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Equilibrium and MHD Stability of Fusion Plasmas


  • Typical spatial scales in plasmas: (1h): Some kinetic and fluid characteristic lengths; recall to the plasma graininess parameters  and to the hypothesis at the basis of the mean field theory.
  • The fluid description of plasmas:  derivation from the kinetic theory (4h): The moments of the transport equation for the distribution function of charged particles  with (Boltzmann) and without  collisions (Vlasov). The closure problem and implications about the thermodynamic equilibrium. Notes about the explicit evaluation of the pressure tensor in the presence of a magnetic field. 
  • The fluid description of plasmas: examples of modelling at different temporal or spatial scales  (3h): Equations of the: two-fluid description; one-fluid description (collisional and collisionless MHD); electron-magnetohydrodynamics  (EMHD). Outline about their relevance to laboratory and astrophysical plasmas. The generalized Ohm’s law in the Alfvènic and whistlers frequency regimes.
  • Topological invariants in a magnetized ideal plasma, and their violation  (3h):Lagrangian and integral invariants in fluid dynamics. Applications to magnetized plasmas in ideal MHD: Alfvèn theorem (frozen-in law) and connection theorem (magnetic lines can not reconnect in an ideal plasma). Conservation of the magnetic helicity and Woltjer invariants in an ideal plasma. Violation of the  topological constraints due to non-ideal terms in the Ohms’ law (magnetic reconnection).
  • Dynamical equilibria in magnetized plasmas (4h): Parameterization of the magnetic field  and the definition of the magnetic stream-function. Magnetic surfaces and the tokamak safety factor. Applications to different geometries:  stream function in slab geometry (notes about its application to the poloidal field in a strong guide field plasma and reduced MHD);  stream function of the magnetic field of a planet and Ferraro’s theorem;  z-pinch; theta-pinch; Lutz-Schluter-Grad-Schafranov equation for a tokamak plasma.
  • Some instabilities in plasmas (3h): The MHD energy principle, destabilizing contributions
  • And: free energy. And Destabilization of near-zero frequency modes. Application to a tokamak plasma: resonant surfaces and magnetic reconnection. Notes about the occurrence of magnetic reconnection in tokamaks (sawtheeth, disruptions) and in astrophysics (solar flares, solar wind-magnetosphere interactions).
  • or, depending on the choice of the students: Outline of some other plasma instabilities in the fluid models. For example: interchange modes,  kink  modes (in tokamaks); Rayleigh-Taylor (in astrophysics and inertial fusion plasmas); Kelvin-Helmoltz  (astrophysical  plasmas); two-stream instability (unconfined plasmas); counter-streaming current filamentation instability (astrophysics, inertial fusion)..
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Université de Lorraine, Nancy, France