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The three-dimensional (3D) self-consistent dynamo problem is quite formidable. It is necessary to follow the evolution of the entropy, pressure, velocity, and magnetic field, all of which are three dimensional and each of which provides nonlinear feedbacks on the others. It is this complexity that precludes any back-of-the-envelope intuition of the spatial structure and dynamics involved in the maintenance and evolution of the Earth's magnetic field.

We have recently produced the first 3D, time-dependent, self-consistent numerical solution of the magnetohydrodynamic equations that describe thermal convection and magnetic field generation in a rapidly rotating spherical fluid shell with a solid conducting inner core. This solution, which serves as an analogue for the geodynamo, is a self-sustaining supercritical dynamo that has maintained a magnetic field for three magnetic diffusion times, roughly 40,000 years. The most exciting feature (that has never been simulated before) is a reversal of the dipolar part of the magnetic field that occurs near the end of our simulation.

We solve the conservation equations for mass, magnetic flux, momentum, and energy with an equation of state and a magnetic induction equation in 3D spherical geometry using a numerical method we developed here at Los Alamos. The challenging numerical aspect of this very nonlinear problem is the dominant influence of the Coriolis and Lorentz forces in the presence of very small viscous forces. Over 2,000,000 numerical timesteps were needed to simulate the 40,000 years. This required over 300 Mbytes of memory and 2000 hours of cpu time on a Cray C-90 and took a year and a half of daily monitoring and analysis.

The simulated fluid velocity in the outer core reaches a maximum of 0.4 cm/s; and at times the magnetic field can be as large as 560 gauss. Magnetic energy is usually about 4000 times greater than the kinetic energy of the convection that maintains it. Viscous and magnetic coupling to both the inner core below and the mantle above cause time-dependent variations in their respective rotation rates; the inner core usually rotates faster than the mantle and decadal variations in the length of the day of the mantle are similar to those observed for the Earth. The pattern and amplitude of the radial magnetic field at the core-mantle boundary (CMB) and its secular variation are also similar to the Earth's. The maximum amplitudes of the longitudinally averaged temperature gradient, shear flow, helicity, and magnetic field oscillate between the northern and southern hemispheres on a time scale of a few thousand years.

Magnetic field diffuses into the solid inner core from the inner core boundary (ICB). Since the field within the inner core can only undergo diffusion, it changes on a magnetic diffusion time scale that is larger than the advective time scale for field in the outer liquid (convecting) core. In addition, the radial gradient of the zonal flow is usually greatest near the ICB; this effectively generates toroidal field from poloidal field, always providing a strong source of new field at the ICB. We find the presence of a finitely conducting inner core diminishes the chaotic tendency the magnetic field otherwise would have in our 3D solutions. The thin conducting layer above the CMB provides magnetic coupling to the mantle but is less effective in stabilizing the evolution of the magnetic field in the upper part of the outer core.

The longitudinally averaged magnetic field in most of the outer core normally has the opposite dipolar polarity to that in the inner core. It also varies on a similar (advective) time scale as the axisymmetric temperature and velocity. A time series shows that this outer core field periodically attempts to reverse its polarity but usually fails and returns to its normal profile because the field in the inner core usually does not have enough time to decay away and allow the reversed polarity to diffuse in before the field in the outer core changes again.

After maintaining the same magnetic polarity for most of the simulation, our solution underwent a field reversal over a very short transition time of about 1000 years. Magnetic energy decreased, on the average, by about a factor of four during the 4000 years prior to the reversal and then quickly recovered after the reversal.

Several model improvements will be needed to obtain more realistic simulations of the geodynamo. Compositional convection, in addition to thermal convection, needs to be included. Viscous and thermal eddy diffusivities need to be tensor diffusivities to capture their local anisotropic properties. Lateral variations in the radial heat flux at the CMB due to large temperature variations in the lower mantle certainly must influence the structure and dynamics of thermal convection in the outer core and possibly the magnetic reversal frequency. Topographic coupling should take the place of the non-axisymmetric viscous coupling. Finally, the effects on core dynamics of luni-solar precession of the mantle also needs to be tested in a 3D model.

Significance:
We have produced the first three-dimensional, time-dependent, self-consistent numerical simulation of the generation of the Earth's magnetic field by convection in the Earth's liquid core. The solution is a self-sustaining dynamo that has maintained a magnetic field for roughly 40,000 years. The most exciting feature (that has never been simulated before) is a reversal of the dipolar part of the magnetic field that occurs near the end of our simulation.
Supercomputing Center after our proposal was favorably reviewed in the national competition for these supercomputing resources. A paper by G.A. Glatzmaier & P.H. Roberts has been reviewed and accepted for publication in Physics of the Earth and Planetary Interiors.

Collaborators:
Paul H. Roberts University of California Los Angles (UCLA)
Publications:
Glatzmaier, G.A. and P.H. Roberts, "A Three-Dimensional Convective Dynamo Solution with Rotating and Finitely Conducting Inner Core and Mantle," Physics of the Earth and Planetary Interiors, submitted (1994).