a lagrangian dynamic model of sea ice for data assimilation
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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A Lagrangian Dynamic Model of Sea Ice
for Data Assimilation
R. LindsayPolar 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 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
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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)
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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)
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SPH Neighbors
A, h, grad(u)
Stress Gradients
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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
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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 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
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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 -> 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 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
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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
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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.