Modelling the thermodynamic growth of sea ice: insights from quasi-static models

The thermodynamic growth of sea ice is governed by heat transfer through the ice together with appropriate boundary conditions at the interfaces with the atmosphere and ocean. Several different representations of this process have been used in climate modelling, including the simplest zero-layer models (Semtner, 1976) and more complex partial-differential-equation-based models (Maykut & Untersteiner, 1971; Bitz & Lipscomb, 1999). Recently, these latter models have been extended to include a representation of the dynamic evolution of the salinity of sea ice based on mushy-layer theory (Turner et al., 2013; Griewank & Notz, 2013; Rees Jones and Worster, 2014). Salinity variation might be expected to have a significant effect on ice growth given that it controls the relative proportions of solid ice and liquid brine, and these materials have different thermal properties. 

In this study, we develop a simplified framework to investigate the effects of variations in the thermal properties of sea ice. We develop and test a quasi-static simplification. In this simplification, we apply a transformation to the underlying heat equation such that the spatial coordinate scales with the ice thickness. We then neglect the explicit time dependence. This procedure reduces the full partial differential equation to an ordinary differential equation. The solution is exact for constant forcing conditions. 

We show that ice salinity has only a modest effect on the growth rate, notwithstanding its large effect on the thermal properties of sea ice. The model allows us to unpick the physical causes, which are related to the trade-off between the effect of salinity on thermal conductivity and latent heat release. We calculate the growth of ice under steady and time-dependent forcing. Under steady forcing, the ice growth equation admits an analytical approximate solution, which compares well to numerical solutions. We show that saltier ice initially grows slightly faster but subsequently grows slightly slower, a further trade-off explaining the relatively weak sensitivity of ice growth to salinity.

Under time-dependent forcing, we show that the quasi-static model compares well to full partial-differential-equation-based models. So our approach offers intermediate complexity between zero-layer Semtner models and full models based on partial differential equations such as Maykut-Untersteiner/Bitz-Lipscomb/mushy-layer models.

References:
Semtner, A. J. (1976) J. Phys. Oceanogr. 6 (3), 379–389.
Maykut, G. A. & Untersteiner, N. (1971) J. Geophys. Res. 76 (6), 1550–1575.
Bitz, C. M. & Lipscomb, W. H. (1999) J. Geophys. Res. – Oceans 104 (C7), 15669–15677.
Turner, A. K., Hunke, E. C. & Bitz, C. M. (2013) J. Geophys. Res. – Oceans 118 (5), 2279–2294.
Griewank, P. J. & Notz, D. (2013) J. Geophys. Res. – Oceans 118 (7), 3370–3386.
Rees Jones, D. W. & Worster, M. G. (2014) J. Geophys. Res. – Oceans 119 (9), 5599–5621.