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Internal Waves and Tides in Stars and Giant Planets Read chapter Read first chapter

2020 | OriginalPaper | Chapter

5. Fluid Dynamics of Earth’s Core: Geodynamo, Inner Core Dynamics, Core Formation

Authors : Renaud Deguen, Marine Lasbleis

Published in: Fluid Mechanics of Planets and Stars

Publisher: Springer International Publishing

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Abstract

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.

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previous chapter Rotational Dynamics of Planetary Cores: Instabilities Driven By Precession, Libration and Tides
next chapter A Brief Introduction to Turbulence in Rotating and Stratified Fluids
Footnotes
1
The temperature actually does not reach the melting temperature because (i) the melting temperature increases with pressure; and (ii) the high-temperature gradient observed in mines is limited to depth of \({\sim } 30\) km or less, the temperature gradient becoming much less pronounced at deeper depth because of convective motions in the mantle.
 
2
Compression waves, called P-waves, and shear waves, called S-waves.
 
3
Proof: consider a small vector \(\varvec{\delta } \) having material end-points, advected by the flow. At time \(t+dt\) this small vector will be equal to
$$\begin{aligned} \varvec{\delta } (t+dt)&= - \mathbf {u}(\mathbf {x},t)dt + \varvec{\delta } (t) + \mathbf {u}(\mathbf {x}+\varvec{\delta } ,t)dt, \end{aligned}$$
(5.31)
$$\begin{aligned}&= \varvec{\delta } (t) + \varvec{\delta } \cdot \varvec{\nabla } \mathbf {u}\,dt + \mathcal {O}(\varvec{\delta } ^{2}). \end{aligned}$$
(5.32)
Taking the \(dt \rightarrow 0\), \(\varvec{\delta } \rightarrow 0\) limit gives
$$\begin{aligned} \frac{D \varvec{\delta } }{Dt} = \left( \varvec{\delta } \cdot \varvec{\nabla } \right) \mathbf {u}. \end{aligned}$$
(5.33)
\(\varvec{\delta } \) therefore evolves according to the same equation as \(\mathbf {B}\) [Eq. (5.25)].
 
4
Proof: The fact that a magnetic tube is material follows directly from Helmholtz theorem. To show that its magnetic flux does not vary with time, write its time derivative as
$$\begin{aligned} \frac{d}{dt} \int _{S} \mathbf {B}\cdot d\mathbf {S} = \int _{S} \frac{\partial \mathbf {B}}{\partial t}\cdot d\mathbf {S} + \oint \limits _{\mathcal {C}} \mathbf {B} \cdot \mathbf {u}\times d\mathbf {l}, \end{aligned}$$
(5.34)
use the diffusion-free induction equation to write the first term on the RHS as
$$\begin{aligned} \int _{S} \frac{\partial \mathbf {B}}{\partial t}\cdot d\mathbf {S} = \int _{S} \varvec{\nabla } \times (\mathbf {u}\times \mathbf {B})\cdot d\mathbf {S}, \end{aligned}$$
(5.35)
and the identity \((\mathbf {A}\times \mathbf {B})\cdot \mathbf {C}=-\mathbf {B}\cdot (\mathbf {A}\times \mathbf {C})\) plus Stokes’ theorem to write the second term as which is equal to the opposite of the first term on the RHS of Eq. (5.34).
 
5
To show this, consider a plane wave propagating in the z-direction, with fluid velocity where \(\mathcal {R}\) indicates the real part. If the fluid is incompressible, \(\varvec{\nabla }\cdot \mathbf {u}=0\), and hence which is equal to 0 since the plane wave assumption implies that the velocity field is a function of z and t only. This implies that the component \(u_{z}\) of the velocity field must be spatially uniform. In other words, only the component of the velocity field perpendicular to the propagation direction can oscillate spatially: the wave must be transverse.
 
6
Consider small perturbations of the velocity and magnetic fields, linearise the Navier–Stokes and induction equations, take the curl of these two equations, and combine them after taking the time derivative of the curled Navier–Stokes equation (vorticity equation).
 
7
The evolution of the parcel is actually even an isentropic process, since the parcel’s evolution is both adiabatic (no transfer of heat and mass with its surrounding) and reversible (no friction).
 
8
Alternatively, Eq. (5.120) is a sufficient condition for stability.
 
9
The CMB efficiency is usually defined in another way, such that \(P_{a} = \left( \epsilon _{\mathrm {icb}} + \epsilon _\mathrm {cmb} \right) Q_\mathrm {cmb}\). This has the advantage of linking \(P_{a}\) directly to \(Q_\mathrm {cmb}\), but then \(\epsilon _\mathrm {cmb}\) is itself a function of \(Q_\mathrm {cmb}\) (it includes a factor \(Q_\mathrm {cmb} ^{>\mathrm {ad}}/Q_\mathrm {cmb}\)).
 
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Metadata
Title
Fluid Dynamics of Earth’s Core: Geodynamo, Inner Core Dynamics, Core Formation
Authors
Renaud Deguen
Marine Lasbleis
Copyright Year
2020
Book
Fluid Mechanics of Planets and Stars

Print ISBN: 978-3-030-22073-0

Electronic ISBN: 978-3-030-22074-7

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-22074-7_5

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