Neil M. Ribe

Three‐dimensional modeling of plume‐lithosphere interaction

In order to understand better the dynamics of hot spots such as Hawaii, we present a three-dimensional numerical model for the interaction of a thermal plume with a moving lithosphere. The model domain is a rectangular box filled with fluid whose viscosity depends upon temperature and pressure. The lithosphere is represented by a layer of cold, highly viscous fluid moving with an imposed horizontal velocity U in the x direction, and a thermal plume is generated by a circular temperature anomaly on the bottom of the box. The steady flow is determined numerically using a hybrid spectral/finite difference technique. The flow is characterized by a “stagnation streamline” of width y ∼ x1/5 that represents the edge of the spreading plume material. We illustrate the detailed behavior of the model using the example of the Hawaiian plume. Our best fitting Hawaiian model is obtained by adjusting the plume buoyancy flux B until the predicted topography anomaly matches the observed Hawaiian swell topography; we find B = 4100 kg s−1, which implies a plume radius of 90 km for an assumed plume/mantle temperature contrast of 300°C. The predicted topography is supported primarily by density anomalies beneath the lithosphere and cannot be explained by lithospheric erosion. We therefore conclude that the classical “lithospheric reheating” model is unable to account for hotspot swells. The horizontal flux of buoyancy associated with the swell exceeds B by up to 80%, suggesting that current estimates of B for mantle plumes are too high. Empirically derived scaling laws for the width of the stagnation streamline and for the topography anomaly exhibit power law dependencies on B and U that agree well with those predicted by the “refracted plume” model of Olson (1990). The principal weakness of the model is that the predicted geoid/topography ratio of 0.010 for the Hawaiian swell is about twice the observed value of 0.004–0.006.

On the relation between seismic anisotropy and finite strain

Because seismic anisotropy in the upper mantle is due primarily to lattice preferred orientation (LPO) of olivine crystals induced by deformation, observations of anisotropy can provide constraints on mantle deformation and flow. In this paper, we use a previously published theory for deformation-induced LPO (Ribe and Yu, 1991) to quantify the relation between seismic anisotropy and deformation for two model compositions: a pure olivine (dunite) aggregate, and a harzburgite comprising 70% olivine and 30% enstatite. Analytical and numerical solutions of the governing equations show that the predicted LPO (and hence also the seismic anisotropy) depends only on the finite strain, and not on the deformation path by which that strain is produced. The magnitude of the anisotropy depends only on the ratios c1/c2 and c2 /c3, where c1 > c2 > c3 are the lengths of the principle axes of the finite strain ellipsoid, and the directions of maximum and minimum compressional wave velocity Vp coincide with the axes c1. and c3, respectively. We calculate the percent anisotropies for compresional and shear waves propagating along the three principle axes of the finite strain ellipsoid, as functions of the axial ratios c1/c2 and c2 /c3, for our two model compositions. However, inferences of the magnitude of finite strain obtained using these results are likely to be minimum estimates because our theory neglects the effects of dynamic recrystallization. Finally, we present, as a byproduct of this study a new direct method for calculating the finite strain produced by an arbitiary deformation history.

Dynamic elevation of the Cordillera, western United States

We introduce a methodology that synthesizes topography, gravity, crustal-scale seismic refraction velocity, and surface heat flow data sets to estimate dynamic elevation, i.e., the topography deriving from buoyancy variations beneath the lithosphere. The geophysical data independently constrain the topographic effects of surface processes, crustal buoyancy, and thermal boundary layer thickness. Each of these are subtracted from raw elevation of the western U.S. Cordillera to reveal dynamic elevation that can exceed 2 km and is significant at >95% confidence. The largest (∼1000 km diameter) of the dynamic elevation anomalies resembles a numerical model of a hypothetical Yellowstone hotspot swell, but the swell model does not account for all of the significant features seen in the dynamic elevation map. Other dynamic elevation anomalies are spatially correlative with Quaternary volcanism, but partial melt can contribute no more than a few hundred meters of elevation. Hence much of the dynamic elevation likely derives from other thermodynamic anomalies. Possible alternative mechanisms include both superadiabatic upwelling and adiabatic phase boundary deflections maintained by latent heat effects. Comparison of seismicity and volcanism to effective viscosity gradients, estimated from lithospheric flexural rigidity to facilitate the numerical swell model, suggests that tectonism focuses where lithosphere with negligible mantle viscosity abuts lithosphere with significant uppermost mantle viscosity.

A kinematic model for recrystallization and texture development in olivine polycrystals

The interpretation of seismic anisotropy in the mantle requires a knowledge of the relationship between the lattice preferred orientation (LPO) of crystals and the convective flow field. In order better to understand this link, we present a model for the evolution of LPO in olivine aggregates that deform by both intracrystalline slip and dynamic recrystallization. Dynamic recrystallization depends on the dislocation density of the grains, which is a function of the applied local stress. Grains with a large density of dislocations lower their bulk strain energy by nucleating strain-free sub-grains at a rate proportional to a dimensionless nucleation parameter λ*. Grains with high energy are then invaded by grains with low energy by grain-boundary migration, at a rate proportional to a dimensionless grain-boundary mobility M*. The value of λ* is constrained by observed LPO patterns in experimentally deformed olivine aggregates, and M* is constrained by the temporal evolution of the strength of the LPO. For M*=125±75 and λ*>3, the model predictions agree well with the experimental results. Numerical calculations of LPO using our model are significantly faster than those based on viscoplastic self-consistent or equilibrium-based theories, making the model especially suitable for applications for complex convective flows.

The dynamics of plume-ridge interaction, 1: Ridge-centered plumes

Abstract We investigate the dynamics of mantle plumes rising beneath mid-ocean ridges. To clarify the physics of this process, we examine first a simple ‘lubrication theory’ model in which a point source (analogous to a plume ‘stem’) of volume flux Q located directly beneath the ridge releases buoyant fluid into a viscous corner flow driven by a velocity boundary condition u(x) = Utanh(x/d), where U is the half-spreading rate and the ‘gap width’ between the diverging plates is ∼ 5 d . Numerical solutions of the differential equation governing the plume head thickness S(x, y) show how the width W of the plume head along the ridge depends on Q , U , d , and σ ≡ gΔϱ/48 n , where Δϱ is the density deficit of the plume and η is the viscosity. In the geophysically relevant ‘narrow gap’ limit (Q/σ) 1/4 ≫ d , W ∼ (Q/U) 1/2 II b 0.053 , where II b =Q/σU 2 is the ‘buoyancy number’. Numerical solutions of a more realistic 3D convection model with strongly temperature and pressure-dependent viscosity obey a nearly identical scaling law, and show no evidence that W is increased by ‘upslope’ flow of plume material toward the ridge along the sloping base of the rheological lithosphere. To apply our model to Iceland, we incorporate into it a melting parameterization that allows prediction of the excess crustal thickness produced by melting in the plume head. This extended model shows that the observed depth anomalies along the Mid-Atlantic Ridge near Iceland cannot be explained by a hot (temperature contrast Δ T ∼ 250° C ) and narrow (radius ∼ 60 km) ridge-centered plume. Instead, the anomalies are consistent with a much cooler and broader upwelling.

A theory for plastic deformation and textural evolution of olivine polycrystals

Seismic anisotropy in the upper mantle is due primarily to preferred orientation of olivine crystals induced by progressive deformation. In order to understand better the origin of seismic anisotropy, we present a simple theory for plastic deformation and textural evolution of olivine polycrystals. Each crystal in the aggregate is assumed to deform by intracrystalline slip on three major slip systems, whose hardnesses and stress exponents are known from experiments. The evolving grain orientation distribution in the aggregate is calculated by minimizing the difference between the local (crystal) deformation and the global (aggregate) deformation subject to the constraint of global strain compatibility. The axial compression texture predicted by our model agrees well to first order with that determined experimentally by Nicolas et al. (1973), although there are significant discrepancies in the details. Our results suggest that the orientation texture developed during progressive plane strain deformation is a nearly unique function of the finite strain, such that the crystallographic axes [100], [010], and [001] are concentrated around the finite strain axes a, c, and b, respectively (a > b > c). This result may allow the state of finite strain at depth to be estimated from observations of seismic anisotropy.

Inferring nonlinear mantle rheology from the shape of the Hawaiian swell

The convective circulation generated within the Earth’s mantle by buoyancy forces of thermal and compositional origin is intimately controlled by the rheology of the rocks that compose it. These can deform either by the diffusion of point defects (diffusion creep, with a linear relationship between strain rate and stress) or by the movement of intracrystalline dislocations (nonlinear dislocation creep). However, there is still no reliable map showing where in the mantle each of these mechanisms is dominant, and so it is important to identify regions where the operative mechanism can be inferred directly from surface geophysical observations. Here we identify a new observable quantity—the rate of downstream decay of the anomalous seafloor topography (swell) produced by a mantle plume—which depends only on the value of the exponent in the strain rate versus stress relationship that defines the difference between diffusion and dislocation creep. Comparison of the Hawaiian swell topography with the predictions of a simple fluid mechanical model shows that the swell shape is poorly explained by diffusion creep, and requires a dislocation creep rheology. The rheology predicted by the model is reasonably consistent with laboratory deformation data for both olivine and clinopyroxene, suggesting that the source of Hawaiian lavas could contain either or both of these components.

The generation and composition of partial melts in the earth's mantle

Abstract A set of equations is presented which combines the constraints of fluid dynamics and multicomponent phase equilibrium to provide a unified description of partial melting in the earths mantle. The equations are applied to a one-dimensional model for pressure-release melting of a simplified mantle material, which contains only two chemical components exhibiting either (a) complete solid solution or (b) a binary eutectic. In both cases, melting occurs over a range of depths. The unmelted crystalline residue (“matrix”) is modeled as a saturated porous medium, through which the melt can migrate because of its differential buoyancy. Since melt interacts continuously with the matrix during ascent, melting occurs by equilibrium rather than fractional fusion. This equilibrium fusion is not the same as batch fusion, however, since material elements are quickly dispersed by migration of melt relative to the matrix. To a first approximation, the temperature profiles (adiabats) in the partially molten zone are independent of melt migration. The slope of the adiabats varies in inverse proportion to the number of degrees of freedom which characterizes the melting. Melting of a complete solid solution occurs along a “wet” adiabat whose slope is controlled by absorption of latent heat. Melting of a eutectic system occurs along a steeper “univariant” adiabat until one solid phase is exhausted, and subsequently along a wet adiabat. The velocity of melt migration can exceed the mantle upwelling velocity by an order of magnitude or more. The volume fraction of melt present is always less than the fraction of the material which has melted, and is unlikely to exceed a few percent. For a wide range of initial conditions, melting of a eutectic system produces erupted melts having constant major element composition and widely varying trace element composition. This result may provide a partial explanation for the characteristic major- and LIL-element patterns observed in MORB. Liquid compositional paths during pressure-release melting cannot be determined from isobaric sections of phase diagrams, and can only be calculated by solving the energy equation subject to the appropriate depth-dependent phase equilibrium constraints.

The dynamics of plume‐ridge interaction: 2. Off‐ridge plumes

A variety of geophysical and geochemical evidence indicates that ascending mantle plumes can interact with ocean ridges located up to 1400 km away. I investigate the dynamics of this interaction using a simple model in which a point source with volume flux Q (analogous to a plume “stem”) releases buoyant fluid into a viscous corner flow driven by the divergence of rigid surface plates with thickness ∼(kx/U)1/2, where U is the half spreading rate. The point source is located at a distance xp from the ridge, and ridge migration is neglected. The buoyant fluid forms a thin sublithospheric layer whose thickness S(x, y) satisfies a nonlinear advection-diffusion equation describing the balance of advection by the corner flow, buoyancy-driven “self-spreading,” flow toward the ridge along the sloping base of the lithosphere, and continuous accretion into the lithosphere. Numerical solutions of this equation yield scaling laws for the lateral extent W (“waist width”) of plume material along the ridge, the fraction R of the plume flux that crosses the ridge, and the maximum value of xp beyond which interaction ceases. The sloping base of the lithosphere has only a minor (few tens of percent) influence on these quantities, which are determined principally by the balance of advection and self-spreading. An extension of the model to include plume-induced lithospheric thinning shows that this process increases the waist width by an amount of order 10%. Finally, the model provides a new explanation for the observation that plumes interact primarily with ridges that are migrating away from them, rather than toward them.

Coiling of viscous jets

A stream of viscous fluid falling from a sufficient height onto a surface forms a series of regular coils. I use a numerical model for a deformable fluid thread to predict the coiling frequency as a function of the threads radius, the flow rate, the fall height, and the fluid viscosity. Three distinct modes of coiling can occur: viscous (e.g. toothpaste), gravitational (honey falling from a moderate height) and inertial (honey falling from a great height). When inertia is significant, three states of steady coiling with different frequencies can exist over a range of fall heights. The numerically predicted coiling frequencies agree well with experimental measurements in the inertial coiling regime.