FINITE 0 e
MED
TUJBULECE AND ALCAIOR SCAING
by
Kim Molvig, S. P. Hirslman and J. C. Whitson
Research Report
PFC/RR-79-4
Februaxy 1979
FINITE 0
UNIVERSAL MODE TURBULENCE AND ALCATOR SCALING
Kim Molvig
Plasma Fusion Center & Nuclear Engineering Dept..
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
and
S. P. Hirshman and J. C. Whitson
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37830
ABSTRACT
An outline for a self-consistent theory of finite
e universal mode
Saturation results from resonance broadening of the
turbulence is given.
electron response due to magnetic shear.
Electron diffusion, for
S> m e/m , is due to the magnetic part of the fluctuations.
sion coefficient, D
=
.1 (Te/(T
4
1
+ T e)
)) (me/miSe )(Ls/Ln)
s n
2
The diffu-
2
p 2/L'n
x v ii
scales inversely with density, is independent of magnetic field, and is
in excellent quantitative agreement with observations on the Alcator tokamak.
2
One of the principal theoretical goals in tokamak research is the
development of a self-consistent turbulence theory for the short wavelength fluctuations thought to be responsible for anomalous transport.
This paper presents a resonance broadening turbulence theory for the
finite $
universal instability.
An approximate analytic solution of the
coupled, non-linear, eigenmode equations is obtained.
this solution is verified by numerical computations.
The accuracy of
The resulting for-
mula for the anomalous electron thermal conductivity, eq. (14), has many
similarities with experimental observations, including absolute magnitude,
and scaling with density, temperature and ion mass.
regimes where 6
> m /m,
For typical tokamak
the calculation constitutes an example of a self-
consistent theory of stochastic
magnetic fluctuations.
Until recently,2 most turbulence theories ignored shear in the equilibrium magnetic field.
Without shear, turbulence mainly effects the ions.
The basic saturation picture, as developed by Dupree,
balanced linear
L
NL
electron growth, y e, against nonlinear (turbulent) ion damping, y .
Y
Taking
= k2 D is the basis for the yL/k2 estimates of the anomalous diffusion
coefficient.
However, recent theory
has found that yNL << k D, because
the ion-wave interaction is weak for low frequency modes, and consequently,
for tokamak parameters, that ion non-linearity is not a viable saturation
mechanism.
With shear and because of their rapid mobility along the field lines, electrons are the species the most strongly effected by turbulence.2
S
ions cause damping linearly, due to the shear, y1 .
Here, the
Electron growth is
Nt
modified by shear induced resonance broadening, to y
.
Saturation,
NL
S
Ye = y , can occur at low turbulence levels, consistent with observations.
3
In a sense, the main point of this paper is to show that shear induced
resonance broadening is applicable to the electromagnetic problem and provides an effective mechanism for the saturation of such instabilities.
2
Of course, as before,
the shear also produces the instability due to the
same effect for finite radial diffusion, D
2
"'
V eipe
The specific example considered here is basically a drift wave, although
finite a
does modify the shear damping in an important way, and when
S> Me/m , most of the transport is due to the magnetic part of the fluctuation.5
We consider
a cylindrical tokamak, with equilibrium field com-
ponents B = B
e + B (r)e
o -z
6-
.
Since 8
4rnT /B
ae
e
<< 1, the compressional
mode of fluctuations may be neglected, and it suffices to consider the parallel
component of the vector potential A
series in poloidalk-nd toroidal angle
wave vector is given by k 11
factor.
=
k
expfime - in
"'
r - r
=
x
- iwtl, the parallel
(m - nq)/Rq, where q = rB 0RB
For each mode, the rational surface is at r
and we let x
k
Expanding the modes in a Fourier
.
,
such that q(r mn
be the distance from the rational surface.
k x/Ls.
The drift wave parity considered here
-ik 1
(D - T) is odd, where T, defined as A
=
=
mn,
Then
As in linear theory the modes have a definite parity
with respect to x.
so that E
is the safety
is P even,A
odd,
w/k 1 c, is even
in x.
The non-linear electron response to 0 and T may be computed in the
manner indicated previously.2
f
e
=
e
-- F
T F
e
Writing the electron fluctuation as
(1
--
w*e
--
)w +he,
where the first term is the adiabatic response (or the k 11 v
the drift kinetic equation),
let
(1)
-Co
limit of
the result of this "renormalization" is equiva-
t
lent to computing h efrom the diffusion equation.
6
4
2
-i(w -
- Dd /dx
2
=
] he
e.
FM
'
w*e)(0 - T )
(-
(2)
Now D contains magnetic as well as electrostatic contributions,
2 2
c k 12
D
-
2
-
A 11
j
R(w - k
v
) ,
(3)
c
m
B
m,n
where R =
v
=$
J
s a turbulently broadened
dt exp[i(w - k Iv,)t -.1 (11 v ) Dt 3
0
'V
c "V
2
1/2
(see eq. (10)), where vA = (B /4 rnmi)
Note, since A =--
resonance function.
2
that when ve /v
2
A
> 1 or a
e
A
> m /m ,
e i
of the fluctuations is dominant.
diffusion due to the magnetic part
The physical properties of shear induced
resonance broadening can be inferred directly from eq. (2) .
It is derived
by detailed renormalization techniques, but represents also a plausible
physical model.
Equation (2) is appropriate when the particle orbits in the presence
of the fluctuations exhibit the stochasticity property, thus justifying diffuTo verify the stochas-
sive behavior on the spatial scale of a wavelength.
ticity property we examine the orbit equations representing the deviation from
the linear or unperturbed trajectory
i
dt "r
d6r
=
ck 6
B(-mn
rmn -n
1
-A
c
n mn ) sin(me -
n$
- wt)
(4)
It;
d66
dt
where 6 =
_
Pq
-
dlnq
dr
t + 66,
v
6r +
r
o =
Pt,
and n6,
of order r
m6e,
is neglected.
8
The shearless contribution to the 6 fluctuation is small and can be discarded .
For small enough amplitude these orbits generate chains of resonances, or
5
islands, in the r 8 phase space, each chain centered at the radius given
by w = k
v
11
.
e < m/
In the electrostatic limit,
the radial island
widths are
L
ES
e
S
while for 8
e
1/2,
e
Te
> me/mi when the perturbation of the orbits is mostly magnetic,
the islands, for the drift wave parity, have width5
A
I
x Ls
=
k
rmn
BW~ 1/3
(6)
.I
where x
to be defined shortly, is essentially the Alfven layer thickness.
The spacing between island chains is basically a geometric condition giving
the distance between zeroes of m - nq(r) (ignoring the frequency in
For modes (m,n) and (m + V, n +n),
w - k 11 v
=
0).
by dq/dr6r
=
(yn - nm)/(n(n + n).
the distance is given
For mode numbers in the 102 range which,
as we find, dominate the spectrum, the numerator Un - nm can be made of
order one, so that approximately,
ARES
as noted previously 1.
or A
/ A
I RES
~
(7)
r/(m dlnq/dlnr),
The Chirikov condition for stochasticity A ES/ARE
I RES
> 1, will be satisfied for electrostatic fluctuation levels
on the order of e n/Te
10,
or magnetic
fluctuations of
rmn /B --10
Thus, for these high mode number fluctuations the rational surfaces are
packed very densely together and the stochasticity condition is, for all
practical purposes, always satisfied.
.
6
Returning to the eigenmode problem, the electron density and current
Finite gyroradius ion fluctuations
fluctuations can be computed from eq. (2).
Combining these, using quasineutrality
are computed from linear theory.
n
n
4w (J
*
= -$
and Ampere's law -V2 A
(d - 0) [A
=
-
-
i+
leads to
+
-
exj),
(8)
dx
i2kc
2
b) k
-
= (
[A
-
vA
dx
k
-
+
22 +
-
e
)].
(I+I
C
We have retained the standard notation wherever possible, defining
b = k 2
2,
d = (r0 - r 1 )(T + We/w),
2 = d
a (xI)
1
= T/T,
r (b) = e-b I (b),
(T + W
/w) r (W* /r)
/ 1XI = (W - W
(1-r
=
2
x
=
vy/k
,
x
-
k c
r- r 1 ), A= d 1 [l+T(l-r )-row*e
(Ln/Ls) 2,
k'
)d1(x /IxI)Z((x +ixc
],
a k /Ls
/IX).
Note that the underlaying electromagnetic modes persist in the presence
of stochasticity, at least for the odd parity of A11
To simplify the solution to Eqs. (8) and (9) we use an approximate algebraic relation between 0 and * (obtained by interpolating
between the asymptotic relations) in Eq. (8) to give a self-adjoint
equation for 0.
ment.9
The approximation is based on the following argu-
We are concerned with what is basically a drift wave, and the
principal dynamics are
(91
7
The primary finite beta effect (as is known from previous
described by eq. (8).
work 1 9 is the
the Alfven
reduction of the parallel electric field, or 0
layer, xA = w/k
as x
+
0 (this follows from eq. (9), by
requiring finiteness of the right hand side).
by combining (8) and (9) to give d 2 /dx2
x k
II
c*/w.
As x
-+
p, within
To represent this
vA, due to inductive effects.
algebraically, note that $ + 0
-
-E0
The large x behavior is found
c/wd )(v /c2 )(d2 /dx2 - b)
= (k
-, the derivative terms are dominant, implying *
'k
x 2d /x2
so by a simple interpolation
(x 2 + x 2
t2" /
2
where x
(10)
2
= x d .
approximate eigenmode equation is then obtained from eqs.
The
(10) and (8).
2
dx
2 2
x (A - y x
2
(D8
-
x
2
2
+ a ) + (x2
e
= 0
(11)
2
+ x 2
2
Equation (11) passes to the usual electrostatic equation as x
0.
+
A quadratic form may be constructed by multiplying eq. (11) by D and
integrating over x to give
J x2
2
0 = +dx
x
~
x2
e
22
x2
x
2
1
(12)
+ x
Requiring the first variation of S to be zero gives eq. (11).
Since we are
interested in the dispersion relation near saturation, we pass to the limit
1'
(k
=W[
'
x (W - W
)TC.
2
)
2
-1/3
D-
< 1 of the electron response
,
or ae = i
=
a/2
-
r(-)d
4
Taking the normalized trial function to be 0 = (ahl/ex(-a2/2),
with a the variational parameter, and doing the integrals for
S
1 1-1
y 2 /2a + v
Vax < 1, yields
Ex y/ + A + a e, which again reproduces the electrostatic
8
form when x
The parameter a is determined by SS/6x = 0 - 1/2 +
+ 0.
+ Vx a-1/2 /2.
2/2a2
Dominance of the first two terms gives the Weber trial function.
> P, or Se > (Ln /Ls)
However, when wx 2
(1 +
T)
-1 (8b)-1/2, which is typi-
Tr
cally satisfied for tokamaks, the last two terms dominate and we obtain
2
al/2
/
x
as the variational parameter.
Here eT
is chosen
uniquely as the cube root of -1, by applying the outgoing wave boundary condition.
The dispersion relation now follows as,
- ( T x V) 2/3 +
- =4(5x
S = 0S
A+i [
Note that the primary finite a
x
(13)
3/2
-1l13
1 1
2/3
2xP) + ir(-) d (w - w*)TI'
313/2
effect is to modify the shear damping.
Finite
pushes out the eigenmode, enhancing the outward convection of wave energy;
the resultant damping involves a geometric mean of x
and i.
At marginal stability, the frequencies are real, and the real and
imaginary parts of eq. (13) give the frequency, w =
coefficient, D = D(k ),
Here D(k ),
respectively.
w(k ),
and diffusion
which is interpreted as
the amount of diffusion necessary to stabilize mode k , is given by
Te
D = .07 (
4 me
)
Te+Ti
'
Ls 2
)
(
e
n
v P12
v
(14)
Ln
This is not yet the transport coefficient since it depends on the wave parameter,
k
.
Actually ion collisions, as well as higher order spatial turbulent
broadening effects and ion non-linearities
act to give
additional damping for b > 1, with the consequence that D(b) will have a maximum near b
= 2.
The dependence of b
scaling of D very weakly.
on plasma parameters will effect the
For convenience,
in practical applications, we simply put b = 2 in eq. (14) which gives the
formula quoted in the abstract.
F
9
The D of eq. (14) is an electron test particle diffusion coefficient.
When inserted in the electron kinetic equation, appropriately using the ambipolar potential12, one can obtain transport equations in the usual way.
The transport equations in Ref. (12) apply to the present case when
For crude estimates D equals the electron thermal
a > m /mi.
e i*
e
conductivity, Xe, and particle diffusion is not anomalous (ion diffusion
being slow for kI p
> 1).
Rewriting (14) to display the scaling (with lengths in cm. and temperatures in eV, V the ion mass in units of the proton mass), gives
6
Dl
1
D = 2.65 x 1016
1
/
np1/2
Te
T +TT I
which is independent of magnetic field.
T 3/2
T(5
T
For T
L
2
[Li
=
(15)
Ln
Ti, eq.
unfavorably with temperature, but if the dependence on T
(15)
=
scales
T IT
is
considered, (15) is consistent with recent observations on PLT13 where
confinement improved with increased T .
The behavior with ion mass,
D T p-1/2 is in agreement with ISX-A measurements. 1
Further, eq. (15)
shows a tendency for improved confinement with decreased aspect ratio,
(note L
ns/
= (Rq/r) dln n/dlnq is approximately R/a) a feature which,
15
qualitatively, has been seen in the data.
Note that
103
m /m
e i
(L /Ln)3I
s
n
2
v
equation (14) can be written D = Le
.
c2
This, to the extent that the factor
is one (it is of this order for tokamaks), is the
[6 x
n brackets
Ohkawa 16formula.
10
Of course, the physics underlying this diffusion is much different, being due
to low frequency modes, as is the scaling with ion mass, the T /T
ratio, q,
and several other parameters.
Finally taking Alcator 17 profiles at r = 5 cm, Ls a 100 cm., Ln a 5 cm.,
and T = 1 KeV, and defining TE
2
aL 2/D,
we find TE
=
2.4 x 10
18
which is within 50% of the Alcator empirical scaling law1 8
-19
2
naL 2
-19
E = 3.77
x
10
We might note that this transport mechanism is also consistent with the interpretation of the soft X-ray anomally given in Ref.
(12),
and that the absolute
value of D from eq. (14) agrees very well with that found empirically from
the X-ray data.
Obviously, the theory as described in this paper is not accurate to
50%, and many omitted effects and questions about the finite
problem remain to be considered.
8
e eigenvalue
However, the main attributes of Eq. (14),
namely absolute magnitude and density scaling, have been verified by the
numerical solution of Eqs. (8) and (9).
The significant point is that,
when saturated by shear induced resonance broadening, the simplest, almost
archetypal, drift instability, the universal mode, gives inverse density
scaling of the anomalous thermal conductivity.
One then feels that this
feature will be retained when complicating effects are added, and that
on this basis, the established empirical scaling, TE c n, can be understood.
2
naL
References
1.
A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38 (1978).
2.
S. P. Hirshman and K. Molvig, ORNL/TM - 6620, Phys. Rev. Lett. (to be
published).
3. T. H. Dupree, Phys. Fluids 10, 1049 (1967).
4.
T. H. Dupree and D. J. Tetrault, Phys. Fluids 21, 425 (1978).
5.
J. D. Callen, Phys. Rev. Lett. 39, 1540 (1977).
6.
K. Molvig and S. P. Hirshman, MIT Plasma Fusion Center Research Report No.
PFC/RR-78-7 (unpublished). This report gives adetailed calculation including
higher orders terms (not contained in eq. (1) ) like the "a" term of Ref. 4.
These terms do not alter the variational calculation given in this paper.
7.
See Refs. 2 and 6 for further explanation.
8.
This is typical for problems considered in the theory of stochasiticity,
e.g. G. M. Zaslavskii and B. V. Chirikov, Sov. Phys. Uspekhi 14, 549 (1972).
[Usp. Fiz. Nauk 105, 3 (1971)].
Motivated by a related but different approximation in S. M. Mahajan
and D. W. Ross, Fusion Research Center Report 159, The University of
Texas at Austin, 1977, (unpublished).
9.
10.
P. J. Catto, A. M. El-Nadi, C. S. Liu, and M. N. Rosenbluth, Nucl.
Fusion 14, 405 (1974).
11.
J. A. Krommes (Private Communication, 1979)
12.
K. Molvig, J. E. Rice, and M. S. Tekula, Phys. Rev. Lett. 41, 1240 (1978).
13.
H. P. Eubank, Bull. APS 23, 745 (1978).
14.
M. Murakami, et. al., Plasma Confinement in the ISX-A Tokamak, paper
IAEA-CN-37-N-4, Innsbruck (1978).
15.
D. L. Jassby, D. R. Cohn, and R. R. Parker, Nucl. Fusion 16, 1045 (1976).
16.
Ohkawa, T., General Atomic Report, GA-Al-4433, May 1977, and
17.
M. Gaudreau, et. al.,
18.
A. Gondhalekar, D. Overskei, R. Parker, and J. West, MIT Plasma Fusion
Center Report PFC/RR-78-15 (1978) (submitted to Nuclear Fusion).
Phys. Rev. Lett. 39, 1266 (1977).