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Plate Tectonics and Convection in the Earth's Mantle

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Plate Tectonics and Convection in the Earth's Mantle E ARTH SYSTEM SCIENCE PLATE TECTONICS AND CONVECTION IN THE EARTH’S MANTLE: TOWARD A NUMERICAL SIMULATION Numerical models of mantle convection are starting to reproduce many of the essential features of continental drift and plate tectonics. The authors show how such methods can integrate a wide variety of geophysical and geological observations. late tectonics is a kinematic description posed to some other mode of tectonics or ther- of Earth that treats the outer shell of its mal convection? mantle as a number of plates or rigid Answering these more subtle questions is com- spherical caps that move with respect plicated by the fact that the primary effect of plate Pto each other (see the “Plate tectonics” sidebar). motion is to consume the old ocean floor and re- The mantle is the outer, solid 3,000-km-thick cycle it into the mantle. The primary evidence of shell that overlies Earth’s fluid outer core. An plate history is therefore limited to the past 100 enormous amount of geological and geophysi- to 200 million years or so (less than 10% of the cal data has gone into determining the motion overall history of plate tectonics). We therefore of the plates,1 and within the last few years di- rely heavily on evidence drawn from theoreti- rect GPS measurements have corroborated the cal and computational models and from the geological constraints on the motions of plates. continents that are not consumed wholesale by A fundamental question in geology has been, plate motions. Ideally these two sources of evi- what drives the plates? This question has largely dence go hand in hand to reinforce each other. been solved—the plates are part of a system of large-scale thermal convection—and geody- namicists have moved on to more difficult ques- Model formulation tions, such as what are the details of the coupling The equations we solve are the standard ones between surface motions and deeper mantle for thermal convection in a fluid where viscos- flow? and why do we have plate tectonics as op- ity is high enough for inertia to be ignored: η∇2 ρ α ∇ u = g 0 T + p (1) 1521-9615/00/$10.00 © 2000 IEEE ∇·u = 0. (2) LOUIS MORESI Commonwealth Scientific and Industrial Research Organization Equation 1 is the equation of motion relating MICHAEL GURNIS the fluid velocity, u, to the gravitational accel- eration multiplied by the fluid density variation California Institute of Technology because of temperature, ρ0αT, and the pressure SHIJIE ZHONG gradients. The coefficient α is the coefficient Massachusetts Institute of Technology of thermal expansion. The reference density, 22 COMPUTING IN SCIENCE & ENGINEERING Plate tectonics Geodynamically, oceanic plates represent the top thermal stress. Because the rheology used in this conceptual model boundary layer of a system of thermal convection called man- does not consider the material’s deformational history, we tle convection, and this boundary layer sinks, or subducts, into refer to this as the instantaneous rheology model. the mantle at converging margins. The motion of plates— Clearly this is one fundamental component generating their speed and direction—is a balance between the buoy- plate tectonics, but the mechanical memory of the crust ancy of cooling and thickening oceanic lithosphere and sub- and lithosphere also plays a fundamental role. An impor- ducted slabs on the one hand and viscous flow on the other. tant piece of geological evidence pointing toward this alter- Plate motion and the associated mantle flow appear to be a native hypothesis is that preexisting faults and long-lived simple mode of thermal convection and have been described zones of preexisting weakness control the location of plate as a fluid-dynamical process since the seminal paper by Don margins. The convecting system reuses old, weak structures Turcotte and Ron Oxburgh.1 because less energy is expended in reactivating a preexist- The process, however, is not solely fluid dynamical. ing structure than in creating an entirely new plate margin Oceanic plates are mostly rigid, with little deformation from pristine, intact lithosphere. within their interior, and an appreciable amount of plate An important goal of geophysics is the formulation of dy- motion occurs by strike-slip. Normally, in buoyancy-driven namically self-consistent, time-dependent models of mantle fluid flow, there would only be diverging and converging convection in which plate tectonics naturally arises. This is surface motions (poloidal flow), but the motions on the an important long-term objective. Resolving this problem Earth’s surface have nearly an equal amount of toroidal will help explain why Earth has plate tectonics, while other motion—the most significant part of the toroidal flow planets, particularly Venus, do not. It will also lead to the de- being strike-slip movement at plate boundaries. A sub- velopment of dynamic models that integrate a wide variety stantial fraction of the total dissipation associated with of geological, geophysical, and geochemical observations. plate motion could occur in the bending of the oceanic Little serious debate remains concerning the dynamics lithosphere as it subducts.2 Why Earth has plate tectonics is controlling first-order features of instantaneous plate kine- a significant unanswered question. matics. The field has now moved on to equally important Ultimately, why a planet has a particular brand of tecton- but unsolved problems in mantle dynamics: ics resides not only in obvious factors such as total mass and surface temperature, but in its rheology as well. Silicates are • Why do we have plate tectonics, as opposed to some the principal building blocks of the solid Earth, which has other tectonic mode that would remove planetary heat? nonlinear and temperature-, pressure-, grain-size-, and • What controls the time dependence of mantle volatility-dependent rheologies. Perhaps the greatest chal- convection? lenge facing the development of realistic computer models • What is the connection between mantle dynamics and of mantle convection—realistic enough to test against the the wide range of observations against which convec- rich array of geological and geophysical observations—is tion models have not traditionally been compared? the incorporation of these complex rheologies. Qualitatively, one solution to the existence of plate tec- The development and use of computational models are tonics lies in a balance between thermal convection, the fundamental components of this endeavor. thermally activated rheology of silicates, and brittle failure of rocks at low pressures and temperatures. Stresses in the References plates are largest at converging margins above down- 1. D.L. Turcotte and E.R. Oxburgh, “Finite Amplitude Convective Cells welling mantle (subducting slabs) so that an otherwise and Continental Drift,” J. Fluid Mechanics, Vol. 28, 1967, pp. 29–42. cold and strong lithosphere fails and gives rise to weak 2. C.P. Conrad and B.H. Hager, “Effects of Plate Bending and Fault plate margins. With this model, the lithosphere rapidly Strength at Subduction Zones on Plate Dynamics,” J. Geophysical fails on geological time scales under the action of tectonic Research, Vol. 104, No. B8, Aug. 1999, pp. 17551–17571. ρ0, can change if the rocks undergo a phase For simplicity’s sake, these equations assume change from one crystal form to another in re- that the viscosity is constant. In practice, however, sponse to increasing pressure and temperature viscosity is a very strong function of temperature, within the planet. The dynamic viscosity, η, for pressure, composition, crystal grain size, and rocks at around 1,300 degrees centigrade is stress, and this cannot be ignored in computer enormous and similar to that of window glass simulations. The stress dependence is particularly at room temperature. important from the computational point of view MAY/JUNE 2000 23 because the equations then become nonlinear. gether with the modeled internal dynamics. The effect is that the viscosity becomes a function Fortunately, the FEM is also well-suited to of itself and we must treat the problem quite dif- problems in which the geometry might be com- ferently. The continuity equation, Equation 2, plicated—for example, if we need an irregular or ensures conservation of mass. In this case, the con- complex mesh—this can be handled quite natu- straint is stronger; it also enforces incompressibil- rally within the standard formulation of the ity on the flow. This equation is tightly coupled to FEM. Using a reasonably structured mesh, the the equation of motion, and the two must be curvature of Earth can be included into models solved as part of the same procedure. with very little additional computational over- A third equation describes the evolution through head. Three-dimensional finite-element models time of the temperature patterns in the fluid: remain time-consuming, and finding optimal so- ∂ lution algorithms requires considerable effort. T+⋅∇∇ = +κ2 (3) ∂ u TQ T t Computing the fluid velocity Here t is time, κ is a thermal diffusivity, and Q is In most dynamic systems, we can formulate a heat source term, which in Earth is largely asso- the FEM in a time-explicit manner that is robust ciated with the energy radioactive decay liberated. and simple to implement. The alternative is to This energy equation is coupled to the equation use implicit methods that are more elaborate and of motion through the fact that the density of often more temperamental but that cover much rocks changes with their temperature, and it is this larger time increments at each step. In our case, density variation that drives the motion. The mov- however, the fact that inertia is negligible leaves ing fluid carries heat with it (advection, which is ac- the equation of motion independent of time. It complished by the u·∇T term), but heat also dif- can only be solved implicitly. 2 fuses independently of the fluid motion (the ∇ T The traditional implicit approach in FEM has term).
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