Passive Thermal Stability in Ignited Tokamaks
by Enhanced Transport and Radial Motion
L. Bromberg and D. R. Cohn
Revised:
May 1980
MIT Plasma Fusion Center Report PFC/RR-79-19-R
ETF Design Center Report ETF-R-8-PS-003
Passive Thermal Stability in Ignited Tokamaks
by Enhanced Transport and Radial Motion*
L Bromberg and D. R. Cohn
A.L T. Plasma FusionCenier
Cambridge, Massachusetts 02139
and
-
fETF Design Cenier
Oak Ridge, Tn 37830
M.I.T. Plasma Fusion Center Report PFC/RR-79-19-R
Revised: May1980
* Supported by U.S. D.o.E. Contract EG-77-S-02-4183.A002
Abstract
A 0-dimensional model with parabolic profiles is employed to assess the use of the combination of enhanced thermal conduction loss and radial motion for passive thermal stability control of an ignited tokamak
plasma. The increases in ignition requirements needed for the achievement of passive stability are determined.
Both the independent thermal stabilization effects of radial motion and enhanced transport and the additional
stabilizing effect of the dependence of the enhanced transport upon radial motion are considered. The enhanced thermal conduction is modeled by the addition of an ion energy loss channel to a plasma described
by neoclassical ion energy transport; this additional ion loss channel is represented by an energy confinement
time scaling ri,ag ~ T;-* where T is the ion temperature and e is a variable parameter. Calculations are
made for tokamak plasmas where the electron energy confinement time is described by the empirical scaling
law re ~ na 2 . The combination of the independent effects of radial motion and an enhanced transport loss
with ri,dd ~ 77/2, as might be the case for increased ion thermal conductivity due to ripple trapping, can
be used to obtain thermal stability for central ion temperatures greater than 15 keV with very small (< 10%)
increases in the value of nr, required for ignition.
I
I. Introduction
Passive thermal stability control in tokamak plasmas by operation at high ion temperature [1,2), by the
addition of ripple-induced thermal conduction losses [3) and by the onset of balloning mode instabilities [4)
has been suggested. Furthermore, it has been found that the effects of plasma motion increase the stability
of the plasma against thermal fluctuations [5,6). This paper examines the effect on ignition requirements of
passive thermal stability control by the use of both enhanced ion energy transport and radial motion. The
enhanced thermal conduction is modeled by the addition of an ion energy loss channel to a plasma described
by neoclassical ion energy transport; this additional loss channel is represented by an energy confinement time
scaling Ti,add ~ T7i where e is a variable parameter.
2
II. Ignition and Stability requirements
Ignition (i.e. thermal equilibrium) is defined in terms of a four dimensional requirement on Ti, r/r,, nme
and T, without reference to specific electron or ion energy confinement laws [2]. Here n is the central plasma
density, re and ri are the electron and ion energy confinement times, and T, and Ti are the central electron and
ion temperatures. A zero-dimensional spatial model is used; the density and temperature profiles are assumed
to be parabolic. No fast alpha losses are included in the analysis. No impurity effects have been included. The
values of nr, and T required for ignition are determined by specifying Ti and i/re
The energy-balance equations for the ions and electrons are
1 dT
2
2-n- dt
=ji- '
1 dT,
2 dt
1 nTi
3
2 -ri - 5
2
'
mT
2 re
3
5
(Ti - T,)
2 TdV-
-n-
3 V dt
rj
n T-r-
T)
je
2 T dV
=
3 V dt
W(2)
Here n 2 W.,i and n 2 Wa,e are the ion and the electron heating rates by the alpha particles. V is the plasma
volume. ri is the electron-ion equilibration rate. The last term in (1) and (2) represents the heating or cooling
rates due to changes in plasma volume. At ignition n dTi/dt = n dT./dt = 0 and dV/dt = 0.
In order to illustrate the approach and to make some projections for future machines, we have assumed
that the electron energy confinement is given by the ALCATOR empirical scaling law r, ~ na2, where a
is the plasma minor radius [7]. It is also assumed that in the absence of additional transport introduced to
achieve thermal stability the ion energy confinement time is given by neoclassical transport [8]. Based upon
these assumptions, ri/ft > 1 for ignited tokamak plasmas. For ri/
> I the main ion loss is through ion-
electron coupling; this condition determines the minimum ignition requirements ((nre)ign,min and #lign,min)
at a given ion temperature [2].
For a tokamak plasma, conservation of toroidal flux implies that the volume varies as V ~ Ra2 ~ R 2 ,
where R and a are the major and minor radii of the plasma respectively [9]. (This relation is not appropiate Ibr
very low aspect ratios and/or high plasma pressures [10]). The last term in equation (1) can be rewritten as
2TidV
3V dT
4 Teff dR
T i (Te 1 dT
3= R dTCI TffC
i dt
OTeiidTe
aT, di
3
4
3
Ti (OTcff dTi
T eAOTi
dt +
8Tef55dT
(3)
)2edt
A similar transformation can be used for the last term of (2). Y1, = Tf11/R dR/dTff where Te, 1 is the spatially
averaged total plasma pressure divided by the plasma density. 17, represents the elasticity of the plasma motion
and is determined by the AtHD properties of the system [6].
Thermal stability requirements are calculated following the approach of reference 2. The system of equations (1) and (2) is augmented with a set of equations for the alpha particles and are linearized around the
ignition point. The stability properties of the linearized system is determined using an eigenvalue analysis. It is
assumed that the total number of particles in the plasma remains constant.
In the presence of the additional losses, r. ~ ri,"d
-
T-. For given values of T and e the thermal
stability analysis provides a maximum value of ri/r, at which neutral stability is obtained. This reduction in the
ratio of ri/re relative to the case where there was no additional losses (Le where r/r, > 1) leads to an increase
in the value of nr, required for ignition. The value of nr, necessary for ignition when r7/re is determined by
the requirement of neutral stability is represented by (nr,)i,,ta.
Figure 1 shows the minimum value of nr,required for ignition, (nr.)n,,mi,, and the values required for
ignition and stability (nrc)i,,ta. The curves shown in Figure 1 are drawn for 17, = 0 (Le. the plasrma is
not allowed to move radially). Results are shown for e =-i3/2 and e = 7/2 (corresponding to ripple plateau
diffusion [11] and ripple trapping diffusion [12], respectively). It should be noted that at T ~ 50 keV,
(nr),,n,mgn = (n7,)i,,tb
At this temperature the plasma is naturally stable without the anomalous loss [1,2].
Figure 2 shows the values of (nre)9n,mn and (nr,)i,&,"
for 7, = 0.3. This value of 17,. is typical of
ETF/INTOR designs [6,13,14] and of ignition test reactors [6,15]. The increase in nr required for stability .is
significantly smaller than in Figure 1. At Tj
-
25 keV the plasma is stable without the anomalous ion loss.
Figures 3 and 4 show calculations for paB2 required for ignition for ALCATOR empirical scaling for the
electron energy confinement time [7]. In this case
nr, ~+
/n
2
T, + Tj + 11/n
where f is the average toroidal beta, B is the toroidal magnetic field and T represents the contribution of the
suprathermal alpha particles to the plasma pressure. Figure 3 has-been drawn for 17, = 0 while Figure 4 has
4
been drawn for 1,. = 0.3.
As mentioned above, Figures 1-4 show the values of nre and j required for the achievement of neutral
stability (that is, where perturbations of the linearized system do not grow or decay). A further increase in the
enhanced transport is needed to make the system truly stable, which leads to somewhat larger values of nr,
and ( than shown in Figures 1-4. For the simple case where T = Ti = T, neutral stability occurs when
P. = P,
and dP0 /dT = dP.,,/dT where P0 is the alpha power and P.. is the total loss power. In going
from neutral stability to true stability, P, = P,, at two different values of T which are in the vicinity of the
one value of T at which neutral stability occured; the equilibrium point with the highest temperature is stable
with dP/dT < dP,../dT.
The effect of the increased thermal conductivity on the temperature and density profiles has been
neglected in the zero-dimensional calculation. If the effect is to fiatten the pressure profiles, then the increases in
nre and P which are shown in Figures 1-4 are an underestimate.
It has been assumed that the additional ion thermal transport which is imposed to obtain stability does not
affect the suprathermal alpha particles. Significant loss of suprathermal alpha particles will further increase the
ignition requirements needed for passive thermal stability ontrol. If the anomalous ion transport is due to nonaxisymmetry of the toroidal field (ie ripple), then some of the alpha particles will be lost. However, in the
case of a tokamak, the effect of the ripple would be to deplete the trapped particle region of phase space. The
passing particles would not be significantly affected by the ripple until most of their energy has been lost [16].
Since the lost fraction of the alpha particle energy is approximately constant, the alpha particle loss has neither
a stabilizing nor destabilizing effect on the thermal properties of the plasma. However, the curves in Figures 1
and 2 are shifted upwards by approximately f, where f is the lost fraction of alpha particles. Figures 3 and 4 are
shifted upwards by
v7.
In addition to the contributrion of the separate effects of enhanced transport and radial motion, radial motion can change the enhanced transport. The effect of radial motion on the enhanced trasport can be modelled
by introducing a parameter
R dn7,.dd
nri,d.d dR
-y; indicates the variation of nri,j.w with major radius; in Figures 1-4 it was assumed that yj = 0. The
linearized system of equations used to calculate the stability properties can be modified to include the effect of
5
nonzero values of -yi. It is found, for example, that for e = 3/2 and -Y= -10
(which might be characteristic
of the spatial dependence of ripple plateau diffusion in a tokamak reactor [3,11]) the increases in nr, required
for stability in Figure 2 are reduced by a factor of - 5 relative to the increases required in the case -i = 0.
6
III. Conclusion
The increase in ignition requirements necessary for passive thermal stability control by enhanced transport
and radial motion has been determined. It is assumed that the enhanced tranport occurs through the addition
of an energy loss channel described by i ~ T7- to a plasma described by empirical electron and neoclassical
ion energy confinement times. As e decreases, the values of (nre)in,tab and An,,t
and T
-
20 keV, mre increases by
-
100% relative to (nr,)ign,min when e
-
increase. For t~r, = 0
3/2 and by
-
30% when
e ~ 7/2; the corresponding increases infl are - 40% for e = 3/2 and - 15% for e = 7/2. For fl, = 0.3,
the increases in nmr and P are much smaller. It appears that macroscopic thermal stability in tokamak plasmas
might be obtained over a wide temperature range with small increases in ignition requirements by utilizing the
combined effects of radial motion and enhanced thermal conduction losses. If the enhanced thermal conduction
loss is provided by toroidal field ripple effects, the effects on the confinement of suprathermal alpha particles
must be minimized.
7
Acknowledgement
The authors wish to thank Dr. J.L Fisher for useful comments.
8
References
[1] CLARKE, J.F., U.S. Department of Energy, Hot Ion Mode Ignition and a Thermally Stable Tokamak
Reactor, submitted for publication to Nucl. Fusion (in press).
[2] BROMBERG, L, COHN, D.R. and FISHER, J.L, Nuci Fusion 19 1359 (1979)
[3] PETRIE, T.W. and RAWLSJ.M., General Atomic Co Report GA-A15218, March (1979)
[4] OKABAYASHI, M., Princeton Plasma Physics Laboratory, private communication (1978)
[5] LACKNER, K. and WILHELM, R., Max Planck Institute fur PlasmaPhysik, Garching, Germany, private
communication (1979)
[6] BROM BERG, L and COHN, D.R., M.I.T. Plasma Fusion Center Report PFC/RR-80-1, (April 1980)
[7] COHN, D.R., R. R. PARKER and D. L JASSBY, Nucl Fusion 16 (1976) 31; JASSBY, D.L., D. R. COHN
and R. R. PARKER, Nucl Fusion 16(1976) 1045
[8] HINTON, F.L and HAZELTINE, R.D., Rev of Modern Phys. 48 (1976) 239
[9] FURTH, H.P. and YOSHIKAWA, S., Phys. Fluids 13 (1970)2593
[10) HOLMES, J.A., PENG, Y.-K. M. and, LYNCH, SJ., Oak Ridge National Laboratory Report ORNL/TM6761 (March 1979)
[11] BOOZER, A., Princeton Plasma Physics Laboratory Report PPPL-1619 (Jan 1980)
[12] STRINGER, T.E., Nucl Fusion 12(1972) 689
[13] STEINER, D., BECRAFT, W.K., BROWN, T.C., et aL, Oak Ridge National Laboratory Report ORNL/TM6720(1979)
[14] U.S. INTOR WORKSHOP TEAM, U.S. INTOR (November 1979); (unpublished report)
[15] For example, BROMBERG, L, COHN, D.R., WILLIAMS, J.E.C. and BECKER, H., Compact Tokamak
Ignition Test Reactors, M.I.T. Plasma Fusion Center Report PFC/RR-78-12-R (March 1979), (to be published in J. of Fusion Energy); Compact Ignition Experiment Internal Status Report, prepared by Max
Plank Institut fur Plasmaphysik, Garching and the Divisone Fusione of CNEN, Frascati (1978).
[16] BOOZER, A., Princeton Plasma Physics Laboratory, in Workshopfor Detenniningthe Ripple Requirements
for ETF, MIT, Cambridge, Ma. (Sept 1979) (unpublished summary).
9
List of Figures
Figure 1.
(nr,)inimin and
(nre)ign,,
as a function of Ti. q, = 0. Curves are drawn for ri ~ T- 3 / 2 and
ri ~ Ti 7/2
Figure 2. Same as Figure 1 but for ,. = 0.3
Figure 3. flign,min and Pign,,a as a function of Ti.
r 4
77/2
Figure 4. Same as Figure 3 but for 17,.= 0.3
10
1,
0. Curves are drawn for; ~ T-3/2 and
0
tO
Ln
0
Ln
c
40-IN
tft
0
>
Q)
*,e'
0
cnc
0
('\j
c
E
O
II
0[
LO
0
0
(S Eu)
',u
iL
0
0
On
+0
0
M
C
>
a)
c
0
c
c
0
CN\
aE
(\J
0
C
II
OI
cL'.
17
0
co-
(s
uJ
G4
tu
L
0
0
.n
mJ
c0
a
>
QQN
CLL
0
0
N
0
(It-
N~
iU W)
0
Z 8 0 S/
0
L0
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C
c
0
>
c
0
~0
.NN
0
N
0
(Y)
Itj
0l
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1W)
e8o U!'
LL