Equations of hydrodynamics

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Table of contents

1 Introduction 
2 Definition of the basic concepts 
2.1 Equations of motion: Navier-Stokes equations
2.1.1 Eulerian and Lagrangian viewpoints
2.1.2 Equations of hydrodynamics
2.1.3 The planetary case: rotating fluid under gravity
2.1.4 Beta-plane approximation and shallow water equations
2.1.5 Importance of waves in the dynamics
2.2 Structure of a giant planet
2.2.1 The gravitational moments
2.2.2 Why are giant planets convective?
2.2.3 Hydrostatic balance
2.3 Short summary
3 Calculation of Jupiter’s gravitational field: testing the Concentric Maclaurin Spheroid method 
3.1 Brief overview of the CMS method
3.2 Evaluation of the uncertainties of the CMS method
3.2.1 Analytical evaluation
3.2.2 Spacing as a power of k
3.2.3 Exponential spacing of the layers
3.3 Numerical calculations
3.3.1 How to match Juno’s error bars
3.3.2 Importance of the first layers
3.3.3 1 bar radius and external radius
3.4 Calculations with a realistic equation of state
3.4.1 Impact of the high atmospheric layers on the CMS method
3.4.2 Errors arising from a (quasi) linear repartition
3.4.3 Intrinsic uncertainties on the Jupiter models
3.5 Taking the 1 bar level as the outer boundary condition
3.5.1 Irreducible errors due to the high atmosphere region (less than 1 bar)
3.5.2 Error from the finite number of spheroids
3.6 Conclusion
4 New models of the interior of Jupiter 
4.1 A brief history of Jupiter
4.2 Method
4.2.1 Concentric MacLaurin Spheroids
4.2.2 Equations of state
4.2.3 Galileo constraints on the composition
4.3 Simple benchmark models
4.3.1 Homogeneous adiabatic gaseous envelope
4.3.2 A region of compositional and entropy variation within the planet
4.4 Locally inward decreasing Z-abundance in the gaseous envelope
4.4.1 Inward decreasing abundance of heavy elements in some part of the outer envelope
4.4.2 Constraints from the evolution
4.5 Models with at least 4 layers and an entropy discontinuity in the gaseous envelope
4.5.1 No entropy discontinuity
4.5.2 Entropy discontinuity in the gaseous envelope
4.6 Discussion
4.6.1 Hydrogen pressure metallization and H/He phase separation
4.6.2 Layered convection
4.6.3 External impacts, atmospheric dynamical effects
4.6.4 Magnetic field
4.6.5 Evolution
4.6.6 Does the observed outer condition lie on an adiabat?
4.7 Conclusion
5 From the core of Jupiter to the atmosphere of hot Jupiters
5.1 Introduction
5.2 The inflated radius: a connection between atmospheric dynamics and the interior structure
5.2.1 Observations
5.2.2 Kinetic or ohmic dissipation
5.2.3 Advection of potential temperature
5.3 Global Circulation Models
5.3.1 Common simplifications
5.3.2 The Unified Model
5.3.3 A step further: clouds and disequilibrium chemistry
5.4 A deep and robust feature: superrotation
5.5 Conclusion
6 ECLIPS3D 
6.1 Introduction
6.2 The algorithm
6.2.1 Linearised equations
6.2.2 Boundary conditions
6.2.3 Energy equation
6.2.4 Method of solution
6.2.5 Maximum Resolution
6.3 Benchmarking
6.3.1 Initial atmospheric rest state
6.3.2 Steady state circulation: unstable jet
6.3.3 Baroclinic instability
6.3.4 Rossby-Haurwitz waves
6.3.5 Linear steady circulation with drag
6.4 Conclusion
7 Equatorial dynamics of hot Jupiters 
7.1 Introduction
7.2 Notations and 2D shallow water equation and solution
7.2.1 Theoretical framework
7.2.2 Non linear accelerations from the linear steady state
7.2.3 Time dependent solutions
7.3 Insensitivity of Matsuno-Gill to the differential heating
7.4 Wave propagation and dissipation
7.4.1 Decay time of damped waves
7.4.2 The particular case of Kelvin waves
7.4.3 Short summary
7.5 Transition to superrotation
7.5.1 Shape of the linear steady states
7.5.2 Order of magnitude analysis
7.5.3 3D GCM simulations with various forcings
7.6 Conclusion
8 Conclusion and perspectives