PFC/JA-86-30
Requirements for Ohmic Ignition
I. H. Hutchinson
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA
02139
May 1986
Submitted to:
Journal of Fusion Energy
This work was supported by the U.S. Department of Energy Contract
No. DE-AC02-78ET51013. Reproduction, translation, publication, use
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Requirements for Ohmic Ignition
I. H. Hutchinson
ABSTRACT
An analysis of ohmic ignition criteria is presented,
quirements on T,
nt and n/j
finement assumptions.
with Spitzer
is required.
For
resistivity
The
value are how much
in a form easily applicable
circular
cross-section
to various con-
'NeoAlcator'
a value of B 2 a approximately equal
outstanding
increase
uncertainties
in
schemes
to
in current density is achievable
shaping and what the exact NeoAlcator coefficient is.
Key Words:
giving the re-
Ignition, Ohmic Heating, Confinement
tokamaks
to 250 T 2 m
lower
this
by plasma
Introduction
It is
generally acknowledged
that to obtain ignition of a magnetically
confined plasma which is heated only by ohmic dissipation, due to DC currents
in the
plasma,
is
very
difficult.
rapid decrease
of
the
resistance
recent unfavorable
on tokamaks have
scalings
reawakened
fact be achieved.
models,
there
to be
required for ignition
of
with
interestill
is
in whether
no clear consensus
because
temperature.
observed with
studies[2,3l
an ohmically
this
increasing
of confinement
Despite previous
appears
Qualitatively
of
the
However,
auxiliary heating
ohmic ignition could in
using
specific confinement
on what are the conditions
heated plasma and
what
these
plasma
requirements demand of the machine engineering under various scaling assumptions.
The purpose of this paper is to set forth these ignition requirements
as clearly and succinctly as possible.
By doing so, the engineering require-
ments and their sensitivity to physics assumptions
can be directly and quan-
titatively deduced in terms of a few simple analytic formulae.
The uniform plasma power balance is analysed first leading to the rather
simple theorem that,
when the confinement time is independent
of temperature
and proportional to density, the optimum ignition point is at the temperature
where Bremsstrahlung and alpha heating balance (T = 4.4 keV).
are then
incorporated
in
order
to
obtain
realistic
machine
Profile effects
requirements.
Brief discussion of other issues and limitations is followed by a summary of
conclusions.
Zero-Dimensional Analysis
Consider initially a uniform plasma in which the power balance equation
can be written in terms of the power densities of ohmic (POH), alpha particle
-2-
(Pa)
dW
- - POH + Pa - Pc
dt
Pb
-
Qa<av>
3nT
-
. ng2 + n2
---
4
= n
H
[F2
2
T-
3
/
2
processes as
and bremsstrahlung (Pb)
conduction loss (Pc)
- Cn2 T1/2
T
3T
nT
-
+ R (T)
- CT1 /2].
Here n is the electron density in an assumed 50-50 DT mixture with temperatures Te
-
TD
=
TT -
T;
j
is
the
current
density,
T the
bremsstrahlung) energy confinement time, n the resistivity, Q
energy (3.5
MeV),
< ov > the DT rate coefficient,
fusion heating power (Qa
< ov >/4),
F the ratio
R(T)
(non-
the alpha
the normalized
of density to current
density (n/j), and H and C are the constant resistance and bremstrahlung
coefficients.
Cyclotron radiation is ignored because, as we shall see,
the temperatures of interest to Ohmic ignition are rather low.
The normal
course dW/dt
=
steady-state
equilibrium power balance condition is of
0, which determines T if the other quantities are known.
One needs to be somewhat cautious in defining the meaning of 'ignition'
for an ohmically heated plasma because normally the ohmic power cannot
be
'switched
off'
in the way that auxiliary power can.
Physically the
most rigorous definition comes about by recognizing that the stability
of the
power-balance
(dW/dt).
a slight
equilibrium
If this sign is negative,
increase
the energy;
i.e.
in
is
determined
by
the
the equilibrium is stable,
temperature leads
to a
a loss of energy (dW/dt<O).
'negative
of 3/aT
sign
because
feedback'
into
On the other hand if the
sign is positive the equilibrium is unstable and a positive temperature
-3-
perturbation will grow with time and lead to an (initially) exponential
growth of temperature.
Clearly, what is required for ohmic ignition is
to reach the point where the usually stable low temperature equilibrium
solution becomes marginally unstable;
because then a small temperature
perturbation will lead to a steadily increasing temperature moving the
plasma into the regime where alpha heating is dominant.
In other words,
the ignition point of interest is the 'thermal runaway' point.
Mathematically, then, the ignition condition is that
3
dW
-(-)
aT dt
dW
= 0
0
dt
Since
simultaneously.
-
,
these
derivatives
2
could equally well replace dW/dt by (1/n
We make
the
assumption,
confinement scalings,
that
T
valid
is not
assume
for
n to
be
constant
we
) dW/dt in these expressions.
most
empirical
explicitly
tokamak
ohmic
on T.
Then
dependent
we can immediately eliminate nT from the ignition condition by writing
3
3
(-)
3T nt
0 =-
a
H
[ aT
F2
= -
5H
-
T-
-
2F
7
2
/
2
+
d
dT
R
T-5/2 + T
R
(-)
T
Since R is a known function of T,
- CT-1/2
n2 T
C
2
+
T-
which
ohmic
3
/
dW
_ ]
dt
2
we can regard this equation as indi-
cating that for a given value of F (the
temperature at
1
-_
-
ignition
n/j
ratio),
occurs.
It
there
is more
is a unique
convenient,
though,
to solve the equation for F 2 with T as the independent variable.
For any
T,
ohmic
5H
F 2 - -/[T
2
7
/2
ignition
can
occur
d
R
C
(-) + - T 2
dT T
2
,
-4-
at
that
temperature
only
when
Having solved for F we can substitute back into the power balance
to find what value of nT is required for ignition at this temperature.
We find immediately:
R
4
nT - 3/[- T - (-) + T
5
dT T
- 5
2
at ignition.
d
R
CT-1/2
Whether or not any confined plasma in fact ignites depends
upon whether its Lawson product nT exceeds this value or not.
Of course there
the confinement
time
scale.
should
a piori
no adequate
is presently
theory for how
a widespread observation
However,
is that the ohmic confinement time is proportional to density.
So let
us adopt a scaling of the form
T - K n
where all
other
dependences
incorporated into K.
on
factors
such
as
field
and
size
are
Then we write
nT . Kj2 F 2
and recognize that, from the engineering viewpoint,
there will generally
be practical constraints on Kj2 which will determine whether this equation can be satisfied,
for a given scaling,
in an achievable machine.
the ignition criteria just discussed prescribe
Put the other way around,
the values of nt and F2 for ignition at any T, and hence demand a value
of Kj2
at
least
required machine
equal
nT/F
to
figure
engineering when ignition
of
2
We
.
merit.
can therefore
The
least
demand
regard
is
-5-
2
placed
occurs at the minimum value of nT/F
our previous equations we can readily evaluate this.
Kj
2
.
as
a
upon
From
2
The minimum value of nT/F
occurs at that temperature which makes
d
1
1 d
- (F 2 ) = 0
nT () + 2
dT nT
F dT
where the
total
derivatives
ignition condition.
d
3
a
3
dT
nT
3T
nT
here mean derivatives
But, using the power balance equation,
F = const
Hence the minimum value of nT/F
HT-3/2
---
in
2
dF2
3
3
dT
3F2
nT
dF2
T
=
const
dT
HT- 5 /2
F4
at ignition occurs where
3T
-
-
F2
or,
along the marginal
0 ,
=
nT
other
words,
where
Note
exactly balance.
that
the
ohmic
since
heating
and
conduction
losses
ohmic
and
conduction
powers
the
balance, so also do the alpha and bremsstrahlung powers.
fore immediately
write down the equation determining
We can there-
the optimum tem-
perature for ignition as
R(Tm) - CT/2
m
=00,
and the required nT as
3F2 T5 /2
m
(nT)m
H
Notice that
the solution for Tm,
in the uniform plasma model is just
the well known minimum possible temperature for idealized ignition (4.4
keV for
Zeff
-
1).
However,
time for ignition there
we
because
do
not
require
we always have
infinite
some
confinement
ohmic heating to
balance the non-bremsstrahlung losses.
In order to obtain explicit solutions for the parameters we have
discussed,
we need to adopt an analytic
-6-
form for R.
Various forms are
available,
however
it
proves
of
considerable
later, in discussing profile effects,
simplifying
advantage
to adopt a power-law expression.
The expression we shall use here is
R - A T 3.5
m 3 s-1 keV-2.5
A - 3.9 x 10-23
;
where we are choosing to measure all energies and temperatures in key.
This form approximates the DT reaction rate to within
to 10 keV and to within ~ 5% from ~ 3 to 7 keV,
of most
interest.
uncertainties in
These
the
errors
are
modelling.
in
20% from T - 2.5
which covers the range
negligible
Also,
-
these
compared
units,
to
the
other
Spitzer
resistivity with in A - 16 and the bremsstrablung coefficient with Zeff
- 1 give
H -
1.64
ms'1A- 2 keV5/ 2 ;
108
x
m 3 s-1 keV1/2
C - 3.3 x 10-21
With these forms we can write explicit expressions:
n2
-
- F2
=
5H
-
C
/ [2.5 AT5 + - T2 ]
102 8 /[0.0024T 5 + 0.040T 2 ]
m-
2
A-2
2
2
j2
-
4
nT
- 3/[2AT 2 .5 -
-1/2]
-
- 102 0 /f0.0026T
2
.5 - 0.088T-1/ 2 ]
m- 3s
5
and also for the optimum ignition point:
Tm
C
()1/3
. 4.4 keV
A
15 (C/A)5 /6
"
-
In Fig
5A(C/A) 5 /3 + C
(C/A)2/3
1 are
these equations,
plotted
together
the
with
values
= 1.6 x 102 1 m-3 s.
of
the machine
the
F2
parameters
figure-of-
and
merit nT/F
2
nT from
(=
Kj 2 ),
-
--
F
2
[0.00 24 T5 + 0.040 T 2 ]
6 [2.5 AT5 + CT 2 /2]
nT
= 10-8
5H
(2AT2 .5
m-'A
[0.00 26
- 4CT-1/2/51
-7-
T2 . 5
+ 0.088 T-1-
2]
2
.
The idealized ignition curve for nT (ignoring the ohmic heating
term in
3/[AT2 . 5 - CT-
the power
1
/2])
balance
obtained by
is also
shown.
The ohmic ignition occurs at a noticeably lower curve than the idealized
value.
The previous discussion is essentially completely general, applying
to any confinement scheme.
Even different assumptions about the n and
T dependence of T could be incorporated and the analysis suitably modified.
However,
should be
there
incorporated
are various
in
important
order to
obtain
considering some of these complicating
additional
realistic
factors
results.
which
Before
factors let us end this discus-
sion of the simple uniform plasma model by adopting a specific tokamak
the NeoAlcator confinementt 4],
transport scaling:
T = Kn
;
K = 2 x 10-21 R 2 a K1/2
Then ignoring shaping (K = 1) and using j = 2B/poRq we get
nT
F2
- Kj2 = 5 x 10-9 (B 2 a)/q
2
(m-3 s)/(m- 2 A- 2 )
with B in Tesla, a in meters and q the safety factor.
optimum value
of nT/F 2 , which is
3T
/H we
get
Substituting the
the minimum
required
machine parameter as
B 2a
-
T2 m.
150
q2
This number is comparable to those quoted elsewhere[5]
noted that
simply to apply this
criterion to the
but it should be
center
of a tokamak
(where q = 1)
leads to a considerably too optimistic view of the ignition
requirements,
because
of
the importance
now discuss.
-8-
of profile
effects,
which we
Profile Effects and Shaping
In order
profiles of
to determine
plasma
quantitatively
parameters
on the
the effects
ignition
of non-uniform
condition
we
adopt
a
model in which the plasma is taken to be cylindrical and all profiles
are taken to be proportional to a parabola to some power:
(1 - r2 /a2 )a
T =T
;
(1 - r2 /a)
n = n
.
The convenience of this model is that the volume average of any quantity
f - fo(1 - r 2/a2 )k is readily shown to be
1
< f >
- fo
-
k+1
This enables us to generalize the previous treatment very rapidly.
simply recognize
that
the correct
total power-balance
We
equation is the
volume average of equation 1, which then becomes
d
--
3nT
- CT 1 /2 n 2 ] >
= < [HT- 3 /2 j2 + AT 3 .5 n 2 -
<W>
dt
H
3
T
/2
A
(3
2+1
00
Fe2
3TO
C
n0 T(a+a+1)
T 3 .5
+
0
=
(3.5c0+28+1)
TI/2
(a/2+2B+1)
Here we have assumed j c T 3 /2 and put Fo
-
n0 /j0 .
Given this equation,
we can see that the previous analysis goes through just as before except
that the coefficients must be replaced by modified values
3a
H
=H/(-
+ 1)
2
A
C"
-
A/(3.5a + 28 + 1)
C/( - + 28 + 1)
2
-9-
So to obtain
the central
ignition condition
equations by
simply
we
their
values
appropriate
replace
the parameters
alternates
primed
defined
2AT 2 .5
previous
For
example:
above.
4CT-1/2
(3.5a+2a+1)
(a/2+26+1)
1/3
3.5a+20+1
C
in all
-
not (a+a+1) - 3/[
Tm -
to the profile-corrected
)
-1/3
A
a/2+20+1
nT
C 5/6
(-)m - 3 (-)
A
F2
1
3.5a+26+ 1
-I
H
a/2+20+1
5/6
3a/2+1
a++1
where we leave the subscript zero as understood from now on.
In Fig.
2 we show the correction
function of a for three
be noted
that
in a
edge to
central
typical values
Tokamak
(3a/2+1)
the density profile
2
-
is equal to qa/qo,
by increasing a helps
)m as a
It should
the
ratio of
Therefore a is strongly
to lower the required
value
.
consider a tokamak with qa - 2,
2/3), peaked density, a -1.
in the Neo Alcator formula,
ff.
and 1).
2
The value of 8 is less certain but peaking
As an optimistic example
(a
of a (0,1/2
safety factor in this model.
contrained by known physics.
of nT/F
factors for Tm and (nT/F
Then we find the nT/F
2
We include the correction no
qo
-
=
1,
1.5 ii
which applies to the chord averaged density
requirement calls for
)
a/2+2a+1
This value is approximately
3a/2+1
5/6
3.5a+2$+1
B2 a - 150(
) 1.5 = 250 T 2m.
(
a/2+2$+1
the lowest plausible value for a
tokamak with no neoclassical resistivity enhancement.
-10-
circular
profile
If we allow shaping,
One
main reasons.
the vertical
scaling is
other
The
hand,
of
effect
stronger
density for fixed
effects
shaping
is
current
unchanged.
is to
increase
is uncertain and may depend
configuration.
In
absence
of
the
other
the
On
mean
current
Quite how that affects
profile factors
to
proportional
decreased
are
edge.
q' at the plasma
the
additional
in
increase
permitted
requirement
profile
presuming other
a
B2 a
the
axis,
the
as K1/2 (c E b/a,
In so far as shaping increases the current
density at fixed q and B.
density on
is
T
base for this
the data
although
elongation);
small.
in NeoAlcator
the increase
is
to deal
are more difficult
to have a favorable effect for two
Vertical elongation appears
with.
J-2,
effects
on details
a well-established
the
of the plasma
model
for
the
profile-effects of shaping, an estimate may be gained by supposing the
shaping simply
to
allow
(qc = 2lra 2 B/1 0RI ).
If
a
lowering
the
of
the
circular
edge
current-density
central
rises
q-value
by only
a
relatively smaller amount, then what shaping allows is a broadening of
the temperature
current profiles,
and
example we might suppose that at qc
=
i.e.
a
modification
1 we should take a - 0,
case we get B2 a = 110 T 2 m in our previous example.
may have the potential to ease the
Further
Tokamaks is essential
to
For
in which
This crude estimate
more than a rough indication that shaping
should be regarded as little
significant factor.
of a.
give
ignition
experimental
a firmer
engineering demands
on
work
basis
for
shaping
shaping
of
by a
ohmic
estimates.
Other Corrections and Limitations
Modifications of
effects are
easily
the
coefficients
incorporated.
due
For example,
-11-
to
additional
physical
trapped-particle
neo-
classical corrections to the resistivity may increase the value of H by
This increases
over
averaged
a factor which,
the F 2
the nT/F
value and decreases
2
the value of nT or of Tm,
ately but does not affect
as 2[3].
as large
may be
the profile,
value proportionsince they depend
only on A and C.
Loss of some fraction of the alpha power due to unconfined drift
orbits can be incorporated as a decrease in A, which increases nT/F
2
as
A-5 /6.
A more complex situation arises when noticeable amounts of impurity
Assuming
are present.
radiative
to
contribution
the
losses
to
be
dominantly bremsstrahlung Zeff modification - an assumption which will
prove false for heavy impurities - we can calculate how a small amount
of impurity
of
three effects:
Z modifies the
charge
ignition
conditions.
There are
increase of ohmic power by a factor which is approxi-
mately (0.65 Zeff + 0.35), decrease of alpha power, due to dilution, by
a factor ((Z - Zeff)/(Z - 1)12, and increase of radiation by Zeff.
All
that we require is to modify the coefficients H, A, C by these factors.
2
In particular the nT/F
Zeff (Z-1) 2
correction factor is
1
5/6
0.65 Zeff + 0.35
(Z - Zeff)
differentiate
If we
Zeff
dfz
--
fz
this
5
=
dZeff
expression
5/6 +
with
Zeff
-
3
respect
to
Zeff
we
Zeff
- 0.65
Z - Zeff
-12-
0.65 Zeff + 0.35
get
0.18 + 1.67/(Z -
1 is always positive:
which at Zeff impurities always
it
make
to achieve
more difficult
1).
Therefore
ignition even
on
this optimistic assumption about radiation.
An issue which so far we have not addressed concerns whether the F
values we
require
'density-limit'
are
Normally
achievable.
tokamaks
The basis of this is that for fixed
q the mean current density is proportional to B/R.
number.
the F parameter
tokamaks,
a
This is often expressed in
which cannot be exceeded.
terms of a 'Murakami number' nR/B.
for circular
experience
Therefore, at least
is proportional to the Murakami
Now the optimum F value is
5H
F.
5A(C/A)5/3
ignoring profile
2
+ C(C/A) /
this value corresponds
x
- 1/2
4.6 x 1013
=
This translates
effects.
number of 2F/Po = 7.3
3
101 9 m- 2 T-1.
to the high range
(for
q
-
m-1 A-1,
1)
to a Murakami
It is interesting to note that
of density operation
for most
Tokamaks but not significantly exceeding typical achievable values.
It
should also be noted that a smaller F-value by a factor of up to 2 does
not greatly alter the engineering factor nT/F
ignition density
generally
2
.
than present
is less
Therefore the required
expectations
of what
should be achievable.
The density limit is not a serious problem.
course,
confinement degradation occurs
if significant
Of
near the density
limit, that is a problem.
Concerning the Beta limits,
if
we operate at the optimum F value
this implies that
2wonT
- -t
B2
4
2vo FmTmj
=
"
B2
FmTm
-
qRB
-13-
using the
circular
q-value.
Substituting
the
zero-dimensional
values
and q - 1 we get
0.13
at M
-
BR
Thus at the optimum ignition point in a high-field machine the central St
value will be quite modest,
field will be essential
1% for Bt = 13T, R
e.g.
=
to achieving ignition anyway,
1m.
Since high
this
shows
that
the beta-limit should not be a problem.
We can
current (4
confirm
5MA)
to
that
an
confine
ohmic
the
ignition
alphas.
Tokamak
Using
has
sufficient
again the
circular
q
value which we take to be 2 we get
27ra 2 B
I - q iioR
a2 B
10 R
=
MA.
Since B2 a is determined by the ignition requirement (which for present
purposes we take to be B2 a = 100) we can eliminate B to get
a
100 al/
1
2
Thus the current
R
MA
is more than enough,
in even quite small machines to
confine the alphas.
Also we should
verify
presumably the irreducible
tion we
Using
require.
that neoclassical
mimimum,
the
ion
transport,
is low enough
to permit
approximate
formula[61
for
which
is
the ignithe
Banana
regime:
nTNC
we find,
107 1 2 T1/
2
(a/R)1/2
for T = 4.4 keV and (a/R)
required nT(= 1.6
x
10
2 1
m-
3
s on the
= 1/4 that in order to achieve
zero-d model)
-14-
the
we need at least 12
MA of
will
This
current.
previous equation.
sized machine
Of
transport
ion
the
of
enhancement
appreciable
there were
confinement.
neoclassical
sufficient
should provide
even a relatively modest
So again,
from the
0.23 m,
a
provided
be present
course,
above
if
Neo-
classical, ignition might be prevented.
Summary of Conclusions
(1)
unique
a
which is
ratio
the
of
function
(F)
of any assumptions about
Independent
density.
takes place
physically defined,
Ohmic ignition,
at a temperature
current
of density
to
confinement
other than
that T is not explicitly a function of temperature.
(2)
The nT required for ignition is similarly a unique
and hence
T,
of
lying
idealized
below the
slightly
function of F,
ignition
fusion
curve (see Fig. 1).
(3)
The quantity nT/F
2
constitutes an engineering figure-of-merit when
It is then proportional to
confinement time scales proportional to n.
and its required value is likewise a
the square of the current-density
unique function of ignition temperature.
(4)
The minimum value
4.4 keV
where
the
of nT/F
NeoAlcator
requirement;
ally
scaling
x
for a uniform plasma,
occurs,
bremsstrahlung
quired to reach this is 1.6
(5)
2
and
alpha
powers
at T
The
balance.
=
nT re-
102 1m- 3 s.
allows
the
nT/F
2
to
be
reexpressed
as
a B2 a
requiring B 2 a > 150 in a uniform plasma but more realistic-
250 in a
circular
tokamak (not
accounting
for trapped particle
resistivity enhancement) when profile effects are accounted for.
-15-
Impurity effects are always detrimental to the ability to achieve
(6)
igntion.
(7)
Density
limits,
beta limits,
alpha drift
orbit
confinement
and
neoclassical ion transport do not appear to be serious limiting factors
in practical
sized
achieve ignition,
(8)
machines.
these
If machine
limits
will
not
parameters
be
are
such
significantly
as
to
stretched.
The most crucial areas of uncertainty in determining the engineer-
ing requirements for igniton are (a) the neoAlcator coefficient and (b)
the degree to which the current density,
increased by
issues,
particularly
to encompass
Our
shaping.
values
shaping,
(from
present
degree
of
uncertainty
the B 2 a uncertainty
leave
100
particularly on axis,
to = 300)
which range
in
large
from
can be
these
enough
plausibly
achievable to presently inconceivable.
Acknowledgements
I am grateful to D.
R.
R.
Parker
for
helpful
R.
Cohn,
J. P.
discussions.
Freidberg,
M.
This
was- supported
Department of Energy Contract DE/AC02-78ET51013.
work
-16-
Greenwald,
and
by
References
1.
D. R. Cohn and L. Bromberg, Submitted to J. Fusion Energy.
2.
B. Coppi, Comments Plasma Phys. Cont. Fusion 3, 47 (1977).
3.
C. E. Wagner, Phys. Rev. Lett 46, 654 (1981).
4.
R. R. Parker, M. Greenwald, S. C. Luckhardt, E. S. Marmar, M.
Porkolab,
5.
and S.
M.
Wolfe,
Nucl.
Fusion 25,
1127 (1985).
D. R. Cohn, L. Bromberg, and D. L. Jassby, 7th Topical Meeting
on Fusion Technology, Reno (1986).
6.
W. M. Stacey Fusion Plasma Analysis, Wiley (1981).
-17-
Figure Captions
Fig. 1
The values of n/j(-F) and nT required for ohmic ignition at a
The idealized ignition nT
temperature T in a uniform plasma.
requirement (ignoring ohmic heating) is shown for comparison.
The ratio nT/F 2 is the machine figure-of-merit requirement.
It is minimized at the optimum temperature Tm
Fig. 2
The profile correction factors for the central optimum temperature and nT/F 2 requirements as a function of the temperature
and density profile indices (OL and 0).
-18-
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