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Motivation

A strongly sheared ExB flow is believed to cause suppress turbulence.

Is it possible to design a stellarator so that this occurs at some pre-defined location?

It seems possible to achieve electron-root plasmas in reactor-sized stellarators (Beidler ISHW 2022).

References

CD Beidler, M Drevlak, J Geiger, P Helander, HM Smith and Y Turkin, Nucl. Fusion 64, 126030 (2024).

P Helander, AG Goodman, CD Beidler, MD Kuczynski and HM Smith, J. Plasma Phys. 90, 175900602 (2024).

B.F. Lee, S.A. Lazerson, H.M. Smith, C.D. Beidler and N.A. Pablant, Nucl. Fusion 64, 106054 (2024).

E Lascas Neto, R Jorge, CD Beidler and J Lion, J. Plasma Phys. 91, E24 (2025).

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Radial electric field

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Radial current in gyrokinetics

According to standard gyrokinetics, the radial current from small-scale fluctuations vanishes to lowest order.

Sugama et al., PoP 1996

Parra & Catto, PPCF 2008

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Radial current from neoclassical transport

Neoclassical radial particle flux of each species σ

In most stellarators, this flux is ambipolar, only for one or a few values of E_r.

This condition determines E_r even if most of the transport is turbulent!

Exceptions:

unnecessarily well neoclassically-optimised fields
axisymmetric and (perhaps) quasisymmetric fields
small scales: zonal flows

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Radial electric field

Neoclassical ambipolarity equation is nonlinear (Mynick & Hitchon 1983)

Usually E_r < 0 (ion root) since D_e < D_i.

Causes strong inward neoclassical transport for highly charged impurities.

E_r > 0 (electron root) has been observed in low-density plasmas with T_e > T_i.

Beneficial for impurity expulsion
Hitherto thought to be impossible in reactors since T_e = T_i.

Hastings, Nucl. Fusion 1986

Ion root

Electron root

Intermediate root is unstable since dJ_r/dE_r < 0.

Unstable root

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Neoclassical transport of electrons and ions

The diffusion coefficient for a particle of speed v depends on two dimensionless parameters:

Small-Ma limit

Larger Ma:

Ma = 0

Ma = 0.003

Beidler et al., Nucl. Fusion 2011

Galeev et al. 1969

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Neoclassical transport of electrons and ions

In the ion root, the diffusion coefficients are given by

where ε_i and ε_eff are coefficients depending only on the B-field geometry.

Ion transport (determined by ε_i) is controlled mostly by shallowly trapped particles.
Electron transport (determined by ε_eff) depends on all trapped particles.

Notation:

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Neoclassical transport of electrons and ions

Ambipolarity condition

implies electric field

which is positive (for dn/dr < 0) if

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Neoclassical theory of electron root optimisation

Estimating

gives a criterion for the onset of electron root approximately when

Explains why the electron root is at all possible despite m_i >> m_e.

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Experimental evidence

In most stellarators, the radial electric field broadly follows the predictions from neoclassical theory.

Electron roots predicted and observed in LHD, CHS, W7-AS and TJ-II at low density when T_i < T_e.

Electron root not expected nor observed

in any present-day stellarator at moderate or high densities, where T_i = T_e,
or in HSX although T_i << T_e.

Further verification of theory underway in W7-X.

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Neoclassical theory of electron root optimisation

The criterion

suggests that the electron root could be achievable even in a large stellarator (small ρ_*i) with T_e = T_i.

Electron root possible in the core by targetted (de)-optimisation

Decrease the ratio .
Improve confinement of shallowly trapped particles, degrade it for deeply trapped ones.

The edge will be in the ion root.

Sudden transition from E_r > 0 to E_r < 0 at some radius.

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Transport barrier?

ExB flow can suppress tubulence when (Waltz 1994, Ivanov et al, 2023)

For electrostatic instabilities with

If the width of the transition region is w and , a transport barrier should arise if

and increase the core temperature by at least

In experiments,

electron roots are often accompanied by steep T_e profiles in the core.
expected hysteresis observed in W7-AS (Stroth PRL 2001).

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A concrete example��

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Optimisation goals

Quasi-isodynamic magnetic field, implying

Good fast-ion confinement
Small neoclassical transport
Negligible bootstrap current

Reduced ITG- and TEM-driven turbulence
MHD stable up to some target β

maximum-J property at this β

Edge islands for divertor operation
Coils simpler than, or comparable to, those of W7-X

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Example of a recently optimised QI stellarator (SQuID)

Aspect ratio = 10

Number of field periods = 4

Ballooning stable for

Excellent alpha-particle confinement

Goodman et al, submitted to JPP (2025)

|B| on a flux surface

Pressure distribution in poloidal cross section

Poloidal cross sections

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Magnetic field

Two different coil sets

Field strength at half radius over the flux surface and along B.

B(l)

distance l along B

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Electron root in W7X-size device with T_e = T_i

Scaled to W7-X volume and field strength
Density and temperature profiles such that

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Electron root in large reactor

Electron root also possible in large reactor. Example

R = 20m, a = 1.9 m, V = 1450 m3

n_e

n_D = n_T

n_D + n_T

n_He

T_e

T_i

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Theoretical issues

The electron-ion-root-transition region cannot be described by standard local neoclassical theory.

In the figures above instead modelled by a cruder model in the NTSS transport code.
Has also recently been calculated with the global gyrokinetic EUTERPE code without turbulence.

In order to assess the strength of a transport barrier, global simulations of simultaneous neoclassical and turbulent transport should be carried out.

Kuczynski et al, 2024

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Testing predictions in W7-X

A central feature of the SQuID electron root with T_i = T_e is the simulataneous presence of three roots in the plasma core.

Will the plasma ”choose” the electron root?

Could be tested in the high-mirror configuration of W7-X with T_i < T_e.

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Summary

In non-quasisymmetric stellarators, the radial electric field is determined by neoclassical transport, even if most of the energy transport is turbulent.

It is possible to tailor the magnetic field so that E_r > 0 in the core and E_r < 0 in the edge, even if T_e=T_i.

Strong ExB shear arises in the transition region, perhaps causing a transport barrier.

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Maximum-J

In an omnigenous field

is constant on flux surfaces and can either increase or decrease with radius. The latter is stabilising for curature-driven instabilities. Thus we desire

Related to grad B and to trapped-particle precession

and requires for the most deeply and most shallowly trapped particles

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Barely trapped particles

In a QI field, the maximum field strength on each flux surface is attained on a curve of contant α and in its vicinity

In a vacuum field,

whence it follows that B cannot attain a local maximum in the interior of any finite domain, and thus

Thus, shallowly trapped particles always satisfy the maximum-J criterion in an omnigenous vacuum field.

However, in practice, these particles are frequently the most non-omnigenous ones.

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Deeply trapped particles

At the minimum of B,

and we desire

which is difficult to attain. The near-axis expansion suggests two helpful ingredients:

A non-zero pressure gradient
Large torsion of the magnetic axis

increasing elongation away from ϕ_min

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Tentative example of max-J in vacuum

Optimised near-axis example shows that it is possible to reverse the precession frequency of deeply trapped particles in vacuum.

Due to non-omnigenity of barely trapped particles, some of these may not precess in the desired direction.

Rodríguez & Helander, unpublished 2023.

Precession frequency vs trapping parameter

(different field lines in grey)

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Summary

QI stellarators are very attractive.

Can be optimised to high degree of accuracy

Pfirsch-Schlüter current is small in QI stellarators.

Many moments vanish exactly, e.g.

Streamlines close within one period.
Tight rigorous upper bound.

Bootstrap current vanishes to the three lowest significant orders in ν*.

The precession of deeply trapped particles can be reversed not only through a non-zero pressure gradient, but also in vacuum.

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The maximum-J property

The “second adiabatic invariant“

can either increase or decrease with radius. The latter, i.e.

has several benefits:

responsible for fast-ion confinement at high β in W7-X
correlated with a magnetic well (good for MHD stability)
stabilising for curvature-driven gyrokinetic instabilities

density-gradient-driven TEMs completely stabilised.

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Bootstrap current: low collisionality

In non-omnigenous stellarators, the boostrap current behaves very differently.

Numerically, no convergence at low collisionality.

Recently explained by Kasilov et al. (unpublished) who showed that, in a generic stellarator,

The displayed coefficients J_n all vanish for QI.

Singular limit of several small parameters

For instance:

Kernbichler et al. PPCF 2016

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Energy loss channels

Energy is lost from the plasma through

Radiation
Collisional “neoclassical“ transport
Turbulent transport

In tokamaks, turbulence nearly always dominates.

In stellarators, neoclassical losses can be substantial at high temperature.

Heat diffusivities scale as

Small prefactor ε_eff for neoclassical transport in modern stellarators.

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Trivial QI

A trivial way of achieving arbitrarily good QI quality in the near-axis expansion:

Choose a magnetic axis and B(l) along the axis.

Choose elliptical flux surfaces that are infinitely elongated perpendicular to the curvature vector.

κ = b.grad b

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The Pfirsch-Schlüter current

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The Pfirsch-Schlüter current

The Pfirsch-Schlüter current satisifies

which implies

In a QI field with I(ψ)=0,

As a result:

No current crosses the B_max-contours. The current streamlines close in each module of the stellarator.

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The Pfirsch-Schlüter current: further properties

The Pfirsch-Schlüter current has zero mean:

implying no net toroidal current, as in any field with I(ψ) = 0.

In addition

for any function f(B).

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Pfirsch-Schlüter transport

The Pfirsch-Schlüter current gives rise to enhanced collisional transport. Larger than classical transport by the factor

which in a tokamak becomes

but is much smaller in a QI stellarator. A rigorous upper bound is given by

which is typically small.

There is almost no PS enhancement of classical transport.

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The bootstrap current

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Bootstrap current

In any exactly omnigenous field, the bootstrap current is of the form

where J₀ is substantial in tokamaks and QA fields. In a perfectly QI field, however,

Goodman et al, JPP 2023

Sanchez et al, Nucl. Fusion 2023

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Physical reason

Consider a tokamak where at t=0 the distribution function of all species is initialised as an exact Maxwellian with a density gradient

Because the particles drift radially, the distribution function will evolve. The radial excursion is

and the distribution function evolves toward

which carries a nonzero current because of the correlation between

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Physical reason

In a QI stellarator with I(ψ) = 0

The final distribution function

carries no net current since there is no correlation between

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Particle orbits

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Corollaries

The contour of maximum magnetic field on a flux surface is straight in Boozer coordinates, which can be taken to be located at ϕ=2nπ/N. Thus, in a QI field

All field lines are “of equal length“.

If , then for some function

only term in QS field

only term in QI field without current drive

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Radial particle drift

Consider a particle travelling along the magnetic field from 1 to 2. The accumulated radial drift is

As a particle travels along the field in a QI device, ψ returns to the same value whenever B does so, since

In particular, a circulating particle returns to the same flux surface whenever

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Pfirsch-Schlüter transport

The Pfirsch-Schlüter current gives rise to enhanced collisional transport. Larger than classical transport by the factor

which in a tokamak becomes

but is much smaller in a QI stellarator. A rigorous upper bound is given by

which is typically small.

There is almost no PS enhancement of classical transport.

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QI, QA, QH

Perfect collisionless orbit confinement requires all level curves of B = |B| on each flux surface to have the same topology.

Three possibilities:

Quasi-isodynamic (QI)

Quasi-axismmetric (QA)

Quasi-helically symmetric (QH)

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Quasi-isodynamic stellarators

Quasi-isodynamic (QI) stellarators are those that are omnigenous and have poloidally closed B-contours.

Subbotin et al, Nucl. Fusion 2006

Beidler et al, Nucl. Fusion 2011.

all maximima the same

all minima the same

B along field line

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Wendelstein 7-X

Wendelstein 7-X is a poor approximation of QI.

Neither the minimum nor the maximum B-contours close poloidally.

W7-X standard

W7-X high-mirror

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Straight sections

In a QI MHD equilibrium, the field-line curvature

vanishes at the points of minimum and maximum B on the magnetic axis. Otherwise the B-contours cannot close poloidally.

Goodman et al, JPP 2023.

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Straight sections

In a QI MHD equilibrium, the field-line curvature

vanishes at the points of minimum and maximum B on the magnetic axis. Otherwise the B-contours cannot close poloidally.

W7-X

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CIEMAT QI

Example of recently optimised QI stellarator

Sanchez et al, Nucl. Fusion 2023.

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“Precise“ QI

Recent vacuum optimisation by Goodman et al. (JPP 2023).

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Near-axis expansion

Near-axis expansion of QI stellarator fields uses the following input:

Axis shape with zero curvature at minima and maxima
B(l) along axis

Not available in QA or QH!

Elongation

Plunk, Simons Annual Meeting, NYC 2023.

Plunk, Landreman & Helander, JPP 2019 Rodriguez & Plunk, PoP 2023

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Physical properties

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Particle trapping wells

Basic theorem (Cary & Shasharina, PoP 1997):

Write the field in Boozer coordinates as

Let δ(ψ,α,B) be the arc length along the field between points of equal field strength B on either side of a minimum. Then, in any omnigenous field

The largest field strength on each flux surface is reaced on the curves

Goodman et al., JPP 2023.

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Radial particle drift

Consider a particle travelling along the magnetic field from l₁ to l₂. The accumulated radial drift is

As a particle travels along the field in a QI device, ψ returns to the same value whenever B does so, since

In particular, a circulating particle returns to the same flux surface whenever

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The Pfirsch-Schlüter current

The Pfirsch-Schlüter current satisifies

In a QI field with no net enclosed toroidal current, I(ψ)=0,

No current crosses the B_max-contours. The current streamlines close in each module of the stellarator. In addition

for any function f(B).

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Pfirsch-Schlüter transport

The Pfirsch-Schlüter current gives rise to enhanced collisional transport. Larger than classical transport by the factor

which in a tokamak becomes

but is much smaller in a QI stellarator. A rigorous upper bound is given by

which is typically small.

There is almost no PS enhancement of classical transport.

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Pfirsch-Schlüter transport: example

Example of “precise“ QI configuration with N=1 by Goodman et al. (JPP 2023).

A very small E_r eliminates the 1/ν-regime
There is hardly any sqrt(ν)-regime
Instead PS regime down to the lowest collisionalities.
Unlike other stellarator regimes, inward grad-T-driven particle transport possible.

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Bootstrap current

In any exactly omnigenous field, the bootstrap current is of the form

where J₀ is substantial in tokamaks and QA fields. In a perfectly QI field, however,

Goodman et al, JPP 2023

tokamak

QI stellarator

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Separating bad curvature from magnetic trapping

Magnetically trapped orbits reside where B is small.

Magnetic curvature is unfavourable where field lines are convex.

In tokamaks, these regions coincide

both on outboard side of the torus

In some stellarators, they are separated.

Wendelstein 7-X from above

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Good and bad curvature

Electrons move quickly along field lines.

bounce frequency >> turbulence frequency

Adiabatic invariant of trapped partices is conserved:

depends on minor radius r and energy E.

Conservation of J implies for convex field lines

It is thus energetically favourable to move particles outward.

vice versa if

plasma

vacuum

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The maximum-J property

The “second adiabatic invariant“

can either increase or decrease with radius. The latter, i.e.

has several benefits:

responsible for fast-ion confinement at high β in W7-X
beneficial for MHD stability
stabilising for curvature-driven gyrokinetic instabilities

density-gradient-driven TEMs completely stabilised.

Possible in QI but not in QA or QH.