Climate reservoirs and budgets

Often we hear phrases like "balance of nature" that imply a natural stasis in the world. It's an old idea, occurring in all human cultures, and running in Western thought back to the ancients and beyond, Plato and Aristotle the most influential. Although the concept of natural stasis has been abandoned by modern science, there is a more limited but precise and up-to-date version of this concept - we've met it before as conservation laws or symmetries.

The role of conservation laws in complex systems is sharply two-faced. On the one hand, what invariant structure they have - what remains over time - is tied directly to these laws. On the other, complex systems typically have such a vast number of variables (degrees of freedom) - think of the weather, or of an ecosystem, or the global economy - that the number of conservation laws is far too few to put much of a constraint on how such system can evolve over time. That is why such systems usually behave in chaotic ways ("chaos is weakly constrained") and exhibit the classic pattern of complexity, the spontaneous formation and dissipation of structures.

Everyday conservation laws. The conservation laws important for the climate (air-ocean system, essentially) are familiar from earlier postings and perhaps from chemistry class. They include conservation of total mass, of individual atoms, of energy. More specialized and qualified forms include conservation of air, water, and heat. For a simple system of a finite number of degrees of freedom, conservation laws take the form

function of variables = constant, or change in conserved function = 0

For a continuous system of flows in space, the conservation law takes the form

flow out of a volume - flow into that volume + change of quantity in volume = 0

For "quantity," substitute mass, number of each atomic element, energy, etc. If you draw the boundaries of the volume the right way, so that there are no flows into or out, the conservation law becomes

change of quantity in the volume = 0

There's nowhere for it to go.

Still useful even if not exact. More generally, "partial" conservation laws are useful for quantities that are not conserved, but can still be tracked by what essentially amounts to an accounting device. For example,

flow out of volume - flow into volume + change of quantity in volume = quantity transformed within volume

For example, chemical or phase changes might make certain quantities exactly conserved (like the number, each, of hydrogen and oxygen atoms and water molecules), while others might be subject to transformations that proceed at a certain rate: for example, particular forms of water - solid, liquid, vapor - which are not separately conserved, but transformed into one another in such a way that the number of H atoms, O atoms, and H2O molecules each remains unchanged.

Symmetry, identity, conservation. And that's why conservation laws are related to symmetries. A symmetry, to a mathematical physicist, says, the system does all sorts of things as it evolves, but certain aspects of it retain a constant identity. The game is then to identify what those "invariant" aspects are. In other cases, certain "almost exact" identities can be picked out and used to make approximations.

Conservation laws and climate. Such conservation laws, in their "spatial flow" forms, are the foundation for understanding climate as an "accounting" system: so much air and water (never leaving or being added to - the climate system is "closed" with respect to air and water), so much heat flowing in and out (the system is "open" with respect to heat flow), so much radiation flowing in and out.

Applying these laws correctly means having to identify "reservoirs" of air, water, radiation, heat, etc., some of them truly closed, if you draw their boundaries correctly; some of them only approximately closed; some of them truly open. It also means identifying flows correctly. For example, "global warming" (enhanced infrared-active gases in the air) changes the flow of heat upward in the atmosphere, but it does not trap heat in a fixed volume in the lower atmosphere. This is perhaps the most exact way to state that fallacy.

Another example is water, both the total amount and flows from one place to another. The amount of water in the air-ocean system is almost conserved; there are some slow geochemical reactions that take water out of the system.* But eventually, that water is recycled, reappears in volcanic eruptions, and gets re-injected into the air-ocean system. That points to another consideration in correct application of conservation laws, that of time scale. Something might be taken out that eventually gets put back in. And the geochemical reactions themselves might be so slow and at such a low level that water in the air-ocean system might as well be considered exactly conserved to high accuracy, over shorter time scales.

A year ago on this blog, conservation law reasoning was used to conclude that an enhanced evaporation rate (from, say, "global warming") would lead, not only to more clear-air water vapor, but necessarily to more condensation, clouds, and precipitation: necessarily, since the water is not escaping into space. So it has to come down, at the same rate it's going up, if we sum over the whole atmosphere.

For truly open systems, on the other hand, it's better to think of the flows as providing a daily or annual "budget" of sorts: so much radiation in per day, so much heat out, and so on. The radiation-heat flow is determined by the Sun's output, which itself is not exactly steady. The climate "works with a budget" that's not exactly the same every day or every year. Another example is water flow, considered not on the scale of the whole planet, but within some limited ecosystem. This system might have water "reservoirs," to use the word in its everyday sense; but these reservoirs are open, not closed, and dependent on direct rainfall and ground flow. Because they're open, such reservoirs will not, in general, be faced with even an approximately steady flow in and flow out. Their "water budgets" vary much more wildly.

POSTSCRIPT: The last solar magnetic activity cycle (the one that peaked in 2001) should have ended last year. Typically, the next couple years see an upswing of activity: sunspots, solar wind "gusts," the new cycle's first solar atmosphere mass ejections. But not this time. The peak should be 2011 or 2012. So far it's unusually quiet.

Such periods of extended or exceptional solar quiescence are almost always associated with somewhat colder temperatures here - which is just what we've been seeing the last year or so. (Hat tip to Instapundit.)
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* An example is the set of reactions mentioned here last year as removing carbon dioxide from the atmosphere into the oceans. It starts with dissolution of CO2 gas into CO2 bubbles in the water by diffusion, over roughly a 10-year time scale, with an almost equally fast outgassing of CO2 bubbles back into the atmosphere. CO2 molecules are conserved at this step, but not the next. The water (H2O) then reacts with the CO2 to form carbonic acid (H2CO3). A CO2 and a water molecule are destroyed in the creation of one carbonic acid molecule. But the number of C, the number of O, and the number of H are separately conserved.

Then the fun starts. The H2CO3 dissociates in the water, as all acids do, into H+ and HCO3-, then into 2H+ and CO3--. (The + and - are electric charges.) The opposite reactions occur at the same rates, but the CO3-- is also slowly but steadily removed altogether by binding over century or so timescales with ocean salts: potassium (K+), calcium (Ca++), and magnesium (Mg++), all with some positive charge.

The resulting minerals - calcium carbonate (CaCO3, or limestone, chalk, etc.), magnesium carbonate (MgCO3, or dolomite), and potassium carbonate (K2CO3, or potash) - sink to the ocean floor, where, many, many millennia later, they end up contributing to the natural release of CO2 and H2O from volcanoes back into the atmosphere.