Gyrosymmetry: Global considerations

Abstract

In the guiding center theory, smooth unit vectors perpendicular to the magnetic field are required to define the gyrophase. The question of global existence of these vectors is addressed using a general result from the theory of characteristic classes. It is found that there is, in certain cases, an obstruction to global existence. In these cases, the gyrophase cannot be defined globally. The implications of this fact on the basic structure of the guiding center theory are discussed. In particular, it is demonstrated that the guiding center asymptotic expansion of the equations of motion can still be performed in a globally consistent manner when a single global convention for measuring gyrophase is unavailable. The latter fact is demonstrated directly by deriving a new expression for the guiding-center Poincaré-Cartan form exhibiting no dependence on the choice of perpendicular unit vectors.

Automation of the guiding center expansion

“…One first writes out the components of the transformed dϑ ǫ and H in a family of fibered coordinate systems on M that cover M. Then one chooses the local representatives of G n (ǫ) to eliminate the θ-dependence in these components in each coordinate system in the covering. For consistency 18 , one also must demand that the local definitions of G n (ǫ) agree when changing from one fibered coordinate system in the covering to another. This last consistency condition is one statement of the principle of gyrogauge invariance.…”

“…(4) are gyrosymmetric 18,19 (see section IV for the precise definition of gyrosymmetric tensors). Because the particle trajectories are periodic in this limit, Kruskal's general theory 11 tells us that we can asymptotically deform, or rearrange, the phase space M using a nonunique ǫ-dependent near-identity transformation T ǫ : M → M such that the transformed…”

From liquid crystal models to the guiding-center theory of magnetized plasmas

“…However, as pointed out by Littlejohn [23], in the general case there is no possible nontrivial choice of a 0 such that ∇ i a 0 ·b 0 ×a 0 = 0 for all i. See also [6] for more recent work on this point. The relations obtained here can be used in higher order theory to replace the term ε 2 µR ·Ẋ, which appears in Littlejohn's Lagrangian upon retaining higher order terms [24].…”

“…As Northrop [28,29] showed, working with dimensionless quantities in (6) leads to identifying ε with the ratio of the Larmor gyroradius to the characteristic distance over which fields change. However, Northrop also noticed that this is equivalent to identifying ε with the dimensional charge-to-mass ratio (that is ε = q/m) and this avoids the necessity of rescaling the Lorentz-force equation (6). We notice that this procedure prevents the variable ρ in (7) from possessing the dimension of a length.…”

A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

“…The usual coordinate for the gyro-angle suffers from several issues, both from a mathematical and from a physical point of view [5][6][7][8][9] : it is gauge-dependent, does not exist globally for a general magnetic geometry, and induces a disagreement between the coordinates and the physical state, since it implies an anholonomic momentum. In order to avoid these issues and to give a more intrinsic framework to the theory, we consider performing the reduction using the initial physical gauge-independent coordinate for the gyro-angle.…”