Geodesic formulation of magnetic field lines in toroidal fusion systems

The magnetohydrodynamics (MHD) instabilities are considered as particularly dangerous because they usually involve large-scale motions and short time scales. However, a realistic picture of dynamic plasma processes requires a nonlinear theory. The understanding of the basic linear instability is usually a very helpful starting point, in particular since linear theory has a solid mathematical foundation. The research work consists of looking for a new nonlinear formulation of the equilibrium MHD equations taking into account nested tori surfaces which represent the hamiltonian magnetic field lines that contain the topological information (turbulence, stochasticity, magnetic islands, reconnection) of the plasma. The authors introduce a new formulation called the geodesic magnetic field line formulation. The geodesic formulation is based on the equivalence between the hamiltonian system and the associated geodesic trajectory. The geodesic formulation is related to the first approximation of the gyroradius, the geodesic evolution of the magnetic field line minimizes the magnetic energy. Using adequate Boozer coordinates, they obtain three coupled geodesic equations by minimizing the magnetic energy. Using adequate Boozer coordinates, they obtain three coupled geodesic equations by minimizing the magnetic energy and the helicity. This geodesic formulation allows them to determine the profile of the safety factor q and the profile of the ratio β = 2microP/B2 related to this special class of MHD equilibrium for toroidal devices. An asymptotic safety factor equation for small aspect ratio tokamak is obtained