Temperature lapsing

The last posting stated a set of imaginary climates, along with the real one, in terms of the temperatures of a few surfaces: the Earth's surface and the cloud tops and bottoms. The entities of the lower atmosphere are treated as a few idealized bodies, with temperature, pressure, and density varying by altitude, but not horizontally. Energy and heat flow in and out in a steady state. The biggest thing left out was the specifics of how temperature varies (and secondarily, pressure and density) with height above the surface, the lapse rate mentioned in an earlier posting. Refer back to the discussions of heat transport and condensation for a refresher.

Why is this variation important? Temperature differences signal heat flow, and in fact, heat flow determines temperature distribution. The specific relationship between how heat flows and how the temperature declines with altitude is all about the details of exactly how heat is transferred.

Bottom line up top: the dry adiabatic lapse rate of 9.8 ^oK/km is modified to a roughly constant 6.5 ^oK/km. Three of these modifications, the non-adiabatic radiative and convective heat flows and the removal of water vapor pressure by condensation, steepen the lapse rate. The fourth, the saturation-adiabatic injection of latent heat, strongly moderates the lapse rate, making it shallower. It is the latent-heat release moderation that wins, at all altitudes in the troposphere. The temperature at the tropopause is a frigid 217 ^oK = -56 ^oC = -69 ^oF, but not as frigid as it would be if the lapse rate were steeper than it actually is.

---

Start with local thermodynamic equilibrium (LTE), which includes mechanical, thermal, and phase equilibrium, supplemented by steady inflows and outflows of radiation and evaporative heat. The real atmosphere is not this, but it never strays far from it. Global equilibrium is violated by the energy flows and by the fact that pressure, temperature, and phase of water vary with altitude. The pressure (mechanical) equilibrium is the closest to exact. Hydrostatic equilibrium (relating the pressure lapse rate to the density and gravitational acceleration) is almost exact at all times and places, except in severe weather like tornadoes.

If we ignore energy flows, the atmosphere is adiabatic, which just means no heat is being added from or released to the outside. The dry adiabatic lapse rate is 9.8 ^oK/km = 5.4 ^oF per 1000 feet, the gravitational acceleration (g = 9.8 m/s²) divided by the heat capacity of dry air at constant pressure (cP = 1004 J/kg·^oK). With a small, realistic addition of water vapor, the heat capacity is slightly raised, and the wet adiabatic lapse rate reduces slightly to 9.7 ^oK/km.

Allowing water vapor to condense into liquid droplets (usually visible as clouds) and phase equilibrium to be established completes the LTE picture, but it also complicates the vertical temperature slope. Schematically, it now looks like:

           g     [1 + vapor-pressure-decrease]
dT/dz = - --- * -------------------------------
         cP           [1 + latent-heat]

This saturation adiabatic lapse rate is not constant with altitude z, but varies from about 4.7 ^oK/km to a steeper 6 or 7 ^oK/km at higher altitudes. The actual lapse rate averages over the whole troposphere to 6.5 ^oK/km, although it varies significantly with time of day, season, and whether or not clouds are present. Why is the lapse rate fairly constant with altitude?

Two critical components are missing from the adiabatic lapse picture, both the effect of radiative and convective heat flows. These flows violate thermal equilibrium,* which is less respected than hydrostatic equilibrium: the real atmosphere is not adiabatic, because heat is continually flowing in and out, at a roughly steady rate. The lapse rate now schematically looks like:

           g     [1 + vapor-pressure-decrease + rad + conv]
dT/dz = - --- * --------------------------------------------
         cP               [1 + latent-heat]

The rad and conv terms, by themselves, tend to steepen the lapse rate. But their effect has to be gauged together with the other terms to see the complete result in dT/dz. Each term is proportional to the heat flowing by that mechanism, radiative or convective. Their presence steepens the slope of T because more heat flowing implies bigger temperature differences.

Although it is less important in the clear air and harder to understand theoretically, start with convection. In this mechanism, parcels of overheated wet air move up, dump their heat at a higher altitude, then float back down - like a waterwheel. There's no net motion of air, just as a waterwheel suffers no net motion; but there is a net flow of heat from lower to higher altitudes. The average convective motion of heat in the clear air well away from the ground is slow, less than a meter per second. The convective mixing length is roughly 20 to 30 meters. Clear-air convection is fairly turbulent and inefficient: in clear air, those overheated parcels tend to lose their heat quickly. In clouds, the mixing length doesn't change much, but the convective velocity roughly triples. Cloud convection is much more efficient, because clouds are opaque: the convective parcels keep their heat until they reach the cloudtops. The conv term in the temperature lapse rate is proportional to the product of the convective velocity and mixing length.

Radiative heat transfer in clear air is diffusive: infrared (IR) or heat photons are absorbed and re-emitted by air molecules many times before they reach the top of the troposphere. Only certain molecules are really good at this, the main one being water vapor. (Oxygen and nitrogen, OTOH, don't do much with photons of that wavelength - about 5-20 microns or millionth of a meter - about 10-40 times the wavelength of visible light.) Any matter that's really good at absorbing radiation at particular wavelength is also good at emitting it.** IR-sensitive molecules like water vapor don't absorb and hold heat; rather, they're exceptionally efficient at passing the heat along. Adding more water vapor to the clear air, for example, makes the rad term in dT/dz larger, steepening the lapse rate, and speeding up heat flow.

In clouds, radiative heat transport is ineffective, and convection takes over the whole burden of transporting heat upwards to the cloudtops. That's why the convective heat transfer velocity jumps considerably in clouds.

The curious thing is that in the lower atmosphere, these factors that control how fast temperature drops all harmonize to make that rate remain close to -6.5 ^oK/km. Much of the explanation rests with the magical properties of water, both vapor and liquid droplets.

Low altitude, clear air: The radiative term rad is at its largest here, because the water vapor is densest near the surface, its source. But the condensation-driven latent-heat and vapor-pressure-drop terms also have their biggest effects here, and the two effects compete. The condensation effect wins out, keeping the lapse rate shallower than the dry adiabatic value of 9.8 ^oK/km. The convective term conv is small but significant.

Higher altitude, clear air: The rad term fades in significance. The conv term is about the same. The condensation-driven terms also get smaller.

Clouds: The condensation-driven terms continue to shrink. (That might seem strange, since clouds are the most visible result of condensation, but we're looking at the heat transfer, not the reflection of visible light that our eyes can see.) The rad term disappears, while the conv term increases and roughly compensates for the loss of rad.

Above the clouds: The amount of water vapor (or crystallized ice) is much smaller now than it was below. The rad term returns, smaller. The condensation terms are smaller but still significant. The conv term is smaller than in clouds and stops altogether at the tropopause, because the temperature stops lapsing at that point.

As the previous posting explained, isolating different mechanisms at work is important to sorting out the physics of climate. But in reality, adding or removing water from the atmosphere is a single-step thing: it has all of those ramifications (radiative, vapor pressure, latent heat) simultaneously. The distinction among the different effects of water in the air is conceptual and verbal, not physical.
---
* The final component of LTE is phase equilibrium, which of the three components, is the most routinely violated in a fairly obvious way by the formation of clouds (through evaporation followed by condensation), the dissipation of clouds by precipitation or evaporation aloft, and by heat flows. LTE allows clouds to have condensed, but not to be condensing; and liquid water to have evaporated but not to be evaporating. OTOH, it is precisely in stable clouds that LTE is most respected: there is a good approximation to phase equilibrium between water vapor and liquid droplets.

** More exactly, at any particular wavelength of radiation, the radiative absorptivity of matter = radiative emissivity of matter, Kirchhoff's law of radiation. You might remember Kirchhoff from his two laws of electrical circuit analysis (which are just energy and charge conservation in a restricted form). Same brilliant Kirchhoff, different branch of physics.