Convection in a compressible fluid with infinite Prandtl number

Jarvis, Gary T.; McKenzie, Dan

doi:10.1017/s002211208000225x

Cited by 269 publications

(201 citation statements)

References 16 publications

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“…The effects of viscous dissipation and adiabatic heating and cooling are included as is the release and consumption of latent heat from the phase transitions. This approach, in a mathematical sense, emerges from the fully compressible anelastic-liquid approximation [6] in the limit ! 1, where is the thermodynamic Grueneisen parameter.…”

Section: Model Descriptionmentioning

confidence: 99%

“…For the Earth's mantle we have taken into account that our model cannot simulate plate-like behavior by introducing a sublithospheric temperature, rather than the actual surface temperature into the model. Thus T 0 then lies between 0.2 and 0.5 [6,9]. For Venus and Mars, when one takes into account the rather thick lithosphere in these planets without plate tectonics [10], T 0 may reach values of around 1.0 and 1.3 respectively.…”

Section: Introductionmentioning

confidence: 97%

“…The surface temperature is commonly expressed as a dimensionless number T 0 , which represents the ratio between the temperature at the top of the convecting layer and the temperature drop across this layer. The quantity T 0 enters into the energy equation in the anelastic-liq-uid [6] and the extended Boussinesq approximation [7,8], since the amount of adiabatic heat released or consumed by compression or expansion is proportional to the absolute temperature. Why is the magnitude of this particular parameter important for planetary convection?…”

Section: Introductionmentioning

confidence: 99%

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The influences of surface temperature on upwellings in planetary convection with phase transitions

Steinbach

Yuen

1998

Earth and Planetary Science Letters

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The importance of surface temperature for mantle convection appears with the presence of adiabatic heating and cooling and the release and consumption of latent heat in the presence of phase transitions. For some planetary bodies these effects cannot be neglected. The dimensionless surface temperature T 0 , which is the ratio between the temperature at the top of the convective region and the temperature drop across the mantle, is close to one for Mars and Venus. For the Earth, T 0 lies between 0.2 and 0.5. The dynamical influence of T 0 is especially poignant for internally heated convection with temperature-dependent viscosity. There is a tight coupling between the magnitude of the temperature field and the viscosity itself. We have studied temperature-dependent viscosity convection for both low-T 0 (0.2) and high-T 0 (1.2) situations and with internal heating in mantle convection with two upper-mantle phase transitions. Our results show that within this range of T 0 there exist two regimes for the evolution of upwellings in the mantle. In transient situations plume-plume collisions lead to the formation of megaplumes for high-T 0 regimes but are less likely to do so for low T 0 . In the long-term regime, plumes with low T 0 are prone to develop from the transition zone with a supply of hot material coming from the shallow lower mantle. In systems with high T 0 , however, long-lived plumes tend to have deeper mantle origins. In quasi-layered situations high T 0 may act as a positive feed-back mechanism in inducing powerful hot upwellings into the upper mantle.

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Section: Model Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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The influences of surface temperature on upwellings in planetary convection with phase transitions

Steinbach

Yuen

1998

Earth and Planetary Science Letters

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“…Constraints on the thermal-mechanical structure of the whole mantle have previously been made for variable viscosity, using the mean-field Ž . theory Yuen and Zhang, 1989 . For constant viscosity convection the interior of EBA and anelastic compressive convection is adia-Ž batic Jarvis and Mc Kenzie, 1980;Machetel and . Yuen, 1989;Steinbach et al, 1989 .…”

Section: Modelling Resultsmentioning

confidence: 99%

Modelling planetary dynamics by using the temperature at the core–mantle boundary as a control variable: effects of rheological layering on mantle heat transport

Berg

Yuen

1998

Physics of the Earth and Planetary Interiors

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In planetary convection, there has been a great emphasis laid on the usage of the Rayleigh number as a control parameter for describing the vigor of convection. However, realistic mantle rheology not only depends on temperature, pressure, strain-rate and composition, but also on the nature of the dominant creep mechanism, which varies with pressure and also with temperature. It is difficult to study the effects of varying influences from the convective strength without also changing the mantle flow law in the process. We have adopted the approach of using as the sole control parameter, the temperature at the core-mantle boundary, T , in modelling planetary dynamics with a composite non-Newtonian and Newtonian CMB rheology, which is temperature-dependent in the upper mantle and both temperature-and pressure-dependent in the lower mantle. Increasing the T strengthens convective vigor and leads to a non-linear increase of averaged temperature, CMB heat-flow and root-mean-squared velocity. The interior viscosity decreases strongly with T and internal heating due to CMB radioactivity. A viscosity maximum is found in the horizontally averaged viscosity profile at a depth around 2000 km. This viscosity hill moves downward with diminishing amplitude in the face of increasing dissipation number and internal heating. The bottom third of the lower mantle appears to be superadiabatic as a consequence of the stiff lower-mantle rheology. The Ž . scaling relationship between the Nusselt Nu number and T shows a relatively insensitive increase of Nu with T . In CMB CMB Ž . terms of an effective Rayleigh number of the whole system, Ra , the power-law exponent of the Nu Ra relationship is E E very low, around 0.12. Strong pressure-dependence of lower-mantle rheology and its large volume relative to the entire mantle would induce a much lower cooling rate of the planet than previous models based on parameterized convection with a temperature-dependent viscosity. q

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“…It is a historical accident that the relationship between adiabatic heating and viscous dissipation was derived by a steady-state analysis of the equation for energy conservation (Turcotte et al, 1974;Backus, 1975;Hewitt et al, 1975) instead of from statements of momentum and mass-conservation, so it was not realized at that time how closely gravitational power-release and viscous dissipation are linked in compressible as well as incompressible flow. (Another reason this was not realized by many early workers is because they usually decided to ignore the fourth term − ρ r gβpu 3 dV of Equation (13) in their numerical experiments-this is called the "Truncated Anelastic Approximation, " so that their numerical experiments never demonstrated an exact balance between viscous dissipation and gravitational power release (Jarvis and McKenzie, 1980;Leng and Zhong, 2008). Note that the above derivation does not depend on the assumption of a 1-D adiabatic reference state.…”

Section: Energy Balance In Compressible Convectionmentioning

confidence: 99%

The Current Energetics of Earth's Interior: A Gravitational Energy Perspective

2016

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The Earth's mantle convects to lose heat (Holmes, 1931); doing so drives plate tectonics (Turcotte and Oxburgh, 1967). Significant gravitational energy is created by the cooling of oceanic lithosphere atop hotter, less dense mantle. When slabs subduct, this gravitational energy is mostly (∼86% for whole mantle flow in a PREM-like mantle) transformed into heat by viscous dissipation. Using this perspective, we reassess the energetics of Earth's mantle. We also reconsider the terrestrial abundances of heat producing elements U, Th, and K, and argue they are lower than previously considered and that consequently the heat produced by radioactive decay within the mantle is comparable to the present-day potential gravitational energy release by subducting slabs-both are roughly ∼10-12 TW. We reassess possible core heat flow into the base of the mantle, and determine that the core may be still losing a significant amount of heat from its original formation, potentially more than the radioactive heat generation within the mantle. These factors are all likely to be important for Earth's current energetics, and argue that strong plume-driven upwelling is likely to exist within the convecting mantle.Keywords: mantle energetics, mantle evolution, Urey ratio, gravitational potential energy, mantle secular cooling, core energetics, core secular cooling INTRODUCTION Roles of Gravitational Energy Transformations in Mantle ConvectionA wonderful realization of the Plate Tectonics revolution was that the surface oceanic plates form the upper thermal boundary layer of a convecting mantle (Turcotte and Oxburgh, 1967). When oceanic plates cool near the Earth's surface, they become denser than underlying mantle. This density contrast provides the gravitational pull that causes plates to sink when they subduct. It is perhaps most familiar to think of convection in terms of cooling-or heating-linked buoyancy forces that cause hot regions to rise and cold regions to sink within a convecting fluid. While less familiar, an equivalent way to think about these phenomena is in terms of gravitational potential energy. Both rising low-density and sinking high-density regions release gravitational potential energy. In a highly viscous fluid like the Earth's mantle where inertial forces are negligible, the gravitational energy released from a sinking thermal density anomaly is completely transformed into viscous dissipation energy within the deforming fluid. For example, the Stokes problem of a sinking heavy ball in a highly viscous medium can be treated either as a force balance between the net buoyancy force on the ball and the viscous resisting force from the surrounding fluid's deformation, or as an energy balance between the gravitational energy released by the ball's sinking Morgan et al.Earth's Current Energetics and the viscous dissipation within the surrounding fluid. (In fluid dynamics, it is well-known that all slow viscous flow involves viscous friction-or viscous dissipation-that generates heat as the material strains. In this perspe...

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Convection in a compressible fluid with infinite Prandtl number

Cited by 269 publications

References 16 publications

The influences of surface temperature on upwellings in planetary convection with phase transitions

The influences of surface temperature on upwellings in planetary convection with phase transitions

Modelling planetary dynamics by using the temperature at the core–mantle boundary as a control variable: effects of rheological layering on mantle heat transport

The Current Energetics of Earth's Interior: A Gravitational Energy Perspective

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