Friday, March 02, 2007

Digression on radiation

The previous postings on thermodynamics talked about matter, radiation, and their thermodynamics - the organization or disorganization of energy at the molecular level. In those postings, I cheated a bit in the discussion of what drives climate change, by the statement that "differences drive climate." That's true, but only for the matter part - the air, ground, and water. In the case of radiation, it's not true, and the distinction between radiation and matter in these respects has important implications for climate and its theory and modeling.

Recall the ideal gas law and the important conclusion of that discussion: you need only three numbers to characterize the physical state of non-moving air (ignoring the qualitative phase of atmospheric water - vapor, condensed, or precipitated.) These are the temperature, pressure, and chemical composition (which in practice means, how much water vapor). The magic of statistical thermodynamics shows that the need for these three numbers is a direct result of the conservation of three summed physical quantities in a closed system of matter particles: energy, volume, and particle number. Think of it like a budget: lots of things can happen in the meantime, but the sum of all the energies and the sum of all the volumes associated with the particles have to remain the same over time, as well as the number of particles. The association between conserved quantity and local thermodynamic measurement is:
  • Conserved energy <-> temperature
  • Conserved volume <-> pressure
  • Conserved number <-> chemical composition
Radiation in every respect is simpler than matter. Matter is made up of molecules of different types that can be transformed by chemical reactions, although the identity and number of each kind of constituent atom remains the same during these reactions. (A related fact is that the amount of matter, measured by its total mass - in grams or whatever - remains constant as well.) Matter can travel, both at the molecular level and in macroscopic bulk, at any speed from zero to the speed of light. OTOH, radiation is made up of photons, which by definition can travel in a vacuum only at the speed of light. (I don't just mean the vacuum of outer space. The space between molecules is a vacuum too.) One consequence is that photons have no mass, so there's no "conserved" mass associated with them. The only distinction between one photon and another is its energy and direction of travel.* The total energy and volume taken up by a "gas" of photons is conserved, but not their number: photons are absorbed and emitted by matter in arbitrary number. All that's conserved is the total energy in the radiation; that fixed energy can be subdivided among many photons, or few; and ditto for the volume.

The result is that a gas of photons needs only two numbers to characterize its physical state, numbers associated with total energy and volume. And if we take a local point of view and consider only densities (such as energy per volume, instead of total energy), we only need one number. That number is usually taken to be the radiation temperature. The radiation pressure and energy density are functions of that temperature alone. Compare with the ideal gas law:
  • Energy density of radiation = universal constant*T3
  • Pressure of radiation = universal constant*T4
For climate, what's even more important, is that the emission and absorption of radiation depend only on the radiation temperature (also as the fourth power). A localized patch of surface on a hot body emits so much power per area (say, watts per cubic centimeter), a quantity that depends just on the local temperature. In particular, it doesn't depend on temperatures somewhere else - that is, it depends on local temperature alone, not on temperature differences.

You can feel this emission of radiant heat from a hot body without touching it, and if its temperature is high enough, you can even see the thermal radiation at frequencies higher than infrared, in the visible part of the electromagnetic spectrum. The heat and light are transmitted as

Hot body (matter) -> radiation -> warm skin and eyes (matter) ,

which drives home the independent physical existence of radiation. If you touch the hot body, the heat is trasmitted more directly:

Hot body (matter) -> warm skin (matter)

Because radiation is so simple, it's easy to not only express the complete theory of how it's emitted and absorbed, it's also easy in many cases to solve that theory exactly or with accurate, controlled approximations. The relationship of radiation to the larger conservation laws (especially for energy) is transparent, unlike the murky case of matter. Photons also have no differential velocity (they all travel at the speed of light), so there's no friction of the kind you see with fluid viscosity, where different fluid layers slide past one another at different velocities. And photons do not interact with each other; they are created and destroyed only by matter.

The part of climate theory dealing with radiation alone is straightforward, and it if it weren't for the complicated nature of matter, climate theory could be solved exactly or in excellent approximation. Simplified textbook treatments of climate for students often use caricatures of climate that emphasize the inflow, absorption, and re-emission of radiation, hiding the complexity of air and ocean - and get results that match observation surprisingly well. But such treatments sweep the hard stuff under the rug by taking important facts as givens from observation instead of explaining them theoretically. It's the mess of matter - its hetereogeneous make-up of different atoms and molecules, its more complex conservation laws and thermodynamics, its complicated flows and phase changes of water vapor - that makes climate theory impossible to solve. That in turn forces climate theorists to turn to simplified models and dubious, uncontrolled approximations.
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* Equivalently, its frequency and wave vector. I'm ignoring the photons' spin or polarization, which for climate is irrelevant. Polarization is important if you're wearing sunglasses on a sunny day, especially on the water 8-)

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