The influence of temperature and depth dependent viscosity on geoid and topography profiles from models of mantle convection

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Abstract

We consider the geoid and topography profiles from convection models with depth and temperature-dependent viscosity models in a 2-D Cartesian geometry. We use a very low viscosity region in the top center of the geometry to generate plate-like viscosity models. We focus on subduction regions, which have narrow (100–200 km) trenches. The narrow trench is only a minor feature in the regional geoid. In depth-dependent viscosity models, we change the geometry of the weak zone around the ocean trench to study the impact of the weak zone on trench topography and the geoid. We also vary the vertical viscosity structure of the models. The weak zone geometry has very little effect on the surface velocity of the ocean plate or the large scale pattern of convection. The width of the weak zone, within a reasonable range, has little effect on topography and geoid profiles. There is a narrow trough in the geoid and topography profiles. This trough becomes wider and deeper with a high viscosity upper mantle. This may suggest that the high viscosity upper mantle is not suitable for our models. In temperature-dependent viscosity models, we vary the viscosity values for different layers. Temperature-dependent viscosity is also important to producing realistic subduction models; however, the activation energy must be weaker than laboratory estimates of olivine under mantle pressures and temperatures. From both depth and temperature-dependent viscosity models, the viscosity in the lower mantle should be at least ten times greater than in the upper mantle, consistent with previous studies of the geoid in regions of subduction. The topography and geoid profiles can match the observations when the viscosity increases with depth.

Introduction

It is generally accepted that convection in the Earth's mantle occurs by solid-state creep flow. The most direct expression of mantle convection is the motion of plates and the associated transport of heat (e.g., Parsons and Daly, 1983), but, equally important to our understanding of mantle convection are the long-wavelength geoid and surface topography anomalies (e.g., Hager and Richards, 1989). Even though the variations of the geoid are not necessary associated with plate velocity (Hager and Clayton, 1989), topography and geoid anomalies can reflect the density contrast that drives mantle convection and plate motion. At wavelengths greater than several thousand kilometers, geoid anomalies can reflect the internal density contrasts that drive mantle convection and, hence, plate motions (e.g., Forte et al., 1993). The long wavelength geoid includes contributions from mass anomalies due to internal density anomalies and deformation of any chemical boundaries, such as the surface and core-mantle boundary. Because the surface and core mantle boundary deform in response to viscous flow, calculating geoid anomalies requires a model of the viscosity structure of the Earth and the resulting surface and core-mantle boundary deflection, generated by viscous flow driven by those anomalies (e.g., Pekris, 1935; Richards and Hager, 1984; Ricard et al., 1984).

In this paper, we will examine a series of models that illustrate the effect of: the initial conditions; the vertical variation in viscosity (depth-dependent); the shallow dip of the slab, parameterized by a geometrically defined weak zone; and the role of temperature-dependent viscosity in our models. We evaluate our models by comparing them with the observations that trenches on the Earth are about 100–200 km in width and that the trench is not the dominant feature in the geoid profile across a subduction zone. In fact, there is generally a long wavelength geoid high across a subduction zone. As we will show, some model parameters have a strong effect on surface observations (i.e., it is well known depth-dependent viscosity can affect the sign of the geoid) while other parameters have little or no effect (i.e., the dip angle of the weak zone).

Much of our understanding of the internal structure of the mantle is based on comparison of viscous flow models with observations made at the Earth's surface. This includes studies of post-glacial rebound (e.g., Mitrovica, 1996), geoid anomalies and plate motions predicted from seismic tomography (e.g., Hager and Richards, 1989; Forte et al., 1993) and the geoid and topography profiles over specific regions, such as subduction zones (e.g., Hager, 1984; Zhong and Gurnis, 1992). Mitrovica (1996)analyzed decay times associated with uplift of two sites in the classic work of Haskell (1935)on mantle rheology. He showed that the average viscosity of the mantle, often called the Haskell value (1021 Pa s), encompasses a region from the base of the lithosphere to a depth greater than 1000 km. Mitrovica's analysis of the data also showed that moderate variations of viscosity with depth are consistent with the data as long as the average value of the upper 1400 km of the mantle is in the range of 0.65–1.10×1021 Pa s.

There have been numerous mantle viscosity models based on fitting the observed geoid using seismic tomography models as the representation of the internal density anomalies (c.f., King, 1995a). These can be grouped into two or three general classes of models. One class of models has a low viscosity layer, at 100–400 km depth, approximately a factor of 30 smaller than the rest of the upper mantle (e.g., Hager and Richards, 1989). Another class has a low viscosity at 400–670 km depth (Forte and Peltier, 1991; King and Masters, 1992). In a recent paper, King (1995b)found that a high viscosity layer at 400–670 km depth, 5–10 times the viscosity of the 100–400 km region, also fits the observed long-wavelength geoid. Almost all viscosity models determined from the geoid require an increase in viscosity at or near 670 km.

Because of computational limitations, most studies of mantle viscosity have only considered depth (pressure) variations in viscosity. Subduction zones are one of the regions of the mantle where lateral variations in rheology are likely to be important. While the models we will consider in this paper represent only a portion of the mantle, and thus are more limited than the 3-D spherical models used in most mantle viscosity studies, they allow us to consider temperature-dependent (i.e., laterally varying) rheology. The key observation is the correlation of long-wavelength (i.e., degree 4–9) geoid highs over subduction zones (Hager, 1984).

Section snippets

Method

One of the important outstanding problems in mantle convection is our poor understanding of the generation of piecewise uniform (plate-like) surface velocities. A number of methods of generating uniform surface velocities have been used, including: imposing uniform velocities as boundary conditions (Davies, 1989), using weak and strong zones to concentrate deformation (e.g., Schmeling and Jacoby, 1981), constraining the surface to behave rigidly while balancing stresses with the mantle (Gable

Results

The models described below are all examined at the same relative non-dimension time and are well past the transient phase from the initial condition. While the flow does not reach steady-state at these Rayleigh numbers, these models reflect a period of time when the flow was relatively steady and well beyond any start-up transients.

From the observations, the geoid shows a long wavelength high across the subduction area and a short wavelength low directly over the trench (Fig. 2). The purpose of

Discussion and conclusion

Neither the calculated geoid nor the topography profiles are affected by variations in the weak zone dip. This implies that the weak zone dip is not an important factor in mantle convection models. The width of the weak zone geometry can be considered a factor that can affect our geoid and topography profiles. If the weak zone geometry is too wide, the geoid and topography profiles near the ocean trench can be changed, becoming even wider and deeper, and the surface velocities (Fig. 4) show a

Acknowledgements

This paper was significantly improved by two careful and thoughtful anonymous reviewers and Editor, B. Romanowicz. Junnan Chen was supported by a research fellowship from the Purdue Research Foundation. Scott D. King acknowledges support from the National Science Foundation through award EAR-9627230.

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