SOME EXACT SOLUTIONS IN LINEARIZED MAGNETO-AERODYNAMICS FOR ARBITRARY MAGNETIC REYNOLDS NUMBERS
Aerodynamic linearized systems for magnetic reynolds numbers
MAGNETOHYDRODYNAMICS↗
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Large-Eddy Simulation of Conductive Flows at Low Magnetic Reynolds Number
In this paper we study the LES method with dynamic procedure in the context of conductive flows subject to an applied external magnetic field at low magnetic Reynolds number R(sub m). These kind of flows are encountered in many industrial applications. For example, in the steel industry, applied magnetic fields can be used to damp turbulence in the casting process. In nuclear fusion devices (Tokamaks), liquid-lithium flows are used as coolant blankets and interact with the surrounding magnetic field that drives and confines the fusion plasma. Also, in experimental facilities investigating the dynamo effect, the flow consists of liquid-sodium for which the Prandtl number and, as a consequence, the magnetic Reynolds number is low. Our attention is focused here on the case of homogeneous (initially isotropic) decaying turbulence. The numerical simulations performed mimic the thought experiment described in Moffatt in which an initially homogeneous isotropic conductive flow is suddenly subjected to an applied magnetic field and freely decays without any forcing. Note that this flow was first studied numerically by Schumann. It is well known that in that case, extra damping of turbulence occurs due to the Joule effect and that the flow tends to become progressively independent of the coordinate along the direction of the magnetic field. Our comparison of filtered direct numerical simulation (DNS) predictions and LES predictions show that the dynamic Smagorinsky model enables one to capture successfully the flow with LES, and that it automatically incorporates the effect of the magnetic field on the turbulence. Our paper is organized as follows. In the next section we summarize the LES approach in the case of MHD turbulence at low R(sub m) and recall the definition of the dynamic Smagorinsky model. In Sec. 3 we describe the parameters of the numerical experiments performed and the code used. Section 4 is devoted to the comparison of filtered DNS results and LES results. Conclusions are presented in Sec. 5.
Knaepen, B.↗
Boundary-value problem for plasma centrifuge at arbitrary magnetic Reynolds numbers
We solve in closed form the boundary-value problem for the partial differential equations which describe the (azimuthal) rotation velocity and induced magnetic fields in a cylindrical plasma centrifuge with ring electrodes of different radii and an external, axial magnetic field. The electric field, current density, and velocity distributions are discussed in terms of the Hartmann number H and the magnetic Reynolds number R. For small Hall coefficients, the induced magnetic field does not affect the plasma rotation. As a result of the Lorentz forces, the plasma rotates with speeds as high as 100,000 cm/sec around its axis of symmetry at typical conditions, so that the lighter (heavier) ion and atom components are enriched at (off) the center of the discharge cylinder.
Wilhelm, H. E.↗
A high magnetic Reynolds number dynamo
A boundary-layer solution to a high magnetic Reynolds number R periodic dynamo model shows that: (1) flux expulsion forces the magnetic field into flux sheets; (2) the principal contribution to the alpha effect arises from regions of flow stagnation along a flux sheet; and (3) the alpha effect scales as R exp-1/2. Arguments for these effects persisting in turbulent dynamos are given.
Perkins, F. W.↗
MHD Turbulence at Moderate Magnetic Reynolds Number
In the present article we will consider the decay of MHD turbulence under the influence of a strong external magnetic field at moderate magnetic Reynolds numbers. Typical values of R(sub m) that are considered here range from R(sub m) approx. 0.1 to R(sub m) approx. 20. As a comparison, the initial kinetic Reynolds number common to all our simulations is Re(sub L) = 199. This means that the range of Prandtl numbers explored is 5 x 10(exp -4) to 10(exp -1). Our motivation is mainly to exhibit how the transition from the QS approximation to FMHD occurs. At the lowest values of R(sub m) studied here, the QS approximation is shown to model the flow faithfully. However, for the higher values of R(sub m) considered, it is clearly inadequate but can be replaced by another approximation which will be referred to as the Quasi-Linear (QL) approximation. Another objective of the present study is to describe how variations in the magnetic Reynolds number (while maintaining all other parameters constant) affect the dynamics of the flow. This complements past studies where variations in either the strength of the external magnetic field or the kinetic Reynolds number were considered. This article is organized as follows. In section 2 we recall the definition of the quasi-static approximation. Section 3 is devoted to the description of the numerical experiments performed using the quasi-static approximation and full MHD. In section 4 we describe the quasi-linear approximation and test it numerically against full MHD. A concluding summary is given in section 5.
Knaepen, B.↗
Magnetic field generation at high magnetic Reynolds number
The lowest-order contribution of finite electrical resistivity to the process of magnetic-field regeneration at high magnetic Reynolds number is calculated. It is found that finite resistivity changes the calculated regeneration rate by less than a factor of 2.
Levy, E. H.↗
Anisotropy of MHD Turbulence at Low Magnetic Reynolds Number
Turbulent fluctuations in MHD flows are known to become dimensionally anisotropic under the action of a sufficiently strong magnetic field. We consider the technologically relevant case of low magnetic Reynolds number and apply the method of DNS of forced flow in a periodic box to generate velocity fields. The analysis based on different anisotropy characteristics shows that the dimensional anisotropy is virtually scale-independent. We also find that, except for the case of very strong magnetic field, the flow is componentally isotropic. Its kinetic energy is practically uniformly distributed among the velocity components.
Zikanov, O.↗
Selective decay hypothesis at high mechanical and magnetic Reynolds numbers
Implications of certain applications of turbulence theory to two-dimensional turbulence and magnetohydrodynamic flow are discussed. It is shown that the use of the Navier-Stokes equation (NSE) for measurements of turbulent fluctuations has been effective only for three-dimensional flows. For two-dimensional flows, used for the study of large-scale motions in the atmosphere or ocean, enstrophy is cascaded to high wave numbers and dissipated at a finite rate even at infinite Re. MHD flows are numerically calculated for the two-dimensional case and analytically for the three-dimensional case, for which discrepancies in the relative rates of energy and cross helicity decay lead to a recommendation that numerical calculations for the three-dimensional case be carried out to determine the precise decayed states.
Matthaeus, W. H.↗
Boundary-value problem for a counterrotating electrical discharge in an axial magnetic field
An electrical discharge between two ring electrodes embedded in the mantle of a cylindrical chamber is considered, in which the plasma in the anode and cathode regions rotates in opposite directions under the influence of an external axial magnetic field. The associated boundary-value problem for the coupled partial differential equations describing the azimuthal velocity and radial current-density fields is solved in closed form. The velocity, current density, induced magnetic induction, and electric fields are presented for typical Hartmann numbers, magnetic Reynolds numbers, and geometry parameters. The discharge is shown to produce anodic and cathodic plasma sections rotating at speeds of the order 1,000,000 cm/sec for conventional magnetic field intensities. Possible application of the magnetoactive discharge as a plasma centrifuge for isotope separation is discussed.
Hong, S. H.↗
Magnetic Flux Compression Experiments Using Plasma Armatures
Magnetic flux compression reaction chambers offer considerable promise for controlling the plasma flow associated with various micronuclear/chemical pulse propulsion and power schemes, primarily because they avoid thermalization with wall structures and permit multicycle operation modes. The major physical effects of concern are the diffusion of magnetic flux into the rapidly expanding plasma cloud and the development of Rayleigh-Taylor instabilities at the plasma surface, both of which can severely degrade reactor efficiency and lead to plasma-wall impact. A physical parameter of critical importance to these underlying magnetohydrodynamic (MHD) processes is the magnetic Reynolds number (R(sub m), the value of which depends upon the product of plasma electrical conductivity and velocity. Efficient flux compression requires R(sub m) less than 1, and a thorough understanding of MHD phenomena at high magnetic Reynolds numbers is essential to the reliable design and operation of practical reactors. As a means of improving this understanding, a simplified laboratory experiment has been constructed in which the plasma jet ejected from an ablative pulse plasma gun is used to investigate plasma armature interaction with magnetic fields. As a prelude to intensive study, exploratory experiments were carried out to quantify the magnetic Reynolds number characteristics of the plasma jet source. Jet velocity was deduced from time-of-flight measurements using optical probes, and electrical conductivity was measured using an inductive probing technique. Using air at 27-inHg vacuum, measured velocities approached 4.5 km/s and measured conductivities were in the range of 30 to 40 kS/m.
Turner, M. W.↗
Inductive Measurement of Plasma Jet Electrical Conductivity (MSFC Center Director's discretionary Fund)
Measurement of plasma jet electrical conductivity has utility in the development of explosively driven magnetohydrodynamic (MHD) energy converters as well as magnetic flux compression reaction chambers for nuclear/chemical pulse propulsion and power. Within these types of reactors, the physical parameter of critical importance to underlying MHD processes is the magnetic Reynolds number, the value of which depends upon the product of plasma electrical conductivity and velocity. Therefore, a thorough understanding of MHD phenomena at high magnetic Reynolds number is essential, and methods are needed for the accurate and reliable measurement of electrical conductivity in high-speed plasma jets. It is well known that direct measurements using electrodes suffer from large surface resistance, and an electrodeless technique is desired. To address this need, an inductive probing scheme, originally developed for shock tube studies, has been adapted. In this method, the perturbation of an applied magnetic field by a plasma jet induces a voltage in a search coil, which, in turn, can be used to infer electrical conductivity through the inversion of a Fredholm integral equation of the first kind. A 1-in.-diameter probe using a light-gas gun. Exploratory laboratory experiments were carried out using plasma jets expelled from 15-g shaped charges. Measured conductivities were in the range of 4 kS/m for unseeded octol charges and 26 kS/m for seeded octol charges containing 2-percent potassium carbonate by mass.
Turner, M. W.↗
The growth of the tearing mode - Boundary and scaling effects
A numerical model of resistive magnetic tearing is developed in order to verify and relate the results of the principal approximations used in analytic analyses and to investigate the solutions and their growth-rate scalings over a large range of primary parameters which include parametric values applicable to the solar atmosphere. The computations cover the linear behavior for a variety of boundary conditions, emphasizing effects which differentiate magnetic tearing in astrophysical situations from that in laboratory devices. Eigenfunction profiles for long and short wavelengths are computed and the applicability of the 'constant psi' approximation is investigated. The growth rate is computed for values of the magnetic Reynolds number up to a trillion and of the dimensionless wavelength parameter down to 0.001. The analysis predicts significant effects due to differing values of the magnetic Reynolds number.
Steinolfson, R. S.↗
Neutron star dynamos and the origins of pulsar magnetism
Neutron star convection is a transient phenomenon and has an extremely high magnetic Reynolds number. In this sense, a neutron star dynamo is the quintessential fast dynamo. The convective motions are only mildly turbulent on scales larger than the approximately 100 cm neutrino mean free path, but the turbulence is well developed on smaller scales. Several fundamental issues in the theory of fast dynamos are raised in the study of a neutron star dynamo, in particular the possibility of dynamo action in mirror-symmetric turbulence. It is argued that in any high magnetic Reynolds number dynamo, most of the magnetic energy becomes concentrated in thin flux ropes when the field pressure exceeds the turbulent pressure at the smallest scale of turbulence. In addition, the possibilities for dynamo action during the various (pre-collapse) stages of convective motion that occur in the evolution of a massive star are examined, and the properties of white dwarf and neutron star progenitors are contrasted.
Thompson, Christopher↗
Magnetic reconnection with large separatrix angles
The magnetic reconnection process is studied here using incompressible MHD simulations with different inflow boundary conditions and different magnetic Reynolds numbers R(m). The angle between the magnetic separatrices is in steady state reconnection depends mainly on the normal magnetic field on the inflow boundary. In steady state nonuniform reconnection with large separatrix angles, field-aligned plasma jets appear slightly downstream of the magnetic separatrices. The field-aligned plasma jet are stronger when R(m) is larger. Each field-aligned plasma jet consists of two parts: a slow shock and a fast-mode compressional wave. The slow shock converts the magnetic energy into plasma kinetic energy by acceleration and heating. The fast-mode compressional wave decelerates the plasma to a smaller outflow speed and heats it further. Nearly all the magnetic energy flowing into the diffusion region is converted into other forms. The length and width of the diffusion region depend on the values of the reconnection rate, R(m), and the normal magnetic field on the inflow boundary.
Yan, M.↗
Anisotropy in MHD turbulence due to a mean magnetic field
The development of anisotropy in an initially isotropic spectrum is studied numerically for two-dimensional magnetohydrodynamic turbulence. The anisotropy develops due to the combined effects of an externally imposed dc magnetic field and viscous and resistive dissipation at high wave numbers. The effect is most pronounced at high mechanical and magnetic Reynolds numbers. The anisotropy is greater at the higher wave numbers.
Shebalin, J. V.↗
Anisotropy in MHD turbulence due to a mean magnetic field
The development of anisotropy in an initially isotropic spectrum is studied numerically for two-dimensional magnetohydrodynamic turbulence. The anisotropy develops due to the combined effects of an externally imposed dc magnetic field and viscous and resistive dissipation at high wave numbers. The effect is most pronounced at high mechanical and magnetic Reynolds numbers. The anisotropy is greater at the higher wave numbers. Previously announced in STAR as N83-12998
Shebalin, J. V.↗
Inverse energy cascades in three-dimensional turbulence
Fully three-dimensional magnetohydrodynamic (MHD) turbulence at large kinetic and low magnetic Reynolds numbers is considered in the presence of a strong uniform magnetic field. It is shown by numerical simulation of a model of MHD that the energy inverse cascades to longer length scales when the interaction parameter is large. While the steady-state dynamics of the driven problem is three-dimensional in character, the behavior has resemblance to two-dimensional hydrodynamics. These results have implications in turbulence theory, MHD power generator, planetary dynamos, and fusion reactor blanket design.
Hossain, Murshed↗
Reconnection-driven oscillations in dwarf nova disks
A class of oscillations observed during eruption of dwarf novae has been interpreted as oscillations of the accretion disks in these systems.These oscillations are quasi-periodic with coherence times typically between three and 15 cycles. It is shown that magnetic field reconnection at high magnetic Reynolds number can drive disk oscillations. The expected stochastic geometry of disk magnetic fields could naturally produce the observed phase incoherency.
Tajima, T.↗