[PPT] - A continuum model for snow and firn on the surface of ice sheets Ian PowerPoint Presentation

SLIDE 1

A continuum model for snow and firn

n the surface of ice sheets

Ian Hewitt (University of Oxford), Colin Meyer (University of Oregon)

SLIDE 2

Motivation - boundary conditions for ice-sheet modelling

Large-scale (long time-scale) evolution of an ice-sheet model requires surface boundary conditions for mass flux and temperature. Accumulation Runoff These effective conditions differ from averages of the actual surface quantities due to melt percolation and refreezing. We develop a continuum model that allows us to calculate the appropriate conditions for given surface forcing. (cf. IMAU-FDM, SNOWPACK, etc) Surface boundary layer The snow/firn layer on the surface transmits the actual surface conditions to effective surface conditions that the ice-sheet model sees. Temperature

SLIDE 3

The near-surface boundary layer

accumulation melt percolation runoff + refreezing runoff ice flows into ice sheet ice flows out of ice sheet Accumulation area Ablation area compaction accumulation

SLIDE 4

Two easy cases

No melting No accumulation accumulation dry compaction mass flux = accumulation

temp. = mean surface temp.

runoff mass flux = surface melt rate

temp. = mean ‘capped’ surface temp.

SLIDE 5

A continuum model

Velocity < 0 Velocity > 0 (a) Accumulation area (b) Ablation area Runoff

0 < S < 1 S = 1

Porosity

p ηf φ

Saturation

S

Conservation equations Darcy flux

qw = k(φ)kf(S) ηw (ρg rpw) S = 1 1 pw = pc(S) =

r

Energy equation

∂φ ∂t + ui · rφ = m ρi C

Compaction / refreezing

C = p ηf φ C = c0,1(T, a) φ

r

+ mass- & energy-conserving jump conditions

T Temperature

Meyer & Hewitt 2017 The Cryosphere see also earlier work by Colbeck, Gray, Morland, and others

capillary pressure compaction rate

∂ ∂t [(1 φ)ρi] + r · [(1 φ)ρiui] = m ∂ ∂t [φSρw] + r · [φSρwui + ρwqw] = m ∂ ∂t [(1 φ)ρiciT] + r · [(1 φ)ρiciTui] = r · [(1 φ)kirT] mL

refreezing

T < Tm

SLIDE 6

Example - surface melting of a cold stratified snow column

‘Movie’

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 7

Example - surface melting of a cold stratified snow column

‘Movie’

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 8

Example - surface melting of a cold stratified snow column

‘Movie’

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 9

Example - surface melting of a cold stratified snow column

‘Movie’

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 10

Example - surface melting of a cold stratified snow column

‘Movie’

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 11

Example - surface melting of a cold stratified snow column

‘Movie’

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 12

Example - surface melting of a cold stratified snow column

‘Movie’ Runoff Melt Temperature / water content Porosity

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

0.5

Porosity

5 10 15 20

Depth, z [m]

20
10

Temperature, T [C]

SLIDE 13

0.5 1 1.5 2

Time, t [y]

20
10

Teff [C]

Teff(t) =

Periodic forcing experiments

prescribed accumulation (constant)

QS(1 a) + QL σT 4 + hT (Ta T) = k∂T ∂z + msL

prescribed forcing temperature

T eff

in linearised surface energy balance

heff(Teff T) =

∂z

heff(Teff T) = k∂T ∂z + msL

SLIDE 14

Runoff Melt Accumulation Forcing temp. Surface temp. Deep temp. Temperature / water content Porosity

Periodic forcing experiments

SLIDE 15

Runoff Melt Accumulation Forcing temp. Surface temp. Deep temp. Temperature / water content Porosity

Periodic forcing experiments

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Runoff Melt Accumulation Forcing temp. Surface temp. Deep temp. Temperature / water content Porosity

Periodic forcing experiments

SLIDE 17

Runoff Melt Accumulation Forcing temp. Surface temp. Deep temp. Temperature / water content Porosity

Periodic forcing experiments

SLIDE 18

14
12
10
8
6
4

Mean Forcing, Teff [C]

14
12
10
8
6
4
2

Mean Temperature, T [C]

Surface Deep

1 2 3 4

Rate, [mwe/y]

Accumulation Melt Runoff

Annual averages (effective conditions)

Effective surface temperature is elevated and runoff is buffered over an intermediate range of thermal forcing. The size and location of that range depend on accumulation rate.

T eff

Mean forcing Temperature Mass fluxes

SLIDE 19

14
12
10
8
6
4

Mean Forcing, Teff [C]

14
12
10
8
6
4
2

Mean Temperature, T [C]

Surface Deep

1 2 3 4

Rate, [mwe/y]

Accumulation Melt Runoff

Annual averages (effective conditions)

Effective surface temperature is elevated and runoff is buffered over an intermediate range of thermal forcing. The size and location of that range depend on accumulation rate.

14
12
10
8
6
4

Mean Forcing, Teff [C]

14
12
10
8
6
4
2

Mean Temperature, T [C]

Surface Deep

1 2 3 4

Rate, [mwe/y]

Accumulation Melt Runoff

T eff

Mean forcing Temperature Mass fluxes

SLIDE 20

Response to a gradual warming

Temperature / water content Porosity Melt Runoff Temperature

SLIDE 21

Summary

We developed a continuum model that describes melt percolation, refreezing and compaction. We calculate average properties (melt, runoff, temperature) for periodic surface forcing. Elevated firn temperature and buffered runoff are found for intermediate thermal forcing, depending on accumulation rate (even in steady state). Gradual changes in forcing produce complicated evolution - water storage and the initiation of runoff are highly sensitive to specifics of the forcing.

SLIDE 22

Future directions

Compare model with other approaches and with observations. Extend to more dimensions

lateral flow naturally occurs in this model

if the surface or water table is sloped. Heterogenous percolation

in principle the model can produce high

permeability ‘pipes’.

Humprey et al 2012

SLIDE 23