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Simulated annealing for three-dimensional low-beta reduced MHD equilibria in cylindrical

geometry

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2017 Plasma Phys. Control. Fusion 59 054001

(http://iopscience.iop.org/0741-3335/59/5/054001)

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Simulated annealing for three-dimensionallow-beta reduced MHD equilibria incylindrical geometry

M Furukawa1 and P J Morrison2

1Graduate School of Engineering, Tottori University, Minami 4–101, Koyama-cho, Tottori 680-8552,Japan2 Physics Department and Institute for Fusion Studies, University of Texas at Austin, TX 78712, UnitedStates of America

E-mail: furukawa@damp.tottori-u.ac.jp

Received 4 September 2016, revised 1 December 2016Accepted for publication 10 January 2017Published 17 March 2017

AbstractSimulated annealing (SA) is applied for three-dimensional (3D) equilibrium calculation of ideal,low-beta reduced MHD in cylindrical geometry. The SA is based on the theory of Hamiltonianmechanics. The dynamical equation of the original system, low-beta reduced MHD in this study,is modified so that the energy changes monotonically while preserving the Casimir invariants inthe artificial dynamics. An equilibrium of the system is given by an extremum of the energy,therefore SA can be used as a method for calculating ideal MHD equilibrium. Previous studiesdemonstrated that the SA succeeds to lead to various MHD equilibria in two-dimensionalrectangular domain. In this paper, the theory is applied to 3D equilibrium of ideal, low-betareduced MHD. An example of equilibrium with magnetic islands, obtained as a lower energystate, is shown. Several versions of the artificial dynamics are developed that can effectsmoothing. The smoothing effect is examined in the numerical results.

Keywords: simulated annealing, stationary state, magnetohydrodynamics

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1. Introduction

The calculation of magnetohydrodynamics (MHD) equilibriais fundamental for fusion plasma research. Axisymmetrictoroidal equilibria are described by the well-known Grad–Shafranov (GS) equation [1–3]. Because the GS equation isan elliptic differential equation, of the same type as Poissonʼsequation, numerical methods for solving it are well-estab-lished [4]. The extension of the GS equation to includeplasma rotation has also received attention [5–8]. The GSequation including toroidal rotation is also an elliptic differ-ential equation that can be solved by the same numericalmethods as the original GS equation. Other extensions such asinclusion of anisotropic pressure are also possible. (See e.g.[9] for a review of MHD equilibrium calculations.) Theinclusion of poloidal rotation, however, can make the

equilibrium equation hyperbolic [5, 6], for which no generalnumerical method has been established.

The calculation of three-dimensional (3D) MHD equili-bria is considerably more involved. The existence of nestedmagnetic surfaces is not guaranteed generally. Variousnumerical codes for the 3D MHD equilibrium have beendeveloped. Variational Moment Equilibrium Code(VMEC)[10, 11] may be the most used one, where nested magneticsurfaces are assumed to exist. In VMEC the energy of thesystem is minimized by the steepest descent method to obtainan equilibrium. Princeton Iterative Equilibrium Solver(PIES)[12] is another type, where nested magnetic surfaces are notassumed. In PIES the solution method consists of an iterationwith the following steps: (i) calculation of the pressure bymagnetic field line tracing, (ii) calculation of current densityby the MHD equilibrium equation for the obtained pressure

Plasma Physics and Controlled Fusion

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0741-3335/17/054001+11$33.00 © 2017 IOP Publishing Ltd Printed in the UK1

and (iii) determination of the magnetic field by the Ampèreʼslaw. Another code Helical Initial value solver for Toroidalequilibria (HINT) [13] and its spawn HINT2 [14] are partlysimilar to PIES. HINT2 solves the MHD evolution equation,instead of steps (ii) and (iii) of PIES, under a fixed pressuregiven by step (i). Inclusion of dissipation leads to an equili-brium. A new type of the equilibrium code is Stepped Pres-sure Equilibrium Code(SPEC) [15], where an equilibrium isconstructed by connecting multiple layers of Taylor relaxedstates (Beltrami fields) under continuity of the total pressure.In each layer the plasma pressure is flat and the existence ofmagnetic surfaces is not assumed. Ideal Perturbed Equili-brium Code (IPEC) [16] calculates 3D MHD equilibriumperturbatively by adding zero-frequency, linear ideal MHDmodes to an axisymmetric equilibrium. As for inclusion ofplasma rotation, an extension of HINT2 to toroidally rotatingequilibrium is on-going [17].

Such 3D structures in magnetically confined plasmas arestrongly related to self-organization and relaxation processes.Magnetic island formation, effects of plasma rotation on themagnetic island, and their interactions with externally appliedmagnetic fields have recently received attention in tokamak aswell as helical [18] plasma research. Also, the transition tohelical equilibria of reversed field pinch plasmas [19–21] is aninteresting self-organization phenomenon that is beinginvestigated. Therefore, 3D MHD equilibrium codes, inaddition to nonlinear evolution codes, are of great importance.For these studies, the existence of magnetic surfaces shouldnot be assumed, and plasma rotation should be included.Moreover, it is important to characterize equilibria in a sys-tematic way in order to understand important physicalphenomena.

The present work concerns an MHD equilibrium code ofanother type based on simulated annealing (SA) [22, 23].Originally the idea was developed for neutral fluids anddemonstrated to work for computing simple equilibria [24](see also [25, 26]). Briefly, we solve an artificial dynamics,which is generated on the basis of the physical dynamics, toreach an energy extremum while preserving conservationproperties of the physical system other than the energy. Theenergy extremum is an equilibrium of the system of idealfluids described by Hamiltonian [27, 28]. Later it was gen-eralized in [29] to apply to a large class of equilibria ofHamiltonian field theories by allowing for smoothing and theenforcement of constraints that select out a broader classequilibria. SA is based on Hamiltonian structure, and can beapplied to fluid and plasma models, in particular ideal MHD,because they are Hamiltonian in terms of noncanonicalPoisson brackets involving a skew-symmetric Poissonoperator [30, 31]. As we will see below, the SA does not relyon the existence of flux surfaces nor assumption of absense ofplasma rotation. Therefore, the SA is a promising candidatefor an equilibrium code of a new type.

As noted above, we consider Hamiltonian systems whichincludes ideal neutral fluid dynamics and MHD. For Hamil-tonian systems the time evolution of the dynamical variablesis determined by the functional derivative of the Hamiltonianmultiplied by a skew-symmetric Poisson operator. The

harmonic oscillator is the simplest finite-dimensional exam-ple, for which the state variable is u q p, T= ( ) , with q and pbeing the usual canonical coordinate and momentum,respectively, and the Hamiltonian is H q p 2.2 2= +( ) Using

uH q p, T¶ ¶ = ( ) and skew-symmetric canonical Poisson

matrix J 0 11 0

,c -⎜ ⎟⎛⎝

⎞⎠≔ gives the equations in the Hamiltonian

form Juut cHd

d= ¶

¶. Because of the skew-symmetry of Jc, the

energy of the oscillator is conserved. In addition to the energyconservation, for more general noncanonical Poisson brack-ets, with Poisson operators J, there exist Casimir invariantsarising from degeneracy. The degeneracy is not because ofvariable change, but is inherent in the Eulerian forms of idealfluid dynamics and MHD and can be seen as arising from theunderlying canonical description in terms of Lagrangianvariables (see e.g. [31]). Such systems evolve on a surfacespecified by its energy and the Casimir invariants in thecorresponding phase space of the dynamical variables. Themagnetic and cross helicities are examples for ideal MHD. Asurface defined by constant Casimir invariants in the phasespace is called a Casimir leaf. An extremum of the energy onthe Casimir leaf gives an equilibrium, a stationary state, asfirst noted in the plasma literature [27] and then later for theneutral fluid [28]. If we solve the physical evolution equation,the system follows a trajectory with a constant energy on theCasimir leaf. However, it does not relax to an equilibrium ifthe initial condition is not an equilibrium.

SA uses an artificial evolution equation obtained from theHamiltonian structure of the physical evolution equation by‘squaring’ the Poisson bracket, i.e. the dynamics is given by

Juut

Hd

d2= ¶¶. For such artificial dynamics, it is easy to see that

the energy monotonically decreases, as can easily be shownfor the harmonic oscillator example. Similarly, for MHD, theenergy of the system changes monotonically; however, as wewill see in the theoretical development, the Casimir invariantsare preserved because the artificial dynamics is constructed onthe basis of the Poisson bracket of the original dynamicswhich generates the Casimir invariants. A schematic pictureexplaining the Casimir leaf, the physical and the SA is shownin figure 1. Because the energy extremum on a Casimir leaf isby definition an equilibrium state, this method can be used asa numerical method for finding equilibria. An advantage ofthis method is that the stationary state is characterized by thevalues of the Casimir invariants.

The original work [24–26] was effective for simpleequilibria, but because of the plurality of equilibria it did notprove effective for equilibria of interest, and needed to bemodified. This was done in [29], where the term SA wasintroduced for this method of the artificial dynamics, byintroducing a general symmetric bracket that allows forsmoothing and the use of Dirac theory to impose constraints.On the basis of these early studies, SA was applied to 2D low-beta reduced MHD [32] in [22], and a method to pre-adjustvalues of the Casimir invariants, by pre-adjusting initialconditions, in order to characterize the sought equilibriumstates was developed in [23].

2

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

The previous studies were performed in a 2D rectangulardomain with periodic boundary conditions in both directions,except for a few cases in [29] where layer models were usedfor describing a third dimension. In the present study a 3Dcode is developed, although the outer boundary of the plasmais still cylindrical. Then a stationary state with magneticislands with multiple helicities can be obtained if it has lowerenergy than a cylindrical symmetric state. The code uses thesymmetric bracket of [29] that can effect smoothing.Although the code development is still in progress in someparts, we have observed that such a smoothing effect isimportant for numerical stability of the code as well as for thecomputational cost. The code can be numerically unstableeasily without the smoothing. Also, the computational costcan be affected by the smoothing because the relaxation pathto the lower energy state can be changed. Therefore, we focuson the smoothing effect in examining numerical results.

The paper is organized as follows. In section 2, the set-ting of the problem is explained, the SA method is summar-ized, with a focus on the 3D low-beta reduced MHD example,and three types of symmetric brackets are introduced.Section 3 presents numerical results, with the choices of thesymmetric brackets examined, and a stationary state withmagnetic islands calculated. Next, section 4 contains discus-sion, where some remaining issues are raised. Finally, thepaper is summarized in section 5.

2. Theory

2.1. Reduced MHD system

In this study, let us consider a cylindrical plasma with minorradius a and length R2 0p . The cylindrical coordinates arer z, ,q( ), with the toroidal angle being z R0z ≔ and theinverse aspect ratio given by a R0e ≔ . Physical quantitiesare normalized by the length a, the magnetic field in the z-direction B0, the Alfvén velocity v BA 0 0 0m r≔ with 0m and

0r being vacuum permeability and typical mass density,respectively, and the Alfvén time a vA At ≔ . Then low-betareduced MHD is given by

U

tU J

J, , , 1j y e

z¶¶

= + -¶¶

[ ] [ ] ( )

t, , 2

yy j e

jz

¶¶

= -¶¶

[ ] ( )

where the fluid velocity is v z j= ´ ˆ , the magnetic field isB z zy= + ´ˆ ˆ, the vorticity is U j^≔ , the currentdensity is J y^≔ , the Poisson bracket for two functions fand g is zf g f g, ´ [ ] ≔ ˆ · , the unit vector in the zdirection is denoted by z, and is the Laplacian in the r–θplane.

2.2. Simulated annealing theory

Now, we briefly review the governing SA system, referringthe reader to [29] for a detailed explanation. The artificialdynamics of SA is given by

uu

tH, , 3

¶¶

= (( )) ( )

where u is a vector of the dynamical variables, uH [ ] is theHamiltonian functional and F G,(( )) is the symmetric bracketfor two functionals uF [ ] and uG [ ], defined by

x x x x

F G x x

F u K u G

, d d

, , , , 4iij

j

3 3

ò ò¢

´ ¢ ¢

(( )) ≔

{ ( )} ( ){ ( ) } ( )

where Kij( ) is a definite symmetric kernel, and

ux

x xux

F G x x

F

uJ

G

u

, d d

, 5i

ijj

3 3

ò òdd

dd

¢

´¢

¢

{ } ≔

[ ]( )

( ) [ ]( )

( )

is the Poisson bracket for two functionals, with denotingthe whole domain of the system. The quantity Jij( ) is theskew-symmetric Poisson operator, and u uFd d[ ] and

u uGd d[ ] are the functional derivatives of F and G, respec-tively. Here a functional derivative u

uFdd

[ ] of an arbitrary

functional uF [ ] is defined by u u uF Fd

d 0

d = + =

=[ ] [ ˜]

uxd ,uu

F3ò

dd

˜[ ] where u is an arbitrary function satisfying thesame boundary condition as u. Note that is a small para-meter and is not the inverse aspect ratio e. The sign of theright-hand side is taken so that energy decreases as timeprogresses. Note that the Dirac theory for imposing con-straints is not used in this paper. As we will describe insection 2.5, we can generate a variety of artificial dynamics bythe choices of the symmetric kernel. We will introduce threetypes of Kij( ) in section 2.5 that can effect smoothing. We willexamine the smoothing effect in the numerical results insection 3.

Here let us discuss briefly the meaning of the symmetrickernel by using the finite-dimensional analogue that appearedin section 1. The artificial dynamics generated by squaring the

Figure 1. Schematic picture explaining Casimir leaf, physical andartificial dynamics.

3

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

Poisson bracket can be rewritten as u u u H, , ,u

ti j jd

d

i

= { }{ }where i j, 1, 2= and ,{· ·} denotes the usual Poisson bracketdefined in classical mechanics (only in this paragraph). Thisequation is already similar to (3) with (4), however, theevolution equation in the finite-dimensional system can be

generalized to u u K u H, , ,u

ti j

jkkd

d

i

= { } { } where k 1, 2= andKjk( ) should have a definite sign. If Kjk( ) is taken to be a unitmatrix, the evolution equation is the same as the one gener-ated just by squaring the Poisson bracket. If Kjk( ) is diagonalbut the diagonal elements are different, the change rates of thevariables in time can be made different. The choice of Kjk( )can generate a variety of artificial dynamics.

Let us come back to the infinite-dimensional MHD sys-tem. The Hamiltonian structure for low-beta reduced MHD,as was first given in [33, 34], has u u u,1 2 T≔ ( ) where u U1 =and u2 y= , and the Hamiltonian

uH x Ud1

263 1 2 2

ò y + ^ ^-

^[ ] ≔ {∣ ( )∣ ∣ ∣ } ( )

with being the whole domain of the cylindrical plasma.This Hamiltonian is the physical one, and is substituted inequation (3) for the SA. The first and the second terms of thethe integrand of (6) correspond to the kinetic energy Ek andmagnetic energy Em, respectively, and the skew-symmetricPoisson operator is given by

x x x x

x x

x

J

U

,

, ,

, 0. 7

ij 3d

y ez

y ez

¢ = ¢ -

´- - +

¶¶

- +¶¶

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

( ( )) ( )

[ ( ) ] [ ( ) ]

[ ( ) ]( )

Here x3d ( ) is the Diracʼs delta function in 3D space, andits explicit form is x r r3d d d q d z e=( ) ( ) ( ) ( ) since

x r rd d d d3 q z e= . In order to write down the evolutionequation of the SA, we need to calculate the Poisson bracketbetween the dynamical variables, and between the dynamicalvariable and the Hamiltonian. The functional derivative of

uH [ ] is straightforward:u

uH

J. 8

dd

j=

--( )[ ] ( )

The functional derivatives of U and ψ can be obtained byconsidering the functionals

x x x xU x Ud , 93 3

ò d= ¢ ¢ - ¢( ) ( ) ( ) ( )

x x x xxd . 103 3

òy y d= ¢ ¢ - ¢( ) ( ) ( ) ( )

The functional derivatives of xU ( ) and xy ( ) are

xu

x xU

0, 11

3dd

d= - ¢⎛⎝⎜

⎞⎠⎟

( ) ( ) ( )

xu x x

0, 123

dyd d

=- ¢

⎛⎝⎜

⎞⎠⎟

( )( ) ( )

hence we obtain

x x x x xU U U, , , 133d¢ = ¢ - ¢{ ( ) ( )} [ ( ) ( )] ( )

x x x x xx x

U , , ,

14

33

y y d ed

z¢ = ¢ - ¢ -

¶ - ¢

¶ ¢{ ( ) ( )} [ ( ) ( )] ( )

( )

x x x x xx x

U, , ,

15

33

y y d ed

z¢ = ¢ - ¢ -

¶ - ¢

¶ ¢{ ( ) ( )} [ ( ) ( )] ( )

( )

x x, 0, 16y y ¢ ={ ( ) ( )} ( )x x x x x

xU H U J

J

, , ,

, 17

j y

ez

= +

-¶ ¶

{ ( ) } [ ( ) ( )] [ ( ) ( )]( ) ( )

x x xx

H, , . 18y y j ejz

= -¶ ¶

{ ( ) } [ ( ) ( )] ( ) ( )

These confirm that the physical evolution equations arewritten as

uu

tH, 19

¶¶

= { } ( )

by the definitions of the Hamiltonian (6) and the Poissonbracket (5).

Using the above Poisson brackets, the symmetricbrackets are obtained as

x x

x xx

U H U

JJ

, ,

, , 20

j

y ez

=

+ -¶¶

(( )) [ ( ) ˜ ( )]

[ ( ) ˜( )]˜( ) ( )

x xx

H, , , 21y y j ejz

= -¶¶

(( )) [ ( ) ˜ ( )] ˜ ( ) ( )

where

x x x x

x x x

x K f

K f

d ,

, , 22

UUU

U

3

òj =-

+ yy

˜ ( ) ( ( ) ( )

( ) ( )) ( )

x x x x

x x x

J x K f

K f

d ,

, , 23

UU3

ò=-

+

y

yyy

˜( ) ( ( ) ( )

( ) ( )) ( )

and

x xx x x x

x x x xK

K K

K K,

, ,

, ,24ij

UU U

U

¢ =¢ ¢ ¢ ¢

y

y yy

⎛⎝⎜⎜

⎞⎠⎟⎟( ( ))

( ) ( )( ) ( )

( )

with f U and f y being defined by the right-hand sides of thephysical evolution equation as

x x x x xx

f U JJ

, , , 25U j y ez

+ -¶¶

( ) ≔ [ ( ) ( )] [ ( ) ( )] ( ) ( )

x x xx

f , . 26y j ejz

-¶¶

y ( ) ≔ [ ( ) ( )] ( ) ( )

Observe, U and ψ are advected by j and J in SA, instead ofjand J in the physical dynamics. A variety of artificialdynamics can be generated by different choices of the kernelKij( ). We will present some choices of Kij( ) in section 2.5, andwill examine the smoothing effects by the choice of Kij( ) inthe numerical results in section 3.

4

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

2.3. Casimir invariants

Before examining the choice of Kij( ), let us introduce theCasimir invariants. A Casimir invariant of the system isdefined as a functional uC [ ] that satisfies

C F, 0 27={ } ( )

for any functional uF [ ]. From the definition of the symmetricbracket (4), it is obvious that the Casimir invariants in theoriginal dynamics are preserved in the artificial dynamics. Ifwe write u uC C C,1 2

Td d =[ ] ( ) and u uF F F,1 2Td d =[ ] ( ) ,

then

x x

x x x

x xx

x x xx

x x x

x xx

x x xx

x x x

x xx

x x xx

C F x x

C U F

FF

C FF

x C U F

FF

C FF

x F C U

CC

F CC

, d d

,

,

,

d ,

,

,

d ,

,

, ,

28

3 3 3

1 1

22

2 11

31 1

22

2 11

31 1

22

2 11

ò ò

ò

ò

d

y ez

y ez

y ez

y ez

y ez

y ez

= ¢ ¢ -

´ ¢ -

- +¶ ¶

+ ¢ - +¶ ¶

= ¢ ¢ - ¢ ¢

- ¢ ¢ +¶ ¢

¶ ¢

+ ¢ - ¢ ¢ +¶ ¢

¶ ¢

= ¢ ¢ - ¢ ¢

- ¢ ¢ -¶ ¢

¶ ¢

+ ¢ - ¢ ¢ -¶ ¢

¶ ¢

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

{ } ( )

( ( )( [ ( ) ( )]

[ ( ) ( )] ( )

( ) [ ( ) ( )] ( )

( ( )( [ ( ) ( )]

[ ( ) ( )] ( )

( ) [ ( ) ( )] ( )

( ( )( [ ( ) ( )]

[ ( ) ( )] ( )

( ) [ ( ) ( )] ( )

( )

where the last equality follows upon integration by parts. Inorder to satisfy C F, 0={ } for any F1 and F2, C1 and C2

must satisfy

x x x xx

C U CC

, , 0, 291 22y ez

+ +¶¶

=[ ( ) ( )] [ ( ) ( )] ( ) ( )

x xx

CC

, 0. 3011y ez

+¶¶

=[ ( ) ( )] ( ) ( )

ChoosingC 01 = andC 12 = , yields xC x Cd3mò y= ( ) ≕ ,

while choosing C 11 = and C 02 = , yields C xd3ò=

xU Cv( ) ≕ . (See [33] for a discussion of how these areremnants of the helicity and cross helicity.) The accuracy of anumerical simulation can be tested by monitoring the con-servation of Cm and Cv.

2.4. Cross helicity

Another conserved quantity of ideal MHD is a cross helicity,which is defined by

v BC xd 31c3

ò≔ · ( )

x Ud . 323

ò y= ( )

The functional derivative of Cc is given by

uC

U. 33cd

dy= ⎜ ⎟⎛

⎝⎞⎠ ( )

Then, x x x xC U C, , 01 2 y+ =[ ( ) ( )] [ ( ) ( )] and x xC ,1 y[ ( ) ( )]0= in (29) and (30), however, ζ-derivative terms remain

finite generally. When uF [ ] in (27) is taken to be uH [ ],F1 j= - and F J2 = - , and we can show that

xx

xx

C H xU

J

x

, d

1

2d

0

34

c3

3 2 2

ò

ò

j ez

eyz

zj y

= ¢ ¢ ¶ ¢

¶ ¢+ ¢ ¶ ¢

¶ ¢

= - ¢ ¶¶ ¢

+

=

^ ^

⎛⎝⎜

⎞⎠⎟{ } ( ) ( ) ( ) ( )

(∣ ∣ ∣ ∣ )

( )

by integration by parts under appropriate boundary condi-tions. Therefore, the cross helicity Cc is also conserved by theSA as well as the physical dynamics.

2.5. Choices for the symmetric kernel

In section 2.2, we obtained a general form of j and J in (22)and (23). Here we introduce three choices for Kij( ). First,let us set the off-diagonal terms of Kij( ) to zero in this

paper for simplicity. Then, x x xJ x K fd ,UUU3

ò= - ˜ ( ) ( )and x x xJ x K fd ,3

ò= - yyy˜ ( ) ( ). Thus we may use the

definition

x x x xh x K fd , 353

ò= - ˜( ) ( ) ( ) ( )

with h chosen to be j or J , K to be KUU or Kyy, and f to bef U or f y, respectively. Below we introduce three choices ofK which will be used in the numerical calculations. Themotivation for the choices is to examine their smoothingeffect. Such smoothing may affect the relaxation path to theenergy minimum, computational cost as well as the numericalstability. Since it is easiler to show the expressions of h in theunified manner, here we introduce the Fourier representationof xh( ) in θ and ζ as

xh h r e . 36m n

m nm n

,

iå= q z+˜( ) ˜ ( ) ( )( )

5

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

Then the Fourier coefficients are given by

x

x x x

x x x

x x

h r h

x K f

x f K

x f K r

1

2d d e

1

2d d d , e

d1

2d d , e

d , .

37

m nm n

m n

m n

m n

2i

23 i

32

i

3

ò

ò

ò

pq z

pq z

pq z

=

= -

= -

= -

q z

q z

q z

- +

- +

- +

∮ ∮

∮ ∮

∮ ∮

˜ ( )( )

˜( )

( )( ) ( )

( )( )

( )

( ) ( )( )

( )

( )

( )

For our first choice of smoothing we consider

x xx x

x xK ,

0

0, 38ij

UU3

3

a da d

=-

- yy

⎛⎝⎜

⎞⎠⎟( ( ))

( )( )

( )

where 0UUa > and 0a >yy are constants that scale theresulting advection fields j and J . Since the Fourier repre-sentation of the 3D delta function is given by x x3d - =( )

r r er m n

m n2 ,

i i2 d - åe

pq q z z- + - ( )

( )( ) ( ) with r rd - ( ) being a

Diracʼs delta function in r, and it is straightforward to obtainh r f rm n m na= -˜ ( ) ( ); namely,

r f r , 39m n UU m nUj a= -˜ ( ) ( ) ( )

J r f r . 40m n m na= - yyy˜ ( ) ( ) ( )

These advection fields are the right-hand sides of the physicalevolution equations (1) and (2) multiplied by UUa- anda- yy, respectively. Note that this is the same situation as the

artificial dynamics generated by squaring the Poisson bracketJ explained in the Introduction. We refer to this version ofsmoothing as ‘SA-1’.

The second choice of smoothing introduced in thispaper is

where gqz is defined by

g , , ,

, 42

2

2

2

2q zq z q z

d q q d z z

¶¶

+¶¶

= - - -

qz

⎛⎝⎜

⎞⎠⎟ ( )

( ) ( ) ( )

i.e., gqz is a Greenʼs function in the θ–ζ plane. By Fouriertransforming (42) in θ and ζ to obtain the Fourier expansioncoefficients of gqz , and we obtain

xK r r rm n r

,2

e .

43

m nm n

2 2 2id

a ep

= - +

q z- + ( ) ( )( ) ( )

( )

( )

Then

h rm n

f r , 44m n m n2 2

a= -

+˜ ( ) ( ) ( )

which gives the advection fields j and J . The symmetricbracket with this smoothing has the effect of reducing short-

wave-length components of the advection fields in the θ–ζ

plane, and thus those modes are not easily excited. It is similarto the one introduced for 2D vortex dynamics [29] and 2Dlow-beta reduced MHD [22, 23]. We refer to this version ofsmoothing as ‘SA-2’.

Lastly, consider our third choice for smoothing,

x xx x

x xK

g

g,

, 00 ,

, 45ijUUa

a =

yy

⎛⎝⎜

⎞⎠⎟( ( ))

( )( ) ( )

where

x x x xg , , 463 d - - ( ) ≔ ( ) ( )

i.e., each diagonal component of the symmetric kernel isproportional to the Greenʼs function in 3D. If we Fourierexpand g in θ, ζ, q and z as

x xg g r r, , e e ,

47m nm n

m n m nm n m n

, ,,

i iåå = q z q z

+ + ( ) ( )

( )

( ) ( )

we can express (46) as

r rr

rg r r

m

rn g r r

rr r

1,

,2

.

48

m n m n

m n m n

,

2

22 2

, 2e

ep

d

¶¶

¶¶

- + =-

-

- -

- -

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

( ( ))

( )( )

( )( )

Here g r r,m n m n, - - ( ) means g r r,m n m n, ( ) withm m = - and n n = - . For a given r, we can solve thehomogeneous equation to obtain the solutions in both

r r0 < and r r 1 < regions. These are actually linearcombinations of the modified Bessel functions. In order todetermine the coefficients of the linear combination, we

require the continuity of gm n m n,- - and the jump condition

rg r r

r

,49m n m n

r

r,

0

0

e¶

¶= -- -

-

+( )( )

at r r= . The jump condition (49) is obtained by integrating(48) from r 0 - to r 0 + . Note that we solved (48) togetherwith (49) numerically, instead of using the modified Besselfunction explicitly. By using this symmetric kernel, we obtain

h r

r r g r r f r

2

d , . 50

m n

m n m n m n

2

0

1

,ò

ape

= -

´ - -

˜ ( ) ( )

( ) ( ) ( )

This version of smoothing can effect not only the behavior inthe θ–ζ plane but also in the r direction, because the advectionfield h at a radial position is obtained by integrating the ori-ginal advection field f over the radius with the weight of the

x xKr r g r

r r g r,

, , , 0

0 , , ,, 41ij

UUa e d q z q z

a e d q z q z =

-

- qz

yy qz

⎛⎝⎜⎜

⎞⎠⎟⎟( ( ))

( ) ( )( ) ( )

( )

6

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

Greenʼs function. We refer to this version of smoothing as‘SA-3’.

Since these choices of the kernels may be different in thesmoothing effect, the relaxation path towards an equilibriumcan be different. If there are several local minima of theenergy on the Casimir leaf, especially if those local minimaare located closely, the system can relax to a different equi-librium. However, we know little about the structure of suchlocal minima in the phase space, so we do not discuss it indetail in this paper. Even if the system goes to a differentequilibrium due to the difference of the relaxation path, it isstill an equilibrium of the original system because u H, 0={ }when u H, 0=(( )) with the kernel Kij( ) with definite sign.

3. Numerical results

Consider now our numerical results obtined from a codedeveloped for solving the artificial evolution equation (3).The code imposes regularity of physical quantities at r=0and 0j y= = at the plasma boundary. The pseudo-spectralmethod is adopted in θ and ζ, which allows for multiplehelicities, while a second-order finite difference method isused in r. For time advancement, fourth-order Runge–Kuttawith adaptive step-size control is used. Starting from an initialcondition, the artificial evolution equation is solved and, inaccordance with theory, the energy of the system decreasesmonotonically. When the relative change rates of both kineticand magnetic energy, E t Ed dk k∣ ∣ and E t Ed dm m∣ ∣ , becomesmaller than a tolerance, the simulation is stopped.

For the numerical results shown below, the inverse aspectratio 1 10e = , while the grid numbers for r, θ and ζ are 100,32 and 16, respectively. The Fourier mode numbers includedin the simulation are m10 10 - and n0 5 ,respectively. The pseudo-spectral method together with two-thirds prescription was used for avoiding aliasing errors. Thetolerance for the convergence was chosen to be 10−6.

We present results for two initial conditions. The firstcorresponds to a trivial stationary state where U U r= ( ) and

ry y= ( ), with corresponding j and J satisfying U j= ^and J y= ^ , also being functions of r only. Clearly, theright-hand side of (3), or (19), becomes zero in this case, andno change of the system occurs. Indeed, the simulation codestopped immediately after initializing.

The second initial condition is given by the stationarystate of the first one, plus a small perturbation that has a radialmagnetic field resonant at a rational surface. The small per-turbation changes the field-line topology by opening amagnetic island. If the stationary state with cylindrical sym-metry is unstable against the associated tearing mode, weexpect the system to evolve and reach a stationary state withmagnetic islands, with its energy decreased by the SA.

Figure 2 shows the safety factor profile q r( ) of the sta-tionary state with cylindrical symmetry. The plasma rotationwas assumed to be zero and ry ( ) was chosen so that thesafety factor q r r re y= - ¢( ) ( ), where the prime denotes rderivative. Specifically, q r q r1 20

2= -( ) ( ) was used,where q0 is the safety factor at r=0, which gives q=2 atr 1 2= . Note that any function q r( ), and no rotation in thiscase, is a stationary state.

The stationary state shown in figure 2 is unstable againsta tearing mode with mode numbers m 2= - and n=1,which has the tearing mode parameter [35] 22.4D¢ » . Wethen expect that the system may evolve to a stationary statewith magnetic islands as a lower energy state if we add them 2= - and n=1 perturbation. Thus a small perturbationwith m 2= - and n=1 was added to the cylindricallysymmetric state. Although the radial profile is not taken to bethe same as the eigenmode structure of the linear tearingmode, it is important to give a radial magnetic field across theq=2 surface to open a tiny magnetic island initially. Theradial profile of the perturbed velocity field, as well as itsphase to the perturbed magnetic field, were chosen also to beconsistent with the linear tearing mode, while it is not thesame as its eigenfunction. The radial profiles of the m 2= -and n=1 components are shown in figure 3. Real or ima-ginary parts of those variables now shown in figure 3 are zerodue to the choice of the relative phase among those variablesfor the initial condition.

Let us compare our three smoothing kernels. For theinitial condition shown in figures 2 and 3, the right-hand sidesof the evolution equations at t=0 are plotted in figure 4. Inthe figure, ‘physical’ shows the m 2= - and n=1 compo-nents of the right-hand side of the physical evolutionequation (19). Only the imaginary part of

U

t

d

d2 1- and the real

part oft

d

d2 1y- are plotted because the real and imaginary parts

of them, respectively, are zero for the initial condition. ForSA-2 and SA-3, the amplitudes of the plotted figures aremultiplied by 10 and 100, respectively for easier comparison.Observe in figure 4(a) how the r-dependence is smoother forSA-3.

Figure 5 shows the radial profile of g r r,m n m n, - - ( ) ofSA-3 for r 0.2 = , 0.4, 0.6 and 0.8. The mode numbers arem 2= - and n=1 in figure 5(a), and m 10= - and n=5in figure 5(b). The range of the vertical axis is the same for

Figure 2. The safety factor profile q r( ) for a stationary state.

7

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

both figures. Note that g r r,m n m n, - - ( ) has smaller ampli-tudes for high m and n. This smoothing effect in 2D is thesame as that observed in [22, 23, 29], and is present also inSA-2. The smoothing effect in r of SA-3 is larger for smallerm and n because the radial extent of g r r,m n m n, - - ( ) islarger for smaller m and n. Note that there is no smoothingeffect in r if g r r r r,m n m n, d = - - - ( ) ( ), as in SA-2. Thusthe smoothing effect in r of SA-3 may disappear if m and n goto infinity, while the smoothing in θ and ζ becomes infinitelylarge.

The time evolution of the energy and the conservedquantities are shown in figure 6 with 100UUa a= =yy forSA-2 and SA-3. SA-1 was numerically unstable, and a sta-tionary state was not obtained. From figure 6(a), we observethat the total energy E Ek m+ decreases monotonically.Figure 6(b) shows the time history of E t Ed dk k∣ ∣ and

E t Ed dm m∣ ∣ . When they became lower than the tolerance10−6, the simulation was stopped. Since the magnetic energyEm is dominant, its change is relatively small from thebeginning. From figures 6(c) to (e), we observe that quantitiesthat should be conserved are well conserved in the simulation.The change of Cm is monitored relative to its initial valueC 0 3.76m =( ) , while Cv and Cc are plotted directly since theirinitial values are zero.

SA-3 requires longer t for convergence. Although thetime t is not physical and depends on UUa and ayy, lateconvergence can also be because SA-3 smooths in r, and thustends to prevent generation of fine structure in r. Magneticislands have current channels localized in r that may be easierto generate with SA-2 than SA-3. This also indicates that astationary state with fine structure in θ and ζ may take moresimulation time using SA-2 and SA-3.

Figure 3.Depiction of m 2= - and n=1 components of (a) UI and jI , (b) yR and JR at t=0. A radial magnetic field exists at the q=2surface.

Figure 4. Depiction of m 2= - and n=1 components of (a) U

t

d

d2 1-

I and (b)t

d

d2 1y-

R at t=0 for physical dynamics and SA. Since the

amplitudes are largely different, those of SA-2 and SA-3 are multiplied by 10 and 100, respectively. The significant smoothing effect in r ofSA-3 is observed in (a).

8

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

Figure 7 shows the real parts of the radial profiles

2 1y-R , 4 2y-R , J 2 1-R and J 4 2-R of the obtained sta-tionary state. The radial magnetic field of the m 2= - andn=1 mode remains at the q=2 surface when the magneticisland exists. These profiles differ greatly from the initialcondition, with larger amplitudes, while the radial profile ofthe m n 2 1= - mode is still similar to the correspondinglinear mode. Therefore, the magnetic island of this stationarystate saturated in a weakly nonlinear sense.

On the other hand, the radial profiles of SA-2 and SA-3 are a bit different. Also, almost no change was observed in

UI and jI . These will be discussed in the next section.

4. Discussion

Firstly, let us investigate why SA-2 and SA-3 differ. Onereason could be the tolerance for stopping the simulation,which was set to E t Ed dk k∣ ∣ and E t Ed dm m∣ ∣ becomingsmaller than 10−6. While the magnetic energy of them n 0 0= component is very large, the relative rate ofchange of Em of the m n 0 0¹ components was very small.Therefore, we may need another criterion for convergence.For example, separating out the energy of the m n 0 0=mode and monitoring the relative change rates of both com-ponents of energy could be an improvement.

Secondly, let us investigate why UI and jI did notchange during the SA evolution; i.e., why ψ relaxed fasterthan U. One possible reason is again the convergence criter-ion. If a longer simulation is performed with a much smallertolerance, U and j may also change. Another possible reasonmay be due to the choice of UUa and ayy, especially theirratio. If we write the evolution equations for SA-1 explicitly,

we have

x x

x xx

U

tU f

ff

,

, , 51

UUUa

a y ez

¶¶

=

+ -¶¶

yyy

y⎛⎝⎜

⎞⎠⎟

[ ( ) ( )]

[ ( ) ( )] ( ) ( )

x xx

tf

f, . 52U

Uya y e

z¶¶

= -¶¶

yy⎛⎝⎜

⎞⎠⎟[ ( ) ( )] ( ) ( )

Therefore, the ratio of UUa to ayy can significantly affect thetime evolution of U. As was studied in [22] for the 2D cases,the relaxation path can change if we change the ratio of UUa toayy. As we observe, the time evolution of U is governed bytwo advection fields f U and f y, while ψ by f U only. Thereforethe relaxation path can change if we change the ratio ofcontributions from f U and f y. This situation is also the samefor SA-2 and SA-3. If the relaxation of U is much slower thanψ, a simple solution is to increase the ratio of UUa toayy. Thenthe right-hand side of the evolution equation of ψ becomessmaller and that of U becomes larger. However, if UUa isincreased in the present code, the simulation tends to beunstable. The dependence of the numerical stability on UUaand ayy, in addition to the choice of the symmetric bracket,needs to be examined more carefully.

The result of section 3 is only one example of a magneticisland stationary state achievable with SA. When the initialperturbation of the m 2= - and n=1 component was chosenlarger, the m=0 and n=0 components of U and ψ werechanged significantly by the nonlinear effects, leading to a dif-ferent stationary state. Incorporating Dirac constraints as in [29]should be explored in the future for selecting out desired states.Also, the effects of the m=0 and n=0 component of theplasma rotation should be investigated because it changes thelinear stability against tearing modes. Details of these issues willbe studied and will be reported on in the near future.

Figure 5. Radial profile plots of the 3D Greenʼs functions g r r,m n m n, - - ( ) of SA-3 with (a) m 2= - , n=1 and (b) m 10= - and n=5,for r 0.2 = , 0.4, 0.6 and 0.8. The amplitudes of g r r,m n m n, - - ( ) are smaller for larger m and n, implying a larger reduction of the short-

wave-length components of the advection fields in the θ–ζ plane. Also, observe the larger smoothing in r for smaller m and n, since the radialextent of g r r,m n m n, - - ( ) is larger for smaller m and n.

9

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

Figure 6. Time evolution of (a) total energy E Ek m+ , (b) relative change rate of energy E t Ed dk k∣ ∣ and E t Ed dm m∣ ∣ , (c) relative changeC t C C0 0m m m-( ( ) ( )) ( ), (d)Cv and (e)Cc. The values 100UUa a= =yy were used. The total energy decreases monotonically and reachesa stationary state. The relative change of Cm is normalized by the initial value C 0m ( ) in (c). Since C 0v = and C 0c = at t=0, just theirvalues themselves are plotted in (d) and (e).

10

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison

5. Summary

The method of SA was developed to obtain a 3D stationarystate of low-beta reduced MHD in cylindrical geometry. Thetheory of SA was explained for low-beta reduced MHD, andthree versions of the symmetric bracket were introduced. Asimulation demonstrated that the energy of the systemmonotonically decreases by SA, while conserving otherinvariants. Starting from a cylindrically symmetric state withthe addition of a perturbation that opens a small magneticisland at a rational surface, SA generated a stationary statewith magnetic islands as a lower energy state. Especially wefocused on the smoothing effects by the symmetric bracketsin examining the numerical results. A symmetric bracket withhigher smoothing may require longer simulation time forconvergence, while it can contribute to numerical stability.Several issues for consideration in the future were discussed.

Acknowledgments

MF was supported by JSPS KAKENHI Grant #23760805and #15K06647. PJM was supported by US DOE Grant#DE-FG02-04ER-54742 and the Humboldt Foundation.

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6 459

Figure 7. Radial profiles of (a) 2 1y-R and 4 2y-R and (b) J 2 1-R and J 4 2-R of the obtained stationary state are plotted. Almost no changesoccur in UI and jI . The m n 2 1= - components have similar structure as the linear mode.

11

Plasma Phys. Control. Fusion 59 (2017) 054001 M Furukawa and P J Morrison