Minimizing Separatrix Crossings through Isoprominence J.W. Burby1N. Duignan2 3aand J.D. Meiss2 1Los Alamos National Laboratory Los Alamos NM 97545 USA

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Minimizing Separatrix Crossings through Isoprominence

J.W. Burby,1N. Duignan,2, 3, a) and J.D. Meiss2

1)Los Alamos National Laboratory, Los Alamos, NM 97545 USA

2)Department of Applied Mathematics, University of Colorado, Boulder,

CO 80309 USA

3)School of Mathematics and Statistics, University of Sydney, NSW 2050,

Australia

(Dated: 20 October 2022)

A simple property of magnetic ﬁelds that minimizes bouncing to passing type transitions of guiding

center orbits is deﬁned and discussed. This property, called isoprominence, is explored through the

framework of a near-axis expansion. It is shown that isoprominent magnetic ﬁelds for a toroidal

conﬁguration exist to all orders in a formal expansion about a magnetic axis. Some key geometric

features of these ﬁelds are described.

I. INTRODUCTION

As a collisionless charged particle moves through a strong inhomogeneous magnetic ﬁeld, B, its motion

comprises three disparate timescales. On the gyrofrequency timescale1,2 the particle’s position along the

ﬁeld line is frozen while it rapidly rotates around the local magnetic ﬁeld vector with gyroradius ρ. This

rapid, nearly-periodic motion gives rise to near conservation of the famous magnetic moment µ, which,

more generally, is an adiabatic invariant.3,4 On a longer timescale, comparable to L/v with Lthe ﬁeld

scale length and vthe characteristic particle speed, the particle moves along magnetic ﬁeld lines while

experiencing the so-called mirror force −µb· ∇|B|where bis the unit vector along B. The perpendicular

kinetic energy µ|B|, when restricted to a particle’s nominal ﬁeld line plays the role of an eﬀective potential

for the particle’s parallel dynamics. When the particle’s energy is low enough that it is trapped in a well

for this potential, it bounces back and forth between a pair of turning points. When the particle is not

bouncing—its energy is larger than the highest potential peak—unbounded streaming along the ﬁeld line

ensues. These two scenarios correspond to bouncing and passing orbits, respectively. Finally, on the longest

timescale the particle drifts from ﬁeld line to ﬁeld line5due to the presence of perpendicular gradients in

the magnetic ﬁeld. If the orbit is initially bouncing and this drift does not cause a sudden change in the

turning points, then the motion approximately conserves the longitudinal adiabatic invariant J=Huds

where u=b·vis the parallel velocity and sthe position along the ﬁeld line. But when the turning points

do suﬀer a sudden change, then Jceases to be well-conserved. If the particle is still bouncing, the value

of Jsuﬀers a quasi-random jump, but if one or both of the turning points suddenly disappears Jceases

to be well-deﬁned altogether. When a bouncing particle has turning points that suﬀer any such abrupt

change, we say the particle suﬀers an orbit-type transition. It is these orbit-type transitions that bear the

responsibility for breakdown of the adiabatic invariance of J.

Since orbit-type transitions correlate with deleterious particle transport in magnetic conﬁnement devices

such as stellarators,6the search for magnetic ﬁeld conﬁgurations that minimize the probability of orbit-type

transitions warrants detailed investigation. Various strategies have been envisioned for identifying ﬁelds

which minimize such transitions, including quasisymmetry7–10 and omnigeneity.11–13 Here we present an

initial study of a diﬀerent strategy that we refer to as isoprominence. An isoprominent ﬁeld is deﬁned as

a nowhere-vanishing, divergence-free, vector ﬁeld Bsuch that the height of each potential peak of µ|B|is

independent of ﬁeld line, as sketched in Fig. 1, below. More precisely, isoprominence requires that |B|is

locally constant when restricted to the surface Σ−deﬁned as the set of points such that b· ∇|B|= 0 and

(b· ∇)2|B|<0.

a)Email correspondence: Nathan.Duignan@sydney.edu.au

arXiv:2210.10218v1 [physics.plasm-ph] 18 Oct 2022

As we will discuss in Sec. II, particles that move in an isoprominent ﬁeld cannot suﬀer orbit-type tran-

sitions to ﬁrst order in guiding center perturbation theory. While isoprominent ﬁelds do not comprise the

most general class of magnetic ﬁelds with this property, they enjoy the beneﬁt of an immediately trans-

parent physical interpretation. Moreover, as we show in Sec. III, isoprominent ﬁelds may be constructed

to all orders in an expansion about a given magnetic axis in powers of distance from the axis. We will

present details of this asymptotic expansion as well as examples of magnetic ﬁelds that are very nearly iso-

prominent in Sec. IV. Though the ﬁelds we construct do not necessarily satisfy the equilibrium conditions

of magnetohydrodynamics, we hope that this initial study motivates further study of isoprominence as a

potentially useful concept for stellarator optimization.

FIG. 1. Sketch of the magnetic topography, |B(s, α, 0)|, for an isoprominent ﬁeld, where sdenotes ﬁeld line arc length and

αa coordinate on Σ−. Note that the maximum values of |B|on each of the two components of Σ−does not change from

one ﬁeld line to the next, even though the position of the maxima on the s-axis and its value along the valley can both vary.

II. ISOPROMINENT MAGNETIC FIELDS

In this section we deﬁne the property of isoprominence for a magnetic ﬁeld. Then, we give an intuitive

argument as to why an isoprominent ﬁeld should mitigate orbit-type transitions in guiding center dynamics.

A formal proof is then given in Prop. II.6.

Let Bbe a smooth, nowhere-vanishing, divergence-free, vector ﬁeld deﬁned on an open region Q⊂R3.

The scalar functions

B=|B|, B0=b· ∇B, and B00 = (b· ∇)2B,

where b=B/B, quantify the magnitude of Band its rate of change along integral curves of the magnetic

ﬁeld—called, for simplicity, B-lines. If B0vanishes at a point σ∈Qthen Brestricted to the B-line passing

through σhas a critical point at σ. We say that σis critical along B. In this case, if B00(σ) is non-zero

then Brestricted to the B-line passing through σhas either a local maximum or local minimum at σ.

We say that Bis locally minimal along Bor local maximal along Bat σaccording to the sign of

B00(σ).

Deﬁnition II.1. The magnetic ridge associated with Bis the smooth submanifold Σ−⊂Qcomprising

points σ∈Qsuch that Bis locally maximal along Bat σ. Note that Σ−may have several connected

components.

Since points σ∈Σ−are not critical points of B:Q→Rin the usual sense, the function B−=B|Σ−:

Σ−→Ris generally non-constant, recall Fig. 1. This general situation may be visualized as follows. Fix

σ∈Σ−and let `σ(s) be the B-line passing through σparameterized by arc length s. The restriction of

Bto `σdeﬁnes a single-variable function Bσ=B|`σ, with a graph (s, Bσ(s)) that we call the magnetic

topography of σ. In general this topography has various peaks and valleys. Upon variation of σ∈Σ−the

magnetic topography will continuously deform, the peaks shifting in sand changing in height. Of course,

a peak may also collide with a valley, and either peaks or valleys may evaporate.

With this general picture in mind, isoprominence is deﬁned as follows.

Deﬁnition II.2. A nowhere-vanishing, divergence-free vector ﬁeld B, deﬁned on an open region Q⊂R3,

is isoprominent if the magnitude of Bis constant when restricted to a component of the magnetic ridge

Σ−.

For an isoprominent ﬁeld, the magnetic topography may still deform as σvaries within Σ−, but only in a

restricted manner - the peak heights Bσ(σ) cannot change. Note also that Σ−for an isoprominent ﬁeld is

a manifold of degenerate critical points for B.

Isoprominence mitigates orbit-type transitions for bouncing particles. This can be understood intuitively

as follows. In the guiding center approximation, a bouncing particle that starts on a ﬁeld line `σinitially

oscillates in a magnetic well that is deﬁned by a pair of magnetic peaks s∗

aand s∗

b, where `σ(s∗

a), `σ(s∗

b)∈Σ−.

As σvaries, so to do s∗

a,and s∗

b. It follows that s∗

a, and s∗

bare functions of σ. Without loss of generality,

let s∗

a(σ)< s∗

b(σ). Generally, the oscillations of a bouncing particle have turning points sa(σ)< sb(σ)

where Bσ(sa(σ)) = Bσ(sb(σ)) = E/µ, for particle energy Eand magnetic moment µ. Since the particle

bounces in the well, s∗

a(σ)< sa(σ)< sb(σ)< s∗

b(σ) and

min (Bσ(s∗

a(σ)), Bσ(s∗

b(σ))) > E/µ. (2.1)

Note that equality is not allowed here because such orbits would asymptotically approach Σ−and not

bounce. In order for the particle to suﬀer an orbit-type transition it must drift onto a ﬁeld line `σ0where

at least one of the bounce points sa(σ0) or sb(σ0) becomes coincident with s∗

a(σ0) or s∗

b(σ0). But if this

were to happen for an isoprominent ﬁeld then, since energy is conserved,

E/µ =Bσ0(s∗

a/b(σ0)) = Bσ(s∗

a/b(σ)),

which contradicts the inequality (2.1).

The weakness of this intuitive reasoning is that it assumes the guiding center Hamiltonian is given

precisely by

H0(x, u) = 1

2u2+µ B. (2.2)

However, in non-constant magnetic ﬁelds the single-particle Hamiltonian in guiding center coordinates

includes an inﬁnite series of higher-order correction terms.2,14,15 To understand the true implications of

isoprominence in the context of higher-order guiding center perturbations, we require a more general argu-

ment with a slightly weaker conclusion. As we will now explain, the higher-order terms can be accounted

for at the price of only approximately ensuring the suppression of type transitions.

Our strategy will amount to ﬁrst characterizing the set in phase space that separates diﬀerent bouncing

orbit types and then analyzing the component of the guiding center vector ﬁeld transverse to this separatrix.

As we will show, the bounce average of the transverse component vanishes through ﬁrst order in the guiding

center expansion parameter =ρ/L for isoprominent ﬁelds. This will imply that the bounce-averaged ﬂux

of particles across type boundaries is at most O(2).

The various classes of bouncing orbit types are deﬁned in terms of the zeroth-order guiding center

(ZGC) equations,

x=ub(x),

˙u=−µb(x)· ∇B(x),

that describe the motion of a particle with guiding center x∈Qand parallel velocity u∈Rin the limit

→0. These equations exactly conserve the energy (2.2). As mentioned above, solutions of these equations

that lie on the boundary of bouncing orbits asymptotically approach Σ−either in the future or the past.

If (x, u)∈Q×Ris the initial condition for such an orbit and σ∈Σ−is the magnetic peak it approaches

asymptotically, then the value of its conserved energy is given by

H0(x, u) = µ B(σ).

We say (x, u) is contained within the separatrix.

The following proposition characterizes the tangent space to the separatrix when the magnetic ﬁeld B

is isoprominent.

Proposition II.3. Suppose Bis isoprominent. If (x, u)lies within the separatrix for the ZGC equations

and x/∈Σ−then the tangent space to the separatrix at (x, u)is spanned by vectors of the form

δx

δu=δx

−µ1

uδx· ∇B(x), δx∈R3.

Proof. First we will argue that when Bis isoprominent, each connected component of the separatrix is

contained in a level set of H0. Equivalently, we will show that H0is locally-constant under the ﬂow of

the ZGC ﬂow along the separatrix. Let (x, u) be a point contained in the separatrix. Without loss of

generality, assume (x, u) has a forward-time limit, and so is a point in the stable manifold of Σ−. Let

Ft:Q×R→Q×Rbe the ZGC ﬂow. Choose an open neighborhood Ucontained in the separatrix

and containing (x, u) such that each point (x0, u0)∈Uhas a forward-time limit on a common connected

component of Σ−. If (x1, u1) and (x2, u2) are separatrix points in Uthen

H0(x1, u1) = lim

t→∞ H0(Ft(x1, u1)) = µ B(σ1,0),

H0(x2, u2) = lim

t→∞ H0(Ft(x2, u2)) = µ B(σ2,0),

where (σi,0) = limt→∞ Ft(xi, ui). Since σ1and σ2are contained in a common connected component of

Σ−and Bis locally constant on Σ−we must have B(σ1) = B(σ2). It follows that H0(x1, u1) = H0(x2, u2),

and that H0is locally constant on the separatrix, as claimed.

Now we will use the isoenergetic property of the separatrix to deduce the form of its tangent spaces. Let

(x, u) be a point in the separatrix with u6= 0. Any smooth curve (x(t), u(t)) contained in the separatrix

with (x(0), u(0)) = (x, u) must satisfy H0(x(t), u(t)) = H0(x, u). Diﬀerentiating in time implies

0 = u˙u+µ˙

x· ∇B,

where ˙u=du(0)/dt and ˙

x=dx(0)/dt. This implies ˙u=−µ˙

x· ∇B/u, which is the desired result.

Remark II.4. Note that this result merely says the tangent space to the separatrix at (x, u) is equal to

the tangent space to the energy level containing (x, u) when the magnetic ﬁeld is isoprominent.

Next we establish the general result that the leading-order energy for a conservative nearly-periodic

system is well-conserved on average.3,4,16,17

Proposition II.5. Let X=X0+ X1+2X2+. . . be a formal power series of a vector ﬁeld in on a

manifold M. Assume that Xadmits a formal energy invariant H=H0+ H1+2H2+. . . and that all

trajectories for X0are periodic with angular frequency function ω0. Then, after averaging along X0-orbits,

the rate of change of H0along Xis O(2).

Proof. For each z∈M, let z(t) be the unique solution to ˙z=X0(z) with z(0) = z. By assumption this

orbit is periodic, so let T(z)=2π/ω0(z) denote the period. The rate of change of H0along Xis

P≡d

dtH0=LXH0=LX0H0+LX1H0+2LX2H0+··· =−LX0H1+O(2),

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MinimizingSeparatrixCrossingsthroughIsoprominenceJ.W.Burby,1N.Duignan,2,3,a)andJ.D.Meiss21)LosAlamosNationalLaboratory,LosAlamos,NM97545USA2)DepartmentofAppliedMathematics,UniversityofColorado,Boulder,CO80309USA3)SchoolofMathematicsandStatistics,UniversityofSydney,NSW2050,Australia(Dated:20October20...

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Minimizing Separatrix Crossings through Isoprominence J.W. Burby1N. Duignan2 3aand J.D. Meiss2 1Los Alamos National Laboratory Los Alamos NM 97545 USA

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