On the validity of the guiding-center approximation in the presence of strong magnetic gradients

Abstract

The motion of a charged particle in a nonuniform straight magnetic field with a constant magneticfield gradient is solved exactly in terms of elliptic functions. The connection between this problem and the guiding-center approximation is discussed. It is shown that, for this problem, the predictions of higher-order guiding-center theory agree very well with the orbit-averaged particle motion and hold well beyond the standard guiding-center limit ǫ ≡ ρ/L ≪ 1, where ρ is the gyromotion length scale and L is the magnetic-field gradient length scale.

Drift-wave stability in the field-reversed configuration

“…These flux surfaces do not include the magnetic null-point, allowing the guidingcenter approximation to remain valid. 67 The average gyrokinetic parameter is q i /L B ' 0.2 in the core and q i /L B ' 0.006 in the SOL which are gyro-kinetically valid (about 1.3% difference between the guiding-center and orbitaveraged positions in case of q i /L B ' 0.2 as shown by Brizard 67 ).…”

GENE-3D: A global gyrokinetic turbulence code for stellarators

“…In the edge region, its appropriateness can be questioned in certain situations (see, e.g., Refs. [33,34]). However, for the physics applications considered here (mainly stellarators and perturbed tokamaks), the gyrokinetic ordering is assumed to be valid.…”

Action-angle coordinates for motion in a straight magnetic field with constant gradient

“…The motion of a charged particle in a straight magnetic field B = B(y) z with constant perpendicular gradient was considered recently [1] in order to explore the validity of the guiding-center approximation [2] in the present of strong gradients. There, a single orbit in the (x, y) plane was solved exactly in terms of elliptic functions and integrals [3], and the orbit-averaged position y and drift velocity ẋ were compared, respectively, with the polarization shift and magnetic-drift velocity predicted by guiding-center Hamiltonian theory [2].…”

“…The purpose of the present paper is to consider all orbit types associated with the straight magnetic field B = B 0 (1 − y/L) z considered in Ref. [1], where B 0 is the field magnitude at y = 0 and |∇ ln B(y)| y=0 ≡ 1/L defines a constant gradient length scale L. The equations of motion in the (x, y)-plane perpendicular to the magnetic field are…”

“…(2) we can now express the y-dynamics in terms of the energy conservation law E = M ẏ2 /2 + V (y, u), where the potential energy V (y, u) ≡ M ẋ2 /2 represents a quartic potential in y, with minima ( ẋ = 0) at y/L = 1 ± √ 2u and a local maximum at y = L, where V (L, u) = 1 2 M (u Ω 0 L) 2 . The study of charged particle motion in the presence of a magnetic neutral sheet in a magnetized space plasma has been carried out elsewhere [4][5][6].…”