Second-order Turbulence Closure Model Research Articles

The Stability Functions and Realizability of the Turbulent Kinetic Energy\u2013Scalar Variance Closure for Moist Atmospheric Boundary-Layer Turbulence

The problem of realizability of the second-order turbulence closure models (parametrization schemes) is addressed through the consideration of the so-called “stability functions”. The emphasis is on the turbulence kinetic energy–scalar variance (TKESV) closure scheme that carries prognostic transport equations for the turbulence kinetic energy (TKE) and for the variances and covariance of two quasi-conservative scalars suitable for describing moist atmospheric boundary-layer turbulence. Stability functions appear within the framework of truncated closure schemes, where (i) the Reynolds-stress and scalar-flux equations (and, within the framework of one-equation TKE schemes, also equations for scalar variances and covariance) are reduced to the diagnostic algebraic formulations by neglecting the substantial derivatives and the third-order transport terms, and (ii) simplified linear parametrizations of the pressure-scrambling terms are used. The stability functions are ill-behaved (tend to infinity or become negative) over a certain range of governing parameters, e.g., mean velocity shear and buoyancy gradient. Using the approach of Helfand and Labraga (J Atmos Sci 45:113–132, 1988), we develop regularized stability functions for the TKESV scheme that reveal no pathological behaviour over their entire parameter space. The physical meaning of the regularization procedure and its relation to non-linear parametrizations of the pressure-scrambling terms and to weak non-equilibrium hypothesis are discussed. Finally, realizability of turbulence closures is considered within a more general framework of the moments problem of the probability theory.

A non-breaking-wave-generated turbulence mixing scheme for a global ocean general circulation model

A novel vertical mixing scheme to describe the influence of the non-breaking surface waves in ocean general circulation models is proposed based on the second-order turbulence closure model and a mathematical relationship fitted between the in situ observations of the turbulence kinetic energy dissipation rate and the calculated velocity shear module of non-breaking surface waves. The non-breaking-wave-generated turbulence mixing coefficients can be calculated empirically in terms of the wave spectral parameters including ΦSW, K, and ω, where ΦSW is the wave number spectrum of the significant wave height, K is the wave number, ω is the surface wave frequency. The effect of the new mixing scheme on global ocean circulation simulation was tested using the MArine Science and NUmerical Modeling ocean model. The results indicate significant improvement in the upper-ocean temperature structure and the mixed layer depth, i.e., the non-breaking-wave-generated turbulence mixing plays an important role in the upper ocean at high latitudes in summer for both hemispheres compared with traditional Mellor-Yamada 2.5 scheme.

Improvement of the Mellor–Yamada–Nakanishi–Niino Planetary Boundary-Layer Scheme Based on Observational Data in China

The Mellor–Yamada–Nakanishi–Niino (MYNN) planetary boundary-layer (PBL) scheme is a second-order turbulence closure model that is an improved version of the Mellor–Yamada scheme based on large-eddy simulation data. It simulates PBL structure and evolution well, particularly over the ocean surface. However, when used with various underlying surfaces in China, the scheme overestimates the turbulent momentum flux and the sensible heat flux. Based on observations of surface fluxes in China, we attempt to improve the MYNN model by modifying the parameters and representation of the turbulence scale. Closure constants and empirical expressions in the diagnostic equation are chosen first, and an additional component of the turbulent heat flux is considered in the potential temperature prognostic equation to improve the surface heat-flux modelling. The modified MYNN scheme is incorporated into a three-dimensional mesoscale model and is evaluated using various underlying surface observations. Amelioration of the surface turbulent fluxes is confirmed at five observational sites in China over different land-use types.

Interpreting three-dimensional spore concentration measurements and escape fraction in a crop canopy using a coupled Eulerian–Lagrangian stochastic model

Plant disease epidemics caused by pathogenic spores are common threat to agricultural crops. Pathogenic spores are often produced and released inside plant canopies but are transported out of the canopy region by turbulent motions and advected longitudinally over long distances so as to infect other fields. Hence, the fraction of spores that “escape” the canopy sets the effective source strength that determines how fast and far plant diseases can spread. To explore the governing variables of this escape and spatial spread of spores, extensive spore release and recapture experiments were conducted in a maize field and interpreted using a coupled Eulerian–Lagrangian stochastic model (LSM). Spores were released from point sources located at three prototypical depths inside the canopy. Concentration measurements were obtained inside and above the canopy with a 3-dimensional grid of spore collectors. The experimental measurements of mean spore concentration were then used to evaluate an LSM for spore dispersion. The drift and dispersion terms of the LSM were predicted employing a conventional second-order closure model of turbulence within plant canopies thereby allowing this combined Eulerian–Lagrangian formulation to be applied to a broad set of agricultural crops. The dispersion model includes spore deposition on and rebound from canopy elements. The combination of experimental and numerical simulations was then used to quantify the fraction of spores that escape the canopy. Effects of release height and friction velocity on the escape fraction of spores were explored with the LSM.

Analytical estimation of mixing coefficient induced by surface wave-generated turbulence based on the equilibrium solution of the second-order turbulence closure model

Based on the equations of motion and the assumption that ocean turbulence is of isotropy or quasi-isotropy, we derived the closure equations of the second-order moments and the variation equations for characteristic quantities, which describe the mechanisms of advection transport and shear instability by the sum of wave-like and eddy-like motions and circulation. Given that ocean turbulence generated by wave breaking is dominant at the ocean surface, we presented the boundary conditions of the turbulence kinetic energy and its dissipation rate, which are determined by energy loss from wave breaking and entrainment depth respectively. According to the equilibrium solution of the variation equations and available data of the dissipation rate, we obtained an analytical estimation of the characteristic quantities of surface-wave-generated turbulence in the upper ocean and its related mixing coefficient. The derived kinetic dissipation rate was validated by field measurements qualitatively and quantitatively, and the mixing coefficient had fairly good consistency with previous results based on the Prandtl mixing length theory.

Assessment of turbulence closure models for resonant inertial response in the oceanic mixed layer using a large eddy simulation model

Large eddy simulation (LES) of the resonant inertial response of the upper ocean to strong wind forcing is carried out; the results are used to evaluate the performance of each of the two second-order turbulence closure models presented by Mellor and Yamada (Rev Geophys Space Phys 20:851–875, 1982) (MY) and by Nakanishi and Niino (J Meteorol Soc Jpn 87:895–912, 2009) (NN). The major difference between MY and NN is in the formulation of the stability functions and the turbulent length scale, both strongly linked with turbulent fluxes; in particular, the turbulent length scale in NN, unlike that in MY, is allowed to decrease with increasing density stratification. We find that MY underestimates and NN overestimates the development of mixed layer features, for example, the strong entrainment at the base of the oceanic mixed layer and the accompanying decrease of sea surface temperature. Considering that the stability functions in NN perform better than those in MY in reproducing the vertical structure of turbulent heat flux, we slightly modify NN to find that the discrepancy between LES and NN can be reduced by more strongly restricting the turbulent length scale with increasing density stratification.

Modeling study of the aspect ratio influence on urban canopy energy fluxes with a modified wall-canyon energy budget scheme

The influence of the aspect ratio (building height/street canyon width) and the mean building height of cities on local energy fluxes and temperatures is studied by means of an Urban Canopy Model (UCM) coupled with a one-dimensional second-order turbulence closure model. The UCM presented is similar to the Town Energy Balance (TEB) model in most of its features but differs in a few important aspects. In particular, the street canyon walls are treated separately which leads to a different budget of radiation within the street canyon walls. The UCM has been calibrated using observations of incoming global and diffuse solar radiation, incoming long-wave radiation and air temperature at a site in São Paulo, Brazil. Sensitivity studies with various aspect ratios have been performed to assess their impact on urban temperatures and energy fluxes at the top of the canopy layer. In these simulations, it is assumed that the anthropogenic heat flux and latent heat fluxes are negligible. Results show that the simulated net radiation and sensible heat fluxes at the top of the canopy decrease and the stored heat increases as the aspect ratio increases. The simulated air temperature follows the behavior of the sensible heat flux.

A new second-order closure model for rough-wall turbulent flows using the Brinkman equation

A second-order turbulence closure is developed for the new rough-wall layer modeling approach using the Brinkman equation for turbulent flows over rough walls. In the proposed approach, we model the fluid dynamics of the volume averaged flow in the near-wall rough layer by using the Brinkman equation. The porosity can be calculated based on the volumetric characteristics of the roughness and the permeability is modeled. Interface stress jump conditions including the Reynolds stress components are also considered. The Reynolds-averaged Navier–Stokes equations are solved numerically above the near-wall rough layer, while a second-order turbulence closure is employed in all regions. The rough-wall second-order closure is developed by adopting an existing smooth-wall model. The computational results, including the skin friction coefficient, the log-law mean velocity, the roughness function, the Reynolds stresses, and the turbulent kinetic energy, are presented and compared with those obtained by using a previously developed two-equation turbulence closure. The results show that the new rough-wall layer modeling approach with the second-order turbulence closure model satisfactorily predicted the skin friction coefficient, the log-law mean velocity, the roughness function, and the Reynolds shear stress. However, the results for the Reynolds normal stresses are different from the measured data in the inner 20–60% of the boundary layer due to the interface stress jump conditions employed in the present rough-wall layer modeling approach.

Comments on “Second-order turbulence closure models for geophysical boundary layers: A review of recent work”

Comments on “Second-order turbulence closure models for geophysical boundary layers: A review of recent work”

Feasible Method for the Assimilation of Satellite-Derived SST with an Ocean Circulation Model

Abstract The surface restoring condition of satellite-derived sea surface temperatures (SSTs) is validated as a feasible assimilation method with an ocean circulation model that incorporates the strongly nonlinear mixed layer model. The restoring treatment is an empirical method for correcting the heat flux in order to pull the predicted SST toward the observed SST; it is referred to as the nudging method in this study. A one-dimensional experiment is conducted to evaluate the skill of the nudging method. The mixed layer model (MLM) used in the experiment is a second-order turbulence closure model that has a strong nonlinearity. The skill of the nudging method is compared with that of an ensemble Kalman filter, which is a statistically optimal method for nonlinear dynamic models. Although the nudging method is statistically suboptimal, the result of the experiment shows that the skill of this method is comparable when using an appropriate restoring time scale. A three-dimensional experiment using an ocean general circulation model (OGCM), which incorporates the same MLM as that used in the one-dimensional experiment, is also conducted to further examine the skill of the nudging method. By applying the nudging method to the OGCM, the model improves the estimated thermal structure not only near the surface, but also in the subsurface layers.

Second-order turbulence closure models for geophysical boundary layers. A review of recent work

Recent contributions to second-order turbulence modelling are reviewed with an emphasis on models of the coastal ocean. Classical and recent turbulence models are reformulated using a unifying notation, making differences in model structure and model parameters more transparent. The essential characteristics of a number of models are compared. It is further demonstrated that the mathematical stability and realisability of the models is crucial for practical applications. The parameter constraints to insure stable computations are discussed. Model performance is evaluated for a number of idealised entrainment scenarios that are typical for shelf seas: entrainment in linearly stratified and two-layer fluids caused by (a) surface wind stress (b) bottom stress due to water motion driven by a barotropic pressure gradient (c) bottom stress due to water motion in a bottom jet. It is shown that the predicted entrainment velocities are in good agreement with available laboratory data, provided the models have been calibrated according to a theoretical constraint. In addition, model behaviour in free convection is re-analysed with the help of recent data from large eddy simulations (LES). It is shown that even the most simple models, which do not solve a differential equation for the turbulent kinetic energy (TKE), can approximately reproduce the entrainment depth, however, they fail completely in reproducing the correct budget of the TKE-balance. In contrast, second-order models solving transport equations for the TKE and a length scale determining variable are in reasonable agreement with the TKE-budgets from LES.

Canopy Architecture and Turbulence Structure in a Coniferous Forest

Synchronous sonic anemometric measurements at five heightswithin a mixed coniferous forest were used to test two different parameterisations ofcanopy architecture in the application of a second-order turbulence closure model. Inthe computation of the leaf drag area, the aerodynamic sheltering was replaced with anarchitectural sheltering, assumed to be analogous to the clumping index defined in radiativetransfer theory. Consequently, the ratio of leaf area density and sheltering factor was approximatedby the effective leaf area or the mean contact number, both obtained from the inversion of non-destructive optical measurements. The first parameter represents the equivalentrandomly dispersed leaf area in terms of shading, the second is the average number of leavesthat a straight line intercepts penetrating the canopy with a certain zenith angle. Theselection of this direction was determined by the analysis of the mean angle of the wind vectorduring sweep events. The drag coefficient values obtained from the inversion of themomentum flux equation, using the two proposed parameterisations, are in good agreement withvalues found in the literature. The predicted profiles of turbulence statistics reasonablymatch actual measurements, especially in the case of the mean contact numberparameterisation. The vertical profile of leaf drag area, obtained by forcing the turbulence modelto match the observed standard deviation of vertical velocity (σw), is intermediatebetween the two empirical ones. Finally, the proposed canopy parameterisations were appliedto a Lagrangian transport model to predict vertical profiles of air temperature, H2O andCO2 concentration.

Calculation of Surface Fluxes under Convective Conditions by Turbulence Closure Models

Abstract A method of deriving new relationships between near-surface turbulent fluxes and vertical differences of wind speed and potential temperature between two levels in the atmospheric surface layer from simplified second-order turbulence closure models is presented. The paper aims at a coherent treatment of fair-weather conditions and the asymptotic free-convection limit. The traditionally used representation of wind and temperature profiles is modified to reflect the three-sublayer structure of the surface layer in accordance with recent proposals; this leads to a generalization of the profile method. The modification suggests a new framework for the analysis of empirical data gathered in statically unstable fair-weather conditions. A general formulation of the problem involves a set of integral equations, which is solved numerically; with this aim, a new “convective” vertical coordinate transformation suitable for the computation of the upper part of the profile is proposed. Two popular turbulence ...

The Adaptation of the Atmospheric Boundary Layer to a Change in Surface Roughness

The adaptation of the atmospheric boundary layer to a change in the underlying surface roughness is an interesting problem and hence much research, theoretical, experimental, and numerical, has been undertaken. Within the atmospheric boundary layer an accurate numerical model for the turbulent properties of the atmospheric boundary layer needs to be implemented if physically realistic results are to be obtained. Here, the adaptation of the atmospheric boundary layer to a change in surface roughness is investigated using a first-order turbulence closure model, a one-and-a-half-order turbulence closure model and a second-order turbulence closure model. Perturbations to the geostrophic wind and the pressure gradients are included and it is shown that the second-order turbulence closure model, namely the standard k - ∈ model, is inferior to a lower-order closure model if a modification to limit the turbulent eddy size within the atmospheric boundary layer is not included within the model.

Numerical Simulation of Flow over Two-Dimensional Hills Using a Second-Order Turbulence Closure Model

We study turbulent flow over two-dimensional hills. The Reynolds stresses are represented by a second-order closure model, where advection, diffusion, production and dissipation processes are all accounted for. We solve a full set of primitive non-hydrostatic dynamic equations for mean flow quantities using a finite-difference numerical method. The model predictions for the mean velocity and Reynolds stresses are compared with the measured data from a wind-tunnel experiment that simulates the atmospheric boundary layer. The agreement is good. The performance of the second-order closure model is also compared withthat of lower level turbulence models, including the eddy-viscositymodel and algebraic Reynolds stress models. It is concluded that thepresent closure is a considerable improvement over the other modelsin representing various physical effects in flow over hills. Thefeasibility of running a finite-difference numerical simulationincorporating a full second-order closure model on an IBM workstationis also demonstrated.

A one-dimensional physical-biological model study of the pelagic nitrogen cycling during the spring bloom in the northern North Sea (FLEX '76)

A one-dimensional model of the pelagic ecosystem was developed and applied to the spring bloom in the northern North Sea making use of the data set obtained during the Fladenground experiment FLEX '76. The physical submodel is the second-order turbulence closure model of level 2 type developed by Mellor and Yamada (1974, 1982). The biological submodel is a depth-resolved version of the nitrogen flux model of the lower trophic levels in the pelagic proposed by Fasham et al. (1990). The parameter set employed by Fasham et al. did not yield satisfying results. However, using a parameter set adapted to the North Sea ecosystem we obtained a realistic overall description of the development of the North Sea ecosystem during the spring bloom. We were able to hindcast successfully the onset, duration, magnitude and daily variability of the net primary production, the magnitude of the PON export flux to the sea bottom, of the bacterial production and of the nitrogen regeneration within the water column. From the results of the simulation a mass budget of nitrogen fluxes within the euphotic zone and the deeper water layers as well as between them was derived. The results of the simulation suggest that strong herbivorous grazing caused the decay of the bloom. The comparison with the grazing by mesozooplankton as estimated from the observations favors the hypothesis that herbivorous microzooplankton was mainly responsible for the breakdown. The depth dependence of the vertical particulate flux obtained from the simulation exhibits the hyperbolic character recently found in different oceanic regions. The vertical particulate nitrogen flux shows a stronger decrease than typically observed for the particulate carbon flux. This is in correspondence with the observation that there was a remarkable increase of the C/N ratio of POM with depth during FLEX '76.

An aerodynamic analysis of turbine cascade by using a second-order closure of turbulence

Numerical predictions of cascade flows within an axial turbine stage are analyzed in this paper. A second-order Reynolds stress turbulence closure model was employed, which was formulated using the body-fitted curvilinear coordinates. The fractional step method of a finite difference scheme was employed for the numerical computations. The computations of the pressure, heat transfer, and turbulence stresses are presented in both stator and rotor flow regions. Comparisons are made between the Reynolds stress model and the k - ε model and demonstrated the superiority of the Reynolds stress model.

Second-Order Closure Study of Open-Channel Flows

A complete second-order closure model of turbulence is used to predict the behavior of fully developed turbulent flows in open channels of both simple and compound cross sections. This level of closure entails the solution of seven differential transport equations for turbulence quantities and is found to reproduce fairly accurately the details of the turbulence-driven secondary motions that occur in the cross-stream planes. In particular, the number, location, and strength of the secondary-flow cells are well predicted, as is their effect on the bulk properties of the mean flow. Data from simple rectangular channels are used to check the model's sensitivity to the effects of a free surface, and data from compound channels, both symmetric and asymmetric, serve to validate the model for a range of parameters typical of those encountered in practice.

An approximate analytical theory to predict the diurnal effects of local advection

An approximate analytical theory that predicts the diurnal effects of local advection is described. Von Karman's integral method for steady-state advection is combined with the one-dimensional (vertical) time-varying heat and vapour transport theory previously derived by the author. The time-average part of the theory is in reasonably good agreement with profiles and flux densities calculated in the literature with a second-order turbulence closure model for two local advection situations. The time-average part of the theory is also in good agreement with the measured profiles of temperature and vapour density for these two situations and the measured latent heat flux density for the second situation. Diurnal variations predicted with the theory for the first situation are presented. Some improvements to the theory are suggested.

A meso-scale plume dispersion model. Preliminary evaluation in a heterogeneous area

A plume dispersion model for meso-γ-scale applications is developed. The model is formulated with the aim to study plume spreading in moderately complex terrain. An eddy diffusivity analogue is used in order to facilitate future inclusion of chemical transformations. The full diffusivity tensor, obtained from a second-order closure turbulence model, is used to described the dispersion process. The model is driven with simulated meteorological fields, obtained by means of a second-order turbulence closure model. An initial model evaluation has been undertaken using model predictions for an area featuring horizontally inhomogeneous dispersion conditions. Comparison with measurements, as well as with predictions of several other model types for distances out to 40 km, shows that the model is able to describe realistically plume widths and concentration in areas with strongly varying dispersion conditions. Non-negligible errors in plume position were however found for all models compared.