Adjoint methods for Transport Equations

Gyrokinetic simulations of plasma transport have been the workhorse of numerical understanding of magnetically confined plasmas for nearly two decades. With increasing focus on designing the core of a fusion pilot plant, the focus of simulations has moved from highly accurate verification and validation simulations to the task of trying to design

and optimize whole devices.

The framework into which advanced gyrokinetic simulations fit, that of multiscale plasma transport, is well-known [Abel et. al. 2013 Rep. Prog. Phys.]. Advanced numerical methods for solving the transport equations are being deployed (see poster by M Kelly, this conference). However, to integrate such transport calculations within an optimisation framework some knowledge of the sensitivity of the solutions to these equations is needed. Typically, we wish to examine a small number of global figures-of-merit from a transport simulation, such as stored energy or fusion yield, and calculate their derivatives with respect to many design parameters. For such problems, where the number of quanties is small and the number of variables is large, adjoint methods are known to be advantageous [see, e.g. Plessix et. al. Geophys. J. Intl. 2006]. Adjoint methods for sensitivity calculations are now well-established in

plasma physics [Paul et. al., Nucl. Fusion 2018; Paul et. al. J. Plasma Phys. 2019], and in this work we apply them to the transport equations of multiscale gyrokinetics.

To use the notion of an adjoint, we first introduce an inner-product structure on the space of perturbations about solutions of the transport equations. From this, we derive fully-general adjoint equations that can be used to analyze generic derivatives in the full transport framework. As a proof-of-concept example, we take classical transport in a cylindrical screw pinch and calculate derivatives of several important global functionals. In particular we show derivatives of the integrated fusion yield with respect to the location of an applied heat source. We also discuss the possible applications of these derivatives in a control system and how efficiencient the various underlying numerical calcualtions would have to be to apply this.