Generation of compressible modes in MHD turbulence

Astrophysical turbulence is magnetohydrodynamic (MHD) in nature. We discuss fundamental properties of MHD turbulence and in particular the generation of compressible MHD waves by Alfvénic turbulence and show that this process is inefficient. This allows us to study the evolution of different types of MHD perturbations separately. We describe how to separate MHD fluctuations into three distinct families: Alfvén, slow, and fast modes. We find that the degree of suppression of slow and fast modes production by Alfvénic turbulence depends on the strength of the mean field. We review the scaling relations of the modes in strong MHD turbulence. We show that Alfvén modes in compressible regime exhibit scalings and anisotropy similar to those in incompressible regime. Slow modes passively mimic Alfvén modes. However, fast modes exhibit isotropy and a scaling similar to that of acoustic turbulence both in high and low plasmas. We show that our findings entail important consequences for star formation theories, cosmic ray propagation, dust dynamics, and gamma ray bursts. We anticipate many more applications of the new insight to MHD turbulence and expect more revisions of the existing paradigms of astrophysical processes as the field matures. PACS 47.65.+a; 52.30.Cv; 52.35.Ra; 95.30.Qd     

Flow stability of a conducting liquid flowing down an inclined plane in the presence of a magnetic field

The stability of a steady flow of incompressible, conducting liquid down an inclined plane in the presence of longitudinal and transverse magnetic fields is studied. Solutions of the linearized magnetohydrodynamic equations with corresponding boundary conditions are found on the assumption that the Reynolds number R_g and the wave number are small. It is shown that the longitudinal magnetic field plays a stabilizing role. It is known [1] that the flow of a viscous liquid over a vertical wall is always unstable. In this article it is shown that the instability effect at small wave numbers may be eliminated if the longitudinal magnetic field satisfies the conditions found. The case when the Alfvén number and the wave number are small and the Reynolds number is finite is also examined.     

Gasdynamic Analogies in Problems of the Oblique Interaction of MHD Shock Waves

Gasdynamic analogies are constructed for the oblique interaction of MHD shock waves (counter colliding or overtaking). These analogies fairly adequately describe the complex dependences of the gas dynamic parameters of the medium on the magnetic field strength and inclination. The complete gas dynamic analogy in which the MHD interaction is simulated by the interaction of two gas dynamic shock waves with Mach numbers calculated on the basis of the fast magnetosonic speeds adequately describe the state of the medium for weak and moderate magnetic fields. The hybrid model, in which the state behind the interacting shock wave is calculated from the MHD relations on discontinuities and the gas dynamic analogy is then used, gives satisfactory results in a stronger field.     

Structure and stability of quasiparallel small-amplitude magnetohydrodynamic shocks

The structure and stability of quasiparallel magnetohydrodynamic shock waves of small but finite amplitude are investigated. Only those waves whose propagation velocities are close to the Alfvén velocity are considered, i.e., fast shock waves in a medium in which the Alfvén velocity is greater than the speed of sound and slow shock waves in a medium in which the Alfvén velocity is less than the speed of sound and, moreover, intermediate (nonevolutionary) shock waves.In conclusion, the author wishes to thank A. A. Barmin for discussing his results and offering useful comments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 153–160, July–August, 1989.     

Steady flow of a liquid metal in a rectangular MHD channel

The known experimental studies of steady flows of a liquid metal in magnetohydrodynamic (MHD) channels of rectangular section [1–4] were performed only for a few values of the Reynolds number, which does not permit a clear delineation of the fundamental governing laws of the flow in the zone of transition from laminar to turbulent flow. In addition, the study of turbulent MHD flows has been limited to two-dimensional channels.Below we present some results of experimental studies of the effect of a transverse magnetic field on the resistance coefficient for mercury flow in an MHD channel with side ratio 1 to 2.5. The choice of a channel with this side ratio was dictated by the need for studies of the intermediate case between flows in two-dimensional and square channels, which differ significantly from one another because of the different effect of the walls parallel to the magnetic field. In our studies, for each value of the Hartmann number the investigations were made for 30–50 values of the Reynolds number.Notation B₀ flux density of the applied magnetic field - M Hartmann number - R Reynolds number - _tm resistance factor of turbulent MHD flow - _* critical value of the resistance factor - geometric parameter of channel - the component of resistance factor in ordinary hydrodynamics due to pulsations - normed function - electric conductivity of metal - viscosity of metal - R₀ shydraulic radius - N smagnetic field parameter     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Motion of a conductive piston in a channel with variable inductance

The motion of a conductive piston in the channel of a magnetohydrodynamic (MHD) generator of the conduction type with compound electrodes is considered. Formulas are obtained for calculation of the energy characteristics of the pulse MHD generator for various operational regimes. It is shown that in an MHD generator at magnetic Reynolds number values Re_m = ₀u₀ 1 (where ₀ is the permeability of a vacuum, is the electrical conductivity of the piston, u₀ is the initial velocity, and is the characteristic dimension), the energy transferred to an ohmic load may significantly exceed the values obtained in [1, 2]. Conditions for high-efficiency transformation of piston kinetic energy to electrical energy are considered for limiting values of the ratio of the latter to initial magnetic field energy in the generator channel.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 41–46, November–December, 1973.The authors thank V. I. Yakovlev for his helpful evaluation.     

倾斜非均匀磁场下导电方管磁流体管流的数值模拟研究

热核聚变反应堆液态金属包层应用中的一个重要问题是液态金属在导电管中流动和强磁场相互作用产生的额外的磁流体动力学压降.这种磁流体动力学压降远远大于普通水力学压降.美国阿贡国家实验室ALEX研究小组,对非均匀磁场下导电管中液态金属磁流体动力学效应进行了实验研究,其实验结果成为液态金属包层数值验证的标准模型之一.液态金属包层在应用中会受到不同方向的磁场作用,本文以ALEX的非均匀磁场下导电方管中液态金属管流实验中的一组参数为基础,保持哈特曼数、雷诺数和壁面电导率不变,采用三维直接数值模拟的方法,研究了外加磁场与侧壁之间的倾角对导电方管内液态金属流动的速度、电流和压降分布的影响.研究结果表明:沿流向相同横截面上的速度、电流以及压力分布均随磁场的倾斜而同向旋转.倾斜磁场均匀段,横截面上的高速区位于平行磁场方向的哈特曼层和平行层交叉位置,压力梯度随磁场倾角的增大先增大后减小.倾斜磁场递减段,在三维磁流体动力学效应作用下,横截面上的高速射流位置向垂直磁场方向偏移.磁场递减段的三维磁流体动力学压降随磁场倾角的增大而增大.随磁场倾斜,截面上的射流峰值逐渐减小,二次流增强,引发层流向湍流的转捩.      

The magnetic field in inhomogeneous turbulent flow

We consider a particular model of magnetohydrodynamic turbulents. The most fundamental assumption we make is that the velocity correlation time is negligible. By using a selective summation of the perturbation theory series an exact equation for the magnetic field is obtained when the mean square value of the velocity depends on coordinates, i.e., when the turbulence isinhomogeneous. The result makes it possible to obtain the macroscopic Maxwell's equations, i.e., the equations for the large-scale components of the electromagnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 12–18, January–February, 1971.In conclusion, the author thanks R. Z. Sagdeev and V. E. Zakharov for a discussion of the results.     

Field-aligned flow of a conducting fluid past a source of magnetism

Summary A prescribed source of magnetism moves at constant speed through a viscous conducting incompressible fluid with an aligned uniform magnetic field. The velocity and magnetic fields induced at a distance from the source are calculated. The induced fields are also calculated for the case in which the applied field is absent. Although no special symmetry or alignment is assumed, the source is ideal in the sense that enclosures (wires or magnets) are infinitesimal in at least two dimensions. Dynamical interactions will occur in a viscous fluid and their effect in the far field is estimated.As a consequence of finite conductivity and viscosity, the usual wakes are present which trail or lead the source depending upon the sign of (1–A ²), where A is the ratio of the source speed to the Alfvén speed in the undisturbed fluid. Outside the wake the total perturbation magnetic field due to the source is the static field plus a monopole field, divided by (1–A ²).An estimate is also made of the rate at which energy is dissipated as a consequence of viscous interactions and ohmic heating throughout the fluid, outside the immediate vicinity of the source.Geo-Astrophysics Laboratory.Plasma Physics Laboratory.     

Dissipative instability of a nonisothermal electrically conducting flow between parallel plates in a transverse magnetic field

Problems of dissipative instability (in particular, overheating) in magnetohydrodynamies has been studied in [1–6]. The Leontovieh mechanism of overheating instability is explained in [I] by the example of a stationary homogeneous plasma in a strong magnetic field along which current flows. The rate of buildup of perttbations is estimated in [2] to explain the effect of overheating instability on the operation of an MHD generator. The effect of inhomogeneity in the temperature field and in the boundaries of the region on the formarion of this instability has been studied by the example of discharge in a stationary medium in the absence of a magnetic field [3], Certain cases of overheating instability in magnetohydrodynamies are considered in [4, 6], where it is shown that it can be aperiodic as well as oseillatery (Alfven and acoustic waves). Finally, the hydro-dynamic and overheating branches of instability in the ease of non-isothermal plasma flow in a plane MHD channel was investigated in [6]. But the overheating instability was examined without allowance for the dependence of the viscosity and thermal-conductivity coefficients on temperature in the limiting case S R_m 1 and only for small perturbation wavelengths. The development of shortwave perturbations is studied below with allowance for viscosity and thermal conductivity and for a wider range of conditions A 1. Overheating instability over the entire range of wavelengths for the ease considered in [6] is also studied.The author thanks Yu. M. Zolotaikin for programming and performing the calculations.     

Steady-State MHD Flows in Nozzles with an External Longitudinal Magnetic Field

Steady-state MHD flows in channels of the nozzle type in the presence of an external longitudinal magnetic field can be divided into two significantly different classes. Subcritical flows, in which the Alfvén velocity calculated from the longitudinal magnetic field is less than the plasma velocity, have mainly the same properties as flows in a transverse magnetic self-field and their quantitative characteristics depend only slightly on the longitudinal magnetic field strength. Supercritical flows with the opposite inequality for the velocities correspond to strong longitudinal magnetic field. The main difference is the transitions between different forms of energy (kinetic, thermal, and electromagnetic). The present study contains a classification of possible flows, namely, sub- and supercritical and sub-, super-, and transonic flows with respect to the fast and slow magnetosonic and Alfvén velocities. Examples of these flows are given. The effect of the problem parameters on the flow properties is investigated.     

Channel flow of an anisotropically conducting medium in a zone of entry into a magnetic field

A considerable number of papers are devoted to the problem of the deformation of a plane flow of a conducting liquid moving through a channel |x| < , 0 y h=const in a zone of entry into a magnetic field B=(0, 0, B. (x)), where (x) is the Heaviside unit function((x)=0 for x < 0 and (x)=i for x < 0). Apparently the first paper in this direction was that of Shercliff [1, 2] in which the asymptotic (for x .o- )profile of a perturbed velocity was. determined for a flow of an isotropic conducting liquid in a channel with nonconducting walls. The flow considered by Shemliff takes place in magnetohydrodynarnic flowmeters. Complete calculation of the perturbed flow of an isotropie conducting liquid in the channel of a magnetohydrodynamic generator is carried out in [3]. Asymptotic velocity profiles in the channel of a magnetohydrodynamic generator, with ideally segmented electrodes and the flow of an anisotropically conducting medium along them, were found in [4]. General formulas for the calculation of the asymptotic velocity profile, from the known distribution of the perturbing forces along the channel, are presented in [5]. In [6, 7] the Green function is proposed for the solution of the equation for the stream function of the perturbed flow. Finally, in [8], the solution for the perturbed flow of an anisotropically conducting liquid in a channel with continuous electrodes is described by means of the Green function, and the asymptotic profiles of the velocity are calculated.In this paper the flow of anauisotropically conducting liquid is determined in a channel with ideally segmented electrodes. The solution is set up with the aid of the Fourier method. The resulting series, in which the slowly converging part can be related to the asymptotic profile [4] calculated from the solution of an ordinary differential equation, make it possible to determine the velocity field rapidly. A detailed deformation pattern of the velocity profile is set up. Certain general properties of the flow in a zone of entry into a magnetic field are noted; with the aid of these it is possible to discover the error in the calculations [8].     

Electrosolenoidal flows in a cylindrical cavity

The main difficulties in investigating three-dimensional magnetohydrodynamic (MHD) flows with vorticity arise, first, because it is necessary to solve an independent boundary-value problem in order to find the field of the electromagnetic forces and, second, because the regimes of these flows are strongly nonlinear for the majority of high-power technological MHD processes and a number of natural phenomena. Particular importance attaches to MHD flows generated by the interaction of an electric current applied to the fluid with the magnetic self-field. This class of MHD flows has become known as electrosolenoidal flows [1]. The presence of a definite symmetry in the distribution of the electromagnetic forces and the geometry of the region of the liquid conductor makes it possible to find a solution in self-similar form. The present paper is devoted to exact solutions of the nonlinear equations for axisymmetric electrosolenoidal flows of a conducting incompressible fluid in infinite cylindrical cavities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 48–53, May–June, 1991.     

Performance investigation of plasma magnetohydrodynamic power generator

magnetohydrodynamic (MHD) power generator system involves several subjects such as magnetohydrodynamics, plasma physics, material science, and structure mechanics. Therefore, the performance of the MHD power generator is affected by many factors, among which the load coefficient k is of great importance. This paper reveals the effect of some system parameters on the performance by three-dimensional (3D) numerical simulation for a Faraday type MHD power generator using He/Xe as working plasma. The results show that average electrical conductivity increases first and then decreases with the addition of magnetic field intensity. Electrical conductivity reaches the maximum value of 11.05 S/m, while the applied magnetic field strength is B = 1.75 T. When B > 3T, the ionization rate along the midline well keeps stable, which indicates that the ionization rate and three-body recombination rate (three kinds of particles combining to two kinds of particles) are approximately equal, and the relatively stable plasma structure of the mainstream is preserved. Efficiency of power generation of the Faraday type channel increases with an increment of the load factor. However, enthalpy extraction first increases to a certain value, and then decreases with the load factor. The enthalpy extraction rate reaches the maximum when the load coefficient k equals 0.625, which is the best performance of the power generator channel with the maximum electricity production.     

Solution of magnetohydrodynamic flow in a rectangular duct by Chebyshev collocation method

In this study, matrix representation of the Chebyshev collocation method for partial differential equation has been represented and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Numerical solution of velocity and induced magnetic field is obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. The Chebyshev collocation method is used with a reasonable number of collocations points, which gives accurate numerical solutions of the MHD flow problem. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H≤1000. Copyright © 2010 John Wiley & Sons, Ltd.     

Theory of a magnetohydrodynamic flow regulator for liquid metal

A theory is developed for unsteady flow of liquid metal in a de MHD flow regulator channel for small magnetic Reynolds numbers. It is shown that it is possible to use the quasistationary approximation for calculating the integral flow parameters.MHD methods, along with their applications in direct transformation of thermal energy into electrical energy, in recent years have been ever more widely used in nuclear energy and metallurgy for transporting and for measuring the parameters of liquid metal flows [1, 2]. Complete sealing, operation over a wide temperature range, and simplicity of control and automation are among the unquestionable advantages of all MHD devices.The existing theory of MHD devices is limited primarily to the stationary regimes of operation of electromagnetic pumps and magnetic flow-meters.The most characteristic operating regimes for the MHD regulator are the transient regimes, which are defined primarily by the hydrodynamics of the liquid metal flow.The present article is devoted to the study of the unsteady flow of liquid metal in the channel of a dc MHD flow regulator with independent excitation. The magnetic Reynolds are low (R _m =U ₀ b1).     

Development of magnetohydrodynamic boundary layers

Many authors have studied the problem of the development of a hydrodynamic boundary layer when a body is suddenly set in motion. The results obtained are presented most fully in the monographs of H. Schlichting [1] and L. G. Loitsyanskii [2]. In magnetohydrodynamics the development of the boundary layer over the surface of an infinite flat plate for uniform oncoming flow has been closely studied (for example [3, 4]). Below, the problem of the development of a plane magnetohydrodynamic boundary layer is considered in a different formulation. We shall suppose that the distributions of velocity U(x) and enthalpy h(x) are given along the body contour for t=0. At that moment the viscosity and thermal conductivity mechanisms are instantaneously switched on. Viscous and thermal boundary layers begin to grow in a direction normal to the body. The medium in the boundary layer interacts with the magnetic field. This formulation corresponds to the development of a magnetohydrodynamic boundary layer on a body which is set in motion with a jerk, in the case where the rate of establishment of magnetohydrodynamic flow of the inviscid, thermally nonconducting fluid around the body is much less than the rate of development of the boundary layer. Then U(x) and h(x) are found by solving the problem of stationary magnetohydrodynamic flow of an inviscid thermally nonconducting fluid around a body, or simply the hydrodynamic flow if the medium interacts with the field only in the boundary layer.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

Oblique interaction of alfvén and contact discontinuities in magnetohydrodynamics

A general method of solving problems of the interaction of stationary discontinuities is proposed. The problem of the oblique incidence of an Alfvén plane-polarized discontinuity on a contact discontinuity is examined in the general formulation. A solution is constructed numerically over the entire range of variation of the governing parameters. A number of effects associated with the magnetohydrodynamic nature of the interaction are explored. For example, the formation in space of sectors in which the density falls by several orders (almost to a vacuum) is detected. The solutions obtained are of interest, for example, for investigating the interaction between Alfvén discontinuities in the solar wind and the magnetopause, plasmopause and other inhomogeneities whose boundary can be approximated by a contact discontinuity [13–15].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 131–142, January–February, 1990.