# Numerical Glacier Modelling Glossary

Sometimes, glacier modellers seem to speak a different language. Here I have started a glossary, aimed at people without expertise in glacier and ice-sheet numerical modelling. I am keen to have contributions and suggestions – please comment, email or tweet for inclusions. I have attempted to keep maths to a minimum to ensure that it is more broadly understandable.

This page was developed by Bethan Davies with contributions from James Lea (@JamesMLea).

## A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z

1-Dimensional (1D). A glacier model where the grid is just one cell wide. Will include a width factor to account for ice flux. See also Flowline Model. They can include shape factors or lateral drag parameterisations to simulate width-averaged glacier velocities.

• Refs: Nick et al., 2009, 2010

2-Dimensional (2D).  2D models can include both flowline and plan-view models. Planform, distributed models use the whole domain; they are 2-D in the horizontal plane. These models may be preferable when the influence of topography is important. Flowline models can also be described as 2D if they include vertical and along-flow calculations.

• Refs: Plummer and Phillips 2003.

3-Dimensional (3D). A kind of spatially distributed model using the whole domain with the ice flow calculated for different depths through the ice column.

## A

Ablation. Ablation is the amount of ice lost through the processes of surface melting, iceberg calving, melting at the base of the ice sheet or under ice shelves, sublimation, or any other process by which ice sheet lose mass.

Accumulation. Accumulation is the process by which glaciers or ice sheets gain mass, through snow or rain, or avalanches. Internal accumulation may include rain and meltwater percolating through the snowpack and then refreezing. Basal accumulation may include freezing on of liquid water at the base of the glacier or ice sheet.

Adaptive Grid. Also known as Adaptive Mesh Refinement (AMR). This technique, which varies the grid size in parts of the model domain, allows modellers to use fine resolution only where needed to improve accuracy (such as at the grounding line). This can save substantially on computer power.

• Refs: Cornford et al., 2013.

Altitude tiling scheme. These are often used in earth system or climate models to provide a sub-grid resolution parameterisation of altitude. Sub-grid tiles are binned into categories of altitude. For example, a single grid cell may have 10% at x altitude bin, 50% at y altitude bin, and 40% at z altitude bin.

Artefact. An uncertainty or error arising from incomplete knowledge of the environmental inputs, equations governing ice flow or computational power. May result in an erroneous ice sheet geometry or velocity.

Asymptotic. Glaciers often display asymptotic (limiting) behaviour, for example, in sensitivity experiments and response time tests.

## B

Balance Velocity. In an idealised glacier that is in equilibrium with climate (so the surface profile does not change), the balance velocity is the velocity required to maintain the surface mass balance. Essentially, as much mass must be moved out of the accumulation zone as is gained by accumulation.

Basal sliding. See Sliding.

Buttressing. Ice shelves buttress the glaciers that flow into them. Removal of ice shelves, for example, during catastrophic collapse, changes the membrane stresses and alters the dynamics of the tributary glaciers. They respond by accelerating, receding and shrinking. For tidewater glaciers, calved ice in front of the glacier can re-freeze during winter to form a solid mass that provides backstress, and allow the glacier to seasonally advance. This fjord ice is frequently called melange or sometimes the Greenlandic word sikussak is used.

• Refs: Hindmarsh, 2006; Nick et al., 2013

## C

Calibration. Glacier and ice-sheet models must be calibrated and validated against observations. These are usually observations of glacier length, thickness, volume, ice-surface geometry or ice velocity under a given climatic state. Model parameters must usually be tuned until the simulation matches observations. However, glaciers and ice sheets are rarely in equilibrium with climate, and glaciers are usually in a state of advance or recession; in this case, it may be necessary to dynamically calibrate the simulation against changes in ice volume over a longer period of time.

Calving law. Models must have a means of removing mass once a glacier moves over the grounding line. Calving of icebergs, which are then considered lost to the system, can be parameterised in a number of ways. One of the simplest is to just define a calving rate, which is tuned to observations. Physically based calving laws tend to relate calving rates to the magnitude of the longitudinal strain rate. In Eigen calving, volume loss through calving at the ice front is proportional to the determinant of the strain rate tensor. Benn et al., 2007, formulate a calving law that predicts calving where the depth of surface crevasses equals the ice height above sea level. Nick et al. (2010) expand on Benn et al.’s calving law to incorporate basal crevassing, thus allowing full-thickness calving behaviour, as observed in Greenland (e.g. James et al., 2014).

Coupling. Glacier or ice-sheet models may be coupled to earth-system models, ocean models or atmospheric climate models. Models may be fully coupled or loosely coupled. Full coupling means that outputs from the climate model feed into the ice sheet model (for example, in the form of temperature and precipitation data); the ice-sheet model evolves in response to this. The changes to albedo and land cover are fed as input into an earth system model, and changes in altitude and land-ice and sea-ice cover may be fed into the climate model as input data. The models therefore work together to provide a holistic, if simplified, simulation of the interactions and feedbacks between ice sheets, the ocean and the atmosphere. This is very computationally complex and requires supercomputer access, but is an exciting and growing area of research.

## D

Degree Day Factor.  A degree day factor (also known as the melt coefficient) is the amount of melt that occurs per positive degree day. Degree day factors can be measured on a glacier (for example, from ablation stakes), and are expressed as millimetres water-equivalent per degree per day (mm w.e. °C-1 day-1). So, for a particular glacier, if the degree day factor is 2 mm w.e. °C-1 d-1, then in one day, if the average daily temperature is +1°C, then 2 mm water-equivalent of melt will be observed on ablation stakes on the glacier’s surface. If the average temperature is +2°C, then 4 mm w.e. of melt will be observed.

• Refs: Golledge and Levy 2011.

Degree Day Model. A Degree Day Model is a means of calculating glacier surface melt. A degree day model assumes that, for each 1°C over 0°C, a certain depth of snow (measured in millimetres water equivalent, mm w.e.) will be melted. A degree day model therefore takes into account the amount of energy available for melting over the course of the year (the total sum of daily average temperatures above 0°C over one year, the positive degree day sum or PDD), as well as how much melt occurs per degree (the degree-day factor).

Depth-dependence. These are model variables that change with depth. They are typically implemented for SIA, higher-order and 3D models. Examples of such processes can include, but are not limited to ice temperature, stress, velocity components, ice crystal fabric and water content.

Depth-integrated. See vertically integrated.

Diagnostic modelling. In a diagnostic simulation, ice-flow mechanics and other factors are explored. They may isolate a small part of an ice sheet (such as the calving front or grounding line) to study it in great detail. They may investigate the physics of a specific process in detail. This serves to highlight the importance of certain mechanisms, improve the calculations of ice flow or mass balance, and provide insight into processes.

• Refs: Huybrechts 2007.

Driving stress. The force making ice flow in the direction of decreasing ice-surface slope.

Domain. The Model Domain is the extent or region covered by the model.

Dynamic. A dynamic model (as opposed to a static model) calculates changes in thickness through time (dh/dt). A static model is a reconstruction for a single point in time.

• Refs: Benn and Hulton (2010)

Dynamic Calibration. In a dynamic calibration, a model is forced with observed climate data and tuned until the glacier replicates observed glacier fluctuations over a specific timescale. This is useful, because glaciers are rarely in balance with the climate, and so tuning a model to fit a specific glacier geometry is unlikely to yield an accurate representation of the glacier.

• Ref: Oerlemans (1997).

## E

E-folding time. In numerical glacier modelling, this usually refers to response-time tests. The E-folding time is the time taken for a glacier to achieve two-thirds of its ice-volume change after receiving an initial perturbation. For example, a modeller may impose a step-change in temperature and observe the time taken for the glacier model to attain a steady-state equilibrium profile. This may be non-linear and the final small changes in ice thickness may be on-going for some time; the e-folding time (the time interval in which an exponentially growing quantity increases by a factor of e; it is the base-e analog of doubling time) is therefore used to compare response time.

• Refs: Anderson et al., 2008; Leysinger Vieli and Gudmundsson, 2004.

Enthalpy. This is the sum of the internal energy and the product of pressure and volume (Enthalpy = energy of the system + (pressure * volume))

## F

Finite Difference. In maths, finite difference methods are numerical solutions for approximating the solutions of differential equations.

Fixed Grid. In a fixed grid, the grounding line is not defined explicitly, but must fall between grid points, generally referred to as a sub-grid parameterisation. Fixed grid models may struggle to generate accurate grounding line positions during transient runs. Can be either Fixed Grid or Fixed Irregular Grid (FIG). Includes nested grid representations of the grounding line.

• Refs: Pattyn et al. 2012.

Flowline model. A type of numerical model that simulates ice flow along a flow line (one grid-cell wide), as opposed to a spatially distributed model. This is a simpler form of model that is computationally less expensive. Flowline models can include a width-dependent shape-factor to ensure conservation of mass as the valley widens or narrows. They can be very useful for modelling valley glaciers or ice streams within a larger ice sheet. There are different kinds of flowline model depending on the physics used to simulate ice flow:

1D: Depth-integrated (SSA or SIA) / vertically integrated

2D: flowline with a grid mesh where the ice-flow solution is calculated for different depths through the ice thickness. Physics may be SIA, SSA, hybrid or Full Stokes. Lateral drag can be parameterised, but not as effectively as a full 3D model.

• Refs: Davies et al., 2014; Jamieson et al., 2012; Enderlin et al., 2013; Golledge et al., 2014a.

Flux. Ice discharge through a certain point or gate, taking into account the thickness of the ice column and velocity profile through the ice column. Also known as change in thickness through time (dh/dt).

Forcing. Glacier models are often forced with prescribed climatic data (as opposed to full coupling to an atmosphere-ocean model). These may take the form of annual temperature and precipitation data (e.g., Davies et al., 2014).

Fortran. A programming language used in many ice sheet and climate models. See also: Python.

Full Stokes. The complete description of all stresses acting on ice, solved across the whole domain. It includes the three normal stresses and six shear stresses. Determining the full stress field and its evolution through time and space requires solving the Stokes equations. Developed by George Stokes (1819-1903).

E.g., Elmer Ice.

## G

Geothermal heat flux.  The amount of energy received by a glacier or ice sheet at its base through geothermal heating is the Geothermal heat flux. This model parameter is essential as it strongly affects the amount of melting at the base of the ice. The amount of water at the ice-bed interface controls basal sliding.

GLINT. A module in the ice sheet model GLIMMER that allows coupling to a climate model or to climate reanalysis data. It automatically handles the various temporal and spatial transformations necessary.

Glen’s Flow Law. Glen’s Flow Law is the corner stone of glaciology and describes how individual ice crystals deform in response to stress. Glen’s Flow Law considers ice as a nonlinear viscoelastic fluid (exhibits both viscous and elastic characteristics when undergoing deformation), and relates strain rates to stresses mostly to the third power. The rate of deformation for a given stress depends on the temperature and fabric of the ice; warmer ice deforms more easily. In ice sheet models, this effect of temperature is normally accounted for by adopting a temperature-dependent rate factor. If the ice temperature is calculated simultaneously with the velocity field, the model is said to be thermo-mechanically coupled.

Grounding Line. The Grounding Line is the point at which a marine-terminating glacier or ice stream begins to float.

## H

Higher-Order Model. See also Second Order Model. These are numerical models generally based on the Shallow Ice Approximation (though theoretically also possible for the Shallow Shelf Approximation) that include additional stress terms, especially longitudinal stresses and transverse stresses (deviatoric stress gradients). These models can be coupled to simulate sheet and ice flow. They are a simplification of the Full Stokes numerical scheme. Also known as Multilayer Longitudinal Stresses (LMLA).

Example: Blatter-Pattyn type models that consider the hydrostatic approximation in the vertical direction, by neglecting the vertical resistive stresses.

• E.g., BISICLES.
• Refs: Pattyn, 2003; Hindmarsh 2004.

Hybrid Models. These models include results from the Shallow Ice and Shallow Shelf approximations, each used / superimposed where most appropriate, coupled across the grounding line. In the hybrid SSA-SIA model, the SSA model is used as a basal sliding law for the SIA model, with zero basal friction for the ice shelf. Also known as “HySSA”.

Hysteresis. This is the dependence of the output of a system (such as a numerical model) not only on its current input, but also on the history of its past inputs. The history affects the value of the internal state of the model. To predict future outputs, either the internal state or history must be known.

• Ref: Durand et al., 2009.

## I

Ice shelf. The floating extent of land ice. Ice shelves buttress onshore glaciers and are important for ice dynamics; the majority of ice ablation in Antarctica is through basal melting and calving of ice shelves.

Inputs. Ice sheet and glacier models need to be initialised with input data, which, depending on model complexity, includes parameters such as the bed topography of the domain in metres above sea level, (including ice thickness where there is ice), temperature and precipitation at sea level (many more additional factors in energy balance models, such as wind speed, insolation, etc), relative sea level, sea surface temperatures, geothermal heat flux, constant parameters such as the thermal conductivity of ice, precipitation and temperature lapse rates, density of ice and water, gravitational acceleration, density of the mantle, and so on. Where input parameters are poorly known, they need to be tuned against observations.

Intercomparison. Ice sheet models often undergo an intercomparison exercise, where they are given the same input data and are assessed as to their output data, often validated against observational data.

• Refs: Pattyn et al., 2012

Isothermal. A model where the ice simulated is all the same temperature (see Glen’s flow law, and thermo-mechanically coupled)

## K

k. Extinction coefficient for light in ice, used in energy balance models.

## L

Longitudinal stress. When basal motion dominates internal ice dynamics, shearing and longitudinal stretching become important. Longitudinal stress modifies basal shear stress and is the pushing or pulling effect of the ice. Accelerating ice imposes tensile stresses, and decelerating ice imposes compressional stresses. This may result in brittle fracture of the ice, and can be seen in crevasses as the ice flows over a steeper part of the bed, for example. Most simple, shallow ice approximation-based numerical models do not solve longitudinal stresses directly, though can have this parameterised.

• Refs: Cuffey and Patterson, 2010

## M

Mass Balance. Mass Balance is simply the gain and loss of ice from the glacier system. It is the sum of the processes of accumulation and ablation.  Mass balance (b) is the product of accumulation (c) plus ablation (a).

b = c + a

Mass balance is usually given in metres water equivalent (m w.e.). It does not include mass changes due to ice flow.

• Surface mass balance is the net balance between the processes of accumulation and ablation on a glacier’s surface (it does not include dynamic mass loss and basal melting).
• Climatic mass balance includes surface mass balance and internal accumulation.
• Ice dynamical changes may include changes to ice discharge and acceleration or deceleration of flow, which can lead to dynamic thinning or thickening, ice-shelf collapse, marine ice sheet instability, and other factors resulting in changes in the glacier beyond surface mass balance.
• Further reading: the excellent Glossary of Mass Balance and Related Terms.

Model. A numerical glacier or ice-sheet model solves a number of sequential partial differential equations governing the processes of mass balance and ice-flow. Models are therefore usually computer programs that can quantitatively simulate ice thickness evolution over time (dh/dt) or simulate a certain process (prognostic or diagnostic modelling). Models can range in complexity, depending on the size of the domain, whether they are flowline or spatially distributed / planform models, the complexity of their mass balance calculations, the size of the domain (small valley glacier versus large ice sheet), and the degree to which the calculations governing ice flow have been approximated. They may be coupled to climate or earth-system models, so that feedbacks and links between the ice sheet and climate can be accounted for, or they may simply have climatic inputs prescribed.

Monotonic. In maths, monotonic means that successive members of a sequence either consistently increase or decrease, but do not oscillate in relative value. Each member of a monotone increasing sequence is greater than or equal to the preceding member; each member of a monotone decreasing sequence is less than or equal to the preceding member.

• Refs: Parizek and Alley, 2004; Durand et al., 2009

Moving Grid. A grid that tracks the grounding line explicitly. Adaptive mesh refinement can improve accuracy.

• Sometimes called stretching grid models.
• Refs: Pattyn et al., 2012; Jamieson et al., 2012

## N

Non-unique solutions. Often numerical models will be capable of providing very similar solutions using different parameter values that are used to scale the model inputs. The solutions provided are therefore non-unique.

## O

Outputs. Prognostic models, numerical models calculate the change in ice thickness through time, so the outputs are essentially the change in glacier extent, thickness, geometry and velocity following a perturbation. This may include time-dependent simulations (temperature, precipitation and other inputs vary through time, perhaps based on proxy archive data), or sensitivity experiments or response time tests, where step-changes in parameters are applied. These outputs may take a range of forms, depending on the model code, but are usually some form of text file that outputs changes in mass balance, velocity and ice thickness.

## P

Parallelised. As in the Parallel Ice Sheet Model (PISM); this means that the code is suitable for running on a parallel processing system. Many calculations can be run simultaneously.

• Refs: Winkelmann et al., 2011

Parameter. Numerical models have multiple inputs or measureable factors, such as temperature, sea level, calving coefficient, sliding coefficient and so on. These parameters form a set that together define the system.

Parameterisation. Part of a model where the precise physics are either not directly calculated, or not known, but are represented by an empirically derived relationship. Examples of this include the incorporation of longitudinal stresses in shallow ice approximation based models, and simulating the grounding line position at a greater accuracy than that of the model itself

Positive degree day (PDD) scheme. Melting is parameterised as a function of the sum of daily air temperatures above the melting point. The positive degree day sum (PDD) is the sum total of daily average temperatures above 0°C in a given time period (normally measured over one year). For example, if over the course of one week, daily temperatures are: 0°C, -2°C, 1°C, 1.5°C, -3°C, 0°C, and 2°C, then the PDD is 4.5°C for that week (1°C + 1.5°C + 2°C). Melt on snow or ice is assumed to be minimal below 0°C, so only temperatures above 0°C are added together. PDD can therefore be thought of as the total energy available for melting snow and ice over the course of one year.

• Refs: Huybrechts and Oerlemans, 1990.

Prognostic Modelling. Prognostic (or predictive) modelling is a time-dependent (time-evolving) simulation where an ice sheet model evolves in response to a forcing. They predict the evolution of ice thickness and geometry over time. The forcing may be past air or ocean temperatures from proxy records or past or future climate model outputs. In coupled simulations, the climate model may evolve in response to outputs from the ice sheet; this is useful for understanding feedbacks and links.

• Refs: Golledge et al., 2014b; Huybrechts 2007.

Python. A scientific programming language often used for coding ice-sheet models. Fortran and C++ are also frequently used.

## R

Regional climate model (RCMs).  Simulate atmosphere and ocean over a limited domain, with higher spatial and temporal resolution than a General Circulation Model (GCM). RCMs are forced at their lateral boundaries with GCM simulations or reanalysis data (such as ERA-40). They are computationally expensive.

Resistive stresses. Resistive forces acting at the boundaries of an ice mass, that resist the Driving Stress. These forces include the bed of the glacier (basal drag), the lateral margins (lateral drag), and the up- and down-glacier ends (gradients in longitudinal stress). In the interior parts of ice sheets, the force balance is essentially between the driving stress and basal drag, as predicted by the Shallow Ice Approximation. In ice shelves, there is zero basal drag and the force balance is between the driving stress and longitudinal stresses and lateral drag, as predicted by the Shallow Shelf Approximation.

• Refs: Huybrechts 2007.

Resolution. The grid size or cell size of a model’s domain. Resolution may vary in adaptive mesh or moving grid models, as a higher resolution is required in areas with complex physics (for example, grounding lines).

Response time. The Response time is the time span needed for a glacier to find a new
steady state after a change in mass balance. It can be defined as the time constant in an exponential asymptotic approach to a final steady state after a sudden change in
climate to a new constant climate. A response time test investigates the time taken for a glacier to attain equilibrium following a perturbation, such as a step change in temperature or precipitation.

• Refs: Anderson et al., 2008; Davies et al., 2014; Leysinger Vieli and Gudmundsson 2004.

## S

Second-Order Model. See Higher Order Model.

Sensitivity experiment. Glacier modellers need to test the parameter space around various glaciological, model and climate parameters. What is the influence of a key variable? How important is it? What effect does it have on the glacier’s simulated output?

• Refs: Doughty et al., 2013; Anderson et al., 2008; Enderlin et al., 2013

Shallow Ice Approximation (SIA). The SIA solves an approximation of the force balance and mass continuity equations. This approximate model assumes a small aspect ratio of vertical to horizontal length and a large ratio of vertical to horizontal stress (longitudinal derivatives of stress, velocity and temperature are assumed to be small compared with vertical derivatives). It is computationally very efficient and good for simulating sheet flow within ice sheets where ice surface slopes are evaluated over horizontal distances at least an order of magnitude greater than ice thickness. The SIA neglects vertical shear.

“Shallowness” refers to the small depth to width ratio of ice sheets, which simplifies equations and reduces computational cost. The SIA is not relevant for key areas, such as ice divides, ice streams or grounding lines, since it excludes membrane stress transfer across the grounding line.

• E.g., GLIMMER.
• Refs: Pattyn et al., 2013, Pattyn et al., 2012, Pattyn 2003, Hindmarsh and Le Meur 2001, Bueler and Brown 2009.

Shallow Shelf Approximation (SSA). The SSA is primarily for modelling large ice shelves or ice streams with zero till strength. In ice sheets, the SIA is poor at modelling rapid flow speeds as it neglects longitudinal stresses. Fast ice flow arises from a combination of sliding over the glacier bed and the deformation of the subglacial sediments and basal ice (basal slip). The SSA may be used to approximate basal sliding and fast ice-flow. This approximation allows vertical integration and results in essentially a 2-D model in plan form, or a 1-D model in flowline. It can be applied to ice streams, ice shelves and other areas of fast ice flow in the margins of an ice sheet.

Shape factor.  Where a model calculates ice flow along a centreline, such as in a flowline model, one needs to account for variations in ice flux resulting from variations in speed across the glacier. As the down-slope weight of the glacier is supported by shearing along the horizontal and vertical planes, we need to account for the boundary conditions. These are usually given by a width-dependent shape factor (f), using tabulated values for a parabola (Cuffey and Patterson, 2010). The shape factor gives the correct value of the surface velocity when the equations governing ice deformation are integrated over depth.

• Ref: Cuffey and Patterson, 2010.

Sliding. Sliding at the ice-bed interface. Basal slip may include ice sliding on a thin layer of lubrication at the ice-bed interface, and deformation of the uppermost glacial sediments. Basal slip is necessary for fast ice flow and is difficult to approximate with the SIA. A realistic basal boundary condition is yet to be developed for use in ice sheet models.

• Refs: Cuffey and Patterson, 2010.

Spatially Distributed Model. This kind of numerical model uses the whole domain; they are 2-D in the horizontal plane (may also be called planform model). These models may be preferable when the influence of topography is important. The kind of model and complexity depends on the physics used.

2D spatially distributed models are depth-integrated (SSA, possibly with deformation parameterised). They average processes over the vertical extent. Depth-integrated models are also known as vertically integrated models. Examples of such processes include ice temperature, stress, velocity components, ice crystal fabric and water content.

3D models calculate the ice flow solution for different depths through the ice column. May use the SIA, SSA/hybrid, or Full Stokes solutions. They incorporate vertical processes explicitly. These are known as 3-D Thermo-mechanically coupled models, and they are at the top end of model complexity and accuracy.

• Refs: Plummer and Phillips 2003.

Spinup. Before conducting sensitivity experiments, response time tests, or forcing an ice sheet model with climate data, modellers may want their ice sheet model to be in equilibrium with its given climate. They will therefore initialise the model under certain climatic conditions and allow it to reach equilibrium before applying different tests. This allows variations in ice surface topography or temperature resulting from the domain and model parameterisation to be smoothed out, and ensures that the model is in equilibrium with its climate, meaning that you can accurately assess its response to the forcing.

Steady-state. Steady-state conditions are often applied during numerical models, for example, during spin-up, sensitivity experiments or during tuning, or perhaps because the modeller wants to investigate a particular process. These simulations may or may not be time-dependent, but input parameters (such as sea and air temperatures, precipitation etc) are held constant and do not vary in time or space.

Strain. Strain is the change in shape of an object (e.g., glacier ice), following the application of stress. More information on glacier stress and strain here.

## T

Thermo-mechanically coupled. As ice viscosity is temperature dependent (ice deforms more readily at warmer temperatures), models that integrate the change in temperature through the ice column (from thermal conductivity of ice) and use this to inform ice deformation in the ice column are said to be thermo-mechanically coupled.

Time-dependent. When modelling ice volume evolution through time, where the model’s climate forcing varies, the ice-sheet model is said to be time dependent. Models may also be called Transient.

Tuning. When a numerical glacier or ice sheet model is parameterised, it will normally be tuned to observations; these may be modern observations of glacier ice thickness, velocity, extent, under certain observed climatic conditions.Glacier and model parameters are systematically varied until the glacier fits observations; there is likely to be only a very small number of combinations of model parameters that allow the glacier to fit observations. Where different combinations of parameter values provide similar outputs, these are referred to as non-unique solutions. The more observations you have, the better-tuned your model is likely to be. Sensitivity experiments are then normally carried out to fully explore the parameter space and understand the influence of these model parameters.

Note that the model is normally tuned so that it is in equilibrium under these climatic conditions. However, glaciers are rarely in equilibrium; they are usually either in a state of advance or recession. Dynamic calibration is a means of tuning a model to a longer-term record of changes in glacier extent and velocity.

## U, V

Velocity. The speed of a glacier. Surface ice velocities are often denoted by ‘u’. May be derived from ice deformation or basal sliding components (ud or us).

Vertically integrated. The SSA is vertically integrated, with ice-flow averaged through the ice column. These depth-integrated models average processes over the vertical extent. Examples of such processes include ice temperature, stress, velocity components, ice crystal fabric and water content.

Viscosity. Ice deforms more readily at warmer temperatures; this is said to be the viscosity of ice.

## W, X, Y, Z

Weertman’s Theory of Sliding. Our modern understanding of how glaciers slide over hard beds dates from Weertman (1957). He proposed two mechanisms by which ice at the pressure melting point moves past bumps in the rigid bed. The first mechanism is regelation. Pressure increases upstream of bedrock bumps, which resist ice movement. Ice is colder upstream than downstream of the bump, and heat flows upstream through the bump. The ice melts on the upstream side and refreezes downstream.

The second mechanism is enhanced creep. Ice deforms viscously, but bumps interfere with the ice flow and produces an excess stress that increases deformation. The ice stretches and compresses to move over and around the bump.