Finite Plasma Pressure Research Articles

Electromagnetic effects in convective cell fluctuations and diffusion

Renormalized statistical theory of low-frequency hydrodynamic fluctuations in magnetized plasmas is applied for the general case of electromagnetic fluctuations. Stationary spectra of electromagnetic field thermal fluctuations are derived with regard to nonlinear interaction between potential and magnetic field excitations. The well known results of potential theories are recovered only in the limit of cold electrons or for plasmas with very low pressure. Substantial broadening of the convective cell potential spectrum and suppression of convective cell diffusion due to magnetic field fluctuations in the plasma of small but finite pressure are predicted.

Stability of ideal MHD modes of a finite pressure plasma in toroidal systems with a complex shape of the magnetic axis

The authors have obtained an equation for small oscillations describing the excitation of ideal small scale MHD modes in a finite pressure plasma with allowance for inertial effects in arbitrary magnetic configurations. This equation is used to analyse the plasma stability in an ASPERATOR NP-4 type trap. It is shown that in the currentless regime both ideal ballooning and Mercier modes may be excited in this trap. It is established that these modes are absolutely unstable because of the growth with increasing plasma pressure of the magnetic ‘hill’ in the device. The growth rates of the excited modes are determined. In unstable regimes, a relative improvement in confinement is observed when, with increasing pressure, a transition may occur from regimes with larger growth rates to regimes with smaller growth rates. Such a transition is connected with finite plasma pressure effects when the destabilizing action of finite pressure, leading to a growth of the ‘magnetic hill’ and to an absolute instability of Mercier modes, is partially compensated for by the stabilizing action connected with the growth of the shear value with increasing plasma pressure.

Suppression and control of magnetic islands in toroidal plasmas

Magnetic islands play an important role in determining the stability and confinement properties of toroidal plasmas. The nonlinear growth of magnetic islands can lead to major disruptions in tokamaks. In stellarators, which do not have a continuous symmetry, the formation of magnetic islands can set equilibrium beta limits. In the first part of this paper, a novel method is proposed for reducing drastically the size of m=2, n=1 islands in tokamaks by introducing a beamlet of energetic ions just outside the q=2 surface via parallel neutral-beam injection. A simple physical picture is given of the nonlinear stability of a tearing mode in the Rutherford regime in the presence of energetic ions. This physical picture is supported by a kinetic calculation in the long mean-free-path regime in which the effect of bootstrap currents and resistive interchanges are retained. Estimates show that the energy of the beamlet required for the suppression is a small fraction of that used for Ohmic or beam-heating of the background plasma. In the second part of the paper, an equilibrium beta limit is obtained for stellarators by calculating the island widths induced by finite plasma pressure; the widths are then constrained by the criterion of island overlap. The theory is applied to the Heliotron-E device [Nakamura et al., Phys. Fluids B 2, 2528 (1990)]. It is shown that some aspects of the experimental observations on internal disruptions in Heliotron-E can be interpreted as signatures of the equilibrium beta limit.

Transition from resistive-G to ηi- driven turbulence in stellarator systems

By an electromagnetic incompressible two-fluid model describing both ion temperature gradient drift modes (ηi modes) and resistive interchange modes (g modes), a new type of ηi mode is studied in cylindrical geometry including magnetic shear and an averaged curvature of Heliotron/Torsatron. This ηi mode is destabilized by the coupling to the unstable g mode. Finite plasma pressure beta increases the growth rate of this mode and the radial mode width also increases with plasma pressure beta indicating large anomalous transport in the Heliotron/Torsatron configuration. The transport from ηi mode exceeds that from resistive g when the mean-free-path exceeds the machine circumference. For plasma beta above two to three times the Suydam limit, the m=1/n=1 growth rate increases from the ηi mode value to the magnetohydrodynamic (MHD) value.

Modified polynomial function model for reversed-field pinches

A generalized analytic representation for the magnetic field and current density profiles in a reversed-field pinch (RFP) is proposed. These profiles have zero current density at the wall and finite plasma pressure. The profiles are characterized by two free parameters here taken to be the field-reversal parameter (F) and pinch parameter (Θ). From the profiles, many useful quantities such as magnetic energy, beta, inductance, resistance, and Ohmic input power are calculated. These quantities provide a basis for analyzing experimental data and performing electrical circuit modeling of RFP discharges.

Characteristics of the toroidal relaxed state of reversed field pinches and comparison with experimental results

A toroidal equilibrium of a reversed field pinch based on Taylor’s Bessel function model (BFM) is investigated. Characteristics of the toroidal relaxed state for a few reversed field pinch devices are presented in detail. It is shown that the toroidal relaxed state agrees with experimental observations not only in the inner region, like BFM, but also to a good extent in the outer region of a reversed field pinch. Inclusion of toroidal effects in the BFM leads to a decrease of the toroidal reversed flux and an increase of the minor radius at which the toroidal field reverses. The toroidal relaxed state has ‘‘μ’’ decreasing in the outer region when calculated using experimental procedure and a saturation of the on-axis value of the pinch parameter Θ. Toroidal effects and finite plasma pressure lead to enhancement of the pinch parameter Θ. The results in this paper show that the observed departure of experiment from the BFM is partly due to the toroidal geometry.

Equilibrium beta limits in Heliotron-E due to magnetic island overlap

Widths of magnetic islands induced by finite plasma pressure are computed for the Heliotron-E configuration [Phys. Rev. Lett. 53, 2242 (1984); Nucl. Fusion 27, 895 (1987)]. Equilibrium beta limits are obtained from the criterion of island overlap. Earlier experimental observations on internal disruptions in Heliotron-E are revisited.

Stability of a cometary ionosphere/ionopause including pressure effects

The stability analysis by Ershkovich, McKenzie, and Axford of a cometary ionosphere/ionopause determined by ion-neutral friction is extended to include the effects of finite plasma pressure. It is shown that, from a formal viewpoint, the inclusion of pressure effects leaves intact the sufficient condition for instability. Application of the results to Comet Halley demonstrates that plasma pressure reduces the instability growth rate of the cavity and therefore that the boundary should not be expected to exhibit any significant mixing arising from overturning. The generalization to include plasma pressure is applicable to a wider range of physical parameters than is available at Halley and is of interest to other comets, particularly during periods of high solar activity. 12 refs.

A turbulent dynamo model for the reversed field pinches of plasma

A dynamo model for reversed field pinches (RFP’s) is proposed from the viewpoint of magnetohydrodynamic (MHD) turbulence. This model aims at investigating the sustainment phase of RFP’s, and consists of the mean magnetic induction equation and three equations for the bulk properties of MHD fluctuations, namely, the MHD turbulent energy, its dissipation rate, and the MHD helicity. In particular, the alpha dynamo effect is composed of the familiar alpha term and a new alpha term closely related to inhomogeneity of the field. This model paves the way for explaining the generation and sustainment of alpha and beta effects. Moreover, it can treat finite plasma pressure gradients and fulfill the vanishing of plasma currents at the plasma edge. This model can also be extended to the Earth’s magnetic dynamo model where the mean motion of a fluid plays an important role.

Electromagnetic drift vortices in a rotating plasma cylinder

This paper is devoted to the studies of the possibility of the existence of strongly localized nonlinear structures in a rotating plasma filament. It was established that electromagnetic, drift-Alfven and drift-balloon waves in cylindrical plasma traps are self-organizing in a form of two-dimensional solitary vortices. The wave field consists of a single monopole or a pair of vortices of nonequal intensity, which move over the circular trajectory along azymuth. The analysis was performed both for the plasma of finite pressure and that of low pressure with account of the effect of finite Larmor radius of ions in the field of the gravity. The vortices contain the trapped particles which, moving in plasma, can affect the processes of the transport in the plasma traps as well as the collective processes in a magnetospheric plasma.

Finite pressure equilibrium effects on helical ripple transport in torsatrons

A three-dimensional equilibrium code (NEAR) is used to study plasma equilibrium in torsatron configurations having various winding laws. The harmonic spectra describing the variation of |B⃗| along magnetic field lines are calculated at finite plasma pressure and then used to calculate the geometric factors that determine the magnitudes of the neoclassical transport coefficients in the helical ripple trapping regime, where the particle and heat fluxes are inversely proportional to the collision frequency. The results show that an appropriate choice of winding law can reduce helical ripple transport by a factor of about 2 in the vacuum field and at high plasma pressure.

Drift instabilities of a relativistic plasma. Part 2. Kinetic theory of low-frequency drift instabilities of a relativistic finite-pressure plasma

The generalization of the kinetic theory of low-frequency long-wavelength drift instabilities (instabilities due to the number density and temperature gradients and an instability of the Kelvin–Helmholtz type) is given for the case of a relativistic plasma. The role of relativistic effects, taking into account the anisotropy of the particle momentum distribution function, is clarified. Particular attention is paid to the investigation of instabilities of these perturbations in a relativistic electron-positron plasma.

Interchange, rotational, and ballooning stability of long-thin axisymmetric systems with finite-orbit effects

The stability of axisymmetric tandem mirror plasmas with respect to interchange, rotational, and ballooning modes is investigated in the paraxial approximation. The stabilizing effects of finite orbits, rigid energetic fast-drifting electrons, nearby conducting walls, and line-tying by a cold plasma halo are incorporated. Numerical calculations are performed to construct equilibria with finite plasma pressure and to determine linear stability by integrating a two-dimensional, initial-value equation. These numerical calculations support and extend analytical results. Analytical and numerical stability criteria are obtained.

Hall Effects on Magnetic Relaxation

A generalized vorticity is introduced whose self-linkage (the hybrid helicity) and flux are invariants of ideal incompressible magnetohydrodynamics (MHD) when the Hall term is included. A model of magnetofluid relaxation is constructed for Hall magnetohydrodynamics by assuming that the energy seeks the minimum value compatible with constrained values of magnetic helicity, hybrid helicity, axial magnetic flux, and fluid vorticity flux. As a result of the coupling of magnetic field to fluid vorticity in the generalized vorticity, it is found that the relaxed magnetic-field configuration need not be force free. The presence of a nonvanishing fluid vorticity is shown to be necessary for the existence of relaxed magnetic-field configurations that confine a finite plasma pressure. The study has potential relevance to the dynamics and morphology of space and cosmic plasmas, as well as to pressure confinement and current drive in fusion plasmas.

The effect of finite β on stellarator transport

A theory of the modification of stellarator transport caused by the presence of finite plasma pressure is developed and applied to a range of stellarator configurations. For typical planar-axis stellarators, transport can change by a factor of about 4–5 in the progression from zero pressure to the equilibrium β limit of the device. For transport-optimized configurations, the factor can be over an order of magnitude. Thus, a stellarator with transport-optimized vacuum fields can have poor confinement at the desired operating β. Without an external compensating field, increasing β tends to degrade confinement, unless the initial field structure is very carefully chosen. The theory permits correct determination of this vacuum structure, in terms of the desired structure of the field at a prescribed operating β. With a compensating external field, the deleterious effect of finite β on transport can be partially eliminated.

Alfvén Waves in Current-carrying Solar Magnetic Flux Tubes

In theories of the heating of the solar corona a number of authors have recently considered the propagation and damping of fast and slow magnetohydrodynamic waves, in the form of surface waves localized on the interfaces of coronal flux tubes (Ionson 1978, Wentzel 1979, Roberts 1981, Cramer and Donnelly 1983). The damping of these waves occurs, in addition to a rather weak global damping due to viscous or resistive dissipation, by means of a localized absorption at a so-called ‘resonance’ in the density or magnetic field profile forming the flux tube. At such a resonance, the wave frequency is equal to the local value of the Alfven wave frequency, in the case of Alfven resonance absorption, or at the local slow mhd wave frequency, in the case of the ‘cusp’ or ‘compressive singularity’ resonance in a finite pressure plasma.

Electromagnetic drift modes driven by pressure gradients in tokamaks

A hybrid of hydrodynamics and kinetics is used to study the effect of finite plasma pressure on the pressure-gradient driven toroidal drift modes. The linear drift modes of the system are given by a fifth-order polynomial describing the coupling of the electron-drift, the ion-acoustic, and the shear Alfvén oscillations. The characteristic frequencies, growth rates, and polarization of the electromagnetic modes are investigated as functions of the parameters of toroidicity, plasma gradients, and plasma pressure.

Reverse Field Configurations with Minimum Potential Energy

The reversed field pinch configurations (RFP) with minimum potential energy are examined by applying the variational principles to the cylindrical plasma of finite plasma pressure in the flux conserving shell, based on the invariance of a volume integral K (Waltjer's invariance).It is found that the obtained analytical formulas of RFP are expressed as the some combinations of Bessel functions and have the permissible maximum beta value of 5% on axis.

Nonlinear, two-dimensional magnetohydrodynamic calculations

A nonlinear, time-dependent, hydromagnetic model is developed. The model is based on the eight partial differential equations of resistive magnetohydrodynamics (MHD). The equations are expressed as a set of conservation laws which are written in general, orthogonal, curvilinear coordinates in two space dimensions. No assumption about the ordering of terms is made. The resulting equations are then solved by the method of finite differences on an Eulerian mesh. We develop spatial finite-difference techniques which guarantee the simultaneous conservation of the desired physical quantities throughout the course of the calculation. Conservative boundary conditions on thermodynamic quantities at a conducting boundary are derived, and special algorithms are developed for advancing the solution at a singular boundary. For the temporal differencing, we use the Alternating Direction Implicit (ADI) method. We apply our model to the difficult case of resistive instabilities. We present results relevant to the nonlinear evolution of these modes in three distinct coordinate systems. One of these cases depends on finite plasma pressure, and can be studied only with a general model such as that presented here.

High-latitude Pc1–2 geomagnetic pulsations and their connection with location of the dayside polar cusp

The result of investigating high-latitude Pc1–2 pulsations are presented in this paper. They show that these unstructured oscillations are typical in intervals of low magnetic activity for regions of projections of the dayside cusp on the Earth's surface. The morphological properties of these pulsations, namely the character of their diurnal variations and dependence of their amplitude and frequency of occurrence on magnetic activity on different latitudes, suggest methods of utilization for tracing the location of the equatorial boundary of the dayside cusp. It is suggested that Pc1–2 pulsations are generated mainly in the dayside magnetosheath on field lines, crossing the magnetopause and entering in the dayside cusp. The possible mechanism of generation is the ion-cyclotron instability of plasma of finite pressure ( β ⊥ ≳ 1) and with anisotropic temperature ( T ⊥ > T ∥).