magnetized particle motion in 3d fields and the ......magnetized particle motion in 3d fields and...

MAGNETIZED PARTICLE MOTION IN 3D FIELDS AND THE NATURE OF 3D EQUILIBRIA

Felix I. ParraRudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK

1

3D magnetic fields in stellarators

• Stellarators have inherent advantages• No current in the plasma ⇒ no current drive, no current instabilities

2

• Magnetic field Bmust be 3D (that is, without direction of symmetry) if we want• steady state• nested flux surfaces

(surfaces || to B)• B (mostly) generated

by external currents

Magnetized particle motion• Assume steady state E and B: E = –∇𝜙 ~ T /eL• Constant total energy

• Motion for 𝜌* = 𝜌/L

Parallel motion• To lowest order, particles move along magnetic field lines

4

dl

dt= vk = ±

s

2

✓E � µB(l)� Ze�(l)

m

◆= ±

p2 (E � U(l))

4 F.I. Parra, I. Calvo, J.L. Velasco and J.A. Alonso

l

µ3

µ2

µ1

µ4

U

E

lL(E , µ3) lR(E , µ3)

UM (µ3)

Um(µ3)

Figure 1. Example of e↵ective potential U for four values of µ: µ1, µ2, µ3 and µ4. Note theminimum and maximum values of U for µ3, Um(µ3) and UM (µ3). We show the bounce pointsfor a particle with magnetic moment µ3 and an energy E that satisfies Um(µ3) < E < UM (µ3).

d ⇣/dr. We assume that r has units of length, and that ↵ is an angle, that is, it doesnot have units. The mapping x(r,↵, l) uniquely determines the position because thedeterminant of its Jacobian is always di↵erent from zero,

@rx · (@↵x ⇥ @lx) =1

rr · (r↵⇥ rl) = 0⇣

2⇡B, (2.5)

To obtain (2.5), we have used (rr ⇥ r↵) · rl = 2⇡( 0⇣)�1B · rl and B = Bb̂ = B @lx.Given E and µ, the particle motion along a magnetic field line is given by

dl

dt= v||( ,↵, l, E , µ) = ±

p

2 (E � U( ,↵, l, µ)), (2.6)

where v||( ,↵, l, E , µ) is the velocity parallel to the magnetic field, and

U( ,↵, l, µ) = µB( ,↵, l) +Ze�( ,↵, l)

m(2.7)

is the e↵ective potential for the motion along the magnetic field. Trapped particles satisfyUm( ,↵, µ) < E < UM ( ,↵, µ), where Um( ,↵, µ) and UM ( ,↵, µ) are the minimumand maximum value of U(l) along the magnetic field line of interest (see figure 1).Trapped particles move periodically between the two bounce points lL( ,↵, E , µ) andlR( ,↵, E , µ), sketched in figure 1. Since the parallel velocity v|| vanishes at the twobounce points, they are determined by the equation

E � U(lL) = 0 = E � U(lR). (2.8)

Note that here we only display the dependence of U on l to shorten the notation. Wewill continue to do this for the rest of the article.

The adiabatic invariant of the quasi-periodic motion in (2.6), known as second adiabaticinvariant, is

J(r,↵, E , µ) =I

v|| dl = 2

Z lR

lL

|v||| dl = 2p

2

Z lR

lL

p

E � U(l) dl, (2.9)

where the factor of 2 is due to the fact that we need to consider both signs of theparallel velocity. The slow average drift across magnetic field lines is related to the second

• ℰ > UM (𝜇) ⇒ v|| does not change sign: passing particles

• ℰ < UM (𝜇) ⇒ v|| vanishes at bounce points: trapped particles

Perpendicular motion• Ignore perpendicular motion for passing particles:

perpendicular drifts give small correction to position• We’ll have to come back to this

• Trapped particles do not leave initial region unless we keep perpendicular drifts

• Use flux coordinates to describe perpendicular motion of trapped particles

5

Flux coordinates• Simple picture: coordinates on plane cutting through B

lines

• r and 𝛼 constant along B

• Magnetic flux across surface dr d𝛼 =

6

r ·B = 0 ) B = (r,↵)rr ⇥r↵

B ·rr = 0 = B ·r↵ ) B = (x)rr ⇥r↵

r↵

dr d↵

Motion of trapped particles• Perpendicular motion in r-direction

• Since parallel motion is periodic with period 𝜏b = ∮dl /v|| ,decompose r into quasiperiodic and secular pieces

• Averaging over parallel trapped orbit

• Integrating in l,

7

dr

dt=

dx

dt·rr = vd ·rr

r(t) = r(t) + er(l(t)) ) drdt

' drdt

+ vk@er@l

dr

dt=

1

⌧b

Ivd ·rr

dl

vk

vk@er@l

= vd ·rr �dr

dt) er ⇠ ⇢⇤r ⌧ r ' r

Final equations• Since 𝛼 is equivalent to r

• We can simplify these equations

8

dr

dt' 1

⌧b

Ivd ·rr

dl

vk⇠ ⇢⇤

vtLr

d↵

dt' 1

⌧b

Ivd ·r↵

dl

vk⇠ ⇢⇤

vtL↵

Second adiabatic invariant• Adiabatic invariant of periodic motion of trapped particles

• Equations based on second adiabatic invariant

• Particles move conserving their second adiabatic invariant

•

9

J(r,↵, E , µ) =I

vk dl = 2

Z lR

lL

s

2

✓E � µB(l)� Ze�(l)

m

◆dl

dJ

dt=

dr

dt

@J

@r+

d↵

dt

@J

@↵= 0

dr

dt' mc

Ze ⌧b

@J

@↵d↵

dt' � mc

Ze ⌧b

@J

@r

Summary of particle motion• Passing particles move mostly along B lines

• Trapped particles move along paths of constant J• In addition, particles have “oscillations” (small in 𝜌*) around the

paths of constant J

• To describe particle motion, we need maps of J• J can be easily constructed for given 𝜙 and B

10

Particle motion in stellarators• Stellarator: confine to have nuclear fusion collisions

• Need flux surfaces• A large number of Coulomb collisions will occur

• r labels flux surfaces and 𝛼 lines within flux surfaces• Alternatively, poloidal and toroidal angles 𝜃 and 𝜁 within flux surface

11

✓

⇣

Passing particles• Passing particles follow B lines ⇒ two types of surfaces

12

• In most surfaces, pitch of B⇒ one B line defines whole flux surface = ergodic

• In a few surfaces, pitch of B⇒ B lines close on themselves = rational

Passing particles on ergodic surfaces• To lowest order, passing particles stay on flux surfaces• Can passing particles move across surfaces (in r-

direction)? No on ergodic surfaces• Passing particles sample ergodic surfaces entirely• r-direction drift ∝ gradients & curvature parallel to flux surface

• Gradients & curvature “average” to zero due to periodicity in 𝜃 & 𝜁⇒ average r-direction drift = 0

• r can be described by quasiperiodic piece

13

vd ·rr = �mc

ZeB

✓Ze

mr�+ µrB + v2kb̂ ·rb̂

◆· (b̂⇥rr)

vk@er@l

= vd ·rr ) er ⇠ ⇢⇤r ⌧ r

vk@er@l

= vkb̂ ·rer = vd ·rr )Z

B linesvd ·rr

dl

vk= 0

Passing particles on rational surfaces• Particles do not sample rational surfaces entirely. Do they

move across flux surfaces (in r-direction)? Not quite • Naively, equation for quasiperiodic piece is singular

• Passing particles are still confined because rational surface is surrounded by ergodic flux surfaces

14

Singular Not satisfied in general

vkb̂+ er

@

@r

⇣vkb̂

⌘�·rer = vd ·rr

) r̃ ⇠ p⇢⇤r ⌧ r

Collisions• Distribution function must be Maxwellian

• Passing particles stay at constant r to lowest order in 𝜌*⇒𝜂 and T must only depend on r

• Trapped particles follow paths of constant J, in general different from flux surfaces ⇒ 𝜂 and T must be constants to be compatible with passing particles

• To achieve confinement, 𝜂 and T must depend on r⇒ Jpaths must coincide with flux surfaces

15

J = J(r, E , µ) ) @J@↵

= 0

f = fM = ⌘(x)

✓m

2⇡T (x)

◆3/2exp

✓� mET (x)

◆

Omnigeneity

Omnigeneous stellarators• Maxwellian with 𝜂 and T only depending on r

⇒ Quasineutrality imposes that 𝜙 depend only on r• Since 𝜙 does not depend on l, convenient to use

variables v and 𝜆 = 2𝜇/v 2

• Bounce points B (lL) = 1/𝜆 = B (lR)• Passing particles 𝜆 < 1/BM, trapped particles 𝜆 > 1/BM

16

n(x) = ⌘(r) exp

✓�Ze�(x)

T

◆

J = 2

Z lR

lL

s

2

✓E � µB(l)� Ze�

m

◆dl = 2v

Z lR

lL

p1� �B(l) dl

”Constructing” omnigeneous fields• Trapped particles need paths that do not close

• Choose 𝜃 and 𝜁 such that B lines are parallel straight lines

17

Nucl. Fusion 55 (2015) 033005 F.I. Parra et al

Figure 5. Contour plot of the omnigeneous magnetic field.

along a field line between two contours with the same valueof magnetic field strength on two opposite sides of a well is afunction only of the flux surface and of the well.

The extra assumption that all the local minima and all thelocal maxima have the same value has been used for simplicityin previous work, such as [11, 12]. The results of these papersonly need to be generalized slightly to account for local minimaand maxima with values different from the global minimumand maximum on the flux surface, but the qualitative resultsare probably unchanged.

In certain classes of stellarators, it may be beneficialfor the minima of the magnetic field strength on a fluxsurface to have similar values [13], but we have shown thatin general this is not a necessary condition for optimizedstellarators. The more general conditions for omnigeneitydiscussed in this article ensure that the neoclassical fluxesdo not scale inversely with collisionality. These conditionshave been derived for a flux surface without considerationof the neighbouring flux surfaces because we have limitedour analysis to particles with small radial orbit widths. Thestudy of long term confinement of energetic particles [14, 15]requires a more careful analysis than the one performed here,but one would expect that optimizing neoclassical fluxes wouldimprove energetic particle confinement.

In practice, the design of an omnigeneous stellaratorexperiment would be based upon the multiple-criteriaoptimization procedures analogous to those used in [1–3]to identify a 3D MHD equilibrium. As part of such adesign, collisionless charged particle losses would need to

be computed directly using the codes based on the guidingcentre drift equations, as in [14, 15]. However, due to the highdimensionality of the optimization problem, simple but robustoptimization criteria are required to use the optimization codeseffectively, for selecting appropriate weighted cost functions,initial configurations, and search algorithms. Our results heregive such criteria. An optimization that imposed that the localminima and maxima must be the same on a given flux surfacecould give a stellarator close to omnigeneity, but it would haveignored a large part of the allowed parameter space, thereforemissing potentially better solutions.

Acknowledgments

The authors would like to thank J.R. Cary and S.G. Shasharinafor helpful discussions. This work has been carried outwithin the framework of the EUROfusion Consortium and hasreceived funding from the European Union’s Horizon 2020research and innovation programme under grant agreementnumber 633053. The views and opinions expressedherein do not necessarily reflect those of the EuropeanCommission. This research was supported in part bythe RCUK Energy Programme (grant number EP/I501045)and by grant ENE2012-30832, Ministerio de Economı́a yCompetitividad, Spain.

References

[1] Gori S. and Nührenberg J. 1999 Proc. Joint Varenna–LausanneInt. Workshop on Theory of Fusion Plasmas (Varenna, Italy,1998) (Bologna: Compositori) p 473

[2] Subbotin A.A. et al 2006 Nucl. Fusion 46 921[3] Kasilov S.V. et al 2013 Plasma Phys. Rep. 39 334[4] Wobig H. 1993 Plasma Phys. Control. Fusion 35 903[5] Nührenberg J. et al 1995 Trans. Fusion Technol. 27 71[6] Anderson F.S.B. et al 1995 Fusion Technol. 27 273[7] Neilson G.H. et al 2002 J. Plasma Fusion Res. 78 214[8] Cary J.R. and Shasharina S.G. 1997 Phys. Rev. Lett. 78 674[9] Cary J.R. and Shasharina S.G. 1997 Phys. Plasmas 4 3323

[10] Boozer A.H. 1982 Phys. Fluids 25 520[11] Helander P. and Nührenberg J. 2009 Plasma Phys. Control.

Fusion 51 055004[12] Landreman M. and Catto P.J. 2012 Phys. Plasmas 19 056103[13] Mynick H.E., Chu T.K. and Boozer A.H. 1982 Phys. Rev. Lett.

48 322[14] Lotz W., Merkel P., Nührenberg J. and Strumberger E. 1992

Plasma Phys. Control. Fusion 34 1037[15] Drevlak M., Geiger J., Helander P. and Turkin Y. 2014 Nucl.

Fusion 54 073002

5

⇣ ⇣

✓

[Cary & Shasharina, PRL 97, PoP 97]

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⇣ ⇣

✓


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⇣ ⇣

✓


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⇣ ⇣

✓

B-contours imply J cannot be constant on flux surface

Maxima BM and minima Bmalong B lines are the same


��1

• Rewrite integral as integral over B

⇒ ∂l /∂B independent of 𝛼⇒ length along B line between two points with same B

Keeping ∂J /∂𝛼 = 0 for all 𝜆

21

B

l

↵

J = 2v

Z lR

lL

p1� �B(l) dl = 2v

Z 1/�

Bm

p1� �B @l

@BdB

dB@l

@BdB


Keeping ∂J /∂𝛼 = 0 for all 𝜆• Rewrite integral as integral over B

⇒ ∂l /∂B independent of 𝛼⇒ length along B line between two points with same B

22

B

l

↵↵+�↵

J = 2v

Z lR

lL

p1� �B(l) dl = 2v

Z 1/�

Bm

p1� �B @l

@BdB

�l(B,↵) = �l(B,↵+�↵)









Acknowledgments


References







Fusion 54 073002

5

Construct omnigeneous B maps

23

B

⇣

✓

⇣

�l(B)

• Choose a rough 1D Bprofile• Number of BM and Bm

• Choose “half” of your contours• From BM to Bm

• Choose Δl (B)• Boozer 𝜃 and 𝜁: B lines

are straight lines, and Δl∝ Δ𝜃, Δ𝜁

• Infinite B (𝜃, 𝜁) that are omnigeneous!


Construct omnigeneous B maps• Choose a rough 1D B

profile• Number of BM and Bm

• Choose “half” of your contours• From BM to Bm

• Choose Δl (B)• Boozer 𝜃 and 𝜁: B lines

are straight lines, and Δl∝ Δ𝜃, Δ𝜁

• Infinite B (𝜃, 𝜁) that are omnigeneous!

24








Acknowledgments


References







Fusion 54 073002

5

B

⇣

✓

⇣


Omnigeneous stellarator equilibria• Omnigeneous stellarators can reach Maxwellian

equilibrium ⇒ satisfy MHD force balance

• Assume known a flux surface r and B (𝜃, 𝜁) on it• MHD equilibrium parallel to flux surface ⇒• ∇∙ B = 0 ⇒ determines surface shape of surface r - ∆r• MHD equilibrium perpendicular to flux surface ⇒ B (r - ∆r, 𝜃, 𝜁)

• Integrate radially inwards to find equilibrium• Regularity at the magnetic axis ⇒ only the shape of flux

surface r (and not B (𝜃, 𝜁) on it) can be specified

25

b̂(✓, ⇣)

B2

4⇡b̂ ·rb̂�r?

✓B2

8⇡

◆= rr@p

@r

Omnigeneous equilibria• Can choose one surface to

be omnigeneous: choose 2D function z (x, y) to obtain B (𝜃, 𝜁)

• In general, omnigeneitylost in other surfaces

• Is there an omnigeneoussurface that propagates radially? For B with all derivatives continuous, only perfect axisymmetry: TOKAMAK

26

✓

⇣

[Garren & Boozer, PoP 91]

Any other omnigeneous equilibria?• Stellarators can be optimized to be near-omnigenous• Some examples

• Boozer 𝜃 and 𝜁: B lines are straight lines, and Δl∝ Δ𝜃, Δ𝜁

27

Nucl. Fusion 51 (2011) 076001 Special Topic

0 π 2πNφ

0

π

2π

θ0 π 2π

Nφ

0

π

2π

θ

0 2π 4πθ

0.95

1.00

1.05

B/B

0

Figure 5. Contours of constant magnetic field strength for one fieldperiod of HSX (with N = 4) are shown for the ρ = 0.5 flux surfaceof the standard vacuum configuration which has !ι = 1.0537.Contours have been plotted following the same scheme employed infigure 1. In the lower frame, the magnitude of B/B0 along the fieldline passing through φ = 0 and θ = 0 is plotted over two poloidalcircuits of the torus.

(QIPC) [69] is depicted in figure 8. The stronger variation of Bwith poloidal angle in the case of QPS is largely due to its smallaspect ratio R0/a = 2.65 in comparison with R0/a = 11.5 forQIPC. Both configurations have a fraction of trapped particleswell in excess of 50% but nevertheless fulfil the requirementof small radial transport coefficients in the lmfp regime as canbe seen in the next section.

The final configuration investigated within the ICNTSis the Wendelstein 7-X (W7-X) device [70, 71], which iscurrently under construction at IPP in Greifswald, Germany.This helias (helical-axis advanced stellarator) [72] emergedfrom an integrated design process which had the goal offinding magnetic fields which simultaneously fulfil a numberof optimization criteria relevant to good plasma performance.Additionally, the dimensions of W7-X (R0 = 5.5 m,a > 0.5 m) along with its heating and support systemswere chosen to enable the high-performance, steady-statedischarges necessary to assess the potential of such a devicefor attractive reactor operation. For experimental flexibilitythe W7-X coil system is capable of numerous magneticconfigurations, among which are examples of the various

0 π 2πNφ

0

π

2π

θ

0 π 2πNφ

0

π

2π

θ

0 2π 4πθ

0.95

1.00

1.05

B/B

0

Figure 6. Contours of constant magnetic field strength for one fieldperiod of NCSX (with N = 3) are shown for the ρ = 0.5 fluxsurface of the reference S3 plasma configuration with Ip = 174 kAand ⟨β⟩ = 4.1% [65] which has !ι = 0.4942. Contours have beenplotted following the same scheme employed in figure 1. In thelower frame, the magnitude of B/B0 along the field line passingthrough φ = 0 and θ = 0 is plotted over two poloidal circuits of thetorus.

neoclassical optimization strategies discussed here, with theexception of quasi-symmetry (as small bootstrap currentwas an optimization criterion for W7-X). At its simplest,this strategy involves nothing more than the large averageelongation of the W7-X flux surfaces, which lies in the range4.5 < κ < 7.0 depending on the configuration. The magneticfield for such a case is exemplified by the W7-X low-mirrorvacuum configuration plotted in figure 9(a), which is obtainedby choosing the coil currents so as to zero the toroidal-mirrorterm (the b0,1 component of B in Boozer coordinates) onthe magnetic axis. Unlike W7-AS, deeply trapped particles‘see’ only a rather small variation of B in this field andthus experience the full benefit of the large reduction inaverage toroidal curvature, leading to a significant decreasein neoclassical radial transport. Further improvements arepossible, however, by introducing a modest toroidal mirrorinto B to simultaneously profit from large elongation andstrong drift optimization, as is done for the W7-X standardconfiguration shown in figure 9(b) (which has equal currentsin all non-planar coils, resulting in b0,1 = 0.046 on axis).By further increasing the toroidal mirror, W7-X has access

11

Nucl. Fusion 51 (2011) 076001 Special Topic

0 π 2πNφ

0

π

2π

θ

0 π 2πNφ

0

π

2π

θ

0 2π 4πθ

0.94

0.96

0.98

1.00

1.02

1.04

1.06

B/B

0

0 π 2πNφ

0

π

2π

θ

0 π 2πNφ

0

π

2π

θ

0 2π 4πθ

0.95

1.00

1.05

1.10

B/B

0

(a) (b)

Figure 9. (a) Contours of constant magnetic field strength for one field period of W7-X (with N = 5) are shown for the ρ = 0.5 flux surfaceof the low-mirror vacuum configuration which has !ι = 0.8623. Contours have been plotted following the same scheme employed in figure 1.In the lower frame, the magnitude of B/B0 along the field line passing through φ = 0 and θ = 0 is plotted over two poloidal circuits of thetorus. (b) Contours of constant magnetic field strength for one field period of W7-X are shown for the ρ = 0.5 flux surface of the standardvacuum configuration which has !ι = 0.8700. Contours have been plotted following the same scheme employed in figure 1. In the lowerframe, the magnitude of B/B0 along the field line passing through φ = 0 and θ = 0 is plotted over two poloidal circuits of the torus.(c) (overleaf) Contours of constant magnetic field strength for one field period of W7-X are shown for the ρ = 0.5 flux surface of thehigh-mirror vacuum configuration which has !ι = 0.8823. Contours have been plotted following the same scheme employed in figure 1. Inthe lower frame, the magnitude of B/B0 along the field line passing through φ = 0 and θ = 0 is plotted over two poloidal circuits of thetorus.

presented in this section, normalizations to the results for anaxisymmetric tokamak with B/B0 = 1−ϵt cos θ in the plateau(p), banana (b) and Pfirsch–Schlüter (PS) regimes have beenchosen:

D⋆11 ≡D11

Dp11

, Dp11 =

π

4v2dR0

v!ι;

D⋆31 ≡D31

Db31, Db31 =

23

vdR0

!ιϵt(1 − fc);

D⋆33 ≡D33

DPS33, DPS33 =

v2

3ν⟨B2⟩B20

.

In these equations, fc is the fraction of circulating(non-reflected) particles which for the large-aspect-ratioapproximation used here is given by fc = 1−1.46

√ϵt . For the

axisymmetric tokamak with B · ∇B = −!ιϵt(B × ∇B) · ∇r ,one can show that a single mono-energetic transport coefficientis sufficient to describe all neoclassical effects [19] as the other

two may be determined from

D11 =1!ιϵt

m

qB0

B20⟨B2⟩

νD31 +83

(vdR0

v!ι

)2ν

D31 =1!ιϵt

m

qB0

(v2

3−

B20⟨B2⟩

νD33

).

Using these expressions, it is straightforward to determine thecollisional and collisionless asymptotes of the three normalizedmono-energetic transport coefficients for the large-aspect-ratio, axisymmetric tokamak

D⋆11(ν⋆ → ∞) = 32

3πν⋆, D⋆11(ν

⋆ → 0) ≈ 2.5 ν⋆

ϵ3/2t

,

D⋆31(ν⋆ → ∞) = 0, D⋆31(ν⋆ → 0) = 1,

D⋆33(ν⋆ → ∞) = 1, D⋆33(ν⋆ → 0) = fc.

Also commonly used as a figure of merit to indicate thelevel of 1/ν transport in stellarators is the so-called effective

13

W7X, IPP-MPI, Germany HSX, U. Wisconsin, USA

[Beidler et al, NF 11]

Some caveats• Description of particle motion based on particles moving

sufficiently far in between collisions

• For “high” 𝜈 ≳ 𝜌*vt /L, radial motion controlled by collisions

• MHD equilibrium equations are not solvable in rationalflux surfaces for non-omnigeneous stellarators

28

⌫ . ⇢⇤vtL

r · (Jkb̂) +r · J? = 0 ) B ·r✓JkB

◆= �r ·

⇣ cBb̂⇥rp

⌘

Singular

�r

r⇠ 1

⌫

dr

dt⇠ mc

Ze⌫⌧b

@J

@↵. ⇢⇤vt

⌫L. 1

What else is there to do?• If we relax continuity of derivatives, are there

omnigeneous solutions other than the tokamak• This is independent of optimization programme: maybe the other

omnigeneous solutions require unbuildable magnets!• If perfect omnigeneity can ever be reached, how do we

model low collisionality regions?• Inherently global problem with complicated particle orbits?• Some attempts of understanding the problem by expanding in

either aspect ratio or “closeness” to omnigeneity• How do we solve the problem of MHD equilibrium?

• More of this in Stellarator sessions!• Is this theory of interest elsewhere?

29

magnetized particle motion in 3d fields and the ......magnetized particle motion in 3d fields and...

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