Equations for nonlinear MHD convection in shearless magnetic systems

A closed set of reduced dynamic equations is derived that describe nonlinear low-frequency flute MHD convection and resulting nondiffusive transport processes in weakly dissipative plasmas with closed or open magnetic field lines. The equations obtained make it possible to self-consistently simulate transport processes and the establishment of the self-consistent plasma temperature and density profiles for a large class of axisymmetric nonparaxial shearless magnetic devices: levitated dipole configurations, mirror systems, compact tori, etc. Reduced equations that are suitable for modeling the long-term evolution of the plasma on time scales comparable to the plasma lifetime are derived by the method of the adiabatic separation of fast and slow motions.     

Adiabatic separation of motions and reduced MHD equations

A variational method for separating fast and slow motions in quasi-Lagrangian continuous media is proposed, which makes it possible to discard fast stable collective degrees of freedom and to derive simpler (reduced) nonlinear equations describing the adiabatic dynamics of quasi-Lagrangian systems. The method is applied to derive an improved version of the reduced Kadomtsev-Pogutse-Strauss MHD equations that describe the dynamics of a tokamak plasma with steady-state sheared flows, as well as adiabatic equations for two-dimensional modeling of MHD plasma convection near the threshold for flute instability in systems like compact tori. __________ Translated from Fizika Plazmy, Vol. 26, No. 6, 2000, pp. 566–576. Original Russian Text Copyright ￠ 2000 by Pastukhov.     

MHD stability of a collisionless anisotropic plasma in a system with an internal conductor

A study is made of the MHD stability of a collisionless anisotropic-pressure plasma in a nonparaxial magnetic configuration with an internal conductor in cylindrical geometry. A stability criterion for flutelike modes is obtained, and the families of marginally stable profiles of the longitudinal and transverse plasma pressures are calculated by using the Chew-Goldberger-Low anisotropic MHD equations. Possible marginally stable plasma states are considered with allowance for the expected turbulent relaxation and self-organization processes, on the one hand, and isotropization processes, on the other. A stability criterion for Alfvén modes is also derived in the Chew-Goldberger-Low model.     

Plasma convection near the threshold for MHD instability in nonparaxial magnetic confinement systems

Using a highly nonparaxial magnetic confinement system with an internal levitated ring as an example, it is shown that, in a plasma near the threshold for ideal MHD instability, the external heating and the original local dissipative processes may give rise to and maintain self-consistent nonlinear MHD convection, which leads to an essentially nonlocal, enhanced heat transport. A closed set of equations is derived that makes it possible to describe such convective processes in a weakly dissipative plasma with β～1. Numerical simulations carried out with a specially devised computer code demonstrate that the quasisteady regime of nonlinear convection actually exists and that the marginally stable profile of the plasma pressure is maintained. A large amount of data on the structure of the nascent convective flows is obtained and analyzed.     

Stability of Alfvén modes in a collisionless anisotropic plasma confined by a highly curved magnetic field

The stability of Alfvén modes in a collisionless plasma with an anisotropic pressure in a highly curved magnetic field is studied. A linearized equation for describing longitudinally nonuniform MHD perturbations with frequencies below the bounce frequency is derived. In this equation, the perturbations of longitudinal and transverse pressures are calculated using a collisionless kinetic equation. It is shown that longitudinal fluxes of the transverse and longitudinal plasma energies give rise to pressure perturbations different from those in the Chew-Goldberger-Low collisionless hydrodynamics. The corresponding energy principle is constructed. A stability criterion for Alfvén modes is obtained and is found to be more stringent than that in the Chew-Gold-berger-Low model.     

Low-frequency instabilities of collisionless plasma and the 16-moment approximation

Using the 16-moment equations that take into account heat fluxes in anisotropic collisionless plasma, the properties of magnetohydrodynamic (MHD) instabilities are investigated. For all instabilities occurring in the MHD approach (the normal incompressible firehose instability, the second compressible almost longitudinal firehose instability, and the almost transverse mirror instability of slow magnetosonic modes, as well as thermal instability caused by the heat flux directed along the magnetic field), their kinetic analogs are considered. The kinetic dispersion relation in the low-frequency range in the vicinity of the ion thermal velocity is analyzed. The flow of plasma ions along the magnetic field is taken into account. The thresholds and instability growth rates obtained in the MHD and kinetic approaches are found to be in good agreement. This indicates that the 16-moment MHD equations adequately describe the dynamics of collisionless plasma.     

On the theory of Cherenkov emission in a plasma

The Cherenkov emission of transverse-longitudinal waves in an anisotropic plasma is considered by applying a Hamiltonian method and by drawing an analogy between the equations for the Cherenkov emission of purely transverse and purely longitudinal waves in isotropic media and the equations for the emission of transverse-longitudinal electromagnetic waves in a highly anisotropic medium (a magnetized plasma). A formula for the emitted power is derived, as well as an expression for the directional pattern of the emitted waves in an anisotropic plasma.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

Formation of self-consistent pressure profiles in simulation of turbulent convection in tokamak plasmas

The formation of pressure profiles in turbulent tokamak plasmas in ohmic heating regimes and transient regimes induced by turning-on of electron-cyclotron resonance (ECR) heating is investigated. The study is based on self-consistent modeling of low-frequency turbulent plasma convection described by an adiabatically reduced set of hydrodynamic-type equations. The simulations show that, in the ohmic heating stage, turbulence forms and maintains profiles of the total plasma pressure corresponding to turbulentrelaxed states. These profiles are close to self-consistent profiles of the total plasma pressure experimentally observed on the T-10 tokamak in ohmic regimes with different values of the safety factor q _L at the limiter. Simulations of nonstationary regimes induced by turning-on of on- and off-axis ECR heating show that the total plasma pressure profiles in the transient regimes remain close to those in the turbulent-relaxed state, as well as to the profiles experimentally observed on T-10.     

MHD stability condition of an anisotropic-pressure plasma in axially symmetric confinement systems formed by a poloidal field

A set of integrodifferential (over the longitudinal coordinate) equations for the transverse components of the plasma displacement minimizing the Kruskal-Oberman functional of the potential energy of MHD perturbations is derived. The stability condition corresponds to the absence of negative eigenvalues of this set for any magnetic surface in plasma.     

Longitudinal structure of MHD perturbations at the boundary of convective stability in the Kruskal-Oberman model

It is shown that, in contrast to the MHD model, a perturbation at the boundary of convective stability of a finite-pressure plasma in confinement systems without an averaged minB in the Kruskal-Oberman model is not generally a purely flute one. The reasons for this discrepancy are clarified. The analysis is carried out for axisymmetric configurations formed by a poloidal magnetic field.     

Nonlinear theory of coupled waves exemplified by beam-plasma instability

The nonlinear stage of instability of an annular electron beam spatially separated from an annular plasma is investigated. The equations describing coupled waves for an arbitrary ratio between the beam and plasma densities are derived. It is shown that instability saturates at distances on the order of several inverse spatial growth rates. The saturation is caused by relativistic nonlinearity, generation of the second harmonic, and low-frequency modulation of the electromagnetic field. At larger distances, resonant generation of low-frequency beam oscillations becomes a dominant factor. In the case of a low-density beam, an expression for the maximum power of the generated plasma wave is obtained in an explicit form.     

Influence of the Hall effect on the plasma dynamics in an MTF/MAGO chamber

The role of the Hall effect in experiments on the magnetic implosion of a D-T plasma in a cylindrical MTF/MAGO chamber fed from a helical explosive magnetic generator is investigated. The plasma dynamics is simulated numerically by a 2D code developed for solving the set of MHD equations with account of the Hall effect. In simulations, the generator, the break switch, and other units were replaced with LR circuits. It is shown that taking into account the Hall effect provides better agreement between numerical simulations and experimental data.     

MHD Waves and Instabilities in Two-Component Anisotropic Plasma

Plasma Physics Reports - Based on the 16-moment MHD transport equations, the propagation of linear waves in an anisotropic homogeneous cosmic plasma is considered. A general dispersion relation is...     

Stabilization of ballooning modes by nonparaxial cells

An analysis is made of the effect of high-curvature stabilizing nonparaxial elements (cells) on the MHD plasma stability in open confinement systems and in confinement systems with closed magnetic field lines. It is shown that the population of particles trapped in such cells has a stabilizing effect not only on convective (flute) modes but also on ballooning modes, which govern the maximum possible β value. In the kinetic approach, which distinguishes between the effects of trapped and passing particles, the maximum possible β values consistent with stability can be much higher than those predicted by the MHD model.     

Modeling of the tearing instability in unreduced two-fluid magnetohydrodynamics

The problem of the tearing instability is solved numerically in cylindrical geometry by using the unreduced two-fluid MHD model. It is shown that the duration of the nonlinear stage of the tearing instability in a hot plasma is rather sensitive to such factors as the initial radial density and temperature profiles, the initial ion-to-electron pressure ratio, and the longitudinal thermal conductivity. Depending on these factors, the two-fluid effects (primarily, the Hall effect) can either greatly hasten the magnetic reconnection process (in comparison to that in the one-fluid MHD model) or greatly slow it. An illustrative explanation of the results obtained is given.     

Constitutive stress--strain relations for the myocardium in diastole

The importance of stress-strain myocardial constitutive relations is that they provide a criterion for behavior in vivo. Our purpose was to develop constitutive equations which are valid in diastole. The myocardium was assumed to be composed of a nonlinear viscoelastic, inhomogeneous, anisotropic (transversely isotropic) and incompressible material operating under adiabatic and isothermal conditions. The expressions contain five moduli. Two are fixed by the restriction of incompressibility, one is estimated, the remaining two refer to directions along and perpendicular to a fiber. Both possess a bimodal variation with intermodal switching occurring in late rapid filling and diastasis. They are functions of time and material constants. These constants can be observed. A dynamic test is suggested. Constitutive statements complete a set of equations sufficient for the solution of a class of boundary value problems. One type is formulated. They also permit the determination of stress from measured strain. Examples are given.     

Reduced MHD equations with allowance for slow magnetosonic waves

The so-called reduced magnetohydrodynamics, which deals with the motion of incompressible fluids and is usually applied to describe plasma flows in a strong toroidal magnetic field, has a number of drawbacks and, in some cases, fails to produce correct results. The equations proposed here are simpler than the original MHD equations and are free of these drawbacks. These equations, like reduced MHD equations, make it possible to remove from consideration fast magnetosonic waves and to introduce the vector potential for the poloidal magnetic field. However, our equations differ from the reduced MHD equations in that they completely incorporate slow magnetosonic waves, the specific features of the toroidal geometry, and the effects of the toroidal velocity.