Spectra of atmospheric water in precipitating quasi-geostrophic turbulence

Atmospheric water has a complex behaviour partly due to the influence of precipitation. Consequently, it is challenging to explain properties of water such as the scale-dependence of its variance, for which a range of spectral exponents has been identified in observational data. Here, a precipitating quasi-geostrophic (PQG) model is explored as a possible prototype for contributing to understanding of water spectra, in an idealised setting. Geostrophic turbulence is examined in numerical simulations, where precipitation is included to explore its effect on the water spectrum, but where phase changes are neglected to allow corresponding theoretical analysis. The water spectral exponent is seen to range from approximately -1.4 to approximately -5 depending on the rainfall speed parameter, , which indicates a significant influence of precipitation on the water spectrum. The limiting values of this range are explained through asymptotic analyses for large and small values of . To obtain this theoretical understanding of the model, a key observation is that water can be written as a linear combination of two other tracers (equivalent potential temperature and a moist variable M), which themselves have theoretically tractable spectra. These two other tracers are linked to distinct modes of the PQG equations-the vortical mode and a moist mode - and the analysis here highlights the usefulness of wave or mode decompositions for understanding water in a saturated domain.