On steady poloidal and toroidal flows in tokamak plasmas

The effects of poloidal and toroidalflows on tokamakplasma equilibria are examined in the magnetohydrodynamic limit. “Transonic” poloidal flows of the order of the sound speed multiplied by the ratio of poloidal magnetic field to total field B₀/B can cause the (normally elliptic) Grad–Shafranov (GS) equation to become hyperbolic in part of the solution domain. It is pointed out that the range of poloidal flows for which the GS equation is hyperbolic increases with plasma beta and B₀/B, thereby complicating the problem of determining spherical tokamakplasma equilibria with transonic poloidal flows. It is demonstrated that the calculation of the hyperbolicity criterion can be easily modified when the assumption of isentropic flux surfaces is replaced with the more tokamak-relevant one of isothermal flux surfaces. On the basis of the latter assumption, a simple expression is obtained for the variation of density on a flux surface when poloidal and toroidalflows are simultaneously present. Combined with Thomson scattering measurements of density and temperature, this expression could be used to infer information on poloidal and toroidalflows on the high field side of a tokamakplasma, where direct measurements of flows are not generally possible. It is demonstrated that there are four possible solutions of the Bernoulli relation for the plasma density when the flux surfaces are assumed to be isothermal, corresponding to four distinct poloidal flow regimes. Finally, observations and first principles-based theoretical modeling of poloidal flows in tokamakplasmas are briefly reviewed and it is concluded that there is no clear evidence for the occurrence of supersonic poloidal flows.This work was jointly funded by the Australian Government through International Science Linkages Grant No. CG130047, the Australian National University, the United Kingdom Engineering and Physical Sciences Research Council, and by the European Communities under the contract of Association between EURATOM and CCFE

Azimuthally symmetric MHD and two-fluid equilibria with arbitrary flows

Magnetohydrodynamic (MHD) and two-fluid quasi-neutral equilibria with azimuthal symmetry, gravity and arbitrary ratios of (nonrelativistic) flow speed to acoustic and Alfven speeds are investigated. In the two-fluid case, the mass ratio of the two species is arbitrary, and the analysis is therefore applicable to electron-positron plasmas. The methods of derivation can be extended in an obvious manner to several charged species. Generalized Grad-Shafranov equations, describing the equilibrium magnetic field, are derived. Flux function equations and Bernoulli relations for each species, together with Poisson's equation for the gravitational potential, complete the set of equations required to determine the equilibrium. These are straightforward to solve numerically. The two-fluid system, unlike the MHD system, is shown to be free of singularities. It is demonstrated analytically that there exists a class of incompressible MHD equilibria with magnetic field-aligned flow. A special sub--class first identified by S. Chandrasekhar, in which the flow speed is everywhere equal to the local Alfven speed, is compatible with virtually any azimuthally symmetric magnetic configuration. Potential applications of this analysis include extragalactic and stellar jets, and accretion disks.Comment: 18 pages, 0 figure

Toroidal ripple transport of beam ions in the mega-ampère spherical tokamak

The transport of injected beam ions due to toroidalmagnetic field ripple in the mega-ampère spherical tokamak (MAST) is quantified using a full orbit particle tracking code, with collisional slowing-down and pitch-angle scattering by electrons and bulk ions taken into account. It is shown that the level of ripple losses is generally rather low, although it depends sensitively on the major radius of the outer midplane plasma edge; for typical values of this parameter in MAST plasmas, the reduction in beam heating power due specifically to ripple transport is less than 1%, and the ripple contribution to beam ion diffusivity is of the order of 0.1 m² s⁻¹ or less. It is concluded that ripple effects make only a small contribution to anomalous transport rates that have been invoked to account for measured neutron rates and plasma stored energies in some MAST discharges. Delayed (non-prompt) losses are shown to occur close to the outer midplane, suggesting that banana-drift diffusion is the most likely cause of the ripple-induced losses.This work was funded by the RCUK Energy Programme under Grant EP/I501045, by the Australian Research Council, and by the European Communities under the Contract of Association between EURATOM and CCFE

Full orbit simulations of collisional impurity transport in spherical tokamak plasmas with strongly-sheared electric fields

The collisional dynamics of test impurity ions in spherical tokamak plasmas with strongly-sheared radial electric fields is investigated by means of a test particle full orbit simulation code. The strength of the shear is such that the standard drift ordering can no longer be assumed and a full orbit approach is required. The effect of radial electric field shear on neoclassical particle transport is quantified for a range of test particle mass and charge numbers and electric field parameters. It is shown that the effect of a sheared electric field is to enhance the confinement of impurity species above the level observed in the absence of such a field. The effect may be explained in terms of a collisional drag force drift, which is proportional to particle charge number but independent of particle mass. This drift acts inwards for negative radial electric fields and outwards for positive fields, implying strongly enhanced confinement of highly ionized impurity ions in the presence of a negative radial electric field.Comment: 16 pages, 6 figures, submitted to Nuclear Fusio

Field-guided proton acceleration at reconnecting X-points in flares

An explicitly energy-conserving full orbit code CUEBIT, developed originally to describe energetic particle effects in laboratory fusion experiments, has been applied to the problem of proton acceleration in solar flares. The model fields are obtained from solutions of the linearised MHD equations for reconnecting modes at an X-type neutral point, with the additional ingredient of a longitudinal magnetic field component. To accelerate protons to the highest observed energies on flare timescales, it is necessary to invoke anomalous resistivity in the MHD solution. It is shown that the addition of a longitudinal field component greatly increases the efficiency of ion acceleration, essentially because it greatly reduces the magnitude of drift motions away from the vicinity of the X-point, where the accelerating component of the electric field is largest. Using plasma parameters consistent with flare observations, we obtain proton distributions extending up to gamma-ray-emitting energies (>1MeV). In some cases the energy distributions exhibit a bump-on-tail in the MeV range. In general, the shape of the distribution is sensitive to the model parameters.Comment: 14 pages, 4 figures, accepted for publication in Solar Physic

The quasi-linear relaxation of thick-target electron beams in solar flares

The effects of quasi-linear interactions on thick-target electron beams in the solar corona are investigated. Coulomb collisions produce regions of positive gradient in electron distributions which are initially monotonic decreasing functions of energy. In the resulting two-stream instability, energy and momentum are transferred from electrons to Langmuir waves and the region of positive slope in the electron distribution is replaced by a plateau. In the corona, the timescale for this quasi-linear relaxation is very short compared to the collision time. It is therefore possible to model the effects of quasi-linear relaxation by replacing any region of positive slop in the distribution by a plateau at each time step, in such a way as to conserve particle number. The X-ray bremsstrahlung and collisional heating rate produced by a relaxed beam are evaluated. Although the analysis is strictly steady state, it is relevant to the theoretical interpretation of hard X-ray bursts with durations of the order of a few seconds (i.e., the majority of such bursts)

Electron Inertial Effects on Rapid Energy Redistribution at Magnetic X-points

The evolution of non-potential perturbations to a current-free magnetic X-point configuration is studied, taking into account electron inertial effects as well as resistivity. Electron inertia is shown to have a negligible effect on the evolution of the system whenever the collisionless skin depth is less than the resistive scale length. Non-potential magnetic field energy in this resistive MHD limit initially reaches equipartition with flow energy, in accordance with ideal MHD, and is then dissipated extremely rapidly, on an Alfvenic timescale that is essentially independent of Lundquist number. In agreement with resistive MHD results obtained by previous authors, the magnetic field energy and kinetic energy are then observed to decay on a longer timescale and exhibit oscillatory behavior, reflecting the existence of discrete normal modes with finite real frequency. When the collisionless skin depth exceeds the resistive scale length, the system again evolves initially according to ideal MHD. At the end of this ideal phase, the field energy decays typically on an Alfvenic timescale, while the kinetic energy (which is equally partitioned between ions and electrons in this case) is dissipated on the electron collision timescale. The oscillatory decay in the energy observed in the resistive case is absent, but short wavelength structures appear in the field and velocity profiles, suggesting the possibility of particle acceleration in oppositely-directed current channels. The model provides a possible framework for interpreting observations of energy release and particle acceleration on timescales down to less than a second in the impulsive phase of solar flares.Comment: 30 pages, 8 figure

A critical Mach number for electron injection in collisionless shocks

Electron acceleration in collisionless shocks with arbitrary magnetic field orientations is discussed. It is shown that the injection of thermal electrons into diffusive shock acceleration process is achieved by an electron beam with a loss-cone in velocity space that is reflected back upstream from the shock through shock drift acceleration mechanism. The electron beam is able to excite whistler waves which can scatter the energetic electrons themselves when the Alfven Mach number of the shock is sufficiently high. A critical Mach number for the electron injection is obtained as a function of upstream parameters. The application to supernova remnant shocks is discussed.Comment: 4 pages, 2 figure, accepted for publication in Physical Review Letter

Surfatron and stochastic acceleration of electrons in astrophysical plasmas

Electron acceleration by large amplitude electrostatic waves in astrophysical plasmas is studied using particle-in-cell (PIC) simulations. The waves are excited initially at the electron plasma frequency $\omega_{\rm pe}$ by a Buneman instability driven by ion beams: the parameters of the ion beams are appropriate for high Mach number astrophysical shocks, such as those associated with supernova remnants (SNRs). If $\omega_{\rm pe}$ is much higher than the electron cyclotron frequency $\Omega_{\rm e}$, the linear phase of the instability does not depend on the magnitude of the magnetic field. However, the subsequent time evolution of particles and waves depends on both $\omega_{\rm pe}/\Omega_{\rm e}$ and the size of the simulation box $L$. If $L$ is equal to one wavelength, $\lambda_0$, of the Buneman-unstable mode, electrons trapped by the waves undergo acceleration via the surfatron mechanism across the wave front. This occurs most efficiently when $\omega_{\rm pe}/\Omega_{\rm e} \simeq 100$: in this case electrons are accelerated to speeds of up $c/2$ where $c$ is the speed of light. In a simulation with $L=4\lambda_0$ and $\omega_{\rm pe}/\Omega_{\rm e} = 100$, it is found that sideband instabilities give rise to a broad spectrum of wavenumbers, with a power law tail. Some stochastic electron acceleration is observed in this case, but not the surfatron process. Direct integration of the electron equations of motion, using parameters approximating to those of the wave modes observed in the simulations, suggests that the surfatron is compatible with the presence of a broad wave spectrum if $\omega_{\rm pe}/\Omega_{\rm e}> 100$. It is concluded that a combination of stochastic and surfatron acceleration could provide an efficient generator of mildly relativistic electrons at SNR shocks