Tokamak Physics Jan Mlyn 2 Magnetic field GradShafranov

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. Fyzika tokamaků 1: Úvod, opakování

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

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!) Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Mg field in arbitrary coordinates All functions of x, t Suppose that never vanishes coordinates Jacobian of the transformation Magnetic field lines: Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Magnetic field Hamiltonian is the magnetic flux in the f direction: From the equations of magnetic field lines: is Hamiltonian, c generalised momentum, q generalised coordinate and f generalised time Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Rotational transform, q Integraton Transformation gives complete topology If canonical transformation leading to (axisymmetry), then rotational transform / 2 p Safety factor Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Axisymmetric tokamak Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Poloidal coordinates Field line is straight if Tokamak Physics 2: Mg. field, Grad-Shafranov equation

q profiles Ampère’s law Circular plasma: in particular model: divertor: Tokamak Physics 2: Mg. field, Grad-Shafranov equation

R, F, z coordinates Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Grad-Shafranov equation We shall work in cylindrical coordinates and assume axisymmetric field p as well as RBf are functions of only. Tokamak Physics 2: Mg. field, Grad-Shafranov equation

Grad-Shafranov equation From (1): two arbitrary profiles I( ), p( ) ; boundary condition 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 Tokamak Physics 2: Mg. field, Grad-Shafranov equation

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 Tokamak Physics 2: Mg. field, Grad-Shafranov equation