Accurate modelling of basal melting beneath ice shelves is key to reducing the uncertainty in forecasts of ice shelf stability and, thus, the polar contribution to sea level rise. Numerical ocean models are important tools to investigate these critical and remote regions. The relative inflexibility of structured grid models means, however, that they can struggle to resolve these processes in realistic cavity geometries.

We present a new nonhydrostatic unstructured mesh model for flow under ice shelves built using the Firedrake finite element framework. Verification tests based on a novel Method of Manufactured Solutions (MMS) test, including melting, shows that the model discretisation is sound and second order accurate. This test may have wider applicability as a rigorous code verification procedure for the ice-ocean modelling community.

In detailed testing our results are shown to compare favorably, in terms of accuracy and robustness, against those obtained with the popular MITgcm model for a series of benchmarks of increasing complexity and different meshing techniques, culminating with a full 3D ice shelf cavity domain using the community standard ISOMIP+ \emph{Ocean0} test case. We build on existing work to show how spurious numerical mixing dictates the sensitivity of the melt rate parameterisation to grid resolution.

A key motivation behind using Firedrake is the availability of an automatically generated adjoint model. Preliminary adjoint sensitivity calculations are promising and point to the ability to address important questions on ocean influence on ice shelf vulnerability in the future.

Finally, we apply the forward model to investigate the dynamics of ocean flow in the presence of complicated basal features. We extend the literature, previously confined to 2D, to consider the importance of crevasse orientation and rotational effects for a suite of 3D experiments.