The initial value problem in Lagrangian drift kinetic theory

TL;DR

This paper introduces a renormalized variational approach to high-order drift kinetic theory that preserves the initial value problem and eliminates unphysical rapidly varying modes present in conventional methods.

Contribution

The paper proposes a novel renormalized variational method that respects the original initial value problem, avoiding unphysical modes in high-order drift kinetic theories.

Findings

The renormalized approach matches conventional accuracy.

It eliminates unphysical rapidly varying modes.

The method is demonstrated on a toy model and extended to full drift kinetic systems.

Abstract

Existing high-order variational drift kinetic theories contain unphysical rapidly varying modes that are not seen at low-orders. These unphysical modes, which may be rapidly oscillating, damped, or growing, are ushered in by a failure of conventional high-order drift kinetic theory to preserve the structure of its parent model's initial value problem (Vlasov-Poisson for electrostatics, Vlasov-Darwin or Vlasov-Maxwell for electromagnetics.) In short, the system phase space is unphysically enlarged in conventional high-order variational drift kinetic theory. I present an alternative, "renormalized" variational approach to drift kinetic theory that manifestly respects the parent model's initial value problem. The basic philosophy underlying this alternate approach is that high-order drift kinetic theory ought to be derived by truncating the all-orders system phase space Lagrangian instead of the usual "field+particle" Lagrangian. For the sake of clarity, this story is told first through the lens of a finite-dimensional toy model of high-order variational drift kinetics, the analogous full-on drift kinetic results are discussed subsequently. The renormalized drift kinetic system, while just as accurate as conventional formulations, does not support the troublesome rapidly varying mode.