Numerical simulation of plasma motion in a magnetic field. Two-dimensional case

A model of dynamics and heating of a plasma cloud in a magnetic field is considered in a two-temperature approximation. Based on a predictor-corrector-type implicit difference scheme, spreading of a plasma cloud in an external magnetic field is numerically simulated, and the influence of this field on spread dynamics is evaluated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 121–132, May–June, 2007.     

Simulation of the motion and heating of an irregular plasma

A physicomathematical model for plasma heating and confinement is formulated on the basis of some assumptions on the behavior of a dense plasma cloud in a magnetic field. The model allows for the ionization and heating of the plasma cloud by the surrounding deuterium plasma due to heat conduction and heating by a superthermal electron current. The expansion of a plasma cloud in an external magnetic field is studied using some simplifications in a magnetohydrodynamic approximation. Plasma heating is modeled by an external source. The basic equations include continuity, motion, energy, and magneticfield equations. For numerical solution of the problem, we developed a finitedifference scheme of the type of a universal algorithm with splitting into physical processes and spatial directions, which allowed us to obtain separate solutions of the equations of magnetic induction and gas dynamics. Calculations of the propagation of a plasma cloud heated by a source in an external magnetic field were performed. The mechanism of the effect of the magnetic field and heat source on plasma cloud expansion is determined. The results agree qualitatively with experimental data.     

Effect of viscosity on the current layers emerging upon propagation of the Alfvén pulse in a hyperbolic magnetic field

The propagation of the Alfvén pulse in the vicinity of the X-point in the presence of viscosity is studied for the first time. It is shown that, in contrast to the case of magnetosonic perturbation, where the dynamic viscosity η (the point is that we are dealing with dimensionless quantities), which is small compared to the magnetic plasma viscosity ν, does not affect the flow, this influence is of primary importance in the Alfvén case. The magnitude of the steady-state current density is proportional to (vνη)^-1/4. It is also shown that at large times the distribution of the z-component of a magnetic field that is close to the distribution obtained in solving a linear problem is established in this significantly nonlinear problem. The effect of the heat conduction on this process is studied. Institute of Computational Technologies, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 11–16, November–December, 1999.     

Effect of viscosity on the propagation of nonlinear Alfvén waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

It is proposed to consider the propagation of surface waves along a tangential magnetohydrodynamic discontinuity in the particular case where the fluid velocities on both sides of the interface are equal to zero. In [1] it was shown that waves called surface Alfvén waves may be propagated along the surface separating a semi-infinite region without a field from a region with a uniform magnetic field. The linear theory of surface Alfvén waves in a compressible medium was considered in [2]. In [3] the damping of surface Alfvén waves as a result of viscosity and heat conduction was investigated. The propagation of low-amplitude nonlinear surface Alfvén waves in an incompressible fluid in the absence of dissipative processes is described by the integrodifferential equation obtained in [4]. By means of a numerical solution of this equation it was shown that a perturbation initially in the form of a sinusoidal wave will break. The breaking time was determined. In this paper the equation derived in [4] is extended to the case of a viscous fluid. It is shown that the equation obtained does not have steady-state solutions. The propagation of periodic disturbances is investigated numerically. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–104, November–December, 1986. The author wishes to thank L. S. Fedorov for assisting with the calculations.     

Vlasov–Maxwell–Boltzmann Diffusive Limit

Inspired by the work (Bastea et al. in J Stat Phys 1011087–1136, 2000) for binary fluids, we study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov–Maxwell–Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDEs, the incompressible Vlasov–Navier–Stokes–Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such an expansion up to any order.     

Numerical Simulation of Plasma Dynamics in a Nonuniform Magnetic Field

An efficient algorithm is proposed enabling numerical simulations of plasma dynamics in a nonuniform magnetic field. The present numerical data are in good agreement with experimental data obtained in a GOL-3 setup and with previous simulations. The experimentally observed effect of fast transfer of energy to ions is confirmed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 35–45, January–February, 2006.     

An investigation of the nonsteady interaction of a conducting gas cloud with a given electrical circuit

The behavior of a cloud of conducting gas obtained from a coaxial plasma gun is investigated as it passes through a constant magnetic field. The way that this cloud interacts by induction with an electrical circuit coupled to an ohmic resistance is also studied. Paticular attention is paid to the study of the energy characteristics of the interaction (the energy generated in the ohmic resistance, relations between the plasma energy and Joule dissipation) as a function of the geometry and certain parameters of the electrical circuit. The process is analyzed theoretically for small values of the magnetic Reynolds number. Experimental and theoretical results are compared.     

Singular Limits of the Equations of Magnetohydrodynamics

This paper studies the asymptotic limit for solutions to the equations of magnetohydrodynamics, specifically, the Navier–Stokes–Fourier system describing the evolution of a compressible, viscous, and heat conducting fluid coupled with the Maxwell equations governing the behavior of the magnetic field, when Mach number and Alfvén number tends to zero. The introduced system is considered on a bounded spatial domain in \mathbbR³{\mathbb{R}^{3}}, supplemented with conservative boundary conditions. Convergence towards the incompressible system of the equations of magnetohydrodynamics is shown.     

Convection of a ferrofluid in an alternating magnetic field

Parametric convective instability of a horizontal layer of a homogeneous ferrofluid under the action of an alternating magnetic field is studied. A case with rigid boundaries is considered. Convection thresholds are found. In an alternating magnetic field with a zero mean value, perturbations are found to have a synchronous character. These perturbations, however, can belong to different classes, because they depend on the temperature difference on the layer boundaries, the layer thickness, the frequency and amplitude of the alternating external field, and the physical properties of the ferrofluid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 18–27, July–August, 2009.     

Poincaré’s Variational Problem in Potential Theory

One of the earliest attempts to rigorously prove the solvability of Dirichlet’s boundary value problem was based on seeking the solution in the form of a “potential of double layer”, and this leads to an integral equation whose kernel is (in general) both singular and non-symmetric. C. Neumann succeeded with this approach for smoothly bounded convex domains, and H. Poincaré, by a tremendous tour de force, showed how to push through the analysis for domains with sufficiently smooth boundaries but no hypothesis of convexity. In this work he was (according to his own account) guided by consideration of a variational problem involving the partition of energy of an electrostatic field induced by charges placed on the boundary of a domain, more precisely the charge distributions which render stationary the energy of the field inside the domain divided by the energy of the field outside the domain. Unfortunately, a rigorous treatment of this problem was not possible with the tools available at that time (as Poincaré was well aware). So far as we know, the only one to propose a rigorous treatment of Poincaré’s problem was T. Carleman (in the two-dimensional case) in his doctoral dissertation. Thanks to later developments (especially concerning Sobolev spaces, and spectral theory of operators on Hilbert space) we can now give a complete, general and rigorous account of Poincaré’s variational problem, and that is the main object of the present paper. As a by-product, we refine some technical aspects in the theory of symmetrizable operators and prove in any number of dimensions the basic properties of the analogue of the planar Bergman–Schiffer singular integral equation. We interpret Poincaré’s variational principle as a non-selfadjoint eigenvalue problem for the angle operator between two distinct pairs of subspaces of potentials. We also prove a series of novel spectral analysis facts (some of them conjectured by Poincaré) related to the Poincaré–Neumann integral operator.     

'T-RANS' Simulation of Deterministic Eddy Structure in Flows Driven by Thermal Buoyancy and Lorentz Force

The paper reports on the application of the Time-dependent Reynolds-Averaged Navier–Stokes (T-RANS) approach to analysing the effects of magnetic force and bottom-wall configuration on the reorganisation of a large coherent structure and its role in the transport processes in Rayleigh–Bénard convection. The large-scale deterministic motion is fully resolved in time and space, whereas the unresolved stochastic motion is modelled by a `subscale' model for which the conventional algebraic stress/flux expressions were used, closed with the low-Re number (k)-(ε)-(θ²) three-equation model. The applied method reproduces long-term averaged mean flow properties, turbulence second moments, and all major features of the coherent roll/cell structure in classic Rayleigh–Bénard convection in excellent agreement with the available DNS and experimental results. Application of the T-RANS approach to Rayleigh–Bénard convection with wavy bottom walls and a superimposed magnetic field yielded the expected effects on there organisation of the eddy structure and consequent modifications of the mean and turbulence parameters and wall heat transfer. This revised version was published online in August 2006 with corrections to the Cover Date.     

Modeling Effective Interface Laws for Transport Phenomena Between an Unconfined Fluid and a Porous Medium Using Homogenization

In this chapter of the special issue of the journal “Transport in Porous Media,” on the topic “Flow and transport above permeable domains,” we present modeling of flow and transport above permeable domains using the homogenization method. Our goal is to develop a heuristic approach which can be used by the engineering community for treating this type of problems and which has a solid mathematical background. The rigorous mathematical justification of the presented results is given in the corresponding articles of the authors. The plan is as follows: We start with the section “Introduction” where we give an overview and comparison with interface conditions obtained using other approaches. In Sect. 2, we give a very short derivation of the Darcy law by homogenization, using the two-scale expansion in the typical pore size parameter ε. It gives us the definition of various auxiliary functions and typical effective properties as permeability. In Sect. 3, we introduce our approach to the effective interface laws on a simple 1D example. The approximation is obtained heuristically using the two steps strategy. For the 1D problem we calculate the approximation and the effective interface law explicitly and show that it is valid at order O(ε ²). Next, in Sect. 4 we give a derivation of the Beavers–Joseph–Saffman interface condition and of the pressure jump condition, using homogenization. We construct the corresponding boundary layer and present a heuristic calculation, leading to the interface law and being based on the rigorous mathematical result. In addition, we show the invariance of the law with respect to the small variations in the choice of the interface position. Finally, there is a short concluding section. The research of A.M. was partially supported by the GDR MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN).     

Transformation of energy and composition of gas mixtures in a collision of rarefied supersonic flows

Transformation of energy and composition of gas mixtures in a convergent flow from a strip source on a cylindrical surface toward the axis is systematically studied with the use of the direct simulation Monte Carlo method. Information on the influence of gas-flow rarefaction and system geometry on the temperature and concentration of the heavy gas in a dense cloud formed on the axis is obtained. The use of convergent supersonic flows is demonstrated to offer new possibilities for research in the field of physical gas dynamics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 198–204, March–April, 2009     

Influence of the hall effect on plasma compressibility in a strong toroidal magnetic field

The development of the tearing instability is studied in the presence of a high toroidal magnetic field and a high plasma conductivity. The variation of the plasma density is shown to be significant in this case. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 3–9, March–April, 1998.     

Direct Conversion of the Energy of Laser and Fusion Plasma Clouds to Electrical Energy During Expansion in a Magnetic Field

The paper deals with the physical and electrotechnical principles of the promising method of direct conversion of the kinetic energy of an expanding plasma cloud to electrical energy by inductive generation of currents in short–circuited load coils that enclose the plasma and are oriented across the external magnetic field. An analysis of plasma deceleration by a magnetic field and transfer of plasma energy to an inductive load gave a solution of the problem in general form and the dimensionless parameters of the problem that determine the deceleration radius, the coil current, and the theoretical conversion efficiency. The role of the basic physical effects, including parasitic ones (plasma instabilities and Joule heating), influencing the real efficiency is assessed. A comparison of the results with data of experiments with laser–produced plasma clouds on a KI–1 facility and with available numerical results shows that in the optimized version of the method for conversion of inertial confinement fusion energy, a 30% efficiency can be achieved.     

Influence of a homogeneous axial magnetic field on Taylor–Couette flow of ferrofluids with low particle–particle interaction

Experimental results concerning the stability of Couette flow of ferrofluids under magnetic field influence are presented. The fluid cell of the Taylor–Couette system is subject to a homogeneous axial magnetic field and the axial flow profiles are measured by ultrasound Doppler velocimetry. It has been found that an axial magnetic field stabilizes the Couette flow. This effect decreases with a rotating outer cylinder. Moreover, it could be observed that lower axial wave numbers are more stable at a higher axial magnetic field strength. Since the used ferrofluid shows a negligible particle–particle interaction, the observed effects are considered to be solely based on the hindrance of free particle rotation.     

Optimization of a ribbon diode with magnetic insulation for increasing the current density in a high-current relativistic electron beam

The geometry of the ribbon diode of the U-2 accelerator is optimized to increase both the current density and the total current of the relativistic electron beam for its subsequent injection into the plasma of a multimirror GOL-3 trap. Beam simulation in the diode was performed using the POISSON-2 applied software modified on the basis of the results obtained using the theory of a planar diode in an inclined magnetic field. As a result of the optimization, the diode geometry and the magnetic field configuration were found that should provide a factor of 1.5–2 increase in the current density in experiments with a small angular divergence of electron velocities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 25–35, May–June, 2009.     

Convex Sobolev Inequalities Derived from Entropy Dissipation

We study families of convex Sobolev inequalities, which arise as entropy–dissipation relations for certain linear Fokker–Planck equations. Extending the ideas recently developed by the first two authors, a refinement of the Bakry–émery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry–émery criterion fails. The main application of our theory concerns the linearized fast diffusion equation in dimensions d ≧ 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating convex Sobolev inequalities, and prove that these bounds are sharp on a specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.     

Numerical investigation of the expansion dynamics and generation of magnetic fields in plasma jets

The processes of interaction between concentrated energy fluxes and solid targets have been investigated in a number of studies. The generation of magnetic fields in erosional plasma formations has been experimentally observed [1–4]. However, the evolution of magnetic fields in plasma jets has not yet been studied in sufficient detail. This paper is devoted to a numerical investigation of unsteady three-dimensional erosion plasma flows and the generation in those flows of magnetic fields resulting from the action of laser radiation on solid targets. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–140, November–December, 1986. The authors are grateful to I. V. Nemchinov and B. T. Fedyushin for useful discussion of their results.     

Effects of heat generation on g-Jitter induced natural convection flow in a channel with isothermal or isoflux walls

This paper discusses the behavior of g-jitter induced free convection in microgravity under the influence of a transverse magnetic field and in the presence of heat generation or absorption effects for a simple system consisting of two parallel impermeable infinite plates held at four different thermal boundary conditions. The governing equations for this problem are derived on the basis of the balance laws of mass, linear momentum, and energy modified to include the effects of thermal buoyancy, magnetic field and heat generation or absorption as well as Maxwell's equations. The fluid is assumed to be viscous, Newtonian and have constant properties except the density in the body force of the balance of linear momentum equation. The governing equations are solved analytically for the induced velocity and temperature distributions as well as for the electric field and total current for electrically-conducting and insulating walls. This is done for isothermal–isothermal, isoflux–isothermal, isothermal–isoflux and isoflux–isoflux thermal boundary conditions. Graphical results for the velocity amplitude and distribution are presented and discussed for various parametric physical conditions.