We describe a new technique for implementing the constraints on magnetic fields arising from two hypotheses about the fluid core of the Earth, namely the frozen-flux hypothesis and the hypothesis that the core is in magnetostrophic force balance with negligible leakage of current into the mantle. These hypotheses lead to time-independence of the integrated flux through certain 'null-flux patches' on the core surface, and to time-independence of their radial vorticity. Although the frozen-flux hypothesis has received attention before, constraining the radial vorticity has not previously been attempted. We describe a parametrization and an algorithm for preserving topology of radial magnetic fields at the core surface while allowing morphological changes. The parametrization is a spherical triangle tesselation of the core surface. Topology with respect to a reference model (based on data from the Oersted satellite) is preserved as models at different epochs are perturbed to optimize the fit to the data; the topology preservation is achieved by the imposition of inequality constraints on the model, and the optimization at each iteration is cast as a bounded value least-squares problem. For epochs 2000, 1980, 1945, 1915 and 1882 we are able to produce models of the core field which are consistent with flux and radial vorticity conservation, thus providing no observational evidence for the failure of the underlying assumptions. These models are a step towards the production of models which are optimal for the retrieval of frozen-flux velocity fields at the core surface.
A major limitation in the analysis of physical quantities measured from a stratigraphic core is incomplete knowledge of the depth to age relationship for the core. Records derived from diverse locations are often compared or combined to construct records that represent a global signal. Time series analysis of individual or combined records is commonly employed to seek quasi-periodic components or characterize the timescales of relevant physical processes. Assumptions that are frequently made in the approximation of depth to age relationships can have a dramatic and harmful effect on the spectral content of records from stratigraphic cores. A common procedure for estimating ages in a set of samples from a stratigraphic core is to assign, based on complementary data, the ages at a number of depths (tie points) and then assume a uniform accumulation rate between the tie points. Imprecisely dated or misidentified tie points and naturally varying accumulation rates give rise to discrepancies between the inferred and the actual ages of a sample. We develop a statistical model for age uncertainties in stratigraphic cores that treats the true, but in practice unknown, ages of core samples as random variables. For inaccuracies in the ages of tie points, we draw the error from a zero-mean normal distribution. For a variable accumulation rate, we require the actual ages of a sequence of samples to be monotonically increasing and the age errors to have the form of a Brownian bridge. That is, the errors are zero at the tie points. The actual ages are modeled by integrating a piecewise constant, randomly varying accumulation rate. In each case, our analysis yields closed form expressions for the expected value and variance of resulting errors in age at any depth in the core. By Monte Carlo simulation with plausible parameters, we find that age errors across a paleomagnetic record due to misdated tie points are likely of the same order as the tie point discrepancies. Those due to accumulation rate variations can be as large as 30 kyr, but are probably less than 10 kyr. We provide a method by which error estimates like these can be made for similar stratigraphic dating problems and apply our statistical model to an idealized marine sedimentary paleomagnetic record. Both types of errors severely degrade the spectral content of the inferred record. We quantify these effects using realistic tie point ages, their uncertainties and depositional parameters. (C) 2002 Elsevier Science B.V. All rights reserved.
We present a statistical analysis of magnetic fields simulated by the Glatzmaier-Roberts dynamically consistent dynamo model. For four simulations with distinct boundary conditions, means, standard deviations, and probability functions permit an evaluation based on existing statistical paleosecular variation (PSV) models. Although none closely fits the statistical PSV models in all respects, some simulations display characteristics of the statistical PSV models in individual tests. We also find that nonzonal field statistics do not necessarily reflect heat flow conditions at the core-mantle boundary. Multitaper estimates of power and coherence spectra allow analysis of time series of single, or groups of, spherical harmonic coefficients representing the magnetic fields of the dynamo simulations outside the core. Sliding window analyses of both power and coherence spectra from two of the simulations show that a 100 kyr averaging time is necessary to realize stationary statistics of their nondipole fields and that a length of 350 kyr is not long enough to full characterize their dipole fields. Spectral analysis provides new insight into the behavior and interaction of the dominant components of the simulated magnetic fields, the axial dipole and quadrupole. Although we find spectral similarities between several reversals, there is no evidence of signatures that can be conclusively associated with reversals or excursions. We test suggestions that during reversals there is increased coupling between groups of spherical harmonic components. Despite evidence of coupling between antisymmetric and symmetric spherical harmonics in one simulation, we conclude that it is rare and not directly linked to reversals. In contrast to the reversal model of R. T. Merrill and P. L. McFadden, we demonstrate that the geomagnetic power in the dipole part of the dynamo simulations is either relatively constant or fluctuates synchronously with that of the nondipole part and that coupling between antisymmetric and symmetric components occurs when the geomagnetic power is high.
Techniques for modelling the geomagnetic field at the surface of Earth's core often penalize contributions at high spherical harmonic degrees to reduce the effect of mapping crustal fields into the resulting field model at the core-mantle boundary (CMB). Ambiguity in separating the observed field into crustal and core contributions makes it difficult to assign error bounds to core field models, and this makes it hard to test hypotheses that involve pointwise values of the core field. The frozen-flux hypothesis, namely that convective terms dominate diffusive terms in the magnetic-induction equation, requires that the magnetic flux through every patch on the core surrounded by a zero contour of the radial magnetic field remains constant, although the shapes, areas and locations (but not the topology) of these patches may change with time. Field models exactly satisfying the conditions necessary for the hypothesis have not yet been constructed for the early part of this century. We show that such models must exist, so testing the frozen-flux hypothesis becomes the question of whether the models satisfying it are geophysically unsatisfactory on other grounds, for example because they are implausibly rough or complicated. We introduce an algorithm to construct plausible fleld models satisfying the hypothesis, and present such models for epochs 1945.5 and 1980. Our algorithm is based on a new parametrization of the field in terms of its radial component B(r) at the CMB. The model consists of values of B(r) at a finite set of points on the CMB, together with a rule for interpolating the values to other points. The interpolation rule takes the specified points to be the vertices of a spherical triangle tessellation of the CMB, with B(r) varying linearly in the gnomonic projections of the spherical triangles onto planar triangles in the planes tangent to the centroids of the spherical triangles. This parametrization of B(r) provides a direct means of constraining the integral invariants required by the frozen-flux hypothesis. Using this parametrization, we have constructed field models satisfying the frozen-flux hypothesis for epochs 1945.5 and 1980, while fitting observatory and survey data for 1945.5 and Magsat data for 1980. We use the better constrained 1980 CMB field model as a reference for 1945.5: we minimize the departure of the 1945.5 CMB field model from a regularized 1980 CMB field model, while constraining the 1945.5 model to have the same null-flux curves and flux through those curves as the 1980 model. The locations, areas and shapes of the curves are allowed to change. The resulting 1945.5 CMB field model is nearly as smooth as that for 1980, fits the data adequately, and satisfies the conditions necessary for the frozen-flux hypothesis.