| 33 | | So this approximation only works under '''slowly varying temperature''' situation.[[BR]][[BR]][[BR]] |
| | 33 | So this approximation only works under '''slowly varying temperature''' situation.[[BR]] |
| | 34 | |
| | 35 | Upon a closer look at the simulations conducted below, we can observe that the energy inside the domain does not conserve. There is a small increase of total energy during the evolution, especially for the force-free field case. This phenomenon is explained below. [[BR]] |
| | 36 | In the case of resistive MHD, the energy can be dissipated in the form of Joule heat, comparing to the infinite conductivity case, where the voltage inside the fluid is everywhere zero, and no heat is generated by the current. [[BR]] |
| | 37 | If we dot the resistive induction equation with the magnetic field B, we obtain the time evolution equation for magnetic energy:[[BR]][[BR]] |
| | 38 | [[latex($\partial (B^2)/\partial t + \nabla \cdot S = - \textbf{J}\cdot\textbf{E}$)]][[BR]][[BR]] |
| | 39 | Since we know that:[[BR]][[BR]] |
| | 40 | [[latex($\textbf{J}=\eta\textbf{E}$)]][[BR]] |
| | 41 | and also:[[BR]][[BR]] |
| | 42 | [[latex($\textbf{E}=\nabla \times \textbf{B}$)]][[BR]][[BR]] |
| | 43 | This leads to an extra energy source term:[[BR]][[BR]] |
| | 44 | [[latex($\rho\frac{\partial e}{\partial t}=\eta (\nabla \times \textbf{B})^2$)]][[BR]][[BR]] |
| | 45 | 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]] |
