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## Wonderland and Radiative Forcing – Part Two

In Wonderland, Radiative Forcing and the Rate of Inflation we looked at the definition of radiative forcing and a few concepts around it:

• why the instantaneous forcing is different from the adjusted forcing
• what adjusted forcing is and why it’s a more useful concept
• why the definition of the tropopause affects the value
• GCM results usually don’t use radiative forcing as an input

In this article we will look at some results using the Wonderland model.

Remember the Wonderland model is not the earth. But the same is also true of “real” GCMs with geographical boundaries that match the earth as we know it. They are not the earth either. All models have limitations. This is easy to understand in principle. It is challenging to understand in the specifics of where the limitations are, even for specialists – and especially for non-specialists.

What the Wonderland model provides is a coarse geography with earth-like layout of land and ocean, plus of course, physics that follows the basic equations. And using this model we can get a sense of how radiative forcing is related to temperature changes when the same value of radiative forcing is applied via different mechanisms.

In the 1997 paper I think that Hansen, Sato & Ruedy did a decent job of explaining the limitations of radiative forcing, at least as far as the Wonderland climate model is able to assist us with that understanding. Remember as well that, in general, results we see from GCMs do not use radiative forcing. Instead they calculate from first principles – or parameterized first principles.

### Doubling CO2

Now there’s a lot in this first figure, it can be a bit overwhelming. We’ll take it one step at a time. We double CO2 overnight – in Wonderland – and we see various results. The left half of the figure is all about flux while the right half is all about temperature:

From Hansen et al 1997

Figure 1 – Green text added – Click to Expand

On the top line, the first two graphs are the net flux change, as a function of height and latitude. First left – instantaneous; second left – adjusted. These two cases were explained in the last article.

The second left is effectively the “radiative forcing”, and we can see that the above the tropopause (at about 200 mbar) the net flux change with height is constant. This is because the stratosphere has come into radiative balance. Refer to the last article for more explanation. On the right hand side, with all feedbacks from this one change in Wonderland, we can see the famous predicted “tropospheric hot spot” and the cooling of the stratosphere.

We see in the bottom two rows on the right the expected temperature change :

• second row – change in temperature as a function of latitude and season (where temperature is averaged across all longitudes)
• third row – change in temperature as a function of latitude and longitude (averaged annually)

It’s interesting to see the larger temperature increases predicted near the poles. I’m not sure I really understand the mechanisms driving that. Note that the radiative forcing is generally higher in the tropics and lower at the poles, yet the temperature change is the other way round.

### Increasing Solar Radiation by 2%

Now let’s take a look at a comparison exercise, increasing solar radiation by 2%.

The responses to these comparable global forcings, 2xCO2 & +2% S0, are similar in a gross sense, as found by previous investigators. However, as we show in the sections below, the similarity of the responses is partly accidental, a cancellation of two contrary effects. We show in section 5 that the climate model (and presumably the real world) is much more sensitive to a forcing at high latitudes than to a forcing at low latitudes; this tends to cause a greater response for 2xCO2 (compare figures 4c & 4g); but the forcing is also more sensitive to a forcing that acts at the surface and lower troposphere than to a forcing which acts higher in the troposphere; this favors the solar forcing (compare figures 4a & 4e), partially offsetting the latitudinal sensitivity.

We saw figure 4 in the previous article, repeated again here for reference:

From Hansen et al (1997)

Figure 2

In case the above comment is not clear, absorbed solar radiation is more concentrated in the tropics and a minimum at the poles, whereas CO2 is evenly distributed (a “well-mixed greenhouse gas”). So a similar average radiative change will cause a more tropical effect for solar but a more even effect for CO2.

We can see that clearly in the comparable graphic for a solar increase of 2%:

From Hansen et al (1997)

Figure 3 - Green text added - Click to Expand

We see that the change in net flux is higher at the surface than the 2xCO2 case, and is much more concentrated in the tropics.

We also see the predicted tropospheric hot spot looking pretty similar to the 2xCO2 tropospheric hot spot (see note 1).

But unlike the cooler stratosphere of the 2xCO2 case, we see an unchanging stratosphere for this increase in solar irradiation.

These same points can also be seen in figure 2 above (figure 4 from Hansen et al).

Here is the table which compares radiative forcing (instantaneous and adjusted), no feedback temperature change, and full-GCM calculated temperature change for doubling CO2, increasing solar by 2% and reducing solar by 2%:

From Hansen et al 1997

Figure 4 – Green text added – Click to Expand

The value R (far right of table) is the ratio of the predicted temperature change from a given forcing divided by the predicted temperature change from the 2% increase in solar radiation.

Now the paper also includes some ozone changes which are pretty interesting, but won’t be discussed here (unless we have questions from people who have read the paper of course).

### “Ghost” Forcings

The authors then go on to consider what they call ghost forcings:

How does the climate response depend on the time and place at which a forcing is applied? The forcings considered above all have complex spatial and temporal variations. For example, the change of solar irradiance varies with time of day, season, latitude, and even longitude because of zonal variations in ground albedo and cloud cover. We would like a simpler test forcing.

We define a “ghost” forcing as an arbitrary heating added to the radiative source term in the energy equation.. The forcing, in effect, appears magically from outer space at an atmospheric level, latitude range, season and time of day. Usually we choose a ghost forcing with a global and annual mean of 4 W/m², making it comparable to the 2xCO2 and +2% S0 experiments.

In the following table we see the results of various experiments:

Hansen et al (1997)

Figure 5 – Click to Expand

We note that the feedback factor for the ghost forcing varies with the altitude of the forcing by about a factor of two. We also note that a substantial surface temperature response is obtained even when the forcing is located entirely within the stratosphere. Analysis of these results requires that we first quantify the effect of cloud changes. However, the results can be understood qualitatively as follows.

Consider ΔTs in the case of fixed clouds. As the forcing is added to successively higher layers, there are two principal competing effects. First, as the heating moves higher, a larger fraction of the energy is radiated directly to space without warming the surface, causing ΔTs to decline as the altitude of the forcing increases. However, second, warming of a given level allows more water vapor to exist there, and at the higher levels water vapor is a particularly effective greenhouse gas. The net result is that ΔTs tends to decline with the altitude of the forcing, but it has a relative maximum near the tropopause.

When clouds are free to change the surface temperature change depends even more on the altitude of the forcing (figure 8). The principal mechanism is that heating of a given layer tends to decrease large-scale cloud cover within that layer. The dominant effect of decreased low-level clouds is a reduced planetary albedo, thus a warming, while the dominant effect of decreased high clouds is a reduced greenhouse effect, thus a cooling. However, the cloud cover, the cloud cover changes and the surface temperature sensitivity to changes may depend on characteristics of the forcing other than altitude, e.g. latitude, so quantitive evaluation requires detailed examination of the cloud changes (section 6).

### Conclusion

Radiative forcing is a useful concept which gives a headline idea about the imbalance in climate equilibrium caused by something like a change in “greenhouse” gas concentration.

GCM calculations of temperature change over a few centuries do vary significantly with the exact nature of the forcing – primarily its vertical and geographical distribution. This means that a calculated radiative forcing of, say, 1 W/m² from two different mechanisms (e.g. ozone and CFCs) would (according to GCMs) not necessarily produce the same surface temperature change.

### References

Radiative forcing and climate response, Hansen, Sato & Ruedy, Journal of Geophysical Research (1997) – free paper

### Notes

Note 1: The reason for the predicted hot spot is more water vapor causes a lower lapse rate – which increases the temperature higher up in the troposphere relative to the surface. This change is concentrated in the tropics because the tropics are hotter and, therefore, have much more water vapor. The dry polar regions cannot get a lapse rate change from more water vapor because the effect is so small.

Any increase in surface temperature is predicted to cause this same change.

With limited research on my part, the idealized picture of the hotspot as shown above is not actually the real model results. The top graph is the “just CO2″ graph, and the bottom graph is the “CO2 + aerosols” – the second graph is obviously closer to the real case:

From Santer et al 1996

Many people have asked for my comment on the hot spot, but apart from putting forward an opinion I haven’t spent enough time researching this topic to understand it. From time to time I do dig in, but it seems that there are about 20 papers that need to be read to say something useful on the topic. Unfortunately many of them are heavy in stats and my interest wanes.

## Clouds & Water Vapor – Part Seven – Upper Tropospheric Models & Measurement

In an earlier article on water vapor we saw that changing water vapor in the upper troposphere has a disproportionate effect on outgoing longwave radiation (OLR). Here is one example from Spencer & Braswell 1997:

From Spencer & Braswell (1997)

Figure 1

The upper troposphere is very dry, and so the mass of water vapor we need to change OLR by a given W/m² is small by comparison with the mass of water vapor we need to effect the same change in or near the boundary layer (i.e., near to the earth’s surface). See also Visualizing Atmospheric Radiation – Part Four – Water Vapor.

This means that when we are interested in climate feedback and how water vapor concentration changes with surface temperature changes, we are primarily interested in the changes in upper tropospheric water vapor (UTWV).

### Upper Tropospheric Water Vapor

A major problem with analyzing UTWV is that most historic measurements are poor for this region. The upper troposphere is very cold and very dry – two issues that cause significant problems for radiosondes.

The atmospheric infrared sounder (AIRS) was launched in 2002 on the Aqua satellite and this instrument is able to measure temperature and water vapor with vertical resolution similar to that obtained from radiosondes. At the same time, because it is on a satellite we get the global coverage that is not available with radiosondes and the ability to measure the very cold, very dry upper tropospheric atmosphere.

Gettelman & Fu (2008) focused on the tropics and analysed the relationship (covariance) between surface temperature and UTWV from AIRS over 2002-2007, and then compared this with the results of the CAM climate model using prescribed (actual) surface temperature from 2001-2004 (note 1):

This study will build upon previous estimates of the water vapor feedback, by focusing on the observed response of upper-tropospheric temperature and humidity (specific and relative humidity) to changes in surface temperatures, particularly ocean temperatures. Similar efforts have been performed before (see below), but this study will use new high vertical resolution satellite measurements and compare them to an atmospheric general circulation model (GCM) at similar resolution.

The water vapor feedback arises largely from the tropics where there is a nearly moist adiabatic profile. If the profile stays moist adiabatic in response to surface temperature changes, and if the relative humidity (RH) is unchanged because of the supply of moisture from the oceans and deep convection to the upper troposphere, then the upper-tropospheric specific humidity will increase.

They describe the objective:

The goal of this work is a better understanding of specific feedback processes using better statistics and vertical resolution than has been possible before. We will compare satellite data over a short (4.5 yr) time record to a climate model at similar space and time resolution and examine the robustness of results with several model simulations. The hypothesis we seek to test is whether water vapor in the model responds to changes in surface temperatures in a manner similar to the observations. This can be viewed as a necessary but not sufficient condition for the model to reproduce the upper-tropospheric water vapor feedback caused by external forcings such as anthropogenic greenhouse gas emissions.

The results are for relative humidity (RH) on the left and absolute humidity on the right:

From Gettelman & Fu (2008)

Figure 2

The graphs show that change in 250 mbar RH with temperature is statistically indistinguishable from zero. For those not familiar with the basics, if RH stays constant with rising temperature it is the same as increasing “specific humidity” – which means an increased mixing ratio of water vapor in the atmosphere. And we see this is the right hand graph.

Figure 1a has considerable scatter, but in general, there is little significant change of 250-hPa relative humidity anomalies with anomalies in the previous month’s surface temperature. The slope is not significantly different than zero in either AIRS observations (1.9 ± 1.9% RH/°C) or CAM (1.4 ± 2.8% RH/°C).

The situation for specific humidity in Fig. 1b indicates less scatter, and is a more fundamental measurement from AIRS (which retrieves specific humidity and temperature separately). In Fig. 1b, it is clear that 250- hPa specific humidity increases with increasing averaged surface temperature in both AIRS observations and CAM simulations. At 250 hPa this slope is 20 ± 8 ppmv/°C for AIRS and 26 ± 11 ppmv/°C for CAM. This is nearly 20% of background specific humidity per degree Celsius at 250 hPa.

The observations and simulations indicate that specific humidity increases with surface temperatures (Fig. 1b). The increase is nearly identical to that required to maintain constant relative humidity (the sloping dashed line in Fig. 1b) for changes in upper-tropospheric temperature. There is some uncertainty in this constant RH line, since it depends on calculations of saturation vapor mixing ratio that are nonlinear, and the temperature used is a layer (200–250 hPa) average.

The graphs below show the change in each variable as surface temperature is altered as a function of pressure (height). The black line is the measurement (AIRS).

So the right side graph shows that, from AIRS data of 4 years, specific humidity increases with surface temperature in the upper troposphere:

From Gettelman & Fu (2008)

Figure 3 – Click to Enlarge

There are a number of model runs using CAM with different constraints. This is a common theme in climate science – researchers attempting to find out what part of the physics (at least as far as the climate model can reproduce it) contributes the most or least to a given effect. The paper has no paywall, so readers are recommended to review the whole paper.

### Conclusion

The question of how water vapor responds to increasing surface temperature is a critical one in climate research. The fundamentals are discussed in earlier articles, especially Clouds and Water Vapor – Part Two - and much better explained in the freely available paper Water Vapor Feedback and Global Warming, Held and Soden (2000).

One of the key points is that the response of water vapor in the planetary boundary layer (the bottom layer of the atmosphere) is a lot easier to understand than the response in the “free troposphere”. But how water vapor changes in the free troposphere is the important question. And the water vapor concentration in the free troposphere is dependent on the global circulation, making it dependent on the massive complexity of atmospheric dynamics.

Gettelman and Fu attempt to answer this question for the first half decade’s worth of quality satellite observation and they find a result that is similar to that produced by GCMs.

Many people outside of climate science believe that GCMs have “positive feedback” or “constant relative humidity” programmed in. Delving into a climate model is a technical task, but the details are freely available – e.g., Description of the NCAR Community Atmosphere Model (CAM 3.0), W.D. Collins (2004). It’s clear to me that relative humidity is not prescribed in climate models – both from the equations used and from the results that are produced in many papers. And people like the great Isaac Held, a veteran of climate modeling and atmospheric dynamics, also state the same. So, readers who believe otherwise – come forward with evidence.

Still, that’s a different story from acknowledging that climate models attempt to calculate humidity from some kind of physics but believing that these climate models get it wrong. That is of course very possible.

At least from this paper we can see that over this short time period, not subject to strong ENSO fluctuations or significant climate change, the satellite date shows upper tropospheric humidity increasing with surface temperature. And the CAM model produces similar results.

### References

Observed and Simulated Upper-Tropospheric Water Vapor Feedback, Gettelman & Fu, Journal of Climate (2008) – free paper

How Dry is the Tropical Free Troposphere? Implications for Global Warming Theory, Spencer & Braswell, Bulletin of the American Meteorological Society (1997) – free paper

### Notes

Note 1 - The authors note: “..Model SSTs may be slightly different from the data, but represent a partially overlapping period..”

I asked Andrew Gettelman why the model was run for a different time period than the observations and he said that the data (in the form needed for running CAM) was not available at that time.

## Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheres

Once we start measuring climate parameters we get a lot of data. To compare datasets, or datasets with models, we can look at means, standard deviations, medians, percentiles, and so on.

I’ve frequently mentioned the problem that climate is nonlinear. If we investigate the underlying physics of most processes we find that the answer to the problem does not scale linearly as inputs change.

Roca et al (2012) say:

The main reason for water vapor to be of importance to the energetics of the climate lies in the nonlinearity of the radiative transfer to the humidity. The outgoing longwave radiation (OLR) is indeed much more sensitive to a given perturbation in a dry rather than moist environment, conferring a central role of the moisture distribution in these regions to the radiation budget of the planet and to the overall climate sensitivity.

The authors demonstrate that with the same mean value of water vapor in a dry climate we can get different values of radiation to space for different distributions. (Note that FTH = free tropospheric humidity. This is the humidity above the atmospheric boundary layer – the boundary layer ranges from between a few hundred meters and one km):

Energy constraints on planet Earth (i.e. applying the first law of thermodynamics) require that, at equilibrium, the Earth emits in the long wave as much radiation as its gets from the Sun. This budget approach is hence focused on the mean values of the OLR over the whole planet and over long time scales corresponding to the global radiative-convective equilibrium theory.

While the mean OLR is the constrained parameter, owing to the nonlinearity of the clear-sky radiative transfer to water vapour (Figs. 2a, 3), the whole distribution of moisture has to be considered rather than its mean in order to link the distribution of humidity to that of radiation.

To illustrate this, the OLR sensitivity to FTH curve (Fig. 2a) and four distributions of FTH for a dry case are considered (Fig. 2bc):  a constant distribution with mean of 14.5%, an uniform distribution with mean of 14.5% bounded within plus or minus 5%, a Gaussian distribution with mean of 14.5% (and a 5% standard deviation) and a generalized log-normal distribution with a mean of 14.5% shown in Fig. 2c. The mean OLR corresponding to the constant distribution is 311 W/m². The uniform and normal distribution yield to a mean OLR larger by 0.7 W/m² in both cases.

The log-normal PDF, on the other hand, gives a 3 W/m² overestimation of the OLR with respect to the constant case. At the scale of the doubling of CO2 problem, such a systematic bias could be significant depending on its geographical spread, which is explored next.

PDF is the probability density function.

And in case it’s not clear what the authors were saying, the same average humidity can result in significantly different OLR depending on the distribution of the humidity from which the average was calculated.

Figure 1

We saw the importance of the drier subsiding regions of the tropics in Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert in that they have much higher OLR than the convective regions.

This paper calculates the results (using the vertical profile of temperature as a multi-year summer average of Bay of Bengal conditions from ERA-40) that with a constant boundary layer humidity (BLH), increasing FTH from 1% to 15% reduces OLR by 23 W/m². Increasing FTH from 35% to 50% reduces OLR by only 8 W/m². The spectral composition of these changes is interesting:

Figure 2

The authors comment that the changes in surface temperature (in the 2nd graph) result in a smaller change in OLR, which seems to be indicated from the brightness temperature graph. I have asked Remy Roca if he has the OLR calculations for this second graph to hand.

Then a statistical test is applied to values of humidity at 500 hPa (about 5.5 km altitude):

Figure 3

We see that the moist areas are more likely to have a normal (gaussian) distribution, while the dry areas are less likely.

Here is an actual distribution from Ryoo et al (2008), for different regions from 250 hPa (about 11km) for both tropical (red) and sub-tropical regions (blue):

Figure 4

The authors use the frequency of occurrence of relative humidity less than 10% as a measure:

The need of handling the whole PDF of humidity instead of only the mean of the field implies the manipulation of the upper moments of the distribution (skewness and kurtosis). While the computations are straightforward, the comparison of two PDFs through the comparison of their 4 moments is not. Assuming a generalized log-normal distribution also requires 4 parameters to be fitted. It can be brought down to 2 parameters by imposing the lower and upper range limit of the distribution (0 and 100% for instance) at the cost of limiting the possible distributions.

The simplified model (Ryoo et al. 2009) also comprises only two parameters, linked to the first two moments of the distribution. Still, the moments-to-moments comparison of PDFs remains difficult.

Here, it is proposed to limit the analysis to a single parameter characterizing the PDF with emphasis on the dry foot of the distribution: the frequency of occurrence of RH below 10%, noted in the following as RHp10.

The paper then provides some graphs of the frequency of RH below 10%. We can think of it as another way of looking at the same data, but focusing on the drier end of the dataset:

From Roca et al 2012

Figure 5

From Roca et al 2012

Figure 6

The authors then consider the source of the driest air at 500hPa. Now this uses what is called the advection-condensation method, something I hope to cover in a later article on water vapor. But for interest, here is their result:

From Roca et al 2012

Figure 7

The middle graph is the first graph with air sourced from the extra-tropics excluded.

The RHp10 distribution of the reconstructed field for the boreal summer 2003 is compared to the RHp10 distribution obtained by keeping only the air masses that experienced last saturation within the intertropical belt (35S–35N) in Fig. 9. Excluding the extra-tropical last saturated air masses overall moistens the atmosphere. The domain averaged RHp10 decreases from 37 to 23% without the extra-tropical influence. While the patterns overall remain similar within the two computations, the driest areas nevertheless appear more impacted and less spread in the tropics only case (Fig. 9 middle). The very dry features in the subtropical south Atlantic is mainly built from tropical originating air with the fraction of extra-tropical influence less than 10% (Fig. 9c).

### Conclusion

Even if a monthly mean value of a climatological value from a model matches the measurement monthly mean it doesn’t necessarily mean that the consequences for the climate are the same.

Small changes in the distribution of values (for the same average) can have significant impacts. Here we see that this is the case for dry regions.

In Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert we saw that these dry regions have a big role in cooling the tropics and therefore in regulating the temperature of the planet. Understanding more about the distribution of humidity and the mechanisms and causes is essential for progress in climate science.

Other articles in this series:

Clouds and Water Vapor – Part One – introducing some ideas from Ramanthan from ERBE 1985 – 1989 results

Clouds and Water Vapor – Part One – Responses - answering some questions about Part One

Clouds and Water Vapor – Part Two - some introductory ideas about water vapor including measurements

Clouds and Water Vapor – Part Three - effects of water vapor at different heights (non-linearity issues), problems of the 3d motion of air in the water vapor problem and some calculations over a few decades

Clouds and Water Vapor – Part Four - discussion and results of a paper by Dessler et al using the latest AIRS and CERES data to calculate current atmospheric and water vapor feedback vs height and surface temperature

Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert - focusing on a 1995 paper by Pierrehumbert to show some basics about circulation within the tropics and how the drier subsiding regions of the circulation contribute to cooling the tropics

### References

Tropical and Extra-Tropical Influences on the Distribution of Free Tropospheric Humidity over the Intertropical Belt, Roca et al, Surveys in Geophysics (2012) – paywall paper

Variability of subtropical upper tropospheric humidity, Ryoo, Waugh & Gettelman, Atmospheric Chemistry and Physics Discussions (2008) – free paper

## Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert

In Atmospheric Circulation – Part One we saw the Hadley circulation: convection in the tropics and subsidence in the subtropics:

From Marshall & Plumb (2008)

Figure 1

The distribution of relative humidity in the atmosphere is a result of this circulation.

The sun heats the tropical ocean surface which both warms the air just above it and also evaporates water into this air. This hot moist air rises. As this air rises it cools, due to adiabatic expansion (see Potential Temperature), and water vapor condenses out, releasing the latent heat stored. The strongest examples are known as deep convection because the convected air rises all the way to the tropopause (the top of the troposphere).

Cold air can hold much less water vapor than hot air – for example, air at 30°C can hold seven times as much water vapor as air at 0°C. Air at the warmest ocean surface can hold about 1,000 times (in g/kg) more water vapor than the coldest point in the atmosphere (the tropical tropopause).

So by the time convected air reaches the very cold tropopause (top of the troposphere) it has become very dry.

Once at the tropopause it slowly subsides, and warms due to compression by the atmosphere [updated sentence Dec  27th]. During this subsidence, the absolute amount of water vapor doesn’t increase (no source of new water vapor), but the temperature does increase. Therefore, the relative humidity (RH) – the amount of water vapor present vs the maximum that could be held – keeps decreasing.

Here is the annual average of relative humidity (originally shown in Clouds and Water Vapor – Part Two):

From Soden (2006)

Figure 2

The tropical troposphere is moist, while the sub-tropics are much drier. Here is the frequency of very low humidity at 500hPa (about 5.8 km altitude) from Roca et al (2012):

From Roca et al 2012

Figure 3

And from the same paper, a longer term average of the free tropospheric humidity (FTH = humidity above the boundary layer) to the left and the frequency of occurrence of very low humidity (<10%) to the right:

From Roca et al 2012

Figure 4

Why are we interested in very low humidity?

### Pierrehumbert 1995

There are a number of climate scientists with a significant contribution to the study of water vapor in climate, and with apologies to people I have missed, my own informal list includes Richard S Lindzen, Kenneth Minschwaner, Kerry Emmanuel, Isaac M Held, Brian J Soden, Raymond T Pierrehumbert, Steven C Sherwood, Andrew E Dessler, Rémy Roca.

Pierrehumbert wrote a 1995 paper, Thermostats, radiator fins, and the local runaway greenhouse, which seems to be somewhat out of date now but a good starting point to illustrate some important concepts. (A more comprehensive paper on the background to this topic is Pierrehumbert’s 1999 paper, reference below).

Our version of the single-cell model is distinguished primarily by a choice of some radical simplifications that allow us to bring out the central behavior transparently. The chief utility of the model is didactic. We introduce it to bring out in concrete terms the repercussions of some of the phenomena discussed in section 3. It has too many adjustable parameters and too much missing physics to enable reliable quantitative projections of climate change to be made, but it will be nonetheless of interest to see whether such a model can be made to yield earthlike conditions..

[Emphasis added]. For those who are unfamiliar with climate models, this is much much much simpler than any real climate model. As an aside Isaac Held has a great article on the rationale for, and problem of, simplifying climate models in The ‘Fruit Fly’ of Climate Models. It’s an article more about making simpler GCM’s than about making 2-box models, but the points are still valid.

Below, the tropics represented in two parts – the convective region with high humidity, and the subsiding region with low humidity.

From Pierrehumbert 1995

Figure 5

The essence of the main part of his paper is that the tropical atmosphere, with high humidity, is not very efficient at radiating away the large amounts of solar heat absorbed, while the low humidity subsiding region is much more effective at this.

Here is a simplified example demonstrating the problem of radiating away high incident solar radiation as relative humidity (RH) increases (very simplified because this atmospheric profile has a constant RH above the boundary layer):

From Pierrehumbert 1995

Figure 6

From Fig. 2 [figure 6 in this article] we see that if the full annual-mean insolation of 420 W/m² were absorbed, T(0) would run away to temperatures in excess of 340K for any relative humidity greater than 25%. Even in Sc [solar radiation] is reduced to 370 W/m² to account for the mean clear sky albedo in the tropics, the temperature would run away for relative humidities as low as 50%.

Considered locally, the present-day tropics would thus be in a runaway state (or nearly so) so long as it is sufficiently close to saturation.

Clouds do not alter this conclusion because insofar as Cs + Cl = 0 in the tropics the reduction in solar absorption is compensated by an equal reduction in OLR. In order to stabilize the tropical runaway, one must appeal to the lateral heat transports out of the moist regions. Satellite observations show OLR of 300 W/m² or less over the warmest tropical oceans, confirming the inability of the warmest oceans to get rid of the absorbed solar radiation locally.

(See Note 1).

So, of course, one well known mechanism for tropical cooling is export of heat to higher latitudes. Basic climate texts demonstrate that this takes place as a matter of course by plotting the absorbed solar radiation vs OLR by latitude. The tropics absorb more energy than they radiate, while the poles radiate more than they absorb. The average poleward transport of energy by latitude can be calculated as a result.

The other mechanism of tropical cooling takes place in the subsiding regions of the tropics.

Pierrehumbert comments (on his simple model):

The warm pool atmosphere cannot get rid of its heat, because of the strong water vapor greenhouse effect; this heat must be exported via zonal and meridional heat fluxes, to drier regions where it can be radiated to space. These dry, non-convective regions act like “radiator fins” stuck into the side of the warm pool atmosphere. The “super greenhouse” shape of the clear-sky OLR curve in the analysis of Raval and Ramanathan (1989) and Ramanathan and Collins (1991) provides direct evidence for radiator fins, since it shows that OLR is generally higher in some cooler SST regions than it is over the warmest tropical waters.

### How does Air at the Tropopause Subside?

The air at the tropopause is very cold. Why doesn’t it sink down below the warmer air underneath?

This question was answered in Potential Temperature. Air that rises cools even without any exchange of heat with the surroundings (due to losing internal energy while doing work expanding against the lower pressure).

Air that sinks warms without any exchange of heat with the surroundings (due to gaining internal energy from work done on it by the compression of the higher pressure atmosphere).

And the formulas for both of these processes are very simple and well-understood. So the important graph is the graph of potential temperature vs altitude (or pressure), which shows what temperature each parcel of air would have if it was moved to the surface without any exchange of heat. It allows us to properly compare air temperature at different heights (pressures).

We see that potential temperature – the real comparison metric – increases with height. This is to be expected – warmer air floats above cooler air:

From Marshall & Plumb (2008)

Figure 7 – Click for a larger image

So, if we take air, warmed by strong solar heating at the surface, and raise it quickly to the tropopause, how does it ever come down?

Consider the air with potential temperature of 360K (almost 87°C if moved adiabatically back to the surface). If it starts to sink it warms (due to compression by the atmosphere) and its natural buoyancy pushes it back up.

The mechanism for air to subside involves losing heat “diabatically”. Adiabatic means no exchange of heat with surroundings, which can happen with rapid air movement during convection. Diabatic means there is an exchange of heat with the surroundings.

And as the air cools it sinks. (Its actual & potential temperature decreases, allowing it to sink, but then compressional warming takes place and its actual temperature increases).

From Minschwaner & McElroy 1992

Figure 8

If there was no radiative cooling there would be no gentle subsidence, at least nothing like the current process we see in the atmosphere.

Skip the next section if you don’t like maths..

### Maths Digression

There is an equation for the subsiding region which relates the heating rate (=-cooling rate), H, with two important parameters:

H ∝ cp.ω.∂θ/∂p

where H = heating rate (=-cooling rate), ∝ is the symbol for “proportional to”, c= heat capacity of air under constant pressure, ω = rate of change of pressure with time following the parcel (how fast the parcel is ascending or descending), ∂θ/∂p = change in potential temperature with pressure, so this is a measure of the atmospheric stratification

The two important parameters are:

• ω – subsidence rate
• ∂θ/∂p – stratification of the atmosphere

The value H is essentially dependent on the amount of radiatively-active gases in the atmosphere in the subsiding region. There is also an effect from any mixing with extra-tropical colder air.

### Results from the Teaching Model

Here is a sample result from Pierrehumbert’s model under some simplified assumptions (no ocean heat transport and no heat transfer between tropics and extra-tropics).

The solid curve is Energy In to the warm pool = absorbed solar – cooling due to atmospheric circulation from the cold pool. The dashed curve is Energy Out from the warm pool:

From Pierrehumbert 1995

Figure 9

Pierrehumbert makes the comment that the stability of the solution depends on the steepness of the solid curve and this is due to the fixed emissivity of the “cold pool” atmosphere. Remember that the region with subsidence has little water vapor above the boundary layer. In fact, as we will see in the upcoming graphs, it is the ability of the subsiding region to cool via radiation that allows the atmospheric circulation.

Here is set of graphs under the same simplified assumptions (and with RH=100% in the warm pool) showing how the surface temperature (Ts1 = warm pool sea surface temperature, Ts2 = cold pool sea surface temperature) varies with emissivity of the cold pool atmosphere. Each graph is a different ratio of surface area of cold pool vs warm pool. Remember that the “warm pool” is the convecting regions and the “cold pool” is the subsiding regions:

From Pierrehumbert 1995

Figure 10

We can see that when the emissivity of the cold pool region is very low (when the amount of “greenhouse” gases is very low) the warm pool regions go into a form of thermal runaway. This is because radiative cooling is now very ineffective in the subsiding regions and so the tropical large-scale atmospheric circulation (the Hadley circulation) is “choked up”. If air can’t cool, it can’t descend, and so the circulation slows right down.

Consider the case where there is much less CO2 in the atmosphere – then the emissivity is governed mostly by water vapor. So the dry subsiding region has little ability to radiate any heat to space – preventing subsidence – but the hot moist convecting region cannot radiate sufficient heat to space because the emission to space is coming from higher up in the atmosphere, e.g. see fig. 6, of the water vapor.

So increasing the emissivity from zero (increasing “greenhouse” gases) cools the climate to begin with. Then as the emissivity increases past a certain point the warm pool surface temperatures start to increase again.

And so long as the cold pool area is large enough compared with the warm pool area the temperatures can be quite reasonable – even without any export of heat to higher latitudes.

This is a very interesting result. We see that climate is not “linear”. In simple terms “not linear” means that just because one area cools down by 1°C doesn’t mean that an equal size area must heat up by 1°C.

Now we see a result with slightly more realistic boundary conditions – heat is exported to higher latitudes (and RH reduced to 75% in the warm pool):

From Pierrehumbert 1995

Figure 11

Overall, the result of the (slightly) more realistic conditions is simply reducing the temperatures. This is not surprising.

### Conclusion

The 1995 paper is quite complex and covers more than this topic (note for keen readers, the end of the paper has a summary of all the terms used in the paper, something I wish I had known while trying to make sense of it).

The model is a very simplified model of the atmosphere and can easily be criticized for any of the particular assumptions it makes.

The reason for highlighting the paper and drawing out some of its conclusions is because there is a lot of value in understanding:

• the large scale circulation
• its effect on water vapor
• what factors allow air near the tropopause to cool and descend
• the non-linearity of climate

Of particular interest might be understanding that more “greenhouse” gases in the subsiding regions allow a faster circulation, which in turn removes more heat from the climate than a slower circulation.

Other articles in this series:

Clouds and Water Vapor – Part One – introducing some ideas from Ramanthan from ERBE 1985 – 1989 results

Clouds and Water Vapor – Part One – Responses - answering some questions about Part One

Clouds and Water Vapor – Part Two - some introductory ideas about water vapor including measurements

Clouds and Water Vapor – Part Three - effects of water vapor at different heights (non-linearity issues), problems of the 3d motion of air in the water vapor problem and some calculations over a few decades

Clouds and Water Vapor – Part Four - discussion and results of a paper by Dessler et al using the latest AIRS and CERES data to calculate current atmospheric and water vapor feedback vs height and surface temperature

Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheres - demonstrating that different distributions of water vapor yet with the same mean can result in different radiation to space, and how this is important for drier regions like the sub-tropics

### References

Atmosphere, Ocean and Climate Dynamics, Marshall & Plumb, Elsevier Academic Press (2008)

Tropical and Extra-Tropical influences on the distribution of free tropospheric humidity over the inter-tropical belt, Roca et al, Surveys in Geophysics (2012)

Thermostats, radiator fins, and the local runaway greenhouse, Pierrehumbert, Journal of the Atmospheric Sciences (1995) – free paper

Subtropical Water Vapor As a Mediator of Rapid Global Climate Change, Pierrehumbert, (1999)

### Notes

Note 1 – The statement:

Clouds do not alter this conclusion because insofar as Cs + Cl = 0 in the tropics the reduction in solar absorption is compensated by an equal reduction in OLR

relates to the fact that in the tropical region the overall cloud effect is close to zero. This is surprising and the subject of much study. For a starting point see On the Observed Near Cancellation between Longwave and Shortwave Cloud Forcing in Tropical Regions, J.T. Kiehl, Journal of Climate (1994)

## Clouds and Water Vapor – Part Four

In this article in the series we will look an interesting paper:

An analysis of the dependence of clear-sky top-of-atmosphere outgoing longwave radiation on atmospheric temperature and water vapor, by Dessler, Yang, Lee, Solbrig, Zhang and Minschwaner, JGR (2008).

I used some results of this paper in Theory and Experiment – Atmospheric Radiation, but only for comparing calculated top of atmosphere radiative fluxes vs measurement.

I think that the “basic physics” of radiative transfer and atmospheric convection is challenging enough, and the question of feedback even harder.

There are hundreds of papers (thousands really) on this confusing subject and so, for me, drawing conclusions requires a lot of research. Luckily, most people interested in the climate debate already know the answer to the question of feedback from water vapor, so this article won’t be so interesting to them.

In fact, the paper under review doesn’t claim any real answers in the subject of water vapor feedback with climate change:

We are looking at regional variations in lapse rate in a fixed climate, rather than variations in the average lapse rate as the climate changes. This result demonstrates the unsuitability of using variations in different regions in our present climate as a proxy for climate change.

But even with the guarded comments of a published paper, the results are very interesting – and help, at the very least, to illuminate some aspects of how water vapor and atmospheric temperature interact to change the radiative cooling from the planet.

So for people looking for a quick answer, it’s not here. For people wanting to understand the interaction between surface temperature, atmospheric temperature, water vapor and outgoing longwave radiation (OLR) – this might provide a few insights on their journey.

### Background

In trying to understand feedback we want to know what happens to the outgoing longwave radiation (OLR) from the climate as surface temperature changes.

Some basic (but hard to calculate) radiative physics – already covered in many places including CO2 – An Insignificant Trace Gas? Part Seven – The Boring Numbers (and the preceding parts of the series) – tells us that, all other things being equal, a doubling of CO2 in the atmosphere from pre-industrial levels will lead to a surface temperature change of about 1°C.

Apart from other variability – how will the climate respond to this increase in surface temperature?

It is only by isolating different causes and effects that we can hope to understand the complexity of the climate. It’s slow but there is more chance of getting the correct answer.

The main miscreant identified as possibly causing a much higher than 1°C increase is water vapor feedback. Water vapor is the dominant “greenhouse” gas, but is variable in space and time as it responds to climate conditions. See, for example, Clouds and Water Vapor – Part Two.

When we think about feedback, one of the most important considerations is how OLR responds to a change in surface temperature.

Let’s consider the change in surface radiation when the temperature increases by 1°C. Most of the earth’s surface has an emissivity very close to 1. At 15°C the increase in surface radiation for this 1°C increase, ΔR = 5.5 W/m². So if the OLR also increased by 5.5 W/m² then the feedback from the climate would be zero. (See comment below for why this is not quite correct).

Why? Because all of the increase in surface radiation has also been emitted from the climate system into space. Picture the scene if instead 10 W/m² was emitted into space after this 1°C increase in surface temperature – this would be negative feedback.

And if the OLR change was 1 W/m² ? This would be positive feedback. Because the increase in radiation from the surface wasn’t matched by radiation from the climate system.

If this doesn’t make sense, ask a question. It’s hard to make progress without grasping this point.

### Measurements and “Model”

Dessler compares the results of over 100,000 measurements of top of atmosphere (TOA) fluxes from the CERES satellite with two band models which provide computational efficiency (see note 1).

Figure 1 – Comparison of a band model with measured results

This is simply to demonstrate that the model for calculating TOA fluxes is reliable and accurate. The results are used for later calculations. Other graphs in the paper compare the results against surface temperature and latitude to confirm that no bias exists in the results.

Atmospheric temperature and water vapor are measured using AIRS – Atmospheric Infrared Sounder flying on the NASA Aqua satellite. CERES = “Clouds and the Earth Radiant Energy System”, which is also flying on the Aqua satellite.

The measurements taken by AIRS and CERES are “virtually simultaneous”.

These measurements were all taken in March 2005 between 70°N and 70°S over the ocean under clear skies.

The measurements were selected from nighttime measurements. Why? To eliminate any contribution around the 4μm wavelength from solar radiation.

### A Basic Equation

Equations aren’t fun for a lot of people and that’s understandable. Stay with me, I will try and explain it in plain English.

What we want to know is how OLR (radiation from the climate to space) changes as surface temperature changes. If we can establish this, we can understand how the climate currently responds to surface temperatures – and what feedbacks are currently in place, at least for the time under consideration:

Figure 2 – The equation

The red term is the main value we want to know – the “rate of change” of OLR with surface temperature

Or, how much does the outgoing longwave radiation change as surface temperature changes?

The orange term = the sum (vertically through the atmosphere) of all the changes in OLR as surface temperature changes, due to the change in atmospheric temperature

The green term = the sum (vertically through the atmosphere) of all the changes in OLR as surface temperature changes, due to the change in water vapor

And before we “dive in”, the basic concepts are, in simple terms:

• if the atmosphere gets warmer it radiates more to space – and this cools the climate
• if water vapor increases it reduces the OLR (because it is a “greenhouse” gas) – and this heats the climate (because less radiation to space takes place)
• if water vapor increases it reduces the “lapse rate” (note 2), making the atmosphere warmer higher up, increasing radiation to space – and this cools the climate

Now let’s take a look at the graphical picture of how atmospheric temperature and humidity vary with surface temperature and height. Think of surface temperature as a proxy for latitude.

Here is how the air temperature vs height, and humidity vs height, vary with surface temperature:

from Dessler (2008)

Figure 3 – Measurements

For those new to humidity measurements in the atmosphere, note the strong dependency on surface temperature and on height in the atmosphere (1000hPa is the surface and 200hPa is around 12km above the surface).

Now we want to plot two of the terms in the equation (figure 2). The colors are matched up with the highlighted terms in the original equation.

From Dessler (2008)

Figure 4 – Calculated – Color text added

These values are calculated by using the model. (We have already seen that this band model accurately calculates the OLR from surface temperature, air temperature and humidity).

As you would expect, when air temperature increases by 1K the OLR increases – because a hotter atmosphere radiates at a higher intensity. This is with all other conditions held the same.

And as you might expect, when the humidity is increased by 10% the OLR decreases – because a more opaque atmosphere has a lower transmittance to surface radiation. This is with all other conditions held the same.

We have been calculating these terms from figure 2:

Now, find how OLR changes due to surface temperature changes we need to also find out these terms:

Or, in English:

• the change in air temperature due to surface temperature changes
• the change in humidity due to surface temperature changes.

From Dessler (2008)

Figure 5 – Measured – Color text added

So, to give an example of what these graphs show, we can see that at around 293-294K, an increase in surface temperature has little or no effect on humidity. Around 300K an increase in surface temperature has a large effect on humidity.

Now we are going to multiply the terms together to find:

• the change in OLR with surface temperature – due to atmospheric temperature changes
• the change in OLR with surface temperature – due to humidity changes

Figure 6 – Results – Color text added

The advantage of this method is that when we look at the summary and say, for example:

Oh that’s interesting, the strongest positive feedback effects are around 302 K, what causes that? The strongest negative feedback effects are around 290 – 295 K, what causes that?

- we can review the terms that created the result and see which dominates – and why.

Looking at the total, we can see that between 298 – 303 K the OLR decreases as surface temperature increases (note that the plot is of the change in OLR as Ts increases versus Ts). And below 298 K the OLR increases as surface temperature increases.

This is in agreement with Raval & Ramanathan’s work based on ERBE data shown in Part One where the positive feedback comes from the tropics, and is reduced by the negative feedback from the sub-tropics and mid-latitudes.

The decrease of OLR as surface temperature increases became known as the super-greenhouse effect. Remember that any effect below an increase of 5.5W/m².K (at 15°C) is a positive feedback. (And at 30°C, this threshold value is 6.3W/m².K). An actual decrease of OLR as surface temperature increases is, therefore, a very strong positive feedback effect.

We can see the result plotted against surface temperature and height – now let’s see the total value against surface temperature, and some comparisons of the actuals vs reference scenarios:

Figure 7 – Color text and highlighting added

The first graph shows the change in OLR with surface temperature – due to atmospheric temperature changes. The blue line shows the result if the lapse rate was fixed. Remember that a lower value of changing OLR with Ts is more towards positive feedback.

This is a quantitative estimate of the effect of the changing lapse rate on dOLR/dTs, and it shows that it is negative for almost all values of Ts. In other words, as Ts increases, so does the lapse rate, and the general effect of this is to reduce dOLR/dTs, and therefore OLR, below what they would be if the atmosphere maintained a constant lapse rate.

The second graph shows the change in OLR with surface temperature – due to humidity changes. The purple line shows the result if relative humidity was constant. (And see the results from Sun & Oort, shown in Part Three).

In the subtropics, the ‘‘changing RH’’ line is positive, meaning that RH decreases with increasing Ts. This relative dryness contributes to high values of OLR here, providing a key pathway for the climate system to lose energy back to space. As Ts crosses the convective threshold, ≈298 K, the RH of the atmosphere abruptly increases, leading to a strong increase in q and a reduction in OLR and its gradient.

The third graph compares the results by using the data graphed in Figure 1 with the results derived through this article – and they are the same.

We also plot in this panel the right-hand side of equation (1):

Σi(∂OLR/∂Ti)(∂Ti/∂Ts) + Σi(∂OLR/∂qi)(∂qi/∂Ts) + ∂OLR/∂Ts,

derived from lines plotted in Figures 8a and 8b. As one can clearly see, the agreement is excellent. Note that this is a stringent test as these two lines are derived from completely independent data: one line is derived entirely from CERES data while the other line is derived entirely from AIRS data and a radiative transfer model. The excellent agreement gives us great confidence that, given observations of Ta and q, the clear-sky OLR budget is well understood inthe present atmosphere. We also see no evidence that neglected terms are important, in agreement with previous work..

### Conclusion

The paper gives us an excellent insight into how atmospheric temperature and humidity vary as surface temperature varies – over the ocean. And how this maps into changes in OLR as surface temperature changes.

We see that the results are similar to Ramanathan’s work shown in Part One.

These are valuable insights.

If surface temperature increases from any cause, does this mean that positive feedback from water vapor will amplify this? If surface temperature reduces from any cause, does this mean that positive feedback from water vapor will amplify this?

Surely that depends.

But ask yourself this – if the results had shown the opposite effect, would you find them significant?

Other articles in this series:

Clouds and Water Vapor – Part One – introducing some ideas from Ramanthan from ERBE 1985 – 1989 results

Clouds and Water Vapor – Part One – Responses - answering some questions about Part One

Clouds and Water Vapor – Part Two - some introductory ideas about water vapor including measurements

Clouds and Water Vapor – Part Three - effects of water vapor at different heights (non-linearity issues), problems of the 3d motion of air in the water vapor problem and some calculations over a few decades

Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert - focusing on a 1995 paper by Pierrehumbert to show some basics about circulation within the tropics and how the drier subsiding regions of the circulation contribute to cooling the tropics

Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheres - demonstrating that different distributions of water vapor yet with the same mean can result in different radiation to space, and how this is important for drier regions like the sub-tropics

### Notes

Note 1: See CO2 – An Insignificant Trace Gas? Part Four for more explanation of ”band models”. A “model” doesn’t mean “GCM”. In this case it simply means a more efficient way of calculating the TOA flux than using “line by line” calculations in the radiative transfer equations.

The HITRANS database contains 2.7M spectral lines and so “doing it the long way” takes a lot of time. Therefore, over time, many band models have been created – and critically evaluated – against the hard way.

Note 2: The lapse rate is the decrease in temperature as you go up through the atmosphere. In a dry atmosphere the temperature reduces at around 10K/km. In a very moist atmosphere the temperature reduces at around 4K/km. And, on average, the lapse rate is 6.5K/km. So the more water vapor there is in the atmosphere, the warmer the atmosphere at any given height.

## Clouds and Water Vapor – Part Three

In Part One we had a look at Ramanathan’s work (actually Raval and Ramanathan) attempting to measure the changes in outgoing longwave radiation vs surface temperature.

In Part Two (Part Zero perhaps) we looked at some basics on water vapor as well as some measurements. The subject of the non-linear effects of water vapor was raised.

Part One Responses attempted a fuller answer to various questions and objections about Part One

Water vapor feedback isn’t a simple subject.

First, a little more background.

### Effectiveness of Water Vapor at Different Heights

Here are some model results of change in surface temperature for changes in specific humidity at different heights:

From Shine & Sinha (1991)

For newcomers, 200mbar is the top of the troposphere (lower atmosphere), and 1000mbar is the surface.

You can see that for a given increase in the mixing ratio of water vapor the most significant effect comes at the top of the troposphere.

The three temperatures: cool = 277K (4°C); average = 287K (14°C); and warm = 298K (23°C).

Now a similar calculation using changes in relative humidity:

From Shine & Sinha (1991)

The average no continuum shows the effect without the continuum absorption portion of the water vapor absorption. This is the frequency range between 800-1200 cm-1, (wavelength range 12-8μm) – often known as the “atmospheric window”. This portion of the spectral range is important in studies of increasing water vapor, something we will return to in later articles.

Here we can see that in warmer climates the lower troposphere has more effect for changes in relative humidity. And for average and cooler climates, changes in relative humidity are still more important in the lower troposphere, but the upper troposphere does become more significant.

(This paper, by Shine & Sinha, appears to have been inspired by Lindzen’s 1990 paper where he talked about the importance of upper tropospheric water vapor among other subjects).

So clearly the total water vapor in a vertical section through the atmosphere isn’t going to tell us enough (see note 1). We also need to know the vertical distribution of water vapor.

Here is a slightly different perspective from Spencer and Braswell (1997):

Spencer and Braswell (1997)

This paper took a slightly different approach.

• Shine & Sinha looked at a 10% change in relative humidity – so for example, from 20% to 22% (20% x 110%)
• Spencer & Braswell said, let’s take a 10% change as 20% to 30% (20% + 10%)

This isn’t an argument about how to evaluate the effect of water vapor – just how to illustrate a point. Spencer & Braswell are highlighting the solid line in the right hand graph, and showing Shine & Sinha’s approach as the dashed line.

In the end, both will get the same result if the water vapor changes from 20% to 30% (for example).

### Boundary Layers and Deep Convection

Here’s a conceptual schematic from Sun and Lindzen 1993:

The bottom layer is the boundary layer. Over the ocean the source of water vapor in this boundary layer is the ocean itself. Therefore, we would assume that the relative humidity would be high and the specific humidity (the amount of water vapor) would be strongly dependent on temperature (see Part Two).

Higher temperatures drive stronger convection which creates high cloud levels. This is often called “deep convection” in the literature. These convective towers are generally only a small percentage of the surface area. So over most of the tropics, air is subsiding.

Here is a handy visualization from Held & Soden (2000):

Held and Soden (2000)

The concept to be clear about is within the well-mixed boundary layer there is a strong connection between the surface temperature and the water vapor content. But above the boundary layer there is a disconnect. Why?

Because most of the air (by area) is subsiding (see note 2). This air has at one stage been convected high up in the atmosphere, has dried out and now is returning back to the surface.

Subsiding air in some parts of the tropics is extremely dry with a very low relative humidity. Remember the graphs in Part Two – air high up in the atmosphere can only hold 1/1,000th of the water vapor that can be held close to the surface. So air which is saturated when it is at the tropopause is – in relative terms – very dry when it returns to the surface.

Therefore, the theoretical connection between surface temperature and specific humidity becomes a challenging one above the boundary layer.

And the idea that relative humidity is conserved is also challenged.

### Relationship between Specific Humidity and Local Temperature

Sun and Oort (1995) analyzed the humidity and temperature in the tropics (30°S to 30°N) at a number of heights over a long time period:

Sun and Oort (1995)

Note that the four graphs represent four different heights (pressures) in the atmosphere. And note as well that the temperatures plotted are the temperatures at that relevant height.

Their approach was to average the complete tropical domain (but not the complete globe) and, therefore, average out the ascending and descending portions of the atmosphere:

Through horizontal averaging, variations of water vapor and temperature that are related to the horizontal transport by the large-scale circulation will be largely removed, and thus the water vapor and temperature relationship obtained is more indicative of the property of moist convection, and is thus more relevant to the issue of water vapor feedback in global warming.

In analyzing the results, they said:

Overall, the variations of specific humidity correlate positively at all levels with the temperature variations at the same level. However, the strength of the correlation between specific humidity variations and the temperature variations at the same level appears to be strongly height dependent.

Sun & Oort (1995)

Early in the paper they explained that pre-1973 values of water vapor were more problematic than post-1973 and therefore much of the analysis would be presented with and without the earlier period. Hence, the two plots in the graph above.

Now they do something even more interesting and plot the results of changes in specific humidity (q) with temperature and compare with the curve for constant relative humidity:

Sun & Oort (1995)

The dashed line to the right is the curve of constant relative humidity. (For those still trying to keep up, if specific humidity was constant, the measured values would be a straight vertical line going through the zero).

The largest changes of water vapor with temperature occur in the boundary layer and the upper troposphere.

They note:

The water vapor in the region right above the tropical convective boundary layer has the weakest dependence on the local temperature.

And also that the results are consistent with the conceptual picture put forward by Sun and Lindzen (1993). Well, it is the same De-Zheng Sun..

### Vertical Structure of Water Vapor Variations

How well can we correlate what happens at the surface with what happens in the “free troposphere” (the atmosphere above the boundary layer)?

If we want to understand temperature vertically through the atmosphere it correlates very well with the surface temperature. Probably not a surprise to anyone.

If we want to understand variations of specific humidity in the upper troposphere, we find (Sun & Oort find) that it doesn’t correlate very well with specific humidity in the boundary layer.

Sun & Oort (1995)

Take a look at (b) – this is the correlation of local temperature at any height with the surface temperature below. There is a strong correlation and no surprise.

Then look at (a) – this is the correlation of specific humidity at any height with the surface specific humidity. We can see that the correlation reduces the higher up we go.

This demonstrates that the vertical movement of water vapor is not an easy subject to understand.

Sun and Oort also comment on Raval and Ramanathan (1989), the source of the bulk of Clouds and Water Vapor – Part One:

Raval and Ramanathan (1989) were probably the first to use observational data to determine the nature of water vapor feedback in global warming. They examined the relationship between sea surface temperature and the infrared flux at the top of the atmosphere for clear sky conditions. They derived the relationship from the geographical variations..

However, whether the tropospheric water vapor content at all levels is positively correlated with the sea surface temperature is not clear. More importantly, the air must be subsiding in clear-sky regions. When there is a large-scale subsidence, the influence from the sea is restricted to a shallow boundary layer and the free tropospheric water vapor content and temperature are physically decoupled from the sea surface temperature underneath.

Thus, it may be questionable to attribute the relationships obtained in such a way to the properties of moist convection.

### Conclusion

The subject of water vapor feedback is not a simple one.

In their analysis of long-term data, Sun and Oort found that water vapor variations with temperature in the tropical domain did not match constant relative humidity.

They also, like most papers, caution drawing too much from their results. They note problems in radiosonde data, and also that statistical relationships observed from inter-annual variability may not be the same as those due to global warming from increased “greenhouse” gases.

Other articles in this series:

Clouds and Water Vapor – Part One – introducing some ideas from Ramanthan from ERBE 1985 – 1989 results

Clouds and Water Vapor – Part One – Responses - answering some questions about Part One

Clouds and Water Vapor – Part Two - some introductory ideas about water vapor including measurements

Clouds and Water Vapor – Part Four - discussion and results of a paper by Dessler et al using the latest AIRS and CERES data to calculate current atmospheric and water vapor feedback vs height and surface temperature

Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert - focusing on a 1995 paper by Pierrehumbert to show some basics about circulation within the tropics and how the drier subsiding regions of the circulation contribute to cooling the tropics

Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheres - demonstrating that different distributions of water vapor yet with the same mean can result in different radiation to space, and how this is important for drier regions like the sub-tropics

### References

How Dry is the Tropical Free Troposphere? Implications for Global Warming Theory,
Spencer & Braswell, Bulletin of the American Meteorological Society (1997)

Humidity-Temperature Relationships in the Tropical Troposphere, Sun & Oort, Journal of Climate (1995)

Distribution of Tropical Tropospheric Water Vapor, Sun & Lindzen, Journal of Atmospheric Sciences (1993)

Sensitivity of the Earth’s Climate to height-dependent changes in the water vapor mixing ratio, Shine & Sinha, Nature (1991)

Some Coolness concerning Global Warming, Lindzen,Bulletin of the American Meteorological Society (1990)

### Notes

Note 1 – The total amount of water vapor, TPW ( total precipitable water), is obviously something we want to know, but we don’t have enough information if we don’t know the distribution of this water vapor with height. It’s a shame, because TPW is the easiest value to measure via satellite.

Note 2 – Obviously the total mass of air is conserved. If small areas have rapidly rising air, larger areas will have have slower subsiding air.

## Clouds and Water Vapor – Part One – Responses

After posting Part Two on water vapor, some people were unhappy that questions from Part One were not addressed.

I have re-read through the many comments and questions and attempt to answer them here. I ignore the questions unrelated to the feedbacks of water vapor and clouds – like the many questions about the moon, answered in Lunar Madness and Physics Basics. I also ignore the personal attacks from a commenter that my article(s) was/were deceptive.

### The Definition

The major point from the perspective of a few commenters (including critics of Part Two) was about the radiometric definition of the “greenhouse” effect.

Ramanathan analyzed the following equation:

F = σT4 – G

where F is outgoing longwave radiation (OLR) at top of atmosphere (TOA), T is surface temperature, and G is the “greenhouse” effect.

For newcomers, F averages around 240 W/m² (and higher in clear sky conditions).

The first term on the right, σT4, is the Stefan-Boltzmann equation which calculates radiation from a surface from its temperature, e.g., for a 288K surface (15°C) the surface radiation = 390 W/m².

If the atmosphere had no radiative absorbers (no “greenhouse” effect) then F= σT4, which means G=0. See The Hoover Incident.

The approach Ramanathan took was to find out the actual climate response over 1988-89 from ERBE scanner data. What happens to the parameters F and G when temperature increases?

Why is it important?

If increasing CO2 warms the planet, will there be positive, negative or no feedback from water vapor? Apparently, Ramanathan thought that analyzing the terms in the equation under changing conditions could shed some light on the subject.

However, the equation itself was brought into question, mainly by Colin Davidson, in a number of comments including:

..In the section “Greenhouse Effect and Water Vapour”, he introduces an equation:

F = σTs^4 – G

I didn’t understand what this equation was trying to say. How are the Surface Radiation and the Outgoing Long Range Radiation linked, noting that there are other fluxes from the Surface into the Atmosphere? And one of these (evaporation) is stronger than the NET Surface radiation, while direct Conduction is also a significant flux?
The sentence “So the radiation from the earth’s surface less the “greenhouse” effect is the amount of radiation that escapes to space.” is not accurate.

Missing from this sentence are the following:
Incoming Solar Radiation Absorbed by the Atmosphere(A);
Evaporated Water from the Surface(E);
Direct Conduction from Surface to Atmosphere(C)
Back-Radiation from Atmosphere to Surface (B)

Writing down the fluxes for the atmosphere as a black box:
F= A+(S-B)+E+C (where S=Stephan-Boltzmann Surface Radiation),
Making G = S-F = B-A-E-C

So G doesn’t appear to me to make much PHYSICAL sense, and is certainly NOT the “Greenhouse Effect”, as the evaporative and conductive species are not greenhouse animals, but B and A certainly belong in the zoo..

And:

..I have shown that both those claims are incorrect. G does not represent the “Greenhouse” effect of an IR active atmosphere, as it contains terms (Evaporation and Conduction) which are plainly IR insensitive, nor does it represent the upward surface flux less the amount of longwave radiation leaving the planet.

What G represents is anyone’s guess, but it is not an easily identifiable physical quantity.

Hence my problem with the equation F=S-G as a starting point for any analysis – it doesn’t seem to represent anything coherent. Why not start with the TOA balance, the Surface balance, or the Atmospheric balance?

I am concerned about this. Is the whole theorem of climate sensitivity based on the incorrect notion that the factor G represents the Greenhouse Effect?

In this post I summarise some of my concerns.

1. F= Sunlight – Reflected sunlight. Unless the earth’s short-wave albedo changes, the Outgoing Long-Wave Radiation(F) is constant, whatever the state of the Greenhouse. So dF/dTs does not represent the Greenhouse Effect, but is a representation of the change of surface temperature with cloudiness.

2. F= S(urface Radiation) + G, but G= E(vaporation) +C(onduction) + A(bsorbed Solar Radiation) – B(ack Radiation). Of these terms, only A and B are Greenhouse dependent. C and E are Greenhouse independent. dG/dTs is therefore not a measure of the Greenhouse Effect.

3. It is unclear if the amount of radiation from the surface escaping “through the window” direct to space is constant. If CO2 concentration increases we expect some tightening of the window, but not much. On the other hand any increase in surface temperature will increase the amount of radiation, so the two processes may balance. Kiehl and Trenberth keep this constant at 40W/m^2 despite raising the surface temperature over time by 1DegC, suggesting that it may be close to constant.

Assuming that is so, the fluxes warming the atmosphere from the Surface are constant, the (B)ack radiation increasing by roughly the same as the sum of the increases in Radiation from the Surface(S) and (E)vaporation. Basically when the surface temperature increases, the increase in Evaporation is balanced by a
decrease in Net Surface Radiation Absorbed by the Atmosphere.
As the heat entering the lower atmosphere is unchanged (though the amounts entering at each height will change), the overall Lapse Rate to the tropopause will be unchanged. So the temperature at the Tropopause will always be the Surface Temperature minus a Constant. The sensitivity of the Tropopause temperature is therefore the same as (and driven by) the sensitivity of the Surface temperature to changes in “forcing” (either solar or back-radiation).

This sensitivity is between 0.095 and 0.15 DegC/W/m^2.

And a search in that post will highlight all the other comments.

My attempts at explaining the concept did not appear successful. I don’t think I will have any more success this time, but clearly others think it is important.

I find Colin’s comments confused, but I’ll start with the main point of Ramanathan (paraphrased by me):

What happens if the climate warms from CO2 (or solar or any other cause) – will water vapor in the climate increase, causing a larger “greenhouse” effect?

That’s the question that many people have asked. These people include well-known figures like Richard Lindzen and Roy Spencer, who believe that negative feedbacks dominate.

Scenarios to Demonstrate the Usefulness of the Definition

If the surface temperature in one location goes from 288K (15°C) to 289K (16°C) the surface radiation will increase by 5.4 W/m². (The Stefan-Boltzmann law). How can we determine whether positive or negative feedbacks exist?

Condition 1. Suppose under clear skies when the temperature was 288K we measured OLR = 265 W/m² and when the temperature increased to 289K we measured OLR = 275 W/m². That means OLR has increased by 10 W/m² for a surface radiation increase of 5.4 W/m². Let’s call this condition Good.

Condition 2. Suppose instead that when the temperature increased to 289K we measured OLR = 265W/m². That means OLR has not changed when surface radiation increased by 5.4 W/m². Let’s call this condition Bad.

• In condition Good we have negative feedback, where the atmospheric “greenhouse” response to higher temperatures is to reduce its absorption of longwave radiation
• In condition Bad we have positive feedback - the situation where more heat has been trapped by the atmosphere – the atmosphere has increased its absorption of longwave radiation

Whether or not more heat also leaves the surface by evaporation or conduction doesn’t really matter for this analysis. It doesn’t tell us what we need to know.

In fact, it’s quite likely that if evaporation increases we might find that positive feedback exists. However, that depends on exactly where the water vapor ends up in the atmosphere (as the absorption of longwave radiation by water vapor is non-linear with height) and how this also changes the lapse rate (as the moist lapse rate is less than the dry lapse rate).

It’s possible that if convective heat fluxes from the surface increase we might find that negative feedback exists – this is because heat moved from the surface to higher levels in the atmosphere increases the ability of the atmosphere to radiate out heat. This is also part of the lapse rate feedback.

But all of these different effects are wrapped up in the ultimate question of how much heat leaves the top of atmosphere as a function of changes in the surface temperature. This is what feedback is about.

So for feedback we really want to know – does the absorptance of the atmosphere increase as surface temperature increases? (see note 3).

That’s as much as I can explain as to why this measure is the useful one for understanding feedback. This is why everyone that deals with the subject reviews the same fundamental equation. This includes those who believe that negative feedbacks dominate.

See Note 2 and Note 3.

Colin Davidson’s points

Colin often makes very sensible statements and points but many of the statements and claims cited earlier suffer from irrelevance, inaccuracy or a lack of any proof.

Missing the point – as I described above – was the main problem. In the interests of completeness we will consider some of his statements.

The third comment cited above indicates one of the main problems with his approach:

..Unless the earth’s short-wave albedo changes, the Outgoing Long-Wave Radiation(F) is constant, whatever the state of the Greenhouse. So dF/dTs does not represent the Greenhouse Effect..

This is not the case. Suppose that absorbed solar radiation is constant. This does not mean that OLR (=”F” in Colin’s description) will be constant. From the First Law of Thermodynamics:

Energy in = Energy out + energy added to the system

In long term equilibrium energy in = energy out. However, we want to know what happens if something disturbs the system. For example, if increased CO2 reduces OLR then heat will be added to the climate system until eventually OLR rises to match the old value – but with a higher temperature in the climate. The same is the case with any other forcing. (See The Earth’s Energy Budget – Part Two).

In fact we expect that for a particular location and time OLR won’t equal solar radiation absorbed. We also have the problem that any “out of equilibrium” signal we might try to measure at TOA is very small, and within the error bars of our measuring equipment.

I didn’t understand what this equation was trying to say. How are the Surface Radiation and the Outgoing Long Range Radiation linked, noting that there are other fluxes from the Surface into the Atmosphere? And one of these (evaporation) is stronger than the NET Surface radiation, while direct Conduction is also a significant flux?

This is a very basic point. The surface radiation and outgoing longwave radiation (OLR) are linked by the equations of atmospheric absorption and emission (see note 4). With no absorption, OLR = surface radiation. The more the concentration of absorbers in the atmosphere the greater the difference between surface radiation and OLR. If we want to find out the feedback effect of water vapor this is exactly the relationship we need to study. Surface radiation and OLR are linked by the very effect we want to study.

A similar problem is suggested in the second comment cited:

..Hence my problem with the equation F=S-G as a starting point for any analysis – it doesn’t seem to represent anything coherent. Why not start with the TOA balance, the Surface balance, or the Atmospheric balance?

How is it possible to extract positive or negative feedback from these?

We expect that at TOA and at the surface the long term global annual average will balance to zero. But we can’t easily measure evaporation or sensible heat. Without carefully placed pyrgeometers we can’t measure DLR (downward longwave radiation) and without pyranometers we can’t measure the incident solar radiation at the surface. In any case even if we had all of these terms it doesn’t help us extract the sign or magnitude of the water vapor feedback.

If we had lots of measurement capability at a particular location it might help us to estimate the evaporation. But then we have the problem of where does this water vapor end up? This is a problem that Richard Lindzen has frequently made – and is also made by Held & Soden in their review article (cited in Part Two). Approaching the problem (from the surface energy balance) without knowing the answer to where water vapor ends up we can’t attempt to calculate the sign of water vapor feedback.

Colin also makes a number of other comments of dubious relevance in the last section of text I extracted.

He states that evaporation and conduction are “greenhouse independent” – but I question this. More “greenhouse” gases mean more surface irradiation from the atmosphere, and therefore more evaporation and conduction (and convection).

The amount of radiation escaping through the so-called “atmospheric window” is not constant (perhaps a subject for a later article). The rest of the statement covers the belief in some kind of simplified atmospheric model where everything is in balance – and therefore a positive feedback is defined out of existence:

Basically when the surface temperature increases, the increase in Evaporation is balanced by a decrease in Net Surface Radiation Absorbed by the Atmosphere.
As the heat entering the lower atmosphere is unchanged (though the amounts entering at each height will change), the overall Lapse Rate to the tropopause will be unchanged. So the temperature at the Tropopause will always be the Surface Temperature minus a Constant. The sensitivity of the Tropopause temperature is therefore the same as (and driven by) the sensitivity of the Surface temperature to changes in “forcing” (either solar or back-radiation).

When surface temperature increases, evaporation is not balanced by a decrease in net surface radiation absorbed by the atmosphere. In fact, when surface temperature increases, surface radiation increases and possible atmospheric absorption of this radiation increases (due to humidity increases from more evaporation). Exactly what change this brings in DLR (atmospheric radiation received by the surface) is a question to be answered. By saying everything is in balance means that the solution about positive feedback is already known. If so, this needs to be demonstrated – not claimed.

The rest of the statement above suffers from the same problem. None of it has been demonstrated. If I understand it at all, it’s kind of a claim of climate equilibrium which therefore “proves” (?) that there isn’t water vapor feedback. However, I don’t really understand what it might demonstrate.

### Other Comments Needing Response from the Original Article

From Leonard Weinstein:

Since the issue is not resolved that the temperature in the upper troposphere has increased, and the relative humidity has not stayed nearly constant (it has clearly decreased) over the period of greatest lower troposphere temperature increase, the argument seems less than resolved. The lack of increased water vapor in the stratosphere pushes that point even further.

The argument isn’t resolved by this piece of work. This is one attempt to measure the effect over a period of good quality data.

Finely, the data and analysis of Roy Spencer seems to lead to different conclusions even on the data interpretation. Can you point out his errors and respond to those issues?

Roy Spencer’s analysis doesn’t address this period of measurement. His paper is about the period from 2000-2008.

From NicL:

However, I take issue with your statement “It should be clear from these graphics that observed variations in the normalized “greenhouse” effect are largely due to changes in water vapor.” The spatial maps referred to merely indicate a correlation between these two things. It is unscientific to infer causation from correlation. Ramathan himself goes no further than to say the graphics suggest that variations in water vapour rather than lapse rates contribute to regional variations in the greenhouse effect.

It’s unscientific to infer causation from correlation in the absence of a theory that links them together. It’s solidly established that water vapor absorbs longwave radiation from the surface, and it’s solidly established that CO2 and other “greenhouse” gases are well-mixed through the atmosphere, while water vapor is not. Therefore, there is a strong theoretical link.

I think, in common with various other repondants, that changes in lapse rates and in the height of the tropopause are key issues in modelling the greenhouse effect, yet they seem rarely discussed. Ramanathan’s chapter does not really cover them.

What makes you say they are rarely discussed? There are many papers discussing the different processes involved in modeling water vapor feedback. However, Ramanathan’s chapter is primarily about measurements. Of course he refers to the different aspects of feedback in the chapter.

### Conclusion

One commenter in part two said:

I want to give him a chance to reflect on whether he wants to defend the Ramanathan analysis in Part 1 or separate himself with dignity, which he can still do..

The primary question seemed to be the approach, and not the results, of Ramanathan.

Ramanathan tested the changes in atmospheric absorptance of longwave radiation with temperature changes. To claim this is inherently wrong is a bold claim and one I can’t understand. Neither can Richard Lindzen or Roy Spencer, at least, not from anything I have read of their work.

There are other possible approaches to Ramanathan’s results. Other researchers may have replicated his work and found different results. Other researchers may have analyzed different periods and found different changes.

There are also theoretical considerations – whether changes in the equilibrium temperature as a result of increased CO2 can be considered as the same conditions under which seasonal changes indicated positive water vapor feedback.

The question for readers to ask is: Did Ramanathan find something important that needs to be considered?

Ramanathan himself said:

However, our results do not necessarily confirm the positive feedback resulting from the fixed relative humidity models for global warming, for the present results are based on annual cycle.
If I someone can point out the theoretical flaw in Ramanathan’s work then I might “separate myself with dignity” otherwise I will be happy to stand by the idea that he has demonstrated something that needs to be considered.

Other articles in this series:

Clouds and Water Vapor – Part One – introducing some ideas from Ramanthan from ERBE 1985 – 1989 results

Clouds and Water Vapor – Part Two - some introductory ideas about water vapor including measurements

Clouds and Water Vapor – Part Three - effects of water vapor at different heights (non-linearity issues), problems of the 3d motion of air in the water vapor problem and some calculations over a few decades

Clouds and Water Vapor – Part Four - discussion and results of a paper by Dessler et al using the latest AIRS and CERES data to calculate current atmospheric and water vapor feedback vs height and surface temperature

Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert - focusing on a 1995 paper by Pierrehumbert to show some basics about circulation within the tropics and how the drier subsiding regions of the circulation contribute to cooling the tropics

Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheres demonstrating that different distributions of water vapor yet with the same mean can result in different radiation to space, and how this is important for drier regions like the sub-tropics

Note 1.

The actual change in emission of radiation for a 1°C rise in temperature depends on the temperature itself, one of the many non-linearities in science. The example in the article was for the specific temperature of 288K, along with the desire to avoid confusing readers with too many caveats.

Here is the graph of radiation change for a 1°C rise vs temperature:

For the mathematicians it is an easy exercise. For non-mathematicians, the change in radiation = 4σT³ W/m².K (obtained by differentiating the Stefan-Boltzmann equation with respect to T).

Note 2.

This article is about a specific point in Ramanathan’s work queried by some of my readers. His explanation of how to determine feedbacks is much more lengthy and includes some important points, especially the demonstration of the relationships in time between the various changes. These are important for the determination of cause and effect. See the original article and especially the online chapter for more a detailed explanation.

Note 3.

The rate of change of surface radiation with temperature, σdT4/dT = 4σT³ W/m².K (see note 1) is 5.4W/m² per K at 288K. However, the rate of change of OLR, dF/dT, for the no feedback condition is slightly more challenging to determine and not intuitively obvious.

Ramanathan, based on his earlier work from 1981, determined the “no feedback” condition (i.e., without lapse-rate feedback or water vapor feedback) was dF/dT=3.3 W/m².K. And for positive feedback this parameter, dF/dT would be less than 3.3.

Roy Spencer and William Braswell in their just-published work in JGR, On the diagnosis of radiative feedback in the presence of unknown radiative forcing has exactly the same value as the determination of the no feedback condition.

Note 4.

There are many different formulations of the solutions to the radiative transfer equations. This version is from Ramanathan’s chapter in Frontiers of Climate Modeling:

This is just to demonstrate that there is a strong mathematical link between surface radiation and OLR, and one that is very relevant for determining whether positive or negative feedbacks exist.

## Clouds and Water Vapor – Part Two

In Part One we covered a lot of ground. In this next part we will take a look at some basics about water vapor.

The response of water vapor to a warmer climate is at the heart of concerns about the effect of increasing the inappropriately-named “greenhouse” gases like CO2 and methane. Water vapor is actually the major “greenhouse” gas in the atmosphere. But unlike CO2, methane and NO2, there’s a huge potential supply of water vapor readily available to move into the atmosphere. And all it takes is a little extra heat to convert more of the oceans and waterways into water vapor.

Of course, it’s not so simple.

Before we dive into the subject, it’s worth touching on the subject of non-linearity - something that doesn’t just apply to the study of water vapor. Some people are readily able to appreciate the problem of non-linearity. For others it’s something quite vague. So before we’ve even started we’ll digress into slightly more familiar territory, just to give a little flavor to non-linearity.

### A Digression on Non-Linearity

People who know all about this can just skip to the next section. For most people who haven’t studied a science or maths subject, it’s a natural assumption to assume that the world is quite a linear place. What am I talking about?

Here’s an example, familiar to regular readers of this blog and anyone who has tried to understand the basic concept of the “greenhouse” effect.

If the atmosphere did not absorb or emit radiation the surface of the earth would radiate at an average of around 240 W/m² (see The Hoover Incident, CO2 – An Insignificant Trace Gas? and many other articles on this blog).

This would mean a surface temperature of a chilly 255K (-18°C).

With the “greenhouse” effect of a radiating atmosphere, the surface is around 288K (+15°C) and radiates 390 W/m².

As one commenter put it (paraphrasing to save finding the quote):

Clearly you haven’t done your sums right. If 240 W/m² means a temperature of 255K, then 390 W/m² means a temperature of (390/240)x255 which is way more than the actual temperature of 288K (15°C).

Now that commenter spelt out the maths but many more people don’t even do that and yet feel instinctively that something is wrong when results can’t be simply added up, or fitted on a straight line.

In the case of that approach, the actual temperature – assuming a linear relationship between radiation and temperature – would be 414K or 141°C. That approach is wrong. The world is not linear.

How much radiation does it take to raise the equilibrium surface temperature by 10°C (or 10K)? This assumes a simple energy balance where more radiation received heats up the surface until it radiates out the same amount.

The answer might surprise you. It depends. It depends a lot. Here’s a graph:

So if the surface is at 100K ( -173°C), it takes only 2.6 W/m² to lift the temperature by 10K (10°C).

• At 200K (-73°C), it takes 20 W/m²
• At 300K (27°C), it takes 65 W/m²
• At 400K (127°C), it takes 151 W/m²

The equation that links radiation to temperature is the Stefan-Boltzmann equation, and the relationship is j=εσT4,where T is temperature.

If the equation was something like j=kT, then it wouldn’t matter what the current temperature was – the same amount of energy would lift the temperature another 10K. For example, if it took 10 W/m² to lift the temperature from 100K to 110K, then it would take 10W/m² to lift the temperature from 300K to 310K. That would be a linear relationship.

But he world isn’t linear most of the time. Here are some non-linear examples:

• radiation from surfaces (and gases) vs temperature
• absorption of radiation by gases vs pressure
• absorption of radiation by gases vs wavelength
• pressure vs height (in the atmosphere)
• water vapor concentration in the atmosphere vs temperature
• convective heat flow

It’s important to try and unlearn the idea of linearity. Intuition isn’t a good guide for physics. At best you need a calculator or a graph.

Digression over.

### Water Vapor Distribution

Let’s take a look at water vapor distribution in the real world (below).

Both graphs below have latitude along the horizontal axis (x-axis) and pressure along the vertical axis (y-axis). Pressure = 1000 (mbar) is sea level, and pressure = 200 is the top of the troposphere (lower atmosphere).

The left side graph is specific humidity, or how much mass of water vapor exists in grams per kg of dry air.

The right side graph is relative humidity, which will be explained. Both are annual averages.

Water Vapor Observations, Soden – “Frontiers of Climate Modeling”, chapter 10

Click for a larger view

As a comparison the two graphs below show the change in specific humidity and relative humidity from June/Jul/August to Dec/Jan/Feb:

Water Vapor Observations, Soden – “Frontiers of Climate Modeling”, chapter 10

Click for a larger view

The most important parameter for water vapor is the maximum amount of water vapor that can exist – the saturation amount. Here is the graph for saturation mixing ratio at sea level:

You can see that at 0°C the maximum mixing ratio of water vapor is 4 g/kg, while at 30°C it is 27 g/kg. Warmer air, as most people know, can carry much more water vapor than colder air.

(Note that strictly speaking air can become supersaturated, with relative humidities above 100%. But in practice it’s a reasonable guide to assume the maximum at 100%).

Here’s the graph for temperatures below zero, for water and for ice – they are quite similar:

Relative humidity is the ratio of actual humidity to the saturation value.

Saturation occurs when air is in equilibrium over a surface of water or ice. So air very close to water is usually close to saturation – unless it has just been blown in from colder temperatures.

### The Simplified Journey of a Parcel of Moist Air

Let’s consider a parcel of air just over the surface of a tropical ocean where the sea surface temperature is 25°C. The relative humidity will be near to 100% and specific humidity will be close to 20 g/kg. The heating effect of the ocean causes convection and the parcel of air rises.

As air rises it cools via adiabatic expansion (see the lengthy Convection, Venus, Thought Experiments and Tall Rooms Full of Gas – A Discussion).

The cooler air can no longer hold so much water and it condenses out into clouds and precipitation. Eventually this parcel of air subsides back to ground. If the maximum height reached on the journey was more than a few km then the mixing ratio of the air will be a small fraction of its original value.

When the subsiding air reaches the ground – much warmer once again due to adiabatic compression – its relative humidity will now be very low – as the holding capacity of this air is once again very high.

Take a look at the graph shown earlier of relative humidity:

Annual averages don’t quite portray the journey of one little parcel of air, but the main features of the graph might make more sense. In a very broad sense air rises in the tropics and descends into the extra-tropics, which is why the air around 30°N and 30°S has a lower relative humidity than the air at the tropics or the higher latitudes.

Why isn’t the air higher up in the tropics at 100% relative humidity?

Because the air is not just made up of air rising, there is faster moving rising air, and a larger area of slowly subsiding air.

Held & Soden in an excellent review article (reference below), said this:

To model the relative humidity distribution and its response to global warming one requires a model of the atmospheric circulation. The complexity of the circulation makes it difficult to provide compelling intuitive arguments for how the relative humidity will change. As discussed below, computer models that attempt to capture some of this complexity predict that the relative humidity distribution is largely insensitive to changes in climate.

### The Complexity

The ability of air to hold water vapor is a very non-linear function of temperature. Water vapor itself has very non-linear effects in the radiative balance in the atmosphere depending on its height and concentration. Upper tropospheric water vapor is especially important, despite the low absolute amount of water vapor in this region.

Many many researchers have proposed different models for water vapor distribution and how it will change in a warmer world – we will have a look at some of them in subsequent articles.

Measurement of water vapor distribution has mostly not been accurate enough to paint a full enough picture.

### Measurements

There are two ways that water vapor is measured:

Radiosondes (instruments in weather balloons) provide a twice-daily high resolution vertical profile (resolution of 100m) of temperature, pressure and water vapor. However, in many areas the coverage is low, e.g. over the oceans.

Radiosondes provide the longest unbroken series of data – going back to the 1940′s.

Measurements of humidity from radiosondes are problematic – often over-stating water vapor higher up in the troposphere. Many older sensors were not designed to measure the low levels of water vapor above 500hPa. As countries upgrade their sensors it appears to have introduced a spurious drying trend.

Comparison of measurements of water vapor between adjacent countries using different manufacturers of radiosonde sensors demonstrates that there are many measurement problems.

Here’s a map of radiosonde distribution:

From “Frontiers of Climate Modeling” (2006)

Satellites provide excellent coverage but mostly lack the vertical resolution of water vapor. One method of measurement which gives the best vertical resolution (around 1km) is solar occultation or limb sounding. The satellite views the sun “sideways” through the atmosphere at a water vapor absorption wavelength like 0.94μm, and as the effective height changes the amount of water vapor can be calculated against height.

This method also allows us to measure water vapor in the stratosphere (and in fact it’s best suited for measuring the stratosphere and the highest levels of the troposphere).

Here are the established satellite systems for measuring water vapor:

From “Frontiers of Climate Modelling” (2006)

Here is a water vapor measurement from Sage II:

From “Frontiers in Climate Modeling” (2006)

There are many disadvantages of solar occultation measurement – large geographic footprint of measurement, knowledge of ozone distribution is required and measurements are limited to sunrise and sunset.

The other methods involve looking down through the atmosphere – so they provide better horizontal resolution but worse vertical resolution. Water vapor absorbs and emits thermal radiation at wavelengths through the infrared spectrum. Different wavelengths with stronger or weaker absorption provide different “weighting” to the water vapor vertical distribution.

The new Earth Observing System, EOS, which began in 1999 has many instruments for improved measurement:

Mostly these provide improvements, rather than revolutions, in accuracy and resolution.

Finally, an interesting picture of upper tropospheric relative humidity from Held & Soden (2000):

Upper tropospheric humidity, Held & Soden (2000)

You can see – no surprise – that the relative humidity is highest around the clouds and reduces the further away you move from the clouds.

### Conclusion

Understanding water vapor is essential to understanding the climate system and what kind of feedback effect it might have.

However, the subject is not simple, because unlike CO2, water vapor is “heterogeneous” – meaning that its concentration varies across the globe and vertically through the atmosphere. And the response of the climate system to water vapor is non-linear.

Measurements of water vapor are not quite at the level of accuracy and resolution they need to be to confirm any models, but there are many recent advances in measurements.

Other articles in this series:

Clouds and Water Vapor – Part One – introducing some ideas from Ramanthan from ERBE 1985 – 1989 results

Clouds and Water Vapor – Part One – Responses - answering some questions about Part One

Clouds and Water Vapor – Part Three - effects of water vapor at different heights (non-linearity issues), problems of the 3d motion of air in the water vapor problem and some calculations over a few decades

Clouds and Water Vapor – Part Four - discussion and results of a paper by Dessler et al using the latest AIRS and CERES data to calculate current atmospheric and water vapor feedback vs height and surface temperature

Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbert - focusing on a 1995 paper by Pierrehumbert to show some basics about circulation within the tropics and how the drier subsiding regions of the circulation contribute to cooling the tropics

Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheres - demonstrating that different distributions of water vapor yet with the same mean can result in different radiation to space, and how this is important for drier regions like the sub-tropics

### References

Frontiers of Climate Modeling, ed. J.T. Kiehl & V. Ramanathan, Cambridge University Press (2006)

Water Vapor Feedback and Global Warming, I.M. Held & B.J. Soden, Annual Review of Energy and the Environment (2000)

## Clouds and Water Vapor – Part One

In the CO2 series we looked at the effect of CO2 without climate feedbacks. The “answer” to the doubling of CO2 was a “radiative forcing” of 3.7W/m^2 and an increase in surface temperature of about 1°C.

There are many ways to introduce this problem. We’ll start with the great Ramanathan, who is always worth reading. This article discusses the ideas in the chapter The Radiative Forcing due to Clouds and Water Vapor (by Ramanathan and Inamdar) from Frontiers of Climate Modeling by Kiehl and Ramanathan (2006). Note that the link allows you to download the chapter. Well worth reading.

And if you have questions about whether CO2 can influence temperature or whether the inappropriately-named “greenhouse” effect exists, take a look at the CO2 series (and ask questions there).

### Preamble

Various papers from the 60′s onwards that attempted to model the change in radiative flux and surface temperature (as a result of changes in CO2 concentrations) usually solved the problem using (at least) two scenarios:

• constant absolute humidity
• constant relative humidity

The reason is that absolute humidity is less realistic than relative humidity – and the concept of relative humidity leads to positive feedback. Why positive feedback? Higher concentrations of CO2 lead to increased radiative forcing and so the surface and tropospheric temperature increases. As a result – under constant relative humidity – the amount of water vapor in the troposphere increases. Water vapor is a greenhouse gas and so further increases “radiative forcing”.

One of the questions that come to people’s minds is whether this leads to thermal runaway. The answer, when considering the “extra” effect from water vapor is no, and this is because there are also negative feedbacks in the system, especially the fact that radiation (a negative feedback) increases as the 4th power of absolute temperature.

But enough of trying to think about the complete solution before we have even begun. Let’s take the time to understand the thinking behind the problem.

### Cloudy Skies

Clouds are one of the toughest problems in climate science, and as a result, many models and experiments differentiate between cloudy and clear skies.

The ERBE experiments clarified the main effects from clouds.

Here is OLR (outgoing longwave radiation) under clear and cloudy (=all skies) averaged over 1985-1989:

ERBE OLR for clear and cloudy skies, 1985-1989

Here is the albedo (%), or % of solar radiation reflected:

ERBE albedo = reflected solar radiation % for clear and cloud skies, 1985-1989

Clouds reflect solar radiation by 48 W/m2 but reduce the outgoing longwave radiation (OLR) by 30 W/m2, therefore the average net effect of clouds – over this period at least – is to cool the climate by 18 W/m2. Note that these values are the global annual average.

Here are the net shortwave (solar reflection) and net OLR effects from clouds over the whole period:

and the two effects combined:

### The “Greenhouse” Effect and Water Vapor

I’ll try and keep any maths to a minimum, but a few definitions are needed.. if you don’t like seeing equations the explanations in the text mean you haven’t missed anything essential.

We will call the “greenhouse” effect of the atmosphere and clouds, G, and the average OLR (outgoing longwave radiation), F:

F = σTs4 – G

The first term on the right-hand side of the equation, σTs4, is just the radiation from the earth’s surface at a temperature of Ts (the Stefan-Boltzmann equation). So the radiation from the earth’s surface less the “greenhouse” effect is the amount of radiation that escapes to space.

G is made up of the clear sky “greenhouse” effect, Gclear, and the (longwave) effect of clouds, Gcloud.

Now as we move from the hotter equator to the colder poles we would expect Gclear to reduce simply because the surface radiation is much reduced – a 30°C surface emits 480 W/m2 and a 0°C surface emits 315 W/m2. A large proportion of the changes in the “greenhouse” effect, Gclear, are simply due to changes in surface temperature.

Therefore, we introduce a normalized “greenhouse” effect, gclear:

gclear = Gclear / σTs4

This parameter simply expresses the ratio between the clear sky “greenhouse” effect and the surface radiation. The variations in this normalized value reflect changes in atmospheric humidity and lapse rates (the temperature profile up through the atmosphere). See especially CO2 – An Insignificant Trace Gas? Part Five for a little more illumination on this.

The global average value for Gclear = 131 W/m2 and for gclear = 0.33 – i.e., the atmosphere reduces the radiation escaping to space by 33%.

Here is how gclear varies around the world (top graphic) compared with water vapor around the world (bottom graphic):

It should be clear from these graphics that observed variations in the normalized “greenhouse” effect are largely due to changes in water vapor. [Note - change of notation from the graphics - ga in the graphic is gclear in my text]

Water vapor decreases from equator to pole due to temperature (lower temperatures mean lower absolute humidity), and increases over ocean compared with land (because of the availability of water to evaporate).

### Feedback

If we can see that the “greenhouse” effect is strongly influenced by water vapor, we want to know how water vapor changes in response to surface and tropospheric temperature changes.

To make sense of this section it’s helpful to follow some maths. However, I recognize that many people would rather skip any maths so this is in the last section for reference.

Here are the results from ERBE for: the tropics (30°N – 30°S) with surface temperature; the “greenhouse” effect + the normalized version; and the change in water vapor in different vertical sections of the atmosphere:

Ramanathan says:

For the tropics, Ts peaks in March/April, while for 90°N–90°S, Ts peaks in July. We can qualitatively interpret the phase of the annual cycle as follows. The tropical annual cycle is dominated by the coupled ocean–atmosphere system and as a result, the temperature response lags behind the forcing by a maximum of about three months (π/2); thus, with the solar insolation peaking in December 21, the temperature peaks in late March as shown in Figure 10.

Now the whole globe as a comparison:

Ramanathan again:

The extra-tropical and global annual cycle is most likely dominated by the hemispherical asymmetry in the land fraction. During the northern-hemisphere summer (June, July, and August), the large land masses warm rapidly (with about a one month lag) which dominates the hemispherical and global mean response; however, during the southern-hemisphere summer, the relatively smaller fraction of land prevents a corresponding response. Thus, the globe is warmest during June/July and is coldest during December/January.

What can we make of the correlation? Correlation doesn’t equal causation.

The best fit is a phase lag of less than a month which implies that water vapor and gclear are not driving Ts – because the feedback in that case would require more than one month. The converse, that Ts is driving water vapor and gclear, is much more likely because convective time scales are very short.

Of course, this is a deduction from a limited time period.

One of the key relationships in understanding feedback is the change in Gclear with Ts (mathematically we write this as dGclear/dTs – which means “the rate of change with Gclear as Ts varies “).

For reasons briefly outlined in the maths section, if dGclear / dTs > 2.2 it implies positive feedback from the climate.

When the data is plotted from the ERBE data we can see that in the tropics the value is the highest, much greater than 2.2, and when the whole globe is included the value reduces significantly. However, the value for the whole globe still implies positive feedback.

In this last graph we see the feedback value for progressively wider latitude ranges – so on the left we are only looking at the tropics, while over on the right (90°N to 90°S) we are looking at the entire planet. This helps to see the contribution from the tropics progressively outweighed by the rest of the plant – so the important point is that without the strong effect from the tropics the feedback might well have moved to negative.

The feedback doesn’t change between clear sky and all sky, implying that the cloud feedback doesn’t impact the climate system feedback (on these timescales).

However, our results do not necessarily confirm the positive feedback resulting from the fixed relative humidity models for global warming, for the present results are based on annual cycle. We need additional tests with decadal time-scale data for a rigorous test. Nevertheless, the analysis confirms that water vapor has a positive feedback effect for global-scale changes on seasonal to inter-annual time scales.

He also comments on other work (including Lindzen) that finds different results for the relative important of water vapor in different vertical sections of the troposphere.

Hopefully, we will get the opportunity to consider these in future articles.

### Conclusion

This is a big subject which has lots of different perspectives, and only one is developed here. Therefore, I hope that this is the first of many articles on the subject.

It should be helpful to see the approach and one way of interpreting the data. There is a theoretical framework behind the concepts, which can be seen in Ramanathan’s paper from 1981: The Role of Ocean-Atmosphere Interaction in the CO2 Climate Problem. (You can find a free copy online). It’s quite involved but perhaps some of the concepts from this paper will be in one of the next posts in this series.

Other articles in this series:

Clouds and Water Vapor – Part One – Responses – answering some questions about Part One

Clouds and Water Vapor – Part Two – some introductory ideas about water vapor including measurements

Clouds and Water Vapor – Part Three – effects of water vapor at different heights (non-linearity issues), problems of the 3d motion of air in the water vapor problem and some calculations over a few decades

Clouds and Water Vapor – Part Four – discussion and results of a paper by Dessler et al using the latest AIRS and CERES data to calculate current atmospheric and water vapor feedback vs height and surface temperature

Clouds & Water Vapor – Part Five – Back of the envelope calcs from Pierrehumbertfocusing on a 1995 paper by Pierrehumbert to show some basics about circulation within the tropics and how the drier subsiding regions of the circulation contribute to cooling the tropics

Clouds & Water Vapor – Part Six – Nonlinearity and Dry Atmospheresdemonstrating that different distributions of water vapor yet with the same mean can result in different radiation to space, and how this is important for drier regions like the sub-tropics

### Maths

I’ll follow the notations from the chapter reasonably closely. But I think they are confusing so I have changed a few of them.

And if you do want to understand the maths it’s definitely worth taking a look at the more detailed explanations in the chapter to understand this beyond the surface.

F = OLR (outgoing longwave radiation)

F = Fclear (1-f) + f.Fcloudy, where f is the fraction of clouds, and Fclear is the clear sky OLR

F = σTs4 – G                [1], where G is the “greenhouse” effect

and G = Gclear + Gcloud [2]

Now the main feedback parameter is dF/dTs, so:

dF/dTs = 4σTs3 – dG/dTs = 4σTs3 – (dGclear/dTs + dGcloud/dTs)           [3]

Note that 4σTs3 = 5.5 Wm-2K-1 (at T=289K)

Background:  dGclear/dTs is affected by water vapor and lapse rate and dGcloud/dTs is affected by cloud feedback and lapse rate

Now dGclear/dTs = 4σTs3 – dFclear/dTs [4]

now for Ts changing with no lapse rate feedback and no water vapor feedback, dFclear/dTs = 3.3 Wm-2K-1 (from Ramanathan 1981, see ref above in conclusion).

Therefore, if there is positive feedback dFclear/dTs < 3.3 Wm-2K-1 and if negative feedback dFclear/dTs > 3.3 Wm-2K-1 – because a lower value of F (OLR) means a higher value of G (greenhouse effect)

And from [4], if dGclear/dTs > 5.5 – 3.3 = 2.2 then there is positive feedback.