Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: steady-state,homogeneous regimes

We propose a new turbulence closure model based on the budget equations for the key second moments: turbulent kinetic and potential energies: TKE and TPE (comprising the turbulent total energy: TTE = TKE + TPE) and vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature). Besides the concept of TTE, we take into account the non-gradient correction to the traditional buoyancy flux formulation. The proposed model permits the existence of turbulence at any gradient Richardson number, Ri. Instead of the critical value of Richardson number separating—as is usually assumed—the turbulent and the laminar regimes, the suggested model reveals a transitional interval, , which separates two regimes of essentially different nature but both turbulent: strong turbulence at ; and weak turbulence, capable of transporting momentum but much less efficient in transporting heat, at . Predictions from this model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulation and large-eddy simulation.     

Interaction of a negatively buoyant line plume with a density interface

Laboratory experiments were carried out to investigate the interaction between turbulent line buoyant plumes and sharp density interfaces, with the aim of using the results to interpret oceanic observations pertinent to crack openings in the polar ice-cap (leads). These openings take the form of long narrow channels, and are often modeled as line bouyant plumes. The plumes descend as in a homogenoous fluid, impinge on the density interface, and then spread horizontally as gravity currents. Depending on the Richardson number , where Δb is the buoyancy jump across the interface, l_D is the half-width of the plume before the impingement and q₀is the buoyancy flux per unit length of the source, different flow patterns were identified. When Ri < 0.5, the plumes penetrate deep into the bottom layer, deflect horizontally and then spread while showing little vertical rise. When 0.6 < Ri < 5, the penetration is significant, but the fluid bounces back after entraining heavy fluid from the lower layer and then spreads horizontally above the interface as a gravity current. Appreciable mixing between this current and the lower layer was detected when Ri <1. When Ri > 10, the penetration was small and a sharp-nosed gravity current emerged some time after the impact. Measurements were made on the penetration depth, the velocities of the gravity current and the subsurface flow towards the plume, the entrainment rate and other wave parameters. Possible implications of the results for oceanic cases are also discussed.     

The role of molecular diffusion in the deepening of the mixed layer

Turbulent mixing across heat-stratified density interfaces was studied in the laboratory using oscillating-grid generated turbulence. The aim was to study the transition between the entrainment regimes dominated by interfacial wave-breaking and molecular diffusion, and to study the characteristics of the latter. It was observed that, above a critical Richardson number Ri_c, which depends on the Peclet number Pe, the mixing due to wave breaking disappears and that Ri_c Pe⁻ⁿ, where the mean value of the exponent n is approximately . Above Ri_c, the entrainment is molecular-diffusion dominated and takes place through a sequence of events: the buoyancy gradient of the initially sharp density interface is weakened by molecular diffusion until the mixed-layer eddies can engulf a portion of the interfacial layer wherefore the interface sharpens again. Thus, the entrainment events are recurrent with a rate-controlling diffusion stage between them. An entrainment law of the form E Ri⁻²Pe⁻², where E is the entrainment coefficient and Ri is the Richardson number, is suggested for the diffusion-dominated entrainment regime.     

Stable Atmospheric Boundary Layer Over a Uniform Slope: Some Theoretical Concepts

Theoretical arguments are developed to describe the effects of a uniform slope on the development of the stably stratified atmospheric boundary layer (SBL). A maximum sustainable surface buoyancy flux exists for the SBL overlying a uniform, non-sloping surface. In this study it is shown that the SBL overlying a uniform shallow slope (with gradient of the order of 1:1000) also supports a maximum sustainable buoyancy flux, B _max, but that the value of B _max is influenced by the gradient of the slope, γ. It is demonstrated that in the limit γ → 0, results for the SBL over a horizontal surface are recovered.     

Experiments on steady buoyancy transfer through turbulent fluid layers separated by density interfaces

A laboratory experiment was performed to investigate mixing across a density interface which separates two turbulent fluid layers and coexists with a stabilizing buoyancy flux. It was found that the buoyancy flux (q₀) across the interface and through the turbulent layers (of depth D) becomes steady and constant in magnitude in the vertical direction, only when , where u is the horizontal r.m.s. velocity at the base of the mixed layers. The results suggest that mixing across the density interface is controlled by a dynamically important buoyancy gradient induced in the turbulent layers and that parameters such as the bulk Richardson number, , where Δb is the interfacial buoyancy jump, are of secondary importance. Measurements are used to infer the mixing mechanism at the interface, the mixing efficiency of stratified fluids and the entrainment law. Some geophysical applications of the results are also discussed.     

Intermittent turbulence in measurements of the temperature structure parameter under very stable conditions

C _T ²measurements taken over a desert in stable conditions indicate that the atmosphere remains intermittently turbulent for Ri numbers as high as 10. This is in contrast to previous results which suggest that the atmosphere is essentially nonturbulent for Ri > 2. These measurements also indicate that time-averaged C _T ²measurements do not scale with the time-averaged mean Ri number in very stable conditions. However, the standard deviation of log₁₀ C _T ²does appear to scale with Ri.     

Buoyancy and The Sensible Heat Flux Budget Within Dense Canopies

In contrast to atmospheric surface-layer (ASL) turbulence, a linear relationship between turbulent heat fluxes (F_T) and vertical gradients of mean air temperature within canopies is frustrated by numerous factors, including local variation in heat sources and sinks and large-scale eddy motion whose signature is often linked with the ejection-sweep cycle. Furthermore, how atmospheric stability modifies such a relationship remains poorly understood, especially in stable canopy flows. To date, no explicit model exists for relating F_T to the mean air temperature gradient, buoyancy, and the statistical properties of the ejection-sweep cycle within the canopy volume. Using third-order cumulant expansion methods (CEM) and the heat flux budget equation, a “diagnostic” analytical relationship that links ejections and sweeps and the sensible heat flux for a wide range of atmospheric stability classes is derived. Closure model assumptions that relate scalar dissipation rates with sensible heat flux, and the validity of CEM in linking ejections and sweeps with the triple scalar-velocity correlations, were tested for a mixed hardwood forest in Lavarone, Italy. We showed that when the heat sources (S_T) and F_T have the same sign (i.e. the canopy is heating and sensible heat flux is positive), sweeps dominate the sensible heat flux. Conversely, if S_T and F_T are opposite in sign, standard gradient-diffusion closure model predict that ejections must dominate the sensible heat flux.     

Buoyancy in tropical cyclones and other rapidly rotating atmospheric vortices

Motivated primarily by its application to understanding tropical-cyclone intensification and maintenance, we re-examine the concept of buoyancy in rapidly rotating vortices, distinguishing between the buoyancy of the symmetric balanced vortex or system buoyancy, and the local buoyancy associated with cloud dynamics. The conventional definition of buoyancy is contrasted with a generalized form applicable to a vortex, which has a radial as well as a vertical component. If, for the special case of axisymmetric motions, the balanced density and pressure distribution of a rapidly rotating vortex are used as the reference state, the buoyancy field then characterizes the unbalanced density perturbations, i.e. the local buoyancy. We show how to determine such a reference state without approximation.The generation of the toroidal circulation of a vortex, which is necessary for vortex amplification, is characterized in the vorticity equation by the baroclinicity vector. This vector depends, inter-alia, on the horizontal (or radial) gradient of buoyancy evaluated along isobaric surfaces. We show that for a tropical-cyclone-scale vortex, the buoyancy so calculated is significantly different from that calculated at constant height or on surfaces of constant σ (σ = (p − p^*)/(p_s − p^*), where p is the actual pressure, p^* some reference pressure and p_s is the surface pressure). Since many tropical-cyclone models are formulated using σ-coordinates, we examine the calculation of buoyancy on σ-surfaces and derive an expression for the baroclinicity vector in σ-coordinates. The baroclinic forcing term in the azimuthal vorticity equation for an axisymmetric vortex is shown to be approximately equal to the azimuthal component of the curl of the generalized buoyancy. A scale analysis indicates that the vertical gradient of the radial component of generalized buoyancy makes a comparatively small contribution to the generation of toroidal vorticity in a tropical cyclone, but may be important in tornadoes and possibly also in dust devils.We derive also a form of the Sawyer–Eliassen equation from which the toroidal (or secondary) circulation of a balanced vortex may be determined. The equation is shown to be the time derivative of the toroidal vorticity equation in which the time rate-of-change of the material derivative of potential toroidal vorticity is set to zero. In analogy with the general case, the diabatic forcing term in the Sawyer–Eliassen equation is shown to be approximately equal to the time rate-of-change of the azimuthal component of the curl of generalized buoyancy.Finally, we discuss the generation of buoyancy in tropical cyclones and contrast the definitions of buoyancy that have been used in recent studies of tropical cyclones. We emphasize the non-uniqueness of the buoyancy force, which depends on the choice of a reference density and pressure, and note that different, but equivalent interpretations of the flow dynamics may be expected to arise if different reference quantities are chosen.     

Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-Eddy Simulations. Part II: Forced Convection

Forced convection in a quasi-steady atmospheric boundary layer is investigated based on a large-eddy simulation (LES) model. The performed simulations show that in the upper portion of the mixed layer the dimensionless (in terms of mixed layer scales) vertical gradients of temperature, humidity, and wind velocity depend on the dimensionless height z/z _i and the Reech number Rn. The peak values of variances and covariances at the top of the mixed layer, scaled in terms of the interfacial scales, are functions of the interfacial Richardson number Ri. As a result expressions for the entrainment rates, in the case when the interfacial layer has a finite depth, and a condition for the presence of moistening or drying regimes in the mixed layer, are derived. Profiles of dimensionless scalar moments in the mixed layer are proposed to be expressed in terms of two empirical similarity functions F _m and F _i , dependent on dimensionless height z/z _i , and the interfacial Richardson number Ri. The obtained similarity expressions adequately approximate the LES profiles of scalar statistics, and properly represent the impact of stability, shear, and entrainment. They are also consistent with the parameterization proposed for free convection in the first part of this paper.     

A Hierarchy of Energy- and Flux-Budget (EFB) Turbulence Closure Models for Stably-Stratified Geophysical Flows

Here we advance the physical background of the energy- and flux-budget turbulence closures based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth’s rotation. In accordance with modern experimental evidence, the closure implies the maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the two principally different regimes: “strong turbulence” at ${Ri \ll 1}$ typical of boundary-layer flows and characterized by the practically constant turbulent Prandtl number Pr _T; and “weak turbulence” at Ri > 1 typical of the free atmosphere or deep ocean, where Pr _T asymptotically linearly increases with increasing Ri (which implies very strong suppression of the heat transfer compared to the momentum transfer). For use in different applications, the closure is formulated at different levels of complexity, from the local algebraic model relevant to the steady-state regime of turbulence to a hierarchy of non-local closures including simpler down-gradient models, presented in terms of the eddy viscosity and eddy conductivity, and a general non-gradient model based on prognostic equations for all the basic parameters of turbulence including turbulent fluxes.     

Processes in the surface mixed layer of the ocean

Models for the evolution of the surface mixed layer need to be improved to include dominant processes such as Langmuir circulation. It is shown that the wave forcing in Langmuir circulation models is much stronger than that due to a surface buoyancy loss, and studies of the erosion by the cells of a pre-existing stratification are described. Mixed layer models will also need to allow for horizontal inhomogeneity. It is shown, for example, that the horizontal buoyancy gradient that may be left behind after a storm produces restratification that can be significant. The nonlinearity of the equation of state is another real-world factor; it gives rise to an annual average surface buoyancy that is misleading as it is compensated by interior cabbeling. Current work linking the mixed layer to water mass formation is also introduced.     

复杂地形强降雪过程中垂直运动诊断分析

利用WRF模式对2018年11月30日伊犁河谷和天山北坡强降雪过程进行数值模拟,并分析复杂地形强降雪过程垂直速度和垂直动能变化机制。研究表明,冷锋过境造成地表气压升高,干空气气柱质量增大,从而导致垂直气压梯度力和干空气气柱浮力发生变化,进而引起垂直运动发生发展。垂直速度局地时间变化主要取决于扰动垂直气压梯度力、水物质拖曳力和扰动干空气浮力。在天山北坡,气流过山时,迎风坡的扰动垂直气压梯度力较大,扰动干空气浮力较小,二者合力促进上升运动;在背风坡,扰动垂直气压梯度力和扰动空气浮力形成向下的合力,产生下沉加速度,导致背风坡下沉大风。扰动垂直气压梯度力做功和扰动干空气浮力做功情况基本相反,背风坡扰动垂直气压梯度力和综合强迫做功项抑制垂直动能,扰动干空气浮力和水物质拖曳力做功项增强垂直动能。此外,扰动垂直气压梯度力和扰动干空气浮力做功项主要出现在中低层,水物质拖曳力做功项主要位于低层,平缓地形处的综合强迫做功明显小于地形复杂处。     

Remote sensing of the dynamic stability of the atmospheric boundary layer

Summary The relative strength of the stabilizing effect of buoyancy and the destabilizing effect of velocity shear in a stratified shear flow, such as a stable atmospheric boundary layer, is measured by the gradient Richardson number, Ri_g. The boundary layer static stability, as described by the buoyancy frequency, N, can be calculated from the virtual potential temperature gradient derived from RASS temperature profiles. The mean wind profiles from a sodar can be used to calculate the mean vertical velocity shear. In combination these profilers are potentially a powerful tool for the remotely sensing the dynamic stability of the boundary layer. However, experience shows that the combinations of two experimentally derived quantities, like N and shear, may give highly variable results. On the other hand, a simple sensitivity analysis shows that reasonable estimates of Ri_g are achievable over a range of conditions in the stable nocturnal boundary layer. To test this conclusion, high spatial and temporal resolution temperature and velocity soundings were obtained above 50m in the stable nocturnal boundary layer using a 920MHz continuous wave Radio Acoustic Sounding System (RASS) and 1.875kHz and 5.00kHz Doppler sodars. Examples of the evolution of Ri_g are presented from 24 hours of observations of the boundary layer in Canberra, on the tablelands in south- eastern Australia. Most of the boundary layer had Ri_g between 0.1 and 1. Thus, it was marginally dynamically stable, even with the gradient Richardson number calculated from finite differences over a vertical interval of 68m. A comparison of the results from the two sodars showed that the velocity shear increased significantly when the vertical differencing interval was decreased from 68m to 20m.     

Localized stirring in a field of salt-fingers

In a series of laboratory experiments, a partially mixed patch was produced in thick linear concentration gradients favorable to salt-finger convection. Salt-fingers, which give rise to an up-gradient flux of buoyancy, can reduce and invert the density gradient in the initial imposed patch. This leads to overturning convection within the patch if (a) the ratio of ambient T and S gradients, R_ρ=αT_z/βS_z, is near one; (b) the initial imposed turbulence results in a nearly well-mixed patch; and (c) the patch thickness is large enough that convective eddies are able to transport T and S faster than salt-fingers. Once overturning occurs, subsequent turbulent entrainment can lead to growth of the patch thickness. Experimental results for one-dimensional patches (layers) agree well with the theoretical prediction. This thickening is in contrast to the collapse that a partially mixed three-dimensional patch would experience due to lateral intrusion in a wide tank.     

COMPARISON OF SOME LIMITS FOR STABILITY CLASSIFICATION

1 INTRODUCTION Used in a number of models for pollution dissipation as the only factor to define the state of atmospheric turbulence or describe the capability of atmospheric diffusion, atmospheric stability is one of the essential parameters in the study on the atmospheric boundary layer. Whether the stability is correctly categorized immediately affects the computations of diffusion models with plumes of various types. Much work has been done at home and abroad on the classification of s…     

On the turbulent Prandtl number in the stable atmospheric boundary layer

This study focuses on the behaviour of the turbulent Prandtl number, Pr _t , in the stable atmospheric boundary layer (SBL) based on measurements made during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA). It is found that Pr _t increases with increasing stability if Pr _t is plotted vs. gradient Richardson number, Ri; but at the same time, Pr _t decreases with increasing stability if Pr _t is plotted vs. flux Richardson number, Rf, or vs. ζ = z/L. This paradoxical behaviour of the turbulent Prandtl number in the SBL derives from the fact that plots of Pr _t vs. Ri (as well as vs. Rf and ζ) for individual 1-h observations and conventional bin-averaged values of the individual quantities have built-in correlation (or self-correlation) because of the shared variables. For independent estimates of how Pr _t behaves in very stable stratification, Pr _t is plotted against the bulk Richardson number; such plots have no built-in correlation. These plots based on the SHEBA data show that, on the average, Pr _t decreases with increasing stability and Pr _t < 1 in the very stable case. For specific heights and stabilities, though, the turbulent Prandtl number has more complicated behaviour in the SBL.     

The influence of stratification on heat and momentum turbulent transfer in antarctica

Data from the Antarctic winter at Halley Base have been used in order to evaluate qualitatively and quantitatively how the stratification in the low atmosphere (evaluated with the gradient Richardson number, Ri) influences the eddy transfers of heat and momentum. Vertical profiles of wind and temperature up to 32 m, and turbulent fluxes ( , and ) measured from three ultrasonic thermo-anemometers installed at 5, 17 and 32 m are employed to calculate Ri, the friction velocity (u _*) and the eddy diffusivities for heat (K _h ) and momentum (K _m ). The results show a big dependence of stability onK _m ,K _h andu _*, with a sharp decrease of these turbulent parameters with increasing stability. The ratio of eddy diffusivities (K _h /K _m ) is also analyzed and presents a decreasing tendency as Ri increases, reaching values even less than 1, i.e., there were situations where the turbulent transfer of momentum was greater than that of heat. Possible mechanisms of turbulent mixing are discussed.     

Direct Numerical Simulation of Stable Channel Flow at Large Stability

We consider a model for the stable atmospheric boundary at large stability, i.e. near the limit where turbulence is no longer able to survive. The model is a plane horizontally homogeneous channel flow, which is driven by a constant pressure gradient and which has a no-slip wall at the bottom and a free-slip wall at the top. At the lower wall a constant negative temperature flux is imposed. First, we consider a direct numerical simulation of the same channel flow. The simulation is computed with the neutral channel flow as initial condition and computed as a function of time for various values of the stability parameter h/L, where h is the channel height and L is related to the Obukhov length. We find that a turbulent solution is only possible for h/L < 1.25 and for larger values turbulence decays. Next, we consider a theoretical model for this channel flow based on a simple gradient transfer closure. The resulting equations allow an exact solution for the case of a stationary flow. The velocity profile for this solution is almost linear as a function of height in most of the channel. In the limit of infinite Reynolds number, the temperature profile has a logarithmic singularity at the upper wall of the channel. For the cases where a turbulent flow is maintained in the numerical simulation, we find that the velocity and temperature profiles are in good agreement with the results of the theoretical model when the effects of the surface layer on the exchange coefficients are taken into account. Frans Nieuwstadt, a recently retired member of the BLM Editorial Board and a well-known member of the boundary-layer/turbulence community, died unexpectedly on 18 May 2005. An obituary will appear in a later issue of BLM.     

Equilibrium theory of the planetary boundary layer with an inversion lid

The maintenance of an elevated inversion in steady flow above a cold, rotating surface is shown to be possible for a certain range of the buoyancy number bfV _g, where b is the buoyant acceleration appropriate to the density deficiency of the fluid above the inversion, f is Coriolis parameter and V _gis geostrophic velocity (so that fV _gis also horizontal pressure gradient in kinematic units). The height of the inversion lid is determined by a balance of surface stress and buoyancy, in a way which may be deduced from laboratory experiments. With the aid of such empirical evidence a theory is constructed for the layer below the inversion lid. The cross-isobar angle of ground-level stress is found to increase with the buoyancy number, to a limiting value of 90, by which time the inversion descends to the ground. Under typical conditions, a temperature difference of order 10C is necessary to eliminate the possibility of an equilibrium, elevated inversion lid and reduce ground level wind stress to a vanishingly small value.Woods Hole Oceanographic Institution Contribution #3011On leave from the University of Waterloo, Ontario