# Modelling mountain glaciers

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## Introduction

The physics governing the behaviour of mountain glaciers is identical to that governing ice sheets - both in terms of the interactions between the atmosphere and the snow/ice surface and the physics of glacier flow. However the complex topography and steep slopes that characterise mountain areas mean that the surface mass balance and flow tend to be treated in slightly different ways.

The evolution of an ice mass can be described $\frac{\partial H }{\partial t} = M - \nabla \cdot q$

where q is the ice flux and $M = M_{acc} + M_{abl}$ is the mass balance. $M_{acc}$ is the amount of accumulation, and $M_{abl}$ the ablation.

## Modelling mass balance

Mass balance depends on the local climate of the glacier or ice sheet. As glaciers and ice sheets are often in remote areas this step can be difficult. Different ways of obtaining relevant climate data are:

• single climate station combined with lapse rates
• weighted average of mulitple stations
• two-dimensional interpolation from multiple stations
• downscaled reanalysis/GCM data
• mesoscale atmospheric model output

Modelling accumulation

The interactions between the atmosphere and the snow/ice surface can be complex. The amount of snow that accumulates has high spatial variability in high-relief terrain which is difficult to model because of the influence of wind and avalanche transport. A scheme which captures the bulk effect of the processes that govern snow accumulation, if not the processes themselves, is to simply use a snow/rain threshold temperature $T_{snow}$ (often ~ $1^oC$) combined with measured or modelled precipitation. The effect of wind and avalanche transport may be determined empirically. $M_{acc} = \left\{ \begin{array}{l l} p \cdot f & \quad {if T \leq T_{snow}}\\ 0 & \quad {if T > T_{snow}}\\ \end{array} \right.$

Modelling ablation

The amount of snow and ice that melts is governed by the energy balance at the surface: $Q_m = I(1-\alpha) + L_{out} + L_{in} + Q_H + Q_E + Q_R + Q_G$

where $Q_m$, the energy available for melt, is balanced by

• $I(1-\alpha)$ the incoming solar radiation $I$, less the amount reflected from the snow/ice surface which is controlled by the surface albedo $\alpha$
• $L_{out}$ the outgoing long-wave radiation
• $L_{in}$ the incoming long-wave radiation
• $Q_H$ the sensible heat flux
• $Q_E$ the latent heat flux
• $Q_R$ the rainfall heat flux
• $Q_G$ the subsurface (ground) heat flux

All of these components can be calculated using empirical or theoretical relationships, and are often calculated at high temporal resolution (e.g. hourly). However, there is a large requirement for detailed climate data which can be hard to meet. See Hock (2005) for a detailed review of how these components are usually calculated.

Positive Degree-day models

Rather than a data intensive energy balance model, many studies use a simpler degree-day model to calculating surface melt. This approach relies on the strong statistical relationship between snow/ice melt and positive temperature sums. This relationship holds even when, as in continental areas, the energy balance is dominated by radiation. The only input data required are temperature. $M_{abl}= \sum_{}^{} T_{pos}\cdot f$

The positive temperature sum PDD is typically calculated monthly (using daily data or an assumption about the distribution of temperature within the month) by summing positive temperatures. A different degree-day factor $f$ is used for snow and ice surfaces, reflecting the different albedo and surface roughness characteristics.

## Modelling glacier flow

While the equations of glacier flow are the same for mountain glaciers as for ice sheets, there are some important differences in the way that they are applied. For example, the thermodynamics of glaciers in temperate areas is usually neglected and a constant ice deformation factor is used. However, the shallow ice approximation which is commonly used in an ice sheet setting is not valid for the majority of mountain glaciers with their steep bed slopes and high aspect ratio. Various types of models have been applied to mountain glaciers to understand the link between climate and glacier response, from simplified analytical models (e.g. Klok and Oerlemans, 2003), one-dimensional flowline models (e.g. Oerlemans, 1997), two-dimensional SIA models (e.g. Plummer and Phillips, 2003), and two-dimensional models with longitudinal stresses (e.g. Golledge and hubbard, 2005).

References

Golledge, N. R., and A. Hubbard. 2005. Evaluating Younger Dryas glacier reconstructions in part of the western Scottish Highlands: a combined empirical and theoretical approach. Boreas 34 (August): 274-286. doi:10.1080/03009480510013024.

Hock, Regine. 2005. Glacier melt: a review of processes and their modelling. Progress in Physical Geography 29, no. 3: 362-391. doi:10.1191/0309133305pp453ra.

Klok, E. J., and J. Oerlemans. 2003. Deriving historical equilibrium-line altitudes from a glacier length record by linear inverse modelling. The Holocene 13, no. 3: 343-351. doi:10.1191/0959683603hl627rp.

Oerlemans, J. 1997. A flowline model for Nigardsbreen, Norway: Projection of future glacier length based on dynamic calibration with the historic record. Annals of Glaciology 24: 382-389.

Plummer, Mitchell A., and Fred M. Phillips. 2003. A 2-D numerical model of snow/ice energy balance and ice flow for paleoclimatic interpretation of glacial geomorphic features. Quaternary Science Reviews 22, no. 14: 1389-1406. doi:10.1016/S0277-3791(03)00081-7.