ALBERT

Notes: It is sometimes desirable to calculate a magnetic field model when only measurements of the direction of the magnetic vector are available. Obviously any such model is indeterminate to an arbitrary constant scalar multiplier; here we investigate the possibility of further ambiguities when the data are perfect and complete. In two dimensions (outside an infinite cylinder with no variation along its axis) we demonstrate the existence of a 2n-parameter family of solutions for the field, where (n + 2) is the number of rotations of the direction of the field vector during one circuit of the bounding circle. In three dimensions (the field outside a sphere) we show by example that many-parameter families of solutions are possible for axisymmetric fields of octupole type (two lines of off-axis dip-poles). These results place the status of the direction problem on the same level as the total intensity problem: in both cases the solutions are known to be non-unique through examples, but no general rule is available to classify the ambiguities. However, the direction problem is homogeneous and we are able to give a method for investigating uniqueness from a local analysis about one particular solution. We argue that field models represented by truncated spherical harmonic expansion with fixed truncation level will be unique (except for some possible special cases), but that severe ambiguities in such models can be investigated by the usual methods of linearized inversion theory to provide a numerical demonstration of qualified ‘uniqueness’. Several‘realistic’field models are studied and all are found to be‘unique’(in the above sense) to within the arbitrary multiplier; it is most probable that none of the field models published so far suffers from the type of non-uniqueness discovered in our example.

Notes: Geomagnetic secular variation is caused by flow of liquid iron in the core. Geomagnetic observations can be used to determine properties of the flow but such calculations in general have non-unique solutions. We prove a uniqueness theorem: the flow is determined uniquely if it is toroidal (zero horizontal divergence), the mantle is an insulator, the core a perfect conductor (the frozen-flux hypothesis), and there is no surface current in the boundary layer at the top of the core, and provided the magnetic field satisfies a simple point condition. The condition of no surface current allows use of the horizontal components of secular variation; previous studies have used only the radial component. Horizontal components allow simultaneous determination of the shear (radial derivatives of horizontal components of velocity).We have devised a new numerical method for determining core flows based on a discrete vector spherical transform (DVST) which exactly transforms between a grid of points on the surface of a sphere and a truncated spherical harmonic series. The transform method is faster than methods involving Gaunt and Elsasser integrals and eliminates most of the lengthy algebra. We determine a flow from spherical harmonic models of main field and secular variation for epoch 1970 which satisfies more than 95 per cent of the weighted radial component of secular variation and more than 94 per cent of the entire secular variation vector. This fit is considerably better than can be achieved by geostrophic flows and only slightly worse than by general steady motions; the improvement in adding poloidal motions is comparable with that in relaxing the frozen flux hypothesis.The main features of the flow are two gyres in the Atlantic hemisphere placed almost symmetrically about the equator with strong equatorial westerly flow and return easterly flows in high latitudes; it is similar to other published steady flows based on only the radial component of secular variation and severe damping. There is very little flow in the Pacific region. The shear indicates a radial length scale of about 600 km. It is very similar in form to the velocity and its sign is such that the velocity weakens with depth, as would be expected if the flow were driven from the core–mantle boundary rather than by convection from below. The overall pattern is consistent with flow in a density-stratified layer driven by lateral temperature gradients in the lower mantle, although the result allows other scenarios.

Notes: We examine thermal convection in a rotating spherical shell with central gravity and a spatially non-uniformly heated outer surface at two values of the Prandtl number: Pr=7.0, appropriate for water at room temperature, and Pr=0.7, appropriate for air at standard temperature and pressure, by numerical calculation. Four calculations are performed in a sequence: the onset of convection with homogeneous temperature boundary condition, nonlinear boundary-forced steady convection, stability of the forced steady convection to infinitesimal disturbances, and time stepping of subsequent secondary convection. Unlike our previous study of the infinite Prandtl number limit [J. Fluid Mech. 250, 209 (1993)] inertial terms in the equation of motion for moderate Prandtl numbers play a key role in the dynamics. The effects of an inhomogeneous temperature boundary condition on nonlinear convection are illustrated by varying the wavelength and strength of the imposed boundary temperature. It is shown that even a slight inhomogeneity in the thermal boundary condition can lock azimuthally drifting convection and make it stationary, or modify the normal drifting convection rolls to a vacillating structure. In the infinite Prandtl number case, when inertial forces are absent from the equation of motion, resonance occurs when the wavelengths of boundary forcing and natural convection coincide. Fluid inertia destroys this resonance for finite Prandtl number fluids. The same effect reduces in size the stability region where steady convection is locked to the boundary, and steady convection becomes unstable to time-dependent convection. The period of the secondary convection is close to that obtained with uniform temperature boundaries but the spatial structure is dramatically changed, exhibiting vacillations between the wavelength of the boundary temperature and that of the natural convection. © 1996 American Institute of Physics.