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E ARTH SYSTEM SCIENCE

PLATE TECTONICS AND IN THE ’S : TOWARD A NUMERICAL SIMULATION

Numerical models of mantle convection are starting to reproduce many of the essential features of and . 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 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 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 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 , 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 plate margins. With this model, the lithosphere rapidly Strength at 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). Much of the interesting physics occurs been to build the matrix equation and solve it di- when the advection and diffusion of heat compete rectly using a method such as Crout elimination. in different directions and strike a dynamic bal- Exploiting the fact that the finite-element ma- ance. The energy equation contains time deriva- trices are quite tightly banded, we can do this tives that the equation of motion lacks, so it is not relatively efficiently. Unfortunately, direct-solu- surprising that these equations require entirely dif- tion methods are limited to “small” problems be- ferent computational methods. cause the solution time scales vary rapidly with the number of unknowns (in the worst case as 3 N , where N is the number of unknowns, and 2 Computational approach even in the best case as N ). Iterative methods We use a finite-element method to solve the can achieve much better performance than this mathematical model outlined earlier. The FEM once the problems start to become larger. For is, in general, robust and accurate when dealing example, preconditioned conjugate-gradient with strong variations of material properties methods can obtain a solution to a given accu- from place to place, such as we expect to find in racy in a time proportional to N log N. In the- our situation. In part, this comes about because ory, the optimal method for our problem is the equations are first integrated to an equiva- multigrid, which when properly formulated can lent weak—or variational—form before being find a solution in a time proportional to N. discretized using the mesh. The multigrid method works by formulating the finite-element problem on several different Flat Earth? scales2—usually a set of grids that are nested one The essence of good modeling is to incorpo- within the other, sharing common nodes. The rate the essential features of the system in ques- solution progresses on all of the grids at the same tion while remaining as simple as possible—to time, with each grid eliminating errors at a dif- make the models easier to interpret and faster to ferent scale. The effect is to propagate informa- calculate. One of the traditional simplifications tion very rapidly between different nodes in the in mantle dynamics has been to work in a “flat- grid that the local support of the element shape Earth” model and then to limit the computation functions would otherwise prevent . In fact, a to two dimensions (one horizontal and one ver- single traverse from fine to coarse grid and back tical). This can prove very limiting when trying can directly connect all nodes in the mesh to to draw observed behaviors of the real Earth to- every other—allowing nodes that are physically

24 COMPUTING IN SCIENCE & ENGINEERING 1 Fine

Fine-grid error

H Coarse Iteration Fine-grid improved 2

Coarse grid

h Coarse-grid improved Iteration 3

Coarse to fine

Fine-grid improved Iteration (a) (b) (c)

Figure 1. Multigrid solution methods: (a) Different grids can reduce errors; the fine grid reduces the short-wavelength error, whereas the coarse grid reduces the long-wavelength error. Correctly combining the two can reduce the errors at both scales in one cycle. (b) Shape functions associated with a node common to the coarse mesh—H—and the fine mesh—h—show the way in which information from different distances can be brought to a given node using different mesh scales. (c) We can devise different schemes for traversing the different meshes: (1) the V cycle, (2) the W cycle, and (3) the full multigrid method.

coupled but remote in the mesh to communicate problem is first to write equations for a com- directly during each iteration cycle. pressible flow. If the continuity equation (Equa- The multigrid effect relies on using an itera- tion 2) is written as tive solver on each of the grid resolutions, which acts like a smoother on the residual error at the ∇·u − p/λ = 0 (4) characteristic scale of that particular grid (see Figure 1a). We use Gauss-Seidel iteration be- then in the limit that λ → ∞, the fluid becomes cause it has exactly this property. On the coarsest incompressible. Equation 4 can be substituted grid, we can use a direct solver because the num- into the equation of motion to give ber of elements is usually very small. η∇2u − λ∇∇·u = ρgz. (5) Incompressibility The multigrid method leads us very quickly to In this way, we can make the otherwise small a fluid velocity, but this might not, in general, be ∇. u term become dominant when λ becomes incompressible. Constraining the multigrid large. We have also succeeded in eliminating p as method to finding only incompressible solutions an independent variable. is a major difficulty. When using variational This can be very effective if direct-solution methods, the usual way to apply a constraint is to methods are used, but for iterative methods it penalize the weak form of the equations. If there generally spells disaster. In our experience, λ of is some property we wish to make vanish (here, 100 or 1,000 is the maximum that can be used any compressible component of the velocity before the penalty terms ruin the convergence field), we multiply it by a large number and find a of the iteration method. This is far from the way to add it back into the equation we want to value of 107 to 109 required to suppress com- solve. By scaling the small constraint term, we pressibility to the level of machine precision. can make it as important as the other terms. A better alternative when using an iterative The easiest way to see how this works for our solver is not to eliminate the pressure parame-

MAY/JUNE 2000 25 ter, p, but to solve it as an unknown on the mesh. for the advection term u·∇T. When velocity and pressure variables are com- Pure advection is a translation of the local bined in the same system of equations, the ap- temperature field from one part of the mesh to proach is known as a mixed method. Numerous another: In a Lagrangian reference frame (one strategies exist for solving the mixed formula- moving with the fluid), there is no change in tion, but we have found only one approach that temperature at any given point of reference, al- is robust enough to use with strongly varying though the coordinate system rapidly becomes material properties while still letting us retain tangled. In an Eulerian reference frame (one the speed of the multigrid velocity solver. fixed in space), the temperature at a given node This is the Uzawa scheme, in which an outer it- point must be updated as fluid moves past it. The eration for pressure corrects a nested set of ve- fluid element that is now on a particular node locity iterations that build up to an incompress- was, in the previous time step, at some general ible flow solution. We use a conjugate-gradient point in the mesh that was almost certainly not scheme for this outer iteration. The nature of associated with a node point. The advection the Uzawa iteration does not let us develop a re- operation therefore requires the equivalent of liable multigrid version of the outer iteration, an interpolation operation to determine this ear- but by using multigrid for the inner (velocity) it- lier off-node temperature. After several time eration, the computation time still scales linearly steps, repeated interpolation operations start to with the number of node points in the mesh. smooth the temperature field as information is lost. The smoothing is akin to a diffusion term, Nonlinear rheology so provided there is a genuine physical diffusion Stress-dependent viscosity makes the equation that operates more rapidly than the erroneous of motion nonlinear. Suddenly, it becomes nec- diffusion of the advection operator, it is possible essary to introduce yet another layer of iteration to solve the problem on an Eulerian mesh. to determine the viscosity at each node as a func- If no diffusion occurs, or the physical diffusion tion of the changing velocity field. A triply coefficient is smaller than the numerical one, then nested iteration loop poses the danger of un- we need special measures. An example might be manageably long solution times. However, we tracking a dye (or other chemically distinct entity have found that using multigrid concepts can such as continental crust embedded in the lithos- again help us avoid wasting CPU time. phere). We often handle this issue by dealing not The viscosity is a function of the stress-field that with mesh-based variables but with a set of La- is distributed across the mesh and is, in turn, a func- grangian points that we usually refer to as tracer tion of the viscosity pattern. At the start of the so- particles. These particles are passively moved with lution, we don’t know what the ultimate viscosity the flow using a Runge-Kutta integration scheme field will look like. For this reason, there seems lit- that operates over the course of a time step. An tle point in using the finest grid resolution avail- interpolation step occurs in obtaining nodal val- able. Instead, we operate on one of the coarser ues from the particles, but the smoothing is not grids, using a suitably smoothed density field as the cumulative because this procedure simply sam- driving term. At this resolution, even the triply ples—rather than disturbs—the particles. nested loop is not very time-consuming, and we Another problem arises from the fluid’s mo- quickly home in on the magnitude of the viscosity tion relative to the grid. Although a particular and stresses everywhere. We can then interpolate discretization might be very accurate based on this information to a finer grid where we need far the temperature and velocity distribution at the fewer iterations to adjust the viscosity and stresses. beginning of a time step, the point at which the This method is very close to what is known as new value is computed has moved slightly with the full multigrid algorithm, although the multiple the fluid by the end of the time step. Upwinding nesting of iterative loops is slightly unusual. is the standard cure for this problem—which is using a discretization scheme that is weighted Time and temperature more strongly in the upstream direction than in We almost always want to see the solution the downstream direction to compensate for the evolve through time in our simulations, so we motion during a finite time step. generally solve this equation explicitly in time. Essentially this involves integrating Equation 3 forward through time from a given initial con- Convection models and plate tectonics dition. This is generally straightforward except The earliest computational solutions to the

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Stokes convection equations made the simplify- entirely stagnant, with little or no observable ing assumption that the viscosity was constant. motion. Vigorous convection continues under- Despite the experimental evidence that suggests neath the stagnant layer with very little surface viscosity variations should dominate in the man- manifestation. This theoretical work demon- tle, agreement with some important observa- strates that the numerical simulations are pro- tions was remarkably good. ducing correct results and suggests that we The simulations could not produce platelike mo- should look for physics beyond pure viscous flow tions at the surface (instead producing smoothly in explaining plate motions. distributed deformation), but the average velocity, The obvious association of plate boundaries the heat flow, and the observed pattern of subsi- with earthquake activity suggests that we’ll find dence of the ocean floor were well matched. relevant effects in the brittle nature of the cold More sophisticated models included the effect plates. Brittle materials have a finite strength, of temperature-dependent viscosity as a step to- and if they are stressed beyond that point, they ward more realistic simulations. In fact, Uli break. This is a familiar enough property of Christensen observed the opposite: Convection everyday materials, but rocks in the lithosphere with temperature-dependent viscosity is a much are nonuniform, subject to great confining pres- worse description of the oceanic lithosphere sures and high temperatures, and they deform than constant viscosity convection.3 over extremely long periods. This makes it diffi- Theoretical studies of the asymptotic limit of cult to know how to apply laboratory results for convection in which the viscosity variation be- rock breakage experiments to plate simulations. comes very large (comparable to values deter- An ideal, very general, rheological model for the mined for mantle rocks in laboratory experi- brittle lithosphere would incorporate the effects ments) find that the upper surface becomes that result from small-scale cracks, faults,

MAY/JUNE 2000 27 Of these approaches, the continuum approach is best able to demonstrate the spectrum of be- haviors as convection in the mantle interacts with brittle lithospheric plates. For studying the evolution of individual plate boundaries and the fine scale of observables such as gravity and sur- face deformation, methods that explicitly include discontinuities work best. The simplest possible continuum formulation τ includes a yield stress, yield, expressed as a non- linear effective viscosity, τ η = yield (6) eff ε˙ . where ε is the strain rate. This formulation can be incorporated very easily into the mantle- dynamics modeling approach that we outlined because it involves making modifications only to the viscosity law. There might be some numerical difficulties, however, because the strongly non- linear rheology can lead to dramatic variations in the viscosity across relatively narrow zones. Figure 2 compares two convecting layers with strongly temperature-dependent viscosity, one having a yield stress specified for cold material. Both simulations were started from the same slightly perturbed conductive temperature gra- dient, and we embedded two blocks of passive Figure 2. A layer of fluid whose viscosity decreases strongly with tracers in the fluids to show the deformation’s temperature is (a) heated from below. Initially, the temperature is evolution. set to increase uniformly from top to bottom with a very mild per- If it cannot yield, the cold boundary layer de- turbation. Two blocks of marker particles are embedded in the velops a stagnant lid. This is shown by the light- fluid—green at the top, black at the bottom. (b) Stagnant lid green block of tracers, which barely deform dur- convection develops when the yield stress is higher than typical ing the entire simulation (whereas the black stresses that motion in the fluid generates (the green block of parti- block of tracers is unrecognizably distorted). cles is virtually undisturbed). Surface motion is smooth and of low The surface velocity is both small and smoothly magnitude—shown by the graph and the horizontal arrows. The varying. There is a slight wobble in the pattern curved arrows show the general sense of the circulation. (c) Mobile- of upwellings and downwellings as time pro- lid convection develops when the yield stress is low enough to be gresses, but the basic pattern persists uninter- overcome by convection. The surface motion is two orders of mag- rupted throughout. nitude larger than in Figure 2b. The hazy purple areas show where By contrast, when the cold boundary layer can the yield stress is exceeded. The more intense the color, the faster yield, no stagnant lid forms and the upper sur- the material is deforming. face is highly mobile.4 This confines the defor- mation at the surface to relatively narrow zones of divergence and convergence. The convection anisotropy, ductile shear localization caused by dy- pattern is very dynamic, and, although the same namic recrystallization, and so on. To date, most number of downwellings and upwellings persists attempts to account for the brittle nature of the through the entire lifetime of the simulation, the plates have greatly simplified the picture. Some shapes and locations of upwellings and down- models have imposed weak zones that represent wellings are constantly shifting. plate boundaries; others have included sharp dis- An obvious limitation of this formulation is continuities that represent the plate-bounding that it does not track the history of deformation faults; and still others have used continuum meth- with the fluid as it moves: Each parcel of fluid is ods in which the lithosphere’s yield properties are considered pristine at the beginning of each time known but not the geometry of any breaks. step. This means that the damage that occurs at

28 COMPUTING IN SCIENCE & ENGINEERING Figure 3. This shows convection in a system with Lagrangian particle tracking that allows the use of a strain-weakening yield stress. Dark blue areas are cool, red areas warm. Viscosity increases strongly as the material cools. The light-blue re- gion indicates that yielding at a high strain rate takes place at the region where the cold material turns back down into the interior. The surface velocity plot shows that the motion is strongly partitioned into broad, nondeforming regions separated by narrow zones of deformation: Typical velocities are a few millimeters to a few centimeters per year. Note how the zone of yielding has changed from Figure 2c; it is now considerably more localized and focused into a single band that separates the two converging regions.

the point of failure does not accumulate—miss- particles in every element that store the mater- ing what we consider to be one of the most im- ial’s deformation and yielding history. portant features of Earth’s lithosphere. The rheological law is, once again, about as To directly incorporate such effects requires simple as can be. The yield stress in the material the advection of the strain history through the decreases as a function of the amount of strain mesh. Even the simplest formulations therefore the material accumulates while yielding takes encounter the advection–interpolation difficulty, place (to a total of about half the initial strength). which rapidly degrades the stored information. This produces a feedback effect in which strain One alternative is to parameterize the rheological and strain rate localize very strongly and domi- weakening associated with the strain history nate the system’s subsequent evolution. The lo- while ignoring the advection of the history—pro- calized zone close to the is light ducing an effective rheology with strain-rate blue; it forms a linear feature that becomes ap- weakening.5 However, this only partially solves proximately locked in place. Despite the rela- the need for a damage model. The simplest com- tively high resolution and the restriction to 2D, plete solution would be to introduce a La- the localization is still far too indistinct to possess grangian framework for the strain variables. the properties of a genuine fault. One approach we recently tried is using the The time evolution of the simulations shown tracer-particle scheme to track the strain history. in Figures 2 and 3 shows very clearly how sub- The tracer particles can carry any amount of in- tle changes in the material properties of the formation, including tensor quantities. Their lithosphere can produce dramatically different values are updated from the mesh-based vari- behavior. ables by standard finite-element interpolation from the shape functions. The Lagrangian ref- erence frame ties back into the mesh by using The spherical Earth the particle locations in place of the usual quad- The spherical shell geometry for Earth’s man- rature points for computing element integrals. tle has an important influence on the convection The method comes from recent work to extend pattern in map view and must be treated realis- the particle-in-cell schemes used in the 1950s tically for global problems. We also must aban- and 1960s.6 Some subtleties are required for don flat-Earth models when modeling anything modeling convecting fluids associated with the other than regional-scale observable data, in- enormous strains, but the method is essentially cluding plate motions, the gravitational field, as simple as outlined earlier. Earth’s rotation and polar wander, and mantle Figure 3 shows the result of such a calculation. seismic structure—all global features. The grid resolution is similar to that in Figure Several recent studies have focused on going 2, but now there are also approximately 8 to 12 beyond the flat-Earth approximation.7 Model-

MAY/JUNE 2000 29 Figure 4. This full-spherical shell model of mantle convection shows (a) low-temperature isosurfaces in blue, (b) high- temperature isosurfaces in yellow, and the core mantle boundary in red. The viscosity depends on both temperature (the cold areas being strong) and position (plate margins being weak). The green lines represent present-day plate margins. The cold downwellings have a distinct sheet morphology reminiscent of the structures imaged by , whereas the hot upwellings have a more complex, plume-like morphology at mid-depth in the simulation.

ing mantle dynamics in a spherical shell geome- quirement of the grid and parallel computing. try ultimately has two important goals. The first As an example, we present a time-dependent goal is to investigate the planform of mantle con- thermal convection calculation with tectonic vection with realistic rheology and geometry and plates, temperature-dependent rheology and lay- to understand the nature of seismically observed ered viscosity, and predominantly internal heat- 8 mantle structure. The second is to explain sur- ing. Combining temperature-dependent and face observables such as plate motion. reduced viscosity at plate margins results in mo- Of necessity, whole Earth simulations are bile surface plates. The reduced viscosity at plate enormously demanding computationally and re- margins simulates the effects of enhanced defor- quire the most efficient computational grids, nu- mation associated with interplate seismicity. merical algorithms, and parallel computing. The The model starts with a small amount of neg- simulations we present here are the product of atively buoyant (cold) material in subduction a computational tool that Shijie Zhong and his zones. The model is then integrated for about colleagues developed.8 One benefit of choosing two turnover times so that the heat transfer and an FEM is that the bones of the spherical code flow field are statistically time-invariant. are essentially identical to those of the flat-Earth Figure 4 shows a snapshot of the characteristic code, and the same efficient solution algorithms thermal structure from this calculation. Figure 4a (full multigrid, Figure 1c3) can be applied. The shows how the cold downwellings originate at mesh comprises 12 logically rectangular sub- plate margins and sink into the mantle as sheet meshes that are used to tile the sphere uniformly. structures. The downwellings impinge the The rectangular meshes are fully subdividable core–mantle boundary and spread over the bound- to allow an efficient implementation of the ary. The sheet structure resembles the morphol- multigrid algorithm. ogy of Benioff zones of deep seismicity as well as Parallel computation is implemented using a tomographic images of seismic velocity below the message-passing algorithm on the structurally termination of seismicity at the 660-km depth.9 enforced communication (between the 12 sub- As shown in Figure 4b, throughout most of grids) and, if enough processors are available, the mantle, the upwellings have plume-like mor- within each of the subgrids. The full multigrid phologies. Near the surface, upwellings are long algorithm resembles the one described earlier and linear and are closely associated with di- but with special features that result from the re- verging plate boundaries. The hot plumes com-

30 COMPUTING IN SCIENCE & ENGINEERING Figure 5. Figures a through f show the results of our 3D time- dependent model of thermal convection in the Australian re- gion from 130 million years ago to the present. The left box for each time frame shows the tec- tonic plates and their velocities, which are imposed as kinematic boundary conditions. Inside the domain (not shown), the full dy- namic coupling of Equations 1 to 3 are solved. The right box for each time frame shows the resulting resulting from the fluid flow. It matches the observed changes in the shape of Earth, inferred from sea-level change, extreme- ly well. (Figure courtesy of the American Association for the Advancement of Science.)

mence at the core-mantle boundary and are through time on a variety of space and time preferentially located below either spreading scales. Moreover, nearly all models of mantle centers or plate interiors, including those in the convection, including those shown in Figures 2 Pacific. The hot plumes, when impinging the and 4, are time dependent, at least for part of surface, might produce extensive volcanic activ- their history. ity and might be responsible for volcanic chains Because convection is a buoyancy-driven phe- such as Hawaii and Iceland. nomenon, it would be beneficial to find other ob- By controlling the locations of downwelling servations that are also sensitive to mantle flow. structures, surface plates determine the scale of Topography and gravity, which are sensitive to in- thermal structure within the mantle. The down- ternal mass anomalies, provide a powerful con- welling sheet and upwelling plume structures straint on mantle dynamics, as do tomographic and their spatial scales are consistent with the images of the mantle constructed from seismic observations of seismic structures from seismic waves. Unfortunately, such observations only pro- tomography. Global models such as this can also vide us with present-day, instantaneous snapshots relate interior dynamics to surface plate motion of convection. However, buoyancy-driven flows and the geoid, and therefore impose important cause deflections of density interfaces, a contribu- constraints on how mantle rheology depends on tion to surface topography called dynamic topogra- temperature.10 phy. Such deflections of Earth’s surface cause rela- tive sea-level fluctuations that leave a decipherable record in the geological history of the continents. Australian case study To understand the vertical motion of mass A fundamental but poorly constrained aspect within the mantle as a function of time, we inte- of mantle convection is its time dependence. grated a variety of geological observations in the This is especially unfortunate, because we know Australian region with the aid of time-dependent that plate kinematics change substantially 3D mantle convection models.11 We have focused

MAY/JUNE 2000 31 on the Australian region because large-scale fea- of this cool mantle were drawn up by the north- tures exist there that have never been adequately wardly migrating diverging plate boundary be- explained by the application of the kinematic rules tween Australia and Antarctica (as show in Figures of plate tectonics but that might be associated with 5e through 5f). This caused a circular dynamic the radial motion of buoyancy within the mantle. topography depression to develop at the present We focused on the subsidence and then emer- position of the AAD. Using the observations to gence of the Australian continent out of the sea constrain how topography changes as a function of during the Cretaceous Period, which was a period time, we can follow the fall and rise of mass within of extensive marine flooding of the other conti- the mantle over a period of about 100 million years. nents worldwide by a sea-level rise. We also fo- The implication of this regionally focused cused on the existence of a present-day cold spot, study is that disparate data types can be under- often interpreted as a convective downwelling, in stood within the context of time-dependent the mantle below the spreading center between mantle convection when convection models are Australia and Antarctica (a region known as the explicitly tailored to observed paleogeography. Australian–Antarctic discordance, or AAD). We As convection models begin to incorporate mul- have shown that the two features are related to tiphysics, there seems to be no limit to the man- the overriding of a downwelling (a long-lived ner in which geological observations, which con- subduction zone) by the Australian plate. strain the time domain, can be integrated with We built a 3D model of the mantle around present-day geophysical observations. Australia using the numerical techniques that we have just described. Figure 5 shows the results. One of our most important geological con- ur goal has been to combine the straints is the observed kinematic history of the Stokes and energy equations with a plates (referred to as paleogeography) over the realistic rheology, thereby letting us last 100 million years or so. In principle, we understand the complex dynamic could develop a dynamic model with the aim of Ocoupling that occurs in the mantle and that gives reproducing observed plate motions. However, rise to plate tectonics and other surface features. this problem is highly nonlinear; our under- This approach holds great promise because it standing of the feedback between plate margin makes a tremendous amount of data relevant to rheology and plate motions is still in its infancy, understanding Earth’s dynamics. The challenge and the computational cost is too formidable for is that the computational models must be inher- us to realistically pursue this approach at pre- ently realistic, particularly when predicting ob- sent. Instead, we imposed observed plate mo- served geography or plate history, so that the tions with velocity boundary conditions onto a models can be connected with observations. We 3D domain while making the flat-Earth as- view this as one of the most exciting future di- sumption described earlier. rections of computational . Starting with Figure 5a, the models show a This raises an important philosophical issue slab—a convective downwelling—about 1,400 km that many areas of computational science face: from the restored eastern margin. The models Should a model attempt an explicit match with suggest that above cold fluid, the viscous stresses observations, or is a match in a statistical sense pulled the surface downward so that a deep trough sufficient? In the case of geophysics, we must ask developed off the coast of Australia. Figures 5a whether we wish to produce a model of Earth as through 5c show that, as Australia moved east in a it really is today or of an Earth that might have fixed reference frame and over the cold fluid from been if the chaotic processes of formation and 130 to 190 million years ago, a broad dynamic evolution happened to work out differently. topography depression of decreasing amplitude Clearly, one important goal is to determine migrated west across the continent. This caused whether a fully dynamic model can predict the the continent to subside and then uplift, as shown kinematic rules of plate tectonics . That is, can we in Figures 5c through 5e. During this period, most create a generic model of Earth that looks cor- of the slab or cold fluid descended into the deeper rect—a model in which plates have about the cor- mantle, but the models show that part of the rect size distribution, have the same degree of cooler mantle became trapped within the so-called sharpness around their edges, and move at about transition zone (a region between 410 and 660 km the same velocity as observed? Obviously, the acid deep, bounded by two phase transitions). test regarding whether such generic models really From 40 million years ago to the present, wisps do look correct is whether they can predict the spe-

32 COMPUTING IN SCIENCE & ENGINEERING cific information that we do have about the evolu- 3. U. Christensen, “Convection with Pressure and Temperature De- pendent on Newtonian Rheology,” Geophysical J., Vol. 77, 1984, tion of our particular “possible Earth.” Hence, our pp. 343–384. conviction that integrated observation and model- 4. L. Moresi and V. Solomatov, “Mantle Convection with a Brittle ing-driven research is of fundamental importance. Lithosphere: Thoughts on the Global Tectonics Styles of the Earth Important developments must occur if we are and Venus,” Geophysical J. Int’l, Vol. 133, No. 3, June 1998, pp. to achieve still more realistic matches with ob- 669–682. 5. P. Tackley, “Self-Consistent Generation of Tectonic Plates in servations: First, in the realm of plate-margin Three-Dimensional Mantle Convection,” Earth & Planetary Sci- and shear-zone physics, for example, we must in- ence Letters, Vol. 157, Nos. 1–2, Apr. 1998, pp. 9–22. corporate explicit, history-dependent rheologies 6. D. Sulsky, Z. Chen, and H.L. Schreyer, “A Particle Method for driven by changes in the grain size or incorpo- History Dependent Materials,” Computer Methods in Applied Me- ration of volatiles (if indeed this is the relevant chanics and Eng., Vol. 118, Nos. 1–2, Sept. 1994, pp. 179–196. 7. P.J. Tackley et al., “Effects of an Endothermic Phase-Transition at controlling physics) to better simulate plate tec- 670-km Depth in a Spherical Model of Convection in the Earth’s tonics. Initially this can be achieved through a Mantle,” Nature, Vol. 361, No. 6414, Feb. 1993, pp. 699–704. flat-Earth approximation in two dimensions. 8. S. Zhong et al., “The Role of Temperature-Dependent Viscosity Eventually this will have to be extended into a and Surface Plates in Spherical Shell Models of Mantle Convec- full 3D spherical geometry, although some of the tion,” to be published in J. Geophysical Research. 9. S.P. Grand, R.D. van der Hilst, and S. Widiyantoro, “Global Seis- fine-scale physics will inevitably need to be pa- mic Tomography: A Snapshot of Convection in the Earth,” GSA rameterized, just as it is in general circulation Today, Vol. 7, No. 4, Apr. 1997, pp. 1–7. models of Earth’s atmosphere and ocean system. 10. S. Zhong and G.F. Davies, “Effects of Plate and Slab Viscosities Second, because it is not yet clear which ob- on the Geoid,” Earth & Planetary Science Letters, Vol. 170, No. 4, servations best constrain the dynamics of plate July 1999, pp. 487–496. 11. M. Gurnis, R.D. Müller, and L. Moresi, “Dynamics of Cretaceous boundaries, predictions from geodynamic mod- Vertical Motion of Australia and the Australian-Antarctic Discor- els should guide the acquisition of the most rel- dance,” Science, Vol. 279, No. 5356, Mar. 1998, pp. 1499–1504. evant observations on plate boundaries. Third, in a problem that areas of computational Louis Moresi is a research scientist with the CSIRO Di- science face as more complex physics is coupled vision of Exploration and Mining in Perth, Australia. He into codes, the software will have to undergo develops finite-element methods for the study of greater levels of benchmarking to prove its verac- highly deforming Earth materials and applies the codes ity. Indeed, computational geodynamics has been to the study of continental deformation, plate tecton- placed on a much firmer footing since the first ics, convection, and heat flow. He has a DPhil from Ox- benchmark comparisons of codes a decade ago. ford University in geophysics and a BA from Cam- Finally, it is not yet possible to fully compute bridge University in physics. Contact him at CSIRO mantle convection in a sphere with the coupled Exploration & Mining, PO Box 437, Nedlands, 6009, physics that gives rise to plate tectonics. How- Western Australia; [email protected]. ever, given the newly developed finite-element codes, such models will achieve sufficient reso- Michael Gurnis is a professor of geophysics in the Seis- lution with an order-of-magnitude increase in mological Laboratory at Caltech. His research interests CPU speeds and memory over the largest mas- are in the application of computational mechanics to sively parallel supercomputers at national and problems related to plate tectonics and the Earth’s international centers. deep interior. He is a fellow of the American Geophys- ical Union and a senior fellow of the Geological Society Acknowledgments of America. He has a PhD in geophysics from the Aus- The Australian Geodynamics Cooperative Research tralian National University. Contact him at MS 252-21, Centre (AGCRC) partially funded Louis Moresi’s work, and Caltech, Pasadena, CA 91125; [email protected]. the AGCRC’s director has given permission for this article’s publication. This article is contribution number Shijie Zhong is a research scientist at MIT. His research 8676 of the Division of Geological and Planetary involves the use of large-scale computer models to Sciences, Caltech. This work is funded in part by the study the deep interiors of the Earth and other terrestrial National Science Foundation and the National planets. He earned a BS in geophysics from the Univer- Aeronautics and Space Administration. sity of Science and Technology of China and a PhD in References geophysics and scientific computing from the Univer- sity of Michigan. He is a member of the American Geo- 1. P. Kearey and F.J. Vine, Global Tectonics, 2nd ed., Blackwell Sci- ence Ltd., London, 1996. physical Union. Contact him at the Dept. of Earth, At- 2. W.L. Briggs, A Multigrid Tutorial, Society for Industrial and Ap- mospheric, and Planetary Sciences, MIT, Bldg. E54-511, plied Mathematics, Philadelphia, 1987. Cambridge, MA 02139; [email protected].

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