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

A continuum model for snow and firn on the surface of ice sheets Ian Hewitt (University of Oxford), Colin Meyer (University of Oregon)

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 . Temperature Accumulation Surface boundary layer Runoff The snow/firn layer on the surface transmits the actual surface conditions to effective surface conditions that the ice-sheet model sees. 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)

The near-surface boundary layer Ablation area Accumulation area accumulation accumulation runoff runoff melt percolation + refreezing compaction ice flows out of ice sheet ice flows into ice sheet

Two easy cases No melting No accumulation accumulation runoff dry compaction mass flux = accumulation mass flux = surface melt rate temp. = mean ‘capped’ surface temp. temp. = mean surface temp.

A continuum model Conservation equations T Temperature ∂ p Porosity ∂ t [(1 � φ ) ρ i ] + r · [(1 � φ ) ρ i u i ] = � m φ η f refreezing Saturation S ∂ ∂ t [ φ S ρ w ] + r · [ φ S ρ w u i + ρ w q w ] = m (b) Ablation area (a) Accumulation area Runoff Darcy flux 0 < S < 1 q w = k ( φ ) k f ( S ) ( ρ g � r p w ) η w capillary pressure or 1 p w = � p c ( S ) = S = 1 Compaction / refreezing ∂φ ∂ t + u i · r φ = m compaction rate � C S = 1 ρ i C = p or φ C = c 0 , 1 ( T, a ) φ η f Energy equation Velocity > 0 Velocity < 0 ∂ T < T m ∂ t [(1 � φ ) ρ i c i T ] + r · [(1 � φ ) ρ i c i T u i ] = r · [(1 � φ ) k i r T ] � mL + mass- & energy-conserving jump conditions Meyer & Hewitt 2017 The Cryosphere see also earlier work by Colbeck, Gray, Morland, and others

Example - surface melting of a cold stratified snow column ‘Movie’ 0 5 Depth, z [m] 10 15 20 0 0.5 -20 -10 0 Porosity Temperature, T [C]

Example - surface melting of a cold stratified snow column ‘Movie’ 0 0 5 5 Depth, z [m] Depth, z [m] 10 10 15 15 20 20 0 0 0.5 0.5 -20 -20 -10 -10 0 0 Porosity Porosity Temperature, T [C] Temperature, T [C]

Example - surface melting of a cold stratified snow column ‘Movie’ 0 0 0 5 5 5 Depth, z [m] Depth, z [m] Depth, z [m] 10 10 10 15 15 15 20 20 20 0 0 0 0.5 0.5 0.5 -20 -20 -20 -10 -10 -10 0 0 0 Porosity Porosity Porosity Temperature, T [C] Temperature, T [C] Temperature, T [C]

Example - surface melting of a cold stratified snow column ‘Movie’ 0 0 0 0 5 5 5 5 Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] 10 10 10 10 15 15 15 15 20 20 20 20 0 0 0 0 0.5 0.5 0.5 0.5 -20 -20 -20 -20 -10 -10 -10 -10 0 0 0 0 Porosity Porosity Porosity Porosity Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C]

Example - surface melting of a cold stratified snow column ‘Movie’ 0 0 0 0 0 5 5 5 5 5 Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 -20 -20 -20 -20 -20 -10 -10 -10 -10 -10 0 0 0 0 0 Porosity Porosity Porosity Porosity Porosity Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C]

Example - surface melting of a cold stratified snow column ‘Movie’ 0 0 0 0 0 0 5 5 5 5 5 5 Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] 10 10 10 10 10 10 15 15 15 15 15 15 20 20 20 20 20 20 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 -20 -20 -20 -20 -20 -20 -10 -10 -10 -10 -10 -10 0 0 0 0 0 0 Porosity Porosity Porosity Porosity Porosity Porosity Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C]

Example - surface melting of a cold stratified snow column Runoff ‘Movie’ Melt 0 0 0 0 0 0 Temperature / water content 5 5 5 5 5 5 Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] Depth, z [m] 10 10 10 10 10 10 15 15 15 15 15 15 Porosity 20 20 20 20 20 20 0 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 0.5 -20 -20 -20 -20 -20 -20 -10 -10 -10 -10 -10 -10 0 0 0 0 0 0 Porosity Porosity Porosity Porosity Porosity Porosity Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C] Temperature, T [C]

Periodic forcing experiments prescribed accumulation (constant) 0 T eff [C] prescribed forcing temperature T e ff ( t ) = T e ff -10 -20 0 0.5 1 1.5 2 � Time, t [y] ∂ z h e ff ( T e ff � T ) = k ∂ T in linearised surface energy balance ∂ z + m s L Q S (1 � a ) + Q L � σ T 4 + h T ( T a � T ) = k ∂ T ∂ z + m s L h e ff ( T e ff � T ) =

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

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

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

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

Annual averages (effective conditions) 4 Mass fluxes Accumulation Melt 3 Runoff Rate, [mwe/y] 2 1 0 0 Temperature Mean Temperature, T [C] -2 -4 -6 -8 -10 -12 Surface Deep -14 -14 -12 -10 -8 -6 -4 Mean Forcing, T eff [C] Mean forcing T e ff 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.

Annual averages (effective conditions) 4 4 Mass fluxes Accumulation Accumulation Melt Melt 3 3 Runoff Runoff Rate, [mwe/y] Rate, [mwe/y] 2 2 1 1 0 0 0 0 Temperature Mean Temperature, T [C] Mean Temperature, T [C] -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 -12 -12 Surface Surface Deep Deep -14 -14 -14 -14 -12 -12 -10 -10 -8 -8 -6 -6 -4 -4 Mean Forcing, T eff [C] Mean Forcing, T eff [C] Mean forcing T e ff 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.

Response to a gradual warming Temperature Melt Runoff Temperature / water content Porosity

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.

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

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