Variational data assimilation in geomagnetism: progress and perspectives

1. Variational data assimilation in geomagnetism: progress and perspectives Andy Jackson, Kuan Li ETH Zurich & Phil Livermore University of Leeds

2. Outline • Rationale for 4DVar • Proof of concept: Hall effect dynamo • (Hardcore Bayes: a prior pdf for core surface field estimation)

4. Visible and hidden parts of the magnetic field Measurements Based only on poloidal magnetic field data on the boundary of the core, can we infer interior properties (including those of the toroidal field)?

5. Technology transfer from meteorology: 4DVar variational data assimilation 2-D observations in time of a time-evolving system can give information about the third (hidden) dimension Applications: Seismology (X=velocity), Mantle convection (X=temperature), Core convection (X=magnetic field) X __ =f (X) t

6. Declination1700-1799 Declination1800-1930

7. 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

8. System Evolution Magnetic Field New Initial Condition Fournier et al 2010

9. A dynamical model for the core Begin by solving this part B = Magnetic field u= Fluid velocity T= Temperature J = Current density Ω = Rotation vector

10. Misfit of poloidal field measurements at core surface Misfit = χ2 = (Observed-Predicted)2

11. Kinematic Induction Equation (u given and constant)and its adjoint B = Magnetic field v= Fluid velocity η= Magnetic Diffusivity   =(vB) + η2B t B = Adjoint Magnetic field .B =0 Same boundary conditions as B -  =(B )v + η2B -p +f(Misfit) t Equation operates in reverse time

12. Dynamo Equations Misfit derivatives with respect to initial condition Adjoint differential equation to integrate in reverse time

13. A neutron star toy problem A toy problem to illustrate the physics • Evolution of the field is given by • The Hall effect is thought to be responsible for the field regeneration   = Rm(BB) + 2B t Induction through Hall effect Ohmic diffusion

14. 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?

15. 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

16. 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]

17. Mie (toroidal-poloidal) representation .B=0 => B = Tr + Pr T=Tlm etc based on spherical harmonics

18. Iterative reconstruction of l=1 toroidal coefficient True initial value Initial guess Trajectory Forward model Adjoint model

19. Reconstructed toroidal field (2-D surface poloidal observations, Rm=20, 7k years) True state Kuan Li

20. Rm=5 30,000 years surface poloidal observations |B|=1 isosurface Iteration True Initial State Reconstructed Initial State Kuan Li & Andrey Sheyko

21. 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

22. 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

23. 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

24. 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

25. Andreas’ geomagnetic inference problem from day 1 44TW total

27. Ohmic dissipation (Watts) from a magnetic field

28. Ohmic dissipation (Watts) from a magnetic field

29. Independence of xlm? CLT => much more certain knowledge than expected Dim(X)=∞? Backus=> much more certain knowledge than expected

30. The B field is not infinite dimensional – magnetic dissipation

31. The prior for Ohmic dissipation P(dissipation) Minimum (or no dynamo for last 3.5 Billion years) Dissipation Maximum (or oceans -> steam)

32. P(x1,x2,x3,….xn) ~ (x12+x22+x32+….xn2)-n/2+1 How to solve this inference problem? Too big for MCMC