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:
where μ0 is the magnetic permeability, p(ψ) is the pressure, F(ψ) = RBφ
and the magnetic field and current are given by
- Δ * is given by
.
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.
