The distribution of freshwater within the Arctic Ocean and its export from it are intimately involved in climate and climate change processes both within and outside the Arctic Ocean. River runoff in the Arctic Ocean constitutes a major part of the Arctic Ocean freshwater budget. Within the Arctic Ocean, variability in the distribution of river runoff will be reflected in the location of the cold halocline that isolates the sea ice from the warm Atlantic Layer. Outside the Arctic Ocean, such variability will impact on the salinity of North Atlantic waters (Great Salinity Anomaly) and on deep convection areas of the North Atlantic Ocean, and thereby potentially on global thermohaline circulation. Rivers entering the Arctic Ocean have high levels of total alkalinity that contribute significantly to the total alkalinity of the surface Polar Mixed Layer. We exploit total alkalinity data to trace river runoff in the surface Polar Mixed Layer and to observe variability in the river runoff distribution in the Eurasian Basin over the period 1987--2001. The river runoff front changed from a position over the Gakkel Ridge in 1987 and 1991 to over the Lomonosov Ridge in 1996, and returned to a midpoint between the two ridges in 2001. Wind field changes as characterized by the Arctic Oscillation index are considered to be a major factor in determining ice and surface water flow. We note a correlation with 4--6 years delay between changes in river runoff distribution and the Arctic Oscillation index. We show that the delay can be inferred from a geostrophic flow calculation.