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David Cébron
Beyond the thermo-solutal dynamo paradigm
Puzzling observations: planetary fields & tidal flows
David Cébron
ISTerre, CNRS, Univ. Grenoble Alpes
Les Houches, October 2018
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David Cébron
Geomagnetism
Geomagnetism, convection & dynamo theory
o Lecture 1. Geomagnetic field: sources & observations
o Lecture 2. Dynamo theory, experiments and simulations
o Lecture 3/4. Earth’s core: rotating fluids and convection
Beyond the thermo-solutal dynamo paradigm
o Lecture 5: Puzzling observations: planetary fields & tidal flows
o Lecture 6: Magnetic fields driven by tides/precession in planets/stars
2
Lectures – Magnetic field of the Earth and its origin
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David Cébron
Outline
1. Questioning the thermo-solutal dynamo paradigm for the Earth
2. Questioning the thermo-solutal dynamo paradigm for the Moon
3. Questioning the thermo-solutal dynamo paradigm beyond the Moon
4. A way out via waves and mechanical forcings?
5. Mechanically driven parametric instabilities
3
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David Cébron
The thermo-solutal convection paradigm
o Currently for the Earth : convection
3
Crystallization of
the inner core
Secular cooling
Core radiogenic
heating
Superadiabatic
gradient CONVECTION
GEODYNAMO
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David Cébron
The thermo-solutal convection paradigm
o Currently: convection
4
Schaeffer,
HDR, 2015
Temperature in the
equatorial plane
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David Cébron
The thermo-solutal convection paradigm
o Today, convection probably powers the geodynamo …
o But possible stratified layer! Debated presence but indirect
clues :
- Geodesy : nutations derived dissipation,
- Geomagnetic: MAC waves detection claimed (Buffett’s works)
5
CMB
ICB
Convective
flows Stably stratified
fluid layer?
(MAC: Magnetic-Archimedes-Coriolis)
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David Cébron
The thermo-solutal convection paradigm
6
CMB
ICB
No convection =>
no dynamo (?)
Reduced convection zone
=> Strongly affects geodynamo
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David Cébron
The thermo-solutal convection paradigm
6
CMB
ICB Reduced convection zone
=> Strongly affects geodynamo
AND revised (debated) larger estimates of the thermal conductivity
question the usual convection model (marginally enough power)
Nimmo, ToG, 2015
Difficulties for the current geodynamo… and in the past?
No convection =>
no dynamo (?)
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David Cébron
The thermo-solutal convection paradigm
7
Time
Present - 4 Ga - 3 Ga - 2 Ga - 1 Ga
Geodynamo
o Rourke et al. (2017): ”how to power convection in the core and thus
a dynamo for the vast majority of the Earth’s history remains one
of the most pressing puzzles in geophysics”
Dynamo mechanism?
(not enough cooling + radiogenic)
Inner core Thermally stratified core? Nimmo,
ToG, 2015
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David Cébron
Precipitation in the core: a speculative scenario
8
O’Rourke & Stevenson (2016)
• SiO2 can also play (better?) this role (Hirose et al., Nature, 2017)
• Speculative scenario requiring impacts
• Other unusual model exist : e.g. with a stratified BMO (Laneuville+18)
A try to save this beloved thermo-solutal model
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David Cébron
A little known double-diffusion in the Earth’s core
9
o Early Earth
requires precipitation
o Double diffusion?
Little known…
but allow convection in a
thermally stratified core!
o How does-it work?
Fingering instability, well
known in oceanography!
C,T Fast diffusion of T wrt. C
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David Cébron
Fingering convection in a rotating sphere
10
m
Ek=
10
-4
Ek=
10
-5
Ek=
10
-6
Double-diffusive convection onset
Large scale,
small Ra
Spherical double-diffusive simulations with RaT <0 can be crucial for the Earth
But only Net et al. (2012) & Bouffard et al. (2017) !
=> On-going work
Earth
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David Cébron
Fingering convection in a rotating sphere
10
Emergence of
strong zonal flows Pr=0.3, Sc=3, Ek=10-5, RaC=1010 , RaT=-RaC/3
m
Ek=
10
-4
Ek=
10
-5
Ek=
10
-6
Large scale,
small Ra
Earth
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David Cébron
Outline
1. Questioning the thermo-solutal dynamo paradigm for the Earth
2. Questioning the thermo-solutal dynamo paradigm for the Moon
3. Questioning the thermo-solutal dynamo paradigm beyond the Moon
4. A way out via waves and mechanical forcings?
5. Mechanically driven parametric instabilities
Beyond the Earth, the Moon is the only body for which we have data
constraining the planetary magnetic field evolution over a long timescale
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David Cébron
A lunar magnetic field
11
Current Earth
Aubert, IPGP
• Global magnetic field
• Dynamical (internal) origin
Introduction
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David Cébron
A lunar magnetic field
11
Current Earth
Aubert, IPGP
• Global magnetic field
• Dynamical (internal) origin
Current Moon
Spatial data, Wieczorek, IPGP
• Local magnetic field
• Static (crustal) origin
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David Cébron
A lunar magnetic field
11
Current Earth
Aubert, IPGP
• Global magnetic field
• Dynamical (internal) origin
Introduction
Current Moon
Spatial data, Wieczorek, IPGP
• Local magnetic field
• Static (crustal) origin
Where does the lunar magnetic field, recorded in the lunar
rocks, come from?
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David Cébron
Paleomagnetic analyses of lunar rocks
12
Fuller & Cisowski, Geomagnetism (1987) Wieczorek et al., Rev. Miner. Geochem. (2006)
Su
rface p
ale
oin
ten
sit
y B
su
rf (
mT
)
Apollo 11
Data from
Apollo rocks
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David Cébron
Paleomagnetic analyses of lunar rocks
12
Su
rface p
ale
oin
ten
sit
y B
su
rf (
mT
)
External origin? internal?
Induction? Dynamo?
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David Cébron
Paleomagnetic analyses of lunar rocks
12
Su
rface p
ale
oin
ten
sit
y B
su
rf (
mT
)
External origin? internal?
Induction? Dynamo?
Cisowski et al., JGR (1983)
Even with a Moon orbiting on the Earth’s surface, the Earth’s
field is 2 times too weak (actually, Roche limit → 1mT)
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David Cébron
External field due to meteoritic impacts?
13
Giant impacts during LHB (Sea of Serenity: 700km diameter)
Moon
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David Cébron
External field due to meteoritic impacts?
13
Moon
Hot plasma
cloud
Hood and Artemieva 2008
Moon Moon
Convergence to
antipode in ~ 1h
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David Cébron
External field due to meteoritic impacts?
13
Moon
Introduction
Hot plasma
cloud
Hood and Artemieva 2008
Lune Moon
Convergence to
antipode in ~ 1h
Antipodal convergence of
the cloud => amplification
of the ambiant field?
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David Cébron
External field due to meteoritic impacts?
13
Moon
Hot plasma
cloud
Hood and Artemieva 2008
Lune Moon
Convergence to
antipode in ~ 1h
Field recorded by shocks (SRM) due to ejecta
impacts (seismic waves too weak/fast)
Yes! Correlation
Antipodal convergence of
the cloud => amplification
of the ambiant field?
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David Cébron
2nde generation of paleomagnetic data
1. Field of external origin due to shock (SRM)? Duration of B ≤ 1d
14
o Non shocked rock, cooling slowly (time > 102 – 103 j)
o Modern paleomagnetic analysis
Introduction
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David Cébron
2nde generation of paleomagnetic data
1. Field of external origin due to shock (SRM)? Duration of B ≤ 1d
14
o Non shocked rock, cooling slowly (time > 102 – 103 j)
o Modern paleomagnetic analysis
Introduction
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David Cébron
2nde generation of paleomagnetic data
1. Field of external origin due to shock (SRM)? Duration of B ≤ 1d
14
o Non shocked rock, cooling slowly (time > 102 – 103 j)
o Modern paleomagnetic analysis
Introduction
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David Cébron
2nde generation of paleomagnetic data
1. Field of external origin (SRM)? Explain certain data
2. Convection driven internal dynamo?
Ingrédients: (i) Liquid core, (ii) Convection in the core
15
Introduction
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David Cébron
The Moon, a differentiated body
o Laser ranging : dissipation at the CMB
=> Liquid outer core (Williams et al. 2001)
o Seismic detection => liquid outer core Weber et al., Science (2011); Garcia et al., PEPI (2011)
16
The Moon is differentiated… but still
debated liquid core boundaries
Crust
Mantle
Noyau
Introduction
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David Cébron
Lunar fluid core boundary: a debated issue…
17
Modified from Wieczorek et al. 2006 (Amir Khan, personnal communication)
Khan et al. 2004, 2013, de Viries et al. 2010
100km
450km 350-450km
Fe Core
Modified from Wieczorek et al. 2006
Williams et al 2001, Nimo et al. 2012
Fe Core
Melt Zone
Introduction
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David Cébron
Lunar fluid core boundary: a debated issue…
17
Khan et al. 2004, 2013, de Viries et al. 2010
100km
450km 350-450km
Fe Core
Modified from Wieczorek et al. 2006
Williams et al 2001, Nimo et al. 2012
Fe Core
Melt Zone
Introduction
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David Cébron
Lunar fluid core boundary: a debated issue…
17
Khan et al. 2004, 2013, de Viries et al. 2010
100km
450km 350-450km
Fe Core
Modified from Wieczorek et al. 2006
Williams et al 2001, Nimo et al. 2012
Melt Zone
Melt zone? Solid inner core?
In any case, an outer liquid core exists!
Introduction
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David Cébron
2nde generation of paleomagnetic data
1. Field of external origin (SRM)? Explain certain data
2. Convection driven internal dynamo?
Ingrédients: (i) Liquid core, (ii) Convection in the core
Introduction
![Page 34: L’instabilité elliptique dans les enveloppes fluides des ... · o Lecture 1. Geomagnetic field: sources & observations o Lecture 2. Dynamo theory, experiments and simulations o](https://reader035.vdocument.in/reader035/viewer/2022062604/5fb3eef45cd4791bdc7a666b/html5/thumbnails/34.jpg)
David Cébron
Paleomagnetic analyses of lunar rocks
Fuller & Cisowski, Geomagnetism (1987) Wieczorek et al., Rev. Miner. Geochem. (2006)
Su
rface p
ale
oin
ten
sit
y B
su
rf (
mT
)
=> BCMB is 10-20 times larger than BCMB of the current
Earth! … For a lunar core volume 103 times smaller?!
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David Cébron
Thermal convection
Thermal convection driven lunar dynamo
18
Dynamo too short and too weak!
Konrad & Spohn, Adv. Space Res. (1997), Wieczorek et al., Rev. Miner. Geochem. (2006)
Introduction
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David Cébron
Thermal convection
Thermal convection driven lunar dynamo
18
Dynamo too weak!
Konrad & Spohn, Adv. Space Res. (1997), Wieczorek et al., Rev. Miner. Geochem. (2006)
Introduction
Inner core
crystallization
Evans et al. (2017)
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David Cébron
Thermal convection
Thermal convection driven lunar dynamo
19
Dynamo too weak!
Konrad & Spohn, Adv. Space Res. (1997), Wieczorek et al., Rev. Miner. Geochem. (2006)
Introduction
Core
crystallization
Evans et al. (2017)
Our MagLune ANR Project
• Accuracy of data? Error bars?
• Alternative dynamo models?
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David Cébron
Our improved data with errorbars
19
0
20
40
60
80
100
120
140
00,511,522,533,544,5
Our MagLune ANR Project
• Improved data with
modern techniques…
• … providing error bars!
Surf
ace
pale
oin
tensity (
mT
)
Time before Present (Billions of years ago)
Courtesy of C. Lepaulard &
J. Gattacceca (CEREGE)
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David Cébron
Outline
1. Questioning the thermo-solutal dynamo paradigm for the Earth
2. Questioning the thermo-solutal dynamo paradigm for the Moon
3. Questioning the thermo-solutal dynamo paradigm beyond the Moon
4. A way out via waves and mechanical forcings?
5. Mechanically driven parametric instabilities
![Page 40: L’instabilité elliptique dans les enveloppes fluides des ... · o Lecture 1. Geomagnetic field: sources & observations o Lecture 2. Dynamo theory, experiments and simulations o](https://reader035.vdocument.in/reader035/viewer/2022062604/5fb3eef45cd4791bdc7a666b/html5/thumbnails/40.jpg)
David Cébron
Usual thermo-solutal model
20
o Current Earth : usual model challenged
o Early Earth : requires unusual models (e.g. precipitation)
o Early Moon : requires unusual models
o And Venus? Earth size => convection model gives a dynamo
but no current dynamo!
- low heat flow ? (O’Rourke & Korenaga 2015)
- solidified core? (Dumoulin+17)
- stratified core? (Jacobson+17)
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David Cébron
Usual thermo-solutal model
21
o Current Earth : usual model challenged
o Early Earth : requires unusual models (e.g. precipitation)
o Early Moon : requires unusual models
o Venus : surprisingly not dynamo. Stratified core?
o Mercury? Dynamo! But ‘enigmatically weak’ (Christensen 2010)
- thin fluid shell & strong (hidden) toroidal field? (Stanley+05)
- deep convecting sublayer covered
by a thick stagnant fluid region? (Christensen 2006)
- stable layer and laterally variable heat flux? (Tian+15)
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David Cébron
Usual thermo-solutal model
22
o Current Earth : usual model challenged
o Early Earth : requires unusual model (e.g. precipitation)
o Early Moon : requires unusual model
o Venus : surprisingly not dynamo. Stratified core?
o Mercury : requires unusual models
o And Ganymede? Dynamo!
‘The origin of the magnetic field at the Jovian Moon Ganymede is
one of the unsolved problems in the outer solar system’
(Ruckriemen+18)
- Fe snow driven convection? (Bland+08)
- Remelting of Fe-crystals? (Christensen15)
- …
NASA/ESA
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David Cébron
Usual thermo-solutal model
o Current Earth : usual model challenged
o Early Earth : requires unusual model (e.g. precipitation)
o Early Moon : requires unusual model
o Venus : surprisingly not dynamo. Stratified core?
o Mercury : requires unusual models
o Ganymede: requires unusual models
And beyond the solar system?! Stars…
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David Cébron
Stellar context
23
= stably stratified layer
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David Cébron
Stellar context
23
= stably stratified layer
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David Cébron
Stellar context
23
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David Cébron
Magnetism of intermediate mass, radiative stars
24
10% with surface magnetic field B0
Strong fields B0 (e.g. Ap stars)
o Typical strength: 102 - 104 G
o Often static fields
o B0 from a fossil origin
Ultra-weak fields B0 (e.g. Vega)
o Typical strength: 0.1 - 1 G
o Often dynamical fields
o B0 generated by dynamos?
Lignières et
al. (2008)
Dynamo in stratified fluids?!
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David Cébron
Usual thermo-solutal model
25
o Current Earth : usual model challenged
o Early Earth : requires unusual model (e.g. precipitation)
o Early Moon : requires unusual model
o Venus : surprisingly not dynamo. Stratified core?
o Mercury : requires unusual models
o Ganymede: requires unusual models
o Radiative stars: requires unusual models
Key point
Dynamo field convection
X flow
How can we drive flows
in planetary cores?
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David Cébron
Outline
1. Questioning the thermo-solutal dynamo paradigm for the Earth
2. Questioning the thermo-solutal dynamo paradigm for the Moon
3. Questioning the thermo-solutal dynamo paradigm beyond the Moon
4. A way out via waves and mechanical forcings?
5. Mechanically driven parametric instabilities
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David Cébron
Alternative mechanisms
o Rotation: huge reservoir of energy!
o E.g. on Earth: - rotational energy >1029 J
- Current geodynamo requires 0.1-2 TW
=> Enough energy for geodynamo during 3-63 Gy !
26
Magnetic energy
(dynamo effect)
Thermal energy
(convection) + Rotational
energy
Glatzmaier &
Roberts 1995
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David Cébron
Alternative mechanisms
o Rotation: huge reservoir of energy!
o E.g. on Earth: - rotational energy >1029 J
- Current geodynamo requires 0.1-2 TW
=> Enough energy for geodynamo during 3-63 Gy !
OK, but how?
(solid body rotations are not dynamo capable)
Using inertial waves and mechanical (or harmonic) forcings!
26
Magnetic energy
(dynamo effect)
Thermal energy
(convection) + Rotational
energy
Glatzmaier &
Roberts 1995
Tides? Precession Tilgner 2005
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David Cébron
Inertial waves?
27
o Core: a rotating incompressible fluid layer => inertial waves
Coriolis force : a restoring force
V = v(r,z) e i(wt+mq)
w : frequency
m : azimuthal wavenumber
* 2v
v v p v vt
Inertial waves : eigenmodes of rotating flows
V
spin x V
Inertial waves frequency: [-2 ; 2]
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David Cébron
Mechanical forcings?
Planet
B
Liquid core = an incompressible
fluid rotating in a moving ellipsoid
o Moving? Tides, libration, precession, etc.
o Rotating? Inertial modes, very small viscosity
o Ellipsoid? Triaxial (a≠b≠c), spheroid (a=b), sphere
15 10
210 10 1
w
ER
w
28
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David Cébron
Librations
Librations in longitude Librations in latitude
Simulated views of the
Moon over one month
29
Librations = periodic perturbations of the inertia axes
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David Cébron
Precession?
30
Precession = periodic perturbations of the rotation direction
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David Cébron
Tides?
Deformation
of all layers
a
def
b
Tides Elliptical
streamlines
31
Tides = periodic perturbations of the shape
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David Cébron
Orders of magnitude
Earth
precession
Earth
tides
Moon libration
Negligible forcings?
No! Because of the presence
of waves in the fluid…
32
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David Cébron
Orders of magnitude
Earth
precession
Earth
tides
Moon libration
Waves + forcings
Direct resonance
Parametric resonance
32
Very small forcing
1st order consequences
Negligible forcings?
No! Because of the presence
of waves in the fluid…
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David Cébron
Direct resonance of a wave
Temporal matching =well
chosen forcing frequency
2
t
vv v p E v v f
t
f(t) : periodic forcing
A direct resonance requires a close spatial and temporal matching
between the forcing and a free eigenmode of the system
Spatial matching = forcing not
orthogonal to the eigenmode Q
* 0V
f Q d
This difficult condition require viscosity
to be matched for librations and tides…
33
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David Cébron
Direct resonance of IW by precession
34
1 ez
Po
d2
d
uu u p u r
t t
Inviscid fluid in the mantle frame
f
1. Spatial resonance condition: eigenmode linear in space coordinates
=> Yes! Only one, called the spin-over mode, of frequency fso
Noir 2
003 Spin-over mode resonance: the fluid rotation
axis is tilted from the container rotation axis!
Basic flow of precession…
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David Cébron
Direct resonance of IW by precession
34
1 ez
Po
d2
d
uu u p u r
t t
Inviscid fluid in the mantle frame
f
1. Spatial resonance condition: eigenmode linear in space coordinates
=> Yes! Only one, called the spin-over mode, of frequency fso
2. Temporal resonance condition
Coriolis frequency
Forcing frequency
~ 1 f
~ 1
soPo
11
so
Pof
2 resonances for
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David Cébron
Direct resonance of IW by precession
35
o Detailed theory for precession resonance? - Inviscid: Poincaré (1910)
- Viscous spheroid: Busse (1968)
o Experimental validation: Noir et al. (2003)
o But only one resonance?!
Fluid rotation
vector direction
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David Cébron
Direct resonance of IW by precession
35
o Detailed theory for precession resonance? - Inviscid: Poincaré (1910)
- Viscous spheroid: Busse (1968)
o Experimental validation: Noir et al. (2003)
o But only one resonance?! Using Noir & Cebron (2013),
we obtain 2 resonances in
a precessing ellipsoid
Vid
al, 2
018
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David Cébron
Direct resonance of IW by libration
No direct resonance in inviscid fluids (Zhang et al. 2013)
But viscous effects allow to verify the spatial condition…
= spin [1 + e sin(wlib t)]
36
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David Cébron
Direct resonance of IW by libration
(1,1) (2,1) (3,1) (4,1)
Sauret, Cebron, Morize, Le Bars, J. Fluid Mech. 2010
Velo
city
in th
e
axia
l dire
ctio
n
pre
ssu
re d
iffe
ren
ce
ce
nte
r / p
ole
spin/wlib
Inertial waves are the
eigenmodes of rotating flows
Aldridge & Toomre (JFM 1969)
= spin [1 + e sin(wlib t)]
36
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David Cébron
Direct resonance of IW by libration
Sauret, Cebron, Le Bars, Le Dizès. Phys. Fluids 2012
= spin [1 + e sin(wlib t)]
37
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David Cébron
Direct resonance of IW by tides
38
Ek=10-3,
10-4, 10-5,
10-6,
Ogilvie,
2014
No direct resonance
in full ellipsoids!
(here: different CMB
and ICB ellipticities)
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David Cébron
Dynamo of direct resonance of waves?
o Yes, it works!
E.g. dynamo of the precession-driven resonant spin-over mode!
(Tilgner 2005)
o But these dynamos are irrelevant for planets…
They disappear in the limit of vanishing viscosity!
(e.g. Herreman & Lesaffre 2011)
o What about combination of waves?
Tubulence via parametric resonances!
39
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David Cébron
Outline
1. Questioning the thermo-solutal dynamo paradigm for the Earth
2. Questioning the thermo-solutal dynamo paradigm for the Moon
3. Questioning the thermo-solutal dynamo paradigm beyond the Moon
4. A way out via waves and mechanical forcings?
5. Mechanically driven parametric instabilities
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David Cébron
An example: longitudinal librations
40
def = [1e sin wt ] Oscillation of a
rigid ellipsoid
Base flow
(mantle frame)
Oscillating
rotation
Elongational
flow
Perturbation of the rotation
- azimuthal wavenumber 2
- frequency w
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David Cébron
Libration Driven Elliptical instability (LDEI)
40
def = [1e sin wt ] Oscillation of a
rigid ellipsoid
Base flow
(mantle frame)
LDEI: 3D
destabilization
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David Cébron
Parametric resonance?
A famous example : the Botafumeiro
of Santiago de Compostela Cathedral
Parametric pendulum
The Mathieu equation
=> Instability for certain w
41
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David Cébron
Introduction – The tidal instability
Inertial instability in rotating solids
An initial rotation around the middle
inertia axis is unstable!
42
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David Cébron
Introduction – The tidal instability
Inertial instability in rotating solids
An initial rotation around the middle
inertia axis is unstable!
I1 d1/dt + (I3-I2) 23 = 0 I2 d2/dt + (I1-I3) 31 = 0 I3 d3/dt + (I2-I1) 12 = 0
42
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David Cébron
Introduction – The tidal instability
Toy experiment representing a tidally deformed spinning body
spin
43
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David Cébron
2 dimensionless numbers: b and E = /FR2 = Re-1
Introduction – The tidal instability
Toy experiment representing a tidally deformed spinning body
b
a
spin
43
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David Cébron
Introduction – The tidal instability
Non linear viscous modelling (Lacaze et al. , 2004)
‘s effects calculated from boundary layers analyses
(Greenspan, Kerswell)
2 dimensionless numbers:
b & E = /FR2 = Re-1
44
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David Cébron
Introduction – The tidal instability
Non linear viscous modelling
‘s effects calculated from boundary layers analyses
(Greenspan, Kerswell)
/ 0.5 2.62 /E b b
Exact theoretical expression!
44
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David Cébron
An experimental evidence
The tidal instability, a kind of inertial instability
Révolutio
n orbitale
Révolutio
n orbitale
Roll 1
Roll 2
Orbital
revolution
Fluid domain:
triaxial ellipsoid
Orbital
revolution
Vortex interactions,
shears, etc.
Local origin
Very small deformation 1st order consequences
45
IRPHE (Lacaze, Le Gal, Le Bars)
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David Cébron
The elliptical instability
o Tidal parametric instability = an elliptical streamline is unstable!
o Elliptical instability can thus be seen as a - a local instability (elliptical streamlines)
- a global instability (parametric resonance of inertial modes)
o Recently identified (70s) but actually present in old literature…
1d
1d
0
yx
U xt
b
b
2 z
ue u u U U u p E u
t
Linearized Navier-Stokes
is unstable!
46
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David Cébron
Mechanical forcings & parametric instabilities
Precession
(m=1)
Tides
(m=2)
Libration
(m=0, m=2)
Direct resonance of an inertial wave Parametric resonance of inertial waves
e e +miq
Inertial wave a Inertial wave b
Mechanical forcing (m=1,2)
e e +miq Mechanical forcing
e e +miq
Inertial wave a
Mechanical forcing
(3,1)
+ Translated inner
core (m=1)?
+ Topography? ...
47
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David Cébron
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David Cébron
Additional slides
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David Cébron
A planet? A star?
o At geological times : a self-gravitating incompressible fluid in rotation
o Riemann (1861) : What if an internal fluid vorticity is present?
Newton (1687)
Slightly deformed sphere Maclaurin (1742)
Spheroid
Jacobi (1834)
Triaxial ellipsoid
Ω Credit: Jos Leys/Etienne Ghys
44
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David Cébron
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
b / a
c / a
Ellipsoidal figures of equilibrium
o Jacobi, 1834
Jacobi ellipsoids
Axes ratio fixed by the rotation
Maclaurin
spheroids
Unstable
Maclaurin
spheroids
def
45
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David Cébron
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
b / a
c / a
Riemann ellipsoids
o Jacobi, 1834
o Riemann, 1861
prograde
retrograde
Even with an internal vorticity, not all axes
ratios are possible
def
45
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David Cébron
Stability of Riemann ellipsoids
o Jacobi, 1834
o Riemann, 1861
o Chandrasekhar, 1965
Unstable ellipsoids for linear pertubations
(i.e. infinitesimal rotations around axes)
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![Page 88: L’instabilité elliptique dans les enveloppes fluides des ... · o Lecture 1. Geomagnetic field: sources & observations o Lecture 2. Dynamo theory, experiments and simulations o](https://reader035.vdocument.in/reader035/viewer/2022062604/5fb3eef45cd4791bdc7a666b/html5/thumbnails/88.jpg)
David Cébron
Stability of Riemann ellipsoids
o Jacobi, 1834
o Riemann, 1861
o Chandrasekhar, 1965
Unstable ellipsoids for linear pertubations
Unstable ellipsoids for quadratic pertubations
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
b / a
c / a
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David Cébron
o Jacobi, 1834
o Riemann, 1861
o Chandrasekhar, 1965
o Lebovitz & Lifschtiz, 1996
Short-wavelength instabilities of Riemann ellipsoids
Very few Riemann ellipsoids can exist…
because of the elliptical instability!
0 0.2 0.4 0.6 0.8 10
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