Magneto-Fluid Dynamics Seminar
An adjoint approach for the shape gradients of three-dimensional magneto-hydrodynamic equilibria
Speaker: Elizabeth Paul, University of Maryland, College Park
Location: Warren Weaver Hall 905
Date: Tuesday, November 19, 2019, 11 a.m.
A continuous adjoint method has been developed for obtaining the derivatives of functions of the magneto-hydrodynamic (MHD) equilibrium equations with respect to the shape of the boundary of the domain or the shape of the electro-magnetic coils . This approach is based on the generalization of the self-adjointness of the linearized MHD force operator. The adjoint equation corresponds to a perturbed force balance equation with the addition of a bulk force, rotational transform, or toroidal current perturbation. We numerically demonstrate this approach by adding a small perturbation to the non-linear VMEC  solution, obtaining an order $10^2-10^3$ reduction in cost in comparison with a finite difference approach. Examples are presented for the shape gradient of the rotational transform and vacuum magnetic well, a proxy for MHD stability. The adjoint solution required for the magnetic ripple, a proxy for near-axis quasisymmetry, requires the addition of an anisotropic pressure tensor to the MHD force balance equation. This modification has been implemented in the ANIMEC  code. We furthermore demonstrate that this adjoint approach can be applied to compute shape gradients of two important figures of merit , the departure from quasisymmetry and the effective ripple in the low-collisionality neoclassical regime, but require the development of new equilibrium solvers. Finally, initial steps toward adjoint solutions with a linearized equilibrium approach will be presented.
 Antonsen, T.M., Paul, E.J. & Landreman, M. 2019 Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Journal of Plasma Physics 85 (2).
 Hirshman, S.P. & Whitson, J.C. 1983 Steepest descent moment method for three-dimensional magnetohydrodynamic equilibria. Physics of Fluids 26 (12), 3553.
 Cooper, W.A., Hirshman, S.P., Merazzi, S. & Gruber, R. 1992 3D magnetohydrodynamic equilibria with anisotropic pressure. Computer Physics Communications 72 (1),1–13.
 Paul, E.J., Antonsen, T.M., Landreman, M., Cooper, W.A. Submitted to Journal of Plasma Physics.