A Lagrangian Dynamic Model of Sea Ice

for Data Assimilation

R. LindsayPolar 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– 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 Celldu/dt = fc + (wind + current + internal/

wind = airCw G2

current = waterCc (uice – uwater)2

internal = grad(

f( grad(u), P*, )• Viscous plastic rheology (Hibler, 1979)

Smoothed Particle Hydrodynamics

• The interpolated value of a scalar:

Ū(xo,yo) = wi ai ui / wi ai

wi = exp( -ri2 / L2 )

• The gradient:

du(xo,yo) /dx = (2 / L2 ) wiai ui / wiai

wixo-xi ) exp( -ri2 / L2 )

grad(u)

SPH Neighbors

A, h, grad(u)

Stress Gradients

Boundary Conditions

1. Coast is seeded every 25 km with cells with 10-m ice, zero velocity. These are included in the deformation rate estimates

2. 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 matrixZ = observed velocity over the interval

Rz = observed velocity errorZ’ = HF = first guess mean velocityG = F + K(Z – Z’) , new guess

PF = (I - KH) PF , new errorK = PHT , Kalman gain

H P HT - Rz

Dynamic Assimilation

• For each displacement observation determine a corrective force

du/dt = fc + (wind + current + internal/ Kfx,cor

fx,cor = 2(xobs - xmod ) / t2

• 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.