Eddies play an important role in the circulation and mixing in the Arctic Ocean. Past observations have shown that anticyclonic eddies dominate over cyclonic eddies under Arctic Sea ice. Current theories of small eddies generated in the open ocean, in the marginal ice zone, and under sea ice give incoherent explanations of this phenomenon. In a recent numerical study using a model with high vertical resolution, the principal investigators demonstrated that a shallow density source of cooling or brine ejection produces anticyclonic eddies under sea ice, contrary to the intuition that cooling produces cyclones. This counterintuitive result suggests two things. First, high vertical resolution is needed to correctly simulate the dynamic behavior of small eddies. Second, the presumption that iceinduced friction will modify an eddy to a submerged state without altering its polarity or stability is highly questionable and should be intensely scrutinized. A thorough examination of existing theoretical ideas including the one mentioned above using qualified numerical models is proposed. The dynamics to be investigated under the Arctic environment include stability of isolated eddies, eddy-eddy interactions, eddy-current interactions, and shallow brine convection. A three-dimensional, hydrostatic, primitive-equation model will be used initially. In addition, the principal investigators will develop a nonhydrostatic model suitable for studying small-scale motions under sea ice. Hopefully, results from this investigation will not only answer the question why anticyclones dominate under sea ice but also shed light on the circulation and mixing processes in the Arctic Ocean.