We derive a sufficient condition for the linear stability of plasma equilibria with incompressible flow parallel to the magnetic field, $\bf B$, constant mass density and anisotropic pressure such that the quantity $σ_d= μ_0(P_\parallel -P_\perp)/B^2$, where $P_\parallel$ ($P_\perp$) is the pressure tensor element parallel (perpendicular) to $\bf B$, remains constant. This condition is applicable to any steady state without geometrical restriction. The condition, generalising the respective condition for MHD equilibria with isotropic pressure and constant density derived in Throumoulopoulos & Tasso (2007), involves physically interpretable terms related to the magnetic shear, the flow shear and the variation of total pressure perpendicular to the magnetic surfaces. On the basis of this condition we prove that if a given equilibrium is linearly stable, then the ones resulting from the application of Bogoyavlenskij symmetry transformations are linearly stable too, provided that a parameter involved in those transformations is positive. In addition, we examine the impact of pressure anisotropy, flow, and torsion of a helical magnetic axis, for a specific class of analytic equilibria. In this case we find that the pressure anisotropy and the flow may have either stabilising or destabilising effects. Also, helical configurations with small torsion and large pitch seem to have more favorable stability properties.
In ideal MHD, the magnetic flux is advected by the plasma motion, freezing flux-surfaces into the flow. An MHD equilibrium is reached when the flow relaxes and force balance is achieved. We ask what classes of MHD equilibria can be accessed from a given initial state via smooth incompressible ideal motion. It is found that certain boundary displacements are formally not supported. This follows from yet another investigation of the Hahm--Kulsrud--Taylor (HKT) problem, which highlights the resonant behaviour near a rational layer formed by a set of degenerate critical points in the flux-function…
Being motivated by the paper [O. I. Bogoyavlenskij, Phys. Rev. 66, 056410 (2002)] we generalise the symmetry transformations for MHD equilibria with isotropic pressure and incompressible flow parallel to the magnetic field introduced therein in the case of respective CGL equilibria with anisotropic pressure. We find that the geometrical symmetry of the field-aligned equilibria can break by those transformations only when the magnetic field is purely poloidal. In this situation we derive three-dimensional CGL equilibria from given axisymmetric ones. Also, we examine the generic symmetry transfo…
We present several numerical solutions to a generalized Grad-Shafranov equation (GGSE), which governs axisymmetric plasma equilibria with incompressible flows of arbitrary direction, using fully connected, feed-forward, deep neural networks, also known as multi-layer perceptrons. Such artificial neural network (ANNs) are trained to approximate tokamak-relevant equilibria upon minimizing the GGSE mean squared residual in the plasma volume and the poloidal flux function on the plasma boundary. Solutions for the Solovev and the general linearizing ansatz for the free functions involved in the GGS…
It is believed that plasma rotation can affect the transitions to the advanced confinement regimes in tokamaks. In addition, in order to achieve fusion temperatures modern tokamaks rely on auxiliary heating methods. These methods generate pressure anisotropy in the plasma. For incompressible rotation with pressure anisotropy the equilibrium is governed by a Generalized Grad-Shafranov (GGS) equation and a decoupled Bernoulli-type equation for the effective pressure, $\bar{p}=(p_\parallel+p_\perp)/2$, where $p_\parallel$ ($p_\perp$) is the pressure tensor element parallel (perpendicular) to the …
We obtain analytic solutions of a generalised Grad-Shafranov equation describing steady states with incompressible plasma flow of arbitrary direction, toroidal current reversal and either nested or non-nested magnetic surfaces. It turns out that the component of the flow velocity non parallel to the magnetic field can result in normal equilibria with central current-reversal, i.e. equilibria with nested magnetic surfaces and monotonically varying pressure profiles.
Plasma rotation in connection to both zonal and mean (equilibrium) flows can play a role in the transitions to the advanced confinement regimes in tokamaks, as the L-H transition and the formation of Internal Transport Barriers. For incompressible rotation the equilibrium is governed by a generalized Grad-Shafranov (GGS) equation and a decoupled Bernoulli-type equation for the pressure. For parallel flow the GGS equation can be transformed to one identical in form with the usual GS equation. In the present study on the basis of the latter equation we have extended HELENA, an equilibrium fixed …
A generalised Grad-Shafranov equation that governs the equilibrium of an axisymmetric toroidal plasma with anisotropic pressure and incompressible flow of arbitrary direction is derived. This equation includes six free surface functions and recovers known Grad-Shafranov-like equations in the literature as well as the usual static, isotropic one. The form of the generalised equation indicates that pressure anisotropy and flow act additively on equilibrium. In addition, two sets of analytical solutions, an extended Solovev one with a plasma reaching the separatrix and an extended Hernegger-Masch…
We study stationary free boundary configurations of an ideal incompressible magnetohydrodynamic fluid possessing nested flux surfaces. In 2D simply connected domains, we prove that if the magnetic field and velocity field are never commensurate, the only possible domain for any such equilibria is a disk, and the velocity and magnetic field are circular. We give examples of non-symmetric equilibria occupying a domain of any shape by imposing an external magnetic field generated by a singular current sheet charge distribution (external coils). Some results carry over to 3D axisymmetric solutions…
We analyze an axisymmetric equilibrium of a plasma endowed with toroidal and poloidal velocity fields, with the aim to characterize the influence of the global motion on the morphology of the magnetic confinement. We construct our configuration assuming that the poloidal velocity field is aligned with the poloidal magnetic field lines and, furthermore, we require that the plasma mass density depend on the magnetic flux function (or equivalently, that the plasma fluid be incompressible). We then derive a sort of Grad-Shafranov equation for such an equilibrium and implement it to tokamak relevan…
In this Master thesis we investigate the influence of pressure anisotropy and incompressible flow of arbitrary direction on the equilibrium properties of magnetically confined, axisymmetric toroidal plasmas. The main novel contribution is the derivation of a pertinent generalised Grad-Shafranov equation. This equation includes six free surface functions and recovers known Grad-Shafranov-like equations in the literature as well as the usual static, isotropic one. The form of the generalised equation indicates that pressure anisotropy and flow act additively on equilibrium. In addition, two sets…
Recent results will be presented on steady states of magnetically confined plasmas obtained by conventional and Hamiltonian methods. The presentation consists of three parts. The first one concerns the derivation of a generalized Grad-Shafranov equation describing helically symmetric equilibria with pressure anisotropy and incompressible plasma flow of arbitrary direction with application to straight-stellarator configurations [1]. The impact of pressure anisotropy and flow on the equilibrium characteristics is also examined. In the second part the axisymmetric equilibrium code HELENA is exten…
Stationary field-aligned MHD flows at astropauses and in astrotails. Principles of a counterflow configuration between a stellar wind and its interstellar medium wind
Context. Observations have shown that twisted magnetic fields naturally occur, and indeed are omnipresent in the Sun's atmosphere. It is therefore of great theoretical interest in solar atmospheric waves research to investigate the types of magnetohydrodynamic (MHD) wave modes that can propagate along twisted magnetic flux tubes. Aims. Within the framework of ideal MHD, the main aim of this work is to investigate small amplitude incompressible wave modes of twisted magnetic flux tubes with m more or equal 1. The axial magnetic field strength inside and outside the tube will be allowed to vary,…
The Multi-region Relaxed MHD (MRxMHD) has been successful in the construction of equilibria in three-dimensional (3D) configurations. In MRxMHD, the plasma is sliced into sub-volumes separated by ideal interfaces, each undergoing relaxation, allowing the formation of islands and chaos. The resulting equilibrium has a stepped pressure profile across sub-volumes. The Stepped Pressure Equilibrium Code (SPEC) [S.R. Hudson et al., Phys. Plasmas 19, 112502 (2012)] was developed to calculate MRxMHD equilibria numerically. In this work, we have extended the SPEC code to compute MRxMHD equilibria with …
TRIMEG-GKX: an electromagnetic gyrokinetic particle code with a Piecewise Field-Aligned Finite Element Method for Micro- and Macro-Instability Studies in Tokamak Core Plasmas
In-situ observations in the Earth's and Saturn's magnetosheaths and in the solar wind reveal the presence of Alfvén vortices as intermittent structures in the range of scales from fluid lengths down to few ion lengths. The density and the magnetic field associated with them appear to be compressible for higher plasma betas. Until now, only incompressible Alfvén vortices have been known. Motivated by space plasma observations we develop a new model of magnetic vortices in high-beta plasmas with anisotropic temperature, possessing compressible density and magnetic field, whose typical size range…
Results of a three-dimensional, flux-driven, electrostatic, global, two-fluid turbulence simulation for a 5-field period stellarator with an island divertor are presented. The numerical simulation is carried out with the GBS code, recently extended to simulate plasma turbulence in non-axisymmetric magnetic equilibria. The vacuum magnetic field used in the simulation is generated with the theory of Dommaschk potentials, and describes a configuration with a central region of nested flux surfaces, surrounded by a chain of magnetic islands, similarly to the diverted configurations of W7-X. The hea…
The evolution of a passive scalar field is considered for a slowly varying stratified medium, which is convected in an incompressible sheared flow with many overlapping static flux islands. Within the quasilinear/random phase approximation, a multiple scale expansion is made. Due to the rapid spatial variation of the temperature, the "ensemble" averaged/ slowly varying part of the solution is not described by the arithmetic average of the oscillatory evolution equation. The standard Markovian and continuum approximations are shown to be invalid. For times of order $N$, where there are $O(N^2)$…
The fundamental difference between incompressible ideal magnetohydrodynamics and the dynamics of a non-conducting fluid is that magnetic fields exert a tension force that opposes their bending; magnetic fields behave like elastic strings threading the fluid. It is natural, therefore, to expect that a magnetic field tangled at small length scales should resist a large-scale shear in an elastic way, much as a ball of tangled elastic strings responds elastically to an impulse. Furthermore, a tangled field should support the propagation of `magnetoelastic waves', the isotropic analogue of Alfvén w…
