Based on anomalies in the pressure of the regions from their long-term averages (computed over a period of over 100 years), the oscillations are assigned parameters, called the AO index and NAO index, respectively, which change with time. The sign of the parameter (whether it is positive or negative) can predict certain features of the climate of much of the northern hemisphere, and are particularly important in the winter.
The Arctic Oscillation index is computed from the pressures of the subtropical and subarctic regions. The pressure gradient between the two latitudes determines the sign of the AO index. If pressures are higher in the subtropics than normal and lower in the subarctic, the AO index is positive, and the AO is said to be in its positive phase, while if subtropical pressures are anomalously low and subarctic pressures anomalously high, the AO index is negative, and the AO is said to be in its negative phase.
The above figure shows the general shape of the path of the jet stream during positive and negative phases of the AO. During positive AO, the jet stream tends to be stronger and more linear in its path during the winter, locking cold air in the Arctic regions and generally leading to warmer winters in the subtropical regions. Since the position of the jet stream allows tropical moisture to venture farther north, the subtropics are also generally wetter during these periods.
When the AO index is negative, the jet stream becomes more sinusoidal, with the amplitude of the variations in the jet stream's latitude generally proportional to the magnitude of the negative phase. Where the jet stream dips south, large masses of cold air can engulf regions for days or weeks, generally resulting in colder, snowier winters. At the same time, however, the upswings in the jet stream can bring warm air to generally cold areas. Winters in the northern hemisphere with a negative AO index generally tend to be more volatile.
The NAO is closely related to the AO, except that the index is determined only by the pressure gradient between latitudes only in a very specific region: the North Atlantic near 30°W longitude, or the subtropical region near the Azores Islands, and the Arctic region near Iceland. The effects of the NAO on the jet stream are similar to the AO, but they are not the same. The NAO index is generally a very good indicator of the winter temperature anomaly in the eastern U.S. and Europe, and though its sign usually agrees with that of the AO, this is not always the case:
The graphs of the AO index (top) and the NAO index (bottom) over the winters of roughly the same time period, from the late 1800's to the early 2000's. The indices clearly are related; both are predominantly negative in the period 1960-1980 and predominantly positive from 1980 to 2000, but on some years, they disagree. For example, if the NAO index is positive and the AO index negative, the jet stream may be straight over the Atlantic, bringing warm air to the eastern U.S. and western Europe, but sinusoidal elsewhere. This happened, for example, in the winter of 2011-2012. Conversely, if the NAO index is negative and the AO positive, there may be a large dip in the jet stream over the U.S. and a weak pressure gradient over the Atlantic, but cold air masses may be fairly well confined to the Arctic at other longitudes. This situation occurred in the winter of 2008-2009.
These oscillations, relative to El Nino and La Nina, are notoriously hard to predict. In addition, while the weather of any given winter is influenced by the El Nino/La Nina, the AO, and the NAO, many other factors also come into play. However, the fairly consistent accuracy with which the AO and NAO have predicted the winter weather of North America and Europe serve to exemplify the importance of atmospheric phenomena even in determining the climate of a region thousands of miles away.
Sources: The Winters of Our Discontent from The Scientific American, December 2012, AO and NAO on Wikipedia
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