The buoyancy boundary condition applied to floating portions of ice sheets and glaciers in Stokes models requires special consideration when the glacier rapidly departs from hydrostatic equilibrium. This boundary condition can manifest in velocity fields that are unphysically (and strongly) dependent on time step size, thereby contaminating diagnostic stress fields. This can be especially problematic for models of calving glaciers, where rapid changes in geometry cause configurations that suddenly depart from hydrostatic equilibrium and lead to inaccurate estimates of the stress field. Here we show that the unphysical behavior can be cured with minimal computational cost by reintroducing a regularization that corresponds to the acceleration term in the stress balance. This regularization provides consistent velocity solutions for all time step sizes.

Stokes simulations are used in glaciology as a tool to determine the time evolution of glaciers

Here we show that a common method used to implement the ice–ocean boundary condition in Stokes models can result in solutions that are unphysically sensitive to the choice of simulation time step size. This behavior manifests in applications that allow for rapid changes in the model domain – a type of change associated with models that allow for instantaneous calving events or crevasses

Denoting the velocity field in two dimensions by

In the Stokes approximation, we drop the acceleration term on the right-hand side of Eq. (

A diagram showing the boundary conditions of an idealized floating ice shelf. The ice–ocean interface is subject to two normal stresses – the depth-dependent water pressure and the numerical damping force for stabilization to hydrostatic equilibrium. The dashed red line illustrates an iceberg that breaks off from the top of the calving front (exaggerated), reducing the freeboard and instantaneously perturbing the ice shelf from hydrostatic equilibrium.

To illustrate an example where the Stokes flow problem rapidly departs from hydrostatic equilibrium, we consider a two-dimensional floating ice shelf (Fig.

Problems arise with this form if the glacier is not

We can more appropriately specify the boundary condition for Stokes flow at the ice–ocean interface by writing it in the form

The additional isostatic adjustment term

Different numerical methods use different techniques to estimate

We illustrate the time step dependence using a floating ice shelf as an example. In this case, global force balance is not guaranteed, leading to a formally ill-posed problem. However, the lack of a solution (which results in an exceptionally large velocity using the sea-spring stabilization) persists even when grounded ice is included in the domain because the velocity (and strain rate) solution can become unphysically sensitive to the position of the ice–ocean interface.

For our test, we implement an idealized rectangular ice shelf of thickness 400 m and length 10 km. This ice shelf is initialized to be in exact hydrostatic equilibrium. We set the inflow velocity for the upstream boundary condition to 4 km a

To emulate the occurrence of a calving event that would perturb the ice shelf from hydrostatic equilibrium, a rectangular section of length 50 m and thickness 20 m is removed from the upper calving front of the glacier (Fig.

The problem is implemented in FEniCS

In Fig.

Maximum

The velocity solution is unphysical because the neglected acceleration term is not actually small relative to the other terms in Eq. (

Restoring the inertial term effectively introduces a Newtonian damping term on the entire body of the glacier where the damping coefficient is

When we include both damping terms, vertical velocity, effective strain rate, maximum shear stress, and greatest principal stress maintain physical values for both small and large time steps (Fig.

Although the sea-spring solution shows a smaller

Our study shows that using a common numerical stabilization method of the ice–ocean boundary in Stokes glacier modeling, there is an explicit time step dependence of the solution that is unphysical for small time steps when the domain rapidly departs from hydrostatic equilibrium. For model applications where changes in the domain are only due to viscous flow, the time step dependence is not problematic as long as domains are (nearly) in hydrostatic equilibrium at the start of simulation. However, for applications where rapid changes to the model domain occur, such as when calving rules are implemented, sudden departure from hydrostatic equilibrium is not only possible but expected. In these cases, time step dependence of the solution will appear. This can contaminate solutions of the stress after calving, potentially leading to a cascade of calving events and an overestimate of calving flux if numerical artifacts are not addressed. However, the time step dependence can be easily cured with little computational cost by reintroducing the acceleration term to the Stokes flow approximation. The acceleration term regularizes the solution for small time step sizes and provides consistent solutions for all time steps.

Code to run the tests in this paper is available at

BB identified the numerical issue with guidance from JB. BB and JB developed the proposed solution to the numerical issue. BB prepared the manuscript with contributions from JB.

The authors declare that they have no conflict of interest.

This work is from the DOMINOS project, a component of the International Thwaites Glacier Collaboration (ITGC). Logistics were provided by NSF's U.S. Antarctic Program and NERC's British Antarctic Survey (ITGC contribution no. ITGC:010).

This research has been supported by the National Science Foundation, Office of Polar Programs (grant no. 1738896) and the Natural Environment Research Council (grant no. NE/S006605/1).

This paper was edited by Valentina Radic and reviewed by Christian Schoof and one anonymous referee.