Viscoelastic Relaxation of Topography on the Earth and Solid Planets

Topography and gravity anomalies on planets including the Earth and the Moon have significant power at long wavelengths. The long wavelength anomalies can be either supported statically by the elastic strength of lithosphere or maintained dynamically through planetary mantle convection. The capacity of elastic lithosphere to support topography increases sharply with decreasing planetary radius due to membrane stresses. Previous elastic shell models have revealed important insights into the origin of long wavelength topography, however, the assumption that the elastic shell overlies an invisid fluid interior in thin elastic shell models has two drawbacks. First, the models are static with no time scales. Second, since the rheology of major constituents of terrestrial planets, silicates, is thermally activated, it is unlikely that a sharp rheological boundary can exist within the lithosphere. Particularly, the different compensation states of lunar basins suggest that thermal history is important for the topographic relaxation.

To overcome these shortcomings we have developed analytical models of topographic relaxation in a spherical geometry for a multi-layer viscoelastic medium with a layer of crust overlying the mantle. The viscosity in our models can be related to temperature with a rheological equation for silicates. The planetary radial temperature profile within the surface conductive thermal boundary layer can be estimated from either surface age with a half-space cooling model or measured heat flow. We solve the time evolution of topography at the crust-mantle boundary (Moho) and the surface for a given initial topography at these boundaries.

The theory and approach are discussed in the following publication:
Zhong, S. and M. T. Zuber, Long-wavelength topographic relaxation for a self-gravitating planets with multi-layer viscoelastic rheology, submitted to J. Geophys. Res., 1999.

We have been pursuing a number of applications of this theory to geodynamical problems of varying wavelength scales. The figures below show one example: the role of membrane stresses in supporting long wavelength topography on planets of different sizes.


Time evolution of degree of compensation (Dc = mass anomaly at Moho/mass anomaly at surface) for three different spherical harmonics l=2, 4, and 8 for planets with different radii.

We find that for a dry olivine rheology, a 100 myr old planetary surface for Earth-like planets can only support long-wavelength non-isostatic topography anomalies for about 1 Ma before the crust reaches approximately an isostatic state. However, as indicated by the above the figures, the same age surface with the same rheology can support more than 0.35 and 0.65 long-wavelength topography, including degree 2, over 4 billion years for Mars and Moon-like planetary radii, respectively. The older the surface, the longer it takes to relax the topography. Other forms of rheology do not change our results significantly.

Since the relaxation times are the same for internal loads, our results also suggest that for relatively small planets, mantle buoyancy is unlikely to produce significant dynamic topography even at degree 2, and even if active mantle convection is present. This has significant implications for the interpretation of long wavelength topography and gravity anomalies on the Moon and Mars.

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Maria Zuber (zuber@mit.edu)