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Let's look at a couple of examples.
Here's a typical situation. An air parcel is displaced upward in an atmosphere that gets cooler with height. As the parcel rises, it cools. In the right-hand figure, the parcel has risen a considerable distance, and has cooled quite a bit. The environmental air at the parcel's new level is cooler than it is lower down, but you can see that the parcel has nevertheless become colder than its environment and should sink back down. This situation is called stable.
Here's a different situation. Same air parcel displaced upward, but now the atmosphere is much cooler aloft. When the air parcel reaches the new level, it has cooled off just as much as in the previous example, but now the surrounding air is so much colder that the air parcel is actually lighter than its surroundings. As a result, it will continue accelerating upwards and will never return to its original level. This situation is called unstable.
What's the critical environmental lapse rate to cause instability? Well, if an air parcel cools at the dry adiabatic lapse rate, the atmosphere would have to get colder with height at a rate greater than the dry adiabatic lapse rate.
Here's what those two situations would look like on a sounding diagram. The arrows represent the change in temperature of an air parcel as it ascends, and the solid line represents the distribution of temperature with height of the environmental atmosphere.
In the stable case, the air parcel rises along a dry adiabat and immediately becomes colder than its surroundings. In the unstable case, the air parcel rises along the same dry adiabat but becomes warmer than its surroundings because the environmental lapse rate is so large.
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