We derive a generalized Grad-Shafranov equation governing helically symmetric equilibria with pressure anisotropy and incompressible flow of arbitrary direction. Through the most general linearizing ansatz for the various free surface functions involved therein, we construct equilibrium solutions and study their properties. It turns out that pressure anisotropy can act either paramegnetically or diamagnetically, the parallel flow has a paramagnetic effect, while the non-parallel component of the flow associated with the electric field has a diamagnetic one. Also, pressure anisotropy and flow affect noticeably the helical current density.

By choosing appropriate deformed Maxwellian ion and electron distribution functions depending on the two particle constants of motion, i.e. the energy and toroidal angular momentum, we reduce the Vlasov axisymmetric equilibrium problem for quasineutral plasmas to a transcendental GradShafranov-like equation. This equation is then solved numerically under the Dirichlet boundary condition for an analytically prescribed boundary possessing a lower X-point to construct tokamak equilibria with toroidal sheared ion flow and anisotropic pressure. Depending on the deformation of the distribution functions these steady states can have toroidal current densities either peaked on the magnetic axis or hollow. These two kinds of equilibria may be regarded as a bifurcation in connection with symmetry properties of the distribution functions on the magnetic axis.

Nonlinear z-independent solutions to a generalized Grad-Shafranov equation (GSE) with up to quartic flux terms in the free functions and incompressible plasma flow non parallel to the magnetic field are constructed quasi-analytically. Through an ansatz the GSE is transformed to a set of three ordinary differential equations and a constraint for three functions of the coordinate x, in cartesian coordinates (x, y), which then are solved numerically. Equilibrium configurations for certain values of the integration constants are displayed. Examination of their characteristics in connection with the impact of nonlinearity and sheared flow indicates that these equilibria are consistent with the L-H transition phenomenology. For flows parallel to the magnetic field one equilibrium corresponding to the H-state is potentially stable in the sense that a sufficient condition for linear stability is satisfied in an appreciable part of the plasma while another solution corresponding to the L-state does not satisfy the condition. The results indicate that the sheared flow in conjunction with the equilibrium nonlinearity play a stabilizing role.

Several classes of analytic solutions to a generalized Grad-Shafranov equation with incompressible plasma flow non-parallel to the magnetic field are constructed. The solutions include higher transcendental functions such as the Meijer G-function and describe D-shaped and diverted configurations with either a single or double X-points. Their characteristics are examined in particular with respect to the flow parameters associated with the electric field. It turns out that the electric field makes the safety factor flatter and increases the magnitude and shear of the toroidal velocity in qualitative agreement with experimental evidence on the formation of internal transport barriers in tokamaks, thus indicating a potential stabilizing effect of the electric field.

In Ref.[1] it was proposed an analytic form to describe the boundary of an axisymmetric plasma. The form determines analytically smooth boundaries, e.g. Dshaped ones, and numerically boundaries having an X-point. Here we propose an alternative form describing boundaries with X-points fully analytically.The two coordinates are (R, z) and they are normalized with respect to the major radius R 0 . For example for ITER we have R 0 = 6.2m. The boundary is up-down asymmetric and it consists of a smooth upper part and a lower part possessing an Xpoint. For the upper part we have four parameters. One is the inverse aspect ratio ǫ 0 = a/R 0 . For example in the case of ITER it is a = 2.1m and thus ǫ 0 = 0.338. The other three are the upper elongation κ, the upper triangularity δ and a parameter n related to the steepness of the triangularity of the upper boundary curve, i.e. to the mean value of the derivative dρ/dθ [see Eqs. (1) and (2) below]. For the ITER case it is κ = 1.86 and δ = 0.5. Introducing the normalized coordinates ρ = R/R 0 and ζ = z/R 0 , we have for the ζ coordinate of the uppermost point (see Fig. 1): ζ u = κǫ 0 and δ = (1 − ρ δ )/ǫ 0 .The form for the upper part of the boundary is given bywhere α = sin −1 (δ). Thus the following relations hold: ρ δ = 1 − δǫ 0 and θ δ = π − tan −1 (κ/δ). The parameter τ is any increasing function of the usual polar angle θ, satisfying τ (0) = 0, τ (π) = π and τ (θ δ ) = π/2. In our form we have taken τ (θ) = t 0 θ 2 + t 1 θ n t 0 = θ n δ −