Chapter 4. Atmospheric Temperature and Stability

Chapter 4. Atmospheric Temperature and Stability 4.1 The temperature structure of the atmosphere Most people are familiar with the fact that the temperature of the atmosphere decreases with altitude. The temperature outside a commercial airliner at 12 km (36,000 ft) is typically -40 C or colder. Mountains are often capped with snow and ice, while adjacent valleys are green and lush. In this section we introduce students to the physical principles that explain the decline of atmospheric temperatures aloft and examine the condensation process in clouds and its effects on atmospheric temperature. We then study the concept of stability: what determines whether air is buoyant and convection can take place, or whether the atmosphere is stable and air parcels tend to return to their original altitude when displaced. The circulation in the atmosphere, which is driven to a large extent by convective motions, is discussed in Chapter 5; radiative processes (absorption and reflection of sunlight, thermal radiation (radiant heat)), which also play important roles in determining temperature and climate, are discussed in Chapter 6. Latitude=30N Altitude (km) 0 10 20 30 40 50 60 mesosphere stratosphere troposphere Fig. 4.1 Average observed temperature distribution with altitude. The temperature of the atmosphere on average in summer is shown for 30 N latitude, along with the names given to the major regions of the atmosphere (troposphere, stratosphere, mesosphere). The upper boundaries of the troposphere and stratosphere ("tropopause", "stratopause", respectively) are indicated as horizontal lines. 220 240 260 280 300 T (K) The lowest region of the atmosphere, up to the first temperature minimum at 12-16 km, is known as the troposphere, a name derived from the Greek words tropos, turning, and spaira, ball. The troposphere is relatively unstable because of the decrease of temperature with altitude. Air in the troposphere is poised to over-turn (to convect) much like water in 1

a kettle heated from below. Most of the weather of the planet is confined to the troposphere. The upper boundary of the troposphere, the altitude corresponding to the temperature minimum, is known as the tropopause. Temperature increases with altitude in the stratosphere (tropopause to 45 km), from the Latin word stratus meaning stretched out or layered. Vertical motions are strongly inhibited in the stratosphere as a consequence of the increase of temperature with altitude; an air parcel that is pushed upwards becomes denser than the air it seeks to displace and is driven back to its point of origin as we discuss below. The temperature maximum near 45 km (the stratopause) marks the upper boundary of the stratosphere. As we shall see, most of the world's ozone (O 3 ) is contained in the stratosphere. Stratospheric ozone is vitally important to life because it absorbs harmful ultraviolet light from the sun, preventing it from reaching the earth's surface. Stratospheric ozone has been studied intensively for the last 30 years, reflecting concerns that global-scale changes have been caused by human activities (Chapter 7). The region of decreasing temperature above the stratopause is known as the mesosphere (from the Greek word mesos, middle). Above the mesopause (85 km) there is a region of very rapidly increasing temperature called the thermosphere (not shown in the figure), which attains the highest temperatures observed anywhere in the atmosphere. Densities are extremely low, and satellites orbit Earth for extended periods of time in the thermosphere. Thousands of kilometers above the surface, the atmosphere merges slowly with the interplanetary medium. 4.2 Displacement of an air parcel with altitude: work done by an air parcel on the atmosphere We now examine one of the basic factors causing atmospheric temperatures to decrease with altitude in the troposphere: work must be done on the atmosphere by a rising air parcel. (The other major factor, atmospheric radiation, is discussed in Chapter 6). Consider taking a parcel of air and moving it vertically from altitude Z 1 to Z 2 (see Fig. 4.2), not allowing heat or mass to exchange between the parcel and the environment. If we initially keep the volume fixed, then the pressure remains equal to P 1. From the barometric law, we know that the ambient pressure P 2 at Z 2 is lower than the initial pressure P 1 that the parcel had at Z 1, and we therefore allow the parcel at Z 2 to expand until the pressure inside is equal to the ambient pressure. As the air parcel expands, it pushes against the force due to the pressure of the surrounding atmosphere, doing work (Chapter 2). To do this work, the parcel requires energy, but we have not allowed heat to be added to the parcel. Where does this energy come from? The only source is the kinetic energy of the molecules inside the air parcel. We know from Chapter 2 that this internal energy is related directly to temperature, E = 3/2 kt. Since energy is conserved ("1 st Law of Thermodynamics") the work done by the parcel on the atmosphere must be accompanied by a drop in temperature. This is a fundamental reason why atmospheric temperature decreases with altitude. 2

We can determine the temperature change of an air parcel with altitude when the parcel moves verticall without exchanging any heat with its surroundings. This quantity, T/ Z, is called the adiabatic lapse rate. (The term adiabatic refers to the fact that the parcel is not permitted to exchange heat with the atmosphere.) Consider an air parcel with mass m Z (km) 0 5 10 15 Pressure and work on a vertically-displaced air parcel P2,Z2 0.0 0.4 0.8 P1,Z2 P1,Z1 P2,Z2 The parcel does work on the atm expanding from P1->P2 at Z2 P1,Z2 P1,Z1 Figure 4.2 Vertical displacement of an air parcel. This diagram illustrates schematically the upward displacement of an air parcel in the atmosphere, from altitude Z 1 to Z 2.. No exchange of heat or mass is allowed with the environment. Since pressure decreases with height (left panel, barometric law), the parcel will expand in volume and pressure will drop inside; in the process, the parcel does work on the atmosphere at Z 2, and the temperature of the parcel decreases by an amount given by the adiabatic lapse rate. P (bar) that is raised distance Z, for example, due to the wind blowing over a mountain. As our parcel goes up the mountain, another parcel goes down. Let us suppose that the distribution of temperature and pressure in the atmosphere is so arranged that no energy has to be supplied to effect the switch--the parcel that is falling gains just enough energy to pull the other parcel up, like a perfectly matched pair of cable cars. What will be the relationship between temperature and pressure, given that pressure and altitude obey the barometric law? Since no energy needs to be supplied to exchange the parcels, and work is done against gravity by the rising parcel, there must be an exact trade between internal energy and gravitational energy. That means that the sum of these two is same in all parcels with the same mass at any altitude. The work done against gravity (mg Z, see work defined in Chapter 2) must exactly balance the amount of energy extracted from the internal kinetic energy of the air molecules in the air parcel. The extracted energy is defined as -m c p T, where c p is the specific heat of air, the amount of energy required to raise the temperature of 1 kg of air by 1 C (c p 1005 J/kg). We therefore know that the two quantities are equal (work against gravity and energy extracted from molecules): which can be re-arranged to yield the remarkably simple mg Z=-m c p T Eq. 4.1a T / Z = -g / c p = -9.8 C / km. Eq. 4.1b T / Z, the adiabatic lapse rate, is the temperature change for an air parcel (cools if it moves up, warms if it moves down) when it suddenly moves from one altitude to another. 3

Our special atmosphere, where all parcels move up or down freely without exchanging energy with the environment, is said to be neutrally stable as discussed below. There are many ways that an air parcel can be forced to move adiabatically in the real atmosphere. If the air blows over an obstacle, such as a mountain, the upward motion on the front side and downward motion on the rear side occur too rapidly for heat or mass to be exchanged with the surroundings. If air near the ground is heated, for example by absorption of sunlight by the ground, then the scale height H increases and the whole air column will be pushed upwards. Air parcels forced to move in this way have no way to know if their motion is balanced by that of other air parcels, or if they are being push up or down some other way. Therefore, air parcels forced to move up or down adiabatically and which maintain pressure equal to the surrounding atmosphere, change temperature with altitude as given by the adiabatic lapse rate (Eq. 4.1b). Note that the ambient atmosphere need not have this lapse rate (usually the atmosphere is not neutrally stable), so the parcel that has moved will usually be at a different temperature than the surrounding air. It will therefore have a different density (Perfect Gas Law). This distinction, between the temperature change in a parcel forced to move vertically and the temperature change with altitude in the ambient atmosphere, represents a key concept in the discussion that follows. 4.3 Stability Altitude (km) 0 2 4 6 8 10 B unstable adibatic A stable Fig 4.3 Ambient lapse rates and parcel temperature changes. Schematic diagram showing stable (Curve A) and unstable (Curve B) temperature profiles for the ambient atmosphere, compared to the lapse rate followed by a parcel subjected to adiabatic vertical displacement. 200 220 240 260 280 300 T (Kelvin) Figure 4.3 shows the adiabatic lapse rate and two hypothetical profiles for atmospheric temperature. In profile A, the atmosphere cools with altitude at a rate T/ Z= -6.5 K/km, less rapidly with altitude than the adiabatic lapse rate. Suppose an air parcel is moved adiabatically upwards a short distance in this atmosphere. The pressure decreases according to the barometric law and temperature decreases according to the adiabatic lapse rate. The temperature of the air parcel will therefore be lower than the temperature of the surrounding atmosphere. The parcel will therefore be denser than the surrounding air. It is negatively buoyant and will tend to sink back to where it came from. The atmosphere in profile A is stable, i.e. parcels given a little push 4

up or down return to where they started. If we wanted to take a parcel to a higher altitude in this atmosphere, we would have to warm it, i.e. supply heat. In profile B, the atmosphere cools with altitude at a faster rate, T/ Z=-10 K/km, than the adiabatic lapse rate. An air parcel moved upwards will warmer than the surrounding atmosphere, therefore less dense and thus buoyant. The parcel will tend to continue to rise due to buoyancy, and the atmosphere is unstable. If the atmospheric temperature profile were the same at the adiabatic lapse rate, an air parcel moved up or down would have the same density as the surrounding air and have zero buoyancy, the case of neutral stability discussed above. Exercise: explain why a parcel given a push down in a stable atmosphere will return to its original altitude. Exercise: Consider an atmosphere with a lapse rate of -10 K/km. Show that an air parcel displaced downward 100m is negatively buoyant and will be accelerated downwards. Find the magnitude of the acceleration. Estimate the velocity of this parcel when it has descended 200m, if the acceleration stayed constant. Do you expect that it would actually be moving faster, or slower, when it reaches this level? 4.4 Convection Dry Convection In class we observe a demo in which a tank of water is heated on one side and allowed to return to room temperature on the other. Small particles are suspended in the water so that we can observe the motion of the fluid. The warm water rises in a column, then spreads out horizontally as it cools off, descends on the cooler side, and flows back towards the heated side to make a loop. Where the water is heated, buoyancy is generated, since the density of the warmer water is lower than the density of the rest of the water in the tank. The rising column of fluid moves mass into the upper part of the tank on the heated side, slightly increasing the depth of the column of water, and thus the pressure, on the heated side. The water spreads horizontally pushed by this pressure gradient, forming the circulation loop (see further discussion of pressure forces in Chapters 5 and 6). The experiment is the analogue of the sea breeze: the heating is provided by sunlight over the land, warming the surface and raising air temperatures. The atmosphere over the land on a sunny afternoon has typically been stirred sufficiently to reach a state where the lapse rate is approximately adiabatic (Eq. 4.1b) through the lowest 1-3 km, i.e. the atmosphere is neutrally stable. Additional heating of the ground raises the temperature and makes the local lapse rate unstable, allowing air parcels to rise rapidly in what is called the atmospheric mixed layer. Over the sea, the surface hardly warms, setting up a situation analogous to that in the tank. 5

Planetary boundary layer During the daytime, solar heating develops a planetary boundary layer over land, in which dry convection vigorously stirs the atmosphere from the ground to altitudes of 1-3 km, as illustrated in Figure 4.4a. Suppose that, at sunrise, the atmosphere initially has a stable lapse rate of -6.8 K/km. Solar heating warms air near the ground rapidly, until the lapse rate reaches -9.8 K/km. Convection then begins, and the warm air parcels mix with the ambient atmosphere until a neutral condition is reached where the lapse rate is approximately adiabatic. As the day progresses, the convective mixed layer, also called the planetary boundary layer, gets deeper and deeper. As day ends, the atmosphere gradually cools by emitting infrared (heat) radiation, restoring a stable lapse rate through the night. The process can then repeat the next day. Figure 4.4a. Schematic diagram of the lapse rate just before sunrise (stable) and in mid-afternoon (neutral or slightly unstable). Dry convection driven by solar heating stirs the lower atmosphere, creating the planetary boundary layer or mixed layer during daytime over land. The convection in the mixed layer rapidly redistributes gases such as water vapor or pollutants that are emitted from the earth's surface. Figure 4.4b. Convection in the atmospheric mixed layer. Photograph showing puffy little clouds, called fair weather cumulus, occurring over land on a typical afternoon. The lapse rate in the mixed layer is approximately adiabatic, and air parcels heated near the ground are buoyant. Each little cloud represents the top of a buoyant plume.(photograph courtesy University of Illinois Cloud Catalog). The generation of convective plumes by solar heating of the ground is often apparent in the afternoon when weather is fine. Little puffy clouds develop at the top portion of these plumes, all with the same altitude at the bottom. Air parcels moving up cool off according to the adiabatic lapse rate, -9.8 K/km, and therefore the cloud bottom is determined by the altitude where the water content of the air brings the relative humidity (Chapter 3) to 6

100%. The association between convection and these "fair weather cumulus clouds" is apparent to the air traveler, since the air is always rough in the mixed layer and the pilot asks passengers to buckle their seat belts shortly before entering the cloud layer. When the plane passes through a cloud, the ride can be very bumpy indeed, because the plane is sampling the core of convection plumes. The air is moving downward between the clouds, as it must to keep the mass balanced, just as it did in the "sea breeze" demo, but the location of the rising and descending motions are distributed over the landscape, not lined up along the beach. Wet convection Wet convection occurs when condensation of water in a rising air parcel adds enough heat to make air parcels significantly more buoyant than they would have been if they were dry. An unstable situation may be created when a parcel moves up, cooling causes condensation, and the latent heat release raises the temperature of the parcel. If the amount of condensation is sufficient, the parcel may become buoyant relative to the surroundings just due to the release of latent heat. In this case the release of latent energy would accelerate the upward motion, leading to more condensation. Wet convection can produce a deep cloud with rain or snow. It is quite typical for the atmosphere to be stable for small displacements of an air parcel, but for condensation that occurs with larger displacements to add enough additional heat to make the parcel buoyant. This situation is called conditional instability, indicating that a given finite displacement is needed to make air parcels buoyant. We examine the effects of condensation of water on atmospheric temperature and buoyancy in two examples--the first representing a relatively dry atmosphere, the second a moist summer day or a day in the moist tropics. 4.5 Conditional Stability We start with a parcel of air at the surface that has a given amount of water vapor, relative humidity less than 100% (see Fig. 4.5). Since the parcel is undersaturated, it cools at the dry adiabatic lapse rate if forced to rise and no condensation occurs. But as the temperature continues to decrease, the temperature may reach the condensation point as given by the Clausius-Clapeyron equation, i.e., the relative humidity of the parcel will reach 100%. Condensation will start at this point, and the parcel will cease cooling at the dry adiabatic rate. Latent heat is being released, thus the process in no longer adiabatic. Addition of heat makes the parcel cool at a slower rate. The resulting lapse rate is called the "moist adiabatic lapse rate", because the heat comes from condensation within the parcel, not from the external environment. This moist lapse rate (or "pseudo-adiabat", since the process is not truly adiabatic) depends on temperature and water vapor content, unlike the dry adiabatic lapse rate that is the same everywhere in the atmosphere. 7

Conditional stability: fair weather clouds (no rain) => Tg 303, dewpt 290 [RH 45%], atm lapse -6.5K/km Conditional instability: deep convective clouds (rain) Tg 308, dewpt 305 [RH84%], atm lapse -8.5K/km ALTITUDE (km) 0 1 2 3 4 5 dry adiabat (parcel) atm. T(Z) wet parcel pseudoadiabat ALTITUDE (km) 0 1 2 3 4 5 Parcel no cond. satd actual (w/cond) ALTITUDE (km) 0 1 2 3 4 5 atm. T(Z) dry adiabat (parcel) wet parcel pseudoadiabat ALTITUDE (km) 0 1 2 3 4 5 Parcel no cond. satd actual (w/cond) Cloud base 270 280 290 300 0 10 20 30 40 270 280 290 300 0 10 20 30 40 50 T (K) Water Vapor Pressure (mbar) T (K) Water Vapor Pressure (mbar) Figure 4.5 Rising air parcel, low water vapor content. Bold red line shows the trajectory of an air parcel forced to rise ("pseudoadiabat"). Also shown are the dry adiabat and the atmospheric profile (upper curve). Figure 4.6 Rising air parcel, high water vapor content. Bold red line shows the trajectory of an air parcel forced to rise ("pseudoadiabat"). Also shown are the dry adiabat and the atmospheric profile (middle curve). Convective cloud over Amazonia 3 Z km 2 latent heat release 1 0 T dew 283 293 303 Temperature K cloud base T dew = T air Figure 4.7. An example of a cloud similar to the diagram in Figure 4.5, showing the updrafts in the center (left panel) and the distribution of temperature and dew point in the interior of the cloud (right panel). The billowing shape, but limited size, indicates that the cloud is probably slightly buoyant. (Photo: Steven C. Wofsy, Manaus, Brazil, 1987.) 8

In the examples in Figures 4.5-4.7, we have atmospheres with stable lapse rates, less negative than the dry adiabatic lapse rate (curve labeled "atm"). In Fig. 4.5, let us examine what happens if we displace an air parcel from the surface with a specific humidity of about 0.016 kg of water vapor per Kg of air. From the plot of the saturated specific humidity on the right we see that at surface temperatures the parcel is undersaturated. Let's raise the parcel to 5 km above the surface. Initially, the parcel cools at the dry adiabatic lapse rate. Near 2 km above the surface condensation begins and the parcel thereafter cools more slowly, at the moist lapse rate. If we take this parcel all the way to 5 km, the parcel remains cooler than the surrounding air, and most of the water has been condensed. Therefore the atmosphere is stable, even for finite displacements of air parcels. Fair weather cumulus will form at about 1.7 km altitude, which will be the cloud base. There will not be any rain showers. Figure 4.6 shows a similar example, but with higher initial temperature and more water vapor in the air parcel. Red lines show the trajectory of the air parcel. We have also assumed a somewhat steeper lapse rate in the atmosphere (less stable, -8 /km vs. -6.5 in Fig. 4.5; this would be typical of the tropics in the Amazon Basin, as illustrated in the photo of a real cloud in Figure 4.7). The parcel starts with a specific humidity of 0.022 kg of water vapor per Kg of air (relative humidity xx%). By the time the air parcel reaches 500 m above the surface, condensation begins. Since we have so much water vapor available for condensation, a lot of heat is added to the parcel the moist lapse rate is only a few degrees per km. By the time the air parcel reaches 5 km, it is warmer than the surrounding air and is thus unstable. In this example, the parcel is stable for small displacements (i.e. less than 500 m), but for larger displacements, it is conditionally unstable. Strong convective rain showers (thunderstorms) could develop in the atmosphere used in this example, reflecting the less stable ambient temperature profile and the high humidity at the ground. The slightly unstable case, pictured in Fig. 4.7, actually developed weak rain showers. 4.6 Main points of Chapter 4 Lapse rates The adiabatic lapse rate (-9.8 K/km) is the temperature change that takes place in an air parcel displaced vertically in the atmosphere, where pressure vs. altitude follows the barometric law. The temperature decline represents the source of energy for the parcel to expand as it goes up, doing work on the surrounding atmosphere. The reverse occurs when an air parcel descends. The moist or saturated lapse rate is the temperature change that takes place in an air parcel with 100% humidity displaced vertically, with condensation maintaining 100% humidity, as the parcel rises. Release of latent heat causes this lapse rate to have smaller magnitude (usually -3 to -7 K/km) than the adiabatic lapse rate. The process may be called pseudo-adiabatic since no heat enters from the environment, but heat is generated by condensation. 9

Stability, instability, and conditional instability The adiabatic lapse rate defines neutral stability (neither stable nor unstable). If the atmospheric lapse rate has smaller magnitude than the adiabatic lapse rate (e.g. -6 K /km), a parcel that is displaced upwards has lower temperature than the surroundings, therefore higher density: parcels tend to fall back where they started and the atmosphere is stable. If the atmospheric lapse rate has larger magnitude than the adiabatic lapse rate (e.g. -10 K /km), a parcel that is displaced upwards has higher temperature, therefore lower density, than the surroundings. Parcels are buoyant when they start to rise (negatively buoyant when they start to descend) and will be accelerated by buoyancy forces. The atmosphere is unstable. The motion of buoyant parcels in an unstable atmosphere tends to bring the ambient temperature gradient to neutral. If parcels contain high amounts of condensable water vapor, the atmosphere may be stable for small displacements of air parcels, but conditionally unstable for larger displacements: a large vertical displacement may release sufficient latent heat to make the parcel warmer than the surroundings and therefore buoyant. When the atmosphere is conditionally unstable, rain showers (often thunderstorms) may occur as heating of the ground causes parcels to rise above the level of conditional instability. Conditional instability is a factor in many weather forecasts. Note You may note that condensation in the first example (Fig. 4.5) does not occur at the dew point of the parcel (295 K) but at a lower temperature, 287 K. As the parcel rises, its pressure goes down (barometric law), and the partial pressure of water also goes down. In this example, at 1.7 km where our parcel started to condense water, atmospheric pressure was.78 bar and the partial pressure of water had been reduced from 16 mb to 16 0.78 = 12.5 mb, corresponding to a dew point of 287 K. Thus, even though water condenses when the partial pressure of water in a parcel (pressure in mb) equals the saturation pressure (in mb) over liquid or solid water; and the saturation vapor pressure depends on temperature only, the partial pressure of water changes as the parcel rises and expands. Specific humidity, defined as the number of kg of water per kg of air, does not change unless condensation occurs. The calculation of the exact altitude at which a parcel becomes saturated with water vapor is consequently rather complicated and lies beyond the scope of this course. Students are expected to understand the underlying process and to be able to perform simple computations where the condensation altitude is supplied. 4.7 Demonstrations 1. Expansion, compression, and temperature in a gas #1. In the first demonstration we compressed air in a pump (syringe), and observed a rise in temperature. The 10

temperature increased when we pushed on the plunger, representing conversion of mechanical work to heat (temperature change), and decreased when we allowed the gas in the syringe to do work on the plunger as we pulled it back. Quite large temperature changes were obtained, up to 100 C! 2. Expansion, compression, and temperature in a gas #2. In the second demo a member of the class discharged a CO 2 fire extinguisher. The expansion of highpressure CO 2 through the nozzle did work on the atmosphere (part was converted to sound, that's why the device makes such a loud noise) and got cold enough to form solid CO 2 (dry ice, -78 C). 3. Buoyancy and circulation in a fluid. We observed a tank of water heated on one side and allowed to return to room temperature on the other. Small particles were suspended in the water so that we could observe the motion of the fluid. The warm water rose in a column, then spread out horizontally as it cooled off, and descended on the cooler side, and flowed back towards the heated side, effectively making a loop. Where the water was heated, buoyancy was generated, since the density of the warmer water is lower than the density of the rest of the water in the tank. Since this process moved mass into the upper part of the tank on the heated side, the column of water increased slightly there and thus the pressure increased there; water spread horizontally pushed by this pressure gradient. As this flow added mass to the upper part of the cooler side, this induced an additional pressure gradient and led to the formation of the circulation loop that you could see. This experiment is the analogue of the sea breeze, where the heating is provided by sunlight over the land, which raises air temperatures there. The sea breeze really does consist of a loop circulation, it does not go very far inland or over the sea (1-10 km) and it does not go very high (about 1 km). Thus when you stand on the beach in Los Angeles and note the wind blowing from the sea, it may be recycled from the city. The sea breeze is one of the simplest examples of convection generated by buoyancy in the atmosphere. CHAPTER 4. ATMOSPHERIC TEMPERATURE AND STABILITY 4.1 The temperature structure of the atmosphere 1 4.2 Displacement of an air parcel with altitude: work done by an air parcel on the atmosphere 2 4.3 Stability 4 4.4 Convection 5 Dry Convection 5 Wet convection 7 4.5 Conditional Stability 7 Note 10 4.6 Main points of Chapter 4 9 Lapse rates 9 Stability, instability, and conditional instability 10 11

4.7 Demonstrations 10 12