Re: The Valles Marineris canyon on Mars. Could it have an atmosphere?

Please excuse a slightly less than authoritative answer from a mad Chemist/Earth Scientist. Our 
astronomers seem to have ducked this one.

On Earth, the pressure of the atmosphere falls off quite quickly with increasing height. Roughly 
speaking, for every 17 kilometres, the atmospheric pressure falls by a factor of 10. 

Most of this fall-off is accounted for by the barometer equation, which relates the pressure at 
height z to the surface pressure: p(z)=p(0) * exp(-Mgz/RT). -- See, for example, Laidler & Meiser, 
Physical Chemistry, 3rd Ed., p. 28. This behaviour is modified in the Earth's atmosphere by 
turbulence, and by temperature variation.

On Mars, the average molar mass of the atmosphere (M) is about 1.5 times that of the Earth. The 
surface gravitational acceleration (g) is about 40% of Earth gravity, and the kelvin temperature (T) 
about 75% of that on Earth. So we would expect the rate of pressure fall-off to be about 80% of 
that on Earth. That means that on Mars, atmospheric pressure is likely to fall by a factor of 10 only 
every 21 kilometres.

The surface atmospheric pressure on Mars is about 1/150 of an Earth atmosphere. If the canyon 
were 21 kilometres deep (which it probably isn't) it might have a pressure up to 1/15 of an Earth 
atmosphere at the floor, which just might be enough to be interesting. That is the good news!

The bad news is this: If the floor of your canyon were indeed warm compared to the ambient 
temperature on the Martian surface, then we would have colder air overlying warmer air, which is a 
mechanically unstable situation. The warmer air would rise, causing great atmospheric turbulence, 
and evening out the pressure distribution. It would also tend to dissipate the heat. (It is this very 
reason that accounts for the turbulent and stormy atmospheric conditions on Earth, Mars, Jupiter, 
and Neptune)

There are only two reasons why a canyon floor on Mars might plausibly be particularly warm. One 
is that it might absorb and trap a lot of solar energy. For a deep canyon this seems unlikely. The 
other is the possibility of geothermal energy (Marsothermal energy?). But the indications are that 
although at one time there was a lot of volcanic activity that helped shape the surface of Mars, it is 
presently quite geothermally incative.

Here is the bottom line: Mars as a whole has an atmosphere, though not much of one. In this 
sense the floor of the Valles Marineris would certainly have an atmosphere, and probably a very 
slightly thicker and warmer one. But it is most unlikely, almost impossible, that its pressure or 
composition or temperature could be interestingly different from that of the rest of the planet. If it 
were warmer, it could not be more than a little thicker; it could be quite a bit thicker, but only if it 
were even colder in the canyon than on the rest of the surface.

The Section on the atmosphere of Mars in R.P. Wayne, Chemistry of Atmospheres, 3rd Edition, is 
reasonably accessible, and well worth checking out. I will not give you an exact page reference, 
because I only have the 2nd (1991) Edition on my bookshelf, and that is now badly out-of-date.