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1
“Profile consistency” features in strongly shaped ohmic tokamak plasmas
H. Weisen, R. Behn, I. Furno, J.-M. Moret, O. Sauter and the TCV team
Centre de Recherches en Physique des Plasmas
Association EURATOM - Confédération Suisse
École Polytechnique Fédérale de Lausanne
CH-1015 Lausanne, Switzerland
Abstract. A large variety of shaped tokamak discharges have been created in the
Tokamak à Configuration Variable (TCV) with main parameters BT<1.5T, R0=0.88m,
a<0.25m, Ip≤1MA (Hofmann F., Lister J.B., Anton M. et al., Plasma Phys. Contr. Fusion 36
(1994) B277). They include limited and diverted discharges with elongations up to 2.58 and
triangularities between -0.7 and 1. Over the entire range of quasi-stationary sawtoothing
ohmic discharges investigated we observe that the widths of plasma pressure profiles,
and the inversion radii scale as the widths of the current profiles, , where
⟨ ⟩ refers to an average over the volume or, practically equivalently, the cross sectional plasma
area. These “profile consistency” features are in apparent agreement with theoretical consider-
ations and can be viewed as an extension of previous studies in circular discharges. The obser-
vations identify the parameter as the appropriate replacement for qa as scaling
parameter for arbitrary cross-sectional shapes. Using a simple current profile model with
0.8≤q0≤1 for the axial safety factor, we show that for sawtoothing ohmic discharges such
observations can be accounted for by the effects of sawtooth activity.
p⟨ ⟩ p 0( )⁄ j⟨ ⟩ j 0( )⁄
j⟨ ⟩ q0j 0( )⁄
2
Experimental observations and theoretical models supporting the idea that electron tem-
perature and/or the pressure and/or current density in tokamaks preferentially assume certain
privileged profile shapes have been reported for more than two decades [1-6]. For instance
“natural” current and pressure profiles from minimum energy models [1,2] are given in
circular geometry by j/j(0)=p/p(0)=(1+(qa/q0-1)ρ2)-2 and have profile widths equal to q0/qa.
Stationary entropy models [3] also lead to the observable prediction j/j(0)=p/p(0). Current
profiles as given by the above expression are also obtained from a model of filamented current
structure, although in that case there is no preferred pressure profile [4]. Arunasalam et al. [5]
have proposed a practical working definition of the profile consistency principle motivated by
tearing mode stability considerations and observations on the Tokamak Fusion Test Reactor
(TFTR) device [7]. It postulates the existence of unique, natural profile shapes for j(r) and
because of ohmic relaxation, for the shape of Te(r) or . The consequences of the postu-
late, as given in [5], are 1) that the normalized sawtooth inversion radius, ρinv= rinv/a, which is
related to the q=1 surface, and 2) the broadness of the temperature profile ⟨Te⟩/Te(0) are device
independent and are functions of 1/qa only. The same authors propose a third consequence
expressed as , which merely combines the postulation of
universal current profiles characterized by and the assumption of Spitzer
conductivity, .
Relations between profile widths, which appear to be consistent with the above theoret-
ical models, are indeed observed in all ohmic L-modes in TCV (Tokamak à Configuration
Variable, [8]), independently of plasma shape. This was borne out in a study of the effect of
plasma shaping on confinement of a set of some 300 discharges reported elsewhere [9]. A few
examples reflecting the variety of plasma shapes in the dataset is presented in Fig. 1. The
dataset comprises plasma currents in the range 0.1-1 MA and average densities ranging from
1.2.1019 to 1.2.1020m-3.
The observation, shown in Fig. 2, is expressed as ,
where the brackets indicate a volume average for the electron pressure and practically equiva-
lently a cross sectional area average for j. The proportionality factor cpe is typically in the
range 0.7-1. The electron temperature and density measurements were obtained using a
multipoint Thomson scattering system with 10-25 measurement locations in the plasma
depending on plasma size. Each data sample is an average over 3-5 individual measurements
obtained at 50 ms intervals (dictated by the laser pulse frequency) during the stationary phase
Te3 2⁄
r( )
Te03 2⁄
IpR0 cσa2Vloop(⁄ F 1 qa⁄( ) )∝
j⟨ ⟩ j0⁄ F 1 qa⁄( )=
σSpit cσTe3 2⁄
=
pe⟨ ⟩ pe 0( )⁄ cpe j⟨ ⟩ q0j0( )⁄=
3
(“flat top”) of the discharges. The times at which these measurements were made were not
synchronized to the sawtooth cycle.
The width of the current profile is known up to a factor q0 from the relation
(1),
where (2) and (3),
is the dimensionless average current density, j0 being the current density on axis, B0 the
toroidal field on axis, B0vac the corresponding vacuum field, R0 the major radius of the
magnetic axis and κ0 the central elongation. The expression on the right hand side of Eq.(1)
reduces to 1/qa in circular discharges when equilibrium corrections to the axial field (due to
paramagnetism and diamagnetism) are neglected and hence appears as a natural candidate
scaling parameter for “profile consistency” in arbitrarily shaped plasmas. We note that this
parameter can be evaluated without knowledge of the still somewhat controversial axial safety
factor. The core elongation and axial field used to evaluate this expression for Fig.2 were
obtained from the equilibrium reconstruction [10] and, for the former, confirmed by x-ray
tomography. For the purposes of predictive scaling they are found to be accurately given (at
least in TCV) by the following scaling expressions relating to global quantities:
κ0 ≈ 1+0.5.(κ95-1)+0.2.(κ95-1)2 (4) and (5),
where κ95 is the elongation at 95% of the poloidal flux.
Although Fig.2 only shows electron pressure, it is unlikely that the relation, which is
observed at all densities, is specific to electrons only. At average densities above about
5.1019m-3, representing 50% of the samples in the database, electron-ion energy equipartition
is such that central electron and ion temperatures cannot differ by more than 20% even if half
of the ohmic input power density is assumed to be transferred from the electrons to the ions.
TCV is equipped with a 200-channel X-ray tomography system which was used to
measure the sawtooth inversion radius in these discharges [11,12]. For shaped plasmas we
define the effective inversion radius as where Ap is the plasma cross
sectional area and Ainv is the area enclosed within the flux surface at which sawteeth invert. In
j⟨ ⟩q0j0---------
j⟨ ⟩ ∗ B0⋅ vac
κ0 1 κ⁄ 0+( ) B0⋅----------------------------------------=
j0
κ0 1 κ0⁄+( )B0
µ0R0q0------------------------------------= j⟨ ⟩ ∗
µ0R0 j⟨ ⟩B0vac
--------------------=
B0 B0vac⁄ 1 0.054 1 βp–( ) j⟨ ⟩ ∗ 2⋅+≈
ρinv Ainv Ap⁄=
4
Fig.3 we see that the inversion radius scales indeed as ρinv≈<j>/q0j0. This parameter provides
a better fit than inverse safety factor based expressions such as 1/q95. In particular the correc-
tion term for central elongation becomes important for κ0>1.6, whereas the equilibrium field
correction is important for the highest normalized current densities which are obtained at
extreme elongation. In the discharges with κa>2.2 of this dataset the paramagnetic increase of
the axial toroidal field is in the range 10-14%.
The increase of the inversion radius, which is linked to the q=1 surface, with average
current density can be understood from the limitation of the core current density by magneto-
hydrodynamic (MHD) stability. Although the exact value of the central safety factor is still
somewhat controversial, it is generally accepted that in ohmic sawtoothing discharges q0 is in
the range 0.8-1 [13,14]. We propose a model current profile given outside the sawtoothing
zone (ρ>ρ1) as jout=Eσneo+jboot where is the externally induced electric
field, σneo is the neoclassical conductivity and jboot is the bootstrap current, which in these
ohmic plasmas only contributes a few percent to the total current. Inside the q=1 surface a pre-
scribed safety factor profile of the form is assumed, where ρ1
is the normalized radius of the q=1 surface. For ρ≤ρ1 the corresponding current profile is
. (6)
Because of uncertainties in the measurement of Zeff we don’t use the absolute value of
jout. Instead Eσneo is renormalized such that jout(ρ1)=jin(ρ1), thereby matching the current
density profiles in the core and in the confinement zone. For any choice of the free parameter
q0, we then determine ρ1 from the condition . For these calculations the relevant
neoclassical expressions were taken from Sauter et al. [15] and evaluated using the experimen-
tal data. The trapped electron fraction was obtained following a formulation by Lin-Liu and
Miller [16]. Examples of the resulting profiles are shown in Fig.4 for κa=1.2 and 2.4. The
radial coordinate is defined as , where A is the cross sectional area of the flux
surface under consideration and Ap is the total cross sectional area of the plasma. The current
densities in Fig.4 are normalized to the axial current density for a circular discharge with q0=1
and B0=B0vac. The figure also illustrates the beneficial effects of elongation on the current
carrying capacity of the discharge. The q=1 radius from the model is consistent with the first
consequence [5] of “profile consistency” ρ1=F1(<j>⁄q0j0)≈<j>⁄q0j0. The scaling of ρ1, shown
in Fig.5 for q0=0.9, is seen to be identical to that of ρinv in Fig.3. The assumption q0=0.9
E Vloop 2πR0( )⁄=
q ρ( ) q0 1 q0–( ) ρ2 ρ12⁄( )+=
jin ρ( ) j0q0
q ρ( )-----------
B ρ( )B0
------------κ ρ( ) 1 κ ρ( )⁄+( )
κ0 1 κ0⁄+----------------------------------------- 1
ρ21 q0–( )⋅
ρ12
q ρ( )⋅-----------------------------–
⋅ ⋅ ⋅ ⋅=
j Ad∫ Ip=
ρ A Ap⁄=
5
provides the best overall match of ρ1 and ρinv, although good agreement is obtained for the
whole range 0.8≤q0≤1.
The effect of sawtoothing is to flatten, or at least limit the peaking of, the profiles of
temperature and density in the core. This suggests that the scalings of the widths of the plasma
profiles other than the current profile are to a large extend also a result of sawtoothing and
depend on the position of the q=1 surface. Fig. 6 shows the measured widths of ne, Te and pe
as function of ρ1. These widths are close to those expected if the profiles were merely trape-
zoidal (broken line) or trapezoidal squared (continuous line) with a flat portion up to ρ1. The
choice of simple flat core model pressure profiles, although not fully realistic, is motivated
both by the measurements and the fact that with ohmic heating alone the core power density is
too low for significant reheating and profile peaking during the sawtooth rise phase. The
profile shapes outside the inversion radius depend on transport. This however only has a
modest effect on the width <p>/p0 as seen when comparing the two model pressure profiles in
fig.6. It appears that profile widths, expressed as <x>/x0 are not very sensitive to details of the
profiles and hence to transport in the confinement zone (ρ>ρ1). We conclude that profile
relations such as <j>⁄j0≈<p>⁄p0 in ohmic plasmas are a consequence of sawtoothing and result
from the limitation of the central current density and from the flattening of the core profiles of
temperature and density.
In summary, the equation for the width of the current profile, Eq.(1), suggests the
scaling parameter <j>/(q0j0) for profile widths in arbitrarily shaped plasmas as a replacement
for 1/qa. Figs 2. and 3. show that this is indeed the appropriate choice for <pe>/pe0 and ρinv.
The proposed model current profile, for which q0 is in the generally accepted range for saw-
toothing discharges (0.8≤q0≤1) and which is determined by neoclassical conductivity and
bootstrap current in the confinement zone leads to a scaling for ρ1 which is identical to the one
observed for ρinv, as shown in Fig.5. Hence we conclude that the limitation of core current
density by the sawtooth instability is responsible for the scaling of ρ1 and ρinv. Moreover the
flattening of the core profiles of other quantities, such as pressure, inside ρ1 leads to profile
widths with values close to the one of ρ1 (and hence of <j>/j0) while being fairly insensitive to
transport and profile shapes in the confinement zone (Fig.6).
It appears as a coincidence that the effect of sawteeth is to mimic the profile consistency
relations in the sense of Aranusalam et al’s working definition [5] and predicted by the above
mentioned theories which are based on very different physics principles and do not include the
effects of internal MHD activity such as sawteeth. Therefore, despite the apparent agreement,
6
the observations reported here should not be considered as supportive of these theories. They
do not however exclude that the processes implied by those theories may also be at work. The
existence of profile-shaping processes other than sawtoothing is suggested for instance by
observations of Te-profile resilience in the confinement zone of sawtoothing discharges [17].
These processes would be expected to determine profile relationships throughout the cross
section of sawtooth-free ohmic discharges, e.g. at sufficiently low <j>* or high q95. To be fair
the above mentioned theoretical predictions should be compared to observations on plasmas
which are not dominated by sawteeth or other forms of strong MHD activity. Unfortunately in
TCV, as in many other devices, ohmic operation with <j>* sufficiently low for sawteeth to
disappear often leads to strong tearing mode activity and an increased likelihood of disrup-
tions, which is why this part of the operational domain remains largely unexplored. Theories
aimed at explaining profile relationships and their scaling therefore should take the effects of
sawtoothing into account and/or make predictions on observable quantities which are not
directly affected by sawtooth activity.
Acknowledgement: This work was partly supported by the Swiss National Funds for Scien-
tific Research.
References:
[1] Biskamp D., Comments Plasma Phys. Control. Fusion 10 (1986) 165 [2] Kadomtsev B., Proc. Int. Conference on Plasma Physics, Kiev, 6-12.4 (1987) 1273[3] Minardi E., Physics Letters A 240 (1998) 70 [5] Arunasalam V. et al., Nuclear Fusion 30 (1990) 2111[6] Coppi B., Comments Plasma Phys. Controlled Fusion 5 (1980) 261[4] Taylor J.B. Phys. Fluids B 5 (1993) 4378[7] Young K.M.,Bell M., Blanchard W.R et al., Plasma Phys. Contr. Fusion 26 (1984) 11[8] Hofmann F., Lister J.B., Anton M. et al., Plasma Phys. Contr. Fusion 36 (1994) B277[9] Weisen H., Alberti S., Barry S. et al., Plasma Physics Contr. Fusion 39 (1997) B135[10] Hofmann F. and Tonetti G., Nuclear Fusion 28 (1988) 1871[11] Anton M., Weisen H., M.J. Dutch, Plasma Phys. Control. Fusion 38 (1996) 1849[12] Furno I., Weisen H., Moret J.M et al., 24th EPS on Controlled Fusion Plasma Physics,
ECA 21A (1997), part II, 545. European Physical Society, Petit-Lancy, Switzerland.[13] Levinton F.M., Zakharov L., Batha S.H. et al., Phys. Rev. Lett 72 (1994) 2895[14] Rice, et al., Fusion Engineering and Design, 34-35 (1997) 135[15] Sauter O., Lin-Liu Y.R., Hinton F.L. and Vaclavik J., Theory of Fusion Plasmas. Proceed-ings of the Joint Varenna-Lausanne Int. Workshop, Varenna, Italy, 1994. Edited by E. Sindoni,F. Troyon and J. Vaclavik. Published for the Società Italiana di Fisica by Editrice Compostori,Bologna, Italy (1994), page 337. ISBN 88-7794-066-2.[16] Lin-Liu Y.R and Miller R.L., Physics of Plasmas 2 (1995) 1666[17] Fredrickson E.D., McGuire K.M., Goldston R.J. et al., Nuclear Fusion 27, (1987) 1897
7
Figures
12819, 0.55s
175 kA
12016, 0.7s
450 kA
12868, 0.72s
725 kA
9452, 0.85s
425 kA
Fig. 1. Examples of plasma cross
sections in the dataset.
0.1 0.2 0.3 0.4 0.5 0.6
0.1
0.2
0.3
0.4
0.5
0.6
<j>/(q0j0)
<p e
>/p
e(0)
1.0 <κa<1.25 1.25<κa<1.5 1.5 <κa<1.75 1.75<κa<2.0 2.0< κa<2.25 2.25<κa<2.6
Fig. 2. Widths of electron pressure
profiles versus widths of current
profiles (up to factor q0) in ohmic
discharges. Symbols refer to
discharge elongation at the last
closed flux surface.
0.1 0.2 0.3 0.4 0.5 0.6
0.1
0.2
0.3
0.4
0.5
0.6
ρin
v fr
om to
mog
raph
y
<j>/(q0j0)
1.0 <κa<1.25 1.25<κa<1.5 1.5 <κa<1.75 1.75<κa<2.0 2.0< κa<2.25 2.25<κa<2.6
Fig. 3. Scaling of inversion
radius from soft-x ray tomography
in ohmic discharges. Symbols refer
to discharge elongation.
8
0 0.5 10
0.5
1
1.5
j(ρ)
µ0
R0/
2B0
vac
effective minor radius ρ
#9888: <j>*=0.86 κa=1.21, κ0=1.16
#11877: <j>*=1.7, κa=2.42 κ0=2.14
q0=1 q0=0.9 q0=0.8
Fig. 4. Model current profiles for
two discharges with elongations
equal to 2.42 (top curves) and 1.21
(lower curves) and same qa=2.9 for
three choices of q0. The arrows
indicate the positions of the q=1
surfaces corresponding to q0.
0.1 0.2 0.3 0.4 0.5 0.6
0.1
0.2
0.3
0.4
0.5
0.6
<j>/(q0j0)
ρ(q
=1)
1.0 <κa<1.25 1.25<κa<1.5 1.5 <κa<1.75 1.75<κa<2.0 2.0< κa<2.25 2.25<κa<2.6
Fig. 5 Scaling of effective radii
of q=1 surface from current profile
model for q0=0.9 in ohmic
discharges. Symbols refer to
discharge elongation.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ρ(q=1)
<p e
>/p
e0,
<T e
>/T
e0, <
n e>
/ne
0
<ne>/ne0 <Te>/Te0
<pe>/pe0
−− trapez. − trapez.2
Fig 6 Widths of electron density,
electron temperature and pressure
profiles versus effective q=1 radius
from model, assuming q0=0.9. The
broken and continuous lines refer to
trapezoidal and trapezoidal squared
model profiles with a flat part
extending to ρ(q=1).