A rigorous treatment of moist convection in a single column
  We study a single column model of moist convection in the atmosphere. We state the conditions for it to represent a stable steady state. We then evolve the column by subjecting it to an upward displacement which can release instability, leading to a time dependent sequence of stable steady states. We propose a definition of measure valued solution to describe the time dependence and prove its existence. 
Bin Cheng, Jingrui Cheng, Michael Cullen, John Norbury and Matthew   Turner
math.AP