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:
Here is the update in 2009 by Trenberth, Fasullo & Kiehl:
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:
- 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
- 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).
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:
Figure 4 – Highlighted orange text added
Here is the regional breakdown:
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
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.
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).
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