Time-dependent kinematic dynamos with stationary flows

1989 ◽  
Vol 425 (1869) ◽  
pp. 407-429 ◽  
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Magnetic Reynolds Number ◽  
Velocity Models ◽  
Stationary Flows ◽  
Induction Equation ◽  
Steady Solutions ◽  
Simple Flows ◽  
Magnetic Induction Equation ◽  
Stationary Velocity

Numerical solutions to the magnetic induction equation in a sphere have been obtained for a number of stationary velocity models. By searching for non-steady magnetic fields and in some circumstances showing that all magnetic field modes decay, the inability of several earlier researchers to find convergent steady solutions is explained. Results of previous authors are generally confirmed, but also extended to cover non-steady fields, different values of magnetic Reynolds number and other parameters, and higher truncation limits. Some non-decaying fields are found where only decaying or non-convergent results have previously been reported. Two flows ∊ s 0 2 + t 0 2 and ∊ s 0 2 + t 0 1 , each consisting of two very simple axisymmetric rolls are seen to sustain growing fields provided that (i) the magnetic Reynolds number R and the poloidal to toroidal flow ratio ∊ are of appropriate magnitudes, and (ii) the meridional s 0 2 flow is directed inwards along the equatorial plane and out towards the poles. An even simpler axisymmetric single roll flow ∊ s 0 1 + t 0 1 is also seen to support growing fields for appropriate ∊ and R . These simple flows dispel the somewhat prevalent belief that dynamo maintenance relies on the supporting flow being complex, and having length scale significantly less than that of the conducting fluid volume.

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.

Planar velocity dynamos in a sphere

2006 ◽  
Vol 462 (2072) ◽  
pp. 2439-2456 ◽  
Author(s):  
A.A Bachtiar ◽  
D.J Ivers ◽  
R.W James
Keyword(s):  
Magnetic Field ◽  
Outer Core ◽  
Magnetic Reynolds Number ◽  
Main Magnetic Field ◽  
Dynamo Action ◽  
Field Growth ◽  
Planar Velocity ◽  
Induction Equation ◽  
Planar Flows ◽  
Magnetic Induction Equation

The Earth's main magnetic field is generally believed to be due to a self-exciting dynamo process in the Earth's fluid outer core. A variety of antidynamo theorems exist that set conditions under which a magnetic field cannot be indefinitely maintained by dynamo action against ohmic decay. One such theorem, the Planar Velocity Antidynamo Theorem , precludes field maintenance when the flow is everywhere parallel to some plane, e.g. the equatorial plane. This paper shows that the proof of the Planar Velocity Theorem fails when the flow is confined to a sphere, due to diffusive coupling at the boundary. Then, the theorem reverts to a conjecture. There is a need to either prove the conjecture, or find a functioning planar velocity dynamo. To the latter end, this paper formulates the toroidal–poloidal spectral form of the magnetic induction equation for planar flows, as a basis for a numerical investigation. We have thereby determined magnetic field growth rates associated with various planar flows in spheres. For most flows, the induced magnetic field decays with time, supporting a planar velocity antidynamo conjecture for a spherical conducting fluid. However, one flow is exceptional, indicating that magnetic field growth can occur. We also re-examine some classical kinematic dynamo models, converting the flows where possible to planar flows. For the flow of Pekeris et al . (Pekeris, C. L., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the Earth's magnetic field. Phil. Trans. R. Soc. A 275 , 425–461), this conversion dramatically reduces the critical magnetic Reynolds number.

Numerical models of hydromagnetic dynamos

1975 ◽  
Vol 67 (4) ◽  
pp. 625-646 ◽  
Author(s):  
S. A. Jepps
Keyword(s):  
Large Scale ◽  
Numerical Models ◽  
Fluid Flows ◽  
Magnetic Reynolds Number ◽  
Small Scale ◽  
Initial Field ◽  
Finite Difference Equations ◽  
Induction Equation ◽  
Scale Turbulence ◽  
Magnetic Induction Equation

The magnetic induction equation is solved numerically in a sphere for a variety of prescribed fluid flows. The models considered are the so-called ‘αω dynamos’, in which both small-scale turbulence and large-scale shearing play a significant role. Solutions are obtained by marching the finite–difference equations forward in time from some initial field. For a critical value of the magnetic Reynolds numberRmsolutions which neither grow nor decay are found.Further calculations are performed with a ‘cut-off effect’ in which an attempt is made to simulate the effect of the Lorentz forces on the turbulence. For supercritical values of R, the magnetic field now stabilizes a t a finite value instead of increasing indefinitely. The form of the final field is compared with that produced at criticalRmin the absence of the cut-off effect.

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.

scholarly journals Azimuthal magnetorotational instability with super-rotation

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
G. Rüdiger ◽  
M. Schultz ◽  
M. Gellert ◽  
F. Stefani
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Numerical Solutions ◽  
Hartmann Number ◽  
Outer Cylinder ◽  
Magnetic Reynolds Number ◽  
Neutral Stability ◽  
Magnetorotational Instability ◽  
Magnetic Prandtl Number ◽  
The Stability

It is demonstrated that the azimuthal magnetorotational instability (AMRI) also works with radially increasing rotation rates contrary to the standard magnetorotational instability for axial fields which requires negative shear. The stability against non-axisymmetric perturbations of a conducting Taylor–Couette flow with positive shear under the influence of a toroidal magnetic field is considered if the background field between the cylinders is current free. For small magnetic Prandtl number $Pm\rightarrow 0$ the curves of neutral stability converge in the (Hartmann number,Reynolds number) plane approximating the stability curve obtained in the inductionless limit $Pm=0$. The numerical solutions for $Pm=0$ indicate the existence of a lower limit of the shear rate. For large $Pm$ the curves scale with the magnetic Reynolds number of the outer cylinder but the flow is always stable for magnetic Prandtl number unity as is typical for double-diffusive instabilities. We are particularly interested to know the minimum Hartmann number for neutral stability. For models with resting or almost resting inner cylinder and with perfectly conducting cylinder material the minimum Hartmann number occurs for a radius ratio of $r_{\text{in}}=0.9$. The corresponding critical Reynolds numbers are smaller than $10^{4}$.

Inversion of the geomagnetic induction problem

1970 ◽  
Vol 315 (1521) ◽  
pp. 185-194 ◽  
Keyword(s):  
Frequency Spectrum ◽  
Real Data ◽  
Real Analytic Function ◽  
Numerical Application ◽  
Geomagnetic Induction ◽  
Induction Equation ◽  
Real Analytic ◽  
Principle Of Causality ◽  
Magnetic Potentials ◽  
Magnetic Induction Equation

An algorithm has been found for inverting the problem of geomagnetic induction in a con­centrically stratified Earth. It determines the (radial) conductivity distribution from the frequency spectrum of the ratio of internal to external magnetic potentials of any surface harmonic mode. The derivation combines the magnetic induction equation with the principle of causality in the form of an integral constraint on the frequency spectrum. This algorithm generates a single solution for the conductivity. This solution is here proved unique if the conductivity is a bounded, real analytic function with no zeros. Suggestions are made regarding the numerical application of the algorithm to real data.

Rotating elliptic cylinders in a viscous fluid at rest or in a parallel stream

1977 ◽  
Vol 79 (1) ◽  
pp. 127-156 ◽  
Author(s):  
Hans J. Lugt ◽  
Samuel Ohring
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Elliptic Cylinder ◽  
The Body ◽  
Oscillatory Behaviour ◽  
Incompressible Fluid Flow ◽  
Transient Period ◽  
Parallel Stream ◽  
Rotating Bodies

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

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.

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