Published by Fusion Energy Division, Oak Ridge National Laboratory
Building 5700 P.O. Box 2008 Oak Ridge, TN 37831 -6169, USA
Editor: James A. Rome Issue 119
E-Mail: jar@ornl.gov Phone (865) 482-5643
On the Web at http://www.ornl.gov/sci/fed/stelnews
April 2009
Numerical investigation of
electron orbits in the Columbia
Non-neutral Torus
1. Introduction
The Columbia Non-neutral Torus (CNT) is a simple stellarator
made of only four planar coils and dedicated to the
study of plasmas of arbitrary neutrality confined on magnetic
surfaces [1,2]. This is not an optimized stellarator,
and it exhibits large variations in magnetic field strength
on surfaces. However, confinement is expected to be good
because of the large radial electric field created by charge
imbalance in non-neutral plasmas [3, 4].
In particular, for pure electron plasmas confined on magnetic
surfaces the electric potential O is dictated by a Poisson-
Boltzmann equation [3]:
V2d) = ^ ( V ) e x p f - £ ®
£0
(1)
where \\i is a label for magnetic surfaces, and N(\\j) is a
function constant on surfaces which, along with the exponential
term, determines the density. The electron temperature,
r eO) is also constant on surfaces due to fast
streaming of electrons along field lines. In CNT TE is typically
around 4 eV. A 3-D code was developed to solve Eq.
(1) in the nontrivial geometry of CNT [5]. This code
shows that the potential varies significantly on magnetic
surfaces when there are few Debye lengths in the plasma
(a < XD), but tends to be constant on surfaces when many
Debye length are present (a > 10^D). However, the equilibria
are quite sensitive to electrostatic boundary conditions.
In CNT, in the absence of the good conductor
surrounding the plasma that was recently installed, and
even for XD = 1.5 cm « a « 15 cm, potential variations
on the outer surfaces are significant. This should be contrasted
with quasineutral plasmas in equilibrium on magnetic
surfaces in which potential variations on surfaces are
essentially absent. Potential variations may only be present
dynamically, and at lower values than those considered
in this article, in the form of electrostatic turbulence.
Because of the electron space charge, all equilibria exhibit
very strong negative electric fields (e|AO| /TE » 1). The
beneficial influence of a moderate ambipolar electric field
on the confinement of quasi-neutral plasmas in stellarators
was established long ago [6]. Here we investigate the
effects of very strong electric fields on confinement. A
purely radial electric field greatly improves the quality of
the orbits, as expected; but because the electric field is so
strong, toroidal resonances appear at moderately low values
of the magnetic field, leading to unconfined orbits. In
addition, potential variations on surfaces add to the complexity
of the orbits and also lead to bad orbits.
In this issue...
Numerical investigation of electron orbits in the
Columbia Non-neutral Torus
The confinement of pure electron plasmas in the
Columbia Non-neutral Torus (CNT) is expected to be
good because of the very large radial electric field,
resulting from space charge, which closes the orbits in
the poloidal direction. However, the confinement is
limited by the presence of unconfined orbits. The
radial electric field is so large that E x B rotation can
lead, at low B fields, to toroidal resonances. In addition,
variations in the electric potential on magnetic
surfaces, inherent to CNT equilibrium, add to the complexity
of the trajectories and can also lead to bad
orbits. We have written a code to investigate electron
orbits in the magnetic and electric fields expected in
CNT. Results of the calculations are presented. A
more detailed discussion will be submitted to Physics
of Plasmas 1
Motojima succeeded by Komori at NIFS
After six years of service, Professor Osamu Motojima
will retire on March 31, 2009. He will be succeeded by
Professor Akio Komori 8
All opinions expressed herein are those of the authors and should not be reproduced, quoted in publications, or
used as a reference without the author's consent.
Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the U.S. Department of Energy.
2. Electron orbits
A. Boozer flux coordinates
We make use of Boozer flux coordinates (\|/, 0, (p) [7]. \|/
is a radial coordinate proportional to the toroidal magnetic
flux, with \\f = \\rb at the plasma boundary. 0 and cp are
poloidal and toroidal angles, respectively. In the (cp,0)
plane, field lines are straight with a slope i(\|/), the rotational
transform. These coordinates are derived in CNT
using simple integration along the field lines, as described
in Ref. [8].
Among the many advantages of using Boozer coordinates
is that the drift motion depends only on the strength of the
magnetic field, not on its direction. CNT has 2-fold periodicity
and stellarator symmetry about the point ((p,0) =
(0,0), meaning that the magnetic field strength can be
described as B = 'Zbmncos(mQ - 2«cp). The n = 0 components
of B are the toroidal components, whereas the n ^
0 components are the helical components. For an accurate
description of the magnetic field we keep about 60 terms
in this Fourier series. A magnetic field strength of 0.1 T is
typical of CNT and is used in this article, unless otherwise
stated.
B. Choice of electric potential
We will focus our attention on three different electrostatic
potentials. In Section 3, we investigate the electron orbits
in the absence of an electric field, 0 = 0. This represents
the situation where there is negligible space charge and
serves to illustrate the quality of the orbits given the magnetic
topology without electrostatic (primarily E x B)
effects. In Section 4, we investigate the orbits in the case
of a strong electric potential, constant on the magnetic surfaces.
This is an idealized situation that should lead to
vastly improved confinement. In Section 5, we investigate
an electrostatic potential that has significant variations on
the magnetic surfaces. The potential chosen is a reasonably
accurate representation of the actual electrostatic
potential in CNT and gives rise to complicated drift orbits.
3. Electron orbits with no electric potential
As mentioned above, CNT has a very simple coil configuration.
Thus its magnetic topology is very different from
those of highly optimized stellarators such as W7X [10].
In particular, there are huge variations in B on surfaces, as
shown in Fig. 1.
0.12
0.10
0.08
0.06
0.04
-0.4 -0.2 0.0 0.2 0.4 Mag. Field (T)
0I2TI
Fig. 1. Map of B on the surface \\i = 0.5\\fb. One clearly distinguishes
the large helical variations in B and the two
troughs centered on 0 = 0. 0 = 0 corresponds to the outboard
(low B region) of the torus, whereas 0 = ±n corresponds
to the inboard (high B region) of the torus.
Because of these large variations in B there exists a very
large fraction of trapped electrons, most which are helically
trapped. Because these electrons are trapped in magnetic
wells, they cannot take advantage of the i-induced
poloidal rotation. These trapped electrons stay localized in
the poloidal direction and magnetically drift out of the
torus in a time
eaRB
mv
(2)
where a is the minor radius and R the major radius. In
CNT, a «13 cm, R& 22 cm. So for an electron with kinetic
energy Wk = 4 eV in a B = 0.1 T magnetic field, this estimate
yields ?ioss « 0.4 ms. The typical orbit of a helically
trapped electron is given in Fig. 2.
Stellarator News -2- April 2009
\
/ / / \ X \ / / ' \ • \ / • \ ' \
A ' / N
/ 1 ^ ' X
I t ' ^ x" \
1 ' ' '
I 1 1 ' il
s / \ \ ^
/ V / \ \
~ - c / N v X
/ v / V \ ^ \
/ / ^ V " 1 \ \ |
I t 1 „ ! ^, ' I 1 1 I » * ^ ** "
I * 1 x '
/ ' I 1
J ' / / \ " \ * ' A V J / - ' /
\ \ ^ * "• -- ~ \ % / /
\ N ' lib y /
\\ \ / \ ^s/ f „ - - "* \ '>£7 /
/ V
Fig. 2. Orbit of an electron in the absence of electric field.
The electron, born at a minimum of B, is helically trapped
and quickly lost. Color corresponds to time: blue is t = 0,
red is t= f|oss.
Because of the large fraction of trapped electrons we
expect direct losses to be large. And because all surfaces
are subject to variations in B, all surfaces have a potentially
large loss cone. To assess how bad the confinement
of single orbits is, 1000 electrons are started on a surface
and followed until they leave the confinement region, i.e.,
they cross the last closed flux surface. The electrons are
sampled with random poloidal angle, toroidal angle, and
pitch, but all have kinetic energy W^ = 4 eV. In Fig. 3 we
plot the fraction of confined electrons as a function of time
for different surfaces. We can observe that even deep in
the plasma, more than half of the electrons are lost.
« 1.0
0.4 0.6
Time (ms)
Fig. 3. Fraction of confined electrons vs time on different
surfaces in the absence of electric field. 1000 4-eV electrons
are initially started on each surface. At right are the
corresponding surfaces seen on a cross section of the
plasma.
An electron born with a pitch X = vy/v at a location
(\|/, 0, cp) is magnetically trapped on its birth surface if
W<xm a x ( V , e , c p ) = / i - y ^ y ) , (3)
V £max(v)
where Bmax(i|/) is the maximum of the B field on the surface
\|/. However, electrons with a pitch close enough to
A,max
a r e o n l y toroidally trapped, not helically trapped.
Toroidally trapped electrons, although lost in general, are
lost less quickly than helically trapped electrons. This is
illustrated in Fig. 4.
Wt = 4T,
1 e-1
1e-2
Ums)
Fig. 4. Loss cone structure in the absence of electric field
at a minimum of B on the surface vj/ = 0.5\i/b The color (log
scale) denotes the time it takes an electron to leave the
confinement region. A 5-ms confinement time is considered
to be infinite confinement time. One can clearly distinguish
the region of helical trapping, \X\ < IA,hell, from the
region of toroidal trapping, IA,hell < \X\ < IA,maxl.
4. Electron orbits with electric potential constant
on surfaces, 0(\j/)
A. Orbit confinement
If one adds a finite electrostatic potential that is constant
on each magnetic surface, but varies from surface to surface,
O = 0(\|/), this adds a purely poloidal E x B drift to
all particles. Because the electric field is negative, this
poloidal rotation is in the positive 0 direction, as is the
poloidal rotation induced by the rotational transform i for
forward-passing electrons. However, for i to provide
poloidal rotation, the electron must travel along the field
lines, whereas the E x B rotation is also effective for
trapped electrons. Hence the E x B drift can help to close
the orbits of trapped electrons. As an illustration we give
in Fig. 5 the orbit of the helically trapped electron in Fig.
2, this time with a strong radial electric field.
This process is effective if rloss, the typical loss time of an
electron, is long compared to tE x B, the half-poloidal transit
time due to the E x B drift:
lE x B
na
E x B
7i a
AO
(4)
Stellarator News -3- April 2009
Fig. 5. Orbit of a helically trapped electron in a strong,
purely radial electric field. The electron stays helically
trapped but the E x B drift effectively closes the orbit in the
poloidal direction.
As mentioned earlier, in CNT eAO » TE so that for a thermal
electron
E x B
^loss
2n a Wk
R MO
« 1 (5)
and closing of the orbits through E x B drift is very effective.
This is illustrated in Fig. 6 where we plot the loss
cone at a minimum of B on the surface \j/ = 0.5\|/^.
Wk = 4T.
Fig. 6. Loss cone structure in the case of a strong, purely
radial electric field at a minimum of B on the surface y =
0.5\|/b. The color (log scale) denotes the time it takes an
electron to leave confinement. A 5-ms confinement time is
considered to be infinite. All trapped electrons are perfectly
confined by the Ex B rotation (compare with Fig. 4).
B. Direct losses
Only electrons that have enough kinetic energy and are
born close enough to the plasma boundary can escape confinement
before closing their orbits in the poloidal direction.
This is confirmed numerically by running a
simulation similar to the one presented in the previous section,
where we start 1000 4-eV electrons on different \j/
surfaces and keep track of the fraction of confined electrons
in time. The initial sampling is as described before.
Results are presented in Fig. 7, showing that on the \j/ =
0.5 surface all particles are confined. In the edge region,
there are some losses of energetic particles; see Fig. 8,
which shows the loss cone for particles starting on the \\j =
0.9 surface, near the plasma edge. The loss cone in this
case is somewhat similar to that without an electric potential,
but now significant kinetic energy is needed even for
the deeply trapped particles before they can escape.
Fig. 7. Fraction of confined electrons vs time on different
surfaces in the case of a strong and purely radial electric
field. 1000 4-eV electrons are initially started on each surface.
Only electrons born very close to the last closed surface
are lost.
Fig. 8. Loss cone structure in the case of a strong, purely
radial electric field at a minimum of B on the surface \j/ =
0.9\|/b. The color (log scale) denotes the time it takes an
electron to leave confinement. A 5-ms confinement time is
considered to be infinite. Only high-energy helically
trapped electrons are lost.
Stellarator News -4- April 2009
C. Resonances
Because E x B rotation is in the positive 0 direction, this
rotation adds to the i-induced poloidal rotation for copassing
electrons (k > 0), whereas it subtracts from the tinduced
poloidal rotation for counter-passing electrons
(k< 0). For very strong electric fields, these two rotations
can cancel out for counter-passing particles, leading to a
toroidal resonance [11]. The E x B velocity is
v Ex B
E
B'
AO
aB
and the i-induced velocity is
v> = v \ \ - b x £ 1 vw
(6)
(7)
where s is the inverse aspect ratio 8 = a/R. Estimating AO
with Poisson's equation and Vy « vth = J f 7m yields the
resonance condition:
NDpL ~ l&a > (8)
where ND = a/XD is the number of Debye lengths in the
plasmas and pL is the electron Larmor radius. The resonance
condition can be fulfilled for typical conditions of
operation in CNT. For the potential we have chosen here, a
resonance is observed with B ~ 0.02 T, which is a typical
magnetic field in CNT.
2.5
2.0
1.5
1.0
0.5
0.0
>
>
>
> -4
2.5
2.0
1.5
1.0
0.5
0.0
Fig. 9. Deterioration of the loss cone resulting from resonant
electrons in the case of a strong and purely radial
electric field. Both plots are at the same location at a minimum
of B on the surface vj/ = 0.7\j/b. Top: B= 0.1 T and is
far from resonance, bottom: resonances with B = 0.01 T.
Resonances degrade confinement because the resonant
particles do not rotate poloidally, see Fig. 9.
Because of the resonance, counter-passing electron orbits
are changed to banana-like orbits, see Fig. 10. These are
understood as follows. In low B regions the positive
E x B drift overcomes the negative i-induced rotation and
the net motion is in the positive 0 direction. But as the
electron explores the surface along the field lines, it goes
to regions of higher B, which reduces the E x B drift. The
parallel velocity, however, does not decrease appreciably
because electrons subject to such effects have very small
magnetic moments. This leads to a net negative 0 motion.
Averaged over a toroidal period, the electron experiences
a slow poloidal motion in one direction. Because of this
slow poloidal motion the radial drifts accumulate and the
electron moves radially out (in) consequently gaining (losing)
kinetic energy. At some point the kinetic energy is
large (small) enough to reverse this toroidally averaged
poloidal motion and the electron closes its banana-like
orbit. The width of these banana orbits is very large, and
electrons can be lost if the orbits intersect with the plasma
boundary.
Fig. 10. Two "banana-like" orbits starting at the same initial
location and with the same initial pitch (X = -1). The only
difference is the initial kinetic energy: 4 eV (left) and 16 eV
(right). Color denotes time.
We must emphasize the fact that although they look similar,
these banana-like orbits are very different from the
ones found in tokamaks. In tokamaks, banana orbits are
due to toroidicity and the direction of the shift from the
flux surface depends on the sign of the initial parallel
velocity of the particle. Here banana-like orbits are due to
l l e - 2
Ums)
i-induced
rotation
Wk = 16eV
ExB
rotation
Stellarator News -5- April 2009
resonances in the poloidal motion, and occur for counterpassing
electrons only; the direction of the shift depends
on the initial parallel kinetic energy of the electron. If the
electron is born with little parallel kinetic energy, its initial
poloidal motion is in the positive 0 direction, whereas if it
is born with a lot of parallel kinetic energy, its initial
poloidal motion is in the negative 0 direction.
5. Electron orbits with potential nonconstant
on surfaces
We now turn to the most complicated case of an electrostatic
potential depending on all three space coordinates,
as is the case in CNT. Although potential variations on
surfaces are essentially absent in quasineutral plasmas in
equilibrium, they are inherent to non-neutral plasmas in
equilibrium on magnetic surfaces. In quasineutral plasmas,
potential variations may be present dynamically and at
lower values in the form of electrostatic turbulence.
Variations in the electric potential in the poloidal direction
create radial E x B drift. This drift does not depend on the
kinetic energy of the electron as magnetic drifts do. Even
low-energy electrons can make significant radial excursions.
And by doing so they can pick up kinetic energy
from the electric field and effectively increase their magnetic
drifts.
Fig. 11. Complicated orbit of an electron in the real potential
of CNT, non-constant on magnetic surfaces. The electron
born at \|/ = 0.5\i/b with = 4 eV ends up being lost.
Color corresponds to time: blue is t= 0, red is t= fout.
The combination of all these effects makes analytical calculation
impractical. The complexity of the orbits is illustrated
in Fig. 11, where we plot the trajectory of the same
electron as in Fig. 2 and Fig. 5 but in a potential
0(\|/, 0, cp) typical of CNT before the installation of conducting
boundaries.
Numerical integration of the orbits of 4-eV electrons
started on different surfaces shows that there exists a large
fraction of unconfined orbits exists even deep inside the
plasma (see Fig. 12).
Fig. 12. Fraction of confined electrons in the case of an
electric potential with significant variations on surfaces.
1000 4-eV electrons are started on each surface. Losses
are significant even deep in the plasma.
Intuitively this is understood as follows. Without electric
field and neglecting magnetic drifts, electrons circulate all
around magnetic surfaces following field lines. Now if
there is an electric field, electrons also experience E x B
drift on equipotential surfaces because
vExB - E x = - V O x B/B2 , so that
v £ x g - V O = 0 . And if the equipotential surfaces do not
match magnetic surfaces, electrons can undergo E x B
drift from one magnetic surface to another. After a few
steps of jumping from one surface to the other an electron
can find its way out of the plasma (see Fig. 13). Simulations
show that this process is quite effective and can
remove electrons from the plasma in tenths of microseconds
(see Fig. 12).
Stellarator News -6- April 2009
Constant Potential
(CD = Const.)
Trajectory of
a particle
Magnetic Surface
(ip = Const.)
Fig. 13. Conceptual sketch of an orbit when there is a mismatch
between equipotentials and magnetic surfaces.
Electrons jump from magnetic surface to magnetic surface
by drifting on equipotentials and can find their way out of
the plasma.
6. Conclusion
The strong radial electric field created by charge imbalance
in CNT has been shown to greatly improve the orbits,
as expected. However, two main mechanisms have been
identified that can create bad orbits in CNT despite this
large radial electric field. First, even when the electric
potential is perfectly constant on surfaces, toroidal resonances
occurring at low magnetic field B that destroy orbit
confinement. Second, nonconstancy of the electric potential
on surfaces also creates bad orbits. The electric potential
used in Section 5 is the potential in CNT with
electrostatic boundary conditions imposed by the coils and
the vacuum chamber. The recent installation of a conducting
boundary surrounding the plasma should then considerably
improve the orbits. Indeed, even if this does not
make the potential perfectly constant on surfaces, it certainly
greatly reduces the variations on surfaces. Experimental
and numerical work is currently being carried out
to characterize more precisely how the potential at the
plasma boundary is affecting the orbits and transport in
CNT.
Acknowledgments
This work is supported by the National Science Foundation-
Department of Energy Partnership in Basic Plasma
Science under Grant NSF-PHY-06-13662.
Benoit Durand de Gevigney, Thomas Sunn Pedersen,
and Allen H. Boozer
Applied Physics and Applied Mathematics Department
Columbia University, New York, New York 10027
References
[1] T. Sunn Pedersen, A. H. Boozer, J. P. Kremer, R. G. Lefrancois,
W. T. Reiersen, F. Dahlgren, and N. Pomphrey,
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[12] J. Fajans, Phys. Plasmas 10, 1209 (2003).
Stellarator News -7- April 2009
People
Motojima succeeded by
Komori at NIFS
Professor Akio Komori was appointed to be the next
Director-General of the National Institute for Fusion Science
(NIFS) as the successor of Professor Osamu
Motojima, who completed his term of six years on March
31,2009.
Professor Komori earned a Ph.D. in nuclear fusion science,
plasma physics, at Tohoku University in 1978, and
performed research at Oak Ridge National Laboratory,
Tohoku University, and Kyushu University. He has
worked at NIFS since 1993, and has been the Director of
the Department of Large Helical Device Project since
His term of office as the Director-General will be 4 years,
from April 1, 2009 to March 31, 2013.
http://www.nifs.ac.jp/en/press/081218.html
2003.
Stellarator News - 8 - April 2009