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

2. 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

3. 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

4. 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

5. 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) =

6. 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)

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

8. 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

9. 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

10. 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

11. One Two-year Trajectory

12. 10-day Trajectories and Ice Thickness

13. Movie:
Trajectories
Thickness
Deformation
Vorticity

14. Validation Against Buoys
Correlation: (0.66)
(N = 12,216)
RMS Error: km/day
Speed Bias: km/day
Direction Bias: degrees

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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.