scholarly journals Taylor state dynamos found by optimal control: axisymmetric examples

2018 ◽  
Vol 853 ◽  
pp. 647-697 ◽  
Author(s):  
Kuan Li ◽  
Andrew Jackson ◽  
Philip W. Livermore
Keyword(s):  
Magnetic Field ◽  
Optimal Control ◽  
Vector Fields ◽  
Inertial Force ◽  
Basis Set ◽  
Viscous Force ◽  
Time Step ◽  
Geostrophic Flow ◽  
Spectral Expansions ◽  
The Magnetic Field

Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported by Wu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.

scholarly journals Diffusion of Particles, Heat and Magnetic Fields in Compressible Turbulent Media

1991 ◽  
Vol 130 ◽  
pp. 71-74
Author(s):  
A.Z. Dolginov ◽  
N.A. Silant’ev
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Vector Fields ◽  
Particle Number ◽  
Particle Number Density ◽  
Turbulent Diffusivity ◽  
Number Density ◽  
Scalar And Vector Fields ◽  
Kinetic Coefficients ◽  
The Magnetic Field

AbstractA new method for the calculation of kinetic coefficients is presented. This method allows us to obtain the distribution of scalar and vector fields (such as the temperature, the admixture particle number density and the magnetic field) in turbulent cosmic media with any value of S = u0т0/R0. The explicit expression for the “turbulent” diffusivity DT is obtained. In some cases DT becomes negative, implying the clustering of the admixture particles in patches (a local increase of the temperature and magnetic fields). The magnetic α-effect is considered for the case S ~ 1.

A formulation for computation of a class of collision-free plasmas in two dimensions

1982 ◽  
Vol 28 (1) ◽  
pp. 141-147 ◽  
Author(s):  
J. W. Dungey
Keyword(s):  
Magnetic Field ◽  
Dispersion Equation ◽  
Two Dimensions ◽  
Higher Moments ◽  
Time Step ◽  
Pressure Anisotropy ◽  
The Magnetic Field

The restrictions imposed are that the magnetic field is everywhere in the x direction, and that no quantity varies with x, but several interesting instabilities can still occur. After some discussion of objectives, a fluid-like formulation is pursued, in which the pressure anisotropy is retained, but higher moments neglected. It shows a resonance at twice the gyrofrequency, and for electrons the constraint on the time step would be unacceptable, so they should be treated more crudely. Then the dispersion equation shows only two modes, which appear sufficiently harmless for us to proceed to computations.

Acoustic End Effects in Magnetohydrodynamic Submerged Vehicular Propulsors

1992 ◽  
Vol 36 (01) ◽  
pp. 69-76
Author(s):  
John S. Walker ◽  
Gita Talmage ◽  
Samuel H. Brown ◽  
Neal A. Sondergaard
Keyword(s):  
Magnetic Field ◽  
Inertial Force ◽  
Zero Field ◽  
Uniform Field ◽  
Field Region ◽  
Characteristic Ratio ◽  
Fringing Field ◽  
Field Lines ◽  
Channel Configuration ◽  
The Magnetic Field

This paper treats the effects near the ends of the channel on the transmission and reflection of periodic acoustic waves generated at some cross section inside a magnetohydrodynamic (MHD) seawater propulsion system. A region of high uniform magnetic field inside the MHD submerged vehicular propulsor is separated from the essentially zero magnetic field outside the channel by a nonuniform, fringing magnetic field at each end of the channel. The channel configuration chosen here is that of a straight, rectangular duct with electrically insulating top and bottom walls perpendicular to the magnetic field and highly conducting sidewalls parallel to the field. In particular, the mathematical analysis focuses on determining the percentage of the incident wave which is reflected by the fringing-field region back into the uniform-field region and the percentage which is transmitted through the fringing-field region into the zero-field region. The key parameter is the acoustic interaction parameter N, which is the characteristic ratio of the electromagnetic body force opposing motions across magnetic field lines to the inertial "force" in the acoustic wave. Solutions are presented for the fundamental, plane acoustic mode for arbitrary values of Ν and for all acoustic modes for Ν < 1. The amplitudes of the reflected and transmitted waves depend on the wave frequency, the length of the fringing-field region, N, and the type of wave mode. The magnetic field introduces a strong anisotropy with strong damping of modes involving transverse motions across magnetic field lines and with weak damping of modes involving transverse motions along field lines. This is the third in a series of articles on MHD marine propulsion from the David Taylor Research Center MHD propulsion program [Brown et al (1990), Tempelmeyer (1990)].

scholarly journals On the fine-scale structure of vector fields convected by a turbulent fluid

1963 ◽  
Vol 16 (4) ◽  
pp. 545-572 ◽  
Author(s):  
P. G. Saffman
Keyword(s):  
Magnetic Field ◽  
Vector Fields ◽  
Scale Structure ◽  
Homogeneous Turbulence ◽  
Small Scale ◽  
Ohmic Dissipation ◽  
Large Wave ◽  
Turbulent Fluid ◽  
Wave Numbers ◽  
The Magnetic Field

This paper is a contribution to the study of statistically homogeneous, dynamically passive vector fields convected by a turbulent fluid and subject to a molecular diffusivity λ that is small compared with the kinematic viscosityv. Two types are considered: the first, denoted byF, has the property that the total flux across a material surface moving with the fluid is conserved if λ = 0 (e.g. magnetic field); and the second, denoted byG, is the gradient of a conserved scalar quantity θ (e.g. temperature gradient). Attention is focused on small-scale variations with length-scale less than$(v^3|\epsilon)^{\frac {1}{4}}$. A theory of Batchelor's in terms of Eulerian correlations for the distribution of θ for the case when λ [Lt ]vis extended and applied to the vector fields, thereby giving equations for the covariance tensors ofFandGappropriate for separations less than$(v^3|\epsilon)^{\frac {1}{4}}$. According to these equations, the effect of convection on small-scale components of the fields is to amplify and also to distort by reducing the scale; the ratio of these two effects is measured by a parameter σ. It is shown that if$\sigma \textless {\frac{5}{2}$, the small-scale structure is stable against perturbations however small λ/vmay be, the amplification being eventually balanced by the dissipation which is increased by the reduction of scale. In the case of the quantityG, σ = 1. The value of σ for the case ofFis not known, but reasons are given for believing that it is less than one, and it is concluded that the behaviour of$\overline{\bf F^2}$and$\overline{\bf G^2}$in a field of homogeneous turbulence is qualitatively the same. In particular,$\overline{\bf F^2}$does not grow indefinitely with time as predicted by previous arguments. The correlation functions for small separations and the corresponding spectrum functions for a statistically steady state are obtained. The relation between this analysis and that for random vector fields in a uniform straining motion of infinite extent is considered in detail, for Pearson has shown that, if the strain is an irrotational distortion, then$\overline{\bf F^2} \rightarrow \infty$with time. It is shown that this divergence is due to the amplification of components with very small wave-numbers or, equivalently, of very large scale, and it is therefore not considered relevant to a study of homogeneous turbulence.The particular case of the magnetic field in a good conductor is considered. If the Lorentz forces are unimportant, it is estimated that the magnetic energy of a weak seed field will be in general amplified by the turbulence by a factor lying somewhere between the Reynolds and magnetic Reynolds numbers of the turbulence before ohmic dissipation as increased by the reduction of scale limits the growth, and it is suggested further that the magnetic field will eventually decay to zero in the absence of external electromotive forces.In an appendix, the theory is applied tentatively to the turbulent vorticity (which satisfies the same equation asFif λ =v) and an expression for the energy spectrum function for very large wave-numbers is deduced. This is compared with an expression given by Townsend, and is found to have a similar qualitative behaviour but gives values about one-half as large.

VISUALIZING THE DISTRIBUTION OF SPIRAL MAGNETIZATION IN THE FERROMAGNETIC LAYER OF A SUPERCONDUCTING SPIN VALVE

2021 ◽  
Vol 0 (1) ◽  
pp. 5-9
Author(s):  
D.I. DZEPAROV ◽  
◽  
N.A. GUSEV ◽  
N.G. PUGACH ◽  
B.G. LVOV ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Vector Fields ◽  
Spin Valve ◽  
Ferromagnetic Layer ◽  
Field Research ◽  
Simulation Software ◽  
Superconducting Film ◽  
Software Environment ◽  
Spiral Magnet ◽  
The Magnetic Field

In this paper, we investigate the switch of a magnet-based superconducting spin valve. As a magnet, we consider the MnSi compound characterized by a complex magnetic structure with its magnetization in the form of a spiral helicoid. A model is given for a double-layered superconducting spin valve based on the spiral magnet with an adjoining superconducting thin layer. As shown previously, critical temperature of such a superconducting film depends on the direction of the spiral magnet. Calculations have been performed for magnetic dynamics of this structure, which in prospect can serve as the basis for creating memory or logic elements of low temperature nanoelectronics. Several problems have been treated with regards to mathematical simulation of switching the spin valve magnetization, research on the spin valve structure using electromagnetic simulation software to change magnetization and rotation of the spiral magnetic vector under the magnetic-field pulse, construction and analysis of 3D magnetization distributions in spiral magnets to trace the process of remagnetization. In this work, we use a Matlab-based software environment and tools for constructing distribution plots of vector fields. It is shown that the direction of the spiral magnet can be switch by pulsed magnetic field. Research has been done on the resultant magnetization distributions visualized in the form of vector fields.

scholarly journals Plasma and fields in the wake of Rhea: 3-D hybrid simulation and comparison with Cassini data

2008 ◽  
Vol 26 (3) ◽  
pp. 619-637 ◽  
Author(s):  
E. Roussos ◽  
J. Müller ◽  
S. Simon ◽  
A. Bößwetter ◽  
U. Motschmann ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Electric Field ◽  
Vector Fields ◽  
Current System ◽  
Direct Consequence ◽  
Asymmetric Distribution ◽  
The Moon ◽  
Simulation Code ◽  
Field Lines ◽  
The Magnetic Field

Abstract. Rhea's magnetospheric interaction is simulated using a three-dimensional, hybrid plasma simulation code, where ions are treated as particles and electrons as a massless, charge-neutralizing fluid. In consistency with Cassini observations, Rhea is modeled as a plasma absorbing obstacle. This leads to the formation of a plasma wake (cavity) behind the moon. We find that this cavity expands with the ion sound speed along the magnetic field lines, resulting in an extended depletion region north and south of the moon, just a few Rhea radii (RRh) downstream. This is a direct consequence of the comparable thermal and bulk plasma velocities at Rhea. Perpendicular to the magnetic field lines the wake's extension is constrained by the magnetic field. A magnetic field compression in the wake and the rarefaction in the wake sides is also observed in our results. This configuration reproduces well the signature in the Cassini magnetometer data, acquired during the close flyby to Rhea on November 2005. Almost all plasma and field parameters show an asymmetric distribution along the plane where the corotational electric field is contained. A diamagnetic current system is found running parallel to the wake boundaries. The presence of this current system shows a direct corelation with the magnetic field configuration downstream of Rhea, while the resulting j×B forces on the ions are responsible for the asymmetric structures seen in the velocity and electric field vector fields in the equatorial plane. As Rhea is one of the many plasma absorbing moons of Saturn, we expect that this case study should be relevant for most lunar-type interactions at Saturn.

Pseudo null curve variations for Bishop frame in 3D semi-Riemannian manifold

2019 ◽  
Vol 16 (03) ◽  
pp. 1950043 ◽  
Author(s):  
Zehra Özdemi̇r
Keyword(s):  
Magnetic Field ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Space Form ◽  
Charged Particles ◽  
Null Curve ◽  
Null Curves ◽  
The Magnetic Field ◽  
Killing Equations

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

scholarly journals In situ magnetotail magnetic flux calculation

2015 ◽  
Vol 33 (6) ◽  
pp. 769-781 ◽  
Author(s):  
M. A. Shukhtina ◽  
E. Gordeev
Keyword(s):  
Magnetic Field ◽  
Solar Wind ◽  
Magnetic Flux ◽  
Empirical Formula ◽  
Calculation Algorithm ◽  
Time Step ◽  
Mhd Simulations ◽  
External Sources ◽  
The Magnetic Field

Abstract. We explore two new modifications of the magnetotail magnetic flux (F) calculation algorithm based on the Petrinec and Russell (1996) (PR96) approach of the tail radius determination. Unlike in the PR96 model, the tail radius value is calculated at each time step based on simultaneous magnetotail and solar wind observations. Our former algorithm, described in Shukhtina et al. (2009), required that the "tail approximation" requirement were fulfilled, i.e., it could be applied only tailward x ∼ −15 RE. The new modifications take into account the approximate uniformity of the magnetic field of external sources in the near and middle tail. Tests, based on magnetohydrodynamics (MHD) simulations, show that this approach may be applied at smaller distances, up to x ∼ −3 RE. The tests also show that the algorithm fails during long periods of strong positive interplanetary magnetic field (IMF) Bz. A new empirical formula has also been obtained for the tail radius at the terminator (at x = 0) which improves the calculations.

scholarly journals Magnetic structures in a dynamo simulation

1996 ◽  
Vol 306 ◽  
pp. 325-352 ◽  
Author(s):  
A. Brandenburg ◽  
R. L. Jennings ◽  
Å. Nordlund ◽  
M. Rieutord ◽  
R. F. Stein ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Vector Fields ◽  
Magnetic Energy ◽  
Three Dimensional ◽  
Dynamo Action ◽  
Transfer Rates ◽  
Magnetic Structures ◽  
Vorticity Vector ◽  
Early Growth Phase ◽  
The Magnetic Field

We use three-dimensional simulations to study compressible convection in a rotating frame with magnetic fields and overshoot into surrounding stable layers. The, initially weak, magnetic field is amplified and maintained by dynamo action and becomes organized into flux tubes that are wrapped around vortex tubes. We also observe vortex buoyancy which causes upward flows in the cores of extended downdraughts. An analysis of the angles between various vector fields shows that there is a tendency for the magnetic field to be parallel or antiparallel to the vorticity vector, especially when the magnetic field is strong. The magnetic energy spectrum has a short inertial range with a slope compatible with k+1/3 during the early growth phase of the dynamo. During the saturated state the slope is compatible with k−1. A simple analysis based on various characteristic timescales and energy transfer rates highlights important qualitative ideas regarding the energy budget of hydromagnetic dynamos.

A three-dimensional, iterative mapping procedure for the implementation of an ionosphere-magnetosphere anisotropic Ohm's law boundary condition in global magnetohydrodynamic simulations

1995 ◽  
Vol 13 (8) ◽  
pp. 843-853 ◽  
Author(s):  
M. L. Goodman
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Electric Field ◽  
Current Density ◽  
Time Step ◽  
Mhd Simulation ◽  
Ohm’S Law ◽  
Field Aligned Current ◽  
Ohm's Law ◽  
The Magnetic Field

Abstract. The mathematical formulation of an iterative procedure for the numerical implementation of an ionosphere-magnetosphere (IM) anisotropic Ohm's law boundary condition is presented. The procedure may be used in global magnetohydrodynamic (MHD) simulations of the magnetosphere. The basic form of the boundary condition is well known, but a well-defined, simple, explicit method for implementing it in an MHD code has not been presented previously. The boundary condition relates the ionospheric electric field to the magnetic field-aligned current density driven through the ionosphere by the magnetospheric convection electric field, which is orthogonal to the magnetic field B, and maps down into the ionosphere along equipotential magnetic field lines. The source of this electric field is the flow of the solar wind orthogonal to B. The electric field and current density in the ionosphere are connected through an anisotropic conductivity tensor which involves the Hall, Pedersen, and parallel conductivities. Only the height-integrated Hall and Pedersen conductivities (conductances) appear in the final form of the boundary condition, and are assumed to be known functions of position on the spherical surface R=R1 representing the boundary between the ionosphere and magnetosphere. The implementation presented consists of an iterative mapping of the electrostatic potential ψ the gradient of which gives the electric field, and the field-aligned current density between the IM boundary at R=R1 and the inner boundary of an MHD code which is taken to be at R2>R1. Given the field-aligned current density on R=R2, as computed by the MHD simulation, it is mapped down to R=R1 where it is used to compute ψ by solving the equation that is the IM Ohm's law boundary condition. Then ψ is mapped out to R=R2, where it is used to update the electric field and the component of velocity perpendicular to B. The updated electric field and perpendicular velocity serve as new boundary conditions for the MHD simulation which is then used to compute a new field-aligned current density. This process is iterated at each time step. The required Hall and Pedersen conductances may be determined by any method of choice, and may be specified anew at each time step. In this sense the coupling between the ionosphere and magnetosphere may be taken into account in a self-consistent manner.

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