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Constrained Transport EMF Kernel

VL+CT Integrator

I worked a lot this week on figuring out the CT electric fields kernel. After rereading Stone & Gardiner 2009, Gardiner & Stone 2008, and Stone et al. 2008 and looking at the source code for Athena I think I have a good handle on how exactly to compute the CT EMFs. I’m still working on implementation as it’s fairly tricky but I anticipate finishing it next week. The biggest hurdle was figuring out exactly where and how to use the magnetic fluxes returned from the HLLD solver. According to the Athena source code this is the correct way to convert them into EMFs.

Note that the directions used here are relative to the internal workings of the HLLD solver. Since the HLLD solver is inherently 1D we run it three times, once for each direction. So in the case where the solver is running in the Y direction the solver’s Y field is actually the Z field and the solvers Z field is actually the X field, cyclically extended for the Z direction

HLLD solve DirectionEquation for Magnetic FluxEqn. as a Cross ProductEMF
$$X$$$$V_x B_y - B_x V_y$$$$(V \times B)_z$$$$-\varepsilon_z$$
$$X$$$$V_x B_z - B_x V_z$$$$-(V \times B)_y$$$$\varepsilon_y$$
$$Y$$$$V_x B_y - B_x V_y$$$$(V \times B)_z$$$$-\varepsilon_x$$
$$Y$$$$V_x B_z - B_x V_z$$$$-(V \times B)_y$$$$\varepsilon_z$$
$$Z$$$$V_x B_y - B_x V_y$$$$(V \times B)_z$$$$-\varepsilon_y$$
$$Z$$$$V_x B_z - B_x V_z$$$$-(V \times B)_y$$$$\varepsilon_x$$

Magnetic field storage side

Magnetic fields were stored on the “left” (i-1/2) face of the cell that was indexed; i.e. the i,j,k cell stored the magnetic fields at the i-1/2, j-1/2, and k-1/2 faces. This is opposite the way Cholla stores the other face centered quantities (interface states and fluxes) so I switched the magnetic fields to be stored on the “right” faces instead. The changes are in PR #140.