Fluid Mechanics of Planets and Stars

Internal waves play an important role in tidal dissipation in stars and giant planets. This chapter provides a pedagogical introduction to the study of astrophysical tides, with an emphasis on the contributions of inertial waves and internal gravity waves.

This chapter begins with the principles determining a star’s structure: hydrostatic and thermal balance, and energy generation and transport. These imply that some stars have stably stratified cores and convective envelopes, whereas other stars have convective cores and stably stratified envelopes. The convection in stars is predominantly low Mach number, but the density at the top of a convection zone can be orders of magnitude smaller than the density at the bottom. We derive the anelastic equations which can model efficient, low Mach number convection. The properties of stars can be inferred by studying the waves at their surface. Here we describe sound and internal gravity waves, both of which have been observed in the Sun or other stars. The second half of this chapter discusses two phenomena at the interface between the convective and stably stratified layers of stars. First we consider convective overshoot, the convective motions which can extend into an adjacent stably stratified fluid. This can lead to substantial mixing in the stably stratified part of stars. Then, we discuss internal gravity wave generation by convection, which can lead to wave-induced energy or momentum transport. These illustrate some important fluid dynamical problems in stellar astrophysics.

This chapter summarizes and extends part of the lectures on internal gravity waves in the atmosphere and ocean by focusing upon the various instabilities associated with vertically propagating internal waves, including breaking due to convective overturning and shear, parametric subharmonic instability, and modulational instability associated with spatially localized wave packets.

In this chapter, we explore how gravitational interactions drive turbulent flows inside planetary cores and provide an interesting alternative to convection to explain dynamo action and magnetic fields around terrestrial bodies. In the first section, we introduce tidal interactions and their effects on the shape and rotation of astrophysical bodies. A method is given to derive the primary response of liquid interiors to these tidally-driven perturbations. In the second section, we detail the stability of this primary response and demonstrate that it is able to drive resonance of inertial waves. As the instability mechanism is introduced, we draw an analogy with the parametric amplification of a pendulum whose length is periodically varied. Lastly, we present recent results regarding this instability, in particular its nonlinear saturation and its ability to drive dynamo action. We present how it has proved helpful to explain the magnetic field of the early Moon.

This chapter is built from three 1.5 h lectures given in Udine in April 2018 on various aspects of Earth’s core dynamics. The chapter starts with a short historical note on the discovery of Earth’s magnetic field and core (section “Introduction”). We then turn to an introduction of magnetohydrodynamics (section “A Short Introduction to Magnetohydrodynamics”), introducing and discussing the induction equation and the form and effects of the Lorentz force. Section “The Geometry of Earth’s Magnetic Field” is devoted to the description of Earth’s magnetic field, introducing its spherical harmonics description and showing how it can be used to demonstrate the internal origin of the geomagnetic field. We then move to an introduction of the convection-driven model of the geodynamo (section “Basics of Planetary Core Dynamics”), discussing our current understanding of the dynamics of Earth’s core, obtaining heuristically the Ekman dependency of the critical Rayleigh number for natural rotating convection, and introducing the equations and non-dimensional parameters used to model a convectively driven dynamo. The following section deals with the energetics of the geodynamo (section “Energetics of the Geodynamo”). The final two section deal with the dynamics of the inner core, focusing on the effect of the magnetic field (section “Inner Core Dynamics”), and with the formation of the core (section “Core Formation”). Given the wide scope of this chapter and the limited time available, this introduction to Earth’s core dynamics is by no means intended to be comprehensive. For more informations, the interested reader may refer to Jones (2011), Olson (2013), or Christensen and Wicht (2015) on the geomagnetic field and the geodynamo, to Sumita and Bergman (2015), Deguen (2012) and Lasbleis and Deguen (2015) on the dynamics of the inner core, and to Rubie et al. (2015) on core formation.

This chapter discusses basic aspects of turbulent flows relevant for the small-scale fluid dynamics of planets and stars. We particularly focus on how geometrical confinement, rotation, and stratification affect the nature of turbulent motions at different spatial scales. We introduce a hierarchy of models from the celebrated theory of Kolmogorov valid for homogeneous and isotropic turbulence to gradually more realistic models including rotation and stratification effects. Emphasis is put on simple physical processes and qualitative observations and not on rigorous mathematical derivations.