts for the loss of magnetic energy due to reconnection.[[BR]] |
| 87 | The total energy change for the resistive step is therefore:[[BR]][[BR]] |
| 88 | [[latex($\partial \epsilon/\partial t + \nabla \cdot \textbf{S} = 0$)]][[BR]][[BR]] |
| 89 | Here the [[latex($j^2/\eta$)]] dissipation term is absent because the dissipation of magnetic energy does not change the total energy: the loss of magnetic energy is converted into thermal energy ([[latex($j^2/\eta$)]] is indeed the heat generated by current [[latex($j$)]] in the plasma). [[BR]] |
| 90 | In the code, the [[latex($\textbf{S}$)]] term is calculated explicitly by calculating [[latex($\textbf{J} \times \textbf{B}$)]] at each face centers. This additional flux is added to the total energy flux, while the [[latex($j^2/\eta$)]] dissipation term is automatically accounted. |
| 91 | |
| 92 | [[BR]][[BR]][[BR]] |