Numerical studies of magnetic field annihilation

1972 ◽  
Vol 7 (2) ◽  
pp. 293-311 ◽  
Author(s):  
J. C. Stevenson
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Solar Flare ◽  
Electrical Resistance ◽  
Numerical Solutions ◽  
Magnetic Reynolds Number ◽  
Numerical Studies ◽  
Order Of Magnitude ◽  
Field Lines ◽  
Magnitude Estimates

The behaviour of a plasma permeated by a magnetic field, where the field possessess a hyperbolic neutural point, is considered. Results from numerical solutions of the magnetohydrodynamic formulation for such flows are reported. Problems are posed with the solar flare models of Dungey, Sweet & Petschek in mind. No evidence is found to support the idea that compression of the field lines near a hyperbolic null, in the presence of electrical resistance, can radically alter the geometry of those field lines (e.g. the formation of switch-off shocks). These computations do show that, for large values of the magnetic Reynolds number, a rate of annihilation, more rapid than that derived from order-of-magnitude estimates, is possible.

A physical description of magnetic helicity evolution in the presence of reconnection lines

1991 ◽  
Vol 46 (1) ◽  
pp. 179-199 ◽  
Author(s):  
Andrew N. Wright ◽  
Mitchell A. Berger
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Magnetic Reynolds Number ◽  
Magnetic Helicity ◽  
Two Dimensional ◽  
Flux Tubes ◽  
Physical Description ◽  
Reconnection Region ◽  
Field Geometry ◽  
Field Lines

The dissipation of relative magnetic helicity due to the presence of a resistive reconnection region is considered. We show that when the reconnection region has a vanishing cross-section, helicity is conserved, in agreement with previous studies. It is also shown that in two-dimensional systems reconnection can produce highly twisted reconnected flux tubes. Reconnection at a high magnetic Reynolds number generally conserves helicity to a good approximation. However, reconnection with a small Reynolds number can produce significant dissipation of helicity. We prove that helicity dissipation in two-dimensional configurations is associated with the retention of some of the inflowing magnetic flux by the reconnection region, vr. When the reconnection site is a simple Ohmic conductor, all of the magnetic field parallel to the reconnection line that is swept into vr is retained. (In contrast, the inflowing magnetic field perpendicular to the line is annihilated.) We are able to relate the amount of helicity dissipation to the retained flux. A physical interpretation of helicity dissipation is developed by considering the diffusion of magnetic field lines through vr. When compared with helicity-conserving reconnection, the two halves of a reconnected flux sheet appear to have slipped relative to each other parallel to the reconnection line. This provides a useful method by which the reconnected field geometry can be constructed: the incoming flux sheets are ‘cut’ where they encounter vr, allowed to slip relative to each other, and then ‘pasted’ together to form the reconnected flux sheets. This simple model yields estimates for helicity dissipation and the flux retained by vr in terms of the amount of slippage. These estimates are in agreement with those expected from the governing laws.

A re-examination of the role of magnetic field lines in electromagnetic induction

1988 ◽  
Vol 66 (3) ◽  
pp. 245-248
Author(s):  
D. H. Boteler
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Field Line ◽  
Electric Fields ◽  
Electromagnetic Induction ◽  
Magnetic Reynolds Number ◽  
Unified View ◽  
Field Lines ◽  
The Magnetic Field

By adopting a view of magnetic fields, originally proposed by Faraday, in which the magnetic field changes by a movement of field lines, it is shown that a changing magnetic field can be described by the relation [Formula: see text] where v is the velocity of the magnetic field lines. These field-line velocities are shown to be the same as material velocities in conditions of infinite magnetic Reynolds number. The "moving field-line" view provides a phenomenological model of a changing magnetic field that is useful in electromagnetic induction studies. It also allows for a unified view of electromagnetic induction in which all induced electric fields can be explained by the v × B force alone.

Flow induced in a cylindrical column by a uniformly rotating magnetic field

1967 ◽  
Vol 297 (1451) ◽  
pp. 558-563
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Reynolds Number ◽  
High Frequency ◽  
Fourier Transforms ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Rotating Magnetic Field ◽  
Cylindrical Column ◽  
Field Lines

Under laboratory conditions, the magnetic Reynolds number is quite small in a conductor, but can be made appreciable if a high frequency rotating field is applied. Moffatt investigated this problem for high magnetic Reynolds numbers and concluded that there existed a magnetic boundary layer due to spiralling of field lines. Applying Fourier transforms and solving the corrected equations, we find that at low magnetic Reynolds numbers the field lines uniformly penetrate the cylindrical column and do not exhibit any appreciable spiralling. The rotation opposes the drift due to conductivity which is evened out as one proceeds from the centre to the surface. This uniform behaviour persists for small magnetic Reynolds number inside and outside. When the magnetic Reynolds number becomes large, of the order of 100 (say), the field lines passing through the axis of the cylinder exhibit spiralling as suggested by Moffatt since the diffusion is unable to counterbalance the rotational effects.

Turbulent dynamo action at low magnetic Reynolds number

1970 ◽  
Vol 41 (2) ◽  
pp. 435-452 ◽  
Author(s):  
H. K. Moffatt
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Dynamo Action ◽  
Ohmic Dissipation ◽  
Magnetic Energy Density ◽  
Magnetic Diffusivity

The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale l of the turbulence is considered. There are no external sources for the field, and in the absence of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number Rm = u0l/λ (where u0 is the root-mean-square velocity and λ the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities l/L and Rm, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order Rm−2l and greater. In the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t−½exp (α2t/2λ3) where α(= O(u02l)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

Minimum-drag shapes in magnetohydrodynamics

2011 ◽  
Vol 467 (2136) ◽  
pp. 3371-3392 ◽  
Author(s):  
Michael Zabarankin
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Optimality Condition ◽  
Boundary Integral Equations ◽  
Integral Formula ◽  
Magnetic Reynolds Number ◽  
Boundary Integral ◽  
The Body ◽  
Mhd Equations ◽  
Minimum Drag

A necessary optimality condition for the minimum-drag shape for a non-magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained. It is assumed that the flow and magnetic field are uniform and parallel at infinity, and that the body and fluid have the same magnetic permeability. The condition is derived based on the linearized magnetohydrodynamic (MHD) equations subject to a constraint on the body’s volume, and generalizes the existing optimality conditions for the minimum-drag shapes for the body in the Stokes and Oseen flows of a non-conducting fluid. It is shown that for any Hartmann number M , Reynolds number Re and magnetic Reynolds number Re m , the minimum-drag shapes are fore-and-aft symmetric and have conic vertices with an angle of 2 π /3. The minimum-drag shapes are represented in a function-series form, and the series coefficients are found iteratively with the derived optimality condition. At each iteration, the MHD problem is solved via the boundary integral equations obtained based on the Cauchy integral formula for generalized analytic functions. With respect to the equal-volume sphere, drag reduction as a function of the Cowling number S= M 2 /( Re m   Re ) is smallest at S=1. Also, in the considered examples, the drag values for the minimum-drag shapes and equal-volume minimum-drag spheroids are sufficiently close.

Critical Reynolds Number Estimates by Thermo-Dynamic (Stochastic) Methods

1974 ◽  
Vol 96 (3) ◽  
pp. 788-794
Author(s):  
P. Hrycak ◽  
M. J. Levy
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Stochastic Methods ◽  
Shear Stresses ◽  
Statistical Probability ◽  
Average Shear ◽  
Order Of Magnitude ◽  
Dynamic Stochastic ◽  
Magnitude Estimates

Methods based on fundamental thermodynamic principles and the notion of statistical probability have been used to estimate the point of instability and the lower critical Reynolds number for a round pipe and an infinite channel. It is also shown that order of magnitude estimates of the ratio of the average shear stresses for each regime allow one to draw definite conclusions about the lower and the upper critical Reynolds number in a variety of geometries.

scholarly journals Magnetized Neutron Stars

2017 ◽  
Vol 45 ◽  
pp. 1760032
Author(s):  
Gibran H. de Souza ◽  
Ernesto Kemp ◽  
Cecilia Chirenti
Keyword(s):  
Magnetic Field ◽  
Neutron Star ◽  
Neutron Stars ◽  
Numerical Solutions ◽  
Internal Magnetic Field ◽  
Shafranov Equation ◽  
Field Lines

In this work we show the results for numerical solutions of the relativistic Grad-Shafranov equation for a typical neutron star with 1.4 solar masses. We have studied the internal magnetic field considering both the poloidal and toroidal components, as well as the behavior of the field lines parametrized by the ratio between these components of the field.

A theoretical study of the effects of wall conductivity, non-uniform magnetic fields and variable-area ducts on liquid-metal flows at high Hartmann number

1978 ◽  
Vol 84 (3) ◽  
pp. 471-495 ◽  
Author(s):  
Richard J. Holroyd ◽  
John S. Walker
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Field Strength ◽  
Hartmann Number ◽  
Reverse Flow ◽  
Free Fluid ◽  
Uniform Field ◽  
Uniform Region ◽  
Field Lines ◽  
Straight Duct

Flows of incompressible, electrically conducting liquids along ducts with electrically insulating or weakly conducting walls situated in a strong magnetic field are analysed. Except over a short length along the duct where the magnetic field strength and/or the duct cross-sectional area vary, the duct is assumed to be straight and the field to be uniform and aligned at right angles to the duct. Magnitudes of the field strength B0 and the mean velocity V are taken to be such that the Hartmann number M [Gt ] 1, the interaction parameter N (= M2/Re) [Gt ] 1 (Re being the Reynolds number of the flow) and the magnetic Reynolds number Rm [Lt ] 1.For an O(1) change in the product VB0 along the duct across the non-uniform region, it is shown that:(i) In the non-uniform region the streamlines and current flow lines follow surfaces containing the field lines satisfying $\int B^{-1}ds = {\rm constant}$, the integration being carried out along the field line within the duct; these surfaces are equipotentials and isobarics. This leads to(ii) a tube of stagnant, but not current-free fluid at the centre of the duct parallel to the field lines around which the flow divides to bypass it. To accommodate this flow,(iii) the usual uniform field/straight duct flow is disturbed over very large distances upstream and downstream of this region, the maximum length O(duct radius × M½) occurring in a non-conducting duct;(iv) a large pressure drop is introduced into the pressure distribution regardless of the direction of the flow, the effect being most severe in a non-conducting duct, where the drop is O(duct radius × (uniform field/straight duct pressure gradient) × M½);(v) in the part of the duct with the lower value of VB0 a region of reverse flow occurs near the centre of the duct and the stagnant fluid.

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