J. Plasma Fusion Res. SERIES, Vol.4 (2001) 167-172

Rational surfaces, ExB sheared Frows and rransport
Interplay in Fusion Plasmas
HIDALGO carlos*, PEDROSA Maria A., ERENTS Kevinr, GONqALVES Bruno2, MATTHEry5
Guy F.r,
CASTELLANO JAViEr, CENCIE Luis3, BALBiN Rosa, SANCHEZ Edilberto and DIAMoND Patrick
H.a

Laboratorio Nacional de Fusi6n, Euratom-Ciemat, Madrid, 2g040 spain

I(IKAEA Fusion, culhamscience Centre, culham, Abingdon, oxon, oXr4 3EA, uK

2Euratom - IST,
3Llniversidada(Jniversity

I04g-001 Lisbon, portugal

Carlos III, Madrid, Spainof carifurnia, san Diego, La Joila, califurnia 92093-0319, usA(Received: 5 December 2000 / Accepted: lg August 2001)

Abstract
Experimental evidence of a strong interplay between magnetic topology (rational surfaces)
and thegeneration of ExB sheared flows has been observed in the plasma edge region ofstellarator (TJ-II) andtokamak (JET) devices. Constant and varying in time,ExB sheared flows are close to thecritical value totrigger the transition to improved confinement regimes. The plasma conditions where this has
been
observed are clearly below the power threshold to trigger the formation of transport barriers.
Flowsdriven by fluctuations are candidates to explain these experimental results.

Keywords:
Ex8-shear flow' turbulence suppression, rational surface, transport barrier, Reynolds stress, magnetic
viscositv

1. lntroductionof the radial electric field is much slower that thecorrelation time of the ambient turbulence t5l. The
theory ofExB shear suppression of turbulence has been

Since the first generation of edge transport barriers

in tokamak plasmas [1], significant progress has been
made to understand the mechanisms that control theirgeneration. One of the great achievements of the fusion

recently extended to include time dependent,ExB flows[6]. The best performance of existing fusion plasma
devices has been obtained in plasma conditions where

community in the last years has been the developmentof techniques to control plasma turbulence based in the^ExB shear stabilization mechanisms are likely playing a

ExB shear stabilization mechanism 12-41.T\e generalityof this mechanism is based on the fact that the drift ofcharged particles in the presence of an electric (E) andmagnetic field (B) does not depend on the mass or
charge of the particles. When the ExB shearing rare
(oprn) approaches the growth rate of the dominant
instability (T), co"," = y, a reduction in the radial
correlation length is predicted. The earliest theory of
ExB shear suppression is valid when the time variation*

key role [7-8]. Both edge and core transport barriers are

related to a large increase in the ExB sheared flows.
Direct measurements of transport and fluctuations
during the generation of edge transport barriers showthat concurrent changes in turbulence amplitudes, spatial
scales, and multi-field phase angles lead to a reductionof turbulent particle flux during the formation of the Hmode transport barrier [9].

Corresponding author's e-mail: carlos.hidalgo@ ciemat.es
r67@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Hidalgo C. et al., Rational Surfaces, ExB Sheared Flows and Transport Interplay in Fusion Plasmas

Confinement properties are strongly linked to themagnetic topology of the magnetic trap. The presence ofrational surfaces can cause the formation of island or
magnetic ergodic regions, which can degrade the qualityof confinement in magnetically confined plasmas in(a)I,c#h,lsome cases. Low order resonances are often associatedwith "hard" MHD events such as disruptions or "soft"i,ffffii

MHD events, which limit the plasma performance intokamaks. On the other hand, the generation of internaland edge transport barriers is linked to plasma regionswith a unique magnetic topology [7'8'10'11]. In
configurations with low or negative magnetic shear,internal transport barriers (ITBs) are formed close to thelocation of 4-in, in the proximity of low order rationalsurfaces. Edge transport barriers are located close to theboundary between the region with open and closed
magnetic field lines, and in this plasma region there arestrong gradients in the level of fluctuations [12]. In thecontext of stellarator physics, global confinement
depends strongly on the value of the edge rotationaltransform [13]. Latest results from LHD stellarator haveshown the possible role of low order resonances in thegeneration of edge transport barriers [14].

R-R0(m)This paper reports the influence of magnetic
topology (rational surfaces) in the generation of ExB
sheared flows in tokamaks and stellarators, andexperimental evidence of DC and time dependent ExB
sheared flows in the range of @s's critical.

Fig. 1 TJ-ll vacuum configuration (a) without and (b)with medium size island lnlm = 8l5l in the plasmaboundary region.

focused in the n = 8lm = 5 and n = 4lm = 2 resonances,which are located near the last closed flux surface.

2. Experimental Set-UP
Experiments were carried out in the plasma edge

Figure 1 shows TJ-II vacuum configuration with andregion of the JET tokamak and TJ-II stellarator. Plasmas
investigated in this paper were ECRH plasmas (/ =53.2GH2, Prcnn = 300kW) in the low magnetic shear
TJ-II stellarator (B = lT, n = 5 x 1018m-3) and X-point
configurations in JET tokamak (B = l-2T, Ip = l-2N{A.,n = 2-3 x 10rem-3). Radial profiles of the ion saturation
current and floating potential have been measured using

without medium size island (n/m = 815).

3. Plasma Profiles and ExB Sheared Flows
Typically, ion saturation current signals show asmooth increase as the probe moves radially inwards inthe plasma edge. This behaviour is shown in figure 2 for
TJ-II and JET edge plasma profiles. This is a charact-a fast movable Langmuir probe system, both in JET and

TJ-II. Plasma potential (@o) profiles have been

eristic behaviour previously reported in tokamak,

estimated from electron temperature (2") and floating
potential (@1) profiles using @o = @r * aT"(a = 2.5).

stellarator and RFPs devices [17-19].
The presence of natural resonances (n = Sltn = 5),
as predicted by vacuum field line calculations, has been
observed as a flattening in the edge plasma profiles in
the TJ-II stellarator (Fig. 3). Experiments carried out in
the JET plasma configurations with B = 1T and 1p =The experimental set up allows to investigate the radialand poloidal structure of fluctuations.

The existence of closed and nested magnetic
surfaces in good agreement with the calculated ones has
been demonstrated by magnetic surface measurements

lMA have also shown evidence of flattening in edge
plasma profiles, which might be assigned to the q = 4
surface (Fig. 3). These observations have been interpreted in terms of the influence of rational surfaces onin the TJ-II stellarator t15,161. To study the effect ofrational surfaces on radial electric field we have chosen
the plasma configurations with iota (a) = 1.6 and 2, and168

Hidalgo C- et al., Rational Surfaces, EXB Sheared Flows and Transport Interplay in Fusion plasmas

0.03
TJJI

0.3

shot 2852

0.02

(a)

(b)

3 o.z

I

0.01

0.1
radially inwards

0

JET
shot 505?3ndially inwards-5 -4 -3 -2 -l

0

0
-2

I

z-zr-r" (cm)

-1 r 0123- rlcrs (cm)Fig. 2 Radial profile of the ion saturation current (a) in the TJ-ll stellarator for a plasma configuration
with iota (a1 = 1.g(i.e. with the 412 surface located out of the plasma boundary region)and (b) for the shot 50673 (B = 2T, tp = 2MAl
in JET.

,-4->0.02E!t)F

rrdiallyi+rm&

rl
rlt'{

()

$o,r

TJ.II
shot 1984

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ta

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a)

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shot

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ta

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tDt (.) - I,ar'ilrl. "'i' '..tlttt

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q)

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s0
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-4-3-2-101

z-2""*, (crn)Fig. 3 Radial profiles of the ion saturation current, floating potential and plasma potential (a) in TJ-ll plasmaaconfiguration with a low order resonance (8/5) located near the plasma boundary (b) for the JET shot 50762 (B=1T, /e = 1MA).

plasma profiles [20,21].

of fluctuations) are linked to rational surfaces well

There is a significant variation in the plasma
potential and poloidal phase velocity of fluctuations

below the L-H power threshold.
These results illustrate the link between significant

linked to the flattening of plasma profiles, The resulting

ExB sheared flows and rational surfaces in stellarator

ExB sheared flow is of the order of lds-l near the radial

and tokamak devices, which might explain the strong

position where there is a flattening in the density profile.

interplay between the magnetic topology and theThis value turns out to be comparable to the inverse of
the correlation time of fluctuations (r = l0ps). Thisobservation implies that ExB sheared flows with decorrelation rates close to the critical value to reduceturbulence (i.e. arsr6 = llr, rbeing the correlation time

generation of transport barriers in fusion plasmas.

4. Dynamical ExB Flow and Transport inTokamak and Stellarator plasmas
Recent gyrofluid and fluid simulations havet69

Hidalgo C. et a/., Rational Surfaces, ExB Sheared Flows and Transport Interplay in Fusion Plasmas

observed small scale fluctuating sheared -ExB flows
122,231. These flows are driven by fluctuations and,according to these simulations, they can substantiallyreduce turbulent transport. The effective sheared rate offluctuating ExB flows is less effective than the slowly
varying component in reducing turbulent transport. This
reduction in the effectiveness is particularly importantwhen the ExB flow patterns changes in a time scale
faster than the eddy tum over time [6].
From this perspective, two open (and connected)
question for experimentalists are the following: What is
the level of fluctuations in the radial electric field?, and:Is the effective shearing rate of fluctuating radial electricfields high enough to control transport?. In order toanswer these questions, it is important not only tomeasure the radial electric field, but also to investigate
the coupling between fluctuating radial electric fields

2500
2000

JET
shot 50573

IrI.r
-rl
I
... 1..

1500

(a)
I

rYl

I
I

1000

I

o
F

r

tt
I
I

500

0

0l

-l

a

r - rlcFs(cm)
250o
2000

d.

1500

' -ltttt,r]sr€

and ExB transport.

The radial profiles of fluctuations in the radial
electric field have been investigated in the plasma edge
region of the JET tokamak and in the TJ-II stellarator,ahneglecting electron temperature fluctuations effects (Fig.
4). The rms level of fluctuations increases as the probe
is inserted into the plasma edge region, reaching values
in the range of 1000 - 2000V/m in the plasma boundary
region. The averaged frequency of E, fluctuations is

TJ.II
shot 3574

1000

r

t,
- rh..
.L(b)

l...trlftd

500

0--5-4-3-2-1012

z - zlcFs (cm)Fig. 4 Root mean square (r.m.s.) value of fluctuations in

the radial electric field in the TJ-ll stellarator (a)and JET tokamak (b) computed neglecting I

comparable to the width of the turbulent spectra. A
rough estimation of the effective shearing rate of thefluctuation effects.

fluctuating radial electric fields can be computed asa,7, -(E*aia)^, (L,,--t-).
fr)p,a=f",-ffi:

the radial electric field at different radial locations.

(1)

where al1 and Acor are the mean frequency of fluctuating

5. Why ExB Sheared Flows?

radial electric field and the width of the turbulent
spectra respectively, ,1" is the radial correlation ofsheared flows is an important physics issue in tokamaks

Understanding the mechanisms which drive -ExB

and in stellarator devices. The present experimental
results show the development of both DC and fluctuating radial electric fields with decorrelation rate(oB,s) comparable to the growth rate of plasma
instabilities (1Q5 - 106s-t) in plasma regimes well below
the power threshold for the transition to improved
confinement regimes. This result strongly suggests therole of turbulent driven fluctuating radial electric field(via Reynolds stress). The Reynolds stress measures thedegree of anisotropy in the structure of fluctuations.
Radially varying Reynolds stress allows the turbulence
to rearrange the profile of poloidal momentum, generating sheared poloidal flows. In the plasma boundary
region, strong gradients in the level of fluctuations canprovide the radial-poloidal non-isotropic turbulence. In

fluctuations and B is the toroidal magnetic field. The
function/(at1 | La,1) takes into account the reduction of
the effective shearing rate when the time scale of thefluctuating radial electric field is faster than thecorrelation time of fluctuations: /(coi / Lar) = I if ari <<Lrr,r;fGor lAa4; = 0 if ar1>> Aa4 [6].

In the case of JET and TJ-II edge plasma
conditions, (4"au),-. = 1000V/m, ar = Lror and theaverage radial correlation is in the range of lcm. Using
expression (1), the effective decorrelation rates are close
to the critical value to regulate turbulent transport ((oExB= 105s-t), well below the L-H power threshold power in

JET and in ECRH plasmas in the TJ-II stellarator. A
more accurate computation of the fluctuating alBts
shearing rate requires the simultaneous measurements oft'70

Hidalgo C. et al., Rational Surfaces, ExB Sheared Flows and Transport Interplay in Fusion plasmas

the plasma core region a modification in the degree ofanisotropy in the radial-poloidal structure of fluctuations
is expected at the rational surfaces, where fluctuations
show maximum amplitude.

-55

-60
-65

5.1 Link between low and high frequencyt'fluctuations via Reynolds stress
A quantitative estimate of the importance offluctuation induced flows requires comparison of thedamping rate of poloidal flows (i.e. ltVs) with thedriving term via Reynolds stress in the equation

-75
-80

-l 106

describing the time evolution of the poloidal flows:+=-p(r,)-*(r,r,)-:+ (E,8,)

FB'or-5 1050.fr, E t ry/m)2
Fig. 5 Correlation between floating potential and theQ)electrostatic Reynolds stress in the plasma edgeof JET tokamak.

Assuming that the frequency of fluctuating poloidal
flows (ale) is much smaller than the damping rate (i.e.
ae << It), that poloidal flows are directty related withradial electric fields (V6 = E,lB = (lt\ dAftr, @ being
the plasma potential) and that the term of the Reynolds
stress tensor can be related to the _ExB velocities. it
follows that(r,) =--70

via Reynolds stress.

5.2 Rational surfaces and Reynolds stressdriven flows
A possible explanation of the flow structure nearrational surfaces is the coupling of flow generation andturbulence (i.e. sheared flows driven by fluctuations via
Reynolds stress). In the absence of ExB sheared flows,(3)

fluctuations are expected to show maximum amplitudea * f= (8, E,)=+* (6,),at the rational surface. The corresponding instability(4)mode width will depend on the nature of the instability

and on the magnetic shear scale length. As a con_sequence, a modification in the degree of anisotropy in

ku and, k, being the poloidal and radial wave-number offluctuations, and yis the correlation parameter between
poloidal and radial electric fields. Expressions (3) andthe radial-poloidal structure of fluctuations (i.e.
Reynolds stress) is expected at the rational surface. Thismechanism can provide sheared poloidal flows linked to
the location of rational surfaces.

(4) show the relation between low frequency

fluctuations in the plasma potential (@) and the highfrequency fluctuations.
The plasma potential is related with the floating
potential and electron temperature by the expression @o

A model has been developed based on the reduced

MHD equations for the evolution of the poloidal
magnetic flux and vorticity plus separated evolutions forthe electron density and temperature [24]. Density andtemperature profiles show the flattening caused by the= (Dt * dT"(u = 2-3). Figure 5 shows the correlation
between the floating potential (@i) and the electrostatic

Reynolds stress in the plasma edge of JET tokamak.

presence of the magnetic island. Another effect of theInterestingly both quantities are clearly correlated.

vacuum magnetic island is the generation of a global

Assuming a linear behaviour and using expression (4), it

poloidal flow through Reynolds stress. This flow
oscillates in time and changes direction in a qua_follows that p= 5 x l0as-r, which turns out to be close
to the expected damping rate due to magnetic pumping

siperiodical manner. The poloidal propagation shows acontinuos change with increasing vacuum magnetic
island width. Below a threshold, the electrostatic andmagnetic components of the Reynolds stress have
similar size and radial profiles, but the opposite sign.
This causes a near cancellation of these two terms, as it
happens in the case of electromagnetic turbulence.

for JET edge parameters. A systematic investigation of
the relation between plasma potential (@) and its
fluctuations, both in the plasma edge region (usingprobes) and in the plasma core region (by means of the

HIBP), is needed to clarify the importance of thecoupling between low and high frequency fluctuations

t7l

Hidalgo C. et aI., Rational Surfaces, ExB Sheared Flows and Transport Interplay in Fusion

Above the threshold, the vacuum magnetic island

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t6l T.S. Hahm, M.A. Beer,Z.Lin et al., Phys. Plasmas

sheared flow is established. These findings are
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experimental results in terms of the rational surface

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