Changes between Version 8 and Version 9 of AstroBearProjects/resistiveMHD
- Timestamp:
- 08/11/11 16:03:01 (13 years ago)
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AstroBearProjects/resistiveMHD
v8 v9 7 7 [[latex($\frac{\partial \textbf{B}}{\partial t}=\nabla \times (\eta \nabla \times \textbf{B})$)]] 8 8 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]]9 The magnetic diffusivity 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 11 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta \nabla^2 \textbf{B}$)]] … … 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]] 14 14 15 In the constant resistivity case, matrices 1, 2, 3 are identical, the expanded matrix is still tridiagonal and can be blocked into three identical parts. [[BR]]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]]15 In the constant diffusivity case, matrices 1, 2, 3 are identical, the expanded matrix is still tridiagonal and can be blocked into three identical parts. [[BR]] 16 Now for the first equation, 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 diffusion, usually heating would occur and surppress the local resistivity and thus the diffusivity. By expanding the first equation, we have the form:[[BR]] 17 17 18 18 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta \nabla^2 \textbf{B} + \nabla \eta \times (\nabla \times \textbf{B})$)]] … … 22 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]] 23 23 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]]24 One may wonder if it is possible to throw away the second term on the right hand side of the diffusion equation to just let the diffusivity 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]] 25 25 26 26 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta (T) \nabla^2 \textbf{B}$)]] 27 27 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]]28 Unfortunately, this does not work because the diffusion equation itself 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 the above equation, we end up getting:[[BR]] 29 29 30 30 [[latex($\frac{\partial \textbf{B}}{\partial t}=\eta \nabla^2 (\nabla \cdot \textbf{B}) + (\nabla \eta \cdot \nabla)\textbf{B}$)]] … … 33 33 So this approximation only works under '''slowly varying temperature''' situation.[[BR]][[BR]][[BR]] 34 34 35 = ====================== = 36 = Real Magnetic Diffusivity and its Scaling = 37 = ====================== = 38 39 In cgs units, the real magnetic diffusivity can be written as:[[BR]] 40 41 [[latex($\eta = c^2/4\pi \sigma$)]] 42 43 where [[latex($\sigma$)]] is the material conductivity which depends on the electron-ion collision rate [[latex($\nu_{ei}$)]] which in term depends on the electron temperature [[latex($T^{-3/2}$)]]. [[BR]] 44 The diffusivity can thus be expressed as function of electron temperature[[BR]] 45 46 [[latex($\eta = 0.14 m_{e}^{1/2} c^2 \ln \Lambda F(Z_{eff})(k_B T_{e}^{3/2})$)]] 47 48 where [[latex($\ln \Lambda$)]] is the Coulomb logarithm:[[BR]] 49 50 [[latex($\ln \Lambda = 16.3+1.5 \ln T - 0.5\ln n$, (for $T < 4.2 \times 10^{5}$ K))]] 51 52 [[latex( $22.8+1.5 \ln T - 0.5\ln n$, (for $T > 4.2 \times 10^{5}$ K))]] 35 53 36 54 = ======================================= =
