Download On ion flow caused by the inductive electric field in a tokamak

On ion flow caused by the inductive electric field in a tokamak
P. Helander
Citation: Phys. Plasmas 8, 4700 (2001); doi: 10.1063/1.1402172
View online: http://dx.doi.org/10.1063/1.1402172
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PHYSICS OF PLASMAS
VOLUME 8, NUMBER 10
OCTOBER 2001
On ion flow caused by the inductive electric field in a tokamak
P. Helander
EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, United Kingdom
共Received 9 July 2001; accepted 16 July 2001兲
It was recently pointed out that the inductive electric field in a tokamak can give rise to toroidal and
poloidal plasma rotation comparable to that sometimes observed in experiments. Here it is shown
that the flow velocity of heavy impurity ions, which is normally what is measured, is lower than that
of the bulk plasma for rotation produced in this way. If the bulk ions and electrons are in the banana
regime, the impurity rotation is at most about two thirds of the bulk plasma rotation and decreases
with increasing effective ion charge (Z eff) and distance from the magnetic axis.
关DOI: 10.1063/1.1402172兴
n e ⯝n i . In the experiments reported in Ref. 5 the impurity
charge is z⫽19, and as is typical in these and many other
experiments, we assume that the bulk ions and electrons are
in the banana regime of collisionality while the impurities
are collisional. Under the influence of a toroidal electric field
each species satisfies the flux-surface averaged 具 ¯ 典 force
balance equations2
Understanding the causes of tokamak plasma rotation—
both toroidal and poloidal—is important since it sheds light
on the transport mechanisms that are operational in the
plasma. For instance, transport barriers are frequently observed to be associated with sheared rotation, and it is widely
believed that sheared flow is the cause of these barriers. Conversely, plasma transport properties affect the rotation by,
e.g., sustaining it against viscous damping.
The rotation velocity of each species a in an axisymmetric tokamak is given by an expression of the form
具 B•“•⌸a 典 ⫽ 具 B 共 R a 储 ⫹n a e a E 储 兲 典 ,
具 B•“•⌰a 典 ⫽ 具 BH a 储 典 .
Va ⫽ ␻ a R ␸ˆ ⫹u a ␪ B,
Following Hirshman and Sigmar, we relate the viscosities ⌸a and ⌰a to the poloidal flux of particles and heat by
where ␻ a and u a ␪ are flux functions, R is the major radius, ␸ˆ
the toroidal unit vector, and B the magnetic field.1–3 This is
the most general expression for an incompressible flow tangential to flux surfaces. Note that the toroidal rotation has
contributions from both ␻ a and u a ␪ while poloidal rotation is
described by u a ␪ alone. Standard neoclassical theory relates
these quantities to linear combinations of radial density and
temperature gradients and 共for ␻ a ) the radial electric field. In
a recent paper,4 however, it was pointed out that the inductive electric field used to drive Ohmic current also contributes significantly to u a ␪ for bulk ions (a⫽i) in a pure hydrogen plasma, and this may be relevant to recent
observations in Alcator C-Mod of spontaneous rotation in
Ohmic plasmas.5 On the other hand, it is the rotation of
impurity ions that is measured in these 共and most other兲 experiments, from Doppler broadening of line radiation, and it
is well known that this velocity can be quite different from
that of the bulk plasma.6 It is the purpose of this Brief Communication to clarify this point by presenting a calculation
which extends that of Ref. 4 to account for the presence of a
heavy impurity species in the plasma and, at the same time,
simplifies the analysis by using the Hirshman–Sigmar moment formalism.2 From this calculation, it is evident how to
make the further generalization to an arbitrary number of
impurities of general charges and masses.
We thus consider a plasma consisting of electrons (e),
hydrogenic bulk ions (i), and heavy impurity ions (z). We
treat the impurity charge as a large expansion parameter, z
Ⰷ1, and assume that the impurity strength parameter ␣
⫽Z eff⫺1⫽nzz2/ne is of order unity so that n z z/n e Ⰶ1 and
1070-664X/2001/8(10)/4700/4/$18.00
冔冉
冓
␮ˆ a1
1
B•“•⌸a
⫽
2
␮ˆ a2
具 B 典 B•“•⌰a
and
冓 冔 兺冉
1
BR a 储
⫽
2
BH
B
具 典
a储
⫽
b
␮ˆ a2
␮ˆ a3
冊冉
ab
l 11
ab
⫺l 12
ab
⫺l 21
ab
l 22
冊
u a␪
⫽Ma •ua
2q a ␪ /5p a
冊冉
u b␪
2q b ␪ /5p b
兺b Lab •ub ,
冊
共1兲
where q a ␪ ⫽qa •“ ␪ /B•“ ␪ is the contravariant component of
the poloidal heat flux and p a ⫽n a T a the pressure of species
a. These expressions are obtained by expanding the distribution functions in Sonine polynomials and truncating after
two terms; as usual this approximation turns out to be accurate within a few percent, i.e., as accurate as the Coulomb
logarithm in the collision operator. In order to isolate the
inductive electric field as the single driving term, we have
assumed in Eq. 共1兲 that all radial gradients vanish, so that
␻ a ⫽0 for each species. 共This assumption is satisfied if the
density and temperature gradients are sufficiently weak and,
for the gradient of the electrostatic potential, i.e., the radial
electric field, if the plasma is viewed from a rigidly toroidally rotating frame of reference.兲 The coefficients
␮ˆ a j ⫽
3 具共 “ 储B 兲2典
␮aj
具 B 2典
and
4700
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Phys. Plasmas, Vol. 8, No. 10, October 2001
l ab
jk ⫽m a n a
冉
j⫺1,k⫺1
M ac
␦ ab 兺
␶ ac
c
⫹
On ion flow caused by the inductive electric field . . .
j⫺1,k⫺1
N ab
␶ ab
冊
Ma •ua ⫽
,
which can be looked up in Refs. 2 and 3, summarize all
kinetic information needed to evaluate the flows. Neoclassiˆ aj ,
cal effects are described by the viscosity coefficients ␮
which thus depend on collisionality and flux-surface geometry. In the banana regime
冉 冊
冉 冊
冉
冉
冊
3/2
2 2
with ␶ ab ⫽3(2 ␲ ) 3/2⑀ 20 m 1/2
a T a /n b e a e b ln ⌳, f t the effective
2
fraction of trapped particles, and f c ⫽1⫺ f t . For a tokamak
with circular cross section and small inverse aspect ratio, ⑀
Ⰶ1, the trapped particle fraction is f t ⯝1.46⑀ 1/2. The friction
coefficients are
m in i
Lii ⫽⫺
␶ iz
冉
1
3
2
13 &
⫹
4
␣
冉 冊
冉 冊
Lzi ⫽
m in i 1
␶ iz 0
Lie ⫽
m en e 1
␶ ee 0
Lee ⫽⫺ 共 1⫹ ␣ 兲
3
2
0
3
2
0
,
冊
冉 冊
冉
冊
Liz ⫽
Lzz ⫽⫺
m in i 1
␶ iz 0
⫽ ␣ ⫺1 Lze ,
m en e
␶ ee
m in i 1
␶ iz 23
,
冉
1
3
2
3
2
13
&
⫹
4
1⫹ ␣
0
0
,
0
␣ &⫹
冊
冉
冊
n ae a具 E 储B 典 / 具 B 2典
.
0
For the electrons, this reduces to
from which the electron flow velocity can be evaluated by
matrix inversion, yielding the usual neoclassical reduction of
electric conductivity due to trapping. The ion flows are then
obtained by solving the 4⫻4 system of equations
␮ˆ i1
0.533⫹ ␣
mn f
␮ˆ i2 ⫽ i i t ⫺0.625⫺1.5␣ ,
␶ ii f c
␮ˆ i3
1.386⫹3.25␣
3
2
Ea ⫽
兺b Lab •ub ⫹Ea ,
共 Me ⫺Lee 兲 •ue ⫽Ee ,
␮ˆ e1
0.533⫹ 共 1⫹ ␣ 兲
m en e f t
␮ˆ e2 ⫽
⫺0.625⫺1.5共 1⫹ ␣ 兲 ,
␶ ee f c
␮ˆ e3
1.386⫹3.25共 1⫹ ␣ 兲
冊
where
4701
15
2
,
冉
Mi ⫺Lii
⫺Liz
⫺Lzi
⫺Lzz
冊冉 冊 冉
冊
ui
Ei ⫺Lie •ue
⫽
,
uz
⫺Lze •ue
where we have noted that the viscosity is negligible for collisional impurities and that the electric force acting on the
impurities is smaller than their friction against electrons,
m e j 储 / ␣␶ ei
R ze 储
⬃
⬃zⰇ1.
n z zeE 储
n z ze 2 E 储
The motion of the impurities is thus simply determined by
the balance of the impurity-electron and impurity-hydrogen
friction forces. The first element of the right-hand side of Eq.
共2兲 is equal to the sum of the electric force on the bulk ions
and their friction against the electron population, and is
sometimes written as
具 共 n i eE 储 ⫹R ie 储 兲 B 典 n i e 具 E * B 典
⫽
,
具 B 2典
具 B 2典
,
where zero entries denote quantities of order (m i /m z ) 1/2Ⰶ1
or (m e /m i ) 1/2Ⰶ1. We do not need Lei and Lez as the electron
flow velocity is much larger than those of the ion species. As
far as the electrons are concerned, the ions are practically
stationary and the electron–ion friction force is independent
of the ion flow velocity.
We thus obtain the following system of equations for
each species:
共3兲
where E ⫽E 储 ⫹R ie 储 /n i e is the ‘‘effective electric field’’ felt
*
by a bulk ion.1 In a pure plasma 共no impurities兲 embedded in
a straight magnetic field this vanishes, E ⫽0, and the ions
*
do not ‘‘feel’’ the electric field; it is exactly cancelled by the
friction force from the electrons. The reason for this cancellation is that the electric force acting on the electrons must
necessarily equal their friction against ions. As pointed out in
Ref. 7, the situation is different in an impure tokamak plasma
since the electric force on the electrons is balanced not only
by friction against bulk ions, but also by the mirror force
共causing trapping兲 and friction against impurity ions. As a
result of these effects, the first row in Eq. 共2兲 equals Eq. 共3兲
with
ee
ee
兲 ␮ e2 ⫺ 共 ␮ e3 ⫺l 22
兲 ␮ e1
共 ␮ e2 ⫺l 12
3.96⫹2.59x⫹ ␣ 共 4.21⫹3.24x 兲 ⫹ ␣ 2 共 1⫹x 兲
␣
x
具 E *B 典
⫽
⫹
,
ee
ee
ee 2 ⯝
2
1⫹ ␣ 2.59共 0.65⫹x 兲共 1.44⫹x 兲 ⫹ ␣ 共 3.24⫹ ␣ 兲共 1⫹x 兲 1⫹ ␣
具 E 储 B 典 共 ␮ e1 ⫺l 11 兲共 ␮ e3 ⫺l 22 兲 ⫺ 共 ␮ e2 ⫺l 12 兲
where x⫽ f t / f c . The first term in this expression reflects the
fact that the total friction on electrons from both ion species
is proportional to Z eff⫽1⫹␣, where the term ␣ represents the
contribution from impurities so that, in a straight magnetic
field, they take up a fraction ␣ /(1⫹ ␣ ) of the momentum
imparted to the electrons by the electric field. The second
共2兲
共4兲
term in Eq. 共4兲 describes the effect of electron trapping. It is
clear that for realistic parameters in a tokamak, ␣ ⬃x⬃0.5,
the bulk ions ‘‘feel’’ a significant fraction 共more than half兲 of
the parallel electric field.
We now turn to the ion rotation, which is obtained by
solving the system of equations 共2兲. The resulting expres-
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4702
Phys. Plasmas, Vol. 8, No. 10, October 2001
P. Helander
sions are very complicated in general; we give only limiting
forms. In the limit of very few impurities, the rotation velocities of bulk and impurity ions, respectively, are
û i ␪ ⫽
⯝
û z ␪ ⫽
m in i具 B 2典
u
e ␶ ii 具 E 储 B 典 i ␪
3.98共 1.02⫹x 兲共 1.53⫹x 兲
,
共 0.65⫹x 兲共 1.44⫹x 兲共 2.17⫹x 兲
␣ →0,
m in i具 B 2典
u
e ␶ ii 具 E 储 B 典 z ␪
0.73共 1.5⫹x 兲共 3.84⫹x 兲
⯝
,
共 0.65⫹x 兲共 1.44⫹x 兲共 2.17⫹x 兲
␣ →0.
FIG. 1. Ratio of bulk ion and impurity ion poloidal flows as a function of
the effective fraction of trapped particles, f t , for different values of the
impurity strength parameter ␣ ⫽n z z 2 /n i . When f t or ␣ is large, the bulk ion
flow is much larger than that of the impurities.
Thus the rotation speed of trace impurities is thus about
2/3 of the plasma rotation near the magnetic axis 共where x
⫽0) and falls off toward the edge. If the impurities are more
numerous, their speed is even smaller when compared with
that of the bulk ions, as shown in Fig 1. In the Lorentz limit
13/4
,
û i ␪ ⯝
1⫹x
兲␣
共
1
,
û z ␪ ⯝
共 1⫹x 兲 ␣ 2
û z ␪ ⯝
3.78⫹3.83␣ ⫹ ␣ 2
,
共 1⫹ ␣ 兲共 2.41⫹ ␣ 兲共 0.75⫹1.95␣ ⫹ ␣ 2 兲
f t →0.
␣ →⬁,
␣ →⬁,
and close to the magnetic axis,
The impurity rotation is always in the same direction as
the bulk ion rotation but is considerably smaller if ␣ 共or, to a
lesser extent, f t ) gets large.
In contrast, the poloidal rotation caused by radial gradients usually calculated in neoclassical theory does not depend strongly on the impurity content and stays finite in the
limit ␣ →⬁. These flows are6
û i ␪ ⯝
3.25共 0.71⫹1.24␣ ⫹ ␣ 2 兲
,
共 1⫹ ␣ 兲共 0.75⫹1.95␣ ⫹ ␣ 2 兲
u i␪⫽
1.17⫹3.64␣ ⫹1.99␣ 2
d ln T i
IT i
,
2
e 具 B 典 1⫹0.46x⫹ 共 2.58⫹1.65x 兲 ␣ ⫹1.33共 1⫹x 兲 ␣ 2 d ␺
u z␪⫽
IT i
d ln p i 1.17⫹0.69x⫹ 共 3.64⫹2.47x 兲 ␣ ⫹1.99共 1⫹x 兲 ␣ 2 d ln T i
⫺
⫹
,
e 具 B 2典
d␺
1⫹0.46x⫹ 共 2.58⫹1.65x 兲 ␣ ⫹1.33共 1⫹x 兲 ␣ 2
d␺
f t →0,
冋
where I⫽RB ␸ and ␺ is the poloidal flux function, so that
B⫽I( ␺ )“ ␸ ⫹“ ␸ ⫻“ ␺ . Hence it was noticed in Ref. 6 that
in the edge pedestal, where the pressure gradient is very
steep, the poloidal rotation of impurities is larger than and in
the opposite direction to the bulk ion rotation.
In conclusion, our results may be summarized as follows. Considering a tokamak plasma consisting of collisionless electrons and ions and collisional impurities, we have
calculated the ‘‘effective electric field’’ 共4兲 acting on the ions
and its effect on ion flow. Although this effective electric
field vanishes for a pure plasma in a straight magnetic field,
this is not at all true in a realistic tokamak discharge. In fact,
for typical tokamak parameters E /E 储 ⲏ0.5. As pointed out
*
in Ref. 4, this electric field drives a poloidal and toroidal ion
flow, but as we have seen here the corresponding flow of
heavy impurities 共which is what is measured兲 is significantly
smaller than that of the bulk plasma. This behavior is the
册
opposite to that of rotation caused in the more orthodox neoclassical way by radial gradients, which tends to be larger for
impurities than for bulk ions, making the detection of rotation driven by the Ohmic field correspondingly more difficult.
The fact that E /E 储 is so large implies that it is not
*
unreasonable to expect that ion runaway may sometimes occur in tokamaks, as suggested by Furth and Rutherford.7
During normal tokamak operation the toroidal electric field
is too low for this, but just like runaway electrons tend to be
produced in tokamak disruptions, one could expect that ion
runaway could occur during reconnecting instabilities where
the toroidal electric field becomes very large. Work is in
progress to assess whether this can explain recent observations of bursts of high-energy ions following internal reconnection events in the Mega-Ampere Spherical Tokamak
共MAST兲.8
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Phys. Plasmas, Vol. 8, No. 10, October 2001
ACKNOWLEDGMENTS
Helpful discussions with P. J. Catto and R. J. Hastie are
gratefully acknowledged.
This work was funded jointly by the U.K. Department of
Trade and Industry and Euratom.
1
2
F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 共1976兲.
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On ion flow caused by the inductive electric field . . .
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4703
P. Helander and D. J. Sigmar, Collisional Transport in Magnetized Plasmas 共Cambridge University Press, Cambridge, in press兲.
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P. J. Catto, R. J. Hastie, I. H. Hutchinson, and P. Helander, Phys. Plasmas
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I. H. Hutchinson, J. E. Rice, R. S. Granetz, and J. A. Snipes, Phys. Rev.
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共1991兲.
7
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