Earth’s Energy Imbalance – Part II – Watts Up With That?

Kevin Kilty

Part I of this series focussed on the sources of data substantiating an energy imbalance, the magnitude of said energy imbalance, and the likely uncertainty of this magnitude. Measurements suggest this magnitude is most likely around 0.76W/m²  but the uncertainty is optimistically stated as being as small as 0.1. Even the authors of scientific reports admit their uncertainty does not include all factors, especially instrumentation and processing biases. Most (89%) of this energy imbalance ends up warming the oceans.

Part II examines climate feedback, its potential magnitude, and likely uncertainty. Part III, when I get around to it, will focus on whether or not the Earth possesses a regulator; that is, whether there is some physical process that will limit response to such an imbalance continuing on into the future. In particular, Part III will explore the curious status of Le Chatelier’s principle, which people often invoke lately to suggest there must be such a regulating mechanism.

The Standard View

It’s best to start with the standard argument for why increasing CO2 will raise surface temperature. Then critique this argument. Figure 1 is from a review article by Held and Soden (hereafter H&S), from 2000[1] but which was largely repeated in 2006.[2] I can do no better to explain than to just paraphrase H&S themselves. Their review article, I think, explains the standard view as clearly, and simply as anything I have read.

To maintain an energy balance, the Earth must radiate back to space the 240 W/m² portion of absorbed solar radiation it receives. To balance this a black body radiator would have to have a temperature of 255 K (240 = σ T_e⁴) which we will call T_e the emission temperature. This temperature occurs at a height above the surface which we call Z_e. As pictured in Figure 1, one might think of the average infrared photon escaping to space from near this level.

As H&S say  “It is an oversimplification to assume that temperature gradients within the troposphere do not change as the climate warms, but this simple assumption has proven to be a very useful point of reference…”

With a fixed T_e and fixed gradient (Lapse rate Γ)  surface temperature then becomes; T_s = T_e + ΓZ_e. In this simple model only changes in Z_e matter. Now the argument takes the following path.

An increased concentration of CO2 in the atmosphere makes the atmosphere more opaque to outgoing infrared radiation from the surface. Thus, to have a CO2-doubled atmosphere equally transparent above Z_e to enable escape of the average photon, Z_e must reside higher in the atmosphere. A doubling of CO2 makes the more opaque atmosphere equally transparent above at Z_e+150m. However, the invariant gradient of 6.5K/km means the temperature at Z_e+150m is lower by about 1K, and according to the Stefan-Boltzmann law this amounts to a reduction in outgoing radiation by about 4W/m²(236.3 = σ 254⁴). There is an energy imbalance that warms the entire atmosphere and surface.

Water Vapor Concentration

The effects do not stop at this point. The entire atmosphere is now 1K warmer, and at the Earth’s surface this higher temperature, according to the Clausius-Clapeyron relationship, will lead to an increase in water vapor pressure at saturation of about 7%. This, in turn, makes the atmosphere more opaque still, and raises Z_e again. The process repeats, but converges to a new equilibrium at surface temperature enhancement of 1/(1-β_H2O), where β_H2O=0.4 is the feedback factor for water vapor.  As H&S say in their 2006 paper, “ a number of important aspects of the hydrological response to warming are a direct consequence of the increase in lower-tropospheric water vapor.”

My Critique

Recognizing that H&S admitted this model is an oversimplification, let’s nonetheless critique its main elements in order of their appearance.

There is no emission surface at 255K

First, people appear to literally believe in an emission surface in the middle troposphere with a temperature of 255K that radiates as a black body. In other words, they view the problem as akin to a typical boundary value problem with the surface acting as one boundary and some imagined layer above acting as the other. While the surface does behave as a near black body (emissivity=0.97), the clear atmosphere is nowhere so emissive that a thin layer will act as a black body.  As H&S, themselves, say in a different publication[3], “…Owing to its much

larger emissivity, the surface contribution is an order of magnitude larger than that from any individual 100-mb atmospheric layer.”  What happens instead is that the compensating outgoing LWIR escapes over a broad vertical region of the atmosphere that begins right at the surface for some wavelengths.[4] The upper boundary of this problem is complex.

Instead of a single degree of freedom, Z_e, establishing surface temperature, there are many different configurations that will do the task. The emission surface has a complicated, and ever changing, configuration. While the idea of an increase in height of the emission surface is one possible response, an atmosphere dehumidfied from above, which is what precipitation accomplishes, could place the average emission surface lower into the atmosphere without changing the surface temperature much if at all.[5]

Radiative-convective equilibrium doctrine

Second, the unvarying 6.5K/km gradient value of radiative-convective equilibrium is not helpful. Anyone who has examined temperature profiles knows that they are hugely more complex than just a constant gradient.  Figure 2 shows a number of model atmospheric temperature profiles drawn from MODTRAN. Note that the only constant 6.5 lapse rate in the set is the U.S. Standard 1976 Atmosphere – a made-up profile of atmospheric non-structure designed by committee.[6] The other examples actually have some structure to them which tell us something about the dynamics of heat transfer in various locales.

Figure 2.

Evaporation

Water vapor assumes primacy in this simplified model, especially in the tropics. Invoking the Clausius-Clapeyron (hereafter CC) relationship means that each 1K rise in surface temperature adds 7% more water vapor into the lower troposphere – it’s a geometrical increase of the most powerful greenhouse gas.

To promote CC scaling, H&S rest their analysis on an atmospheric dynamics relationship for the evaporation process.[7] I have argued in a few instances here and elsewhere that this CC scaling is wrong by reason that the distribution of water vapor throughout the atmosphere is non-equilibrium and dependent on transport processes. It is energy constrained; whereas atmospheric dynamics models of evaporation simply assume the energy constraint vanishes.[8]

Engineering hydrology concerns itself with evaporation from surface storage.[9] Of the expressions for evaporation which they have developed from this focus, some are atmospheric dynamics based; others are energy balance based; others are a combination of the two.[10] Atmospheric dynamics based expressions work well enough, but must have a scale-size issue because they don’t consider energy balance. Without energy balance the process is unphysical.

As Landsay, et al, say in regard to energy used for evaporation.

“In Deep lakes with capacity for considerable heat storage, sudden changes in wind and humidity have longer lasting effects; heat into or from storage assists in balancing energy demands. Thus by using stored excess energy excessive evaporation during a dry, windy week can reduce evaporation which would otherwise occur in subsequent weeks.”[10]

Energy balance provides a constraint. I think atmospheric dynamics is a weak argument. Evaporation (depth of open water evaporated per unit time) based on an energy balance would look something like this:

E=(Q_n+Q_v-Q₀)/(ρH_v(1+R) ); where,

 R is Bowen’s ratio, Q_n=net all wave radiation, Q₀ is energy going into storage, Q_v=energy advected, and H_v is latent heat of vaporization.

Ignoring observations in favor of theory

Regarding the feedback enhancement from water vapor, H&S, say this:

“…There is no simple physical argument of which we are aware from which one could have concluded beforehand that β_H2O was less than unity. The value of β_H2O does, in fact, increase as the climate warms if the relative humidity is fixed. On this basis, one might expect runaway conditions to develop eventually if the climate warms sufficiently.”

One might respond this way. There is no simple physical argument except that the precursor to water vapor, liquid water, has covered a majority of the surface of Earth for 4 billion years, under widely varying conditions, including enhanced CO2, and we have not observed anything remotely like a runaway greenhouse effect. In fact, we more commonly observed excursions into exceptional cold.

Climate modellers seem more impressed with agreement among their models than they seem to be with observations. I am not a climate modeller, but I am not impressed with proof of correctness through consistency among models. I have some experience modeling heat transport. I have translated complex codes for all sorts of purposes from one programming language to another, and debugged the results. It was common enough for me to find the same mistakes in different platforms to suggest common ancestry of codes; sometimes agreement is just lack of independence.

Closing the water vapor controversy

Fourth, as H&S say, closing the water-vapor controversy requires comparison with data.

“Given the acceleration of the trends predicted by many models, we believe that an additional 10 years may be adequate, and 20 years will very likely be sufficient, for the combined satellite and radiosonde network to convincingly confirm or refute the predictions of increasing vapor in the free troposphere and its effects on global warming.”

How well do we know the underlying physics?

Bob Irvine wrote about feedback two years ago. He showed data similar to, but independent of that in Figure 3. Figure 3 shows Era5 reanalysis from the tropics plotting 2m temperature and dew point data against one another. There is a rise in dew point temperatures, specific humidity or mixing ratio, as one prefers, all show a modest rise in absolute humidity of about 3% over the past two decades. The observed rise does not support CC scaling and certainly not a constant relative humidity.

Figure 3.

Irvine’s essay provided a Table comparing AR4 to AR6 feedback values. Of particular interest are the large changes in the combined feedback values for water vapor+lapse rate.  In AR4 (2007) this value is 0.96 ± 0.08 W/Km2. In AR6 (2019) it is stated as 1.30 (1.15 to 1.47). Perhaps noting that water vapor has not kept pace with CC scaling in the two decades from 2000 to 2019 caused the revision. Of greater interest is the stated uncertainty.

Consider AR4 uncertainty of 0.08 as pertaining to a coverage factor of 1.0, and the interval for AR6 as the 90% confidence interval. This places central values four standard deviations apart, meaning that each estimate is highly unlikely in view of the other. In addition, a graphic in reference [3] shows the water vapor and lapse rate feedback values separately. As was often noted in Part I with regard to energy imbalance, the uncertainty of combined quantities becomes smaller than uncertainty of its components. How does this happen? Possibly model biases in estimates of water vapor feedback are anticorrelated with biases in lapse rate feedback.

Cloud Feedback

Everyone recognizes that clouds are a weakness of global climate models. Everyone may not recognize the tremendous variability of clouds day-to-day. Figure 4 below shows what total downward welling solar radiation looks like on two closely spaced days along the Colorado front range. The raw data is by UTC day, so these plots are patched together. Yet what they show is patently clear. The partially cloudy day has enhanced downward radiation when cumulus or cumulonimbus north and northwest of the observatory redirect scattered light toward the observatory, but more often reduced radiation to winter-time subarctic conditions when they shade the observatory. The change in daily received solar (downwelling total solar) is from 34,000kJ/m² on the clear day, to 23,000kJ/m² on the other. That’s huge by anyone’s estimation. Even the surface albedo (ratio of blue curve to red) changes from 18.8% to 19.3% simply because of the redirection of sunlight.

Figure 4.

Figure 5 is from data taken just north of Laramie, Wyoming, at an elevation of 2200m on a clear summer day. There is no SURFRAD site here, but I own numerous radiometers and was testing/calibrating one. By pure serendipity, I caught the sheerest of clouds – Subvisual cirrus so insubstantial that I could not see them by eye. However, the radiometer detected them, and occasionally when the cirrus formed a wisp that could be discerned by eye which passed in front of the Sun, I could correlate it with the radiometer.

Lynch [12] suggests these clouds have an optical depth near τ=0.03, which would translate into a power density variation in the neighborhood of 1000e^-0.03=970 or decline of 30 W/m². Just about what Figure 5 shows.  Thus, even in this instance of the least substantial clouds one could imagine, the effect is ten times as large as that of a degree K change in surface temperature.

Clouds present a large climate forcing.

Figure 5

Nevertheless, this climate forcing is not what “feedback” means in the context of climate science. Feedback is the effect a warmer surface has on the radiative difference between clear sky and total sky. It is the change in cloud forcing (clear sky less total sky), and complicating the matter is that clear sky is a calculation from theory again. AR4 lists the feedback effect of clouds as 0.69 ± 0.38 W/K-m²; AR6 lists 0.42 (-0.1 to 0.94).

Exploring Feedback

Models result generally in positive feedback. I have no basis for arguing with that. What I disbelieve is that a relatively tiny differential quantity, calculated as the difference between two other large variable quantities which are, themselves, differences of large variable quantities, aren’t swamped by uncertainty. This is especially so given the lack of resolution in climate models, plus the parameterizations of things like clouds, convection and precipitation that aren’t calculated directly from physics.

I am not an opponent of climate modeling, but I do wish the results of modeling could be grounded by comparison with observations. I fully recognize that observations can be so encumbered with problems of calibration and data reduction schemes that what results has large uncertainty also. Nonetheless, I want to see a comparison now and then. So, what do we do about feedback?

Let’s consider feedback calculation schemes. They’re done with models. The challenge with doing the same things with observations is having long enough runs of days to approximate climate. Can any schemes be mimicked with observations? Soden et al, outline schemes based on modeling.[3] How would they translate to observations? 

Scheme 1: Think of the net energy imbalance at the top of the atmosphere (TOA) as a function of just a few items. Figuratively call this R( w, T, a, c); where w stands for water vapor, T for surface temperature, a for surface albedo, and c for clouds. Run multiple models changing only one item, c for example, at a time and compare to the unperturbed state. This is very difficult to do with observations because it is difficult to search for and find extended runs of days that are identical in all respects except for one item.

Scheme 2: Separate feedback into two factors. The first, the “radiative kernel,” depends only on the radiative algorithm and the other is simply the change in the climatology of the feedback of interest of two comparator states. The product of the two is the feedback. This method is not pertinent to observation. Yet, radiative kernels are interesting for a different reason in Part III.

Scheme 3: Perturb the climate model with a step change in sea surface temperature. Then infer the climate sensitivity from resulting computed changes in radiative fluxes.

The scheme most amenable to observations is scheme 3. The sea surface perturbations available naturally are ENSO, PDO, AMO, and so forth. In addition, we might think about organizing the effort like the factorial experiments we do in engineering; we purposely change multiple variables in each successive run because that is what the weather will do. Build a table of contrasts with four factors (w,T,a,c) and note how each changes in each successive change of PDO or ENSO. Eventually we will fill in the entire table of contrasts and have a rough idea not only of factors of feedback but also interactions among them.[13]

Conclusion

Just as in Part I, in Part II I find numbers applied to fundamental concepts that are small but with uncertainty estimates I cannot reconcile nor find entirely credible. The entire topic of climate change appears to be like this; guided by numbers and measurements that need to be certain within 0.1% but are often 10 or 100 times worse. Is the cloud feedback positive? I don’t know. There is a lot of weather to sort through to find out. However, if it is positive that is not necessarily a bad thing as explained in Part III.

References and Notes

1-Held, I. M., and B. J. Soden, 2000: Water vapor feedback and global warming. Annu. Rev. Energy Environ., 25, 441–475.

2-Isaac Held, Brian J. Soden, 2006, Robust Responses of the Hydrological Cycle to Global Warming, J. Climate, V. 19, p.5686

3-Brian J. Soden, et al, 2008, Quantifying Climate Feedbacks Using Radiative Kernels, J. Climate, V. 21, p3504

4-Even in the moist tropical atmosphere, clear sky is over 80% transmissive to many segments of the IR spectrum as wide as 2 inverse centimeters in wavenumber.

5- This is an element of Lindzen’s iris hypothesis.

6- This mean profile may be a case of an average that is never actually observed – like the 3.5 average of dice rolls. Of further interest, the band 8,9, and 10 water vapor satellite images are reduced using the U.S. standard atmosphere. Perhaps use of this model in this context could be explored in some future post.

7-An atmospheric dynamics model from H&S: “ … evaporation E from the ocean can be

modeled as proportional to the difference between the saturation vapor pressure at

the surface temperature T* and the vapor pressure in the atmosphere at some small

convenient reference height…”

8-An analogous situation occurs in electronic feedback circuits. The power supply is almost never shown explicitly in such circuits under the assumption that the power supply is capable of providing whatever the feedback circuitry demands. Clipping occurs when output reaches close to the power supply rails. This doesn’t occur in models even though it happens physically.

9-I worked as a USGS hydrologist for three years in the 1970s. Employment is a capable teacher.

10-Ray K. Lindsay, Jr., et al, 1975, Hydrology for Engineers, 2nd ed, Mcgraw-Hill

11- Lynch, Subvisual Cirrus, Aerospace Report number TR-93(3308)-1, 1994, available online at https://apps.dtic.mil/sti/tr/pdf/ADA289329.pdf

12- I had a bit of fun with generative AI. I asked if ENSO could be used as an analog of climate change. It said “no”. I then asked if El Niño could be. AI said “yes”.

13-At a conference in 2002 I suggested this as a method of making history itself, and historical sciences, look more like experimental science. No one, to my knowledge, has taken up the idea.