Einat Lev

Anisotropic viscosity in flow models

Most of the materials that comprise the Earth are inherently anisotropic in properties such as strength and seismic wave speed. However, most geodyamical models treat the earth as an isotropic viscous fluid. One of my central research activities currently is studying the effect of anisotropic viscosity on geodynamical flow models. We examined this influence in a number of test cases --

    Rayleigh-Taylor instabilities                |            Subduction              |          strain localization in the asthenosphere

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        Lithospheric instabilities

We employed analytical solution and numerical models to study the effect of anisotropic rheology on the development of Rayleigh-Taylor instabilities in a dense layer overlying a buoyant half-space. This setup simulates the development of lithospheric instabilities and the viscously-mobile deep part of the mantle lithosphere. We find that the wavelength and growth rate of the instabilities was strongly affected by the initially prescribed fabric of the dense layer. Also the interplay between regions with different initial fabric causes changes in the location and geometry of the downwellings.

                     

Growth-rate curves, plotting the growth rate of Rayleigh-Taylor instabilities versus the perturbation wave number, for models with varying degrees and orientations of anisotropy of a dense upper layer overlaying an isotropic half-space: black - isotropic upper layer with viscosity equal to the viscosity of the bottom half-space; green curves - isotropic dense layer with viscosity equal to the average of the normal and shear viscosities of the anisotropic cases (dark green - geometric average, light green - arithmetic average); blue curves - horizontal easy shear direction (dark blue - ηs/ηN = 0.1, dashed light blue - ηs/ηN = 0.01); red curves - easy shear direction dipping at 45◦ (maroon - ηs/ηN = 0.1, dashed pink - ηs/ηN = 0.01 ). For the anisotropic cases, ηN = ηbottom. The minimum point of each curve, indicating the most unstable wave number for each configuration, is also shown.

Material distribution in models with different configurations of initial anisotropic fabric taken after the fastest downwelling sinks over half the box depth. Panel A shows the results for an isotropic model. The black cosine curve at a depth of 0.15 marks the original interface between the dense and buoyant layers. The vertical dashed black lines show the deepest points of the original density interface, where the dense layer was thickest. Red material starts with a horizontal fabric; Yellow material starts with a fabric dipping at 45◦. Blue materials are isotropic. Interestingly both panels 3-5B and 3-5G, which start with distinctly different material arrangements, show large downwellings comprised of both anisotropic materials, while others do not.

This animation shows an example of different drips developing at a layer with laterally varying anisotropy.

Diapirism

 We placed a dense layer on top of a buoyant layer, in an unstable setup. We looked at the flow fields that develop during the initial upwelling of the buoyant material through the dense layer. We compared the isotropic case with a case in which the bottom(light) layer is anisotropic and a case in which the top layer is anisotropic . While the flows in all cases share the same major characteristics, the strain rate fields (top right panel of the animations) is very different - the strain in the isotropic and top-anisotropic cases is diffuse, and in the bottom-anisotropic case it is localized at the edged of the light upwelling. This may be of major importance to the dynamics of salt diapirs, for example.

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Slab Subduction

Subduction of slabs was demonstrated to lead to development of strong anisotropic fabric when slabs buckle due to a viscous boundary such as the 660km discontinuity or the core-mantle boundary (e.g. McNamara et al 2003). We looked at the effect of anisotropic upper mantle viscosity on the geometry of subducting slabs. In the following examples we demonstrate that while slabs subducting in an Isotropic upper mantle do indeed buckle, an Anisotropic upper mantle enables the slab to slide on top of the boundary and the geometry is very different.

We ran also kinematic subduction models - models where we prescribe the shape and the velocity of the sinking slab. These models enabled us to carefully investigate how anisotropic viscosity changes the thermal structure of the mantle wedge (the corner between the sinking slab and the over-riding upper plate). These changes imply, for example, that in the presence of anisotropic viscosity, the rocks in the mantle wedge will melt at shallower depth. This may change our understanding of the nature of volcanic eruptions and earthquakes in subduction zones.

A comparison of the thermal fields and resulting melting regions for the isotropic and anisotropic models. The background color shows the difference in temperature (Taniso-Tiso) at t=15 Ma. Circles mark elements that reach temperatures above the dry solidus (black=isotropic, pink=anisotropic). Diamonds along the slab interface show regions prone to wet melting – warmer than the vapor-saturated solidus but colder but still within the chlorite stability field. The solid curves show the boundary between the overriding lithosphere and the mantle wedge. The black line is a thermal definition of the lithosphere-asthenosphere boundary, using TLAB = 0.7 Tmax. The pink line is a based on a mechanical definition of the LAB, marking the line of strain rate= 10e−14  1/sec.

A comparison of thermal quantities from the isotropic (solid blue line) and anisotropic (dashed red line) models.

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Strain localization in the asthenosphere

Alignment of mechanically anisotropic rocks by shearing leads to effective weakening in the direction of deformation. A ikely result is localization of the deformation in a narow shear zone. We use this phenomena to better constrain the degree of anisotropic viscosity and the grain size of upper mantle minerals -- two important rheological parameters that are generally poorly constrained. We use numerical models of asthenospheric flow (simple shear) to determine the grain size and anisotropic viscosity required to explain the observed confinement of seismic anisotropy to a layer at the top of the convecting upper mantle. We find that a grain size larger than 10 mm gives the best fit to the observations. The ratio of shear viscosity to normal viscosity is 0.3 or more, depending on grain size.

Thickness of the zone with sufficient as a function of the the grain size d and viscosity contrast δ. Areas in warm colors indicate thicknesses of 40 km or more, which are consistent with seismic observations. Cool colors indicate parameter values that give a layer that is either too thin (mostly for to low δ values) or that the strain rate is not high enough (grains too small).	Strain rate profiles with depth for models with grain size d = 10mm. The vertical dashed line shows the cut-off criteria defining the zone with sufficient strain. The curves reveal the correlation between δ and the thickness of the zone of localized strain.

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Sinking dense sphere in a light medium:

we placed a dense sphere in a light medium, and allow it to sink. In the animations we show the time evolution of the flow field: the left panel shows the density of the particles tracked (blue=dense, red=light); the middle panel shows the strain rate field; the right panel shows the alignment of particles with the flow field. As can be clearly seen from the left-hand panels, the sphere deforms as a result of the flow field in and around it. The deoformation of and anisotropic sphere is very different from that of an isotropic sphere in an anisotropic medium, and both deviate significantly from a purely isotropic case.

    Thermal convection

The flow in the anisotropic case stabilizes faster after the initial overturn than in the isotropic case, and the down/upwellings in it are more focused.

Funding note:

This material is based upon work supported by the National Science Foundation under Grant No. 0409564 (Modeling) and EAR- 0337697 and CD- 6892042 (Tibet)

Legal note:

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.