Abstract The existence of multiple equilibria (ice-covered and ice-free states) is explored using a set of coupled, nondimensional equations that describe the heat and salt balances in basins, such as the Arctic Ocean, that are subject to atmospheric forcing and two distinct water mass sources. Six nondimensional numbers describe the influences of atmospheric cooling, evaporation minus precipitation, solar radiation, atmospheric temperature, diapycnal mixing, and the temperature contrast between the two water masses. It is shown that multiple equilibria resulting from the dependence of albedo on ice cover exist over a wide range of parameter space, especially so in the weak mixing limit. Multiple equilibria can also occur if diapycnal mixing increases to O(10−4) m2 s−1 or larger under ice-free conditions due to enhanced upward mixing of warm, salty water from below. Sensitivities to various forcing parameters are discussed. Significance Statement The purpose of this study is to better understand under what circumstances high-latitude seas, such as the Arctic Ocean, can exist in either an ice-covered or an ice-free state. The temperature and salinity of the ocean, as well as the heat exchange with the atmosphere, are drastically different depending on which state the ocean is in. The theory presented here identifies how forcing from the atmosphere and ocean dynamics determines whether the ocean is ice covered, ice free, or possibly either one.