The Science of Doom

Many curiosity values in atmospheric physics take on new life in the blogosphere. One of them is the value in Kiehl & Trenberth 1997 for the “atmospheric window” flux:

From Kiehl & Trenberth (1997)

Figure 1

Here is the update in 2009 by Trenberth, Fasullo & Kiehl:

From Trenberth, Fasullo & Kiehl (2009)

Figure 2

The “atmospheric window” value is probably the value in KT97 which has the least attention paid to it in the paper, and the least by climate science. That’s because it isn’t actually used in any calculations of note.

What is the Atmospheric Window?

The “atmospheric window” itself is a term in common use in climate science. The atmosphere is quite opaque to longwave radiation (terrestrial radiation) but the region from 8-12 μm has relatively few absorption lines by “greenhouse” gases. This means that much of the surface radiation emitted in this wavelength region makes it to the top of atmosphere (TOA).

The story is a little more complex for two reasons:

1. The 8-12μm region has significant absorption by water vapor due to the water vapor continuum. See Visualizing Atmospheric Radiation – Part Ten – “Back Radiation” for more on both the window and the continuum
2. Outside of the 8-12 region there is some transparency in the atmosphere at particular wavelengths

The term in KT97 was not clearly defined, but what we are really interested in is what value of surface emitted radiation is transmitted through to TOA – from any wavelength, regardless of whether it happens to be in the 8-12 μm region.

Calculating the Value

One blog that I visited recently had many commenters whose expectation was that upward emitted radiation by the surface would be exactly equal to the downward emitted radiation by the atmosphere + the “atmospheric window” value.

To illustrate this expectation let’s use the values from figure 2 (the 2009 paper) – note that all of these figures are globally annually averaged:

• Upward radiation from the surface = 396 W/m²
• Downward radiation from the atmosphere (DLR or “back radiation”) = 333 W/m²
• These commenters appear to think the atmospheric window value is probably really 63 W/m² – and thus the surface and lower atmosphere are in a “radiative balance”

This can’t be the case for fairly elementary reasons – but let’s look at that later.

In Visualizing Atmospheric Radiation – Part Two I describe the basics of a MATLAB line by line calculation of radiative transfer in the atmosphere. And Part Five – The Code gives the specifics, including the code.

Running v0.10.4 I used some “standard atmospheres” (examples in Part Twelve – Heating Rates) and calculated the flux from the surface to TOA:

• Tropical – 28 W/m² (52 W/m²)
• Midlatitude summer – 40 W/m² (58 W/m²)
• Midlatitude winter – 59 W/m² (62 W/m²)
• Subarctic summer – 50 W/m² (61 W/m²)
• Subartic winter – 55 W/m² (56 W/m²)
• US Standard 1976 – 65 W/m² (72 W/m²)

These are all clear sky values, and the values in brackets are the values calculated without the continuum absorption to show its effect. Clear skies are, globally annually averaged, about 38% of the sky.

These values are quite a bit lower than the values found in the new paper we discuss in this article, and at this stage I’m not sure why.

This paper is: Outgoing Longwave Radiation due to Directly Transmitted Surface Emission, Costa & Shine (2012):

This short article is intended to be a pedagogical discussion of one component of the KT97 figure [which was not updated in Trenberth et al. (2009)], which is the amount of longwave radiation labeled ‘‘atmospheric window.’’ KT97 estimate this component to be 40 W/m² compared to the total outgoing longwave radiation (OLR) of 235 W/m²; however, KT97 make clear that their estimate is ‘‘somewhat ad hoc’’ rather than the product of detailed calculations. The estimate was based on their calculation of the clear-sky OLR in the 8–12 μm wavelength region of 99 W/m² and an assumption that no such radiation can directly exit the atmosphere from the surface when clouds are present. Taking the observed global-mean cloudiness to be 62%, their value of 40 W/m² follows from rounding 99 x (1 – 0.62).

They comment:

Presumably the reason why KT97, and others, have not explicitly calculated this term is that the methods of vertical integration of the radiative transfer equation in most radiation codes compute the net effect of surface emission and absorption and emission by the atmosphere, rather than each component separately. In the calculations presented here we explicitly calculate the upward irradiance at the top of the atmosphere due to surface emission: we will call this the surface transmitted irradiance (STI).

In other words, the value in the KT97 paper is not needed for any radiative transfer calculations, but let’s try and work out a more accurate value anyway.

First, how the clear sky values vary with latitude:

Figure 3 – Clear sky values

Note that the dashed line is “imaginary physics”. The water vapor continuum exists but it is very interesting to see what effect it contributes. This is seen by calculating the effect as if it didn’t exist.

We see that in the tropics STI is very low. This is because the effect of the continuum is dependent on the square of the water vapor concentration, which itself is strongly dependent on the temperature of the atmosphere.

The continuum absorption is so strong in the tropics that STIclr in polar regions (which is only modestly influenced by the continuum) is typically 40% higher than the tropical values.Figure 3 shows the zonal and annual mean of the STIclr to emphasize the role of the continuum. The STIclr neglecting the continuum (dash-dotted line) is generally more than 80 W/m² at all latitudes, with maxima in the northern subtropics (mostly associated with the Sahara desert), but with little latitudinal gradient throughout the tropics and subtropics; the tropical values are reduced by more than 50% when the continuum is included (dashed lines). The effect of the continuum clearly diminishes outside of the tropics and is responsible for only around a 10% reduction in STIclr at high latitudes.

Interestingly, these more detailed calculations yield global-mean values of STIclr of 66 and 100 W/m², with and without the continuum, very close to the values (65 and 99 W/m²) computed using the single global-mean profile, in spite of the potential nonlinearities due to the vapor pressure–squared dependence of the self-continuum.

For people unfamiliar with the issue of non-linearity – if we take an “average climate” and do some calculations on it, the result will usually be different from taking lots of location data, doing the calculations on each, and averaging the results of the calculations. Climate is non-linear. However, in this case, the calculated value of STIclr on an “average climate” does turn out to be similar to the average of STIclr when calculated from climate values in each part of the globe.

We can appreciate a little more about the impact of the continuum on this atmospheric window if we look at the details of the calculation vs wavelength:

From Costa & Shine (2012)

Figure 4 – Highlighted orange text added

Here is the regional breakdown:

From Costa & Shine (2012)

Figure 5 – Clear and All-sky values – Orange highlighted text added

Note that conventionally in climate science clear sky results are the climate without clouds (i.e., a subset), whereas ‘cloudy sky’ results are the results with both clear and cloudy (i.e., all values).

The authors comment:

When including clouds, the STI is reduced further (Fig. 2c) because clouds absorb strongly throughout the infrared window. In regions of high cloud amount, such as the midlatitude storm tracks, the STI is reduced from a clear-sky value of 70 W/m² to less than 10 W/m². As expected, values are less affected in desert regions. The subtropics are now the main source of the global mean STI. The effect of clouds is to reduce the STI from its clear-sky value of 66 W/m² by two-thirds to a value of about 22 W/m²

Method

They state:

Clear-sky STI (STIclr) is calculated by using the line by line model Reference Forward Model (RFM) version 4.22 (Dudhia 1997) in the wavenumber domain 10–3000 cm^-1 (wavelengths 3.33–1000 mm) at a spectral resolution of 0.005 cm^-1. The version of RFM used here incorporates the Clough–Kneizys–Davies (CKD) water vapor continuum model (version 2.4); although this has been superseded by the MT-CKD model, the changes in the midinfrared window (see, e.g., Firsov and Chesnokova 2010) are rather small and unlikely to change our estimate by more than 1 W/m²..

..Irradiances are calculated at a spatial resolution of 10° latitude and longitude using a climatology of annual mean profiles of pressure, water vapor, temperature, and cloudiness described in Christidis et al. (1997). Although slightly dated, the global-mean column water amount is within about 1% of more recent climatologies.

Carbon dioxide, methane, and nitrous oxide are assumed to be well mixed with mixing ratios of 365, 1.72, and 0.312 ppmv, respectively. Other greenhouse gases are not considered since their radiative forcing is less than 0.4 W/m² (e.g., Solomon et al. 2007; Schmidt et al. 2010); we have performed an approximate estimate of the effect of 1 ppbv of chlorofluorocarbon 12 (CFC12) (to approximate the sum of all halocarbons in the atmosphere) on the STIclr and the effect is less than 1%.

Likewise, aerosols are not considered. It is the larger mineral dust particles that are more likely to have an impact in this spectral region; estimates of the impact of aerosol on the OLR are typically around 0.5 W/m² (e.g., Schmidt et al. 2010). The impact on the STI will depend on, for example, the height of aerosol layers and the aerosol radiative properties and is likely a larger effect than the CFCs if they are mostly at lower altitudes; this is discussed further in section 4. The surface is assumed to have an emittance of unity.

And later in assumptions:

Our assumption that the surface emits as a blackbody could also be examined, using emerging datasets on the spectral variation of the surface emittance (which can deviate significantly from unity and be as low as 0.75 in the 1000–1200 cm^-1 spectral region, in desert regions; e.g., Zhou et al. 2011; Vogel et al. 2011). Some decision would need to made, then, as to whether or not infrared radiation reflected by surfaces with emittances less than zero should be included in the STI term as this reflection partially compensates for the reduced emission. Although locally important, the effect of nonunity emittances on the global-mean STI is likely to be only a few percent.

The point here is that if we consider the places with emissivity less than 1.0 should we calculate the value of flux reaching TOA without absorption from both surface emission AND surface reflection? Or just surface emission? If we include the reflected atmospheric radiation then the result is not so different. This is something I might try to demonstrate in the Visualizing Atmospheric Radiation series.

As is standard in radiative transfer calculations, spherical geometry is taken into consideration via the diffusivity approximation, as outlined in this comment.

Why The Atmosphere and The Surface cannot be Exchanging Equal Amounts of Radiation

This is quite easy to understand. I’ll invent some numbers which are nice round numbers to make it easier.

Let’s say the surface radiates 400 and has an emissivity of 1.0 (implying Ts=289.8 K). The atmosphere has an overall transmissivity of 0.1 (10%). That means 360 is absorbed by the atmosphere and 40 is transmitted to TOA unimpeded. For the radiative balance required/desired by the earlier mentioned commenters the atmosphere must be emitting 360.

Thus, under these fictional conditions, the surface is absorbing 360 from the atmosphere. The atmosphere is absorbing 360 from the surface. Some bloggers are happy.

Now, how does the atmosphere, with a transmissivity of 10%, emit 360? We need to know the atmosphere’s emissivity. For an atmosphere – a gas – energy must be transmitted, absorbed or reflected. Longwave radiation experiences almost no reflection from the atmosphere. So we end up with a nice simple formula:

Transmissivity, t = 100% – absorptivity

 Absorptivity, a = 90%.

What is emissivity? It turns out, explained in Planck, Stefan-Boltzmann, Kirchhoff and LTE, that emissivity = absorptivity (for the same wavelength).

Therefore, emissivity of the atmosphere, e = 90%.

So what temperature of the atmosphere, Ta, at an emissivity of 90% will radiate 360? The answer is simple (from the Stefan Boltzmann equation, E=eσTa⁴, where σ=5.67×10^-8):

Ta = 289.8 K

So, if the atmosphere is exactly the same temperature as the surface then they will exchange equal amounts of radiation. And if not, they won’t. Now the atmosphere is not at one temperature so it makes it a bit harder to work out what the right temperature is. And the full calculation comes from the radiative transfer equations, but the same conclusion is reached with lots of maths – unless the atmosphere is at the same temperature as the surface then they will not exchange equal amounts of radiation.

Conclusion

The authors say:

This study presents what we believe to be the most detailed estimate of the surface contribution to the clear and cloudy-sky OLR. This contribution is called the surface transmitted irradiance (STI). The global- and annual- mean STI is found to be 22 W/m². The purpose of producing the value is mostly pedagogical and is stimulated by the value of 40 W/m² shown on the often-used summary figures produced by KT97 and Trenberth et al. (2009).

References

Earth’s Annual Global Mean Energy Budget, Kiehl & Trenberth, Bulletin of the American Meteorological Society (1997) – free paper

Earth’s Global Energy Budget, Trenberth, Fasullo & Kiehl, Bulletin of the American Meteorological Society (2009) – free paper

Outgoing Longwave Radiation due to Directly Transmitted Surface Emission, Costa & Shine, Journal of the Atmospheric Sciences (2012) – paywall paper

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In Part Two we covered quite a bit of ground. At the end we looked at the first calculation of heating rates. The values calculated were a little different in magnitude from results in a textbook, but the model was still in a rudimentary phase.

After numerous improvements – outlined in Part Five – The Code, I got around to adding some “standard atmospheres” so we can see some comparisons and at least see where this model departs from other more accurate models.

First, what are heating rates? Within the context of this model we are currently thinking about the longwave radiative heating rates, which really means this:

If the only part of climate physics that was actually working was “longwave radiation” (terrestrial radiation) then how fast would different parts of the atmosphere heat up or cool down?

As we will see this mechanism (terrestrial radiation) mostly results in a net cooling for each part of the atmosphere.

The atmosphere also absorbs solar radiation – not shown in these graphs – which acts in the opposite direction and provides a heating.

Lastly, the sun warms the surface and convection transfers heat much more efficiently from the surface to the lower atmosphere – and this makes up the balance.

So, with longwave heating (cooling) curves, we are consider one mechanism of how heat is transferred.

Second, what is “longwave radiation”? This is a conventional description of the radiation emitted by the climate system, specifically the fact that its wavelength is almost all above 4 μm. The other significant radiation component in the climate system is “shortwave radiation”, which by convention means radiation below 4 μm. See The Sun and Max Planck Agree – Part Two for more.

Third, what is a “standard atmosphere”? It’s just a kind of average, useful for inter-comparisons, and for evaluation of various climate mechanisms around ideal cases. In this case, I used the AFGL (air force geophysics lab) models which are also used in the LBLTRM (line by line radiative transfer model).

Here is a graph for tropical conditions of heating rate vs height – and with a breakdown between the rates caused by water vapor, CO2 and O3:

Figure 1

Notice that the heating rate is mostly negative, so the atmosphere is cooling via radiation – which means for this atmospheric profile water vapor, CO2 and ozone have a net effect of emitting more terrestrial radiation out than they absorb via these gases.

Here is a textbook comparison:

From Petty (2006)

Figure 2

And a set of graphs detailing the tropical condition for temperature, pressure, density and GHG concentrations:

Figure 3 – Click to enlarge

Now some comparisons of the overall heating rates for 3 different profiles:

Figure 4

Here is a textbook comparison:

From Petty (2006)

Figure 5

So we can see that the MATLAB model created here from first principles and using the HITRAN database of absorption and emission lines is quite close to other calculated standards.

In fact, the differences are small except in the mid-stratosphere and we may find that this is due to slight differences in the model atmosphere used, or as a result of not using the Voigt profile (this is an important but technical area of atmospheric radiation – line shapes and how they change with pressure and temperature in the atmosphere – see for example Part Eight – CO2 Under Pressure).

Pekka Pirilä has been running this MATLAB model as well, has helped with numerous improvements and has just implemented the Voigt profile so we will shortly find out if the line shape is a contributor to any differences.

For reference, here are the profiles of the other two conditions shown in figure 4: Midlatitude summer & Subarctic summer:

Figure 6 – Click to enlarge

Figure 7 – Click to enlarge

Related Articles

Part One - some background and basics

Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code - code can be downloaded, includes some notes on each release

Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..

Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased

Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases

Part Nine – Reaching Equilibrium - when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system

Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased

Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases

References

AFGL atmospheric constituent profiles (0.120 km), by GP Anderson et al (1986)

A First Course in Atmospheric Radiation, Grant Petty, Sundog Publishing (2006)

The data used to create these graphs comes from the HITRAN database.

The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)

The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)

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Understanding atmospheric radiation is not so simple. But now we have a line by line model of absorption and emission of radiation in the atmosphere we can do some “experiments”. See Part Two and Part Five – The Code.

Many people think that models are some kind of sham and climate scientists should be out there doing real experiments. Well, models aren’t a sham and climate scientist are out there doing lots of experiments. Various articles on Science of Doom have outlined some of the very detailed experiments that have been done by atmospheric physicists, aka climate scientists.

When you want to understand why some aspect of a climate mechanism works the way it does, or what happens if something changes then usually you have to resort to a mathematical model of that part of the climate.

You can’t suddenly increase the amount of a major GHG across the planet, or slow down the planetary rotation to ½ its normal speed. Well, not without a sizable investment, a health and safety risk, possible inconvenience to a lot of people and, at some stage, awkward government investigations.

You can’t stop the atmosphere emitting radiation or test a stratosphere that gets cooler with height. But you can attempt to model it.

Mathematical models all have their limitations. We have to understand what the model can tell us and what it can’t tell us. We have to understand what presuppositions are built into the model and what can change in real life that is not being modeled in the maths. It’s all about context.

(Well-designed) models are not correct and are not incorrect. They are informative if we understand their limitations and capabilities.

In contrast to mathematical models built around the physics of climate mechanisms, many people commenting in the blog world (or even writing blogs) have a vague mental model of how climate works. This of course is way way ahead of a climate model built on physics. It has the advantage of not being written down in equations so that no one can challenge it and seemingly plausible hand-waving argument 1 can be traded against hand-waving argument 2. Unfortunately, on this blog we don’t have the luxury of those resources and – where experiments are not available or not possible – we will have to evaluate the results of mathematical models built on physics and observations.

All the above is not an endorsement of what GCMs tell us. And not an indictment. Hopefully no one reading the above paragraphs came to either conclusion.

When I first built the line by line model it had more limitations than today. One early problem was the stratosphere. In real life the temperature of the stratosphere increases with height. In the model the temperature decreased with height.

This was expected. O2 and O3 absorb solar radiation (primarily ultraviolet) and warm mainly the middle layers of the stratosphere. But the model didn’t have this physics. The model, at this stage, primarily modeled the absorption and emission of terrestrial (aka ‘longwave’) radiation by the atmosphere.

So, after a few versions a very crude model of solar absorption was added. Unfortunately, this solar absorption model still did not create a stratosphere that increased with temperature. This was quite disappointing.

Then commenter Uli pointed out that the model had too much stratospheric water vapor and I added a new parameter to the model which allowed stratospheric water vapor to be set differently from the free troposphere. (So far I’ve been using a realistic level of 6ppmv).

The result was happily that the stratosphere, left to its own (model) devices, started increasing with temperature. The starting point is simply a temperature profile dictated to the model, and the finish point is how the physics ends up calculating the final temperature profile:

Figure 1 – A warmer stratosphere and a happier climate model

At the same time, I’ve been updating the model so that it can run to some kind of equilbrium and then various GHGs can be changed.

This was to calculate “radiative forcing” under various scenarios, and specifically I wanted to show how energy moved around in the climate system after a “bump” in something like CO2. This is something that many many people can’t get right in their heads. One of the objectives of the model is to show bit by bit how the increased CO2 causes a reduction in net outgoing radiation, and how that in turn pushes up the atmospheric and surface temperature.

On this journey, once the model stratosphere was behaving a little like its real-life big brother it occurred to me that maybe we could answer the question of why the stratosphere was expected to cool with increased CO2.

See Stratospheric Cooling for some background.

Previously I have worked under the assumption that there are lots of competing “terms” in the energy balance equation for how the stratosphere responds to more CO2 and so simple conceptual models are not going to help.

Now the Science of Doom Climate Model (SoDCM) comes to the rescue.

In fact, while I was waiting for lots of simulations to finish on the PC I was reading again the fascinating Radiative Forcing and Climate Response, by Hansen, Sato & Ruedy, JGR (1997) – free paper – and in a groundhog day experience realized I didn’t understand their flux graphs resulting from various GCM simulations. So the SoDCM allowed me to solve my own conceptual problems.

Maybe.

Let’s take a look at stratospheric cooling.

Understanding Flux Curves

In this simulation:

• CO2 at 280 ppm
• no ozone, CH4 or NO2 for longwave absorption
• boundary layer humidity at 80%
• free tropospheric humidity at 40%
• stratospheric water vapor at 6 ppmv
• tropopause at 200 hPa
• top of atmosphere (TOA) at 1 hPa
• solar radiation at 242 W/m² with some absorbed in the stratosphere and troposphere as shown in figure 1 of Part Nine – Reaching Equilibrium

The surface temperature reached equilibrium at 281K and the tropopause was at 11 km:

Figure 2

The equilibrium was reached by running the model for 500 (model) days, with timesteps of 2 hours. The ocean depth was only 5 meters simply to allow the model to get to equilibrium quicker (note 1).

Then at 500 days the CO2 concentration was doubled to 560 ppm and we capture a number of different values from the timestep before the increase and the timestep after the increase.

Let’s take a look at the up and down fluxes through the atmosphere. See also figure 6 of Part Two. In this case we can see pre- and post-2xCO2, but let’s first just try and understand what these flux vs height graphs actually mean:

Figure 3 – Understanding the Basics

If flux just stays constant (vertical line) through a section of the atmosphere what does it mean?

It means there is no net absorption. It could mean that the atmosphere is transparent to that radiation. It could mean that the atmosphere emits exactly the same amount that it absorbs. Or some of both. Either way, no change = no net radiation absorbed.

Take a look in figure 3 at the (pre-CO2 doubling) upward flux above 10km (in the stratosphere). About 237 W/m² enters the bottom of the stratosphere and about 242 W/m² leaves the top of atmosphere. So the stratosphere is 5 W/m² worse off and from the first law of thermodynamics this either cools the stratosphere or something else is supplying this energy.

Now take a look at the (pre-CO2) downward flux in the stratosphere. At the top of atmosphere there is no downward longwave radiation because there is no source of this radiation outside of the atmosphere. So downward flux = 0 at TOA.

At the bottom of the stratosphere, about 27 W/m² is leaving. So zero is entering and 27 W/m² is leaving – this means that the stratosphere is worse off by 27 W/m².

If we add up the upward and downward longwave fluxes through the stratosphere we find that there is a net loss of about 32 W/m². This means that if the stratosphere is in equilibrium some other source must be supplying 32 W/m².

In this case it is the solar absorption of radiation.

If we were considering the troposphere it would most likely be convection from the surface or lower atmosphere that would be balancing any net radiation loss from higher up in the troposphere.

So, to recap:

• think about the direction radiation is travelling in:
  • if it is reducing in the direction it is travelling then energy is absorbed into that section of the atmosphere
  • if it is increasing in the direction it is travelling then energy is being lost from that section of the atmosphere
• if plots of flux against height are vertical that means there is no change in energy in that region
• if flux vs height is constant (vertical) then it either means
  • the atmosphere is transparent to that radiation, OR
  • the atmosphere is isothermal in that region (emission is balanced by absorption)

Take another look at figure 3 below 10km:

1. The upward radiation is reducing with height – energy is absorbed by each level of the atmosphere. This is a net heating.
2. The downward radiation is increasing – energy is lost from each level of the atmosphere. This is a net cooling.
3. The slope of the curves is not equal. This is because energy is transferred via convection in the troposphere.

Understanding these concepts is essential to understanding radiation in the atmosphere.

Upward Flux from Changes in CO2

Let’s take a closer look at the upward and downward changes due to doubling CO2. So the “pre” curve is the atmosphere in a nice equilibrium condition. And the “post” curve is immediately after CO2 has been doubled, long before any equilibrium has been reached.

Let’s zoom in on the upward fluxes in the stratosphere pre- and immediately post-CO2 doubling:

Figure 4

Even though the curves are roughly parallel from 10km through to 30km you should be able to see that there is a larger gradient on the post-2xCO2 curve. So pre-CO2 increase, the stratosphere loses a net upward of about 5 W/m², and after CO2 increase the stratosphere loses a net upward of about 6 W/m².

This means more CO2 increases the cooling of the stratosphere when we consider the upward flux. So now the question is, WHY?

If we want to understand the answer, the most useful ingredient is to look at the spectral characteristics of pre- and post. Here we take the radiation leaving at TOA and subtract the radiation entering at the tropopause. So we are considering the net energy lost (why lost? because this calculation is energy out – energy in), and as a function of wavenumber.

Here is the spectral graph of energy lost by the stratosphere due to upwards radiation, before the CO2 increase:

Figure 5

The post-CO2 doubling looks very similar so here is a comparison graph, with a slight smoothing (moving average window) just to allow us to see a little more clearly the main differences:

Figure 6

So we see that in the case of post-2xCO2, the energy lost is a little higher, and it is in the wavenumber region where CO2 emits strongly. CO2′s peak absorption/emission is at 667 cm^-1 (15 μm).

Just to confirm, here is the difference – post-2xCO2 minus pre-2xCO2 and not smoothed:

Figure 7

We can see that the main regions of CO2 absorption and emission are the reason. And we note that the temperature of the stratosphere is increasing with height.

So the reason is clear – due to principles outlined earlier in Part Two. Because the stratospheric temperature increases with height, the net emission (i.e., emission less absorption) of radiation, as we go up through the stratosphere will be a progressively higher value. And once we increase the amount of CO2, this net emission will increase even further.

This is what we see in the spectral intensity – the net change in stratospheric emission [(out-in)_2xCO2 - (out-in)_1xCO2] increases due to the emission in the main CO2 bands.

Downward Flux from Changes in CO2

Here is what we see when we zoom in on the downward flux in the stratosphere:

Figure 8

Of course, as already mentioned, the downward longwave flux at TOA must be zero.

This time it is conceptually easier to understand the change from more CO2. There’s one little fly in the understanding ointment, but let’s come to that later.

So when we think about the cooling of the stratosphere from downward flux it’s quite easy. Coming in at the top is zero. Coming out of the bottom (pre-CO2 increase) is about 27 W/m². Coming out of the bottom (post-2xCO2) is about 30 W/m². So increasing CO2 causes a cooling of about 3 W/m² due to changes in downward flux.

Here is the spectral flux (unsmoothed) downward out of the bottom of the tropopause, pre- and post-2xCO2:

Figure 9

And as with figure 7, below is the difference in downward intensity as a result of 2xCO2. This is post less pre, so the positive value overall means a cooling – as we saw in the total flux change in figure 8.

The cause is still due to the CO2 band but the specifics are a little different from the upward change. Here the center of the CO2 band has zero effect. But the “wings” of the CO2 band – around 600 cm^-1 and 700 cm^-1 are the places causing the effect:

Figure 10

The temperature is reducing as we go downwards so the emission from the center of the CO2 band cannot be increasing as we go downward. If we look back at figure 7 for the upward direction, the temperature is increasing upward so the emission from the center of the CO2 band must be increasing.

And the conceptual fly in the ointment alluded to earlier – this one can be confusing (or simple..) – if the starting flux at TOA is zero and the temperature decreases downward surely the downward flux only gets less? Less than zero? Instead, think of the whole stratosphere as a body. It must emit radiation due to its temperature and emissivity. It can’t absorb any radiation from above (because there is none), so it must emit some downward radiation. As its emissivity increases with more GHGs it must emit more radiation into the troposphere. It’s simple really.

Let’s now finalize this story by considering the net change in flux with height due to CO2 increases. Here if “net” is increasing with height it means absorption or heating. And if “net” is reducing with height it means emission or cooling. See note 2 where the details are explained.

So the blue line (upward flux) decreasing from the tropopause up to TOA means that the change in flux is cooling the stratosphere. And likewise for the green line (downward flux). This is just the results already shown as spectral changes now shown as flux changes:

Figure 11

Net Effect

If we combine figure 11 for the total net effect of doubling CO2:

Figure 12

From the tropopause at 11km through to TOA we can see that the combined change in flux due to CO2 doubling causes a cooling of the stratosphere. (And from the surface up to the tropopause we see a heating of the troposphere).

By comparison, here is an extract from Hansen et al (1997):

From Hansen et al (1997)

Figure 13

The highlighted instantaneous graph is the one for comparison with figure 12.

This is the case before the stratosphere has relaxed into equilibrium. Note that the “adjusted” graph – stratospheric equilibrium – has a vertical line for ΔF vs height, which simply means that the stratosphere is, in that case, in radiative equilibrium.

Notice as well that the magnitude of my graph is a lot higher. There may be a lot of reasons for that, including that fact that mine is one specific case rather than some climatic mean, and also that the absorption of solar radiation in my model has been treated very crudely. (Lots of other factors include missing GHGs like CH4, N2O, etc).

Reasons

So we have seen that the net emission of radiation by CO2 bands is what causes the cooling from upward radiation and the cooling from downward radiation when CO2 is increased.

For further insight, I amended the model so that on the timestep before and just after equilibrium the stratosphere was:

A) snapped back to an isothermal case, with the temperature set at the tropopause temperature just calculated

B) forced into a cooling at 4 K/km (c.f. the troposphere with a lapse rate of 6.5 K/km)

Case A, temperature profile just before and after equilibrium:

Figure 14

And the comparison to figure 11:

Figure 15

We can see that the downward flux change is similar to figure 11, but the upward flux is different. It is fairly constant through the stratosphere. This is not surprising. The flux from below is either transmitted straight through, or is absorbed and re-emitted at the same temperature. So no change to upward flux.

But the downward flux only results from the emission from the stratosphere (nothing transmitted through from above). As CO2 is increased the emissivity of the atmosphere increases and so emission of radiation from the stratosphere increases. The fact that the stratospheric temperature is isothermal has a small effect as can be seen by comparing the green curve on figures 15 & 11. But it isn’t very significant.

Now let’s consider case B. First the temperature profile:

Figure 16

Now the net flux graph:

Figure 17

Here we see that the effect of increased CO2 on the upward flux is now a heating in the stratosphere. And the net change in downward flux still has a cooling effect.

Figure 18

Here we see that for a stratosphere where temperature reduces with altitude, doubling CO2 would not have a noticeable effect on the stratospheric temperature. Depending on the temperature profile (and other factors) there might be a slight cooling or a slight heating.

Conclusion

This is a subject where it’s easy to confuse readers – along with the article writer. Possibly no one that was unclear before made it the whole way and said “ok, got it”.

Hopefully, if you only made it only part of the way through, you now have a better grasp of some of the principles.

The reasons behind stratospheric cooling due to increased GHGs have been difficult to explain even for very knowledgeable atmospheric physicists (e.g., one of many).

I think I can explain stratospheric cooling under increasing CO2. I think I can see that other factors like the exact temperature profile of the stratosphere on any given day/month and the water vapor profile (not shown in this article) will also affect the change in stratospheric temperature from increasing CO2.

If the bewildering complexity of up/down, in-out, net of in-out, net of in-out for 2xCO2-original CO2 has left you baffled please feel free to ask questions. This is not an easy topic. I was baffled. I have 4 pages of notes with little graphs and have rewritten the equations in note 2 at least 5 times to try and get the meaning clear – and am still expecting someone to point out a sign error.

Related Articles

Part One - some background and basics

Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code - code can be downloaded, includes some notes on each release

Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..

Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased

Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases

Part Nine – Reaching Equilibrium - when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system

Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased

Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.

References

The data used to create these graphs comes from the HITRAN database.

The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)

The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)

Radiative Forcing and Climate Response, by Hansen, Sato & Ruedy, JGR (1997) – free paper

Notes

Note 1: The relative heat capacity of the ocean vs the atmosphere has a huge impact on the climate dynamics. But in this simulation we were interested in reaching an equilibrium for a given CO2 concentration & solar absorption – and then seeing what happened to radiative balance immediately after a bump in CO2 concentration.

For this requirement it isn’t so important to have the right ocean depth needed for decent dynamic modeling.

Note 2: The treatment of upward and downward flux can get bewildering. The easiest approach is to just consider the change in flux in the direction in which it is travelling. But because upward and downward are in opposite directions, F↑ is in the direction of z, and F↓ is in the opposite direction to z, so heating and cooling are in opposite directions.

Due to changing GHGs:

If F↑(z)_2xCO2 – F↑(z) < 0 => Heating below height z (less flux escaping);

F↑(z)_2xCO2 - F↑(z) > 0  => Cooling below height z

If F↓(z)_2xCO2 – F↓(z) < 0 => Cooling below height z (less flux entering);

F↓(z)_2xCO2 - F↓(z) > 0  => Heating below height z

So for example for figure 11 – the net upward = F↑(z) - F↑(z)_2xCO2 & net downward = F↓(z)_2xCO2 - F↓(z)

Flux “divergence”

dF↑(z)/dz < 0  => Heating of that part of the atmosphere (upward flux is reducing due to being absorbed)

dF↓(z)/dz < 0  => Cooling of that part of the atmosphere (downward flux is increasing as we go down due to more being emitted, or rewritten is very strange English to match the equation: downward flux is decreasing in the upward direction)

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We have mostly looked at the upward spectra at the top of atmosphere (TOA) as various conditions are changed. There’s a good reason for this focus – the outgoing longwave radiation (OLR) determines how much the climate system cools to space.

Over a given timescale this either matches absorbed solar radiation or the planet is heating or cooling. So it is changes in OLR (or absorbed solar) that really affect the heat balance in the climate.

By comparison, the trend in downward longwave radiation (DLR) at the surface is more a result of overall planetary heating and cooling. But of course, the climate is a lot more complex than indicated by that last statement.

Let’s take a look. Note that Part Four – Water Vapor already has some graphs of how the DLR or “back radiation” changes with water vapor concentration.

Here is the DLR for 4 different surface temperatures. In each case there is a lapse rate of 6.5 K/km, the boundary layer humidity (BLH) = 100%, the free tropospheric humidity (FTH) = 40% and there were 10 atmospheric layers in the model with a top of atmosphere at 50 hPa. More about the model in Part Two and Part Five – The Code.

The top graph is the real case, the bottom graph is without the effect of the water vapor continuum:

Figure 1

The continuum operates over the whole range of terrestrial wavelengths of interest, but its main impact is in the “atmospheric window region” between 800-1200 cm^-1. This window region doesn’t have many strong absorption lines so absorption from any other cause has a big effect.

As we can see, the “window” is very dependent on temperature – which is mainly a result of the amount of water vapor. It’s clearer when we look at the spectral difference between the two cases for each of the temperatures:

Figure 2

Notice that the 273 K (0 °C) condition is almost unaffected by the continuum. This is because the effect is dependent on the amount of water vapor squared. And the amount of water vapor is strongly dependent on temperature.

Let’s look at the total flux for both cases and compare with a reference of blackbody emission from the bottom layer of atmosphere (in this case 400m above the surface so about 2.6°C cooler than the surface, and see note 1):

Figure 3

This shows that once we are above a surface temperature of 300 K (27 °C) with high boundary layer humidity the radiation from atmosphere to surface is getting close to blackbody emission. The graph also demonstrates that most of that change is due to the continuum.

Now good emitters are also good absorbers. So here is another way of looking at the same effect - the % of surface radiation in the 800-1200 cm^-1 window region that makes it to the top of atmosphere (without being absorbed anywhere along the way):

Figure 4

These were all with CO2 at 360ppm (and N2O at 319 ppbv, CH4 at 1775 ppbv and no ozone).

Let’s look at how changing CO2 concentration affects these results.

Figure 5

This is a very important graph – what does it show?

• while different surface temperatures have quite different TOA radiation to space – the change in CO2 causes a fairly constant change in this radiation
• changing CO2 has much less effect on the DLR (radiation from the atmosphere to the surface), and as the temperature increases this effect is even more reduced

Let’s look at the “delta”:

Figure 6 – [Corrected Jan 23]

This shows clearly how the change in atmospheric DLR due to doubling CO2 is very much a function of surface temperature. And at the same time, the change in TOA radiation (“OLR”) is almost independent of surface temperature.

From the information presented in this article on how DLR is affected by water vapor at high temperatures the first point shouldn’t be surprising. And from the explanation in Part Four – Water Vapor both points shouldn’t be surprising.

For interest, here are the two DLR spectrum for 280 ppm & 560 ppm at 288 K, and below, the difference:

Figure 7

Conclusion

The surface energy balance is very important for determining the dynamics of surface heat transfer, including initiating convection. As the temperature gets up to 30°C the ability of the surface to radiate to space is reduced to a very low value.

“Deep convection” which drives the tropical circulation is mostly initiated in these very hot surface conditions.

The effect of changing CO2 on atmospheric radiation to the surface (DLR) is small. With high boundary layer relative humidity, water vapor masks out most of the effect of changing CO2 in hotter surface conditions.

But the effect of increasing CO2 on the TOA radiation balance is completely different. High surface humidities have little or no effect on this TOA balance. And there, doubling CO2 has a significant impact (all other things being equal) as shown in figure 12 of Part Seven – CO2 increases.

Working out radiation balance through the atmosphere in your head is difficult. Most people attempting it don’t have the right “calibration points”.

The fundamental physics is straightforward, at least in terms of the values of absorption and emission of radiation (not the “why”). But calculating the result requires computing effort and an integration (summation) across:

• multiple layers at different temperatures and concentrations
• the hundreds of thousands of absorption/emission lines of multiple GHGs
• a large range of wavenumbers

Related Articles

Part One - some background and basics

Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code - code can be downloaded, includes some notes on each release

Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..

Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased

Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases

Part Nine – Reaching Equilibrium - when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system

Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases

Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.

References

The data used to create these graphs comes from the HITRAN database.

The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)

The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)

Notes

Note 1: This model looks at the range of wavenumbers 200-2,500 cm^-1, which equates to 4-50μm, to ease up the calculation effort required. This means that when we sum up the contribution from all calculated wavelengths we are missing some bits. So for example, if we calculate the emission of thermal radiation by a surface at 288K with an emissivity of 1.0 we calculate 390 W/m² – the “blackbody flux”.

But with our “restricted view” of the spectrum we will instead calculate 376 W/m².

Almost all of the “missing spectrum” is in the far infra-red (longer wavelengths/lower wavenumbers), and is subject to relatively high absorption from water vapor.

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In the series so far we have seen how radiation interacts with the atmosphere for a given surface/atmospheric condition.

That is, if the temperature is say 288K (15°C) and the atmospheric temperature decreases at 6.5 K/km (the “lapse rate”) and the concentration of water vapor is this, and the concentration of CO2 is that.. then:

• what is the outgoing radiation, OLR, at the top of the atmosphere (TOA)?
• what is the surface downward radiation, DLR?
• what do the spectra look like and why?
• how do surface temperatures, water vapor concentrations and CO2 concentrations change these values?

These are all important questions, and the necessary first step. Because if we don’t understand these points then it is impossible to work out how the atmosphere reaches a steady state under those conditions, and of course, impossible to work out how a new steady state will be reached if something changes.

The atmospheric model is described in brief in Part Two and in a comment, then in detail in Part Five – The Code.

The earlier model had some ability to step forward in time and calculate temperature change (but with many limitations and some flaws). Specifically I wanted to be able to track the change of energy in each layer and also account for convective heat flow.

A recent commenter asked (about the effect of doubling CO2):

All what you’re saying would show is *if* the surface temperature were to increase by 1.1C (independent of mechanism) it would restore radiative balance for both +3.7 W/m^2 of post albedo solar power and +3.7 W/m^2 of GHG absorption (which I agree with). In no way does this prove or demonstrate that +1.1C at the surface (for no feedback) is itself a requirement to restore balance at the TOA from +3.7 W/m^2 of GHG absorption. This is because you’ve made no accounting for cause and effect – you’ve only shown that the outcome of one potential effect (i.e. +1.1C at the surface) would restore balance at the TOA.

This is an interesting question, and this kind of question is part of the reason for this series.

Let’s first look at the simple question of how any steady state is reached. I add more specifics about the model (v.0.10.1) at the end of the article and have added the code to Part Five – The Code.

This update of the model now includes an “ocean” and some very simple solar heating of the atmosphere:

Figure 1

This is shown in °C/day but is easily converted to W/m² by dividing by 86,400 (number of seconds in a day) and multiplying by the heat capacity of that layer of the atmosphere. Each of the atmospheric layers in the model have roughly the same number of molecules so the graph of W/m² absorbed in a given layer looks quite similar in shape.

I introduced this solar heating partly because the old model had bad accounting at the top of atmosphere, where solar absorption just (magically) kept the stratosphere isothermal (the same temperature). That constraint is now gone in this update of the model.

Here is a model run:

Figure 2 – Click to expand

What’s going on here? Well, let’s first take a look at the energy balance at the surface and for the whole planet (TOA):

Figure 3 – Positive downward (so positive flux imbalance at TOA means the planet is heating up)

The starting point for this model was an ocean temperature of 288K and a lapse rate of 6.5 K/km, up to a tropopause of 200 hPa with an isothermal stratosphere. The solar radiation absorbed by the climate was 242 W/m², with about 100 W/m² absorbed in the atmosphere and the balance absorbed by the surface.

Each time step was 2 hours, and the model was run for 800 days (10,000 time steps).

Why is the model out of balance to begin with?

There’s no reason it should be in balance. I’ve simply prescribed a surface temperature and an atmospheric temperature profile and humidity and CO2 concentration. Why should that happen to be the steady state condition for the solar absorbed radiation of 242 W/m² with 100 W/m² absorbed in the atmosphere?

[Update Jan 22nd - A good point added from Pekka: "It’s perhaps not clear enough to every reader that this thread is describing what happens when the starting point is an initial state that you happened to choose rather than a state that the system could have reached at different external conditions. Thus the Figure 4 tells what happens initially for this specific case rather than values somehow applicable to the real atmosphere."]

So the model works by energy accounting in each time step for each layer in the model. Energy cannot be created or destroyed. Radiation emitted and absorbed is calculated by the relevant equations (already explained). Convection moves heat if the atmosphere above is too cold (see Potential Temperature and Density, Stability and Motion in Fluids). Energy retained increases the temperature of the layer. Energy lost reduces the temperature of the layer.

Let’s consider the surface. On timestep 1 the surface is radiating 376 W/m² (note 1). All of the surface fluxes are shown (in W/m²) in the diagram:

Figure 4

The net is 18 W/m² and so the “ocean” absorbs this energy which means it heats up. In 2 hours (one timestep) this comes to 129 kJ/m² and as the “ocean” in this model is just 10m deep (to allow quicker progress to any equilibrium) this equates to a temperature increase of 0.0031 °C.

If the net heat absorbed is 18 W/m² why doesn’t figure 3 show that? Ok, let’s zoom into the first month of figure 3 and we can see it clearly:

Figure 5

How did the convective flux get calculated? Why isn’t it higher?

The model calculates all the radiative fluxes up and down through each layer, works out the absorbed energy and the resulting temperature increase. Then it checks between each layer to see if the lapse rate is exceeded (see Temperature Profile in the Atmosphere – The Lapse Rate). This means the atmosphere would be unstable, resulting in convection.

The model then calculates the transfer of heat which would satisfy the lapse rate – the layer below loses X Joules, the layer above gains X Joules and new temperatures are calculated based on their respective heat capacities. This is what the model calculates and then adjusts temperatures, logs the convective heat moved and adjusts the energy change in each layer for that timestep.

The graph below zooms in on the first 30 days of the bottom right graph in figure 2:

Figure 6

So convection in this particular instance isn’t any higher at the start simply because of the respective temperatures. Then the first atmospheric layer starts cooling via radiation (it loses more heat via radiation than it gains via solar heating) and this means that convection increases from the surface with each timestep – until a more steady condition is reached.

Now a key point is that the surface imbalance changes over time - which we see in figure 3.

Now there’s no magic “model driver” that makes this happen. It’s just basic heat transfer laws. The model just reflects, in a simplistic way, how heat is transferred between an ocean, an atmosphere and space.

Now let’s look at the TOA balance – look back at figures 3 and 5. This is the balance for the whole climate. What might be interesting is to see that the climate is initially out of balance – losing heat.

But why doesn’t the cooling of the climate mean that the imbalance just reduces until a steady state is reached? How is it possible for the climate to start heating at day 10 and peak somewhere around day 50 and then gradually reduce?

This is very typical of complex dynamic scenarios. Readers familiar with dynamic heat transfer (and any kind of dynamic  physics/chemistry/engineering problems) will have seen these kind of graphs before – overshoot, decay to equilibrium.

What is completely unsurprising though is that the ocean and atmosphere end up in a steady state where cooling to space matches solar absorption – that is, the balance at TOA is ultimately zero.

Here’s a summary of the energy change, in kJ per timestep of 2 hours, of ocean, energy and TOA:

Figure 7

In the first few days the ocean and atmosphere are very much out of balance and so a big “reshuffle” of energy takes place where the ocean absorbs energy and the atmosphere loses energy until they are in much closer balance. Then there is a gradual cooling of the system (primarily via ocean cooling) which eventually leads to an overall TOA balance – which can be seen in figure 2.

In a subsequent article we will take this steady state condition, then increase the CO2 concentration and see what happens.

Conclusion

We’ve seen via one specific example how heat transfer, via radiation and convection, lead to a new equilibrium condition. This can include some oscillation on the way to equilibrium.

This particular case has no claim to be the “definitive median atmospheric condition”. It’s just a sample atmosphere that wasn’t in perfect balance for its conditions.

Many people have conceptual models of how heat moves in the atmosphere and often these mental models are wrong. The purpose of this article is to illustrate how the basic heat transfer mechanisms work. As we can see with this simple example, it would be surprising to get the right answer about dynamic and final temperatures from some hand-waving arguments.

If you have questions please ask. We can examine the energy transfer from many different perspectives.

Some Model Specifics

This update to the model has removed the constraint of keeping the stratosphere isothermal (see note 2).

Instead solar radiation is absorbed in the atmosphere according to the standard heating curves, for example, those found in Petty 2006 p.315:

From Grant Petty (2006)

Figure 8

The stratosphere is not well modeled because the higher levels of the stratosphere are not included. These absorb most of the solar radiation via O2 & O3 and consequently keep the lower levels warmer than the equilibrium reached in this model.

This model had 12 layers, with the TOA at 20 hPa (most previous models had 10 going up to 50 hPa) – the reason for going higher was just curiosity about the resulting temperature profile.

Figure 9

Clearly the solar absorption in the atmosphere should be calculated via the absorption characteristics of the various molecules but this will take some work, and the current model is just an interesting starting point (or resting place depending on my interest level in this aspect of the model). The main flaw in the current approach is that increasing water vapor in the lower atmosphere should increase heating via solar radiation but the model has a static absorption profile shown in figure 1. The main advantage is that it is a lot more accurate than having zero atmospheric absorption.

The convective accounting is also a little challenging. The problem is first that to calculate a convective adjustment we can’t just change layer 2 temperature to make it layer 1 temperature + lapse rate x height. Because after we work that out we have to move the right amount of heat from layer 1 to layer 2 to make layer 2 heat up enough. This reduces layer 1 heat by an equal amount and reduces layer 1 temperature dependent on its heat capacity and the temperature difference is now incorrect. This first problem is easily solved with a formula (see note 3).

Quick people unlike myself will immediately realize that we have not solved the problem at all because when we now consider layer 2- layer 3, the result will move layer 2 temperature and now layer 1-layer 2 is incorrect.

A bigger simultaneous equation might do the trick, but I’m pretty sure that it would come unstuck without some careful thinking about layers where the actual lapse rate in a given time step is less than the prescribed lapse rate. A quick solution was to do multiple loops (iterate towards a solution) and check the lapse rates via a graph and some printing out. Matlab is truly the friend of the mathematically lazy.

There are some checks and balances in my coding. Each time step the model calculates the difference between the TOA balance and the energy absorbed in all layers of the model. This should be zero otherwise I have not implemented the first law of thermodynamics. In this model run the maximum absolute error in any time step was 3 nJ/m² – or less than 4 pW/m². This is just rounding errors in the maths.

The model currently is tasked with printing an error message if ever more than 1J/m² goes missing in a time step.

The Matlab function returns the temperature profiles vs time, energy changes vs time for each layer, convective energy vs time for each layer, along with surface balance, TOA balance and lots of other parameters. The energy vs time graphs in figure 2 are shown as W/m² so they can be related to other fluxes, but they are stored as Joules per m² – it’s just difficult to consider whether 1.67 x 10⁵ J/m² is the kind of value we are expecting or not – and of course the (Joules) numbers change as the time step is adjusted.

Related Articles

Part One - some background and basics

Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code - code can be downloaded, includes some notes on each release

Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..

Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased

Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases

Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased

Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases

Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.

References

The data used to create these graphs comes from the HITRAN database.

The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)

The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)

Notes

Note 1:  The emission of thermal radiation by a surface at 288K with an emissivity of 1.0 is 390 W/m². This is across all wavelengths. The model looks at the range of wavenumbers 200 – 2500 cm^-1 (equates to 4-50 μm) to ease up the calculation effort required. Across this range the emission is 376 W/m².

Note 2:  This was partly to avoid what would look like confusing energy accounting, where solar absorption = the amount prescribed in the model + what we find necessary to keep the stratosphere isothermal.

Note 3: If T1 and T2 are the unadjusted temperatures (found via radiative energy movement), and T1′ and T2′ are the temperatures that should result from lapse rate Γ and height difference z, then:

Convective heat, CE = [(T1-T2) - zΓ]/(1/Cp1 + 1/Cp2)

and then T2′ = T2 + CE/Cp2, T1′ = T1 – CE/Cp1

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In Part Seven we looked at progressively wider wavelength ranges to see where doubling of CO2 had an impact.

We also compared the top layer of the atmosphere (in that model) with the bottom layer – again seeing very significant differences.

This series is aimed at helping people understand how “greenhouse” gases have the effect that they do.

Now we have a model (described in Part Five) which incorporates the actual line by line absorption and emission of various GHGs in a realistic atmosphere we can play around with “what if” scenarios.

Usually as we go up in altitude and pressure drops, the absorption lines get narrower. Around the tropopause the lines are almost 1/5 of their surface width.

I introduced a new on/off parameter into the code which “turns off” this physics and allows us to keep the lines the same width as at the surface. Then compared 280 to 560 ppm of CO2 under one clear sky condition for the case with and without absorption line narrowing.

The top graph is the difference in TOA spectrum with correct physics. The bottom graph is the same but with all absorption lines at their surface width:

Click to expand

The results surprised me. I expected that the effects of absorption lines narrowing would be more significant for this “increased GHG” scenario.

The difference in outgoing radiation (OLR) across this band (which is most but not all of the CO2 effect) is only 0.1 W/m².

[Update shortly afterwards inspired by comment from Hockey Schtick]

The original article might give the false impression that the narrowing of lines has little effect on TOA flux. Actually it has a significant effect. For example, for the case 280 ppm the difference in total TOA flux for correct – incorrect physics = 23.5 W/m². It’s just that for 560 ppm the difference in total TOA flux for correct – incorrect physics = 24.8 W/m². 

The values annotated on the graph are the flux for that wavenumber region only.

TOA flux in the list below is across all wavelengths.

CO2 ppm      Line Width                                    TOA flux

280             Narrows with lower pressure       268.4 W/m²

560             Narrows with lower pressure       262.2 W/m²

280             Constant                                 244.9 W/m²

560             Constant                                 237.4 W/m²

Difference 280 ppm correct – incorrect physics = 23.5 W/m²
Difference 560 ppm correct – incorrect physics = 24.8 W/m²

Difference 280 ppm – 560 ppm (correct physics) = 6.2 W/m²
Difference 280 ppm – 560 ppm (incorrect physics) = 7.5 W/m²

DLR (downward longwave radiation = atmospheric longwave radiation incident on surface) for the same conditions:

CO2 ppm      Line Width                                    DLR flux

280             Narrows with lower pressure       374.7  W/m²

560             Narrows with lower pressure       376.9 W/m²

280             Constant                                 378.1 W/m²

560             Constant                                 380.6 W/m²

Difference 280 ppm correct – incorrect physics = -3.4 W/m²

Difference 560 ppm correct – incorrect physics = -3.7 W/m²

Difference 280 ppm – 560 ppm (correct physics) = -2.2 W/m²
Difference 280 ppm – 560 ppm (incorrect physics) = -2.5 W/m²

Related Articles

Part One - some background and basics

Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database

Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions

Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed

Part Five – The Code - code can be downloaded, includes some notes on each release

Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..

Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased

Part Nine – Reaching Equilibrium - when we start from some arbitrary point, how the climate model brings us back to equilibrium (for that case), and how the energy moves through the system

Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased

Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases

Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.

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