1. Fyzika tokamaků1: Úvod, opakování1 Tokamak Physics Jan Mlynář 2. Magnetic field, Grad-Shafranov Equation Basic quantities, equilibrium, field line Hamiltonian, rotational transform, axisymmetric tokamak, q profiles, Grad-Shafranov equation.

2. Tokamak Physics2: Mg. field, Grad-Shafranov equation2 Revision of basic quantities Magnetic field (magnetic induction) B Magnetic flux Ampère’s law Faraday’s law Maxwell’s equations A.... Magnetic vector potential G.... Gauge (~ particular choice) Faraday’s law

3. Tokamak Physics2: Mg. field, Grad-Shafranov equation3 Field line, equilibrium Magnetic field line “nested surfaces” Equilibrium: Axisymmetry  nested mg. flux surfaces Magnetic field lines and j lie on the magnetic flux surfaces (but can not overlap otherwise the pressure gradient would be zero!)

4. Suppose that never vanishes Tokamak Physics2: Mg. field, Grad-Shafranov equation4 Mg field in arbitrary coordinates All functions of x, t  coordinates Jacobian of the transformation Magnetic field lines:

5. Tokamak Physics2: Mg. field, Grad-Shafranov equation5 Magnetic field Hamiltonian is the magnetic flux in the  direction: From the equations of magnetic field lines:  is Hamiltonian,  generalised momentum,  generalised coordinate and  generalised time

6. Tokamak Physics2: Mg. field, Grad-Shafranov equation6 Rotational transform, q Safety factor Integraton  Transformation    gives complete topology If canonical transformation leading to (axisymmetry), then rotational transform / 

7. Tokamak Physics2: Mg. field, Grad-Shafranov equation7 Axisymmetric tokamak

8. Tokamak Physics2: Mg. field, Grad-Shafranov equation8 Poloidal coordinates Field line is straight if

9. Tokamak Physics2: Mg. field, Grad-Shafranov equation9 q profiles Ampère’s law Circular plasma: in particular model: divertor:

10. Tokamak Physics2: Mg. field, Grad-Shafranov equation10 R, , z coordinates

11. Tokamak Physics2: Mg. field, Grad-Shafranov equation11 Grad-Shafranov equation We shall work in cylindrical coordinates and assume axisymmetric field p as well as RB  are functions of  only.

12. Tokamak Physics2: Mg. field, Grad-Shafranov equation12 Grad-Shafranov equation two arbitrary profiles I (  ), p (  ) ; boundary condition From (1): Notice: The form on the title slide (copy from Wesson) is different as many authors use a different definition of flux, while here we defined

13. Tokamak Physics2: Mg. field, Grad-Shafranov equation13 Grad-Shafranov equation Something to think about: Why is it not similar to a magnetic dipole field? Next lecture: Solovjev solution of the Grad-Shafranov equation, Shafranov shift, plasma shape, poloidal beta, flux shift in the circular cross-section, vacuum magnetic field, vertical field for equilibrium, Pfirsch-Schlüter current