The barometer reading measures the atmospheric pressure at the
observing location. Atmospheric pressure is the force that the air exerts on
an object. The barometric pressure can be thought of as the weight of the air
above you. The less air that is above you, the less force it exerts on you.
The more air above you, the stronger the force of air pressure. So as the amount
of air above you decreases, so does the barometric pressure. As the weight of
a column of air above you increases, so does the pressure.
Nature always likes to equalize differences in nature. Hot flows to cold, and high pressure flows to low pressure. As the relatively higher pressure air flows in to the lower pressure area (to equalize the difference) at the surface, something called convergence results. Obvioulsy, the air cannot force its way into the ground, therefore it is forced to rise. As the air rises, it cools off as a result of the decreased pressure (remember the ideal gas law?). As the air cools, it cannot hold the same amount of water vapor as it could before, thus water condenses into clouds. Clouds can only form, however, if there is enough water vapor in the air. Sometimes air can rise wihtout ever forming clouds because there is not alot of water in the air.
When the barometric pressure falls, a storm is usually on the way. For us here in the Northeast, a falling barometer with winds out of the Southwest usually spell inclement weather. The faster the pressure falls, the stronger the storm that is potentially on the horizon.
Please note that although a certain barometric pressure reading at the surface can indicate stormy weather, the actual relative pressure of different layers in the atmosphere can vary. It is possible to have high pressure at the surface, and low pressure aloft. The term barometric pressure, however, is used to describe the surface pressure at the observation station.
The pressure observed by a barometer can be affected in many ways. First and foremost is the elevation of the observer. As you go higher and higher into the atmosphere, the pressure drops substantially. In order to standardize the barometric pressure readings taken all over the globe, the readings are converted with a formula to the "Sea Level Pressure." The sea level pressure takes into account the elevation of the observing station, the temperature of the air, and also the latitude of the observer. The temperature comes into play, since colder temperatures result in air that is more dense, thus increasing its weight. The latitude of the observer matters because the effects of gravity change as one travels from latitude to latitude. The Earth is not a perfect sphere, but rather an oblate spheroid. That means the Earth is a little squished, being compressed from pole to pole. This compression is analogous to pressing down on the top of a basketball with your foot. The side-to-side measurement becomes greater than the top-to-bottom measurement. Since barometric pressure can be though of as the weight of the air above you, it varies as the gravity varies. After all, if it weren't for gravity objects would have mass, but not weight!
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