Ulrich Hansen

Mantle convection simulations with rheologies that generate plate-like behaviour

A long-standing problem in geodynamics is how to incorporate surface plates in numerical models of mantle convection. Plates have usually been inserted explicitly in convection models as rigidrafts, as a separate rheological layer, or as a high-viscosity region within weak zones. Plates have also been generated intrinsically through the use of a more complex (non-newtonian) rheology for the entire model, but with a prescribed mantle flow. However, previous attempts to generate plates intrinsically and in a self-consistent manner (without prescribed flow) have not produced surface motions that appear plate-like. Here we present a three-dimensional convection model that generates plates in a self-consistent manner through the use of a rheology that is temperature and strain-rate dependent, and which incorporates the concept of a yield stress. This rheology induces a stiff layer on top of a convecting fluid, and we find that this layer breaks at sufficiently high stresses. The model produces a style of convection that contains some of the important features of plate tectonics, such as the subduction of the stiff layer and plate-like motion on the surface of the fluid mantle. However, the model also produces some non-Earth-like features, such as episodic subduction followed by the slow growth of a new stiff layer, which may be more consistent with the style of convection found on Venus.

Dynamical consequences of depth-dependent thermal expansivity and viscosity on mantle circulations and thermal structure

The effects of both depth-dependent thermal expansivity and depth-dependent viscosity on mantle convection have been examined with two-dimensional finite-element simulations in aspect-ratio ten boxes. Surface Rayleigh numbers between 107 and 6 × 107 have been considered. The effects of depth-dependent properties, acting singly or in concert, are to produce large-scale circulations with a few major upwellings. The interior of the mantle is cooled by the many cold instabilities, which are slowed down and eventually swept about by the large-scale circulation. The interior temperature of the mantle can be influenced by the trade-off between depth-dependent properties and internal heating. For chondritic abundance of internal-heating and depth-dependent thermal expansivity, the viscosity increase across the mantle can be no greater than a factor of around ten in order to keep the lower mantle adiabatic. The thermal contrasts between the cold blobs and the surrounding mantle are strongly reduced by depth-dependent properties, whereas the lateral differences between the hot upwelling and the ambient lower mantle can be significant, over several hundred degrees. Depth-dependent properties also encourage the formation of a stronger mean-flow in the upper mantle, which may be important for promoting long-term polar motions.

The Bent Hawaiian-Emperor Hotspot Track: Inheriting the Mantle Wind

Bends in volcanic hotspot lineaments, best represented by the large elbow in the Hawaiian-Emperor chain, were thought to directly record changes in plate motion. Several lines of geophysical inquiry now suggest that a change in the locus of upwelling in the mantle induced by mantle dynamics causes bends in hotspot tracks. Inverse modeling suggests that although deep flow near the core-mantle boundary may have played a role in the Hawaiian-Emperor bend, capture of a plume by a ridge, followed by changes in sub-Pacific mantle flow, can better explain the observations. Thus, hotspot tracks can reveal patterns of past mantle circulation.

Boundary layer control of rotating convection systems.

Turbulent rotating convection controls many observed features of stars and planets, such as magnetic fields, atmospheric jets and emitted heat flux patterns. It has long been argued that the influence of rotation on turbulent convection dynamics is governed by the ratio of the relevant global-scale forces: the Coriolis force and the buoyancy force. Here, however, we present results from laboratory and numerical experiments which exhibit transitions between rotationally dominated and non-rotating behaviour that are not determined by this global force balance. Instead, the transition is controlled by the relative thicknesses of the thermal (non-rotating) and Ekman (rotating) boundary layers. We formulate a predictive description of the transition between the two regimes on the basis of the competition between these two boundary layers. This transition scaling theory unifies the disparate results of an extensive array of previous experiments, and is broadly applicable to natural convection systems.

On the Rayleigh number dependence of convection with a strongly temperature-dependent viscosity

The strong dependence of the rheology of a fluid on temperature has a great impact on the style of thermally driven convection. When the viscosity contrast is sufficiently large, the viscosity of the coldest fluid at the top of a bottom-heated box is so high that this fluid layer becomes very stiff and a so-called cold “stagnant lid” develops on top of a hot convecting layer. Studying this style of convection is relevant for planetary mantles since the rheology of mantle material is likely to be very strongly temperature dependent. In this paper, the Rayleigh number dependence of stagnant-lid convection with a viscosity contrast of 106 is studied numerically in two and three dimensions in wide Cartesian domains. Like in constant-viscosity cases, the convection in the layer underneath the stagnant lid undergoes the typical transition from steady to time-dependent with the thinning of plumes and with the appearance of boundary layer instabilities as the Rayleigh number increases. A stagnant-lid style of con...

Trench migration and subduction zone geometry

Seismic tomography results show a large variety in upper mantle structure along convergent plate boundaries. We numerically investigate the effect of trench migration on the evolution of a slab (with temperature dependent viscosity) encountering a viscosity interface. We find that subduction zone geometry is sensitive to even small rates (1 cm/yr) of retrograde motion: increase in trench migration rate decreases the dip angle of the slab and its ability to penetrate the lower mantle. Upon including a background mantle flow it turns out that trench migration relative to the upper mantle flow is more decisive than the absolute plate velocities. Subduction zone geometry appears to be strongly time-dependent. We conclude that individual tectonic setting and time-dependent slab behaviour can account for many different types of observable subduction zone anomalies. Our model temperature can adequately account for magnitudes and patterns of seismic anomalies as obtained from seismic tomography.

Mixing in vigorous, time‐dependent three‐dimensional convection and application to Earth's mantle

An understanding of the mechanism of mixing in highly viscous convecting fluids is of crucial importance in explaining the observed geochemically heterogeneous nature of Earths mantle. Using constant viscosity numerical experiments, we describe the mixing mechanism of time-dependent Rayleigh-Benard convection with an infinite Prandtl number in a three-dimensional (3-D) rectangular container. Mixing is observed by following the positions of passive tracers advected by the flow. The major mixing mechanisms may be described in terms of the within-cell mixing and the cross-cell mixing. The flow structure in which tracers move on toroidal surfaces, that was previously observed in steady state 3-D convection systems is perturbed by boundary layer instabilities in the time-dependent experiments. This flow structure allows a very efficient exchange of mass between the boundary layers and the core of the convection cell even in the absence of time dependence. We compare this result with calculations carried out in two spatial dimensions. In similar two-dimensional (2-D) experiments, exchange of mass between boundary layers and core of the convection cell is entirely effected by the boundary layer instabilities. Mixing between neighboring cells appears much slower in three dimensions than in similar 2-D experiments, perhaps because the 3-D cell structure is more stable relative to the boundary layer instabilities. The inferred mixing rates are observed to be relatively insensitive to initial tracer location, but the timescale for mixing, tm, decreases with increasing Rayleigh number (tm goes approximately as Ra(−3/2)). The timescale of mixing is an important constraint on the large scale structure of Earth, because large-scale geochemical heterogeneities persist to the present day, implying that the mantle is not well mixed.

The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with a variable viscosity

Abstract In this paper we discuss the application of a finite-volume multigrid method to solve three-dimensional thermally driven convection in a highly viscous, incompressible fluid with a variable viscosity. The conservation laws are solved in the primitive variable formulation, A second-order control volume method is used as discretization. Two schemes are used for time stepping, a second-order implicit-explicit scheme based on the Crank-Nicolson and Adams-Bashforth method, and a fully-implicit θ-method. The implicit system of nonlinear equations are solved using multigrid iteration with the SIMPLER method as smoother. In this paper, we describe the implemented multigrid method and investigate its efficiency and the robustness for different viscosity contrasts. Convergence tests showed that with a small modification of the SIMPLER method, the multigrid method exhibits a satisfactory convergence rate even for viscosity contrasts up to 109. Three cases of time-dependent thermally driven convection with v...

Transition to hard turbulence in thermal convection at infinite Prandtl number

Direct numerical simulations of two‐dimensional high Rayleigh (Ra) number, base‐heated thermal convection in large aspect‐ratio boxes are presented for infinite Prandtl number fluids, as applied to the Earth’s mantle. A transition is characterized in the flow structures in the neighborhood of Ra between 107 and 108. These high Ra flows consist of large‐scale cells with strong intermittent, boundary‐layer instabilities. For Ra exceeding 107 it is found that the heat‐transfer mechanism changes from one characterized by mushroom‐like plumes to one consisting of disconnected ascending instabilities, which do not carry with them all the thermal anomaly from the bottom boundary layer. Plume–plume collisions become much more prominent in high Ra situations and have a tendency of generating a pulse‐like behavior in the fixed plume. This type of instability represents a distinct mode of heat transfer in the hard turbulent regime. Predictions of this model can be used to address certain issues concerning the mode of time‐dependent convection in the Earth’s mantle.

Hard turbulent thermal convection and thermal evolution of the mantle

This article summarizes the results of hard turbulent convection obtained in laboratory experiments and numerical simulations. Its applications to mantle convection are illustrated by two-dimensional numerical solutions to (1) Newtonian, (2) non-Newtonian convection and (3) Newtonian convection with multiple phase transitions. In Newtonian mantle convection the transition from soft to hard turbulence is marked by the appearance of disconnected plumes. Spectral analysis of the time series of the Nusselt number reveals the presence of a spectral scaling subrange for hard turbulence but not for soft turbulence. In hard turbulence there is correspondence between the spectra in frequency and wavenumber domains. The slope of the seismic wave spectra measured from seismology suggests that the mantle convection today is strongly time-dependent. The transition to hard-turbulence takes place at much lower Nusselt numbers for non-Newtonian than for Newtonian rheology. For the mantle this would have important ramifications. Non-Newtonian plumes behave quite differently from Newtonian ones in that large curvatures are developed in their trajectories in the hard turbulent regime. Mantle convection with phase transitions tends to become more layered with increasing Rayleigh numbers. The style of mantle convection might have changed from a layered to a more whole mantle type of flow with time. Catastrophic overturns associated with strong gravitational instabilities in the transition zone could be responsible for superplume events.