Lecture 1: The Fluid Dynamics of Mantle Convection

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Conservation Laws and Heat Transfer:

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The fundamental principle of heat transfer is based on conservation laws. Consider a volume V of material enclosed by a surface S1.

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Heat Content: The heat contained within a small volume element dV is given by ρCPT dV, where ρ is the density, CP is the specific heat capacity, and T is the temperature. The total heat H within volume V is the integral of this quantity over the entire volume: H = ∫V ρCPT dV1.

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Heat Flow: Changes in heat content H can only occur if heat flows across the surface S. If Q is the rate at which heat flows outward, then the rate of change of H is given by -dH/dt = Q. The negative sign indicates that the volume cools if Q is positive2.

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Heat Flux and Temperature Gradient: The heat flux depends on the temperature gradient, denoted by ∇T, and heat flows down the temperature gradient. The heat flux across a surface element dS is given by dQ = -k∇T · dS, where k is the thermal conductivity. The dot product accounts for the direction of the temperature gradient. No heat flows across dS if the gradient of T is parallel to the surface, in which case ∇T · dS = 02.

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Divergence Theorem: The heat flow equation can be expressed as a volume integral using the divergence theorem, resulting in: -d/dt ∫V ρCPT dV = -∫S k∇T · dS = ∫V ∇ · (k∇T) dV3.

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Heat Equation: This leads to the partial differential equation describing heat transfer, which is: ∂(ρCPT)/∂t - ∇ · (k∇T) = 0. If ρ, CP, and k are constant, this equation simplifies to the heat equation: ∂T/∂t = κ∇2T, where κ is the thermal diffusivity, given by κ = k/ρCP3.

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Advection: In a moving fluid, heat transport includes the advection term, ∫S ρCPTv · dS, where v is the fluid velocity4. The heat equation then becomes ∂T/∂t + (v · ∇)T = κ∇2T. This term represents the heat carried by the fluid's motion and makes fluid dynamics problems much more complex4.

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Scaling and Non-dimensionalization

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Importance of Scaling: Laboratory experiments are often used to understand the behavior of the Earth. Scaling the equations allows us to relate laboratory experiments to the Earth's behavior4.

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Non-dimensionalization: By introducing dimensionless variables z’ = z/a and t’ = t/t0 and t0 = a2/κ, the heat equation transforms into ∂T/∂t’ = ∇’2T where a is the thickness of the slab5. This non-dimensionalized equation is the same for all bodies, irrespective of their size, simplifying analysis5.

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Cooling Time: The time it takes for the temperature within a plate to change by 1/e is given by t' = 1/π25. The real cooling time is τc = a2/π2κ, which is a very important equation in geology6.

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Convection and the Rayleigh Number

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Convection Conditions: A fluid will convect if there are horizontal temperature variations7. The advection of heat must exceed its diffusion rate.

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Buoyancy Force: Temperature variations cause density variations of ±ραT, where α is the thermal expansion coefficient. The total buoyancy force within a region of temperature +T is gραTa2. This produces a shear stress of gραTa on the boundary7.

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Viscosity: Viscosity, denoted by η, is defined as the ratio of shear stress to the velocity gradient. Thus η = gραTa/(v/a). This gives an advection time τa of a/v = η/gραTa8.

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Rayleigh Number: The ratio of the cooling time to the advection time, τc/τa, is dimensionless and is defined by the Rayleigh number (Ra). τc/τa = gραTa3/π2κη. The Rayleigh number is usually written as Ra = ρgαT0a3/κη8.... * A large Rayleigh number indicates vigorous convection, while a small Rayleigh number means feeble or absent convection9.

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Scaling the Advection-Diffusion Equation: The advection-diffusion equation can be scaled using T = T0T’, t = a2/κRa t’ and v = κRa/a v’, resulting in Ra (∂T’/∂t’ + v’ · ∇’T’) = ∇’2T’. Unlike the diffusion equation, the Rayleigh number cannot be removed by scaling, making circulations with large and small Ra different9....

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Convection Experiments

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Laboratory Experiments: Convection experiments, often using shadowgraph methods, are used to study mantle dynamics. A convecting layer is heated from below and cooled from above using hot and cold water passed between glass sheets11.

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Shadowgraph Technique: Parallel rays of light are shone through the convecting liquid. Hot, rising fluid is marked by dark lines, while cold, sinking fluid is marked by bright lines12.

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Convection Cells: Convection cells tend to be as wide as they are deep12.

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Rayleigh Number and Flow Geometry: The geometry of the flow changes with the Rayleigh number. Just above the critical Rayleigh number, flow is in steady cylindrical rolls. As the Rayleigh number increases, the flow transitions to bimodal patterns, and then to spoke patterns at very high Rayleigh numbers13....

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Bimodal convection is where a new set of rolls appears at right angles to the original set of rolls13.

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Spoke patterns consist of rising and sinking sheets that meet at points, resembling the hub of a wheel14.

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Visualizing Flow: Different visualization methods are used in experiments, such as using lasers to observe interference patterns, or floating aluminum particles in the fluid to track motion13....

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Internal Heating: Internal heating alters convection patterns making downwellings more prominent16.

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Temperature-Dependent Viscosity

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Non-constant Viscosity: The viscosity of some fluids varies with temperature, which can lead to a wider variety of stable planforms than fluids with constant viscosity17.

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Corn Syrup Example: Corn syrup's viscosity changes significantly with temperature, making it a useful substance to use for experiments17.

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Numerical Simulations: Numerical simulations reveal that temperature-dependent viscosity can affect the flow pattern. Heating from below with variable viscosity leads to thinner hot rising sheets compared to cold sinking sheets18.

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Effect of Shear

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Shear Forces: Applying shear to a convection pattern, for example by moving a Mylar sheet across the surface of the fluid, can convert spoke patterns into rolls whose axes are parallel to the shear direction19.

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Key Concepts in Fluid Dynamics

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Dimensionless Numbers: Dimensionless numbers like the Rayleigh number are important because the behavior of a fluid is controlled by these values20.

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Rayleigh Number (Ra): Ra = ρgα∆Ta3/κη. It measures the ratio of heat carried by fluid movement to that carried by conduction. It is large when convection is vigorous20.

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Buoyancy Forces: These forces maintain flow due to density differences21.

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Advective Transport: Transport of energy, momentum, or a chemical species by fluid motion, as opposed to diffusion or conduction21.

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Dynamic vs. Kinematic Viscosity: The dynamic viscosity η is related to the kinematic viscosity ν by ν = η/ρ21.

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Peclet Number (Pe): Pe = Vd/κ is the ratio of heat transport by advection to that by diffusion. Here, V is fluid velocity, d is plate thickness, and κ is thermal diffusivity22....

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Prandtl Number (Pr): Pr = ν/κ measures the ratio of momentum diffusion to heat diffusion23.

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Reynolds Number (Re): Re = Va/ν measures the ratio of momentum advection to momentum diffusion23....

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Nusselt Number (Nu): Measures the ratio of heat transported through a convecting layer to that carried by conduction without convection24.

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Vorticity (ω): Defined as ω = ∇ × v. For a rotating rigid body, the vorticity is twice the angular velocity. It also applies to fluids undergoing general deformation25.

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Newtonian vs. Non-Newtonian Fluids: A Newtonian fluid has a stress that is directly proportional to strain rate. A non-Newtonian fluid's stress is proportional to a power of the strain rate25....

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Geometry of Convection

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Planform: The geometry of the convection when viewed from above (e.g. square, hexagonal, rolls)26.

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Aspect Ratio: Ratio of the distance between rising and sinking regions to the depth of the layer26.

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Linear vs. Nonlinear Equations

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Linear Equations: Have the form dx/dt = -λx, where x appears only as x27. They are easily solved by standard methods.

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Nonlinear Equations: Have the form dx/dt = -λx2, where x can have powers. These are difficult to solve27.

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Solving Convective Equations

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Linearization: If velocities are small, approximate linear equations can be derived by ignoring terms including v². This is used to determine whether a layer will convect or not by introducing small temperature disturbances27.

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Stability Analysis: If disturbances grow exponentially or by growing oscillations, the layer is unstable or overstable, respectively. Layers can also be stable to small disturbances but not to larger ones (finite amplitude instability)28.

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Nonlinear Equations: Most fluids must be studied with nonlinear equations because the Rayleigh number is large. Solutions are obtained using numerical techniques and properly scaled laboratory experiments29.

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The temperature structure often consists of thin boundary layers and interior flow29.

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Rising regions may be called plumes or jets depending on their origin29.

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Turbulence: It is best to avoid the term laminar and turbulent when discussing convection as it relates more to the type of mathematical analysis. "Turbulence is that which the speaker does not understand"29.

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Adiabatic Processes

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Adiabatic vs Isentropic: An adiabatic process involves no heat exchange, while an isentropic process is adiabatic and reversible30.

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Mantle Conditions: The movement of fluid in the interior of a convecting layer is usually too rapid for heat conduction to matter. Thus temperature changes are mainly due to density changes resulting from pressure. The temperature gradient under such conditions is called an isentropic or adiabatic temperature gradient30.

Lecture 2: Plate Tectonics as Convection

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Mantle Rheology

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Deformation of Mantle Rocks: Understanding mantle convection requires understanding how mantle rocks deform when stressed over long time scales, which is known as rheology31.

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Crystalline Structure and Defects: Rocks are composed of crystals with regular arrangements of atoms, but also contain defects such as point defects, line defects (dislocations), and planar defects (grain boundaries)31....

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Dislocation Glide: Dislocations facilitate deformation, requiring only bonds in the neighborhood of the dislocation to be broken32.

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Yield Stress: The stress required to move dislocations depends on the spacing between dislocations. At low temperatures, crystals deform only by dislocation movement, but at higher temperatures, diffusion of atoms and vacancies allows creep33.

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Diffusion Creep

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Nabarro-Herring Creep: This is volume diffusion through the solid crystal, which dominates at high temperatures and low stress. The stress-strain rate relationship is σ = Ad2ϵ̇, where d is grain size34.

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Coble Creep: Grain boundary diffusion dominates at lower temperatures. The stress-strain rate relationship is σ = A'd3ϵ̇, also linearly dependent34....

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Dislocation Creep

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Nonlinear Relationship: At high stress and temperature, dislocations can move by climb, where atoms are added to or removed from the plane of atoms of an edge dislocation. This leads to a nonlinear relationship between strain rate and stress: ϵ̇ = Aσn, 3.5 < n < 5.5, which is independent of grain size35....

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Preferred Crystal Orientations: Dislocation creep produces preferred crystal orientations36.

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Deformation Maps

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Homologous Temperature: θ = T/Tm, where T is the temperature and Tm is the melting temperature in Kelvin37....

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Homologous Stress: τ = σ/μ, where σ is the shear stress and μ is the shear modulus38.

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Deformation Maps: Created by Frost and Ashby (1982), these maps plot creep behavior as a function of homologous stress and temperature. Diffusion creep dominates at low stresses and high temperatures, while dislocation creep dominates at higher stress and lower temperatures38....

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Grain Size Reduction: Dislocation creep can produce new grain boundaries, reducing grain size to the point where diffusion creep becomes important40.

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Independent Constraints on Mantle Rheology

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Postglacial Rebound: Analysis of postglacial rebound following the last ice age provides independent constraints on mantle rheology41.

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Uplift Timescale: The characteristic timescale of the uplift is related to the size of the ice sheet and the viscosity of the mantle. The time constant τ = 4πη/ρgλ relates viscosity to the wavelength of the ice sheet42.

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A typical timescale is τ ~ 4000 years resulting in a viscosity estimate η = 1x10²¹ Pa s43.

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Forces Maintaining Plate Motions

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Challenges in Modeling: Numerical models have not been able to self-consistently generate plate tectonics due to issues such as sinking regions being plume-like rather than planar43. The behavior of faults on plate boundaries is also a challenge to reproduce43.

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Driving and Resistive Forces: Plate motions are driven by forces such as ridge push and slab pull and resisted by forces such as basal drag and slab resistance44.

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Driving Forces

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Ridge Push: Results from the elevation of spreading ridges above old seafloor. Oceanic plates are in isostatic equilibrium, and as they cool and density increases their elevation reduces45.... This process stops at about 80 Ma when the lithosphere stops thickening.

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Small-scale convection beneath the plate removes the cold and dense base of the lithosphere as it gets too thick (a Rayleigh-Taylor instability)47.

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Pressure beneath mid-ocean ridges is greater than that beneath old seafloor, leading to a force of 2 to 3 x 10¹² N per meter48.

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Slab Pull: Results from the sinking of cold material into the mantle at subduction zones. The buoyancy force can exceed 10 x 10¹² N m^-1. The net force is limited by resisting forces such as drag, friction on the megathrust and the work done in bending the plate at the outer rise49.

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Resistive Forces

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Basal Drag: Moving plates involves shearing the mantle, leading to viscous drag forces50.

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Asthenosphere: Under fast-moving plates, seismic anisotropy is aligned with the direction of plate motion, suggesting that basal drag is resistive50. The magnitude depends on the plate velocity, the thickness of the asthenosphere, and its viscosity50.

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Mountain Ranges

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Buoyancy: Even though mountains are isostatically compensated, they exert a force on the surrounding lowlands due to pressure differences which can be up to at least 6 x 10¹² N m^-151.

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Dominant Forces

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Slab Pull Dominance: Circumstantial evidence suggests that slab pull may be the dominant driving force52.

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Force Balance Calculations: Used to calculate the magnitudes of forces such as ridge push, slab pull, and tractions on the base of the plate by balancing torques53.

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Mantle Flow Calculations: Simple calculations use mass conservation to relate deep return flow to ridges from subduction zones54....

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The forces on the plates should balance54.

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Pressure differences between the mantle beneath slabs and ridges are small54.

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Low Viscosity Layer: Estimates of mantle viscosity from mantle flow calculations agree with those from postglacial rebound if a low viscosity layer under the plates is included54....

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Complex Mantle Flow Calculations: Numerical models simulate plate motion using mantle flow, though they are sensitive to chosen parameters and treatment of faults57. * Faults are often manually introduced, and model behavior is dependent on chosen rheology57.

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Creation of New Plate Boundaries

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Spreading Ridges: Thought to form by stress concentrations on transform faults or by uplift over an upwelling mantle plume58.

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Subduction Zones: Require a certain length to generate driving buoyancy forces and become self-sustaining. Convergence of at least 20 mm/yr for at least 10 Ma is required59. Out-of-plane forces or growth along-strike can initiate subduction59.

Lecture 3: Mapping Mantle Convection

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Models of Mantle Convection

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Small and Large Scale Flow: Models of mantle convection include small-scale cells with an aspect ratio of about one and larger-scale plate motions60.

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If the plates are coupled to the deeper mantle, small-scale convection might form rolls parallel to plate motion. If decoupled by a low viscosity zone, small-scale flow will be three-dimensional60.

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Mapping Mantle Convection in the Oceans

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Challenges in Observation: The planform of mantle flow is difficult to observe due to the overlying plates. Thermal disturbances at the base of the plate take about 60 Ma to appear as heat flow changes at the surface61.

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Gravity and Surface Elevation: Measurements of the gravity field and surface elevation directly reflect mantle motion beneath the plates. The surface moves upwards over rising limbs and downwards over sinking regions61.

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Gravity Field: Gravity is the sum of two effects: gravity due to the temperature distribution (negative above rising regions) and gravity from surface deformation (positive above rising regions)62.

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The net gravity is positive above a rising region, but is smaller than expected from the surface deformation.

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Correlation between Gravity and Topography: Convection models show a strong correlation between gravity and topography62.

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Corrections for Topography: Surface topography is affected by various factors, including crustal thickness variations. Large areas of the ocean have uniform crustal thickness, requiring correction for the age of the lithosphere by calculating residual depth dr = dt - do63.... do is observed depth and dt is expected depth based on the depth-age relationship63.

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Gravity Corrections: The gravity field must be corrected for the age of the plate, with the magnitude of the correction dependent on the spreading rate and plate age64....

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Mapping Mantle Convection Using Gravity and Residual Depth: The correlation between gravity and topography can be used to map convection. Regions with positive residual depth and positive gravity anomaly often indicate rising mantle material65....

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Convective Gravity Anomalies

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Distinguishing Convective Anomalies: These are distinct from shorter wavelength anomalies due to plate flexure66.

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North Atlantic: Early studies in the North Atlantic showed a correlation between gravity and residual depth, suggesting a region with upwelling mantle material65.

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Elastic Response of Plates

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Flexure: The elastic response of a plate to stresses unrelated to mantle convection must be understood before observing convection effects. For example the weight of the Hawaiian ridge causes a depression of the sea floor67.

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Pacific Ocean Mapping

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Satellite Measurements: Maps of the Pacific from satellite measurements of gravity and ship measurements of residual depth show positive and negative anomalies from mantle convection. Wavelength of these anomalies is 2000-3000 km68.

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Convection occurs at a smaller scale than the plate motions, and the planform is mostly three dimensional rather than rolls69.

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Swells and Basins: Swells such as Hawaii are correlated with positive gravity and residual depth anomalies70. Basins are also correlated with each other.

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Atlantic Ocean Mapping

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Similar Behavior: The Atlantic shows similar patterns, with positive gravity and residual depth anomalies not associated with spreading ridges69.

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Ratio of Gravity to Topography

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Testing Convection Models: The ratio of gravity to topography is around 30 mGals/km as predicted by convection models, and there is good correlation between the two, with no significant evidence of changes with age71....

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Mapping Mantle Convection in the Continents

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Crustal Thickness Variations: Continental topography is mainly controlled by crustal thickness rather than mantle convection73.

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Long-Wavelength Anomalies: Gravity anomalies with wavelengths > 500 km are used to map convective circulation in continents, because the lithosphere can't support these with elastic stresses73.

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African Plate Example: The convective circulation under the African plate is unrelated to surrounding ridges and instead reflects rising and sinking plumes73. There is a correlation between the gravity anomalies and topography similar to what is seen in models.

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Ratio in Africa: The ratio of gravity to topography is about 45 mGals/km at long wavelengths, which is greater than that for the Pacific due to differences in density contrast between rock and water, and rock and air74.

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Other Continents: Convective support of topography is expected globally, but can be complicated by active tectonics74.

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Global Maps of Dynamic Topography

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Combined Measurements: Residual depth estimates are combined with gravitational estimates to create global maps of dynamic topography75.

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Consistency with Geological Observations: The global map is consistent with geologic observations, for example, active hotspot volcanism occurs in regions of positive dynamic support75.

Lecture 4: Whole Mantle Convection?

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Debate Over Mantle Convection

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One vs. Two Layer Convection: A major debate is whether the mantle convects as a single layer or two layers separated by a boundary at 660 km depth.

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Earthquake Observations

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Sinking Slabs: Earthquakes in subducted lithosphere show how cold slabs sink through the mantle. Stress distribution reveals the influence of the phase change at 660 km76....

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Focal Mechanisms: Mechanisms of earthquakes show that slabs undergo down-dip extension then compression around the 660 km boundary78. ‘Extension’ means failure by faulting in such a way as to lengthen the slab, and ‘compression’ the converse.

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Slab Behavior at 660 km: Some slabs become horizontal at 660 km, with mechanisms suggesting extension as they spread out horizontally on the interface79.

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Dynamics of the 660 km Discontinuity

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Phase Transition: The phase transition at 660 km is endothermic, causing a deepening of the transition in cold slabs which creates an upward buoyancy force, which can cause slab material to pond. The ratio of the downward to upward forces can be written as ρ2αgh/γ∆ρ80.

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Viscosity Contrast: Studies of postglacial uplift suggest that the viscosity of the lower mantle is 10-30 times greater than that of the upper mantle. This contrast alone can account for the seismic focal mechanisms even if slabs penetrate the lower mantle81.

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Geochemical Constraints

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Layered Convection: Geochemistry suggests layered convection to preserve a region of primitive mantle not processed by ridges and island arcs. A depleted upper mantle also exists where MORB is formed82.

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Isotopic Systematics: Isotopic ratios are used as constraints, requiring estimates of parent-daughter ratios. Geochemical arguments constrain mass of material in 'reservoirs' and how long they have been isolated83.

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Key isotopic systems are ⁴⁰K→⁴⁰Ar, ⁸⁷Rb→⁸⁷Sr, ¹⁴⁷Sm→¹⁴³Nd, ²³²Th→²⁰⁸Pb, ²³⁵U→²⁰⁷Pb, ²³⁸U→²⁰⁶Pb, ¹⁷⁶Lu→¹⁷⁶Hf, ¹⁸⁷Re→¹⁸⁷Os, ¹²⁹I→¹²⁹Xe, ¹⁸²Hf→¹⁸²W, and ²⁴⁴Pu→heavy Xe isotopes84....

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Primitive Mantle Composition: Estimated using ratios of involatile elements and abundances of Ca and Al, with proportions from carbonaceous chondrite meteorites. K/U ratios in the continental crust are used because they are incompatible elements in solid phases during melting85.

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Geochemical Reservoirs:

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Atmosphere: Important for rare gasses except He86.

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Continental crust: Has a mean age of 2 Ga, calculated from Nd isotopic ratio and Sm/Nd of the continents86.

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MORB source: Uniform and depleted in incompatible elements compared to primitive mantle87.

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OIB source: Associated with ocean island basaltic volcanism with varying numbers of different reservoirs hypothesized88.

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Some OIB sources such as Hawaii have larger ³He/⁴He ratios (R/Ra) than MORB89.

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High R/Ra requires a primitive source that has undergone less degassing than the MORB source89.

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Constraints on Convective Flow

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Preservation of MORB Reservoir: Convection should not remove the MORB reservoir from the upper mantle on a timescale of 2 Ga. Convection should also not destroy plume sources90.

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Subduction and the MORB Source: Subducted slabs sink to the base of the upper mantle, possibly into the core-mantle boundary. It is unclear how the MORB source can be depleted if it has upwelled from the lower mantle91.

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Hot plumes can only arise from a boundary heated from below.

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Recycled Material: Parent-daughter systems like U-Pb, Sm-Nd, and Rb-Sr in plume sources suggest they contain recycled material, although this is debated. Melting under ridges does not significantly fractionate these elements, but melting under oceanic islands does.92...

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If the plume reservoir is not primitive, there is no need to preserve a large reservoir unaffected by convective stirring93.

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Two-Layer Convection: Two-layer convection would not require the same constraints94.

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Boundary between Reservoirs: Whether this boundary is at 660 km is not confirmed by geochemistry, although that is generally assumed94.

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Seismic Tomography

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Method: Seismic travel times are inverted to determine 3D velocity structure using P and S waves, providing information on mantle structure95.

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Travel times are measured to about 0.1 seconds, and then used to calculate velocity96.

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Inversion Artifacts: The inversion process can create artifacts that may appear to be real features96.

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Checkerboard Test: A checkerboard model is used to test the inversion process, revealing that it is less accurate under the oceans97. * This test involves calculating travel times through a model with regular variations in seismic velocity, and then inverting it97.

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Slab Penetration: Early tomographic images showed that slabs may penetrate the lower mantle. For example, high velocity slabs were seen in the lower mantle beneath the Americas98.

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Grand (1994) argued that these slabs resulted from subduction of the Farallon plate 90-120 Ma ago98.

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More Recent Studies: Later studies showed that some slabs remain in the upper mantle and others do penetrate99....

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Subduction Zone Variability: Studies show that slab behavior varies by location, with some slabs stagnating above the 660 km discontinuity, some penetrating the 660, and some being trapped in the uppermost lower mantle100....

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One or Two Layers?: The conflict between layered convection as suggested by geochemistry and seismology vs whole mantle convection suggested by early seismology is showing signs of being resolved102. Limited mass exchange between the upper and lower mantle may satisfy both geochemical and seismological constraints102.

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The dynamical problem, of why some slabs penetrate the lower mantle and some do not remains102.