Changes between Version 27 and Version 28 of AstroBearProjects/resistiveMHD

Timestamp:

01/28/13 00:52:32 (12 years ago)

Author:

Shule Li

Comment:

—

AstroBearProjects/resistiveMHD

@@ -26 +26 @@
 In AstoBEAR, we explicitly calculate the resistivity induced current on the cell edges, following equation: [[BR]]
 [[latex(\textbf{J} = \eta \nabla \times \textbf{B})]] [[BR]]
-The magnetic field is represented by the aux field, which is centered on the cell faces. Its curl therefore reside on the cell edges. Here is an example on  [[BR]]
-calculating the diffusive current on the x direction, notice that the red arrow is where we are calculating the diffusive current, the green arrows are  [[BR]]
-where the magnetic field originally resides: [[BR]]
-[[Image(resistive_diagram.png, 30%)]] [[BR]]
+The stencil for this explicit solver is a 3 by 3 cube surrounding the cell we want to update. The magnetic field is represented by the aux field, which is [[BR]]
+centered on the cell faces. Its curl therefore reside on the cell edges. Here is an example on calculating the diffusive current on the x direction, notice [[BR]]
+that the red arrow is where we are calculating the diffusive current, the green arrows are where the magnetic field originally resides: [[BR]]
+[[Image(resistive_diagram.png, 20%)]] [[Image(http://www.pas.rochester.edu/~shuleli/res_dia1.png, 30%)]] [[BR]]
 The actual code looks like the following in 2D (jy are initialized to 1): [[BR]]
 {{{
@@ -148 +148 @@
 Notice that in our island formation problem, the perturbation is anti-symmetric about the center. So the density pattern is also anti-symmetric. It is trivial to use [[BR]]
 a symmetric perturbation in order to obtain a symmetric island pattern. [[BR]]
+
+'''Microphysical Resistivity'''
+
+In cgs units, the real magnetic diffusivity can be written as:[[BR]]
+
+[[latex(\eta = c^2/4\pi \sigma)]]
+
+where sigma is the material conductivity which depends on the electron-ion collision rate mu which in term depends on the electron temperature to the three halves. [[BR]]
+The cross field diffusivity can thus be expressed as function of electron temperature if we assume the plasma is fully ionized:[[BR]]
+
+[[latex(\eta = 8.2243 \times 10^5 Z_{eff} F(Z_{eff}) \ln \Lambda T_{e}^{-3/2})]]
+
+where T is the electron temperature in eV, Zeff is the effective ion charge.The Coulomb logarithm for interested number density is plotted below: [[BR]]
+[[Image(http://www.pas.rochester.edu/~shuleli/CoulombLog.png, 30%)]][[BR]]
+[[BR]][[BR]]
+
+Here, T is the same as before, n is the electron number density in the units of per cubic centimeter. F is a function of effective ion charge:[[BR]][[BR]]
+
+[[latex(F(Z) = \frac{1+1.198Z+0.222Z^2}{1+2.996Z+0.753Z^2})]][[BR]][[BR]]
+
+Since the electron's maxwellian will be distorted when there is a strong outer magnetic field applied, the realistic resistivity can depend on the local field orientation. [[BR]]
+The anisotropicity is around 2 for the parallel and cross field resistivity. This requires the electron mean free path to be much longer than the gyroradius under such a field: [[BR]]
+[[latex(R_B << \lambda_{mfp})]][[BR]][[BR]]
+or, the gyro frequency to be much greater than the mean electron ion collision frequency. [[BR]]
+This effect is not considered currently. [[BR]][[BR]]
+
+'''How to Turn on Different Types of Resistivity'''
+
+Currently in the code, the microphysical resistivity gets calculated at each cell edges by averaging the temperatures and density information surrounding that edge. [[BR]]
+The resistive source term can be turned on in astrobear by putting the resistive flag to true:[[BR]]
+{{{
+lResistive     = .false.    ! Turns on resistivity [.false.]
+}}}
+The second parameter '''ResType''' determines which type of resistivity one requires: 1 for constant computational resistivity; 2 for artificial resistivity, 3 for microphysical 
+resistivity. [[BR]]
+{{{
+ResType        = 2          ! Resistivity type. 1-constant, 2-user defined distribution, 3-Spitzer. [1]
+}}}
+The third parameter '''resistivity''' is the user defined constant resistivity. When ResType = 2, the resistivity is determined by the routine "realResistivity". The [[BR]]
+distribution can be manually determined. In this routine, all variables are in computational units. If ResType = 3, the resistivity is microphysical and is automatically [[BR]]
+calculated using the scaling parameters. the "resistivity" parameter will be ignored. [[BR]]
+{{{
+resistivity    = 1.2d-1     ! resistivity value [0d0]
+}}}
+The F function gets calculated automatically via and is a global const. If ResType = 3, The microphysical resistivity automatically scales with the input scaling parameter.[[BR]]