A New Polyhedral Model

The Glaciation model began as a single geological column, and then multiplied into 18 columns at 5º latitudes in the northern hemisphere, and has now become a polyhedral model with 80 triangular faces, 24 of them representing continents, and 56 oceans.

At present, it is intended to only use the 56 oceanic areas to initially find ocean air temperatures throughout the year. It is assumed that snow does not build up on the oceans, and so ocean albedo remains constant at about 0.1.

The remaining 24 continental column heat flows are then calculated at 1-day, 5-day, or 73-day intervals. As snow settles at high latitudes (usually when air temperatures fall below 0º C), the change of albedo is used to find new air temperatures and surface temperatures. And from this a new global mean air temperature can be found.

There is no attempt to model air movement (winds) but instead an Air Mixing Factor is used as an approximation. When Air Mixing Factor equals 0, there is assumed to be no air movement, and the air temperature at any location is whatever is calculated from sunshine, surface albedo, etc. If Air Mixing Factor equals 1, all regions are assumed to have the same global mean air temperature. Values between 0 and 1 are used to provide air temperatures between local air temperature and global mean air temperature.

With no Air Mixing, the regions behave independently from each other, with snow building up and melting at different rates.. When there is some degree of Air Mixing, the different regions tend to become entrained, and to grow and contract at the roughly the same rate.

The result, with an Air Mixing Factor of 0.25, is that there are cycles of glaciation followed by deglaciation at roughly 7,000 year intervals. Snow depth in metres is shown in each region.

The graph below shows air temperatures and surface rock temperatures in the triangular “Europe” region. And it also shows the number of  glaciated regions::

This shows subglacial surface rock temperatures slowly rising during the glaciated period, and melting the overlying snow after about 7,000 years. During the brief interglacial period, subglacial surface rock temperatures rapidly fall back to a low level, eventually allowing snow to start settling again, and a new period of glaciation to commence.

The snowfall regime in this model is at present the same constant amount in every region, If air temperatures are above 0º C, the snow is assumed to fall as rain, and to run off the surface. Precipitation, as snow or water, continues throughout both glacial and interglacial periods, and is about 25 m every 300 years in order to not to have too many snow layers in the snow sheets.

In reality, terrestrial glacial cycles last about 100,000 years, with 10,000 year interglacials. It’s hoped that with improvements in the model, glaciation cycles can be extended.

This is a new model, that was only got working in January 2020, and needs to be further refined.

[Glaciation.java Source Code]

[y120k.hcm (standard continental geological column)]

[y120kO.hcm (standard ocean geological column)]

Adding Milankovitch Cycles

I’ve now included a Keplerian orbital model in the Glaciation model code, and used it to generate numerically mean daily solar power at latitude 65º N on summer solstice over 310 kyears using Earth orbit eccentricity, obliquity, and perihelion longitude at 1000 year intervals provided by J Laskar, and a solar constant of 1367 W/m² (from M Kharseh) at 1 AU radius from Sun..

The values calculated compare quite well with Laskar’s own figures (below, using Milankovitch Orbital Data Viewer).

The period covers the current interglacial period (present day shown as 0), and the two preceding ones. Laskar has orbital elements that extend back one million years.

So I’m now able to generate Milankovitch cycle solar power at all Earth latitudes over this period. I currently only use daily mean values of solar power, but I could also generate hourly values should I need to do so.

Steady Snowfall on Multi-layer Snow Sheets

Real snow doesn’t appear overnight in sheets hundreds of metres deep. Instead it builds up slowly in layers.

And so in the next, slightly more realistic model, snow is dropped at a constant steady rate to form layers of snow.

Once again, as snow depth builds up, subglacial surface rock temperatures rise. At first the rate of melting of the snow at the base of the ice sheet is less than the rate at which snow is added on the surface of the ice sheet, and the ice deepens. But as the surface rock temperature rises, the rate of snow melting increases until it equals – and then exceeds – the rate of surface snow deposition. The snow sheet first gets deeper, and then gets shallower, even though snow falls at a constant rate.

snowfall_on_multilayer_snow_sheet But after a while the snow sheep gets no shallower. This because, as the snow sheet thins, its thermal resistance falls, and heat flow out of the snow sheet increases until it equals the heat flow from the rocks beneath. When heat flow in equals heat flow out, the snow sheet stops melting.

Even when the snow stops falling after 100,000 years, and most of it melts, there still remains a thin layer of snow.

If thin layers of snow don’t melt, how then did the ice age end?

Different Depths of Ice at a Single Latitude

Next looking at how single ice layers of different depths behave at latitude 62.5ºN, southern Greenland. These are just large blocks of ice dropped on the surface of the Earth.

Black and green lines show surface rock temperatures in degrees K. Red line shows ice temperature in degrees K. Orange/yellow shows snow depth in metres (off scale). Cyan line shows mean annual air temperature in degrees K. Timescales are in years with 1e5 years equal to 100,000 years.

The ice is more thermally conductive than snow, so a lot more is needed to have the same effect as snow. If ice depth is greater than about 3 km. surface rock temperatures rise, and so do ice temperatures. If ice depth is less than 2 km, subglacial surface rock temperatures fall.

Ice melting is only not shown because it takes too long to happen.

Interestingly, the current depth of Greenland glaciers is 2 – 3 km, which is in accord with the results above.

One thing this suggests is that ice ages don’t start with heavy snowfall, but instead with the deposition of thin layers of snow over very large areas of land. The effect of the snow is to raise the albedo of the surface of the Earth to 0.8 or 0.9, reflecting 80 – 90% of sunlight back into space, and causing the temperature of the atmosphere above the snow to fall precipitately. If the snow layer is only a few metres deep, the subglacial surface rock has little thermal resistance above it, and heat is rapidly conducted through the snow to the cold atmosphere. Both the snow/ice and the surface surface rocks beneath them get colder. A deep freeze sets in. There is no melting of snow or ice.

Once there’s a layer of deep-frozen snow or ice in place, subsequent depositions of snow simply result in the snow and ice getting deeper. If 1 metre of snow falls every year, after 1000 years the snow will be 1 km deep, and after 5000 years 5 km deep

As the snow or ice gets deeper, the surface rocks cease cooling and start warming, because there is sufficient thermal resistance in the overlying snow/ice to slow the heat loss to a rate lower than the geothermal heat gain by surface rocks from deeper (and hotter) rock layers.

Different Depths of Snow at a Single Latitude

Next looking at how single snow layers of different depths behave at latitude 47.5ºN.

Black, red, and green lines show surface rock temperatures in degrees K. Orange/yellow shows snow depth in metres. Cyan line shows air temperature above snow in degrees K. Timescales are in years with 1e5 years equal to 100,000 years.

The model uses the same initial temperature profile of the Earth at all latiudes. This means that surface rock temperatures will initially fall at the pole, and rise at the equator. So an initial period of 12,000 years is used to stabilise surface rock temperatures before snow deposition.
single_snow_layers_at_47_5N

As snow deepens above 175 m in depth, melt time grows longer.

But below 175 m depth, melt time also grows longer. And with 5 m snow surface rock temperatures actually fall rather than rise.

This happens because as the snow layer thins, its thermal resistance falls, and it loses heat more rapidly to the colder atmosphere above. At some point the rate of heat loss to the atmosphere is equal to the rate of heat gain from the surface rocks below, and the snow doesn’t melt at all. And when the snow is still thinner, the rate of heat loss to the atmosphere is even greater than the rate of heat loss from the surface rocks, and the snow gets colder, and the surface rocks get colder too.

If this doesn’t happen with the very thinnest layers (e.g. 1 m snow), it’s probably because there’s enough heat in the surface rocks to melt the overlying snow before the surface rocks cool down enough to stop the snow melting.

All of which prompts the suggestion that ice ages don’t start with blizzards that build snow sheets hundreds of metres deep, but instead begin with shallow snow sheets over large areas of land which are only slightly too deep for them to be melted by the surface rocks beneath them, and which cause a deep freeze to set in with permafrost beneath the snow. This snow never goes away, and slowly gets deeper with subsequent snow depositions on top of it, gradually turning thicker, and compressing to form ice.

Dropping 200 Metres of Snow at Different Latitudes

The next thing, using this simple model, was to see what happened when a 200 m layer of snow was deposited at different latitudes.

Near the North Pole, the snow took many thousands of years to melt, but did eventually melt after surface rock temperatures gradually kept on rising beneath the snow. After the snow melted, surface rock temperatures fell far more rapidly than they had risen.

Black, red, and green lines show surface rock temperatures in degrees K. Orange/yellow shows snow depth in metres. Cyan line shows air temperature above snow in degrees K. Timescales are in years with 1e5 years equal to 100,000 years.

The model uses the same initial temperature profile of the Earth at all latiudes. This means that surface rock temperatures will initially fall at the pole, and rise at the equator. So an initial period of 12,000 years is used to stabilise surface rock temperatures before snow deposition.

Unsurprisingly, snow melted fastest nearer the equator than the pole. Below latitude 15º N, air temperatures were too high for snow.

What it means is that if 200 m of snow were to fall simultaneously at all latitudes, the snow would melt first at low latitudes, and then at higher latitudes until ultimately it had all melted right the way to the North Pole.

So if there’s still deep snow and ice in Antarctica and Greenland, it may simply be that it hasn’t had a long enough time in which to melt.

How long would it take for a geothermal heat flow rate of 50 milliWatts per square metre to melt 1000 metres of ice? This is a volume of 1000 cubic metres of ice, with a mass of 917,000 kg, which would require, given a latent heat of fusion of 334,000 Joules per kilogram, 3.06E11 Joules of heat to melt it. With a heat flow rate of 50 milliWatts per square metre, this would take 6.12E12 seconds, or 194,063 years.

Source: List 1

A Theory of Ice Ages

What happens when a few hundred metres of snow lands on the surface of the Earth at the start of an ice age? Does the snow melt? Does the ground beneath the snow cool down? Does anything happen at all?

There’s only one way to find out: Build a model. Build a heat flow model of the snow, and the air above it, and the ground beneath it. Add in a Sun that shines daily onto the surface of the snow. Given some set of initial temperatures of air, snow, and ground, figure out the heat flows between these layers, and their new temperatures after a short interval (like one hour or one day). And then start the model running, and watch what happens.

And that’s just what I did. I built a computer simulation model of the heat flow below and inside and above a block of snow. And then I set it running. I used to build dynamic heat flow models of buildings 40 years ago, as a post-graduate and research assistant. Back then we were building electronic analogues heat flow models. I don’t know whether anyone does that any more.

I’m not a climate scientist. I’m not a glaciologist. I’m not a geologist. In fact, I’m not a scientist at all. I’m just an interested layman, and nobody has paid me to build and run my model. I just like building models of things, and watching how they behave. I’ve even got my own orbital simulation model in which I cam watch what planets and moons and asteroids do (and, needless to say, I’m not an astronomer either).

So I’m not claiming to be any kind of expert on this stuff. And I’m not asking anybody to trust me. In fact, I strongly suggest that you shouldn’t trust me at all. Really what I would suggest is that you build your own model.

If you only want to listen to experts, don’t read this blog, because I’m not an expert.

So do you want to know what happened when I dropped 400 m of superfine snow onto the surface of the Earth at latitude 47,5º N?

What happened was rather surprising.

The ground beneath the snow started to warm up. Its temperature rose steadily for thousands of years. And finally it melted all the snow.

400m_snow_meltThe 5 superimposed graphs on the right were generated at runtime by my model. They show how the temperature of three layers of granite surface rocks (black line topmost, red line, green line) varied over about 220,000 years. And how the air temperatures above the snow (cyan line) varied over the same period. All temperatures in degrees Kelvin, in which water’s freezing point is 273º K and its boiling point is 373º K. And it also shows snow thickness (yellow line) in metres. Time goes from left to right.

The model run starts without any snow present, when the surface rocks are allowed to cool down over about 12,000 years, which is about the length of a terrestrial interglacial.

And then at the 12,000 year mark, I drop 400 metres of superfine snow on top of the surface rocks, which is when snow depth jumps from 0 to 400 m.

At the same time the air temperature above the snow drops by about 60º K. That happens because the surface albedo (or reflectivity) has changed from about 0.4 for granite to 0.8 for snow, and a lot of sunlight is being reflected back out into outer space without warming either the snow or the air above it. The result is a rapid drop in air temperatures in the air column above the snow.

But the really surprising and interesting thing is the way that the surface rock temperatures start rising, and carry on rising for the next 200,000 years. Why is that happening?

Here’s the explanation. Before the snow landed, the surface rocks had reached an equilibrium temperature where they were gaining as much heat from the interior of the Earth as they were losing to the atmosphere (and to outer space) above them. But when the snow landed, while the top surface rocks carried on gaining heat from warmer layers below, they could no longer lose heat at the same rate to the atmosphere above because the snow layer offers considerable resistance to any heat flow. Snow is a very good thermal insulator. And the superfine snow I’m using here has the same thermal resistivity as the expanded polystyrene or polyurethane foam that’s found in the air gaps inside brick walls. So the topmost surface rocks are continuing to gain heat at the same rate as before from the rocks beneath them, but are no longer losing heat at the same rate through the insulating snow above them. So the surface rocks start gaining heat, and when they gain heat, their temperature rises. And that’s why we’re seeing their temperature rising. The temperature rises slowly because the heat flow rate into them is very small – only a few milliWatts/m².

And the snow above the surface rocks can’t rise above oº C (273º K). Or it takes a great deal of heat to turn solid frozen water (snow) at at 0º C to liquid water at 0º C. It’s called a phase change. One phase change is from ice to water, or water to ice. Another from water to steam, or steam to water.

So the 400 m of snow starts changing phase from ice to water as soon as its temperature rises to 0º C. And it then takes many tens of thousands of years to complete the phase change.

And that’s what happens after 212,000 years. The snow turns to water. And the water runs away. And when all the snow turns into water and runs away, the surface rocks (whose temperatures have risen 70º C over the previous 200,000 years) cease to be covered in a deep layer of thermal insulation, and so they start to rapidly cool down. And within 10,000 to 20,000 years they’re almost back to the temperatures at which they started 200,000 years earlier.  And at the same time that the snow all melts, the albedo of the Earth falls from 0.8 to 0.4, and it stops reflecting most of the sunlight back into space, and air temperatures return to what they had been before the snow appeared.

So if 400 m of snow lands on the surface of the Earth, the rocks beneath the snow will warm up and melt all the overlying snow. The Sun doesn’t melt the snow. Neither does carbon dioxide in the atmosphere melt the snow. Nor do variations in the Earth’s orbit over many thousands of years (Milankovitch cycles) melt the snow. Nor does anything else. The snow is melted by the warming rocks beneath it.

So that’s what my simulation model says what happens during an ice age, when large areas of land are covered in deep snow or ice. The rocks beneath the snow and ice heat up, and eventually they melt the snow and ice.

And this might explain the last ice age lasted about 100,000 years, and the interglacial before it only lasted about 10,000 years. It’s because the surface rocks beneath the snow heat up very slowly, but cool down fast.

In fact, it may explain how ice ages work, and why there have been these long ice ages punctuated by brief interglacial periods. During the long ice ages, the rocks beneath the snow heat up very slowly, and eventually melt the snow. And during the interglacials the surface rocks cool down again much more rapidly, and cool to the point where snow can start building up again, and the next ice age start, in a repeat never-ending cycle.

But that’s just my explanation. And I’m not a climate scientist or a glaciologist or a geologist. So you don’t have to believe me.

And it’s not an explanation you’ll find in any textbook. I’ve never seen this explanation anywhere. And I’ve never seen any graphs like the one above anywhere, showing surface rock temperatures slowly rising and then sharply falling.

In my next post, whenever I get round to writing it, I’ll discuss today’s model some more, and maybe a few other matters as well.