Symmetry Invariant Solutions in Atmospheric Boundary Layers

Abstract

Abstract The symmetries of the governing equations of atmospheric flows constrain the solutions. The present study applies those symmetries identified from the governing equations to the atmospheric boundary layers under relatively weak stratifications (stable and unstable). More specifically, the invariant solutions are analyzed, which conserve their forms under possible symmetry transformations of a governing equation system. The key question is whether those invariant solutions can rederive the known vertical profiles of both vertical fluxes and the means for the horizontal wind and the potential temperature. The mean profiles for the wind and the potential temperature in the surface layer predicted from the Monin–Obukhov theory can be recovered as invariant solutions. However, the consistent vertical fluxes both for the momentum and heat no longer remain constant with height, as assumed in the Monin–Obukhov theory, but linearly and parabolically change with height over the dynamic sublayer and the above, respectively, in stable conditions. The present study suggests that a deviation from the constancy, though observationally known to be weak, is a crucial part of the surface-layer dynamics to maintain its symmetry consistency. Significance Statement The atmospheric flows are governed by a differential equation system, which is often difficult to solve in any satisfactory manner, either analytically or numerically. However, without solving them explicitly, many insights can be obtained by examining the “symmetries” of the governing equations. The study suggests that basic vertical profiles of the mean state of the atmospheric boundary layer is more strongly constrained by the symmetry consistency than suggested by standard similarity theories.