Rotation-driven Magnetohydrodynamic Flow Using Local Ensemble Transform Kalman Filtering

Transkript

1 Rotation-driven Magnetohydrodynamic Flow Using Local Ensemble Transform Kalman Filtering Kayo Ide 2 ide@umd.edu Sarah Burnett 1 burnetts@math.umd.edu Nathanaël Schaeffer 4 nathanael.schaeffer@ univ-grenoble-alpes.fr Daniel Lathrop 3 lathrop@umd.edu 1 Department of Mathematics, University of Maryland 2 Department of Atmospheric and Oceanic Sciences, University of Maryland 3 Department of Physics, University of Maryland 4 ISTerre lab of the CNRS, Grenoble, France Sarah Burnett (UMD) LETKF on Dynamo flow 1 / 23

2 Purpose Dynamo action The Sun The Earth Other celestial bodies NASA s SDO AIA 171 Telescope Sarah Burnett (UMD) LETKF on Dynamo flow 2 / 23

3 Background E = ν Ω o L 2, Ro = Ω i Ω o, Re = Ro Ω o E, Rm = (Ω i Ω o ) L 2 η Sarah Burnett (UMD) LETKF on Dynamo flow 3 / 23

4 Three-Meter Diameter Spherical Couette Experiment1 liquid sodium filled constant temperature (125 ± 0.5 C) magnetic field and pressure measurement on the boundary. 1 D.S. Zimmerman, S.A. Triana, H.-C. Nataf and D.P. Lathrop. A turbulent, high magnetic Reynolds number experimental model of Earth s core. Journal of Geophysical Research (Solid Earth). 119: (2014). Sarah Burnett (UMD) LETKF on Dynamo flow 4 / 23

5 Magnetohydrodynamic flow in spherical shells 2 Governing equations t u + (2Ω o + u) u = 1 ρ p + ν u + 1 ( b) b, µ 0 ρ t b = (u b η b), Boundary conditions No slip condition u = 0, b = 0, b matches a vacuum field at r = r o, r i Initial conditions Differential rotation and Imposed dipole magnetic field. 2 A Tilgner. Magnetohydrodynamic flow in precessing spherical shells. Journal of Fluid Mechanics, 379:303318, Sarah Burnett (UMD) LETKF on Dynamo flow 5 / 23

6 Nathanaël Schaeffer s XSHELLS code 3 Radial component of the surface magnetic field Re = , E = 10 4, Ro = 1.75, Rm = Nathanael Schaeffer. Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochemistry, Geophysics, Geosystems, 14(3):751758, Sarah Burnett (UMD) LETKF on Dynamo flow 6 / 23

7 Data assimilation Sarah Burnett (UMD) LETKF on Dynamo flow 7 / 23

8 Toy problem: The Lorenz model 4 Nonlinear model: Ẋ(t) = σ (Y X), Ẏ (t) = X (r Z ) Y, Ż (t) = XY bz Solve with 4th order Runge-Kutta method. Figure: Lorenz attractor with parameters σ = 10, r = 28, and b = 8/3. 4 Lorenz, Edward N. Deterministic nonperiodic flow. Journal of the atmospheric sciences 20.2 (1963): Sarah Burnett (UMD) LETKF on Dynamo flow 8 / 23

9 Data Assimilation x b (t n+1 ) = M [ x a (t n ) ], y o (t n ) = H n [ x t (t n ) ] + ɛ n Lorenz example: M solves ẋ = d X Y dt Z σ (Y X) = X (r Z ) Y XY bz Sarah Burnett (UMD) LETKF on Dynamo flow 9 / 23

10 Kalman Filtering Given observations at time t n, ɛ N (0, R), x a N ( x a, P a), x b N ( x b, P b) Goal: Obtain the most likely trajectory of x(t) that fits with the model at times t 1 < t 2 < < t n in the least squares sense. This is done by minimizing a cost function, J o t (x) = n i=1 [y o i H i (M t,ti (x))] T R 1 i [y o i H i (M t,ti (x))] How do we compute this? Sarah Burnett (UMD) LETKF on Dynamo flow 10 / 23

11 Kalman Filtering Given observations at time t n, ɛ N (0, R), x a N ( x a, P a), x b N ( x b, P b) Goal: Obtain the most likely trajectory of x(t) that fits with the model at times t 1 < t 2 < < t n in the least squares sense. This is done by minimizing a cost function, J o t (x) = n i=1 [y o i H i (M t,ti (x))] T R 1 i [y o i H i (M t,ti (x))] How do we compute this? KALMAN FILTER Sarah Burnett (UMD) LETKF on Dynamo flow 10 / 23

12 Kalman Filtering ITERATIVE METHOD Forecast step x b n = M tn 1,t n x a n 1, P b n = M tn 1,t n P a n 1 MT t n 1,t n. Analysis step (Jt o (x) min) x a = x b + P a H T R 1 (y o H x b ), P a = (I + P b H T R 1 H) 1 P b. Forecast step x b (t n+1 ) = M tn,t n+1 x a (t n ). Sarah Burnett (UMD) LETKF on Dynamo flow 11 / 23

13 Local Ensemble Transform Kalman Filtering 5 Cost function, J(x), we want to minimize. J(x) = [x x b ] T (P b ) 1 [x x b ] + [y o H(x)] T R 1 [y o H(x)]. Analysis ensemble (k members) with square-root filters (deterministic approach). 5 Brian R Hunt, Eric J Kostelich, and Istvan Szunyogh. Efficient data assimilation for spatiotemporal chaos: A local ensemble transform kalman filter. Physica D: Nonlinear Phenomena, 230(1):112126, Sarah Burnett (UMD) LETKF on Dynamo flow 12 / 23

14 Local Ensemble Transform Kalman Filtering Background state estimate and error covariance matrix with k ensemble members. x b = 1 k k i=1 x b(i), P b = 1 k 1 Xb ( X b) T Sarah Burnett (UMD) LETKF on Dynamo flow 13 / 23

15 Local Ensemble Transform Kalman Filtering Transform x such that X b w S for some w S. True state is now ( ) x = x b + X b w, where w N 0, (k 1) 1 I Note: H ( x b + X b w ) ȳ b + Y b w Cost function J(w) = (k 1)w T w + [y 0 ȳ b Y b w] T R 1 [y o ȳ b Y b w] Analysis equations where w a = P a (Y b ) T R 1 (y o ȳ b ) W a = [(k 1) P a ] 1/2 P a = [(k 1/ρ)I + (Y b ) T R 1 Y b ] 1 Sarah Burnett (UMD) LETKF on Dynamo flow 14 / 23

16 Local Ensemble Transform Kalman Filtering Transform back to S (original model space) Calculate the ensembles, Apply the forecast step, x a = x b + X b w a, X a = X b W a. w a(i) = W a (i) + w a, x a(i) = x b + X b w a(i). x b(i) (t n+1 ) = M ( ) x a(i) (t n ). Sarah Burnett (UMD) LETKF on Dynamo flow 15 / 23

17 Validation: Perfect model test Sarah Burnett (UMD) LETKF on Dynamo flow 16 / 23

18 Lorenz model What if we use complete versus partial observations? 3 members ρ = l = 3 X y = Y Z RMS error: X = Y = Z = Sarah Burnett (UMD) LETKF on Dynamo flow 17 / 23

19 Lorenz model 3 members ρ = l = 2 [ ] Y y = Z RMS error: X = Y = Z = Sarah Burnett (UMD) LETKF on Dynamo flow 18 / 23

20 Lorenz model 3 members ρ = l = 1 y = [ X ] RMS error: X = Y = Z = Sarah Burnett (UMD) LETKF on Dynamo flow 19 / 23

21 Lorenz model: LETKF, EnKF 6, and EKF 7 RMS error for every 8 time steps LETKF (no localization) 3 members 6 members (ρ = ) (ρ = ) EnKF 3 members 6 members 0.30 (δ = 0.04) 0.28 (δ = 0.02) EKF 0.32 (µ = 0.02, δ = 0) RMS error for every 25 time steps LETKF (no localization) 3 members 6 members (ρ = ) (ρ = ) EnKF 3 members 6 members 0.71 (δ = 0.39) 0.59 (δ = 0.13) EKF 0.63 (µ = 0.1, δ = 0.05) 6 Kalnay, Eugenia, et al. 4DVar or ensemble Kalman filter?. Tellus A 59.5 (2007): Yang, Shu-Chih, et al. Data assimilation as synchronization of truth and model: Experiments with the three-variable Lorenz system. Journal of the atmospheric sciences 63.9 (2006): Sarah Burnett (UMD) LETKF on Dynamo flow 20 / 23

22 Current work: Quasi-2D unsteady fast dynamo flow 8 Governing Equations: Velocity field u(x, y, t) = 2 cos 2 t (0, sin x, cos x) + 2 sin 2 t (sin y, 0, cos y) Induction equation t b x + ( ) u x x + u y y + iku z bx = ( ) b x x + b y y ux + η 2 b x, t b y + ( ) u x x + u y y + iku z by = ( ) b x x + b y y uy + η 2 b y Initial condition B 0 = (i, 1, 0)e ikz where B(x, y, z, t) = b(x, y, t)e ikz 2π periodic boundary conditions for x and y. 8 Childress, Stephen, and Andrew D. Gilbert. Stretch, twist, fold: the fast dynamo. Vol. 37. Springer Science & Business Media, Sarah Burnett (UMD) LETKF on Dynamo flow 21 / 23

23 Future Work: Phase 3 & 4 Apply the LETKF to xshells. Run data assimilation trial and compare this with the purely numerical run. Trials assimilating b data with various numbers of observations, assimilation windows, and number of ensemble members Study the effect on the velocity field. Is it reliable? Sarah Burnett (UMD) LETKF on Dynamo flow 22 / 23

24 Thank you! Sarah Burnett (UMD) LETKF on Dynamo flow 23 / 23