J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Uniﬁed Theory of Ghost and Quadratic-Flux-Minimizing Surfaces
Robert L. DEWAR, Stuart R. HUDSON1) and Ashley M. GIBSON
Plasma Research Laboratories, Research School of Physics & Engineering, The Australian National University, Canberra
ACT 0200, Australia
1)
Princeton Plasma Physics Laboratory, PO Box 451, Princeton NJ 08543, USA
June 22, 2010

(Received: 14 November 2009 / Accepted: 14 March 2010)

A generalized Hamiltonian deﬁnition of ghost surfaces (surfaces deﬁned by an action-gradient ﬂow) is givenand specialized to the usual Lagrangian deﬁnition. Numerical calculations show uncorrected quadratic-ﬂuxminimizing (QFMin) and Lagrangian ghost surfaces give very similar results for a chaotic magnetic ﬁeld weakly
perturbed from an integrable case in action-angle coordinates, described by L = L0 + �L1 , where L0 (θ̇ ) (withθ̇ denoting dθ/dζ) is an integrable ﬁeld-line Lagrangian and � is a perturbation parameter. This is explainedusing a perturbative construction of the auxiliary poloidal angle Θ that corrects QFMin surfaces so they are also
ghost surfaces. The diﬀerence between the corrected and uncorrected surfaces is O(� 2 ), explaining the observed
smallness of this diﬀerence. An alternative deﬁnition of ghost surfaces is also introduced, based on an actiongradient ﬂow in Θ, which appears to have superior properties when uniﬁed with QFMin surfaces.
Keywords: Toroidal magnetic ﬁelds, Hamiltonian dynamics, Lagrangian dynamics, almost-invariant tori

1. Introduction

provided we modify the deﬁnition of ghost surface slightly,
there being some freedom also in the deﬁnition of actiongradient ﬂow. This raises the further question, acceptingthat ghost surfaces need to be redeﬁned to achieve uniﬁcation with QFMin surfaces, is the deﬁnition used in Ref. 4
optimal, or is there a still better class of uniﬁed ghost andQFMin surfaces?
In Secs. 2–4 we review the QFMin and ghost surface
concepts, and in Sec. 5 summarize the uniﬁcation proposed
in Ref. 4. In Sec. 6 we summarize its perturbative implementation in a test problem.
In Sec. 7 we propose a new deﬁnition of ghost surfaces
based on an action-gradient ﬂow in the auxiliary angle Θ,rather than the ﬂow in the unperturbed canonical coordinate θ used in Ref. 4. The new deﬁnition is not only simpler than that in Ref. 4, in that it does not require a variable
rate of ﬂow along pseudo ﬁeld lines, it also leads naturally
to straight pseudo-ﬁeld-line orbits in the Θ-ζ plane. This is
illustrated perturbatively for the same case used in Sec. 6.

Recent calculations [1] of heat diﬀusion along chaotic
ﬁeld lines show that the isotherms correspond very closelywith the “approximate” magnetic surfaces, associated withmagnetic island chains, known as ghost surfaces [2]. Thesesurfaces include the “X-point” and “O-point” closed ﬁeld
lines of their associated islands. (By “O-point” ﬁeld line
we mean either the elliptically stable ﬁeld line at the centerof an island or its hyperbolically unstable continuation if it
has undergone a period-doubling bifurcation.)
Closed ﬁeld
�lines extremize the magnetic action, A·dl, the hyperbolic
X-point ﬁeld lines in the chaotic separatrices being minima
and the O-point ﬁeld lines being minimax or saddle pointsof the action. Ghost surfaces are constructed by interpolating smoothly between these two closed-ﬁeld-line classes
by evolving the O-point ﬁeld lines into the X-point ﬁeldlines along paths of steepest descent of action, thus generating a family of “pseudo-orbits,” i.e. paths that comeclose to extremizing action.
An alternative approach to deﬁning approximate magnetic surfaces passing through magnetic islands, is to
use the quadratic-ﬂux-minimizing (QFMin) surfaces introduced by Dewar, Hudson and Price [2, 3]. Ghost surfaces
have nice mathematical properties but are diﬃcult to construct and have no obvious physical interpretation. QFMin
surfaces, on the other hand, have the computational attraction of being easy to construct, and the physical attractionof being deﬁned in terms of a measure of the magnetic ﬂux
transport through the surface, but have been found to exhibit undesirable distortions in some circumstances.
However the deﬁnition of QFMin surfaces is notunique, as it depends on the choice of an auxiliary poloidal
angle Θ; which raises the question: Can Θ be chosen sothat QFMin surfaces coincide with ghost surfaces? Thisquestion has recently [4] been answered in the aﬃrmative,

Fig. 1 A sketch of the general curvilinear toroidal coordinate
system described in the text.

author’s e-mail: robert.dewar@anu.edu.au487©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

R.L. Dewar et al., Unified Theory of Ghost and Quadratic-Flux-Minimizing Surfaces

2. Coordinates and Fluxesrate of increase of θ along the path is the rational rotationaltransform -ι p,q = p/q.
For a given ν there are generally two distinct p, qperiodic orbits. In particular, when ν = 0 we ﬁnd the true
periodic orbits associated with the X- and O-points of ap, q island chain.
The family of p, q-periodic QFMin pseudo-orbits
parametrized by ν deﬁnes an “almost invariant” surface
with rotational transform -ι p,q , which passes through thep, q island chain as depicted in Fig. 2.

We use a general curvilinear coordinate system ψ, θ, ζ,
as shown in Fig. 1, where θ and ζ are poloidal and toroidal
angles, respectively, and ψ is a label for the coordinate surfaces that has the dimension of magnetic ﬂux and is suchthat B = ∇ψ×∇θ + ∇ζ×∇χ.1 �
The standard linear ﬂux dS n·B through surface Γ:ψ = ψΓ (θ, ζ), where dS is an element of surface area andn is the unit normal, is independent of choice of coordinates and vanishes identically. Thus it is independent ofthe choice of Γ also, whether it be a magnetic surface (invariant torus of the ﬁeld-line ﬂow) or otherwise.

3. Action and ghost surfaces

Using the vector potential representation B = ∇×A,
the magnetic action of a closed p, q-periodic path C isk�1

1.0

S[C] ≡0.8

I�2Π
0.4
0.2

0.2

0.4

Θ�2Π

0.6

0.8

CA·dl ≡� 2πq0A·ṙ dζ,(3)where ṙ ≡ dr/dζ along the path. Hamilton’s Principle,
δS = 0, gives the Euler–Lagrange equation δS/δr =ṙ×B = 0, which implies ṙ is parallel to B as required.
Closed paths making S stationary are isolated, but thePoincaré–Birkhoﬀ theorem implies there are at least two
associated with each island chain, as illustrated in Fig. 2,with the X-point orbits being minima of S, while the Opoint orbits are minimax (saddle) points of S. The ghostcurve strategy for “joining the dots” is to generate a familyof pseudo-orbits, labelled by a continuous parameter τ, byﬂowing down the action gradient from minimax orbits tominimizing orbits,

0.60.00.0�

1.0Fig. 2 Poincaré plot for the Standard Map with the p, q =1, 3, 1,2 and 2,3 island chains, with associated actionminimizing (blue dots) and -minimax (red dots) periodicorbits. Also shown are almost-invariant curves interpolating between them.

δS
Dr= − ·Pghost ,Dτδr(4)

where Pghost is a symmetric nonnegative dyadic whose
choice is discussed below. The union of this set of pseudoorbits deﬁnes a ghost surface (a ghost curve being thePoincaré section of a ghost surface).

To measure the amount by which Γ departs from beinga magnetic surface, we are instead led to deﬁne the positivedeﬁnite quadratic ﬂux [3]� �n·B
1 2π 2πn·B, (1)dθdζ
ϕ2 [Γ] ≡2 0 0n·∇θ×∇ζ n·∇Θ×∇ζwhere Θ is an auxiliary poloidal angle, not necessarily
the coordinate angle θ. Setting δϕ2 = 0 for arbitrary
variations ψΓ we ﬁnd the Euler–Lagrange equation for
QFMin pseudo-orbits [3, 4] is Bν ·∇ν = 0, where ν ≡n·B/n·∇Θ×∇ζ and the “pseudo magnetic ﬁeld” Bν is deﬁned byBν ≡ B − ν∇Θ×∇ζ.

(2)

Thus ν parametrizes a continuous family of pseudo ﬁeldlines, which can be thought of as orbits of a dynamicalsystem. We deﬁne a p, q-periodic pseudo-orbit (p and qbeing integers) as a path that is a closed loop, with θ increasing by 2πp when ζ increases by 2πq, so the averageFig. 3 A comparison of uncorrected (Θ = θ) QFMin curves(thick lines) and ghost curves (thin lines) for the morestrongly chaotic case described in Ref. 4. Some cases
where QFMin curves violate the graph property are seen.

1 Note that χ = χ(ψ, θ, ζ) cannot in general be made a surface functionin three-dimensional systems due to the generic nonintegrability of themagnetic ﬁeld-line ﬂow.

488

R.L. Dewar et al., Unified Theory of Ghost and Quadratic-Flux-Minimizing Surfaces

where Θ0 is, as yet, an arbitrary label, satisfy the Euler–Lagrange equation

Ghost curves and uncorrected QFMin curves (i.e.
curves for which the auxiliary angle Θ has been taken
to be the same as the coordinate angle θ) are comparedin Fig. 3. For the lower-order p, q cases the two deﬁnitions of almost-invariant curves are almost indistinguishable. However, higher-order uncorrected QFMin curves
can cease to be graphs over θ and do not agree with theisotherms found by Hudson and Breslau [1]. Thus we are
led to seek a way to choose Θ so that the two approaches
may be reconciled, keeping the physical appeal and easeof implementation of the QFMin approach while retaining
the superior mathematical properties of ghost surfaces.

δSν δS=− ν(Θ0 )Θθ (θ, ζ) = 0 ,δθδθwith Θθ (θ, ζ) ≡ ∂Θ(θ, ζ)/∂θ.
To reconcile QFMin and ghost orbits we require thatthe family of pseudo-orbits deﬁned by Eqs (9) and (10) is
the same family as is generated by Eq. (7). Thus the labels Θ0 and τ must be functionally dependent: τ = τ(Θ0 ),dτ = τ� (Θ0 )dΘ0 . Eliminating δS/δθ between Eq. (10) andEq. (7) and observing that Θθ ≡ (DΘ/DΘ0 )/(Dθ/DΘ0 ),where Dθ/DΘ0 ≡ ∂θ(ζ|Θ0 )/∂Θ0 , we ﬁnd the reconciliation condition
�2�Dθμ(ζ)
DΘ=− �.
(11)DΘ0τ (Θ0 )ν(Θ0 ) DΘ04. Lagrangian approach

We ﬁrst map this problem onto one of Lagrangian dynamics in order to make better contact with the mathematical literature, e.g. Ref. 5.
The vector potential A = ψ∇θ − χ(ψ, θ, ζ)∇ζ gives
our assumed
form for B and leads to the action integral�S[C] = (ψdθ − χdζ), the ﬁrst variation being
� 2πq ����� �θ̇ − ∂ψ χ δψ − ψ̇ + ∂θ χ δθ dζ . (5)δS =We now deﬁne Θ0 so that, for all Θ0 ,τ� (Θ0 )ν(Θ0 )θ(ζ|Θ0 + 2π)

Setting δS = 0 for all δψ and δθ leads to Hamiltonian
equations of motion with θ as generalized coordinate, ψas canonical momentum, χ as Hamiltonian, and ζ as time:θ̇ = ∂ψ χ and ψ̇ = −∂θ χ.
The transition to the Lagrangian approach is made in
the usual way, by solving θ̇ = ∂ψ χ� to give ψ as a function ofθ̇ and writing the action as S = L dζ, where L(θ, θ̇ , ζ) ≡ψ(θ, θ̇ , ζ) θ̇ − χ(ψ, θ, ζ). The ﬁrst variation now becomes� 2πqδS = 0 δθ (δS/δθ) dζ, the functional derivative (actiongradient) beingd ∂L
δS ∂L=−.
(6)δθ∂θ dζ ∂θ̇Specialization of the general ghost-orbit gradient ﬂow
Eq. (4) to the Lagrangian form is eﬀected by taking
Pghost = � −1 eψ eψ + μ−1 eθ eθ and taking the limit � → ∞to enforce the constraint θ̇ = ∂ψ χ. Then the gradient ﬂow
becomesDθ
1 δS= −,
(7)Dτμ(ζ) δθ
where μ(ζ) = O(1) allows us to generalize the form assumed in Ref. 2 slightly, which we shall ﬁnd necessary forreconciliation to be possible.

(9)θ(ζ|Θ0 ) + 2π ,(12)For example, consider
L=2∞�θ̇−�Vm,n exp(imθ − inζ) ,2 m,n=−∞(14)∗
with the reality condition Vm,n= V−m,−n , and � the expansion parameter. (Deﬁning the largest perturbation coeﬃcient, V2,1 , to be 1, � = 10−3 for the case shown in Fig. 3.)
As the unperturbed system is integrable, the expansions of ν(Θ0 ), μ(ζ), and Θ(θ, ζ) are of the formν=�ν1 + � 2 ν2 + . . . ,(15)1 + �μ1 + � 2 μ2 + . . . ,���(1)
�Θm,n + . . . exp i(mθ − nζ) ,= θ+μ =Θ

m,nand the (p, q) QFMin pseudo-orbits are expanded as���(1)(2)+ � 2 θm,n+ ...
�θm,nθ(ζ|Θ0 ) = -ι p,q ζ + Θ0 +m,n�
�
× exp i m -ι p,q − n ζ + mΘ0 , (16)��We now seek to choose Θ so that QFMin pseudoorbits are also the Lagrangian ghost pseudo-orbits deﬁned
above. We require [4] ∂ψ Θ = 0. That is,θ = θ(ζ|Θ0 ) ,≡−1 ,6. Perturbative construction of QFMin-ghost
surfaces

5. Reconciling QFMin and Lagrangian Ghost
surfaces(8)≡choosing μ(ζ) so that Eq. (11) satisﬁes the solvability condition that the integral of both sides with respect to Θ0 overthe interval [0, 2π] must be 2π, giving⎡� 2π��2 ⎤−1⎢⎢dΘ0 ∂θ(ζ|Θ0 ) ⎥⎥⎥⎥(13)μ(ζ) = ⎢⎢⎢⎣⎥⎦ .
2π∂Θ000Θ ≡ Θ(θ, ζ) .

(10)

where -ι p,q ≡ p/q.
At O(�) we ﬁnd μ1 = 0. Also, Θ(1)m,n is not used in the(1)calculation of θm,n . Thus, to ﬁrst order, uncorrected ghostand QFMin pseudo-orbits are identical. This explains why
the low-order uncorrected ghost and QFMin orbits appear
almost identical in Fig. 3.

Then Ψν = Ψ, Lν = L − νΘ, ∂Lν /∂θ̇ = ∂L/∂θ̇ , and members of our family of QFMin pseudo-orbits489

R.L. Dewar et al., Unified Theory of Ghost and Quadratic-Flux-Minimizing Surfaces�ϑΘ = 1 + m,n imθm,n exp[imΘ0 + i(m -ι − n)ζ]. Insertingthese into Eq. (22), and equating the LHS to zero at O(�),we get(1)(2)
Explicit expressions for ν1 , μ2 , θm,n, θm,nand Θ(1)m,n are(2)
given in Ref. 4, but, as there is a misprint in θm,n , we give
below a corrected version,
(2)
θm,n=

θ
�� Nm,m
� ,n,n�iδ̄mp,nq(m -ι p,q − n)2 m� ,n� (m� -ι p,q − n� )2ν1 (Θ0 ) = −(1)θm,n=ν2(24)−i=

�m,n

δmp,nq��m� ,n�ΘNm,m
� ,n,n�(m� -ι p,q − n� )2× Vm+m� ,n+n� Vm∗ � ,n� eimΘ0 , (25)

and(18)(2)
θm,nwhere T is the timelike label for Θ ghost orbits [so dT =T � (Θ0 )dΘ0 ] and the inverse function ϑ(Θ, ζ) is deﬁned bysolving Eq. (9) for θ as a function of Θ.
From Eq. (10) we also have, for corrected QFMin orbits,=Θ�� Nm,m
� ,n,n�iδ̄mp,nq2
�(m -ι p,q − n) m� ,n� (m -ι p,q − n� )2× Vm+m� ,n+n� Vm∗ � ,n� ,(26)��Θ����
where Nm,m
� ,n,n� ≡ m (m + m ) m + m − m δ(m+m� )p,(n+n� )q(the sign of the δ(m+m� )p,(n+n� )q term in this expression being
the opposite of that for θ ghost orbits).

(19)

8. Conclusion

Eliminating δS/δθ between Eq. (18) and Eq. (19) we ﬁnd
the new reconciliation condition

We have reviewed the motivation and formulation ofa recently published [4] uniﬁcation of ghost and quadraticﬂux-minimizing surfaces, and have presented a new formulation that appears more natural and more useful in thatit could be used to form a straight pseudo-ﬁeld-line coordinate system.
More work remains to be done to develop a numerical algorithm for constructing such ghost/QFMin surfaces
at arbitrary order of perturbation, and to show they preserve the desirable properties of our previous formulationof ghost surfaces.
The relation to heat transport in chaotic ﬁelds also
needs to be investigated more deeply.

(20)

where, in the second equality, we have deﬁned Θ0 so thatT � (Θ0 )ν(Θ0 ) ≡ −1, which satisﬁes the solvability condi� 2πtion that 0 dΘ0 DΘ/DΘ0 = 2π without the need to introduce a factor like the μ(ζ) required in Sec. 5.
Furthermore, Eq. (20) is satisﬁed by choosing the ζdependence of the pseudo-orbit labeling so thatΘ(ζ|Θ0 ) = Θ0 + -ιζ ,imδ̄mp,nq Vm,n,(m -ι p,q − n)2which are the same results as in Ref. 4, showing that θand Θ ghost orbits (and, by deﬁnition, reconciled QFMinorbits) are equivalent at this order.
At O(� 2 ) we ﬁndIn Sec. 4 we deﬁned ghost surfaces in terms of anaction-gradient ﬂow in θ, but deﬁned QFMin surfaces interms of Θ. Here we further unify the formulation by deﬁning the action-gradient ﬂow also in terms of Θ,DΘ= −T � (Θ0 )ν(Θ0 ) = 1 ,DΘ0(23)and7. Θ ghost surface formulation

δS= ν(Θ0 ) .
δΘimδmp,nq Vm,n eimΘ0 ,m,n=−∞× Vm+m� ,n+n� Vm∗ � ,n� ,(17)
��
θ��
�
�
where Nm,m� ,n,n� ≡ m (m + m ) m + m + m δ(m+m� )p,(n+n� )q .
Also, δ̄mp,nq ≡ 1 − δmp,nq , where δmp,nq ≡ 1 when mp =nq, is 0 otherwise, and the prime on the sum over m� andn� indicates that the resonant terms, m� p = n� q, are to bedeleted.

δS
δSDΘ
=−
≡ −ϑΘ,DTδΘδθ

∞
�(21)

conjugating the θ dynamics to straight-ﬁeld-line dynamics(cf. Ref. 6). Thus the Θ ghost surface approach appears
to be much superior to the θ ghost surface approach, butneeds to be tested in practice.
As a ﬁrst test we adapt the perturbative calculation inSec. 6 to Θ ghost surfaces. First, expanding ϑ(Θ, ζ) = Θ +�m,n θm,n exp(imΘ − inζ) and using Eq. (21), we recoverEq. (16). This ansatz is to be inserted in Eq. (19), written
in the form��dLθ̇(22)− Lθ ϑΘ + ν(Θ0 ) = 0 .
dζ�From Eq. (14), dLθ̇ /dζ = − m,n (m -ι − n)2 θm,n exp[imΘ0 +�i(m -ι − n)ζ], Lθ = −� m,n imVm,n exp[imθ(ζ|Θ0 ) − inζ], and[1] S. R. Hudson and J. Breslau, Phys. Rev. Lett. 100, 095001(2008).
[2] S. R. Hudson and R. L. Dewar, J. Plasma Phys. 56, 361(1996).
[3] R. L. Dewar, S. R. Hudson, and P. F. Price, Phys. Lett. A
194, 49 (1994).
[4] S. R. Hudson and R. L. Dewar, Phys. Lett. A 373, 4409
(2009). (Figures 1 and 3 of the present paper are reproduced
from [4] with permission from Elsevier.)[5] C. Golé, Symplectic Twist Maps : Global Variational Techniques (World Scientiﬁc, Singapore, 2001).
[6] R. L. Dewar and J. D. Meiss, Physica D 57, 476 (1992).

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