From: Bunge, Hagelberg and Travis, GJI (2003), 152, 1-22
Mantle convection models require an initial condition some time
in the past. Because this initial condition is unknown for Earth,
we cannot infer the geologic evolution of mantle flow from forward
mantle convection calculations even for the most recent Mesozoic
and Cenozoic geologic history of our planet. Here we introduce a
fluid dynamic inverse problem to constrain unknown mantle flow
back in time from seismic tomographic observations of the mantle
and reconstructions of past plate motions using variational data
assimilation. We derive the generalized inverse of mantle convection
and explore the initial condition problem in high-resolution, 3-D
spherical mantle circulation models for a time period of 100
Myrs, roughly comparable to half a mantle overturn. We present
a synthetic modeling experiment to demonstrate that mid-Cretaceous
mantle structure can be inferred accurately from fluid dynamic
inverse modeling, assuming present-day mantle structure is well-known,
even if an initial first guess assumption about the mid-Cretaceous
mantle involved only a simple 1-D radial temperature profile. We
also demonstrate that convecting present-day mantle structure back
in time by reversing the time-stepping of the energy equation is
insufficient to model mantle structure of the past. The difficulty
arises, because such backward convection calculations ignore thermal
diffusion effects, and therefore cannot account for the generation
of thermal buoyancy in boundary layers as we go back in time. Inverse
mantle convection modeling should make it possible to infer a number
of flow parameters from observational constraints of the mantle.
Key Figures of this paper: For details please refer to the publication.
Figure 2: (a) Cut-away of the 3-D temperature initial condition field for the reference mantle circulation model (see text) seen from the Pacific hemisphere. The model is obtained by imposing (assimilating) mid-Mesozoic plate motions (c) until quasi steady-state is reached. Blue is cold, and red is hot and the linear color scale ranges from from 0 to 2300 deg C. The upper 100 km of the mantle are removed to show the convective planform. Narrow hot zones near the surface reflect passive mantle upwelling at the Izanagi (IZA), Farallon (FA), Pacific (PA) and Phoenix (PH) spreading centers. The cold downwelling in the cross-sectional view under the northwestern Pacific results from subduction of the Izanagi and Farallon plates. (b) Same as (a) but after 100 Myrs of present-day plate motion (d) have been imposed. (c) Map of plate boundaries and velocities for the 119-100 Myrs stage from Lithgow-Bertelloni and Richards [1998]. The ancient Izanagi, Farallon and Phoenix plates occupy most of the Pacific basin. (d) Same as (c) but for the present-day from Gordon and Jurdy [1986]. The Izanagi, Farallon and Phoenix plates have largely disappeared.
Figure 4: Temperatures for a perturbed mantle circulation model where the assumed initial condition is a 1-D radial temperature profile (see text) shown for First Guess (a/b) and Best Guess (c/d) case at the initial and final state. Blue is cold, red is hot and the color scale and view angle are identical to Figures 2a/b, for comparison. The upper 100 km of the mantle are removed to show the convective planform. The initial condition of the First Guess case shows the assumed 1-D profile. At the final state the First Guess case shows slabs reaching into the mid mantle but not below, because the assimilated 100 Myrs plate motion history is insufficient to structure deep mantle flow, which preserves a memory of the initial condition. In the Best Guess case temperatures at the initial and final state agree closely with the `true' temperatures of the reference model in Figures 2a/b.
