Just off the top of my head:
You could make the emitter emit both heat and density, then with a positive gravity value (the default) You could set the density buoyancy to a negative value and temperature buoyancy to a (probably considerably higher) positive value, and play with temperature dissipation to make the content loose heat over time.
If I'm right, this should the content rise initially, then fall as the temperature lowers.
To make it stay above the ground, if it isn't enough to have the boundary stop it from falling, you could either try a collission plane a bit above the lower boundary, or you could have a plane emit heat only at the bottom.
I haven't tried any of this though, so I don't know if it works.