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Journal of Fusion Energy ISSN 0164-0313 J Fusion EnergDOI 10.1007/s10894-014-9801-7
Fast Ion Trajectory Calculations inTokamak Magnetic Configuration UsingSymplectic Integration Algorithm
Majid Khan, Abdullah Zafar &M. Kamran
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ORIGINAL RESEARCH
Fast Ion Trajectory Calculations in Tokamak MagneticConfiguration Using Symplectic Integration Algorithm
Majid Khan • Abdullah Zafar • M. Kamran
� Springer Science+Business Media New York 2014
Abstract A numerical scheme based on Symplectic
Integration Algorithm (SIA) has been used to develop an
orbit following code to calculate fast ion trajectories in
tokamak magnetic configuration. For the purpose of dem-
onstrating the expediency of symplectic schemes, the
algorithm has been applied to the Henon-Heiles system and
compared with non- symplectic Runge–Kutta Algorithm
(RKA) for numerically integrating the Hamiltonian equa-
tions. In contrast to RKA, the long-time stability of SIA has
been highlighted. Furthermore, the SIA has been used to
find the exact trajectories of, trapped and passing, fast ions
in tokamak. In particular, the effect of an intrinsic magnetic
field perturbation has been investigated, i.e. toroidal field
ripples. This perturbation is toroidal field ripple (TFR)
which are there due the discrete number to toroidal coils
around the torus. The numerical scheme used shows
excellent conservation of particle energy as well as of
angular momentum (in axi-symmetric case). The effect of
TFR on these trajectories has been simulated and it is
shown that the resonance between toroidal precession of
bananas and the field ripples results in spread of the tra-
jectories for banana particles, whereas the passing fast ions
are unaffected.
Keywords Symplectic integration algorithm � Toroidal
field ripples � Tokamak � Fast ions � Resonance condition
Introduction
The ever growing population and rapid urbanization have
resulted in a still increasing demand for energy. Consid-
ering the rising energy demand and the constrained avail-
ability of energy resources, one needs more efficient
utilization of energy and to develop new technologies as a
long term solution to this problem. Reactors based on the
fusion of nuclei of light elements may provide an almost
unlimited supply of energy in the future. Controlled nuclear
fusion can be realized on earth as well and made available
for energy production. This is thanks to the relatively large
fusion reaction cross-section in a deuterium—tritium
mixture at temperatures around 200 million Kelvin. This
reaction produces neutron and alpha particles along with
lot of energy, these alphas (subject of this study) have
energies much greater than the bulk plasma and refer here
as fast ions. Fast or energetic ions, produced by neutral
beam injection, ion cyclotron resonance heating or/and by
fusion reactions, require to be well confined in order to
provide sufficient plasma heating. These ions play a crucial
role in achieving and sustaining favorable burning plasma
conditions. However, losses of fast energetic ions from
tokamak plasma occur through different mechanisms, e.g.
first orbit losses, via scattering, through diffusive transport
mechanism and collective plasma instabilities [1, 2]. Pre-
cise knowledge of fast ions trajectories over long time
periods is utmost important for predicting their transport in
present and future tokamaks. Hence, modeling of energetic
ions in toroidal magnetic confinement devices is an
M. Khan (&)
Department of Physics, Kohat University of Science
& Technology, Kohat, Pakistan
e-mail: [email protected]; [email protected]
M. Khan � A. Zafar
Department of Physics, Quaid-i-Azam University, Islamabad,
Pakistan
M. Kamran
Research in Modeling and Simulation (RIMS) Group,
Department of Physics, COMSATS Institute of Information
Technology, Islamabad, Pakistan
123
J Fusion Energ
DOI 10.1007/s10894-014-9801-7
Author's personal copy
important pre-requisite in the development, design and
operation of future nuclear fusion reactors.
One requires numerical tools which do not accumulate
errors at each time step and can be used for exact trajectory
calculations of fast ions in tokamak plasmas, in particular,
in the presence of magnetic field perturbations. The Sym-
plectic Integration Algorithm (SIA) based numerical sim-
ulations has been used here to find the trajectories of fusion
alphas in tokamak. In such numerical scheme—due to
satisfying symplectic condition at each time step—the
numerical dispersion produced by the iteration routine is
negligible [3, 4]. Hence, it demonstrates the preservation of
the canonical structure of the equation of motion in the
frame-work of Hamilton’s equations. The symplectic
schemes are best suited for the dynamical systems which
need long time scale assessment [4, 5]. These algorithms
preserve exactly the symplectic differential form, i.e. all
Poincare invariants [6] are conserved.
Various magnetic perturbations like Toroidal field (TF)
ripples can degrade fast ions confinement. In the outer
region of the tokamak plasmas—due to loose winding of
toroidal coils—TF ripples have strongest effect and can
cause fast ions losses very quickly which in turn can also
damage the first wall of the tokamak [7, 8]. Here in this
work a SIA has been used to find the trajectories of ener-
getic ions and the effect of TFR on their orbits. To dem-
onstrate the benefit of integrating the canonical equations
of motion by a symplectic algorithm we referred here, for
simplicity, to circular magnetic flux surfaces.
Symplectic Integration of a General Hamiltonian
The symplectic integration, at each time step, conserves the
Hamiltonian structure of equations of motion and satisfies
the symplectic condition
MT p; qð ÞJM p; qð Þ ¼ J; ð1Þ
where M is the Jacobian of transformation, q and p are
vectors of canonical conjugate coordinates and J is the
standard symplectic matrix given by
J ¼ 0 I
�I 0
� �ð2Þ
Here I is the identity matrix. Sympletic condition for
canonical transformation is an important tool to check
whether a given transformation is canonical or not.
Canonical transformations play a key role in Hamiltonian
dynamics. In non-symplectic numerical integration algo-
rithm (NSIA) there is no such limitation even if the energy
conservation is built into the algorithm. The only limitation
on the NSIAs is to investigate for short interval of time
which is the reason that we miss global structure. In SIA,
time evolution is by a canonical transformation, i.e.,
p tið Þ; q tið Þ|fflfflfflfflfflffl{zfflfflfflfflfflffl}oldcoordinates
)Canonical
p ti þ dtð Þ; q ti þ dtð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}newcoordinates
To describe equation of motions of a system for general
Hamiltonian function H (p, q), the transformation equa-
tions can be carried out by using generating function, G(p,
Q) = G(p1, …, pi; Q1, …, Qi), with
Pi ¼ �oG
oQi
ð3Þ
and
qi ¼ �oG
opi
ð4Þ
The generating function can be discretized and expan-
ded in terms of the time step, dt, as
Gðp;QÞ ¼X1l¼0
dtl
l!Glðp;QÞ ð5Þ
where G0 = - p � Q gives the identity transformation.
Using Eq. (5) in Eq. (3) follows
Pi ¼ pi þX1m¼1
dtm
m!Pmðp;QÞ ð6Þ
with
Pm ¼ �oGmðp;QÞ
oQm
ð7Þ
For Hamiltonian function the canonical equations of
motion are
_Pi ¼�oH
oQi
; and _Qi ¼�oH
oPi
ð8Þ
Furthermore, the time derivative of P is given by
_Pi ¼oPi
otþ oPi
oQi
_Qi ð9Þ
Upon using Eq. (9) in Eq. (8) one finds [3]
X1m¼1
Pmdtm�1
ðm� 1Þ!þX1m¼1
dtm
m!
oPm
oQm
Q�m
¼ � oH
oQi
pþX1m¼1
Pmdtm
m!;Q
!ð10Þ
Defining
DP ¼X1m¼1
dtm
m!Pmðp;QÞ ð11Þ
the argument Pi = pi ? … = pi ? DP in the Hamiltonian
can be expanded using Taylor series to write
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X1m¼0
dtm
m!Pmþ1 ¼
X1m¼1
dtm
m!
o2Gm
oQmoQm
X1l¼0
DPo=oPð Þl
l!
" #oH
oP
�X1l¼0
DPo=oPð Þl
l!
oH
oQl
Using binomial theorem and equating the coefficients of
same powers of dt yields [3]
oGmþ1
oQ¼ �m!
Xs
s¼1
1
s!
o2Gs
oQsoQs
Xm�s
l¼0
Xm�s
r1;...;rm¼0;P
ri¼l;P
iri¼m�s;
1
r1!...rm!
P1
1!
o
oP
� �r1
� � � Pm
m!
o
oP
� �rmoH
op
þ m!Xm�s
l¼0
Xs
r1;...;rm¼0;P
ri¼s;P
iri¼m;
P1
1!
o
oP
� �r1
� � �
Pm
m!
o
oP
� �rmoH
oq
For m = 0, 1, 2, … we can calculate G to any desired
order, e.g.
G1 ¼ H;
G2 ¼ �Xn
i¼1
oH
opi
� oG1
oQi
;
G3 ¼ �Xn
i¼1
oH
opi
� oG2
oQi
� oG1
oQi
�Xn
j¼1
oG1
oQj
o2H
opiopj
( )
and so on. Using Eq. (5) in Eq. (4) to write
Qi ¼ qi þX1l¼1
dtl
l!
oGlðp;QÞopi
; i ¼ 1; . . .; n ð12Þ
This equation has to be solved by iterations. During each
iteration step the precedent Q is introduced again on the
right hand side of the equation. The iteration terminates
when Q changes by less than a defined tolerance. Then the
value of P has been evaluated using Eq. (6).
Next we compare the simulations based on SIA and
RKA by solving a simple model Hamiltonian function
introduced by Henon and Helies to discuss star motion in
an axial symmetric galaxy [9]. The corresponding Hamil-
tonian function can be written in the form [3, 9]
H p;Qð Þ ¼ 1
2Q2
1 þ Q22 þ p2
1 þ p22
� �þ Q2
1Q2 �1
3Q3
2
Here we evaluate and analyze the star trajectories for same
initial conditions by using SIA and RKA, and will dem-
onstrate why we prefer the former. Instead of plotting the
whole three dimensional trajectories we will show the
Poincare maps by plotting the normalized space coordinate
q2 and normalized momentum p2 only when the trajectory
intersects the hyper plane q1 = 0. Poincare maps of star
trajectories are shown in Fig. 1, both of these figures result
from simulations using 4th order SIA and 4th order RKA
by using exactly same initial conditions and for same total
energy. Looking at Fig. 1 it is clear that for large time
simulations the Poincare map of star trajectory evaluated
by using SIA is very fine and well defined sharp bound-
aries. In contrast, calculations based on RKA show fuzzy
image with no well-defined orbit boundary. These precise
contours indicate the periodic star motion, whereas in this
case RKA is only capable of providing indistinct truncated
orbits. Similar results are displayed in Fig. 2 where the
simulations were carried out for a different kind of tra-
jectories having the initial conditions (q1, q2, p2)ini-
tial = (0, 0, 0) and the total energy H = 0.150 (a.u). Here
again the plot which results from SIA are precise and well
defined while in case of RKA the plot is blurred and
boundaries are cloudy. We have also inspected the sensi-
tivities of SIA and RKA calculations to a shift in total
energy which can affect transition of geometric structures
in phase-space, e.g. from stable to chaotic motion. It has
been found that the change in energy brings about a phase-
space trajectory change, in contrast to that RKA simula-
tions did not produce these trajectory characteristics.
Trajectory Simulations of Fast Ions in Tokamak Using
SIA
Hot plasma, of some specific high temperature, is required
to achieve fusion reaction. One of the important mecha-
nisms in this regard is the self-heating or internal-heating
process in which charged fusion products like fast alpha
particles transfer their energy to plasma species through
collisions. There are two categories of charge particles in
tokamak: (i) passing particles which move around the
tokamak having large parallel component of velocity vk
and (ii) trapped particles that are trapped in the magnetic
well having small vk. Note that the magnetic well arises
due to difference of magnetic fields on the inboard and
outboard of the tokamak [10]. The Hamiltonian of fast
alphas in a large aspect ratio and low plasma pressure can
be written as [11–13]
H p; qð Þ ¼ bðp1; qÞg
w p1ð Þ þ p2f g� �2
þlb ð13Þ
where w is the poloidal flux, g = qL A/a is the normalized
Larmor radius with a and A denoting, respectively, the
minor radius and aspect ratio, qL defines the Larmor radius
of fast alphas, b(p1, q) is the magnetic field strength. While
taking TF ripples into account the magnetic field strength
b(p1, q) can be written as the superposition, i.e.,
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b p1; qð Þ ¼ bas p1; q1ð Þ þ bTFRðp1; qÞ ð14Þ
where bas gives the field strength of non-perturbed part,
bTFR represents the TF ripples strength and is supposed to
be of the from [11, 14, 15]
bTFR ¼ dTF0 I0 Fð Þ cos Nq2ð Þ ð15Þ
Here N gives number of toroidal field coils and the
parameter dT F 0 indicates minimumripples amplitude,
while I0 (F) is the modified Bessel function with argument
[16, 17]
F ¼ N
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRr � Rð Þ2þZ2
RRr
sð16Þ
The symbol Rr gives the radial position at which ripple
amplitude has its minimum value, R and Z are the usual
cylindrical coordinates. In Eq. (13) the canonical coordinates
q1 and q2 denote, respectively, the poloidal and toroidal angles
with their conjugate canonical momentum p1 and p2. More-
over, the poloidal flux is calculated from the safety factor qs
profile, whose radial dependence is chosen to be of the form
qs rð Þ ¼ qs0
1þ r2=a2
qs0
qsa
� 1
� ��1
ð17Þ
where qs0 0 and qs
a are the values of safety factor at the
magnetic axis and at the plasma boundary, respectively.
The poloidal flux is then calculated from the expression
w ¼Z r
0
B � rhr0dr0 ¼Z r
0
r0dr0
qs r0ð Þ ð18Þ
Simulation Results and Discussions
Trajectories of Fast Alphas in An Axi-Symmetric
Configuration
Now using SIA developed in Sect. 2 and Hamiltonian
defined by Eq. (13), we simulate the trajectories of fast
alpha particles in an axi-symmetric magnetic configuration,
i.e., bT F R = 0. For our simulations we have used the
following parameters for an ITER size tokamak:
Rr = 5.6 m, R0 = 6.3 m, dT F 0 = 2.5 9 105, B0 = 5T,
qs = 4.1 and qs0 = 1.
Figures 3 show the trajectories of trapped and passing
alpha particles with different speeds. Note that alpha par-
ticle birth energy is E = 3.52 MeV which corresponds to a
speed V0 = 1.3 9 107 m/s and has been used as reference.
From Fig. 3a we conclude that width of banana orbit is
Fig. 1 Comparison of Poincare
maps for star trajectories with
initial conditions (q1, q2, p1,
p2)initial = (0.10, 0.10, 0.10,
0.10) and H = 0.02133 (a.u).
a SIA based calculations.
b RKA based calculations
Fig. 2 Poincare maps
comparison for Henon–Helies
system using (a) 4th order SIA
and (b) 4th order RKA for (q1,
q2, p2)initial = (0, 0, 0) and
H = 0.150 (a.u)
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directly proportional to its speed which is consistent with
[18]
dBa /mV
qBhe1=2; ð19Þ
where dBa represents width of banana orbit, Bh poloidal
magnetic field, m mass of alpha particles and e ¼ r=R is the
inverse aspect ratio. In Fig. 3b trajectories of passing par-
ticles having different energies are shown where slight shift
of the orbits has been depicted and the shift is proportional
to particles speed.
Zoom tip of banana orbit of fusion born alpha has been
displayed in Fig. 4 in the absence of magnetic perturbation,
i.e., bT F R = 0, and we note that there is no alteration or
shift of the trapped particles orbits. Figure 5 elucidate that
even in long-time simulations of trapped alpha motion
(more than 103 bounce periods), the 3rd order SIA applied
is seen to conserve perfectly the toroidal angular
momentum and impressively well the energy of any fast
alpha test particle, namely with a relative accuracy of
10–10 .
Effect of TFR Perturbation on Trajectories of Alpha
Particles
If toroidal separation between two adjacent TF coils or its
multiple is equal to the toroidal distance between succes-
sive lower or upper banana tips, the particle’s bounce-
motion is said to be in resonance with the toriodicity of TF
ripples. This resonance between the toroidal precession of
banana orbits and the TF ripples periodicity bring about a
banana guiding center motion and traces out a so-called
super banana orbit with a new type of radial excursion [19].
The resonance condition reads
Nxt vð Þ ¼ lxb vð Þ
where N is the number of TF coils, l denotes the resonance
level, xt and xb represent, respectively, the toroidal pre-
cession frequency and the bounce frequency. In the pre-
sence of magnetic TF-ripples with dmax = 1 % the
trajectories of fast ions are displayed in Fig. 6. The orbit
excursions as indicated by ripple resonance are shown in
the plots. Note again that the width of banana orbit and
orbit shift both are proportional to speed of fast alphas.
Under the influence of TF-ripples, zoomed tip of banana
orbits are shown in the Fig. 7.
It is clear that in the presence of TF-ripples the tips of
banana orbits are very much dispersed which clearly indi-
cates that in the presence of magnetic perturbation like TFR
it is easy for trapped particles to get free form the trapped
region. Moreover, after including the TFR magnetic per-
turbation the toroidal angular momentum is no more con-
served as depicted in Fig. 8. This non-conservation of p2
Fig. 3 The poloidal projection of (a) trapped and (b) passing alphas orbits in an axi-symmetric magnetic field configuration, i.e. in the absence
of TF-ripples for different starting energies. The starting point of each trajectory is R - R0 = 0, z = - 1.3, q2 = 0 and v} = 0
Fig. 4 Zoom tip of banana orbit in the absence of TFR magnetic
perturbation
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Fig. 5 a Temporal evaluation
of toroidal angular momentum,
p2 when a TFR perturbation is
switched off and b the variation
of total energy with time for
different alpha particle speeds
Fig. 6 Projection of (a) trapped and (b) passing alphas orbits; on to
the poloidal plane in the presence of TFR perturbation for different
starting speeds. The starting point is defined by R - R0 = 0, z =
- 1.3 m, q2 = 0 and v} = 0
Fig. 7 Zoom tip of a banana orbit showing the radial excursion as
caused by TFR
Fig. 8 Time variation of toroidal angular momentum, p2 in the
presence of TFR magnetic perturbation. Clearly indicating that it is no
more a constant of motion
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results in enhanced radial diffusion coefficient of fast ions
from the tokamak [11, 20].
Summary
The method of symplectic integration has been employed
in the development of the orbit following code for the
calculation of the trajectories of energetic alpha particles
confined in a tokamak magnetic configuration. Due to the
perfect conservation of the constants-of-motions in SIA,
longer-time evolutions could be computed without loss of
accuracy. Resonances between the periodicity of toroidal
field ripples and the toroidal precession of the banana orbits
are shown to result in a so-called superbanana motion,
which can substantially enhance their radial transport since
even small pitch angle scattering of those particles can
result in large radial excursions. At first the SIA has been
shown to be superior over the non-symplectic one, by
solving Henon-Heiles Hamiltonian. For Poincare maps of
star trajectories it is concluded that trajectories evaluated
by SIA are very clear and well defined with no dispersion,
while in case of RKA the same maps are not well-defined,
blurred, cloudy, dispersive and have unclear boundaries.
Based on SIA, guiding center Hamiltonian of fast alpha
particles in tokamak has been simulated to find the tra-
jectories of alphas. For different energies, the trajectories
of these ions were evaluated and it is shown that for
trapped (passing) particles trajectories the banana width
(orbit shift) is directly proportional to their speed. When
the TFR magnetic field perturbation is introduced in the
ode, similar dependence is observed. However, for specific
alpha particle energies (resonance values), spreading in the
banana orbits has been highlighted. Moreover, the toroidal
angular momentum is no more conserved and can cause the
radial diffusion of energetic ions from tokamak.
This work has been supported by the Higher Education
Commission of Pakistan under project No. PM-IPFP/HRD/
HEC/2012/3563. The views and opinions expressed herein
do not necessarily reflect those of the Higher Education
Commission.
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