A glacier is a flowing body of ice. Glaciers in Antarctica, including Thwaites, flow out into the sea just as rivers do. The technical term for this is a marine-terminating glacier. Broadly speaking, the glacier can be split into two sections: the part which is in contact with the bedrock (ice sheet) and the part which floats on the ocean (ice shelf). The boundary between these sections is known as the grounding line. Different physical processes affect these two sections of the glacier, which makes the position of the grounding line a very important factor in the dynamics of the glacier.

Mathematical models are used across many disciplines of science to investigate how systems change over time. Ice flow models can be used to predict how glaciers will behave in the future, or how they evolved in the past. We use powerful computers to perform the complex calculations required quickly and efficiently.

The model we use at Northumbria is called Úa. It calculates the ice velocities and thickness using a range of inputs we provide from both observational data and mathematical parameterisations representing physical processes. Before starting experiments, we set up a mesh which represents the area we are interested in. This mesh is made up of triangular elements, which gives us a lot of flexibility to control the size of elements in different parts of the domain. A calculation is performed for each mesh element, so the smaller the elements in an area the more detailed our results will be. The triangular structure also means the boundaries can be any shape we choose.

Once the domain is set up, we give the model information about the initial state of the ice and what is happening at the boundaries of our computational domain, then we run it forward in time. The model solves the system of equations which describe ice flow over a series of time steps, each time updating the relevant values before running the calculation again further into the future. We can instruct it to produce output for any particular time we are interested in.

An important part of the ice system is how it interacts with the ocean. Floating ice shelves at the ends of glaciers are the site of major factors influencing ice flow. The temperature of the ocean can cause ice to melt, or new ice to form from freezing seawater. Many simplistic parameterisations exist which allow ice flow models to account for this, but the best way to get an accurate idea of how ocean interactions affect the ice over time is to use coupled models.

Just as models are used to predict ice flow, there are also models used to simulate ocean circulation. We can use these to obtain values for the velocities, temperature and salinity of the water which comes into contact with ice shelves. Coupling ice and ocean models together can be a difficult process, and uses more computational time and resources than running an individual ice flow model. However, the results are generally more realistic.

At Northumbria, we couple the ice flow model Úa with the ocean circulation model MITgcm when performing coupled model runs.

Inverse modelling, or inversion, is an important part of initialising our model. This is an iterative process by which we derive values for properties of the ice flow which cannot be directly observed or measured. As an example, we use inversion to work out the basal friction coefficient, C, based on observed surface velocities.

The process begins with an initial guess at the C field. This can be anything from a single value applied across the entire ice sheet to a complex distribution based on previous knowledge of the area. This guess is then used as an input with which the model calculates the velocity field. After the calculation, the inversion algorithm finds the difference between the calculated velocities and the known measurements, known as the misfit. It will then adapt the initial guess for C with the aim of making the misfit smaller, and calculate velocities using its new version of C. This process repeats until the misfit is sufficiently reduced. The final C field, which provides a good match to observed velocities, can be used as a boundary condition when running the model forward in time to predict the future.

A simplified sketch of the results of this procedure is shown in the figure to the right, with C₀ being the initial guess, and the adapted guesses C₁ and C₂ producing results which get closer to the measurements. It should be noted that while this illustrates the basic concept, the inversion processes used in models are more complicated than the curve fitting illustrated here.