Fluid Mechanics of Planets and Stars (CISM International Centre for Mechanical Sciences (595))

This book explores the dynamics of planetary and stellar fluid layers, including atmospheres, oceans, iron cores, and convective and radiative zones in stars, describing the different theoretical, computational and experimental methods used to study these problems in fluid mechanics, including the advantages and limitations of each method for different problems. This scientific domain is by nature interdisciplinary and multi-method, but while much effort has been devoted to solving open questions within the various fields of mechanics, applied mathematics, physics, earth sciences and astrophysics, and while much progress has been made within each domain using theoretical, numerical and experimental approaches, cross-fertilizations have remained marginal. Going beyond the state of the art, the book provides readers with a global introduction and an up-to-date overview of relevant studies, fully addressing the wide range of disciplines and methods involved. The content builds on the CISM course “Fluid mechanics of planets and stars”, held in April 2018, which was part of the research project FLUDYCO, supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program. Preface 6 Contents 8 1 Internal Waves and Tides in Stars and Giant Planets 9 Introduction to Internal Waves 9 Plane Inertial Waves 9 Plane Internal Gravity Waves 10 Properties of Internal Waves 11 Internal Wave Beams 12 Reflections and Singularities 14 Inertial Waves in a Sphere 15 Instabilities of Internal Waves 17 Forced Internal Waves 18 Interiors of Stars and Giant Planets 20 Interior Models 20 Waves and Instabilities in a Stratified, Rotating Body 22 Approaches to Internal Waves in Stars and Giant Planets 28 Tidal Interaction of Two Bodies 28 Introduction 28 Gravitational Interaction of Particulate Systems 29 Gravitational Interaction of Fluid Bodies 31 Incompressible Fluid Ellipsoids 32 Internal Waves and Astrophysical Tides 34 Equilibrium and Dynamical Tides 34 Tidal Dissipation via Inertial Waves 35 Tidal Dissipation via Internal Gravity Waves 36 Future Perspectives 37 References 37 2 Waves and Convection in Stellar Astrophysics 39 Stellar Structure and Evolution 39 Lane–Emden Equation 40 Stellar Energy Transport 42 Stellar Evolution Equations 44 Modeling Stellar Convection 47 Sound and Internal Gravity Waves in Stars 51 Uniform Background 52 Isothermal Background 52 General Background 54 Helioseismology 56 Convective–Radiative Interfaces: Convective Overshoot 59 Convective Penetration Length 60 Parameterizations of Convective Overshoot 61 Measuring the Turbulent Diffusivity 62 Physical Mechanism of Convective Overshoot 65 Convective–Radiative Interfaces: Internal Gravity Wave Generation 65 IGW Transport and Damping 66 Spectrum of IGWs Excited by Convection 69 Dynamical Influence of IGW 72 Conclusion 73 References 74 3 Internal Waves in the Atmosphere and Ocean: Instability Mechanisms 79 Overview 79 Dispersion and Polarization Relations 80 Boussinesq Internal Waves 82 Anelastic Internal Waves 84 Breaking Conditions 85 Overturning 85 Convection 85 Shear Instability 86 Summary of Breaking Instabilities 86 Triad Resonant Instability 87 Modulational Stability and Instability 89 Future Directions 96 References 96 4 Rotational Dynamics of Planetary Cores: Instabilities Driven By Precession, Libration and Tides 98 Introduction: From Planetary Magnetic Fields to Core Turbulence 98 Tidal Forcings in Planetary Cores: The Primary Response to Tides 101 The Shape of a Planet Undergoing Tidal Distortion 102 Flow Driven By Differential Spin and Orbit 102 Perturbation of the Rotation Rate: Libration 106 Perturbations of the Rotation Axis: Precession 108 Instabilities Driven By Mechanical Forcings: From Parametric Resonance to Turbulence 111 Parametric Sub-harmonic Resonance of a Pendulum 111 Oscillators in Planetary Cores: Inertial Waves 116 Parametric Excitation: The Case of Tidally-Driven Instabilities 118 Quantifying the Growth Rate: A Global Approach 119 Quantifying the Growth Rate: Short Wavelength Approximation 122 The Elliptical Instability in Planetary Cores 123 An Overview of Some Ongoing Research 124 Present Tools for Investigating Mechanical Forcings and Instabilities in the Laboratory 124 The Saturation of the Instability 124 Dynamo Driven By Mechanical Forcing 126 Instabilities in Presence of an Inner Core 126 The Mysterious Magnetic Field of the Early Moon 127 Conclusion—Beyond the Elliptical Instability in Planetary Cores 129 References 130 5 Fluid Dynamics of Earth's Core: Geodynamo, Inner Core Dynamics, Core Formation 135 Introduction 136 The Birth of Geomagnetism 136 The Discovery of the Core 138 A Short Introduction to Magnetohydrodynamics 140 Classical Electromagnetism 140 The Charge Density in an Electrically Conducting Material 141 The Non-relativistic Limit 142 The Induction Equation 143 The Rmll1 Limit 144 The Rmgg1 Limit 144 An Example of a Kinematic Solution of the Full Induction Equation 146 The Lorentz Force 146 The Non-relativistic Limit 147 Magnetic Pressure and Tension 147 Alfvèn Waves 149 The Geometry of Earth's Magnetic Field 152 Spherical Harmonics Decomposition 152 Demonstrating the Internal Origin of the Geomagnetic Field 154 The Magnetic Field at the Core-Mantle Boundary 156 The Field Within the Core: Poloidal–Toroidal Decomposition 156 Core Flow Inversion 157 Basics of Planetary Core Dynamics 157 The Geodynamo Hypothesis 157 Self-exciting Dynamos 158 What Drives the Geodynamo? 161 Rotating Convection 163 Governing Equations and Non-dimensional Parameters 163 Geostrophy 164 Onset of Thermal Convection 166 Compressibility Effect 169 Application to Earth's Core 172 Convective Dynamos 174 Governing Equations 174 Successes and Challenges 176 Energetics of the Geodynamo 177 How Dissipative is the Geodynamo? 180 Estimating the Available Power 180 Inner Core Dynamics 184 Constitutive Equations 185 Boundary Conditions 187 Non-dimensionalisation and Final Set of Equations 188 Unstable or Stable Stratification in the Inner Core? 189 Deformation Induced by the Lorentz Force 191 Neutral Stratification 192 Stable Stratification 194 Core Formation 195 Problem Set-Up and Non-dimensional Numbers 197 Preliminary Considerations 198 Terminal Velocity 198 Maximal Stable Size of a Falling Drop 199 The Low Reynolds Limit: Diapirism 200 The High Reynolds Limit: Metal–Silicates Separation in a Magma Ocean 202 Observations from Laboratory Experiments 202 Large-Scale Dynamics: Turbulent Entrainment Model 205 Fragmentation 207 Chemical and Heat Transfer at the Drop Scale 211 References 212 6 A Brief Introduction to Turbulence in Rotating and Stratified Fluids 219 Introduction 219 3D Homogeneous Isotropic Turbulence 220 The Zeroth Law of Turbulence 222 Richardson's Cascade and Kolmogorov's 2/3rd Law 222 Spectral Statistics 224 2D Homogeneous Isotropic Turbulence 226 Governing Equations 226 Forward Cascade of Enstrophy 227 Inverse Cascade of Energy 228 Experimental and Numerical Evidences 228 Fate of the Energy at Large Scales 229 Thin-Layer Turbulence 229 Geostrophic Turbulence 230 Fundamental Concepts 230 Quasi-geostrophy (QG) 232 2D Turbulence on a β Plane 232 3D Homogeneous Turbulence in Rotating Fluids 235 Governing Equations 236 Inertial Waves 236 Phenomenological Observations 237 Anisotropic Energy Transfers 239 Inverse Cascade 240 3D Homogeneous Turbulence in Stratified Fluids 241 Equations 241 The Zig-Zag Instability and the Buoyancy Scale 242 The Buoyancy Reynolds Number 243 Conclusion 244 References 245 Front Matter ....Pages i-vii Internal Waves and Tides in Stars and Giant Planets (Gordon I. Ogilvie)....Pages 1-30 Waves and Convection in Stellar Astrophysics (Daniel Lecoanet)....Pages 31-70 Internal Waves in the Atmosphere and Ocean: Instability Mechanisms (Bruce R. Sutherland)....Pages 71-89 Rotational Dynamics of Planetary Cores: Instabilities Driven By Precession, Libration and Tides (Thomas Le Reun, Michael Le Bars)....Pages 91-127 Fluid Dynamics of Earth’s Core: Geodynamo, Inner Core Dynamics, Core Formation (Renaud Deguen, Marine Lasbleis)....Pages 129-212 A Brief Introduction to Turbulence in Rotating and Stratified Fluids (Benjamin Favier)....Pages 213-241