Data assimilation in problems of mantle dynamics: Methods and applications
We present and compare several methods (backward advection, adjoint, and quasi-reversibility) for assimilation of geophysical and geodetic data in geodynamical models. These methods allow for incorporating observations and unknown initial conditions for mantle temperature and flow into a three- dimensional dynamic model in order to determine the initial conditions in the geological past. Once the conditions are determined the evolution of mantle structures can be restore. Using the quasi-reversibility method we reconstruct the evolution of the descending lithospheric slab beneath the south-eastern Carpathians. We show that the geometry of the mantle structures changes with time diminishing the degree of surface curvature of the structures, because the heat diffusion tends to smooth the complex thermal surfaces of mantle bodies with time. Present seismic tomography images of mantle structures do not allow definition of the sharp shapes of these structures in the past. Assimilation of mantle temperature and flow instead provides a quantitative tool to restore thermal shapes of prominent structures in the past from their diffusive shapes at present.
Quantifying the uncertainty of 3-D backward mantle convection models
Reconstructing the past thermal evolution of Earth's mantle from present-day mantle heterogeneity is one of the most numerically challenging tasks in geodynamics. A successful reconstruction of the time-dependent, 3- D mantle convective structure in the geological past provides invaluable insight into the origin and evolution of a number of fundamental surface processes that include epeirogeny, eustatic sea level change, state of stress in the lithosphere, and true polar wander. A critical question is: what are the limitations of backward mantle convection simulations and, most importantly, can we quantify the uncertainties inherent in such time-reversed models? To address this question, we use a pseudo-spectral formulation of mantle convection in 3-D spherical geometry where the equations of conservation of mass and momentum are solved only once in terms of spectral Green functions while the time-integration of the energy equation is solved in the combined spectral and spatial domains. The model incorporates geodynamically constrained variations of viscosity with depth as well as surface plates and their reconstructed history. The initial thermal heterogeneity of the mantle is inferred from a recent high-resolution seismic tomography model derived from the joint inversion of both seismic and geodynamic data sets. We reconstruct the past evolution of mantle heterogeneity and flow using the simplest approach to backward mantle convection, namely by reversing the time variable in the energy equation and omitting thermal diffusion. We consider time intervals in which our convection simulations are in steady-state thermal equilibrium and we evaluate the effects of neglecting physically irreversible processes (e.g. thermal diffusion) by comparing present-day solutions with forward solutions obtained from reconstructed mantle heterogeneity. Understanding the limitations of this simplest approach to backward convection will aid in evaluating the effects of various convection parameters in other formulations of backward convection such as the 4-D variational and the quasi-reversible approaches.
The Influence of Evolving Plate Boundaries in 3D Mantle Convection Simulations
A variety of mantle convection
simulations featuring model plates have
shown that plate-like surface motion
underlying convection patterns, surface heat
flux, system temperature and plume
stability. However, plate
boundaries move at comparable speeds to
the velocities associated with
convection driven flow in the mantle,
therefore the shapes and sizes of the
tectonic plates change considerably in just
one mantle transit time. In order to properly
assess the influence of plate-like surface
motion on mantle convection, it is thus
necessary to investigate convection in
systems featuring evolving plate geometries.
In the calculations presented here we
behaviour of systems featuring both evolving plate
velocities and shapes. The plate velocities evolve
that the shear stress on the base of
each of the finite thickness high
viscosity plates sums to zero at all
times, to ensure that the plates
neither drive nor resist the
convection. We compare the evolution of
the surface and basal heat flow and
changes in convection planform in two
sets of calculations in which simple
plate geometries change with time while
plate velocity responds dynamically to
the evolving driving forces in the
plate-mantle system. The models feature
either 4 or 9 polygon-shaped plates
resulting in a surface characterised by
piece-wise continuous uniform velocities
corresponding to each plate interior. In the
first study we
compare the difference in evolution of a pair
of models featuring 9 plates in a 6×6×1
system. In one calculation the plate
boundaries are held fixed and in the second
the plate boundaries evolve dynamically in
response to motion of the plate geometry
triple junctions. In addition to thermal
compare the time-dependence of the plate
velocities in these models.
In a second set of calculations using 4×4×1
solution domains we prescribe the evolution
of four plates, rotating the initial
plate geometry through 90 degrees at a
constant rate in each experiment.
The influence of the plate boundary motion on
the convection is compared
in models featuring different viscosity
Onset of Time-Dependent 3-D spherical Mantle Convection using a Radial Basis Function-Pseudospectral Method ; Spectral-Finite Volume ; Spectral Higher-Order Finite- Difference Methods
Many numerical methods, such as finite-differences, finite-volume, their yin-yang variants, finite-elements and spectral methods have been employed to study 3-D mantle convection. All have their own strengths, but also serious weaknesses. Spectrally accurate methods do not practically allow for node refinement and often involve cumbersome algebra while finite difference, volume, or element methods are generally low-order, adding excessive numerical diffusion to the model. For the 3-D mantle convection problem, we have introduced a new mesh-free approach, using radial basis functions (RBF). This method has the advantage of being algorithmic simple, spectrally accurate for arbitrary node layouts in multi-dimensions and naturally allows for node-refinement. One virtue of the RBF scheme allows the user to use a simple Cartesian geometry, while implementing the required boundary conditions for the temperature, velocities and stress components on a spherical surface at both the planetary surface and the core-mantle boundary. We have studied time- dependent mantle convection, using both a RBF-pseudospectral code and a code which uses spherical- harmonics in the angular direction and second-order finite volume in the radial direction. We have employed a third code , which uses spherical harmonics and higher-order finite-difference method a la Fornberg in the radial coordinate.We first focus on the onset of time-dependence at Rayleigh number Ra of 70,000. We follow the development of stronger time-dependence to a Ra of one million, using high enough resolution with 120 to 200 points in the radial direction and 128 to 256 spherical harmonics.
Effects of Inner Core Conductivity on Planetary Magnetic Field Reversal Frequency
Although the Earth's inner core occupies only 4 percent of the total core volume, it has significant influences on core dynamics. When simulating the geodynamo, it is necessary to model a spherical shell rather than a full sphere in order to capture the effects of the inner core on the fluid dynamics of a rapidly rotating sphere. In addition, the inner core's conductivity can provide stability to dynamo processes because of the anchoring effect of a solid conductor on magnetic field lines. Various studies have investigated the influence of inner core conductivity in modeling planetary dynamos. Some suggest that the finite conductivity of the inner core reduces the frequency of reversals and affects magnetic field morphology while others suggest the inner core conductivity does not have a significant effect on the resulting magnetic fields. Here we investigate numerical dynamo models in both thick and thin shells in order to understand the influence of inner core conductivity and initial conditions. We show that a conducting inner core reduces the frequency of reversals and results in axial-dipolar dominated fields.
Earth thermal history simulations with layering at various depths in the mantle
Understanding Earth's heat energy budget over all of geological time presents a number of challenges. The current measured surface heat flow is significantly greater than the geochemically estimated internal heating rate which requires a significant degree of mantle secular cooling. This degree of secular cooling is difficult to obtain if the mantle lost heat efficiently at early times. One possible mechanism to decrease convective efficiency at early times is mantle layering at 660-km depth. Also, the persistence of Earth's magnetic field over the last 3.5 Gyrs combined with the relatively high estimates for the current core temperature require that either the core was initially much hotter than the mantle, or that there is radioactive internal heating in the mantle, or that a mechanism, such as a stagnant lower mantle layer acts to significantly decrease the heat flow efficiency from the core to the mantle. In this contribution we will present simulations with mantle layering at 660-km depth and demonstrate that mantle layering is not an effective mechanism for storing mantle heat at early times. Simulations with a stagnant layer in D', that persists over much of Earth's history, will also be presented and we will demonstrate that as long as this layer is not strongly enriched in radioactive elements, it can act to substantially increase the predicted age of the inner core and allow a long lived geodynamo.