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Revisiting Entrainment Relationships for Shear‐Free and Sheared

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Revisiting Entrainment Relationships for Shear‐Free and Sheared Received: 16 November 2017 Revised: 4 April 2018 Accepted: 9 May 2018 Published on: 18 October 2018 DOI: 10.1002/qj.3330 RESEARCH ARTICLE Revisiting entrainment relationships for shear-free and sheared convective boundary layers through large-eddy simulations Cheng Liu1,2 Evgeni Fedorovich2 Jianping Huang1 1Yale-NUIST Center on Atmospheric Environment, International Joint Laboratory on Entrainment is critical to the development of the atmospheric convective bound- Climate and Environment Change (ILCEC), ary layers (CBLs) and for the exchange of energy and substances between the Nanjing University of Information Science and boundary layer and free atmosphere. In this study, entrainment relationships are sys- Technology, Nanjing, China 2School of Meteorology, University of Oklahoma, tematically evaluated from large-eddy simulations (LESs) of shear-free and sheared Norman, Oklahoma CBLs under a broad variety of atmospheric conditions. A total of 36 LES runs Correspondence are conducted for varying free-atmosphere stratifications, surface heat fluxes, and Evgeni Fedorovich, 120 David L. Boren Blvd, height-constant geostrophic winds. It has been found that entrainment flux ratios NWC/SoM, University of Oklahoma, Norman, OK 73072, USA. have a common value of 0.2 regardless of the stratification of free atmosphere and the Email: [email protected] surface heat-flux value when analysed within the zero-order-model (ZOM) frame- Funding information work for the shear-free CBLs, whereas the actual (LES) entrainment ratios are not National Natural Science Foundation of China, only stratification-dependent, but also surface heat flux-dependent. For the sheared 41575009, Grant 41575009. Postgraduate Research and Practice Innovation Program of Jiangsu CBLs, the entrainment-flux ratios are substantially increased when the geostrophic Province, KYLX16_0945, Grant KYLX16_0945. winds are strong while they have a slight reduction under weak wind conditions. Priority Academic Program Development of In the free-encroachment regime, the dimensionless entrainment rates have been Jiangsu Higher Education Institutions, PAPD. found to be approximately constant and surface heat-flux independent, with the Ministry of Education of China, PCSIRT. Priority Academic Program Development of Jiangsu value in the range from 0.4 to 0.7 depending on the method used to evaluate the Higher Education Institutions (PAPD). National CBL depth. Key Research and Development Program of China, 2017YFC0210102. KEYWORDS convective boundary layer, entrainment ratio, entrainment relationships, free encroachment, thermal stratification, wind shear 1 INTRODUCTION the shear-generated turbulence. Due to this mixing, the CBL is often called the convectively mixed layer. The clear atmospheric convective boundary layer (CBL), Most commonly, the CBL develops through the process driven primarily by the heat (buoyancy) transfer from the of convective entrainment – a continual turbulent exchange warm underlying surface and secondarily affected by wind of air and transported atmospheric substances between the shears, is commonly observed in the atmosphere during CBL and the quiescent free-atmospheric region aloft. Since fair weather daytime conditions (Stull, 1998; Holtslag and the free atmosphere is typically stably stratified and has Duynkerke, 1998; Fedorovich et al., 2004). In the presence its (virtual) potential temperature increasing with height, it of wind shear, the layer is referred to as sheared CBL, and is represents the heat source for the CBL, and the warmer, called shear-free (or shearless) CBL when the shear effect is more buoyant air from the troposphere is transported down- absent or negligible in comparison with the buoyancy contri- ward into the mixed layer as a result of entrainment. The bution. The CBL is typically much more turbulent than the entrainment layer (zone) thus acts as an interface between main portion of the troposphere within which it develops. The the mixed core of the CBL and the free atmosphere. Because CBL air is vertically well mixed due to the vigorous turbulent the (virtual) potential temperature grows so sharply through- motions resulting from the buoyant turbulence production and out the entrainment layer that the corresponding absolute Q J R Meteorol Soc. 2018;144:2182–2195. wileyonlinelibrary.com/journal/qj © 2018 Royal Meteorological Society 2182 LIU ET AL. 2183 different degrees of complexity. The first of such models, the zero-order jump bulk model (ZOM) was originally proposed by Lilly (1968) for the shear-free CBL entrainment. In the ZOM, the entrainment zone is considered, in a horizontally average sense, as a zero-order discontinuity interface which is propagating upwards in time. Across this interface, the mete- orological variables, averaged over horizontal planes, change in a jump-like manner. In the ZOM, the entrainment heat flux is given by Qi =−Δ(dzi/dt), where Δ is the (virtual) potential temperature jump across the zero-order interface at zi (Zilitinkevich, 1991); see Figure 1. The ZOM approach was later generalized also for the sheared CBL (e.g. Fedorovich, FIGURE 1 Sketch of (a) potential temperature, (b) kinematic heat (potential temperature) flux, and (c) x-component of wind in the horizontally 1995; Conzemius and Fedorovich, 2006b). Details of entrain- homogeneous CBL. Thin black lines indicate actual (LES) profiles, and ment parametrizations developed within the ZOM framework black bold lines present their ZOM counterparts. Grey horizontal lines are presented in the next sections of our article. According to indicate the lower, zil, and upper, ziu, limits of the entrainment zone methodology adopted in our study, the smoothed profiles of potential temperature and wind velocity, which are approxi- mated using the ZOM assumptions, are retrieved from obser- air temperature increment across the layer is often positive, vations or numerical simulations by horizontal averaging in the entrainment layer is called also the capping inversion a fixed coordinate system. As discussed in Lilly (2002), the layer or simply capping inversion. Convective entrainment is actual interface between the mixed-layer and free-atmosphere a key process that determines the vertical redistribution of air is rather convoluted, so if a coordinate transformation is heat, momentum and scalars in the lower atmosphere dur- applied by normalizing the vertical coordinate by the local ing daytime. The entrainment happens on scales of motion interface height, with horizontal averaging being performed that are typically not resolved by meso- and larger-scale in the normalized coordinate system, the averaged profiles atmospheric models, so the entrainment effects in such mod- would retain the sharpness of the interface. els need to be parametrized. Usually such parametrizations Following the ZOM, higher-order bulk models of con- are constructed in terms of the so-called integral parameters vective entrainment have been developed that allowed for a of entrainment (Fedorovich et al., 2004). Among the basic finite inversion-layer thickness. Performance of the first-order entrainment parameters considered are the capping-inversion model of entrainment with linear potential temperature pro- (entrainment-zone) thickness, the entrainment rate dzi/dt, file across the capping inversion, originally proposed by Betts that is the rate of change in time of inversion height zi in (1974), was studied in Van Zanten et al. (1999), Conzemius the absence of large-scale subsidence (here zi is commonly and Fedorovich (2006b; 2007), Pino et al. (2006), Kim et al. defined as the level of the heat-flux minimum within the (2006), and Sun and Wang (2008) using large-eddy simu- entrainment zone), and the entrainment (flux) ratios for var- lation (LES) data. A higher-order general-structure model ious entrained substances. For heat, the entrainment ratio of entrainment originally outlined in Deardorff (1979) was is given by Ai =−Qi/Qs,whereQi is the kinematic flux further advanced in Fedorovich and Mironov (1995), who of heat at zi (the entrainment heat flux; note that this flux employed a self-similar representation of the potential tem- is directed downwards, so it is negative), and Qs ≥ 0isthe perature profile within the entrainment zone. The model was surface heat flux (see schematic in Figure 1). In the entrain- tested against LES data in Fedorovich et al. (2004). Both ment parametrizations developed for atmospheric models, the first-order and general-structure approaches found only lim- integral parameters are usually sought as functions of the ited applicability in the atmospheric modelling due to their free-atmosphere stratification, typically assumed to be linear relative complexity, where ZOM-based parametrizations of and continuous, and surface fluxes of heat, momentum and entrainment were used rather extensively, primarily because scalars. Historically, many parametrizations of this kind have of their simplicity and practicality. been proposed and discussed in the literature (e.g. Lilly, 1968; Relationships between the integral parameters of the Betts, 1973; 1974; Carson, 1973; Stull, 1973; Tennekes, 1973; shear-free entrainment resulting from ZOM were thoroughly Zeman and Tennekes, 1977; Deardorff, 1979; Driedonks, evaluated through LES in Fedorovich et al. (2004). One of 1982; Driedonks and Tennekes, 1984; Zilitinkevich, 1991; the goals of that study, focused on the convective entrain- Fedorovich, 1995; 1998; Fedorovich and Mironov, 1995; ment in a linearly stratified atmosphere,
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