Normal Modes of Rotating Earth Models

Habilitation a Diriger des RecherchesYves Rogister

October 1st, 2012

Scientific activities (1/2)

• 20 authored or co-authored papers

• Co-supervisor of 2 PhD students

1. Anthony Memin 2007-11Modelisation des variations geodesiques produites par la fonte

de glaciers. Separabilite des effets des deglaciations passee et

actuelle.

2. Yann Ziegler 2012-15Modelisation de la rotation de la Terre et analyse conjointe des

donnees du mouvement du pole et de gravimetrie

Scientific activities (2/2)

• Leader of• Polar gravity program (IPEV 2011-15)• Earth rotation research projects

(GRAM 2011 and INSU-PNP 2012)

• Member of• 2 IAG study groups 2007• SCAR research program and expert group 2011• CNFRA 2012• IAU (Commission 19 Earth rotation) 2012

Motivation

Comments from colleagues

Normal modes? Earth rotation?

• What is it used for?

• It looks interesting, but it’s complicated!

Normal mode = Free oscillation

Example 1

x = A sinωt

ω =

√

k

m

Normal mode = Free oscillation

Example 2

i = i0 sinωt

ω =1√LC

Earth rotation

flattening = 1

300

Variations of Earth rotation (1/3)

(Secular) Variation of the length of the day

Denis et al. 2011

Variations of Earth rotation (2/3)

Variations of the position of the rotation axis

Variations of Earth rotation (3/3)

Polar motion

Phenomenon Causes

Secular polar motion Post-glacial rebound, melting of glaciersDecadal variations Global mass redistribution, core-mantle couplingMarkowitz wobble ∼ 30 yearsChandler wobble Ocean-bottom and atmosphere pressure changesAnnual wobble Seasonal air and water mass redistributionsDiurnal motion Ocean tides

Adapted from Hopfner 2004

Structure of the Earth

Constraints from Earth’s rotation

• Departure from spherical symmetry ⇒ flattening andflattening variation

• Deformability of the Earth ⇒ rheological parameters

• Differential rotation of inner core, outer core and mantle⇒ couplings: electromagnetic, viscous...

Normal modes of a rotating elastic Earth model

Theory : Clairaut coordinates

Seismic modes, rotational modes, core spectrum

Chandler wobble and core spectrum

Normal Modes of Rotating Earth Models

Theory : Clairaut coordinates

Seismic modes, rotational modes, core spectrum

Chandler wobble and core spectrum

Spherical Earth model

Dziewonski & Anderson 1981

Hydrostatic figures of equilibrium of rotating planets

∇p = −ρ∇V − ρΩ0 × (Ω0 × r)

PREM Inverse flattening

300

320

340

360

380

400

420

0 1500 3000 4500 6000

radius (km)

Clairaut coordinates q, χ, ν

r = q + h(q, χ)

θ = χ

ϕ = ν

r , θ, ϕ = spherical polar coordinates

Local equations of motion

ρd2s

dt2= ∇ · δt − ρ∇φ′ + ρ (∇ · s) ∇φ − ρ∇ (s · ∇φ) − 2ρΩ × ds

dt

δt = λ∇ · s I + µ[

∇s + (∇s)T]

φ = φ − 1

2|Ω × r|2

φ (r) = −G

∫

V

ρ

|r − r′| dV ′

∇2φ′ = −4πG ∇ · (ρs)

Local equations of motion in Clairaut coordinates

r = q + h(q, χ)

θ = χ

ϕ = ν

Method

1. Spherical harmonics expansion of displacement field s

2. Equations developed up to 2nd order in h

Normal Modes of Rotating Earth Models

Theory : Clairaut coordinates

Seismic modes, rotational modes, core spectrum

Chandler wobble and core spectrum

Normal modes of a rotating elastic Earth model (1/2)

3 families

1. Seismic modes

2. Rotational modes

3. Spectrum of the liquid outer core

Normal modes of a rotating elastic Earth model (2/2)

Effect of rotation on seismic modes

Example 1 viewed in a rotating frame

X = a1 sin(ω + Ω)t + a2 sin(ω − Ω)t

Y = a1 cos(ω + Ω)t − a2 cos(ω − Ω)t

ω =

√

k

m

Splitting of the seismic mode 0S2

Eigenfunctions Eigenfrequencies (mHz)

0.295

0.300

0.305

0.310

0.315

0.320

-2 -1 0 1 2

Harmonic order m

−30

−20

−10

0

10

20

30

40

0 1000 2000 3000 4000 5000 6000

Radius (km)

RadialTangential

× observed

+ computed

Rotational modes

Feature : motion of rotation axis

i. Tilt-over Mode (TOM)ii. Free Core Nutation (FCN)ii. Free Inner Core Nutation (FICN)iv. Chandler Wobble (CW)v. Inner Core Wobble (ICW)

Tilt-over mode

Tilt-over mode

s = β × r = ∇ × (W r) = τ 11

β = β [cos (Ω0t) ex − sin (Ω0t) ey ]

W = rβ sin θ cos (Ω0t + ϕ)

1

0.8

0.6

0.4

0.2

0 0 2000 4000 6000

W

Radius (km)

Tcomp = 86164.22 s

TTOM = 86164.10 s

Free Core Nutation

s = τ 1

1+ σ1

2+ τ 1

3

-5

-4

-3

-2

-1

0

1

0 2000 4000 6000

W

Radius (km)

TFCN = 1 / (1 + 1/458.4) sidereal dayTVLBI = 1 / [1 + 1/(429.0±0.3)] sdTGGP = 1 / [1 + 1/(429.7±2.9)] sd

Free Inner Core Nutation

s = τ 1

1+ σ1

2+ τ 1

3

-600

-500

-400

-300

-200

-100

0

0 2000 4000 6000

W

Radius (km)

TFICN = 1 / (1 - 1/472.9) sidereal day

Chandler Wobble

s = τ−1

1+ σ−1

2+ τ−1

3

-0.2

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000

W

Radius (km)

TCW = 404.7 solar daysTobs = 434 solar days

Inner Core Wobble

s = τ−1

1+ σ−1

2+ τ−1

3

0

1000

2000

3000

4000

5000

0 2000 4000 6000

W

Radius (km)

TICW = 4008 solar daysTAM = 2750 solar days

for PREM (Rochester & Crossley 2009)

Core Spectrum

• Confined in the liquid outer core

• Restoring forces:gravity (buoyancy) + inertia (Coriolis)

• Unobserved

Non-rotating spherical model

Key parameter: N2 = −g(

1

ρ

dρ

dr+ g

v2p

)

N2

sup = 6 · 10−8rad

2s−2

N2

inf = −8 · 10−8rad

2s−2

(Valette & Lesage 2007)

Schwarzschild criterion

N2 ω2

> 0 0 ≤ ω2 ≤ N2sup

< 0 N2

inf ≤ ω2 ≤ 0

Rotating model

S ⊆ Essential spectrum ⊆ S’

(Valette 1989)

If N2 < 0, stable modes

Planetary (Rossby) modes

Incompressible liquid shell

s = τmm = ∇ × (W m

m r)

W mm = Arm

ωm =2Ω

m + 1

m = 1 → tilt-over mode

0 1 2 3 4 5 6 7 8 9

10 11

0 2 4 6 8 10 12 14 16 18 20

Per

iod

(sid

erea

l day

s)

Harmonic order m

(m+1)/2

PREM

Normal Modes of Rotating Earth Models

Theory : Clairaut coordinates

Seismic modes, rotational modes, core spectrum

Chandler wobble and core spectrum

Rotational modes and core spectrum

Observed Chandler wobble (solar days)

Pan (2007)

Guo et al. (2005)

Hopfner (2003)

Schuh et al. (2001)

Vicente & Wilson (1997)

Kuehne et al. (1996)

Furuya & Chao (1996)

Lenhardt & Groten (1985)

Chao (1983)

Dickman (1981)

Carter (1981)

Wilson & Vicente (1980)

Ooe (1978)

McCarthy (1974)

400 420 440 460

+ single periodN multiple period- - variable period

Theoretical Chandler wobble (solar days)

(Smith & Dahlen 1981)

Observed Chandler Wobble Q

Vicente & Wilson 1997

Kuehne et al. 1996

Furuya & Chao 1996

Lenhardt & Groten 1985

Wilson & Vicente 1980

Ooe 1978

200 400 600 800 1000

Search for a double Chandler wobble

Coupled LC circuits

ω2

± =ω2

1+ ω2

2±

√

(ω2

1+ ω2

2)2 − 4(1 − α1α2)ω

2

1ω2

2

2(1 − α1α2)

ω1 =1√

L1C1

ω2 =1√

L2C2

α1 =M

L1

α2 =M

L2

Avoided crossing

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5ω1/ω2

ω-/ω2ω+/ω2ω1/ω2ω2/ω2

Diatomic molecule

Lewis and Hougen 1968

Chandler Wobble and core modes

N2 = constant in the liquid cores = τ−1

1+ σ−1

2+ τ−1

3

40

35

30

25

20−15 −12 −9 −6 −3 0

289.4

330.7

385.8

462.3

578.7

Eig

enfr

eque

ncie

s (n

Hz)

Rel

ativ

e pe

riods

(so

lar

days

)

N2 (10−9 rad2/s2)

—— Core Modes—— CW

Chandler Wobble and core modes

N2 = constant in the liquid cores = τ−1

1+ σ−1

2+ τ−1

3

29.5

29.0

28.5

28.0

27.5

-1.8 -1.78 -1.76 -1.74 -1.72

392.3

399.1

406.1

413.4

420.9Eig

enfr

eque

ncie

s (n

Hz)

Rel

ativ

e pe

riods

(so

lar

days

)

N2 (10-9 rad2/s2)

Eigenfunctions at avoided crossing

N2 = −1.7625 10−9 rad2 s−2

−0.8−0.6−0.4−0.2

0 0.2 0.4 0.6 0.8

1

0 1000 2000 3000 4000 5000 6000

Radius (km)

Toroidal 1Spheroidal 2Toroidal 3

−1

−0.5

0

0.5

1

1.5

0 1000 2000 3000 4000 5000 6000

Radius (km)

Toroidal 1Spheroidal 2Toroidal 3

T = 404.6 solar days T = 407.1 solar days

Two Chandler wobbles!

Summary

• Unified approach of Earth rotation and long-period seismology

• Three families of normal modes

• Splitting of seismic modes by rotation

• Influence of core spectrum on rotational modes

Projects

• Influence of thermo-chemical structure in the lower mantle onEarth normal modes

• Polar gravimetry

Influence of thermo-chemical structure in the lower mantleon Earth normal modes

Collaborative research project

• Taipei• Frederic Deschamps (Academia Sinica)• Cheng-Yin Chu (PhD. student, National Central University)

• IPGS• Severine Rosat• Yann Ziegler

How do density anomalies affect rotational modes?Depth (km)

670 - 1200

1200 - 2000

Trampert et al. 2004

2000 - 2891

Polar gravimetry

• Reference values for gravity(geoid, airbourne and terrestrial surveys)

• Gravity variations due to• post-glacial rebound• current ice melting

Arctic

Ny-Alesund (Svalbard)

Courtesy of A. Memin

Antarctica

Collaborative project (F, US, I, NZ, DK, GB, AR, CL, D)

DdUMZS

McM+SB

O’Higgins

Rothera

Palmer

0o

180o

90o E270o

30o

150o210o

330o

60o

120o240o

300o

-80o

-70o

-60o

Antarctica

Gravity variations (µGal)

0

10

20

30

40

50

60

70

80

90

100

1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

MZS (TNB AB)

MZS (IAGS)

MCM (Satgrav)

MCM (Thiel-1)

SB

DDU

Merci !