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!
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
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
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)
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
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
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)
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
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
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
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
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
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