We study the physical origins of parallel guiding center drift and parallel velocity-dependent effective magnetic field (B*) appearing in the Hamiltonian formulation of a particle motion in a curved magnetic field. The magnetic twist term is shown to be responsible for both of them. The parallel guiding center drift originates from the change in the effective parallel velocity due to the combined effect of the initial perpendicular motion (v⊥0) and the magnetic twist. The effective magnetic field arises from the collated effect of the change in v⊥0 due to its initial parallel guiding center motion and the conservation of magnetic moment. This understanding clearly shows that the appearance of B* is a consequence of magnetic moment conservation in curved magnetic fields.

For the Hamiltonian description of a guiding center motion of a charged particle in strongly magnetized fusion plasmas, such as in tokamaks, one must properly consider the impact of an inhomogeneous confining magnetic field. In the presence of such a prescribed inhomogeneous magnetic field, a guiding center exhibits parallel free streaming and perpendicular drift motion. The physics origin of perpendicular guiding center drift is well-known: due to the change in the Larmor radius during gyration by the spatial variation of the background magnetic field. A rather peculiar drift motion in a curved magnetic field is parallel drift (ud), which can arise even without ∇B and/or a field line curvature. In the Hamiltonian formulation of the guiding center motion in the curved magnetic field, ud appears as the first order in the expansion parameter ϵ = ρi/LB, where ρi and LB1=B/B are the ion thermal Larmor radius and the typical length scale of the magnetic field variation, respectively.1 For instance, it appears after gyro-averaging of dρ/dt in a drift kinetic description of a plasma.2 The first derivation of an expression for ud was made by Sivukhin, who obtained it from projection of the local particle velocity to the direction of the magnetic field at the gyrocenter position.3 Then, some authors re-derived ud using various approaches, leading sometimes to different expressions.1–6 While it is mathematically straightforward to derive ud, the physics picture leading to ud has been elusive or unavailable in the existing literature.

In addition to parallel drift motion, the effective parallel magnetic field, B*, appears in the description of parallel dynamics. For instance, the equation of motion parallel to an instantaneous magnetic field is given by1 
dudt=emB*B*E*,
(1)
where u is the parallel velocity; B*=B+(mc/e)u×b̂ is the effective magnetic field, with B being the prescribed confining field and b̂ being the unit vector parallel to B; B*=b̂B*=B+(mc/e)ub̂×b̂; and E* is the effective electric field. B* also appears when one constructs a guiding center coordinate system composed of (vμ, ϕ) from the particle phase space variables. Here, v, μ, and ϕ denote the parallel velocity, magnetic moment, and gyroangle, respectively. This B* is a Jacobian for the guiding center variables, enabling phase space volume conservation.

Traditionally, B* has been introduced as a natural consequence of the mathematical formulation of the Hamiltonian guiding center theory.1 The use of B* in the gyrokinetic formulation enables phase space volume conservation.1 Similar to ud, the physics picture of how B* arises in the guiding center theory has not been well clarified in the plasma physics literature. A connection of ud to the first order correction of gyroangle-independent μ is indicated formally in Ref. 1, without illuminating the physics picture behind it. Moreover, since B* is in the denominator in Eq. (1), a singularity issue may arise when the parallel velocity is strong enough to satisfy B*=0.7 

In this paper, we present physics pictures that produce ud and B*. They are shown to originate from the change in effective parallel and perpendicular guiding center velocities combined with the magnetic field twist contained in the dyadic expression ∇B. In particular, one will see that the combination of the magnetic moment conservation and the change in the effective perpendicular velocity naturally results in B* in a curved magnetic field. From this physics picture, we show that the potential singularity in Eq. (1) is a spurious one. In fact, the physics picture allows us to obtain an asymptotic value of B* at the value of parallel velocity where singularity is supposed to arise. Therefore, this fact should be consolidated by a formal gyrokinetic theory in the future.

To proceed, we consider a positively charged ion moving in curved magnetic fields. Initially, the particle is assumed to move with parallel (v‖0) and perpendicular (v⊥0) velocities with respect to the magnetic field at the guiding center position, which is designated by R = R0 = (0, 0, z). Without loss of generality, we take the magnetic field at R0 to be entirely in the ẑ direction, i.e., BR0=B0ẑ. For simplicity, we also assume that ∂Bz/∂x = ∂Bz/∂y = 0, while ∂Bz/∂z ≠ 0. This is to avoid the perpendicular guiding center drift motion due to the ∇B drift. The important magnetic field configuration that we retain is the magnetic twist, represented by ∂Bx/∂y and ∂By/∂x. Here, Bx and By are x and y components of the magnetic field, respectively. Figure 1 shows a schematic illustrating the magnetic field configuration and coordinates being used in this analysis.

FIG. 1.

Geometry of curved magnetic fields and the particle gyromotion being considered in this work.

FIG. 1.

Geometry of curved magnetic fields and the particle gyromotion being considered in this work.

Close modal
First, we elucidate the physics origin for the parallel guiding center drift. Consider a particle orbiting around a guiding center with gyroangle ϕ, as shown in Fig. 1. In its local coordinate system, the particle exhibits a local parallel motion due to the presence of local Bx and By. Here, the local magnetic fields are given by
Bx=Bx|R0+ρ0B|x=ρ0sinϕByx+cosϕBxx,
(2)
By=By|R0+ρ0B|y=ρ0cosϕBxy+sinϕByy.
(3)
Note that Bx and By contain terms representing the magnetic field twist and ∇ · B. Terms representing ∇ · B is of importance in the magnetic moment conservation and give rise to an effective parallel magnetic field. However, it plays no role in the guiding center parallel drift, as will be shown shortly. It is sufficient to consider only the magnetic twist terms to produce parallel guiding center drift. In this circumstance, the magnetic moment, μ=v2/2B0, is strictly conserved.
Owing to the field line twist, the ion gyro-motion generates a guiding center motion that is parallel to B0ẑ. This can be seen by decomposing v⊥0 into x̂ and ŷ components,
vx=v0sinϕ,vy=v0cosϕ.
(4)
Then, the local parallel velocity of a particle due to v⊥0 is given by
v=vBB=vρ0Bsin2ϕByxcos2ϕBxy+sinϕcosϕBxxByy=v2ΩiBsin2ϕByxcos2ϕBxy+sinϕcosϕBxxByy,
(5)
where Ωi is the ion gyro-frequency with respect to B0ẑ. The projection of the local parallel velocity in the ẑ direction gives parallel guiding center motion at R0. Since we assume ∂Bz/∂x = ∂Bz/∂y = 0, Eq. (5) coincides with the projection.
One can then obtain the gyroangle-independent guiding center parallel velocity after one gyration by gyro-averaging Eq. (5),
v=12π02πdϕvcosθ=v22ΩiBBxyByxBzB0=v22Ωib×b.
(6)
Here, the “−” sign is to account for the diamagnetic nature of the ion gyromotion, cos θ represents projection of the local parallel velocity into the ẑ direction, and b=ẑ is the unit vector in the parallel direction at R0. When deriving Eq. (6), we have used the relation b · ∇ × B = b · ∇B × b + Bb · ∇ × b and B0 = Bz. Equation (6) is exactly same as the parallel drift term that has been derived in Refs. 1 and 2 and the third term in Eq. (8.3) in Ref. 8. More complicated parallel drift terms could appear in a general guiding center theory, such as those shown in Eq. (8.3) in Ref. 8. Although a further analysis on a physics picture on those terms is not carried out in this paper, we conjecture that they can also be derived from a similar picture when one considers a general B0 instead of B0ẑ.

At this point, it is instructive to compare Eq. (6) with Eq. (2.26) in Ref. 3. The main difference between them is that all the parameters in Eq. (6) (i.e., v and Ωi) are evaluated with respect to the prescribed magnetic field at the guiding-center position in the former case, while a gyro-averaged value of v2/Ωi (i.e., v=v2/2Ωib×b) appears in the latter one. Therefore, Eq. (6) is consistent with the expression for ud presented in Ref. 8. Summarizing, the physics picture presented in this work clearly demonstrates that the parallel guiding center drift originates from the change in the local parallel velocity due to the presence of the magnetic field twist.

Next, we consider the origin of B* in a curved magnetic field. In this case, the point is that the initial non-zero parallel (i.e., v0ẑ) velocity gives rise to a change in the effective local perpendicular velocity. The initial vb yields vx and vy, which change the perpendicular velocity with respect to R0. Since
vxv0=BxB0,vyv0=ByB0,
(7)
one obtains
vx=v0ρ0B0sinϕByx+cosϕBxx,
(8)
vy=v0ρ0B0sinϕByx+cosϕBxx.
(9)
We note that a change in v⊥0 arises from v‖0 combined with the magnetic field twist. The magnitude of the change in v⊥0 due to v‖0, δv⊥0, is then given by
δv0=vxsinϕvycosϕ=v0ρ0B0sin2ϕByxcos2ϕBxy+sinϕcosϕBxxByy=v0v0ΩiB0sin2ϕByxcos2ϕBxy+sinϕcosϕBxxByy,
(10)
whose gyro-average gives rise to
δv0=v0v02Ωib×b.
(11)
Since the perpendicular velocity has changed with respect to the initial one, the conservation of magnetic moment (μ) requires a corresponding change in the effective amplitude of magnetic field at the guiding center, which is the origin for the appearance of B*. Thus, one demands
v02B0=v2B*=v02B*1+v02Ωib×b2.
(12)
From Eq. (12), one can easily find that B* is given by
B*=B01+v0Ωib×b+v02Ωib×b2=B01+v0Ωib×b+Oρi/LB2.
(13)
The first term in Eq. (13) is exactly the same as the well-known form of B* that has been used in the Hamiltonian dynamics and gyrokinetic theories in curved magnetic field geometry. Note that B* has been introduced for convenience in mathematical formulation and/or as a natural result of the construction of a phase space Jacobian. In this work, we have shown that it is actually a consequence of the change in effective perpendicular velocity with respect to the magnetic field at the guiding center in the presence of the magnetic field twist. Combined with the conservation of magnetic moment of a particle moving in the curved magnetic field, this yields the appearance of B*.

An important consequence of Eq. (13) is that an exact conservation of magnetic moment requires B* up to the second order in ρi/LB. This implies that the singularity appearing in parallel dynamics of a guiding center is actually a spurious singularity, not a genuine one. B* must approach B0/4 asymptotically as v‖0 increases to satisfy the singularity condition, v‖0 = −Ωi/b · ∇ × b. Thus, the exact conservation of magnetic moment in modern gyrokinetic theory9 will preclude the spurious singularity in parallel guiding center dynamics, although a systematic removal of the singularity is not a scope in this work.

After elucidating its physical origin, it is easy to understand that B* naturally appears as a Jacobian in the coordinate transformation from phase space to guiding center variables to conserve the magnetic moment of a charged particle. In addition, it is now trivial to assess the limitation of conservative gyrofluid moment equations that have been developed using B rather than B* as the Jacobian. The time scale for the GF simulations is restricted up to the time where /dt is conserved up to the first order.10,11 This study also shows that any mathematical attempt to modify B* in the parallel equation of motion for removal of possible singularity7 should comply with the physics picture presented in this work.

In summary, we have clarified the physics origins for the parallel guiding center drift (ud) and the parallel velocity-dependent effective magnetic field (B*) appearing in the kinetic formulation of plasma dynamics in curved magnetic fields. The magnetic twist term in ∇B is shown to be responsible for both of them. The parallel guiding center drift originates from the change in the net parallel velocity due to the combined effect of the initial perpendicular motion (v⊥0) and the magnetic twist. The effective parallel magnetic field arises from the collated effect of the change in v⊥0 due to initial parallel guiding center motion and the conservation of magnetic moment. This understanding elucidates the reason why v-dependent B* appears as a Jacobian in the guiding center variables. It is a natural consequence of μ conservation in a curved magnetic field. In addition to this, we found that the possible singularity in parallel dynamics of a guiding center is actually spurious. The physics pictures presented in this study, therefore, will be useful for an understanding of guiding center kinetics in a general confining magnetic field in magnetic fusion devices.

This work was supported by the R&D Program through the Korea Institute of Fuson Energy (KFE) funded by the Ministry of Science, ICT, and Future Planning of the Republic of Korea (Grant No. KFE-EN2141-10).

The author has no conflicts to disclose.

Hogun Jhang: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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