Stochastic loss of alpha particles in a helias reactor
Abstract. It is shown that collisionless orbit transformation of the locally trapped particles to the locally passing ones and vice versa in the Wendelstein-line optimized stellarators results in stochastic diffusion of energetic ions. This diffusion can lead to the loss of an essential fraction of...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Stochastic loss of alpha particles in a helias reactor / Ya.I. Kolesnichenko, C.D. Beidler, V.S. Marchenko, I.N. Sidorenko, H. Wobig // Вопросы атомной науки и техники. — 2000. — № 3. — С. 22-24. — Бібліогр.: 9 назв. — рос. |
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| author | Kolesnichenko, Ya.I. Beidler, C.D. Marchenko, V.S. Sidorenko, I.N. Wobig, H. |
| author_facet | Kolesnichenko, Ya.I. Beidler, C.D. Marchenko, V.S. Sidorenko, I.N. Wobig, H. |
| citation_txt | Stochastic loss of alpha particles in a helias reactor / Ya.I. Kolesnichenko, C.D. Beidler, V.S. Marchenko, I.N. Sidorenko, H. Wobig // Вопросы атомной науки и техники. — 2000. — № 3. — С. 22-24. — Бібліогр.: 9 назв. — рос. |
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| description | Abstract. It is shown that collisionless orbit transformation of the locally trapped particles to the locally passing ones and vice versa in the Wendelstein-line optimized stellarators results in stochastic diffusion of energetic ions. This diffusion can lead to the loss of an essential fraction of energetic ion population from the region where the characteristic diffusion time is small compared to the slowing down time. The loss region and the magnitude of the loss can be minimized by shaping the plasma temperature and density profiles so that they satisfy certain requirements. The predictions of the developed theory are in agreement with the results of numerical modelling of confinementof a-particles in a Helias reactor, which was carried out in this work with the use of the orbit following code. The considered diffusion seems to represent the dominant mechanism of classical losses of a-particles in a Helias reactor.
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UDC 533.9
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 22-24 22
Stochastic Loss of Alpha Particles in a Helias Reactor
Ya.I. Kolesnichenko 1), C.D. Beidler 2), V.S. Marchenko 1), I.N. Sidorenko 2),
H. Wobig 2)
1) Scientific Centre "Institute for Nuclear Research", Kyiv, 03680,Ukraine
2) Max-Planck-Institut für Plasmaphysik, EUROATOM Association
D-85740 Garching bei München, Germany
e-mail contact of main author: ftd@nucresi.freenet.kiev.ua
Abstract. It is shown that collisionless orbit transformation of the locally trapped particles to the locally passing
ones and vice versa in the Wendelstein-line optimized stellarators results in stochastic diffusion of energetic ions.
This diffusion can lead to the loss of an essential fraction of energetic ion population from the region where the
characteristic diffusion time is small compared to the slowing down time. The loss region and the magnitude of the
loss can be minimized by shaping the plasma temperature and density profiles so that they satisfy certain
requirements. The predictions of the developed theory are in agreement with the results of numerical modelling of
confinementof α-particles in a Helias reactor, which was carried out in this work with the use of the orbit following
code. The considered diffusion seems to represent the dominant mechanism of classical losses of α-particles in a
Helias reactor.
1.Introduction
The lack of the axial symmetry is known to lead to
the loss of all locally trapped α -particles in
conventional stellarators. Therefore, a reactor based
on such systems is not possible. Several ways are
suggested to make acceptable the confinement of
alphas. One of them is to optimize the magnetic
configuration. This way is implemented in a recently
constructed stellarator Wendelstein 7-X and the
development of a Helias reactor [1,2]. In the
optimized stellarators (which can also be refered to as
Helias configurations) the effects of plasma
diamagnetism are sufficiently strong to make closed
and weakly deflecting from the magnetic flux surfaces
the contours of the longitudinal adiabatic invariant,
∫= dlvJ || . This implies that superbanana orbits will
not arise, and thus, the locally trapped particles, which
constitute the main fraction of escaping alphas in
conventional stellarators, will be confined in the
optimized systems. However, we will show in this
work that a considerable fraction of energetic ions can
be lost even in the optimized stellarators. The reason
for this is the stochastic diffusion of the so-called
"transitioning particles", i.e., particles whose orbits
are transformed from locally trapped to locally
passing ones and vice versa.
2. Diffusion of transitioning particles
We study the diffusion arising due to the following.
The adiabaticity of the particle motion in the phase
space breaks down near the separatrix between the
regions of the locally trapped and locally passing
orbits (see Fig. 1). Because of this the adiabatic
invariant J acquires a phase dependent jump each time
when a particle crosses the separatrix [3]-[5]. The
phases of the motion do not correlate for successive
transitions.
Therefore, the multiple crossings of the separatrix are
accompanied by the random walk of particles in the J
space, resulting in spatial diffusion. Note that stochastic
diffusion having the same nature may take place also in
tokamaks, where, however, it plays a minor role, being
associated with the presence of the ripple wells [6].
The corresponding diffusion coefficient can be written
as follows:
τ
〉∆〈
=
2)( r
D , (1)
here r∆ is the change of the particle radial coordinate
Fig. 1. Sketch of level contours of J for the locally
trapped particles (thin lines) and the 1=κ contour
(bold line inside the plasma) in a Helias. Locally
trapped particles moving along the constJ l =
contours become locally passing ones after crossing
the bold line.
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 22-24 23
caused by the orbit transformation, means
ensemble averaging, r is the effective flux surface
radius defined by the equation 2/2rB=ψ ; τ is the
characteristic time,
)(
2
1 pl τττ += , (2)
τl and τp are the characteristic times of the particle
motion in the locally trapped and passing states. The
factor 1/2 takes into account that a particle crosses the
separatrix twice per a full period of a hybrid passing-
localized orbit. The time τl is essentially the
precession time of a localized particle, whereas
∫
=
=−
=
)1(
)1(
2
2
/)/2(
κθ
κθ
θθτ &dPp where θ& is the frequency
of the poloidal motion of a passing particle, P is the
probability of the orbit transformation resulting in the
trap of a passing particle. The probability P and the
radial jump caused by the separatrix crossing can be
expressed in terms of the longitudinal invariant [3]-
[7]. All ingredients in Eq.(1) were calculated with
using the bounce-averaged equations of the particle
motion in the magnetic field
]cos)()cos()(
cos)()(1[ 0
θψεφθψε
φψεψε
th
m
N
NBB
−−−
++=
(3)
where ψ, θ, φ are the magnetic flux coordinates with
ψ the toroidal magnetic flux; ε0 describes the change
of the vacuum magnetic field due to finite β; εm, εh,
and εt are the amplitudes of the mirror, helical, and
toroidal harmonics, respectively, εm being dominant in
the plasma core; N >> 1 is the number of the field
periods along the large azimuth of the torus. As a
result, a diffusion coefficient was obtained. Its
magnitude for mht εεε <<<< , which is the case in
the plasma core of a Helias, can be evaluated as
'
0
2
32
424
ε
ε
ερω
π m
hBB
arN
R
D ≈ , (4)
where ωB is the energetic ion gyrofrequency, ρB is the
gyroradius; a, R are the minor and major radius of the
torus, respectively; dxd /0
'
0 εε ≡ with x=r/a.
Equation (4) is relevant to particles with the pitch-
angle parameter 0~1/ εεµα +−Ε≡ mB .
The condition that an energetic ion will be lost
because of diffusion (rather than displaced within the
plasma) is sd ττ << , where τs is the characteristic
slowing down time, and τd is the diffusion time
defined by
D
ra
rd
2)(
~)(
−
τ . (5)
It is of importance to know the fraction of
transitioning particles, which is essentially the
stochastic-diffusion-induced loss fraction of alpha
particles when the condition sd ττ << is satisfied.
This quantity relevant to a flux surface is given by
min
min
max
max
11
)(
α
α
α
α
ν
+
−
+
=r , (6)
where
.
,
0max
0min
thm
thm
εεεεα
εεεεα
+++=
−−+=
3. Alpha particle loss in a Helias reactor: predictions of
a theory and numerical simulations
Using Eqs. (4)-(6) let us evaluate the diffusion time and
the fraction of transitioning α-particles in a Helias reactor
with N=5, R/a =10, and β ~ 5% [8]. At first, we make a
simple estimate by approximating Fourier harmonics of
the magnetic field in Eq. (1) as follows:
.08.0,05.0
,08.0,1.0 2
0
xx
x
ht
m
==
==
εε
εε
(7)
Then at r=a/2 we obtain:
.%15)2/(,
5
)2/(
4
≈
≈ a
a
a
BB
d ν
ρω
τ (8)
When B=5 T, Eq. (8) yields sd 02.0≈τ for
30/ =Ba ρ and sd 06.0=τ for 40/ =Ba ρ . On the
other hand, the slowing down time of a 3.5 MeV α -
particle in a plasma with the electron
density 3201032 −×−= mne and the temperature
keVT 1510 −= is sec1.0~sτ . This means that an
essential fraction of alphas will be lost to the wall from a
flux surface of the radius r/a=0.5 in a system with
30/ =Ba ρ , but particles can hardly escape from the
r/a=0.5 surface when 40/ =Ba ρ . Furthermore, if
profile shapes of the plasma density and temperature
change in a way that τs strongly decreases with the radius,
the condition sd ττ << may violate near the plasma edge.
Then energetic ions will diffuse to the periphery and
thermalize in that region. In this case the main effect of
the stochastic diffusion will be the broadening of the
radial profile of the power deposition of energetic ions
rather than their loss. As es nT /2/3∝τ , this will be the
case when the temperature strongly decreases with r,
whereas the ne(r) profile is flat.
Now we calculate numerically the diffusion
coefficient, the fraction of transitioning particles and the
diffusion time, using corresponding equilibrium data. The
obtained dependence of τd on α at r/a =0.5 is presented in
Fig. 2. We observe that the dependence of τd on α has a
rather flat minimum around 0εεα += m where D was
expected to be maximum; the magnitude of τd is in
qualitative agreement wi th the estimates above.
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 22-24 24
Fig. 2. The diffusion time versus the pitch
parameter 1/ −≡ BE µα at ar 5.0= in a Helias
reactor with %7.4=β .
These results are in agreement with the
confinement time and the fraction of lost particles
calculated numerically using a guiding center code in
Ref. [9]. In the present work, similar calculations
were carried out for a Helias reactor [8]. The
simulations began with an ensemble of 250 α-
particles, all of which have the same initial energy E0
=3.52 MeV but are assigned to different starting radii
according to the appropriate birth profile; the
remaining spatial and velocity coordinates were
chosen randomly. The plasma parameters and profiles
were taken from 1-D numerical simulations for two
scenarios: first, a high-density (central electron
density 320103)0( −×= mne ), low-temperature
(central temperature T(0)=15 keV ) case similar to
that considered previously [9]; second, low-density
( 320105.1)0( −×= mne ), high-temperature
(T(0)=25 keV ) case. These scenarios lead to nearly
identical birth profiles for the α-particles but slowing-
down times which differ by a factor of four. Given the
arguments above, one must therefore expect greater
losses of fast α-particles for the low-density, high-
temperature scenario as even those particles born near
the plasma center (with ss 46.0≈τ ) can diffuse to
the plasma edge before slowing down. The carried out
calculations confirmed this. It was found that (i) for
the high-n, low-T scenario 19 particles are lost during
a simulation time of 0.12 s leading to a lost-energy
fraction of 0.02, 60% of which is due to 5 particles
with Eloss>1MeV; (ii) in the low-n, high-T scenario
losses increase to 54 particles during a simulation
time of 0.44 s. The lost-energy fraction is 0.09, of
which 85% can be attributed to 32 particles with
Eloss>1MeV.
4. Conclusions
We have shown that transitioning energetic particles in
advanced stellarators of Wendelstein line undergo the
stochastic diffusion associated with the orbit
transformation of localized and passing particles. This
diffusion may lead to the loss of α-particles and other
energetic ions from the plasma core of a Helias reactor
and Wendelstein 7-X for the time ~ 0.01 s. A key
parameter affecting the magnitude of the diffusion
coefficient is the ratio a/ρB ( 4)/( aD BB ρω∝ ). The
fraction of escaping α-particles can be of the order of 10%
for α-particles produced at 2/~ ar . It can be even
more for the injected ions when their pitch angles
correspond to transitioning particles,
3.0~~/|| mvv ε . However, the diffusion process is
relatively slow. Therefore, the diffusion not necessarily
leads to the loss of ions of high energy to the wall. When
the electron temperature is characterized by strongly
peaking radial distribution, whereas the electron density
profile is flat, the energetic ions may be thermalized near
the edge before being lost.
A general conclusion which follows from our work is
that the stochastic diffusion of transitioning particles may
represent the dominant mechanism of the loss of energetic
ions in optimized stellarators. The dependence of the
obtained diffusion coefficient on plasma parameters and
the relatively large diffusion time indicate that the loss
region and the loss fraction of energetic ions in Helias
configurations can be minimized by shaping the plasma
temperature and density profiles so that they satisfy
certain requirements.
References
[1] F. HERRNEGGER, F. Rau, H. Wobig (Ed.).
Contributions to Wendelstein 7-X and Helias Reactor
1991-1998. Report IPP 2/343 (1999).
[2] F. Wagner. Stellarators and optimised stellarators.
Transactions of Fusion Technology 33 (1998) 67.
[3] A.V. Timofeev. On conservation of an adiabatic
invariant with the change of the character of the motion.
Sov. Phys.-JETP 48 (1978) 656.
[4] J.R. Cary, D.F. Escande, J.L. Tennyson. Adiabatic
invariant change due to separatrix crossing. Phys. Rev. A
34 (1986) 4256.
[5] A.I. Neishtadt. On the change of adiabatic invariant
due to separatrix crossing. Sov. J. Plasma Phys. 12 (1986)
992.
[6] V.S. Marchenko. Collisionless diffusive fluxes of
locally trapped ions in tokamaks with rippled magnetic
field. Nucl. Fusion 35 (1995) 69.
[7] P.N. Yushmanov. Ripple-induced diffusion transport
processes in tokamaks. Reviews of Plasma Physics,vol.16,
Consultants Bureau, New York (1990) 55.
[8] H. Wobig, et al.. Power balance in stellarator
reactors. Fusion Energy 1998 (Proc. 17th IAEA
Conference, Yokohama) IAEA 4, (1999) 1235.
[9] W. Lotz, et al.. Collisionless α -particle confinement
in stellarators. Plasma Phys. Control. Fusion 34 (1992)
1037.
|
| id | nasplib_isofts_kiev_ua-123456789-82355 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:31:24Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kolesnichenko, Ya.I. Beidler, C.D. Marchenko, V.S. Sidorenko, I.N. Wobig, H. 2015-05-28T19:16:17Z 2015-05-28T19:16:17Z 2000 Stochastic loss of alpha particles in a helias reactor / Ya.I. Kolesnichenko, C.D. Beidler, V.S. Marchenko, I.N. Sidorenko, H. Wobig // Вопросы атомной науки и техники. — 2000. — № 3. — С. 22-24. — Бібліогр.: 9 назв. — рос. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82355 533.9 Abstract. It is shown that collisionless orbit transformation of the locally trapped particles to the locally passing ones and vice versa in the Wendelstein-line optimized stellarators results in stochastic diffusion of energetic ions. This diffusion can lead to the loss of an essential fraction of energetic ion population from the region where the characteristic diffusion time is small compared to the slowing down time. The loss region and the magnitude of the loss can be minimized by shaping the plasma temperature and density profiles so that they satisfy certain requirements. The predictions of the developed theory are in agreement with the results of numerical modelling of confinementof a-particles in a Helias reactor, which was carried out in this work with the use of the orbit following code. The considered diffusion seems to represent the dominant mechanism of classical losses of a-particles in a Helias reactor. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Magnetic confinement Stochastic loss of alpha particles in a helias reactor Article published earlier |
| spellingShingle | Stochastic loss of alpha particles in a helias reactor Kolesnichenko, Ya.I. Beidler, C.D. Marchenko, V.S. Sidorenko, I.N. Wobig, H. Magnetic confinement |
| title | Stochastic loss of alpha particles in a helias reactor |
| title_full | Stochastic loss of alpha particles in a helias reactor |
| title_fullStr | Stochastic loss of alpha particles in a helias reactor |
| title_full_unstemmed | Stochastic loss of alpha particles in a helias reactor |
| title_short | Stochastic loss of alpha particles in a helias reactor |
| title_sort | stochastic loss of alpha particles in a helias reactor |
| topic | Magnetic confinement |
| topic_facet | Magnetic confinement |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82355 |
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