# Grad-shafranov equation

**Grad-shafranov equation**

(Science: radiobiology) Reduced magnetohydrodynamic-equilibrium equation for an axisymmetric, toroidal plasma. (similar reduced equations can be derived for the cases of helical symmetry and for the straight cylinder.)

analytic and numerical studies of these equations are important in exploring potential plasma configurations. The lowest order force balance in the plasma is simply that the lorentz force must be balanced by the pressure force. This balance, combined with Maxwell's equations, determines the equilibrium configuration of the magnetic field.

When the toroidal configuration is axisymmetric, and the equilibrium plasma flow is zero, the magnetic field may be written in terms of a stream function psi that satisfies the **grad-shafranov equation** Deltapsi = - mu_0 R^2 p'(psi) - FF'(psi). Here p is the plasma pressure and f = R B_phi. (R is the radial distance from the axis of the machine).

in an axisymmetric torus, in the absence of equilibrium plasma fluid flows, the magnetic field may be written in terms of a scalar potential. When the plasma is in equilibrium (forces balance and the plasma is stationary), this scalar potential obeys a non-linear elliptic equation known as the **grad-shafranov equation**.