Nonlinear Magnetic Diffusion Research Articles

Stability analysis of a class of nonlinear magnetic diffusion equations and its fully implicit scheme

We studied a class of nonlinear magnetic diffusion problems with step-function resistivity $ \eta(e) $ in electromagnetically driven high-energy-density physics experiments. The stability of the nonlinear magnetic diffusion equation and its fully implicit scheme, based on the step-function resistivity approximation model $ \eta_\delta(e) $ with smoothing, were studied. A rigorous theoretical analysis was established for the approximate model of one-dimensional continuous equations using Gronwall's theorem. Following this, the stability of the fully implicit scheme was proved using bootstrapping and other methods. The correctness of the theoretical proof was verified through one-dimensional numerical experiments.

Unconditional superconvergence analysis of an energy-stable C–N fully discrete scheme for the nonlinear magnetic diffusion model with memory

The unconditional superconvergence behavior of an energy-stable Crank–Nicolson (C–N) fully discrete scheme with nonconforming EQ1rot finite element method (FEM) is investigated for the nonlinear integro-differential system associated with the diffusion of the magnetic field. The stability of the numerical solution in the broken H1-norm is proved by the Gagliardo–Nirenberg (G–N) inequality, and the unique solvability is demonstrated by the Brouwer fixed point theorem. In which the boundness of the L∞-norm and the inverse inequality are eliminated. The combination of the Green formula and two special properties of EQ1rot element leads to the unconditional superconvergence estimate without any restriction between the time step and space partition size. Finally, the validity of the theoretical findings is supported by some numerical results.

Turbulent magnetic helicity fluxes in solar convective zone

ABSTRACT Combined action of helical motions of plasma (the kinetic α effect) and non-uniform (differential) rotation is a key dynamo mechanism of solar and galactic large-scale magnetic fields. Dynamics of magnetic helicity of small-scale fields is a crucial mechanism in a non-linear dynamo saturation where turbulent magnetic helicity fluxes allow to avoid catastrophic quenching of the α effect. The convective zone of the Sun and solar-like stars, as well as galactic discs, are the source for production of turbulent magnetic helicity fluxes. In the framework of the mean-field approach and the spectral τ approximation, we derive turbulent magnetic helicity fluxes using the Coulomb gauge in a density-stratified turbulence. The turbulent magnetic helicity fluxes include non-gradient and gradient contributions. The non-gradient magnetic helicity flux is proportional to a non-linear effective velocity (which vanishes in the absence of the density stratification) multiplied by small-scale magnetic helicity, while the gradient contributions describe turbulent magnetic diffusion of the small-scale magnetic helicity. In addition, the turbulent magnetic helicity fluxes contain source terms proportional to the kinetic α effect or its gradients, and also contributions caused by the large-scale shear (solar differential rotation). We have demonstrated that the turbulent magnetic helicity fluxes due to the kinetic α effect and its radial derivative in combination with the non-linear magnetic diffusion of the small-scale magnetic helicity are dominant in the solar convective zone.

MHD instabilities developing in a conductor exploding in the skin effect mode

The results of experiments with exploding copper conductors, performed on the MIG facility (providing currents of amplitude of about 2.5 MA and rise time of 100 ns), are analyzed. With an frame optical camera, large-scale instabilities of wavelength 0.2–0.5 mm were detected on the conductor surface. The instabilities show up as plasma “tongues” expanding with a sound velocity in the opposite direction to the magnetic field gradient. Analysis performed using a two-dimensional MHD code has shown that the structures observed in the experiments were formed most probably due to flute instabilities. The growth of flute instabilities is predetermined by the development of thermal instabilities near the conductor surface. The thermal instabilities arise behind the front of the nonlinear magnetic diffusion wave propagating through the conductor. The wavefront on its own is not subject to thermal instabilities.

Dynamics of large-scale instabilities in conductors electrically exploded in strong magnetic fields

The growth of large-scale instabilities during the propagation of a nonlinear magnetic diffusion wave through a conductor was studied experimentally. The experiment was carried out using the MIG terawatt pulsed power generator at a peak current up to 2.5 MA with 100 ns rise time. It was observed that instabilities with a wavelength of 150 μm developed on the surface of the conductor hollow part within 160 ns after the onset of current flow, whereas the surface of the solid rod remained almost unperturbed. A system of equations describing the propagation of a nonlinear diffusion wave through a conductor and the growth of thermal instabilities has been solved numerically. It has been revealed that the development of large- scale instabilities is obviously related to the propagation of a nonlinear magnetic diffusion wave.

Diffusion of fast rising strong magnetic fields into conductors

The basic processes occurring in a conductor exploding in a current skinning mode are the propagation of a nonlinear magnetic diffusion wave in the conductor and the formation of low-temperature plasma at its surface. An experimental study of the phenomenon of nonlinear magnetic diffusion into conductors in magnetic fields of induction rising at a rate up to 3·109 T/s was carried out on the MIG generator capable of producing a peak current up to 2.5 MA within a rise time of 100 ns. It has been found experimentally that the average velocity of a nonlinear magnetic diffusion wave in an aluminum conductor placed in a strong magnetic field (up to 300 T) rising at a high rate (on average, 3·109 T/s) is (2.7÷3.3)·105 cm/s. This is comparable to the velocity of sound in aluminum under normal conditions and reasonably agrees with predictions of numerical simulations.

Mechanism and Simulation of Generating Pulsed Strong Magnetic Field

A strong magnetic field (over 1000 T) was recently experimentally produced at the Academy of Engineering Physics in China. The theoretical methods, which include a simple model and MHD code, are discussed to investigate the physical mechanism and dynamics of generating the strong magnetic field. The analysis and simulation results show that nonlinear magnetic diffusion contributes less as compared to the linear magnetic diffusion. This indicates that the compressible hydrodynamic effect and solid imploding compression may have a large influence on strong magnetic field generation.

Mathematical simulation of helical flux compression generator physics

Helical flux compression generators (HFCG) are used as sources of powerful peak currents and magnetic field strengths. We propose a numerical method of the HFCG resistance calculation, based on nonlinear magnetic diffusion. The method has been implemented with a two-dimensional HFCG model and demonstrated good agreement with experiments both at small and high conductor heating.

Dynamics of the formation of intense magnetic flux tubes on the sun by a converging plasma flow

We study the dynamics of the formation of an intense magnetic flux tube (IMFT) from the background magnetic field by a converging plasma flow with nonlinear magnetic diffusion due to the Cowling conductivity. We confirm the formation of a stationary profile of the magnetic field that was obtained earlier and estimate its onset time. We also consider relaxation of the background field inhomogeneities in the course of their advection into the magnetic flux tube. In the case of nonzero projection of the electrical current on the magnetic field, we found a new mode of the field inhomogeneity relaxation in plasmas with a highly anisotropic resistivity tensor. In this mode, unlike the case of absence of currents along the magnetic flux tube axis, which was studied earlier, an impulsive component of the magnetic energy release is observed, and the magnetic field outside the impulsive component is nearly force-free. A qualitative description of this phenomenon is proposed, analytical estimates are given, and the results of numerical calculations are presented.

Nonlinear turbulent magnetic diffusion and effective drift velocity of a large-scale magnetic field in two-dimensional magnetohydrodynamic turbulence

We study a nonlinear quenching of turbulent magnetic diffusion and effective drift velocity of large-scale magnetic field in a developed two-dimensional MHD turbulence at large magnetic Reynolds numbers. We show that transport of the mean-square magnetic potential strongly changes quenching of turbulent magnetic diffusion. In particularly, the catastrophic quenching of turbulent magnetic diffusion does not occur for the large-scale magnetic fields B >> B(eq)/square root[Rm] when a divergence of the flux of the mean-square magnetic potential is not zero, where B(eq) is the equipartition mean magnetic field determined by the turbulent kinetic energy and Rm is the magnetic Reynolds number. In this case the quenching of turbulent magnetic diffusion is independent of magnetic Reynolds number. The situation is similar to three-dimensional MHD turbulence at large magnetic Reynolds numbers whereby the catastrophic quenching of the alpha effect does not occur when a divergence of the flux of the small-scale magnetic helicity is not zero.

Nonlinear Magnetic Diffusion in a Bi2212 Hollow Cylinder: Measurements and Numerical Simulations

This paper compares numerical simulations and measurements of the nonlinear magnetic diffusion phenomenon in a Bi2212 high temperature superconducting (HTS) hollow cylinder. Although the measurements were done many years ago, it is only recently that computer power has become sufficient to reproduce the experiment by simulation on a personal computer in a reasonable time. Observations of flux density (magnitude and phase) both inside and outside the cylinder were carried out at different frequencies and magnitudes of a sinusoidal applied field. In addition, the computed current and flux density profiles inside the superconductor were obtained. The superconductor was modeled by E - J characteristics that depend on Jc and a power law exponent n. Both parameters are isotropic functions of B. The paper shows that numerical methods are useful to extract model parameters and to predict the behavior of HTS elements entering in the conception of real applications, such as the inductive fault current limiters.

Magnetic actuators and sensors

Preface. PART I MAGNETICS. 1. Introduction. 1.1 Overview of Magnetic Actuators. 1.2 Overview of Magnetic Sensors. 1.3 Actuators and Sensors in Motion Control Systems. References. 2. Basic Electromagnetics. 2.1 Vectors. 2.1.1 Gradient. 2.1.2 Divergence. 2.1.3 Curl. 2.2 Ampere's Law. 2.3 Magnetic Materials. 2.4 Faraday's Law. 2.5 Potentials. 2.6 Maxwell's Equations. Problems. References. 3. Reluctance Method. 3.1 Simplifying Ampere's Law. 3.2 Applications. 3.3 Fringing Flux. 3.4 Complex Reluctance. 3.5 Limitations. Problems. References. 4. Finite-Element Method. 4.1 Energy Conservation and Functional Minimization. 4.2 Triangular Elements for Magnetostatics. 4.3 Matrix Equation. 4.4 Finite-Element Models. Problems. References. 5. Magnetic Force. 5.1 Magnetic Flux Line Plots. 5.2 Magnetic Energy. 5.3 Magnetic Force on Steel. 5.4 Magnetic Pressure on Steel. 5.5 Lorentz Force. 5.6 Permanent Magnets. 5.7 Magnetic Torque. Problems. References. 6. Other Magnetic Performance Parameters. 6.1 Magnetic Flux and Flux Linkage. 6.1.1 Definition and Evaluation. 6.1.2 Relation to Force and Other Parameters. 6.2 Inductance. 6.2.1 Definition and Evaluation. 6.2.2 Relation to Force and Other Parameters. 6.3 Capacitance. 6.3.1 Definition. 6.3.2 Relation to Energy and Force. 6.4 Impedance. Problems. References. PART II ACTUATORS. 7. Magnetic Actuators Operated by Direct Current. 7.1 Solenoid Actuators. 7.1.1 Clapper Armature. 7.1.2 Plunger Armature. 7.2 Voice Coil Actuators. 7.3 Other Actuators Using Coils and Permanent Magnets. 7.4 Proportional Actuators. 7.5 Rotary Actuators. Problems. References. 8. Magnetic Actuators Operated by Alternating Current. 8.1 Skin Depth. 8.2 Power Losses in Steel. 8.2.1 Laminated Steel. 8.2.2 Equivalent Circuit. 8.2.3 Solid Steel. 8.3 Force Pulsations. 8.3.1 Force with Single AC Coil. 8.3.2 Force with Added Shading Coil. 8.4 Cuts In Steel. 8.4.1 Special Finite-Element Formulation. 8.4.2 Loss and Reluctance Computations. Problems. References. 9. Magnetic Actuator Transient Operation. 9.1 Basic Timeline. 9.2 Size, Force, and Acceleration. 9.3 Linear Magnetic Diffusion Times. 9.3.1 Steel Slab Turnon and Turnoff. 9.3.2 Steel Cylinder. 9.4 Nonlinear Magnetic Diffusion Time. 9.4.1 Simple Equation for Steel Slab with B-H. 9.4.2 Transient Finite-Element Computations for Steel Slabs. 9.4.3 Simple Equation for Steel Cylinder with B-H. 9.4.4 Transient Finite-Element Computations for Steel Cylinders. Problems. References. PART III SENSORS. 10. Hall Effect and Magnetoresistive Sensors. 10.1 Simple Hall Voltage Equation. 10.2 Hall Effect Conductivity Tensor. 10.3 Finite-Element Computation of Hall Fields. 10.3.1 Unsymmetric Matrix Equation. 10.3.2 2D Results. 10.3.3 3D Results. 10.4 Toothed Wheel Hall Sensors for Position. 10.5 Magnetoresistance. 10.5.1 Classical Magnetoresistance. 10.5.2 Giant Magnetoresistance. 10.5.3 Newest Forms of Magnetoresistance. 10.6 Magnetoresistive Heads for Hard-Disk Drives. Problems. References. 11. Other Magnetic Sensors. 11.1 Speed Sensors Based on Faraday's Law. 11.2 Inductive Recording Heads. 11.3 Proximity Sensors Using Impedance. 11.3.1 Stationary Eddy Current Sensors. 11.3.2 Moving Eddy Current Sensors. 11.4 Linear Variable Differential Transformers. 11.5 Magnetostrictive Sensors. 11.6 Flux Gate Sensors. 11.7 Magnetometers and Motes. Problems. References. PART IV SYSTEMS. 12. Coil Design and Temperature Calculations. 12.1 Wire Size Determination for DC Currents. 12.2 Coil Time Constant and Impedance. 12.3 Skin Effects and Proximity Effects for AC Currents. 12.4 Finite-Element Computations of Temperatures. 12.4.1 Thermal Conduction. 12.4.2 Thermal Convection and Thermal Radiation. 12.4.3 AC Magnetic Device Cooled by Conduction, Convection, and Radiation. Problems. References. 13. Electromagnetic Compatibility. 13.1 Signal-to-Noise Ratio. 13.2 Shields and Apertures. 13.3 Test Chambers. 13.3.1 TEM Transmission Lines. 13.3.2 TEM Cells. 13.3.3 Triplate Cells. Problems. References. 14. Electromechanical Finite Elements. 14.1 Electromagnetic Finite-Element Matrix Equation. 14.2 0D and 1D Finite Elements for Coupling Electric Circuits. 14.3 Structural Finite-Element Matrix Equation. 14.4 Force and Motion Computation by Timestepping. 14.5 Typical Electromechanical Applications. 14.5.1 DC Solenoid with Slowly Rising Input Current. 14.5.2 DC Solenoid with Step Input Voltage. 14.5.3 AC Clapper Solenoid Motion and Stress. 14.5.4 Transformers with Switches or Sensors. Problems. References. 15. Electromechanical Analysis Using Systems Models. 15.1 Electric Circuit Models of Magnetic Devices. 15.1.1 Electric Circuit Software Including SPICE. 15.1.2 Simple LR Circuits. 15.1.3 Tables of Nonlinear Flux Linkage and Force. 15.1.4 Analogies for Rigid Armature Motion. 15.1.5 Maxwell SPICE Model of Bessho Actuator. 15.1.6 Simplorer Model of Bessho Actuator. 15.2 VHDL-AMS/Simplorer Models. 15.2.1 VHDL-AMS Standard IEEE Language. 15.2.2 Model of Solenoid Actuator. 15.3 MATLAB/Simulink Models. 15.3.1 Software. 15.3.2 MATLAB Model of Voice Coil Actuator. 15.4 Including Eddy Current Diffusion Using a Resistor. 15.4.1 Resistor for Planar Devices. 15.4.2 Resistor for Axisymmetric Devices. Problems. References. 16. Coupled Electrohydraulic Analysis Using Systems Models. 16.1 Comparing Hydraulics and Magnetics. 16.2 Hydraulic Basics and Electrical Analogies. 16.3 Modeling Hydraulic Circuits in SPICE. 16.4 Electrohydraulic Models in SPICE and Simplorer. 16.5 Hydraulic Valves and Cylinders in Systems Models. 16.5.1 Valves and Cylinders. 16.5.2 Use in SPICE Systems Models. 16.6 Magnetic Diffusion Resistor in Electrohydraulic Models. Problems. References. Appendix: Symbols, Dimensions, and Units. Index.

Small-scale magnetic buoyancy and magnetic pumping effects in a turbulent convection

We determine the nonlinear drift velocities of the mean magnetic field and nonlinear turbulent magnetic diffusion in a turbulent convection. We show that the nonlinear drift velocities are caused by three kinds of the inhomogeneities; i.e., inhomogeneous turbulence, the nonuniform fluid density and the nonuniform turbulent heat flux. The inhomogeneous turbulence results in the well-known turbulent diamagnetic and paramagnetic velocities. The nonlinear drift velocities of the mean magnetic field cause the small-scale magnetic buoyancy and magnetic pumping effects in the turbulent convection. These phenomena are different from the large-scale magnetic buoyancy and magnetic pumping effects which are due to the effect of the mean magnetic field on the large-scale density stratified fluid flow. The small-scale magnetic buoyancy and magnetic pumping can be stronger than these large-scale effects when the mean magnetic field is smaller than the equipartition field. We discuss the small-scale magnetic buoyancy and magnetic pumping effects in the context of the solar and stellar turbulent convection. We demonstrate also that the nonlinear turbulent magnetic diffusion in the turbulent convection is anisotropic even for a weak mean magnetic field. In particular, it is enhanced in the radial direction. The magnetic fluctuations due to the small-scale dynamo increase the turbulent magnetic diffusion of the toroidal component of the mean magnetic field, while they do not affect the turbulent magnetic diffusion of the poloidal field.

New mechanism of generation of large‐scale magnetic fields in merging protogalactic and protostellar clouds

AbstractA new mechanism of generation of large‐scale magnetic fields in colliding protogalactic clouds and merging protostellar clouds is discussed. Interaction of the colliding clouds produces large‐scale shear motions which are superimposed on small‐scale turbulence. Generation of the large‐scale magnetic field is due to a “shear‐current” effect (or “vorticity‐current” effect), and the mean vorticity is caused by the large‐scale shear motions of colliding clouds. This effect causes the generation of the mean magnetic field even in a nonrotating and nonhelical homogeneous turbulence. There is no quenching of the nonlinear shear‐current effect contrary to the quenching of the nonlinear alpha effect, the nonlinear turbulent magnetic diffusion, etc. During the nonlinear growth of the mean magnetic field, the shear‐current effect only changes its sign at some value of the mean magnetic field which determines the level of the saturated mean magnetic field. Numerical study shows that the saturated level of the mean magnetic field is of the order of the equipartition field determined by the turbulent kinetic energy. The estimated large‐scale magnetic field for merging protogalactic clouds is about several microgauss, and for merging protostellar clouds is of the order of several tenth of microgauss. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear theory of a “shear-current” effect and mean-field magnetic dynamos

The nonlinear theory of a "shear-current" effect in a nonrotating and nonhelical homogeneous turbulence with an imposed mean velocity shear is developed. The shear-current effect is associated with the W x J term in the mean electromotive force and causes the generation of the mean magnetic field even in a nonrotating and nonhelical homogeneous turbulence (where W is the mean vorticity and J is the mean electric current). It is found that there is no quenching of the nonlinear shear-current effect contrary to the quenching of the nonlinear alpha effect, the nonlinear turbulent magnetic diffusion, etc. During the nonlinear growth of the mean magnetic field, the shear-current effect only changes its sign at some value B (*) of the mean magnetic field. The magnitude B (*) determines the level of the saturated mean magnetic field which is less than the equipartition field. It is shown that the background magnetic fluctuations due to the small-scale dynamo enhance the shear-current effect and reduce the magnitude B (*) . When the level of the background magnetic fluctuations is larger than 1/3 of the kinetic energy of the turbulence, the mean magnetic field can be generated due to the shear-current effect for an arbitrary exponent of the energy spectrum of the velocity fluctuations.

Finite-Element Computation of Nonlinear Magnetic Diffusion and Its Effects When Coupled to Electrical, Mechanical, and Hydraulic Systems

Finite elements are used to compute eddy currents and magnetic diffusion times in steel cylinders with nonlinear B--H curves. Computations are made of diffusion times versus current in a typical axisymmetric magnetic actuator, showing good agreement with recently published approximate analytical formulas. The computed nonlinear diffusion times are then used to derive an equivalent resistor that is used in a model of an electrohydraulic system. The diffusion time causes delays in the coupled mechanical and hydraulic responses.

Irreversibility line and magnetic field dependence of the critical current in superconducting MgB2 bulk samples

The third harmonic components of the ac susceptibility of MgB2 bulk samples have been measured as a function of applied magnetic fields, together with standard magnetization cycles. The irreversibility line (IL) of the magnetic field has been extracted from the onset of the third harmonic components. Using a (1 − t)α glass/liquid best fit where α = 1.27 IL shows a coherent length ξ divergence with exponent ν = 0.63, which indicates a 3D behaviour. Moreover, using the numerical solution of the non-linear magnetic diffusion equation, considering the creep model in a 3D vortex glass, a good description of the vortex dynamics has been obtained. The behaviour of the magnetization amplitude (∼Hz) and the ac susceptibility signals (kHz), at different applied magnetic fields, 3.5 T < Hdc < 4.5 T, and at the reduced temperature 0.86 < t < 0.93 (T = 22 K), shows that the superconducting dynamic response of vortices in the MgB2 samples is not evidently dependent on the frequency.

Nonlinear turbulent magnetic diffusion and mean-field dynamo.

The nonlinear coefficients defining the mean electromotive force (i.e., the nonlinear turbulent magnetic diffusion, the nonlinear effective velocity, the nonlinear kappa tensor, etc.) are calculated for an anisotropic turbulence. A particular case of an anisotropic background turbulence (i.e., the turbulence with zero-mean magnetic field) with one preferential direction is considered. It is shown that the toroidal and poloidal magnetic fields have different nonlinear turbulent magnetic diffusion coefficients. It is demonstrated that even for a homogeneous turbulence there is a nonlinear effective velocity that exhibits diamagnetic or paramagnetic properties depending on the anisotropy of turbulence and the level of magnetic fluctuations in the background turbulence. The diamagnetic velocity results in the field being pushed out from the regions with stronger mean magnetic field, while the paramagnetic velocity causes the magnetic field to be concentrated in the regions with stronger field. Analysis shows that an anisotropy of turbulence strongly affects the nonlinear turbulent magnetic diffusion, the nonlinear effective velocity, and the nonlinear alpha effect. Two types of nonlinearities (algebraic and dynamic) are also discussed. The algebraic nonlinearity implies a nonlinear dependence of the mean electromotive force on the mean magnetic field. The dynamic nonlinearity is determined by a differential equation for the magnetic part of the alpha effect. It is shown that for the alphaOmega axisymmetric dynamo the algebraic nonlinearity alone (which includes the nonlinear alpha effect, the nonlinear turbulent magnetic diffusion, the nonlinear effective velocity, etc.) cannot saturate the dynamo generated mean magnetic field while the combined effect of the algebraic and dynamic nonlinearities limits the mean magnetic field growth.

Magnetic diffusion and dissipation in high- Tc superconductors subjected to sinusoidal applied fields

Solution of the linear magnetic diffusion equation of a superconductor slab with a sinusoidal applied field is used to explain recently reported experimental results on the transient response of a high- T c superconductor tube. The developed linear model can qualitatively explain very well the experimentally observed delay between the applied field and the penetrated field, and other salient features of the experiments. The analysis shows that the linear magnetic-flux-density profile of Bean's critical-state model is not approached, even under quasi-steady-state conditions. A direct consequence of magnetic diffusion is that, under AC steady-state condition, dissipation is always associated with ρJ 2, in addition to the hysterestic loss. The results of the analysis can also be used to explain qualitatively the characteristics of a superconductor-shielded core reactor. The effects of nonlinear magnetic diffusion and heating are discussed.

Frequency dependence of HTS AC harmonic susceptibility in the Kim-Anderson and collective pinning vortex glass models

By numerically solving the nonlinear magnetic flux diffusion equation, the temperature dependence of the AC magnetic susceptibility have been computed at different frequencies in the framework of the collective pinning vortex glass and of the Kim-Anderson models. In particular the frequency dependence of the amplitude of the third harmonic have been analyzed. Such dependence is determined by the dynamic curvature of the E-J characteristics, normalized to the dynamic resistivity.