Melting snow

Marcel Minnaert has a lot to say about melting snow. But he gives no formulas for the way a heap of snow melts. It is evident that a heap of snow melts quicker when it gets smaller. Then it has more surface for less volume. But is the melting exponential? And is it indeed the case - as the pictures suggest - that the concentration of dirt increases as the surface gets smaller?

161. Melting snow (by Marcel Minnaert)

A story about Benjamin Franklin: When then the sun was shining after it had snowed, he put pieces of differently coloured fabric on the snow, and noted how quickly they began to sink into the snow. The black piece sank first and the deepest, then the red, then the white. Indeed this was to be expected: black, which absorbs all colours, will be heated most by the sun - white, which reflects all the rays will be heated least, and the red in between these two. This argument is not entirely satisfactory, because both the black and white patches absorb the infra-red part of the radiation.

Each time when snow falls and then the sun shines, it can be seen that every stone and every twig lying on the snow, will soon be surrounded by a cavity, where the snow has melted. This is the experiment of Franklin as done by Nature! The snow around stones will melt first, because stones absorb radiation well and thus are much warmer.
Nature scholars have often noted that when hares lie in the snow, the snow does not melt under their bodies. This shows that their fur must be a really bad conductor of heat.

Probably not the parallel grooves mentioned by MM. More likely the sun melting the snow.

If one looks carefully at a layer of fallen snow, one sees that they often show a  shading of parallel grooves, and sometimes there are two systems of stripes. These are probably the traces of the wind, because their direction is the same almost everywhere, and in places that are well protected from the wind, these systems of stripes are less pronounced.
When wind follows a short frost period, hard snow grains and possibly rain drops could make these systems of grooves.

A different mechanism probably causes the remarkable parallel grooves that run downhill in sloping terrain. On a horizontal surface, they have all sorts of quirky directions. This phenomenon occurs only when a layer of old snow is covered by young snow and then is followed by rain. Rainwater seeps through the porous young snow and accumulates in streams inside the old-new interface; all these streams flow in the direction of the sloping ground and carve out small ditches. The young snow settles into the ditches, and then shows the peculiar appearance that we described.

Snow seems to melt slower under the dirt. But ... this conflicts with dirt absorbing more heat (see above).

Another typical form of melting snow occurs when the snow is dirty, and there is bright sunshine. Then needles, pointed shapes, erratic peaks and miniature reefs up to 15 cm high will form that face the sun and follow the sun in the course of the day. Apparently the snow was protected here and there against the radiation by a little piece of dirt and it melted more slowly in the shadow cone of each of these sunscreens. In the Andes these shapes are created in the same way on a much grander scale, the famous "penitent snow."

It is always worth noting the order in which the snow melts. In one case it stayed longer on the sleepers of rail tracks than on the earth surrounding them. Obviously the ground was  warmer and provided enough heat to melt the snow, while the wood of the sleepers conducts heat poorly, so the sleepers cool at their tops and cannot deliver a constant flow of heat. In another case, the snow stayed only on the sharply defined squares or circles around a few trees of an avenue: the trees had recently been planted, and the disturbed earth apparently had different thermal properties than the surrounding soil.

In general, the snow stays longer in shady places where it is protected against the warm thaw wind, where perhaps the soil is slightly higher than its surroundings, so the melt water flows away.Such observations can be performed when the snow begins to fall; the white layer forms first on wood, grass, loose soil, and only much later on the highly conductive asphalt of the roadway.At what point in your environment, do you find the longest laying snow? What influence does the nature of the soil have, or vegetation, proximity of water or swamp?

Most likely a combination of all the factors above creates the surrealistic shapes of the molten snow: