A Lagrangian Dynamic Model of Sea Ice for Data Assimilation R. Lindsay Polar Science Center University of Washington

Why Lagrangian? Displacement measurements (RGPS and buoys) are for specific Lagrangian points on the ice Assimilation can force the model ice to follow these points Model resolution can be made to vary spatially New way of looking at modelling ice may bring new insights

Outline of the Talk Description of the model Two Year Simulation Structure Numerical scheme Boundary conditions Forcings Two Year Simulation Validation against buoys Data Assimilation Kinematic Dynamic

Lagrangian Cells Defined by: [ x, y, u, v, h, A ] Position Velocity Mean Thickness Compactness (2-level model) These properties vary smoothly in a region Cells have area for weighting, but no shape…area is not conserved Initial spacing is 100 km Smoothing scale is 200 km Cells are created or merged to maintain approximately even number density

Force Balance at each Cell du/dt = fc + (twind + tcurrent + tinternal) / r twind = rair Cw G2 tcurrent = rwater Cc (uice – uwater)2 tinternal = grad( s ) s = f( grad(u), P*, e ) Viscous plastic rheology (Hibler, 1979) =

Smoothed Particle Hydrodynamics The interpolated value of a scalar: Ū(xo,yo) = S wi ai ui / S wi ai wi = exp( -Dri2 / L2 ) The gradient: du(xo,yo) /dx = (2 / L2 ) S wi ai ui / S wi ai wi = (xo-xi ) exp( -Dri2 / L2 )  grad(u)

SPH Neighbors s A, h, grad(u) Stress Gradients

Boundary Conditions Coast is seeded every 25 km with cells with 10-m ice, zero velocity. These are included in the deformation rate estimates An additional perpendicular repulsive force ( 1/r2 ) is added away from all coasts with a very short length scale

Integration Solve for [x, y, u, v] by integrating the time derivatives [u, v, du/dt, dv/dt] Find h and A from the growth rates and divergence Basic time step is ½ day Adaptive time stepping used for baby steps using the Fehlberg scheme, 4th and 5th order accurate, error tolerance specified, steps size reduced if needed 50 to 500 calls for the derivatives per time step (1.5 to 15 min per call) Cells added and dropped, and nearest neighbors found once per day

Forcings IABP Geostrophic winds Mean currents from Jinlun’s model Thermal growth and melt from Thorndike’s growth-rate table Initial thickness from Jinlun’s model Coast outline is from the SSMI land mask

One Two-year Trajectory

10-day Trajectories and Ice Thickness

Movie: 1997-1998 Trajectories Thickness Deformation Vorticity

Validation Against Buoys Correlation: 0.72 (0.66) (N = 12,216) RMS Error: 6.67 km/day Speed Bias: 0.59 km/day Direction Bias: 11.8 degrees

Data Assimilation Methods Cells are seeded at the initial times and positions of observed trajectories buoys or RGPS We seek to reproduce in the model the observed trajectories and extend their influence spatially. Use a two-pass process (work in progress) Kinematic or Dyanamic

Kinematic Assimilation Use a simple Kalman Filter to find the corrections for the positions of seeded cells at each time step Extend corrections to all cells at each time step with optimal interpolation Compute x, y -> then u, v, divergence, A, and h for all cells

Kalman Filter Assimilation of Displacement F = vector of first guess velocities PF = error covariance matrix Z = observed velocity over the interval Rz = observed velocity error Z’ = HF = first guess mean velocity G = F + K(Z – Z’) , new guess PF = (I - KH) PF , new error K = PHT , Kalman gain H P HT - Rz

Dynamic Assimilation For each displacement observation determine a corrective force du/dt = fc + (twind + tcurrent + tinternal) / r + Kfx,cor fx,cor = 2(Dxobs - Dxmod ) / Dt2 Extend fcor to all cells with optimal interpolation Rerun the entire model for the DA interval, including fcor

Conclusions A new fully Lagrangian dynamic model of sea ice for the entire basin has been developed. SPH estimates of the strain rate tensor Adaptive time stepping Solves for x ,y, u, v, h, A Good correspondence with buoy motion Assimilation work well under way

Work to be done Complete dynamic assimilation. Will it work? If it does, examine fcor . What can it tell us about forcings and internal stress? Implement kinematic refinement to get exact reproduction of the observed trajectories. Develop model error budget and time & space dependent error estimates. Minimize model bias in speed or direction Assimilate buoy and RGPS data for a two year period with output interpolated to a regular grid.