28 | | 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]] |
29 | | 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]] |
30 | | where the magnetic field originally resides: [[BR]] |
31 | | [[Image(resistive_diagram.png, 30%)]] [[BR]] |
| 28 | 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]] |
| 29 | 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]] |
| 30 | that the red arrow is where we are calculating the diffusive current, the green arrows are where the magnetic field originally resides: [[BR]] |
| 31 | [[Image(resistive_diagram.png, 20%)]] [[Image(http://www.pas.rochester.edu/~shuleli/res_dia1.png, 30%)]] [[BR]] |
| 150 | |
| 151 | '''Microphysical Resistivity''' |
| 152 | |
| 153 | In cgs units, the real magnetic diffusivity can be written as:[[BR]] |
| 154 | |
| 155 | [[latex(\eta = c^2/4\pi \sigma)]] |
| 156 | |
| 157 | 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]] |
| 158 | The cross field diffusivity can thus be expressed as function of electron temperature if we assume the plasma is fully ionized:[[BR]] |
| 159 | |
| 160 | [[latex(\eta = 8.2243 \times 10^5 Z_{eff} F(Z_{eff}) \ln \Lambda T_{e}^{-3/2})]] |
| 161 | |
| 162 | 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]] |
| 163 | [[Image(http://www.pas.rochester.edu/~shuleli/CoulombLog.png, 30%)]][[BR]] |
| 164 | [[BR]][[BR]] |
| 165 | |
| 166 | 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]] |
| 167 | |
| 168 | [[latex(F(Z) = \frac{1+1.198Z+0.222Z^2}{1+2.996Z+0.753Z^2})]][[BR]][[BR]] |
| 169 | |
| 170 | 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]] |
| 171 | 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]] |
| 172 | [[latex(R_B << \lambda_{mfp})]][[BR]][[BR]] |
| 173 | or, the gyro frequency to be much greater than the mean electron ion collision frequency. [[BR]] |
| 174 | This effect is not considered currently. [[BR]][[BR]] |
| 175 | |
| 176 | '''How to Turn on Different Types of Resistivity''' |
| 177 | |
| 178 | Currently in the code, the microphysical resistivity gets calculated at each cell edges by averaging the temperatures and density information surrounding that edge. [[BR]] |
| 179 | The resistive source term can be turned on in astrobear by putting the resistive flag to true:[[BR]] |
| 180 | {{{ |
| 181 | lResistive = .false. ! Turns on resistivity [.false.] |
| 182 | }}} |
| 183 | The second parameter '''ResType''' determines which type of resistivity one requires: 1 for constant computational resistivity; 2 for artificial resistivity, 3 for microphysical |
| 184 | resistivity. [[BR]] |
| 185 | {{{ |
| 186 | ResType = 2 ! Resistivity type. 1-constant, 2-user defined distribution, 3-Spitzer. [1] |
| 187 | }}} |
| 188 | The third parameter '''resistivity''' is the user defined constant resistivity. When ResType = 2, the resistivity is determined by the routine "realResistivity". The [[BR]] |
| 189 | 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]] |
| 190 | calculated using the scaling parameters. the "resistivity" parameter will be ignored. [[BR]] |
| 191 | {{{ |
| 192 | resistivity = 1.2d-1 ! resistivity value [0d0] |
| 193 | }}} |
| 194 | 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]] |