Cylindrical scaling for dynamical cooling models of the Earth

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Abstract

A detailed comparison is presented between axisymmetric spherical shell and cylindrical geometry for use in mantle convection modeling. If a rescaling of the mantle and core radii is adopted, such that the curvature of the cylindrical model approximates that of the spherical Earth, the heat and mass transport properties turn out to be very similar. Without the scaling, the volume of the lower mantle and the surface area of the core are overestimated, which leads to incorrect estimates of heat production and heat flow. This is particularly important for thermal evolution models. An explicit comparison with parameterized convection calculations shows that the scaling used here provides a good approach to dynamical model calculations.

Introduction

Thermal cooling models are essential for our understanding of the thermal and chemical evolution of the Earth. Previous studies have used parameterized convection simulations (e.g. Spohn and Schubert, 1982, Christensen, 1985, McNamara and Van Keken, 2000) that include the appropriate effects of radiogenic heating and secular cooling in a spherical geometry. The main drawback is the necessary parameterization of the relationship between heat loss and dynamical parameters, which makes it difficult to assess the influence of many aspects that are considered important for the dynamics of the Earth. These include: (1) spatially variable properties such as rheology, expansivity, and conductivity; (2) compressible effects such as viscous heating and adiabatic heating and cooling; (3) the influence of the mantle transition zone; (4) the effects of continents and changes in tectonic style. Due to improved computer and software availability, full dynamical modeling by solution of the governing mass and heat transfer equations has become an important alternative to parameterized convection.

We have made significant progress in the last decade with 3D spherical models (e.g. Bercovici et al., 1989, Tackley et al., 1993, Zhang and Yuen, 1996, Bunge et al., 1996), but the use of this geometry for calculations at realistic convective vigor are still very expensive, and currently prohibitively so for evolution models that by necessity must span a significant portion of the age of the Earth. This explains the popularity and widespread use of simplifying geometries which include those of the axisymmetric spherical shell, cylinder, and 2D or 3D Cartesian box (Fig. 1). In most cases, the approximation involves a straightening of the curved state of the Earth, either in only one dimension, which leads to a cylinder, or in both to yield a Cartesian geometry. These simplifications come at a cost. The axisymmetric spherical geometry has found some use in convection modeling (Solheim and Peltier, 1994, Van Keken and Yuen, 1995), as it best approximates the curvature of the spherical Earth, but due to the rotational symmetry a rather awkward asymmetry between polar regions and equatorial plane exists. This causes the dynamics to be governed by the volumetrically more important equatorial region. In addition, the poles form artificial boundaries where up- or down-welling plumes tend to get trapped. Cartesian models arise from the complete straightening of the curved geometry and are very popular (e.g. Tackley, 2000) due to the simpler form of the equations, ease of discretization and widespread availability of programs or libraries for the solution of partial differential equations in this geometry. However, the lack of curvature has some important consequences. For example, Cartesian models overestimate the volume of deep layers as well as the surface area of the core–mantle boundary. This makes this geometry ill-suited for accurate evaluation of thermal evolution models. The cylinder geometry is based on an intermediate approach, where the spherical model is straightened in only one direction. The assumption of cylinder symmetry reduces the geometry to 2D with associated lower computational cost. The advantage of this approach is that similar to the spherical case the geometry is a simply connected region without artificial boundaries.

In this paper, we will explore the use of a particular scaling of the cylindrical geometry which arrives from making the ratio of surface area of core and mantle the same as in the spherical geometry, which provides a closer approximation to the curvature of the Earth. A simple boundary layer analysis for bottom-heated convection with constant properties showed that this leads to an improved approximation to the heat-transport characteristics of spherical models (Vangelov and Jarvis, 1994). Here, we will provide further evidence that the heat and mass transfer of cylindrical models is very similar to that of spherical models using a number of simple benchmarks that incorporate internal heating and compressible convection. This similarity opens up the possibility of using 2D cylindrical models for studying the thermal cooling of the Earth, and provides therefore an ideal alternative to offset the high cost of 3D spherical models, while avoiding problems with the interpretation of results obtained on Cartesian grids.

Section snippets

Model formulation and numerical approach

In this paper, we are concerned with the solution of the equations governing convection in the Earth’s mantle, assuming that the mantle can be described as an anelastic and weakly compressible fluid at infinite Prandtl number. Assuming the extended Boussinesq approach, we can write the equations of motion as−∇P+∇·(ηϵ̇)=RaαρTĝand the mass conservation equation as∇·u=0The heat equation incorporates terms that describe viscous heating and adiabatic cooling and heating, and can be written as∂T∂t+(u

Comparison of heat and mass transport characteristics

In this section, we will demonstrate the strong similarities of the heat and mass transport of the spherical and rescaled cylindrical geometries. We will first explore steady state bottom and internally heated convection models, followed by a time-dependent benchmark. In all cases, we will use Di=0.5 and α and k are depth-dependent following , . We will first make the additional simplifying assumptions that ρ=1 in the right-hand side of Eq. (1) and the volumetric heating rate Q a constant. This

Discussion and conclusions

The results presented above clearly illustrate that the use of a scaled cylindrical geometry is beneficial to enhance the curvature and make the cylindrical geometry more useful for comparison with a fully spherical model. For steady state results, obtained at moderate convective vigor, the derived quantities such as heat flow and rms velocity agree to within a few percent from the axisymmetric spherical results. This conclusion holds for bottom and internal heating, isoviscous and moderately

Acknowledgements

Discussions with Scott King, Geoff Davies, Henry Pollack, and Allen McNamara are greatly appreciated. Constructive reviews from Ulrich Christensen, the editor Ken Creager, and an anonymous reviewer are gratefully acknowledged.

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