Determining the magnetic field in the core–mantle boundary zone by non-harmonic downward continuation

Summary The length of day and the geomagnetic field are clearly correlated over decadal periods (10–100 yr). Provided the electrical conductivity of the lower mantle is sufficiently high, a considerable part of this correlation can be explained by electromagnetic core–mantle coupling. Investigating the associated core–mantle coupling torque and fluid velocity fields near the core surface, as well as the interpretation of the observed time lag between length of day and geomagnetic field variations requires the calculation of the temporally variable magnetic field near both sides of and on the core–mantle boundary by solving the magnetic induction equation. Such a solution presents a downward field continuation that has a non-harmonic character if the electrical conductivity is accounted for. In this paper the Earth's mantle is assumed to be a two-layer spherical shell, whose inner layer is electrically conductive. We only consider the poloidal part of the magnetic field with boundary values that are conventionally given by a spherical harmonic expansion of the observed geomagnetic potential field at the Earth's surface. Thus, we are concerned with a one-side-data supported problem, analogous to the inverse heat conduction problem (or sideways heat equation problem), well-known as an ill-posed inverse boundary value problem for a parabolic partial differential equation (diffusion equation). We develop a several-step solution procedure for this inverse problem in its integral form and use a special regularization method for the final solution. The capabilities of this downward continuation method (which includes varying the mantle conductivity model, quality of data approximation in the regularization and two different depths) are presented and discussed in comparison with the perturbation approach and the usual harmonic downward continuation. The data series used are the single magnetic field components (Gauss coefficients) of the spherical harmonic field expansion (Bloxham & Jackson 1992, J. geophys. Res., 97, 19 537–19 563) beginning from the year 1840 to 1990. In addition, to investigate the spectral effects (changing amplitudes, phase shifting), an artificial data series is used. The main result is the downward continuation of the radial component of a global (8,8) field onto the core–mantle boundary, produced using the different methods and for the two epochs, 1910 and 1960. Comparing the results with the perturbation solution reveals temporally and locally variable differences up to the order of 5000 nT, while the difference to the harmonic downward continuation amounts to 15000 nT.

[1]  R. Gorenflo,et al.  A noncharacteristic cauchy problem for the heat equation , 1991, Acta Applicandae Mathematicae.

[2]  H. Greiner-Mai.  Decade Variations of the Earth's Rotation and Geomagnetic Core-Mantle Coupling , 1993 .

[3]  P. Roberts,et al.  Time-dependent electromagnetic core-mantle coupling , 1984 .

[4]  E. R. Benton,et al.  Rapid diffusion of the poloidal geomagnetic field through the weakly conducting mantle: a perturbation solution , 1983 .

[5]  Lars Eldén,et al.  The Numerical Solution of a Non-Characteristic Cauchy Probelm for a Parabolic Equation , 1983 .

[6]  J. Achache,et al.  Long-period geomagnetic variations and mantle conductivity: an inversion using Bailey's method , 1981 .

[7]  D. Hào.  A Noncharacteristic Cauchy Problem for Linear Parabolic Equations I: Solvability , 1995 .

[8]  M. Matsushima,et al.  Electromagnetic response of the mantle to long-period geomagnetic variations over the globe , 1998 .

[9]  S. Constable.  Constraints on Mantle Electrical Conductivity from Field and Laboratory Measurements , 1993 .

[10]  Rak-Joong Kim.  NON-CHARACTERISTIC CAUCHY PROBLEM FOR A PARABOLIC EQUATION , 1993 .

[11]  D. Stromeyer,et al.  Downward continuation of heat flow data by means of the least squares method , 1984 .

[12]  J. Poirier,et al.  Electrical conductivity of the Earth's lower mantle , 1989, Nature.

[13]  B. Buffett,et al.  STRATIFICATION OF THE OUTER CORE AT THE CORE-MANTLE BOUNDARY , 1998 .

[14]  Jeremy Bloxham,et al.  Time‐dependent mapping of the magnetic field at the core‐mantle boundary , 1992 .

[15]  H. Greiner-Mai.  The influence of the electromagnetic core-mantle coupling torques on earth's rotation , 1987 .

[16]  A. Tsutsumi.  A remark on the uniqueness of the non-characteristic Cauchy problem for equations of parabolic type , 1965 .

[17]  Heinz W. Engl,et al.  Stability estimates and regularization for an inverse heat conduction prolem in semi - infinite and finite time intervals , 1989 .

[18]  D. Barraclough.  Spherical harmonic models of the geomagnetic field , 1978 .

[19]  Peter Knabner,et al.  Stabilization of ill-posed Cauchy problems for parabolic equations , 1987 .

[20]  V. M. Fishman,et al.  Spectral-statistical spatial analysis of 60- and 30-year geomagnetic field variations and conductivity of the lower mantle. , 1985 .

[21]  P. Knabner,et al.  The optimal stability estimate for some ill‐posed Cauchy problems for a parabolic equation , 1988 .

[22]  Mioara Mandea,et al.  Ørsted Initial Field Model , 2000 .

[23]  Jean-Paul Poirier,et al.  Does infiltration of core material into the lower mantle affect the observed geomagnetic field , 1992 .

[24]  M. Rochester,et al.  Geomagnetic westward drift and irregularities in the Earth's rotation , 1960, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[25]  Christoph Reigber,et al.  Status of the CHAMP Mission , 2000 .

[26]  L. Rosenhead.  Conduction of Heat in Solids , 1947, Nature.

[27]  M. G. Mcleod.  Magnetospheric and ionospheric signals in magnetic observatory monthly means: Electrical conductivity of the deep mantle , 1994 .

[28]  Gauthier Hulot,et al.  An estimate of average lower mantle conductivity by wavelet analysis of geomagnetic jerks , 1999 .

[29]  S. Braginsky.  Dynamics of the stably stratified ocean at the top of the core , 1999 .

[30]  J. Bloxham.  Simple models of fluid flow at the core surface derived from geomagnetic field models , 1989 .

[31]  D. E. Smylie.  Magnetic Diffusion in a Spherically-Symmetric Conducting Mantle , 1965 .