Ever heard of the Chinook? No, not the salmon or the helicopter, the weather phenomenon. The chinook wind is a hot, dry wind that blows from the west in the Montana, Wyoming, and other areas during the colder parts of the year. It is caused by moist air flowing over the Rocky Mountains. Other parts of the world experience similar winds.
Suppose air approaches the west coast of the United States at sea level (say 1000 mb) with a temperature of 5 C and a relative humidity of 100%. Now it runs into the Rocky Mountains. What is the snow level?
On the West Coast, the snow level is usually found at surface temperatures of 2-3 C. Our surface parcel will reach that temperature once it ascends to about 950 mb. (A more detailed diagram would yield a more accurate answer.) Using the conversion near sea level of 100 mb = 1 km, this would be a height of 0.5 km, or 1500 ft.
Now suppose the Rocky Mountains are 10,000 feet high (3000 m) in its path. That means the air will have to ascend to a pressure of about 700 mb to get over them. What will its temperature be at 700 mb?
The air parcel has ascended moist adiabatically, so there will have been lots of condensation. Since it's below freezing most of the way, the water will fall out as snow.
Now that the air parcel has reached the crest of the Rockies, it begins to descend the other side. As it descends, it warms up at the dry adiabatic lapse rate and is no longer saturated. What will be its temperature when it reaches Great Falls, MT (elevation: 1118 m = 3670 ft)?
Hint: the pressure at 1118 m, using our approximate rule of thumb, will be 890 mb.
The air parcel which was producing snow on the windward side of the mountains is now 5 C on the leeward side. As the air parcel continues down the slope, it continues to warm; by the time it hits Bismarck, ND (500 m), it will have warmed to 11 C!
The chinook forms only with a strong pressure gradient, because it takes a strong wind to replace the cooler air sitting on the leeward side of the mountains with the warmer air coming over the top. This wind is a great way to remove snow cover (it's known as the "snow eater" in the Dakotas), and central and eastern Montana are sometimes warmer in the wintertime than any place within 1000 miles of them.
Want more? Try out the wonderful Mountain Simulation created by Doug Yarger at Iowa State University. You can specify the starting characteristics of an air parcel, and watch what happens to it (and the weather) as it ascends a mountain and descends the other side.