M. Onofri, F. Malara, P. Veltri Compressible magnetohydrodynamics simulations of the RFP with anisotropic thermal conductivity Dipartimento di Fisica, Università della Calabria

Compressible MHD equations is the stress tensor Cylindrical coordinates

Compressible MHD equations include heat production and heat transport, which are not present in the incompressible case. Case 1: Adiabatic Case 2: Heat production but Case 3: Isotropic thermal conductivity Case 4: Anisotropic thermal conductivity

Boundary conditions Boundary conditions at r=a for the other variables are calculated using a characteristic wave decomposition Periodic conditions in and directions In the r direction the conducting wall gives: (Poinsot et al., J Comp. Phys.,1992)

Initial conditions Poloidal field Toroidal field Equilibrium field (force-free magnetic field): Perturbations: (Robinson, Nuclear Fusion 1978)‏

To test the numerical code, we verified that it reproduces previous results obtained when the density and pressure dynamics are neglected. (Cappello et al, Phys. Plasmas, 1996) In this case, the results are similar to those obtained in previous works, for example the reversal parameter F becomes positive in the beginning of the simulation, but, due to the action of the unstable modes the reversal is recovered ad later times.

Test: comparison with previous results Field reversal parameter:

Time evolution of the modes m=1,n=-4, (solid line), m=1,n=-3 (dashed line) and m=1,n=-5 (dotted line)‏ The initial perturbations destabilize the system and the unstable modes begin to grow exponentially in the linear phase. After, the modes m=1,n=-4 and m=1,n=-3 have comparable amplitudes, indicating the formation of a MH state. Adiabatic case

Spectral spread Reversal parameter

Isosurface of the density in a SH state at t=400 (left) and in a MH state at t=1000 (right)‏

Non-adiabatic case Time evolution of the modes m=1,n=-4, (solid line), m=1,n=-3 (dashed line) and m=1,n=-5 (dotted line)‏ In the linear phase, the unstable modes grow exponentially and the m=1,n=-4 mode becomes the most energetic. After t=180 most of the energy is contained in the modes m=1,n=-3 and m=1,n=-5. The evolution is faster than in the adiabatic case.

Spectral spread Reversal parameter

Isosurface of the density in a SH state at t=100 (left) and in a MH state at t=200 (right).

Contours of temperature in a SH state (left) and in a MH state (right) and Poincaré maps of magnetic field lines. In the SH state, a magnetic island characterized by high temperature is present. In the MH state the magnetic field is chaotic

Simulation with thermal conductivity Boundary conditions at r=a Boundary conditions at r=a for the density is derived using a characteristic wave decomposition R=4

Spectral spread Reversal parameter

P Radial profile of density, pressure

t=50 t=1000 Contours of temperature in a SH state and in a MH state and Poincaré plots of magnetic field lines.

Anisotropic thermal conductivity The thermal conductivity in a magnetized plasma is anisotropic with respect to the direction of the magnetic field and for a fusion plasma the ratio may exceed ( Fitzpatrick, Phys. Plasmas,1995 )‏ Thermal conduction occurs on different time scales in the parallel and perpendicular direction, so that magnetic field lines tend to become isothermal. In a simulation it is not possible to use a realistic value of because the time step would become too small

Multiple-time-scale analysis ( Frieman J. Math. Phys., 1963) We separate the evolution on fast time scales from the evolution on slow time scales (1) (2) At each time step we look for an asymptotic solution of (1) and use it in (2) Extend the number of time variables....

t=50 t=1000 Contours of temperature in a SH state and in a MH state and Poincaré plots of magnetic field lines.

Reversal parameter Spectral spread

Radial profile of density and pressure P P P

.. Energy flux

Conclusions The density and pressure dynamics is important for the evolution of the system. Using an anisotropic thermal conductivity we can produce hot structures corresponding to closed magnetic surfaces and almost flat temperature when the magnetic field is chaotic.