We study the physical origins of parallel guiding center drift and parallel velocity-dependent effective magnetic field $(B\Vert *)$ 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 $B\Vert *$ is a consequence of magnetic moment conservation in curved magnetic fields.

## I. INTRODUCTION

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 (**u**_{d}), 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, **u**_{d} appears as the first order in the expansion parameter *ϵ* = *ρ*_{i}/*L*_{B}, where *ρ*_{i} and $LB\u22121=\u2207B/B$ 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 *d***ρ**/*dt* in a drift kinetic description of a plasma.^{2} The first derivation of an expression for **u**_{d} 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 **u**_{d} using various approaches, leading sometimes to different expressions.^{1–6} While it is mathematically straightforward to derive **u**_{d}, the physics picture leading to **u**_{d} has been elusive or unavailable in the existing literature.

^{1}

**B**being the prescribed confining field and $b\u0302$ being the unit vector parallel to

**B**; $B\Vert *=b\u0302\u22c5B*=B\Vert +(mc/e)ub\u0302\u22c5\u2207\xd7b\u0302$; and

**E*** is the effective electric field. $B\Vert *$ also appears when one constructs a guiding center coordinate system composed of (

*v*

_{‖},

*μ*,

*ϕ*) from the particle phase space variables. Here,

*v*

_{‖},

*μ*, and

*ϕ*denote the parallel velocity, magnetic moment, and gyroangle, respectively. This $B\Vert *$ is a Jacobian for the guiding center variables, enabling phase space volume conservation.

Traditionally, $B\Vert *$ has been introduced as a natural consequence of the mathematical formulation of the Hamiltonian guiding center theory.^{1} The use of $B\Vert *$ in the gyrokinetic formulation enables phase space volume conservation.^{1} Similar to **u**_{d}, the physics picture of how $B\Vert *$ arises in the guiding center theory has not been well clarified in the plasma physics literature. A connection of **u**_{d} to the first order correction of gyroangle-independent *μ* is indicated formally in Ref. 1, without illuminating the physics picture behind it. Moreover, since $B\Vert *$ is in the denominator in Eq. (1), a singularity issue may arise when the parallel velocity is strong enough to satisfy $B\Vert *=0$.^{7}

In this paper, we present physics pictures that produce **u**_{d} and $B\Vert *$. 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 $B\Vert *$ 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 $B\Vert *$ 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.

## II. PHYSICS OF PARALLEL DRIFT AND THE EFFECTIVE PARALLEL MAGNETIC FIELD

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** = **R**_{0} = (0, 0, *z*). Without loss of generality, we take the magnetic field at **R**_{0} to be entirely in the $z\u0302$ direction, i.e., $BR0=B0z\u0302$. For simplicity, we also assume that *∂B*_{z}/*∂x* = *∂B*_{z}/*∂y* = 0, while *∂B*_{z}/*∂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 *∂B*_{x}/*∂y* and *∂B*_{y}/*∂x*. Here, *B*_{x} and *B*_{y} 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.

*ϕ*, as shown in Fig. 1. In its local coordinate system, the particle exhibits a

*local parallel*motion due to the presence of local

*B*

_{x}and

*B*

_{y}. Here, the local magnetic fields are given by

*B*

_{x}and

*B*

_{y}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, $\mu =v\u22a52/2B0$, is strictly conserved.

**v**

_{⊥0}into $x\u0302$ and $y\u0302$ components,

**v**

_{⊥0}is given by

_{i}is the ion gyro-frequency with respect to $B0z\u0302$. The projection of the local parallel velocity in the $z\u0302$ direction gives parallel guiding center motion at

**R**

_{0}. Since we assume

*∂B*

_{z}/

*∂x*=

*∂B*

_{z}/

*∂y*= 0, Eq. (5) coincides with the projection.

*θ*represents projection of the local parallel velocity into the $z\u0302$ direction, and $b=z\u0302$ is the unit vector in the parallel direction at

**R**

_{0}. When deriving Eq. (6), we have used the relation

**b**· ∇ ×

**B**=

**b**· ∇

*B*×

**b**+

*B*

**b**· ∇ ×

**b**and

*B*

_{0}=

*B*

_{z}. 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

**B**

_{0}instead of $B0z\u0302$.

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 Ω_{i}) are evaluated with respect to the prescribed magnetic field at the guiding-center position in the former case, while a gyro-averaged value of $v\u22a52/\Omega i$ (i.e., $v\Vert =v\u22a52/2\Omega ib\u22c5\u2207\xd7b$) appears in the latter one. Therefore, Eq. (6) is consistent with the expression for **u**_{d} 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.

*v*

_{‖}

**b**yields

*v*

_{x}and

*v*

_{y}, which change the perpendicular velocity with respect to

**R**

_{0}. Since

*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

*μ*) requires a corresponding change in the effective amplitude of magnetic field at the guiding center, which is the origin for the appearance of $B\Vert *$. Thus, one demands

*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 $B\Vert *$.

An important consequence of Eq. (13) is that an exact conservation of magnetic moment requires $B\Vert *$ up to the second order in *ρ*_{i}/*L*_{B}. This implies that the singularity appearing in parallel dynamics of a guiding center is actually a *spurious singularity*, not a genuine one. $B\Vert *$ must approach *B*_{0}/4 asymptotically as *v*_{‖0} increases to satisfy the singularity condition, *v*_{‖0} = −Ω_{i}/**b** · ∇ × **b**. Thus, the exact conservation of magnetic moment in modern gyrokinetic theory^{9} 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 $B\Vert *$ 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 $B\Vert *$ as the Jacobian. The time scale for the GF simulations is restricted up to the time where *dμ*/*dt* is conserved up to the first order.^{10,11} This study also shows that any mathematical attempt to modify $B\Vert *$ in the parallel equation of motion for removal of possible singularity^{7} should comply with the physics picture presented in this work.

## III. CONCLUSIONS

In summary, we have clarified the physics origins for the parallel guiding center drift (**u**_{d}) and the parallel velocity-dependent effective magnetic field $(B\Vert *)$ 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 $B\Vert *$ 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.

## ACKNOWLEDGMENTS

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).

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

## REFERENCES

*Plasma Confinement*

*Reviews of Plasma Physics*

*Transport Processes in Plasmas 1*

*Classical Transport Theory*