incompressible fluid flows in rapidly rotating cavities - Institut de ...

INCOMPRESSIBLE FLUID FLOWS IN RAPIDLY ROTATING CAVITIES Alexandre Fournier

A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF GEOSCIENCES January 2004

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© Copyright 2004 by Alexandre Fournier. All rights reserved.

Abstract The subject of incompressible fluid flows in rapidly rotating cavities, relevant to the dynamics of the Earth’s outer core, is addressed here by means of numerical modeling. We recall in the introduction what makes this topic fascinating and challenging, and emphasize the need for new, more flexible numerical approaches in line with the evolution of today’s parallel computers. Relying upon recent advances in numerical analysis, we first introduce in chapter 2 a spectral element model of the axisymmetric Navier-Stokes equation, in a rotating reference frame. Comparisons with analytical or published numerical solutions are made for various test problems, which highlight the spectral convergence properties and adaptivity of the approach. In chapter 3, we couple this axisymmetric kernel with a Fourier expansion in longitude in order to describe the dynamics of three-dimensional convective flows. Again, several reference problems are studied. In the specific case of a rotating fluid undergoing thermal convection, this so-called Fourier-spectral element method (FSEM) proves to be as accurate as standard pseudo-spectral techniques. Having this numerical tool anchored on solid grounds, we study in chapter 4 fluid flows driven by thermal convection and precession at the same time. A new topic in the vast field of fluid mechanics, convecto-precessing flows are of particular importance for the Earth’s core, and the equations governing their evolution are derived in detail. We solve these using the FSEM; results seem to indicate that to first order, thermal convection and precession ignore each other. We discuss the relevance of these calculations for the Earth’s core and outline directions for future related research.

iii

Acknowledgments The next few paragraphs have been written down quite hastily. This was not a wise thing to do, since they are the only ones that everyone actually reads. Were you not to find your name, blame it on the inherently erratic behaviour of a Ph.D. candidate in his final hours. I am grateful to my advisers, Hans-Peter Bunge and Rainer Hollerbach, for their stimulating remarks and their continuous support over the past five years. Needless to say, I hope that our fruitful interaction will continue in the future. I thank Tony Dahlen, Guust Nolet, and Geoff Vallis for agreeing to check on my philosophical abilities during my final public oral examination. I would especially like to thank Tony for the quality of his graduate course on Theoretical Geophysics and his –inspiring– quintessential display of scientific attitude. Among the Geosciences faculty, I am grateful to Jason Morgan and Allan Rubin for showing a continuous interest in my work. During these five years of graduate school, I have been lucky enough to live with outstanding roommates. Let them be praised here (in chronological order): Rupinder ’Nietzsche’ Singh, Thierry ’method man’ Huck, Emmanuel and Ariane, Nicolas and Samia, and Raffaella ’defense coach’ Montelli. In particular, Emmanuel taught me almost everything I know about the spectral element method. Sharing his office for two years had a major beneficial influence on what I was able to accomplish during my thesis. Thank you Manu! Guyot Hall has been a very enjoyable workplace. In terms of probability of presence, the army of graduate students comes first, with (in alphabetical order): Sigal Abramovich, Richard Allen, Adam Baig, Sara Carena, Meredith Galanter-Hastings, Ramon Gonzalez, Sergei Lebedev, Tarje Nissen-Meyer, Ben Phillips, Li-Fan Yue, Ying Zhou, and Alon Ziv. They are followed closely by the –disordered– legion of postdocs: Shu-Huei Hung, Ludovic Margerin, Brian Schlottmann, Jean-Paul ’Pablo’ Ampuero and Shafer Smith. The staff of Guyot Hall is remarkably efficient in assisting graduate students. I have a thought for Scott Sibio and the library staff. I wish to thank Debbie Smith for her continuous administrative assistance, as well as Nancy Janos and Sheryl Rickwell for helping me out on a number of occasions. I thank Laurel Goodell, the undergraduate lab manager, for her thoughtful advices when I was in charge of introductory geology and geophysics labs for undergraduates. Over at the Princeton Materials Institute, Bill Wichser took very good care of the Bladerunner cluster on which I ran the calculations presented in this thesis. Bill certainly agrees with me on one thing: Linux rocks! This cluster was built by Arch Davies, whom I would like to thank for the quality of his work and the enjoyable discussions we had together. I am grateful to the Graduate School of Princeton university for awarding me a Charlotte iv

Elizabeth Procter honorific fellowship to finance my fifth year of study. My last Princetonian thoughts go to my soccer teammates from the Princeton United Football Club and the fun I had kicking the ball around with them. Parlando di calcio, vorrei ringraziare la signora Montelli per gli autografi e per farmi assaggiare la sua cucina eccezionale. I spent a substantial amount of time in Paris during these past five years. My very special thanks go to Jean-Pierre Vilotte. Jean-Pierre kindly provided me with some office space at the Institut de Physique du Globe de Paris, and he showed a lot of interest for my research work. He has been deeply involved in the development of the numerical model that is presented in this thesis, and he is logically a co-author of the two numerical papers that I wrote. I thank Emmanuel Dormy, Cinzia Farnetani, Claude Jaupart, Stéphane Labrosse, and Yvon Maday for their repeated encouragements. In a recent discussion we had in Paris, Einar Rønquist also gave me several useful tips regarding the optimization of my code. My stays in Paris were made enjoyable by the remarkable atmosphere of the lab I was visiting. I learned all I know about LATEXand most of what I know about Linux from Geneviève Moguilny. IPGP students are very friendly, and are always ready to share a drink. Santé à Buckounet, Rico, Riton, Julien, Carène, Stéphanie, Lydie, Elena, Gaetano, Élise, Padre Diego et Papa Fred. Je leur souhaite à tous bonne chance pour la suite, ainsi qu’au chimiste en herbe Kevin. Je remercie très profondément mes parents pour m’avoir donné le goût de la connaissance et de l’apprentissage, et pour leur soutien sans faille tout au long de mon parcours. Je remercie mes grands-parents pour leur amour. J’ai une pensée émue pour ma grand-mère paternelle que je n’ai malheureusement pas beaucoup connue. Toute mon affection pour mes deux petites sœurs, en leur souhaitant de connaître les mêmes joies que leur grand frère. En fermant cette parenthèse de cinq ans, je pense finalement avec amour à la femme de ma vie, Julie, qui, Pénélope des temps modernes, a enduré ces années de séparation sans se plaindre, en souffrant sans doute en silence mais en m’encourageant constamment, surtout quand l’affaire semblait mal engagée. Pour ça, et pour bien d’autres choses encore, je lui dédie ce travail.

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Contents Abstract

iii

Acknowledgments

iv

List of figures

ix

List of tables

xi

1

Introduction

1

2

Application of the spectral element method to the axisymmetric Navier-Stokes equation 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Spectral element methodology . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 SEM vs. analytic solutions: Steady and unsteady Stokes problems . . . . . 2.6.1 Steady Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Unsteady Stokes problem . . . . . . . . . . . . . . . . . . . . . . 2.7 SEM vs. existing numerical solutions: The Proudman-Stewartson problem . 2.7.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Reference numerical solution . . . . . . . . . . . . . . . . . . . . 2.7.3 SEM solution to the Proudman-Stewartson problem . . . . . . . . . 2.7.4 Adaptivity and enhanced convergence . . . . . . . . . . . . . . . . 2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

A Fourier-spectral element algorithm for thermal convection in rotating axisymmetric containers 36 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 vi

8 8 11 12 14 19 22 22 25 27 27 29 29 32 33

CONTENTS 3.3 3.4 3.5 3.6

3.7

3.8

3.9 4

Three-dimensional weak form . . . . . . . . . . . . . . . . . . . . . . . . Strong cylindrical form - Problem reduction by a Fourier expansion in longitude . . . . . . . . . . . . . . . . . . . . . Cylindrical weak form and axial conditions . . . . . . . . . . . . . . . . . Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Truncation of Fourier expansion . . . . . . . . . . . . . . . . . . . 3.6.2 Spectral element discretization of the meridional problems . . . . . Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Timemarching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Temperature solve . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 A discrete decoupling scheme for the velocity-pressure subproblem 3.7.4 Initialization of the algorithm . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Analytical Stokes flow in a spherical shell . . . . . . . . . . . . . . 3.8.2 Rayleigh-Bénard convection in a vertical circular cylinder . . . . . 3.8.3 Thermal convection in a rotating spherical shell . . . . . . . . . . . Discussion - Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fluid flows driven by thermal convection and precession 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 The model and method . . . . . . . . . . . . . . . . 4.2.1 Spherical shell approximation . . . . . . . . 4.2.2 Governing equations . . . . . . . . . . . . . 4.2.3 Scaling – Expression of the Poincaré force . 4.2.4 Numerical method . . . . . . . . . . . . . . 4.2.5 Choice of parameters . . . . . . . . . . . . . 4.3 Convection without precession . . . . . . . . . . . . 4.3.1 Critical Rayleigh number . . . . . . . . . . . 4.3.2 Finite amplitude convection . . . . . . . . . 4.4 Basic precessing flows . . . . . . . . . . . . . . . . 4.4.1 Reference studies . . . . . . . . . . . . . . . 4.4.2 Basic precessing flows in a spherical shell . . 4.5 Precession and convection . . . . . . . . . . . . . . 4.5.1 Velocity fields . . . . . . . . . . . . . . . . 4.5.2 Temperature fields . . . . . . . . . . . . . . 4.5.3 Heat transport . . . . . . . . . . . . . . . . . 4.6 Summary and discussion . . . . . . . . . . . . . . .

vii

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44 46 48 51 51 51 57 58 60 60 61 62 62 66 70 76 80 80 83 83 83 85 87 88 89 89 92 95 95 96 99 99 99 103 103

CONTENTS 5

Afterwords

106

A Quadrature formulas and polynomial interpolation A.1 Orthogonal polynomials in L2 (Λ) . . . . . . . . A.2 Standard Gauss-Lobatto-Legendre formula . . . A.3 Orthogonal polynomials in L21 (Λ) . . . . . . . . A.4 Weighted Gauss-Lobatto-Legendre formula . . .

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107 107 107 108 109

B Derivation of the algebraic system

111

C Local form of stiffness matrices and singularity removal

115

D A multilevel elliptic solver based upon an overlapping Schwarz method

118

Bibliography

120

viii

List of Figures 1.1

The preliminary reference Earth model. . . . . . . . . . . . . . . . . . . .

2.1 2.2 2.3 2.4

Geometry of the problem and notations. . . . . . . . . . . . . . . . . . . Axial and non-axial basis functions for velocity and pressure. . . . . . . . Meridional spectral element mesh. . . . . . . . . . . . . . . . . . . . . . Relative error (in a L21 sense) of spectral element solution to steady Stokes problem, as a function of polynomial order N . . . . . . . . . . . . . . . . Relative error (in a L21 sense) of spectral element solution to unsteady Stokes problem, as a function of timestep ∆t. . . . . . . . . . . . . . . . Structure of the Proudman-Stewartson flow, and reference numerical solution for E = 10−2 , 10−3 , and 10−4 . . . . . . . . . . . . . . . . . . . . . . Spectral element solution to the Proudman-Stewartson problem for E = 10−2 , 10−3 , and 10−4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of the resolution of the macro- spectral elements mesh on the convergence rate of the SEM for the Proudman-Stewartson problem. . . . . . Example of a three-dimensional Fourier-spectral element mesh. . . . . . .

. 11 . 15 . 19

Geometry and notations. . . . . . . . . . . . . . . . . . . . . . . . . . . Tiling of the meridional domain in a collection of n e = 6 non-overlapping elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrangian GLL bases for velocity, temperature, and pressure. . . . . . . Lagrangian WGLL bases for velocity, temperature, and pressure. . . . . . Meridional spectral element grid. . . . . . . . . . . . . . . . . . . . . . . Reference analytical Stokes velocity fields. Left panel: l = 2 reference solutions, with angular order varying from 0 to 2. Right panel: l = 11 reference solutions, with angular order varying also from 0 to 2. . . . . . L21 norm of error versus polynomial order for analytical Stokes benchmark of the Fourier-spectral element method (FSEM) in a spherical shell for spherical harmonic degree 2, and angular order 0, 1, and 2 (left to right). . Same as figure 3.7, for spherical harmonic degree 11. . . . . . . . . . . .

. 43

2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6

3.7

3.8

ix

2

. 24 . 26 . 28 . 31 . 32 . 35

. . . .

52 53 54 55

. 63

. 65 . 65

LIST OF FIGURES 3.9

3.10

3.11 3.12 3.13 3.14 3.15 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13

Left: Difference between the numerical (σh ) and analytical (σa ) values of growth exponent of axisymmetric convective instability in a shear free cylinder for a = 0.2899, an adiabatic sidewall, R=1000 and P r = 1, as a function of the numerical timestep ∆t. Right: N = 8 mesh. . . . . . . . Left: isocontours of normalized horizontal velocity in the horizontal midplane (z = 0) for a = 1, P r = 6.7, R = 17, 500 (top) and R = 50, 000 (bottom), in the rigid cylinder case. Right: vertical cross-sections along the two symmetry planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equatorial temperature (left) and pressure and velocity (right). . . . . . . Three-dimensional representation of Fourier-spectral element solution to the rotating convection problem in a spherical shell. . . . . . . . . . . . . Example of a Fourier-spectral element mesh used to compute the rotating Rayleigh-Bénard flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of Fourier-spectral element method results for case 0 of numerical dynamo benchmark. . . . . . . . . . . . . . . . . . . . . . . . . Minimal grid spacing h for Fourier-spectral element mesh, as a function of polynomial order N . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 68

. 69 . 72 . 72 . 73 . 75 . 78

Schematic representation of the precessing Earth. . . . . . . . . . . . . . . Geometry of the problem and notations. . . . . . . . . . . . . . . . . . . . Spectral element grid used in this study (rotated here through a 90° angle). Critical Rayleigh number Rc (k) for k ranging from 1 to 6 . . . . . . . . . . Temperature anomalies, z-component of vorticity, radial velocity, and mean zonal flow for R = 1.7, 3.4 and 5.7 Rc . . . . . . . . . . . . . . . . . . . . . Timeseries of modal kinetic energy ekkin contained in modes k = 3 (dashed line) and k = 6 (solid line) for R = 3.4Rc (top) and R = 5.7Rc (bottom). . Kinetic energy density isosurfaces of steady precessing flow for P o = 0.01. Vertical velocity uz in the equatorial plane for P o = 0.01, in a spherical shell (a), and in a full sphere (b). . . . . . . . . . . . . . . . . . . . . . . . Vertical velocity in the equatorial plane, after removal of the solid-body rotation component of the flow, for P o = 0.001 (a), 0.01 (b), and 0.1 (c). . . Temperature spectra for convecto-precessing flows. . . . . . . . . . . . . . Temperature fields in convecto-precessing flows. . . . . . . . . . . . . . . Relative difference of heat flux for convecto-precessing flows and heat flux of the reference convective solution, as a function of P o. . . . . . . . . . . Critical Rayleigh number Rc (k) for E = 10−3 and 10−4 . . . . . . . . . . .

81 83 87 90 91 94 97 97 98 100 102 103 105

D.1 Convergence history for pressure calculation, with and without the overlapping Schwarz preconditioner. . . . . . . . . . . . . . . . . . . . . . . . 119

x

List of Tables 1.1

Rossby and Ekman numbers for the core, the oceans, and the atmosphere. .

2.1

Summary of the Proudman-Stewartson problem results. . . . . . . . . . . . 30

3.1

Summary of results obtained by contributors to case 0 of numerical dynamo benchmark, and their numerical method. . . . . . . . . . . . . . . . . . . . 73 Summary of results obtained with the Fourier-spectral element method for case 0 of numerical dynamo benchmark. . . . . . . . . . . . . . . . . . . . 74

3.2 4.1

4

Ratio of the mean kinetic energy ekin of convecto-precessing flows to the mean kinetic energy of the reference convective state e0kin . . . . . . . . . . 99

xi

Chapter 1 Introduction “Some of them hated the mathematics that drove them, and some were afraid, and some worshiped the mathematics because it provided a refuge from thought and from feeling." John Steinbeck, The Grapes of Wrath (1939)

The solid Earth is a succession of concentric layers of different composition and temperature, brought to our eyes by seismic waves (see figure 1.1). A 2260 km thick shell filled mostly with liquid iron, the outer core constitutes by far the widest ocean of our planet. Its dynamics are governed by the same physical laws that control the evolution of the oceans. Given the tormented life of these, we expect the core to be characterized by a wealth of phenomena occurring on a wide range of time and length scales (see e.g. Hollerbach, 2003). Unfortunately, core dynamicists are not as lucky as their colleagues oceanographers, in the sense that the data catalog they have at hand is, in comparison, dramatically sparse. The catalog comprises timeseries of the length of day, indicative of the exchange of angular momentum between core and mantle (Bloxham, 1998, and references therein), and more importantly, measurements of the magnetic field of the Earth. It is now indeed well accepted that the geomagnetic field is generated and sustained by electric currents stemming from the circulation of liquid iron in the core (Larmor, 1919). This process, known as the geodynamo, has been operating for at least the past three billion years (Merrill et al., 1996). In terms of energy budget, the prevalent idea is that thermo-chemical convection occurring in the core provides enough energy to quench the geodynamo’s thirst (Gubbins & Roberts, 1987). Only recently have we started to monitor geomagnetic activity on a daily basis. This effort was initiated by German scientist Carl Friedrich Gauss in the middle of the Nineteenth 1

I NTRODUCTION

17.5

PSfrag replacements

20

2

density (Mg/m3 )

pressure (kbar)

P-wave speed (km/s) S-wave speed (km/s)

4000 3500

15

3000

12.5

2500

10

2000

7.5

Mantle

1500

Inner Core

5

1000

Outer Core

2.5 0 0

500 1000

2000

3000

4000

5000

6000

0

depth (km)

Figure 1.1: The preliminary reference Earth model of Dziewonski & Anderson (1981). Shown are the radial profiles of density, seismic body waves velocities, and pressure (1 kbar = 0.1 GPa). Two major discontinuities are located at 2890 and 5150 km depth, corresponding to the core-mantle boundary (CMB) and the inner core boundary (ICB), respectively. Despite the tremendous ambient pressure (varying from 130 GPa at the CMB to 330 GPa at the ICB), the outer core must be hot enough to be liquid, as it disallows the propagation of shear waves. Century, through the development of a network of geomagnetic observatories. The data collected in the observatories are now supplemented with satellite data (Hulot et al., 2002), which provide a better coverage and yield the best information concerning the current morphology of the field, and its recent fluctuations (termed the secular variation). On another timescale, paleomagnetists study the field frozen in sedimentary or igneous rocks to infer the magnitude and direction of the geomagnetic field over geological times. Their fundamental finding was to discover that the field (which is predominantly dipolar) reversed polarity every once in while, following a mechanism that remains to be explained. Collecting more and more samples, they showed that the reversal rate varied substantially with time. In particular, there is an interval of 40 million years, known as the cretaceous superchron, during which the field did not reverse at all (Merrill et al., 1996). In spite of their sparsity, geomagnetic and paleomagnetic records indicate by proxy that the core is indeed characterized by rich dynamics, occurring on several timescales. Matters are actually more complicated than they are in the oceans, since the metallic character of

I NTRODUCTION

3

iron requires to add Maxwell’s equations to the set of equations that must be solved if one wishes to build a prognostic model of core dynamics. To be more specific, the nondimensional equations governing convection in the Earth’s core and the geodynamo are, in their simplest, Boussinesq form (Hollerbach, 1996) ∇·u Ro (∂t + u · ∇) u + 2ˆz × u (∂t + u · ∇) Θ ∇·B ∂t B

= = = = =

0, −∇p + E∇2 u + (∇ × B) × B + qRΘr, q∇2 Θ, 0, ∇2 B + ∇ × (u × B) .

(1.1a) (1.1b) (1.1c) (1.1d) (1.1e)

We have written successively in equations (1.1a)-(1.1c) the conservations of mass, momentum, and energy. Equations (1.1d)-(1.1e) are Maxwell’s equations under the magnetohydrodynamic approximation (Gubbins & Roberts, 1987). Here u is the fluid flow, p is pressure, B is the magnetic field, and Θ is the buoyancy field (either thermal or compositional). For simplicity, we will assume henceforth that convection has a purely thermal origin and that Θ refers to the thermal buoyancy field. A more sophisticated model of the geodynamo should include both sources of convection, but this level of sophistication is beyond the scope of this general introduction. Set (1.1) follows from a specific choice of the scales that characterize the problem: Length is scaled by the thickness of the liquid core L = 2260 km, time is scaled by the magnetic diffusion timescale T = L2 /λ, where λ is the magnetic diffusivity. For the core, an estimate for λ of 1.6 m2 /s (Gubbins, 2001, and references therein) makes T approximately equal to 105 years. The fluid velocity is scaled by U = L/T , around 7 10−7 m/s. The magnetic field is scaled by B = (ωρµ0 λ)1/2 , in which ω is the Earth’s rotation rate, ρ is the density of the core, and µ0 is the permeability. This scaling follows from the assumption that the Lorentz force and the Coriolis force are comparable in equation (1.1b), a situation which seems likely in the core (Hollerbach, 1996). Let us now focus on the various nondimensional parameters that appear in set (1.1), starting with the Rayleigh number gαβL2 R= , (1.2) ωκ in which g is gravity, α is the coefficient of thermal expansion, β is the radial temperature gradient that drives convection, and κ is the thermal diffusivity. The Rayleigh number is effectively a measure of the vigor of convection. This particular expression of the Rayleigh number, which involves the rotation rate ω, is appropriate in a rapidly rotating fluid such as the core (Hollerbach, 1996). According to Gubbins (2001), the value of the Rayleigh number for the core should be highly supercritical (by at least ten orders of magnitude), provided that its estimate is based upon the molecular values of the thermal diffusivity κ. The Roberts number q=

κ , λ

(1.3)

I NTRODUCTION

4

Table 1.1: Rossby and Ekman numbers for the core, the oceans, and the atmosphere. The molecular kinematic viscosity ν of each fluid is also given for reference, along with the velocity and length scales associated with each flow. Values for the kinematic viscosity of the ocean and atmosphere taken from Byrd et al. (1960). Note that the velocity and (horizontal) length scales for the atmosphere and oceans are representative of large-scale eddy motion in both systems.

Kinematic viscosity ν Length scale L Velocity scale U Ekman number E Rossby number Ro

Earth’s core

Oceans

Atmosphere

10−6 m2 /s 2260 km 7 10−7 m/s 10−15 10−8

10−6 m2 /s 10−5 m2 /s 100 km 1000 km 0.1 m/s 10 m/s −12 10 10−13 10−2 10−1

is the ratio of thermal to magnetic diffusivities. The molecular values of κ and λ make this number very small: q ≈ 5.4 10−6 (Gubbins, 2001, and references therein). We conclude this nondimensional analysis with the Rossby and Ekman numbers Ro =

λ , ωL2

(1.4)

and

ν . (1.5) ωL2 Bearing in mind that U = λ/L, the Rossby number appears to be the ratio of the nonlinear term in the momentum equation to the Coriolis force. The Ekman number measures the ratio of viscous to Coriolis forces. These numbers are both extremely small in the core, with approximate values of 10−8 and 10−15 for Ro and E, respectively. In particular, the smallness of the Ekman number follows from the low value of the kinematic viscosity ν, on the order of 10−6 m2 /s (de Wijs et al., 1998). These two estimates are summarized in table 1.1, in which we have added for comparison the values of these parameters for large-scale oceanic and atmospheric circulation. E=

After this brief survey of geodynamo theory, we realize that a prognostic model of the geodynamo should advance in time a set of coupled nonlinear equations, a task made even more difficult by the smallness of some of its parameters. Not surprisingly then, the development of such a model has been substantially delayed, in particular with respect to global circulation models of the oceans and the atmosphere, the foundations of which were laid in the mid-sixties. Instead, geophysicists interested in the geodynamo devoted their time to the more tractable kinematic dynamo problem, described

I NTRODUCTION

5

by equations (1.1d)-(1.1e), wondering what kind of core circulation u was able to sustain dynamo action against Ohmic decay. An impressive amount of theoretical work was also pursued to study linearized problems, among which the onset of convective flows in rapidly rotating cavities. These efforts led to a better understanding of the basic mechanisms at work in the core; they are thoughtfully summarized in Gubbins & Roberts (1987). More recently, the increase in compute power enabled Glatzmaier & Roberts (1995) to solve numerically problem (1.1) and to present a computer simulation of a geomagnetic reversal in a self-sustained, convection-driven numerical model of the geodynamo. This seminal breakthrough enabled for the first time comparisons of geodynamo model outputs with data. The comparison was exceptionally favorable, given the gap existing between model and ‘Earth-like’ values of some physical parameters (for example, the Ekman number was set to 10−6 and the Roberts number to 1). Besides, a fortunate consequence of Glatzmaier & Roberts’ paper was to prove to geophysicists that geodynamo theory was not just a refuge for mathematicians deprived of thought and feeling as, subsequently, studies carried out by these authors and others investigated geophysical issues of primary interest, including the differential rotation of the inner core (Glatzmaier & Roberts, 1996b), the angular momentum budget of the Earth (Bloxham, 1998), the secular variation of the Earth’s magnetic field (Bloxham, 2000a) and, in a paleomagnetic perspective, the validity of the geocentric axial dipole hypothesis (Bloxham, 2000b). Dormy et al. (2000) reviewed current geodynamo models in great detail and compared their results to geomagnetic and paleomagnetic observations. As stated above, they are remarkably successful, but questions remain on their ability to reproduce the properties of the field over long time intervals, as models integration times are limited to a few hundreds of thousands of years at the most. This limitation arises from the high spatial resolution that is needed, which limits the size of the numerical timestep, and the lack of parallel scalability of the models –they rely on the expansion of the field variables in spherical harmonics, a global basis not well suited for parallel computing. In a recent paper, Glatzmaier (2002) stressed the need for a new generation of geodynamo models that would enable to reach higher resolutions while allowing for longer integration times. Such models have to rely on grid-based methods –such as the finite-element or finite-volume methods–, since they only require local communications among processors, thereby providing a better parallel efficiency. Moreover, from a practical standpoint, gridbased models should benefit by the current trend in high performance computing, which favors low-cost, off-the-shelf clusters of personal computers against more traditional (and expensive) supercomputers (Bunge & Tromp, 2003). At the same time, spectral methods have proved so far to be more efficient than local methods for achieving a given accuracy, and an effective, grid-based model of the geodynamo is yet to appear. In this thesis, we wish to explore the potential of the spectral element method (SEM) to provide a good numerical approach to the geodynamo problem. A variational method akin to the the finite-element method, the SEM relies on high-order basis functions, which con-

I NTRODUCTION

6

fers it the spectral convergence properties of standard pseudo-spectral methods (Rønquist, 1988). The SEM is flexible in terms of geometry, and its parallel implementation has proved to be highly efficient to solve fluid mechanics problems related to engineering applications (Fischer & Rønquist, 1994; Fischer, 1997). Over the past ten years, the SEM has also been applied to geophysical flows, and SEM models of oceanic and atmospheric circulations have flourished (Ma, 1993; Taylor et al., 1997; Levin et al., 2000; Iskandarani et al., 2003). Their performance holds great promise regarding the application of the method to the inner ocean of the Earth. Consequently, we introduce in this thesis the application of the SEM to core dynamics. We shall leave aside Maxwell’s equations and focus on the modeling of the dynamics of a rapidly rotating neutral fluid. This work can therefore be considered as the first step toward a spectral element model of the geodynamo. Chapters 2 and 3 provide an extensive description of the model. In chapter 2, we present the application of the SEM to the axisymmetric Navier-Stokes equation. In chapter 3, this axisymmetric kernel is coupled with a Fourier expansion in longitude in order to tackle three-dimensional problems. In both chapters, comparisons with analytical and published reference numerical solutions are performed. The method is found to be as accurate as standard spectral techniques. Next, we address in chapter 4 the problem of fluid flows driven by thermal convection and precession at the same time. A long time disregarded source of energy for the geodynamo since the pioneering work of Malkus (1968), precession has recently regained some popularity (Kerswell, 1996; Tilgner & Busse, 2001; Noir et al., 2001; Lorenzani & Tilgner, 2001; Noir et al., 2003; Lorenzani & Tilgner, 2003), and its real influence on core dynamics remains to be assessed. In this chapter, we derive the equations that govern convectoprecessing flows, and use our newborn tool to simulate them. We find that, to first order, precession and thermal convection ignore each other and discuss the relevance of these results for the core. We outline future directions of research to pursue in order to refine these preliminary results.

I NTRODUCTION

7

Notes to the readers The three chapters that follow are written in a self-contained fashion, as they correspond to articles which have been, or are about to be, submitted to scientific journals. Consequently, readers should be prepared to find occasionally redundant ideas and concepts. Chapter 2, “Application of the spectral element method to the axisymmetric Navier-Stokes equation”, has been accepted for publication in the Geophysical Journal International, and it is in press. Chapter 3, “A Fourier-spectral element algorithm for thermal convection in rapidly rotating axisymmetric containers”, has been submitted for publication in the Journal of Computational Physics. Finally, chapter 4, “Fluid flows driven by thermal convection and precession”, is considered for publication in the Journal of Geophysical Research. Coauthors are Hans-Peter Bunge, Rainer Hollerbach and Jean-Pierre Vilotte, for chapters 2 and 3, and Rainer Hollerbach and Hans-Peter Bunge for chapter 4. I am (will be) the first and corresponding author of each paper.

Chapter 2 Application of the spectral element method to the axisymmetric Navier-Stokes equation 2.1

Introduction

As the Earth sheds its heat, its interior undergoes large-scale convective motions. Inside its liquid metallic outer core, these motions generate in turn the geomagnetic field, as was originally proposed by Larmor (1919). More than eighty years after his founding hypothesis it is now widely accepted that thermo-chemical convection provides indeed enough energy to power the geodynamo (Gubbins & Roberts, 1987). Modelling this complex magnetohydrodynamic process is made difficult by the low molecular viscosity of iron under core conditions (Poirier, 1988; de Wijs et al., 1998). In fact, the ratio of viscous stresses to the Coriolis force in the force balance of the core, measured by the Ekman number E, is very small (10−12 at most) resulting in sharp viscous boundary layers (called Ekman layers) of a few meters. Thus, we have little hope in the near future of resolving these small length scales numerically in a computer model of the geodynamo, even if we account for the impressive rise in (parallel) compute power expected over the next years. Despite these difficulties great insight into the working of the geodynamo has been gained over the past decade thanks to progress made jointly by laboratory and numerical modellers (Busse, 2000). As a matter of fact, Glatzmaier & Roberts (1995) simulated the magnetohydrodynamics of an artificially hyperviscous core and presented the first computer simulation of a geomagnetic field reversal using a three-dimensional (3-D) spherical dynamo model. Although far from the appropriate parameter regime, their model produced a magnetic field remarkably similar to the magnetic field of the Earth. This seminal result led subsequently these and other authors (Glatzmaier & Roberts, 1996a; Kuang & Bloxham, 8

B UILDING A SPECTRAL ELEMENT KERNEL FOR AXISYMMETRIC FLOWS

9

1997) to investigate a range of geophysical problems related to the dynamics of the Earth’s core, including the differential rotation of the inner core (Glatzmaier & Roberts, 1996b), the angular momentum budget of the Earth (Bloxham, 1998), the secular variation of the Earth’s magnetic field (Bloxham, 2000a) and, in a palaeomagnetic perspective, the validity of the geocentric axial dipole hypothesis (Bloxham, 2000b). From a numerical standpoint, current dynamo models are based on spherical harmonics to describe the horizontal dependency of the variables (Glatzmaier, 1984; Kuang & Bloxham, 1999; Hollerbach, 2000). The method is certainly the most natural one to consider when attacking the problem of modelling the circulation of a convecting (or precessing) Boussinesq liquid metal in spherical geometry (see also Tilgner, 1999). For instance, the analytic character of spherical harmonics permits to perform a poloidal-toroidal decomposition both of the magnetic and the velocity field, thus satisfying exactly the solenoidal requirements upon these vector fields (Glatzmaier, 1984). Moreover, their use leads to a weak numerical dispersion, and they achieve an almost uniform resolution of the spherical surface. They also circumvent the pole problem that arises when using spherical (r, θ, φ) coordinates. Unfortunately, the main drawback of spherical harmonics originates from their global definition, which requires a rather expensive pseudo-spectral calculation of the nonlinear terms, and consequently gives rise to a difficult processing on parallel computers. As a result, current dynamo simulations are not performed at Ekman numbers smaller than 10−4 (Christensen et al., 1999) for simulations that span several magnetic diffusion timescales, unless one uses a controversial hyperviscosity (Zhang & Jones, 1997; Grote et al., 2000). Questions remain on the ability of these smooth models to reflect turbulent motions in the Earth’s core and to reproduce long-term features of the geomagnetic field, as pointed out by Dormy, Valet & Courtillot (2000). There is hope, however, that if one is able to prescribe an Ekman number small enough, one will reach a parameter regime asymptotically appropriate for the Earth’s core. Indeed, from a theoretical standpoint the core has two options as to how to operate its dynamo, commonly referred to as the weak and strong field regimes (Roberts, 1978). The dynamo inside the Earth may fluctuate between these states (Zhang & Gubbins, 2000), but looking at computer models of the dynamo we have yet to discover how large rotation has to be before a dynamo has the choice between these two distinct regimes. St. Pierre (1993) found that E = O(10−5 ) was sufficiently small to obtain a subcritical strong field dynamo in his plane layer study. However, before applying these results to the real Earth one would have to repeat them in spherical geometry, and vary the Ekman number (and other relevant parameters) enough to be able to determine whether or not there are these two distinct regimes. Indeed, that is precisely the ultimate objective of this work. Nevertheless, St. Pierre’s results suggest that the O(10 −4 ) Ekman number currently being used may need to be reduced by an order of magnitude before one is even qualitatively in the right regime. A reduction in Ekman number could be attained by using numerical methods that execute efficiently on modern parallel computers via domain decomposition and explicit message-


10

passing. In fact, domain decomposition methods based on explicit message-passing have already proven to be successful in finite-element models simulating flow inside the Earth’s mantle at high convective vigor (Bunge & Baumgardner, 1995). Moreover, these methods are well suited to the growing trend of using cost-effective, off-the-shelf PC-clusters in geophysical modelling (Bunge & Dalton, 2001). Consequently, our long-term effort aims at developing a numerical dynamo model that retains the accuracy and robustness of spectral methods while performing well on modern parallel computers such as clusters of PCs. Our approach is based upon the use of the spectral element method (SEM), a variational technique that relies on high-order local shape functions (Patera, 1984; Bernardi & Maday, 1992). The SEM, in fact, combines the geometrical flexibility of the finite element method with the exponential convergence and weak numerical dispersion of spectral methods (Maday & Patera, 1989). In addition, its local character lends itself naturally to domain decompositions, and allows for non-uniform resolution inside the computational domain, i.e. for grid-refinements in localized regions such as the narrow Ekman boundary layers inside the core. Recent geophysical applications of the SEM include ocean-atmosphere modelling (Taylor et al., 1997; Levin et al., 2000; Giraldo, 2001) as well as regional and global seismic wave propagation (Komatitsch & Vilotte, 1998; Komatitsch & Tromp, 1999; Capdeville et al., 2002; Chaljub et al., 2003). To our knowledge, however, the SEM has not yet been applied to models of deep Earth flows, neither in the mantle nor in the core. While Chan et al. (2001) already investigated the implementation of a finite-element method to solve the spherical kinematic dynamo problem, we present and validate here the application of the SEM to the Navier-Stokes equation in an axisymmetric, non-magnetic context. This axisymmetric case can readily be generalized to fully 3-D applications by coupling the SEM in the meridional plane with a Fourier expansion in the longitudinal direction. In this so-called Fourier-spectral element approach (Bernardi et al., 1999), the 3-D problem is broken into a collection of meridional subproblems, which in turn may be parallelized into a number of spatial subdomains. We use cylindrical (s, φ, z) coordinates and solve for primitive variables. We thus do not rely on the expansion of the velocity in terms of a poloidal and a toroidal field: A poloidal-toroidal decomposition generates highorder differential operators which can in turn lead to a substantial numerical dispersion. We therefore show explicitly in this paper how the divergence-free requirement on the velocity field is satisfied with our method. We show furthermore, how we handle the singularities at the axis of rotation by using a weighted Gauss-Lobatto quadrature (Bernardi et al., 1999). The outline of this paper is as follows: Section 2.2 recalls the system of equations of interest, and its detailed variational treatment is presented in section 2.3. We then describe the spatial and temporal discretizations of the variational problem in sections 2.4 and 2.5. The validation of the implementation proceeds by comparing SEM results with analytical solutions for steady and unsteady Stokes problems (section 2.6), and with published spectral solutions in a rapidly rotating context (section 2.7). The SEM is shown in all cases to exhibit the spectral convergence properties of standard spectral methods and to provide numerical accuracy of better than one per mil relative to the reference solution. A concluding discussion follows in section 2.8.


11

Figure 2.1: The approach we describe aims primarily at solving the Navier-Stokes equation in spherical/spheroidal shells (right). Its flexibility allows however to handle axisymmetric containers of more complicated shape (left) that one could use in a laboratory experiment. ˘ follows from the revolution of its meridional In each case, the three-dimensional domain Ω section Ω around its axis of symmetry Γ. ∂Ω is the boundary of Ω.

2.2

Governing equations

As illustrated in figure 2.1, we are interested in describing the axisymmetric motion of an incompressible Newtonian fluid filling an axisymmetric container of arbitrary meridional shape Ω. The revolution of Ω around the axis of symmetry Γ gives rise to the full 3-D ˘ We assume that the rotation rate ω is constant and that the rotation vector ω is domain Ω. parallel to Γ. The unit vector along this axis is denoted by zˆ . Under these conditions, the flow of the fluid is governed by the following non-dimensional equations (e.g. Gubbins & Roberts, 1987): ∂t u + 2ˆ z × u = −∇p + E∆u + f in Ω, ∇ · u = 0 in Ω,

(2.1a) (2.1b)

where u is the velocity of the fluid, p is its pressure augmented of the centrifugal acceleration, and f denotes the body forces which include potentially the nonlinear interactions.


12

The actual treatment of the nonlinearities is beyond the scope of the present paper. But let us mention that they may be dealt with in an explicit fashion, by absorbing them into f . The relative importance of viscous to rotational effects is measured by the non-dimensional Ekman number: ν , (2.2) E= ωL2 in which ν represents the kinematic viscosity of the fluid, and L the depth of the container. For problem (2.1) to be well-posed, we specify boundary conditions ub (t) on the domain boundary ∂Ω (which does not include the intersection of Ω with Γ), as well as conditions on the initial state u0 (x).

2.3

Variational formulation

The spectral element method, like the standard finite-element method, relies on the variational formulation of the equations of interest. At any time t ∈ [0, T ], we consider the velocity and pressure field that we denote by ut (x) = u(x, t) and pt (x) = p(x, t). Using cylindrical coordinates (s, φ, z), the three vector components of u t will subsequently be indicated by (ut,s , ut,φ , ut,z ). The variational formulation of problem (2.1) is obtained by multiplying equations (2.1a) and (2.1b) with appropriate trial functions and integrating the resulting system over the domain Ω. An elementary volume of integration dΩ is a torus, obtained by the revolution of a rectangular meridional section of area dsdz around Γ (see figure 1). It is thus given by: dΩ = 2πsdsdz. (2.3) Following Bernardi et al.(1999), we define the space of square integrable functions L21 (Ω): ) ( Z 1 L21 (Ω) =

w : Ω → R, kwk =

2

w2 dΩ

0. Christensen et al. (2001) used this definition to pick a point whose position is fixed in the drifting frame of reference. Results are plotted on figure 3.14, as a function of the spatial resolution, defined as the third root of the number of degrees of freedom for each scalar variable. We used table 1 of Christensen et al. (2001) to plot contributors’ results. We superimposed results obtained

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Figure 3.14: Convergence of results for rotating convection calculation. Results for ACD, CWG, GJZ, and TMH plotted after table 1 of Christensen et al. (2001). For details on the methods used by these groups, see table 3.1 and references in Christensen et al. (2001). Results obtained by the FSEM in blue, with right triangles. Global data on top row: mean kinetic energy ekin (left) and drift frequency ωd (right). Local data on bottom row: total temperature Ttot (left) and azimuthal velocity uφ (right). Resolution is defined as the third root of the number of degree of freedoms for each scalar variables.


76

via the FSEM. These are also listed in table 3.2, along with the suggested standard values of Christensen et al. (2001). Christensen et al. (2001) pointed out that the different results obtained by the contributors converge to the same values within better than 2%. The rate of convergence being fast in the spectral (Chebyshev) case, and slow when finite differencing is used in the radial direction. It appears that the number of grid points in the Ekman boundary layers is critical to properly resolve the solution. The clustering of the Chebyshev points near the boundaries is clearly an advantage in this situation. Also, as pointed out by Christensen et al. (2001), this case is simple “in the sense that the spatial spectra drop off rapidly with wavenumber and time dependence is weak, which makes it ideally suited for spectral methods”. As a matter of fact, table 3.2 shows that the results obtained here are in excellent agreement with the suggested values (to within better than 0.05% in all cases). As seen on figure 3.14, the rate of convergence is also very satisfactory, and the FSEM is in this respect equivalent to the Chebyshev-Ylm approach. The smooth character of the sought solution prompted us to obtain finer resolutions by increasing the polynomial order N to large values and by keeping the total number of elements ne very small. We benefited therefore from the clustering of the GLL points near the boundaries and the associated good resolution of the Ekman boundary layers. In more chaotic, time-dependent situations, timestepping issues could however lead us to keep the polynomial order low (typically between 6 and 12) and to increase ne to obtain finer resolutions.

3.9

Discussion - Conclusion

We have presented a Fourier-spectral element model of thermal convection for a fluid filling an axisymmetric container, in a rapidly rotating reference frame. This model, which relies heavily on the recent theoretical work of Bernardi et al. (1999), is based upon a Fourier expansion of the field variables in the periodic direction, and the resolution of the associated meridional problems via the spectral element method. A weighted GaussLobatto-Legendre quadrature has been introduced to treat those elements sharing an edge with the axis of symmetry of the three-dimensional domain. Inside a meridional element, velocity and temperature are approximated by polynomials of order N in each direction of space, and a slightly lower order (N − 2) is used to discretize pressure. The resulting semi-discrete system is timestepped using second-order schemes: the Coriolis and viscous forces are discretized either semi-implicitly or fully-implicitly. Nonlinear terms are in both cases approximated by an explicit third-order Adams-Bashforth formula. Three examples have illustrated the accuracy of this approach. The steady Stokes problem in a spherical shell (§3.8.1) proved the validity of our implementation of the meridional bricks of the method (mass matrix, divergence matrix, stiffness matrix, mask arrays), and it highlighted its spectral convergence properties. Not surprisingly, to achieve acceptable accuracy, it is much more efficient to increase N , keeping the number of elements n e con-


77

stant, rather than to increase ne at constant N – this is precisely the difference between the exponential convergence of spectral methods and the algebraic convergence of finite difference and finite element methods. We considered next (§3.8.2) standard Rayleigh-Bénard convection in a cylinder heated from below and found good agreement between our calculations and the analytical results of Rosenblat (1982) and the finite difference model of Neumann (1990), varying the aspect ratio of the cylinder and the velocity boundary conditions. Finally (§3.8.3), we studied the rotating Rayleigh-Bénard benchmark case in a spherical shell of Christensen et al. (2001). This problem combined all the ingredients of the governing equations (3.5) and the smoothness of the sought solution prompted us to use a minimal ne and large N to reproduce the reference solution with an excellent agreement. The previous enthusiastic remarks praising spectral vs. algebraic convergence have to be confronted with practical considerations: when dealing with a nonlinear, unsteady problem (highly supercritical convection for instance), a high enough resolution is needed, which has a drastic influence on the timestep size ∆t. The explicit treatment of nonlinearities implies indeed that ∆t can not exceed a value which is proportional to the smallest distance h between two grid points. In our case, the clustering of the GLL and WGLL points near the element boundaries is such that the meridional minimal grid spacing h med is proportional to 1/(ne N 2 ). Figure 3.15 shows the minimal grid spacing hmed for the mesh of figure 3.13 (consisting of four spectral elements) for different values of N , along with the overall h that results from the use of a Fourier grid of 64 equidistant points in longitude. We observe that in this case hmed ≈ 5.70h. This follows from the clustering of points in the polar regions, which is the root of the so-called pole problem (Boyd, 2001, §18.10). This might not be so crucial for our purposes, since calculations at large C for which this model is ultimately aimed require to resolve very thin Ekman boundary layers in the meridional plane. For larger and larger Coriolis numbers, the increase in meridional resolution is likely to outweigh the increase in K, and therefore hmed should be closer to h. In a strongly nonlinear context, one should anyhow use a constant (moderate) value of N (on the order of 10) and refine the resolution by increasing ne , which allows for the model to be run on more processors. As far as performances are concerned, let us stress that the current version of the code has yet to be optimized (at the serial and parallel levels), following for instance the precepts of Deville et al. (2002, chap. 8). This task has been recently initiated, now that the model is anchored on robust (accurate) foundations. In particular, we wish to develop a new preconditioner for the modified Helmholtz operator (3.80) which, unlike the diagonal preconditioner, does not restrict the timestep to a fraction of C −1 (a fraction of ω −1 if one restores dimensions). The FSEM has an overall complexity of O(Kne N 3 ), and a complexity per processor of O(Kne N 3 /np ), if np denotes the number of processors. It is fundamental to obtain good scaling performances and even more efficient solvers for this approach to allow to tackle challenging problems in planetary dynamics. An already very positive news in this respect is that the FSEM does not require a lot of memory, thanks to the problem reduction in longitude and the tensorized formulation, which preserves us from storing large matrices. In particular, switching from the semi-implicit to fully-implicit scheme has a minimal im-


78

0, {}|~A {

0

:

:

0.8

0.9

1

1.1

1.2

1.3

Figure 3.15: Minimal grid spacing h for a three-dimensional mesh of the northern hemisphere of a spherical shell consisting of 4 spectral elements of varying polynomial order N . The Fourier grid has 64 equally spaced points in longitude. Also indicated is the minimal grid spacing in the meridional plane hmed . Both decrease as N −2 , as indicated by the dashed line, and are proportional: hmed ≈ 5.70h. Note that the points for N = 14 (log N ≈ 1.15) correspond to the mesh depicted on figure 3.13. pact on the memory cost compared to what happens in case one uses spherical harmonics. Whereas we just have to deal with a couple of extra vector/scalar fields, Hollerbach (2000) reports that treating the Coriolis force implicitly instead of explicitly multiplies the memory requirements by at least a factor of K, if K is the number of azimuthal modes. This can be quite prohibitive if one wishes to use a large number of these modes. We can not claim at this stage that the FSEM is definitely advantageous with respect to a fully three-dimensional SEM, which is an appealing alternative to simulate the problem of interest here, especially when the number of Fourier modes becomes large. We can stress, however, that the Fourier expansion in longitude greatly simplifies the implementation because of the dimension reduction. In particular, the overlapping Schwarz method is much easier to implement in two dimensions than it is in three dimensions (Fischer et al., 2000). To conclude, let us emphasize that the excellent agreement obtained for the rapidly rotating Rayleigh-Bénard flow is the first of the kind obtained by a method which does not rely on spherical harmonics and it is truly encouraging for future planetary applications of the model. These include the study of flows driven by precession (Malkus, 1968; Lorenzani & Tilgner, 2001; Noir et al., 2001), for which it is particularly important to take the ellipticity of the planet into account. The Fourier-spectral element approach is well suited for this application, as the shape of the meridional domain Ω is arbitrary. Also, as stated in the introduction, another very interesting and related application lies in the modelling of the


79

dynamo process at work for instance in the Earth’s outer core. The main difference between the Navier-Stokes equation and the so-called induction equation which governs the evolution of the magnetic field lies in the boundary conditions. On the outer boundary of the region filled by the convecting metallic liquid, the magnetic field has to be connected with an exterior potential field. This connection is straightforward if one is using spherical harmonics (Glatzmaier, 1984) but less amenable to a local method like ours. We are currently investigating this issue.

Chapter 4 Fluid flows driven by thermal convection and precession 4.1

Introduction

Aside from its engineering applications related to the control of space vehicles (Vanyo & Likins, 1971), precession has also attracted the attention of geophysicists, since Malkus (1968) advocated it as a plausible source of energy for the geodynamo, the process at work in the Earth’s core, and responsible for the generation of its magnetic field. Figure 4.1 illustrates the mechanism that controls the precession of the Earth: The instantaneous axis of rotation of the Earth is inclined with respect to the normal to the ecliptic plane, and the gravitational action of the Sun (and Moon) on the Earth’s equatorial bulge exerts a torque that tends to pull the bulge into alignment with the instantaneous Earth-Sun (or Moon) axis. This torque causes the retrograde precession of the spin axis of the Earth about the normal to the ecliptic plane, with a period of 25770 yr (Yoder, 1995). Malkus (1968) conducted a series of experiments involving a spheroidal cavity filled with water, and precessing at different precession rates. He first observed a steady solution, of the form of a solid-body rotation organized around an axis tilted with respect to the direction of the ‘experimental’ daily rotation vector ω d (Malkus, 1968, figure 1.1). Increasing the precession rate, the flow became unstable (Malkus, 1968, figures 1.2-1.3) and eventually fully turbulent (Malkus, 1968, figure 1.4). Assuming that the turbulent regime was appropriate for the core, Malkus derived an estimate of the energy that could be fed into the geodynamo and reached a positive conclusion regarding the ability of precession to sustain the dynamo. More precisely, he found the input coming from turbulent precession to be close to 2.5 1010 W. With retrospect, this figure still appears quite small when compared with recent estimates of the needs of the geodynamo, which lie in the 10 11 W − 2 1012 W range (Buffett, 2002; Roberts et al., 2003). 80

F LUID FLOWS DRIVEN BY THERMAL CONVECTION AND PRECESSION

81

Figure 4.1: Schematic representation of the precessing Earth. The angle β between the daily rotation vector ω d and the normal to the ecliptic plane is 23.5 °. The precession of the Earth is retrograde, as indicated by the orientation of the precession vector ω p . The period of precession is 2π/ωp = 25770 yr (Yoder, 1995). Some years later, Rochester et al. (1975) rejected Malkus’ proposition, arguing that the precession of the Earth had not net effect on the core, since the forcing it induces has a diurnal periodicity in the reference frame rotating with the Earth. Contrary to Malkus’ view, their argument was based on the assumption that the response of the core was laminar, thereby ignoring possible longer timescales that could arise from turbulent motions. They estimated the power originating from precession to be close to 10 8 W, a number which was, even at that time, far too small. We should also mention the work by Loper (1975), who included electromagnetic effects in his –laminar– study, and found an input even smaller, of 3 107 W. Loper’s estimate was used for a long time to disregard precession as a plausible source of energy for the geodynamo. More recently, Kerswell (1996) derived an upper bound of the energy that could be dissipated in turbulent precession. He found a dissipation rate fourteen orders of magnitude larger than Loper’s value! Of course, this upper bound is unlikely to be achieved in the core, but Kerswell argued that “even a weakly stirred outer core could extract sufficient energy to quench the geodynamo’s thirst of 1010 /1011 W.” (As an aside, it is interesting to note that over the last 10 years, the estimated thirst of the geodynamo has increased by one order of magnitude.) Furthermore, in their numerical simulation of a precessing plane layer, Mason & Kerswell (2002) demonstrated the possible occurrence of ‘slow’ dynamics –slow with respect to a diurnal periodicity. If these findings could be applied to the core, they would certainly be in contradiction with the laminar hypothesis of Rochester et al. (1975) and Loper (1975).


82

This brief historical introduction shows how crucial it is to assess the laminar or turbulent character of the precession-induced flow inside the Earth’s core. This remains an open question, owing to the difficulty to reproduce the parameters regime relevant for the core, both in laboratory experiments (Noir et al., 2003), and numerical simulations (Lorenzani & Tilgner, 2001; Noir et al., 2001; Lorenzani & Tilgner, 2003). The interest in precession arises in part because of the well-known forcing it induces, called the Poincaré force (see below). A far less constrained source of energy, convection has received most of the attention of geodynamo theoreticians and modelers, especially after the publication of the studies by Rochester et al. (1975) and Loper (1975). There is no doubt that convective currents are taking place inside the core, due to the cooling of the core from the top (controlled by the mantle), and the freezing of the inner core (see e.g. Gubbins & Roberts, 1987). This last phenomenon releases light elements at the bottom of the core and gives a thermo-chemical character to the convective currents. The relative importance of thermal vs. compositional convection is uncertain (Buffett et al., 1996; Labrosse et al., 1997) and current dynamo models usually focus on a purely thermal forcing, the strength of which is artificially increased in order to account somehow for the chemical component of convection (e.g. Glatzmaier & Roberts, 1995). We do not have a definitive idea as to how strong the forcing should be, and it enters models through a non-dimensional parameter, the Rayleigh number R (see below). We refer the reader to Kono & Roberts (2001) for a general definition of this number well-suited for geodynamo simulations, and to Gubbins (2001) for estimates on the magnitude of R in the core. Convection-driven numerical dynamo models proved the existence of self-sustained planetary dynamos (Glatzmaier & Roberts, 1995); some models actually produced magnetic fields that exhibited remarkable similarities with the geomagnetic field (see e.g. Dormy et al., 2000; Glatzmaier, 2002, for recent reviews), despite parameters sometimes several orders of magnitude larger than their ‘Earth-like’ values – the geodynamo problem is a very stiff one (e.g. Hollerbach, 2003). These recent and substantial achievements in dynamo modeling did not serve the cause of precession, and its effects are most of the time ignored. Because we know that the precession of the Earth is definitely felt to some degree by the core, our purpose in this paper is to study the joint effects of thermal convection and precession on a rotating neutral fluid, and in particular to see to which extent each phenomenon influences the overall circulation of the fluid. Strangely enough, this topic has never been addressed in the past, at least up to our knowledge. In the next section, we consequently derive in detail the equations governing the evolution of the fluid and describe the numerical method we apply to solve these, along with our choice of parameters. Next, the two end members of this hybrid problem are studied separately: the convective end member in section 4.3, and the precessing one in section 4.4. In section 4.5, convecto-precessing flows are presented. Results are summarized and discussed in section 4.6, in which we also outline future directions of research relevant to this topic.


83

Figure 4.2: Geometry of the problem and notations. The z axis is chosen parallel to the daily rotation ω d and the x axis is such that at t = 0, ω p belongs to the (xOz) plane. ˆ is the unit vector parallel (anti-parallel) to the precession vector, in case of prograde k (retrograde) precession. The outer and inner boundaries are held at constant temperatures To and Ti = To + ∆T , respectively.

4.2

The model and method

4.2.1 Spherical shell approximation Even though the mechanism that drives the precession of the Earth is intimately related to its ellipticity (see figure 4.1), we shall ignore the flattening the Earth at the poles and will concentrate our efforts on a spherical shell configuration. This approximation is justified in part by the smallness of the departure from a perfectly spherical shell, measured by the ellipticity of the different layers that constitute the Earth. The ellipticity e is equal to 1/300 at the surface of the Earth, and it decreases to 1/373 and 1/416 at the core-mantle and inner core boundaries, respectively (Yoder, 1995). We shall discuss this approximation further in the discussion (§4.6).

4.2.2 Governing equations We consider therefore a Newtonian fluid of density ρ filling a spherical shell with inner and outer radii ri and ro , rotation vector ω d , precession vector ω p , and fixed inner and outer temperatures Ti and To = Ti − ∆T (see figure 4.2).


84

In the frame rotating with ω = ω d + ω p , the conservation of momentum writes ρ [∂t u + u · ∇u + 2ω × u + ω × (ω × r)] = −∇p+η∇2 u+ρ(ω d ×ω p )×r+ρg, (4.1) in which u is the fluid velocity, r is the position vector, p is pressure, η is the viscosity of the fluid, and g is the gravity field. This equation is supplemented with no-slip boundary conditions on the inner and outer spherical boundaries. The term ρ(ω d × ω p ) × r has been christened the Poincaré force by Malkus (1968). As stated in the introduction, it is a well-known forcing field, the expression of which can be computed exactly at every point r. We make the Boussinesq approximation (Chandrasekhar, 1961) and replace ρ by a constant density ρ0 everywhere except in the buoyancy force ρg. This yields ρ ∂t u + u · ∇u + 2ω × u = −∇P + ν∇2 u + (ω d × ω p ) × r + g, (4.2) ρ0 having introduced the modified pressure field P = p/ρ0 + |ω × r|2 /2 and the kinematic viscosity ν = η/ρ0 . We now expand the equation of state about the reference temperature T = T o and obtain ρ = 1 − α(T − To ) = 1 − αT 0 , (4.3) ρ0 if α and T 0 denote the coefficient of thermal expansion and the reduced temperature T −T o , respectively. When inserted into (4.2), the first term of (1 − αT 0 )g can be written as a gradient of the gravitational potential and absorbed into the modified pressure gradient term (which we shall still denote with ∇P ). We obtain ∂t u + u · ∇u + 2ω × u = −∇P + ν∇2 u + (ω d × ω p ) × r − αT 0 g.

(4.4)

We now turn our attention to the conservation of energy, which, when expressed in terms of temperature T , writes ∂t T + u · ∇T = κ∇2 T, (4.5)

in which κ denotes the thermal diffusivity. The boundary conditions are T (r = r i ) = Ti and T (r = ro ) = To . Considering now the reduced temperature T 0 = T − To , the energy equation becomes ∂t T 0 + u · ∇T 0 = κ∇2 T 0 , (4.6)

and it is subject to the boundary conditions T 0 (r = ri ) = Ti −To = ∆T and T 0 (r = ro ) = 0. In their dimensional form (dropping the primes), the equations governing convection and precession for a Boussinesq fluid are therefore ∇ · u = 0, ∂t u + u · ∇u + 2ω × u = −∇P + ν∇2 u + (ω d × ω p ) × r − αT g, ∂t T + u · ∇T = κ∇2 T,

(4.7) (4.8) (4.9)


85

supplemented by the boundary conditions T = ∆T, u = 0 T = 0, u = 0

if r = ri , if r = ro .

(4.10) (4.11)

We do not specify the initial conditions for the time being.

4.2.3 Scaling – Expression of the Poincaré force We choose the lengthscale L of our problem to be outer radius of the shell r o . In a purely convective context, the timescale that is usually chosen is the thermal diffusive timescale (Gubbins & Roberts, 1987). However, for our hybrid problem, we shall pick a different timescale, which originates from the explicit expression of the Poincaré force f P = (ω d × ω p ) × r. In order to get this expression, we resort to Cartesian coordinates and pick a set of unit ˆ , zˆ) such that zˆ is parallel to ω d and such that at t = 0, ω p belongs to the vectors (ˆ x, y ˆ which is either parallel (xOz) plane (see figure 4.2). We also introduce a unit vector k, to the precession vector ω p if precession is prograde, or anti-parallel to ω p if precession is retrograde. Recalling that the time derivative of any vector v expressed in an inertial frame I and in a frame R rotating at the instantaneous angular frequency ω are related by dv dv = + ω × v, (4.12) dt I dt R we note that the precession vector, when seen from the inertial frame, is not varying with time. We thus have dω p dω p =0= + ω × ωp. (4.13) dt I dt R After simplification, we get the equation governing the evolution of the precession vector ω p in R (omitting the subscript R): dω p = ωp × ωd. dt

(4.14)

Given our choice of axes (see figure 4.2), the three Cartesian components of ω p are therefore ωpx = ωp sin β cos ωd t, ωpy = − ωp sin β sin ωd t, ωpz = ωp cos β.

(4.15) (4.16) (4.17)


86

In the rotating frame, the equatorial components of ω p oscillate with a pulsation ωd . This oscillation also appears in the expression of the Poincaré force f P , the components of which are in turn fP x = ωd ωp z sin β cos ωd t, fP y = − ωd ωp z sin β sin ωd t, fP z = ωd ωp sin β (y sin ωd t − x cos ωd t) .

(4.18) (4.19) (4.20)

Consequently, we choose ωd−1 as the timescale for this problem, instead of the thermal diffusive timescale, which does not appear explicitly in the buoyancy force. Scaling velocity with Lωd and temperature with ∆T , one gets the following set of nondimensional equations: ∇ · u = 0, (4.21) ˆ × u = −∇P ∂t u + u · ∇u + 2(ˆz + P ok) ˆ × r + RT r, (4.22) +E∇2 u + P o(ˆz × k) ∂t T + u · ∇T = EP r −1 ∇2 T. (4.23) We have introduced the Poincaré number P o, defined by Po = ±

ωp , ωd

(4.24)

in which the plus and minus signs correspond to prograde and retrograde precession, reˆ = spectively. The nondimensional precession vector ω 0p writes therefore ω 0p = P ok ω p /ωd . The Ekman number ν (4.25) E= ω d L2 is a non-dimensional measure of the magnitudes of the viscous force compared to the Coriolis force. The Rayleigh number R is given by R=

α∆T go . Lωd2

(4.26)

We have assumed here that the radial profile of gravity followed the law g = g o r/ro , with go the magnitude of the gravity field on the outer surface of the shell. This corresponds to a self-gravitating shell of uniform density. Finally, the Prandtl number Pr =

ν κ

(4.27)

is the ratio of the viscous diffusivity to the thermal diffusivity. The boundary conditions have now become T = 1, u = 0 T = 0, u = 0

if r = ri /ro , if r = 1.

(4.28) (4.29)


87

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Figure 4.3: Spectral element grid used in this study (rotated here through a 90° angle). The meridional plane is decomposed in 32 elements of polynomial order 10. The Rayleigh number (4.26) is different from its classical definition Rclas which corresponds to Rayleigh-Bénard convection occurring in a non-rotating plane layer (Chandrasekhar, 1961): α∆T go L3 . (4.30) Rclas = κν R and Rclas are related via R = E 2 P r−1 Rclas . (4.31) This relationship explains why the values of R that appear throughout this study are abnormally small.

4.2.4 Numerical method We solve equations (4.21)-(4.23) using a Fourier-spectral element algorithm described in detail elsewhere (Fournier et al., 2003b, and references therein). We resort to (s, φ, z) cylindrical coordinates and firstly expand the variables of interest in Fourier series in longitude {u, P, T } (s, φ, z) =

k=+K X

k=−K

n

o k ˜k ˜ ˜ , P , T (s, z)eikφ , u k

(4.32)

where K is the order of the expansion (we used K = 32 in most of the calculations that are presented in the next three sections). Given the symmetry in complex space, this procedure gives rise to K + 1 meridional problems (functions of s and z only) for k ∈ {0, . . . , K}. These problems are in turn solved using a parallel spectral element method (SEM). Within the SEM framework, the meridional plane is decomposed in a collection


88

of non-overlapping elements (see figure 4.3). For each Fourier mode k, {˜ uk , P˜ k , T˜k } are approximated locally within each element by means of tensorized bases of polynomials of degree N for velocity and temperature, and N − 2 for pressure. Each meridional problem is cast in its variational form, for which the trial functions are taken to be equal to the basis functions we just described. More specifically, these basis functions are the tensor product of the Lagrangian interpolants defined over the Gauss-Lobatto-Legendre quadrature points of order N . Inside elements sharing an edge with the z axis, a weighted Gauss-LobattoLegendre quadrature is introduced, which incorporates the cylindrical radius in its weight, in order to avoid a potential degeneracy of the discrete set of equations due to nodes located on the z axis (Bernardi et al., 1999). Figure 4.3 shows a spectral element grid consisting of 32 spectral elements of degree 10, which was used for the most of the calculations presented in the following. The resulting semi-discrete problem is timestepped using a second order scheme, which treats the viscous and Coriolis forces implicitly and the nonlinear interactions in a fully explicit fashion, using the 3/2 rule (Orszag, 1971). Regarding the precessing situation of interest here, we should add that the extra Coriolis term 2ω 0p × u is treated explicitly and that the Poincaré force fP is treated implicitly thanks to its well-known expression, see equations (4.18)-(4.20). The resulting algebraic systems are solved using preconditioned conjugate (or biconjugate) gradient algorithms. Jacobi (diagonal) preconditioning is used to solve for velocity and temperature, whereas a multilevel overlapping Schwarz method is used to solve for pressure (Fournier et al., 2003a). Domain decomposition is performed in the meridional plane and parallel processing is implemented following the message passing paradigm (Gropp et al., 1999).

4.2.5 Choice of parameters Equations (4.21)-(4.23) show that the problem of interest is controlled by 6 parameters: the radius ratio ri /ro , the Ekman number E, the Poincaré number P o, the angle β, the Rayleigh number R, and the Prandtl number P r. These define a very wide parameter space, and we shall in the following fix the radius ratio to .35, the Ekman number to 10 −3 , the Prandtl number to 1, and β to π/2. A radius ratio of .35 corresponds to today’s Earth radius ratio (Yoder, 1995). The Ekman number we choose is larger by several orders of magnitude to its Earth value of O(10−12 ), based on different estimates of the viscosity of liquid iron under core conditions (Poirier, 1988; de Wijs et al., 1998). Taking 10−3 is not entirely foolish though, since flows computed for this number already exhibit characteristic properties of rapidly rotating flows, such as narrow Ekman boundary layers, whose thickness goes like E 1/2 L (see e.g. Greenspan, 1990). Moreover, dealing with a neutral fluid, and thereby omitting to consider any magnetic effects, we can as well set the value of P r to 1, instead of the O(10−2 ) value appropriate for liquid metals (Gubbins & Roberts, 1987).


89

Regarding the value of β, let us first split the dimensionless precession vector ω 0p into its component parallel to the axis ω 0pz and its equatorial component ω 0pxy ω 0p = ω 0pz + ω 0pxy .

(4.33)

The Coriolis force fC and the Poincaré force fP can now be written fC = −2(ˆz + ω 0pz + ω 0pxy ) × u, fP = (ˆz × ω 0pxy ) × r.

(4.34)

As long as P o is sufficiently small so that ω 0pz can be neglected with respect to zˆ, the main effect of precession is due to the action of the equatorial component of the precession vector ω 0pxy , the amplitude of which is given in nondimensional units by P o sin β , see equations (4.15)-(4.17). In this respect, all the (β, P o) pairs corresponding to the same P o sin β are equivalent. We therefore choose to set β = π/2 and have P o vary.

4.3

Convection without precession

Being left with two free parameters, we make matters even simpler in this section by restricting our attention to purely convective flows, by varying R and setting P o to 0. As stated in the introduction, the convective end member of our problem has received considerable attention in the past (Gubbins & Roberts, 1987, and references therein). The goal of this section is to recall the main properties of rotating, convective flows and to define a reference convective state for the convecto-precessing calculations of section 4.5.

4.3.1 Critical Rayleigh number First of all, we wish to determine the so-called critical Rayleigh number R c . It corresponds to a critical temperature difference ∆T maintained across the shell, for which thermal conduction is not efficient enough in removing heat from the fluid; convective currents develop in order to get rid of the excess heat. In a milestone paper, Busse (1970) studied the associated linearized problem and showed that the convective instability took the form of a drifting sequence of columns parallel to the axis of rotation, located at a given cylindrical radius. The columns are alternatively hot and cold, relative to the conductive temperature profile. They correspond to a thermal Rossby wave, which propagates in the prograde direction in the case of a sphere. The dependence of the critical longitudinal wavenumber kc and the angular velocity of the wave on E and P r has have been studied in spherical geometry by several authors (see e.g. Yano, 1992; Jones et al., 2000). The main idea is that ones gets an increasing number of pairs as the Ekman number is decreased: for the viscous force to have a substantial effect in the force balance (and eventually induce a secondary, non-geostrophic convective flow), decreasing E makes it necessary for velocity gradients to be localized over shorter and shorter scales.


90

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Figure 4.4: Numerically determined value of the critical Rayleigh number R c (k) for Fourier modes ranging from k = 1 to k = 6. The curve reaches its minimum for k = 3 which is the critical azimuthal wavenumber. Values of Rc (k) are also tabulated for reference. In our case, for a moderately small E = 10−3 , we expect the scale of these instabilities to be quite large, and consequently a small value for kc . The determination of kc is done in a trial and error fashion: the set of governing equations (4.21)-(4.23) is linearized about the basic, conductive state. For a given Fourier mode k, we start from a random meridional temperature distribution, and the linearized set is timestepped for different values of R. The growth rate σ(k) of the instability is monitored, which enables to determine the value of Rc (k) – defined by the marginal condition σ(k) = 0 – by dichotomy. The critical Rayleigh number Rc is given by Rc = min Rc (k). k

(4.35)

Results obtained following this procedure are plotted on figure 4.4, for k ∈ {1, . . . , 6}. The critical convective mode has a threefold symmetry in longitude and is obtained for a Rayleigh number of 0.08817, or equivalently Rclas = 8.817 104 .

F LUID FLOWS DRIVEN BY THERMAL CONVECTION AND PRECESSION ÄÆÅÈÇÉ+ÊËÄÍÌ Ö

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×

Ø

Ù

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91

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à

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Figure 4.5: From top to bottom: Temperature deviation in the equatorial plane, zcomponent of vorticity in the equatorial plane, radial velocity at mid-depth, as seen from the North pole, and mean zonal flow in the northern hemisphere. Results obtained for R = 1.7, 3.4 and 5.7 Rc (from left to right). All plots are normalized. Solid lines (red regions) and dashed lines (blue regions) represent positive and negative values, respectively.


92

4.3.2 Finite amplitude convection Introduction Fully nonlinear solutions to the rotating convection problem are computed starting from an initial temperature perturbation having the critical, threefold symmetry in longitude, and a zero velocity field. Calculations are carried out until a steady, or quasi-steady state is reached. We consider a quasi-steady state when the running average F of R t realized 0 0 a given field f , defined by F (t) = (1/t) 0 f (t )dt , has converged. In what follows, three distinct and representative Rayleigh numbers will be considered, corresponding to 1.7, 3.4, and 5.7Rc . We will refer to the temperature deviation, T − Tcond , in which Tcond is the static, conductive radial temperature profile i ri h ro Tcond (r) = −1 . (4.36) r −r r o

i

A quick tour of rotating convective flows Several snapshots of interest are displayed on figure 4.5. Each column corresponds to a given Rayleigh number. From top to bottom, we plot the temperature deviation in the equatorial plane z = 0, the z-component of the vorticity ∇ × u in the equatorial plane, a view from the North pole of the radial velocity at mid-depth, and the mean zonal flow u˜ 0φ . The first two snapshots aim at describing the the potential periodicity of the characteristic patterns of the flow, while the radial velocity contours should provide additional information on the transport of heat at work in the shell. Finally, the mean zonal flow provides an extra meridional information on the overall circulation of the fluid and in particular, it could reveal the existence of differential rotation. • R = 1.7Rc : For this value of R, the solution reaches a steady state and keeps its original threefold symmetry in longitude. As can be seen on figure 4.5a, the temperature anomalies are located in large Busse rolls, located outside of the tangent cylinder C, the imaginary cylinder that circumscribes the inner sphere and is parallel to the axis of rotation. Figure 4.5d shows that the mean zonal flow is mostly concentrated in tropical latitudes, meaning that there is very little activity occurring in C. Figure 4.5c outlines the fact that radial flow (and hence heat transport) is accordingly concentrated in the equatorial regions. The shape of the Busse rolls does not vary with time, and the solution is steady, except for a retrograde azimuthal drift ωb = −1.39 10−3 . In finite amplitude convection, the slope of the spherical boundary induces via the nonlinear term u · ∇u a retrograde zonal wind which can eventually overcome the prograde propagation of the thermal Rossby waves (e.g. Cardin & Olson, 1994), and this is what is happening here. The


93

spiraling aspect of the contours is related to the slope of the spherical boundary being steeper and steeper as one approaches the outer boundary of the shell. For a detailed analysis of this effect, the reader is referred to Jones et al. (2000). Vorticity contours (figure 4.5b) indicate that the maximum vorticity is located inside the inner Ekman boundary layers. We can still discern three pairs of cyclones and anticyclones in the bulk of the shell. Cyclonic motion draws fluid from the northern and southern hemisphere into the equator, following a mechanism similar to Ekman suction (Greenspan, 1990), while anticyclones expel fluid from the equator. This explains why the cyclones are in comparison wider than the anticyclones. • R = 3.4Rc : Increasing the Rayleigh number, the spiraling effect becomes strong enough to break the Busse rolls, and this ultimately generates a sixfold symmetry in longitude (figures 4.5e and 4.5f). The shape of the resulting structures vary with time and the solution is strongly time-dependent. In order to get a better idea of the fluctuations of the flow, we plot on figure 4.6 (top) the modal kinetic energy ekkin contained in the k = 3 and k = 6 modes with respect to time, in the [500,1000] dimensionless time window. k = 6 is clearly dominant over time, but there are occasional periods of time, such as the [700,800] interval, during which the original threefold symmetry is recovered. Even if it now occurs on a k = 6 azimuthal lengthscale, heat transport is still concentrated in equatorial regions, as can be seen on figure 4.5g. Furthermore, we note the presence of some activity inside C, as differential rotation is now visible on figure 4.5h. • R = 5.7Rc : Increasing the Rayleigh number even further, one might expect an even more chaotic situation. On the contrary, we note on figure 4.5i that the flow seems, on least on this snapshot, to have recovered in part its original symmetry. Cold, k = 3, thermal anomalies are concentrated around the inner core (in a region where the spiraling effect is small). Furthermore, the outer, hot anomalies are not sliced into two distinct halves. The restored threefold symmetry is also noticeable on the vorticity contours (figure 4.5j), where three equatorial anticyclones are visible. We attribute this change in morphology to the existence of a polar transport of heat (figure 4.5k) which has a threefold symmetry and stabilizes the equatorial flow. This polar stabilization effect is also seen on figure 4.6 (bottom): the k = 3 mode is now roughly three times more energetic than the k = 6 mode, and, interestingly enough, the time dependence has a pronounced oscillatory character. Zonal winds (figure 4.5l) are concentrated inside C: this is yet another manifestation of a major change in the character of convection. The development of a polar circulation when a sufficiently supercritical Rayleigh number is prescribed has already been observed in numerical simulations, for instance by Tilgner & Busse (1997).


94

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600

700

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Figure 4.6: Timeseries of modal kinetic energy ekkin contained in modes k = 3 (dashed line) and k = 6 (solid line) for R = 3.4Rc (top) and R = 5.7Rc (bottom). Reference convective state From the three cases considered above, it appears that the structure of a rotating convective flow becomes more and more interesting (and complicated) as we increase the Rayleigh number R, a noteworthy feature being the development of polar convection currents. For simplicity (and computational convenience), we shall ignore time-dependent flows and restrict our attention to the R = 1.7Rc case, which is steady (aside from its azimuthal drift), and define this convective state as the reference convective state, bearing in mind that this flow is characterized by large-scale equatorial features. The reference convective state sets the initial conditions for the problem which mixes both the effects of precession and thermal convection, which we address in section 4.5. Before this, we will review the basic properties of precession-driven flows (setting R = 0 and ignoring thermal effects) in the next section.


4.4

95

Basic precessing flows

4.4.1 Reference studies The problem of precession-driven flows in a fluid embedded in a rigid container was first considered by Poincaré (1910) and a summary of his work can be found for instance in Lamb (1932); Rochester et al. (1975); Malkus (1994). Working in the precessing frame (the frame rotating at ω p only), Poincaré showed that the inviscid steady flow driven inside the cavity was a flow of constant vorticity, corresponding to a rigid rotation of the fluid about a rotation vector ω f , the orientation of which is controlled by the precession rate (viz. the Poincaré number P o), and the ellipticity of the container e. For a small P o, the angle χ between ω d and ω f is given by (Rochester et al., 1975, equation (23)) χ≈

P o sin β , e

(4.37)

an expression which becomes singular in the spherical case e = 0. This singularity can be explained by considering the mechanism responsible for the coupling between the rigid container and the fluid, namely the hydrodynamic pressure torque imposed on the fluid by the container on the ellipsoidal boundary. This torque vanishes in the spherically symmetric case, and viscous effects must be introduced in order for the fluid to be coupled to the container. In another of his milestone papers, Busse (1968) solved the same problem for a viscous fluid, extending the boundary layer formulation of Stewartson & Roberts (1963) to the treatment of the nonlinear terms in the boundary layer equation. Working in the precessing frame also, he derived an implicit expression for ω f , which writes in the spherical case 1/2 0 0 2 1/2 0 ˆ ˆ ˆ ˆ 0.259(E/ω ) + ω · z + z × (ω × z )ω 2.62(Eω ) z × ω f f p p p f ω f = zˆωf2 + , 2 2 2.62 Eωf + 0.259(E/ωf )1/2 + ω 0p · zˆ (4.38) in which E is the Ekman number. Busse found this expression following a rather convoluted scheme, seeking the solution as an expansion in powers of E 1/2 . More recently, Noir et al. (2003) derived the same expression following a much simpler argument based upon a torque balance between precessional, pressure, and viscous torques. The peculiar coefficients that appear in equation (4.38) originate from the excitation of the so-called spinover mode of the sphere by the Poincaré force. The spinover mode represents a rigid rotation about an axis other than the rotation axis of the sphere (or spheroid). As pointed out by Greenspan (1990), “it is the easiest mode to excite, for all that is required is a slight impulsive change in the direction of the rotation axis of a rigidly rotating sphere". This k = 1 mode is definitely excited by the precession of the container, and it appears in the analysis of Noir et al. (2003) in the calculation of the viscous torque: 0.259E 1/2 and


96

2.62E 1/2 are in fact the nondimensional frequency and decay rate of the viscously modified spinover mode, respectively.

4.4.2 Basic precessing flows in a spherical shell In our case, the presence of an inner core of substantial size complicates matters a little bit, and no analytical solution exists for this configuration. We should however retrieve similar flow patterns. To confirm this, we computed using our numerical code the steady solution obtained for a Poincaré number of 0.01, starting from a zero initial velocity field (again, thermal effects are not considered here). The flow reaches a steady state within a few spinup times; on figure 4.7a, we plot a collection of three-dimensional isosurfaces of the kinetic energy density of the solution, in the precessing frame. In this frame, the solution corresponds to a flow of almost uniform vorticity, whose rotation vector ω f is slightly tilted with respect to the z axis. Departure from uniform vorticity occurs in the Ekman boundary layers and is visible through a kink in the isosurfaces near the outer boundary. If we now express this solution in our working frame (figure 4.7b), we find that it is no longer steady and seems to be now associated with a solid-body rotation organized around a vector ω eq located on the equator. This axis rotates about the z axis with a nondimensional unity retrograde angular velocity, as it follows ω 0p – see equations (4.15)-(4.17). Note that to plot figure 4.7b, we have removed the outer Ekman boundary layers, which would otherwise mask the inner cylinders. In order to get a better understanding of the flow, we provide on figure 4.8a a view of the vertical velocity uz in the equatorial plane, looking from the north pole. We observe one upwelling and one downwelling owing to the cylindrical symmetry that is exhibited on 4.7b. The inner core manifests itself through the Ekman layer that is generated in its vicinity. On figure 4.8b, we show for completeness the solution to the same problem obtained in a full sphere. We note that outside of the inner Ekman layer, the two solutions are in very close agreement. In both cases, they exhibit a k = 1 symmetry in longitude. Figures 4.8a and b can be interpreted as images of the viscously modified spinover mode in a spherical shell and in a sphere, respectively. The flow is not, however, exactly organized as a solid body rotation around a global ω eq ; a more precise description could be obtained for instance by considering ω eq a function of the radius r. Precession-driven flows is a wide topic, which goes well beyond the simple description we have made here. Of particular interest are the actual limits of validity of formula (4.38). Tilgner & Busse (2001) made a systematic study of precession-driven flows in a spherical shell with a minuscule inner core (for numerical commodity), and they found that equation (4.38) was in fact valid for a range of (P o, β) broader than expected. Nevertheless, for large values of P o, the angle between ω f and zˆ becomes too large, which can trigger hydrodynamic instabilities. On figure 4.9, we plot the vertical velocity uz in the equatorial plane for obtained for P o = 0.001, 0.01, and 0.1, once the solid body component ω eq × r


! "

97

$%

& ' , ")(+* -

#

Figure 4.7: a: Kinetic energy density isosurfaces of steady precessing flow corresponding to P o = 0.01. The three Cartesian axes are color coded, red is (Ox), green is (Oy), and blue is (Oz). The solution is displayed here in the precessing frame, which is the frame rotating at ω 0p . It is steady in this frame. b: same plot for the solution now seen from the frame rotating at ω 0p + zˆ (our working frame). In this frame, the solution corresponds roughly to a solid-body rotation organized around an axis located in the equator and it is not steady, as this axis rotates about the z axis with a nondimensional unity retrograde angular velocity. We have removed on this plot the outer Ekman boundary layer. .

/

Figure 4.8: a: Isocontours and isosurfaces of vertical velocity uz of steady precessing flow in a spherical shell in the equatorial plane, looking from the north pole. Red regions and solid lines for positive uz , blue regions and dashed lines for negative uz . This plot corresponds to the flow displayed on figure 4.7b. Red and green lines correspond to (Ox) and (Oy) axes, respectively. b: same plot for flow obtained in a full sphere for the same value of the Poincaré number (0.01).


P o = 10−3 a

P o = 10−2

98

P o = 10−1 c

b

Figure 4.9: Isocontours of vertical velocity uz in the equatorial plane, once the solid-body rotation component of the flow has been removed. Plots are normalized, with conventions as in figure 4.8. Results are shown for P o = 0.001 (a), 0.01 (b), and 0.1 (c). has been removed (we are using a globally averaged ω eq ). As stated above, the description by a solid body rotation of angular velocity ω eq gives only a partial description of of the flow. We still observe that the (normalized) residual fields plotted here are remarkably similar for P o = 0.001 and P o = 0.01. On the other hand, the residual field obtained for P o = 0.1 (the largest value we shall use when computing convecto-precessing flows) looks quite different (figure 4.9c), especially in the interior of the shell. This substantial discrepancy does not, however, correspond to a hydrodynamic instability. First, the solution for P o = 0.1 is still steady, aside from the unity retrograde drift. Second, no symmetry has been broken. Noting that the Poincaré force possesses the symmetry fP (−r) = −fP (r),

(4.39)

the three cylindrical components of the basic precessing flow must have the following symmetry properties in Fourier space: if k is odd, if k is even. (4.40) In their paper, Tilgner & Busse (2001) noticed that the instabilities they found displayed the opposite symmetry. We should emphasize that these instabilities occurred for Ekman numbers smaller than the one considered here, the largest being 7 10 −4 (Tilgner & Busse, 2001, table 2). Decomposing the flow corresponding to figure 4.9c into its symmetric and antisymmetric components, we find that it belongs entirely to the symmetric class, which along with the steady character of the global properties of the flow (such as its mean kinetic energy density) indicates that it is still a basic precessing flow, even if it is poorly described by a solid body rotation about an average ω eq . u˜ks (−z) = −˜ uks (z), u˜kφ (−z) = −˜ ukφ (z), u˜kz (−z) = u˜kz (z), u˜kφ (z), u˜kz (−z) = −˜ ukz (z), u˜ks (z), u˜kφ (−z) = u˜ks (−z) =


99

Table 4.1: Ratio of the mean kinetic energy ekin of convecto-precessing flows to the mean kinetic energy of the reference convective state e0kin .

Po ekin /e0kin

4.5

0.005

0.01

0.03

0.05

0.08

0.1

6.60

23.3

1.92 102

4.85 102

1.01 103

1.32 103

Precession and convection

Having in mind the basic properties of convection-driven and precession-driven flows, we now turn to the full problem of flows driven by these two phenomena at the same time. As stated in section 4.3.2, the initial conditions are defined by the convective solution obtained for R = 1.7Rc . We shall consider 6 different Poincaré numbers here: 0.005, 0.01, 0.03, 0.05, 0.08, and 0.1 – all cases reflect a prograde precession of the shell. Precession is impulsively turned on, and the model is run until a new quasi-steady state is obtained.

4.5.1 Velocity fields We begin by looking at the mean kinetic energy ekin characterizing the set of solutions. On table 4.1, they are given as a function of the Poincaré number and normalized with respect to the mean kinetic energy e0kin of the initial conditions. The mean kinetic energy ekin is always larger than e0kin , by one up to three orders of magnitude. The reference convective state being only slightly supercritical, its weight is negligible compared to the input that originates from the Poincaré force, even for small values of P o. The various velocity fields obtained resemble closely the ones depicted in the previous section. As a matter of fact, when we increase the Poincaré number, the morphology of the velocity field evolves from the one depicted on figures 4.7b, 4.9a and 4.9b to the one corresponding to figure 4.9c. Having described these fields previously, we will focus on temperature in the remainder of this section.

4.5.2 Temperature fields On figure 4.10, we plot the temperature spectra S(k), defined by Z 1 |T˜k |2 sdsdz, S(k) = 2V (2K + 2)2 Ω

(4.41)


100

-1

1×10

0213415617981 :;9@ >F>FD :;9@ >9B :;9@ >FG :;9@ >ED :;9@ >FH :;=A@CB L

-2

1×10

-3

1×10

-4

1×10

J IK

-5

1×10

-6

1×10

-7

1×10 0

1

2

3

4

M

5

6

7

8

9

Figure 4.10: Temperature spectra (truncated at k = 9) for convecto-precessing flows. The spectrum of the initial convective state is represented with large empty circles, while convecto-precessing solutions are displayed with solid symbols of varying color and shape.


101

where V is the volume of the shell, and Ω denotes the meridional plane. The (2K + 2) 2 factor in the denominator originates from the normalization of the discrete fast Fourier transform used in the simulation (Frigo & Johnson, 1997). T˜k in equation (4.41) is the total temperature (not the temperature deviation); accordingly, the temperature spectrum is dominated in all cases by the axisymmetric temperature mode, whose energy remains constant to within 1% when the Poincaré number is increased. Leaving this mode aside, we notice not surprisingly that the reference spectrum (plotted with large circles) is dominated by the k = 3 mode. As seen in section 4.3, the Busse rolls are a stable feature, and very little energy is found in the harmonics (less than 2.5% of the total non-axisymmetric energy). As soon as P o > 0, the velocity field is dominated by its order 1 component (see section 4.4.2). The nonlinear term in the heat equation (4.23) couples each temperature mode k with modes k ± 1, thereby populating the entire spectrum. When P o ≤ 0.05, most of the non-axisymmetric energy is nevertheless still confined in the k = 3 mode. The Busse rolls, which are almost stationary in space (|ωb | = 1.39 10−3 1), are not really sensitive to the equatorial solid-body rotation which characterizes the precession-dominated flow. The drift of the cylindrical motion is too fast to have a net effect on the temperature patterns. Roughly speaking, a given Busse roll is not given enough time to wrap itself around the rotating equatorial cylinder. What is happening here is in essence similar to the argument that led Rochester et al. (1975) to contest Malkus’ view: when (correctly) envisioned in the frame rotating at zˆ + ω 0p , a laminar precessing flow does not have a substantial, permanent effect. Figures 4.11a and 4.11b show isosurfaces of non-dimensional temperature deviations of ±0.2 (red for hot, blue for cold) obtained for P o = 0.01 and 0.05, respectively. By temperature deviations, we still mean deviation from the conductive profile (4.36), even if there is no static, conductive solution to the convecto-precessing problem, due to the non-conservative character of the Poincaré force. However, looking at departures from this hypothetical state eases the understanding of the results and their comparison with the purely convective situation. In both cases, the k = 3 symmetry is preserved. The Busse rolls have been rotated with respect to a rotation axis located in the equator, the inclination being more pronounced for P o = 0.05 (figure 4.11b). It takes quite a large value of P o for the original symmetry to be at least partially lost. As indicated on figure 4.10, mode k = 3 is no longer dominant for P o = 0.08 and P o = 0.1. For P o = 0.08, modes k = 1, 2, 3 are equally energetic, while mode k = 1 is dominant for P o = 0.1. Figure 4.11c (P o = 0.08) reveals that secondary cold features have appeared. Three small hot regions can be seen which fill the space between cold regions. For P o = 0.1 (figure 4.11d), cold anomalies are now predominantly of order 2; the three hot anomalies of figure 4.11c have become a pair of substantially larger size.


N

102

P

O

Q

Figure 4.11: Isosurfaces of temperature deviations of amplitude ±0.2 (red for hot, blue for cold) for convecto-precessing flows computed for P o = 0.01 (a), 0.05 (b), 0.08 (c) and 0.1 (d). Axes color-coded as in figure 4.7.


103

0.05 0.04

V

0.03

S VW X S

R SUT

0.02 0.01 0 -0.01 -0.02

0

0.01 0.02 0.03 0.04 YUZ\[ ]^6_\0.05 `ba ]bdfeh0.06 g ` 0.07 0.08 0.09

c

c

0.1

Figure 4.12: Relative difference of heat flux computed at the surface of the shell Φ for convecto-precessing flows and heat flux Φ0 of the reference convective solution, as a function of the Poincaré number.

4.5.3 Heat transport Trying now to relate these observations to integrated quantities, and in particular to the efficiency of convecto-precessing flows in terms of heat transfer, we plot on figure 4.12 the relative difference between the heat flux coming out of the shell Φ for a given P o and the reference heat flux Φ0 characterizing the initial conditions (Φ and Φ0 include the conductive flux). We could have anticipated from the results of the previous sections that weak precessional forcing (P o ≤ 0.03) has a marginal impact on the heat flux. For intermediate values, the heat flux is smaller than Φ0 , indicating that the temperature patterns of figure 4.11b and 4.11c actually inhibit the transport of heat. For the largest Poincaré number (P o = 0.1), which is characterized by a dominant order 1 topology (see figure 4.11d), we note a substantial improvement in the efficiency of the convective machine. The convective contribution to the heat flux is in this case larger by 20% than what it is for Φ 0 .

4.6

Summary and discussion

In this paper, we have laid the theoretical framework necessary to the determination of fluid flows driven by thermal convection and precession, a hybrid problem relevant to the dynamics of the Earth’s core. We have used a parallel Fourier-spectral element code to compute the associated fluid flows, in a spherical shell geometry and for a mild value of


104

the Ekman number (10−3 ). For this value, and a reference convective state obtained choosing a Rayleigh number 1.7 times critical, calculations indicate that, to first order, convection and precession ignore each other. In the cases considered here, the velocity field is primarily a k = 1 precessioninduced field, of unity retrograde drift, and it affects only marginally the k = 3 temperature patterns (and related heat transport). It takes a large value of the Poincaré number for the (almost) stationary Busse rolls that define the reference convective state to be altered by the precessing flow, and for convective transport to be enhanced by precession. This happens in situations for which the flow is substantially different from a solid-body rotation organized around an equatorial axis. We would like to emphasize again that the flow is still stable and that no hydrodynamic instability is present, due to the large value of the Ekman number. Having chosen a steady convective pattern –the only time-dependence is the retrograde drift of the rolls ωb , whose amplitude is 1, 000 times smaller than the drift of the precessing flow–, we have set ourselves in the worst case situation in terms of timescales discrepancy. This shows the need to repeat this first exercise for time-dependent reference convective states, corresponding for instance to R = 5.7Rc (§4.3). Using a more supercritical value of R, the flow is characterized by timescales closer to the rotation period, see figure 4.6 (bottom). This observation, along with the new morphology of convection (figures 4.5k and 4.5l), suggests that the interaction of precession with convection will, in this case, bring up new interesting physics with respect to the situations we have considered in this paper. The computational price to pay to study this regime is not too expensive in terms of resolution: to compute the convective solution for R = 5.7R c , we used the mesh of figure 4.3, taking (N = 13, K = 32) –instead of (N = 10, K = 32) for R = 1.7Rc . The cost comes really from the need for long integration times to perform accurate statistics, but it is not prohibitive (we are currently investigating this new regime). Another path to explore is the one going toward an Earth-like situation, by addressing the same problem for smaller values of the Ekman number. On figure 4.13, we have reproduced (on a log scale) the critical Rayleigh number curve of figure 4.4 and added the same curve obtained for E = 10−4 (shown with diamonds). The critical Fourier mode is now the k = 6 mode, indicating that a convective flow develops smaller scales as the Ekman number is decreased. This broadens the spectral gap between the Poincaré force and the buoyancy force, and should reveal a wide variety of flow behaviors. Also, for smaller values of E, precessing flows are more prone to instabilities (Tilgner & Busse, 2001; Lorenzani & Tilgner, 2001). In their paper, Lorenzani & Tilgner (2001) report one instability occurring at E = 10−4 for P o = −0.05 and β = 40 °(P o sin β = −0.032). Quite interestingly, they report that instabilities at E = 10 −4 have predominantly a k = 7 and k = 8 azimuthal periodicity, a region in Fourier space that overlaps with the convective instability spectrum (figure 4.13), suggesting that the basic convective flow


105

Figure 4.13: Critical Rayleigh number as a function of Fourier mode k for E = 10 −3 (circles) and 10−4 (diamonds). might actually trigger a new kind of convecto-precessing instabilities. Related calculations have a tremendous computational cost but they are within the reach of our code. It is crucial to perform a systematic survey of the (numerically attainable) parameter space before even attempting to say anything about what could happen in the Earth’s core. In particular, the trend we will get when decreasing the Ekman number is particularly important. E for the core is estimated between 10−12 and 10−15 . The value for P o and β are −10−7 and 23.5 °, respectively. We can not reject the possibility a turbulent flow of precessional origin, potentially able to power the dynamo, only on the basis of the smallness of P o. Ideally, we would like to map out the changes in flow properties as we vary the triplet (E, P o, R). This unexplored and promising problem must ultimately incorporate two effects to be relevant for the Earth. First, magnetic effects have to be included. This is not a trivial task, and we just started to work on the implementation of the related induction equation following a Fourier-spectral element methodology. Second, the ellipticity e of the outer boundary (1/373 at the core-mantle boundary) should be taken into account. Noir et al. (2003) show that the pressure torque coupling the rigid container and the fluid is proportional to e, while the viscous torque is proportional to E 1/2 . In the Earth, the latter is consequently smaller than the former. The geometrical flexibility of our numerical approach enables to consider boundaries of arbitrary ellipticity, since there is no constraint on the shape of the meridional section of the computational domain (Fournier et al., 2003a). For values of the Ekman number in the 10−3 −10−4 range, values of e in the 0−0.3 range will have to be considered.

Chapter 5 Afterwords The discussion of chapter 4 outlined the interest of pursuing the study of convecto-precessing flows, by exploring the parameter space more systematically. Highly supercritical convection should be considered, along with higher rotation rates (smaller Ekman numbers) in order to get closer to the Earth’s core regime. Also, the ellipticity of the core-mantle boundary should be taken into account and the spherical shell assumption relaxed. The flexible numerical approach we have presented in this thesis will enable this, but, beforehand, we will have to optimize our code. Not a rewarding exercise for an apprentice geophysicist developing a numerical model from scratch, optimization is a key issue to address in order to get anywhere close to interesting geophysical regimes, as was proved in the past for instance by Bunge & Baumgardner (1995) and Clune et al. (1999). Once this task has been performed, future work will focus on turning this hydrodynamic code into a magnetohydrodynamic one. The equation governing the evolution of the magnetic field (termed the induction equation) is similar to the Navier-Stokes equation, but the boundary conditions that must be prescribed are different. As a matter of fact, the magnetic field at the core-mantle boundary must be connected to an exterior potential field which decays to zero at infinity. This connection is straightforward if one uses spherical harmonics (Glatzmaier, 1984), but is less amenable to a local method. It can still be achieved through the projection of the spectral element solution onto the basis of spherical harmonics, which can be performed rigorously via the introduction of a Dirichlet-to-Neumann (DtN) operator –Chaljub & Valette (2003) successfully applied this technique to model the propagation of low-frequency seismic waves at the global scale. The DtN formalism is readily applicable to the induction equation, and holds great promise for future applications of the research presented in this thesis.

106

Appendix A Quadrature formulas and polynomial interpolation Throughout this appendix, Λ refers to [−1, +1]. The majority of the relationships that follow are from Bernardi et al. (1999) and Deville et al. (2002). Formula (A.21) is our own.

A.1

Orthogonal polynomials in L2(Λ)

Let us denote the Legendre polynomials of order N with LN . The Legendre polynomials are orthogonal in L2 (Λ), that is Z 0 if N1 6= N2 ∀(N1 , N2 ), LN1 (ξ)LN2 (ξ)dξ = (A.1) 1/(N1 + 1/2) if N1 = N2 Λ

They satisfy the conditions LN (1) = 1 and LN (−1) = (−1)N . Each LN satisfies the following differential equation: 0 1 − ξ 2 L0N + N (N + 1)LN = 0. (A.2)

Legendre polynomials are computed by means of the induction formula L0 (ξ) = 1 and L1 (ξ) = ξ, (N + 1)LN +1 (ξ) = (2N + 1)ξLN (ξ) − N LN −1 (ξ), N ≥ 1.

A.2

(A.3)

Standard Gauss-Lobatto-Legendre formula

We recall here the main properties of the standard Gauss-Lobatto-Legendre formula. Let us N set ξ0N = −1 and ξN = 1. Then there exists a unique set of N − 1 nodes ξiN , 1 ≤ i ≤ N − 1 107

Q UADRATURE FORMULAS AND POLYNOMIAL INTERPOLATION

108

in Λ and of N + 1 weights ρN i , 0 ≤ i ≤ N , such that the following exactness property holds ∀Φ ∈ P2N −1 (Λ), The ξi , 1 ≤ i ≤ N − 1 are the zeroes of ρN i =

Z

1

Φ(ξ)dξ = −1

L0N

N X

N ρN i Φ(ξi ).

(A.4)

i=0

and the

ρN i

2 , N (N + 1)L2N (ξiN )

can be expressed as follows

0 ≤ i ≤ N.

(A.5)

A basis for PN is made of the Lagrangian interpolants hN i , 0 ≤ i ≤ N given by (1 − ξ)L0N (ξ) , N (N + 1) 1 (1 − ξ 2 )L0N (ξ) hN (ξ) = , i N (N + 1)LN (ξiN ) ξiN − ξ (1 + ξ)L0N (ξ) . hN (ξ) = N N (N + 1) N −1 hN 0 (ξ) = (−1)

(A.6) 1 ≤ i ≤ N − 1,

(A.7) (A.8)

The derivatives of these interpolants, which appear for instance in the divergence and the Laplacian bilinear forms can be estimated using equation (A.2). They are given by  LN (ξj ) 1  i 6= j,  LN (ξi ) ξj −ξi       −N (N +1) i = j = 0, 4 0 hi (ξj ) = (A.9)   N (N +1)  i = j = N,  4     0 otherwise.

We can derive as well the pressure basis functions as the set of Lagrangian interpolants hiN −2 defined by the interior Gauss-Lobatto points ξiN , 1 ≤ i ≤ N − 1: 2

hiN −2 (ξ) =

A.3

L0N (ξ) 1 − ξiN , N (N + 1)LN (ξiN ) ξiN − ξ

1 ≤ i ≤ N − 1.

(A.10)

Orthogonal polynomials in L21(Λ)

The weighted quadrature we use is based upon a class of polynomials M N defined by MN (ξ) =

LN (ξ) + LN +1 (ξ) , 1+ξ

N ≥ 0.

(A.11)

Q UADRATURE FORMULAS AND POLYNOMIAL INTERPOLATION

109

They are orthogonal in L21 (Λ), that is with the weighted measure (1 + ξ)dξ: Z 0 if N1 6= N2 ∀(N1 , N2 ), MN1 (ξ)MN2 (ξ)(1 + ξ)dξ = 2/(N + 1) if N1 = N2 1 Λ

(A.12)

Each MN satisfies MN (1) = 1 and the differential equation (1 + ξ)2 (1 − ξ)MN0

0

The induction formula on the MN is ( M0 (ξ) = 1 and M1 (ξ) = 12 (3ξ − 1), N +2 1 MN (ξ) − MN +1 (ξ) = ξ − (2N +1)(2N 2N +3 +3)

A.4

(A.13)

+ N (N + 2)(1 + ξ)MN = 0.

N MN −1 (ξ), 2N +1

N ≥ 1.

(A.14)

Weighted Gauss-Lobatto-Legendre formula

We can now define a weighted quadrature formula of the Gauss-Lobatto kind. Let us set ζ0N = −1 and ζNN = 1. Then there exists a unique set of N − 1 nodes ζiN , 1 ≤ i ≤ N − 1 in Λ and of N + 1 weights σiN , 0 ≤ i ≤ N , such that the following exactness property holds ∀Φ ∈ P2N −1 (Λ),

Z

Φ(ξ)(1 + ξ)dξ = Λ

N X

σiN Φ(ζiN ).

(A.15)

i=0

The ζiN , 1 ≤ i ≤ N − 1 are the zeroes of MN0 and the σi can be expressed as follows σ0N =

8 , N (N + 2)MN2 (−1)

σiN =

4 , N (N + 2)MN2 (ζiN )

(A.16)

and 1 ≤ i ≤ N.

(A.17)

Again, under these circumstances, we can set as a basis for PN (Λ) the Lagrangian interpolants liN , 0 ≤ i ≤ N : (ξ − 1)MN0 (ξ) , 2MN0 (−1) + N (N + 2)MN (−1) (1 − ξ 2 )MN0 (ξ) 1 liN (ξ) = , N (N + 2)MN (ζiN ) ζiN − ξ (1 + ξ)MN0 (ξ) N lN (ξ) = . N (N + 2)

(A.18)

l0N (ξ) =

1 ≤ i ≤ N − 1,

(A.19) (A.20)

Q UADRATURE FORMULAS AND POLYNOMIAL INTERPOLATION The values of the derivatives of these interpolants at the WGLL nodes are:  −N (N +2)   i=j=0  6     2(−1)N MN (ζj )  i = 0, 1 ≤ j ≤ N − 1  (1+ζj )(N +1)    N  (−1)   i = 0, j = N  N +1    N +1  (−1) (N +1)  1 ≤ i ≤ N − 1, j = 0  2MN (ζi )(1+ζi )      1 MN (ζj ) 1 ≤ i ≤ N − 1, 1 ≤ j ≤ N − 1, i 6= j ζj −ζi MN (ζi ) 0 li (ζj ) =  −1  1 ≤ i ≤ N − 1, j = i  2(1+ζi )     1  1 ≤ i ≤ N − 1, j = N   MN (ζi )(1−ζi )    N +1 (−1) (N +1)   i = N, j = 0  4     −MN (ζj )  i = N, 1 ≤ j ≤ N − 1  (1−ζj )      N (N +2)−1 i = N, j = N

110

(A.21)

4

The pressure basis functions liN −2 , the interior nodes:

1 ≤ i ≤ N − 1 are the Lagrangian interpolants on 2

liN −2 (ξ) =

MN0 (ξ) 1 − ζiN , N (N + 2)MN (ζiN ) ζiN − ξ

1 ≤ i ≤ N − 1.

(A.22)

Appendix B Derivation of the algebraic system We explain here in detail how the algebraic system (2.31), resulting from the spatial discretization by axisymmetric spectral elements, is derived. Let us recall that the expansion of the velocity using the elemental basis functions anchored at the Gauss-Lobatto points write nΓ X N X N X

ut,h (s(ξ, η), z(ξ, η)) =

e=1 i=0 j=0

+

N eij eij N ueij t,s , ut,φ , ut,z li (ξ)hj (η)

ne N X N X X

e=nΓ +1 i=0 j=0

N eij eij N ueij t,s , ut,φ , ut,z hi (ξ)hj (η).

(B.1)

eij eij The (ueij t,s , ut,φ , ut,z ) are the nodal velocities at the collocation points in the e-th element, and ΩΓ (Ω∅ ) refers to the collection of elements which are (not) adjacent to the axis Γ.

The triplet (e, i, j) corresponds to a local elemental ordering of the nodes. It is useful to define a global numbering of the velocity nodes numv : numv : {1, 2, 3} × {1, · · · , ne } × {0, · · · , N }2 → {1, · · · , Nv } (α, e, i, j) 7→ numv (α, e, i, j),

(B.2)

in which α refers to a component of the velocity field and Nv is the total number of degrees of freedom for the velocity field (Nv = Nv,s + Nv,φ + Nv,z ). Note that numv is a surjection as some nodes can belong to more than one element. Nv can be estimated only when the exact topology of the mesh is known.

111

D ERIVATION OF THE ALGEBRAIC SYSTEM

112

The discrete pressure is expanded according to nΓ N −1 N −1 X X X

pt,h (s(ξ, η), z(ξ, η)) =

N −2 peij (ξ)hjN −2 (η) t li

e=1 i=1 j=1

+

−1 N −1 N ne X X X

N −2 (ξ)hjN −2 (η), peij t hi

(B.3)

e=nΓ +1 i=1 j=1

and we can define equivalently a global numbering of the pressure nodes num p : nump : {1, · · · , ne } × {1, · · · , N − 1}2 → {1, · · · , Np } (e, i, j) 7→ nump (e, i, j),

(B.4)

As pressure is defined elementwise, nump is a bijection. The total number of pressure degrees of freedom is Np = ne (N − 1)2 . Following a Galerkin procedure (Reddy, 1993), the trial spaces X,h and Yh are built with the nodal shape functions associated with the velocity and pressure degrees of freedom, respectively. Velocity trial functions ΨI , 1 ≤ I ≤ Nv can be conveniently chosen so that, when one computes the integrals involved in problem (2.13), one gets a set of three scalar equations, one for each component of the original momentum equation (2.1a). In other words, we set    (ΨIs , 0, 0), Is = I − ∆Ns   ΨI = (0, ΨIφ , 0), Iφ = I − ∆Nφ ,     (0, 0, Ψ ), I = I − ∆N Iz z z

if if if

1 ≤ I ≤ Nv,s Nv,s + 1 ≤ I ≤ Nv,s + Nv,φ , Nv,s + Nv,φ + 1 ≤ I ≤ Nv ,

(B.5)

where ∆Ns = 0, ∆Nφ = Nv,s , and ∆Nz = Nv,s + Nv,φ . For any α ∈ {1, 2, 3}, the scalar function ΨIα is defined by its restrictions on the collection of the subdomains Fv (Iα ) which contain the Iα -th nodal value for component α. Let us define ∀I ∈ {1, · · · , Nv,α }, α Fv (Iα ) = e ∈ {1, · · · , ne } , ∃(iαe , jαe ) ∈ {0, N }2 , numv (α, e, iαe , jαe ) = Iα + ∆Nα .

The definition of the velocity trial function can now be properly written in terms of its elemental restrictions:   lN (ξ)hN (η) if Ω ∈ Ω , e Γ jαe iαe (B.6) ∀e ∈ Fv (Iα ), ΨIα |Ωe ◦F e (ξ, η) =  hN (ξ)hN (η) if Ω ∈ Ω . e ∅ jαe iαe


113

The pressure test functions are on the contrary defined elementwise. We can define similarly Fp (I) as ∀I ∈ {1, . . . , Np }, Fp (I) = e ∈ {1, · · · , ne }, ∃!(ie , je ) ∈ {1, N − 1}2 , nump (e, ie , je ) = I .

As nump is a bijection, Fp (I) is a singlet. Again, a pressure trial function ΦI is defined by its restriction on Fp (I)   lN −2 (ξ)hN −2 (η) if ΩF (I) ∈ ΩΓ , p jFp (I) iFp (I) e (B.7) ∀I ∈ {1, · · · , Np }, ΦI |Ωe ◦F (ξ, η) =  hN −2 (ξ)hN −2 (η) if Ω ∈Ω . iFp (I)

jFp (I)

Fp (I)

∅

Having defined both the shape and trial spaces for velocity and pressure, we can use the quadrature rules (2.26) and (2.28) to compute the integrals involved in the variational formulation (2.13). The semi-discrete problem then writes: At any time t ∈ [0, T ] , find (ut , pt ) solution of M∂t ut + Cut + EKut − DT pt = Mf t , −Dut = 0. M is the Nv × Nv mass matrix which can be expressed as   Ms 0 0 M =  0 Mφ 0  , 0 0 Mz

(B.8a) (B.8b)

(B.9)

where, for each component, the Nv,α × Nv,α mass matrix Mα is defined as [Mα ]Iα Jα = (ΨIα , ΨJα )1 ,

(Iα , Jα ) ∈ {1, · · · , Nv,α }2 .

(B.10)

As the basis functions are defined by the Lagrange interpolants upon the quadrature points, the mass matrix has the remarkable property of being diagonal. ( s(ζiNαe ,ξjNαe ) X σiαe ρjαe 1+ζ |J e |(ζiNαe , ξjNαe ) if Ωe ∈ ΩΓ , N iαe [Mα ]Iα Jα = δIα Jα , (B.11) e N N N N )ρ , ξ s(ξ ρ ) if Ω ∈ Ω . , ξ |J |(ξ i j e ∅ αe αe iαe jαe iαe jαe e∈Fv (Iα ) where δ is the Kronecker symbol.


114

C is the Nv × Nv Coriolis antisymmetric matrix,   0 −2Ms 0 0 0 . C =  2Mφ 0 0 0

K is the Nv × Nv stiffness matrix 

 Ks + Ws 0 0 0 K φ + Wφ 0  , K= 0 0 Kz

(B.12)

(B.13)

in which we have used the following notations:

(Iα , Jα ) ∈ {1, · · · , Nv,α }2 ,

[Kα ]Iα Jα = a0 (ΨIα , ΨJα ), and [Wα ]Iα Jα =

Ψ I α Ψ Jα , s s

, 1

(Iα , Jα ) ∈ {1, · · · , Nv,α }2 .

(B.14)

(B.15)

We are using high-order polynomials ( typically N ≥ 6), and the long-range interactions between nodes makes Kα dense. We have also introduced the Nv × Np rectangular gradient matrix DT given by  T  Ds T  0 . D = DTz

(B.16)

The two non-zero components of this matrix are T Ψ Is D s I s J = ∂s Ψ I s + , ΦJ , (Is , J) ∈ {1, · · · , Nv,s } × {1, · · · , Np }, s 1

(B.17)

and

DTz

Iz J

= (∂z ΨIz , ΦJ )1 ,

(Iz , J) ∈ {1, · · · , Nv,z } × {1, · · · , Np }.

(B.18)

Appendix C Local form of stiffness matrices and singularity removal We provide here a detailed description of the implementation of the elementary scalar and vectorial stiffness matrices Kke and Kek of section 3.6. The case of an element not in contact with Γ is standard: GLL quadrature is used in the two directions of space and no singularity has to be removed. We will therefore restrict our attention on the axial element case. The local representation of a scalar field T in such an element e takes the form e

e

T |Ωe (s (ξ, η), z (ξ, η)) =

N X N X

Teij liN (ξ)hN j (η),

(C.1)

i=0 j=0

where the liN and hN j are the Lagrangian interpolants defined over the WGLL and GLL points of order N , respectively (we will omit the superscript N in what follows). Applying Kke to Te means evaluating (see the definition of ak in equation (3.54)) (Kke · Te )i0 j 0 = (K0e · Te )i0 j 0 + k 2 (Mse · Te )i0 j 0 where (K0e

e

· T ) i0 j 0 =

N X N X i=0 j=0

Teij

Z

Ωe

{∂s (li0 hj 0 ) ∂s (li hj ) + ∂z (li0 hj 0 ) ∂z (li hj )} d Ωe ,

and (Mse

e

· T ) i0 j 0 =

N N X X

Teij

i=0 j=0

115

Z

Ωe

l i0 h j 0 l i h j d Ωe . s s

(C.2)

(C.3)

L OCAL FORM OF STIFFNESS MATRICES AND SINGULARITY REMOVAL

116

Each of these integrals is computed in the parent element Λ2 . For instance, the first term on the right-hand side of (C.2) gives rise to Z ∂s (li0 hj 0 ) ∂s (li hj ) d Ωe Ωe Z (∂η z e ∂ξ − ∂ξ z e ∂η ) (li0 hj 0 ) (∂η z e ∂ξ − ∂ξ z e ∂η ) (li hj ) |J e |−1 (ξ, η)se dξdη = 2 ZΛ ∂η z e li00 hj 0 − ∂ξ z e li0 h0j 0 ∂η z e li0 hj − ∂ξ z e li h0j |J e |−1 (ξ, η)se dξdη. (C.4) = Λ2

Each of the four terms involved in this sum is computed using the quadrature rules described in paragraph 3.6.2. When developing the product in equation (C.4), one gets for instance a term which is Z (∂η z e )2 li00 hj 0 li0 hj |J e |−1 (ξ, η)se dξdη =

Λ2 N N XX

σp ρq li00 (ζp )li0 (ζp )hj 0 (ξq )hj (ξq )

p=0 q=0

= ρj δjj 0

N X p=0

σp li00 (ζp )li0 (ζp )

se (ζp , ξq ) e −1 |J | (ζp , ξq )(∂η z e (ζp , ξq ))2 1 + ζp

se (ζp , ξj ) e −1 |J | (ζp , ξj )(∂η z e (ζp , ξj ))2 . 1 + ζp

When p = 0, σ0 = −1 (or equivalently s = 0): the singularity in the term s e (ζp , ξj )/(1+ ζp ) is removed by applying L’Hospital rule and replacing this term by ∂ ξ se (ξ = −1, η = ξj ), a quantity which is derived from the knowledge of the mapping F e . The same logic applies to each term in equation (C.4) above, and, more generally, to the second part of the right-hand side of (C.2), which we expand here for the sake of completeness: Z ∂z (li0 hj 0 ) ∂z (li hj ) d Ωe Ωe Z (−∂η se ∂ξ + ∂ξ se ∂η ) (li0 hj 0 ) (−∂η se ∂ξ + ∂ξ se ∂η ) (li hj ) |J e |−1 (ξ, η)se dξdη = 2 ZΛ −∂η se li00 hj 0 + ∂ξ se li0 h0j 0 −∂η se li0 hj + ∂ξ se li h0j |J e |−1 (ξ, η)se dξdη. = Λ2

Expressions for the derivatives h0 (ξq ) can be found for instance in Deville et al. (2002, pg. 462). We provide the expressions for the derivatives l 0 (ζp ) in appendix A.4. Note that the actual cost of the total calculation K0e ·Te goes like N 3 instead of the expected N 4 thanks to the tensorized formulation and associated partial summation technique –see for instance Boyd (2001, pg. 184). Let us now turn our attention to Mse in equation (C.3), which is of interest for k > 0 only. The associated axial condition is that T must vanish on Γ. A mask array is therefore applied

L OCAL FORM OF STIFFNESS MATRICES AND SINGULARITY REMOVAL

117

e prior to the calculation of (Mse · Te )i0 j 0 to ensure that T0j = 0 for all j. For the same reason 0 the result of this operation needs to be evaluated for i > 0 only (a subsequent application of the same mask array sets the axial values to zero anyway). A straightforward development of equation (C.3) then leads to

(Mse · Te )i0 j 0 = ρj 0

N X i=1

Teij 0

N X p=0

σp

li (ζp ) li0 (ζp ) se (ζp , ξj 0 ) e |J |(F e (ζp , ξj 0 )). (C.5) se (ζp , ξj 0 ) se (ζp , ξj 0 ) 1 + ζp

The result of Mse · Te is the sum of two contributions, (Mse · Te )1 + (Mse · Te )2 . If p 6= 0 in (C.5), there is no singularity and one gets the first term: (Mse · Te )1i0 j 0 = σi0 ρj 0 Tei0 j 0

1 1 |J e |(F e (ζi0 , ξj 0 )). 1 + ζi0 se (ζi0 , ξj 0 )

(C.6)

On the other hand, if p = 0, the application of L’Hospital rule gives rise to the second contribution which includes non-diagonal terms: N

(Mse · Te )2i0 j 0 = σ0 Rie0 j 0 ρj 0

X se (ζ0 , ξj 0 ) e e Teij 0 Rij |J |(F e (ζ0 , ξj 0 )) 0, 1 + ζ0 i=1

(C.7)

e 0 e 0 in which Rij 0 = li (ζ0 )(1 + ζ0 )/s (ζ0 , ξj 0 ). The expression for li (ζ0 ) is given in appendix e A.4. Again, terms of the form s (ζ0 , ξj )/(1 + ζ0 ) are practically replaced in the implementation by the quantity ∂ξ se (ξ = ζ0 , η = ξj ) whose exact expression depends on the chosen mapping F e (analytical or subparametric).

As far as the elementary vectorial stiffness matrix Kek is concerned, one can show that it is of the form  e  K0 + (1 + k 2 )Mse 2ikMse 0 −2ikMse K0e + (1 + k 2 )Mse 0  , Kek =  (C.8) 0 0 Kke and its implementation follows the lines of the scalar case detailed above.

Appendix D A multilevel elliptic solver based upon an overlapping Schwarz method The pressure increment at each timestep is computed by inverting ∆t E= DM−1 DT . 3/2

(D.1)

This matrix is symmetric positive definite, and is inverted by means of a preconditioned conjugate gradient algorithm. The preconditioner P−1 is an additive Schwarz preconditioner of the form (Fischer, 1997; Fischer et al., 2000) ne X RTe E−1 (D.2) P−1 = RT0 E−1 R + 0 e Re . 0 e=1

It is the sum of a global coarse grid operator (subscript 0) and local subdomains operators(subscript e). R0,e and RT0,e denote restriction and extension operators. The method has a natural parallel aspect in that the subdomains problems can be solved independently. It is based upon the same ideas as the more classical multigrid approach. The local Laplacian operators Ee are defined over overlapping regions centered on each spectral element and aim at removing the high-frequency components of the residual field. The coarse grid operator, E0 , is constructed as the linear finite element Laplacian derived from triangles which vertices are coincident with the spectral elements vertices. It aims at removing the large scale component of the residual field, hence its fundamental global character, which theoretically makes the iteration count independent of n e (e.g. Smith et al., 1996). Both local and coarse problems are small enough so that the Ee and E0 are factorized and inverted using standard linear algebra libraries. The efficiency of the preconditioner is illustrated on figure D.1, for a typical calculation of the pressure increment. With respect to a non-preconditionned case, the iteration count is decreased roughly by a factor of ten. The coarse grid solver contributes in itself to a decrease by a factor of two, and has a very modest computational cost. 118

OVERLAPPING S CHWARZ PRECONDITIONER

xy

w F¡ £¢¥¤ ¦ F¡ £¢¥¤ ¦ «¬!®=¯A°± ¬²=³µ´l¶´6¬²´·²¸

x y

vu \s t j mr mn q mp o mn i jlk

119

§ ¢ § ¢

©¨

¢&¤ª ¦ ª

¢

ª

xy h xy xy { xy h xy w

z

w{w

| ww{w |

z

w w { }w{w{w } 6

z

w{w

~ ww{w ~

z

w{w

ww{w

Figure D.1: Typical iteration count for the calculation of the pressure increment δp in the case of the Proudman-Stewartson problem. The poor conditioning of the pseudo-Laplacian to invert leads to a very slow convergence when no preconditioner is used in the conjugate gradient algorithm (circles). The local component of the preconditioner removes efficiently the high-frequency content of the residual field (squares). Adding its coarse component removes the large-scale components of the error and leads to an extra factor of two reduction of the iteration counts (triangles).

Bibliography Alpert, B. A. & Rokhlin, V., 1991. A fast algorithm for the evaluation of Legendre expansions, SIAM J. Sci. Comput., 12, 158–179. Arrow, K., Hurwicz, L., & Uzawa, H., 1958. Studies in Nonlinear Programming, Stanford University Press, Stanford. Baumgardner, J. & Frederikson, P., 1985. Icosahedral discretization of the two-sphere, SIAM J. Numer. Anal., 22, 1107–1115. Bernardi, C. & Maday, Y., 1988. A collocation method over staggered grids for the Stokes problem, Int. J. Num. Meth. Fluids, 8, 537–557. Bernardi, C. & Maday, Y., 1992. Approximations spectrales de problèmes aux limites elliptiques, vol. 10 of Mathématiques & Applications, Springer-Verlag, Paris. Bernardi, C., Maday, Y., & Patera, A. T., 1994. A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear partial differential equations and their applications, edited by H. Brezis & J. L. Lions, pp. 13–51, Pitman and Wiley, New york. Bernardi, C., Dauge, M., & Maday, Y., 1999. Spectral Methods for Axisymmetric Domains, vol. 3 of Series in Applied Mathematics, Gauthier-Villars, Paris, Numerical algorithms and tests due to Mejdi Azaïez. Blair Perot, J., 1993. An analysis of the fractional step method, J. Comput. Phys., 108, 51–58. Bloxham, J., 1998. Dynamics of angular momentum of the Earth’s core, Annu. Rev. Earth Planet. Sci., 26, 501–517. Bloxham, J., 2000a. The effect of thermal core-mantle interactions on the palaeomagnetic secular variation, Phil. Trans. R. Soc. Lond. A, 358, 1171–1179. Bloxham, J., 2000b. Sensitivity of the geomagnetic axial dipole to thermal core-mantle interactions, Nature, 405, 63–65. Boyd, J. P., 2001. Chebyshev and Fourier Spectral Methods, Dover, New York, 2nd edn. 120

BIBLIOGRAPHY

121

Buffett, B. A., 2002. Estimates of heat flow in the deep mantle based on the power requirements for the geodynamo, Geophys. Res. Lett., 29(12), 10.1029/2001GL014649. Buffett, B. A., Huppert, H. E., Lister, J., & Woods, A., 1996. On the thermal evolution of the Earth’s core, J. Geophys. Res., 101, 7989–8006. Bunge, H.-P. & Baumgardner, J. R., 1995. Mantle convection modeling on parallel virtual machines, Computer in Physics, 9, 207–215. Bunge, H.-P. & Dalton, M., 2001. Building a high-performance linux cluster for largescale geophysical modeling, in Linux Clusters: The HPC revolution, NCSA Conference Proceedings. Bunge, H.-P. & Tromp, J., 2003. Supercomputing moves to universities and makes possible new ways to organize computational research, Eos, 84(4), 30,33. Busse, F. H., 1968. Steady fluid flow in a precessing spheroidal shell, J. Fluid. Mech., 33, 739–751. Busse, F. H., 1970. Thermal instabilities in rapidly rotating systems, J. Fluid. Mech., 44, 441–460. Busse, F. H., 2000. Homogeneous dynamos in planetary cores and in the laboratory, Annu. Rev. Fluid. Mech., 32, 383–408. Byrd, R. B., Stewart, W. E., & Lightfoot, E. N., 1960. Transport phenomena, John Wiley & Sons, New York. Capdeville, Y., Chaljub, E., Vilotte, J.-P., & Montagner, J.-P., 2002. Coupling the spectral element method with a modal solution for elastic wave propagation in global Earth models, Geophys. J. Int., 152, 34–67. Cardin, P. & Olson, P., 1994. Chaotic thermal convection in a rapidly rotating spherical shell: consequenced for flow in the outer core, Phys. Earth Planet. Inter., 82, 235–259. Chaljub, E. & Valette, B., 2003. Spectral element modeling of three dimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core, submitted to Geophys. J. Int.. Chaljub, E., Capdeville, Y., & Vilotte, J.-P., 2003. Solving elastodynamics in a fluid-solid heterogeneous sphere: A parallel spectral element approximation on non-conforming grids, J. Comput. Phys., 152, 457–491. Chan, K. H., Zhang, K., Zou, J., & Schubert, G., 2001. A non-linear, 3-D spherical α 2 dynamo using a finite element method, Phys. Earth Planet. Inter., 128, 35–50. Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford.

BIBLIOGRAPHY

122

Chorin, A. J., 1968. Numerical solution of the Navier–Stokes equations, Math. Comput., 22, 745–762. Christensen, U., Olson, P., & Glatzmaier, G. A., 1999. Numerical modeling of the geodynamo: A systematic parameter study, Geophys. J. Int., 138, 393–409. Christensen, U., Aubert, J., Cardin, P., Dormy, E., Gibbons, S., Glatzmaier, G., Grote, E., Honkura, Y., Jones, C., Kono, M., Matsushima, M., Sakuraba, A., Takahashi, F., Tilgner, A., Wicht, J., & Zhang, K., 2001. A numerical dynamo benchmark, Phys. Earth Planet. Inter., 128, 25–34. Clune, T., Elliott, J., Miesch, M., Toomre, J., & Glatzmaier, G., 1999. Computational aspects of a code to study rotating turbulent convection in spherical shells, Parallel Comput., 25, 361–380. Couzy, W., 1995. Spectral Element Discretization of the Unsteady Navier-Stokes Equations and its Iterative Solution on Parallel Computers, Ph.D. thesis, École Polytechnique Fédérale de Lausanne. Dahlen, F. A. & Tromp, J., 1998. Theoretical Global Seismology, Princeton University Press, Princeton, New Jersey. de Wijs, G. A., Kress, G., Vo˘cado, L., Dobson, D., Alfè, D., Gillan, M. J., & Price, G. D., 1998. The viscosity of liquid iron at the physical conditions of the Earth’s core, Nature, 392, 805–807. Deville, M. O., Fischer, P. F., & Mund, E. H., 2002. High-Order Methods for Incompressible Fluid Flow, vol. 9 of Cambridge monographs on applied and computational mathematics, Cambridge Univ. Press, Cambridge. Dormy, E., Cardin, P., & Jault, D., 1998. MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field, Earth Planet. Sci. Lett., 160, 15–30. Dormy, E., Valet, J.-P., & Courtillot, V., 2000. Numerical models of the geodynamo and observational constraints, Geochemistry Geophysics Geosystems, 1(62). Dziewonski, A. & Anderson, D., 1981. Preliminary reference Earth model (P.R.E.M.), Phys. Earth. Planet. Int., 25, 297–356. Fischer, P. F., 1997. An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations, J. Comput. Phys., 133, 84–101. Fischer, P. F. & Rønquist, E. M., 1994. Spectral element methods for large scale parallel Navier–Stokes calculations, Comp. Meth. Appl. Mech. & Engng., 116, 69–76.

BIBLIOGRAPHY

123

Fischer, P. F., Miller, N., & Tufo, H., 2000. An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in Parallel Solution of Partial Differential Equations, edited by P. Bjørstad & M. Luskin, pp. 159–181, Springer Verlag. Fornberg, B., 1995. A pseudospectral approach for polar and spherical geometries, SIAM J. Sci. Comput., 16, 1071–1081. Fornberg, B. & Merrill, D., 1997. Comparison of finite difference- and pseudospectral methods for convective flow over a sphere, Geophys. Res. Lett., 24(24), 3245–3248. Fournier, A., Bunge, H.-P., Hollerbach, R., & Vilotte, J.-P., 2003. Application of the spectral element method to the axisymmetric Navier-Stokes equation, Geophys. J. Int., in press. Fournier, A., Bunge, H.-P., Hollerbach, R., & Vilotte, J.-P., 2003. A Fourier-spectral element algorithm for thermal convection in rotating axisymmetric containers, submitted to J. Comput. Phys.. Frigo, M. & Johnson, S. G., 1997. The Fastest Fourier Transform of the West, Tech. Rep. MIT-LCS-TR-728, Massachusetts Institute of Technology Laboratory for Computer Science. Gerritsma, M. & Phillips, T., 2000. Spectral element methods for axisymmetric Stokes problems, J. Comput. Phys., 164, 81–103. Gilman, P. M. & Miller, J., 1986. Nonlinear convection of a compressible fluid in a rotating spherical shell, Astrophys. J. Suppl., 61, 585–608. Giraldo, F. X., 2001. A spectral element shallow water model on spherical geodesic grids, Int. J. Num. Meth. Fluids, 35, 869–901. Giraldo, F. X., Perot, J., & Fischer, P. F., 2003. A spectral element semi-lagrangian (SESL) method for the spherical shallow water equations, J. Comput. Phys., 190, 623–650. Glatzmaier, G. A., 1984. Numerical simulations of stellar convective dynamos I. The model and method, J. Comput. Phys., 55, 461–484. Glatzmaier, G. A., 2002. Geodynamo simulations – How realistic are they?, Annu. Rev. Earth Planet. Sci., 30, 237–257. Glatzmaier, G. A. & Roberts, P. H., 1995. A three-dimensional self-consistent computer simulation of a geomagnetic reversal, Nature, 377, 203–209. Glatzmaier, G. A. & Roberts, P. H., 1996a. An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection, Physica D, 97, 81–94.

BIBLIOGRAPHY

124

Glatzmaier, G. A. & Roberts, P. H., 1996b. Rotation and magnetism of Earth’s inner core, Science, 274, 1887–1891. Greenspan, H. P., 1990. The Theory of Rotating Fluids, Breukelen Press, Brookline, MA, 2nd edn. Gropp, W., Lusk, E., & Skjellum, A., 1999. Using MPI: Portable Parallel Programming with the Message-Passing Interface, MIT Press, Cambridge, MA, 2nd edn. Grote, E., Busse, F. H., & Tilgner, A., 2000. Effects of hyperdiffusivities on dynamo simulations, Geophys. Res. Lett., 27, 2001–2004. Gubbins, D., 2001. The Rayleigh number for convection in the Earth’s core, Phys. Earth Planet. Inter., 128, 3–12. Gubbins, D. & Roberts, P. H., 1987. Magnetohydrodynamics of the Earth’s core, in Geomagnetism, edited by J. A. Jacobs, vol. 2, Academic Press, London. Hof, B., Lucas, P., & Mullin, T., 1999. Flow state multiplicity in convection, Phys. Fluids, 11(10), 2815–2817. Hollerbach, R., 1994. Magnetohydrodynamic Ekman and Stewartson layers in a rotating spherical shell, Proc. R. Soc. Lond. A, 444, 333–346. Hollerbach, R., 1996. On the theory of the geodynamo, Phys. Earth Planet. Inter., 98, 163–185. Hollerbach, R., 2000. A spectral solution of the magneto-convection equations in spherical geometry, Int. J. Num. Meth. Fluids, 32, 773–797. Hollerbach, R., 2003. The range of timescales on which the geodynamo operates, in Earth’s Core: Dynamics, Structure, Rotation, edited by V. Dehant, K. Creager, S.-I. Karato, & S. Zatman, Geodynamics series, pp. 181–192, AGU. Hollerbach, R. & Jones, C., 1993. Influence of the Earth’s inner core on geomagnetic fluctuations and reversals, Nature, 365, 541–543. Hughes, T. J. R., 1987. The Finite Element Method : Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey. Hulot, G., Eymin, C., Langlais, B., Mandea, M., & Olsen, N., 2002. Small-scale structure of the geodynamo inferred from Oersted and Magsat satellite data, Nature, 416, 620– 623. Iskandarani, M., Haidvogel, D., & Levin, J., 2003. A three-dimensional spectral element basin model for the solutions of the hydrostatic primitive equations, J. Comput. Phys., 186, 397–425.

BIBLIOGRAPHY

125

Jakob-Chien, R. & Alpert, B. K., 1997. A fast spherical filter with uniform resolution, J. Comput. Phys., 136, 580–584. Jones, C., Soward, A., & Mussa, A., 2000. The onset of thermal convection in a rapidly rotating sphere, J. Fluid. Mech., 405, 157–179. Kageyama, A., Watanabe, K., & Sato, T., 1993. Simulation study of a magnetohydrodynamic dynamo: Convection in a rotating spherical shell, Phys. Fluids. B, 5(8), 2793– 2805. Karniadakis, G. E., Israeli, M., & Orszag, S. A., 1991. High-order splitting methods for the incompressible Navier–Stokes equations, J. Comput. Phys., 97(2), 414–443. Kerswell, R., 1996. Upper bounds on the energy dissipation in turbulent precession, J. Fluid. Mech., 321, 335–370. Komatitsch, D. & Tromp, J., 1999. Introduction to the spectral element method for threedimensional seismic wave propagation, Geophys. J. Int., 139, 806–822. Komatitsch, D. & Vilotte, J.-P., 1998. The spectral element method: An effective tool to simulate the seismic response of 2D and 3D geological structures, Bull. Seism. Soc. Am., 88, 368–392. Kono, M. & Roberts, P., 2001. Definition of the Rayleigh number for geodynamo simulations, Phys. Earth Planet. Inter., 128, 13–24. Kono, M. & Roberts, P., 2002. Recent geodynamo simulations and observations of the geomagnetic field, Rev. Geophys., 40(4), 1013, doi:10.1029/2000RG000102. Kostelec, P. J., Maslen, D. K., Healey, Jr., D. M., & Rockmore, D. N., 2000. Computational harmonic analysis for tensor fields on the two-sphere, J. Comput. Phys., 162, 514–535. Kuang, W. & Bloxham, J., 1997. An Earth-like numerical dynamo model, Nature, 389, 371–374. Kuang, W. & Bloxham, J., 1999. Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: Weak and strong field dynamo action, J. Comput. Phys., 51, 51–81. Labrosse, S., Poirier, J.-P., & el, J.-L. L. M., 1997. On the cooling of Earth’s core, Phys. Earth Planet. Inter., 99, 1–17. Lamb, H., 1932. Hydrodynamics, Dover, New York, 6th edn. Larmor, J., 1919. Possible rotational origin of magnetic fields of Sun and Earth, Elec. Rev., 85, 412.

BIBLIOGRAPHY

126

Levin, J. G., Iskandarani, M., & Haidvogel, D. B., 2000. A nonconforming spectral element ocean model, Int. J. Num. Meth. Fluids, 34, 495–525. Loper, D. E., 1975. Torque balance and energy budget for the precessionally driven dynamo, Phys. Earth Planet. Inter., 11, 43–60. Lopez, J., Marques, F., & Shen, J., 2002. An efficient spectral-projection method for the Navier–Stokes equations in cylindrical geometries II. Three-dimensional cases, J. Comput. Phys., 176, 384–401. Lorenzani, S. & Tilgner, A., 2001. Fluid instabilities in precessing spheroidal cavities, J. Fluid. Mech., 447, 111–128. Lorenzani, S. & Tilgner, A., 2003. Inertial instabilities of fluid flow in precessing spheroidal shells, J. Fluid. Mech., 492, 363–379. Ma, H., 1993. A spectral element basin model for the shallow water equations, J. Comput. Phys., 109, 133–149. Maday, Y. & Patera, A. T., 1989. Spectral element methods for the incompressible Navier– Stokes equations, in State-of-the-Art Surveys on Computational Mechanics, edited by A. K. Noor & J. T. Oden, pp. 71–143, ASME. Maday, Y., Patera, A. T., & Rønquist, E. M., 1990. An operator-integration-factor splitting method for time-dependent problems: Applications to incompressible fluid flow, J. Sci. Comp., 5(4), 263–292. Maday, Y., Meiron, D., Patera, A. T., & Rønquist, E. M., 1993. Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations, SIAM J. Sci. Comput., 14(2), 310–337. Malkus, W., 1968. Precession of the Earth as the cause of geomagnetism, Science, 160, 259–264. Malkus, W., 1994. Energy sources for planetary dynamos, in Lectures on Solar and Planetary Dynamos, edited by M. Proctor & A. Gilbert, pp. 161–179, Cambridge Univ. Press, Cambridge. Mason, R. & Kerswell, R., 2002. Chaotic dynamics in a strained rotating flow: a precessing plane fluid layers, J. Fluid. Mech., 471, 71–106. Matsui, H. & Okuda, H., 2002. Thermal convection analysis in a rotating shell by a parallel finite-element method – development of a thermal-hydraulic subsystem of GeoFEM, Concurrency and computation: Practice and Experience, 14, 465–481, doi:19.1002/cpe.625. Merilees, P., 1973. The pseudospectral approximation applied to shallow water equations on a sphere, Atmosphere, 11, 13–20.

BIBLIOGRAPHY

127

Merrill, R., McElhinny, M., & McFadden, P., 1996. The magnetic field of the Earth, Academic Press, New York. Müller, G., Neumann, G., & Weber, W., 1984. Natural convection in vertical Bridgman configurations, J. Cryst. Growth, 70, 78–93. Neumann, G., 1990. Three-dimensional numerical simulation of buoyancy-driven convection in vertical cylinders heated from below, J. Fluid. Mech., 214, 559–578. Noir, J., Jault, D., & Cardin, P., 2001. Numerical study of the motions within a slowly precessing sphere at low Ekman number, J. Fluid. Mech., 437, 283–299. Noir, J., Cardin, P., Jault, D., & Masson, J.-P., 2003. Experimental evidence of nonlinear resonance effects between retrograde precession and the tilt-over mode within a spheroid, Geophys. J. Int., 154, 407–416. Orszag, S. A., 1971. On the elimination of aliasing in finite difference schemes by filtering high-wavenumber components, J. Atmos. Sci., 28, 1074. Orszag, S. A., 1974. Fourier series on spheres, Mon. Weather Rev., 102, 56–75. Orszag, S. A., 1980. Spectral methods for problems in complex geometries, J. Comput. Phys., 37, 70–92. Patera, A. T., 1984. A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys., 54, 468–488. Poincaré, H., 1910. Sur la précession des corps déformables, Bull. astr., 27, 321–356. Poirier, J.-P., 1988. Transport properties of liquid metals and viscosity of the Earth’s core, Geophys. J. Roy. Astron. Soc., 92, 99–105. Proudman, I., 1956. The almost-rigid rotation of viscous fluid between concentric spheres, J. Fluid. Mech., 1, 505–516. Ran˘cić, M., Purser, R. J., & Mesinger, F., 1996. A global shallow-water model using expanded spherical cube: gnomonic versus conformal coordinates, Quart. J. R. Met. Soc., 122, 959–982. Reddy, J. N., 1993. An Introduction to the Finite Element Method, Engineering Mechanics, McGraw-Hill, New York. Roberts, P., Jones, C., & Calderwood, A., 2003. Energy fluxes and Ohmic dissipation in the Earth’s core, in Earth’s core and lower mantle, edited by C. Jones, A. Soward, & K. Zhang, pp. 130–152, Taylor and Francis. Roberts, P. H., 1978. Magnetoconvection in a rapidly rotating fluid, in Rotating Fluid in Geophysics, edited by P. H. Roberts & A. M. Soward, pp. 421–435, Academic Press.

BIBLIOGRAPHY

128

Rochester, M., Jacobs, J., Smylie, D., & Chong, K., 1975. Can precession power the geomagnetic dynamo?, Geophys. J. R. Astr. Soc., 43, 661–678. Ronchi, C., Iacono, R., & Paolucci, P. S., 1996. The "cubed sphere" : A new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys., 124, 93–114. Rønquist, E. M., 1988. Optimal Spectral Element Methods for the Unsteady ThreeDimensional Incompressible Navier–Stokes Equations, Ph.D. thesis, Massachusetts Institute of Technology. Rosenblat, S., 1982. Thermal convection in a vertical circular cylinder, J. Fluid. Mech., 122, 395–410. Sadourny, R., 1972. Conservative finite-difference approximations of the primitive equation on quasi-uniform spherical grids, Mon. Weather Rev., 100, 136–144. Sadourny, R., Arakawa, A., & Mintz, Y., 1968. Integration of the non-divergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere, Mon. Weather Rev., 96, 351–356. Smith, B., Bjørstad, P., & Gropp, W., 1996. Domain Decomposition, Cambridge Univ. Press. Spotz, W. F. & Swarztrauber, P. N., 2001. A performance comparison of associated Legendre projections, J. Comput. Phys., 168, 339–355. Spotz, W. F., Taylor, M. A., & Swarztrauber, P. N., 1998. Fast shallow water equations solvers in latitude-longitude coordinates, J. Comput. Phys., 145, 432–444. St. Pierre, M. G., 1993. The strong field branch of the Childress–Soward dynamo, in Solar and Planetary Dynamos, edited by M. R. E. Proctor, P. C. Matthews, & A. M. Rucklidge, pp. 295–302, Cambridge Univ. Press. Stewartson, K., 1957. On almost rigid rotations, J. Fluid. Mech., 3, 299–303. Stewartson, K., 1966. On almost rigid rotations part II, J. Fluid. Mech., 26, 131–144. Stewartson, K. & Roberts, P., 1963. On the motion of a liquid in a spheroidal cavity of a precessing rigid body, J. Fluid. Mech., 17, 1–20. Swartztrauber, P. N., 1993. The vector harmonic transform method for solving partial differential equations in spherical geometry, Mon. Weather Rev., 121, 3415–3437. Swarztrauber, P. N. & Spotz, W. F., 2000. Generalized discrete spherical harmonic transforms, J. Comput. Phys., 159, 213–230.

BIBLIOGRAPHY

129

Taylor, M., Tribbia, J., & Iskandarani, M., 1997. The spectral element method for the shallow water equations on the sphere, J. Comput. Phys., 130, 92–108. Tilgner, A., 1999. Spectral methods for the simulation of incompressible flows in spherical shells, Int. J. Num. Meth. Fluids, 30, 713–724. Tilgner, A. & Busse, F., 1997. Finite-amplitude convection in rotating spherical shells, J. Fluid. Mech., 332, 359–376. Tilgner, A. & Busse, F., 2001. Fluid flows in precessing spherical shells, J. Fluid. Mech., 426, 387–396. Tomboulides, A. G., 1993. Direct and Large-Eddy Simulation of Wake Flows: Flow past a sphere, Ph.D. thesis, Princeton University. Touihri, R., Ben Hadid, H., & Henry, D., 1999. On the onset of convective instabilities in cylindrical cavities heated from below. I. Pure thermal case, Phys. Fluids, 11(8), 2078– 2088. van der Vorst, H., 1992. BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statis. Comput., 13, 631–644. Vanyo, J. & Likins, P., 1971. Measurement of energy dissipation in a liquid-filled, precessing, spherical cavity, J. Appl. Mech., 38, 674–682. Wathen, A. J., 1989. An analysis of some element-by-element techniques., Comp. Meth. Appl. Mech. & Engng., 74, 271–287. Williams, G. P., 1969. Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow, J. Fluid. Mech., 37, 727–750. Yano, J.-I., 1992. Asymptotic theory of thermal convection in rapidly rotating systems, J. Fluid. Mech., 243, 103–131. Yarvin, N. & Rokhlin, V., 1998. A generalized one-dimensional fast multipole method with application to filtering of spherical harmonics, J. Comput. Phys., 147, 594–609. Yoder, C. F., 1995. Astronomic and geodetic properties of Earth and the solar system, in Global Earth physics : a handbook of physical constants, edited by T. J. Arhens, vol. 1 of AGU reference shelf, pp. 1–31, American Geophysical Union. Zhang, K. & Gubbins, D., 2000. Scale disparities and magnetohydrodynamics in the Earth’s core, Phil. Trans. R. Soc. Lond. A, 358, 899–920. Zhang, K. & Jones, C. A., 1997. The effect of hyperviscosity on geodynamo models, Geophys. Res. Lett., 24, 2869–2872.