The initial value problem in Lagrangian drift kinetic theory

Burby, J. W.

doi:10.1017/s0022377816000532

Cited by 3 publications

(5 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, as we can see, the previously discussed discrepancies in TP parallel flow and TP theta flow have been resolved; we note that the 'Jacobian shift' is responsible for resolving the discrepancies. Importance of an analogous shift in a time dependent problem has been recently shown by Burby (2016). The CP flow can now be calculated from (5.6) assuming that the lowest-order distribution function is a Maxwellian.…”

Section: Rh Flowsmentioning

confidence: 89%

Trapped particle precession and sub-bounce zonal flow dynamics in tokamaks

Sengupta

Hassam

2018

J. Plasma Phys.

View full text Add to dashboard Cite

A drift-kinetic calculation in an axisymmetric tokamak, with super-diamagnetic flows, is presented to elucidate the relation between the radial electric field, $E_{r}$, zonal flows and the rapid precession of the trapped particle (TP) population. It has been shown earlier (Rosenbluth & Hinton, Phys. Rev. Lett., vol. 80(4), 1998, p. 724, hereafter RH) that an initial radial electric field results in geodesic acoustic mode oscillations which subsequently Landau damp, resulting in a much smaller final residual electric field, and accompanying parallel zonal flows. We observe an apparent paradox: the final angular momentum in the RH parallel zonal flow is much smaller than the angular momentum expected from the well-known rapid precession of the trapped particle population in the RH residual electric field. We reconcile this paradox by illuminating the presence of a population of reverse circulating particle flows that, dominantly, are equal and opposite to the rapid TP precession. Mathematically, the calculation is facilitated by transforming to an energy coordinate shifted from conventional by an amount proportional to $E_{r}$. We also discuss the well-known RH coefficient in the context of effective mass and show how the TP precession and the opposite circulating flows contribute to this mass. Finally, we show that in the long wavelength limit, the RH flows and RH coefficient arise as a natural consequence of conservation of toroidal angular momentum and the second adiabatic invariant.

show abstract

Section: Rh Flowsmentioning

confidence: 89%

Trapped particle precession and sub-bounce zonal flow dynamics in tokamaks

Sengupta

Hassam

2018

J. Plasma Phys.

View full text Add to dashboard Cite

show abstract

“…At least four such Vlasov-Maxwell systems exist and can be used in numerical modeling of plasmas in various branches of science. These are the guiding-center [1], the drift-kinetic [2,3], the gyrokinetic [3,4], and the spin-Vlasov-Maxwell system [5]. They all have a structure similar to equations (1).…”

Section: Introductionmentioning

confidence: 99%

“…Over the years, several papers discussing action principles for the Vlasov-Maxwell system or related ones 7 have been presented [1][2][3][4][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and many of them [1,8,9,[14][15][16][17][18]20] discuss the local energy and momentum conservation laws. Nevertheless, to our knowledge the only documented work dealing with the conservation laws that has been carried out in the spirit of Euler-Poincaré formalism is the recent paper by Sugama et al focusing on the guiding-center Vlasov-Darwin model [20].…”

Section: Introductionmentioning

confidence: 99%

Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems

Hirvijoki¹,

Burby²,

Pfefferlé³

et al. 2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The action principle by Low [Proc. R. Soc. Lond. A 248,[282][283][284][285][286][287] for the classic Vlasov-Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy-and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler-Poincaré formulation of Vlasov-Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while [J. Math. Phys., 39,6,, it is hard to come by a documented derivation of the related energy-and momentum-conservation laws in the spirit of the Euler-Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guidingcenter Vlasov-Darwin system [Phys. Plasmas 25, 102506]. The present exposition discusses a generic class of local Vlasov-Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler-Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear-and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given -the classic Vlasov-Maxwell and the drift-kinetic Vlasov-Maxwell -and the results expressed in the language of regular vector calculus for familiarity.We start with a slightly modified version of Low's action principle [6]. The purpose of the modification is to introduce the capability to handle a wider class of Vlasov-Maxwell-type 1 By systems related to Vlasov-Maxwell models, we mean 1) genuine Vlasov-Maxwell models that form an infinite-dimensional initial-value problem for the dynamical variables, and 2) the so-called Vlasov-Poisson-Ampère models which provide an initial value problem for the distribution function only and constraint equations for the electromagnetic potentials.

show abstract

“…Some somewhat subtle problems then arise in the specification and solution of such equations, and their relationship to standard gyrokinetic theory (Burby 2016); the 'auxiliary field' φ appears as a dynamical variable requiring an initial condition, and unphysical rapid variation of the solutions to the Euler-Lagrange equations appears. This is odd because the field φ may be written in terms of the particle distribution function in Vlasov-Maxwell theory, so should not appear as an extra degree of dynamical freedom.…”

mentioning

confidence: 99%

“…This is odd because the field φ may be written in terms of the particle distribution function in Vlasov-Maxwell theory, so should not appear as an extra degree of dynamical freedom. Burby (2016) explains a method to resolve this; we will explain a related approach to removing these unphysical degrees of freedom, as well as how to implement a more direct approach that simply imposes regular behaviour in the small-perturbation limit. We will first explain how these complications arise in an electrostatic theory (Sharma & McMillan 2015).…”

mentioning

confidence: 99%

Solving gyrokinetic systems with higher-order time dependence

Sharma

McMillan

2020

J. Plasma Phys.

View full text Add to dashboard Cite

We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

show abstract

The initial value problem in Lagrangian drift kinetic theory

Cited by 3 publications

References 31 publications

Trapped particle precession and sub-bounce zonal flow dynamics in tokamaks

Trapped particle precession and sub-bounce zonal flow dynamics in tokamaks

Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems

Solving gyrokinetic systems with higher-order time dependence

Contact Info

Product

Resources

About