PLUTO Test Problems
4.4-patch2
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Diffusion of a linear force-free magnetic field in cylindrical coordinates. More...
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void | InitDomain (Data *d, Grid *grid) |
Solve the diffusion equation for a linear force-free magnetic field in cylindrical coordinates. A force-free field satisfies
which means that and
must be parallel. If
is constant then the solution is given by the Bessel functions:
with vector potential given by
For constant resistivity and zero velocity the induction equation simplifies as follows:
which admits the exact analytical solution meaning that the field remains force free also at subsequent times.
In the test we set .
Note that if pressure and density are initially constant and the velocity is also initially zero everywhere, the previous solution is also an exact solution of the isothermal MHD equations but not of the adiabatic MHD equations because of the Ohmic dissipation term (magnetic energy transforms into heat). However, using a large density makes pressure effects negligible.