Canonicalization and SymplecticSimulation of the Gyrocenter Dynamics inTime-Independent Electromagnetic Fields

by

Ruili Zhang, Jian Liu, Yifa Tang, Hong Qin, Jianyuan

Xiao and Beibei Zhu

Report No. ICMSEC-13-04 December 2013

Research Report

Institute of Computational Mathematics

and Scientific/Engineering Computing

Chinese Academy of Sciences

Canonicalization and symplectic simulation of the gyrocenter dynamics intime-independent electromagnetic fields

Ruili Zhang,1 Jian Liu,2 Yifa Tang,1 Hong Qin,2, 3 Jianyuan Xiao,2 and Beibei Zhu11)LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,China2)Department of Modern Physics and Collaborative Innovation Center for Advanced Fusion Energyand Plasma Sciences, University of Science and Technology of China, Hefei, Anhui 230026,China3)Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543,USA

(Dated: 17 December 2013)

The gyrocenter dynamics of charged particles in time-independent electromagnetic fields is a non-canonicalHamiltonian system. The canonical description of the gyrocenter has both theoretical and practical essen-tiality. We provide a general procedure of the gyrocenter canonicalization, which is expressed by the series ofa small variable ε depending only on the parallel velocity u and can be expressed in a recursive manner. Weprove that the truncation of the series to any given order deduces an exact canonical coordinates for a system,whose Lagrangian approximates to that of the original gyrocenter system in the same order. If flux surfacesexist for the magnetic field, the series stops simply at the second order term and the exact canonical formof the gyrocenter system is feasable. With the canonicalization schemes, the canonical symplectic simulationof gyrocenter dynamics is realized for the first time. The canonical symplectic algorithm have the advantageof good conservation properties and long-term numerical accuracy, while avoiding numerical instability. It isworth mentioning that explicitly expressing the canonical Hamiltonian in new coordinate is usually difficultand impractical. We give an iteration convenient for computing under the original coordinates associated withthe coordinate transformation. This is crucial for modern large-scale simulation studies in plasma physics.The dynamics of gyrocenters in the dipole magnetic field and in the toroidal geometry configuration aresimulated using the canonical symplectic algorithm as well as the higher-order non-symplectic Runge-Kuttascheme for comparison. The overwhelming superiorities of the symplectic method for the gyrocenter systemare evidently exhibited.

I. INTRODUCTION

The dynamics of charged particles in magnetized plas-mas consists of the fast gyromotion and the slow guid-ing center motion. To deal with low frequency phenom-ena, the gyrokinetic theory has been developed to re-solve the multi-scale problem by averaging out or sep-arating the fast gyromotion from the slow gyrocentermotion15,16,22. Gyrokineics provides powerful analytictools and effective simulation models for the study ofmagnetized plasmas2–5,11–16,19–22,26. The gyrocenter sys-tem, rooting from the gyro-symmetry, are usually rep-resented by non-canonical gyrocenter coordinates. Itscanonicalization can benefit the fundamental theory aswell as the advanced simulation techniques. In thispaper, we provide a general procedure of constructingcanonical coordinates for the gyrocenter dynamics intime-independent electromagnetic fields in a series form.The canonical coordinates thus can be achieved recur-sively, without solving the differential equations as inthe traditional proof of Darboux’s theorem. We provethat the truncation of this series to any given order canlead to an exact canonical form for a system, while theLagrangian of this system approximates to that of theoriginal gyrocenter system in the same order of the trun-cation. Moreover, if the magnetic fields can be furtherexpressed in the form of flux surfaces, the series is nat-

urally truncated to the second order term and the exactcanonical coordinates of the gyrocenter system can beobtained directly. This canonicalization scheme can beapplied to canonical symplectic simulation of gyrocenterdynamics conveniently. We design the symplectic algo-rithm for gyrocenter system for the first time, on thebasis of the canonicalization. We simulated the gyrocen-ter dynamics in a dipole magnetic field and a toroidalconfiguration using the canonical symplectic algorithmand a higher-order Runge-Kutta method for compari-son. The long-term conservation properties, numericalaccuracy, and numerical stability of canonical sympleciticsimulation for gyrocenter systems are well unfolded in thecomparison.

The symplectic method is a well-known numerical in-tegrator with global conservation properties for canonicalHamiltonian system. It has been successfully applied tothe test particle simulation and even the PIC methodin the study of plasma physics10. The standard sym-plectic scheme requires the canonical structure of the dy-namical system. Unfortunately, because the gyrocentercoordinates are non-canonical, the symplectic simulationfor gyrocenter system is beset with difficulties. Recently,many efforts have been devoted to the symplectic simu-lation of gyrocenter dynamics. One plan is to make useof the variational symplectic scheme17,23,24. However,because of its multi-step nature, the variational symplec-

2

tic algorithm for gyrocenter dynamics sometimes processunexpected numerical instability. A securer method is todevelop a general and practical procedure of the canon-icalization of the gyrocenter system. Then the standardcanonical symplectic scheme for gyrocenter dynamics canbe applied. According to Darboux’s theorem, the canon-ical coordinates can be theoretically found through solv-ing differential equations. For the gyrocenter dynamicsin magnetic fields, several attempts have been made tofind the canonical coordinates. Meiss and Hazeltine dis-cussed the existence of the canonical coordinates of thegyrocenter systems, but his canonical scheme is imprac-tical for numerical simulations18. White, Zakharov andGao studied the canonical form of the gyrocenter motionin magnetic fields with toroidal flux-surfaces in detail8,27.Different from the previous methods, we provide a set ofdirect formulae in which the coordinate transformationis expressed by the series of a small quantity ε, which de-pends on the parallel velocity u, and acheived in a recur-sive manner. In the process, only matrix multiplicationis required in this scheme, instead of solving differentialequations. In numerical calculation, the manipulationof matrices is much more effective than solving differen-tial equations. Furthermore, there is no assumption onproperties of the magnetic fields in this canonicalizationscheme. Because the series is formally infinite, it shouldbe truncated to contain only finite terms when applyingto a specific manipulation. Surprisingly, the truncatedsystem still preserve good lagrangian structure. We provethat the truncation of the series to any given order de-duces an exact canonical coordinates for a system, whoseLagrangian approximates to that of the original gyrocen-ter system to the same order of the truncation.

Commonly, the magnetic fields we dealt with in plasmaphysics are not chaotic. This kind of magnetic field obeythe flux surface assumption, that is B · ∇Ψ = 0. Withthis assumption, the series form of canonicalization canbe naturally truncated to the second order. So the ex-act canonical coordinates of the gyrocenter system in amagnetic with flux surfaces can be achieved conveniently.In this situation, the canonicalization of the gyrocentersystem and the canonical symplectic simulation of gyro-center dynamics can be performed without any approxi-mation.

Once the canonical coordinates of the gyrocenter sys-tem are obtained, we can apply standard canonical sym-plectic methods to the study of gyrocenter dynamics.The symplectic Runge-Kutta methods are a class ofbroadly used symplectic implicit methods. When theyare applied to the canonicalized gyrocenter systems, theproblem of expressing the Hamiltonian function in newcoordinates is revealed. That’s because generally speak-ing, unless in some special cases, the inverse of this coor-dinates transformation and hence the new Hamiltonianare quite difficult to express. Here, we give a convenientiteration for numerical computing with original coordi-nates, in which the calculation of the gradient of the newHamiltonian is avoided. Then the canonical symplectic

simulation of gyrocenter dynamics can be realized.To verify the correctness of the canonicalization and

the advantage of the canonical symplectic simulation,the numerical examples of gyrocenter dynamics in themagnetic fields with dipole and toroidal configuration arecarried out respectively. We apply the mid-point rule (a2nd-order symplectic Runge-Kutta method) to simulatethe particle’s motion, while observing the evolution ofthe energy of the gyrocenter systems. The numerical re-sults form the canonical symplectic method have betteraccuracy and conservation properties than that from theimplicit Runge-Kutta method of order 3 applied directlyto the non-canonical gyrocenter system in long-term sim-ulation. The canonicalization procedure developed hereis convenient and practical when applying to canonicalsymplectic schemes. It can be utilized to modern large-scale gyrokinetics simulation in both space plasmas andfusion plasmas, where the long-term accuracy and fidelityof algorithms are critical.

The paper is organized as follows. In Sec. II, we dis-cuss the canonicalization of the gyrocenter system in theseries form in general. The truncation of the series arealso investigated. In Sec. III, we discuss the exact canon-ical scheme for magnetic fields with flux surfaces assump-tion. Some useful conclusion are introduced. In Sec. IV,we focus on how to construct the canonical symplecticmethod of the gyrocenter system. Then in Sec. V, nu-merical example of gyrocenter dynamics using canonicalsymplectic algorithms are carried out. By comparison,the correctness and advantage of the canonicalization andsymplectic scheme of the gyrocenter dynamics are testi-fied. Finally in Sec. VI, we offer a brief summary andour future plan on this topic.

II. THE CANONICALIZATION OF GYROCENTERDYNAMICS

In this section, we discuss how to canonicalize the gy-rocenter system in the general sense. The Lagrangian ofthe gyrocenter system can be written as16

L(X, X, u, u) = [A(X)+ub(X)]·X−[1

2u2+µB(X)+φ(X)],

(1)where B = ∇×A is the magnetic field, b = B/B(X) =(b1, b2, b3)> is the unit vector along the direction of mag-netic field, X and u are the position coordinate of thegyrocenter and parallel velocity respectively, and µ isthe magnetic moment, which is a adiabatic invariant.A = (A1, A2, A3)> is the vector potential normalizedby cm/e, and φ is the scalar potential normalized bym/e. The electromagnetic field is assumed to be time-independent in Eq. (1). The Euler-Lagrange equationsof L with respect to X = (x, y, z)> and u give the gyro-center motion equations

K(v)v = ∇H(v) (2)

3

where v = (X>, u)>, H(v) = 12u

2 + µB(X) + φ(X) andK(v) is an antisymmetric matrix

K(v) =

0 a12 a13 −b1−a12 0 a23 −b2−a13 −a23 0 −b3b1 b2 b3 0

,a12 = (

∂A2

∂x− ∂A1

∂y) + u(

∂b2∂x− ∂b1∂y

),

a13 = (∂A3

∂x− ∂A1

∂z) + u(

∂b3∂x− ∂b1

∂z),

a23 = (∂A3

∂y− ∂A2

∂z) + u(

∂b3∂y− ∂b2

∂z).

(3)

In the gyrocenter motion equations, the determinant ofthe matrix K is

det(K(v)) = |b · ∇ × (A + ub)|2 .

If the condition det(K(v)) 6= 0 holds, Eq. (2) can bewritten as a general Hamiltonian system

v = K(v)−1∇H(v). (4)

The system Eq. (4) is a non-canonical Hamiltonian sys-tem. At the same time, a canonical Hamiltonian system,which we struggle to seek, should take the form of

Z = J−1∇H(Z), J =

(0 In−In 0

), (5)

where Z ∈ R2n. According to Darboux’s theorem, forthe non-canonical Hamiltonian system we can find the-oretically the canonical coordinates through solving dif-ferential equations, which is an uneconomic method fornumerical purpose. To avoid solving differential equa-tions, we don’t follow the steps given in the traditionalproof of Darboux’ theorem. Instead, we explore anotherprocedure to realize the canonicalization of the gyrocen-ter system. Let Z = Φ(v) be the transformation from R4

to R4, according to the chain rule, Eq. (4) can be writtenas

Z = (∂Φ

∂v)K(v)−1(

∂Φ

∂v)>∇H(Z) (6)

where H(Z) = H(v). If we demand(∂Φ∂v )K(v)−1(∂Φ

∂v )> = J−1, i.e., K(v) = (∂Φ∂v )>J(∂Φ

∂v ),Eq. (6) becomes a canonical Hamiltonian system in newcoordinates Z through this transformation. DenotingΦ : x 7→ p1(v), y 7→ p2(v), z 7→ q1(v), u 7→ q2(v), wefind the coordinates transformation Φ should satisfy the

following conditions:

(∂p1

∂x

∂q1

∂y− ∂q1

∂x

∂p1

∂y) + (

∂p2

∂x

∂q2

∂y− ∂q2

∂x

∂p2

∂y) = a12,

(∂p1

∂x

∂q1

∂z− ∂q1

∂x

∂p1

∂z) + (

∂p2

∂x

∂q2

∂z− ∂q2

∂x

∂p2

∂z) = a13,

(∂p1

∂y

∂q1

∂z− ∂q1

∂y

∂p1

∂z) + (

∂p2

∂y

∂q2

∂z− ∂q2

∂y

∂p2

∂z) = a23, (7)

(∂p1

∂x

∂q1

∂u− ∂q1

∂x

∂p1

∂u) + (

∂p2

∂x

∂q2

∂u− ∂q2

∂x

∂p2

∂u) = −b1,

(∂p1

∂y

∂q1

∂u− ∂q1

∂y

∂p1

∂u) + (

∂p2

∂y

∂q2

∂u− ∂q2

∂y

∂p2

∂u) = −b2,

(∂p1

∂z

∂q1

∂u− ∂q1

∂z

∂p1

∂u) + (

∂p2

∂z

∂q2

∂u− ∂q2

∂z

∂p2

∂u) = −b3.

We reorganize Eq. (7) to a concise form and get thefollowing theorem.

Theorem 1 The canonical coordinates Z =(p1, p2, q1, q2)> for the gyrocenter dynamics sat-isfy the following equations:

∇p1 ×∇q1 +∇p2 ×∇q2 = ∇×A + u∇× b

∇p1∂q1

∂u−∇q1

∂p1

∂u+∇p2

∂q2

∂u−∇q2

∂p2

∂u= −b

(8)

where ∇ = ( ∂∂x ,

∂∂y ,

∂∂z )>, A is the vector potential and

b is the unit vector along the direction of magnetic fieldin the gyrocenter dynamics.

The theorem is an equivalent description for the canon-ical coordinates in the general sense, which enable us toobtain the canonical form of the gyrocenter system. Con-sidering the first equation of Eq. (8) is linear with respectto u, we expand the new coordinates p1, p2, q1, q2 in theseries of u as

p1(v) = p10(X) +

+∞∑i=1

uip1i(X),

p2(v) = p20(X) +

+∞∑i=1

uip2i(X),

q1(v) = q10(X) +

+∞∑i=1

uiq1i(X),

q2(v) = q20(X) +

+∞∑i=1

uiq2i(X).

(9)

Submitting the expanding series to Eq. (8) and compar-ing the coefficients of uk, we can obtain the equations for

4

the coefficients,

∇p10 ×∇q10 +∇p20 ×∇q20 = ∇×A,

q11∇p10 − p11∇q10 + q21∇p20 − p21∇q20 = −b,q1k∇p10 − p1k∇q10 + q2k∇p20 − p2k∇q20 =

−1

k

( k−1∑i=1

iq1i∇p1,k−i −k−1∑i=1

ip1i∇q1,k−i

+

k−1∑i=1

iq2i∇p2,k−i −k−1∑i=1

ip2i∇q2,k−i

).

(10)The canonical coordinates are expressed in series formand can be calculated recursively. In the process of nu-merical calculation, only matrix multiplication, insteadof solving differential equations, is involved.

To explicit write out the expression of the canonical co-ordinate with the old gyrocenter coordinate, we have toappeal to the concrete expression of the magnetic field.Generally speaking, the vector potential A can be ex-pressed in any coordinates (α, β, γ), that is obeyingA = Aα∇α+Aβ∇β+Aγ∇γ. This formula can be trans-formed to another form as

A = ∇η + p10∇q10 + p20∇q20, (11)

where q10 = β, q20 = γ, η =∫ α

Aα(α′, β, γ)dα

′, p10 =

Aβ − ∂η∂β and p20 = Aγ − ∂η

∂γ . Then the first formula

in Eq. (10) is satisfied spontaneously. Without loss ofgenerality, ∇p10, ∇p20, ∇q20 are linearly independent.Apparently, there are too many freedoms to determinethe coefficients of the series. Further restriction shouldbe claimed. Setting q2k = 0, for k ≥ 1 artificially, we canget q11, p11, p21 from the second formula in Eq. (10)using matrix multiplication

q11

p11

p21

= (∇p10,−∇q10,−∇q20)−1(−b). (12)

Then p1k, q1k, p2k, for k > 1 can be got recursivelyfrom the third equation in Eq. (10) in a similar manner.In this process, no solving of differential equations areinvolved, which is different from the Darboux’s theorem.

The canonical coordinates of gyrocenter dynamics areexpressed by series of the parallel velocity u. To furtherstudy the property of the series, we can rewrite it as aseries of a dimensionless variable ε. We define the di-mensionless variable ε = mu

qBLB, where LB = B

|∇B| is the

characteristic length denoting the change of the magneticfield. So the canonical coordinates can be expressed byseries of a small variable ε depending only on the parallel

velocity u,

p1 = p10 +

∞∑i=1

εip1i, p2 = p20 +

∞∑i=1

εip2i,

q1 = q10 +

∞∑i=1

εiq1i, q2 = q20 +

∞∑i=1

εiq2i.

(13)

We can say that the truncation for the series is anasymptotic approximation to the exact canonical coor-dinates as ε → 0. Actually, the first order truncationp1 = p10 +εp11, p2 = p20 +εp21, q1 = q10 +εq11, q2 = q20

satisfy the following equations

∇p1 ×∇q1 +∇p2 ×∇q2

=B + u∇× b− ε2

2∇× (q11∇p11 − p11∇q11),

(14)

and

∇p1∂q1

∂u−∇q1

∂p1

∂u+∇p2

∂q2

∂u−∇q2

∂p2

∂u

=−[b− ε(q11∇p11 − p11∇q11)

].

(15)

The truncation to the first-order terms of the series leadto are the exact canonical coordinates for a gyrocentersystem with the lagrangian L1

L1 =[A(X)+ub(X)− ε

2

2(q11∇p11−p11∇q11)

]·X−H(v).

(16)Similarly, the k-th order approximate canonical coordi-nates

p1 =

k∑i=0

uip1i, q1 =

k∑i=0

uiq1i, p2 =

k∑i=0

uip2i, q2 = q20,

(17)

satisfy

∇p1∂q1

∂u−∇q1

∂p1

∂u+∇p2

∂q2

∂u−∇q2

∂p2

∂u

=( k∑i=0

ui∇p1i

)( k∑i=1

iui−1q1i

)−( k∑i=0

ui∇q1i

)( k∑i=1

iui−1p1i

)+( k∑i=0

ui∇p2i

)( k∑i=1

iui−1q2i

)−( k∑i=0

ui∇q2i

)( k∑i=1

iui−1p2i

)=− b +

k−1∑l=1

ul[( l+1∑m=1

mq1,m∇p1,l+1−m −mp1,m∇q1,l+1−m

+mq2,m∇p2,l+1−m −mp2,m∇q2,l+1−m)]

+O(uk)

=− b +O(uk) = −b +O(εk).

(18)

In Eq. (18), the third equality holds because(q1i, p1i, p2i) are chosen to satisfy the third equation

5

in Eq. (10). On the other hand, p1, q1, p2, q2 alsosatisfy

∇p1×∇q1 +∇p2×∇q2 = B+ u∇×b+O(εk+1). (19)

So we can say the truncation to k-order for canonicalcoordinates are the exact canonical coordinates for thegyrocenter system with the lagrangian

Lk = (A(X) + ub +O(εk+1))X−H(v), (20)

which is k-order closed to the original lagrangian. Ef-fectively, in the theory of gyricenter dynamics, the corre-sponding Lagrangian is obtained through approximation.The truncating for the series of the canonical coordinatescan be seen as the exact canonical coordinates for theapproximate Lagrangian. This is not only convenient innumerical simulations, but also has a physical meaning.

III. THE CANONICALIZATION OF GYROCENTERS INMAGNETIC FIELDS WITH FLUX SURFACES

In this section, we give the canonicalization of gyro-center systems in magnetic fields with flux surfaces. Thiskind of magnetic field obey the flux surface assumption,that is B · ∇Ψ = 0, where Ψ is called flux label. It canbe written in two forms as{

B = ∇p1 ×∇q10 +∇p2 ×∇q20,

B = q11∇p1 + q21∇p2.(21)

According to the theorem in Section. II, if the magneticfield can be expressed in the above two forms, the re-cursive process searching for the coefficients of the seriesstops at the second term. Then the canonical coordinatescan be explicitly given

p1, q1 = q10 −u

B(x)q11,

p2, q2 = q20 −u

B(x)q21,

(22)

and the gyrocenter dynamics become canonical Hamilto-nian system in the new coordinates. The sufficient condi-tion Eq. (21) coincides the starting point of the previouswork18,27.

In the following, we provide a detailed analysis forthe canonicalization in the magnetic fields with fluxsurface. With the assumption, we can find two func-tions p1(x, y, z) and p2(x, y, z) such that ∇p1 · ∇Ψ =∇p2 · ∇Ψ = 0. Thus, we have the well-defined three-dimensional coordinates (p1, p2, Ψ), which is a curvi-linear coordinate system. In this flux surface coordinatesystem, the magnetic field B has the covariant represen-tation:

B = B1∇p1 +B2∇p2, (23)

The curvilinear coordinates (p1, p2, Φ) have the follow-ing properties

∇p1 ×∇p2 = gΨ∇Ψ,

∇p1 ×∇Ψ = f1∇p1 + f2∇p2,

∇p2 ×∇Ψ = g1∇p1 + g2∇p2,

(24)

where f1, f2, g1, g2, gΨ are corresponding related func-tions. They are important for our following coordinatetransformation. According to Eq. (24), we rewrite B inanother form as,

B = ∇p1 ×∇F1(p1, p2, Ψ) +∇p2 ×∇F2(p1, p2, Ψ)

=

(∂F1

∂Ψf1 +

∂F2

∂Ψg1

)∇p1 +

(∂F1

∂Ψf2 +

∂F2

∂Ψg2

)∇p2

+ (∂F1

∂p2− ∂F2

∂p1)gΨ∇Ψ.

(25)

Generally, the three directions ∇p1, ∇p2 and ∇Ψ areindependent. So the functions F1 and F2 should satisfythe following conditions

∂F1

∂Ψf1 +

∂F2

∂Ψg1 = B1,

∂F1

∂Ψf2 +

∂F2

∂Ψg2 = B2,

∂F1

∂p2− ∂F2

∂p1= 0.

(26)

To let F1 and F2 satisfy Eq. (26), we defineF1 =

∫ Ψ B1g2 −B2g1

g2f1 − g1f2dΨ

′,

F2 = −∫ Ψ B1f2 −B2f1

g2f1 − g1f2dΨ

′.

(27)

According to the definition of Jacobian J ≡ ∇p1 · (∇p2×∇Ψ) and Eq. (24), we can get

g2f1 − g1f2

=− Jg2

∇p1 · ∇p2=

Jg1

|∇p2|2=

Jf1

∇p1 · ∇p2= − Jf2

|∇p1|2.

(28)

If ∇p1 · ∇p2 = 0, Eq. (28) degenerates to −g1f2 =Jg1|∇p2|2 ,= −

Jf2|∇p1|2 . We know that g2f1 − g1f2 6= 0, which

guarantees F1 and F2 in Eq. (27) are meaningful. Ob-viously, F1 and F2 are well defined and automaticallysatisfy the first two equations in Eq. (26). Furthermore,using Eq. (28), we get

∂F1

∂p2= − ∂

∂p2

∫ Ψ B · ∇p2

JdΨ

′,

∂F2

∂p1=

∂

∂p1

∫ Ψ B · ∇p1

JdΨ

′.

(29)

6

It can be verified that Eq. (26) is equivalent to the non-divergence property of the magnetic field,

0 = ∇ ·B = J

[∂

∂p1

(B · ∇p1

J

)+

∂

∂p2

(B · ∇p2

J

)].

(30)

This is physically correct and self-consistent. Thus ac-cording to the sufficient condition mentioned in the be-ginning of this section, we get the canonical coordinates, p1, q1 = F1 −

u

BB1,

p2, q2 = F2 −u

BB2.

(31)

where F1 and F2 is defined as Eq. (27). If ∇p1 and ∇p2

is orthogonal, the transformation also holds. So the ex-act canonical coordinates of the gyrocenter system in amagnetic with flux surfaces can be achieved conveniently.In this situation, the canonicalization of the gyrocentersystem and the canonical symplectic simulation of gyro-center dynamics can be performed without any approxi-mation.

IV. THE CANONICAL SYMPLECTIC SIMULATION OFGYROCENTER DYNAMICS

With the effective canonicalization procedure for gy-rocenter systems discussed in Set. II and Sec. III, weexplicitly gained a transformation Z = Φ(v) that bringsnon-canonical Hamiltonian system v = K−1(v)∇H(v)

to the standard Hamiltonian system Z = J−1∇H(Z).Then we can set foot in the canonical symplectic simula-tion of gyrocenter dynamics and proceed as follows: (i)compute Zn = Φ(vn); (ii) apply a symplectic method tothe standard system which yields Zn+1 = ψh(Zn); (iii)compute finally vn+1 from Zn+1 = Φ(vn+1). The sym-plectic method is a well-known numerical integrator withappropriate global conservation properties for Hamilto-nian systems with canonical structure. This integratorconserves the canonical symplectic structure exactly andguarantees that the energy error is bounded by a smallnumber for all the time steps6,7,9. One kind of convenientand useful symplectic methods is the symplectic Runge-Kutta method. A standard Runge-Kutta method can beexpressed as

Zn+1 = Zn + h

s∑i=1

biJ−1∇H(Ki),

Ki = Zn + h

s∑j=1

aijJ−1∇H(Kj).

(32)

where Ki are intermediate variables. If the coefficientssatisfy

biaij + bjaji = bibj , for all i, j, (33)

the Runge-Kutta method is a symplectic method25. Inthe numerical method, the corresponding Hamiltonian

function H should be expressed as H(Z) = H(v), where

H = H ◦Φ−1. Generally speaking, it’s difficult to expressthe inverse of this coordinate transformation and the newHamiltonian in new coordinates. Here, to overcome the

difficulty, we express the Hamiltonian function H(Z) and

the right hand side J−1∇H(Z) in the original coordinatesv H(Z) = H(v),

J−1∇H(Z) = (∂Φ

∂v)K(v)−1∇H(v),

(34)

The second equation in Eq. (34) holds because of thechain rule

J−1∇H(Z) = Z =∂Φ

∂vv = (

∂Φ

∂v)K(v)−1∇H(v). (35)

The fact that the coordinates transformation is reversibleguarantees that for every Ki, there is a corresponding

Wi such that J−1∇H(Ki) = ∂Φ∂v

)K(v)−1∇H(v)

∣∣∣∣v=Wi

.

Then the symplectic simulation for the gyrocenter dy-namics Eq. (32) can be rewritten as

Φ(vn+1) = Φ(vn) + h

s∑i=1

bi(∂Φ

∂v

)K(v)−1∇H(v)

∣∣∣∣v=Wi

,

Φ(Wi) = Φ(vn) + h

s∑j=1

aij(∂Φ

∂v

)K(v)−1∇H(v)

∣∣∣∣v=Wj

.

(36)The iteration is convenient for computing and the calcu-lating of the gradient of the new Hamiltonian function

H is avoided. The iteration is expressed normally in theoriginal coordinates, but it’s effectively the symplecticsimulation for the canonicalized gyrocenter dynamics innew coordinates, which means that symplectic simulationof the gyrocenter dynamics is realizable. For example,when we apply the mid-point rule which is a symplecticand reversible method of order 2

Zn+1 = Zn + hJ−1∇H(Zn + Zn+1

2), (37)

to the changed canonical Hamiltonian system, the follow-ing implicit iterations in the coordinate vn

Φ(vn+1) = Φ(vn) + h(∂Φ

∂v

)K(v)−1∇H(v)

∣∣∣∣v=W

,

Φ(W ) = Φ(vn) +1

2h(∂Φ

∂v

)K(v)−1∇H(v)

∣∣∣∣v=W

,

(38)should be solved according to the above discussion.

V. APPLICATIONS TO EXAMPLES

In this section, we give two examples to expoundour canonicalization method for the gyrocenter dynam-

7

ics and apply symplectic method to the obtained Hamil-tonian system. The numerical results demonstrate thesuperb properties of symplectic methods applied to thecanonicalized gyrocenter equations in preserving the en-ergy in long-time integration compared to non-symplecticRunge-Kutta method applied directly to guiding-centerequations itself.

A. The example in dipole magnetic field

For the gyrocenter dynamics in the dipole magneticfield, we give the canonical coordinates following canoni-calization method discussed in Section.II. Then the mid-point rule is applied to the obtained canonical Hamilto-nian system. The numerical results show its significantadvantage in preserving the energy over long-time simu-lation compared to non-symplectic Runge-Kutta methodof order 3 applied to gyrocenter equations.

1. The canonical coordinates

The dipole magnetic field B is chosen to be

B(X) = (−M 3xz

r5,−M 3yz

r5,−M 2z2 − x2 − y2

r5), (39)

where r =√x2 + y2 + z2, M is constant with its usual

meaning. The corresponding vector potential, the fieldstrength B(X) and unit magnetic field b(X) can be writ-ten as

A(X) = (My

r3,−Mx

r3, 0),

B(X) = Mr√r2 + 3z2

r5,

b(X) = (− 3xz

r√r2 + 3z2

,− 3yz

r√r2 + 3z2

,−2z2 − x2 − y2

r√r2 + 3z2

).

(40)

Following the steps given in Section.II, the vector poten-tial can be written as

A = Mx2 + y2

r3∇arctan(

x

y), (41)

We set p10 = M x2+y2

r3 , q10 = arctan(xy ), p10 =

any function, q20 = 0. Then solving the second equationin Eq. (10), q11∇p10−p11∇q10+q21∇p20−p21∇q20 = −b,we get q11 = p11 = p21 = 0, p20 = − z

r3 and q21 =r4√

r2+3z2. Because of q11 = p11 = p21 = 0, we don’t need

to solve the third equation in Eq. (10) any more. Theexact coordinates transformation is linear with respectto u and can be obtained

p1 = Mx2 + y2

r3, p2 = − z

r3,

q1 = arctan(x

y), q2 = u

r4

√r2 + 3z2

.

(42)

A well-known fact is that the energy H(v) is an invariantin original equation, and in new coordinates the Hamil-

tonian function H(v) = H(Z) is also invariant. So in nu-merical experiments, whether the energy can be boundedbecome an important criterion for us.

2. The numerical results

For the gyrocenter dynamics in the dipole magneticfield, we apply a non-symplectic implicit Runge-Kuttamethod of order 3

vn+1 = vn +h

2J−1∇H(K1) +

h

2J−1∇H(K2),

K1 = vn +h

2J−1∇H(K1)−

√3h

6J−1∇H(K2),

K2 = vn +

√3h

6J−1∇H(K1) +

h

2J−1∇H(K2).

(43)to the non-canonical Hamiltonian system to simulate par-ticle’s motion. Then we apply the mid-point rule whichis a symplectic method of order 2 to the canonicalizedgyro-center equations. Displayed in Fig. (1) is the com-parison of particle’s orbit in dipole magnetic field calcu-lated by the RK3 method and by mid-point rule with thesame initial conditions. In these numerical examples, theparameters M = 1000 and µ = 0.01. The initial condi-tions are X0 = (1, 1, 1), u0 = 0.01. In Fig. 1(a), theorbit got by RK3 applied to the gyro-center equationsis not accurate, while the orbits calculated by mid-pointrule applied to the canonicalized gyro-center equations inFig. 1(b) is accurate over long integration time. Fig. 1(c)shows the evolution of the energy got by the two methodsand demonstrate the significant advantage of symplecticmethods applied to the changed Hamiltonian system inpreserving energy in long-time integration.

B. The example in Tokamak magnetic configuration

For the gyrocenter dynamics in axisymmetric toka-mak geometry, we give first-order approximate canoni-cal coordinates and exact canonical coordinates. Mid-point rule which is a symplectic method of order 2 isapplied to canonicalized gyrocenter equations in both co-ordinates. The numerical results demonstrate the supe-riority of symplectic methods applied to canonicalizedsystem in preserving the energy over long-time simula-tion compared to non-symplectic Runge-Kutta methodof order 3 applied to guiding-center equations itself.

1. The canonical coordinates

Follow the way given in Section.II, we can get thefirst-order approximate canonical transformation and the

8

FIG. 1. (Color online) Fig. 1(a) orbit numerically obtainedby using the standard RK3 method and Fig. 1(b) that ob-tained by the mid-point rule. Fig. 1(c) Normalized energyH/H0 as a function of time for both methods, where H0 isthe initial energy. The time-step size is h=0.01≈ T/100. Theintegration time is 1000 periods of the orbit in the poloidal(azimuthal) plane.

exact canonical coordinates for the guiding-center equa-tions. So that we can apply symplectic methods to thecanonical Hamiltonian system in new coordinates. In thisgeometry, there are two useful coordinate systems, thecylindrical coordinate system (R, ζ, z) and the toroidalcoordinate system (r, θ, ζ = −ζ). The magnetic field ischosen to be

B =B0r

qReθ +

B0R0

Reζ =

B0r2

qR∇θ −B0R0∇ζ, (44)

where B0, R0, q are constant with their usual meaning.The corresponding vector potential A can be written as

A =B0r

2

2Rqeζ − ln(

R

R0)R0B0

2ez +

B0R0z

2ReR, (45)

and the corresponding magnetic strength B(x) and unitmagnetic field b can be expressed as

B(x) =B0

qR

√r2 +R2

0q2,

b(x) = (−xz −R0qy

R√r2 +R2

0q2,−yz +R0qy

R√r2 +R2

0q2,

R−R0√r2 +R2

0q2

).

(46)

Following the steps given in Section. II, the vector po-tential can be written as

A = −B0r2

2q∇ζ−B0R0log(R)∇z+∇(

B0R0z

2log(RR0)),

(47)where ζ = arctan(xy ). Then setting p10 =

−B0R0log(R), q10 = z, p20 = −B0r2

2q , q20 = ζ, q21 = 0

and solving q11∇p10 − p11∇q10 − p21∇q20 = −b, we canget the first-order approximate canonical coordinates

p1 = −B0R0log(R) + uR−R0√r2 +R2

0q2,

p2 = −B0r2

2q− u qR0R√

r2 +R20q

2,

q1 = z − u Rz

R0R0

√r2 +R2

0q2,

q2 = arctan(x

y).

(48)

The approximate canonical coordinates can be easily ob-tained and we don’t need to solve differential equationsany more. The canonical coordinates are not unique, we

9

can also get another approximate canonical coordinates

p1 = 2z

√−B0R0

R0log(R)−R+R0

(R−R0)2

− u z√r2 +R2

0q2

√− 1

B0R0

(R−R0)2

R0log(R)−R+R0,

p2 = −B0r2

2q− u qR0R√

r2 +R20q

2,

q1 = (R−R0)

√−B0R0

R0log(R)−R+R0

(R−R0)2

− u (R−R0)

2√r2 +R2

0q2

√− 1

B0R0

(R−R0)2

R0log(R)−R+R0,

q2 = arctan(x

y),

(49)and following the discussed in Section.II obtain the exactcanonical coordinates

p1 = 2z

√− u√

r2 +R20q

2−B0R0

R0log(R)−R+R0

(R−R0)2,

p2 = −B0r2

2q− u qR0R√

r2 +R20q

2,

q1 = (R−R0)

√− u√

r2 +R20q

2−B0R0

R0log(R)−R+R0

(R−R0)2,

q2 = arctan(x

y).

(50)The transformation changes the guiding-center motion to

canonical Hamiltonian system Z = J−1∇H(Z).

2. The numerical results

For the gyrocenter equations in the axisymmetric toka-mak geometry, we use the non-symplectic implicit Runge-Kutta method of order 3 to simulate the motion ofcharged particles. Then we apply the mid-point ruleto solve the gyrocenter equations in first-order approxi-mate canonical coordinates and the exact canonical coor-dinates. We denote ACT+S method meaning symplecticmethod applied to guiding-center equations in first-orderapproximate canonical coordinates and ECT+S methodmeaning the symplectic method applied to guiding-centerequations in the exact canonical coordinates.

In these numerical examples, the parameters for themodel tokamak geometry and simulation particles arenormalized by R0 and B0 with safety factor chosen tobe q = 2. Displayed in Fig. (2) is the compar-ison of banana orbits calculated by the RK3 method,by ECT+S method and by ACT+S method with thesame initial conditions. In these numerical examples,the parameters µ = 2.25 × 10−6. The initial conditions

for X0 = (1.05, 0, 0), u0 = 0.0004306. In Fig. 2(a),the orbit is not accurate at the long time scale, whilethe orbits calculated by ECT+S method Fig. 2(b) andthat by 1-order ACT+S method Fig. 2(c) are both ac-curate over long integration time and form closed ba-nana orbit. Displayed in Fig. (3) is the comparisonof a transit orbit calculated by the RK3 method, byECT+S method and by the ACT+S method with thesame initial conditions. In this numerical example, theparameters µ = 2.448× 10−6. The initial conditions arex0 = (1.05, 0, 0), u0 = 0.0008117. In Fig. 3(a), the orbitis not accurate at the long time scale, while the orbitscalculated by ECT+S method Fig. 3(b) and by ACT+Smethod Fig. 3(c) are both accurate over long integrationtime and forms a closed transit orbit.

The long-term energy got by RK3 method graduallydecreases without bound. However, for the symplectic in-tegrator applied to canonical Hamiltonian system eitherin the approximate coordinates or in the exact canonicalcoordinates, the energy error is bounded by a small num-ber for all time steps. This fact is clearly demonstrated inFig. 2(d) and Fig. 3(d), where charged particle’s energynormalized by the initial energy is plotted against time.In fact, the curve of discrete energy got by mid-point rulein approximate canonical coordinates overlap the curvegot in exact canonical coordinates. The numerical re-sults clearly show that the symplectic integrator globallybounds the numerical energy error and maintains the ac-curacy of the orbit for arbitrarily long simulation time.In these numerical examples, for each time step of thesymplectic method, five Newton iterations are used tosearch for the root. Though mid-point rule is of order2, its numerical results show the superb properties forthe guiding-center equation in the new coordinates thanthat of the standard RK3 method in long-time simula-tion. The result displayed in the Fig. (2) and Fig. (3)therefore provide an appropriate comparison in maintain-ing the accuracy of the orbit and in conserving energy.

VI. CONCLUSION

In this paper, we have deduced the general formula forthe canonical coordinates of the guiding center dynamicsin time-independent electromagnetic fields. Interest arethat the series expansion of the coordinates transforma-tion is obtained recursively, and in numerical simulationswe usually use only require approximate canonical coor-dinates by truncating high order terms to a certain accu-racy, such as the machine accuracy. We apply symplecticmethods to the gyrocenter system and offer a convenientway under the original coordinates associated with thetransformation. The examples in dipole magnetic fieldand in axisymmetric Tokamak magnetic field show thesignificant advantages of symplectic method applied tothe canonicalized system in preserving energy in long-term integration.Therefore, we can say the canonizationmethod discussed in this paper is not only easily obtained

10

(a) (b)

(c) (d)

FIG. 2. (Color online) Fig. 2(a) Banana orbit numerically obtained by using the standard RK3 method and Fig. 2(b) thatobtained by the ECT+S method. Fig. 2(c) describes the orbit got by ACT+S method. The integration time is 250 periodsof the closed orbit in the poloidal (azimuthal) plane. Fig. 2(d) Normalized energy H/H0 as a function of time for the threemethods, where H0 is the initial energy. The time-step size is h=100≈ T/400.

but also practical in numerical experiments. In the futurework, we will it to broader cases of gyrocenter dynamics.

The paper focus only on the guiding-center motionindependent of time. For the guiding center motionin a time-dependent electromagnetic field, the corre-sponding Lagrangian will depend on time explicitly, i.e.,L = L(X, X, u, u, t). In this case, the expression ofthe Euler-Lagrange equation are unchanged, which isstill 4-dimensional systems. But the equations are time-dependent. Then we can extend the guiding-center mo-tion to 6-dimensional system and investigate its canonicalcoordinates in the similar way. The canonical transfor-mation and the numerical properties in comparison withstandard integrators need to be further examined in thefuture.

VII. ACKNOWLEDGEMENTS

This research is supported by the ITER-China Pro-gram (Grant No. 2014GB124005) and by the Na-tional Natural Science Foundation of China (Grant Nos.

11371357 and 60931002).

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