G. N. Throumoulopoulos

Symmetric and asymmetric equilibria with non-parallel flows

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.

Nonlinear translational symmetric equilibria relevant to the L–H transition

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.

Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria

The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electromagnetic surfaces.

New classes of exact solutions to the Grad-Shafranov equation with arbitrary flow using Lie-point symmetries

Extending previous work [R. L. White and R. D. Hazeltine, Phys. Plasmas 16, 123101 (2009)] to the case of a generalized Grad-Shafranov equation (GGSE) with incompressible flow of arbitrary direction, we obtain new classes of exact solutions on the basis of Lie-point symmetries. This is done by using a previously found exact generalized Solovev solution to the GGSE. The new solutions containing five free parameters describe D-shaped toroidal configurations with plasma flow non-parallel to the magnetic field. In addition, the full symmetry group is obtained and new group-invariant solutions to the GGSE are presented.

Analytical up-down asymmetric equilibria with non-parallel flows

Generic linear axisymmetric equilibria with plasma flow nonparallel to the magnetic field are obtained on the basis of a generalized Grad-Shafranov equation by employing an ansatz reducing the problem to a set of ordinary differential equations which can be solved recursively. In particular, an ITER like equilibrium with reversed magnetic shear and peaked current density is constructed and its characteristics are studied in connection with the flow. Also for parallel flows, the linear stability is examined by means of a sufficient condition. The results indicate that the flow may have a stabilizing effect.

Equilibria with incompressible flows from symmetry analysis

We identify and study new nonlinear axisymmetric equilibria with incompressible flow of arbitrary direction satisfying a generalized Grad Shafranov equation by extending the symmetry analysis presented in [G. Cicogna and F. Pegoraro, Phys. Plasmas Vol. 22, 022520 (2015)]. In particular, we construct a typical tokamak D-shaped equilibrium with peaked toroidal current density, monotonically varying safety factor and sheared electric field.

An alternative method of constructing axisymmetric toroidal equilibria with nonparallel flow

An alternative method based on an inverse aspect ratio (ϵ) expansion which reduces the axisymmetric equilibrium problem to a set of ODEs containing terms of arbitrary order in ϵ is employed to solve a generalized Grad-Shafranov equation with incompressible sheared flow nonparallel to the magnetic field. The method is applied to construct equilibria with either circular magnetic surfaces and reversed magnetic shear or D-shaped magnetic surfaces and normal magnetic shear. From the former equilibrium, it turns out that the electric field results in an increase of the reversed magnetic shear, thus indicating potential synergetic effects of the sheared flow and the magnetic shear in the formation of an internal transport barrier in consistent with experimental evidence.

Vlasov tokamak equilibria with shearad toroidal flow and anisotropic pressure

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 Grad-Shafranov-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.

A comparison of Vlasov with drift kinetic and gyrokinetic theories

A kinetic consideration of an axisymmetric equilibrium with vanishing electric field near the magnetic axis shows that ∇f should not vanish on axis within the framework of Vlasov theory while it can either vanish or not in the framework of both a drift kinetic and a gyrokinetic theories (f is either the pertinent particle or the guiding center distribution function). This different behavior, relating to the reduction of phase space which leads to the loss of a Vlasov constant of motion, may result in the construction of different currents in the reduced phase space than the Vlasov ones. This conclusion is indicative of some limitation on the implications of reduced kinetic theories, in particular, as concerns the physics of energetic particles in the central region of magnetically confined plasmas.

A generalized Grad-Shafranov equation with plasma flow under a conformal coordinate transformation

We employ a conformal mapping transformation to solve a generalized Grad-Shafranov equation with incompressible plasma flow of arbitrary direction and construct particular up-down asymmetric D-shaped and diverted tokamak equilibria. The proposed method can also be employed as an alternative quasi-analytic method to solving two dimensional elliptic partial differential equations.