Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

A Package of Momentum and Heat Transfer Coefficients for the Stable Surface Layer Extended by New Coefficients over Sea Ice

ingredients of numerical weather prediction and climatemodels. They are needed for the calculation of turbulent fluxes in the surface layer and often rely on the Monin–Obukhov similarity theory requiring universal stability functions. The problem of a derivation of transfer coefficients based on different stability functions has been considered by many researchers over the years but it remains to this day. In this work, dedicated to the memory of S.S. Zilitinkevich, we also address this task, and obtain transfer coefficients from three pairs of theoretically derived stability functions suggested by Zilitinkevich and co-authors for stable conditios. Additionally, we construct non-iterative parametrizations of these transfer coefficients based on earlier work. Results are compared with state-of-the-art coefficients for land, ocean, and sea ice. The combined parametrizations form a package in a universal framework relying on a semi-analytical solution of the Monin-Obukhov similarity theory equations. A comparison with data of the Surface Heat Budget of the Arctic Ocean campaign (SHEBA) over sea ice reveals large differences between the coefficients for land conditions and the measurements over sea ice. However, two schemes of Zilitinkevich and co-authors show, after slight modification, good agreement with SHEBA although they had not been especially developed for sea ice. One pair of the modified transfer coefficients is superior and is compatible to earlier SHEBA-based parametrizations. Finally, an algorithm for practical use of all transfer coefficients in climate models is given

On a Solution of the Closure Problem for Dry Convective Boundary Layer Turbulence and Beyond

We consider the closure problem of representing the higher-order moments (HOMs) in terms of lower- order moments, a central feature in turbulence modeling based on the Reynolds-averaged Navier–Stokes (RANS) approach. Our focus is on models suited for the description of asymmetric, nonlocal, and semiorganized turbulence in the dry atmospheric convective boundary layer (CBL). We establish a multivariate probability density function (PDF) describ- ing populations of plumes that are embedded in a sea of weaker randomly spaced eddies, and apply an assumed delta-PDF approximation. The main content of this approach consists of capturing the bulk properties of the PDF. We solve the clo- sure problem analytically for all relevant HOMs involving velocity components and temperature and establish a hierarchy of new non-Gaussian turbulence closure models of different content and complexity ranging from analytical to semianalyti- cal. All HOMs in the hierarchy have a universal and simple functional form. They refine the widely used Millionshchikov closure hypothesis and generalize the famous quadratic skewness–kurtosis relationship to higher order. We examine the performance of the new closures by comparison with measurement, LES, and DNS data and derive empirical constants for semianalytical models, which are best for practical applications. We show that the new models have a good skill in predict- ing the HOMs for atmospheric CBL. Our closures can be implemented in second-, third-, and fourth-order RANS turbu- lence closure models of bi-, tri-, and four-variate levels of complexity. Finally, several possible generalizations of our approach are discussed

On a solution of the closure problem for atmospheric convective boundary-layer turbulence

К решению проблемы замыкания для турбулентного конвективного атмосферного пограничного слоя. Гряник В. М. и Хартманн Й. Институт физики атмосферы им. А.М. Обухова РАН, Москва, Россия Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany Модели замыкания играют центральную роль в теории турбулентности и её приложениях. Они широко используются в моделях климата, общей циркуляции, пограничного и приземного слоев атмосферы и океана. Наиболее популярны, из-за своей простоты, одно-точечные и двух-точечные (спектральные) замыкания. Одна из первых моделей была предложена Обуховым в его знаменитой сейчас работе, где впервые был получен закон пяти третей для спектра однородной и изотропной турбулентности. Турбулентность в атмосфере и океане может рассматриваться как однородная и изотропная только на малых маштабах в так называемом инерционном интервале. В докладе проблема замыкания рассматривается для сухого конвективного пограничного слоя накрытого инверсией. Турбулентное перемешивание в слое связано с вихрями малых масштабов вблизи поверхности и в зоне вовлечения, а также с крупномасштабными когерентными структурами - плюмами - пронизывающими всю толщу погранслоя. Традиционные гипотезы замыкания плохо работают в этих условиях: 1) потоки тепла, кинетической и потенциальной энергий и моментов старшего порядка не описываются обычными законами диффузии, 2) гипотеза Миллионщикова о гауссовости моментов четвертого порядка противоречит данным наблюдений и численного моделирования, 3) скорость диссипации энергии и потока тепла (и старших моментов) не описывается гипотезой Ротта-Монина об однородной релаксации. Главная причина в том, что предположение о слабом отклонении функции распределения флуктуаций скорости и температуры от гауссовой не выполняется. Чтобы прояснить ситуацию, мы рассматриваем упрощённую модель конвективной турбулентности, приняв гипотезу что не-гауссовость функции распределения может быть описана в так называемом Delta-PDF приближении, которое предполагает аппроксимацию реальной функции распределения суммой Делта-функций. Оно наилучшим образом приспособлено для описания влияния ансамблей когерентных структур. Приближение не ново и использовалось ранее в работах [1]. Главное отличие новой модели состоит во включении в функцию распределения вклада в от популяций вихрей малых маштабов, помимо вклада от когерентных структур. Второе отличие в том, что ранее в [1] анализ проводился асимптотическими методами, новая же модель решена точно. Все моменты высших порядков выражены через неприводимые, среди которых содержится только один момент четвертого порядка - коэффициент корреляции компонент скорости и температуры. Получены явные аналитические параметризации для не-диффузионных потоков импульса, тепла, кинетической и потенчиальной энергии. В частном случае моментов четвертого порядка новые замыкания обобщают гипотезу Миллионщикова на случай сильно ассиметричной турбулентности. Тестирование замыканий на основе данных самолётных измерений (ARTIST), LES и DNS показало хорошее согласие между теорией и данными. В предельных случаях новые замыкания совпадают с полученными в [1], и уже проверенными в [2-4]. [1] Gryanik V.M. and J. Hartmann, 2002: J. Atmos. Sci., 59, 2729; Gryanik, V.M., J. Hartmann, S. Raasch and M. Schröter, 2005: J. Atmos. Sci., 62, 2632. [2] Kupka F. and F. Robinson 2007: Mon. Not. Roy. Astron. Soc. 374, 305, 79. [3] Lenschow, D.H., M. Lothon, S.D. Mayor, P.P. Sullivan and G. Canut, 2011: Boundary-Layer Meteorol. 143, 107. [4] Waggy S., A. Hsieh and S. Biringen, 2016: Geophys. Astrophys. Fluid Dyn. doi: 10.1080/03091929.2016.1196202

A package of momentum and heat transfer coefficientsfor the stable atmospheric surface layer

The polar atmospheric surface layer is often stably stratified, which strongly influences turbulent transport processes between the atmosphere and sea ice/ocean. Transport is usually parametrized applying Monin Obukhov Similarity Theory (MOST) which delivers transfer coefficients as a function of stability parameters (see below). In a series of papers (Gryanik and Lüpkes, 2018; Gryanik et al., 2020,2021; Gryanik and Lüpkes, 2022) it has been shown that differences between existing parametrizations are large, especially for strong stability. One reason is that they are based on different data sets, for which the origin of differences is still unclear. In this situation Gryanik et al. (2021) as well as Gryanik and Lüpkes (2022) proposed a numerically efficient method, which can be used for most of the existing data sets and their specific stability dependences. A package of parametrization resulted that is suitable for its application in weather prediction and climate models. Especially, calculation of fluxes over sea ice were improved. Combined with latest parametrizations of surface roughness it has a large impact on large scale fields as shown recently by Schneider et al. (2021) who applied some members of the package

Динамика бароклинных вихрей с нулевой суммарной интенсивностью (хетонов)

Приводится обзор работ, посвященных изучению свойств вихревых движений в устойчиво стратифицированной быстро вращающейся жидкости и описываемых уравнением эволюции потенциального вихря в квазигеострофическом приближении. Основное внимание уделено вихрям с нулевой суммарной циркуляцией — так называемым хетонам. Рассматриваются задачи самодвижения дискретных хетонов, устойчивости единичного распределенного хетона, взаимодействия двух хетонов конечных размеров. Приводятся также новые решения задач о трех и более дискретных вихрях хетонной структуры. Даются примеры возникновения хаотических режимов. Обсуждаются область применимости хетонной теории и перспективыее возможных приложений, в частности, — для анализа динамической стадии развития глубокой конвекции в океане

Parametrization of Turbulent Fluxes over Leads in Sea Ice in a Non-Eddy-Resolving Small-Scale Atmosphere Model

Leads (open-water channels in sea ice) play an important role for surface-atmosphere interactions in the polar regions. Due to large temperature differences between the surface of leads and the near-surface atmosphere, strong turbulent convective plumes are generated with a large impact on the atmospheric boundary layer (ABL). Here, we focus on the effect of lead width on those processes, by means of numerical modeling and turbulence parametrization. We use a microscale atmosphere model in a 2D version resolving the entire convective plume with grid sizes in the range of L/5 where L is the lead width. For the sub-grid scale turbulence, we developed a modified version of an already existing nonlocal parametrization of the lead-generated sensible heat flux including L as parameter. All our simulations represent measured springtime conditions with a neutrally stratified ABL capped by a strong temperature inversion at 300 m height, where the initial temperature difference between the lead surface and the near-surface atmosphere amounts to 20 K. We found that our simulation results obtained with the new approach agree very well with time-averaged results of a large eddy simulation (LES) model for variable lead widths with L ≥ 1 km and different upstream wind speeds. This is a considerable improvement since results obtained with the previous nonlocal approach clearly disagree with the LES results for leads wider than 2 km. In conclusion, considering L as parameter in a nonlocal turbulence parametrization seems to be necessary to study the effect of leads on the polar ABL in non-eddy-resolving small-scale atmosphere models

Parameterization of drag coefficients over polar sea ice for climate models

A parameterization of drag coefficients has been developed in recent years that accounts for the impact of edges at ice floes, leads, and melt ponds on momentum transport. Melt ponds are a common feature in the inner Arctic during summer while drifting ice floes and their edges influence the surface roughness especially in the marginal sea ice zones during all seasons. Governing parameters in the parameterization that can be easily applied to climate models are the sea ice concentration and aspect ratio h/D where h is the ice freeboard and D is the characteristic length of floes and ponds/leads. When these parameters are not available from a sea ice model, the aspect ratios can also be parameterized as a function of the sea ice concentration so that the new schemes can also be used in stand-alone atmospheric models using observed sea ice concentration. The parameterization is evaluated for idealized meteorological forcing and prescribed sea ice and melt pond concentration in the Siberian Arctic and in parts of the Central Arctic. The required sea ice data are available from remote sensing. The distributions of drag coefficients obtained from traditional parameterizations and from the new one show large differences in this test scenario especially in the region south of 80°N