Climate refers to the long-term weather phenomena associated with a region. It includes the average temperature, the type and frequency of precipitation and the expected range of variability in the weather. Humidity is both a component of climate and a moderating effect in climate. For example, the tropical rainforest has a climate dictated by its relatively constant exposure to sunlight throughout the year, but the high precipitation caused by high average temperatures is just as much a part of the tropical climate. So separating humidity from climate is not simple, but it's still possible to identify some of the climatological effects of humidity levels.
Geography and Climate
Humidity goes a long way toward defining a climate, but it doesn't control everything. Because solar energy drives the Earth's weather, you would expect locations at the same latitude -- which see identical sun exposure -- to have identical climates. You can see this in the average temperatures, for example, of Minneapolis and Bucharest, which are both at about 44.5 degrees north. Minneapolis has an average temperature of about 7 degrees Celsius (44 degrees Fahrenheit), while Bucharest's average is 11 degrees Celsius (51 degrees Fahrenheit). But Mount Everest and the Sahara Desert also are at the same latitude, yet have wildly differing climates. A significant part of that is due to their difference in elevation. But even places at the same latitude and elevation can have quite different climates, and the largest additional factor is humidity.
Air is full of energy. Even in still air, the molecules are constantly shooting around, bumping into each other. Although it's cheating a little bit, you can think of the energy of air as being represented by its temperature -- the hotter the air, the more energy it holds. When water vapor gets tossed into the situation, it suddenly gets a bit more complicated. At "normal" temperatures, water can exist as solid ice, liquid water and gaseous water vapor -- not only can it exist as all three in the same location, it usually does. You can see this yourself by closely observing a glass of ice water. Even though the water is cooled by the ice, some molecules have enough energy to escape the liquid phase and rise off the surface as "fog." Meanwhile, some water vapor molecules already in the air hit the cold sides of the glass and condense back into liquid water. In any environment, water is seeking a balance between the solid, liquid and gaseous states.
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Water and Energy
The reason humidity -- which is a measure of water vapor suspended in the air -- is such an important factor in weather and climate is because water contains extra energy at everyday temperatures. Water is constantly converting among its three forms, but each conversion consumes or releases energy. Put another way, water vapor at room temperature is different from liquid water at the same temperature because it has acquired some extra energy. Even though the temperature is the same, the vapor has more energy because it has converted from a liquid to a gas. In meteorological circles, that energy is called "latent heat." What it means is that a mass of warm, dry air contains much less energy than a mass of humid air at the same temperature. Because climate and weather are functions of energy, humidity is a critical factor in climate.
Water -- and Energy -- Circulation
Virtually all the energy that drives the Earth's climate comes from the sun. Solar energy heats the air and -- more importantly -- the water. Ocean water in the tropics is far warmer than water at the poles, but the water doesn't just sit in one spot. Density differences in water and air, along with the Earth's rotation, drive currents in both air and water. Those currents distribute energy around the Earth, and the energy distributions drive climate. Rainstorms are a very visible manifestation of these currents. Air above warm ocean waters contains a relatively high percentage of water vapor. When that air moves to colder regions, the balance among the three phases of water shifts -- leaning more toward the liquid than to the gas phase. That means the water vapor condenses and rain comes down. Rain is the most visible manifestation of humidity.
Because water carries latent heat, it acts to moderate temperature swings. For example, in the summer humidity of the Midwest, the air cools at night. In turn, the balance of liquid water and water vapor shifts, so some of the water condenses. But when water condenses, it releases its latent heat to the air around it -- actually warming the air even as the lack of sunlight cools the air. When the sun rises, the process reverses. Sunlight heats the air, leading to the evaporation of liquid water to water vapor. But that takes extra energy -- energy that would otherwise go into heating the land and air -- so the temperature doesn't rise that rapidly. So Chicago -- right next to Lake Michigan -- doesn't see anywhere near the daily swing in temperatures that are seen in Phoenix -- in the middle of the dry desert.