[lyner]

This question was asked on a recent broadcast and I have been thinking about it ever since. Not exclusively, you understand; I do have a life too.
Often, the air temperature, at night, can be as high as 4 Celcius yet frost can form on the ground and on cars etc..
Why does the warmer air not melt the ice?
I think I have a coherent answer to this now.
It relies on two things. Firstly, the specific heat capacity of most gases in dry air (that is the amount of heat energy needed to raise the temperature of on kilogram by one degree Celcius) is significantly higher than most materials, like earth and metals found on the ground  (the only  really common substance  with a higher SHC is, in fact, water). Also, remember, the actual MASS of air near the ground is quite small - the ground is much more dense. Secondly, clear air is fairly transparent to infra red radiation; the greenhouse effect is due to the fact that it is getting only slightly less transparent these days.

On a cold, clear, night (and it has to be a very clear night for this to work) the earth and the air are both radiating heat off into space, which is much colder,  at a faster rate than they are absorbing it from space. They will both cool down, as a consequence. The air, however, will cool down slightly slower than the ground because of its higher SHC. You might expect heat from the air to go into the ground and heat it up; well, some heat does BUT the air is pretty transparent and lets a significant amount of radiated heat, from the ground, straight through it. So the ground  'sees' a lot of very cold space and a small effective amount of warmer air above it.  A bit like standing out in the cold with a slightly warm fishing net  around you;  Over all, you'd be losing heat to the surroundings despite this flimsy net. So there is an overall  loss of heat from the ground. After a while, the ground can get below zero Celcius and, when there is just a thin layer of water on it, due to condensation, perhaps,  ( but not enough to affect it's effective SHC significantly) the water can freeze and give a frost.
Evidence:
We know this doesn't happen on cloudy nights and it doesn't cause ice to form on the surface of lakes or on really wet soil, either. I suspect it works better when the air is relatively low in humidity too. It is a common effect in the desert.

[lightarrow]

I think you are right in everything you said.
Just a little to add: you forgot convection. If a body on the ground is hotter than the air, it looses heat also by convection, because hotter air goes up. In the case you describe, the body is colder, so colder air goes down...and the body doesn't loose heat with this mechanism.

[lyner]

yep; another reason for the ground not to heat up..

I couldn't get my head around a complete answer until I thought about SHC!