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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: majid_k665@yahoo.com; Majid.Khan@uibk.ac.at

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

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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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