Does the magnetic field in the fluid core contribute a lot to earth nutation?

Cheng-li Huang Veronique Dehant Xin-hao Liao Oliver de Viron

Tim Van Hoolst

Shanghai Astron. Obs.(China) Royal Obs. of Belgium

Prague, 2006/08/23

Outline

• Background: status & problem in non-rigid Earth nutation studies

• Concept of coupling between nutation and electro-magnetic field (EMC)

• Incorporate EMC into numerical integration approach

• Results & discussion

Status & problems (1):Non-rigid Earth nutation study approaches

• semi-analytic (SOS+VLBI):Mathews,Buffett,Herring(1991,2002): MHB2000

• Generalized Hamiltonian:Getino & Ferrandiz (1995, 2000…)

• Numerical integration (displacement):Smith (1974,1977)Wahr(1981…)Dehant & Defraigne (1997…)Huang et el. (2000…)

Status & problems (2)

• All recent models are compared very well with each other;

• All suffer a big problem: significant discrepancy with VLBI obs. in the out-of-phase (op) part of retrograde annual (-1.0yr) term;

• The op part relates to dissipation; -1.0yr term is strongly influenced by FCN resonance;

• But the contributions from mantle anelasticity, ocean & atmosphere can not reconcile this problem.

IAU2000A nutation model

Mathews :(1991 2002)Herring :(1991 2002)Buffett :(1992 2002)

• Basic theory ( SOS)

• fit 9 key compliance par. to VLBI

• EMC to solve dissipation (OP)

residual remains: -1.0yr op: 0.4 mas

Table 3 of Buffett(1992): Contribution of EMC at CMB to MHB nutations (mas) (profile B)

Period ar ai

-182.6 -0.003 0.003-365.3 -0.068 0.069-6798.4 -0.083 0.0836798.4 0.009 -0.009365.3 0.000 0.000182.6 0.014 -0.01413.7 0.000 0.000

To explain the discrepancy, the magnetic energy is required to be 5 times bigger (l 12)≤

0.4mas

Calculation of the contribution of EMC to nutation in displacement approach

A model for EMC near CMB and ICB

SIC

Conductinglayer

Viscositylayer

ICB

CMB

Mantle

FOC

Taking the FOC as steadily rotating reference frame

The motion equations in displacement approach

+ Lorentz force (& viscosity)

• Lagrange ME for an anelastic, rotating, slightly elliptical, self-gravitational Earth (Smith,1974):

ρ=density; s=displacement; Ω=mean angular velocity; Γ=Cauchy stress tensor; γ=pressure field; φ=gravitational potential field; φ1=incremental Euler potential field caused by the displacement;. Dt=full derivative to epoch t;

2

21

2 ( ) ( )

[ ( ) ] ( ] [ )

tT

t

tD

L D

s D s s

ss

s

s

ρ ρ ρ γ

ρ φ ρ γ

ρν

φ

+ Ω × = − Ω × Ω × + ∇ ⋅ Γ − ∇ ∇ ⋅

− ∇ − ⋅ ∇∇ + ∇

+

⋅ ∇

∇+

Other eqs. & Boundary conditions

• Possion Equation:• Cauchy Stress-strain relationship for isotropic medium

• Boundary conditions– continuous across any boundary:

– continuous across welded boundary: s

)(412 sG ρπφ ⋅∇−=∇

])([)( Tsss ∇+∇µ+⋅∇λ=Γ I

11 ),4(ˆ ,ˆ ,ˆ φρπφ sGnnns +∇⋅Γ⋅⋅

Lorentz Force density (L)

BB1BJL ∇⋅=×≡µ

),( , ˆˆb

φθ

φθ φθ

bbbbb

r <<

+≅)B (b b,BB 00 <<+=

bb1b 1 rr r∂<<∇≡∇

b1 L 0 rrB ∂≈

µ

Induction Equation & solution (b)

B)(-B)V(Bt ×∇×∇××∇=∂ η

b)Bv(b 201t ∇+××∇≈∂ η

.)/(1 const=≡ µση10 vvV +=

Solution (1)

Buffett et al. (1992):

Solution (2)

Buffett et al. (2002):

( )( )( ) ( ) ( ) ( ) ( )( ) ( )

,

1 cos 1 cosi t i t

b t b ib

b r e b r eθ φ

ω φ ω φθ θ±

− − −+ −± ±

≡ ±

= ± +

r

m

B1992

Incorporate L into scalar ME & BC

( )( )20 2 0 1 2 3

emDρ ρ η η η∇ = + −⋅ + +T L

Results (1)

Method: iteration & approximation:

Model for K: near CMB only

ηm = ηf = 1.6 m2/s

(σm= σf = 5*105 S/m,º 200 m)

B1992 k=

0.26 , 0.64 ,dipole uniformr rB mT B mT= =

mTBBRMS uniformr

dipoler 69.0)()( 22 =+=

(same as in Buffett(1992) & Buffett et al. (2002) )

The EMC contribution maybenot so significant as stated ?

Tab.: Contributions of EMC at CMB to nutation (µas) & FCN

(Huang et al., 2006) (Buffett,1992)(Buffett,2002)

MBH2000

Consistency between the change of FCN period & nutation

∆(nutation) ≈

∆FCN=0.4day(this work)

∆FCN=1.2day(Buffett, 1992)

-1.0yr op +41 µas(+40 µas)

+125 µas(+68 µas)

-18.6yr op +26 µas(+20 µas)

+79 µas(+83 µas)

• The results (op of –1.0yr) in Buffett et al.(2002) (450µas) is about one order of magnitude bigger than ours & his 1992 paper.

• Will the polar magnetic field (B0r) itself & the

Coriolis force (f) change k (or KCMB) so much?

Comparing kB1992 & kB2002

Buffett et al.(2002)kbnutation r ∝∂∝∆ )(

Buffett(1992)

k =

Comparing kB1992 & kB2002

±20%

SummaryCalculated the effects of EMC at the CMB onnutations by numerical integration. For the OP of –1yr term, it is 40 µas. It’s smaller than, but at the same order as, that of Buffett(1992) (68µas) , and they are not enough to reconcile the discrepancy from VLBI obs. (~ 400 µas) However, our results for the changes of –1yr & –18.6yr nutations are more consistent with the change of the FCN period (~ 0.4 day).The results in Buffett(2002) (450µas) is about one order of magnitude bigger than ours & his 1992 paper, but the polar magnetic field itself & the Coriolis force are not the dominant factors as he stated.

other candidates? torques at CMB/ICB

gravitationalelectro-magneticviscoustopographic

•Deviation from HSE:mantle convectionlateral heterogeneitydifferential rotation…