Edward R. Benton

Influence of an Axial Magnetic Field on the Steady Linear Ekman Boundary Layer

A hydromagnetic version of the Ekman boundary layer is developed in a simple form in order to study how the geophysically important Ekman suction velocity is affected by magnetic fields. The problem treated consists of a viscous, incompressible, conducting fluid in the presence of an infinite, flat, insulating boundary which rotates at speed Ω0. Outside the boundary layer, the fluid rotates uniformly with speed Ω1 = Ω0 (1 + e), and there is a uniform magnetic field aligned with the rotation axis. An expansion in powers of e, the Rossby number, together with von Karman similarity, leads to an exact solution which to first order in e, describes a continuous transition between pure Ekman flow and a rotating analog of Hartmann flow. The magnetic field is found to inhibit Ekman suction; yet, such a boundary layer may still exert a strong influence on the outer flow because of a new feature that replaces the suction, namely an induced axial current outside the boundary layer. This “Hartmann current,” not presen...

On the spin-up of an electrically conducting fluid Part 2. Hydromagnetic spin-up between infinite flat insulating plates

The linear spin-up of a homogeneous electrically conducting fluid confined between infinite flat insulating plates is analyzed for the case in which a uniform magnetic field is applied normal to the boundaries. As in part 1 (Benton & Loper 1969), complete hydromagnetic interaction is embraced even within linearized equations. Approximate inversion of the exact Laplace transform solution reveals the presence of several flow structures: two thin Ekman–Hartmann boundary layers (one on each plate), which are quasi-steady on the time scale of spin-up, two thicker continuously growing magnetic diffusion regions, and an essentially inviscid, current-free core, which may or may not be present on the spin-up time scale, depending upon the growth rate of the magnetic diffusion regions. When a current-free core exists, it is found to spin-up at the same rate as the fluid within magnetic diffusion regions, although different physical mechanisms are at play. As a result, a single hydromagnetic spin-up time is derived, independently of the thickness of magnetic diffusion regions; this time is shorter than in the non-magnetic problem.

Magnetic probing of planetary interiors

Abstract The following general question is addressed: what can be learned about a planetary interior from measurements of the global planetary magnetic field at (or near) its surface? The discussion is placed in the context of Earth, for clarity, but the considerations apply to terrestrial planets in general (so long as the observed magnetism is either predominantly of internal origin, or else external source effects can be successfully filtered out of the observations). Attention is given to the idealized but typical situation of a rotating but spherically symmetric planet containing a highly conducting uniform fluid core surrounded by a nearly insulating rigid mantle, whose conductivity, a function of at most radius only, falls monotonically from its largest value at the base of the planetary mantle to zero at the planetary surface; the largest value of mantle conductivity as well as the mean value for the whole mantle and the mantle conductance are assumed small compared to the corresponding values of the core. Exterior to the planet is vacuum in the sense of an electrically uncharged insulator. The core fluid is inviscid, Boussinesq and gravitationally driven. Complete and perfect observations of either the instantaneous internal vector magnetic field together with its secular variation at a single epoch, or more realistically, the instantaneous internal vector magnetic field alone at two separated epochs are presumed available; the time separation between measurement epochs is long compared the Ohmic diffusion time of the planetary mantle, but small compared to that of the liquid core. Under such circumstances we describe how information about each of the following planetary properties can, in principle (though not without practical difficulty) be retrieved from the observations: (1) depth of the core-mantle boundary (a result of Hide); (2) depth to the current and motion sources responsible for the planetary dynamo; (3) presence or absence of small-scale turbulence in the upper reaches of the core; (4) large-scale horizontal fluid motion at the top of the core; (5) strength of horizontal currents, zonal magnetic fields, Coriolis and Lorentz forces at the top of the core; and (6) current system in the mantle and strength of electromagnetic core-mantle coupling.

On the strength of electric currents and zonal magnetic fields at the top of the Earth's core: Methodology and preliminary estimates

Abstract A new method is described for estimating: (a) the meridional electric current density, j θ , (b) the vertical growth rate of the zonal magnetic field, ∂B φ / ∂r , or its scale-height, B φ / ∂B φ / ∂r ) and (c) the vertical growth rate of the vertical current density, ∂j r / ∂r , at a few isolated points on the top surface of the Earths core from observations of the internal geomagnetic field at the Earths surface. The theoretical technique rests on combining unaccelerated, gravitationally-driven Boussinesq fluid dynamics of the core with frozen-flux electromagnetism, the mantle being treated as a spherically symmetric insulator. Insertion into this theory of main field models for epochs 1965, 1975 leads to preliminary values for these quantities of magnitude: (a) j θ ∼ 1 A / m 2 , (b) ∂B φ / ∂r ∼ 10 −6 T/m or B φ /( ∂B φ / ∂r ) ∼ 10 m, (c) ∂j r / ∂r ∼ 10 −6 A/m 3 . Some geophysical implications of these estimates are discussed.

On the coupling of fluid dynamics and electromagnetism at the top of the earth's core

Abstract A kinematic approach to short-term geomagnetism has recently been based upon pre-Maxwell frozen-flux electromagnetism. A complete dynamic theory requires coupling fluid dynamics to electromagnetism. A geophysically plausible simplifying assumption for the vertical vorticity balance, namely that the vertical Lorentz torque is negligible, is introduced and its consequences are developed. The simplified coupled magnetohydrodynamic system is shown to conserve a variety of magnetic and vorticity flux integrals. These provide costraints on eligible models for the geomagnetic main field, its secular variation, and the horizontal fluid motions at the top of the core, and so permit a number of tests of the underlying assumptions.

Geomagnetic field modeling incorporating constraints from frozen-flux electromagnetism

Abstract A spherical harmonic representation of the geomagnetic field and its secular variation for epoch 1980, designated GSFC(9/84), is derived and evaluated. At three epochs (1977.5, 1980.0, 1982.5) this model incorporates conservation of magnetic flux through five selected patches of area on the core-mantle boundary bounded by the zero contours of vertical magnetic field. These 15 nonlinear constraints are included like data in an iterative least squares parameter estimation procedure that starts with the recently derived unconstrained field model designated GSFC(12/83) (Langel and Estes). Convergence is approached within three iterations. The constrained model is evaluated by comparing its predictive capability outside the time span of its data, in terms of residuals at magnetic observatories, with that for the unconstrained model. The new model demonstrates significantly improved predictability. Next, it is established that the flux of magnetic secular variation out of the northern (or southern) geographic hemisphere of the core-mantle boundary is nearly conserved by a remake of the field model designated GSFC(9/80). The GSFC(9/84) model is then examined and found to satisfy this independent linear constraint on secular variation very well.

Inviscid, frozen-flux velocity components at the top of earth's core from magnetic observations at earth's surface: Part 1. A new methodology

Abstract Two alternative methods are described for obtaining inviscid velocity components at the surface of earths liquid core, assumed perfectly conducting, given perfect and complete measurements of the geomagnetic main field and secular variation at earths surface. The mantle is treated as a spherically-symmetric insulator for purposes of extrapolation to the core. Both methods utilize the vertical component of the induction equation and require one-dimensional interpolation along special curves on the core surface as the initial step. For the next step, the first method then utilizes the two horizontal components of the induction equation, whereas the second method relies on the horizontal components of Ohms law. Both methods work within the confines of the ambiguity elucidated by Backus (1968) but nonetheless can still yield results of value, because the two horizontal velocity components are determined separately and at distinct locations. A brief comparison of the two methods suggests that the o...

Vorticity dynamics in spin‐up from rest

Impulsively started spin‐up from an initial state of rest for incompressible fluid in a circular cylinder is examined from the viewpoint of the vorticity field. Existing theoretical models of this flow are thought to have implausible vorticity dynamics.

Testing recent geomagnetic field models via magnetic flux conservation at the core-mantle boundary

Abstract The 28 candidate main field models submitted for the International Geomagnetic Reference Field Model Revision 1985 reveal that a dramatic increase in the rate of decline of the Earths total pole strength began around 1960. Nevertheless, all but one field model conserves the absolute magnetic flux linking the core-mantle boundary (CMB) fairly well during the 40 year time span 1945–1985. This supports the frozen-flux approximation on global length and decade time scales. Geostrophic fluid motion at the top of a frozen-flux core conserves the total magnetic flux linking either geographic hemisphere of the CMB. This constraint is poorly satisfied by the candidate secular variation (SV) models. Yet other SV models truncated at higher degree, and models based upon temporal interpolation rather than extrapolation, satisfy this constraint very well. These results lead us to make several recommendations for the 1985 and future IGRF revisions.

A simple method for determining the vertical growth rate of vertical motion at the top of earth's outer core

Abstract A simple new method is described for extracting, from magnetic observations taken at Earths surface, the vertical growth rate of vertical motion, ∂u/∂r, at special isolated points on the top surface of Earths liquid core. The technique utilizes only the radial component of the frozen-flux induction equation and it requires information only on the radial magnetic field, Br, its horizontal gradient, and its secular variations, ∂Br/∂t, at the core-mantle boundary.