Changes between Version 7 and Version 8 of AstroBearProjects/resistiveMHD
- Timestamp:
- 08/11/11 15:26:44 (13 years ago)
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AstroBearProjects/resistiveMHD
v7 v8 5 5 The resistive equation can be written as:[[BR]] 6 6 7 [[ Image(http://www.pas.rochester.edu/~shuleli/mceq1.png, 10%)]]7 [[latex($\frac{\partial \textbf{B}}{\partial t}=\nabla \times (\eta \nabla \times \textbf{B})$)]] 8 8 9 9 The resistivity eta is a function of temperature as we all know. Now one simplification can be made that eta is a constant across the domain. This will simplify the above equation into the following form:[[BR]] 10 10 11 [[ Image(http://www.pas.rochester.edu/~shuleli/mceq2.png, 10%)]]11 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta \nabla^2 \textbf{B}$)]] 12 12 13 13 This equation holds true for each separate field component, and the form resembles the linear isotropic thermal conduction equation. We can solve the elliptic equation for three times or we can write a expanded linear system so that the solver is only called once.[[BR]] … … 16 16 Now for Eq.(1), what if we want to include the temperature dependence into the code? From the next section, you can see that when a field configuration in equilibrium is subject to strong resistivity, usually heating would occur and drive up the local resistivity. By expanding Eq.(1), we have the form:[[BR]] 17 17 18 [[ Image(http://www.pas.rochester.edu/~shuleli/mceq3.png, 20%)]]18 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta \nabla^2 \textbf{B} + \nabla \eta \times (\nabla \times \textbf{B})$)]] 19 19 20 Now the second term can be treated similarly in the code: using the CR scheme, as shown below.[[BR]]20 Now the second term depends on all of the three components of B. So we end up with equations in which the time variance of Bx, By and Bz depend on each other.[[BR]] 21 21 22 [[Image(http://www.pas.rochester.edu/~shuleli/mceq4.png, 30%)]] 23 24 So the coefficient array is changed into the following form. The second term is new comparing to the thermal conduction case. [[BR]] 25 26 [[Image(http://www.pas.rochester.edu/~shuleli/mceq5.png, 20%)]] 22 So the coefficient array is thus no longer a tri-diagonal matrix. For cases where resistive speed is slow (which is usually the case), we can use explicit solver to treat the problem instead.[[BR]] 27 23 28 24 One may wonder if it is possible to throw away the second term on the right hand side of Eq.(3) to just let the resistivity to vary with position but ignore its own spacial variation. This would give us a form of resistive MHD similar to that of the thermal diffusion case but with temperature dependence built in. [[BR]] 29 25 30 [[ Image(http://www.pas.rochester.edu/~shuleli/mceq6.png, 10%)]]26 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta (T) \nabla^2 \textbf{B}$)]] 31 27 32 28 Unfortunately, this does not work for Eq.(6) because Eq.(1) has to be divergence free. Our roughest approximation, treating the resistivity as a constant satisfies the requirement as long as the divergence and the laplacian are commutable. If we try to do the same thing to Eq.(6), we end up getting:[[BR]] 33 29 34 [[ Image(http://www.pas.rochester.edu/~shuleli/mceq7.png, 20%)]]30 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta \nabla^2 (\nabla \cdot \textbf{B}) + (\nabla \eta \cdot \nabla)\textbf{B}$)]] 35 31 36 32 The first term on the right hand side is zero but the second term is not, especially at sharp temperature fronts where grad(T) is large.[[BR]]
