What Term is Used to Describe the Continuous Heating and Cooling of Magma Within the Mantle

Mantle Convection

Mantle convection involves competition between diffusion of heat (thermal conductivity), resistance to motion (viscosity), and buoyancy forces (thermal expansivity).

From: Treatise on Geophysics (Second Edition) , 2015

Tectonophysics

Donald L. Turcotte , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VIII Conclusions

Mantle convection and plate tectonics provide a general framework for understanding tectonophysics. Transport of heat from the interior of the earth drives solid-state convection. Plate tectonics is a direct consequence of this convection. The relative velocity between plates causes crustal deformation at the boundaries between plates. In some cases this deformation is diffuse and is spread over a broad area. Volcanism occurs at most plate boundaries and is also responsible for crustal deformation.

Although we now have a general understanding of tectonophysics, we are still not able to predict earthquakes. Deformation on a local scale is extremely complex. In fact, it is quite likely that local deformation is so complex and chaotic that it is fundamentally impossible to make predictions of earthquakes. Only risk assessments will be possible. There is increasing evidence that scale-invariant, fractal statistics are applicable to a variety of tectonophysics problems. One possible application is the direct association of large earthquakes with small earthquakes; a risk of a great earthquake is present only where small earthquakes are occurring and the level of local seismicity can be used to assess the seismic hazard.

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Mantle Dynamics

A. Davaille , A. Limare , in Treatise on Geophysics (Second Edition), 2015

7.03.6.3.1 Free slip boundaries

Mantle convection is characterized by mobile plates on its top surface, which have no chance to be recovered in experiments with a rigid top boundary since the latter imposes a zero fluid velocity on the surface. Moreover, 2-D numerical simulations had shown that the heat transport strongly depends on the mobility of the surface boundary layer ( Christensen, 1985), which is controlled by the viscosity contrast. Giannandrea and Christensen (1993) and Weinstein and Christensen (1991) therefore carried out experiments in syrups in a large aspect ratio tank ( Figure 37 ) to study the effect of a stress-free boundary ( Figure 5(b) ). They observed two regimes. For γ  ≤   1000, the surface layer is mobile. At Ra  =   10⁵, the morphology of downwellings changes to a dendritic network of descending sheets, with a wavelength more than three times that of the spoke pattern ( Figure 37 ). The existence of this 'whole-layer' or 'sluggish lid' mode with an increased wavelength is in agreement with earlier predictions (Jaupart and Parsons, 1985; Stengel et al., 1982) and 3-D numerical calculations (Ogawa et al., 1991). The Nusselt number drops by 20% for viscosity contrasts between 50 and 5000. For γ  ≥   1000, a stagnant lid forms on the top of an actively convecting region; and both the Nusselt number and the size of the convection cells are nearly identical for both rigid and free boundaries.

Figure 37. Shadowgraphs of the convection planforms found for the case of rigid boundaries (a) and for the case of a stress-free upper boundary (b).

Reproduced from Weinstein SA, Christensen UR (1991) Convection planforms in a fluid with a temperature-dependent viscosity beneath a stress-free upper boundary. Geophysical Research Letters 18: 2035–2038.

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Mantle Dynamics

A. Davaille , A. Limare , in Treatise on Geophysics, 2007

7.03.6.2 Plate Tectonics in the Laboratory?

Mantle convection is characterized by mobile plates on its top surface, which have no chance to be recovered in experiments with a rigid top boundary since the latter imposes a zero-fluid velocity on the surface. Moreover, 2-D numerical simulations had shown that the heat transport strongly depends on the mobility of the surface boundary layer ( Christensen, 1985), which is controlled by the viscosity contrast. Giannandrea and Christensen (1993) and Weinstein and Christensen (1991) therefore carried out experiments in syrups in a large aspect ratio tank ( Figure 39 ) to study the effect of a stress-free boundary ( Figure 4(b) ). They observed two regimes. For γ   ≤   1000, the surface layer is mobile. At Ra  =   10⁵, the morphology of downwellings changes, from the spokes obtained with a rigid boundary, to a dendritic network of descending sheets, with a wavelength more than 3 times that of the spoke pattern ( Figure 34 ). The existence of this 'whole-layer' or 'sluggish lid' mode with an increased wavelength is in agreement with earlier predictions (Stengel et al., 1982; Jaupart and Parsons, 1985) and 3-D numerical calculations (Ogawa et al., 1991). The Nusselt number drops by 20% for viscosity contrasts between 50 and 5000. For γ   ≥   1000, a stagnant lid forms on the top of an actively convecting region; and both the Nusselt number and the size of the convection cells are nearly identical for both rigid and free boundaries.

Figure 34. Shadowgraphs of the convection planforms found for the case of (a) rigid boundaries and (b) a stress-free upper boundary. From Weinstein SA and Christensen UR (1991) Convection planforms in a fluid with a temperature-dependent viscosity beneath a stress-free upper boundary. Geophysical Research Letters 18   : 2035–2038.

Figure 35 shows the regime diagram established from experimental, numerical, and theoretical analysis (Solomatov, 1995). With a viscosity contrast across the lithosphere ∼10⁷, the Earth should be in the stagnant-lid regime, that is, a one-plate planet like Mars. So, if temperature-dependent viscosity is clearly a key ingredient for plate formation, this ingredient alone is not sufficient to generate 'plate tectonics' convection. To make plate tectonics work, a failure mechanism is needed and several options have recently been proposed (see Bercovici et al., 2000; Bercovici, 2003; see Chapters 7.02 and 7.05). The simplest is probably pseudo-plastic yielding (Moresi and Solomatov, 1998; Trompert and Hansen, 1998b; Tackley, 2000; Stein et al., 2004; Grigné et al., 2005): moving plates and thin weak boundaries appear but subduction remain symmetric. Gurnis et al., (2000) also pointed out the importance of lithosphere 'memory': the lithosphere can support dormant weak zones (faults or rifts) over long time periods, but those weak zones can be preferentially reactivated to become new plate boundaries. Another ingredient could be damage (e.g., Bercovici et al., 2001), which introduces some memory in the rheology and allows development over time of weak zones. This approach has been used with some success (e.g., Ogawa, 2003) but a physical understanding of the damage process from the grain to the lithosphere scale is still lacking. Anyway, experimentalists are still looking for a laboratory fluid presenting the 'right' kind of rheology to allow plate tectonics. However, the study of the stagnant-lid regime is still relevant for the dynamics of plate cooling, to which we turn now.

Figure 35. Different regims of thermal convection in a strongly temperature-dependent fluid with free-slip boundaries. From Solomatov S (1995) Scaling of temperature- and stress-dependent viscosity convection. Physics of Fluids 7: 266–274. The gray area is the mantle parameters range.

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Advances in Geophysics

Stephen A. Miller , in Advances in Geophysics, 2013

3.2 Sedimentation on the Ocean Floor

As mantle convection drives oceanic lithosphere towards its subduction demise, the volatile package created at the MOR is covered in pelagic carbonate sediments and pore- and chemically-bound ${\text{H}}_2\text{O}$ . Carbonate is abundant in two main pelagic marine sediments: (1) siliceous limestones and (2) clay-carbonates (marls) (Kerrick & Connolly, 2001a). The additional pelagic sediments and terrigenous sediments transported from continental margins via turbidity currents contain large quantities of ${\text{CO}}_2$ and ${\text{H}}_2\text{O}$ , but the thickness of these deposits is variable and generally not well-constrained. The depth of sediment deposits varies along subduction zones, where sediments are mostly derived from the continents, while sedimentation in deep oceanic basins is poorly known because of limited sampling from ocean drilling.

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Heat transport processes on planetary scales

Anne M. Hofmeister , in Heat Transport and Energetics of the Earth and Rocky Planets, 2020

3.4 The physics of convection: how planets differ from the laboratory

Whole mantle convection is not supported by most observations, nor does mantle tomography provide compelling support (see Chapter 1 for discussion and references). Even proponents of whole mantle convection acknowledge that first order difficulties remain with their models, despite ~50 years of effort. For example, Bercovici (2015) noted that:

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the mechanism moving plates has not been identified;

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known energy sources are too weak to drive convection;

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evidence for thin, thermal plumes is lacking; and

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persistence of the known chemical heterogeneities in the upper mantle is not explained.

Evidence for whole mantle convection rests on the large magnitude of the dimensionless Rayleigh number (Ra ~10⁸) for Earth's ~3000   km thick mantle. However, the historic derivation of the Rayleigh number assumes conditions quite different from those inside the rocky Earth, which motivated Hofmeister and Criss (2018) to probe the underlying physics. This section summarizes their theoretical assessment of stability criteria and arguments for stability of the lower mantle.

3.4.1 The Rayleigh number

The dimensionless number Ra derived by Rayleigh (1916) describes fluids in a box in a uniform gravity field, where:

(3.18) $Ra=\frac{{\alpha }_{vol}\Delta Tg{h}^3}{D\upsilon },$

α _vol is volumetric thermal expansivity, ΔT is the temperature difference across the system, g is the constant gravitational acceleration, D is thermal diffusivity, and υ is kinematic viscosity. Minor modifications to Ra made in geodynamic studies are described below. Non-dimensional numerical simulations of mantle convection use Ra as a variable (e.g., Zhong et al., 2015).

Estimates of supercritical Ra are insufficient evidence for whole mantle convection, because applying Eq. (3.18) to the continental lithosphere suggests that this region should convect (Hofmeister and Criss, 2018), contrary to geologic evidence. Regarding the mantle, the very slow motions of the plates are unlike the strong, time-dependent, turbulent conditions suggested by its high (10⁸) values for Ra. Experimentally, convective flow has only been observed in rather inviscid materials (υ up to 22000   mm²/s as in syrup: White, 1988). That the experimental constraints on stability involve υ that is 17 orders-of-magnitude lower than estimates of mantle viscosity has not been questioned when using Ra to describe Earth's mantle. In contrast, a much smaller discrepancy between strain rates of 10⁻³ to 10⁻⁶  s⁻¹ in laboratory experiments with those of 10⁻¹² to 10⁻¹⁶  s⁻¹ estimated from perceived mantle circulation has been a long-standing concern in deformation studies (e.g., Cordier et al., 2012).

3.4.2 Existing adaptations of the Ra to planets and their pitfalls

3.4.2.1 Accounting for the adiabat?

In atmospheric studies, the adiabatic gradient is substracted from ΔT (e.g., Tritton, 1977). The equation used [ΔT*=ΔT(1−γ_th B _T ⁻¹ΔP)] actually depicts an isentrope. Adiabats and isentropes are equivalent for ideal gas and reasonable for real gas (Hofmeister and Criss, 2019a,b), but not for solid planets. This correction amounts to about 30% of Ra and is thus unimportant to stability arguments.

Instead, thermal stability hinges on whether the material remains solid under the applied temperature gradient. Solids conduct heat up to their melting temperature. This response is irrespective of the thermal gradient, but rather depends on the thermodynamic characteristics of the material. The classical Ra number presumes constant physical properties and thus cannot describe a two-phase system, e.g., a system with co-existing solids and fluids.

3.4.2.2 Incorrect assessment of internal heating

The current adaptation of Ra in geodynamics for internal heating depends more strongly on h than on its third power as in Eq. (3.18) (e.g., Schubert et al., 2001). That the dependence is stronger signals an obvious problem, because internal heating supplies heat towards the top of a box, rather than just at its bottom, and this distribution is stabilizing against convection, not invigorating.

In detail, over a long time, the middle of an internally heated slab becomes increasingly hotter than its bottom, which makes it difficult for the lower half of the slab to convect. Short times are more relevant due to the immensity of Earth (Chapter 2). For this case, internal heating would destabilize the upper half of the slab, at the expense of stabilizing the lower half. Because the internal heating of the Earth is weak, this destablizing effect on the upper half should be neglected, especially insofar as ΔT is an estimate.

3.4.3 Mass conservation violation due to spherical geometry

The classical derivation of the Rayleigh number considers vertical instability of a tabular system, which greatly differs from radial instability in a planet. The different shapes of the volume elements in Cartesian, cylindrical, and spherical-polar coordinate systems lead to different rules for mass conservation during motions of these elements.

Equal thickness (δh) layers in a box have the same mass, obviously. However, concentric shells of equal thickness (δs) in a sphere have different mass because shell area varies with radius (Fig. 3.10). For other geometries, length scales require amending to conserve mass during its transfer between layers. For the sphere, h in Eq. (3.18) must be replaced with L:

Figure 3.10. Schematics in 2-dimensions illustrating how geometry affects scale length in dimensional analysis. (a) Mass balance of parallel layers with equal thickness and density in Cartesian geometry. (b) Imbalanced mass in concentric layers with equal thickness and density. M₁ is the smallest mass, progressing to M₅, the largest mass. If heat energy is considered in terms of heat density (energy per volume), then for equal thickness layers, the same imbalance occurs as for mass. For cylindrical symmetry, the missing factor is (1−h/R _out), rather than that given in Eq. (3.19).

(3.19) $L=h\frac{R_{in}^2}{R_{out}^2}=h\frac{{\left(R_{out}^{}-h\right)}^2}{R_{out}^2}=h{\left(1-\frac{h}{R_{out}^{}}\right)}^2$

To conserve momentum and heat during flow (the continuity equations) requires the same scaling (Hofmeister and Criss, 2018). Thus, for planets, which are nearly spherical, Ra must be multiplied by (1−h/R _out)⁶. This correction is ~0.01 which changes the value of Ra by only two orders of magnitude. More importantly, the correction of Eq. (3.19) changes the character of non-dimensionalized mantle convection equations, which do not currently incorporate the size of the planet.

Size greatly affects heat conduction in spheres (Criss and Hofmeister, 2016; Chapter 2) and therefore must greatly affect convection. Not considering the shapes of the elements has led to excluding a key descriptor of a celestial body, its size, from convection models.

3.4.4 Stability criteria for flow of plastic solids

Infinitesimal stress causes liquids to flow, but the response of solids to stress is much more complicated and has been studied in great detail in engineering and other fields. The Ra number assumes that flow is viscous, but for solids, viscous behavior is limited to certain materials above the glass transition. The much different process of creep can occurs in solids that are hot, but below melting (Meyers and Chawla, 2009). Certain mathematical descriptions of creep (e.g., Nabarro-Herring or Coble) can be recast to mimic equations that depict Newtonian viscous behavior (e.g., Kohlstedt and Hansen, 2015), lending support to models that treat the solid mantle as if it were fluid. However, more realistic equations for creep cannot be made equivalent to viscous behavior. On this basis, Hofmeister and Criss (2018) considered the idealized behavior of a Bingham plastic to develop criteria for flow in the mantle.

Bingham's model requires a minimum shear stress (σ_yield) for plasticity, in order to describe materials with both liquid- and solid-like behavior (see Davis and Selvadurai, 2005; Selhke et al., 2014 for applications to mud and lavas). Regarding mantle materials, multiple types of plasticity are observed, including motion of dislocations, atomic diffusion, and dynamic recrystallization (Kohlstedt and Hansen, 2015). That substantial energy is needed to break and reform atomic bonds in these mechanisms, explains why significant differential stress (δσ) is needed to deform oxides and silicates in the laboratory. For example, Jin et al. (2001) applied ~90–200   MPa at hydroscatic pressures of ~30–50,000   MPa to deform synthetic ecolgites and gabbros.

Thus, for solids to exhibit the liquid-like behavior that underlies the Rayleigh number requires:

(3.20) $\frac{\delta \sigma }{{\sigma }_{\mathrm{yield}}}>1.$

Except for locations close to the moving slabs, conditions in the mantle are hydrostatic (δσ~0), so creep is unexpected.

3.4.5 Derivation of Gr and Ra from force balance

Assuming Newtonian flow allowed Hofmeister and Criss (2018) to evaluate the ratio of the buoyancy to drag forces under constant gravity (g) and physical properties using dimensional analysis as:

(3.21) $\frac{F_{\mathrm{buoyancy}}}{F_{\mathrm{drag}}}\cong \frac{{\rho }_0{\alpha }_V\Delta Tgh\AA }{\AA {\rho }_0\upsilon (\partial u_{\mathrm{flow}}/\partial y)}=\frac{{\alpha }_V\Delta Tgh}{\upsilon (u_{\mathrm{flow}}/h)}=\frac{{\alpha }_V\Delta Tg{h}^3}{\upsilon \left(u_{\mathrm{flow}}h\right)}=\frac{{\alpha }_V\Delta Tg{h}^3}{{\upsilon }^2}.$

where Å is area and the adiabatic gradient is substracted from ΔT. The RHS is identical to Grashof's dimensionless number, Gr, which equals υRa/D.

Criteria for instability were developed by recognizing that heat and mass flow are decoupled in highly viscous liquids (Hofmeister and Criss, 2018). Hence, each force in Eq. (3.21) is mitigated by the relevant diffusion coefficient. Instability exists when:

(3.22) $\frac{F_{\mathrm{bouyancy}}}{F_{\mathrm{drag}}}\frac{D_{\mathrm{mass}}}{D_{heat}}>1;\text{i}\mathrm{.e}.,\frac{\mathrm{Gr}}{\mathrm{Le}}>1;\text{or}\frac{\mathrm{Ra}}{\mathrm{Sc}}>1$

where Sc=υ/D _mass is the Schmidt number and Le=D _heat/D _mass is the Lewis number. Presumably, the minimum stress of Eq. (3.20) is met. Eq. (3.22) provides Ra_critical ~2000 from Sc for water. For gas, Le ~Sc ~1 provides critical numbers of unity for both Gr and Ra. Both deductions are supported by experiments: gas convects when the adiabatic gradient is exceeded (i.e., near Ra of unity) whereas water has a critical number of ~1760.

For a solid close to the yield point, the ratio of buoyant to resistive forces is near unity. Thus, the flow criterion simplifies to Le<1, which is not observed in solids, since mass diffusion is miniscule. The much faster process of thermal diffusion overwhelms creep in solids.

Although mantle convection is unexpected, an alternative exists to instability in a high thermal gradient, namely, when a solid becomes too hot, it melts. In planets, melting produces a phase that is highly buoyant and far less viscous than the solid. Thus, the key heat-transfer processes expected in planets are conduction (radiative diffusion: Section 3.1) and the advective, one-way ascent of magma and volatiles (Section 3.3).

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Mantle Convection and Plumes

Peter Olson , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I Heat Production and Heat Transfer in the Mantle

The process of mantle convection can be regarded as a thermal engine on a planetary scale, which transports heat from the earth's deep interior to the surface. The rate at which the solid earth evolves is dictated by the rate at which this heat transport occurs. In terms of magnitude, thermal energy is by far the most important form of energy available in the mantle for convection and other dynamical processes, and heat flow is far and away the largest form of energy transport in the earth's interior. The primary sources of thermal energy for mantle convection are three: (1) internal heating due to the decay of the radioactive isotopes of uranium, thorium, and potassium; (2) the long-term secular cooling of the earth; and (3) heat from the core. The total heat flow from the mantle is 30–35  TW (1   TW   =   10¹² watts of heat flow), which amounts to roughly 75–80% of the total heat loss from the earth's interior. The remaining 20–25% of the surface heat loss is due to radioactivity in the continental crust, which does not contribute much to mantle convection. The relative importance of these three mantle heat sources is somewhat uncertain, but a reasonable estimate is that radioactive heating contributes about half of the total, secular cooling about 40% and the remaining 10% comes from the core.

There are several mechanisms for heat transfer from the earth's interior to the surface, including thermal conduction, hydrothermal and magmatic circulation within the mantle, and subsolidus convection of the mantle itself. Heat conduction is important in the lithosphere, the layer comprising the surface tectonic plates, and it is also important at the base of the mantle, where heat is conducted into the mantle from the earth's core. Hydrothermal circulation is largely a crustal phenomenon, and magmatism is also restricted to the upper levels of the mantle.

On a global scale, the heat transport by subsolidus convection dominates the other mechanisms. The relative importance of heat transport by mantle convection compared to the other heat transport mechanisms is conveniently expressed in terms of the Nusselt number Nu, the ratio of total heat flow to conductive heat flow in a medium with the same overall thermal gradient. In the mantle, Nu  ≃   20–30, so that, because of mantle convection, the earth's interior looses heat 20–30 times faster than it would if conduction were the primary heat transfer mechanism.

The variation of conductive heat flow over earth's surface and the pattern of plate motions provide information on the planform of mantle convection. Surface heat flow is highest at the midocean ridge spreading centers, where upwellings bring hot mantle to shallow depths and the geothermal gradient is steep. The surface heat flow is observed to decrease with increasing crustal age of the seafloor. This decrease is a consequence of the conductive cooling of the lithosphere with time, as predicted by convection theory.

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Mantle Dynamics

D. Bercovici , in Treatise on Geophysics (Second Edition), 2015

7.01.4 Mantle Properties

The study of mantle convection has a boundless appetite for information on the properties of the convecting medium. Indeed, what caused the theory of continental drift to be marginalized for decades was a material property argument by Sir Harold Jeffreys that the Earth was too strong to permit movement of continents through ocean crust. Later measurements of mantle viscosity by postglacial rebound were an important key element in recognizing that the mantle is fluid on long time scales (see Chapter 7.02 ).

Fluid dynamics is rife with dimensionless numbers, and one of the most important such numbers in the study of convection is the Rayleigh number (see Chapter 7.02 ). The Rayleigh number defines the vigor of convection in terms of the competition between gravitationally induced thermal buoyancy, which acts to drive convective flow, and the dissipative or resistive effects of both fluid viscosity, which retards convective motion, and thermal diffusion, which acts to diminish thermal anomalies. The Rayleigh number is written generically as

[1] $\mathit{Ra}=\frac{\mathit{\rho g\alpha }\mathrm{\Delta }T{d}^3}{\mathit{\kappa \mu }}$

where ρ is the density, g is the gravitational acceleration, α is the volumetric thermal expansivity, ΔT is the typical temperature contrast from the hottest to coldest parts of the fluid layer, d is the dimension of the layer such as the layer thickness, κ is the thermal diffusivity, and μ is the dynamic viscosity (again, see Chapter 7.02 ).

Estimate of the mantle Rayleigh number by itself requires knowledge of the material responses to various inputs such as heat and stress. The response to heat input involves heat capacity, thermal expansivity α, and heat conduction or thermal diffusion (κ) (see Chapter 7.02 , 2.08 , and 2.23 ). The density structure inferred from high-pressure and high-temperature experiments ( Chapter 2.08 ) and seismology ( Chapters 1.01 and 1.21 ) constrain how mantle density ρ responds to pressure changes as upwellings and downwellings traverse the mantle, undergoing simple compression or decompression and solid–solid phase transitions; both effects can have either stabilizing or destabilizing effects on mantle currents ( Chapters 7.02 and 7.09 ).

One of most important factors within the Rayleigh number and in the overall study of mantle convection is viscosity μ. That the mantle is viscous at all was one of the key elements in determining the viability of the mantle convection hypothesis. That continents were inferred to be in isostatic balance implied that they are floating in a fluid mantle; but isostatic equilibrium does not indicate how fluid the mantle is since it gives no information about how long it takes for the floating continents to reach an isostatic state. However, measurements of this approach to isostasy could be taken by examining postglacial rebound, that is, the uplift of high-latitude continental masses such as Scandinavia and Canada, following the melting of the glacial ice caps after the end of the last ice age (Haskell, 1937; see Chapter 3.07 ). From these analyses came one of the most crucial and well-known material properties: the average viscosity of the mantle of μ  =   10²¹  Pa   s. Today, the analysis of the mantle's response to changing loads (e.g., melting ice caps or a decrease in the Earth's rotation rate) is done through an increasingly sophisticated combination of geodetic satellite and field analyses, which further refine the viscosity structure of the mantle ( Chapters 1.27 , 3.07 , 7.02 , and 7.04 ).

The viscosity of the mantle is so large that it is called a slowly moving or creeping fluid and thus does not suffer the complexities of classical turbulence ( Chapter 7.02 ). However, one of the greatest of all complexities in mantle dynamics is associated with the various exotic rheological behaviors of mantle rocks, which are almost exclusively inferred from laboratory experiments ( Chapter 2.18 ). Mantle viscosity is well known to be a strong function of temperature, and the dramatic increase in viscosity toward the surface leads to various conundrums about how plate tectonics forms or functions at all and/or how subduction zones can ever initiate from such a cold strong lithosphere ( Chapters 2.18 , 7.07 , and 7.09 ). The mantle's rheology is also complicated by the various deformation mechanisms it can assume, prevalently diffusion creep at 'low' stress and dislocation creep at higher stress, although other creep and slip mechanisms are also possible ( Chapters 2.18 , 7.02 , and 7.10 ). In diffusion creep, viscosity is a function of mineral grain size, and this effect can induce dramatic changes if grain growth or grain reduction mechanisms exist. In dislocation creep, viscosity is non-Newtonian and is a function of stress itself, thereby undergoing pseudoplastic behavior in which the material softens the faster it is deformed. These and many more complexities (see Chapters 7.02 and 7.07 ) continue to keep mantle convection a rich field.

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Mantle Dynamics

S.J. Zhong , ... M.G. Knepley , in Treatise on Geophysics (Second Edition), 2015

7.05.1 Introduction

The governing equations for mantle convection are derived from conservation laws of mass, momentum, and energy. The nonlinear nature of mantle rheology with its strong temperature and stress dependence and nonlinear coupling between flow velocity and temperature in the energy equation require that numerical methods be used to solve these governing equations. Understanding the dynamic effects of phase transitions (e.g., olivine to spinel phase transition) and multicomponent flow also demands numerical methods. Numerical modeling of mantle convection has a rich history since the late 1960s (e.g., McKenzie et al., 1974; Torrance and Turcotte, 1971). Great progress in computer architecture along with improved numerical techniques has helped advance the field of mantle convection into its own niche in geophysical fluid dynamics (e.g., Yuen et al., 2000).

In this chapter, we will present several commonly used numerical methods in studies of mantle convection with the primary aim of reaching out to students and new researchers in the field. First, we will present the governing equations and boundary and initial conditions for a given problem in mantle convection and discuss the general efficient strategy to solve this problem numerically (see Section 7.05.2 ). We will then briefly discuss finite-difference (FD), finite-volume (FV), and spectral methods in Section 7.05.3 . Since finite elements (FEs) have great utility and have become very popular in the user community, we will discuss FEs in greater details as the most basic numerical tool (see Section 7.05.4 ). We will also include discussions about accelerators and the usage of libraries for massively parallel computing in preparation for the coming technical advances in hardware. For simplicity and clarity, we will focus our discussion on homogeneous, incompressible fluids with the Boussinesq approximation. However, we will also describe methods for more complicated and realistic mantle situations by including compressibility, non-Newtonian rheology, solid-state phase transitions, and thermochemical (i.e., multicomponent) convection ( Section 7.05.5 ). Finally, in Section 7.05.6 , we will discuss some new developments in computational sciences, such as recent software developments, especially dealing with libraries and visualization, which may impact our future studies of mantle convection modeling in a multiscale setting.

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Tectonics and Volcanic and Igneous Plumbing Systems

Benjamin van Wyk de Vries , Maximillian van Wyk de Vries , in Volcanic and Igneous Plumbing Systems, 2018

7.1 Introduction

7.1.1 What is the Role of Plate Tectonics in Volcanic and Igneous Plumbing Systems?

Plate tectonics and mantle convection set the scene for magmatism, and magma is an essential product of plate tectonics. Magma ultimately produces the lithospheric portion of plates. The different tectonic environments creating magmatism are illustrated in Fig. 7.1.

Figure 7.1. Plate tectonics and magma production.

Conceptual figure showing the Earth's plate tectonic system, with the principal magma generation, transport and storage systems.

Magma is produced principally at divergent plate boundaries by spreading at rifts and mid-ocean ridges (about 90% of all magmatic activity is at mid-ocean ridges). New oceanic lithosphere is generated by magma accumulation and eruption in a continually growing and expanding plumbing system.

Continental rifts can also produce magma and eventually evolve into full oceanic spreading systems. Many of these rifts start as passive rifts, with no initial asthenospheric convection. These produce no magma until stretching thins the lithosphere sufficiently for the asthenosphere to melt through decompression. Active rifts, which are associated with early asthenospheric convection, begin with large-scale volcanism and uplift followed by rifting.

Hotspots are zones of rising asthenospheric convection or circulation and/or zones where the mantle is especially prone to melting (Foulger, 2010). At hotspots asthenospheric or lithospheric melts are produced through excess heat and create a variety of volcanic fields ranging from the continental flood basalt provinces (Colobia River, USA), huge oceanic islands (Hawai'i) and large volcanic fields (El Pinacate, Mexico) to small, disperse monogenetic fields (southeast Australia).

At convergent plate boundaries, subduction of oceanic lithosphere causes slab dehydration, volatile release and melting of the overlying mantle. This process has worked to create continental crust since at least 2.4 billion years ago.

In orogenic belts, melting of the crust and mantle lithosphere occurs by heating (from basaltic magma underplating), decompression (by removal of load through erosion or rifting/crustal stretching) or from the addition of volatiles that lower the melting point (solidus).

Subduction of continental material does not cause the same scale of melting as oceanic subduction; instead, the subducted continental material is assimilated and may be recycled at a later date. The mantle circulation perturbed by the subduction may also lead to the asthenosphere rising and decompressing, thus generating additional melt.

Once magma is produced, it collects and moves through the lithosphere in bodies that make up an igneous plumbing system (see, e.g. Chapter 2). Depending on the relevant tectonic environment, the configuration of these plumbing systems varies widely. We will consider magma plumbing at mid-ocean ridges, continental rifts, hotspots, oceanic subduction zones (both oceanic and ocean–continental), continental convergence and transform zones.

We will then take a closer look at very shallow plumbing systems, where topographic stresses and sedimentary/volcanic lithology and rheology are important. This includes superficial structures, which only become significant at very shallow levels. Finally, we will consider the possibility of a unified system for tectonics and igneous plumbing systems.

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Decompression of majoritic garnet: an experimental investigation of mantle peridotite exhumation

Larissa F. Dobrzhinetskaya , ... Krassimir N. Bozhilov , in Advances in High-Pressure Technology for Geophysical Applications, 2005

1 Introduction

The concept of mantle convection suggests that some peridotites occurring as xenoliths in kimberlites and lamproites or as lenses within ultra high-pressure metamorphic (UHPM) belts related to continental collisions were previously at depths corresponding to the mantle transition zone or even in the lower mantle (e.g. Moore and Gurney, 1985; Dobrzhinetskaya et al., 1996; Harte et al., 1999; Stachel, 2001; Gillet et al., 2002). Their rock-forming minerals, equilibrated at high pressures and temperatures, break down during decompression when tectonic processes or explosive kimberlitic and/or lamproitic events transport them to Earth's surface.

Because uplift of mantle xenoliths occurs very quickly, in many cases no more than a few days, these rocks partially preserve the mineral chemistry and microstructural relations formed at great depth and record obvious disequlibrium features such as hydrous-phase-bearing alteration rims around garnet, or some subtle ones such as diffusion profiles (e.g. Smith, 1999) and odd isotopic systematics (e.g. Shimizu, 1999). It is known that evaluations of the depths at which mantle peridotites originate from are always uncertain. On the one hand, a comparison of chemical elements partitioning between co-existing mineral pairs in natural peridotites and high P and T experimental studies demonstrates that such xenoliths were residing at depths as great as ~150–200 km when they were extracted by their volcanic hosts. On the other hand, some inclusions in diamonds (e.g. Moore and Gurney, 1985; Harte et al., 1999) and microstructural relations between intracrystalline lamellae of pyroxenes and their host-garnet (Haggerty and Sautter, 1990; Sautter et al., 1991) strongly suggest that these rocks retain microstructural features corresponding to decompression due to their uplift from depths 300 to 800 km. Discoveries of microdiamonds within metamorphic terranes related to continental collisions have demonstrated that continental material has been subducted to depths of about 120–150 km (Sobolev and Shatsky, 1990; Xu et al., 1992; Dobrzhinetskaya et al., 1995; Nasdala and Massonne, 2000; van Roermund et al., 2002; Yang et al., 2003) and returned to the Earth's surface by a tectonic exhumation.

Moreover, several recent studies now suggest that peridotites tectonically incorporated into crustal lithologies within UHPM terranes may carry microstructural patterns indicating an even deeper recrystallization history. These microstructural patterns include intracrystalline exsolution lamellae of pyroxenes in former majoritic (supersilicic) garnet in garnet peridotites from the Norwegian Caledonides, from Sulu and Qaidam regions of Central Chinese Orogenic Belt (COB), suggesting exhumation of these rocks from 185 km to >250 km (van Roermund and Drury, 1998; Ye et al., 2000; Song et al., 2004), high-pressure clinoenstatite lamellae in clinopyroxene and a high concentration of oriented rods of ilmenites and plates of chromite in olivine from Alpe Arami garnet peridotites from the Swiss Alps and Qaidam from COB, suggesting exhumation from depths >250 km (Dobrzhinetskaya et al., 1996; Bozhilov et al., 1999, 2003; Song et al., 2004), and magnetite lamellae in olivine in garnet peridotite from the Sulu region of COB suggesting exhumation from >400 km (Zhang et al., 1999). Importantly, the peridotites and eclogites of these orogenic belts are enveloped in highly deformed continental gneisses, often showing extensive migmatization. These gneisses have been thought to contain no evidence of a high-pressure history and a controversy has raged for some time about whether or not they have also experienced deep subduction. However, recent careful studies of zircons extracted from these felsic gneisses have shown them to contain inclusions of coesite (e.g. Ye et al., 2000; Liu et al., 2002, 2003). This observation confirms that the crustal gneisses have, indeed, experienced deep subduction but have otherwise lost the microstructural evidence of the high-pressure events that have been preserved in the more refractory mafic and ultramafic rocks. Although the pyroxene exsolutions from garnet in some eclogite intercalated with continental metasediments from COB terranes preserve evidence of subduction to >200 km (Ye et al., 2000), a discussion that such rocks might originate from much shallower depth (~80 km) still continues (e.g. Hirajima and Nakamura, 2004). Such considerable conflicts in interpretation arise from a lack of experimental studies of what kind of microstructures are formed during decompression of the majoritic garnet at varying pressures and temperatures.

We present here experimental data on majoritic garnet synthesized from a mineral mix of garnet peridotite bulk chemistry that has been decompressed at high temperatures and now features exsolution of Mg₂SiO₄ and pyroxenes from majoritic garnet. A brief description of the results of this experimental program has been recently reported (Dobrzhinetskaya et al., 2004). The microstructural patterns of garnet peridotites that we have produced in the laboratory during high-temperature decompression from 14 to 13, 12, or 7 GPa and from 8 to 5 GPa are consistent with chemical changes of the re-equilibrated mineral phases. Therefore, they may be valuable for establishing a standard against which to evaluate microstructures in natural rocks, where chemical characteristics related to their earlier high-pressure history are obliterated by superimposed shallow thermal events.

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