Quasisymmetric toroidal plasmas with large mean flows

Geometric conditions for quasisymmetric toroidal plasmas with large mean flows on the order of the ion thermal speed are investigated. Equilibrium momentum balance equations including the inertia term due to the large flow velocity are used to show that, for rotating quasisymmetric plasmas with no local currents crossing flux surfaces, all components of the metric tensor should be independent of the toroidal angle in the Boozer coordinates, and consequently these systems need to be rigorously axisymmetric. Unless the local radial currents vanish, the Boozer coordinates do not exist and the toroidal flow velocity cannot take any value other than a very limited class of eigenvalues corresponding to very rapid rotation especially for low beta plasmas. VC 2011 American Institute of Physics. [doi:10.1063/1.3624483]


I. INTRODUCTION
There have been numerous studies on plasma flows as an attractive means for improving plasma confinement. 1 Sheared E Â B flows are considered to cause the reduction of transport observed in internal transport barriers (ITBs) 2 and high-confinement modes (H-modes). 3Microscopic oscillatory flow structures called zonal flows also play important roles in regulating microturbulence. 4,5For axisymmetric systems, large mean flows on the order of the ion thermal velocity v T can be produced in the toroidal direction, and the toroidal flows are counted on to avoid dangerous instabilities such as resistive wall modes in tokamaks. 6Similarly, in nonaxisymmetric or helical systems such as stellarators and heliotrons, 7 plasma mean flows and zonal flows are closely connected to both neoclassical and turbulent transport processes [8][9][10][11][12][13][14][15] and it is important to investigate conditions for magnetic geometry to allow large flow velocities.
It was shown by Helander 16 that, in order for the large mean flows of O v T ð Þ to occur in general three-dimensional configurations, the magnetic field strength should be given by a function of the flux surface label and the arc length along the field line.This condition is called isometry 17 or omnigenity 18 and it is satisfied by the quasisymmetric magnetic configurations 19,20 where the magnetic field strength is given as a function of the radial coordinate and one of or a linear combination of the toroidal and poloidal angle coordinates.If flow velocities are of O dv T ð Þ, where the drift ordering parameter d is given by the ratio of the thermal gyroradius q T to the equilibrium gradient scale length L, bounce-averaged radial drift velocities of ripple-trapped particles vanish and accordingly neoclassical transport is significantly reduced in quasisymmetric and quasi-omnigenous systems. 182][23][24][25][26] It is also shown for quasisymmetric toroidal configurations with mean flows of O dv T ð Þ that neoclassical particle fluxes are intrinsically ambipolar, 9,27,28 which implies that the radial electric field and accordingly the mean flow velocity in the direction of quasisymmetry can freely take any values even if the background density and temperature gradients are fixed.Recently, Simakov and Helander 29 treated rapid plasma rotation in quasisymmetric systems although they still assumed the plasma flow Mach number to be low in order to neglect effects of the inertia term due to the rapid flow on the equilibrium force balance.
In this paper, geometric conditions for quasisymmetric toroidal plasmas with large mean flows on the order of the ion thermal speed are investigated in detail by including the flow inertia term in the force balance.We hereafter consider quasi-axisymmetric systems although results shown in the present work can be straightforwardly extended to general quasisymmetric systems.Our previous study 30 showed that, for such rapidly rotating quasisymmetric plasmas, an additional symmetry condition for a component of the metric tensor need to be satisfied in order for the Boozer coordinates 31 to exist.
In Sec.II, it is shown that, for quasisymmetric magnetic configurations with the large mean flows of O v T ð Þ in the direction of quasisymmetry, the equilibrium densities, temperatures, electrostatic potential, angular flow velocity, and the Jacobian associated with the flux coordinates are all independent of f, where f is used as the angle coordinate associated with quasisymmetry.Here, the temperatures and electrostatic potential should be flux-surface functions to the lowest order in d and the potential needs to be independent of f up to the next order, which shows a contrast to the case of Ref. 29 where the potential is assumed to break this symmetry condition.We also find that local currents crossing flux surfaces should vanish or large mean flows cannot take any values except for impractical eigenvalues.For the case of no local radial currents, we can use the Boozer coordinates to represent several geometric constraints imposed by the equilibrium momentum balance equations including the inertia due to the large flows.
In Sec.III, additional conditions for the metric tensor are obtained by using formulas of differential geometry 32 for toroidal flux surfaces.The geometric equations from Sec. III are solved in Sec.IV to show that, for quasisymmetric toroidal plasmas with large mean flows of O v T ð Þ, all components of the metric tensor should be independent of f.This consequence is strongly restrictive and it leads to the conclusion that the rapidly rotating quasisymmetric systems should be rigorously axisymmetric.Finally, Sec.V summarizes results from the present work and Appendices A, B, and C are added to give explanations about eigenvalues of the flow velocities, transformation of flux coordinates, and basics of differential geometry for surfaces, respectively.

II. EQUILIBRIA OF QUASISYMMETRIC ROTATING PLASMAS
We consider toroidal plasmas, in which the magnetic field B is written in terms of the flux coordinates (s, h, f) as where s is an arbitrary label of a flux surface, h and f represent the poloidal and toroidal angles, respectively, and 0 : @/@s denotes the derivative with respect to s.The toroidal and poloidal fluxes within the volume inside the surface with the label s are given by 2pw(s) and 2pv(s), respectively.The contravariant components of the magnetic field B are given from Eq. (1) as where ffiffi ffi g p ½rs Á ðrh Â rfÞ À1 represents the Jacobian for the flux coordinates (s, h, f).
Hereafter, we investigate quasisymmetric toroidal systems with large mean flows on the order of the ion thermal velocity v T .The O v T ð Þ, equilibrium flow should be tangential to the direction of quasisymmetry, in which the field strength B is uniform.Here, for simplicity, we restrict our consideration to the quasi-axisymmetric case, in which the magnetic field strength B is independent of the toroidal angle f, @B=@f ¼ 0; B ¼ Bðs; hÞ; (3) although we can treat general quasisymmetric cases such as quasi-poloidally symmetric and quasi-helically symmetric ones in the same way as shown below.When using the perturbative expansion in terms of the drift ordering parameter d defined by the ratio of the ion thermal gyroradius q T to the equilibrium gradient scale length L, the lowest-order momentum balance equation reduces to Here, the lowest-order electric field is given by , where e a is regarded as a quantity of O d À1 À Á .In the same way as shown in Refs.16 and 33, we find from the lowest-order kinetic equation that the equilibrium flow velocity V 0 of O v T ð Þ, which consists of the E Â B drift and the parallel flow components, should be represented by and that the following incompressibility condition and other constraints hold, where n a and T a are the lowest-order density and temperature of the particle species a, respectively.In the presence of the large flow velocity V 0 , n a is not uniform over the flux surface while T a is still a flux-surface function as in the case of the conventional equilibrium without V 0 .We also see from Eq. ( 6) that the density n a and the Jacobian ffiffi ffi g p are independent of f like the field strength B. Using Eqs. ( 4) and ( 5) we have which leads to Equation ( 8) is also derived from which are immediately obtained from Eq. ( 2) and @ ffiffi ffi g p =@f ¼ 0 in Eq. (6).Then, using @B=@f ¼ 0 and Eq. ( 9), we find where q B f =B h ¼ w 0 s ð Þ=v 0 s ð Þ represents the safety factor.Each term of the lowest-order momentum balance equation shown in Eq. ( 4) has no component in the direction parallel to the magnetic field.To the next order, the parallel momentum balance equation for the particle species a is written as where m a denotes the mass of the particle species a, the electric field E 1 ¼ ÀrU 1 is given by the first-order electrostatic potential U 1 , and the pressure is represented by p a ¼ n a T a .Here, e a U 1 =T a ¼ O 1 ð Þ, and we generally have B Á rU 1 6 ¼ 0 in contrast to B Á rU 0 ¼ 0. Using Eqs. ( 6), (8), and (11), we get where g ff @x=@f j j 2 .For electrons (a ¼ e), Eq. ( 12) is written by neglecting the electron mass m e as which implies that ln n e À eU 1 =T e ð Þis a flux-surface function.Therefore, U 1 is independent of f like n e , @U 1 =@f ¼ 0; U 1 ¼ U 1 ðs; hÞ: (14)   This result is contrastive to the study by Simakov and Helander 29 for rotating quasisymmetric plasmas with low Mach number flows, in which non-intrinsically ambipolar particle fluxes due to the symmetry-breaking electrostatic potential are considered.We now define the f-averaged part A ¼ ð2pÞ À1 Þ Adf and the f-dependent part Ã ¼ A À A for an arbitrary function A of f.Then, Eq. ( 12) is rewritten as which is separated into the f-averaged and f-dependent parts, The species summation of the equilibrium force balance is written to the lowest order as where the current density J is given by Ampe `re's law r Â B ¼ 4p=c ð ÞJ and p ¼ P a p a is the total kinetic pressure.The inertia term resulting from the flow velocity V 0 is included in Eq. (17).Then, taking the inner product between Eq. ( 17) and @x/@f, we obtain 1 2 In the rigorous axisymmetric case, @g ff /@f ¼ 0 holds although this condition is not trivially satisfied for the quasi-axisymmetric case.If @g ff /@f = 0, then Eq. ( 18) leads to nonzero local radial current J s = 0.This gives rise to a serious problem because the quasisymmetric system is considered usually by using the Boozer coordinates while the Boozer coordinates cannot be constructed for the case of J s = 0. Let us consider this point below in more detail.Differentiating Eq. ( 18) with respect to f and using Eqs.(10) and ( 16), we obtain It is shown in Appendix A that nontrivial solutions of Eq. ( 19) exist only if the toroidal flow velocity V f (s) takes one of eigenvalues determined by Eq. (A9) for each flux surface.As dis-cussed in Appendix A, this constraint on V f is so restrictive that we hereafter assume a trivial solution f g ff ¼ 0 of Eq. ( 19) and accordingly @g ff =@f ¼ 0: (20)   Then, we find from Eq. ( 18) that the radial current vanishes Appendix B shows that, for the case of J s ¼ 0, there exist the Boozer coordinates in which the covariant poloidal and toroidal components, B h (s) and B f (s), of the magnetic field are flux-surface functions.It is also shown in Appendix B that, without loss of generality, we can regard the flux surface (s, h, f) as the Boozer coordinates from the beginning for systems considered in this section.Thus, we hereafter assume (s, h, f) to be the Boozer coordinates.Using Eq. ( 20) and the relations between the covariant and contravariant components of the magnetic field, we find that the components g hh , g hf , and g ff of the metric tensor are all independent of f, Then, the contravariant metric tensor component Using Eq. ( 23) and taking the covariant poloidal and radial components of Eq. ( 17), we obtain and respectively.Furthermore, taking only f-dependent part f Á Á Á of Eq. ( 26) yields Noting that B s ¼ g s h B h þ g s f B f , we regard Eq. ( 27) as the constraint imposed on the components g sh and g sf of the metric tensor.

III. GEOMETRIC CONDITIONS FOR TOROIDAL FLUX SURFACES
In this section, we further investigate the conditions considered in Sec.II for quasisymmetric toroidal plasmas with large mean flows on the order of the ion thermal speed.Here, the quasi-axisymmetric case is considered again as an example

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Quasisymmetric toroidal plasmas Phys.Plasmas 18, 082505 (2011)  even though general quasisymmetric cases can be treated similarly.We also recall that the Boozer coordinates (s, h, f) are used and that the field strength B, the Jacobian ffiffi ffi g p ¼ ½rs Á ðrh Â rfÞ À1 and the components g ab (a, b ¼ h, f) of the metric tensor are all independent of f as shown in the previous section.Accordingly, the contravariant component g ss ½g hh g ff À g hf ð Þ 2 =g of the metric tensor is independent of f, too.
In order to examine properties of toroidal flux surfaces, we hereafter use several quantities defined in the differential geometry of surfaces such as the Christoffel symbols and the Riemann curvature tensor. 32Definitions of these quantities for general surfaces are briefly described in Appendix C. Using the components g ab (a, b ¼ h, f) of the metric tensor for each flux surface, which are independent of f, the Christoffel symbols C c ab a; b; c ¼ h; f ð Þ , are given from Eq. (C2) as where @ h ¼ @/@h and gg ss ¼ g hh g ff -(g hf ) 2 are used.The nonzero component R hfhf of the Riemann curvature tensor is derived from using Eq.(C3) and the Christoffel symbols in Eq. ( 28) as We now represent the Fourier series expansion of an arbitrary function Q of f by The components L, M, and N of the second fundamental form for flux surfaces are evaluated by using Eq.(C7).The Fourier coefficients of their f-dependent parts L, M, and Ñ are written as Since the n ¼ 0 Fourier component of the f-dependent part f Á Á Á vanishes, we hereafter consider only the case of n = 0. We find from Eq. (C9) that Ln , Mn , and Ñn satisfy the Fourier-transformed version of the Codazzi-Mainardi equations which impose constraints on ðf g sh Þ n and ðf g sf Þ n through the expressions in Eq. ( 31).Substituting Eq. ( 31) into the second equation in Eq. ( 32), we obtain the first-order ordinary differential equation in h, where Another first-order differential equation is given by substituting Eq. ( 31) into the first equation in Eq. ( 32) and using Eq. ( 33) as where We also find that another condition for ðf g sh Þ n and ðf g sf Þ n is derived from the Fourier transform of the radial equilibrium force balance in Eq. ( 27) as where Combining Eqs. ( 33) and ( 38) gives the first-order ordinary linear differential equation for ðf g sf Þ n with ðf g sh Þ n eliminated.We can also obtain the first-order ordinary linear differential equation for ðf g sh Þ n without ðf g sf Þ n by using Eqs.( 35) and (38).Now, it is questioned whether nontrivial solutions ðf g sh Þ n and ðf g sf Þ n of these linear ordinary differential equations exist or not.Since these equations contain the components g ab (a, b ¼ h, f) of the metric tensor and the Jacobian ffiffi ffi g p in very complicated ways, we first consider the neighborhood of the magnetic axis s ¼ 0 to simplify the equations in the next section, and then discuss the solutions in the whole toroidal volume region.

IV. SEARCH FOR THE METRIC TENSOR IN QUASISYMMETRIC ROTATING PLASMAS
Here, we examine whether or not there exist nontrivial solutions ðf g sh Þ n and ðf g sf Þ n of the linear ordinary differential equations derived in the previous section.For this purpose, we first investigate the neighborhood of the magnetic axis and then consider the whole volume region of the toroidal system.It should be recalled that we assume quasi-axisymmetry @B/@f ¼ 0 with the Boozer coordinates (s, h, f) used for simplicity even though general quasisymmetric cases can be treated in a similar way.
As shown in Eq. (B11), the Jacobian associated with the Boozer coordinates is given by where V (s) represents the volume within the flux surface labeled s.We find in Sec.II that ffiffi ffi g p is independent of f when there exist toroidal flows on the order of ion thermal speed.
Along the magnetic axis s ¼ 0, ffiffi ffi g p is independent of h, too, and it takes a constant value which is denoted by R 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ff ðs ¼ 0Þ p .Then, the total length of the magnetic axis is represented by 2pR 0 .The field strength B(s ¼ 0) : B 0 at the magnetic axis is another constant.In addition, g ab (a, b ¼ h, f) and g ss rs j j 2 are independent of f so that they take constant values along the magnetic axis.Therefore, toroidal cross sections of flux surfaces near the magnetic axis are concentric circles, the center of which is located at the magnetic axis.The unit vector b ¼ B/B along the magnetic field line at s ¼ 0 is in the toroidal direction and it is written In the present section, we define the radial coordinate s by w ¼ 1 2 B 0 s 2 , from which g ss (s ¼ 0) ¼ 1 is derived.Therefore, s is the radius length of the circular toroidal cross section of the flux surface.We now consider the neighborhood of a certain point on the magnetic axis.This small local region is represented by r < and Dz j j < .Here, the cylindrical coordinates r; #; z ð Þis chosen such that the magnetic field is written as where r, #, and ẑ are the orthogonal unit vectors in the r-, #-, and z-directions, respectively, and ẑ coincides with b at the origin (r, z) ¼ (0, 0).An arbitrary function defined in the local region can be expanded with respect to the small radial coordinate s or r, where s $ r $ and Flux surfaces are approximately given by r ¼ const.
It is shown by using the solenoidal field condition r Á B ¼ 0 and Ampe `re's law r Â B ¼ 4p=c ð ÞJ that the components of the magnetic field in the cylindrical coordinates are represented by (41) while the components of the current density are written as Using Eq. ( 41) and r $ s, we obtain We see that V(s) is approximated by the volume of the tube which has the length 2pR 0 and the circular cross section with the radius s.Then, we get where Eq. ( 40) is used.The definition of s immediately gives dw/ds ¼ B 0 s and the safety factor is written as The poloidal and toroidal contravariant components of the magnetic field given in Eq. ( 2) are rewritten as where B h 0 ¼ B 0 =q 0 R 0 and B f 0 ¼ B 0 =R 0 are constants representing the values at the magnetic axis s ¼ 0.
Since g ss (s ¼ 0) ¼ 1 and g ff (s ¼ 0) ¼ (R 0 ) 2 , we can write g ss and g ff near the magnetic axis as with respect to s and recall that, in the Boozer coordinates, the covariant poloidal and toroidal magnetic field components B h (s) and B f (s) are independent of the angle coordinates h and f.Then, we get g hf ¼ const and finally obtain g (45) we find that g hh and g hf are written as Here, using the relation hf ¼ const in Eq. ( 46).It should be noted that the relations h 0 ¼ hh 0 (s) and f 0 ¼ ff 0 (s) give another set of the Boozer coordinates (s, h 0 , f 0 ) from the original (s, h, f) by using arbitrary functions h 0 (s) and f 0 (s).Under this transformation, we see that g sh 0 6 ¼ g sh and g sf 0 6 ¼ g sf hold generally although g h 0 h 0 ¼ g hh , g h 0 f 0 ¼ g hf , and g f 0 f 0 ¼ g ff .Here, we generally have However, we can choose h 0 (s) and f 0 (s) to satisfy g sh 0 ¼ O s 2 ð Þ and g sf 0 ¼ O s ð Þ, from which we find that, in the local region, the new Boozer coordinates (s, h 0 , f 0 ) are related to the local cylinder coordinates r; with the origins of both coordinates coinciding with each other.Hereafter, we use this new set of the Boozer coordinates (s, h 0 , f 0 ) although they are represented by (s, h, f) with the prime omitted.Thus, we now write g sh and g sf as We find from Eqs. (44), (46), and (47) that and that the contravariant basis vectors @x/@s, @x/@h, and @x/@f are orthogonal to each other to the lowest order in s and they are related to the contravariant basis vectors rs, rh, and rf by @x=@s ¼ rs þ OðsÞ; Using Eqs. ( 41) and ( 46)-(48), the covariant magnetic field components are shown to be written as From the relations g sh ¼ g s f g hf À g s h g ff À Á =g and g sf ¼ g s h g hf À g s f g hh À Á =g, we obtain where Eqs. ( 42), ( 44), (46), and (47) are used.Now, Eq. ( 33) is rewritten by using Eqs.( 34), ( 42), (44), and (46) as Then, using Eqs.( 35), (36), ( 42), (44), and (46), we obtain The relation between ðf g sh Þ n and ðf g sf Þ n is derived from Eqs. ( 34), (36), ( 38), (39), ( 42), (44), and (46) as ff Þðf g sh Þ n =ðn 2 q 0 sÞ; where n a0 , V f 0 , and q 0 represent the density, toroidal angular velocity, and safety factor at the magnetic axis s ¼ 0, respectively.Eliminating ðf g sf Þ n from Eq. ( 53) by using Eq.(54) gives the solution of which is written as where C n (s) is a function of s that still remains to be determined.From Eqs. (54) and (56), we have where it is assumed that À1 þ ð4p P a n a0 m a = B 2 0 ÞR 2 0 ðV f 0 Þ 2 6 ¼ 0 holds generally.Then, substituting Eqs. ( 56) and (57) into Eq.( 52), we see that, to the lowest order in s, which is found to reduce to C n (s) ¼ 0 by noting that the component of the Riemann curvature tensor in Eq. ( 29) is proportional to the Gaussian curvature of the toroidal flux surface which is not identical to zero.Therefore, we finally obtain Now, we see that all components g ab and accordingly g ab (a, b ¼ s, h, f) of the metric tensor are independent of f.For such a case, an arbitrary closed f-curve defined by x ¼ x(s, h, f) (0 f 2p) with s and h fixed is a circle because the curvature and the torsion of the f-curve, which can be derived from g ab , need to be independent of f and accordingly constants along the closed curve.Furthermore, two circular closed f-curves, constant interval given by dh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g hh À ðg hf Þ 2 =g ff q between each other.Thus, toroidal surfaces, which the f-curves lie on, are axisymmetric in the local region near the magnetic axis and the magnetic axis itself is a circle of the radius R 0 on a flat plane.Note that ds= ffiffiffiffiffi ffi g ss p ð¼ ds=jrsjÞ represents the local distance between neighboring flux surfaces with the radial coordinates s and s þ ds.Since @g ss =@f ¼ 0 is assumed to be satisfied globally in the whole volume region, all flux surfaces, which encircle the magnetic axis of the circular shape, should be axisymmetric.Now, it is concluded that quasi-axisymmetric rotating plasmas considered here need to be axisymmetric.The same result is obtained for general quasisymmetric rotating plasmas in a similar way to the above one.
Here, it is instructive to rewrite the component R hfhf of the Riemann curvature tensor in Eq. ( 29) by using Eqs.( 42) and (44) as and the components L, M, and N of the second fundamental form are calculated from g ab (a, b ¼ s, h, f) as From Eqs. ( 61) and (C11), we have ff .Thus, we obtain g ð1Þ ff / R hfhf / cos h, where the origin of the poloidal angle h is chosen such that the Gaussian curvature / LN À M 2 ð Þof toroidal flux surfaces near the magnetic axis vanishes for h ¼ 6p/2.This form of g ð1Þ ff is a well-known result for large-aspect-ratio axisymmetric toroidal surfaces, where the Gaussian curvature is positive (negative) for jhj <p/2 (jhj > p/2).

V. CONCLUSIONS
In this work, the conditions for quasisymmetric toroidal systems to allow large flows on the order of the ion thermal velocity are investigated.Taking the case of rotating quasi-axisymmetric plasmas, in which the field strength B does not depend on the toroidal angle f, the equilibrium momentum balance equations are used to show that the component g ff of the metric tensor is shown to be independent of f in the systems with no local radial current, where the Boozer coordinates exist.It is also shown that, unless @g ff =@f ¼ 0, the toroidal flow velocity cannot take any value other than a very limited class of eigenvalues corresponding to very rapid rotation especially for low beta plasmas.In the Boozer coordinates, where @g ff /@f ¼ 0, the metric tensor components g ab (a, b ¼ h, f) associated with the angle coordinates for each flux surface are shown to be independent of f.We finally find that, in order to globally satisfy the equilibrium momentum balance, all metric tensor components g ab (a, b ¼ s, h, f) need to be independent of f, and therefore toroidal flux surfaces are axisymmetric.General quasisymmetric cases can be treated similarly.Thus, quasisymmetric toroidal equilibrium plasmas with large mean flows should be axisymmetric or they have local radial currents which do not allow existence of the Boozer coordinates and impose strong constraints on the toroidal flow velocity.Since, in the present work, quasisymmetry is assumed to be rigorously satisfied with the magnetic field strength B as an analytical function of the flux coordinates, we have not definitely answered yet whether large mean flows on the order of the ion thermal velocity can be compatible with nonaxisymmetric toroidal plasma equilibria such as quasiomnigenous configurations.This remains as a future task.

ACKNOWLEDGMENTS
The authors thank Dr. M. Yokoyama for useful discussions on advanced concepts of helical devices.This work is supported in part by the Japanese Ministry of Education, Culture, Sports, Science, and Technology, Grant No. 21560861 and in part by the NIFS Collaborative Research Programs, NIFS10KNTT003 and NIFS10KDAT020.
APPENDIX A: SOLUTIONS OF EQ. (19)   In this Appendix, we investigate solutions of Eq. ( 19).As a function of f, f g ff is expanded by the Fourier series where ðf g ff Þ 0 ¼ 0.Then, Eq. ( 19) is represented in terms of ðf g ff Þ n by @ @h Introducing new variables n n e Àinqh ðf g ff Þ n and g n , Eq. (A2) is given in the form of the fist-order ordinary differential equations, where the radial coordinate s is included as a parameter in n n , g n , ffiffi ffi g p , n a , V f , and v 0 although it is not shown explicitly.
The solution of Eq. (A3) is written as where Quasisymmetric toroidal plasmas Phys.Plasmas 18, 082505 ( On the right-hand side of Eq. (A4), H represents the h-ordered product of matrix functions of h and Generally, A n (h) and A n (h 0 ) are noncommutative, A n (h)A n (h 0 ) = A n (h 0 )A n (h), for h = h 0 .If A n (h) and A n (h 0 ) are commutative for arbitrary values of h and h 0 , Eq. (A6) reduces to the well-known exponential form Since ðf g ff Þ n and dðf g ff Þ n =dh are periodic functions of h with the period 2p, e inqh n n , and e inqh g n are periodic, too.These periodic conditions give Then, the condition for [n n (0), g n (0)] = [0, 0] to satisfy Eq. (A8) is written as det e 2pinq H exp where I denotes the 2 Â 2 unit matrix.Thus, for a given n and a given radial coordinate s, V f included in the matrix A n (h) needs to take one of eigenvalues determined by the condition shown in Eq. (A9).
In the neighborhood of the magnetic axis s ¼ 0 as discussed in Sec.IV, A n (h) is approximated as, where the radial coordinate s is defined by w 1 2 B 0 s 2 in the same way as in Sec.IV.Here, k is a constant defined by where n a0 , q 0 , and V f 0 are the density, the safety factor, and the toroidal angular velocity on the magnetic axis s ¼ 0, respectively.From Eqs. (A9) and (A10), we obtain det e 2pinq 0 expð2pA n0 ÞÀI À Á ¼det e 2pinq 0 cosð2pnkÞÀ1 e 2pinq 0 sinð2pnkÞ Àe 2pinq 0 sinð2pnkÞ e 2pinq 0 cosð2pnkÞÀ1 !¼ðe 2pinðq 0 þkÞ À1Þðe 2pinðq 0 ÀkÞ À1Þ¼0: Equations ( A11) and (A12) yield where b 0 8pð P a n a0 T a0 Þ=B 2 0 is the central beta and v T0 ½2ð P a n a0 T a0 Þ=ð P a n a0 m a Þ 1=2 represents a characteristic value of the central temperature.Eigenvalues which V f 0 should take are given by Eq. (A13), from which we obtain For the regions other than the neighborhood of the magnetic axis, eigenvalues of V f need to be determined by Eq. (A9).Equation (A14) shows that, unless g ff is independent of f, the toroidal flow velocity of O v T 0 ð Þ cannot take any value other than a very limited class of eigenvalues corresponding to very rapid rotation (which could be supersonic) especially for low beta plasmas.Therefore, the case of @g ff /@f ¼ 0 is considered in the main text of the present paper.

APPENDIX B: TRANSFORMATION OF FLUX COORDINATES
In terms of arbitrary flux coordinates (s, h, f), the magnetic field, which forms nested toroidal surfaces, is represented by where h and f are the poloidal and toroidal angles, respectively, and s is a radial coordinate or a label to specify a flux surface.As explained after Eq. ( 1), the toroidal and poloidal fluxes within the volume inside the surface with the label s are given by 2pw(s) and 2pv(s), respectively, and 0 : @/@s denotes the derivative with respect to s.Consider two sets of flux coordinates (s, h, f) and (s, h A , f A ), where the same radial coordinate s is used.Then, there exists a generating function G A (s, h, f) by which the coordinate transformation is written as 35 We find from Eqs. (B1) and (B2) that the generating function G A satisfies the magnetic differential equation, where ffiffi ffi g p ½rs Á ðrh Â rfÞ À1 and ffiffiffiffiffi g A p ½rs Á ðrh A Ârf A Þ À1 are the Jacobians associated with the flux coordinates (s, h, f) and (s, h A , f A ), respectively.Using Eqs.(B1) and (B2), we also obtain where B h and B f are the covariant poloidal and toroidal magnetic field components, respectively.If the conventional equilibrium condition J Â B=c ¼ rp s ð Þ is assumed, J s ¼ 0 is immediately derived.However, it is emphasized that, in the present paper, we consider the equilibrium force balance given by Eq. (17) where the inertia term associated with the large flow velocity V 0 is included.Therefore, J s ¼ 0 is not trivial here but it imposes additional constraints on the magnetic geometry as discussed in Sec.II.It is shown from Eq. (B5) that B h and B f are written in terms of a certain function g(s, h, f) as @gðs; h; fÞ @f ; (B6) where represents the toroidal (poloidal) current flowing inside (outside) the flux surface with the label s, and g is periodic in h and f.Then, the magnetic field is written as Now, the Boozer coordinates (s, h B , f B ) are obtained from the coordinate transformation in Eq. (B2) with the generating function G A replaced by The Boozer coordinates (s, h B , f B ) are characterized by the properties that the covariant poloidal and toroidal components of the magnetic field are independent of h B and f B .In fact, using Eqs.(B2), (B7), and (B8), we see Thus, we have proved that there exist the Boozer coordinates (s, h B , f B ) for the case of J s ¼ 0 even though the conventional equilibrium condition p , and Eq.(B9), shows is the radial derivative of the volume V(s) within the flux surface labeled s.Thus, in the Boozer coordinates, the Jacobian ffiffiffiffiffi g B p depends on the angle coordinates (h B , f B ) only through 1/B 2 .
In the quasisymmetric toroidal plasma with large mean flows, the flow velocity V 0 is parallel to the direction of quasisymmetry of the magnetic field strength B and the Jacobian ffiffi ffi g p of the flux coordinates (s, h, f) does not vary in the symmetry direction as explained in Sec.II.Let us take the case of quasi-axisymmetry for simplicity although we can treat general quasisymmetric cases in the same way as shown below.Then, @B/@f ¼ 0 and @ ffiffi ffi g p =@f ¼ 0, where we should note that (s, h, f) does not need to be the Boozer coordinates but they should be at least flux coordinates satisfying Eq. (B1).Now, in addition to the above quasisymmetry conditions, we consider again the case of J s ¼ 0 in which there exist the Boozer coordinates (s, h B , f B ) satisfying Eqs.(B9) and (B11) as we have shown.
Since @B/@f ¼ 0 and @ ffiffi ffi g p =@f ¼ 0, the generating function G B yielding the coordinate transformation from (s, h, f) to (s, h B , f B ) can be obtained by directly integrating the magnetic differential equation in the form of Eq. (B3) as where K(s) is independent of (h, f) and appears as a constant of integration.Using Eq. (B4) and noting that G B is independent of f, we find from which @ ffiffiffiffiffi g B p =@f B ¼ 0 is immediately derived.Equation (B13) implies that, if a rapid rotating toroidal plasma with no local radial current is quasisymmetric in terms of a certain set of flux coordinates (s, h, f), then it is so in the Boozer coordinates (s, h B , f B ), too.Thus, as remarked after Eq. ( 21) in Sec.II, we can use the Boozer coordinates from the beginning to investigate quasisymmetric toroidal systems with large mean flows if J s ¼ 0 is satisfied.
In passing, for the present case of @ ffiffi ffi g p =@f ¼ 0, the coordinate transformation in Eq. (B2) using another generation function defined by yields the new flux coordinates (s, h A , f A ) for which the Jacobian is given by ffiffiffiffiffi g A p ¼ V 0 ðsÞ 4p 2 : (B15) Equation (B15) shows that the Jacobian ffiffiffiffiffi g A p is a flux-surface function, which is a characteristic of the Hamada coordinates. 34Under the conventional equilibrium condition J Â B=c ¼ rp s ð Þ, we find that J h A and J f A are flux-surface 082505-9 Quasisymmetric toroidal plasmas Phys.Plasmas 18, 082505 (2011)  functions, where J h A and J f A represent the contravariant poloidal and toroidal components of the current density vector J, respectively.However, it should be noted that, since the large-flow inertia term is included in the equilibrium force balance here, this property of the Hamada coordinates is not generally satisfied in the flux coordinates (s, h A , f A ), where J h A and J f A may not be functions of s alone while ffiffiffiffiffi g A p ¼ V 0 ðsÞ=ð4p 2 Þ.

APPENDIX C: DIFFERENTIAL GEOMETRY OF SURFACES
In this Appendix, we briefly review the definitions of basic quantities used in the differential geometry for surfaces. 32ere, in order to easily derive these quantities in Sec.III for the case of quasisymmetric toroidal surfaces, we represent an arbitrary surface locally by x ¼ x(h, f) with h and f used as two parameters even though they do not need to be angle coordinates in this Appendix.
The first fundamental form is the metric tensor for the surface x ¼ x(h, f), the components of which are defined by E g hh j@ h xj 2 ; F g hf ð@ h xÞ Á ð@ f xÞ; G g ff j@ f xj 2 ; (C1) where simplified notations @ h ¼ @/@h and @ f ¼ @/@f are used.Then, the Christoffel symbols C cab and C c ab (a, b, c ¼ h, f) are defined by using the components of the first fundamental form in Eq. (C1) as ð@ a g bc þ @ b g ca À @ c g ab Þ: (C2) We take the inverse of the 2 Â 2 matrix consisting of g ab (a, b ¼ h, f) when deriving C c ab from C cab .This inverse matrix should not be identified with g ab used in Secs.III-IV because the latter is obtained by taking the inverse of the 3 Â 3 matrix which includes additional components g ss , g sh , and g sf .Using the Christoffel symbols, the components R c dab and R cdab (d, c, a, b ¼ h, f) of the Riemann curvature tensor are defined by The components of the Riemann curvature tensor have the symmetry properties written as Then, it is found that nonzero covariant components of the Riemann tensor for the surface x ¼ x(h, f) are only R hfhf and those obtained from R hfhf by using the symmetric properties in Eq. (C4).
The unit normal vector e to the surface is written as e rs jrsj @x=@h Â @x=@f j@x=@h Â @x=@fj : (C6) Then, the components L, M, and N of the second fundamental form for the surface are defined by using the normal vector e as L h hh @ 2 x @h 2 Á e; M h hf @ 2 x @h@f Á e; N h ff @ 2 x @f 2 Á e: (C7) In terms of the components of the first and second fundamental forms in Eqs.(C1) and (C7), the Gaussian curvature of the surface is given by The components (E, F, G) and (L, M, N) of the first and second fundamental forms must satisfy the Codazzi-Mainardi equations which are written as where we should recall that the Christoffel symbols C c ab contain the components of the first fundamental form as shown in Eq. (C2).Furthermore, (E, F, G) and (L, M, N) must satisfy the Gauss equation given by (C10) which is rewritten by using Eq.(C8) as Here, note that R hfhf is calculated from the components of the first fundamental form or the metric tensor by using Eqs.(C2) and (C3).The Gauss equation in Eq. (C10) [or (C11)] is the basis for Gauss's theorema egregium which states that the Gaussian curvature of a surface can be determined by residents confined onto the surface who measure the dependence of the metric tensor (g ab ) on the parameters (h, f) without even knowing about the three-dimensional space in which the surface is embedded.Bonnet's fundamental theorem of surface theory states that, if given functions (E, F, G) and (L, M, N) satisfy E > 0, EG -F 2 > 0, and the Gauss-Codazzi-Mainardi equations in Eqs.(C9) and (C10) [or (C11)], then there exists a surface for which the components of the first and second fundamental forms are given by (E, F, G) and (L, M, N), respectively, and that such a surface is determined uniquely up to congruence. 37) Downloaded 30 Jan 2012 to 133.75.110.124.Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissionsSubstituting Eqs.(33) and (35) into Eq.(37) yields the linear relation between ðf g sh Þ n and ðf g Þ¼0 where the h integral is taken with s and f fixed.Consequently, g hf ¼ O s 2 ð Þ is derived.From Eqs.(42) and (44), and the relation has a Downloaded 30 Jan 2012 to 133.75.110.124.Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions