{'fraction_non_alphanumeric': 0.04492929292929293, 'fraction_numerical': 0.017616161616161616, 'mean_word_length': 4.078375051292573, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
MHD equilibria with incompressible flows: symmetry approach

We identify and discuss a family of azimuthally symmetric, incompressible, magnetohydrodynamic plasma equilibria with poloidal and toroidal flows in terms of solutions of the Generalized Grad Shafranov (GGS) equation. These solutions are derived by exploiting the incompressibility assumption, in order to rewrite the GGS equation in terms of a different dependent variable, and the continuous Lie symmetry properties of the resulting equation and in particular a special type of"weak"symmetries.


I. INTRODUCTION
There 15-17, but we will restrict our study to incompressible flows. The rationale behind the incompressibility assumption stems from the fact that it has been known in the literature, Refs.18-22, that in this limit the equation that takes the role of the GS in the presence of plasma flows, Eq.(1) in Sec.II, can be transformed into a much simpler equation of the GS-type. This is made possible by a redefinition of the dependent variable (the magnetic flux function ψ) and can be applied as long as the poloidal flow is sub-alfvènic 23 .
This method allows us to generalize to a plasma with flows a class of solutions identified in Ref.10 using this symmetry approach.
The present paper is organized as follows. In Sec.II we introduce the Generalized Grad Shafranov (GGS) equation which includes the effect of poloidal and toroidal flows, take the incompressible plasma limit, define the new dependent variable χ = χ(ψ) and obtain a modified GS equation with an additional term that has an r 4 dependence of the cylindrical radius r.
In Sec.III we apply the concept of weak conditional symmetry to the GS equation. First we recover the D-shaped solution of the GS equation described in Ref.10 as a special case of a wider class of solutions. The common geometrical structure in the poloidal r-z plane of this class of solutions is discussed, and shown to arise directly from its symmetry properties.
On the other hand, determining the specific spatial shape of these solutions involves the (numerical) integration of a nonlinear second order ODE whose coefficients depend of the physical parameters of the specific equilibrium configuration. A qualitative classification of the different types of solutions of this ODE is given.
In Sec.IV A we extend this analysis to the GGS equation and describe a novel class of equilibria solutions with flows. These solutions share the same common geometrical structure of the solutions without flows because the symmetry properties are the same. Specific examples of equilibria with flows are then shown, including solutions that exhibit three magnetic axes along the toroidal direction. These latter solutions correspond to configurations made of three separate sets of tori, each set nested around one of the magnetic axes. By imposing appropriate boundary conditions corresponding to ψ=ψ=const surfaces, each of these sets can be considered as an independent equilibrium configuration.
In the Conclusions we note that in the case where poloidal flows are present the reconstruc-

II. GENERALIZED GRAD SHAFRANOV EQUATION
We consider an axisymmetric Magnetohydrodynamic plasma equilibrium with poloidal (v p ) and azimuthal (v ϕ ) flows and write the corresponding generalized Grad Shafranov equation (see Refs. [15][16][17] in the form Here the notation of Ref.17 has been adopted with coordinates (r, ϕ, z) where the azimuthal angle ϕ is an ignorable coordinate. In Eq.(1) a prime denotes differentiation with respect to the magnetic flux function ψ, ρ(r, z) is the plasma density and the magnetic field B is expressed in terms of its azimuthal and poloidal components, B ϕ and B p through the flux function ψ(r, z) as whereφ = r∇ϕ is the unit vector in the azimuthal direction. The flux functions in Eq. (1) are defined as where S is the plasma entropy, U is the plasma internal energy ad U ρ its derivative at constant entropy.
In the following we consider an incompressible equilibrium with ρ = const. in which case ρ(J − U S I ) = (p + ρv 2 /2 − ρrv ϕ G) and define the Alfvènic Mach number M (ψ) = F/(4πρ) 1/2 . Inverting Eqs.(4-5) we obtain The r.h.s. of Eq.(1) contains functions of ψ and powers of r. By substituting for rv ϕ its expression from Eq.(9) and collecting powers of r 2 , we can rewrite the r.h.s. of Eq.(1) as In the limit of zero plasma flow the 1/r 2 term reduces to the toroidal magnetic field the r-independent term. In, addition they introduce a new term proportional to r 2 . In the remaining part of this article we will consider only sub-alfvènic poloidal flows so that The fact that the alfvènic Mach number M in a constant density equilibrium is a flux function allows us to bring back the operator on the l.h.s. of Eq.(1) to its standard GS form by redefining the dependent variable ψ. Following Refs.18-22 we define the new dependent where M 2 (η) < 1. Then, we obtain Finally, re-expressing the flux functions of ψ on the r.h.s. of Eq.(10) as functions of χ through their dependence on ψ(χ) and using cylindrical coordinates, we obtain (see also

SYMMETRIES.
A. GS equation with no flow: A 4 = 0 Let us start by considering the "standard" GS equation (i.e. with no flow so that in The has revealed to be especially convenient (see Ref. 10 for a discussion and other details).
Writing the GS equation in terms of the Y -invariant variable one obtains the following equation, where the variable r still appears (playing here the role of a parameter, as happens in the case of weak conditional symmetries) This equation can be naturally split into two ODE's involving only the new variable s: Clearly, these equations admit solutions only if there is a precise relationship between F (ψ) and G(ψ). A simultaneous solution of these equations can be found of the form for any real q = 0, with Let us notice that with the choice (22) for F, G, the GS equation admits the Lie point scaling symmetry and that the above solution (21) is invariant under this symmetry.
It is possible to generalize the D-shaped solution described above by taking where a 0 and a 2 are free real coefficients and by setting ψ(r, z) = s 1/4 φ(y), y = r 2 /s ≥ 1.
The variable y is an invariant variable under the symmetry (23) which can be rewritten in a geometrically more transparent way as ψ(r, z)/r 1/2 = φ(y λ , a 2 )/y where we have made explicit the dependence of the solution on the parameter a 2 , and with R 2 = r 2 + z 2 . Notice that y = const. corresponds in the r ≥ 0 half-plane to the lines r = k|z|, with k > 1, whereas y λ = const. corresponds to the borders of the lunes (crescents) defined by the circles as shown in Fig.1 (here the variable z has been translated, see below Eq.(28)). Note that the shape of these lunes does not depend on the value of a 2 and that the function φ(y λ , a 2 )/y 1/4 λ is constant on their borders. Thus the shape in the r-z plane of the class of solutions discussed below can be visualized as given by the level curves of r 1/2 times a function that depends on a 2 and that is constant on the borders of the lunes in Fig.1 whose shape depends only on the transformation parameter λ.

IV. EQUILIBRIA WITH FLOWS
A. Symmetries of the GGS equation We now consider the case of the generalized GS equation (13).
It is easy to show that the two symmetries X 1 and X 2 given above (23) and (26) are still valid: more precisely, the symmetry X 1 is admitted by any equation of the form and, similarly, the exceptional symmetry X 2 is admitted by any equation of the form where at the r.h.s. any combination of terms r χ 1+ 2+ 2q (resp. r χ −3−2 ) with arbitrary constants a and -in principle -any real value of is admitted Restricting to the cases relevant to Eq.(13), we can write instead of (37) and finally, with q = −1/4,

B. Solutions of the GGS equation
We proceed as in Sec. III C by defining For a 4 + ρ(a 2 + 3/2) 0 and a 4 < 0, region B of Fig.2, the solution reaches zero at a finite value y =ȳ, as shown in Fig.3 in terms of the variable 1 − 1/y = z 2 /r 2 . In this case, φ(y) behaves locally as φ(y) ∝ (ȳ − y) 1/6 and, as in the case without flow, a complementary solution can be found forȳ < y < ∞ which starts as φ(y) ∝ (y −ȳ) 1/6 and behaves for For a 4 +a 2 +3/2 < 0 and a 4 > 0, region C of Fig.2, we have a solution which remains strictly positive but does not behave monotonically, and asymptotically grows as ∝ y 1/4 , as shown in Fig.4 in terms of the variable 1 − 1/y. For a 4 + ρ(a 2 + 3/2) 0, there are "separating solutions" which behave asymptotically as y 1/12 . Notice the particular case a 4 = 0, a 2 = −3/2 where one recovers the known solution φ ≡ 1.
In order to illustrate the different shape in the r-z plane of the solutions belonging to region A and to regions B-C, when transformed according to Eq.(43) and shifted in z, we may use simple analytical fitting formulae of the numerical solutions of φ(y, a 2 , a 4 ). The solution from region C with a 2 = −3.5, a 4 = 0.5 (the same of Fig.4) and λ = 1 is shown in Fig.6. It exhibits three magnetic axes along the toroidal direction. Thus this solution corresponds to a configuration made of three sets of surfaces, each set nested around one of the magnetic axes. Each set can be considered as an independent equilibrium configuration.
The configuration centered at the middle axis is D shaped while the outer configuration has a crescent shape. A similar structure, but with steeper gradients, is obtained for solutions from region B. In this latter case the three sets of surfaces are separated by the χ = 0 surface.  This aspect of the problem, together with possible restrictions it might bring on the physical range over which the coefficients a 2 and a 4 can vary, has not been addressed explicitly in the present article. However if physical boundary conditions are imposed, for example by external conductors, that restrict the solution to the "flat" central domain of e.g., the configuration shown in Fig.5, that is to say to a domain inside χ = const (and thus ψ = const) surfaces within which χ is sufficiently larger than zero and the Mach number sufficiently smaller than one, Eq.(11) can be inverted by a simple perturbation procedure. On the contrary, the inverse power dependence on χ of the terms of the terms on the r.h.s. of GGS equation (40) implies that the equilibrium quantities, including the flow profiles, will develop large gradients, and/or the Mach number will approach unity, at the borders of the lunes in Fig.1 where χ → 0 (and, for solutions in region B, near the internal χ = 0 curves). This can be seen by combining d/dψ = [1 − M 2 (ψ)] 1/2 d/dχ with the χ derivatives of the flux functions A 0 (χ), A 2 (χ) and A 4 (χ) in Eqs. (14)(15)(16).
Furthermore, it is instructive to compare the D shaped equilibrium configuration without flows discussed in Ref. 10 with two corresponding equilibria (at fixed J (ψ), see Eq. (7)) with toroidal flows but without poloidal flows so that the functions χ and ψ coincide. This comparison is shown in Fig.7 where 33 the shaded contour plot of the D-shaped equilibrium of Ref.10 is shown in the top frame (a 2 = −3/2, a 4 = 0), that of a corresponding equilibrium with angular velocity increasing with ψ (a 2 = −3/2, a 4 > 0) is shown in the center frame and one with angular velocity decreasing with ψ (a 2 = −3/2, a 4 < 0) in the bottom frame.
We see that a positive gradient of the centrifugal term due to the toroidal flow (a 4 > 0) leads to a shift of the maximum of the flux function ψ(r, z) towards larger values of r and to the formation of large gradients at the outer border, as characteristic of the equilibrium solutions in region A (see e.g., Fig.5). On the contrary, a negative gradient of the centrifugal term (a 4 < 0) causes ψ(r, z) to vanish not only at the boundary of the solution domain but also along two curves inside the domain as is the case for the solutions in region C. This splits the solution into "separate" solutions with different magnetic axes, as characteristic of the equilibrium solutions in regions B and C (see e.g., Fig.6). From the bottom frame of Fig.7 we see that the inner solution is, for the chosen parameters, vanishingly thin, the central one has a rather flat radial profile while the outer solution has large gradients.
As a concluding remark we observe that the procedure developed in the present paper