The electric field produced in the mantle by the dynamo in the core

A rigorous singular perturbation theory is developed to estimate the electric field E produced in the mantle M by the core dynamo when the electrical conductivity σ in M depends only on radius r, and when |r?_rln σ| ? 1 in most of M. It is assumed that σ has only one local minimum in M, either (a) at the Earth's surface ?V, or (b) at a radius b inside the mantle, or (c) at the core-mantle boundary ?K. In all three cases, the region where σ is no more than e times its minimum value constitutes a thin critical layer; in case (a), the radial electric field E_r ≈ 0 there, while in cases (b) and (c), E_r is very large there. Outside the critical layer, E_r ≈ 0 in all three cases. In no case is the tangential electric field E_S small, nearly toroidal, or nearly calculable from the magnetic vector potential A as ??_tA_S. The defects in Muth's (1979) argument which led him to contrary conclusions are identified. Benton (1979) cited Muth's work to argue that the core-fluid velocity u just below ?K can be estimated from measurements on ?V of the magnetic field B and its time derivative $\text{?}{}_{\mathrm{t}}\text{B}$. A simple model for westward drift is discussed which shows that Benton's conclusion is also wrong.In case (a), it is shown that knowledge of σ in M is unnecessary for estimating E_S on ?K with a relative error |r?_r 1nσ|^?1from measurements of E_S on ?V and knowledge of ?_tB in M (calculable from ?_tB on ?V if σ is small). Then, in case (a), u just below ?K can be estimated as $\text{?}\text{r}\text{×}\text{E}{}_{\mathrm{S}}\text{/B}{}_{\mathrm{r}}$. The method is impractical unless the contribution to E_S on ?V from ocean currents can be removed.The perturbation theory appropriate when σ in M is small is considered briefly; smallness of σ and of |r?_r ln σ|^?1 a independent questions. It is found that as σ → 0, B approaches the vacuum field in M but E does not; the explanation lies in the hydromagnetic approximation, which is certainly valid in M but fails as σ → 0. It is also found that the singular perturbation theory for |r?_r ln σ|^?1 is a useful tool in the perturbation calculations for σ when both σ and |r?_r ln σ|^?1 are small.     

A six-parameter statistical model of the Earth's magnetic field

A six-parameter statistical model of the non-dipole geomagnetic field is fitted to 2597 harmonic coefficients determined by Cain, Holter & Sandee (1990) from MAGSAT data. The model includes sources in the core, sources in the crust, and instrument errors. External fields are included with instrument errors. The core and instrument statistics are invariant under rotation about the centre of the Earth, and one of the six parameters describes the deviation of the crustal statistics from rotational invariance. The model treats the harmonic coefficients as independent random samples drawn from a Gaussian distribution. The statistical model of the core field has a correlation length of about 500 km at the core-mantle boundary, too long to be attributed to a white noise source just below the boundary layers at the top of the core. The estimate of instrument errors obtained from the statistical model is in good agreement with an independent estimate based on tests of the instruments (Langel, Ousley & Berbert 1982).     

Propagator matrices in elastic wave and vibration problems

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order and ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerica methods for their computation. When the dispersion relation is somem ^th order minor of the integral matrix it is possible to deal withm ^th minor propagators so that the dispersion relation is a single element of them ^th minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations forSH and forP-SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations forP-SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley-Hamilton theorem and the Lagrange-Sylvester, interpolation formula, are derived.