Flux Richardson Number Research Articles

Resolution Sensitivity and Scaling of Large-Eddy Simulations of the Stable Boundary Layer

Large-eddy simulations (LES) of the continuously turbulent quasi-equilibrium stable boundary layer (SBL) are conducted with grid lengths in the range of 12.5 m to 2 m, in order to explore resolution sensitivity, and determine at what point grid convergence occurs. The structure of the mean potential temperature, winds, and turbulent fluxes varies significantly over this resolution range. The highest resolution simulations show a significant degree of convergence. The dimensionless momentum diffusivity asymptotes to a value of 0.06, corresponding to a limiting flux Richardson number of 0.15. Using the converged simulations, some scaling hypotheses underpinning first-order and second-order closure models are revisited. The effective Richardson number stability functions of the LES are compared with the forms often used in numerical weather prediction (NWP). The mixing implied by the LES is less than that used in NWP. The commonly used similarity profiles for heat and momentum fluxes, and the scalings for dissipation and pressure covariances are compared with the LES. This information could provide guidance for the next generation of SBL parametrization schemes.

Modelling of sediment suspensions in oscillating grid turbulence

A k– ε model is used to describe the steady state of fine-grained sediments maintained in suspension by purely diffusive turbulence, as generated in oscillating grid turbulence experiments. The behaviour is shown to depend both on the bulk Rouse number Rou 0 and the product of the bulk Rouse number and the bulk Richardson number Ri 0 Rou 0 , built on oscillating grid parameters. For Rou 0<0.01, concentrated suspensions are observed with a homogeneous particle concentration in the suspension layer. An interface, called lutocline, separates the suspension layer from the clear water at a distance z m from the grid. The depth of the suspension layer is found to vary as z m/z 0∝(Ri 0 Rou 0) −1/4 . For Ri 0 Rou 0⪡1 the decay of turbulence is affected by the particle concentration only in a region very close to the interface. In this case the flux Richardson number approaches the value of 1 near the interface. The lutocline is seen to vanish for large values of Rou 0. For Rou 0>0.01 the mean sediment concentration and turbulence decay simultaneously with increasing distance from the grid, and no sharp interface is observed.

Turbulent energy production and entrainment at a highly stratified estuarine front

Rates of turbulent kinetic energy (TKE) production and buoyancy flux in the region immediately seaward (∼1 km) of a highly stratified estuarine front at the mouth of the Fraser River (British Columbia, Canada) are calculated using a control volume approach. The calculations are based on field data obtained from shipboard instrumentation, specifically velocity data from a ship mounted acoustic Doppler current profiler (ADCP), and salinity data from a towed conductivity‐temperature‐depth (CTD) unit. The results allow for the calculation of vertical velocities in the water column, and the total vertical transport of salt and momentum. The vertical turbulent transport quantities (, ) can then be estimated as the difference between the total transport and the advective transport. Estimated production is on the order of 10−3 m2 s−3, yielding a value of ɛ(νN2)−1 on the order of 104. This rate of TKE production is at the upper limit of reported values for ocean and coastal environments. Flux Richardson numbers in this highly energetic system generally range from 0.15 to 0.2, with most mixing occurring at gradient Richardson numbers slightly less than . These values compare favorably with other values in the literature that are associated with turbulence observations from regimes characterized by scales several orders of magnitude smaller than are present in the Fraser River.

Turbulence Closure, Steady State, and Collapse into Waves

A new simple two-equation turbulence closure is constructed by hypothesizing that there is an extra energy sink in the turbulent kinetic energy (k) equation representing the transfer of energy from k to internal waves and other nonturbulent motions. This sink neither contributes to the buoyancy flux nor to dissipation, the nonturbulent mode being treated as inviscid. The extra sink is proportional to the squared ratio between the turbulent time scale t ; k/«, with turbulent dissipation rate «, and the buoyancy period T 5 2p/N. With a focus on high‐Reynolds number, spatially homogeneous, stably stratified shear flow away from boundaries, the turbulence is described by equations for a master length scale L ; k3/2/« and the master time scale t. It is assumed that the onset of the collapse of turbulence into nonturbulence occurs at t 5 T. The new theory is almost free of empirical parameters and compares well with laboratory and numerical experiments. Most remarkable is that the model predicts the turbulent Prandtl number, which is generally s 5 s 0/[1 2 (t/T) 2], with s 0 5 1/2, and hence is not a unique function of mean flow variables. Only in structural equilibrium ( 5 0) is the Prandtl t ˙ number a unique function of the gradient Richardson number Rg: s 5 s 0/(1 2 2Rg). These forms of the Prandtl number function immediately determine the flux Richardson number R f 5 Rg/s. Steady state occurs at 5 1/ s Rg 4 with R f 5 1/4, and within structural equilibrium the collapse of turbulence is complete at Rg 5 1/2.

High resolution DNS of jet stream generated tropopausal turbulence

Tropopausal turbulence (TT), generated by jet streams, is examined, for different nonhomogeneous basic state forcing profiles, through high resolution three‐dimensional (3D) direct numerical simulations (DNS). Turbulent core regions, on either side of the jet, are characterized by a dramatic decrease in Brunt‐Väisälä frequency (N), minima in gradient Richardson number (Rig) and shear length scales, and maxima in buoyancy Reynolds number and buoyancy length scales. The edges of turbulent mixing layers are marked by N2‐bulges, which provides an alternate mechanism for multiple tropopauses. While temperature variance peaks in the vicinity of these N2‐bulges, velocity variances peak within the turbulent core. Multiple branches in the scaling of flux Richardson number with Rig are found typical for TT, a result having implications to the problem of turbulence parameterization.

Modification Of The Standard ∈-Equation For The Stable Abl Through Enforced Consistency With Monin–Obukhov Similarity Theory

A condition is derived for consistency of the standard∈-equation with Monin–Obukhov (MO) similarity theory of thestably-stratified surface layer. The condition is derivedby extending the procedure used to derive the analogous condition forneutral theory to stable stratification. It is shown that consistencywith MO theory requires a function of flux Richardson number, Rif, to be absorbed into either of two closure parameters, c∈ 1 or c∈ 2.Inconsistency, on the other hand, results if constant values of these are maintained for all Rif, as is done in standardapplication of the equation, and the large overpredictions ofturbulence found in such application to the one-dimensionalstable atmospheric boundary layer (1D-SBL) are traced to thisinconsistency. Guided by this, we formulate a MO-consistent∈-equation by absorbing the aforementioned function intoc∈ 1, and combine this with a Level-2.5 second-orderclosure model for vertical eddy viscosity and diffusivities.Numerical predictions of the 1D-SBL by the modified model converge to a quasi-steady state, rectifying the predictive failure of the standard∈ -equation for the case.Quasi-steady predictions of non-dimensional variables agree stronglywith Nieuwstadt's theory. Qualitative accuracy of predictionsis inferred from comparisons to field data, large-eddy simulationresults and Rossby-number similarity relationships.

Observations of Flow and Turbulence in the Nocturnal Boundary Layer over a Slope

Measurements were conducted on an eastern slope of the Salt Lake Basin (SLB) as a part of the Vertical Transport and Mixing Experiment (VTMX) conducted in October 2000. Of interest was the nocturnal boundary layer on a slope (in particular, katabatic flows) in the absence of significant synoptic influence. Extensive measurements of mean flow, turbulence, temperature, and solar radiation were made, from which circulation patterns on the slope and the nature of stratified turbulence in katabatic winds were inferred. The results show that near the surface (,25‐50 m) the nocturnal flow is highly stratified and directed downslope, but at higher levels winds strongly vary in magnitude and direction with height and time, implying the domination of upper levels by air intrusions. These intrusions may peel off from different slopes surrounding the SLB, have different densities, and flow at their equilibrium density levels. The turbulence was generally weak and continuous, but sudden increases of turbulence levels were detected as the mean gradient Richardson number ( ) dropped to Rig about unity. With a short timescale fluctuated on the order of a few tens of seconds while modulating with Rig a longer (along-slope internal waves sloshing) timescale of about half an hour. The mixing efficiency (or the flux Richardson number) of the flow was found to be a strong function of , similar to that found in laboratory Rig experiments with inhomogeneous stratified shear flows. The eddy diffusivities of momentum and heat were evaluated, and they showed a systematic variation with when scaled with the shear length scale and the rms Rig vertical velocity of turbulence.

Flux Richardson number measurements in stable atmospheric shear flows

The flux Richardson number Rf (also known as the mixing efficiency) for the stably stratified atmospheric boundary layer is investigated as a function of the gradient Richardson number Rig using data taken during two field studies: the Vertical Transport and Mixing Experiment (VTMX) in Salt Lake City, Utah (October 2000), and a long-term rural field data set from Technical Area 6 (TA-6) at Los Alamos National Laboratory, New Mexico. The results show the existence of a maximum Rf (0.4–0.5) at a gradient Richardson number of approximately unity. These large-Reynolds-number results agree well with recent laboratory stratified shear layer measurements, but are at odds with some commonly used Rf parameterizations, particularly under high-Rig conditions. The observed variations in buoyancy flux and turbulent kinetic energy production are consistent with the concept of global intermittency of the atmospheric stable boundary layer.

Response of turbulent boundary layers to multiple strain rates

Many practical applications, such as in blade cascades and turbomachinery, involve inhomogeneous turbulent shear flows subjected simultaneously to multiple strains. In principle, the applied strain can be combined to yield an effective strain. However, no simple stress–strain relation is capable of establishing turbulent stress or energy balance in the mean or on an instantaneous basis. In the current investigation, a turbulent boundary layer is examined in the presence of convex curvatures of different strengths combined with streamwise (favourable and adverse) pressure gradients, with various values of pressure gradient ratio, (∂P/∂s)/(∂P/∂n). Measurements of the mean and turbulent parameters and flux Richardson number show appreciable changes, mainly in the outer portion of the boundary layer (y+ &gt; 100). The turbulent burst frequency, particularly at the location of application of the additional strain rate, also changes relative to its value with wall curvature alone.Three primary observations from these experiments are as follows: (i) in all cases, the mean velocity profile and all of the measured Reynolds stresses collapse in the near-wall region using standard inner scaling; (ii) the applied strains combine nonlinearly, with one of the strains dominating the local flow during its development; (iii) the ratio of the radial to axial pressure gradient magnitude influences both classical turbulence correlations and mean flow, as well as the physical production cycle of turbulence; and (iv) application rate of newly introduced strain rates is at least as important as their magnitudes.

Mixing Efficiencies in Patchy Turbulence

Abstract The efficiency of mixing in stably stratified systems where the turbulent mixing is confined to intermittent patches is investigated theoretically. It is possible to define two different flux Richardson numbers for mixing in such a system. One, the small-scale flux Richardson number, Rft, is based on the initial potential energy increase caused by small-scale turbulent mixing within the patches. This is the parameter that is obtained from laboratory and numerical experiments intended to determine turbulent mixing efficiencies. The other, the large-scale flux Richardson number, Rf, is based on the final potential energy increase, obtained after the mixed fluid has spread out laterally in the system. This is the relevant parameter for determining large-scale, irreversible, changes in the stratification caused by mixing. It is shown that the large-scale flux Richardson number is always smaller than the small-scale flux Richardson number, and that the difference can be almost a factor of 2. The commo...

Modelling the Leipzig Wind Profile with a (k-ε) model

The Leipzig Wind Profile is generally known as a typical neutral planetary boundary layer flow. But it became clear from the present research that it was not completely neutral but weakly stable. We examined whether we could simulate the Leipzig Wind Profile by using a (<TEX>$k-{\varepsilon}$</TEX>) turbulence model including the equation of potential temperature. By solving analytically the Second Moment Closure Model under the assumption of local equilibrium and under the condition of a stratified flow, we expressed the turbulent diffusion coefficients (both momentum and thermal) as functions of flux Richardson number. Our (<TEX>$k-{\varepsilon}$</TEX>) turbulence model which included the equation of potential temperature and the turbulent diffusion coefficients varying with flux Richardson number reproduced the Leipzig Wind Profile.

Modelling Thermal Stratification in the North Sea: Application of a 2-D Potential Energy Model

The spatial and temporal dynamics of the North Sea ecosystem are dependent upon vertical mixing processes which modify the availability of light and limit nutrients for phytoplankton production. In order to examine the effects of inter and intra annual variations in stratification on ecosystem dynamics we have developed and tested a potential energy model of thermal stratification based on the energy equation (for turbulence). The energy equation relates the temporal and spatial changes of turbulent kinetic energy (TKE), the production of TKE and the dissipation of TKE to the change of potential energy as water masses of different densities are mixed in the field of gravity. A constant ratio between the gain in potential energy and the production of TKE is assumed, known as the flux Richardson number. The model is comprised of 0·5m vertical layers with a temporal time step of 1 day. The model is forced with wind, dew point temperature from Ekofisk oilfield in the central North Sea, and tidal current and atmospheric radiation. The model is used to simulate the seasonal cycle of stratification in the central North Sea in the years 1988, 1989 and 1990 and is compared to density profiles in these years available from the ICES hydrographic database. We find that the model is able to simulate variations in thermal stratification including the seasonal onset and breakdown of stratification, the thermocline depth, and the effects of discrete wind and cooling events. For the years 1988–1990 we find an R2=0·97 between observed and predicted upper layer temperatures. However, the model is less successful in the prediction of temperatures of the intermediate and deep layers (R2=0·46 and 0·14) due to small deviations in thermocline depth and variations in tidal amplitude. The model was then applied to examine potential differences in stratification between the years 1990 and 1996. Simulations suggested that the development of stratification is very rapid in 1990 and that the spring of 1996 is very cold. Both of these observations having the potential to impact on the efficiency of lower trophic level coupling and production.

Energetics of small scale turbulence in the lower stratosphere from high resolution radar measurements

Abstract. Very high resolution radar measurements were performed in the troposphere and lower stratosphere by means of the PROUST radar. The PROUST radar operates in the UHF band (961 MHz) and is located in St. Santin, France (44°39’ N, 2°12’ E). A field campaign involving high resolution balloon measurements and the PROUST radar was conducted during April 1998. Under the classical hypothesis that refractive index inhomogeneities at half radar wavelength lie within the inertial subrange, assumed to be isotropic, kinetic energy and temperature variance dissipation rates were estimated independently in the lower stratosphere. The dissipation rate of temperature variance is proportional to the dissipation rate of available potential energy. We therefore estimate the ratio of dissipation rates of potential to kinetic energy. This ratio is a key parameter of atmospheric turbulence which, in locally homogeneous and stationary conditions, is simply related to the flux Richardson number, Rf .Key words. Meteorology and atmospheric dynamics (turbulence) – Radio science (remote sensing)

Vertical Mixing and Transports through a Stratified Shear Layer

Abstract A stratified shear layer was generated in the laboratory by driving a turbulent mixed layer of depth D over a quiescent, deep dense layer. As a result, a density interface of thickness δb across which the buoyancy jump is Δb was formed between the upper and lower layers. This density interface was embedded in a velocity shear layer of thickness δs across which the velocity jump was ΔU. Detailed velocity, density, and average local Richardson number (Rig) measurements were made through the stratified shear layer, from which the fluxes of momentum and density through the interface as well as energetics of the stratified shear layer were evaluated as a function of Rig. The quantities measured included the flux Richardson number (Rif), the dissipation flux coefficient (Γ), and the eddy diffusivities of momentum and density (Km and Kρ), averaged across the shear layer. The results were compared with various deep and coastal oceanic data as well as common oceanic eddy diffusivity and flux parameterizat...

The Energetics and Tidally Induced Reverse Renewal in a Two-Silled Fjord

A laterally integrated, two-dimensional numerical model is used to examine the influence of the M2 tide on the circulation in the Saguenay Fjord, a two-silled fjord (with a “large” inner and a “small” outer basin) located on the north shore of the St. Lawrence Estuary. It is found that the M2 tide is more vigorous in the outer than in the inner basin and that more vertical mixing occurs in the outer basin. Therefore, the density at depth in the outer basin decreases faster than it does in the inner basin, and the resulting horizontal pressure gradient causes a bottom flow of water from the inner to the outer basin across the inner sill. This “reverse renewal” is evident in both the available observations and the simulation. According to the model, much of the M2 energy withdrawn from the surface tide is fed into the internal tide. Significant tidal energy is also advected by the mean flow velocity. Approximately 25% of the net energy flux into the fjord is dissipated within 2 km of the outer sill. Because of the baroclinic pressure (i.e., the total pressure but with the influence of the surface displacement removed) the subtidal circulation is associated with very large energy fluxes within the fjord. These fluxes are much larger than the net rate at which tidal energy enters the fjord and they are greater than the subtidal, advective energy flux. They represent a large redistribution of energy within the fjord and their horizontal divergence along the fjord is almost in balance with the time rate of change of potential energy. When a flux Richardson number is calculated for the inner basin below sill depth, a value of 0.075 is obtained, which is close to values found for other fjords.

Turbulent flow over a wavy surface: Stratified case

Stably stratified turbulent flow over wavy terrain is investigated using large‐eddy simulation for laboratory‐scale domains. Basic near‐wall turbulence structure and production mechanisms appear similar to the neutral case, though dampened. The state of turbulence in the stratified fluid is analyzed by graphing the relationships between key nondimensional parameters. A shifting and smearing of the curve of flux Richardson number Rif versus turbulent Froude number Frt caused by the wavy boundary (compared with a flat boundary) demonstrates the higher level of turbulence near the wavy wall and shows strong mixing in the detached shear layer over the trough. The relationship describing the state of turbulence in a stratifed fluid given by Ivey and Imberger [1991] is approximately satisfied. Conditional sampling is used to analyze the nature of Reynolds stress events.

Budgets of turbulent kinetic energy and scalar variance in the continental shelf bottom boundary layer

A local turbulent kinetic energy (TKE) budget, in which dissipation balances production minus buoyancy flux, and a nonlocal turbulent scalar variance (TSV) budget, in which dissipation balances production minus TSV flux divergence, are tested observationally. The calculations are based on 6 week long records of velocity and sound speed measured between 0.7 and 5.4 m above bottom on the New England shelf. Estimates of fluxes and gradients are direct, while estimates of dissipation rates are indirectly obtained from inertial subrange spectra. Within 2 m of the bottom, buoyancy flux is not an important component of the TKE budget (flux Richardson number Rf&lt;0.05, on average). At 4.35 m above the bottom, the highest resolved elevation, the direct effect of stratification is more pronounced (Rf = 0.13, on average). The local TKE balance is essentially closed by the measurements within 2 m of the bed (r2&gt;0.8 and regression coefficient = 0.89 ×/÷ 1.06 and 1.08 ×/÷ 1.05 at 1.65 and 0.74 m above the bottom, respectively). The local TKE budget at 4.35 m above the bed is not closed, however. TSV production and dissipation estimates are strongly correlated at all heights (r2&gt;0.64), but dissipation is systematically greater than production, particularly at 0.74 m above the bed (regression coefficient = 2.42 ×/÷ 1.28). Including TSV flux divergence in the balance reduces, statistically significantly, the observed discrepancy. The implications of the results for turbulence closure models and microstructure diffusivity estimates are discussed.

Maximal mixing rate in turbulent stably stratified Couette flow

A rigorous upper bound on the long-time-averaged vertical buoyancy flux is derived from the Navier–Stokes equations for a Boussinesq fluid confined between two parallel horizontal plates a distance d apart, maintained at a constant statically stabilizing temperature difference ΔT and driven at a constant relative velocity ΔU. The upper bound on the volume and long-time-averaged vertical buoyancy flux B≔limt→∞1/t∫0t 〈ρu3〉g/ρ0 dt̃ is B⩽Bmax=(1−16√/Re)(ΔU)3/(64√d), where Re=ΔUd/ν and ρ0 is some reference density. Significantly, Bmax is independent of the bulk Richardson number of the flow and is achieved by an optimal solution with a mixing efficiency (or flux Richardson number) which approaches 0.5 as the Reynolds number becomes large. The time-averaged turbulent dissipation of kinetic energy and the time-averaged vertical buoyancy flux are then in equipartition for the optimizing flow.

Microstructure Observations of Turbulent Mixing in a Partially Mixed Estuary. Part II: Salt Flux and Stress

Abstract Turbulent mixing in the water column was observed with a microstructure profiler in the Hudson River estuary during two cruises in summer and fall 1995. The focus is on the estimation of turbulent salt flux and turbulent stress from measured viscous dissipation rates (e), and on the tidal and fortnightly variability of these fluxes. In estimating eddy viscosity (Km) and eddy diffusivity (Kρ), the authors follow measurement/modeling techniques of Busch and Osborn, while prescribing a variable flux Richardson number (Rf) dependent upon the gradient Richardson number (Ri). It is argued that a steady-state production–dissipation balance holds in the turbulent kinetic energy budget. All turbulence characteristics varied strongly over semidiurnal tidal cycles and over the fortnightly cycle. Subject to complications arising from nontidal flows, the “strongest” mixing occurred during flood on neap tides, and during ebb on spring tides. In the lower part of the water column during floods and spring ebb Km...

Turbulence closure modelling of sediment transport beneath large waves

The development of reliable coastal sediment transport and morphological models depends upon accurate parameterizations of sand transport processes. Here, an existing, local, one-dimensional vertical (1DV), one-equation turbulent kinetic energy ( k) numerical model is extended to study the effects of (i) graded sediment sizes, (ii) turbulence damping and (iii) hindered settling, on the suspended sediment concentration beneath large symmetrical waves, and also on sediment transport rates beneath large asymmetrical waves. The aim of the paper is to explore these effects initially in isolation, and then in combination with one another. Size gradation is treated simply by dividing the bed material into volumetrically equal fractions, and modelling the response of each fraction; turbulence damping is represented primarily by a flux Richardson number correction; while hindered settling is represented by a simplified correction to the settling velocity for grains in isolation. The inclusion of all three physical processes is shown to lead to an improvement in predictive capability of the extended model compared with the one-equation k-closure model discussed by Davies and Li (Continental Shelf Res. 17 (5) (1997) 555–582).