Justifying the WKB approximation in pure katabatic flows

Abstract

Pure katabatic flow is studied with a Prandtl-type model allowing eddy diffusivity/conductivity to vary with height. Recently we obtained an asymptotic solution to the katabatic flow assuming the validity of the WKB method, which solves the fourth-order governing equation coupling the momentum and heat transfer. The WKB approximation requires that eddy diffusivity may vary only gradually compared to the calculated quantities, i.e., potential temperature and wind speed. This means that the scale height for eddy diffusivity must be higher than that for the calculated potential temperature and wind speed. The ratio between the maximum versus the mean eddy diffusivity should be less than that for the scale heights of the diffusivity versus the calculated quantities (temperature and wind). Here we justify (a posteriori) the WKB method independently based on two arguments: (i) a scaling argument and (ii) a philosophy behind a higher-order closure turbulence modeling. Both the eddy diffusivity maximum and the level of the relevant maximum turbulent kinetic energy are above the strongest part of the near- surface inversion and the low-level jet which is required for the WKB validity. Thus, the numerical modeling perspective lends credibility to the simple WKB modeling. This justification is important before other data sets are analyzed and a parameterization scheme written.