We incorporate a physically derived parameterization of gravity drainage, in terms of a convective upwelling velocity, into a one-dimensional, thermodynamic sea-ice model of the kind currently used in coupled climate models. Our parameterization uses a local Rayleigh number to represent the important feedback between ice salinity, porosity, permeability, and desalination rate. It allows us to determine salt fluxes from sea ice and the corresponding evolution of the bulk salinity of the ice, in contrast to older, established models that prescribe the ice salinity. This improves the predictive power of climate models in terms of buoyancy fluxes to the polar oceans, and also the thermal properties of sea ice, which depend on its salinity. We analyze the behavior of existing fixed-salinity models, elucidate the physics by which changing salinity affects ice growth and compare against our dynamic-salinity model, both in terms of laboratory experiments and also deep-ocean calculations. These comparisons explain why the direct effect of ice salinity on growth is relatively small (though not always negligible, and sometimes different from previous studies), and also highlight substantial differences in the qualitative pattern and quantitative magnitude of salt fluxes into the polar oceans. Our study is particularly relevant to growing first-year ice, when gravity drainage is the dominant mechanism by which ice desalinates. We expect that our dynamic model, which respects the underlying physics of brine drainage, should be more robust to changes in polar climate and more responsive to rapid changes in oceanic and atmospheric forcing.