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Factors
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Low air temperature is the factor that first comes to mind when one thinks of arctic climate. Scientifically speaking, temperature is the measure of heat storage. More specifically, it is a measure of the average kinetic energy of individual atoms or molecules.
Temperature is usually measured by a thermometer in degrees Celsius or degrees Fahrenheit. For accurate measurements of air temperature, a thermometer should be adequately ventilated and shielded from precipitation and direct sunlight. For official temperature measurements, thermometers are usually enclosed in a white louvered wooden shelter. By convention, "surface" air temperature is measured at two meters above the surface. At temperatures below -39 degrees Celsius, an alcohol thermometer is used in place of a mercury thermometer because mercury freezes below this temperature.
Some of the local factors influencing air temperature are solar radiation, latitude, surface albedo, the movement (advection) of air masses, and the local distribution of land and water. Of these, solar radiation and latitude are the most influential (and inseparable) factors.
The annual temperature cycle (see figure below) clearly reflects the systematic variation in incoming solar radiation over the course of a year. In high latitudes, poleward of the Arctic circle, the seasonal difference in solar radiation is extreme. Radiation varies from zero in midwinter to a maximum of around 350 - 400 W/m2 in summer. A description of the seasonal cycle of surface air temperature has been drawn from the data section of the Arctic Meteorology and Climate Atlas.

The annual cycle of global radiation (brown line) and surface air temperature (blue line) at a grid cell location in the central Beaufort Sea. Values were drawn from the Arctic Meteorology and Climate Atlas gridded fields for global radiation and two-meter air temperature.
High latitude and resulting low levels of solar radiation, and high albedo combine to keep arctic air temperatures low. However, much of the Arctic is ocean or near the ocean. Lands closest to the ocean have a marine climate with more moderate temperatures than continental regions. Temperature varies widely from place to place across the Arctic. Local topography, and advection of warm or cold air masses over a region, cause local variations in temperature.
Over the course of a day air temperature usually rises from an early morning minimum to an early or midafternoon maximum. This typical pattern can change, however, if an influx of cold air occurs during the same period. Depending on how cold the incoming air is, air temperatures may climb more slowly than usual, may remain steady, or may even fall during daylight hours. Air temperatures may climb through the evening hours as a consequence of strong warm air advection or if cloud cover increases.
A daily pattern of rising and falling air temperature only holds true during the period of the year when the Arctic receives sunlight. Diurnal temperature variation is most pronounced around the equinox, when the differences in solar radiation from day to night are greatest. The figure below shows the daily cycle of temperatures at three times of the year, using data from a Russian drifting station in the central Arctic Ocean. Note the near-constant temperature of 0 degrees Celsius in midsummer. On bare land, solar radiation is absorbed by soil and re-radiated as heat, but in the Arctic Ocean, energy from solar radiation is spent melting ice and snow, and the air temperature stays near freezing. During polar darkness, when solar radiation is absent, air temperature is strongly controlled by cloud cover. It tends to be colder under clear conditions and warmer under cloudy conditions. Advection of warm or cold air, however, can change these relationships.

Surface air temperature over the course of a day (data from drifting station NP-30). Top: Temperature on the day of the summer solstice, under constant sunlight. Middle: Temperature at the spring equinox. Diurnal variation is evident. The maximum occurs at about 2 p.m. local time (or 0040 GMT in this case). Bottom: Temperature on the day of the winter solstice, under constant darkness.
The mean seasonal cycle of temperature is simply described. During winter, temperatures decrease sharply from the northern North Atlantic to the central Arctic Ocean. The high temperatures over the Atlantic sector arise from the moderating influence of ice-free waters in this region, coupled with the frequent cyclone activity associated with the Icelandic Low transporting warm air polewards. Similarly, high temperatures are found south of Alaska associated with the Aleutian Low. The lowest winter temperatures occur over Siberia in association with the cold Siberian High. Standard deviations are also high over Siberia due to variations in the strength of the Siberian High from year to year. The low temperatures over Greenland reflect the high elevation of the ice sheet. While temperatures are of course higher everywhere in summer, the spatial variability seen in winter is much less pronounced in summer, and standard deviations are lower. Temperatures are highest over land areas in summer because the surface is snow free and can be easily heated by the strong solar radiation flux. Over the Arctic Ocean, temperatures are close to zero. This occurs because the sea ice cover is at its melting point, which keeps air temperatures near freezing. The autumn months illustrate the transition back to the winter pattern, with higher temperatures over the Atlantic and south of Alaska, and low temperatures over Siberia and the Greenland ice sheet.