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Nom original: poster-sedi.pdfTitre: Data assimilation in a Quasi-Geostrophic Model of the Earth CoreAuteur: F. Labbé, D. Jault and N.Gillet

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Data assimilation in a
Quasi-Geostrophic Model of the Earth Core
F. Labb´e, D. Jault and N.Gillet
ISTerre, Universit´e Joseph Fourier, Grenoble, France.

Linearization of the forward problem

I We develop a Quasi-Geostrophic (QG) approach to model the fast dynamics
inside the core (internal magnetic field and flow changes).
I Our goal is to image the underlying magnetic forces responsible for the magnetic
changes recorded from satellites and observatories.
I Applications : forecast and re-analysis of rapid changes in the core.

I We linearize : B2 = b20 + b21
b20 is the static part, b21 are small variations (b21 b20).
This linearization limits us to short timescales (e.g. satellite era).
I We filtrate Rossby waves by dropping the
term in (4), then flow changes

∂ b1
are enslaved to changes in the internal magnetic field
I Following Canet et al (2009) we can rewrite the vorticity and induction equations
(4) and (1):

= F b1


b1 = G b0 , ψ
which we combine to give:

∂ 2ψ
= FG( b0 , ψ)

= M(ψ) b0

Quasi-Geostrophic Model
I Induction equation (neglecting diffusion) :
∂tB = ∇ × (u × B)


I We use a viscousless momentum equation, neglecting thermal effects :
Dtu + (ez × u) = −∇Π + (∇ × B) × B
where Π is pressure and λ is the non-dimensionnal Lehnert number. λ gives the
ratio between inertial waves period and (magnetic) Alfv`en wave period.
I The small value of λ inside the core indicates an invariance of rapid large
length-scales flows in direction parallel to rotation axis (Jault 2008)
From there, we can assume 2D flows in the core and study a projection of
motions and magnetic fields in the equatorial plan f the core.
The existence of the inner core prevents us from study phenomenons inside the
tangent cylinder ( tangent to the inner core).

I Taking Y =

∂ 2ψ

, A = M(ψ) and X = b0 , and introducing an error ε we

have now a linear system :

AX = Y + ε


From data of ψ and its time
we can imagine inverting this forward

problem to get an image of b20 in the core.
I Satellite data (good coverage over the past decade) are inverted through a
non-conducting mantle to give us the radial component at core surface Br,CMB(t)
I We follow an ensemble inversion scheme (Gillet, Pais & Jault 2009) to retrieve,
from the radial induction equation at the CMB
∂tBr = −∇H · (uBr)


and several possible realisations of the unresolved field B0r, an ensemble of
possible flow models ψ(t).

Figure: schematic view of a geostrophic
column and the equatorial plane.

Figure: A possible reconstruction of the
equatorial stream function in 2005.

Let ψ be the equatorial stream function. We split the equatorial velocity into a
static part u0 and variations u1 :

u0 + u1
u = u0 + ∇ × (ψez)



factor ensure the conservation of mass in the volume.

I To ensure non-prenetration condition at core boundary, we assume a linear

z-dependancy of velocity field, with uz =
us, where β is the slope of the core
surface. By taking the curl of (2) and averaging over the axial direction (Hide
1966), we obtain:

d 2 ψ
1 ∂ψ

2H − ∇E
λH3 ∂φ
I The term derived from the Lorentz force j × B = (∇ × B) × B introduces
quadratic quantities of the magnetic field :
Z Hh
hB2i =
B2s, B2φ, BsBφ dz

(cf posters by D. Jault and E. Canet)

R.Hide Phil. Transac. of the Royal Soc. of London, A, vol. 259, no. 1107,
pp. 615–647, 1966.
E. Canet, A. Fournier, and D. Jault JGR solid Earth, no. 114, p. B11101, 2009.
N. Gillet, M. A. Pais, and D. Jault Geochem. Geophys. Geosyst., no. 10, 2009.
W. V. R. Malkus J. Fluid Mech., vol. 28, part 4, pp. 793–802.
N. Gillet, D. Jault, E. Canet, and A. Fournier Nature, vol. 465, pp. 74–77, 2010.

Figure: sample of 8 possible equatorial maps of ∂tψ in 2005.

From those we build the data Y in (9). The dispersion in the ensemble of
is representative of the large error bars on flow changes, due to
ill-constrained small scales.
Code and Tests
I A fortran code for the forward problem is curently under development testing
I Possible validation criteria are :
. Good resolution of a nul space , with B parallel to u..
. Compare to analytic results of Malkus (1967) for

. Twin experiment: try to find back the original B from the result of a forward
Objectives and perspectives


I Obtain map of magnetic forces through B
I Timestepping of (8) and (10) in order to deduce the evolution of Br,CMB
NB: Br,CMB is considered as a passive tracer of the surface flow (Br,CMB is 10
times weaker than B inside the core, see Gillet et al, 2010).
. Application to IGRF
. Magnetic field forecast on short timescales (due to linearization)
. Use of data from futur european Swarm mission, and comparison to forecast.

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