Modelling the relative time-scales of the Rayleigh-Taylor Instability and delamination, using Underworld

Why model sub-continental gravitational instabilities?

Within the plate tectonics framework, continents are generally considered to have a much lower density than the asthenosphere below and therefore avoid the kind of recycling that the oceanic crust and lithosphere undergoes. The mantle lithosphere below some continents however can become denser than the asthenosphere when it cools to a steady-state geotherm, which is an unstable configuration. Some parts of the lower crust can also have compositions which are anomalously dense, for example forming through partial-melting or underplating and then transforming into dense eclogite.

Dense sub-continental material however will not necessarily sink in <100-1000 Ma, for example if its strength resists the downward buoyancy forces or if it is not sufficiently perturbed. Sub-continental instabilities do appear to occur, as they can be observed in tomography and their evolution (typically over ~3-50 Ma) can be inferred by variations in volcanism and dynamic topography (e.g. Sierra Nevada, Jones and Saleeby 2013). How weak must lithospheric mantle must be, in order to be recycled at this time-scale? In order to answer this, we need to be able to predict what this time-scale should be for a given set of material properties. This is well understood for the Rayleigh-Taylor Instability (RTI), which describes the growth of a perturbation through non-linear feedbacks in material thickening and thinning. Another mechanism, delamination, involves peeling away of the entire dense layer. How its time-scale compares to the RTI is unclear. This means that if dense lower crust and mantle lithosphere is predominately peeled away by delamination, then current estimates of instability time-scales using the RTI model could be under-estimated, over-estimated, or this time-scale may not significantly depend on which mechanism occurs.

Modelling instabilities with Underworld

Underworld was used to model the RTI (informally called drips) and delamination, in order to compare their fundamental time-scales (Beall et. al. 2017). To generate a drip, the only requirement is that the unstable material has a greater density than the material below (the asthenosphere) and that any non-zero perturbation to its thickness exists. In numerical modelling, the latter can generally not be prevented, due to small amounts of numerical noise. In our models, small sinusoidal perturbations are inserted at the base of the dense material, with different wavelengths, so that the growth of small perturbations of particular wavelengths can be studied in a controlled way. A simple drip / RTI Underworld example can be found here. Because drips are so easily triggered, they can be thought of as the 'default' instability mechanism. To instead trigger delamination, there are an additional two requirements: 1) a weak decollement layer should separate the dense material from the strong upper crust above, allowing a pathway for channel (Poiseuille) flow and 2) the dense material should be thinned completely in a localised region, so that the asthenosphere is connected to the decollement layer.

There are therefore two parameters (see below) which control whether dripping or delamination occurs: the viscosity contrast between the dense material and the decollement layer ($\eta'_c = \eta_c / \eta$) and the size of a finite perturbation (ignoring our additional approximately 'infinitesimal' perturbations) relative to the dense material thickness ($D' = D/L$). If $D'=1$ and $\eta'_c$ is small (we found that it should be $\eta'_c < 10^{-1}$), delamination is triggered. If $D'=0$, then dripping dominates and the growth time-scale agrees with RTI analytical solutions.

Here is a notebook, which reproduces the setup described. You can choose the values of $D'$ and $\eta'_c$ which are expected to trigger dripping or delamination and vary this. How can you actually measure if the instability in a model is more like a drip or delamination? The fundamental difference between the two mechanisms is the kind of deformation which drives their growth. Drips grow because where the dense material is slightly thickened, there exists an anomalous normal stress which drives further thickening, which in turn generates even larger stresses. Therefore dripping material needs to be able to thicken and thin, which results in significant internal shear-strain. Delamination develops because of the peeling which occurs, which involves bending of the dense material, ideally with no thickening and no internal shear-strain.

Shear-strain is measured on particles inside the dense material, which are advected with flow. Specifically, we measure the shear-strain in an orientation parallel to what would be the neutral axis in a bending beam. In the Euler-Bernoulli theory of slender bending beams, there should be no shear-strain in that orientation. The particles in the example notebook contain vectors which are initially shaped like crosses in the schematic above. At each time-step, the shear-strain is calculated at each particle location, projected onto the vectors, which are then rotated (see below). In the delamination end-member, the vectors rotate, but remain orthogonal. For dripping, many of the vectors are sheared so much that they are sub-parallel.

References

Beall, A. P., Moresi, L., & Stern, T. (2017). Dripping or delamination? A range of mechanisms for removing the lower crust or lithosphere. Geophysical Journal International, 210(2), 671-692.

Bird, P. (1979). Continental delamination and the Colorado Plateau. Journal of Geophysical Research: Solid Earth, 84(B13), 7561-7571.

Jones, C. H., & Saleeby, J. B. (2013). Introduction: Geodynamics and consequences of lithospheric removal in the Sierra Nevada, California. Geosphere, 9(2), 188-190.

Originally published at http://www.underworldcode.org on June 29, 2017.