Heat Generation and Transport in the Earth 1st Edition by Claude Jaupart, Jean Claude Mareschal 0521894883 9780521894883

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ISBN 10: 0521894883

ISBN 13: 9780521894883

Author: Claude Jaupart, Jean-Claude Mareschal

Heat provides the energy that drives almost all geological phenomena and sets the temperature at which these phenomena operate. This book explains the key physical principles of heat transport with simple physical arguments and scaling laws that allow quantitative evaluation of heat flux and cooling conditions in a variety of geological settings and systems. The thermal structure and evolution of magma reservoirs, the crust, the lithosphere and the mantle of the Earth are reviewed within the context of plate tectonics and mantle convection – illustrating how theoretical arguments can be combined with field and laboratory data to arrive at accurate interpretations of geological observations. Appendices contain data on the thermal properties of rocks, surface heat flux measurements and rates of radiogenic heat production. This book can be used for advanced courses in geophysics, geodynamics and magmatic processes, and is a reference for researchers in geoscience, environmental science, physics, engineering and fluid dynamics.

Heat Generation and Transport in the Earth 1st Table of contents:

1 Historical notes

1.1 Introduction

1.2 Kelvin and the age of the Earth

1.3 The discovery of radioactivity

1.4 The debate on the cooling mechanism of the Earth

1.5 Heat flux measurements

1.6 Energy budget of the Earth

1.7 Plate tectonics

2 Internal structure of the Earth

Objectives of this chapter

2.1 Introduction

2.2 Gravity and geodesy

2.2.1 Moment of inertia, angular momentum and energy of rotation

2.2.2 Gravitational potential energy of a self-gravitating sphere

2.2.3 Shape of the Earth

2.2.4 Gravity anomalies. Isostasy

2.2.5 Post-glacial rebound. Viscosity of the mantle

2.3 Seismology

2.3.1 Seismicity

2.3.2 Reference Earth model

2.4 Petrology, mineral physics and seismology: Composition and state of the Earth’s interior

2.4.1 Constraints on the thermal structure of the Earth

2.4.2 Attenuation: Q factor

2.5 Lateral variations of seismic structure

2.5.1 Topography of the main seismic discontinuities

2.5.2 Seismic tomography

2.5.3 Anisotropy

2.6 Core and magnetic field

2.7 The shallow Earth

2.7.1 The seismic lithosphere

2.7.2 Oceanic crust

2.7.3 Continental crust

2.7.4 Melting and melt transport

Exercises

3 Basic equations

Objectives of this chapter

3.1 Heat transport mechanisms

3.1.1 Heat conduction: Fourier’s law

3.1.2 Radiation

3.2 Definitions. Thermodynamic relationships

3.2.1 Definitions

3.2.2 Latent heat. Clausius Clapeyron equation

3.3 Conservation of mass

3.4 Conservation of momentum

3.4.1 Equilibrium condition

3.5 Energy equation

3.5.1 General form

3.5.2 Specialized form for no net contraction or expansion

3.6 Radial variations of density in the Earth

3.7 Equations for fluid flow

3.7.1 Viscosity

3.7.2 Navier–Stokes equations

3.7.3 Reynolds number

3.7.4 Other dimensionless numbers

4 Heat conduction

Objectives of this chapter

4.1 Heat conduction: Generalities

4.1.1 Time and distance scales

4.1.2 Superposition of solutions

4.2 Steady-state heat equation

4.2.1 Poisson’s and Laplace’s equations

4.2.2 Steady-state heat equation in one dimension

4.2.3 Steady-state heat equation with sources in a half-space

4.2.4 Steady-state heat equation for the sphere

4.2.5 Variations in thermal conductivity

4.3 Diffusive heat transport: Basic principles

4.3.1 Heating a half-space from its surface

Sudden change in temperature

Contact between two materials with different thermal properties

The air–ground interface

Temperature pulse on a plane

Temperature pulse at the surface of the half-space

Fixed heat flux at the boundary

4.3.2 Point source

Heat pulse

4.3.3 Damping of thermal fluctuations

Spatial variations

Time variations

4.4 General solutions to the steady-state heat equation

4.4.1 Laterally varying boundary conditions

Downward continuation of surface temperature variations

4.4.2 Steady-state heat equation in a layer

Varying heat .ux at the base of a layer

Varying temperature at the base of a layer

4.4.3 Horizontal variations in heat production

4.4.4 Changes in thermal conductivity: Refraction of heat

Two semi-in.nite layers with different thermal conductivities

The region |x| < a with thermal conductivity lambda1

Vertical cylinder embedded in a medium

4.4.5 Non-symmetric temperature in a sphere

4.5 Transient problems

4.5.1 Half-space with time-varying surface boundary condition

Varying surface temperature

Heat flux pulse at the surface

Half-space below a blanketing layer

4.5.2 Moving half-space with constant temperature on the plane z = 0

4.5.3 Cooling of a layer in infinite or semi-infinite medium

4.5.4 Cooling of a layer

Constant initial temperature. Surface and base at temperature 0

Constant initial temperature. Surface at temperature 0, no heat flux at the base

Layer with time-varying boundary conditions

Constant surface temperature. Varying temperature at the base

Constant temperature at the surface. Varying heat flux at the base

Periodic variations in the boundary condition at the base of the lithosphere

Layer with horizontal variations in temperature

4.5.5 Transient heat equation in spherical geometry

Spherical symmetry

Cooling of a sphere with surface kept at .xed temperature

Cooling of a spherical inclusion within a conductive region

Heating of a sphere by heat production

4.6 Thermal stresses

Exercises

5 Heat transport by convection

Objectives of this chapter

5.1 Isolated heat sources: Plumes and thermals

5.1.1 Steady plumes

Laminar plumes

Turbulent plumes

5.1.2 Thermals

Laminar thermals

Turbulent thermals

5.2 Rayleigh–Benard convection

5.2.1 Heuristic argument

5.2.2 Dimensional analysis

5.2.3 The different convective regimes

5.2.4 Convective heat transport

5.2.5 The convective heat flux in a cooling layer

5.2.6 The “4/3” law for the convective heat flux at high Rayleigh number

5.3 Scaling laws for heat flux and velocity in Rayleigh–Benardconvection: General theory

5.3.1 The dissipation equations

Kinetic dissipation

Thermal dissipation

5.3.2 Rigid boundaries

Intermediate Prandtl number

Large Prandtl number

5.3.3 The convective regimes of magma reservoirs

5.3.4 Convection with free boundaries

5.3.5 Summary

5.4 Convection in porous media

5.4.1 Fluid motion in a porous medium. Darcy’s law

5.4.2 Thermal convection in porous media

Energy conservation

Fluid convection in a porous medium. Rayleigh number.

Heat transport

Sea-floor hydrothermal systems

Cooling of intrusions by hydrothermal circulation

Hydrothermal convection in radioactive intrusions

5.4.3 Pipe flow

5.4.4 Topography-driven convection

Exercises

6 Thermal structure of the oceanic lithosphere

Objectives of this chapter

6.1 Continental and oceanic heat flow

6.1.1 Introduction

6.1.2 Lithosphere and thermal boundary layer structure

6.1.3 Basal boundary conditions

6.2 Cooling models for oceanic heat flux and bathymetry

6.2.1 The oceanic heat flux and bathymetry

6.2.2 Cooling half-space model

Bathymetry

Geoid

6.2.3 Heat flux and bathymetry data

Heat flux data

Young sea floor

Old ocean basins. Flattening of the heat flux versus age curve

6.2.4 Plate models for the oceanic lithosphere

Plate with flxed temperature at the base

Plate with fixed heat flux at the base

Thickening of the lithosphere with time

6.2.5 Large-scale variations of mantle temperature

6.2.6 Hydrothermal circulation

6.3 Hot spots and thermal rejuvenation of the oceanic plates

6.3.1 Thermal model: reheating of a half-space

6.3.2 Thermal model of hot spot: reheating of a plate

6.3.3 Heat flux of hot spots

6.4 Other effects of oceanic plate cooling

6.4.1 Seismic tomography of the oceanic plates

6.4.2 Rheological profiles

6.4.3 Seismicity

6.4.4 Effective elastic thickness

Exercises

7 Thermal structure of the continental lithosphere

Objectives of this chapter

7.1 Continental heat flux

7.2 Continental lithosphere in steady state

7.2.1 Vertical temperature distribution

7.2.2 Crustal heat production

7.2.3 Estimating mantle heat flux

Crustal heat production and Moho heat flux

7.2.4 Regional variations of heat flux and lithospheric temperatures

7.3 Long-term transients: Stabilization and secular cooling of the continental lithosphere

7.3.1 Archean conditions

7.3.2 Rundown of the heat producing elements. Secular coolingof the lithosphere

7.3.3 Secular decrease in heat flux from the mantle

7.3.4 Secular cooling and lithospheric thickening

7.4 Thermal perturbations in compressional orogens

7.4.1 General features

7.4.2 Compressional orogens

Effect of overthrusting: stacking of two slabs

7.4.3 Metamorphism. P-T-t paths

Temperature in moving half-space with heat production

Thermal relaxation of thick continental lithosphere

7.5 Thermal regime in regions of extension

7.5.1 Thermal models of rifts and zones of extension

7.5.2 Underplating

7.5.3 Mantle delamination

7.5.4 Delamination and extension

7.5.5 Extension and magma intrusions

7.5.6 Lithospheric extension. Discussion

7.6 Passive continental margins. Sedimentary basins

7.6.1 Cooling of the lithosphere after a heating event

7.6.2 The lithospheric stretching models

7.6.3 Horizontal transport of heat

7.7 Geophysical constraints on thermal structure

7.7.1 Constraints from seismology

Direct calculation of the velocity profile

Inverse calculations

7.7.2 Other geophysical constraints

Strength, seismicity, elastic thickness and thermal regime of the lithosphere

Depth to the Curie isotherms

Thermal isostasy

Electrical conductivity

Concluding remark

Exercises

8 Global energy budget

Objectives of this chapter

8.1 Thermodynamics of the whole Earth

8.1.1 The global energy budget

8.1.2 Changes in gravitational energy: Contraction due to secular cooling

8.1.3 Other energy sources: Tidal heating, crust–mantle differentiation

8.1.4 Secular cooling equation

8.2 Heat loss through the ocean floor

8.2.1 Oceanic heat flux data

8.2.2 Estimating the total oceanic heat loss

8.3 Heat loss through continents

8.3.1 Estimating the continental heat loss

8.3.2 Various contributions to the surface heat flux in continental areas

8.3.3 Mantle heat loss through continental areas

8.4 Heat loss of the Earth

8.5 Radiogenic heat sources in the mantle

8.6 Heat flux from the core

8.7 Mantle energy budget

Exercises

9 Mantle convection

Objectives of this chapter

9.1 Introduction

9.2 Elongated convection cells

9.3 The impact of continents on convection

Continental heat flux and a new boundary condition

Main features of sub-continental convection

9.4 Convection with internal heat sources

9.4.1 Pure internal heating: no heat supplied from below

9.4.2 Layer heated from below and from within

9.4.3 Dynamics of the upper thermal boundary layer

9.5 Temperature-dependent viscosity

9.5.1 Moderate viscosity contrasts

9.5.2 Large viscosity contrasts

9.5.3 Arrhenius viscosity dependence on temperature

9.6 Non-Newtonian rheology

9.6.1 Small viscosity contrasts due to temperature: …

9.6.2 Large viscosity contrasts due to temperature: …

9.7 Mantle plumes as part of a large convective system

9.7.1 Plumes in Rayleigh–Benard convection

9.7.2 Heat flux carried by plumes

9.8 Two scales of convection

9.8.1 Large-scale convective motions

9.8.2 Small-scale convection beneath the lithosphere

9.9 Conclusion

10 Thermal evolution of the Earth

Objectives of this chapter

10.1 Initial conditions

10.1.1 Accretion of the Earth. Differentiation of the core

10.1.2 Magma ocean evolution

10.1.3 Average secular cooling rate

10.1.4 Convective heat flux

10.2 Thermal evolution models

10.2.1 The Urey ratio

10.2.2 “Parameterized” cooling models

10.3 Fluctuations of the mantle heat loss

10.3.1 Vagaries of sea-floor spreading

10.3.2 Heat flow out of the core

10.3.3 Time-dependent fluctuations in mantle temperature?

10.4 Continental growth and cooling of the Earth

10.5 Conclusion

11 Magmatic and volcanic systems

Objectives of this chapter

11.1 A few features of crustal magma reservoirs

11.1.1 Dimensions and time scales

11.1.2 Evolution of magma in a reservoir

11.1.3 Structure of magmatic boundary layers

11.1.4 Convection

11.2 Initial conditions: Super-heated magma?

11.3 Cooling and crystallization of magma sheets: Conduction

11.3.1 Pure substance

Zero super heat. Fixed temperature at the boundary

Zero superheat. Phase change moving at a constant velocity

Finite superheat. Fixed temperature at the boundary

11.3.2 Multi-component melts: Mushy layers

Structure and properties of a mushy layer

Cooling by conduction

Contact temperature at the boundary of a magma body

11.4 Cooling by convection

11.4.1 Convection in superheated melt

11.4.2 Zero superheat: Convection in undercooled melt

The amount of undercooling

Crystallization at the roof and at the floor

11.4.3 The floor and side-walls of an intrusion

Cooling at the side-walls

The floor thermal boundary layer

11.5 Kinetic controls on crystallization

11.6 Conclusion

12 Environmental problems

Objectives of this chapter

12.1 The record of past climate in temperature profiles

12.1.1 General formulation. The direct problem

12.1.2 The inverse problem

The inverse problem discretized

Regularization by singular value decomposition

12.1.3 Examples

12.1.4 Discussion

12.2 Ice sheets and glaciers

Exercises

13 New and old challenges

Appendix A: A primer on Fourier and Laplace transforms

A.1 Impulse response and Green’s functions

A.1.1 Dirac’s delta function

Definition

Some properties of the delta function

A.1.2 Impulse response of a linear time invariant system: the convolution theorem

A.1.3 Other useful functions

The Heaviside function

Boxcar (gate) function

A.1.4 The Laplace transform

Useful Laplace transforms

Operational calculus

A.1.5 Evaluating inverse Laplace transforms

Example: The cooling half-space

A.1.6 Asymptotic expansions

A.2 Fourier series and transforms

A.2.1 Periodic function: Fourier series

A.2.2 Useful Fourier series

A.2.3 Non-periodic functions: Fourier transforms

Scaling in space and wave-number

Translation

Fourier transform of the delta function

Fourier transform of the Heaviside function

Fourier transform of the boxcar (gate) function

Fourier transform of the Gaussian distribution function

Fourier transform of the trigonometric functions

A.2.4 Space and wave-number domains

A.2.5 Differential operators in frequency domain

Derivatives and integrals in Fourier domain

A.2.6 The impulse response in frequency domain. The convolution theorem

A.2.7 Solution of boundary value problems. Steady-state solutionsfor the heat equation

A.2.8 Applications to initial value problems for the heat equation

The infinite rod, periodic initial conditions

Cooling of an infinite layer. Uniform temperature at both surfaces

A.2.9 Cooling of an infinite layer. Fixed temperature on uppersurface, fixed heat flux on lower boun

A.3 Cylindrical symmetry. Hankel transform

Appendix B: Green’s functions

B.1 Steady-state heat equation

B.1.1 Full-space 3D

B.1.2 Half-space 3D

B.1.3 2D Green’s functions

B.1.4 2D problems. Application of complex variables theory

B.2 Transient heat equation

B.2.1 1D Green’s function

Appendix C: About measurements

Measuring heat flux

C.1 Land heat flow measurements

C.1.1 Conventional land heat flow measurements

C.1.2 Bottom hole temperature (BHT) data

C.1.3 Corrections

Effect of topography

Lakes. Horizontal changes in surface boundary conditions

Convective heat transport

Erosion sedimentation

Climatic effects

C.2 Oceanic heat flux measurements

Measurements in lakes

Appendix D: Physical properties

D.1 Thermal conductivity

Variation with temperature

Variation with pressure

Thermal properties of magmas

Mixing laws for thermal conductivity

Tensor of thermal conductivity

D.1.1 Heat capacity

D.1.2 Thermal diffusivity

D.1.3 Latent heat

D.1.4 Porosity–Permeability

D.1.5 Thermal expansion

D.1.6 Viscosity

Appendix E: Heat production

E.1 Heat production rate due to uranium, thorium and potassium

E.1.1 Heat production in the continental crust

E.1.2 Large-scale averages

E.1.3 Heat production in sediments and sedimentary rocks

E.1.4 Heat production variations

E.1.5 Crustal stratification

E.1.6 Subcontinental lithospheric mantle

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