1. 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

2. 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:

3. 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:

4. 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

5. 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)

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

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

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

9. Tokamak Physics9 Internal inductance 3: Tokamak field equilibrium

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

11. 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.

12. 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

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

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

15. Tokamak Physics15 Tokamak discharge 3: Tokamak field equilibrium