Energy Decompositions for Moist Boussinesq and Anelastic Equations with Phase Changes

Abstract

Abstract To define a conserved energy for an atmosphere with phase changes of water (such as vapor and liquid), motivation in the past has come from generalizations of dry energies—in particular, from gravitational potential energy ρgz. Here a new definition of moist energy is introduced, and it generalizes another form of dry potential energy, proportional to θ2, which is valuable since it is manifestly quadratic and positive definite. The moist potential energy here is piecewise quadratic and can be decomposed into three parts, proportional to bu2Hu, bs2Hs, and M2Hu, which represent, respectively, buoyant energies and a moist latent energy that is released upon a change of phase. The Heaviside functions Hu and Hs indicate the unsaturated and saturated phases, respectively. The M2 energy is also associated with an additional eigenmode that arises for a moist atmosphere but not a dry atmosphere. Both the Boussinesq and anelastic equations are examined, and similar energy decompositions are shown in both cases, although the anelastic energy is not quadratic. Extensions that include cloud microphysics are also discussed, such as the Kessler warm-rain scheme. As an application, empirical orthogonal function (EOF) analysis is considered, using a piecewise quadratic moist energy as a weighted energy in contrast to the standard L2 energy. By incorporating information about phase changes into the energy, the leading EOF modes become fundamentally different and capture the variability of the cloud layer rather than the dry subcloud layer.