Modelling Glacier Dynamics
Modelling Glacier Flow: Glacier Mechanics
Force balance in a glacier gives:
Gravitational driving stress
Basal stress
Longitudinal stress
(e.g., extension)
Lateral stress
(horizontal shear)
Modelling Glacier Flow: Glacier Mechanics
Glen’s flow law relates stress to strain rates in ice:
where effective viscosity
This allows stresses to be written in terms of strain rates – ice velocity – giving a system of equations for ice flow, e.g. 3D full solutions are possible where you know a lot (e.g. ice and bed topography, climate fields)
3D full solutions are possible for individual glaciers where you know a lot (e.g. ice and bed topography, climate fields)
ice thickness (m)
ice velocity (m/yr)
Modelling Glacier Dynamics
Glacier statistics, BC and AB
1985 2005
Area (km2) 30,063 26,728
Count 14,329 17,595 Bolch et al., 2010
Challenges in Glacier Modelling
Aerial photos and satellite images provide a fabulous view of the surface, but it is only a 2D view.
How to evaluate the third dimension? ‐ glacier thickness
‐ glacier thinning or thickening
Challenges in Glacier Modelling
Aerial photos and satellite images provide a fabulous view of the surface, but it is only a 2D view.
How to evaluate the third dimension? ‐ glacier thickness
‐ glacier thinning or thickening
Hence: Ice volume changes & river runoff
Estimating Ice Volume
Best is to get out there and drag an ice radar system around. For the Canadian Rockies, n = 4
Estimating Ice Volume
Best is to get out there and drag an ice radar system around. For the Canadian Rockies, n = 4
In the absence of data, there are various techniques out there, most commonly  volume‐area scaling: V = cA
d
 local slope‐thickness relations: H = gs
2 km
Glacier extents: 1955 (black) 1985 (yellow) 2005 (red) (Landsat 2005 image)
The basis of V‐A scaling for glacier volume:
Chen & Ohmura (1990), Bahr et al. (1997), Radic
and Hock (2007, 2010), etc. etc.   1.36
We need something like this because ice thickness H is known for less than 0.1% of the world’s glaciers. Short of elaborate methods to estimate H (see Clarke et al., 2015), we are stuck with this empirical relation.
Tests of V‐A scaling: Generate an ensemble of mountain glaciers spanning a wide range of topographic, glaciological, and climatic space
Ice thickness (m)
Ice thickness (m)
Main findings:
Possible to do better: V‐A parameters as a function of topographic and climatic setting.
Global steady‐state value of   1.46
But we are not in equilibrium. It turns out that   1.36 is actually pretty good for glaciers that are 100 years into a sustained retreat.
Alternatives to 3D models or V‐A Scaling: Simple Methods to Model Ice Dynamics
Haig Glacier Hypsometry
Subgrid Ice Dynamics
Qk
Conservation of mass follows: hk
… k
k-1 …
Marshall et al., 2011
Modelling Ice Dynamics
1.5‐ or 2.5‐D flowline models are also possible Modelling Valley Glacier Dynamics:
Improving Flowline Models
‘Form factors’ for different channel geometries
Subgrid Ice Dynamics
Qk
Adhikari and Marshall, 2012a,b
Velocity fields for different channel geometries
Adhikari and Marshall, 2012a,b
Glacier flowline modelling
Ice velocity u (x) along the glacier centreline can be adjusted to represent the ice flux integrated over the valley cross‐section.
Glacier flowline modelling
Similarly, planform shape can be accommodated
Adhikari and Marshall, 2012a,b
Back to Haig Glacier:
Modelling response to climate sensitivity with full and reduced physics
ice thickness (m)
ice velocity (m/yr)
Summary
Glacier dynamics can be modelled for ensembles of glaciers, within a grid cell or a catchment
‐ ‘lumped’
‐ flowline or flowband models of individual glaciers
‐ some combination of these two:
‐‐ ‘representative glacier elements’ (i.e. by aspect class within a grid cell)
‐ can also be fully 2d/3d distributed, but not clear that this makes sense with respect to limited resolution
and knowledge of boundary conditions