Proceedings of ITC/ISHW2007

On rapid rotation in stellarators
Per HELANDER
Max-Planck-Institut für Plasmaphysik, 17491 Greifswald, Germany
The conditions under which rapid plasma rotation may occur in a three-dimentional magnetic field,
such as that of a stellarator, are investigated. Rotation velocities comparable to the ion thermal speed
are found to be attainable only in magnetic fields which are approximately isometric. In an isometricmagnetic field the dependence of the magnetic field strength B on the arc length l along the field is thesame for all field lines on each flux surface ψ. Only in fields where the departure from exact isometry,
B = B(ψ, l), is of the order of the ion gyroradius divided by the macroscopic length scale are rotation
speeds comparable to the ion thermal speed possible. Moreover, it is shown that the rotation must bein the direction of the vector ∇ψ × ∇B.
Keywords: plasma rotation, kinetic MHD, drift kinetic equation

1 Introduction
It is well known in tokamak research that the plasma
tends to rotate faster in the toroidal direction thanin the poloidal direction, particularly when there isstrong neutral-beam injection. It is also well known
that magnetic field ripples damp toroidal rotation.
Theoretically, it is expected that an axisymmetricplasma should be free to rotate in the toroidal direction, but that the poloidal rotation should be damped.
To be more precise, if the gyroradius ρ is small compared with the macroscopic scale length L, so thatδ = ρ/L  1, then the toroidal rotation can be comparable to the ion thermal speed vT = (2Ti /mi )1/2 ,but the poloidal rotation velocity is of order δvT . This
result follows from the drift-kinetic equation, and will
be rederived below. The purpose of the present paper is to clarify under what conditions rapid rotation(V ∼ vT ) is possible if the magnetic field is not axisymmetric.
We find that rapid rotation can only occur ina certain class of magnetic fields, namely, in fields
that are approximately “isometric”. The definition of
an isometric magnetic field is that the field strength
depends on the arc length l along B in the sameway for all field lines on the same flux surface [1].
Thus in an isometric field B = B(ψ, l), where ψ isa flux-surface label. Interestingly, such fields haveattracted attention because of their favourable confinement properties. They are an important subclassof “omnigenous” magnetic fields, which are fields
where the time-averaged cross-field drift vanishes for
all particle orbits [2, 3]. Quasi-axisymmetric [4] andquasi-helically symmetric [5, 6] fields are examples ofisometric fields, but isometry is a weaker condition
than quasisymmetry. We also find that the therotation velocity vector must point in the direction
author’s e-mail: xxxyyyzzz@u-plasma.ac.jp∇ψ × ∇B, so that the streamlines coincide with lines
of constant magnetic field strength. These contraints
follow from the drift kinetic equation in the limitof zero gyroradius (so-called kinetic MHD), and are
therefore independent of the cross-field transportcollisionality. A somewhat different version of thiscalculation is being published in Physics of Plasmas(October 2007).

2 Expansion of the kinetic equation
First of all, it is important to choose the correct
plasma model from which to proceed. Plasma equilibrium is usually described by ideal MHD, which is
however not sufficient for our present purposes. Ideal
MHD neglects transport altogether, both within andacross flux surfaces, and therefore permits arbitrary
toroidal and poloidal rotation as well as temperature variation within flux surfaces. In reality, paralleltransport is many orders of magnitude larger than perpendicular transport, which implies that flux surfaces
must be approximately isothermal and also constrains
plasma rotation. These features are incorporated inso-called kinetic MHD, which follows from the terogyroradius limit of the ion kinetic equation
e∂V∂f+ (V + v) · ∇f +E0 + v × B −∂tm∂t−(V + v) · ∇V) ·∂f= C(f ) + S,∂v

(0)where v = u − V(r, t) is the velocity vector measured
relative to velocity field V, E0 = E + V × B the electric field in the moving frame, e and m the ion chargeand mass, respectively, C the collision operator andProceedings of ITC/ISHW2007
∂fv⊥ ∂fb × v ∂fS represents any sources present in the plasma. Al.
(4)· b++2∂vkv⊥ ∂v⊥v⊥
∂ϑthough V is in principle arbitrary, we shall choose
it to be equal to the lowest-order mean ion velocity.
Taking the ϑ-average gives the desired drift kinetic
As in MHD, the electic field is ordered to be so large
equationthat the E × B velocity is comparable to the therΛ̄(f0 ) = C(f0 ) + S,
(5)
mal speed, E ∼ vT B, while the collsion frequency
is taken to be comparable to the transit frequency,
wherevT /L, in order to allow for all conventional collisionaleẼk ∂f0
ity regimes. The dependent variables f = f0 +f1 +. . .,Λ̄(f0 ) = (V + vk b) · ∇f0 +m ∂vk
E = E0 + E1 + . . . and, unconventionally, also themagnetic field B = B0 + B1 + . . . are expanded in thevk ∂f0v2∂f0smallness of δ = vT /ΩL  1, where Ω = eB/m. In−+ ⊥ (∇ · b)2
∂vkv⊥ ∂v⊥
order to study equilibrium (rather than the approachto it), the time derivatives of zeroth-order quantities∂f0−(V + vk b) · ∇V · b
are assumed to be small, ∂f0 /∂t  (vT /L)f0 , whilst∂vkhigher-order quantities may vary more rapidly (to alv⊥∂f0
low for turbulence, for instance). The electric field is+ (b · ∇V · b − ∇ · V),2∂v
thus electrostatic in lowest order, E0 = −∇Φ0 .
⊥The largest terms in Eq. (2) are of order Ωf , andwith Ẽk = b · E01 = b · (E1 + V × B1 ). This equation,
the others of order δΩf = vT f /L or smaller. In lowestand its derivation, agrees with that derived in Ref. [9]order, then, our kinetic equation becomes simply
except for the addition of the term containing B1 . Ite∂flooks
simpler if one of the independent variables is0(E0 + v × B0 ) ·= 0,chosen to be the magnetic moment measured in them 0∂v
moving frame, and then agrees with Refs. [7, 10].
and can only hold for all v if
(v × B0 ) ·∂f0= 0,∂v(1)

3 Constraints on the flow velocity(2)

In equilibrium, the source term balances transport
losses and is therefore also relatively small, usuallyof order δ or δ 2 . The solutions to the resulting equilibrium equation (5),and E00 = 0, so thatV⊥ = V − V · bb =B0 × ∇Φ0B02and b · ∇Φ0 = 0, where b = B0 /B0 . We shall assume
that the magnetic field at least approximately (i.e.,in lowest order) possesses flux surfaces, so that Φ0 =
Φ0 (ψ, t).
A drift kinetic equation can now be derived in theconventional way [7, 8, 9, 10] by writingv = vk b + v⊥ (e1 cos ϑ + e2 sin ϑ),
where e1 , e2 are unit vectors with e1 × e2 = b, andaveraging over the gyro-angle ϑ. Equation (1) first
implies that f0 is independent of gyro-angle, ∂f0 /∂ϑ =0, and Eq. (2) becomesΩ∂f∂fe∂f=+Λ(f )+ (v×B1 )·= C(f )+S(3)
∂ϑ∂tm
∂v

where
vk ∂f∂f
Λ(f ) = (v + V) · ∇f + ∇b · v−∂vkv⊥ ∂v⊥
v ∂fv⊥ ∂ϑ 0
eE1∂V+−− (V + v) · ∇Vm∂t+ (cos ϑ∇e2 − sin ϑ∇e1 ) ·Λ̄(f0 ) = C(f0 )(6)

are found from a familiar H-theorem argument. Multiplying the equation by ln f0 , integrating over velocity
space, and taking a flux surfaces average gives
Zln f0 C(f0 ) 2πv⊥ dv⊥ dvk = 0,(7)and it follows that f0 must be a Maxwellian, whose
density n and temperature T may vary over each flux
surface. Substituting this Maxwellian into Eq. (6)gives the equation Λ̄(f0 ) = 0, or
3mv 2−∇ ln T(V + vk b) · ∇ ln n +2T2−eẼk vkmvk+(V + vk b) · ∇V · bTT−2mv⊥(b · ∇V · b − ∇ · V) = 0,2T

which can only be satisfied if the following relations
are satisfied [9, 10]:b · ∇ ln n −eẼkm+ V · ∇V · b = 0,TT

Proceedings of ITC/ISHW2007b · ∇T = 0,
3V · ∇ ln n − ln T = 0,2where Ḃ0 = ∂B0 /∂l. Hence∇ · (nV) = 0,and it follows that Ḃ0 must be expressible as a function of ψ and B0 , i.e., Ḃ0 = Ḃ0 (ψ, B0 ), at least locally.
This implies, in turn, that B0 is isometric. To see thisformally, we note that Eq. (11) can be written as∂ ln Ḃ0∂
∂B0=ln,∂l∂l∂α∂B0 ∂ Ḃ0
∂B0 ∂ Ḃ0−= 0,∂α ∂l∂l ∂α

1b · ∇V · b − ∇ · V = 0.
3
These equations imply
V · ∇ ln n = ∇ · V = b · ∇V · b = 0.

(8)

and integrated once, to yield

We now recall Eq. (2) and note that∂B0∂B0= F (ψ, α),∂l∂α0 = ∇ × (V × B0 ) = B0 · ∇V − V · ∇B0 ,which combined with Eq. (8) leads towith F an arbitrary function. The general solution isV · ∇B0 · B0 = 0.

B0 = B0 (ψ, l0 ),Since (∇B0 ) · B0 = B0 ∇B0 we thus conclude thatV · ∇B0 = 0.
In other words, the streamlines of the flow are givenby the intersection between flux surfaces and surfacesof constant B0 . This means that the velocity field canbe written asV(r) = g(r)∇ψ × ∇B0for some function g(r) of the spatial coordinates r.
The parallel component of the flow is thus
Vk b = g(r)∇ψ × ∇B0 −dΦ0 b × ∇ψ.
dψB0

Taking the scalar product of this equation with b×∇ψgives an expression for g,gb · ∇B0 +1 dΦ0= 0,B0 dψ

and thus enables us to write down an explicit expression for the lowest-order flow velocity,
V=−dΦ0 ∇ψ × ∇B0.
dψ B0 · ∇B0(9)If B0 is written in Clebsch coordinates, B0 = ∇ψ ×
∇α, then V becomesV=∇Φ0 × ∇B0.
(∇ψ × ∇B0 ) · ∇α

The requirement (8) that this flow field should beincompressible now implies a constraint on the spatial
variation of the magnetic field strength,(∇ψ × ∇B0 ) · ∇(B0 · ∇B0 ) = 0.

(10)

If B0 is expressed in coordinates (ψ, α, l), where l isthe arc length along B0 then it follows from Eq. (10)that(∇ψ × ∇B0 ) · ∇Ḃ0 = 0,(11)where l0 = l − l0 (ψ, α) is an arc length coordinate
whose origin l0 is related to F by F ∂l0 /∂α = −1. We
conclude that rotation at a speed comparable to thethermal speed is only possible if the magnetic field isisometric in lowest order. The converse is also true:
the flow field (9) satisfies the conditions (8) if B0 is
isometric, and our theorem can thus be stated in thefollowing way. The lowest-order drift kinetic equation
admits solutions where the mean flow velocity is comparable to the thermal speed if, and only if, the magnetic field is approximately isometric.
Another way of stating this result is that a sufficiently large radial electric field is only possible if themagnetic field is isometric. “Sufficiently large” in thiscontext refers to fields that are strong enough to produce flow velocities comparable to vT (sonic rotation),and it is worth noting that this may occur for fields
that are in fact much smaller than E ∼ vT B (thoughformally of this order, in the sense of the gyroradiusordering assumed). The result (9) can be written asn × ∇B0
V=,
E/B0b · ∇B0(12)where n = ∇ψ/|∇ψ| is the unit vector normal to theflux surfaces and E = −n · ∇Φ0 is the electric field.
The point is that the right-hand side of (12) can be relatively large (but not infinite in an isometric field), in
which case the parallel component of the velocity (9)is significantly larger than the perpendicular one. Intokamaks, for instance, |n × ∇B0 |/(b · ∇B0 ) ∼ q/ 1, where q is the safety factor and  the inverse aspectratio. As is well known, sonic rotation thus occurs already for radial electric fields of order E ∼ vT B/q.
In a stellarator, a similar estimate tends to hold approximately, but the details depend of course on thespecific magnetic configuration. Importantly, sonic rotation can occur at roughly the same electric field as
when the poloidal E × B drift cancels the poloidal

Proceedings of ITC/ISHW2007
component of vk for a thermal ion. This “resonance”
condition is thought to strongly affect the neoclassicaltransport [11].
It is interesting to note that Eq. (9) implies a simple expression for the neoclassical polarisation current.
When combined with the momentum equation
ρdV= j × B − ∇ · P,
dt

where P is the pressure tensor, it yields the perpendicular current asj⊥ −B × (∇ · P)
ρB dVρ ∂∇Φ
= 2 ×'− 2,2B
B
dt
B ∂t

where the last, approximate, equality refers to lowspeeds, ∂V/∂t  V · ∇V.
The result that the eletric field cannot be large
unless the magnetic field is isometric suggests a paradox in the low-density limit, since any electric fieldstrength is possible in vacuum. The resolution lies
in our ordering of the collision frequency, νi ∼ vT /L.
This is the standard neoclassical ordering, and is usually followed by a subsidiary ordering where the collsion frequency is taken to be smaller or larger than
the transit frequency, but usually not as small asν ∼ δvT /L. At extremely low densities, this lattercase must be allowed, in which case the lowest-order
drift kinetic equation becomes Λ̄(f0 ) = 0 and does notconstrain f0 to be Maxwellian or B0 to be isometric.

4 Conclusions
It is well established that plasma rotation tends tohave a beneficial influence on plasma confinement.
Therefore it is of interest to establish under which conditions rotation is allowed to occur. We have considered this question for general three-dimensional magnetic confinement systems, and found that sonic rotation is only possible in isometric magnetic fields,
and can only occur in the direction of constant magnetic field strength. In the special case of a tokamak,
plasma rotation must therefore be purely toroidal inlowest order, as is well known both theoretically andexperimentally. (Although the poloidal rotation in experiments has been reported to exceed its neoclassicalprediction, it is still far smaller than the toroidal rotation [12, 13].)
In stellarators, the radial electric field and rotation velocity are set by the condition of ambipolarcross-field transport, and is usually fairly slow in experiments, V  vT . It is often the case that neoclassical transport dominates, and the magnitude anddirection of the rotation then depend on the collisionality and heating channel. What we have shown in thispaper is that rapid rotation can only occur in isometric magnetic fields and only in the direction ∇ψ ×∇B.

These constraints are approximate, in the sense thatthey only need to be satisfied to lowest order in gyroradius, but are independent of the cross-field transport.
They therefore hold in all (conventional) collisionality regimes, and also in the presence of gyro-kinetic
turbulence.

Acknowledgment
It is a pleasure to acknoledge useful discussions withC. Beidler, P. Catto, H. Maaßberg, J. Nührenberg,and Yu. Turkin.
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