Variational data assimilation in geomagnetism: progress and

perspectives

Andy Jackson, Kuan LiETH Zurich

& Phil Livermore

University of Leeds

Outline

• Rationale for 4DVar

• Proof of concept: Hall effect dynamo

• (Hardcore Bayes: a prior pdf for core surface field estimation)

Visible and hidden parts of the magnetic field

Based only on poloidal magnetic field data on the boundary of the core, can we infer interior properties (including those of the toroidal field)?

Measurements

Technology transfer from meteorology: 4DVar variational

data assimilation2-D observations in time of a time-evolving systemcan give information about the third (hidden) dimension

=f (X)__Applications: Seismology (X=velocity),Mantle convection (X=temperature), Core convection (X=magnetic field)

Xt

Declination1700-1799

Declination1800-1930

Why data assimilation? (PDE constrained optimisation)

• Data• Dynamical model

• Magnetic part (induction equation)• Momentum part (Navier Stokes)• Energy part (temperature equation)

• Unknown is the initial state of the model

System Evolution

Fournier et al 2010

New Initial Condition

Magnetic Field

A dynamical model for the core

B = Magnetic fieldu = Fluid velocityT = TemperatureJ = Current densityΩ = Rotation vector

Begin by solving this part

Misfit of poloidal field measurements at core surface

Misfit = χ2 = (Observed-Predicted)2

Kinematic Induction Equation (u given and constant)

and its adjoint

t = (vB) + η2B

B = Magnetic fieldv = Fluid velocityη = Magnetic Diffusivity

B = Adjoint Magnetic field

.B =0

Same boundary conditions as B

-t

= (B )v + η2B -p +f(Misfit)

Equation operates in reverse time

Misfit derivatives with respect to initial condition

Adjoint differential equation to integrate in reverse time

Dynamo Equations

A neutron star toy problem

• Evolution of the field is given by

• The Hall effect is thought to be responsible for the field regeneration

= Rm(BB) + 2B t

Induction through Hall effect Ohmic diffusion

A toy problem to illustrate the physics

• The initial condition B(t=0) determines the subsequent evolution

• Can we determine the initial condition, and thus the 3-D field at all times?

The adjoint method• Forward problem based on current estimate of B(0)

• Calculate residuals

• Backward propagation (reverse time) of adjoint equation

• Use gradient vector to update estimate of B(0)

• Go again

A closed-loop proof of concept

• Observations of Br are taken every 100 years for 30k years (~1 magnetic decay time)

• Note – no constraints at all on toroidal field• In our simulation Rm=5 [advection is weak]

Mie (toroidal-poloidal) representation

.B=0 => B = Tr + PrT=Tl

m etc based on spherical harmonics

Iterative reconstruction of l=1 toroidal coefficient

Forward model

Adjoint model

True initial value

Trajectory

Initial guess

Reconstructed toroidal field (2-D surface poloidal observations, Rm=20, 7k years)

Kuan Li

True state

Rm=5 30,000 years surface poloidal observations

True Initial State Reconstructed Initial State

Iteration

|B|=1 isosurface

Kuan Li & Andrey Sheyko

Convergence

• There is no proof of uniqueness• For the case of perfect data, we have never

been trapped in local minima• In the case of real, noisy data this will need

testing

What we can learn about the Earth

• When a version of the Navier-Stokes equation has been implemented one will obtain an estimate of the magnetic field structure and temperatures in 3-D

• The unknowns are temperature and magnetic field strength; both affect the dynamics

Summary & Outlook• Variational data assimilation presents itself as a useful technique

for interrogating the core

• We have a differential form for the adjoint of all the dynamo equations, which can be efficiently used with a pseudospectral method

• We have demonstrated convergence on a very nonlinear toy problem

• Left to do: Core’s dynamical evolution T > few decades– We work towards an inviscid formulation of core dynamics– Controls are magnetic field and temperature– 300 years of data seems to be almost sufficient

The challenges• Preconditioners for quasi-Newton updates – have made no

efforts yet. Perhaps we can usefully use the Fichtner/Trampert methodology… but need H-1 not H

• No efforts made on BFGS

• The prior probability

Andreas’ geomagnetic inference problem from day 1

44TW total

Ohmic dissipation (Watts) from a magnetic field

Ohmic dissipation (Watts) from a magnetic field

Independence of xlm?

CLT => much more certain knowledge than expected

Dim(X)=∞?

Backus=> much more certain knowledge than expected

The B field is not infinite dimensional – magnetic

dissipation

The prior for Ohmic dissipation

P(dissipation)

Minimum (or no dynamo for last 3.5 Billion years)

Maximum (or oceans -> steam)

Dissipation

P(x1,x2,x3,….xn) ~ (x12+x2

2+x32+….xn

2)-n/2+1

How to solve this inference problem? Too big for MCMC