On the extrapolation to ITER of discharges in present tokamaks

0.17 MB·English

by  A. G. Peeters

On the extrapolation to ITER of discharges in pressent tokamaks
A.G. Peeters, C. Angioni, A.C.C. Sips, and the ASDEX Upgrade Team
Max Planck Institut fu¨r Plasmaphysik, Botzmannstrasse 2
D-85748 Garching bei Mu¨nchen, Germany
7
0
An expression for the extrapolated fusion gain G=Pfusion/5Pheat (Pfusion being the total fusion
0
power and Pheat thetotal heating power) of ITER in terms of theconfinement improvement factor
2
(H) and the normalised beta (βN) is derived in this paper. It is shown that an increase in nor-

n malised betacanbeexpectedtohaveanegativeorneutralinfluenceonGdependingonthechosen
a confinementscalinglaw. FiguresofmeritlikeHβN/q925 shouldbeusedwithcare,sincelargevalues
J of this quantity do not guarantee high values of G, and might not be attainable with the heating

6 power installed on ITER.
1
PACSnumbers: 52.25.Fi,52.25.Xz,52.30.Gz,52.35.Qz,52.55.Fa

]
h
p I. INTRODUCTION the H-mode scenarios will be shown The latter scenarios
-
m are an active area of research with many contributions
s Inmanytokamakdevicesdischargescenariosarestud- from different machines, for instance, ASDEX Upgrade
a ied with the aim of improving the performance of fu- [2], DIII-D [3], JET [4], and JT60-U [5]. (Although the
pl ture reactor experiments over the current design val- original reference [2] refers to an internal transport bar-
. ues. Essentiallytwoingredientsenterintheoptimisation: rier it was shown in a later paper [6] that the ion tem-
s
c the energy confinement time and the Magneto-Hydro- perature profiles follow the same scaling as those of the
i Dynamic(MHD) stabilitylimit,representedbyacritical standard H-mode).
s
y pressure. Both energy confinement time and obtainable An improvement in confinement or MHD stability
h pressure are measured in current experiments and then wouldallowtooperatethenextsteptokamakexperiment
p for scaling purposes expressed in dimensionless parame- ITER[7, 8] ata higher energymultiplication factor Q,a
[ ters. The confinement improvement is given by the so- higher fusion power, or a higher bootstrap current frac-

1 called H-factor which measures the energy confinement tion. We define the energy multiplication factor Q and
v time (τE) relative to a scaling law (τE,scaling). The ob- the fusion gain G as
5 tainable volume averaged pressure ( p ) is expressed in
8 the normalised quantity βN h i Q= Pfusion, G= Pfusion (2)
1 P 5P
aux heat
1 aB
H =τ /τ β = β . (1)
0 E E,scaling N h i Ip where Pfusion is the totalfusion power,Paux is the exter-
7 nallyappliedheatingpower,andP isthetotalheating
heat
0 In the equations above β is the volume average of poweroftheplasma,whichinsteadystateisequaltothe
s/ 2µ0p/B2 measured in %,h ai is the minor radius in m, losspowerPloss. Sinceonefifthofthefusionpowerheats
ic B isthemagneticfieldinT,andIp istheplasmacurrent theplasma,Pheat =Pfusion/5+Paux andthereisadirect
s inMA.Forthe extrapolationoneassumesaconstantH- relation between G and Q
y factorandthenusestheconfinementscalingtodetermine
h
theconfinementtimeinanextstepdevice. Furthermore, Q
p G= (3)
it is assumed that the working point of the reactor is at Q+5
:
v the same normalised pressure β . The H-factors de-
N
i scribe our imperfect knowledge of the scaling of confine- The currently proposed next-step experiment ITER [8]
X
ment, i.e. the confinementin currentday experiments in isdesignedtoreachQ=10,andanyrealisticscenarioto
r
a someareasofthe parameterspacescalesdifferentlythan be tested in this experiment should reach a Q value sig-
the developed scaling laws suggest. Of course,the use of nificantly larger than one. Scenarios at sufficiently large
a constant H-factor is a daring approach to correct for Q might be further optimised to reach a higher fusion
this lack of knowledge, but nevertheless can give some power, or to reach a higher bootstrap current fraction.
idea of how confinement could be different in the next The latter optimisation is known as the hybrid scenario.
step experiment. It aims at the extention of the pulse length at similar
Different scenarios for improved performance have performance as the design value. It is important for the
been proposed (see for an overview [1]). These scenar- results presented in this paper to stress that improve-
ios include the internal transport barriers (ITB) as well ments in fusion power or bootstrap fraction can only be
as the different scenarios for the improvement of the H- of interest if the the energy multiplication factor is suf-
mode. The results presented in this paper can in prin- ficiently high. In the opinion of the authors a good rep-
ciple be applied to all scenarios, but as examples only resentation of the extrapolated performance of current
2
discharges towards ITER can only be obtained if one of withoutconsideringascalingofthisdensitywithplasma
the figures of merit is directly connected with Q. Other current)since the density dependence that enterscan be
figures could be used to measure the bootstrap current easilyidentified throughthe coefficientα , which canbe
n
and fusion power. Although Q plays a central role in set to zero to obtain the result for a given fixed density.
the ITER experiment their is no published simple scal- Note that we cannot assume P to be constant, since
loss
ing that allows a direct assessment of the extrapolated discharges at different beta will extrapolate to a differ-
value of current discharges. A simple way to obtain a ent fusion power and, hence, to different plasma heating
rough estimate can be extremely useful in assessing the powers. Since for fixed magnetic field and plasma shape
progress made in this large area of research. Finally, we I q−1, with q being the safety factor at 95% of the
p ∝ 95 95
would like to stress that our extrapolation formula aims plasmaradius,andredefiningtheconstantC toinclude
X
atjudgingthe performanceinITERandis notnecessar- all constant design quantities one obtains
ilyapplicableformoregeneralpurposes. Inparticularwe
use a fixed size and magnetic field. A reactor design can τE =HXτX =HXCXq9−5αI−αnPhαePat (8)
also be optimised through changes in these parameters.
CombiningEq.(8)withEq.(6)oneobtainsanexpression
for G. Indicating all quantities of the ITER standard
II. SCALING OF G scenario with an index S, one can build the ratio
G H qαI+αn PαP p
The fusion power is proportional to = X 95S heat (9)
GS HXS q9α5I+αn PhαePat,SpS
σv
Pfusion ∝ hT2iV p2, (4) Tp2hen(βrec/aqllin)g2 tohnaetaPrrhievaets =at Pfusion/5G and Pfusion ∝
N 95
∝
whereT is the plasmatemperature,V is the plasmavol-
ume,and σv isthecrosssectionforthefusionreactions G H G −αp q 1+2αp+αI+αn β 1+2αp
h i = X 95S N
averagedoverthevelocitydistribution. Overthetemper-
G H (cid:18)G (cid:19) (cid:18) q (cid:19) (cid:18)β (cid:19)
S XS S 95 NS
ature range of interest σv T2, such that the fusion (10)
h i ∝
power scales with the pressure (p) squared. The power Combining the terms containing G we finally derive at
loss from the plasma (Ploss) is measured by the energy the desired expression
confinement time τ , and under stationary conditions it
E
is balanced by the total heating power 1 1+2αp+αI+αn 1+2αp
G HX 1+αp q95S 1+αp βN 1+αp
=
W 3pV GS (cid:18)HXS(cid:19) (cid:18) q95 (cid:19) (cid:18)βNS(cid:19)
P = = =P , (5)
loss heat (11)
τ 2τ
E E
The figuresof merit for different scalingscan now be de-
where W is the stored energy. Combining the Eqs (2), rived directly. Here the explicit expressions are given
(4), (5), and (3) one obtains for four different scaling lows. The most commonly
used IPB98(y,2)[7] indicated by τH, the L-mode scaling
E
Q Pfusion ITER89-P denoted by τL, a newly derived scaling from
G= = nTτ , (6) E
Q+5 5P ∝ E Ref.[10]denotedbyτC,andanelectro-staticgyro-Bohm
heat E
scaling law derived in [11] denoted by τEGB
E
where p = nT was used, and n is the plasma density.
This is of course the famous nTτE product. τEH =0.145HHIp0.93B0.15P−0.69n0.41M0.19R1.39a0.58κ0.78
Toproceedwewritetheconfinementtimeasaproduct (12)
oftheimprovementfactorH overtheconfinementtime
X
of an arbitrary scaling law (τX) τEL =0.048HLIp0.85B0.2Ph−e0a.t5n0.1M0.5R1.2a0.3κ0.5 (13)
τE =HXτX =HXCXIpαI BαBPlαosPsnαnMαM aαaκαk.
(7) τC =0.092H I0.85B0.17P−0.45n0.26M0.11R1.21a0.39κ0.82.
E C p heat
Here CX is a constant, M is the effective mass in AMU, (14)
andκistheplasmaelongation. Theexponentsαarethe
exponentsofthescalinglaw. FortheprojectiontoITER τEGB =0.0865H I0.83B0.07P−0.55n0.49M0.14R1.81a0.30κ0.75.
E EGB p heat
the plasma size (a,R,κ) as well as the magnetic field B (15)
andtheeffective massM areassumedtobegivenbythe In the equations above n is the density in units of 1020
designvalues. For the density a fixedratio ofthe Green- m−3, R is the major radius in m, a is the minor radius
wald limit (n ) [9] will be used, i.e. n n I . In in m, M is the averaged ion mass in AMU, κ = A/πa2
Gr Gr p
∝
ourfinalresultitwillnotbe difficult toobtainthe result is the plasma elongation, A is the area of the poloidal
at constant density (i.e. the design value of the density cross section, and P is in MW. The new scaling laws
heat
3
(τC,τEGB) have been obtained after designed experi-
E E TABLEI:ValuesoftheconstantsofEq.(18)forthedifferent
ments have shown a small and possibly absent β depen-
scalings. Both coefficients assuming ITER operation at fixed
dence of the confinement [11, 12, 13, 14, 15] in contrast
absolute density as well as at fixed Greenwald fraction (n =
with the IPB98(y,2) scaling which, when expressed in
normalised quantities (normalised pressure β nTB−2, 0.85nGr) are given. The latter are indicated by the letters
“Gr”.
∝
normalised Larmor radius ρ∗ √T/Ba, and normalised C X Y Z
collisionality ν∗ naT−2) ha∝s an unfavourable beta [7]
∝ IPB98(y,2) 9.62 3.22 -1.77 -1.23
dependence
IPB98(y,2) Gr 41.15 3.22 -3.19 -1.23
BτE ρX∗ βYν∗Z ρ−∗2.7β−0.9ν∗−0.01. (16) ITER89-P 0.892 2.0 -1.70 0
∝ ∝
For the electro-static gyro-Bohm scaling, therefore, zero ITER89-P Gr 1.11 2.0 -1.90 0
beta dependence as well as τ ρ−3 were imposed to Cordey 3.53 1.82 -1.72 0.18
E ∗
∝
derive Cordey Gr 5.94 1.82 -2.20 0.18
Bτ ρ−3β0ν−0.14. (17) EGB 7.41 2.22 -1.62 -0.22
E ∗ ∗
∝ EGB Gr 24.52 2.22 -2.71 -0.22
Severalpapershavepointedoutthefactthattheabsence
of the beta dependence in τ leads at high normalised
E
pressure to more optimistic projections for ITER com- in the ideal condition if it is assumed that there is no
pared with IPB98(y,2) scaling [10, 15]. power degradation. We note that one can define a fig-
ToobtainthescalingexpressionsforGastandardsce- ure of merit in different ways and that HβN/q925 can
nario must be defined. Here q = 3 and β = 1.8 be thought of as a combination of good confinement,
95S NS
will be used. The H-factors (H ) can be calculated by and high fusion power [16]. Also for the generic scal-
XS
dividing the targetconfinementtime by the confinement ing (αP = 0.5) the bootstrap current in a reactor close
−
timesofthescalingcalculatedusingEqs.(12),(13),(14), to ignition scales as HβN [17]. It is, however, clear that
and (15). In the latter equations the ITER parameters HβN/q925 does not provide a figure of merrit for G. This
(I =15 MA, B =5.3 T, R=6.2 m, κ=1.75, n=1020 point is important because it shows that one must be
p
m−3, a = 2 m, M = 2.5, P = 87 MW, τE = 3.68 s) are careful in using HβN/q925. From the scaling Eq. (19)
used, yielding H = 1., H = 2.2, H = 1.07, and it follows that the difference with the figure of merit
HS LS CS
HEGB,S =0.8. One then directly finds HβN/q925 is ∝ (1.8HH/βN)2. The latter quantity is for
discharges with normal confinement H = 1 but high
H
G=CHXqY βZ. (18) normalised pressure β = 3.6 as small as 0.25. This
95 N N
makes a large difference in G. The above example with
with the values of the constantC and the scaling poten-
H =1, β =3.6, and q =4.2 reaches a value for the
H N 95
tialgivinginTableI. Fromthistable itcanbe seenthat
parameter H β /q2 that suggest that the ITER tar-
β has a strongly negative effect in the IPB98(y,2)scal- H N 95
N get (Q = 10) could be reached, whereas in reality such
ing, and a rather small effect in all the other scalings. It
a discharges would extrapolate to Q = 1. Not only is
is clear from Eq. (11) that G β0 occurs only for the
generic scaling τ P−0.5. ∝ N this value of little interest, it also requires a rather large
E ∝ heatingpower,sincethefusionpowerisfourtimeslarger
For the derivation of a figure of merit one often ap-
and energy multiplication is much smaller. The other
proximates the coefficients of the scaling law. For a bet-
scalinglaws suffer fromthe same problem, althoughit is
ter comparisonwe canmake the similarapproximations,
less dramaticdue to the different exponents ofH as well
i.e. α = 1, α = 0, and α = 2/3 for the IPB98(y,2)
I n p − as βN.
and α = 1/2 for all other scaling laws. This yields
p
−
H3
G=10.8 H , (19) III. DIMENSIONLESS VARIABLES
β q2
N 95
and Some confusioncanarisewhen consideringthe scaling
H2 in terms of the dimensionless parameters β, normalised
G=γX q2X, (20) Larmor radius ρ∗ =ρ/a and normalised collisonality ν∗.
95 The scaling of G in terms of dimensionless parameters
with yields [15]
γL =1.24 γC =5.25 γEGB =9.375. (21) G β(BτE)B ρ−∗1.5+Xβ1.25+Yν∗−0.25+ZR−1.25 (22)
∝ ∝
The figureofmeritHβN/q925 canbe consideredascaling whereX,Y,Z arethe coefficientoftheρ∗,β andν∗ scal-
for G if the exponents in the scaling law for the con- ing of confinement as defined in Eq. (16). In the equa-
finement are αI = 1, αn = 0, and αp = 0, i.e. only tion above the dependence on ρ∗ as well as ν∗ has been
4
H β / q2 even for the electro-static gyro-Bohm scaling. Therefore
H N 95
0.45 wearriveatthe conclusionthatforagivendesignreach-
ingthebetalimitdoesnothelpinincreasingG. Ofcourse
the density scaling is more hidden in our approachsince
0.4
itisconsideredtobeadesignvalue. Onecanderivethat
G increases strongly with density for all scalings.
0.35
Finally,itisnotedherethatforconstantheatingpower
β H and all figures of merit have the same form
N
∝
0.3
Hβ H3 H2
2 / qN950.25 q925N → βNq925 → q925 (24)
β
H
H

ITER
0.2 IV. PERFORMANCE DIAGRAM
0.15 Having derived a simple expression that directly al-
lows to evaluate G, a suggestion is made in this section
for a diagram that should allow for an easy assessment
0.1
of the extrapolated performance of any discharge. From
the discussion above it is clear that a better representa-
0.05
1 1.5 2 2.5 3 tion of the data can be obtained by plotting the scaling
q β / 5.4
N for G = Q/(Q+5). against either the scaling of the fu-
sion power, i.e. β2 /q2 , or the scaling of the bootstrap
N 95
current, i.e. q β . Since both are of importance it is
FIG. 1: (color online) Figure of merit HHβN/q925 of the useful to mark95thNe different q values by different sym-
advanced scenario discharges from ASDEX Upgrade. The 95
bols in whatever plot one chooses. This makes that all
symbols correspond to different values of q: circles (mag-
important parameters can be estimated from the same
neta) q < 3.5, squares (blue) 3.5 < q < 4.0, stars (black)
4.0<q<4.5, diamonds red 4.5<q<5.0, crosses q>5.0 graph.
The IPB98(y,2) scaling
1.2
explicitly added compared with Ref. [15]. Using the
scaling G β1.25+Y it was concluded [15] that for the
∝
IPB98(y,2) scaling (X = 0.9) there is no large bene-
− Q = ∞
fit of going to high β since Q increases only moderately 1
with β (β0.35), whereas for the energy confinement scal-
ings that have no beta dependence it would be largely
advantageous to go to the β limit since G β1.25. This 0.8
∝
conclusionseemsindisagreementwiththeresultsderived
in this paper, which rather point at a decreasing G with 2q95 ITER Q = 10
N
β fortheIPB98(y,2)scalingandaGindependentofβ β
foNr the other scaling laws. Eq. (22) is the correctdimenN- 3H / H0.6
8 Q = 5
sionlessexpression,butitmustbe notedthatthescaling 0.
1
withbetaG β1.25+Y holdsonlyatconstantnormalised
Larmorradiu∝s(ρ∗)aswellascollisionality(ν∗). The dif- 0.4 Q = 3
ference with the results in this paper is that the results
are evaluatedfor a fixed machine size, density, and mag-
netic field. With these assumptions it is not possible to 0.2 Q = 1
change β independently of ρ∗ and ν∗. At fixed density
theβ scalingisessentiallyatemperaturescaling,leading
to changes in the normalised Larmor radius as well as
0
thecollisionality. Usingthescalingsofρ∗ andν∗ onecan 1 1.5 q β 2 / 5.4 2.5 3
N
derive from Eqs. (16), (17), and (22)
G T−1.23 G T−0.22. (23)
H EGB
∝ ∝ FIG.2: Fig. 2(coloronline)Scalingfortheadvancedscenario
These scalings are consistent with the diagrams of Ref. discharges based on the IPB98(y,2) scaling. Symbols reflect
[15] where G can be seen to decrease with increasing T theq95 valuesas in figure 1.
5
Figure 1 shows a dataset of advanced scenario dis- attainable beta. Nevertheless for the representation of
charges from ASDEX Upgrade in the representation us- The IPB98(y,2) scaling
ing H β /q2 versus q β . In this figure the different 1.2
H N 95 95 N
q valuesareindicatedwithwithdifferentsymbols(and
95
colours in the online version). In the representation us-
Q = ∞
ing Hβ /q2 even the points at highest q β reach the 1
N 95 95 N
ITER target.
Figure2showsthescalingderivedfromEq.(19). This
canbedirectlycomparedwithFig.1sincethesamescal- 0.8
ing law is used. The obtained picture is different in the
sense that the highest q95βN values no longer reach the 2q95 ITER Q = 10
ITER target for Q. These discharges have only moder- 2 / HC0.6
aretleatciovnefilynesmmeanllt(iHmpr/oβvem)2e.ntIsnatnhdehdiigahgrβaNmsleaQdicnagntoexa- 5.24* Q = 5
H N
ceed infinity, which is obviously unphysical. For those
0.4 Q = 3
discharges for which Q > , the temperature will rise
∞
until the fusion cross section no longer scales quadrat-
ically with T, violating the original assumptions in the
derivation,andleadingtoasmallerincreaseofthefusion 0.2 Q = 1
12 4 8
power with T. This stabilises the solution and leads to
Q= .
∞
In the diagrams presented so far the external heating 0
0 0.5 1 1.5 2 2.5 3
power is still implicit. A better insight of how much ex- 2.77(β /q )2
N 95
ternalheatingpowerisneededtorunacertaindischarge
under reactor conditions can be obtained from the dia-
gram that has the scaling of the fusion power Pfusion FIG. 3: (color online) Extrapolated values of Q/(Q+ 5)
(βN/q95)2 on the x-axis. Because Paux = Pfusion/Q on∝e as a function of the normalised fusion power 2.77(βN/q95)2.
obtains Symbols reflect the q95 values as in figure 1. The dotted
linesindicatetheamountofexternalheating(Paux)necessary
1 β 2 to run the discharge in ITER. From left to right the curves
N
P . (25)
aux correspond to power levels 1, 2, 4, and 8 times the nominal
∝ Q(cid:18)q (cid:19)
95
ITER value
Forafixedauxiliaryheatingpowertherelationabovede-
termines a curvein the G=Q/(Q+5)versus(β /q )2
N 95
large data sets these diagrams are certainly useful. Also
diagram. Figure 3 shows the same data as the dia-
theygiveanideaofwhatparametersareimportantwhen
grams before, plotting G = Q/(Q+5) versus the fusion
powernormalisedtotheITERvalue2.77(β /q )2. The developing scenarios.
N 95
dashedlinesinthisdiagramarethedifferentvaluesofthe
auxiliaryheating power. Fromleft to right1, 2,4, and 8
V. CONCLUSIONS
times the ITER design value. This diagram shows that
high pressure discharges at low Q values would require
a large amount of installed heating power to be run. It In this paper a scaling for tokamak discharges is de-
is this diagram that carries the largest amount of infor- rivedthatdirectlymeasuresthefusiongainG=Q/(Q+
mation and we propose it to be used when representing 5), and which are consistent with the underlying scaling
larger datasets of advanced scenario discharges. laws. It is shown that β does not have a positive in-
N
Thediagramspresentedinthispaperarefarfromper- fluence on G, although it does of course extrapolate to a
fect, with severaleffects not properlyaccountedfor: Ra- larger fusion power. Care is to be taken with figures of
diated power due to Bremsstrahlung as well as dilution merit like Hβ /q2 . Although this figure of merit does
N 95
of the fuel are not properly scaled. The approach with measureacombinationofgoodconfinementandhighfu-
constantH-factorandβ isalwaysdaringforanextrap- sionpower,theITERtargetvalueofsuchaquantitydoes
N
olation. If better MHD stability is, for instance, reached not automatically imply a discharge with a sufficiently
throughcurrent profile shaping, then one should investi- high Q, and might not be attainable with the limited
gateifsuchashapingisextrapolatabletoreactorparam- heating power that will be installed. A proposalis made
eters. Also, although β is a good scaling quantity for for a graphical representation in which both the extrap-
N
ideal MHD instabilities, it does not provide a very good olated Q as well as the fusion power or the bootstrap
scaling for the NTM, which is often found to limit the fraction can be directly assessed.
6
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