Changes between Version 7 and Version 8 of AstroBearProjects/multiphysics

Timestamp:

09/04/12 17:49:21 (12 years ago)

Author:

Shule Li

Comment:

—

AstroBearProjects/multiphysics

@@ -82 +82 @@
 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]] 
 If we dot the resistive induction equation with the magnetic field B, we obtain the time evolution equation for magnetic energy:[[BR]][[BR]]
-[[latex($\partial (B^2)/\partial t  +  \nabla \cdot S =  - \textbf{J}\cdot\textbf{E}$)]][[BR]][[BR]]
-Since we know that:[[BR]][[BR]]
-[[latex($\textbf{J}=\eta\textbf{E}$)]][[BR]]
-and also:[[BR]][[BR]]
-[[latex($\textbf{E}=\nabla \times \textbf{B}$)]][[BR]][[BR]]
-This leads to an extra energy source term:[[BR]][[BR]]
-[[latex($\rho\frac{\partial e}{\partial t}=\eta (\nabla \times \textbf{B})^2$)]][[BR]][[BR]]
-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]]
+[[latex($\partial (B^2)/\partial t  +  \nabla \cdot \textbf{S} =  - \textbf{J}\cdot\textbf{E} = -j^2/\eta$)]][[BR]][[BR]]
+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]]
+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]]
+The total energy change for the resistive step is therefore:[[BR]][[BR]]
+[[latex($\partial \epsilon/\partial t  +  \nabla \cdot \textbf{S} = 0$)]][[BR]][[BR]]
+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]]
+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.
+
+[[BR]][[BR]][[BR]]
 
 Here I overview two approaches to solve the resistive MHD equations. [[BR]][[BR]]
@@ -301 +302 @@
 
 
-