A three-dimensional kinematic dynamo

1975 ◽  
Vol 344 (1637) ◽  
pp. 235-258 ◽  
Keyword(s):  
Magnetic Field ◽  
Three Dimensional ◽  
Characteristic Size ◽  
Steady Flows ◽  
Dynamo Action ◽  
Ohmic Dissipation ◽  
The Earth ◽  
Induction Equation ◽  
High Degree ◽  
The Magnetic Field

A number of steady (marginal) solutions of the induction equation governing the magnetic field created by a particular class of threedimensional flows in a sphere of conducting fluid surrounded by an insulator are derived numerically. These motions possess a high degree of symmetry which can be varied to confirm numerically that the corresponding asymptotic limit of Braginsky is attained. The effect of altering the spatial scale of the motions without varying their vigour can also be examined, and it is found that dynamo action is at first eased by decreasing their characteristic size. There are, however, suggestions that the regenerative efficiency does not persistently increase to very small length scales, but ultimately decreases. It is further shown that time varying motions, in which the asymmetric components of flow travel as a wave round lines of latitude, can sustain fields having co-rotating asymmetric parts. It is demonstrated that, depending on their common angular velocity, these may exist at slightly smaller magnetic Reynolds numbers than the corresponding models having steady flows and fields. The possible bearing of the integrations on the production of the magnetic field of the Earth is considered, and the implied ohmic dissipation of heat in the core of the Earth is estimated for different values of the parameters defining the model.

scholarly journals Estimation of Electrical Conductivity and Magnetization Parameter of Neutron Star Crusts and Applied to the High-Braking-Index Pulsar PSR J1640-4631

Universe ◽  
2020 ◽  
Vol 6 (5) ◽  
pp. 63
Author(s):  
Hui Wang ◽  
Zhi-Fu Gao ◽  
Huan-Yu Jia ◽  
Na Wang ◽  
Xiang-Dong Li
Keyword(s):  
Magnetic Field ◽  
Neutron Star ◽  
Neutron Stars ◽  
General Expression ◽  
Equations Of State ◽  
Rotation Period ◽  
Ohmic Dissipation ◽  
Hartree Fock ◽  
Induction Equation ◽  
The Magnetic Field

Young pulsars are thought to be highly magnetized neutron stars (NSs). The crustal magnetic field of a NS usually decays at different timescales in the forms of Hall drift and Ohmic dissipation. The magnetization parameter ω B τ is defined as the ratio of the Ohmic timescale τ O h m to the Hall drift timescale τ H a l l . During the first several million years, the inner temperature of the newly born neutron star cools from T = 10 9 K to T = 1.0 × 10 8 K, and the crustal conductivity increases by three orders of magnitude. In this work, we adopt a unified equations of state for cold non-accreting neutron stars with the Hartree–Fock–Bogoliubov method, developed by Pearson et al. (2018), and choose two fiducial dipole magnetic fields of B = 1.0 × 10 13 G and B = 1.0 × 10 14 G, four different temperatures, T, and two different impurity concentration parameters, Q, and then calculate the conductivity of the inner crust of NSs and give a general expression of magnetization parameter for young pulsars: ω B τ ≃ ( 1 − 50 ) B 0 / ( 10 13 G) by using numerical simulations. It was found when B ≤ 10 15 G, due to the quantum effects, the conductivity increases slightly with the increase in the magnetic field, the enhanced magnetic field has a small effect on the matter in the low-density regions of the crust, and almost has no influence the matter in the high-density regions. Then, we apply the general expression of the magnetization parameter to the high braking-index pulsar PSR J1640-4631. By combining the observed arrival time parameters of PSR J1640-4631 with the magnetic induction equation, we estimated the initial rotation period P 0 , the initial dipole magnetic field B 0 , the Ohm dissipation timescale τ O h m and Hall drift timescale τ H a l l . We model the magnetic field evolution and the braking-index evolution of the pulsar and compare the results with its observations. It is expected that the results of this paper can be applied to more young pulsars.

Modelling the core magnetic field of the Earth

1982 ◽  
Vol 306 (1492) ◽  
pp. 179-191 ◽  
Keyword(s):  
Magnetic Field ◽  
Spherical Harmonics ◽  
Secular Variation ◽  
Long Wavelength ◽  
Core Field ◽  
The Core ◽  
The Earth ◽  
High Degree ◽  
Current Systems ◽  
The Magnetic Field

The magnetic field generated in the core of the Earth is often represented by spherical harmonics of the magnetic potential. It has been found from looking at the equations of spherical harmonics, and from studying the values of the spherical harmonic coefficients derived from data from Magsat, that this is an unsatisfactory way of representing the core field. Harmonics of high degree are characterized by generally shorter wavelength expressions on the surface of the Earth, but also contain very long wavelength features as well. Thus if it is thought that the higher degree harmonics are produced by magnetizations within the crust of the Earth, these magnetizations have to be capable of producing very long wavelength signals. Since it is impossible to produce very long wavelength signals of sufficient amplitude by using crustal magnetizations of reasonable intensity, the separation of core and crustal sources by using spherical harmonics is not ideal. We suggest that a better way is to use radial off-centre dipoles located within the core of the Earth. These have several advantages. Firstly, they can be thought of as modelling real physical current systems within the core of the Earth. Secondly, it can be shown that off-centred dipoles, if located deep within the core, are more effective at removing long wavelength signals of potential or field than can be achieved by using spherical harmonics. The disadvantage is that it is much more difficult to compute the positions and strengths of the off-centred dipole fields, and much less easy to manipulate their effects (such as upward and downward continuation). But we believe, along with Cox and Alldredge & Hurwitz, that the understanding that we might obtain of the Earth’s magnetic field by using physically reasonable models rather than mathematically convenient models is very important. We discuss some of the radial dipole models that have been proposed for the nondipole portion of the Earth’s field to arrive at a model that agrees with observations of secular variation and excursions.

scholarly journals General relativistic magnetic perturbations and dynamo effects in extragalactic radiosources

2010 ◽  
Vol 6 (S274) ◽  
pp. 393-397
Author(s):  
L. C. Garcia de Andrade
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Magnetic Fluctuations ◽  
Dynamo Action ◽  
Magnetic Contrast ◽  
Mhd Dynamo ◽  
Curvature Energy ◽  
Induction Equation ◽  
The Universe ◽  
The Magnetic Field

AbstractBy making use of the MHD self-induction equation in general relativity (GR), recently derived by Clarkson and Marklund (2005), it is shown that when Friedmann universe possesses a spatial section whose Riemannian curvature is negative, the magnetic energy bounds computed by Nuñez (2002) also bounds the growth rate of the magnetic field given by the strain matrix of dynamo flow. Since in GR-MHD dynamo equation, the Ricci tensor couples with the universe magnetic field, only through diffusion, and most ages are highly conductive the interest is more theoretical here, and only very specific plasma astrophysical problems can be address such as in laboratory plasmas. Magnetic fields and the negative curvature of some isotropic cosmologies, contribute to enhence the amplification of the magnetic field. Ricci curvature energy is shown to add to strain matrix of the flow, to enhance dynamo action in the universe. Magnetic fluctuations of the Clarkson-Marklund equations for a constant magnetic field seed in highly conductive flat universes, leads to a magnetic contrast of ≈ 2, which is well within observational limits from extragalactic radiosources of ≈ 1.7. In the magnetic helicity fluctuations the magnetic contrast shows that the dynamo effects can be driven by these fluctuations.

scholarly journals Data-driven model of the solar corona above an active region

2019 ◽  
Vol 624 ◽  
pp. L12 ◽  
Author(s):  
J. Warnecke ◽  
H. Peter
Keyword(s):  
Magnetic Field ◽  
Active Region ◽  
Extreme Ultraviolet ◽  
Three Dimensional ◽  
Data Driven ◽  
Diffuse Emission ◽  
Qualitative Level ◽  
Induction Equation ◽  
Hot Corona ◽  
The Magnetic Field

Aims. We aim to reproduce the structure of the corona above a solar active region as seen in the extreme ultraviolet (EUV) using a three-dimensional magnetohydrodynamic (3D MHD) model. Methods. The 3D MHD data-driven model solves the induction equation and the mass, momentum, and energy balance. To drive the system, we feed the observed evolution of the magnetic field in the photosphere of the active region AR 12139 into the bottom boundary. This creates a hot corona above the cool photosphere in a self-consistent way. We synthesize the coronal EUV emission from the densities and temperatures in the model and compare this to the actual coronal observations. Results. We are able to reproduce the overall appearance and key features of the corona in this active region on a qualitative level. The model shows long loops, fan loops, compact loops, and diffuse emission forming at the same locations and at similar times as in the observation. Furthermore, the low-intensity contrast of the model loops in EUV matches the observations. Conclusions. In our model the energy input into the corona is similar as in the scenarios of fieldline-braiding or flux-tube tectonics, that is, energy is transported to the corona through the driving of the vertical magnetic field by horizontal photospheric motions. The success of our model shows the central role that this process plays for the structure, dynamics, and heating of the corona.

Characterization of the flow past a truncated square cylinder in a duct under a spanwise magnetic field

2011 ◽  
Vol 691 ◽  
pp. 341-367 ◽  
Author(s):  
Vincent Dousset ◽  
Alban Pothérat
Keyword(s):  
Magnetic Field ◽  
Lorentz Force ◽  
Topological Analysis ◽  
Three Dimensional ◽  
Homogeneous Magnetic Field ◽  
Steady Flows ◽  
Square Cylinder ◽  
Electrically Conducting ◽  
Truncated Cylinder ◽  
The Magnetic Field

AbstractWe study the flow of an electrically conducting fluid past a truncated square cylinder in a rectangular duct under the influence of an externally applied homogeneous magnetic field oriented along the cylinder axis. Our aim is to bridge the gap between the non-magnetic regime, where we previously found a complex set of three-dimensional recirculations behind the cylinder (Dousset & Pothérat, J. Fluid Mech., vol. 653, 2010, pp. 519–536) and the asymptotic regime of dominating Lorentz force analysed by Hunt & Ludford (J. Fluid. Mech., vol. 33, 1968, pp. 693–714). The latter regime is characterized by a remarkable structure known as Hunt’s wake in the magnetohydrodynamics community, where the flow is deflected on either side of a stagnant zone, right above the truncated cylinder as if the latter would span the full height of the duct. In steady flows dominated by the Lorentz force, with negligible inertia, we provide the first numerical flow visualization of Hunt’s wake. In regimes of finite inertia, a thorough topological analysis of the steady flow regimes reveals how the Lorentz force gradually reorganizes the flow structures in the hydrodynamic wake of the cylinder as the Hartmann number $\mathit{Ha}$ (which gives a non-dimensional measure of the magnetic field) is increased. The nature of the vortex shedding follows from this rearrangement of the steady structures by the magnetic field. As $\mathit{Ha}$ is increased, we observe that the vortex street changes from a strongly symmetric one to the alternate procession of counter-rotating vortices typical of the non-truncated cylinder wakes.

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.

scholarly journals Magneto-convection

2012 ◽  
Vol 370 (1970) ◽  
pp. 3070-3087 ◽  
Author(s):  
Robert F. Stein
Keyword(s):  
Magnetic Fields ◽  
Equations Of State ◽  
Three Dimensional ◽  
Dynamo Action ◽  
Spatial And Temporal Scales ◽  
Solar Observations ◽  
Wide Range ◽  
Induction Equation ◽  
The Magnetic Field ◽  
Three Dimensional Simulations

Convection is the transport of energy by bulk mass motions. Magnetic fields alter convection via the Lorentz force, while convection moves the fields via the curl( v × B ) term in the induction equation. Recent ground-based and satellite telescopes have increased our knowledge of the solar magnetic fields on a wide range of spatial and temporal scales. Magneto-convection modelling has also greatly improved recently as computers become more powerful. Three-dimensional simulations with radiative transfer and non-ideal equations of state are being performed. Flux emergence from the convection zone through the visible surface (and into the chromosphere and corona) has been modelled. Local, convectively driven dynamo action has been studied. The alteration in the appearance of granules and the formation of pores and sunspots has been investigated. Magneto-convection calculations have improved our ability to interpret solar observations, especially the inversion of Stokes spectra to obtain the magnetic field and the use of helioseismology to determine the subsurface structure of the Sun.

scholarly journals A finite‐difference, time‐domain solution for three‐dimensional electromagnetic modeling

Geophysics ◽  
1993 ◽  
Vol 58 (6) ◽  
pp. 797-809 ◽  
Author(s):  
Tsili Wang ◽  
Gerald W. Hohmann
Keyword(s):  
Magnetic Field ◽  
Finite Difference ◽  
Execution Time ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Three Dimensional ◽  
Staggered Grid ◽  
Transient Electromagnetic ◽  
The Earth ◽  
The Magnetic Field

We have developed a finite‐difference solution for three‐dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell’s equations in time using a staggered‐grid technique. The time‐stepping uses a modified version of the Du Fort‐Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first‐order Maxwell’s equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence‐free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times. An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth‐order difference method at early times and a second‐order method at other times. Numerical checks against analytical, integral‐equation, and spectral differential‐difference solutions show that the solution provides accurate results. Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains [Formula: see text] grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Ω ⋅ m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much‐needed insight about TEM responses in complicated 3-D situations.

scholarly journals Steady plasma flows in a periodic non-symmetric domain

2021 ◽  
Vol 87 (6) ◽  
Author(s):  
Harold Weitzner ◽  
Wrick Sengupta
Keyword(s):  
Magnetic Field ◽  
Three Dimensional ◽  
Symmetric Domain ◽  
Steady Flows ◽  
Continuous Symmetry ◽  
Plasma Flows ◽  
Mhd Equilibrium ◽  
Mhd Model ◽  
Rotational Transform ◽  
The Magnetic Field

Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Phys. Plasmas, vol. 21, 2014, p. 022515) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by Hameiri (J. Math. Phys., vol. 22, 1981, pp. 2080–2088, § III), to obtain a generalized Grad–Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we can obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

Numerical solutions of the kinematic dynamo problem

1973 ◽  
Vol 274 (1241) ◽  
pp. 493-521 ◽  
Keyword(s):  
Magnetic Field ◽  
Numerical Solutions ◽  
Expansion Method ◽  
Magnetic Reynolds Number ◽  
Axially Symmetric ◽  
Dynamo Action ◽  
Rotating Magnetic Fields ◽  
Induction Equation ◽  
Steady Solutions ◽  
The Magnetic Field

The expansion method of Bullard & Gellman is used to find numerical solutions of the induction equation in a sphere of conducting fluid. Modifications are made to the numerical methods, and one change due to G. O. Roberts greatly increases the efficiency of the scheme. Calculations performed recently by Lilley are re-examined. His solutions, which appeared to be convergent, are shown to diverge when a higher level of truncation is used. Other similar dynamo models are investigated and it is found that these also do not provide satisfactory steady solutions for the magnetic field. Axially symmetric motions which depend on spherical harmonics of degree n are examined. Growing solutions, varying with longitude, 0, as e1^, are found for the magnetic field, and numerical convergence of the solutions is established. The field is predominantly an equatorial dipole with a toroidal field symmetric about the same axis. When n is large the problem lends itself to a two-scale analysis. Comparisons are made between the approximate results of the two-scale method and the numerical results. There is agreement when n is large. When n is small the efficiency of the dynamo is lowered. It is shown that the dominant effect of a large microscale magnetic Reynolds number is the expulsion of magnetic flux by eddies to give a rope-like structure for part of the field. Physical interpretations are given which explain the dynamo action of these motions, and of related flows which support rotating magnetic fields.

Sign in / Sign up

Export Citation Format

Share Document