Exact solutions of the Grad–Shafranov equation via similarity reduction and applications to magnetically confined plasmas

• Similarity reduction of the Grad–Shafranov equation
• Linear and non-linear equilibria with plasma flow
• Tokamaks and compact toroids

We derive exact solutions of a linear form of the Grad–Shafranov (GS) equation, including incompressible equilibrium flow, using ansatz-based similarity reduction methods. The linearity of the equilibrium equation allows linear combinations of solutions in order to obtain axisymmetric MHD equilibria with closed and nested magnetic surfaces which are favorable for the effective confinement of laboratory plasmas. In addition, employing the same reduction methods we obtain analytical solutions for several non-linear forms of the GS equation. In this context analytic force-free solutions in both linear and nonlinear regimes are also derived.