Solidification of sea water leads to patterns of varying sea ice porosity and flow. Features develop on multiple scales, including narrow brine channels which mediate salt fluxes into the ocean. Consequently, adaptive mesh refinement techniques provide a valuable tool for efficiently simulating multiscale flow during initial sea ice growth. These processes are relevant across an increasing area of the world's ocean, as a result of the rapid retreat of the summer sea ice extent and more resilient winter extent. Sea ice formation is a dynamical process, during which convective cells form within a porous layer composed of ice crystals and an interstitial brine. Downwelling at the edge of such cells leads to the development of narrow, entirely liquid channels, through which cold saline brine is efficiently rejected into the underlying ocean. Thus it provides an important buoyancy forcing, which contributes to ocean mixing, the formation of deep water masses and the maintenance of the Arctic halocline. Previous attempts to numerically simulate such processes have struggled to resolve the narrow brine channels, which occupy just a few percent of the domain and evolve in time, at acceptable computational cost. To overcome this limitation we use the Chombo software framework to implement a computational mesh which adapts to provide additional resolution near brine channels. We present initial results that model the early growth of a reactive porous sea ice layer, using simulations with adaptive mesh refinement. Where possible, we make comparisons with previous numerical, experimental and observational studies. This method is also suitable for investigating the more general problem of solidification of a binary alloy, with applications in fields such as magma dynamics and metal casting.