That Snow Calculation

I remain perturbed about the possibility that the recent rapid rate of Arctic sea ice melt is at least partly due to a natural cycle.  My hypothesis is that warming causes sea ice melt which causes cooling which restores the sea ice and so on.

Rather alarmingly, you can’t just subtract the natural cycle to obtain the global warming trend.  Instead, global warming interacts with the mechanisms driving the natural cycle, with uncertain but quite likely destabilising consequences.

If the hypothesis is correct, then there would (obviously) have to be mechanisms for the Earth to lose more heat when the Arctic sea ice extent is reduced.  This could happen in several ways.  One is that the Arctic may simply be warmer in the autumn and winter than it “needs” to be for the Earth to be in thermal equilibrium.  That is, without the insulating effect of the sea-ice at the end of summer, enough heat may simply be radiated away from the ocean waters into space to make a difference.

But it may also be the case that the absence of sea-ice changes weather patterns, in particular by causing cold winters in Europe and North America.  Essentially, instead of cold air remaining in the Arctic all winter, the circumpolar circulation breaks down and cold Arctic air cools the Northern mid-latitudes.  Obviously the cold air can’t cool everywhere at once – maybe one way of looking at the effect is to imagine air masses being cycled through the Arctic “fridge” – but it does tend to produce colder winters in the east of North America and in Europe.

During a cold Northern Hemisphere (NH) winter, the southerly winds which are the counterpart of northerlies tend to pass to the west of Greenland, and of North America, so Alaska for instance is warmer.  Or, to put it another way, the continental highs over Greenland, North America and Europe have more effect on the winter weather than usual.

The result is a lot more snow.  For example, the US eastern seaboard is affected by “nor-easters” – depression systems moving up the coast – dragging cold air down from the north inland, the resultant mixing leading to heavy snowfalls.  Heavy snow can also occur in Europe and indeed Asia.

How much effect could this extra snow have, compared to a normal winter?  The purpose of the following calculation is not to quantify the effect with any accuracy, merely to determine whether it could be significant.  It seems it could.

I asked in a previous post:

“What if 10m km2 snow cover persists for just one extra week?  Besides taking extra energy to melt (which turns out to be relatively insignificant), such a surface would reflect around 50% of incident sunlight relative to a year when the snow cover melted earlier.  At the latitudes (between about 60N and 40N) we’re talking about, a rough, order of magnitude, estimate is that at least 100W/m2 extra energy could be reflected (or used just in melting the snow) for a week.  10m km2 is about 1/25th of the total NH surface, so the snow effect alone is of the order of a negative forcing of around 4W/m2 over the entire NH surface, that is, more than the additional forcing of greenhouse gases, but only for one week of the year.   But if my calculation is too conservative, and in fact it’s several weeks over 20m km2 then we could be talking about a serious feedback.”

I now wonder if this is the right way to look at the problem.   The thing is, sunlight is reflected from snow whether it falls in London in December or Barcelona in March.  It might be possible to calculate the effect of all that snow without having to estimate how much longer snow cover remains in a cold winter.  All we have to assume is that the energy to melt the snow comes from sunlight falling on it.  This will be true for a large area of snow – only the border of such an area will be melted (or sublimed) by heat imported from surrounding land or especially sea (since the sea stores far more heat than the land).

Let’s take our extra 10 million km2 of snow in a cold winter and assume there’s an average of an extra 1m of snow over this area.  Warmer parts – London and Barcelona – will only receive an extra 10cm or so, but further north far more than an extra metre is conceivable.  I’ve had anecdotal accounts of snow depths of more than that, but the point is that this is the total over the winter – some will melt (or sublime) before spring and the snow will be replenished.

The “sublime”s I’ve put in brackets are important:

To melt 10m km2 of snow 1m deep takes: 10*10^6*1000*1000 (for km2 to m2)*100*100 (to cm2) *10 (estimating snow as 10% water)*334J = 3.34*10^20J.

But to sublime the same amount of snow takes ~2.6*10^21J because the latent heat of vaporisation of water is 2270J/g whereas the latent heat of fusion is only 334J/g (I’ve added the two latent heats to find the number for sublimation).

Now, a lot of snow sublimes, e.g. as a result of Chinook winds.  In general snow will sublime rather than melt if the air temperature is below 0C.  Let’s assume that half our snow sublimes as a result of incident sunlight during winter and spring.  This will absorb ~1.3*10^21J directly.

But, as I said in my previous post, this is not the major effect.  The big deal is the sunlight reflected while this process is going on.  The albedo of snow is 80-90% – call it 85%.  So only ~15% of the energy of sunlight is available to melt or sublime the snow.  The albedo of the ground absent snow is around 20% on average.  So even rounding down, 4x as much energy is reflected (85%-20% rounded down to 60%, divided by 15%) as goes into melting the snow.  This calculation is independent of the snow depth in any given location as well as how often lying snow disappears only to return over the course of the winter and spring.

The total energy cost to the planet of 5m km2 of on average 1m total snow cover is therefore about 5*1.3 – call it 6*10^21J, assuming all the energy to sublime it comes from incident sunlight.   This is equivalent to a continuous forcing over the ~250m km2 of the Northern Hemisphere (NH) of 6*10^21 / (250*10^6*10^6 to get metres squared*33*10^6 seconds in the year) = 6^10^21/8*10^21, i.e. about 0.75W/m2.

And we haven’t yet allowed for the other 5m km2 of snow that merely melts!

Since the forcing of greenhouse gases (GHGs) totals around 2.5W/m2, a 0.75W/m2 negative forcing is significant.  In fact, given that the Earth will have warmed to compensate for the GHG forcing, the albedo feedback of a cold NH winter may be enough to slow warming* and could even be enough to produce cooling against the warming trend. And this is in addition to the additional heat loss from the Arctic because of reduced sea-ice cover, which I discussed in one of my earlier posts on this topic.

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* Note that 2010 is an El Nino year, so the global average surface temperature may be warmer than in previous years despite the cold NH winter.