A similarity reduction of the generalized Grad-Shafranov equation

We extend previous work [Y. E. Litvinenko, Phys. Plasmas 17, 074502 (2010)] on a direct method for finding similarity reductions of partial differential equations such as the Grad-Shafranov equation, to the case of the generalized Grad-Shafranov equation (GGSE) with arbitrary incompressible flow. Several families of analytic solutions are constructed, the generalized Solov\'ev solution being a particular case, which contain both the classical and non-classical group-invariant solutions to the GGSE. Those solutions can describe a variety of equilibrium configurations pertinent to toroidal magnetically confined plasmas and planetary magnetospheres.


Introduction
The analysis of similarity reductions of partial differential equations plays a crucial role in many physical applications.Some time ago in [1] a direct method for finding similarity reductions of the Grad-Shafranov equation (GSE) has been introduced, thus generalizing previous work [2] on the subject.This was then generalized [3] for the case of the generalized Grad-Shafranov equation (GGSE) (equation (1) below), i.e. using classical Lie-group methods to find a one-parameter group admitted by the equation and the corresponding group-invariant solution.
The Solovév solution [4] and the Herrnegger-Maschke solution [5] are among the most widely employed analytical solutions of the GSE, the former corresponding to toroidal current density non-vanishing on the plasma boundary while the later to toroidal current density vanishing thereon.In the presence of flow, which plays a role in the transition to improved confinement modes in tokamaks, the equilibrium satisfies a GGSE, in general coupled with a Bernoulli equation involving the pressure [6,7].For incompressible flow, the density becomes a surface quantity and we obtain the GGSE [7,8].Generalized Solovév solutions of the GGSE were derived in [8,9] and asymptotic expansion solutions to the general linearized GGSE in [10,11].Also, some analytical solutions were recently constructed for the axisymmetric Hall magnetohydrodynamics GSE with incompressible flows [12].
A symmetry group of a partial differential equation (PDE) is a set of transformations of the independent or/and dependent variables that allows to generate new classes of solutions from a single solution known.According to the classical Lie group method, a similarity reduction of a PDE is to determine a one parameter group admitted by the equation.Then one seeks the group-invariant solutions called self-similar solutions (cf.[13] for a recent description of the method).Symmetry properties of a system of Euler-type equations were studied in [14].Symmetry properties and solutions to the GSE and the GGSE were studied in [15,16], whereas symmetry methods for differential equations are exposed in [13].Translationally symmetric force-free states are constructed in [17].The literature on exact solutions to the GSE is extensive and a brief reference list includes [18,19], while other symmetry and similarity reduction methods were employed in [20,21,22,23].
Aim of the present study is to construct new families of analytic solutions to the GGSE by using an alternative method of similarity reduction along the lines of [1], describing equilibrium configurations relevant to experimental fusion and space plasmas.Accordingly, a PDE can be reduced to an ordinary differential equation by employing an appropriate functional relation involving the dependent and independent variables, as Eq. ( 4) below, after imposing certain conditions.
The structure of the paper is as follows: In section 2 we present the direct method for finding the similarity reduction of the GGSE and show that the generalized Solovév solution is a particular case of the above mentioned set of solutions.In section 3 we extend the set of solutions to the GGSE and show that all group-invariant solutions of [3] are in fact particular cases of the extended set of solutions.In section 5 we present a new similarity reduction of the GGSE, not considered in [1], associated with solutions which can describe part of planetary magnetospheres.Finally, in section 5 we present a brief discussion on the novel results of this paper.

Axisymmetric equilibria with non-parallel flow from similarity reduction
We consider the GGSE in its completely dimensionless form [7,8] Here, employing cylindrical coordinates(z, r, ϕ), the function u(r, z) relates to the poloidal magnetic flux function and labels the magnetic surfaces; the flux function X(u) relates to the toroidal magnetic field (cf.Eq. (3) below); P s (u) is the pressure in the absence of flow; ρ(u) is the density; M p (u) is the Mach function of the poloidal velocity with respect to the respective Alfvén velocity; and Φ(u) is the electrostatic potential related to the electric field.The flux functions X(u), M p (u), P s (u), ρ(u) and Φ(u) remain arbitrary.Also, the magnetic field and the velocity are given by the relations where Compared with the GSE the GGSE has the additional R 4 -term related to the non parallel to the magnetic field component of the velocity through the electrostatic potential Φ.This term can significantly modify the equilibrium; for example, in the presence of this term, the well known Solovév equilibrium [4], which is similar to the configuration of Fig. 1, acquires an addition X-point outside the separatrix (cf.Fig. 11 of [8]).Also, although plasma rotation in magnetic confinement systems is low, i.e.M p ∼ 10 −2 , the respective electric field and its shear play an important role in the transitions to improved confinement regimes, as the L-H-transition [24].
Choosing the flux function terms to be constant, i.e., (( we have the following form of the GGSE that will be the focus of this study Equation ( 4) is satisfied by the generalized Solovév-type solution [8] The respective equilibrium configuration inside the separatrix, up-down symmetric with a pair of X-points, is plotted in Fig. 1.For E = 0, (5) reduces to the well known Solovév solution [4].
We consider now the similarity reduction of Eq. ( 4) via the ansatz [1] u(r, z) = α(r, z) Substituting ( 6) into ( 4) we obtain Let us choose as a first illustration of the method the case where α = α(r), β = β(r) and x = x(z).Then Eq. ( 7) becomes This equation can be satisfied by choosing x(z) = z 2 /2, provided that the functions α, β satisfy the ODEs These equations are solved for and The solution u(r, z) of the GGSE (4) by means of ( 6), ( 11) and ( 12) is plotted in Fig. 2, Returning to Eq. ( 7), setting x = z/r and assuming α = α(r, z) and β = β(r) we obtain This equation is satisfied by β(r) = r ν and by any solution of the following equations Equations ( 14) and ( 15) constitute a new similarity reduction of the GGSE.Here we note an error in Eq. ( 15) of [1] which in not a solution of Eq. ( 13) of [1] neither for ν = 1/2 nor for ν = 3/2.Eq. ( 14) has the analytic solution [1] w in terms of the associated Legendre functions P (ix).These functions involve infinite converging series; only for particular values of ν, e.g.ν = 0, 1, 2 to be considered below, the solution is expressed in closed form, i.e. in terms of Legendre polynomials.For this reason, in the former case, instead of employing the analytic solution we preferred to solve Eq. ( 14) numerically.
The same procedure applies for the case ν = 3/2.We integrated numerically Eq. ( 14) Figure 5: Up-down asymmetric equilibrium determined by (6) with the aid of the numerical solution of Eq. ( 14) for w(x) with ν = 1/2, shown in Fig. 4.
Figure 7: Up-down asymmetric equilibrium with a lower X-point determined by (6) with the aid of the numerical solution of Eq. ( 14) for w(x) with ν = 3/2, shown in Fig. 6. is plotted in Fig. 7, where u = u Sol + cr 3/2 w with c = 5 × 10 −3 .This is a diverted equilibrium with a lower X-point.
For ν = 3 we obtain the explicit solution of Eq. ( 14) as 3 More equilibria from similarity reduction New non-trivial solutions to the GGSE can be generated from any solution to Eq. (15).For example, we will employ the following solutions to Eq. ( 15): Using Eq. ( 16) for ν = 3 and (20) we have the solution Yet another way to produce new solutions is to consider α = α(r) in Eq. ( 15) leading to the solution In this case using again the solution ( 16) for ν = 3 we find from ( 6) Moreover if u g (r, z) is a solution to the GGSE (4) then u g (r, z) + β(r, z)w[x(r, z)] is also a solution, thus making it possible to combine solutions with different ν.As an example, combining the solutions for ν = 3 with those for ν = 1 presented previously and Eq. ( 20) we find This is plotted in Fig. 8, for The bounding flux surface corresponds to U b = −0.05while at the center we have U c = −0.2.Note that this equilibrium configuration has inverse D-shape.Such equilibria has been the subject of recent tokamak studies because they may exhibit improved confinement properties [25,26].The family of solutions above contains the classical and non-classical group-invariant solutions of the GGSE so that it greatly extends the range of available analytical solutions.It is interesting that all group-invariant solutions to the GGSE are in fact particular cases of the above similarity reduction.Indeed, Eq. ( 14) with ν = 0 is solved for Adding a particular solution by Eq. ( 23) we are led to This is the U inv (x, y)−solution of [3] given by Eq. ( 23) therein.For ν = 1 a solution of ( 14) is the second of (17): while for ν = −1 we solve Eq. ( 14) for Taking a particular linear combination of (28)) and (29) and a(r) from Eq. ( 23) leads to This is the U inv (x, y)−solution of [3] given by Eq. ( 21) therein.Finally, Eq. ( 14) with ν = 2 is solved for Using α(r, z) from Eq. ( 21) we have which is the U inv (x, y)−solution of [3] given by Eq. ( 22) therein.

Conclusions
We have considered a similarity reduction of the GGSE along the lines presented in [1].New classes of exact solutions to the GGSE were produced, with the generalized Solovév solution being a particular case, describing a variety of D-shaped equilibrium configurations exhibiting up-down symmetry or asymmetry and either positive or negative triangularity and diverted configurations with a lower X-point.All groupinvariant solutions of the GGSE presented in [3] are shown to be particular cases of the above formalism.In addition, a new class of solutions, not considered in [1], has been constructed which can describe part of a planetary magnetoshphere for short distances from the planet surface.

( 5 )Figure 3 :
Figure 3: The up-down symmetric equilibrium configuration with smooth bounding surface in connection to the solution (18) of the GGSE (4).

Figure 9 :
Figure 9: Equilibrium configuration in connection with the solution (36) pertinent to part of a planetary magnetosphere.