PFC/JA-92-20
Why O, and ii cannot be Separately
Measured in a Near Circular Tokamak
J. P. Freidberg, M. Graf, A. Niemczewski, S. Schultz, A. Shajii
July 1992
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139 USA
Submitted for publication in:
Plasma Physics and Controlled Fusion
This work was supported by the US Department of Energy under contract DE-FG02-91ER54109. Reproduction, translation, publication, use, and disposal, in whole or in part, by
or for the US Government is permitted.
Abstract
An analytic calculation is presented demonstrating why a separate determination of
poloidal beta, Op, and plasma internal inductance, l, in near circular tokamak plasmas
is not possible from experimental magnetic data. A second order expansion in terms
of the inverse aspect ratio, e, of the Grad-Shafranov equation provides expressions for
the measured fields with explicit dependence on ,. This dependence, however, is very
weak and is shown to be insufficient for determining , when any reasonable magnetic
measurement error estimates are taken into account.
i
1
Introduction
Quantities of particular interest in tokamak fusion plasmas include the plasma current, I, the plasma elongation, n, the Shafranov shift, A, the poloidal beta, p,, and
the internal inductance, li. Based on magnetic measurements, a sufficient amount of
information is generally obtained to, in principle, determine all five of these quantities
[1]. It is found, however, that in near circular plasmas it is not possible to separately
determine beta poloidal and the internal inductance on the basis of these magnetic
measurements alone. The purpose of the work presented here is to provide an analytic
demonstration of this reality which is well known both experimentally and numerically
[5, 2, 7]. By expanding and solving the Grad-Shafranov equation up to and including second order in the inverse aspect ratio, e, it is possible to show that, (i) zeroth
order magnetic data determines the plasma current, (ii) first order data provides the
Shafranov shift as well as information about the linear combination (O, + 1), and (iii)
second order measurements provide the elongation and O, explicitly. The dependence
of the second order field measurements on , , however, is extremely weak. If reasonable measurement error estimates are taken into account, it quickly becomes apparent
that an explicit determination of beta poloidal is not possible. The calculation is made
analytically tractable by using a simple zeroth order equilibrium, first introduced by
Shafranov, which is characterized by the two independent parameters f, and li.
To obtain the magnetic measurements necessary to demonstrate this point, consider
a tokamak plasma in toroidal (r,0,0) coordinates as shown in Fig. 1, with major
radius Ro and minor radius a. The plasma is surrounded by magnetic probes located
at r = b relative to the origin of the coordinate system (see Fig. 2). It is assumed
that the plasma surface is also centered about the origin through the appropriate use
of an externally applied vertical magnetic field. This assumption is not critical and is
introduced primarily for convenience. The probes are capable of measuring magnetic
field components in the azimuthal and radial directions on the measurement surface
which is, in general, not a flux surface. Assume that there are N such probes, where
N > 1. During flat-top operation, it is further assumed that Fourier analysis of
the probe signals is capable of decomposing the fields into various spatial harmonic
components, up to second order. That is, an experimental measurement of Bo, Be1,
Be2 , B, 1 , and B, 2 is possible, where
Be(b, 0)= Beo(b) + Be1 (b) cos 0 + B62 (b) cos 20 +
B,(b, 0) = B,1 (b) sin 0 + B, 2 (b) sin 20 + ...
...
(1)
(2)
The solution that follows demonstrates why, despite knowledge of these field components up to and including second order, it is still not possible to separately determine
1
O,
2
and 1i if the plasma is near circular.
Equilibrium Equation
Consider a tokamak plasma equilibrium satisfying the Grad-Shafranov equation [6]
A*O = R 2 V (
v )
=
-toRj(0, R)
(3)
where b is the poloidal flux function defined by
B, = -ee
R ar
- --
r 90
r,(4)
B, is the poloidal magnetic field and jo is the toroidal current density;
dp
F fdF(
jg(b,R)=R-- +
I(5)
db
Rpto dk
The function p(?) is the plasma pressure, and by definition F(0) = RBO, with Bk
as the toroidal magnetic field. We can transform Eq. (3) into toroidal coordinates as
follows:
V 2 0 = -to
dp
dF
1
(Ro + r cos 0)2 d -F dF +
R
TV)
dO
8o,
(ar
cos -
16p
TO 0
r 8sin
(6)
By definition the poloidal flux function is required to satisfy the following boundary
condition:
'(r,,9) = constant
(7)
where r,(O) is the location of the plasma surface. Without loss of generality we choose
the constant appearing in Eq. (7) to be zero. The poloidal flux function is also required
to remain finite for all r and 9 interior to the measurement surface.
3
Asymptotic Expansion
An asymptotic series is obtained by expanding the equilibrium quantities in orders
of the inverse aspect ratio, E = -, considering the case in which e < 1. We use
an ordering characteristic of the ohmically heated tokamak regime [3]. These can be
summarized as
2
B6
B e
(8)
B = BO + O(E2)
(9)
2po f pdV
2to
~
f BdV zB
2
(10)
We proceed by expanding the poloidal flux function;
0 (r,0)
= 00 (r) + 01 (r, 0) + 02 (r, 0) + ..
(11)
where
Ok(r, 0)/Oo(r) ~ O(e )
The plasma pressure can be Taylor expanded to give
(12)
(13)
P(O) = P(0o) + dP(01 + 0 2 ) +
The free function F(O) introduced earlier is replaced by an equivalent flux function,
B2(0), as follows:
= RO B2 + 2RoB 2 (b)]
(14)
This is convenient for keeping track of the ordering. Eq. (14) can now be expanded to
give
2 dob
-R B
dB2
=0-&B
+
d2 B 2 (1
)Id
3
2
d
B 2 02
,03
++
(15)
The zeroth order equation in the asymptotic expansion of the Grad-Shafranov equation is just an expression of the pressure balance relation:
ld
drdr
\
dp
dORJ) =
d-
__
d
_d
(R BoB 2)
(16)
or, equivalently,
1d
-
~
(rB) =
top
1 d
(BoB 2)
Br (4), BrBe
where from Eq. (4),
3
-
--
(17)
dQ
1
doo
dQ
(18)
RoBe dr
in general for any function Q.
Consider now the first order equation. Note that the left hand side of Eq. (17) is
just the zeroth order contribution to pojo. Thus we are able to write
rdr
dr
/
-
+
r2
Bodr/
1
= -
(19)
B0edr
The second order equation for the poloidal flux function is given by
2 (r) +
f232
BO -R
R 2 (r) cos 20
(20)
where R2 and R 2 are
R2(r) =
2R0
r
--2pory -
Be
1
Be
p'
\Be
2Be
( pojP
Be
]
(21)
(22)
R 2 (r) = R2(r) + r&
In the above equations the primes indicate differentiation with respect to r.
The problem considered here concerns a near circular cross-sectional plasma. We
consider an elliptic cross-section for the plasma such that the ellipticity is of second
order in e. The plasma surface is given by
r = a [1 + K
(1 - cos 26)
;
- 1- e2
(23)
Here, K is the elongation.
The boundary condition on 0 given by Eq. (7) is expanded in powers of E, and the
result is a set of new boundary conditions for the zeroth, first and second order poloidal
flux functions. These boundary conditions become
Oo(r = a) = 0
(24)
V1(r = a, 0) =0
(25)
02(r = a, 0) = -aRoBe(a)
4
I (1 - cos 20)
(26)
4
First Order Solution
To render the problem analytically tractable, consider the following simple zeroth order
profiles first introduced by Shafranov, [8] for plasma pressure and current density in
terms of a shape factor, a, where a = c.
plasma pressure:
p)
for 0 < r < c
0
(27)
for r > c
current density:
jo(r)=
0 <r < c
jo
0 for
for r > c
(28)
The zeroth order poloidal magnetic field can therefore be written as
B,(r) = Be(r) =
(Bo,
for0<r<c
B ec f
for r > c
(29)
where B0 , =
is the value of the field at r = c. The profiles under consideration
are depicted in Fig. 3. This simple model provides a convenient description of a broad
range of plasma profiles. The shape factor, a, is a measure of the peakedness of these
profiles.
Using these profiles, it is now possible to calculate expressions for the poloidal beta,
O,, and the internal inductance, i, of the plasma in terms of a, po, and I. From the
standard definition,
O# =
6
prdr
-r
(30)
substitution of the pressure profile yields
I'OPO
(31)
Also, from
Li
47r
27rRo
(
where
1
2
rB2
B2o dr
Li
2po
5
(33)
we obtain the result that 1i =
the useful asymmetry factor,
-2 In a. We can now combine these quantities to define
1.
1
1
#
Op
In
a
(34)
2 2
(34
Now consider the first order Eq. (19) in more detail. We note that the substitution
A
=Pr
+
01 (r, 0) = 01 (r) cos 0
(35)
reduces the equation for b1 (r,9) to one which is independent of 0. In particular, consider
the solution independently in two regions of space, (I) r < c and (II) r > c. In region
(I), substitution of the appropriate profiles gives
r
r2
-(rV))1
=
+
'1
r(36)
c
Bec
which can also be cast in the form
B ±C4 popo r3
11c
Be
This is recognizable as a standard Euler equation whose solution is
r 2 41 + rO' - 01 =
01 =1 81
c + Becc r3 + Cr
(3cc7)
(38)
where C1 is an arbitrary constant to be determined from the boundary conditions.
In region (II), the first order equation becomes
r 2 0" + rO' - 01 = Bccr
(39)
which is also an Euler equation having the solution
Beecr In
-+
C 2r +
(40)
2
a
r
with arbitrary constants C2 and C3. Note that Of and Of, satisfy the appropriate
regularity conditions.
To determine the appropriate jump conditions to apply to this system, we recognize
from Eq. (19) that, in the vicinity of r = c, a singularity in the form of a Dirac delta
function, S(r - c), exists in the derivative of the current density. In order for this term
to be balanced by other terms in the equation, and to avoid introducing any higher
order singularities, we must restrict the function 0 1 to being continuous at r = c. The
first boundary condition is therefore
6
(41)
,0kf(c) = 0M'(c)
If we then integrate Eq. (19) over an infinitesimal interval about r = c assuming that
V)1 is continuous, we obtain a second condition requiring that
(644'
20,(c)
_64
r
(42)
Or_r=c
We also recall the boundary condition given in Eq. (25) which states that V51(a) = 0.
Using these three conditions, it is possible to determine the unknown coefficients, C1,
C 2 , and C3 . In region I, the solution is found to be
f
SIIoI
?P1 =1 tor
47rI 0+ 4
r2
A
c2
a2
1
2
(43)
We are more interested, however, in the solution in region II, since that is where the
magnetic probes are located. We find that the solution there can be written as
0
=L1
Ir n -+
A
(44)
1 -
To calculate the fields measured at the probe locations, r = b, we use Eq. (4)
together with the large aspect ratio expansion. The first order harmonic terms in each
component of the poloidal field are then
B,1(r) =
(45)
'r0
Ro 0
Be, (r) = 1 (0/'
Ro
(46)
RO r
In the above analysis we have assumed that the Shafranov shift of the plasma has been
zeroed by an externally applied vertical field (i.e. 0 1 (a, 9) = 0). It can be shown that if
a finite shift is included, its effect could be eliminated by taking the difference Bei - Br,
without otherwise changing the functional dependence with respect to the asymmetry
factor, A. For this reason, such a shift is neglected for the remainder of the analysis
without loss of generality. Using the 44' solution at r = b, we find that the measured
fields can be expressed as
4r Ro
andI
a2
1
and
7
+A(1+
)]
(47)
B,.1(b)
In b +A
-
I-
a)(48)
The linear combination B1 = B 1 (b) - B,1 (b) now gives
B
[a 2 A - 1
g
(49)
Under the practical assumptions that R0 ,b, and a are known from direct geometric
measurements, and that I is known from the zeroth order field measurements, it follows
that B 1 is a function of A only. As a result of the first order measurements, therefore,
A is now a determined quantity. Terms in O, and 1i do not appear separately in
Eqs. (47) and (48), but rather only as a combination in A. This is the first part of the
demonstration that 3, and i cannot be separated by first order magnetic measurements
in near circular plasmas, a result first pointed out by Shafranov.
5
Second Order Solution
The second order poloidal flux function is determined from Eq. (20), with the boundary
condition imposed on
as follows
02
given by Eq. (26). We proceed by separating the flux function
0
2
9)
(r,
Substitution of Eq. (50) into Eq. (20) yields two separate equations for
These equations are given by
1dr
dr
B4
dr
dr
dr
B
((
(50)
+ 0 2 (r) cos 29
= 12(r)
IR
2 (r)
0 2
and
2.
(51)
dT
(-(52)
The boundary condition given by Eq. (26) also separates as follows;
2(a)
0
2
cRoBoc
= - (
cRoBec
(a) =
(53)
(54)
Now consider the function _ 2 (r) which contributes to the second harmonic of the magnetic field. The function b2 is unimportant, since its contribution appears only in the
8
zeroth order harmonic and thus adds only a term of order e2 to Bo. It should be noted
however, that as a consistency condition it is required to show that b2 represents the
solution of a well-posed problem. This condition is easily verified.
The function 02 is obtained by substituting the Shafranov profiles into Eq. (52).
The solution of this equation in each region satisfying the appropriate regularity conditions is found to be
c(O, - 4)r + C4r
8
1,=_ cBec
(r < c)
2 +C
rl
F~2 + r 2
=Aa2
ln( ) + Csr +r
(55)
r>C
[Ro
(r>c)
(56)
where C4, C and C6 are constants to be determined from the boundary condition given
by Eq. (54) and from the jump conditions at r = c. In order to determine the jump
conditions we must first recognize the types of singularities that appear in the function
R 2 (r) in the vicinity of r = c. In order to balance the singularities that appear in R 2 (r)
we assume the following convenient form for the poloidal flux function in the region
r - c;
0
2
(r) = Alji + A 2 p + '02"(c) + f(r)
(r
- c)
(57)
where A1 and A 2 are constants to be determined, and f(r) is a continuous function
with a continuous first derivative which by definition satisfies f(c) = 0. The functions
jo and p are chosen to represent the correct singular behaviour of 0 2 at r ' c. Any
other functions with step discontinuities in their value and first derivative at r = c
could have equivalently been used in place of j6 and p respectively. After substitution
of Eq. (57) into the differential Eq. (52) and setting r = c we obtain the values of the
constants A 1 and A 2 ;
A =1
A2 =
where
4,
LC
4
(58)
4&o
,(C)
YON,
PO
RoBer
(59)
is defined as follows;
- b
01
1 (c)
c2Bec:
1 A
2 a2
1
(60)
4)
(60)
The jump conditions are easily determined from Eq. (57) and are given by
(c) - V44(c) = -Aljo
9
(61)
(c) = 2Po A2
(62)
c
dr
dr
Equations (26),(61) and (62) are sufficient to determine the constants C4,C5 and
Ce appearing in Eqs. (55) and (56). The determination of these constants ultimately
yields the second order solution for the poloidal flux function. The function '2 can then
be used to evaluate the second order harmonics of the magnetic field in the vacuum
region. We begin by expanding Eq. (4) to obtain the second order harmonics, Br2 and
B 9 2 in terms of 02. After a straightforward calculation we obtain
02
(C) -
1
Br2 (b)
B02 (b) =
dO
2
i
2
-
bi
2Ro ,
(63)
+
+V
(64)
Next, after a somewhat lengthy calculation that determines the constants C4, Cs and
C6, and then evaluates the second order harmonics we can write the normalized fields
as
a)
b2 = -In(
8
b8 2 =
- 3 In( )-
a +C
A+ (
4
)(4 )(P
V
2
4)(
)(
1)+ (a 4 +C)(
4 b4 )Y(Af,)
+ (a
4
)(C4
)-y (A, O,)
(65)
(66)
where
-y(A, Op) = p2 - 20p,'
+
ln(a) +
(67)
and the normalized fields, br2 and b62 , are defined as
4-2
27M(68)
= B,2(b)/ (polb\
b82 = B02 (b)/ ( pjb)(69)
The a appearing in 1/ and the In a term can be expressed in terms of A and p, by
means of Eq. (34). Equations (65) and (66) are the desired equations for the second
harmonic information.
10
Discussion
6
The results of the analysis carried out in Section 4 indicate that first order Fourier
harmonics do not allow a separate determination of O, and 1j, only of the combination,
A. As demonstrated in Section 5, second order fields (both radial and azimuthal components) show explicit dependence on poloidal beta. However, the following analysis
shows that this dependence is too weak to be of practical experimental value.
Expressions developed in Section 5 can be evaluated using a specific choice of parameters. For example, Figs. 4 and 5 show normalized second order fields (azimuthal
and radial components respectively) as a function of beta poloidal with ellipticity as a
2
parameter. Consistent with our initial assumptions, r - 1 is of order E . Measurement
surface position with respect to the plasma boundary is chosen to be b/a = 1.3. An
important feature of these calculations is that they are carried out at a fixed value of
A, which is assumed to be known from the first order Fourier harmonic fields (see Eq.
(49)). A fixed value of A = 1 is used for this calculation. Figures 4 and 5 demonstrate
that the dependence of second harmonic fields on poloidal beta is very weak, especially
for small values of ,.
To render the problem more intuitive, specific analytic limits can be examined.
For example, the results depend on the distance of the measurement surface from the
plasma boundary. For b/a > 1 the second harmonic expressions can be expressed in
the following form:
b, 2
b6 2
=
b
3
-- In( ) +
a
8
b
3 3
=
3
6
K-
1
3ln(b ) +
a
8
E2
a4
1+
1I+
+
1+4
1ar-3
4
1 + a4 18
62
a4
- 201 + 3-ln(a) + 16
8
0)
K
1
11
ac-3
+k
- 2
1
+ ln(a) + 16
(70)
(71)
The above expressions are identical except for a constant numerical term. Therefore a separate determination of the two independent parameters, elongation and beta
poloidal is not possible. On the other hand, if measurements are taken on the plasma
surface (i.e. b/a -+ 1), then b,-2 depends only on the elongation and b92 depends on
both parameters, as shown below:
b,.2 = (
3
162
A
be24 -1
1 - a7
-
+a4
e2
+
1(72)
2a4
1+a4
11
3
1
- 20p,)1 + -In(a) + 1
3
(73)
The features indicated by the above equations are presented in Fig. 6, where the
normalized fields are plotted for both limits of b/a for A = 1. Even when the measurement surface coincides with the plasma surface, a separate determination of O, is barely
possible only for sufficiently high O, ; 0.8. Furthermore, this case is not practically
feasible except through the use of magnetic probes inserted directly into the plasma.
These however cannot sustain heat loads present in the modern high performance tokamaks. For the sake of further analysis, therefore, assume that there is a minimum
practical location of the measurement coils (eg. b/a = 1.3).
Figure 2 shows one of the possible experimental schemes used to measure the magnetic field. A set of discrete magnetic coils uniformly distributed around the plasma
poloidal cross-section measures radial and azimuthal components of the poloidal magnetic field. These measurements are made with some finite error, ABO and AB,. These
errors are introduced into the Fourier harmonic amplitudes through the discrete Fourier
transformation. The Fourier harmonic amplitudes are given by
N
1
Bok = -Z B, cos kj
N =1
1 N
B,.k = -
Bo sin k,
(74)
(75)
where k denotes the order of the harmonic, N is the number of measurement coils,
and B, and Be3 are the measurements made by the jth coil. Consider an error of
magnitude AB 9 to exist in only one coil; we denote this coil as the m" coil. The error
in the harmonic amplitude of the azimuthal magnetic field becomes
zAB 6
N cos k6m
(76)
N
The square of the total error in the kh harmonic due to all of the coils is obtained
by squaring Eq. (76) and then summing over all m from 1 to N. Note that we have
considered the measurement errors of all the coils to be the same, ABO. Thus the
square of the error in the k h harmonic becomes
A BOk
=
cos2 km
e
N
my=
A similar analysis for the radial measurements gives
(ABek)
2
=
(AB,.k)
2
=
N
For k = 2, Eqs. (77) and (78) indicate that
12
1
m=1
sin 2 kO
(77)
(78)
AB, 2 = AB,(79)
AB
v/2N
V"2N
A relative measure of the errors in the measurement coils can be defined as
ABe2
-
B,.
Bo
For the assumed ohmic tokamak ordering, error propagation through the Fourier analysis can be expressed as:
b
Sb=AB
62
2
6 2
B192
1
_
=
2
_A~R.
N
=v72M2
bo
8b,
B,
B,-22
2
1
_
=
b,
/2~NC2
(81)
Having determined the relative errors in the Fourier harmonic amplitudes, a more
quantitative estimate of the accuracy of the final measurement result, i.e. of ,, can
now be obtained. From Section 5 we can approximate the error propagation by first
Taylor expanding the expression for the normalized field (shown here for b,-2 ):
+ Ab,-2 =±b,-2.
b,-2 = b,.2,
+
b,.2
AN +
80P
1
2
2
OP23
-
bN 2
.
(82)
Ignoring small terms in A 2 or higher orders, the relative error in a measurement
of Op as a function of the relative error in the field measurement can be given by
60
(3,
-
1b./i b,.2 ]
lb,.2
OP,-2/190pl 0, Amca
b.
(83)
Defining the multiplier in front of bb2 as an error propagation factor, E,., we obtain
a useful quantitative gauge. From the above definition, the calculated O, can be related
to the exact 3, in the following way:
(84)
The error propagation factor can be calculated for radial and azimuthal field components separately. Since its behavior is similar for both of these components, only
results for b,-2 are presented here. This factor, E,, is plotted for different parameter
values in Fig. 7. As demonstrated on these plots, E,. is typically a number between
10 and 1000. Error propagation is less severe for beta poloidal close to 1, but becomes
very high for , -+ 0. Dependence of the error propagation on the plasma profile is also
examined. For example, the results from two different values of the assymetry factor are
PCdIC,.tdted
= Neact(1
± E,.Sb,. 2 )
presented in Fig. 7; A = 1 corresponds to relatively flat profiles, (a
-
0.3 - 0.8), while
A = 3 describes steep profiles (a - 0.05 -0.1). It is noted that the steeper profiles lead
to more greatly enhanced error propagation. Taking into account the fact that typical
13
experimental accuracies are, at best, on the order of .2 -. 3% [4], the combination of the
discrete Fourier transformation and the error propagation factor result in final errors
on the order of several hundred percent. Since these errors are largely inherent in the
measurement and analysis process employed, a separate determination of Op and li in
low beta, near circular plasmas remains beyond the capability of current experimental
techniques.
References
[1] Brahms B.J. (1991) Plasma Physics and Controlled Fusion 33, 715.
[2] Brusati M. et al. (1984) Computer Physics Reports I (North-Holland, Amsterdam),
p. 3 4 5 .
[3] Freidberg J.P. (1987) Ideal Magnetohydrodynamics (Plenum, New York).
[4] Granetz R. (1992) private communication.
[5] Lao L.L. et al. (1985) Nuclear Fusion 25, 1421.
[6] Mukhovatov V.S. and Shafranov V.D. (1971) Nuclear Fusion 11, 605.
[7] Ninomiya H. and Suzuki N. (1982) Japanese Journal of Applied Physics 21, 1323.
[8] Shafranov V.D. (1970) Zh. Tekh. Fiz. 40,241. [Sov. Phys.-Tech. Phys. 15, 175].
14
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