Computational Study of the Grad--Shafranov Equation with Flow in Helical Symmetry

We present a numerical solution of the modified Grad--Shafranov equation in cylinder geometry and in the presence of macroscopic rotation. The main challenge is the full inclusion of helical symmetry, a problem relevant in a variety of scenarios. Cylindrical symmetry is often used as lowest-order approximation of the Grad--Shafranov equation in Tokamaks. However, in many other situations in astrophysical and laboratory plasmas, helical symmetry is a better assumption, in particular in Reversed Field Pinch (RFP) configurations in Single-Helicity states. The ``Grad--Shafranov'' equation is written in helical symmetry with a convective term expressed by toroidal and poloidal velocities. Two equations are obtained: a non-linear PDE for the magnetic flux $\Psi(r,u)$ and an algebraic equation for the density $\rho(r,u,\Psi)$ ($u$ is the variable along the helix). Here, we report on our progress in the development of a numerical method and computational code to solve the coupled non-linear system. Examples of applications are described.