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## 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 (35S–35N) 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.

### Articles in the Series

Part One – introducing some ideas from Ramanathan from ERBE 1985 – 1989 results

Part One – Responses – answering some questions about Part One

Part Two – some introductory ideas about water vapor including measurements

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

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

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

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

Part Seven – Upper Tropospheric Models & Measurement – recent measurements from AIRS showing upper tropospheric water vapor increases with surface temperature

Part Eight – Clear Sky Comparison of Models with ERBE and CERES – a paper from Chung et al (2010) showing clear sky OLR vs temperature vs models for a number of cases

Part Nine – Data I – Ts vs OLR – data from CERES on OLR compared with surface temperature from NCAR – and what we determine

Part Ten – Data II – Ts vs OLR – more on the data

### 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

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## 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.

### Articles in the Series

Part One – introducing some ideas from Ramanathan from ERBE 1985 – 1989 results

Part One – Responses – answering some questions about Part One

Part Two – some introductory ideas about water vapor including measurements

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

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

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

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

Part Seven – Upper Tropospheric Models & Measurement – recent measurements from AIRS showing upper tropospheric water vapor increases with surface temperature

Part Eight – Clear Sky Comparison of Models with ERBE and CERES – a paper from Chung et al (2010) showing clear sky OLR vs temperature vs models for a number of cases

Part Nine – Data I – Ts vs OLR – data from CERES on OLR compared with surface temperature from NCAR – and what we determine

Part Ten – Data II – Ts vs OLR – more on the data

### 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)

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