Do we really have a “33 °C Greenhouse effect”
Written by Jan Zeman, Czech Technical University, Prague
The Wikipedia entry for the Greenhouse Gas Effect states:
“If an ideal thermally conductive black-body was the same distance from the Sun as the Earth is, it would have a temperature of about 5.3 °C. However, since the Earth reflects about 30% of the incoming sunlight, this idealized planet’s effective temperature (the temperature of a black-body that would emit the same amount of radiation) would be about −18 °C. The surface temperature of this hypothetical planet is 33 °C below Earth’s actual surface temperature of approximately 14 °C. The mechanism that produces this difference between the actual surface temperature and the effective temperature is due to the atmosphere and is known as the greenhouse effect.”
The statement is almost completely untrue. For instance not even the math adds up: the difference between the two temperatures +14 °C and -18 °C is not 33 °C but 32 °C. But it is not important, what is important here is the fact that there’s not a difference of 33 °C, nor of 32 °C between the hypothetical and real Earth surface temperature. In short, there is clearly a confusion about what is meant scientifically when describing the “surface” of Earth.
I don’t want to rewrite astronomic customs, but for such purposes as a black-body radiation flux equation to and from the planet using the Stefan-Boltzman law, we would think the surface of the Earth should be considered to be “the atmosphere”- not the surfaces of the sea and land. The reason being that it is only the uppermost layer of the planet’s mass that is capable of radiation – in the sense as defined by the Stefan-Boltzman equation – unlike the boundary of the vacuum of space beyond.
This confusion is a result of our human perspective. In the case of big gas planets like Jupiter we observe from the outside and hardly anybody would suggest the immediate exterior of its uncertain small diameter core was the “surface.”
Indeed, there’s an even stranger boundary custom to consider whereby we could discern the “surface” and atmosphere arbitrarily just at the point where Jupiter’s immense atmospheric pressures crosses 10 bar. Nonetheless, when talking about the Stefan-Boltzman law (i.e. about black-body radiation) as applied to Earth and it’s radiation budget, we should consider the gaseous atmosphere as being the Earth’s surface, not the actual surfaces of sea and land below.
The average temperature of Earth’s surface
The surface of Earth’s gaseous atmosphere has an ‘average’ temperature of about −18 °C or even a bit lower. The simplified calculation using data from standard ISA atmospheric model you can find here and although it is still rather a very raw estimation – with maybe over one degree uncertainty and given the imperfection of the standard atmospheric model – the average value resulting from it comes out at about 255 Kelvin or lower. The actual value coming from the simplest calculation using averaging over 1,000 meter steps over the lower 50,000 meters of the atmosphere (99.9% of its mass) is 255.024 Kelvin (–18.13 °C) and it looks even lower if we refine the averaging using a larger sample with a 500 meters step – the resulting average temperature value is then 253.86 Kelvin (-19.29 °C).
So 255 Kelvin is the temperature which usually occurs at 5,000-5,500 m above sea level in the standard atmosphere – which is also the point where the atmosphere has about half pressure, which also more or less half the atmospheric mass; half above, half below this altitudes. But remember, it is an ISA atmospheric model average, in the real atmosphere it varies with multiple factors, mainly with latitude.
From the Stefan-Boltzman law we can calculate that for a black-body which has say the <=255 Kelvin average temperature, the radiation of such a black-body would be:
0.0000000567*255^4 <= 239.76 W per square meter [Eq. 1]
Now, let’s compare the figure to the average absorbed solar irradiance (measured since the beginning of the SORCE satellite experiment in 2003 using TIM instrumentation). The total solar irradiance at the top of the atmosphere was about 1361.1 W ±0.47 per square meter at 1 AU (TIM satellite data here).
Therefore if we assume that the 30% reflection figure from the Wikipedia statement is exactly true, then the solar irradiance of one square meter of the Earth surface (atmosphere) which isn’t immediately reflected to space is:
1361.1 / 4 x 0.7 = 238.19 W (±0.08) per square meter [Eq.2]
The resulting value is clearly even lower than what comes as the maximum from Eq. 1. It could suggest that the average incoming solar radiation flux per square meter, since the beginning of the SORCE-TIM operation in 2003, was even a bit lower (1.57 W/m^2) than what the atmosphere would in average radiate according to its average temperature coming from the standard atmospheric model and Stefan-Boltzman law. This would appear consistent with the global warming hiatus during the period.
But not so fast. The value of the resulting incoming radiation is very sensitive to the exactitude of the “30%” reflection (albedo) figure – which very much looks being a rounded rough estimation – so we can’t jump to the above conclusion until we know the value for sure and to several decimal places.*
But what we can anyway conclude with sufficient certainty is that the surface body of the real Earth – the atmosphere – behaves more or less the way Stefan-Boltzman law predicts for a black-body and so it very much looks there’s in fact by far not a “33 °C” difference as purported by the Wikipedia statement. In other words the Earth surface temperature – the average temperature of the atmospheric mass – is relatively close what the Stefan-Boltzman law predicts for such a body.
The 33 °C difference between the so called global surface air temperature and the average atmosphere temperature is most probably given by another physical atmospheric phenomenon rather than a “Greenhouse effect” and this phenomenon is called the moist adiabatic lapse rate. And this graph below, describes the phenomenon more or less confirming the assumption: Fig. 1
The moist adiabatic lapse rate
The moist adiabatic lapse rate phenomenon is co-caused by considerable atmospheric water content, carrying as water vapor (being significantly lighter than air it is buoyant and rises quickly). It has a considerable amount of latent heat of vaporization, transporting this heat from sea and land surfaces where it is evaporated to the higher levels of the atmosphere. I’ll add that on its way up it is further heated by the sun during the day before condensing. But it is not so important, what is important is the considerable heat leaving the sea and land surfaces.
What needs to be immediately emphasized here is the fact that the latent heat of vaporization doesn’t add the slightest bit to radiation and only when this latent heat is again released during condensation it changes the temperature of the surrounding air at the places where it was released.
The evaporation and evapo-transpiration of heat transport from sea and land surfaces up the atmosphere works much like a compressor-less refrigerator or heat pump (using solar energy and gravity as the energy source and “thermostat”).
It is estimated that at least 505,000 cubic kilometers of water falls as precipitation each year, which is so much water it would cover the whole Earth to a depth of one meter. And this water must be evaporated each year from the sea and land surfaces. Given the water latent heat of vaporization (2.26×106 J/kg) it means that the evaporation of water from sea and land surfaces transports very considerable amount of heat from the sea and land surfaces up to the higher layers of the atmosphere. There the vapor condenses, releases the latent heat, forms clouds and eventually falls back to the ground and into the sea as rain and other forms of condensed liquid or solid (snow) water. We can estimate the amount of heat transported this way by a calculation:
5.05×1017 kg x 2.26×106 J = 1.14×1024 Joules [Eq.3]
(~70.8 W.m-2 average energy flux from the sea and land surfaces as latent heat – good to note that this figure is still significantly lower than the estimation made in the so called Kiehl-Trenberth energy budget.)
Let’s now compare this figure to an estimation of how much heat Earth’s surface receives yearly on average from the sun according to the Eq.2. Result:
238.19W x 5.10072×1014 Joules x 31556926 [seconds in year] = 3.83×1024 Joules [Eq.4]
The comparison shows that about 30% of the total heat converted from the solar shortwave radiation extinction is transported as latent heat via evaporation away from the sea and land surfaces somewhere high in the atmosphere. There it again condenses and releases its latent heat to the atmosphere.
This atmospheric water phenomenon dramatically changes the adiabatic lapse rate from the dry adiabatic lapse rate (~9.8 °C/km) to moist adiabatic lapse rate (varying with air temperature and humidity ~3-9 °C/km) The ISA standard value of adiabatic lapse rate used for aviation is 6.5 °C/km, which is almost exactly the 33 °C difference between sea level mean surface air temperature and the temperature at altitude; where the atmosphere has half of the sea level pressure and where the average temperature of the atmosphere occurs (5 to 5.5 km).
The water vapor as it ascends changes atmospheric temperature profile considerably, and as we have seen in the above graph, it is then more or less consistent with the temperature difference between the average temperature of the atmospheric mass and the average surface air temperature.
Stefan-Boltzman law and sea
After showing that the Earth’s atmosphere as a whole behaves close to how the Stefan-Boltzman law predicts for a black-body we can now look how it is with the sea.
It very much looks like the sea surface temperature anomaly is quite intimately linked with the surface air temperature anomaly. Although they seemed to diverge since the 1970s with the warming up to some ~0.15 °C in the mid 2000s and since then they seem again to converge with the cooling by ~0.1 °C. It would seem intuitive to assume the temperature of the air drives the temperature of the sea surface, but it is definitely not the case and the opposite is true.
Why? Because the average absolute sea surface temperature (~17 °C) is considerably higher than the average absolute surface air temperature (~14 °C). Therefore the sea must warm the air and not vice versa.
And not just due to the special 2nd thermodynamic law concerning heat, which says that heat can’t move from the colder body to the warmer one. But simply because the warmer body always releases more energy per section (say the square meter) than a colder one. Athough, as we have already seen, the sea doesn’t release the energy just by the mid-IR radiation resulting from its temperature, but to a considerable extent by evaporation/latent heat transport. Generally one should assume that the sum of all energy form fluxes released per section, from a warmer body to colder bodies or space surrounding it, should be equal to the radiation energy per section given by the warmer body temperature as the Stefan-Boltzman law predicts. At least until the Second Law of thermodynamics is falsified, which seems unlikely.
Question: What then warms the sea surface if not air? Answer: mainly the sun.
Question: Can’t the atmosphere warm the sea surface instead of the sun by so called atmospheric backradiation? Answer: No.
Why? Besides the fact that the air is on average colder than the sea, the other reason is that the water is extremely opaque to the long-wave mid-IR spectra (at which only the atmosphere can radiate given its average temperature**); when compared to the shortwave solar spectra. This is in order of million of times as this graph shows (mind the y axis is logarithmic): Fig 2
99% of the mid-IR spectra at which the atmosphere radiates is absorbed in the uppermost ~30 micrometers of the sea surface skin and cannot significantly penetrate and warm the water any deeper. Therefore it chiefly contributes not to the sea surface warming, but to the evaporation from the sea surface ‘skin’ we were talking about in the previous section.
On the other hand over 50% of the shortwave solar radiation spectra which penetrated the sea ‘skin’ still gets deeper than 20 meters and still over 5% gets deeper than 100 meters. Unlike for the mid-IR the water is exceptionally transparent to the solar spectra – as we’ve already seen at Fig.2.
How the solar spectra penetrates the sea surface is also suggested by this graph:
The solar radiation still significantly penetrates the sea to the depth of over 100 meters. This is an order of magnitude million of times deeper than the atmospheric mid-IR radiation. It becomes extinct on the way – converted to heat which warms the sea water throughout the so called epipelagic sea surface layer, much, much deeper than it can be warmed by the atmospheric mid-IR radiation. Meanwhile, the ocean, if not covered by ice, has extremely low average albedo of ~0.033 (- value for zero waviness calculated here – for wavy ocean it could be even slightly less due to refractive properties of the air/water interface if wavy, but I omit it). This means that over 96% of the solar radiation reaching the sea surface gets below the surface, becomes extinct/converted to heat in the surface layer and then warms it. It can be estimated that about 90% of the solar radiation reaching sea and land surfaces is combined and converted to heat in the oceans and seas. This is despite the fact the oceans and seas cover only about two thirds of the Earth. But such estimations are not trivial so I will not go into the details as they are not the subject of this article.
A “Greenhouse effect“ in ocean instead of atmosphere?
But what happens with the heat in the sea?
This is even more interesting. Because most of the sea surface is warmer than 3.98 °C and the warmer water being lighter stays at the surface, the temperature of the sea surface layer is on average considerably higher (~17 °C) than sea water in the depths (~4 °C). If we go back to the Stefan-Boltzman law the water in the surface layer of the ocean body on average radiates:
0,0000000567*290.15^4 = 401.86 W per square meter [Eq.5.]
Now, this number appears much higher than the number resulting from the Eq.2. But does the ocean really radiate the 401.86 W per square meter out to the atmosphere as mid-IR radiation?
No, because the ocean surface properties are contrary to the atmospheric properties – very considerably different than the properties of a black-body to which the Stefan-Boltzman law applies.
While the ocean is exceptionally transparent to the incoming solar radiation, as we have seen it is much harder for the mid-IR spectra resulting from the sea surface layer temperature to get out. **
But it is not just because the sea water is extremely opaque to the mid-IR spectra allowing the mid-IR photons to travel in it at just several dozens of micrometers. It is chiefly because the refractive properties of the water/air interface raise the reflectivity away from the water dramatically, with the rising temperature and incidence angles. This is depending on the sea’s surface ‘skin’ temperature – which changes the water radiation spectra up to the radiation spectrum with peak wavelength 7.765 μm (water radiation spectrum of near water boiling point) – the water/air interface at average sea surface temperature reflects typically ~30% and with rising surface skin temperature under insolation up to 40%. This calculation is by using standard Fresnel reflectivity model data here. It is of the mid-IR spectra back to the sea, adding to the simultaneous solar irradiance flux and the atmospheric mid-IR radiation flux and facilitating water evaporation from the sea surface ‘skin’ we have already talked about.
But to evaporate water which has on average the 17 °C and sea level pressure one needs not only 2.26×10^6 J/kg but also the energy to heat the water to the temperature at which it evaporates. This energy can be calculated as:
4.1813 x 1000 x (100-17) = 3.47047×105 J/kg [Eq.6.]
and if we go back to the Eq.3 it means at least:
5.05×1017 kg x 3.47047×105 J = 1.75×1023 Joules of additional heat which must be taken from the water body for the evaporation to take place.
(This then means at least another ~10.9 W.m-2 on average is transported from the sea and land surfaces upwards.)
Now we see that evaporation and evapo-transpiration takes from the sea and land up to the atmosphere at least 81.7 W per square meter, which is already 1.7 W.m-2 higher number than the Trenberth et. al latent heat estimation. Moreover this value together with mid-IR radiation from the sea would raise the water temperature – which rose according to the sea surface anomaly rise in 20th century ~0.65 °C, most probably due to the rising solar activity.
Can we estimate how much this 0.65 °C sea surface anomaly rise would add to the surface radiation+evaporation energy budget?
[0,00000005670373*290,15^4=]401.885 – [0,00000005670373*(290,15-0.65)^4=] 398.296 = 3.589 W per square meter [Eq.7.]
Which for whole the Earth surface would mean surface air temperature forcing of:
3.589 x (36113200 / 51007200) = 2.54 W.m-2 [Eq.8.]
The larger part of it is in direct mid-IR radiation (~~70% – 1.78 W.m-2) – warming first the air immediately above the sea – and in smaller part by latent heat of vaporization (~~30% – 0.76 W.m-2) – transported up warming the atmosphere where the vapor condenses.
Now we see that the rising surface air temperature forcing was due to the raised sea surface temperature, caused chiefly by the rising TSI trend. It is not due to CO2 emissions enhancing any GHE in the atmosphere. And even if something like that was happening, it cannot significantly affect the sea surface temperature. This is because the bulk of the atmospheric mid-IR radiation is absorbed in the very surface ‘skin’ of the sea and immediately effects surface water evaporation, not to the epipelagic sea layer warming.
Closing remarks: What is more likely in the future: Ice age or a catastrophic global warming?
Now, we have the Sun, which releases radiation, this then travels through space and then hits our Earth. The minor part is reflected, the larger part absorbed and then re-radiated as it causes the temperature to rise intermittently.
But because the physical body energy exchange with the surrounding bodies or space rises by the fourth power of the body temperature, the radiation of the body rises dramatically. By rule of thumb this is by 5.5W.m-2 with every degree of temperature rise. So how on Earth can anyone claim 1.8 to 4 degrees temperature rise in a hundred years? – For such a feat we would need 9.9-22 W.m-2 additional forcing to overcome the Stefan-Boltzman law. As we see, such forcing would about equate to the energyneeded to make the water cycle on Earth possible.
So, what makes the average surface air temperature appear ~33°C higher is a forcing of:
[0,00000005670373*288,15^4=] 390.92 – [0,00000005670373*255^4=] 239.76 = 151 W.m-2 [Eq.9.]
The vast majority of this forcing originates from Sun heating the epipelagic layer of the ocean. This then heats the atmosphere by direct mid-IR radiation, latent heat transportation system, and then by convection and heat conduction. It is also why rising sea surface temperatures, due to rising insolation, caused the period of late 20th century warming.
It could be estimated that for triggering a runaway glaciation and ice age is sufficient a forcing of minus couple of Watts (up to 10 W.m-2), which will inevitably occur when (northern) summer solstice will get in the phase with Earth perihelion due to the slow Earth axial precession. The ocean absorbing the bulk of the incoming solar energy reaching sea and land surfaces (which wasn’t immediately reflected due to albedo or absorbed by atmosphere), the ocean, bulk of which is at the southern hemisphere, would become less insolated due to the Earth’s tilt/orbit unfavorable phase.
It will inevitably happen some time up to ~10000 years from now. And it will be paradoxically the warm ocean which will help trigger the ice age – it will still produce wast amounts of atmospheric water content by evaporation, which will then precipitate more and more as snow at higher latitudes, more and more will get deposited at lower and lower latitudes and not melted away during the melt seasons, changing the Earth’s albedo considerably and expedite the ice age triggering process with descending solar forcing due to phasing of southern summer with Earth’s aphelion. (The difference between TSI in aphelion and perihelion – most distant and closest points to Sun during the Earth year’s orbit cycle – is almost 100 W per square meter). The snow and ice albedo rise in lower latitudes is a powerful positive feedback for ice-age triggering. There’s nothing like that for warming, because even if the ice recedes in polar regions, it recedes to higher and higher latitudes, where the insolation due to Earth’s curvature is less and less significant. Even if all sea-ice and ice-sheets would melt – which even at current rates (if they would last) is impossible happen sooner than in several dozens of thousands years – it would not trigger any runaway warming, simply because it would not add sufficient forcing not nor any crucial positive feedback.
So what humankind should really once worry about (not now) is a global cooling, not the warming. Due to the Stefan-Boltzman law a runaway global warming which people as James Hansen threaten us with using analogies with Venus (much closer to the sun and receiving almost double TSI) getting so much attention by inept media and politicians is physically patently impossible without major external forcing source (e.g. ever-rising solar activity at the end of stellar life-cycle), because the energy radiation of any physical body rises with fourth power of its temperature and for every degree of the global temperature anomaly rise one would need an additional forcing of ~5.5 W.m-2. Such forcing patently cannot be caused by the minute amounts of CO2 atmospheric content. Do we really believe man is so powerful that we can avoid ice ages just by burning ‘fossil fuels’?
* – if the Wikipedia figure would be not 30%, but say 29.54% then the result of equation would be: 1361.1/4*0.703449 = 239.76 W/m^2 – same figure as the theoretical outgoing radiation of the atmosphere based on Stefan-Boltzman law and the average atmosphere temperature, which as we seen is rather the upper bound estimation.
The Earth Bond albedo which in fact the Wikipedia entry figure comes from is listed with values ranging from 28.1 to 33.8 (-in the Trenberth et al. 2009 they apply value 29.8 which would result in 238.87 W/m^2 incoming ASR according to the TSI value measured by SORCE-TIM instrument (surely way better TSI measurement device one generation beyond what Trenberth is referring to) and the resulting planetary energy imbalance would in fact come out as negative – Earth radiating more energy than it receives from sun, which would be consistent with the slightly cooling trend in the last decade, but the uncertainties are so high that we can’t really tell) Given the sensitivity of the Eq. 2 outcome to exactness of the Bond albedo value and the uncertainty of the Earth atmosphere average temperature (which is extremely difficult to really measure with sufficient exactness) we cannot decide whether the Earth radiation budget is positive, negative or neutral until we know the Bond albedo value for sure and with sufficient exactitude.
The current state of our knowledge about this things which is rather poor clearly doesn’t warrant any political action and mitigation spending of any scale and the fact that UN keeps attempting to impose them despite this crucial uncertainties tells more about their desire for power and our money than anything about climate.
** – according to Wien’s displacement law the peak wavelength of atmospheric radiation spectra is:
2897.7685 μm.K / 255 K = 11.36 μm. For such spectrum the the water is order of million of times more opaque than for the solar spectrum as the Fig. 2 shows.
For the sea surface radiation the peak wavelength is 2897.7685 μm.K / 290 K = 9.98 μm
Jan Zeman is Associate Professor at the Czech Technical University, Prague