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

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

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

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

David Cébron

The thermo-solutal convection paradigm

o Currently: convection

4

Schaeffer,

HDR, 2015

Temperature in the

equatorial plane

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)

David Cébron

The thermo-solutal convection paradigm

6

CMB

ICB

No convection =>

no dynamo (?)

Reduced convection zone

=> Strongly affects geodynamo

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 (?)

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

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

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

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

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

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

David Cébron

A lunar magnetic field

11

Current Earth

Aubert, IPGP

• Global magnetic field

• Dynamical (internal) origin

Introduction

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

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?

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

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?

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)

David Cébron

External field due to meteoritic impacts?

13

Giant impacts during LHB (Sea of Serenity: 700km diameter)

Moon

David Cébron

External field due to meteoritic impacts?

13

Moon

Hot plasma

cloud

Hood and Artemieva 2008

Moon Moon

Convergence to

antipode in ~ 1h

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?

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?

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

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

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

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

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

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

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

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

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

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?!

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

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)

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?

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)

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

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)

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)

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

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…

David Cébron

Stellar context

23

= stably stratified layer

David Cébron

Stellar context

23

= stably stratified layer

David Cébron

Stellar context

23

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?!

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?

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

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

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

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]

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

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

David Cébron

Precession?

30

Precession = periodic perturbations of the rotation direction

David Cébron

Tides?

Deformation

of all layers

a

def

b

Tides Elliptical

streamlines

31

Tides = periodic perturbations of the shape

David Cébron

Orders of magnitude

Earth

precession

Earth

tides

Moon libration

Negligible forcings?

No! Because of the presence

of waves in the fluid…

32

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…

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

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…

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

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

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

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

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

David Cébron

Direct resonance of IW by libration

Sauret, Cebron, Le Bars, Le Dizès. Phys. Fluids 2012

= spin [1 + e sin(wlib t)]

37

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)

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

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

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

David Cébron

Libration Driven Elliptical instability (LDEI)

40

def = [1e sin wt ] Oscillation of a

rigid ellipsoid

Base flow

(mantle frame)

LDEI: 3D

destabilization

David Cébron

Parametric resonance?

A famous example : the Botafumeiro

of Santiago de Compostela Cathedral

Parametric pendulum

The Mathieu equation

=> Instability for certain w

41

David Cébron

Introduction – The tidal instability

Inertial instability in rotating solids

An initial rotation around the middle

inertia axis is unstable!

42

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

David Cébron

Introduction – The tidal instability

Toy experiment representing a tidally deformed spinning body

spin

43

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

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

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

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)

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

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

David Cébron

David Cébron

Additional slides

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

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

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

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)

def

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

b / a

c / a

45

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

def

45

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

0.2

0.4

0.6

0.8

1

b / a

c / a

def

45