J. Plasma Fusion Res. SERIES, Vol.4 (2001) ti.3_l7g

On Viscous Damping and Rotation in Low-Shear Stellarators
WOBIG Horst
Max-pranck Institut filr prasmaphy sik, EUMT)M As sociation
D-85740 Garching bei Miinchen, Germany(Received: 5 December 2000

/ Accepted: 16 August 2001)

Abstract
one of the main results of the Wendelstein 7-AS stellarator (major radius
2 m, average plasma
radius 0' l8 m, magnetic field 2.5 T' low shear) is the achievementof the H-mode confinement in ECRand NBl-heated plasmas. The features of the H-mode aresimilar to those of a tokamak: reduction of the
Ho-emission, onset of poloidal rotation anda30vo improvementof the energy confinement time, which is
acompanied by a reduction of the turbulence level. However, thecritical dependence on the rotationaltransform is a particular feature of this low shear stellarator and in
the following paper an attempt will bemade to understand this phenomenon by investigating thespecific influence of the magnetic geometry on
the onset of poloidal shear flow. In the frame of a dissipativemodel of a steady-state plasma equilibriumwith flow the poloidal and toroidal force balance on magnetic
surfaces are investigated. The neoclassicalpoloidal viscosity has a strong dependence on the external rotationaltransform. The H-modes windows
found in wendelstein 7-AS can be identified with regions of highorder rational magnetic surfaces, where
viscous damping is small. The paper discusses the conditions forconvective plasma flow, which inconjunction with the inertial forces is providing the spin-up mechanismof poloidal shear flow.

Keywords:
stellarator, plasma rotation, viscosity, convection, bifurcation

1. Introduction

magnetic surfaces close to t = l/2 and r 5/6. The=
regions where in Wendelstein 7-AS L-H transition have

L-H transitions in the low-shear Wendelstein 7_AS
stellarator exhibit all characteristic features of L_H
transitions found in tokamak experiments: a sudden
decay of Ho-emission, increase of plasma confinement
accompanied by a quiescent phase without significant
MHD-activity and plasma fluctuations [l]. In the phaseprior to the L-H transition ELMs have been observed

been observed are around r(a) = 9.43, = 0.52 and= 0.55.

In the H-mode window neither r = l/2-surface nor t 5/=
6 surface exists inside the plasma body. The upperboundary of the H-mode window is characterised by r=ll2, t = 10/19 and t = 516 at the plasma boundary. On
these magnetic surfaces magnetic islands exist, which

which also exhibit intermittency leading to a steepeningof the electron temperature gradient in the quiescentphase between ELMs. However, in contrast to tokamak
experiments these L-H transitions only occur above adensity threshold [2] and only in some narrow gaps on

are the natural result of the 5-fold symmetry of themagnetic field and the toroidal curvature. At the lower
boundary we find the limit at t(0) 5/ll,t 5/10 and r=== 10/19. Magnetic surfaces inside the H_mode window
have a rotational transform L = mlnwith m > 1, n > l.
Such surfaces are called ,,high order rational surfaces..,the iota-scale [3].

A particular feature of the H-mode windows is thatthese exist in the neighbourhood of low-order rational

while those surfaces with low m and n are ..low order@2001 by The Japan Society of plasma
Science and Nuclear Fusion Research

C orre spondin g author' s e -mail : wobi g @ ipp. mp g. de173

WobigH.,OnViscousDampingandRotationinLow-ShearStellarators

rational surfaces". Magnetic islands on the plasmaboundary provide an enhanced mechanism of radial
particle and momentum transfer and thus inhibit thebuild-up of a poloidal shear flow, which is considered asa basic feature of the H-mode physics. The effect ofrotational transform and magnetic field line topology on
poloidal viscosity has been investigated elsewhere [4]'Braginskii viscosity exhibits only a weak dependence on
the rotational transform t5l. The viscosity in the plateau
regime, however, does strongly depend on the rotationaltransform. On low order rational surfaces strong maxima
arise which may lead to strong damping of a poloidal
shear flow. The plateau viscous damping is small in
those regions where the H-mode windows exist' In thefollowing the model of a non-ideal plasma in theboundary region will be discussed. The Ho-emission
indicates the presence of neutral hydrogen in the Lphase, while the H-phase the neutrals apparently havebeen reduced to a low level. Neutrals in the boundary
region affect the parallel momentum balance and leads

to small pressure gradients along the field lines' A
similar effect arises due to parallel viscous forces andthe inertial forces of the plasma rotation' Although theseforces are small the resulting pressure gradients along
fields lines can lead to large perpendicular gradients inpoloidal direction and thus to large convective cells'
Such a convective state can be laminar or turbulent,source' The

1. S is the mass source and Q is the energy
viscous forces are given by the Braginskii viscosity orby the neoclassical viscosity depending on thecollisionality. The interaction with the neutral
background is F, = m1nsNsl6v)rt, ns is the plasma

density and N6 the density of neutral hydrogen, o is the
charge exchange cross section. Here we have assumed

that the neutrals are at rest. The role of neutral gas
background in the physics of L-H transition of tokamaks
has been reviewed by Fukuda [6], experimental studies
have been reported inl'l-91, however there seems to be

be no consensus on how important the interaction withneutrals is. Theoretical studies on the effect of neutral
background on L-H transition are described in refs' [10has
121 The main result is that small neutral interaction

little effect on the L-H transition, however it was shownin ref. [13], that a small friction parallel to magnetic
field lines can give rise to bifurcation and convective
solutions. There is no continuous transition of a modelwith parallel momentum loss by charge exchange to amodel without charge exchange, on rational magnetic
surfaces singularities occur.

The model outlined in eqs. 1-4 is appropriate todescribe a quiescent plasma with small diffusive
velocity, which has no feedback on the momentum
balance. Laminar convective solutions are possible and aselfconsistent rotating plasma can also be described bythis model. Reduced versions of this model have been

showing all modes of transition between these states'
The H-mode is characterised by the absence of a large
convective state, leading to a steepening of density andanalysed in ref. tl4l, in particular existence anduniqueness of solutions have been investigated' In ref'it was shown that the friction model (neglect of[14]viscosity and inertial terms) has a unique solution if
resistivity and plasma neutral are large enough'temperature gradients.

2. Plasma Equilibrium

Convergence of an iterative scheme can be proven under

To describe the plasma equilibrium in the boundary
region we consider the momentum balance of a singlefluid model taking into account dissipative effects asviscosity, resistivity and plasma neutral interaction' The

these conditions. The iterative procedure starts with amagnetic fiels Bs and computes pressure, density'
temperature, electric potential and the plasma current
from eqs. 1-5. The current generates a magnetic field 81
and the procedure is repeated. The sequence of magnetic

momentum balance and Ohm's law are

Yp = j xB -Fn- V'n(v) -YPv

-VO+ vxB=4i

"v

(1)

fields {86} converges if the sequence of current

densities { j"} is uniformly bounded and satisfies some(2)

continuity conditions (ref. t14l)' In the following we
consider the magnetic field as given and analyse how
the solutions of the eqs. l-3 depend on the structure of
the magnetic field.
In order to describe stability and turbulencp the firstorder derivatives have to be added in eqs' 1, 3,4 together

Futhermore, we retain the equation of continuity Vp'r =S and VY = 0. The energY equation isv'\e*/,r,1*oo'v=Q(3)

The ideal gas law p = kTplmr correlates density totemperature. The heat flux is poportional to thetemperature gradient q = -a'YT with a strong anisotropy

with Faraday's law. This will introduce the turbulent
Reynolds stresses <pviv> and the turbulent fluxes <pu>and <pv>. The brackets <> denote the average over the

between parallel and perpendicular thermal conductivity
174

Wobig H., On Viscous Damping and Rotation in Low-Shear Stellarators

turbulent time scale.

i o = P6V) e, : Vo = E(tlD eo + A(V)

3. Momentum Balance
In the limit of ideal magnetohydrodynamics
resistivity, charge exchange interaction and the inertial
forces are neglected. This is the standard model tocompute stellarator equilibria. Since there is no feedback

of a plasma motion on the momentum balance anarbitrary rotation within the magnetic surfaces can besuperimposed. In this case there are two free surfacefunctions <D6(y) and \(y) and the rotation velocity isVO"xB
vo=--;a+i"oB+AoB; V.%=0eox B =Vty(6)

eo is the poloidal Hamada vector on the magnetic
surface, this vector is divergence-free. EVy is the radial
electric field in lowest order. In this order in e we haveno tangential forces on magnetic surfaces, also in firstorder the average inertial forces are zero. In second
order of e the averaged poloidal force balance iso = (ro,,) * (",.r."V)). (r,o (uo* :vo+poar'%*e.%'6r)) e)(4)

In lowest order pressure surfaces, density surfaces,
temperature surfaces and magnetic surfaces coincide. In

and the parallel force balance becomes Eq. g(n r',) + (B v. n(v))

this limit the plasma motion is incompressible.

o=However, even under these conditions several stationary
solutions can exist, in addition to the classical case of a

+ ( u. v.(uou.,vo + po6u, vo + p.4,sr)

slowly diffusing plasma also convective solutions
without net toroidal or poloidal fluxes can exist.

\

ll)

\

(S)

These relation state that in case of an rotating
equilibrium the dissipative forces in direction of themagnetic field and the plasma current are balanced by

Furthermore a superposition ofrotation, convective anddiffusive solutions can exist.

In order to clarify the force balance in more detailthe inertial forces. If the convective velocity is known
these relations can be used to compute the rotation

we assume that a steady state solution exists and that thedominating part of the plasma velocity is a tangential
velocity given by eq. 4. We make the following ansatz v

velocity Ve. In a stellarator without net toroidal current
the current lines are poloidally closed if they sray on= Vo t 6y. Pressure surfaces and magnetic surfaces nolonger coincide. The force balance on pressure surfaces
determines the size of the rotation velocity, which canclosd surfaces. Thus eq. 7 basically describes a poloidal
force balance while eq. 8 describes the toroidal force
balance. The friction and viscous forces tend to slow

be seen by averaging the momentum balance equation

down the plasma rotation, the driving mechanism has toon pressure surfaces. This averaging process on abe provided by the inertial forces. This driving

surface P = c is defined as a surface integral with theweight function lVpl. For any periodic scalar function gmechanism is the Stringer spin-up mechanism
t151. If
the plasma becomes turbulent the averaged momentum
balance eqs. 7 and 8 will be slightly modified and thetime-dependent part of 6y leads to an additional driving
mechanism, which has become known as the Reynolds

we get <B.Vg> = 0 and <,t.Vg> 0. If the magnetic=
surfaces and the pressure surfaces coincide, thisaveraging is taken over magnetic surfaces. To proceedfurther we introduce a formal expansion with respect todissipative and inertial forces. Introducing the expansion
parameter e we consider the friction and viscous forces
as of order e2 and the inertial forces of the order/ttlstress mechanism.

The quadratic term of the inertial forces is thecentripetal forces while the othe two are the Coriolis
forces. The centripetal forces are important at large
velocities, in L-mode they can be neglected. Inserting
V0 into the driving forces yields these forces as a linear
combination of E, A and its radial derivatives[4]. Thee. Themagnetic field is a fixed field, which has closed
magnetic surfaces labeled by the flux function \r.
The plasma functions are expanded in powers of e

j= jo+ejt+... ; v=Vo+e6y+...

convective velocity is written as a sum of aperpendicular and parallel component

p = P(\D + e6p + .. ; p = po (V) + e6p + ..

(5)
In lowest order the current is the current of an ideal

6v=6yr+l,B(e)
The parallel velocity usually is larger than theequilibrium and the velocity is a linear combination of
the lowest order plasma current and the magnetic field

perpendicular velocity t4l. The poloidal force due
I I)

to theWobig H., On Viscous Damping and Rotation in Low-Shear Stellarators

Lfr= _#0,(,i_it)v,

parallel velocity is(o,u'' (","'.)) =RttE

L=vrl 8., *v,'Y - cn',=(poIB'(r,tv"",))(10)

All other coefficients depend on the perpendicular partof dv. This parallel velocity associated to any diffusive

perpendicular convective velocity would lead to driving
forces, which are also proportional to the first orderradial drivatives of E and A. So far a time-independent
convective motion has been assumed. However, if theconvection is unstable and a turbulent state arises' theaveraging procedure must be extented over the timescale of the turbulence. The extension of the previous
analysis to the turbulent plasma is described in [16]' theclassical Stringer spin-up of a quiescent plasma isreplaced by the anomalous Stringer spin-up [17]' The
poloidal spin-up term in this case is,forces yields(1;;.;i)=(il r:)(r)p,=#(l ,,r.r,,0,")u,=#(l otuf r,,ua'u)
tr,=#(l ',r'o',0'"1',=(u,i(l l)',=(ufr-L,',)u Y(16)

variation of the magnetic field strenght, which leads to a
proportionality of all viscous coefficients

The friction forces do not depend on the structure

It,=A2*, ) Vt=AVpof the magnetic field, therefore the H-mode windows(J7)

There is no viscous damping force in toroidal direction'
In this case the matrix of viscosity coefficients in eq' 15
is singular. The same property is valid in any quasi-

can be affected only indirectly by the neutral gas
background. Viscosity - and in particular neoclassicalviscosity - exhibits strong dependence on the topologyof the magnetic field. For a stellarator plasma thesymmetric system. In standard and advanced stellarators
the matrix is non-singular and any poloidal and toroidal
rotation will be slowed down by viscosity'
In summary we may conclude that at small radial
electric field the poloidal viscous force grows linearily
with the plasma rotation velocity and exhibits a strong

surface averaged viscous forces are(,, o .n; = ((r,, - o,)",.YF)#)-t,1),,'nF

In axisymmetric tokamaks there is only a poloidal

4. Viscous Forces(" .v. n; = ((r,,- r,)"

(1s)

and

(o6v (e, >< ".)/ -- R,,E

" \'

(14)

where the viscous coefficients are,\n,,=Ioxn'(r-tVt, t\")I
The overbar denotes the time average'(13)

where Z is the operator and vp the particle drift velocity
including the electric drift. The evaluation of the viscousor convective plasma motion is the main driving term of
the Stringer spin-up. Taking into account the

t-

ry

(r2)

resonant character on rational magnetic surfaces' Strong

The formulation of the viscous forces given in eqs'
12 is valid in any regime ofcollisionality' In a collisiondominated plasma the Braginskii viscosity leads toforces, which show no strong dependence on theradial electric fields reduce the neoclassical viscousdamping and the poloidal force becomes a decreasing
function of the electric field. In this case the resonanceon rational magnetic surface has nearly vanished' The
effect of the electric field grows with poloidal mode

rotational transform (ref. t5l)' In a neoclassical regime'
however, the viscous forces are dependent on therotational transform and the radial electric field [5]'Solving the linearized kinetic equations allows one tonumber n, which implies that the E-field is more
effective on high order rational surfaces than on low

compute the anisotropic pressure in eq' 12' The

order rationals.

linearized equation is

5. Plasma Convection
It must be expected that the poloidal and toroidal
176

Wobig H., On Viscous Damping and Rotation in Low-Shear Stellarators

The feedback of the pressure on the electric potential is
obtained from the parallel component of Ohm's law andthe perpendicular plasma current given by theperpendicular component of eq. l.

. YpxB^VO N(y)xBlL=-ru-mtnofoi-*-t,

Ql)

The first term is the standard diamagnetic current andthe second one the current caused by the electric fietd.

In the following we neglect the contribution fromviscosity and inertia and approximate the equation of theelectric potential by

Rotational Transform

Fig. 1 Poloidal viscous coefficient pe vs rotationaltransform. Collisionality and electric field are kept
fixed. Additional resonant Fourier coefficients at l
= 10/19 and 10121 have been introduced-v (^,noyo+g.*o,..| =, ('o=\ ery\g',l\'"""a',n"/This is a second order elliptic equation for the electric
potential. In the limitfi -+ 0 this property is lost and theelectric potential cannot be computed uniquely. Viscousand inertial forces would add another fourth orderrotation have a feedback on the onset of plasma
convection. The simplest case occurs if 6y is thedifferential operator

classical diffusion velocity plus the pfirsch-Schliiter
effect. This velocity is independent of the superimposed

M(Q) :=

V6 and it leads to the classical Stringer spin-up.

(,\

lB'\B'llv I -1. 1y[ VO<-^, r ) x n I

and instead of eq.22 we get

However, this effect is computed without the dissipativeand inertial terms in the equilibrium equation and only
based on the resistivity in Ohm's law. Taking into
account that neutral particle interaction and viscosity-(^vo,--\- n'l-Y-lm,nofo -oi +r_ V,,<D | +M(iD)\=v (vp:n\ rrot

\B'l.

allow for a small pressure variation along magnetic fieldlines this can give rise to convective solutions, which do
not exist in the limit without these dissipative terms. It is
obvious that a convective thermal flux is minimized if
the pressure stays constant on magnetic surfaces. The
parallel pressure balance yields

B 'Yp =- mrnofoB .v - B .N(v)In the limit of zero neutral density the term M(@)
must be retained. As a non-linear equation eq.24 canhave more than one solution and a bifurcation point canexist, where these solutions merge. The bifurcation point

is determined by linearizing the non-linear equation
around one of the two solutions. Without the viscous(18)

and inertial terms the eq. 24 is linear and has only one/s is the charge exchange collision frequency. The
viscous term and the inertial term have beensummarized in the term N(r). The perpendicularsolution at a given pressure gradient on the right handside. The viscous operator in M would not change thisresult, however the inertial terms I(@) in N can give rise
to multiple solutions. Expanding the potential in Fourier
harmonics yields

velocity is given by Ohm's lawu,-=V%tZ

-\V

(1e)

The first term in eq. 19 is a convective velocity

ir* -perperdicular to the magnetic field and the second termis the classical diffusion velocity. Inserting this into theequation of continuity provides one with an equation for-v.lnp Y:! +-P -v,,pI

B' mtnofo"ll+V. o'

nr)2

e*

[*" (*,,,=loo\^uB' v+lB'l*I

the pressure

ln\

e3)

r,*)],.
(2s)

This is a system of coupled differential equations
for the coefficients @-,(y). Any Fourier mode describes
a convective cell, which due to the boundary condition

Vo;<B=s (20)B

177

Wobig H., On Viscous Damping and Rotation in Low-Shear Stellaratorsof another bifurcation. Furthermore, the non-linear terms
in the energy equation can lead to bifurcations. In
general, the bifurcation point is defined as the point

O.n = 0 on both sides of the slab does not calry particles
or heat across the boundary, but it strongly steepens thetemperature and pressure gradients and thus leads tolarge radial transport. The first term in eq. 25 with theresistivity is the largest one and retaining only this term
would yield an approximation, which is valid outside asingular layer. This singular layer occurs in the vicinityof rational surfaces with m-nr = 0, where the first term

where the linarized equation have a non-trivial solution.

With respect to stability, this point is a point of marginal
stability (ro = 0). If a system approaches this point by aslow parameter variation, the plasma can jump from one
stable solution to another stable solution.

is zero and the second term with the radial derivatives of@.,(V) provides an upper limit to the @.,(ty).

6. Conclusions

Expanding the rotational transform around the resonantsurface allows one to derive an estimate for the width of
the singular layer. The result isfeatures of the H-mode transition in(,/(vl)'(av)'**#

The analysis has shown that a single fluid model ofplasma flow in the boundary region can describe several
Wendelstein 7-AS. The force balance on magnetic
surfaces requires that in L-mode - here defined as astate of small poloidal rotation- large convective motion
occurs in order to compensate the viscous or charge
exchange damping by means of the Coriolis forces. This
convective motion also leads to enhanced plasma and(26)

At the boundary of the singular layer the first andthe second term are of the same order. Retaining theviscosity term in the operator M(Q) would lead to asimilar effect: the width of the singular layer would bedetermined by the resistivity and the viscosity instead of
the plasma neutral interaction. Eq.26 shows that the
singular layer may be large in low-shear systems; it also
grows with plasma resistivity, which implies that any
increase of the temperature in the boundary region

energy losses. In particular, this convection can arise on
low order rational magnetic surfaces, since the singularlayer, caused by resistivity, viscosity and charge
exchange interaction is large on these surfaces- Theconvection is driven by the pressure and temperature
gradients. As in the case of the B6nard convection,

would narrow the width of the singular layer.
Furthermore, the width of the singular layer and thewhen the viscosity is small, this solution can be unstable
leading to a state of turbulent convection, which also

amplitude of the resonant components @.,(V) decrease
with rising poloidal mode number n.
Eqs. 20, 24 and the energy equation eq. 3 represent
a closed system of elliptic equations, which together
with appropriate boundary conditions allow one tocompute plasma pressure, temperature and the electric
potential. A basic issue of these non-linear equations isdrives the poloidal rotation.

Neoclassical viscosity is small on high orderrational magnetic surfaces and it is further reduced by
the radial electric field associated with the poloidal
shear flow. Therefore in those regions with high orderrational surfaces - the H-mode windows - small
convective motion is needed to provide the necessary
inertial drive. This also implies a reduction of plasma
losses. This well-balanced equilibrium between driving

the problem of uniqueness. If only one solution of thesystem exists, bifurcaation and transition between
different states cannot occur. However in anology to theand damping can be modified by the additional dampingclassical B6nard problem several solutions can be
expected. The conditions for unique solutions ofequations 20 and22have been analysed leading to theresult that large dissipative effects (resistivity andcharge exchange losses) make the solution unique.

due to charge exchange losses. By reducing thisdamping term - which happens at density rise - thisdamping is reduced, thus facilitating the H-mode
transition. Qualitavely this explains the experimental
result of a threshold in density. In this model described

However, in a real plasma these effects are small andmore than one solution can exist' As a linear elliptic
equation 22has only one solution, the convective termin eq. 20, however, couples convective motion to thedensity variation and thus can give rise to convective
solutions. This is the equivalent case to the classicalabove, the energy equation only plays a minor role. The
shear flow is governed by the interplay of the continuity

equation and the potential equation.
Since the present theory does not take into accounttime derivatives of plasma parameters and the magnetic
field the issue of stability of these various stationary

thermal convection.

Retaining the inertial forces in eq. 26 would

branches remains open.

introduce another non-linearity, which can be the causet78

Wobig H., On Viscous Damping and Rotation in Low-Shear Stellarators

References[9] A.G. Peeters, J. Stober, D. Cosrer, R. Schneider and

tll M. Hirsch er a/., Plasma phys. Control. Fusion 42,

ASDEX upgrade team, plasma phys. Control.

(2000) A 23r-A236.

Fusion 40, (1998) 717-720.

[2] P. Gridull et al., J. of Nucl. Mat. To be published.
[3] P. Grigull et al., Proc. of Int. Stellarator Workshop,

ll0l K.C. Shaing, Phys. Fluids B 5, (1993), 3841.
l11l K.C. Shaing and C.T. Hsu, phys. plasmas 2,

Madison 1999.
t4l H. Wobig and J. Kisslinger, plasma phys. Control.

(1995), 1801.

[12] B.A. Carreras, P.H. Diamond and G. Vetoulis,

Fusion 42, (2000) 823-841.

Phys. Plasmas 3, (1996), 4106.

l5l H. Wobig and J. Kisslinger, plasma phys. Control.

t13l H. Wobig, Z. f. Naturforsch. 4la, il01 (1986).
t14l M. Spada and H. Wobig, J. phys. A: Math. Gen.
2s, (1992) r57s-rsgr.
ll5l T.A. Stringer, Phys. Rev. Lett.22, (1969) 1770.
[6] H. Wobig, Plasma Phys. Control. Fusion 3g,

Fusion 37, (1995) 893-922.

t6l T. Fukuda, Plasma Phys. Control. Fusion 40,(1998) 543.

t7l K. Tsuchiya et al.,Plasma phys. Control. Fusion
40, (1998) 7r3-716.

(1996), 1053-1081.

[8] L.W. Owen et al.,Plasma Phys. Control. Fusion[17] A.B. Hassam, M. Antonsen, J.F. Drake and C.S.
Lui, Phys. Rev. Leu.66, (1991) 309 ITCtl Toki

40, (r998) 7r7-72O.

2000.

r79