Grad-Shafranov equation

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Prerequisites
MHD
 Plasma

The Grad-Shafranov equation (H. Grad and H. Rubin (1958) Shafranov (1966) ) is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry (the case relevant in a tokamak. Interestingly the flux function ψ is both a dependent and an independent variable in this equation:

\Delta^{*}\psi = -\mu_{0}R^{2}\frac{dp}{d\psi}-F\frac{dF}{d\psi}

where μ0 is the magnetic permeability, p(ψ) is the pressure, F(ψ) = RBφ

and the magnetic field and current are given by

\vec{B}=\frac{1}{R}\nabla\psi\times \hat{e_{\phi}}+\frac{F}{R}\hat{e}_{\phi}

\mu_0\vec{J}=\frac{1}{R}\frac{dF}{d\psi}\nabla\psi\times \hat{e_{\phi}}-\frac{1}{R}\Delta^{*}\psi \hat{e}_{\phi}

The elliptic operator

Δ * is given by

\Delta^{*}\psi = R\frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial \psi}{\partial R}\right)+\frac{\partial^2 \psi}{\partial Z^2}.

The nature of the equilibrium, whether it be a tokamak, reversed field pinch, etc. is largely determined by the choices of the two functions F(ψ) and p(ψ) as well as the boundary conditions.

[edit] References

  • Grad.H, and Rubin, H. (1958) MHD Equilibrium in an Axisymmetric Toroid. Proceedings of the 2nd UN Conf. on the Peaceful Uses of Atomic Energy, Vol. 31, Vienna: IAEA p.190.
  • Shafranov, V.D. (1966) Plasma equilibrum in a magnetic field, Reviews of Plasma Physics, Vol. 2, New York: Consultants Bureau, p. 103.
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