S. S. Zilitinkevich

On the determination of the height of the Ekman boundary layer

The heighthτ of the Ekman turbulent boundary layer determined by the momentum flux profile is estimated with the aid of considerations of similarity and an analysis of the dynamic equations. Asymptotic formulae have been obtained showing that, with increasing instability,hτ increases as ¦μ¦1/2 (where μ is the non-dimensional stratification parameter); with increasing stability, on the other hand,h decreases as μ−1/2. For comparison, a simple estimate of the boundary-layer heighthu determined by the velocity profile is given. As is shown, in unstable stratification,hu behaves asymptotically as ¦μ¦−1, i.e., in a manner entirely different from that ofhτ.

Turbulence energetics in stably stratified geophysical flows: Strong and weak mixing regimes

Traditionally, turbulence energetics is characterised by turbulent kinetic energy (TKE) and modelled using solely the TKE budget equation. In stable stratification, TKE is generated by the velocity shear and expended through viscous dissipation and work against buoyancy forces. The effect of stratification is characterised by the ratio of the buoyancy gradient to squared shear, called the Richardson number, Ri. It is widely believed that at Ri exceeding a critical value, Ric, local shear cannot maintain turbulence, and the flow becomes laminar. We revise this concept by extending the energy analysis to turbulent potential and total energies (TPE, and TTE = TKE + TPE), consider their budget equations, and conclude that TTE is a conservative parameter maintained by shear in any stratification. Hence there is no ‘energetics Ric’, in contrast to the hydrodynamic-instability threshold, Ric−instability, whose typical values vary from 0.25 to 1. We demonstrate that this interval, 0.25 < Ri < 1, separates two different turbulent regimes: strong mixing and weak mixing rather than the turbulent and the laminar regimes, as the classical concept states. This explains persistent occurrence of turbulence in the free atmosphere and deep ocean at Ri ≫ 1, clarifies the principal difference between turbulent boundary layers and free flows, and provides the basis for improving operational turbulence closure models. Copyright

A multi-limit formulation for the equilibrium depth of a stably stratified boundary layer

Currently no expression for the equilibrium depth of the turbulent stably-stratified boundary layer is available that accounts for the combined effects of rotation, surface buoyancy flux and static stability in the free flow. Various expressions proposed to date are reviewed in the light of what is meant by the stable boundary layer. Two major definitions are thoroughly discussed. The first emphasises turbulence and specifies the boundary layer as a continuously and vigorously turbulent layer adjacent to the surface. The second specifies the boundary layer in terms of the mean velocity profile, e.g. by the proximity of the actual velocity to the geostrophic velocity. It is shown that the expressions based on the second definition are relevant to the Ekman layer and portray the depth of the turbulence in the intermediate regimes, when the effects of static stability and rotation essentially interfere. Limiting asymptotic regimes dominated by either stratification or rotation are examined using the energy considerations. As a result, a simple equation for the depth of the equilibrium stable boundary layer is developed. It is valid throughout the range of stability conditions and remains in force in the limits of a perfectly neutral layer subjected to rotation and a rotation-free boundary layer dominated by surface buoyancy flux or stable density stratification at its outer edge. Dimensionless coefficients are estimated using data from observations and large-eddy simulations. Well-known and widely used formulae proposed earlier by Zilitinkevich and by Pollard, Rhines and Thompson are shown to be characteristic of the above interference regimes, when the effects of rotation and static stability (due to either surface buoyancy flux, or stratification at the outer edge of the boundary layer) are roughly equally important.

Calculation Of The Height Of The Stable Boundary Layer In Practical Applications

Currently used and newly proposed calculation techniques for the heightof the stable boundary layer (SBL), including the bulk-Richardson-numbermethod, diagnostic equations for the equilibrium SBL height, and a relaxation-typeprognostic equation, are discussed from the point of view of their physical basis andrelevance to experimental data. Among diagnostic equations, the best fit to data exhibits an advanced Ekman-layer height model derived recently with due regard to the role of the free-flow stability. Its extension to non-steady regimes provides a prognostic equation recommended for use in practical applications.

Resistance Laws and Prediction Equations for the Depth of the Planetary Boundary Layer

Abstract A further discussion of the resistance laws for barotropic planetary boundary layers is presented. Concrete expressions are obtained for universal functions A(μ0), B(μ0) C(μ0)of Rossby number similarity theory for the Ekman boundary layer as well as of analogous functions for non-steady boundary layers. Prediction equations are proposed, which describe variations of the boundary layer depth for unstable and stable stratifications.

Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layer

In this paper we revise the similarity theory for the stably stratified atmospheric boundary layer (ABL), formulate analytical approximations for the wind velocity and potential temperature profiles over the entire ABL, validate them against large-eddy simulation and observational data, and develop an improved surface flux calculation technique for use in operational models.

Diagnostic and prognostic equations for the depth of the stably stratified Ekman boundary layer

Refined diagnostic and prognostic equations for the depth of the stably stratified barotropic Ekman boundary later (SBL) are derived employing a recently developed non-local formulation for the eddy viscosity. In well-studied cases of the thoroughly neutral SBL, the nocturnal atmospheric SBL and the oceanic SBL dominantly affected by the static stability in the thermocline, the proposed diagnostic equation reduces to the Rossby–Montgomery, Zilitinkevich and Pollard–Rhines–Thompson equations, respectively. In its general form it is applicable to a range of regimes including long-lived atmospheric SBLs affected by the near-surface buoyancy flux and the static stability in the free atmosphere. Both diagnostic and prognostic SBL depth equations are validated against recent data from atmospheric measurements. Copyright

Enhanced haze pollution by black carbon in megacities in China

Aerosol-planetary boundary layer (PBL) interactions have been found to enhance air pollution in megacities in China. We show that black carbon (BC) aerosols play the key role in modifying the PBL meteorology and hence enhancing the haze pollution. With model simulations and data analysis from various field observations in December 2013, we demonstrate that BC induces heating in the PBL, particularly in the upper PBL, and the resulting decreased surface heat flux substantially depresses the development of PBL and consequently enhances the occurrences of extreme haze pollution episodes. We define this process as the “dome effect” of BC and suggest an urgent need for reducing BC emissions as an efficient way to mitigate the extreme haze pollution in megacities of China.

A Total Turbulent Energy Closure Model for Neutrally and Stably Stratified Atmospheric Boundary Layers

This paper presents a turbulence closure for neutral and stratified atmospheric conditions. The closure is based on the concept of the total turbulent energy. The total turbulent energy is the sum of the turbulent kinetic energy and turbulent potential energy, which is proportional to the potential temperature variance. The closure uses recent observational findings to take into account the mean flow stability. These observations indicate that turbulent transfer of heat and momentum behaves differently under very stable stratification. Whereas the turbulent heat flux tends toward zero beyond a certain stability limit, the turbulent stress stays finite. The suggested scheme avoids the problem of self-correlation. The latter is an improvement over the widely used Monin–Obukhov-based closures. Numerous large-eddy simulations, including a wide range of neutral and stably stratified cases, are used to estimate likely values of two free constants. In a benchmark case the new turbulence closure performs indistinguishably from independent large-eddy simulations.

Third-Order Transport and Nonlocal Turbulence Closures for Convective Boundary Layers*

The turbulence closure problem for convective boundary layers is considered with the chief aim to advance the understanding and modeling of nonlocal transport due to large-scale semiorganized structures. The key role here is played by third-order moments (fluxes of fluxes). The problem is treated by the example of the vertical turbulent flux of potential temperature. An overview is given of various schemes ranging from comparatively simple countergradient-transport formulations to sophisticated turbulence closures based on budget equations for the second-order moments. As an alternative to conventional ‘‘turbulent diffusion parameterization’’ for the flux of flux of potential temperature, a ‘‘turbulent advection plus diffusion parameterization’’ is developed and diagnostically tested against data from a large eddy simulation. Employing this parameterization, the budget equation for the potential temperature flux provides a nonlocal turbulence closure formulation for the flux in question. The solution to this equation in terms of the Green function is nothing but an integral turbulence closure. In particular cases it reduces to closure schemes proposed earlier, for example, the Deardorff countergradient correction closure, the Wyngaard and Weil transport-asymmetry closure employing the second derivative of transported scalar, and the Berkowicz and Prahm integral closure for passive scalars. Moreover, the proposed Green-function solution provides a mathematically rigorous procedure for the Wyngaard decomposition of turbulence statistics into the bottom-up and top-down components. The Green-function decomposition exhibits nonlinear vertical profiles of the bottom-up and top-down components of the potential temperature flux in sharp contrast to universally adopted linear profiles. For modeling applications, the proposed closure should be equipped with recommendations as to how to specify the temperature and vertical velocity variances and the vertical velocity skewness.