Fyzika tokamaků1: Úvod, opakování1 Tokamak Physics Jan Mlynář 3. Tokamak field equilibrium Solovjev solution of the Grad-Shafranov equation, Shafranov.

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Fyzika tokamaků1: Úvod, opakování1 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

Tokamak Physics3: Tokamak field equilibrium2 Grad-Shafranov equation where Normalised coordinates Inverse aspect ratio Unit flux Dimensionless profiles  ... Shafranov shift, see later Iterative numerical solutions: In: Out:

Tokamak Physics3 Soloviev solution Soloviev solved the special case of the linearised Grad-Shafranov equation: 3: Tokamak field equilibrium Take i.e. Grad-Shafranov eq. Analytical solution: Dimensionless:

Tokamak Physics4 Role of dimensionless quantities 3: Tokamak field equilibrium features the up-down symmetry (but not a HFS-LFS symmetry!!) Soloviev solution of G-S equation

Tokamak Physics5 Plasma shape 3: Tokamak field equilibrium Usual form: elongation: triangularity: General form: i.e. any general shape is decomposed in Fourier series (and no higher m terms)

Tokamak Physics6 Poloidal beta 3: Tokamak field equilibrium Circular cross-section: Large aspect ratio explains why tokamaks cannot reach very high beta

Tokamak Physics7 Flux shift in circular cross-section 3: Tokamak field equilibrium Displaced flux surface: Substituting the Grad-Shafranov equation, integrating…

Tokamak Physics8 Shafranov shift, vacuum mg. field internal inductance 3: Tokamak field equilibrium separatrix: Vacuum magnetic field

Tokamak Physics9 Internal inductance 3: Tokamak field equilibrium

Tokamak Physics10 Vertical field for equilibrium 3: Tokamak field equilibrium Hoop force Self-inductance outside and inside the plasma Equilibrium:

Tokamak Physics11 Pfirsch-Schlüter current 3: Tokamak field equilibrium Total current density 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 gradB and curvature drifts.

Tokamak Physics12 Pfirsch-Schlüter current 3: Tokamak field equilibrium Shafranov shift.  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

Tokamak Physics13 Components of the tokamak field 3: Tokamak field equilibrium

Tokamak Physics14 Total field, vertical stability 3: Tokamak field equilibrium

Tokamak Physics15 Tokamak discharge 3: Tokamak field equilibrium