RationalClimate and friends: Exactly what do we mean when we say a group of molecules are in Local Thermodynamic Equilibrium?

Petty (p126) tells us that LTE exists in a group of molecules when they are exchanging kinetic energy with each other via molecular collisions much faster than they are exchanging energy with the radiation field or other source of energy. I – perhaps incorrectly – therefore conclude that LTE exists if and only if molecular collisions have produce a Boltzmann distribution of excited and ground states in a group of colliding molecules. Then I look at the “kinetic definition” of temperature – proportional to the mean kinetic energy of a group of rapidly colliding gas molecules – more precisely a group of molecules with an average kinetic energy of (3/2)kT, where k is the Boltzmann constant. And I go a step further and conclude that a system where LTE doesn’t exist, doesn’t have a “temperature” according to this definition. (There is a second definition of temperature that involves entropy.)

The semi-classical derivation of Planck’s Law postulates the existence of a Boltzmann distribution of energy between ground and excited states. The exp(-E/kT) term from the Boltzmann distribution appears in the form of exp(-hv/kT) or exp(-hc/lambda*kT) in Planck’s Law.

Petty tells us that LTE doesn’t exist in LED lights, lasers, and fluorescent lights. These devices manage to create excited states without relying on molecular collisions so they can emit visible light without being above 1000 K. These are systems where it isn’t clear that heat always flows from hot to cold – unless you assert they don’t have a properly-defined temperature. And LTE doesn’t exist above the stratosphere, a location where some scientists refer to a “Boltzmann temperature” (based on kinetic energy) and a Planck temperature (based on the rate of photon emission).

When Planck derived his law, he went one step further and postulated thermodynamic equilibrium between quantized oscillations (today molecules) and radiation. In other words, the rate of absorption of photons equals the temperature-dependent rate of emission of photons (and that temperature dependence arises from a Boltzmann distribution of excited states). The atmosphere doesn’t emit radiation with a blackbody spectrum, because temperature changes with altitude and radiation doesn’t always travel far enough or interact strongly enough with gas molecules for absorption and emission to reach equilibrium. This is why applying a black- or gray-body model to the atmosphere can be problematic. Most liquids and solids are much denser than the GHGs in the atmosphere, so absorption and emission are more likely to be in equilibrium at the local temperature, and these materials tend to emit radiation of near blackbody intensity.

When absorption and emission are not in equilibrium, Planck’s Law doesn’t apply and we must fall back on Schwarzschild’s equation for radiation transfer, which is what most climate scientists use to calculate radiative forcing and transfer. o is the cross-section for absorption of a photon and o*B(lambda,T) is the cross-section for emission of a photon – when LTE exists.

dI = n*o*B(lambda,T)*ds – n*o*I*ds

We are lucky that the only portion of our atmosphere that absorbs and emits a significant number of photons (the troposphere and some of the stratosphere) is in LTE, so radiative transfer calculations are relatively simple. And the radiation field in the atmosphere is weak enough that we can ignore stimulated emission. We are unlucky because our education system usually stops with Planck’s law, which provides an inadequate explanation for the behavior of radiation in the atmosphere.

]]>F + lambda*ECS = 0

ECS = -F_2x/lambda

As best I can tell, lambda can be thought of as the planet’s radiative response to a change in GMST – the increase in emission of LWR plus the increase in reflection of SWR per degC increase in GMST (when warming has eliminated the radiative imbalance at the TOA created by a forcing). OSR = outgoing SWR = reflected SWR.

lambda = dOLR/dTs + dOSR/dTs

F_2x is the change in radiative cooling to space (after the stratosphere adjusts) accompanying an instantaneous doubling of CO2 determined by radiative transfer calculations through today’s troposphere – a change that is independent of temperature.

Some people use effective radiative radiative forcing (ERF) instead of traditional radiative forcing, but the later is simpler in this context.

If correct, Dewitte and Clerbaux tell us dOLR/dTs is -2.93 +/− 0.3 W/m2/K (a negative value for heat loss. If ECS were 3.6 K and F_2x were 3.6, then lambda would be -1.0 W/m2/K and dOSR/dTs would need to be +1.9 W/m2/K. If ECS were 1.8 or 1.2 K and F_2x were 3.6 K, then lambda would be -2.0 or -3.0 W/m2/K respectively. In those cases, dOSR/dTs would need to be +0.9 W/m2/K or -0.1 W/m2/K respectively.

Therefore, even if dOLR/dTs were -2.9 W/m2/K, ECS can take on almost any value – if cloud SWR feedback is positive enough.

In his paper on the IRIS effect, Stevens suggests dOLR/dTs could be as large as -4 W/m2/K in the tropics, but only if fewer clouds let more LWR escape to space, with the consequence that less SWR would be reflected at the same time. Lindzen and Choi claimed dOLR/dTs in the tropics is about -5 W/m2/K. Tsushima and Manabe find that dOLR/dTs is -2.2 W/m2/K in response to seasonal warming. This response is weighted towards the regions of the planet with the largest seasonal change in temperature – ie outside the tropics.

Unfortunately, we don’t know dOSR/dTs, in part because monthly changes in OSR (unlike OLR) don’t have a simple linear relationship with changes in Ts. Some of the response is clearly lagged and (changes sign from positive to negative feedback with longer lags). I expect dOSR observed from space through clear skies due to changes in surface albedo (seasonal snow cover and sea ice) to lag behind dTs by several months, but the same phenomena is observed from cloudy skies. (This could be because (poorly-modeled) marine boundary layer clouds are non-local phenomena produced by descending warm dry air imported from long distances and cold ocean currents and upwelling.)

Let’s suppose dOLR/dTs is -2.9 W/m2/K, dOSR is +1.9 W/m2/K, lambda is -1.0 W/m2/K and ECS is 3.6 K/doubling. After 1.25 K of warming, dOLR is -3.6 W/m2 and the LWR imbalance created by 2XCO2 at the TOA has been reduced to 0 W/m2. However OSR has increased by +1.9*1.25 = 2.4 W/m2. So the imbalance has only been reduced by only 1/3 and it is now observed in the SWR channel, despite initially having been created by GHGs in the OLR channel. IIRC, most abrupt 2X and 4X experiments show OLR returning to balance about halfway to steady state warming, with further warming being driven by the increasing amount of SWR absorbed.

In other words, a value of -2.93 +/− 0.3 W/m2/K for dOLR/dTs is not grossly inconsistent with the behavior of AOGCMs. In my very limited experience, modelers simply focus on ECS and F_2X derived from 4X experiments and don’t spend much time discussing its long and short-wavelength components, dOLR/dTs + dOSR/dTs.

]]>FYI. I thought I had inadvertently deleted this post so I resubmitted it. That is why there is a duplicate further down. Apologies for that.

]]>Thanks much both for the explanation and for what looks to be an excellent reference. If I need to get too much deeper into this question I may well track it down.

Your explanation points to a different way for me to think about this, and also suggests several different conceptualizations.

The one that makes the most intuitive sense (but of course I would need to try to do a reasonableness check on the numbers of molecules) is that there are so many excited CO2 molecules in so many different energy states any one instant, that even though the probability of any *one* of them making it through the complete release “process” (as outlined in my above reply to SoD) is vanishingly small, there are so many of them that they simply swamp the small probability of individual success by collective overwhelming brute force, so to speak. I dunno.

But I like your temperature-based explanation (regardless of whatever the detailed emissions process underlying it turns out to be) because it does resolve how downwelling radiation can persist so well even at night.

]]>Yes, thanks. I can pin down my dilemma more precisely.

If it takes at least one millisecond for a CO2 molecule to “unwind”, so to speak, its mechanical stress in such a way as to release a photon, and during that millisecond a number of other molecules collide with it, does it ever actually complete that process, or, does each new collision amount to being the logical equivalent of pressing “restart” on a PC?

To use a baseball analogy: the pitcher is winding up to deliver a pitch to home plate when the opposing team starts throwing baseballs at the pitcher from their dugout. Does the pitcher ever get the pitch off?

Dewitt’s response following looks to address this question from a slightly different perspective so I will think about his response as well.

]]>RC,

Collisional excitation. At LTE, the fraction of excited CO2 molecules depends *only* on the local temperature, not the incoming or outgoing radiation at that wavelength. Or, more properly, over the wavelength range of the excited molecule populationo, as there is a range of vibrational energy levels that can be excited to and emitted from, not to mention the broadening of the individual lines by the Doppler effect and pressure broadening. The intensity of the emitted radiation depends *only* on the number (fraction x number density) of excited molecules, not the lifetime of the excited state for an individual molecule. Btw, LTE applies at altitudes well above the tropopause, i.e. to about 100km.

If you really want to learn about atmospheric radiation transfer, then you should obtain a copy of A First Course in Atmospheric Radiation by Grant W. Petty. ( https://sundogpublishingstore.myshopify.com/products/a-first-course-in-atmospheric-radiation-g-w-petty ). It’s currently on sale for $36, which is only $2 higher than when I bought my copy many years ago.

]]>RationalClimate,

I do not doubt that radiation from that layer exists, BTW. The question is: How is it being generated, if not from CO2?

Assuming I understand your dilemma correctly..

The molecule of CO2 that aborbs radiation at some wavelength (lets say 15.00um) doesn’t re-emit that radiation immediately. Instead, the energy absorbed is transferred by collision to surrounding molecules, so increases the local temperature.

Emission of radiation at 15.00um is from CO2 (it could be from water vapor, but CO2 overwhelms at that wavelength).

Just not from the specific CO2 molecule that absorbed the radiation ( it’s possible, just a vanishingly small probability).

Hopefully that helps?

]]>1. From this column, “Each layer of the atmosphere radiates according to its temperature.”

2. From one of your other columns (hoping, here, that my memory is accurate enough to skip the effort of tracking down the exact quote), a statement that a specific portion of the atmosphere can only radiate according to the extent that it contains GHGs, and then only according to the absorption lines of those GHGs.

3. From Raymond T. Pierrehumbert, regarding CO2, a statement that for the “transitions that are most significant in the thermal IR, the lifetimes tend to range from a few milliseconds to a few tenths of a second.”

and (expanding out, here, the scientific notation because I cannot do superscripts), “In contrast, the typical time between collisions for, say, a nitrogen-dominated atmosphere at a pressure of 10000 Pa and temperature of 250 K is well under 0.0000001 seconds. Therefore, the energy of the photon will almost always be assimilated by collisions into the general energy pool of the matter and establish a new Maxwell–Boltzmann distribution at a slightly higher temperature. That is how radiation heats matter in the LTE limit.”

Note: LTE is local thermodynamic equilibrium (LTE).

(Source: “Infrared radiation and planetary temperature”, Raymond T. Pierrehumbert, January 2011 Physics Today, American Institute of Physics.

Link– https://t.co/9rWJBi0MZ8 )

Barring a typo, it looks like Pierrehumbert’s statement applies to the atmosphere up around the Tropopause. Separately, in a lecture, William Happer made a comparable statement for near surface conditions.

Analysis

I realize that the CO2 situation might not apply to other GHGs, but CO2 is so significant to GHE theory that it deserves it’s own analysis.

So here is the dilemma presented by 1, 2, and 3 above:

IR is emitted at the surface. Some escapes to space, some is absorbed by CO2, some… etc. The IR energy absorbed by the CO2 serves to warm the surrounding packet of air by collisions between the energized CO2 and other air molecules, but the CO2 itself never has a chance to re-radiate a photon, at least anywhere near the surface.

So if CO2 were, in this thought experiment, the only GHG present, and if CO2 can never radiate for itself by emitting an IR photon, how is it possible for the layer of atmosphere containing that CO2 to radiate according to the temperature of that layer?

I do not doubt that radiation from that layer exists. The question is: How is it being generated, if not from CO2?

]]>1. From this column, “Each layer of the atmosphere radiates according to its temperature.”

2. From one of your other columns (hoping, here, that my memory is accurate enough to skip the effort of tracking down the exact quote), a statement that a specific portion of the atmosphere can only radiate according to the extent that it contains GHGs, and then only according to the absorption lines of those GHGs.

3. From Raymond T. Pierrehumbert, regarding CO2, a statement that for the “transitions that are most significant in the thermal IR, the lifetimes tend to range from a few milliseconds to a few tenths of a second.”

and (expanding out here the scientific notation because I cannot do superscripts), “In contrast, the typical time between collisions for, say, a nitrogen-dominated atmosphere at a pressure of 10000 Pa and temperature of 250 K is well under 0.0000001 seconds. Therefore, the energy of the photon will almost always be assimilated by collisions into the general energy pool of the matter and establish a new Maxwell–Boltzmann distribution at a slightly higher temperature. That is how radiation heats matter in the LTE limit.”

Note: LTE is local thermodynamic equilibrium (LTE).

(Source: “Infrared radiation and planetary temperature”, Raymond T. Pierrehumbert, January 2011 Physics Today, American Institute of Physics.

Link– https://t.co/9rWJBi0MZ8 )

Barring a typo, it looks like Pierrehumbert’s statement applies up at around the Tropopause. Separately, in a lecture, William Happer has made a comparable statement for near surface conditions.

I realize that the CO2 situation might not apply to other GHGs, but CO2 is so significant to GHE theory that it deserves it’s own analysis.

So here is the dilemma:

IR is emitted at the surface. Some escapes to space, some is absorbed by CO2, some… etc. The IR energy absorbed by the CO2 serves to warm the surrounding packet of air by collisions between the energized CO2 and other air molecules, but the CO2 itself never has a chance to re-radiate a photon, at least anywhere near the surface.

So if CO2 were, in this thought experiment, the only GHG present, and if CO2 can never radiate for itself by emitting an IR photon, how is it possible for the layer of atmosphere containing that CO2 to radiate according to the temperature of that layer?

I do not doubt that radiation from that layer exists, BTW. The question is: How is it being generated, if not from CO2?

]]>