fluid flow in sea iceseaice.alaska.edu/gi/publications/pringle/06p_fluidflowseaice.pdf · fluid...
TRANSCRIPT
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Fluid flow in Sea Ice
Daniel PringleARSC / Geophysical Institute
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Outline of talk
Sea ice background and motivation
Sea ice porosity and permeability
Introduction to Lattice Boltzmann Methods
Sea ice work so far..
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Sea Ice
Thermal and mechanical barrier between Ocean and Atmosphere: complex ocean-ice-atmosphere interactions.
Indicator and agent of local and global climate change.
Studied via numerical modelling:
needs to be faithful to sea ice physics and processes
Arctic Antarctic Habitat/9 - 16 million km2 4 - 19 million km2 Ecosystem
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Figure: Thomas & Dieckmann, ‘Sea, Blackwell Science, 2003
Sea Ice length scales
Antarctic cover Pack ice Pancake Biological Pores microbiologylayering (diatom chain)
‘microstructure’
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Albedo and Permeability
Arctic Ocean, SHEBA study
α= SW ↑ / SW ↓αOcean ~ 0.07
αSnow ~ 0.85
αPonds < 0.4
αbare ice~ 0.65
Cold ice: impermeable
-ponding
Warm ice: permeable
-drainage
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Other effects..
Heat flow
porosity changes allow discharge of head of cold dense brine
contribution to total heat flow ~ several percent.
Nutrient Delivery to resident biology
Sediment and pollutant transport
Brine expulsion during growth
April 26
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Sea ice = ‘fresh ice’ + inclusions of brine, air, (salts)
Salinity and temperature as ‘state variables’
(S,T,ρ) → vair, vbrine,vice
Lake Ice
vs.
Sea Ice
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Inclusions and Porosity
Constitutive super-cooling at growth front: ‘lamellar interface’
Brine incorporated in inter-lamellar spaces
1 cm
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Porosity Evolution
1 mm
- 15 C
- 5 C
- 2 C
Light et al, 2003
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Porosity Evolution10 cm
Jeremy Miner1 mm
- 15 C
- 5 C
- 2 C
Light et al, 2003
Dual porosity: 0.1 mm / 1 cm scales; geological analogs.
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Permeability
Pk∇−=
μv
Darcy’s Law
-empirical -low flow
v - discharge velocity μ - dynamic viscosity
∇P -Pressure gradient k - permeability
1 Darcy ≈ 1 x 10-12 m2
Gravel ~ 105 – 102 D; Oil reservoir rocks 10 – 10-1 D; Granite 10-6 - 10-7 D
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Permeability of sea ice: percolation?
Few experiments - field or lab
Developing percolation theory (Golden)
-critical brine volume fraction, vb = 5 %
Arctic field data (Eicken)
Now: numerical modeling of flow in imaged sea ice samples
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Obtain 3D internal structure from X-ray Computed Tomography.
Characterize pore space
Apply Lattice Boltzmann method to model flow: calculate k
Numerical Modeling Overview
e.g. Fontainebleau sandstone, Martys and Hagedorn, 2002
LBM
Sea ice ‘dynamic’: porosity and permeability depend on S,T
XCT
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Sea Ice Summary, Specific Objectives
Sea ice porosity from brine, air inclusions.
Dual porosity: primary (0.1 mm) & secondary (1 cm)
Porosity and permeability depend on S,T and history
1.Does sea ice really undergo a ‘percolation’ transition?If so, at what porosity and with what critical exponent?
2. What role do primary and secondary porosity play?
3. Sea ice as an analog to inaccessible geophysical materials near melt transitions: volcanic conduit, lower mantle materials.
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Imaged Sea Ice Structures
Lab-grown sea ice: reconstructions of X-ray CT of 1 cm cores
Heaton, Miner, Eicken, Zhu,Golden, in prep (2006)
Brineinclusions
LBM: good at handling complex porous geometries
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Microscopic Macroscopic
Lattice Boltzmann Modeling
Thermodynamics
Fluid dynamics
Statistical
treatment
Kinetic theory
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Microscopic Macroscopic
Lattice Boltzmann Modeling
Thermodynamics
Fluid dynamics
Statistical
treatment
0u
)u(1u)u(u 2
=⋅∇
+∇+−∇=∇⋅+∂∂ Fp
tμ
ρTraditional Comp. Fluid Dynamics
Discretize macroscopic equations(solve for u, ρ, T)
Kinetic theory
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Microscopic Macroscopic
Lattice Boltzmann Modeling
Thermodynamics
Fluid dynamics
Statistical
treatment
0u
)u(1u)u(u 2
=⋅∇
+∇+−∇=∇⋅+∂∂ Fp
tμ
ρTraditional Comp. Fluid Dynamics
Discretize macroscopic equations(solve for u, ρ, T)
Kinetic theory
Lattice Boltzmann MethodsDiscretize kinetics and recover behavior of macroscopic equations
( f(x,t), particle distribution function )
13 9
2
4
56
7 8
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Lattice Boltzmann Modeling
),v,x( tf rrLudwig Boltzmann
Maxwell-Boltzmann distribution
Kinetic theory
statistical approach for micro → macro
Particle distribution function
Probability of finding a particle
with position x, velocity v, at time t
Equilibrium Distribution ),v,x( tf eq rr
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Lattice Boltzmann Modeling
),x( tfi1
2
3
4
56
7 8
9
D2Q9 lattice i =1:9
particle distribution function in
each lattice direction
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Lattice Boltzmann Modeling
),x( tfi
ρ
ρ
∑
∑=
=
ii
ii
f
f
ivu
1
2
3
4
56
7 8
9
D2Q9 lattice i =1:9
macroscopic output
particle distribution function in
each lattice direction
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Lattice Boltzmann Modeling
),x( tfi
ρ
ρ
∑
∑=
=
ii
ii
f
f
ivu
1
2
3
4
56
7 8
9
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx(
D2Q9 lattice i =1:9
macroscopic output
stream + collide
particle distribution function in
each lattice direction
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LBM ↔ Fluid dynamics
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx(
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LBM ↔ Fluid dynamics
Single relaxation time (BGK)
Equilibrium distributions fieq from specific lattice
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx( ( )),(),(1 txftxf ieq
i −τ
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LBM ↔ Fluid dynamics
Single relaxation time (BGK)
( ) ( )t
xΔ
Δ−=
2
21
31υ τ
Equilibrium distributions fieq from specific lattice
These kinetics give incompressible Navier-Stokes dynamics with:
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx( ( )),(),(1 txftxf ieq
i −τ
τ ↔ kinematic viscosity, ν
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Lattice Boltzmann Modeling
Matlab Central file exchange Iain Haslam (U.Durham -1 page 2D)Google: matlab lattice Boltzmann Google: Iain Haslam
LBM RecipeRead obstacle fileIntialize density distributionLoop for time steps / iterations{Stream fluid ‘particles’Check for obstacles (bounce back)Calc. ρ and vα components, and fα
eq
Collide: ie. calculate relaxationcheck convergence
}Output velocity and density distributions.
e.g. single relaxation time
e.g. simple bounce back
Starting simple!
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Set parameters
Generate porous medium
Stream - calculate fi(x,t)
Calculate ρ and u(x,t)
Calculate fieq(x,t)
Collide/relax towards eqm
Convergence test
Haslam 1-page 2D matlab code
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Take care with obstacle handling and imaging!
Validate against pipe flow and sphere pack geometries.
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~ 110 x 130 pixels here; need features ~ 6+ pixels
Balance resolution vs. sample size: 2-scale problem in sea ice!
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LBM Summary‘Lattice-land’ kinetics lead (quite amazingly!) to Navier-Stokes behavior
(explicit, O(Δt), O(Δx2), finite difference approx to incompressible N/S)
Ingredients:1. lattice2. Collision operator (feq) 3. Obstacle handling conditions
Pros: Complex geometries, highly parallelizable, multi-phase flow, suspension/tracer transport
Care: needed with implementation: lattice choice, boundary handling, collision operator,validation
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1. Obtain 3D internal structure from X-ray Computed Tomographyprimary and secondary pores.
- Jeremy Miner, Hajo Eicken
2. Characterize pore space and connectivity, network modeling- JM, DP, R. Glantz (John Hopkins)
Sea Ice Modeling
3. Lattice Boltzmann Modeling to come: - permeability
- nutrient transport
X-CT Network Modeling
-connectivity
-critical path analysis
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SummarySea ice
> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?
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SummarySea ice
> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?
Lattice Boltzmann Methods> lattice-land kinetics which recover Navier-Stokes behaviour
(~ explicit, O(Δt), O(Δx2), finite difference approx to incomp. N/S)> computationally intensive, but amenable to parallelization
(some caution with boundary handling etc.)
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SummarySea ice
> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?
Lattice Boltzmann Methods> lattice-land kinetics which recover Navier-Stokes behaviour
(~ explicit, O(Δt), O(Δx2), finite difference approx to incomp. N/S)> computationally intensive, but amenable to parallelization
(some caution with boundary handling etc.)
Numerical approach> model flow in imaged real structures - permeability
- nutrient transport> apply method to volcanic samples, insights to other materials
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Acknowledgements
Hajo Eicken, Jeremy Miner
Roland Glantz, John Hopkins University
Greg Newby
ARSC
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Barrow Sea Ice Observatorywww.gi.alaska.edu/BRWICE
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Lattice Boltzmann Modeling
⎥⎦
⎤⎢⎣
⎡−⋅+⋅+=23u)v(
29u)v(31
22 uwf ii
eqi αρ
Single relaxation time (BGK)
( ) ( )t
xΔ
Δ−=
2
21
31υ τ
Equilibrium distribution (D2Q9 lattice)
1
2
3
4
56
7 8
9
( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx( ( )),(),(1 txftxf ieq
i −τ
τ ↔ kinematic viscosity
1/9 a = 1,2,3,4
wα, = 1/36 a = 5,6,7,8
4/9 a = 9
Recover Navier-Stokes with:
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s =
g (s
alt)
/ g (w
ater
)
Temperature [°C]
Effective Medium Approach
)(1
1
bia
i
sii
w
sib
vvvT
v
Tv
+−=
⎟⎠⎞
⎜⎝⎛ −−=
=
ασσ
ρρ
ασ
ρρ
State variables (to be specified)
Salinity
Temperature T
Density ρ (air content)
seaice
salt
mm
=σ
Effective medium property
Component
properties+ +
Geometric model
Weeks & Ackley, 1986