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Warming induced by the latent heat of snow

According to UAH, January 2010 will almost certainly be their warmest January on record, and by its anomaly (which is likely to exceed 0.70 °C), it will be one of the 4 warmest months.

Recent NASA MODIS pictures of the United Kingdom look like an ice age.

I was thinking how it was possible that such an unusually cool January is so warm according to this global methodology. Snow was almost everywhere on our hemisphere, wasn't it? Well, it may actually be a reason.

First, I thought that a problem could exist with the satellite measurements of the solar microwave radiation reflected from the snow. But the solar microwave and even infrared radiation is actually negligible.

The solar surface temperature is around 6000 °C and is about 20 times higher than the surface temperature of the Earth. In Planck's law, these low frequencies get only suppressed by a power law but that's enough. The integrated radiance (below a certain low frequency) goes like the frequency to the fourth power, and (1/20)^4 is a really small part of the energy (6 ppm or so). This is just an order-of-magnitude estimate. UAH AMSU measures the microwaves - and for them, the solar contribution is even smaller when reflected.

Latent heat: numbers

However, the actual latent heat stored in the snow itself is not negligible. First, let us calculate the heat capacity of the whole atmosphere. Using some available figures,

heatCapacityAtmosphere =
510.*10^{12} meter^2 * 1.34*10^7 Joule/meter^2/kelvin = 6.834*10^{21} Joule / Kelvin
The first factor is the surface area of the Earth and the second factor is the specific heat of a squared-meter-sized "column" of the atmosphere. Now, we need the total heat that was extracted from the creation of some excess snow - imagine that more snow was created in January 2010 than in other Januaries.

First, you should remember that the latent heat of melting snow is 334 000 Joule per kilogram, i.e. one thousand times more than William Connolley thinks it is. For the sake of completeness, let me also remind you that the specific heat of water is 4200 Joule per kilogram and Kelvin for water above the freezing point and 2100 Joule per kilogram and Kelvin for ice (or snow) right below the freezing point.

You only need 1/2 of the water's thermal energy to heat up the same amount (mass) of ice or snow. And melting of 1 kilogram of snow/ice into water needs as much energy as warming 80 kilograms of water or 160 kilograms of snow by 1 degree Celsius.

Now, we must know how much extra snow was created. Let's assume that the extra snow was created above 1/4 of the Earth's surface (between 30 °N and the North Pole; you can also use 14.5% between 45 °N and the pole), and it was 3 cm of water equivalent (which translates to 50 cm of snow at -5 °C or so). So the total mass of the excessively created snow is
massOfSnow = 510.*10^{12} meter^2 * 1/4*0.03 meter*1000 kg/meter^3 = 3.825*10^{15} kg
The last factor is the density of water. If you multiply the two figures above, you get
meltingEnergy = massOfSnow*latentHeat = 1.27755*10^{21} Joule
and you may now distribute this released heat (by the water that became solid) to the whole atmosphere in order to find its average warming:
meltingEnergy/heatCapacityAtmosphere = 0.187 kelvin
That's a substantial warming. Once again, if snow is created at 0 °C, its water molecules are losing energy (just like if they're cooling) which means that the environment must be gaining this energy (and it must therefore be warming up).

So with the snow totals above - and I don't claim that they were obtained too rigorously - the atmosphere would warm up by nearly 0.2 °C. It should be obvious but let me just mention that the existence of snow that is "stuck" at 0 °C is enough to cool the environment around. So when it melts, it will cool the atmosphere back, by those 0.2 °C.

Yes, I do predict that the January 2010 UAH temperature anomaly will be above 0.65 °C and probably above 0.70 °C but by May 2010, the anomalies will be below 0.60 °C again, despite the continuing El Nino. Feel free to check my predictions later. ;-)

There are many technical points that you may - and you should - object to. For example, I was assuming that the snow precipitation is created also above the oceans (which is OK) but that its expected early melting won't cool the atmosphere quickly (just the ocean). You should try to think how these things are important and correct the numbers above.

Also, I was neglecting the latent heat of condensation/vaporization which is actually 7 times higher for H2O than its freezing/melting latent heat. The justification of this negligence is the assumption that the total precipitation wasn't too unusual - it just happened to include a higher snow component than is usual.

At any rate, I think that a robust conclusion is that if you want to be able to estimate the temperature at a given day with the accuracy of 0.1 °C, you must also be interested in the amount of snow - at least uniform dozens of centimeters above 1/4 of the Earth's surface.

There are dozens of effects that can pretty quickly change the temperature of the Earth by 0.1 - 1.0 °C - and the CO2 greenhouse effect is just one of them, and one of the slowest ones (although one with a uniform sign, assuming that we won't destroy our economy).

Bonus: adjustments of temperature for latent heat and specific heat of ice

By the way, because of the latent heat of snow, it could be sensible to adjust the local temperatures at individual weather stations by the amount proportional to the local snow cover. Recall that the column of the atmosphere has the heat capacity of 1.34 Joule/(meter^2 Kelvin).

Looking at one squared meter and a 1 °C of warming, the heat is equivalent to the freezing latent heat of (40 kilograms per squared meter) of water. And 40 kg of water per meter^2 is 40 liter = 40 decimeter^2 per 100 decimeter^2 which is 0.4 decimeter = 4 centimeter of water equivalent.

So each 4 centimeters of water equivalent of snow (something like half a meter or more, depending on temperature) are able to cool the whole column of the atmosphere by 1 °C. It might be sensible to consider a snow-correlated local temperature which would be the local temperature minus 1 °C times the height of water equivalent of the average local snow cover divided by 4 cm.

There is one more point that explains some observations that I have made when I analyzed the local weather records. In Prague, I noticed that the seemingly "normal" distribution of the temperatures is highly asymmetric. The temperatures are much more likely to visit highly sub-normal temperatures than the highly above-normal temperatures: the real distribution decreases much more slowly at the cold tail.

The histogram of all January daily average temperatures in Prague in 1973-2009.

In that article, I mentioned that the temperatures can change really quickly in both directions because of an ice-albedo feedback. That's why it's easier to visit "further" cold temperatures.

However, when we look at the specific heat of ice and water, we see another possible explanation - or a contribution to the overall explanation. The specific heat of ice is 1/2 of the specific heat of water. So when water in the rivers, lakes, and the soil freezes, it becomes twice as easy (as far as the required energy flows go) to lower the ice's temperature by another degree.

Moreover, ice is three times as good thermal conductor as liquid water (in the units of W/(m.K) - so that it enters Fourier's law for heat conduction). So in general, I do expect that the temperature swings become much easier below the freezing point as long as there is any ice around.

Such an observation sheds a completely new light upon the warming in the polar regions. The faster changes over there may be due to the higher heat conductivity and lower specific heat of ice, relatively to liquid water. However, this "enhanced speed" disappears as soon as you jump above the freezing point.

A special answer to a retarded author of a blog called "Climate Progress"

I think that I have never said that AGW was a “fad”. I usually use the word “fad” as a neutral description of memes and research programs that are based on some merits but that are overstudied – because their importance is overstated.

This is clearly not the case of AGW which is not based on anything. One could call AGW a “scam” or a “hoax” but not a “fad”.

My headline chosen for this article was not “Emissions caused warm January” because only hopeless crackpots would be able to write something as nonsensical as that. We are discussing why the UAH temperature anomaly jumps by something like 0.45 °C from December 2009 to January 2010.

Even according to the IPCC forecasts which are artificially inflated by a factor of 3 or more, the CO2-induced warming rate is comparable to 3 °C per 100 years which means less than 0.003 °C per month – and this enhanced greenhouse contribution is nearly uniform in time because the CO2 "anomalous" concentration in the atmosphere increases uniformly with our continuous emissions.

This hypothetical 0.003 °C warming per month has clearly nothing whatsoever to do with our physics problem which is a month-on-month anomaly-wise warming by 0.45 °C, i.e. 150 times higher a temperature change than the IPCC-calculated CO2-induced warming during the same time, and whoever is incapable to understand this not-so-subtle point should ask his health insurance company for the money for a brain transplant.

The carbon dioxide is clearly completely irrelevant for all questions about the weather and climate dynamics during one year or as a function of location. It's really irrelevant for pretty much anything.

An alternative possible cause

The UAH temperature team doesn't directly measure the temperatures poleward of 82 degrees North or South. In particular, the far Arctic above 80 degrees North recently cooled - by about 14 °C in the first three weeks of January 2010.

When you add this temperature of the North Pole disk to the global average, you will remove about 0.1 °C from the global mean temperature - the very recent UAH warming would be an underestimate (because they effectively omit the far polar areas) and should be naturally lowered. See a TRF article on this topic.

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