Boundary-Layer Turbulence and Clouds in the Atmosphere: Prospects for Closures

Climate models are based on the numerical solutions of partial differential equations on a finite grid. Computational constraints restrict the horizontal resolution in current global atmosphere models to order 10-100 km. Processes that act on smaller scales cannot be explicitly resolved and have to be parameterized in terms of the resolved large-scale fields. Most turbulent processes in the atmospheric boundary layer, which spans the lowest 1-2 km of the atmosphere, fall into this category and are the focus of this work. The accuracy of parameterization schemes is crucial, as the parameterized processes feed back onto the large-scale motion. For instance, the inaccuracies and differences among climate models in the representation of convection and the low clouds that often top boundary layers translate into uncertainties in cloud radiative effects and their response to climate changes. The cloud radiative feedback contributes to the global energy budget and represents a significant feedback in a changing climate. As a result, the parameterization of low clouds has emerged as a key source of uncertainty in the prediction of climate sensitivity. The work presented in this PhD thesis investigates two different approaches for the parameterization of subgrid scale processes in the boundary layer. Part I investigates the potential and limitations of obtaining parameterizations by truncating the hierarchy of statistical moments in turbulent flows at the certain order. Here, a truncation at second order is considered, which takes into account mean and (co-)variance values of at the second order. In such a second-order closure moments of third order and higher are neglected, such that nonlinear interactions among turbulent eddies are suppressed but nonlinear eddy-mean flow interactions are retained. Part II investigates `assumed-PDF' schemes using mixtures of Gaussian probability distributions. This method assumes that the subgrid scale fields are distributed according to a certain family of probability distributions of which the parameters are determined from the large-scale fields. The two methods are related, since for Gaussian random variables like for second-order closures all third- and higher-order cumulants vanish; hence, a flow that obeys Gaussian statistics can be captured by a second-order closure. In Part I, the dynamics represented by second-order turbulence closures are studied, with focus on two main questions: How well can such a closure, using the complete set of first- and second-order moments, approximate the dynamics in atmospheric boundary layers? Does the inclusion of spatially non-local correlations ameliorate some of the deficiencies of traditional second-order closures, despite the omission of all third- and higher-order moments? In conventional second-order schemes, only a subset of the second-order moments, such as the turbulent kinetic energy (TKE), is treated prognostically. All other moments, as for instance the variance or the turbulent flux of heat and moisture, are treated diagnostically. Treating all second-order moments as prognostic variables allows one to systematically single out errors introduced by omitting certain second-order terms (e.g., non-local correlation terms) from errors introduced by omitting third- and higher-order moments. The dynamics that are captured by the full set of first- and second-order cumulant equations, including local and non-local correlations, can be studied in terms of the quasi-linearized equations of motion. Both sets of equations represent the same dynamics. To study the quasi-linear (QL) dynamics, a large-eddy simulation (LES) code is modified to describe the quasi-linear transport of scalars and momentum. The patterns and statistics of flow fields resulting from these QL LES and the original, non-linear LES are compared against each other. In a dry convective boundary layer, the first-order statistics, such as the mean temperature profile and the growth of the boundary layer, are well represented by the QL LES. However, the QL simulation fails to capture the negative vertical heat flux at the top of the boundary layer and underestimates the vertical transport of turbulent kinetic energy. This underlines the non-linear nature of these processes. Quasi-linear simulations of moist boundary layer convection exhibit excessive growth of higher-order moments, which can be traced back to a violation of local conservation for scalar variables like entropy and total water. Similar difficulties are known for third-order closures, while they seem to have received less attention as a limitation of second-order closures. Part II focuses on the subgrid scale statistics of thermodynamic variables in atmospheric boundary layers and clouds, which are essential for the representation of the cloud cover. A correct estimation of the cloud fraction and the liquid water path is crucial for an accurate computation of cloud radiative feedbacks and precipitation. Building on previous work, this study tries to answer the following question: What degree of complexity is necessary for a probability distribution to represent cloud cover statistics? Can the subgrid scale distribution in the environment of convective updrafts be approximated by a normal distribution, as is required by many parameterization schemes? The present study investigates Gaussian mixture models as assumed probability distribution functions (PDFs) of subgrid scale variability of liquid potential temperature and total water. Data from a suite of LES are used as training and reference data for fitting and testing the PDF models. The results support previous findings that a single Gaussian PDF provides a good approximation in stratiform cloud layers, while up to three Gaussian components are necessary for a good estimate in shallow and deep cumulus convection. The same analysis is applied after decomposing the LES data into convective updrafts and their environment. Conditional statistics indicate that the thermodynamic variables in the environment do not follow a normal distribution due to the presence of passive clouds that remain near the top of convective clouds. In the shallow convection case considered in this study, this can lead to an error of up to 30\% in total cloud fraction and liquid water. With regard to scale-aware cloud closure schemes, the PDF models are tested for LES sampling domains of varying horizontal and vertical extent. The models are robust with regard to the horizontal extent of the sampling domain, as long as the sampling domain covers a representative part of the cloud cover. In contrast, the results show a strong sensitivity to the vertical coarse-graining of the sampling domain, which points to the importance of vertical resolution for parameterization schemes in large-scale models