Quasi-geostrophic MHD waves in the

Earth core

Nathanael Schaeffer and Dominique Jault

Porquerolles, May 2009

Outline

● Magnetic field and core flows (why we care)● Inertial waves, Rossby waves, QG motions● MC waves, torsional waves● Impulsive spin-up● Oscillatory forcing

Earth magnetic field at the surface

year 2000

Hulot et al., 2002

Earth magnetic field at the CORE surface

Gillet, Pais & Jault 2009

VSQG project

Find the velocity field that explains magnetic field variations recorded by both satellites and ground observatories.

Assumption : velocity field consists of columnar motion (QG).

Inertial waves

● Frequency determined by the angle between k and rotation axis

● Special case : motion invariant along the rotation axis have zero frequency => Geostrophic motion.

d=0.01

Vs

Poloidal forcing (constant)E = 1e-7t = 1.6 days

d=0.03

* Same aspect-ratio* smaller size takes longer time to reach boundaries* decay is slower than exp()

z

Vs

Vg = Ω.d

Inertial wavesand Proudman-Taylor theorem

Localized forcing, t = 1.6 days

Inertial wavesand Proudman-Taylor theorem

Vs VzMOVIES !

Rossby waves

When boundaries are taken into account and for

Still there can be zero frequency motions ...

MC waves

without rotation OR k perpendicular to rotation axis

In uniform magnetic field B

Torsional Alfvén waves

Relevant quantity:Bs2 integratedover a cylinder.

Elsasser and Lehnert numberscompare rotation and magnetic field

Elsasser number : quasi-steady assumption, neglecting ∂b/∂t to estimate the current density from the induction equation

Lehnert number uses a a diffusionless assumption for fast time-scales, neglecting η∆b :

Liquid iron

Solid iron

Insulator

Model of the Earth Core

Magnetic diffusion time = 10 000 years

Rotation T : 1 day

B : 3 mT

R = 3000 km

Elsasser # : 1

Lehnert # : 0.0001

Jault, D. (2008) Physics of the Earth and Planetary Interiors, 166

Imposed axisymmetric magnetic field lines used throughout this work.

Axisymmetric case

● Global rotation● Impuse differential

rotation of inner core

● Axisymmetric magnetic field

Travelling Jet = Torsional wave

E=5.7 10-8, Λ=0.52, λ=1.72 10-4

t = 1.3 year

Weak magnetic field is an essential feature

Importance of magnetic field

B=0 Weak B

Non-axisymmetric B field

Azimuthal velocity field at t=18 monthsE=5.7 10-8 , Λ=0.52, λ=1.72 10-4

Same B(m=0) + B(m=32) at same amplitude

Imposed B field has a strong axial dipole component

Br

Bz

Bphi

Champ equatorial

Meridian cut

Worst case : m=1 B field

E = 5.7 e-8 Lehnert ≈ 2.0 e-4 Elsasser ≈ 0.5

Worst case : m=1 B field

t

t

E = 5.7 e-8 Lehnert ≈ 2.0 e-4 Elsasser ≈ 0.5

Worst case : m=1 B field

=> Still strong columnar motion !

Inner Core OscillationsHow does this picture depend on Time-scale ?

=> harmonic oscillation of the inner core.

T = 1 week

Instant field dominatedby inertial wave

Field averaged over 1Ttorsional wave is still there.

Field averaged over 1Tm=0 component is columnar

70%10%

B0 = constant axial field

Slower oscillations

T = 2 months

Inertial wave is now quasi-vertical Axisymmetric (m=0) component=> torsional wave

B

Forcing period = 2 months

Axisymmetry does not dominate

Conclusions● Short time motions tend to be geostrophic

(20 000 years >> t > 10 days

● Magnetic field is crucial in this process !

● Worst case described (m=1) has still a large geostrophic part.

Coming soon :

● Local forcing with magnetic field

● Forcing BY magnetic field

● Interaction with background velocity field

● Random forcing of wave (turbulence ?)