The ice shelf-ocean boundary current has an important control on heat delivery to the base of an ice shelf. Climate and regional models that include a representation of ice shelf cavities often use a coarse grid and results have a strong dependence on resolution near the ice shelf-ocean interface. This study models the ice shelf-ocean boundary current with a non-hydrostatic z-level configuration at turbulence-permitting resolution (1 m). The z-level model performs well when compared against state-of-the-art Large Eddy Simulations showing its capability in representing the correct physics. We showthat theoretical results from a one-dimensional model with parameterised turbulence reproduce the z-level model results to a good degree, indicating possible utility as a turbulence closure. The one-dimensional model evolves to a state of marginal instability and we use the z-level model to demonstrate how this is represented in three-dimensions. Instabilities emerge that regulate the strength of the pycnocline and coexist with persistent Ekman rolls, which are identified prior to the flow becoming intermittently unstable. When resolution of the z-level model is degraded to understand the grid-scale dependencies, the degradation is dominated by the established problem of excessive numerical diffusion. We show that at intermediate resolutions (2-4 m), the boundary layer structure can be partially recovered by tuning diffusivities. Lastly, we compare replacing prescribed melting with interactive melting that is dependent on the local ocean conditions. Interactive melting results in a feedback such that the system evolves more slowly, which is exaggerated at lower resolution.