Proceedings of ITC/ISHW2007

Physics of Flows and Turbulence in Fusion Plasmas
Ulrich Stroth1)1)Institut für Plasmaforschung, Universität Stuttgart, 70569 Stuttgart, Germany(Received 2007 / Accepted 2007)

Results on flows and turbulent transport as obtained from stellarators, tokamaks and non-fusion devices arehighlighted to discuss the complex interaction between them. Both neoclassical and turbulent transport processes
are needed to cover the rich spectrum of experimental observations. The magnetic configuration plays an important role for both processes. Therefore, a better understanding of this important field could lead – after the
first neoclassical optimisation of stellarators – to a second optimisation step which envisages improved control
on turbulent transport.
Keywords: Plasma confinement, turbulence, rotation, zonal flows
DOI: 10.1585/pfr.1.001

1 Introduction
The interplay between flows and turbulence is a fascinating and rich topic of fusion research. Background flowsaligned with the flux surfaces are a prerequisite for confinement of high-temperature plasmas. But even in globally stable plasmas, micro-instabilities lead to the development of turbulence. The interaction between flows andturbulence is twofold. Sheared flows can reduce turbulence
and trigger the development of transport barriers while turbulence is able to power large scale zonal flows in a selfadjusting manner.
The physical description of flows and turbulence depends on details of the magnetic configuration used to confine the plasma: Magnetic curvature and trapped-particle
drifts enter the calculation of growth rates and determine
the character of turbulence. The topology of magnetic ripples not only defines the trapped-particle populations but
also the parallel viscosity and hence the damping of theflows. Rational values of the rotational transform, magnetic islands and magnetic shear are further configurational
elements with relevance for both flows and turbulent transport.

2 Physics of flows
In a simple magnetised torus, the physics of flows is relatively straightforward. The toroidal coordinate is identicalwith the symmetry axis and the plasma can flow freely parallel to the magnetic field B. The poloidal plasma flow is
decoupled from the poloidal one. It can only be generated by the diamagnetic and the E×B Drift. Even in atokamak with ideal toroidal symmetry, due to the helicalfield lines the parallel flow is coupled to the poloidal one.
The toroidal flow now is a superposition of a parallel flow
and drifts due to the poloidal magnetic field component
author’s e-mail: stroth@ipf.uni-stuttgart.deBθ . Furthermore, due to the toroidal magnetic field ripple,
poloidal flows are damped by a viscous force.
Equilibrium flows in toroidal geometry follow fromthe condition ∇p = jdia × B. As it can be seen in Fig. 1,
diamagnetic flows have poloidal and toroidal components.
Due to their mass, the ions are the relevant species for momentum studies. The toroidal component of the ion diamagnetic flow points into the same direction a plasma current has to flow in order to generate the rotational transform(call co-direction).

Er

+

Iϕ

diamagn.

diamagn.

θ

B
ExB

Er xB

θ

Fig. 1 Ion flows in toroidal geometry.

The poloidal flow uθ is subject to a viscous force
which, for qualitative studies, can be approximated by√Fθvisc ≈ mµ̂θ uθ . The force depends on the mass mof the species and the parallel viscosity coefficient µ̂θ .
The neoclassical transport can be understood as a radial
drift vD = Fvisc × B/qB 2 in response to the viscous
force. Since the ion viscosity is higher, ion losses would bestronger than electron losses (assuming the same pressure
gradient). This results in non-ambipolar radial transport
and thus a negative radial electric field. This process goeson until the E×B drift, which slows down the poloidal ion
flow and increases the electron flow, is strong enough tomake the viscous forces on the two species equal and therefore transport ambipolar. What remains is a slow poloidal
ion flow into the ion-diamagnetic direction, which has asmall component into the toroidal co-direction. Even for
a complex stellarator field as the one of W7-AS, the thusProceedings of ITC/ISHW2007
estimated ambipolar radial electric field
Eramb ≈ ∇r pi /en(1)agrees rather well with measurements and fully neoclassical calculations [1, 2].
Experiments on plasma rotation rely on measurements
on impurities and not the mains ions. The impurity ionpressure pI , their density nI and flow velocity perpendicular to the magnetic field uI⊥ have to be measured to allow
for the calculation of the radial electric field according to
the radial force balance for the impurity ionsp0I
Er
+ uI⊥ −
= 0.
(2)
B
ZenB
With expression (1) inserted for ambipolar electric field
one findsuI⊥ = −Er /B + ∂r pI /ZI enI .

(3)

Er is set by the main ions. Since the pressure gradientof the impurities is in general smaller than the one of themain ions, the impurity-ion flow points into the E×B (ctr.)
direction.
very fast

fast

slow

Fig. 2 Components of the ion-momentum equation.

The equilibrium flows can be modified by external
and internal torques. As shown in Fig. 2, forces Fθ andFϕ act through the ion momentum equation on the ion
flow. The ion and electron equations are coupled throughthe radial electric field. External momentum input canstem from neutral-beam injection (NBI), ion or electroncyclotron-resonance heating (ICRH and ECRH). Momentum can also be redistributed internally by Reynolds stress.
Locally this can be treated as a force. Furthermore, radial
momentum transport is also a local source or sink of momentum, which can be represented by the forces in Fig. 2.
The characteristic response time of flows to toroidal and√√poloidal forces are τϕ = µ̂ϕ / m  µ̂θ / m = τθ . Sinceµ̂ϕ is small, the toroidal flows adjust rather slowly to equilibrium changes. Furthermore, due to their small masselectrons react much faster than ions. If the collisionality changes with time, the viscosities can also be modified
leading to a distortion of the equilibrium and changes inflows and radial electric field. These changes happen on afast time scale as far as the electron viscosity is concerned.
Otherwise, the modifications in the flows should happen
slower.
Modifications in the equilibrium act on the ambipolarity condition through the Lorentz force (indicated by arrows). Due to the fast reaction of the electrons, the reconstitution of ambipolarity through the radial component is avery fast process, too. Slow changes are also connected tothe pressure gradient, which can only be modified on theconfinement-time scales. A more direct manipulation of
the radial electric field occurs if radial losses of fast electrons or ions are generated by e.g. ECRH or ICRH.

3 Toroidal flows
The recent interest in toroidal flows arises from their stabilising influence on MHD modes like the resistive wallmodes in tokamaks [3]. In present day devices, toroidal
plasma flows can readily be driven by NBI heating. In a fusion reactor, however, external momentum sources will beabsent or insufficient. Hence, it is worthwhile to study thepossibilities of an intrinsic generation of toroidal flows. In
principle, there are two mechanisms which can drive coreplasma rotation: a momentum inward pinch with the momentum source located at the plasma edge, and the toroidal
component of the turbulent Reynolds stress. While the existence of a momentum pinch can be shown in transientphases of the plasma, the turbulent drive can only be assessed if all neoclassical effects are taken into account correctly. Alternatively, it can be attempted to measure theReynolds stress directly.
In order to asses the importance of pinches or turbulent drives, the measured flow velocities have to be carefully compared to the neoclassical predictions addressed
above. In Ohmically heated (OH) discharges on Alcator
C-mod [4], the impurity ions flow into the direction opposite to the plasma current and the flow reverses sign withplasma current. The same is true for Ohmic plasmas in
TCV [5]. The measured toroidal flow of C6+ in the core isinto the ctr-direction, too. Little flow was observed close
to the separatrix. These results agree with the expectations
from neoclassical theory. The scaling of core flow velocity is uϕ ∼ Ti (0)/Ip [5]. The dependence on the central
ion temperature can be understood in terms of the simple
estimate of Eq. (1). Surprising is the spontaneous change
in flow direction observed in TCV [6]: at a critical density,
the flow changes from ctr to co-direction. This could bean effect of changing collisionality or due to the onset of apinch or a turbulent drive.
On the DIII-D tokamak, a comparison of the rotation
profiles with neoclassical theory has been carried out, too.
Ohmic and ECRH H-mode discharges have been analysed[7]. In OH discharges, slow impurity core rotation was
found. In the H-mode edge, where the impurity pressure
gradient might become strong, the flow was stronger andco-directed. When ECRH was added leading to Te > Ti ,ctr-rotation became evident in the plasma core. The strong
rotation coincided with the ECRH deposition profile. The
comparison with neoclassical theory points to an additional
toroidal torque.
Externally driven radial currents e.g. due to fast electron losses generated by ECRH must lead to a fast changeProceedings of ITC/ISHW2007
in the ambipolar electric field. If electrons are lost, Er becomes less negative and impurity rotation should become
more co-directed. Ion losses, on the other hand, would
lead to faster impurity rotation into the ctr-direction. A
study of the effect of ion-ripple losses on toroidal rotation
has been carried in JT-60U [8]. Although the plasma washeated with co-NBI, the impurity rotation at the plasmaedge was into the ctr-direction. This is consistent with thethe expected effect of ion losses due to the toroidal field
ripple at the plasma edge. Consequently, a reduction of themagnetic ripple due to the insertion of ferritic steel tiles
changed the edge rotation into the co-direction. Also in Alcator C-mod, a change of the toroidal rotation from ctr toco was observed, when ICRH was added onto Ohmic plasmas [9]. Although this behaviour is qualitatively consistent
with neoclassical theory, a detailed analyses revealed thata momentum pinch is required to explain the data. Similar
experiments carried out in ASDEX Upgrade, where ICRH
was added to NBI discharges, yielded a reduction of therotation velocity in both cases, with co and ctr-NBI. Hereconfinement changes were made responsible for the reduction in toroidal momentum [10].
If additional torques are applied due to NBI or wave
momentum input parallel to the magnetic field, the ambipolar electric field is going to change on the slow timescale. The parallel flow contributes an additional flow
component into the poloidal direction which now readsuθ = (|uE×B | − |udia |) cos α ± uk sin α(4)Here α is the pitch angle of the field lines. If the argumentsfrom above still apply and neglecting changes in the pressure profiles, the poloidal ion flow must remain unchanged.
Consequently according to the discussion of Fig. 1, an external co-torque must lead to a reduction of the radial electric field, while a ctr-torque makes Er more negative. Thechange happens slowly as plasma speeds up. The resulting
electric field will be such that the flow mainly aligns withthe direction of symmetry.
Here differences between stellarators and tokamaks
can be expected. In an ideal tokamak the driven flow willalign with the toroidal direction. Since the viscosity in thisdirection is small, neoclassical transport will not change
substantially. In case of a stellarator, the resulting direction of rotation will depend on the specific magnetic configuration. In quasi-symmetric devices, strong flows canbe driven without strongly increasing neoclassical transport. In the absence of symmetry, the momentum input
will not drive the same rotation velocity and the influenceon neoclassical transport will be more important.
In the CHS heliotron, the alignment of the flows withthe direction of minimum ∇B was experimentally demonstrated [11]. In discharges with combined ECRH and
NBI, an internal electron-transport barrier with positive radial electric field developed. The measured impurity flow
pointed opposite to the direction NBI. It was shown thatthe net flow of the main ions is such that it aligns with thedirection of minimum ∇B.
The presented survey shows that there exists no uniform picture of toroidal momentum transport yet . To
disentangle neoclassical effects from turbulent ones is not
simple. But it has to be done properly in order to conclude
on spontaneous turbulent drives. Clearer than steady state
analyses are transient transport experiments. They have already been successfully applied to particle transport, wherean anomalous pinch has been observed in tokamak [12, 13]
and stellarator experiments [14, 15]. Recently, a pinch has
also been discovered in the momentum transport channel[16] and for the explanation, theoretical models related tosymmetry breaking of turbulence [17] and a inwards drift
due to the Coriolis force [18] have been put forward. If
momentum is generated at the plasma edge, the pinch cantransport it to the core. Hence the momentum pinch is one
possible source of toroidal rotation for the plasma core.
However, the momentum source needed for a complete
model remains still unexplained.
Furthermore, there exists also first direct evidence fora turbulent drive of toroidal rotation. Measurement of theparallel Reynolds stress have been carried out in TJ-II [19].
Above a critical density, net energy transfer from turbulence to parallel flows has been found by means of probe
measurements in the plasma edge.

4 Poloidal flows
Sheared poloidal flows are widely accepted to be beneficial
for the reduction of turbulent transport. Of special interestare zonal flows, which are radially localised E×B flows
with poloidal and toroidal mode numbers m = n = 0.
Recent reviews of the vast experimental and theoreticalwork can be found in Refs. [20, 21, 22]. Still open arequestions addressing the nature of the momentum sources,
which can have neoclassical and turbulent elements, andthe detailed mechanism of turbulence reduction. It is also
well known, that the zonal flow couples to the Geodesic
Acoustic Mode (GAM) [23], which appears at the characteristic frequency fGAM ∼ cs /R. Hence, zonal flows mostly
appear in the form of GAMs as it has been demonstrated
in many devices using heavy-ion beam probes, beam emission spectroscopy and Doppler reflectometry (see Ref. [24]for a survey).
Zonal flows are also observed in fluid and gyro-fluid
turbulence simulations. In the numerical models, thegeneration of zonal flows is clearly due to the turbulent
Reynolds stress ∂r hṽθ ṽr i. Also in simulations they are
oscillatory (GAMs) and radially localised. If interchange
turbulence with radially elongated streamers is dominant,one can visually observe the decorrelation of the large turbulent structures [25]. Fluid simulations using the driftAlfvén turbulence code DALF3 [26] have been carried out[27] to test analyses techniques for the study of the turbu-

Proceedings of ITC/ISHW2007
lent drive of zonal flows. The averaged poloidal flow fromthe simulations exhibited typical GAM structures. Using across-bispectrum technique, evidence for Reynolds-stressdrive of the flows was found and correlation analyses also
showed the reduction of turbulence in regions of maximum
flow shear.
In the TJ-II heliac, the Reynolds stress has been measured directly with a probe array [28]. Gradients in the
Reynolds stress were found in the vicinity of a shear layer.
Also in linear devices, evidence for turbulent flow drive
was found [29]. Poloidal rotation was observed in a plasma
without net momentum input. The poloidal momentum
balance was successfully checked using measured flow andReynolds stress profiles. The generation of a zonal flow
can also be interpreted as sign of an inverse turbulent cascade, which is expected to be active in magnetised plasmas.
First direct evidence for the inverse cascade were was obtained from the TJ-K torsatron applying bispectral analyses
methods on 2-dimensional probe data [30].
Hence there is ample evidence of the existence ofpoloidal flows and the importance of a turbulent drive.

5 Flows acting on turbulence
Zonal flows play a key role in the regulation of turbulent
transport. Once generated through Reynolds stress, theflows can act back to reduce turbulence [20]. This interplay can be observed in turbulence simulations, but israther difficult to be unambiguously demonstrated in experiment. Although the shear-decorrelation mechanism
has been pointed out already in 1990 [31], there is still noclear experimental evidence on the reduction of the turbulent scales due to sheared flows.
Turbulent transport, which can be written asΓ̃ ∼ |ñ| φ̃ sin δnφ ,(5)

can be manipulated by different means. The density andpotential fluctuation amplitudes ñ and φ̃ can be reduced.
According to the mixing-length argument, this would bedone by a reduction of the turbulent radial correlation
length. On the other hand, a change in the cross phaseδnφ can influence transport, too. Which of the two effects is more important can sensitively depend on the dominant type of turbulence present in the plasma. While interchange turbulence is characterised by a cross phase of
δnφ ≈ π/2, the phase of drift-wave turbulence is close
to zero. In the first case, a small change of δnφ does
not substantially alter transport while in the latter casesmall changes can already have strong influence on the its
magnitude. Hence, in order to predict the role of shear
flows in future devices, a knowledge of the expected dominant type of turbulence is of great importance. In IonTemperature-Gradient (ITG) or Trapped-Electron Modes(TEM), which both are of the interchange type, the characteristic streamer-like turbulent structures can efficiently

be reduced by sheared flows while the effect of the crossphase might be negligible. This kind of turbulence is expected to be dominant in the core of tokamak plasmas. In
the plasma edge of both stellarators and tokamaks, driftwave turbulence might be more important. Drift waves do
not show the radially elongated structures but they havesmall cross phases, hence the phase might be the key parameter to reduce transport at the plasma edge. Therefore
related important questions are what is the character of turbulence in the core of stellarators and what is the dominant
instability in the plasma edge of both stellarator and tokamak plasmas.
Already with the discovery of the H-mode in the ASDEX tokamak it was shown that the region of the transportbarrier is related to reduced fluctuation amplitudes [32].
This has been confirmed in many different devices since
where a edge transport barrier could be achieved. But
sheared flows can also act on the cross phases as shown inbiasing experiments on the small stellarator TJ-K, where
drift waves have been shown to be the dominant instability[33, 34]. In this case no decorrelation of turbulent structures was detected. Confinement improvement happened
rather due to cross-phase modifications [35].
Less is known about core transport in stellarators. In
the core of larger tokamaks, ITG and TEM modes arelikely to be dominant. This kind of turbulence can accountfor one major property of the temperature profiles, namely
the profile resilience as, e.g. shown for the tokamak ASDEX Upgrade [36]. The fact that∇Te
1=
≈ const.
LT
Te

(6)

results in an offset-linear scaling of the electron heat flux
vs. electron-temperature gradient as, e.g. shown for ToreSupra [37]. Although the electron temperature profiles
in stellarators can become flat in the centre when ECRH
is deposited off-axis, in the W7-AS stellarator there wasalso evidence for profile resilience when the central heating
power was different from zero [38]. It should be stressed,that in a tokamak there is always residual Ohmic heating in
the centre. Hence, similar concepts may also apply to stellarators plasmas and a quantitative comparison of the W7AS results with tokamak models gave a quite good agreement, although the critical gradient in W7-AS is steeper[39].
If profile resilience is also present in stellarators, ITG
and TEM turbulence may also be candidates to explain turbulent transport. Both instabilities are driven in the badcurvature regions of the plasma. Furthermore, to develop
the TEM needs trapped particles in the region of bad curvature. Therefore, a optimisation of the magnetic configuration can also be carried out in view of a stabilisationof these modes. The W7-X stellarator has been optimised
with respect to neoclassical properties. For neoclassicaltransport, the separation of curvature and trapped particles
is also a criterium. In this respect, W7-X has also achieved

Proceedings of ITC/ISHW2007
some degree of optimisation against TEM turbulence.
First gyrokinetic simulations of TEM turbulence in
W7-X geometry show [40] that the spatial distribution of
local transport maxima reflects the fivefold symmetry of
the magnetic configuration. The largest transport levels arefound in regions where magnetic wells and bad curvature
still overlap to some extent.

6 Summary and Conclusions
The paper intended to demonstrate the richness of the phenomena related to the interplay between flows and turbulence in toroidal plasmas and their importance for thedevelopment of future fusion devices. The importancearises from their ability to stabilise MHD modes and to
reduce turbulent transport. Since future devices will not be
equipped with sufficient means to drive strong flows, thespontaneous generation of flows is especially important.
The radial electric field, which is closely related toneoclassical transport, is the key player of the flow system. In plasmas without external momentum input, qualitative agreement with neoclassical expectations is found inmany cases. But there also exist observations like the spontaneous flow reversal in TCV, that cannot readily be explained by neoclassical theory and need more sophisticatedcodes for interpretation. And there exists also clear evidence for momentum pinches which cannot be explainedby standard neoclassical theory. The pinch can act as a momentum source in the plasma core, but a momentum sourceis still needed at the plasma edge. First evidence also existsthat Reynolds stress can redistribute toroidal momentum togenerate sheared toroidal flows.
Radially localised poloidal flows and their characteristics have been well documented in many experiments.
There also exist strong indications for the importance ofReynolds-stress drive of these flows. Due to the toroidal
field ripple, zonal flows also should interact with neoclassical transport. This aspect needs further attention and global
turbulence simulations including neoclassical effects aredesirable. There is also agreement that zonal flows havethe ability to reduce turbulent transport. Details on the
suppression mechanisms, however, are not yet fully understood and can be expected to depend on the type of thedominant instability.
The magnetic configuration enters in different aspects
the flow-turbulence system. It sets the values of the different components of the parallel viscosity and therefore determines the amount of neoclassical transport for a given
flow on the flux surface. In the same way the configuration
determines the ability of the plasma to respond to torques
and to develop zonal flows. On the other hand, curvature and trapped-particle populations are leading ingredients for driving turbulence. Both parameters are predefinedby the magnetic configuration. A better understanding of
the flow-turbulence system could therefore help to attempt

an optimisation of the stellarator configuration with respect
to both neoclassical and turbulent transport.
[1] H. Ehmler et al., Nucl. Fusion 43, L11 (2003).
[2] U. Stroth, in Plasma Physics: Confinement, Transport andCollective Effects, Lect. Notes Phys., edited by A. Dinklage(Springer, Berlin, Heidelberg, 2005), Chap. 9, p. 213.
[3] E. J. Strait et al., Phys. Rev. Lett. 74, 2483 (1995).
[4] J. E. Rice, E. S. Marmar, F. Bombarda, and L. Qu, Nucl.
Fusion 37, 421 (1997).
[5] A. Scarabosio et al., Plasma Phys. Controll. Fusion 48, 663(2006).
[6] A. Bortolon, B. P. Duval, A. Pochelon, and A. Scarabosio,
Phys. Rev. Lett. 97, 235003 (2006).
[7] J. S. deGrassie et al., 11, 4324 (2004).
[8] M. Yoshida et al., Plasma Phys. Controll. Fusion 48, 1673
(2006).
[9] J. E. Rice et al., 45, 251 (Nucl. Fusion).
[10] D. Nishijima et al., Plasma Phys. Controll. Fusion 47, 89(2005).
[11] K. Ida et al., Phys. Rev. Lett. 86, 3040 (2001).
[12] D. R. Baker et al., Nucl. Fusion 38, 485 (1998).
[13] V. Mertens et al., Plasma Phys. Controll. Fusion 32, 965(1990).
[14] U. Stroth et al., Phys. Rev. Lett. 82, 928 (1999).
[15] H. Nozato et al., (2004).
[16] W. D. Lee et al., Phys. Rev. Lett. 91, 205003 (2003).
[17] Ö. D. Gürcan, P. H. Diamond, T. S. Hahm, and R. Singh,
Phys. Rev. Lett. 14, 042306 (2007).
[18] A. G. Peeters, C. Angioni, and D. Strintzi, Phys. Rev. Lett.
98, 265003 (2007).
[19] B. Goncalves et al., Phys. Rev. Lett. 96, 145001 (2006).
[20] P. H. Diamond, S. Itoh, K. Itoh, and T. S. Hahm, Plasma
Phys. Controll. Fusion 47, R35 (2005).
[21] K. Itoh, S. Itoh, P. H. Diamond, and T. S. Hahm, .
[22] A. Fujisawa et al., Phys. Rev. Lett. 98, 165001 (2007).
[23] N. Winsor, J. L. Johnson, and J. M. Dawson, Phys. Fluids
11, 2448 (1968).
[24] A. Fujisawa et al., Plasma Phys. Controll. Fusion (2007).
[25] Z. Lin et al., Nature 281, 1835 (1998).
[26] B. Scott, Plasma Phys. Controll. Fusion 39, 1635 (1997).
[27] M. Ramisch, U. Stroth, S. Niedner, and B. Scott, New J.
Phys. 5, 2012 (2003).
[28] C. Hidalgo et al., Plasma Phys. Controll. Fusion 42, A153
(2000).
[29] G. R. Tynan et al., Plasma Phys. Controll. Fusion 48, S51(2006).
[30] P. Manz et al., Plasma Phys. Controll. Fusion (2007), submitted.
[31] H. Biglari, P. H. Diamond, and P. W. Terry, Phys. Fluids 2,
1 (1990).
[32] F. Wagner et al., Phys. Rev. Lett. 49, 1408 (1982).
[33] U. Stroth et al., Phys. Plasmas 11, 2558 (2004).
[34] N. Mahdizadeh et al., Plasma Phys. Controll. Fusion 47,
569 (2005).
[35] M. Ramisch et al., Plasma Phys. Controll. Fusion 49, 777(2007).
[36] F. Ryter et al., Phys. Rev. Lett. 86, 5498 (2001).
[37] G. T. Hoang et al., Phys. Rev. Lett. 87, 125001 (2001).
[38] U. Stroth et al., Phys. Scr. 51, 655 (1995).
[39] F. Ryter et al., Plasma Phys. Controll. Fusion 48, B453
(2006).
[40] P. Xanthopoulos, F. Merz, T. Görler, and F. Jenko, Phys.
Rev. Lett. 99, 035001 (2007).