Tokamak Physics Jan Mlyn 3 Tokamak field equilibrium

Tokamak Physics Jan Mlynář 3. Tokamak field equilibrium Solovjev solution of the Grad-Shafranov equation, Shafranov shift, plasma shape, poloidal beta, vertical field for equilibrium, Pfirsch-Schlüter current Fyzika tokamaků 1: Úvod, opakování

Grad-Shafranov equation where Normalised coordinates Unit flux Inverse aspect ratio Iterative numerical solutions: In: Dimensionless profiles Out: d. . . Shafranov shift, see later Tokamak Physics 3: Tokamak field equilibrium

Soloviev solution Soloviev solved the special case of the linearised Grad-Shafranov equation: Take i. e. Grad-Shafranov eq. Analytical solution: Dimensionless: Tokamak Physics 3: Tokamak field equilibrium

Role of dimensionless quantities Soloviev solution of G-S equation features the up-down symmetry (but not a HFS-LFS symmetry!!) Tokamak Physics 3: Tokamak field equilibrium

Plasma shape General form: i. e. any general shape is decomposed in Fourier series Usual form: elongation: triangularity: (and no higher m terms) Tokamak Physics 3: Tokamak field equilibrium

Poloidal beta Circular cross-section: Large aspect ratio explains why tokamaks cannot reach very high beta Tokamak Physics 3: Tokamak field equilibrium

Flux shift in circular cross-section Displaced flux surface: Substituting the Grad-Shafranov equation, integrating… Tokamak Physics 3: Tokamak field equilibrium

Shafranov shift, vacuum mg. field separatrix: internal inductance Vacuum magnetic field Tokamak Physics 3: Tokamak field equilibrium

Internal inductance Tokamak Physics 3: Tokamak field equilibrium

Vertical field for equilibrium Hoop force Self-inductance outside and inside the plasma Equilibrium: Tokamak Physics 3: Tokamak field equilibrium

Pfirsch-Schlüter current diamagnetic current Pfirsch-Schlüter current is the component of the current that is parallel to the magnetic field line. It short-cuts the plasma polarisation which would occur due to grad. B and curvature drifts. Tokamak Physics Total current density 3: Tokamak field equilibrium

Pfirsch-Schlüter current • In tokamaks, Shafranov shift results from the Grad-Shafranov equation that describes equilibrium. • It is shown that for this shift to appear, vertical field is required. • The vertical field is balanced in plasma by the Pfirsch-Schlüter current. • This current is identical to the current that results due to the shift of the particle trajectory in a toroidal system with field helicity Shafranov shift . Tokamak Physics 3: Tokamak field equilibrium

Components of the tokamak field Tokamak Physics 3: Tokamak field equilibrium

Total field, vertical stability Tokamak Physics 3: Tokamak field equilibrium

Tokamak discharge Tokamak Physics 3: Tokamak field equilibrium