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We use passive tracers in a one-dimensional numerical model of the Arctic Ocean to determine the residence time in the mixed layer and the cold halocline. When run to a steady state, the model successfully reproduces most of the observed distribution of salinity and temperature in the Arctic above the Atlantic layer. Comparison of model-calculated tritium concentrations with observational data also indicates that the transient properties are correct. An important component of the model is the implementation of a “shelf circulation” of about 0.8 Sv that simulates the observed production and interleaving of cold, highly saline shelf waters. We use the model to derive bulk residence times, which are about 25 years in the mixed layer and about 100 years in the halocline. These values are higher than those published in the literature. We explain how the model residence time is related to different tracer ages by generating age distributions with the model. It is shown that the weighted mean of these distributions corresponds well with published tracer age data. The model dynamics that correctly reproduce Arctic mixed-layer and cold halocline vertical structure in salinity, temperature, density, and various tracers also appear to simulate the natural processes that filter out interannual fluctuations in the freshwater influx from runoff and Bering Strait flow. Under these conditions, the period of the variation must be over 30 years to get 50% of the signal through to Fram Strait. In accord with other investigations, this would suggest that short-term fluctuations in Arctic river runoff are not the direct cause of freshwater anomalies in the northern North Atlantic.