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Tokamak Physics Jan Mlynář 2. Magnetic field, Grad-Shafranov Equation

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. Revision of basic quantities. Magnetic field (magnetic induction) B

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Tokamak Physics Jan Mlynář 2. Magnetic field, Grad-Shafranov Equation

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  1. 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. 1: Úvod, opakování

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

  3. Field line, equilibrium Magnetic field line Equilibrium: “nested surfaces” 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!) 2: Mg. field, Grad-Shafranov equation

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

  5. 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 2: Mg. field, Grad-Shafranov equation

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

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

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

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

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

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

  12. 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 authorsuse a different definition of flux, while here we defined 2: Mg. field, Grad-Shafranov equation

  13. 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 2: Mg. field, Grad-Shafranov equation

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