To understand the origin of tropical cyclones, it is necessary to begin with a discussion of the physics underlying the normal state of the tropical maritime atmosphere. The average vertical thermal structure of the atmosphere is illustrated in Figure 2.1. The lowest layer, the troposphere, extends from the surface upward to 8-15 km, depending mostly on the latitude and season. It is characterized by temperature decreasing with altitude, at an average rate of about . This upward decrease of temperature ends, often abruptly, at the tropopause, above which the temperature is constant or increases with altitude. This layer is called the stratosphere. At an altitude of between 40 and 50 km., the temperature lapse rate again changes sign, at the stratopause, and in the mesosphere the temperature decreases upward. Above the mesopause lies the thermosphere, in which the temperature once again increases with altitude, and here the mean free path between molecular collisions becomes appreciable. The high temperatures of the upper thermosphere are caused by collisions between atmospheric constituents and high energy particles originating in the solar wind; the actual temperature of this region is a strong function of the level of solar substorm activity. But below the thermosphere, the temperature is determined by transfer of electromagnetic radiation, and in the troposphere convective heat transfer is important as well.
The two major constituents of the atmosphere, molecular oxygen and nitrogen, are very nearly transparent to both solar and terrestrial radiation, although some scattering of solar radiation does occur. Oxygen and nitrogen are diatomic molecules with a very limited number of degrees of freedom, thus they are poor absorbers of radiation. If these were the only gases in the atmosphere, then it would be almost completely transparent and the earth's surface would radiate at very nearly the black body temperature necessary to balance the input of solar radiation. Let's calculate what this temperature would be. At the mean radius of the earth's orbit about the sun, the flux of solar radiation, , is known as the solar constant and has a value of about . By elementary geometry, the amount of this radiation intercepted by the earth, per unit surface area of the earth, is just . A fraction of this radiation, is reflected back to space by the earth's surface and by clouds; this fraction is known as the planetary albedo. By the Stefan-Boltzmann law, the black body temperature is given by
where is . Using a planetary albedo of 0.3, this gives a black body temperature of 255 K or -18 C, very much colder than the average surface temperature of the earth, which is 288 K or 15 C.
The surface temperature of the earth is higher than its black body temperature owing to the presence of a few principally triatomic species, notably water vapor, carbon dioxide and ozone. While these represent very small fractions of the atmosphere by mass, they are critical absorbers and emitters of radiation, because their structure allows for many more degrees of freedom; especially in rotation and vibration. Condensed water, in the form of clouds, is also very important in reflecting, absorbing and emitting radiation.
Figure 2.2 shows two black body curves, for the sun and earth, respectively, as well as atmospheric absorption characteristics near the surface and near 11 km. First note that, because the earth's effective emitting temperature is about 255 K whereas that of the sun is close to 6000 K, their two spectra have almost no overlap. So we may usefully talk about solar and terrestrial radiation, or shortwave and longwave radiation, without ambiguity. Also note that the solar spectrum peaks in a region in which there is very little atmospheric absorption, so that most sunlight that is not reflected or absorbed by clouds reaches the surface. Virtually all of the very high energy, ultraviolet radiation is absorbed in the middle stratosphere in a photochemical reaction in which molecular oxygen breaks into 2 oxygen atoms, some of which combine with other available oxygen atoms to form ozone (). Once formed, ozone itself absorbs some ultraviolet radiation. Were most of the incoming ultraviolet spectrum to reach the surface, life as we know it would not be possible. The high temperatures of the upper stratosphere are owing to the absorption of ultraviolet radiation by ozone.
In the terrestrial spectrum, there are large numbers of absorption lines (which are broadened by both Doppler and pressure effects), owing mostly to the triatomic molecules water vapor and carbon dioxide, but with some notable contributions from other trace species such as methane. Water vapor content decreases sharply with height, but there are still large water absorption effects at 11 km. Taken together, the relative opacity of the atmosphere to terrestrial radiation makes for a strong ``greenhouse effect".
The greenhouse effect is central to understanding why the tropical atmosphere supports tropical cyclones. In particular, it necessitates a degree of thermodynamic disequilibrium between the ocean and atmosphere. It is this disequilibrium which supplies energy to tropical cyclones.
To understand the fundamental point, let's have a look at a very simple model of the greenhouse effect, as shown in Figure 2.3. In the first instance, consider 2 layers of gas each with an emissivity of unity in the terrestrial band but which are completely transparent to solar radiation. They overly a surface of emissivity unity. In radiative equilibrium, each layer emits upward and downward, and the surface emits upward. The energy balance for the upper layer is
where is the temperature of the upper layer and is the temperature of the lower layer. Likewise, the energy balance of the lower layer is given by
where is the surface temperature. The surface energy balance is simply
where is the effective blackbody emission temperature, defined
Now we may solve (12) - (14) for the three temperatures, giving
The surface temperature is substantially greater than the blackbody equilibrium temperature. Note also that since , the surface actually received more radiation from the atmosphere than it does from the sun! More to the point, we can show that the air in immediate contact with the ground is not in thermal equilibrium with the ground. We can do this by inserting a vanishingly thin layer of air between the surface and the first, opaque layer. This thin layer has vanishing emissivity, , and its energy balance is
where is the temperature of the thin layer next to the surface. Since the layer is vanishingly thin, it does not affect the energy balance of the surface and other layers. The solution to (16) is then
So clearly, in radiative equilibrium, the surface air is not in thermal equilibrium with the surface. This creates the potential for convective heat transport away from the surface. This will be the subject of the next lecture.