A column model of moist convection: some exact equilibrium solutions



A radiative-convective model for a moist single-column atmosphere is presented. A test problem is considered in which exact, explicit analytical solutions are found by using a form for the saturation specific humidity that closely models that for Earth's atmosphere. Simple equations for the conservation of mass, energy, and moisture are solved with a new type of weak solution, where the fluid satisfies Lagrangian versions of these equations. A numerical method using a Lagrangian-based hard convective adjustment step calculates solutions which converge to the analytical solutions as the vertical mesh size decreases, and is stable in time. When the thermodynamic paths followed by an ascending and descending parcel do not overlap in a potential temperature–pressure diagram, steady equilibrium solutions are found; otherwise, non-equilibrium behaviour of the numerical solution occurs. This work suggests that it is useful to interpret column models in a Lagrangian sense as weak solutions of systems of partial differential equations. Copyright © 2011 Royal Meteorological Society