5.2: Atmospheric Stability and Lapse Rates

Adiabatic Processes

When discussing stability in atmospheric sciences, we typically think about air parcels, or imaginary blobs of air that can expand and contract freely, but do not mix with the air around them or break apart. The key piece of information is that movement of air parcels in the atmosphere can be estimated as an adiabatic process. Adiabatic processes do not exchange heat and they are reversible.

Imagine you have a parcel of air at the Earth’s surface. The air parcel has the same temperature and pressure as the surrounding air, which we will call the environment. If you were to lift the air parcel, it would find itself in a place where the surrounding environmental air pressure is lower, because we know that pressure decreases with height. Because the environmental air pressure outside the parcel is lower than the pressure inside the parcel, the air molecules inside the parcel will effectively push outward on the walls of the parcel and expand adiabatically. The air molecules inside the parcel must use some of their own energy in order to expand the air parcel’s walls, so the temperature inside the parcel decreases as the internal energy decreases. To summarize, rising air parcels expand and cool adiabatically without exchanging heat with the environment.

Now imagine that you move the same air parcel back to Earth’s surface. The air parcel is moving into an environment with higher air pressure. The higher environmental pressure will push inward on the parcel walls, causing them to compress, and raise the inside temperature.

The process is adiabatic, so again, no heat is exchanged with the environment. However, temperature changes in the air parcel can still occur, but it is not due to mixing, it is due to changes in the internal energy of the air parcel.

Dry Adiabatic Lapse Rate

As long as an air parcel is unsaturated (relative humidity < 100%), the rate at which its temperature will change will be constant. A decrease in temperature with height is called a lapse rate and while the temperature decreases with altitude, it is defined as positive because it is a lapse rate. Recall from chapter 3 that the dry adiabatic lapse rate, \(\Gamma_d\), is equal to 9.8 K·km^-1 = 9.8 °C·km^-1. This drop in temperature is due to adiabatic expansion and a decrease in internal energy.

Air rises, expands, and cools at the dry adiabatic lapse rate, approximated as a 10°C decrease per km (created by Britt Seifert).

Let’s get back to the topic of atmospheric stability. Stability in the atmosphere refers to a condition of equilibrium. As discussed with the example of the boulder on a hill or valley, some initial movement resulted in either more (unstable), less (stable), or no change (neutral). Given some initial change in the elevation of an air parcel, if the air is in stable equilibrium, the parcel will tend to return back to its original position after it is forced to rise or sink. In an unstable equilibrium, an air parcel will accelerate away from its initial position after being pushed. The motion could be upward or downward, but generally unstable atmospheres favors vertical motions. Finally, in a neutral equilibrium, some initial change in the elevation of an air parcel will not result in any additional movement.

Determining Stability

How do you know if an air parcel will be stable after some initial displacement? Stability is determined by comparing the temperature of a rising or sinking air parcel to the environmental air temperature. Imagine the following: at some initial time, an air parcel has the same temperature and pressure as its environment. If you lift the air parcel some distance, its temperature drops by 9.8 K·km^-1, which is the dry adiabatic lapse rate. If the air parcel is colder than the environment in its new position, it will have higher density and tend to sink back to its original position. In this case, the air is stable because vertical motion is resisted. If the rising air is warmer and less dense than the surrounding air, it will continue to rise until it reaches some new equilibrium where its temperature matches the environmental temperature. In this case, because an initial change is amplified, the air parcel is unstable. In order to figure out if the air parcel is unstable or not we must know the temperature of both the rising air and the environment at different altitudes.

One way this is done in practice is with a weather balloon. We can get a vertical profile of the environmental lapse rate by releasing a radiosonde attached to a weather balloon. A radiosonde sends back data on temperature, humidity, wind, and position, which are plotted on a thermodynamic diagram. This vertical plot of temperature and other variables is known as a sounding.

Dry Stability

If an air parcel is dry, meaning unsaturated, stability is relatively straightforward. An atmosphere where the environmental lapse rate is the same as the dry adiabatic lapse rate, meaning that the temperature in the environment also drops by 9.8 K·km^-1, will be considered neutrally stable. After some initial vertical displacement, the temperature of the air parcel will always be the same as the environment so no further change in position is expected.

If the environmental lapse rate is less than the dry adiabatic lapse rate, some initial vertical displacement of the air parcel will result in the air parcel either being colder than the environment (if lifted), or warmer than the environment (if pushed downward). This is because if lifted, the temperature of the air parcel would drop more than the temperature of the environment. This is a stable situation for a dry air parcel and a typical scenario in the atmosphere. The global average tropospheric lapse rate is 6.5 K·km^-1, which is stable for dry lifting.

Finally, if the environmental lapse rate is greater than the dry adiabatic lapse rate, some initial vertical displacement of the air parcel will result in the air parcel either being warmer than the environment (if lifted), or colder than the environment (if pushed downward). This is because if lifted, the temperature of the air parcel would drop less than the temperature of the environment. This is an unstable situation for a dry air parcel.

In general for a dry air parcel, the following is true.

\[\Gamma_d=\Gamma_{e n v} \quad \text { NEUTRAL } \nonumber \]

\[\Gamma_d<\Gamma_{e n v} \quad STABLE \nonumber \]

\[\Gamma_d>\Gamma_{e n v} \quad UNSTABLE \nonumber \]

Moist Adiabatic Lapse Rate

When moisture is added, everything gets more complicated. In Chapter 4 we learned that whether or not an air parcel is saturated depends primarily on its temperature and, of course, its moisture content. The graph of the Clausius-Clapeyron relationship shows us that given the same amount of moisture, air is more likely to be saturated at a lower temperature.

We know that as an air parcel is lifted, its temperature drops according to the dry adiabatic lapse rate. So what happens when the air parcel is cold enough that the air becomes saturated with respect to water vapor? The short answer is that if it continues to cool, water vapor will condense to liquid water to form a cloud.

When water vapor condenses, it goes from a higher energy state to a lower energy state. Energy is never created nor destroyed, especially in phase changes, so what happens to all that excess energy? The energy gets released in the form of latent heat. The latent heat of condensation is approximately equal to 2.5 * 10⁶ J·kg^-1, which means that for every kg of water vapor that condenses to form liquid water, 2.5 *10⁶ Joules of energy are released.

This has large consequences for the lapse rate of an air parcel and distinguishes the dry adiabatic lapse rate from the moist adiabatic lapse rate. As latent heat is added from the process of condensation, it offsets some of the adiabatic cooling from expansion. Because of this, the air parcel will no longer cool at the dry adiabatic lapse rate, but will cool as a slower rate, known as the moist adiabatic lapse rate. To summarize, a parcel will cool at the dry adiabatic rate until it is saturated, after which it won’t cool as quickly due to condensation. The moist adiabatic lapse rate varies a little by temperature, but in this class we will consider it a constant for simplicity: \[\Gamma_m=4.5 K \cdot km^{-1}=4 .5^{\circ} C \cdot km^{-1} \nonumber \]

Moist Stability

The effects of moisture change the lapse rate of the air parcel and, therefore, affects stability. However, the concepts are still the same and we still compare the air parcel temperature to the environmental temperature. We have just one added complication to worry about—we need to know whether the air parcel is dry or moist. Some definitions are included below, which take into account both dry and moist adiabatic lapse rates.

A thermodynamic diagram showing the stability of the atmosphere based on the dry (\(\Gamma_d\) = 9.8 K km^-1) and moist (\(\Gamma_m\) = 4.5 K km^-1) adiabatic lapse rates (Created by Britt Seifert).

The atmosphere is said to be absolutely stable if the environmental lapse rate is less than the moist adiabatic lapse rate. This means that a rising air parcel will always cool at a faster rate than the environment, even after it reaches saturation. If an air parcel is cooler at all levels, then it will not be able to rise, even after it becomes saturated (when latent heating will counteract some cooling).

The atmosphere is said to be absolutely unstable if the environmental lapse rate is greater than the dry adiabatic lapse rate. This means that a rising air parcel will always cool at a slower rate than the environment, even when it is unsaturated. This means that it will be warmer (and less dense) than the environment, and allowed to rise.

The atmosphere is said to be conditionally unstable if the environmental lapse rate is between the moist and dry adiabatic lapse rates. This means that the buoyancy (the ability of an air parcel to rise) of an air parcel depends on whether or not it is saturated. In a conditionally unstable atmosphere, an air parcel will resist vertical motion when it is unsaturated, because it will cool faster than the environment at the dry adiabatic lapse rate. If it is forced to rise and is able to become saturated, however, it will cool at the moist adiabatic lapse rate. In this case, it will cool slower than the environment, become warmer than the environment, and will rise.

Hawaiian Focus Box

Around Hawaii, the atmosphere is almost always conditionally unstable, meaning that the environmental lapse rate lies somewhere between the dry and moist adiabatic lapse rates. For this reason, Hawaii almost always has convective clouds. Convective clouds are clouds where the edges are bumpy and cumuliform, like cauliflower. The clouds are convective because the atmosphere is stable to dry lifting and unstable to moist lifting. Once the air is saturated, instability sets in and vertical motion takes off. This is especially common as air is lifted over our mountainous islands. The forced lifting from the terrain creates clouds and rain right over the mountains! In scientific terms, the initial lifting of the stable low level dry air by the terrain causes the air to adiabatically expand and reach saturation, at which point the environment is unstable to moist lifting and convection is the result.