Justifying the WKB approximation in pure katabatic flows

Abstract

Pure katabatic flow is studied with a Prandtl-type model allowing eddy diffusivity/conductivity to varywith height. Recently we obtained an asymptotic solution to the katabatic flow assuming the validity ofthe WKB method, which solves the fourth-order governing equation coupling the momentum and heattransfer. The WKB approximation requires that eddy diffusivity may vary only gradually compared tothe calculated quantities, i.e., potential temperature and wind speed. This means that the scale heightfor 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 thescale 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 maximumand the level of the relevant maximum turbulent kinetic energy are above the strongest part of the nearsurfaceinversion and the low-level jet which is required for the WKB validity. Thus, the numericalmodeling perspective lends credibility to the simple WKB modeling. This justification is importantbefore other data sets are analyzed and a parameterization scheme written.