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What Is Wind and How Does it Form?

What exactly is wind? And how does it form, strengthen, and move across the country? 

Gasses always move from areas of higher air pressure to areas of lower pressure. If the difference in pressure is greater, the gasses move quicker - that is the wind you experience outside. 
Difference between colder and warmer graphics. Image:

Wind is actually caused by temperature differences. When gasses heat up, the atoms speed up, spread apart and rise upwards. By contrast, colder atoms move slower and closer together.

The sun will heat up the air but won't do it evenly (due to any number of factors). This causes differences in temperature, with pockets of warmer and colder air. High pressure results from a more densely packed atmosphere, while low pressure comes from gasses that are more spread out.

So when heat rises, it leaves behind cooler and lower-pressured air behind it. As mentioned before, gasses move from higher pressure to lower pressure, and therefore, air moves from hotter and higher-pressured air above to cooler and lower-pressured air, which is lower down. 
The sun heats up the air and leaves lower pressure behind. Image:

This is all for the general wind in the atmosphere. But what brings that from the atmosphere to the surface? 

Stronger winds at the surface are a result of atmospheric mixing. This is a process in which stronger winds get brought down to the surface from several thousand feet above. It is caused by strong wind shear, which means there are sharp differences in wind speeds as you move vertically. 

Those differences, and the resultant shear, are caused by the temperature differences discussed before (uneven heating by the sun). Stronger winds move downwards as a result of the shear. The amount of shear is a key factor in how windy it will be that day. If there is a sharp change in wind directions or speeds as a storm departs, winds will get stronger.

The bigger the temperature differences in the atmosphere as compared to the ground-level weather, the stronger the winds will be at the surface, since the ground will be at a lower pressure than the atmosphere. That will cause gasses to be transported down to ground-level.