A Lagrangian Dynamic Model of Sea Ice for Data Assimilation

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A Lagrangian Dynamic Model of Sea Ice for Data
Assimilation

• R. Lindsay
• Polar Science Center
• University of Washington

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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

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Outline of the Talk

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

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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 shapearea
  is not conserved
• Initial spacing is 100 km
• Smoothing scale is 200 km
• Cells are created or merged to maintain
  approximately even number density

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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)

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Smoothed Particle Hydrodynamics

• The interpolated value of a scalar
• U(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)

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SPH Neighbors
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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

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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

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Forcings

• IABP Geostrophic winds
• Mean currents from Jinluns model
• Thermal growth and melt from Thorndikes
  growth-rate table
• Initial thickness from Jinluns model
• Coast outline is from the SSMI land mask

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One Two-year Trajectory
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10-day Trajectories and Ice Thickness
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Movie 1997-1998

• Trajectories
• Thickness
• Deformation
• Vorticity

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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

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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

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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 -gt then u, v, divergence, A, and h
  for all cells

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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

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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

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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

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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.