Fluid flow in Sea Ice

Daniel PringleARSC / Geophysical Institute

Outline of talk

Sea ice background and motivation

Sea ice porosity and permeability

Introduction to Lattice Boltzmann Methods

Sea ice work so far..

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

Figure: Thomas & Dieckmann, ‘Sea, Blackwell Science, 2003

Sea Ice length scales

Antarctic cover Pack ice Pancake Biological Pores microbiologylayering (diatom chain)

‘microstructure’

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

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

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

Inclusions and Porosity

Constitutive super-cooling at growth front: ‘lamellar interface’

Brine incorporated in inter-lamellar spaces

1 cm

Porosity Evolution

1 mm

- 15 C

- 5 C

- 2 C

Light et al, 2003

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.

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

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

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

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.

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

Microscopic Macroscopic

Lattice Boltzmann Modeling

Thermodynamics

Fluid dynamics

Statistical

treatment

Kinetic theory

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

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

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

Lattice Boltzmann Modeling

),x( tfi1

2

3

4

56

7 8

9

D2Q9 lattice i =1:9

particle distribution function in

each lattice direction

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

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

LBM ↔ Fluid dynamics

( ) ttftftttf iiii ΔΩ+=Δ+Δ+ ),x(),x(),vx(

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

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

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!

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

Take care with obstacle handling and imaging!

Validate against pipe flow and sphere pack geometries.

~ 110 x 130 pixels here; need features ~ 6+ pixels

Balance resolution vs. sample size: 2-scale problem in sea ice!

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

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

SummarySea ice

> porous geophysical medium with variable permeability> permeability important to: albedo, nutrient delivery, heat transfer> k(φ) relationship not established; percolation threshold?

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.)

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

Acknowledgements

Hajo Eicken, Jeremy Miner

Roland Glantz, John Hopkins University

Greg Newby

ARSC

Barrow Sea Ice Observatorywww.gi.alaska.edu/BRWICE

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:

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