J. Plasma Fusion Res. SERIES, Vol.5 (2002) 318-323

Numerical Simulations of Turbulence in RTP Discharges
with Dominant Off-Axis ECH
DE BAAR Marco R., THYAGARAJA Anantanarayanan', KNIGHT Peter J.',
HOGEWEIJ Pick G.M. and LOPES CARDOZO Niek J.
FOM instituut voor plasmafysica Rijnhuizen, Trilateral Euregio Cluster, Nieuwegein,
The Netherlands, Association Euratom-FOM
1

EUMfOU/\JKAEA Fusion Assoc., Culham Science Centre, Abingdon, OXl4 3DB, UK(Received: ll December 2001 / Accepted:29 May 2002)

Abstract
A two-fluid computer model, CUTIE, of saturated global tokamak turbulence is used to study thetransition from an Ohmic to an Rijnhuizen Tokamak Project (RTP) type-D discharge (a discharge withdominant, off-axis ECH featuring steady state hollow temperature profiles). The simulations were done
for a few current diffusion times. In particular the role of the evolving q-profile in the self-organizationof the plasma has been studied. The dynamics of the bootstrap current, the turbulence drive terms, the
ExB flow and the dynamo terms in the induction equation will be discussed. The effect of MHD activity
on turbulent transport has been investigated. It will be explained why CUTIE 1) produces barriers near
simple rational 4 values as in RTP, and 2) naturally generates the required advective transport to support
off-axis maxima in 2".

Keywords:
anomalous plasma transport, numerical simulation, turbulence, ITB, 2 fluid code, global simulation

1. lntroduction

ExB sheared velocity TinlTn"n < 1, and the turbulent
vortices can be decorrelated. Type II discharges show
barriers in Z" and 7i when PNsr < 5 MW and q-1nInternal Transport Barriers (ITBs) are being
considered for reactor relevant advanced tokamak
scenarios. In recent years, considerable progress has
been made both on the development of diagnostics for
ITB relevant parameters and on the modeling of ITB
dynamics. However, much of the physics of tokamak
turbulence and its suppression is currently not wellunderstood. Notably the role of the q-profile in the ITB

approximately 2I21. In JET, ITBs form in the vicinityof q - l, e = 2 and 4 = 3, andmaybe near q = 312 [31.
Joffrin [4] suggests that in JET, due to toroidal coupling,magnetic braking can increase the local sheared velocity
and yield co-evolution between the rotation profile andthe simple rationals in the q-profile. Power thresholdexperiments suggest a synergy between details in the qprofile and the rotational shear. The JT-60 team reports[5] that in negative central shear discharges barriers may
form at the q - 3 surface. In RTP, a series of barriers
was disclosed near mln = l, 413,312,2ll, 5/2 and 3ll

dynamics remains obscure,
Barriers have been observed in the profiles for theelectron temperature Z"(r), the ion temperature l(r) andthe density n"(r) [1-8]. In TFTR, so-called Enhanced
reversed shear type I discharges show barriers in n"(r)[]. These are excited when the ratio between the linear
srowth rate of the dominant turbulent mode and thet6t.
@2002 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: mdebaar@jet.uk
318

De Baar M.R. et al., Numerical Simulations of Turbulence in RTP Discharees with Dominant Off-Axis ECH

transport on so-called mesoscales. Mesoscale times, r
are between the Alfv6n time and the resistive timescale(/erruen ( T 1 tr"") and mesoscale length-scales L arebetween the ion Larmor radius and the minor radius of
the tokamak (A < L < a). When subject to externalsources (in our case a localised ECH power source), thesystem of equations gives rise to turbulence, whichyields "corrugations" (ie., regions of relatively highradial gradients; ITB's are a special case ofthese) in the
profiles. An essential feature of CUTIE is that these
comrgations are allowed to feed-back on the turbulence
itself. Thus, in CUTIE, the non-linear co-evolution of
instabilities and corcugated profiles is studied.
RTP t2l was a small tokamak (R = 0.72 rrt, 4 =0.164 m) with dominant Electron Cyclotron heating.
RTP's diagnostics were suited to follow small scale
structures in Z"(r) such as barriers and islands. Moving
the ECH power deposition radius in steps of approx. I
mm from the plasma centre to half radius revealed 7
distinct plateaux of constant central I". The plateaux are
separated by sharp transitions. The transitions areassociated with the loss of transport barriers in the

These cases illustrate the key profiles associatedwith barrier dynamics: the ExB zonal flow and 4. The
theory ofExB sheared decorrelation and suppression of
turbulence is u'ell developed. Zonal flows can be taken

into account numerically and the subtle interaction
between the driving and stabilizing terms of the
turbulence can be investigated [7]. The role of the qprofile in turbtrlence suppression and ITB formation is
not well understood. Beklemishev [8] and Garbet [9]mention that low magnetic shear widens the gap
between two a(Uacent rationals in real space. This effectis most easily observed where the density of rationals is

lowest in the q-profile, viz. in the vicinity of simple
rationals.

In this paper, we use the two fluid code CUTIE to
study the transition from an Ohmic to an RTP type-D
discharge [6]. In particular the role of the evolving qprofile in the self-organization of the plasma has beenstudied. The dynamics of the bootstrap current, theturbulence drive terms, the ExB flow and the dynamo
terms in the induction equation will be discussed. The
effect of MHD activity on turbulent transport has been
investigated. Implicitly, this means that we need tosimulate ITB dynamics for several resistive timescales.
The dynamics of the bootstrap current, the turbulence
drive terms (Vp and Vj), the ExB flow and the dynamo
terms will be followed. The effect of MHD activity onelectron transport channel. The plateaux (and theassociated self similar T" profiles) have been labeled as

A, A', A", B, C, D, and E.
Interestingly, the profiles of type D and E (ECH
deposition approximately at half radius) show strong
steady-state off-axis maxima in 2.. Neither the electronion exchange nor the total radiated power can explain
this phenomenon, which therefore implies that an

turbulent transport will be investigated. We areinterested in the formation of mesoscale structures suchas islands, barriers and other comrgations in the profilesof density and temperature and the way these structures

outward heat pinch must exist for these discharges Il].
Off-axis saw-tooth like events implied that the barriers
are close to, but not at, the simple rationals [12] in the
q-profile. The sharp transitions between the consecutive
plateaux are thought to be associated with the loss of the
4 = 7, 413, 312, 2ll, 5/2, 3ll surfaces respectively. A
phenomenological "q-comb" model of the X@) profile

influence the turbulence.

2. CUTIE and RTP Overview
CUTIE [10] is a quasi-neutral, electromagnetic,
two-fluid coml)uter model to simulate saturated global
tokamak turbulence. The partial differential equations(resulting from conservation laws and Maxwell's

and an outward heat pinch up to the ECH power
deposition radius reproduced the experimental

equations, suitably reduced in tokamak ordering) for n.,
T",Tythe potential vorticity 0, the electrostatic potential

observations [3].
RTP is, in many ways, the ideal candidate for ourf, parallel vector potential ry and the ion parallel
velocity v1 arc solved numerically using a pseudospectral scherne. CUTIE uses a periodic cylinder

CUTIE simulations. RTP has small a/R = 0.23,

justifying a periodic cylinder approximation. Its small

minor radius results in not too small a value for p* = p"/equilibrium model, but field-line bending and curvatureeffects on the turbulence are included. Electron inertia
and trapped particle turbulent dynamics are neglected,
apart from the use of the neo-classical transportcoefficients. In particular, the bootstrap-current jbs, hasbeen included, and will be shown to play a crucial rolein the barrier rlynamics. We use CUTIE to model thea. Medium density RTP plasmas were relatively cold
(upto 2 keV), which limits the resistive time and hence
the 'macro' time scale of our simulations. Most
importantly, RTP's phenomenology is non-trivial andrather well diagnosed [6, ll-13].

3t9

De Baar M.R. et al., Numerical Simulations of Turbulence in RTP Discharges with Dominant Off-Axis ECH

3. CUTIE Ssimulations and Comparisonswith RTP Observations

0.50
o.+o
i 0.5o, " 0.20
0.10
4.5

r7t-t\!"t{f

We now report on a CUTIE study of the transition
from an Ohmic to an RTP type-D discharge [6]. This isa discharge with dominant, off-axis ECH. The plasma
parameters for this case are 1p = 80 kA, Bo = 2.24 T,

neav - 2.7x70te m-3. We start our calculations fromarbitrary (Gaussian) profiles and switch-on the ECH at r
= 0 s. The ratio P66g/Ps2 is typically 350 kWl00 kW.
The power is deposited at p = 0.55 with a localization of
approximately I cm, as indicated by the experiment.
The time step in the simulations is At = 50 ns. The gridconsists of 100 radial gridpoints, 32 poloidal and 16
toroidal harmonics.
In the early (between / = 0 and 48 ms) "switch-on
phase" of the ECH power (not shown here), we observe

!.1

4.O

I

E
O)

--*/\

3.5

L.J

o

c

5.0
2.7

,r'

o3

a direct increase of T"(p = 0.55, t). The Z" profile
broadens on an electron energy confinement time-scale(approx. 4 ms). On the longer resistive time scale(approx. 15 ms), the current profile evolves such that thecentral Ohmic input power is reduced. Consequently
T"(P = 0, t) decreases at a rate associated with thecurrent diffusion time of the system. The central q rises
steadily from 4 = I at the start of the simulation to 4 =2.5 at 48 ms through a series of 'resonances' associatedwith rational values at which the profile acquires verysmall local shear (4' - 0), while remaining monotonic

245

50 55 60 65

70

time [ms]Fig. 1 Time traces for [, n" and g. The full lines indicate

the central values, the dashed line indicate thevalues evaluated al rla = 0.55. Note the in version
in [, the strong correlation with q and the episodic behavior.

throughout.
We consider in some detail the processes during r =48 ms to / = 70 ms when the q-profile changes frombeing monotonic to a type-D 'reversed shear' profile. In
the described time window, the central 4 value rises
from 2.5 to 3.3 and the q-profile becomes nonmonotonic before finally settling into a steady state.
Figure 1(a) shows the simulated traces of central
temperature T"(p = 0,1) (full) and the temperature at theECH deposition radius T"(p = 0.55, r) (dashed) from 48

is of interest and can be compared qualitatively with theexperiment. Figure 1(b) shows the evolution of thecentral density n.(0, t) (full) and the density at the ECH
deposition radius n"(r = 0.55, t) (dashed) over the same
period as in Fig. 1(a). Around / = 55 ms, a change ofslope is observed for the two ne traces, indicating asudden improvement in the central particle transport.
Figure l(c) shows 4(r = 0, t) and 4(r = 0.55, t) (fulland dashed respectively) over the same period as in Fig.
1(a). Up to / = 55ms the q-profile is monotonic featuring
a large region of low shear (q = 3) which is extended out
to the power deposition radius (see also Fig. 3). At r =56 ms, the shear is reversed. The two q-traces show thetendency to be clamped by the simple rationals in the qprofile, viz. 5/2,3/l and even l3l4 = 3.25.

ms to 70 ms. As explained above, due to a redistributionof the Ohmic input power, T"(r = 0, r) decreases. At 48
ms, the Z. profile is almost flat out to the ECH
deposition radius. We observe that pronounced off-axis
maxima develop subsequently.
Unlike the power source, the particle source S is
not strongly localised. We use a simple source profile S,
featuring a maximum at half radius [11]. In addition, adensity feedback system injects particles into the plasma
when z"uu < 2.7xlOte m-3. Given the experimental
uncertainties in the particle source profile, the simulatedInterestingly the evolution of the q-profile isepisodic: periods of sudden changes are alternated byperiods of slow evolution. Apparently, there is not asingle time-scale associated with the evolution of thesystem. The sudden changes in q are also reflected in
the other traces. Note the strone correlation betweenn"-profiles are not expected to be quantitatively
accurate. The evolution of the n.-profile shape however320

De Baar M.R. e, a/., Numerical Simulations of Turbulence in RTP Discharges with Dominant Off-Axis ECH

q(r, t) and the other two parameters. Z"(0, t) shows analmost stepwise decrease, fully correlated with changes

0.6

nq

in q(0, Z"). Also, the sudden improvement in the local
particle confinement is correlated with the sharp

c)
-Y
t-J

transition in q6. This resembles the experiment [6,.
The simulated Z"-profiles (r = 48 (full), 50 (dotted)
and 70 ms (dashed)) are presented in Fig. 2(b). As in theFexperiment, steady state hollow Z" profiles are obtainedin these simulations. The transition from flat Z" tohollow ?" proliles occurs between 48 and 50 ms, after
which the hollow 7" profiles are maintained up to 70
ms, for approximately 5 electron energy confinement

c)

o.4

n1

0.2
0.1

nq
v.v
v'

I

o
_Y
n<
v.v

times rs". The experimental Z"(r) profile for the D-

n-r

,oV.Z

plateau is presented in Fig. 2(a). The match between the

t1
u.I
nr\
U. TJ
4

experimental irnd the simulated profiles is reasonableconsidering many assumptions in the theoretical modeland experimental uncertainties. Radiation losses have

0.0 0 .2 0.4 0.6 0.8 1 .0
Normolised position r/o

not been included in the code and the ion-electron
exchange (calculated to be somewhat smaller than local

Ohmic power) is not sufficient by itself to explain theseFig. 2 Experimental (a) and simulated 7i profiles. The

profiles. If the heat conduction in this system were
purely diffusire, this would have resulted inflatT"(r, t)evolution from flat to inverted is shown for thesimulations.The full line shows the I profile for t= 48 ms, dotted for t= 50 ms and dashed for t=

profiles out to the power deposition radius. Clearly, inthe code an outward advective heat flow is active,
supporting the off-axis maxima in Z"(r).

70 ms.

In Fig. 3 the q-profiles at 50 and 70 ms arepresented. (full and dashed respectively). At 50 ms the

q-profile is m:rrginally below 2.5 with large low sheararegion out to the deposition radius This flattening is dueto the dynam() term. In addition, pronounced elbows)-

form in the q-Jrrofile. Following the broadening of the Z"
profile, current is being expelled from the center. This
induces a slo',v inward movement of the elbows. As
soon as the elbow reaches the center of the system, it
can no longer be supported. Consequently, a fasttransition occurs to a new q-profile with flat inner qregion and an elbow at the next simple rational. The
transition fronr a monotonic q-profile to reversed shear
occurs at 4min approx. 3 with 4" > 3.
The total current density (70 ms) is presented inFig. 4(a). The local minimum of central current density
is clearly bonr out in this graph. The region of shear
reversal is restricted to the very center of the plasma.
The regions of constant current density are associatedwith the regions of low shear in the q-profiles. The
elbows are not reflected in the j-profile at 70 ms.
Elbows are transients, showing up in the dynamic phasefrom 48-55 s. They correspond to regions ofhighj. The
profiles for the bootstrap current j5, (70 ms) are shownin Fig. 4(b). Ihe bootstrap current is localized, and

la

N
!

c

o4

O
Lro

tl

.P

\--- -----a

OJ
!-

q

z

0.0 0.2 0.4 0.6 0.8 1.0
Normolised position r/o
Fig. 3 q-profiles at 50 (full) and 70 ms (dashed). Note theq-profile is clamped at simple rationals.
321

De Baar M.R. et al., Numerical Simulations of Turbulence in RTP Discharses with Dominant Off-Axis ECH

peaks just outside the power deposition radius, where

4i
th
l.V

Vp has a maximum value.
Figure 5 shows the poloidal zonal flow (-cE,lB) in
km/s at I = 70 ms. The zonal flow is self-consistently
calculated and shows pronounced spatial variations
outside the deposition radius. These are indicative ofstrong shearing of the radial electric field.

N

t

4ta

t.z

4

= 0.8
g 0.4
F

4. DiscussionnnWhat is the role of the bootstrap current and the
dynamo term in the evolution of the q profile? Bothcomponents lower the local magnetic shear, but thedifference between these quantities is their degree of
localization. The dynamo term <6yxdB> is extended

\J.\J

N

tr
L

over the whole central region and leads to a flattening of
the current density profile within the power deposition
radius. As the dynamo term extends over the central low

0.2

0.1
o
o
c0

F

shear region in the plasma, it allows for radiallyextended magnetic modes. The dynamo is responsible

0.0

for the clamping of the q-profile at the simple rationals.
The modes drive the dynamo term and the dynamo term

0.0 0.2 0.4 0.6 0.8 1 .0

extends the mode. Sometimes even two low (m, n)modes form within the power deposition radius. In suchcases, the two modes start a competition for maximum
size. The two extended modes form a current sheet
within the thin layer separating the modes.
The bootstrap current leads to a local flattening of
the q-profile at the barrier. In CUTIE, the bootstrap
current plays an essential role in the barrier dynamics.

Normolised position r/o

Fig.4 Total (a) and bootstrap current (b). The total current is inverted in the very center of the discharge.

The bootstrap current peaks just outside thepower deposition radius.

Due to the intense heating, a mode forms in the vicinityof the power deposition radius. Asymmetric fluxes overthe mode give rise to the formation of a high Vp region.

ac\-j5" flattens q and changes the driving terms of
turbulence. The mode is extended, and simultaneously
the density ofmodes is decreased. The barrier and the qprofile are thus coupled via the bootstrap current and the

J

3
o

mode.

Another dynamical process going in parallel is thezonal flow which is rather corrugated and therefore
decorrelates the turbulence (see Fig. 5). In CUTIE two
feed-back loops (associated with 4' and 4' respectively)
operate. It is essential that these loops can interact
synergistically. In fact, as both profiles are correlatedwith Vp, the zonal flow and7i, are well aligned and can
co-evolve. We note that the zonal flows are driven by
the ion pressure gradients as well as by turbulent
Reynolds stresses [10] and therefore are also closely
related to the dynamo term in the induction equation.
What is the mechanism leading to the formation of
the pronounced off-axis maxima in the 7" profiles? In

\F

o
.E

o

-z

o_

-4

0.0 0.2 0.4 0.6 0.8 1 .0
Normolised position r/o

Fig. 5 Zonal flow at 70 ms. Note the corrugations in the
profile outside the ECH deposition radius.

322

De Baar M.R. et a/., Numerical Simulations of Turbulence in RTP Discharses with Dominant Off-Axis ECHThis work done under the Euratom-FOM andthe RTP model, these are sustained by an outward heatflow up to the ECH-power deposition radius. In CUTIE
the outward heat flow naturally correlates with theswitch-on of the off-axis ECH: Directly after switch-on,
the the fluctuation level within the deposition radius isenhanced. The interplay of the EM-and ES-componentof the fluctuations gives rise to the outward heat-flow.
No extra loss term such as radiation was rnodelled.
Clearly, this outbound flow is sufficient for supporting
pronounced off-axis maxima in CUTIE. The 'ears' are

Euratom-UKAEA agreements. It was jointly funded by
Euratom, NWO and the UK department of Trade andindustry.

References

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t3l C. Challis et al.,PPCF 43, 861 (2001).
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comparable to the experimental observations.

5. Conclusions
CUTIE simulations of RTP discharges withdominant off-axis ECH have been carried out. The
electromagnetic, quasi neutral two-fluid model with the
dynamo-term, self-consistent zonal flows and neoclassical effects reproduced many of the non-trivial(1992).

t9l X. Garbet et al., Phys. Plasmas 8, 2793 (2001).
tl0l A. Thyagaraja, PPCF 42, 8255 (2000).
ll ll G.M.D. Hogeweij et al., Phys. Rev. Lett. 76, 632

features observed in the experiments. We mention theepisodic co-evolution of n",T" and 4. The code positions
barriers at rationals in q and naturally generates an(1996).

[2] M.R. de Baar et al., Phys. Rev. Lett.78, 4573

outward heat advection for off-axis ECH discharges.
The outward heat flow is sufficient for supporting offaxis.maxima in the 7", which are quite comparable to(1997).

ll3l G.M.D. Hogeweij er al., Nucl. Fusion 38, 1881
(1998).

the experimental observations.

323