Calculation of Large-Aspect-Ratio Tokamak and Toroidally-Averaged Stellarator Equilibria of High-Beta Reduced Magnetohydrodynamics Via Simulated Annealing M

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Citation: Physics of Plasmas 25, 082506 (2018); doi: 10.1063/1.5038043 View online: https://doi.org/10.1063/1.5038043 View Table of Contents: http://aip.scitation.org/toc/php/25/8 Published by the American Institute of Physics PHYSICS OF PLASMAS 25, 082506 (2018)

Calculation of large-aspect-ratio tokamak and toroidally-averaged stellarator equilibria of high-beta reduced magnetohydrodynamics via simulated annealing M. Furukawa,1,a) Takahiro Watanabe,1 P. J . Morrison,2 and K. Ichiguchi3,4 1Faculty of Engineering, Tottori University, Minami 4-101, Koyama-cho, Tottori-shi, Tottori 680-8552, Japan 2Physics Department and Institute for Fusion Studies, University of Texas at Austin, Austin, Texas 78712, USA 3National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan 4SOKENDAI, The Graduate University for Advanced Studies, Toki 509-5292, Japan (Received 30 April 2018; accepted 16 July 2018; published online 6 August 2018) A simulated annealing (SA) relaxation method is used for the calculation of high-beta reduced magnetohydrodynamics (MHD) equilibria in toroidal geometry. The SA method, based on artiﬁcial dynamics derived from the MHD Hamiltonian structure, is used to calculate equilibria of large- aspect-ratio and circular-cross-section tokamaks as well as toroidally averaged stellarators. Tokamak equilibria including incompressible poloidal rotations are obtained and the Shafranov shift is seen to increase nearly quadratically in the rotation speed. A mapping procedure between non- rotating and poloidally rotating equilibria is shown to explain the quadratic dependence of equilibria shift on rotation. Calculated stellarator equilibria are seen to agree reasonably with previous results. The numerical results demonstrate the ﬁrst successful application of the SA method to obtain toroidal equilibria. Published by AIP Publishing. https://doi.org/10.1063/1.5038043

I. INTRODUCTION The SA method was applied to low-beta reduced MHD9 in Ref. 10, using the reduced MHD Hamiltonian structure of The calculation of magnetohydrodynamics (MHD) equi- Refs. 11 and 12. Various equilibria were successfully libria is an indispensable element of basic plasma physics, as obtained in a two-dimensional rectangular domain with peri- well as the application to magnetic fusion research. Many odic boundaries in both dimensions. Moreover, a method methods for equilibrium calculations have been developed was developed for pre-adjusting values of the Casimir invari- for two-dimensional and three-dimensional toroidal plasmas. ants of an initial condition for the SA.13 More recently, low- The present work utilizes and develops the alternative simu- beta reduced MHD equilibria in cylindrical geometry were lated annealing (SA) method of Ref. 1 for the calculation of calculated by SA, where the inside of the plasma was heli- equilibria. This method is based on the fact that ideal fluid cally deformed, even including magnetic islands.14 The sym- dynamics, including ideal MHD, possesses a Hamiltonian 2 metric bracket with a smoothing effect was adopted in these form in terms of a noncanonical Poisson bracket. Because of the Hamiltonian form, ideal MHD conserves the energy of studies, demonstrating the usefulness of SA for the MHD the system and has so-called Casimir invariants that arise equilibrium calculation. For an overview of structure and from degeneracy of the Poisson bracket. With the SA structure-preserving algorithms for plasma physics from method, we solve an artificial dynamics that is derived so much wider view point, including SA, we refer to Ref. 15. In the present paper, SA is applied to calculate high-beta that the energy of the system changes monotonically while 16 the Casimir invariants are preserved. Given that equilibria reduced MHD equilibria in toroidal geometry, which is possible because high-beta reduced MHD is also a Hamiltonian are obtained by extremizing the energy at fixed Casimir 11 invariants,3–5 we can obtain an MHD equilibrium by solving system. Large-aspect-ratio and circular-cross-section toka- the artificial SA dynamics as an initial-value problem. maks as well as toroidally averaged stellarator equilibria are A SA method was first developed for two-dimensional calculated. The results obtained take into account the effect 6–8 of toroidicity, where a proper accounting of the Shafranov neutral fluid flows. The method was significantly general- 17 ized in Ref. 1 by developing several kinds of brackets that shift is observed. We compare our numerical results with monotonically change the energy of the system while preserv- the previous studies and obtain reasonable agreement. For ing the Casimir or other invariants of the original system. tokamak equilibria, we also include incompressible poloidal There, a symmetric bracket with the possibility of various rotation. This highlights one of the advantages of the SA effects such as smoothing through a symmetric kernel and the method. The results given here comprise a significant next preservation of chosen constraints, ones not inherent in the step towards ultimate goal of calculating MHD equilibrium original dynamics, was introduced by making use of a Dirac in fully toroidal geometry. bracket, and the general use of such brackets for obtaining Furthermore, in the present paper, we use a mapping equilibria was termed as “simulated annealing.” procedure between non-rotating and poloidally rotating equilibria for high-beta reduced MHD. This is an extension to high-beta reduced MHD of the low-beta reduced MHD map a)[email protected] first given in Ref. 18, which is a special case of the

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generalization of Refs. 19 and 20. This map nicely explains We also note that the reduced MHD equations (1)–(3) how equilibria change with poloidal rotation. have the same form as the reduced MHD equations for heli- This paper is organized as follows. In Sec. II, the theory cal plasmas derived under the stellarator expansion.21 of the SA is described for high-beta reduced MHD. Next, we However, three changes of Eqs. (1)–(3) are needed for stella- will explain numerical schemes for solving the equations of rators. First, the toroidal angle f is understood as a toroidal the SA and the convergence criterion in Sec. III. Then, equi- angle for long wavelength structures; short wavelength struc- libria of tokamaks and toroidally averaged stellarators are tures are averaged out in the toroidal direction. Second, the presented in Sec. IV. The results include tokamak equilibria poloidal flux function w is replaced by a total poloidal flux with poloidal rotation. There these numerical results are function W :¼ Wh þ w, where Wh is a helical flux generated compared and contrasted with the previous studies. The map- by external coils. Third, the curvature term h is replaced by 1 ping procedure is also presented in Sec. IV. Conclusions are 2 X, where X is the sum of the curvature of the toroidal mag- given in Sec. V. netic field and that of the helical magnetic field. Details of Wh and X will be given in Sec. IV C. II. FORMULATION A. High-beta reduced MHD and normalization B. Hamiltonian formulation The high-beta reduced MHD equations16 are given by For the high-beta reduced MHD equations, the Hamiltonian form was presented in Refs. 11 and 12. The @U @J ¼ ½U; u þ ½w; J e þ ½P; h ¼: f 1; (1) Hamiltonian is given by @t @f ð 1 1 @w @u H½u :¼ d3x jr ð᭝1UÞj2 þ jr wj2 hP ; (6) ¼ ½w; u e ¼: f 2; (2) 2 ? ? 2 ? @t @f D T @P where the state vector u :¼ðU; w; PÞ and the domain is ¼ ½P; u ¼: f 3: (3) @t denoted by D. The functional derivative of H½u is defined through Here, the fluid velocity v and the magnetic field B are ð dH½u expressed by ½ 3 i lim ðH u þ du H½u Þ¼ d x i du ; (7) du!0 D du v ¼ ^z ru; (4) where i ¼ 1, 2, 3 and we sum over the repeated indices. B ¼rw ^z þ ^z; (5) Then, dH½u=du is obtained as where the unit vector in the toroidal direction is denoted by 0 1 u ^z. A coordinate system (r, h, f) is used, where r is the minor dH½u ¼ @ J A: (8) radius, h is the poloidal angle, and f is the toroidal angle. du h The length in the toroidal direction is measured by z: ¼ R0f with R0 being the major radius of the toroidal plasma and By defining the Poisson operator as x :¼ r cos h. The stream function and the magnetic flux func- 0 1 tion are denoted by u and w, respectively, with the vorticity @ ᭝ B ½U; ½þw; e ½P; C in the f-direction defined by U :¼ ?u and the current den- B @f C ᭝ B C sity in the negative f-direction defined by J :¼ ?w. The J :¼ B @ C; (9) B ½þw; e 00C two-dimensional gradient operator r? is defined in the r–h @ @f A plane, and the corresponding Laplacian is defined as usual ½P; 00 by ᭝? :¼r? r?. The Poisson bracket between any functions f and g is defined by ½f ; g :¼ ^z rf rg. Quantities high-beta reduced MHD can be written in Hamiltonian form appearing in the equations above are normalized by the as plasma minor radius a, a typical mass density q0, a typical pressure p , a toroidal magnetic field B , an Alfven velocity @u dH½u 0 pffiffiffiffiffiffiffiffiffiffi 0 ¼J ¼ fgu; H ; (10) defined by vA :¼ B0= l0q0, where l0 is the vacuum perme- @t du ability, and an Alfven time defined by s :¼ a=v .An A A where in the last equality the Poisson bracket is used. A gen- inverse aspect ratio is defined by e :¼ a=R0. Finally, P 2 eral noncanonical Poisson bracket has the form :¼ b0p with b0 :¼ 2l0p0=B0 and p being the normalized 1 2 3 ð ð pressure, and h :¼ ex. The right-hand sides f , f , and f are dF½u dG½u F; G : d3x0 d3x00 Jij x0; x00 ; (11) defined for later use. fg¼ i 0 ð Þ j 00 D D du ðx Þ du ðx Þ Note that if the f-derivative terms in Eqs. (1) and (2) are dropped, then the system reduces to axisymmetric dynamics. where Jij denoting the components of a Poisson operator. In The effect of the toroidal geometry appears only through the our case, J ¼ d3ðx0 x00ÞJ , with J being given by Eq. (9). last term of Eq. (1). (See Ref. 5 for further details.) 082506-3 Furukawa et al. Phys. Plasmas 25, 082506 (2018)

C. Casimir invariant convection fields u~ ; J~, and h~, respectively. The right-hand sides f~1; f~2, and f~3 are again defined for later use. A Casimir invariant is defined to be a functional C½u Finally, we explain the boundary conditions. We will that satisfies {C, F} ¼ 0 for any functional F½u, where the use a Fourier mode expansion in the poloidal and toroidal curly bracket denotes a noncanonical (degenerate) Poisson directions. For the poloidal mode number m ¼ 0 components, bracket. It is easily shown that the high-beta reduced MHD the radial derivatives are set to be zero at r ¼ 0, for the other bracket defined by (9) has the following three Casimir m components, their values are set to be zero. At the plasma invariants: edge, all the Fourier components are set to be zero. ð 3 Cv :¼ d xU; (12) III. NUMERICAL SCHEMES AND CONVERGENCE ðD CRITERION 3 Cm :¼ d x w; (13) A. Numerical schemes D ð In addition to the Fourier mode expansion in h and f, the 3 Cp :¼ d xP: (14) numerical scheme we adopt for solving Eqs. (20)–(22) D employs a second-order finite difference method in r. Here, we only consider axisymmetric configurations; consequently, the mode number n for f only takes the value 0. The second- D. Artificial dynamics for relaxation order implicit Runge-Kutta method is used for the time step- ping, which is time-reversal symmetric and symplectic. The In this paper, we adopt an artificial dynamics defined by implicit equation associated with this procedure is solved by the following double bracket:1,15 the Newton–Raphson method that uses the generalized mini- ð ð mum residual (GMRES) method22 implemented in the 3 0 3 00 i 0 0 00 j 00 ððF; GÞÞ :¼ d x d x fF; u ðx ÞgKijðx ; x Þfu ðx Þ; Gg ; Jacobian-free form. D D We assume that the symmetric kernel (K ) in Eq. (15) is (15) ij diagonal, as in Ref. 14, and we choose each diagonal term to 0 00 0 00 defined for any functionals F½u and G½u, where i, j ¼ 1, 2, 3 have the form aigðx ; x Þ, where ai is a constant and gðx ; x Þ is the three-dimensional Green’s function defined through and (Kij) is a symmetric kernel with a definite sign. The curly bracket in Eq. (15) is, in general, any Poisson bracket of the the Poisson equation form of (11). When we take (K ) as positive definite, then ij ᭝gðx; x00Þ¼d3ðx x00Þ: (23) the energy of the system decreases monotonically by the artificial dynamics Here, the three-dimensional Laplacian is written as ᭝ and 3 00 @u d ðx x Þ is the three-dimensional Dirac delta function. ¼ððu; HÞÞ: (16) This choice of (K ) introduces a smoothing effect in all @t ij directions. For the case at hand, it is convenient to introduce the following artificial convection fields: B. Convergence criterion ð The code used for the present calculations evolves the 3 0 0 i 0 u~ ðxÞ :¼ d x K1iðx; x Þf ðx Þ; (17) Fourier components of U, w,andP. Thus, the right-hand side ðD of the evolution equations (20)–(22) is also Fourier decom- ~i ~ 3 0 0 i 0 posed. We write the Fourier coefficients of ððu; HÞÞ,i.e.,f as JðxÞ :¼ d x K2iðx; x Þf ðx Þ; (18) ~i D f mn, where m and n denote the poloidal and toroidal mode ð numbers, respectively. We also write the Fourier coefficients ~ 3 0 0 i 0 i i hðxÞ :¼ d x K3iðx; x Þf ðx Þ; (19) of f of the original evolution equations (1)–(3) as fmn.Then, D the convergence criterion is set such that the maximum abso- i ~i in terms of which the artificial evolution of Eq. (16) can be lute values among fmn and f mn become smaller than a thresh- 6 written compactly as old value. We took 10 , or smaller in some cases, as the threshold value in the numerical results in this paper. @U ÂÃ@J~ ÂÃ ¼ ½U; u~ þ w; J~ e þ P; h~ ¼: f~1; (20) @t @f IV. NUMERICAL RESULTS @w @u~ A. Axisymmetric tokamak equilibrium without rotation ¼ ½w; u~ e ¼: f~2; (21) @t @f Let us turn to our first example, the calculation of large- @P aspect-ratio, circular-cross-section tokamak equilibria using ¼ ½P; u~ ¼: f~3: (22) @t SA. The initial conditions are chosen to be cylindrically symmetric with the profiles of the safety factor q and the As we see, the convection fields u, J, and h of the original pressure p as plotted in Fig. 1. We take P ¼ b0p and consider evolution equations (1)–(3) are replaced by the artificial three cases with b0 ¼ 0.1%, 0.5%, and 1%. The rotation 082506-4 Furukawa et al. Phys. Plasmas 25, 082506 (2018)

FIG. 2. Time evolution of the kinetic energy Ek, the magnetic energy Em, FIG. 1. The initial safety factor q and the pressure p profiles. the internal energy Ep, and the total energy Ek þ Em þ Ep. The magnetic energy increases as the time proceeds, while the internal energy decreases more, resulting in the total energy decreasing monotonically. velocity is chosen to be zero and, consistently, the stream function u 0. In Subsection IV B, we will consider the case of finite poloidal rotation. Throughout, the inverse From the peaks of the w profiles, we evaluated the aspect ratio is set to be e ¼ 1/10. Shafranov shift D(0), which is shown in Fig. 5. The result of The time evolution of the total and various components of our calculations denoted by “SA” is compared with the the energy is shown in Fig. 2 for the case of b0 ¼ 1%. The Shafranov shift obtained from the analytic solution using the 23 kinetic, magnetic, and internal energies are denoted by Ek, Em, large-aspect-ratio expansion, denoted by “Shafranov eq.”. and Ep, respectively. The magnetic energy increases as the In the latter, the Shafranov shift remains at b0 ¼ 0 since time proceeds, while the internal energy decreases with the the toroidicity remains even at zero beta; however, the total energy decreasing monotonically. Note that the kinetic Shafranov shift seems to be zero at b0 ¼ 0 for SA. This is energy remains zero, a consequence of ((U, H)) being zero for because the toroidal effect fully disappears at zero beta in the chosen initial condition, resulting in U remaining zero. high-beta reduced MHD. Note that the time evolution for the cases b0 ¼ 0.1% and 0.5% We also note that w and P are convected by the same is similar to those of Fig. 2. Also note that the Casimir invari- convection field u~ in the two-dimensional case. Therefore, the ants Cv, Cm,andCp are well conserved in the simulation. initial relation between w and P is retained, i.e., if we give an Figure 3 shows contour plots of w for the equilibria initial condition in a form P ¼ P(w), the same relation holds in obtained in our three cases. The horizontal axis x is the the obtained equilibrium. This is shown in Fig. 6.Weobserve distance along the major radius measured from the plasma that the initial relation between w and P is retained for each center R0. The vertical axis y is the height from the mid- b0. This means, in a sense, that we can specify the pressure plane. Observe, the Shafranov shift increases as b0 is profile of the two-dimensional equilibrium to be obtained, as increased. is usual when solving the Grad-Shafranov equation. Figure 4 shows plots of the w profiles on the midplane With SA, the total poloidal flux is conserved because of of the three obtained equilibria, along with the initial w pro- the Dirichlet boundary condition at the plasma edge. file common to each case. As b0 is increased, the peak of w Therefore, the maximum value of w does not change from moves outward. This is due to the increase in the m ¼ 61 the initial condition, as we see in Fig. 6 as well as in Fig. 4. and n ¼ 0 components of w. For these numerical results, the However, the safety factor profile changes from the initial m ¼ 62 and n ¼ 0 components are very small for the inverse condition. In tokamak equilibrium calculations, it may be aspect ratio e ¼ 1/10. The pressure profiles also have the sim- necessary to control the safety factor profile. Such local pro- ilar outward shift of their peaks. file control using SA will be a future issue.

FIG. 3. Contour plots of w for (a) b0 ¼ 0.1%, (b) 0.5%, and (c) 1%. The horizontal axis x denotes the distance along the major radius direction measured from the center of the plasma R0. The vertical axis y is the height from the midplane. 082506-5 Furukawa et al. Phys. Plasmas 25, 082506 (2018)

FIG. 6. The relation between w and P ¼ b0p is shown in a scatter plot using FIG. 4. The w proﬁles on the midplane for three equilibria obtained by simu- 4 w and P at 10 points on the poloidal cross section. For each b0, the initial lated annealing, along with the initial w proﬁle common to each case. relation between w and P is retained.

B. Axisymmetric tokamak equilibrium with poloidal rotation initial condition to meet the equilibrium condition. Figure 7 shows that the Shafranov shift of the magnetic axis D(0) One of the advantages of SA is that it can handle various increases approximately quadratically in the poloidal rotation types of plasma rotation. Substantial literature exists for axi- speed. This is due to the dynamic pressure of the flow. Here, symmetric tokamak equilibria with plasma rotations and we the poloidal rotation speed is normalized by the toroidal mention Refs. 24–27 as examples, and there are recent results Alfven velocity, but if we normalize it by the poloidal Alfven on a numerical code for equilibrium calculations of two-fluid velocity at the plasma edge, v ¼ 102 corresponds to 3.5% of equations,28 where both toroidal and poloidal rotations can h the poloidal Alfven velocity. The increase in the Shafranov exist in the equilibrium. In the present model, the rotation is shift is about 5%–7% compared to the non-rotating equilib- in the poloidal direction. Note that the purely poloidal rotation rium at v 102; however, the flux surface shape does is possible only in the large aspect ratio limit, under which the hmax ¼ not change much. reduced MHD model is derived. Because the flow is incom- Here, we explain a mapping procedure between an equi- pressible, there is no difficulty in calculating such equilibria. We performed SA starting from the same initial conditions in librium without rotation and poloidally rotating equilibria for Subsection IV A for w and P,butnowwechooseau that the high-beta reduced MHD and then show that the quadratic dependence on the poloidal rotation speed can be explained gives vh ¼ 4vhmaxrð1 rÞ. Therefore, vh has its maximum by the map. The map is an extension of a map first derived value vhmax at r ¼ 1/2 when t ¼ 0. The rotation profile can change as time proceeds. Indeed, u must be a function of w in for the low-beta reduced MHD in Ref. 18 and a special case the equilibrium according to the Ohm’s law (2) with axisym- of that of Refs. 19 and 20. The equilibrium equations includ- i metry. The flux surfaces shift outward in the major radius ing the poloidal rotation are given by f ¼ 0(i ¼ 1, 2, 3). direction at finite beta, and thus u must also change from the From the equation for i ¼ 2, or [w, u] ¼ 0, we obtain u ¼ G(w), where G is an arbitrary function. Also from the

FIG. 5. The Shafranov shift D(0) vs. b0. Here, “SA” denotes the simulated annealing result, while “Shafranov eq.” denotes the analytic solution FIG. 7. The effect of poloidal rotation on the Shafranov shift for b0 ¼ 0.1% obtained by large-aspect-ratio expansion. and 1%. The Shafranov shift increases quadratically in vhmax. 082506-6 Furukawa et al. Phys. Plasmas 25, 082506 (2018) equation for i ¼ 3, or [P, u] ¼ 0, we obtain P ¼ KðuÞ of Sec. II A, the reduced MHD equations (1)–(3) can be used ¼ K GðwÞ¼: LðwÞ, where K is an arbitrary function. for stellarators under an appropriate replacement of variables. Noting that J ¼ ᭝?w and Here, we further assume that the dependence of the equilibrium quantities on f, the toroidal angle for the long wavelength ᭝ 0 ᭝ 00 2 U ¼ ?u ¼ G ðwÞ ?w þ G ðwÞjr?wj ; (24) structure, vanishes. Then, the f-derivative terms drop out. The poloidal flux function w is replaced by a total poloi- f1 ¼ 0 reads dal flux function W :¼ W þ w, where W is independent of hi h h time. Thus, @Wh/@t ¼ 0 and only w on the right-hand side G02ðwÞ᭝ w þ G0ðwÞG00ðwÞjr wj2; w ? ÂÃ? needs to be replaced. The poloidal flux function due to heli- 0 cal coils, derived under the cylindrical approximation, is ½᭝?w; w hL ðwÞ; w ¼ 0: (25) given in normalized form by Therefore, we obtain12 i- ðaÞ FðMerÞ W :¼ h ; (32) 02 0 00 2 h 0 ð1 G ðwÞÞ᭝?w G ðwÞG ðwÞjr?wj M F ðMeÞ þ hL0ðwÞþFðwÞ¼0; (26) where i-hðaÞ is a rotational transform generated by the helical coils, evaluated at the plasma edge, M is the pitch number, e where F(w) is an arbitrary function. This is an extended Grad-Shafranov equation for a poloidally rotating equilib- is the inverse aspect ratio, and rium. By setting G(w) 0, we obtain the Grad-Shafranov ‘ FðMerÞ :¼ I ðMerÞI0 ðMerÞ: (33) equation without rotation as Mer ‘ ‘ ᭝ w hL0 w F w 0: (27) ? þ ð Þþ ð Þ¼ Here, ‘ is the pole number, I‘(z) denotes the ‘-th order modi- fied Bessel function of the first kind, and I0 ðzÞ denotes a z Now, if we define ‘ derivative of I (z). ð ‘ wqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The curvature term X, derived under cylindrical approx- v ¼ XðwÞ :¼ 1 G02ðw0Þ dw0; (28) imation, is given in normalized form by

GðMerÞ the equilibrium equation can be rewritten as X :¼ 2er cos h þ ei- ðaÞ ; (34) h F0ðMeÞ ᭝ v þ hL0ðvÞþFðvÞ¼0; (29) ? where F0ðMeÞ denotes the derivative of F with respect to where Mer and is evaluated at r ¼ 1, and

0 1 ÀÁ‘2 þðMerÞ2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL ðX ðvÞÞ 0 2 2 L ðvÞ :¼ ; (30) GðMerÞ :¼ I‘ðMerÞ þ 2 ðÞI‘ðMerÞ : (35) 1 G02ðX1ðvÞÞ ðMerÞ

FðX1ðvÞÞ The first and the second terms of X are the toroidal magnetic FðvÞ :¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (31) 1 G02ðX1ðvÞÞ field curvature and the helical curvature, respectively.

Note that X1(v) ¼ w. Equation (29) has the same form as Eq. (27) for an equilibrium without rotation. Therefore, an equilibrium without rotation can be mapped to equilibria with poloidal rotation. We see thatpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the source terms of Eq. (29) are changed by the factor 1= 1 G02ðwÞ. Here, G0ðwÞ expresses the poloidal rotation velocity normalized by the poloidal Alfven velocity. WhenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijG02ðwÞj 1, the factor can 02 1 02 be approximated as 1= 1 G ðwÞ ’ 1 þ 2 G ðwÞ that indicates effective increase in the source term, such as the pressure gradient, quadratically in the poloidal rotation speed. This explains the quadratic dependence of the Shafranov shift on the poloidal rotation speed. We also point out that any equilibria without rotation can be identified with one with rotation by a choice of G. This, of course, includes the equilibria obtained here by SA.

C. Heliotron equilibrium averaged over toroidal direction For our next example, we consider heliotron equilibria averaged over the toroidal direction. As mentioned at the end FIG. 8. The magnetic axis position Rax plotted against b0. 082506-7 Furukawa et al. Phys. Plasmas 25, 082506 (2018)

FIG. 9. Contour plots of W for (a) b0 ¼ 3%, (b) b0 ¼ 7%, and (c) b0 ¼ 11%.

We seek to reproduce simulations of the Heliotron E the plasma edge. In our simulation, it is exactly circular, device published in Ref. 29. We assume the inverse aspect while it is slightly elongated in Ref. 29. The aspect ratio is ratio e ¼ 1/10, which is similar to that of the Heliotron E also not exactly matched. Despite these reasons, we observe device, and that the poloidal cross section has a circular reasonable agreement between our SA results and those of shape. Also, we set the pole number ‘ ¼ 2 and the pitch num- Ref. 29. ber M ¼ 19. Figure 8 shows the magnetic axis position Rax as a func- 2 For the initial pressure profile, we take P ¼ b0(1 – s) tion of b0. Observe that the tendency and the magnitude of with s :¼ (W – W(0))/(W(a)–W(0)) being a normalized total the magnetic axis shift agree reasonably with Fig. 2(a) of poloidal flux function, where W(0) and W(a) are evaluated at Ref. 29, despite the differences explained above. the magnetic axis and at the plasma edge, respectively. This TheshapesofthefluxsurfacesareshowninFig.9. is the same profile as that of Ref. 29. Note that this pressure A similar figure of flux surfaces is presented in Fig. 3 of profile remains unchanged during the artificial SA dynamics, Ref. 29 for b0 ¼ 7%. The shapes of the flux surfaces in the as explained in the last part of Sec. IV A. central region of the plasma, especially the deformation to It is difficult to completely reproduce the results of Ref. the D shape at high beta, show behavior similar to that of 29 for several reasons. First, the net plasma current in the Ref. 29. toroidal direction on each flux surface cannot be kept zero Figure 10 depicts the rotational transform. Here, in Fig. with the form of SA used here, while in Ref. 29 it is set to be 10(a), the radial profiles ofi - for b0 ¼ 3%, 7%, and 11% are zero. The second reason is that there is difficulty in control- plotted as functions of the normalized total poloidal flux s. ling the local profile of the rotational transformi - with SA. Alsoi - at the magnetic axisi -ð0Þ, at its minimumi -min, and at Because of these two reasons, we cannot perfectly match the the plasma edgei -ðaÞ as functions of b0 are shown in Fig. rotational transform with that of Ref. 29. A third reason is 10(b). Although it is difficult to identify from the figure,i -ðaÞ that the expressions for the helical flux Wh in Eq. (32) and slightly decreases andi -ð0Þ slightly increases as b0 is the magnetic curvature X in Eq. (34) are derived under the increased. We do not observe a difference betweeni -ð0Þ and cylindrical approximation, while those in Ref. 29 are calcu- i-min, as was observed in Ref. 29. The main reason for this dif- lated numerically from helical coil currents in toroidal geom- ference is that the net current free condition in the equilibria etry. The last reason is due to the difference of the shape of of Ref. 29 is not imposed in our equilibria. The difference

FIG. 10. (a) The radial profiles ofi - for several values of b0 as functions of the normalized total poloidal flux s. (b) The rotational transform at the magnetic axisi -ð0Þ, at its minimumi -min, and at the plasma edgei -ðaÞ plotted against b0. 082506-8 Furukawa et al. Phys. Plasmas 25, 082506 (2018) betweeni -ð0Þ andi -min arises because of the magnetic shear Institute for Fusion Science and JSPS KAKENHI Grant No. reversal that is caused by forcing the net zero current. 15k06651. In summary, because of some differences in the equilibrium parameters, e.g., the net current, we do not obtain per- 1G. R. Flierl and P. J. Morrison, Physica D 240, 212 (2011). fect agreement with Ref. 29. However, we obtain reasonable 2P. J. Morrison and J. M. Greene, Phys. Rev. Lett. 45, 790 (1980). agreement, especially in the Shafranov shift and the flux sur- 3M. D. Kruskal and C. R. Oberman, Phys. Fluids 1, 275 (1958). 4 face shapes. V. I. Arnol’d, Prikl. Math. Mech. 29, 846 (1965), [J. Appl. Math. Mech. 29, 1002–1008 (1965)]. 5P. J. Morrison, Rev. Mod. Phys. 70, 467 (1998). V. CONCLUSIONS 6G. K. Vallis, G. F. Carnevale, and W. R. Young, J. Fluid Mech. 207,133 (1989). We have developed the theory of the simulated anneal- 7G. F. Carnevale and G. K. Vallis, J. Fluid Mech. 213, 549 (1990). ing for calculating high-beta reduced MHD equilibria in 8T. G. Shepherd, J. Fluid Mech. 213, 573 (1990). toroidal geometry. Large-aspect-ratio circular-cross section 9H. R. Strauss, Phys. Fluids 19, 134 (1976). 10 tokamak and toroidally averaged stellarator equilibria were Y. Chikasue and M. Furukawa, Phys. Plasmas 22, 022511 (2015). 11P. J. Morrison and R. D. Hazeltine, Phys. Fluids 27, 886 (1984). successfully calculated. These equilibria possess the proper 12J. E. Marsden and P. J. Morrison, Contemp. Math. 28, 133 (1984). Shafranov shift according to the toroidal effect. We obtain 13Y. Chikasue and M. Furukawa, J. Fluid Mech. 774, 443 (2015). reasonable agreement of our results with the previous stud- 14M. Furukawa and P. J. Morrison, Plasma Phys. Controlled Fusion 59, ies. Especially for the tokamak equilibrium calculation, we 054001 (2017). 15P. J. Morrison, Phys. Plasmas 24, 055502 (2017). obtained equilibria including poloidal rotation. For this case, 16H. R. Strauss, Phys. Fluids 20, 1354 (1977). the Shafranov shift of the magnetic axis increased quadrati- 17V. D. Shafranov, Sov. Phys. - JETP 6, 545 (1958). cally in the poloidal rotation speed. The quadratic depen- 18P. J. Morrison, Bull. Am. Phys. Soc. 31, 1609 (1986). 19 dence is explained by the mapping procedure between non- G. N. Throumoulopoulos and H. Tasso, J. Plasma Phys. 62, 449 (1999). 20T. Andreussi, P. J. Morrison, and F. Pegoraro, Phys. Plasmas 19, 052102 rotating and poloidally rotating equilibria for high-beta (2012). reduced MHD. This achievement highlights one of the 21M. Wakatani, Stellarator and Heliotron Devices (Oxford Univ. Press, advantages of the simulated annealing method, i.e., it can New York, Oxford, 1998). 22 handle equilibria with rotation. Y. Saad and M. H. Schultz, SIAM J. Sci. Stat. Comput. 7, 856 (1986). 23V. D. Shafranov, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1966), Vol. 2, p. 103. 24 ACKNOWLEDGMENTS H. P. Zehrfeld and B. J. Green, Nucl. Fusion 12, 569 (1972). 25B. J. Green and H. P. Zehrfeld, Nucl. Fusion 13, 750 (1973). 26H. R. Strauss, Phys. Fluids 16, 1377 (1973). M.F. was supported by the JSPS KAKENHI Grant No. 27 JP15K06647, while P.J.M. was supported by the U.S. Dept. E. Hameiri, Phys. Fluids 26, 230 (1983). 28L. Guazzotto and R. Betti, Phys. Plasmas 22, 092503 (2015). of Energy Contract No. DE-FG05-80ET-53088. K.I. was 29Y. Nakamura, M. Wakatani, and K. Ichiguchi, J. Plasma Fusion Res. 69, supported by the budget NIFS17KLTT006 of the National 41 (1993).