A semi-analytical solver for the Grad-Shafranov equation

Abstract

In toroidally confined plasmas, the Grad-Shafranov equation, in general a non-linear PDE, describes the hydromagnetic equilibrium of the system. This equation becomes linear when the kinetic pressure is proportional to the poloidal magnetic flux and the squared poloidal current is a quadratic function of it. In this work, the eigenvalue of the associated homogeneous equation is related with the safety factor on the magnetic axis, the plasma beta and the Shafranov shift, then, the adjustable parameters of the particular solution are bounded through physical constrains. The poloidal magnetic flux becomes a linear superposition of independent solutions and its parameters are adjusted with a non-linear fitting algorithm. This method is used to find hydromagnetic equilibria with normal and reversed magnetic shear and defined values of the elongation, triangularity, aspect-ratio, and X-point(s). The resultant toroidal and poloidal beta, the safety factor at the 95% flux surface and the plasma current are in agreement with usual experimental values for high beta discharges and the model can be used locally to describe reversed magnetic shear equilibria.

The force-free twisted magnetosphere of a neutron star

“…In particular, for σ = 1, the Grad-Shafranov equation becomes linear. In this case, the homogeneous part is of the same form as in Ciro & Caldas (2014), so, in principle, the same analytical solutions can be used for the toroidal region. These should then be matched to the vacuum solutions outside the toroidal region, which, in general, will contain any number of unknown multipoles.…”

Magnetic field modeling for non-axisymmetric tokamak discharges

“…In the following, we choosē (3.46) This corresponds to the so called dissimilar sources introduced in [McCarthy (1999)]. In [Ciro and Caldas (2014)] we studied this type of solutions and determined some global properties of the current density in relation to the parameters space. We also identified the relation between the magnetic axis safety factor and the eigenvalue of the associated homogeneous problem.…”

“…In the following we revisit some of these calculations introducing new physical restrictions to the parameters and important improvements to the numerical optimization methods. This allowed us to search for solutions that match both the magnetic axis safety factor and the total plasma current, which was not included in the numerical optimization of the previous research paper [Ciro and Caldas (2014)]. Requiring F 2 to be less than one (diamagnetic plasma) we obtain from (3.46) a couple of restrictions for the parameters defining the dimensionless poloidal current c > 0 , 2c > a.…”

“…This suppresses the gradients towards negative values and the optimization process won't evolve to nonphysical solutions where poloidal current and magnetic axis safety factor are not defined. This new feature is an improvement to the method presented in [Ciro and Caldas (2014)] where the numerical method could eventually evolve to such negative values, giving complex solutions that reduce the error but do not have necessarily a physical meaning.…”

“…As we are interested in simple models which encompasses relevant dynamics we will take a semi-analytical approach. For the axisymmetric part of the magnetic field we will develop a numerical routine based on analytical solutions of the Grad-Shafranov equation [McCarthy (1999); Ciro and Caldas (2014)]. This allows us to approximate efficiently a realistic MHD equilibrium configuration.…”