Large eddy simulation of stratified mixing in a three-dimensional lock-exchange system

Accurate and computationally-efficient modeling of stratified mixing processes are of paramount importance in both coastal and large-scale ocean circulation. In this study, our main objective is to investigate the feasibility and accuracy of large eddy simulation (LES) as a possible tool to study small-scale oceanic processes. To this end, LES is evaluated in a 3D lock-exchange problem, which contains shear-driven mixing, internal waves, interactions with boundaries and convective motions, while having a simple domain, initial and boundary conditions, and forcing.Two general classes of LES models are tested, namely eddy viscosity (EV) models based on constant–coefficient and dynamic Smagorinsky models, and an approximate deconvolution (AD) model. By noting that the dynamic Smagorinsky and AD models have different strengths in that the former is good in providing appropriate dissipation while the latter in preserving the detail of coherent structures on coarse resolution meshes, a hybrid approach combining EV and AD models is also evaluated. A direct numerical simulation (DNS) is performed as the benchmark solution, and all LES models are tested on three coarse meshes. The main measure of mixing is taken as the temporal evolution of background potential energy.It is found that constant-coefficient Smagorinsky models can only provide a marginal improvement over under-resolved simulations, while both dynamic Smagorinsky and AD models lead to significant improvements in mixing accuracy. The primary accomplishment of this study is that it is shown that the hybrid approach attains the best agreement with the mixing curve from DNS, while being computationally approximately a thousand times faster.     

On the turbulent prandtl number in a stably stratified atmospheric boundary layer

The results obtained from both atmospheric and laboratory measurements and from LES data show that, in the stably stratified flows of the atmospheric boundary layer, turbulent mixing occurs at gradient Richardson numbers Ri _g that significantly exceed one: the inverse turbulent Prandtl number Pr _t ⁻¹ decreases with an increase in the thermal flow stability. The decreasing trend of the inverse turbulent Ptandtl number is reproduced in a stably stratified atmospheric boundary layer in agreement with measurement data with the aid of an improved three-parameter turbulence model. In this model, a modified model that takes into account the effect of stratification in the expression for the time scale of the scalar field is used for the pressure-scalar correlation.     

The mixing efficiency and the processes of restratification in a stably stratified medium

The mechanism of the effect of a collapsing turbulent eddy on diapycnal transport in a stably stratified fluid is considered. It is shown that at small Richardson turbulent numbersRi ₀ the mixing efficiency increases asRi ₀, and at large numbers it decreases in proportion toRi ₀ ^–1/2 .Translated by Mikhail M. Trufanov. UDK 551.465.15.     

On the eddy mixing and energetics of turbulence in a stable atmospheric boundary layer

Some changes in the eddy mixing in the atmospheric boundary layer (ABL) are investigated with the use of the mesoscale RANS turbulence model. It is found that the behavior of parameters of the eddy turbulence mixing is in compliance with the recently obtained data of laboratory and atmospheric measurements. In particular, the flow Richardson number (Ri _f ) during the transient flow to a strongly stable state can behave nonmonotonically, growing with the increasing gradient Richardson number (Ri _g ) to the state of saturation at a certain gradient Richardson number (Ri _g ? 1), which separates two different turbulent regimes: the regimes of strong mixing and weak mixing. An analysis of the energetics based on the balance equations of kinetic and potential turbulence energies shows, in particular, that the weak mixing (Ri _g > 1) is quite capable of transferring momentum. This phenomenon can be explained not only by the fact that the flow is sustained by propagating internal waves, which effectively transfer momentum under strong stratification conditions, but also by the fact that turbulence permanently arises in the free atmosphere and in the deep ocean at Ri _g ? 1.     

Reynolds number dependence of mixing in a lock-exchange system from direct numerical and large eddy simulations

Turbulent mixing of water masses of different temperatures and salinities is an important process for both coastal and large-scale ocean circulation. It is, however, difficult to capture computationally. One of the reasons is that mixing in the ocean occurs at a wide range of complexity, with the Reynolds number reaching , or even higher.In this study, we continue to investigate whether large eddy simulation (LES) can be a reliable computational tool for stratified mixing in turbulent oceanic flows. LES is attractive because it can be times faster than a direct numerical simulation (DNS) of stratified mixing in turbulent flows. Before using the LES methodology to compute mixing in realistic oceanic flows, however, a careful assessment of the LES sensitivity with respect to Re needs to be performed first. The main objectives of this study are: (i) to investigate the performance of different LES models at high Re, such as those encountered in oceanic flows; and (ii) to study how mixing varies as a function of Re. To this end, as a benchmark we use the lock-exchange problem, which is described by unambigous and simple initial and boundary conditions. The background potential energy, which accurately quantifies irreversible mixing in an enclosed system, is used as the main criterion in a posteriori testing of LES.This study has two main achievements. The first is that we investigate the accuracy of six combinations of two different classes of LES models, namely eddy-viscosity and approximate deconvolution types, for 3×10³Re3×10⁴, for which DNS data is computed. We find that all LES models almost always provide significantly more accurate results than cases without LES models. Nevertheless, no single LES model that is persistently superior to others over this Re range could be identified. Then, an ensemble of the four best performing LES models is selected in order to estimate mixing taking place in this system at Re=10⁵ and 10⁶, for which DNS is presently not feasible. Thus the second achievement of this study is to quantify mixing taking place in this system over an Re range that changes by three orders of magnitude. We find that the background potential energy increases by about 67% when Re is increased from Re=10³ to Re=10⁶, within the computation period, with the most significant increase taking place from Re=3×10³ to Re=10⁵.     

2012年夏季海南岛东岸上升流区的混合观测

The turbulent mixing in the upwelling region east of Hainan Island in the South China Sea is analyzed based on in situ microstructure observations made in July 2012. During the observation, strong upwelling appears in the coastal waters, which are 3℃ cooler than the offshore waters and have a salinity 1.0 greater than that of the offshore waters. The magnitude of the dissipation rate of turbulent kinetic energy ε in the upwelling region is O(10–9 W/kg), which is comparable to the general oceanic dissipation. The inferred eddy diffusivity K_ρ is O(10–6 m~2/s), which is one order of magnitude lower than that in the open ocean. The values are elevated to K_ρ≈O(10–4 m~2/s) near the boundaries. Weak mixing in the upwelling region is consistent with weak instability as a result of moderate shears versus strong stratifications by the joint influence of surface heating and upwelling of cold water.The validity of two fine-scale structure mixing parameterization models are tested by comparison with the observed dissipation rates. The results indicate that the model developed by Mac Kinnon and Gregg in 2003 provides relatively better estimates with magnitudes close to the observations. Mixing parameterization models need to be further improved in the coastal upwelling region.     

Comparison of laminar model,RANS, LES and VLES for simulation of liquid sloshing

Liquid sloshing in storage tank is a fundamental problem of great engineering importance. Sloshing motion can be laminar or turbulent. However, the necessity for inclusion of turbulence in CFD simulation of sloshing flows has not yet been established. In this paper, three roll–induced sloshing cases are studied to assess the merits and shortcomings of the laminar model and three most–commonly used turbulence models (RANS k–ε, LES and Very LES). To overcome the deficiencies in the RANS and LES, the new Very LES (VLES) model, which combines the RANS k–ε and LES, is developed in this paper. The free surface profiles are reconstructed by a coupled Level–Set and Volume–of–Fluid (CLSVOF) method. To the authors’ knowledge, the comprehensive and systematical assessment of the effect of turbulence on sloshing simulation has not been reported in the literature. The numerical results are evaluated using experimental measurements from Delorme and Souto−Iglesias. The present study indicates that the inclusion of an appropriate turbulence model has a profound influence on the simulations of violent and non–violent sloshing flows. The VLES and LES models can provide accurate predictions of free surface profiles and impact pressures, whereas the laminar flow assumption and the RANS model cannot adequately capture the energy dissipation in the sloshing simulation and lead to the inaccurate flow predictions.     

Towards a mesoscale eddy closure

A turbulence closure for the effect of mesoscale eddies in non-eddy-resolving ocean models is proposed. The closure consists of a prognostic equation for the eddy kinetic energy (EKE) that is integrated as an additional model equation, and a diagnostic relation for an eddy length scale (L), which is given by the minimum of Rhines scale and Rossby radius. Combining EKE and L using a standard mixing length assumption gives a diffusivity (K), corresponding to the thickness diffusivity in the [Gent, P.R., McWilliams, J.C. 1990. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20, 150–155] parameterisation. Assuming downgradient mixing of potential vorticity with identical diffusivity shows how K is related to horizontal and vertical mixing processes in the horizontal momentum equation, and also enables us to parameterise the source of EKE related to eddy momentum fluxes.The mesoscale eddy closure is evaluated using synthetic data from two different eddy-resolving models covering the North Atlantic Ocean and the Southern Ocean, respectively. The diagnosis shows that the mixing length assumption together with the definition of eddy length scales is valid within certain limitations. Furthermore, implementation of the closure in non-eddy-resolving models of the North Atlantic and the Southern Ocean shows consistently that the closure has skill at reproducing the results of the eddy-resolving model versions in terms of EKE and K.     

Numerical simulation of stably stratified turbulent flows over flat and urban surfaces

Stably stratified turbulent flows over surfaces with explicit roughness elements simulating an urban built-up area have been calculated using an LES model. A method of conducting numerical experiments allowing turbulent flows with specified values of the Obukhov scale L to reach a quasisteady state at the surface has been proposed. It has been shown that, to calculate both temperature and wind-velocity profiles over such objects, one can use the same universal dependences that are used for a flat surface. It has been found that stable stratification does not affect the roughness parameter z _0u and the displacement height D.     

Diapycnal mixing by meso-scale eddies

The mean available potential energy released by baroclinic instability into the meso-scale eddy field has to be dissipated in some way and Tandon and Garrett [Tandon, A., Garrett, C., 1996. On a recent parameterization of mesoscale eddies. J. Phys. Oceanogr. 26 (3), 406–416] suggested that this dissipation could ultimately involve irreversible mixing of buoyancy by molecular processes at the small-scale end of the turbulence cascade. We revisit this idea and argue that the presence of dissipation within the thermocline automatically requires that a component of the eddy flux associated with meso-scale eddies must be associated with irreversible mixing of buoyancy within the thermocline. We offer a parameterisation of the implied diapycnal diffusivity based on (i) the dissipation rate for eddy kinetic energy given by the meso-scale eddy closure of Eden and Greatbatch [Eden, C., Greatbatch, R.J., 2008. Towards a meso-scale eddy closure. Ocean Modell. 20, 223–239.] and (ii) a fixed mixing efficiency. The implied eddy-induced diapycnal diffusivity (κ) is implemented in a coarse resolution model of the North Atlantic. In contrast to the vertical diffusivity given by a standard vertical mixing scheme, large lateral inhomogeneities can be found for κ in the interior of the ocean. In general, κ is large, i.e. up to o(10) cm²/s, near the western boundaries and almost vanishing in the interior of the ocean.     

Modeling the eddy transport of momentum and heat: Comparison with direct measurements in free atmosphere

Direct measurements of eddy diffusivities for momentum K _m and heat K _h by Doppler radar and by a radio acoustic sounding system in the upper troposphere and lower stratosphere were used to examine the applicability of three Reynolds-averaged Navier-Stokes (RANS) schemes of stratified turbulence in the environment: the E — ? turbulence scheme modified for stratified flows, the algebraic two-parameter E — ? Reynolds-stress scheme, and the three-parameter \(E - \varepsilon - \overline {\theta ^2 } \) turbulence scheme. All turbulence parameters-the turbulent kinetic energy (E), the dissipation rate (?), and vertical profiles of potential temperature (atmospheric stability) and mean wind velocity-were derived from direct measurements for all three turbulence schemes. It is shown that the profile of the vertical diffusivity of momentum (K _m ) obtained from the three-parameter RANS turbulence scheme agrees well with its directly measured analog. The profile of K _m calculated by the two-parameter turbulence schemes fits measurements rather qualitatively.     

A hybrid spectral/finite-difference large-eddy simulator of turbulent processes in the upper ocean

A three-dimensional numerical model for large-eddy simulation (LES) of oceanic turbulent processes is described. The numerical formulation comprises a spectral discretization in the horizontal directions and a high-order compact finite-difference discretization in the vertical direction. Time-stepping is accomplished via a second-order accurate fractional-step scheme. LES subgrid-scale (SGS) closure is given by a traditional Smagorinsky eddy-viscosity parametrization for which the model coefficient is derived following similarity theory in the near-surface region. Alternatively, LES closure is given by the dynamic Smagorinsky parametrization for which the model coefficient is computed dynamically as a function of the flow. Validation studies are presented demonstrating the temporal and spatial accuracy of the formulation for laminar flows with analytical solutions. Further validation studies are described involving direct numerical simulation (DNS) and LES of turbulent channel flow and LES of decaying isotropic turbulence. Sample flow problems include surface Ekman layers and wind-driven shallow water flows both with and without Langmuir circulation (LC), generated by wave effects parameterized via the well-known Craik–Leibovich (C–L) vortex force. In the case of the surface Ekman layers, the inner layer (where viscous effects are important) is not resolved and instead is parameterized with the Smagorinsky models previously described. The validity of the dynamic Smagorinsky model (DSM) for parameterizing the surface inner layer is assessed and a modification to the surface stress boundary condition based on log-layer behavior is introduced improving the performance of the DSM. Furthermore, in Ekman layers with wave effects, the implicit LES grid filter leads to LC subgrid-scales requiring ad hoc modeling via an explicit spatial filtering of the C–L force in place of a suitable SGS parameterization.     

Turbulence measurements in stratified and well-mixed estuarine flows

From mean velocities measured in estuarine flows it has been found that the velocity distributions are log-linear in stratified flows and logarithmic in well-mixed flows. The results of salinity measurements reveal that the mean salinity profiles are geometrically similar and expressible as a power law. The buoyancy parameters, such as the Monin-Obukhov length scale, the gradient and the flux Richardson numbers, are independent of the flow state. The gradient and the flux Richardson numbers are almost equal, indicating the existence of a local equilibrium layer. The non-dimensional parameter describing dissipation rates of turbulent kinetic energy is a constant of 0·2 and 0·3 for stratified and well-mixed flows respectively. In well-mixed flow the drag coefficient varies with time approaching a constant of about 3·2 × 10^?3 when the flow is stratified. The shape of the turbulent energy spectra are generally flatter and broader in stratified as compared with those of well-mixed flows.     

Modeling water quality and hypoxia dynamics in Upper Charlotte Harbor,Florida, U.S.A. during 2000

Hypoxia has occurred in Upper Charlotte Harbor, a shallow (∼3 m) estuary in Southwest Florida, during moderate to high freshwater flows from the Peace and Myakka Rivers and after hurricanes, due to nutrient loading and vertical stratification. This paper studies the annual hypoxia and water quality dynamics in Upper Charlotte Harbor in 2000, using CH3D-IMS, an integrated modeling system which includes coupled models of circulation, wave, sediment transport, and water quality. The CH3D-IMS simulations showed that bottom-water hypoxic conditions occur during periods with relatively steady moderate to high (5–40 m³/s) freshwater inflows and sediment oxygen demand (SOD). During periods of relatively steady moderate to high river discharge, strong vertical salinity stratification results in reduced vertical mixing which prevents surface water from supplying dissolved oxygen (DO) to bottom water where SOD continuously consumes DO. There was significant temporal fluctuation of the hypoxic water volume, as a result of significant temporal variation in vertical turbulent mixing associated with combinations of spring-neap tides and river discharge. The validated modeling system could be used to forecast hypoxia.     

Dissipation of baroclinic tidal energy and diapycnal mixing in the White Sea

The modeling results obtained using the original version of the three-dimensional finite-element hydrostatic model QUODDY-4 testify that the spatial distributions of dissipation of baroclinic tidal energy and the related coefficient of diapycnal mixing in the deepwater stratified subdomain of the White Sea (the Basin and Kandalaksha and Dvina bays together) are highly similar to those found for low- and midlatitude oceans. It is in the open part of the sea that their values remain equal to the minimum possible values determined by the molecular kinematic viscosity; at its lateral boundaries (not all boundaries, but only individual segments (sites of mixing)), their values increase. In the shallow homogeneous subdomain of the White Sea, the dissipation of baroclinic tidal energy is considerably larger than in the deep stratified subdomain. Accordingly, the vertical eddy viscosity in the first subdomain is a few orders of magnitude higher than the coefficient of diapycnal mixing in the second subdomain. This is caused by an increased tidal velocity due to reduced depths.     

Modeling vertical motion at ocean fronts: Are nonhydrostatic effects relevant at submesoscales?

Through a suite of three-dimensional, high-resolution numerical modeling experiments, we examine the role of nonhydrostatic effects on O(1 km) submesoscale processes at ocean fronts, with particular focus on the vertical velocity field. Several differences between nonhydrostatic and hydrostatic models are pointed out using a framework that enables precise comparison, but it is difficult to identify categorical differences between the model solutions at the grid resolutions afforded. The instantaneous vertical velocity structure is sensitive to the model choice and, even more so, to grid resolution, but the average vertical flux is similar in both hydrostatic and nonhydrostatic cases.When a frontal region with horizontal density gradients is perturbed by wind, a profusion of submesoscale, O(1 km), secondary circulation features develops in the upper 50 m. Narrow, elongated cells of intense up- and down-welling are found to occur close to the surface, overlying broader regions of weaker up- and down-welling associated with the mesoscale meanders of the baroclinically unstable front. The submesoscale down-welling is considerably stronger than up-welling and is concentrated in 1–2 km width filaments within which velocities can attain magnitudes as high as 200 m day⁻¹. The submesoscale features are found to be robust at horizontal grid resolutions varying between 1 and 0.25 km and exist even in the hydrostatic model. Submesoscale circulation is difficult to observe or resolve in coarser resolution circulation models, but is likely to play a significant role in the exchange of energy and properties between the surface ocean and thermocline. Possible mechanisms for the generation of these features are investigated in a follow-on paper.     

Mixing and biological production at eddy margins in the eastern Gulf of Alaska

We performed a multi-day shipboard experiment in June 2001 to test whether combining water from within an anticyclonic mesoscale eddy in the eastern Gulf of Alaska with water from outside could result in enhanced phytoplankton growth and to determine how mixing might influence planktonic assemblages. Initially, the eddy had lower standing stocks of algal pigments (chlorophyll a [chl a] and accessory pigments), nutrients, phytoplankton, and particulate organic carbon/nitrogen compared to waters outside of the eddy. The eddy possessed a greater diversity and abundance of coastal diatoms while the outside waters had a greater proportion of oceanic species, including the endemic pennate diatom, Nitzschia cylindroformis. After one week of incubation, rates of primary production were significantly higher in the mixed water compared to both the eddy and outside treatments. Pigment concentrations (except chl c₃, alloxanthin, and zeaxanthin) and the proportion of large diatoms (mainly Pseudo-nitzschia spp.) and heterotrophic dinoflagellates were greater in the mixed water than would be expected from the simple combination of inside and outside waters. Nutrient limitation (most likely by trace metals) appeared to be less severe in the mixed water. Chl a was enhanced in the mixed water, particularly when compared to the eddy water. The mixing of eddy and outside water masses stimulated primary production by ∼20%, but more importantly, the mixing resulted in a distinct planktonic assemblage. The biomass enrichment was short-lived, indicating that the maintenance of elevated chl a would require further mixing events in a physical setting that also permits an accumulation of biomass. We note that submesoscale processes, including the intensification of ageostrophic circulation that elicits strong vertical mixing in the presence of strain, might explain observed patterns of high phytoplankton standing stocks at the inner edges of Haida eddies in the field.     

北太平洋涡旋对基于细尺度参数化的海洋内部混合的影响

基于Vector Geometry方法对2016—2018年的高度计资料进行涡旋识别,并使用细尺度参数化方法和Argo数据计算了涡旋附近的海洋内部扩散率,分析了北太平洋的涡旋对海洋内部混合的影响。结果显示,研究区域在涡旋影响下的平均扩散率比无涡旋影响下的值大6%,并且气旋涡增强了600—1200m深度的混合,对600—900m深度的混合影响最大,可达18%;反气旋涡明显增强了300—900m深度的混合,但对900—1200m深度的混合没有明显影响。随着与涡旋中心距离的增大,涡旋外围混合扩散率缓慢减小,涡旋内部混合扩散率变化不明显,此结果与2014年3—10月在24°—36°N、132°—152°E区域的一个个例分析结果一致。此外,随着涡旋强度的增大,海洋内部混合明显增强。统计结果表明,在研究区域, 90%的扩散率值在10~(-5.5)—10~(-4)m~2/s范围内。