On the energetics of maximum‐entropy temperature profiles

The problem of determining a maximum‐entropy state in an atmospheric column has different solutions depending on the constraints imposed. As was first shown by J. W. Gibbs, maximization of entropy in a thermally‐isolated, i.e. energy‐conserving, atmosphere yields an isothermal temperature distribution. It has since been argued that conservation of the mass‐weighted integral of potential temperature, or of the column potential enthalpy, is more appropriate for certain dynamical processes. The solution of the maximization problem satisfying this second constraint is quite different: an adiabatic temperature profile. Here, the comparative energetics of entropy maximization is examined in more detail. The main new result is that some exact statements may be formulated as inequalities. It can be rigorously shown, for example, that maximization of entropy under the conservation of potential enthalpy in an initially stable stratification always requires additional energy, and so cannot be accomplished in an isolated layer. Conversely, a temperature profile satisfying the second condition always possesses less entropy than an isothermal state with the same total energy, and so cannot be in thermodynamic equilibrium. Starting with these fundamental results, this paper discusses basic properties, and specific examples, of dynamical processes transporting heat vertically and generating entropy under various constraints, and demonstrates the importance of their physically‐consistent parametrization in atmospheric models. Copyright © 2008 Royal Meteorological Society