Gradient Richardson Number Rig Research Articles

Growth laws and self-similarity in the confined mixing zone of unstratified and strongly stably stratified isokinetic mixing-layers past a splitter plate

The velocity U and the mixing scalar field Θ of turbulent wake flows developing downstream of a slender splitter plate were studied experimentally for unstratified and strongly stably stratified conditions in a square channel with water as fluid. The experimental program comprises five different Reynolds numbers spanning Re0=10 000–60 000 for u0=u1=u2 (isokinetic conditions) between the two planar, initially separated streams and five relative density stratifications from Δρ=0 to 10%, since mixing studies for these stratification strengths are rather limited. This full-channel Reynolds number uses the hydraulic diameter dh=50 mm of the mixing section as the characteristic length scale. These experiments challenge the corresponding numerical calculations based on Reynolds-averaged Navier–Stokes approaches relying on the Boussinesq approximations of the first and second kind. The development of the mixing scalar field Θ was measured by two fundamentally different approaches: laser induced fluorescence (LIF) in the downstream x–y mid-plane and wire-mesh sensors (WMSs) in discrete lateral y–z planes. This allows for a direct comparison of both sensing techniques at locations where both planes intersect. For the unstratified conditions, it is shown that the downstream developing wake velocity deficit decay agrees well with the theoretical approach. The concentration fields based on LIF and WMS profiles of the mean and fluctuating (root mean square, RMS) data confirm self-similarity by introducing a mixing scalar Θ for both techniques. In the approximate limit, it is shown that the mean concentration field can be well described by an error function and the corresponding RMS data by a Gaussian profile in self-similar coordinates. By using the momentum thickness θm at the splitter plate as the normalization parameter, the downstream developing width of the mixing zone δ(x) becomes independent for Re0≥30 000. The mean and the RMS values of the concentration field Θ for the stably stratified experiments show self-similarity in the core of the concentration field but also a departure from the error function and Gaussian profile in the outer parts. Introducing a local gradient Richardson number Rig, it is shown that both the growth and growth-suppression of the mixing zone are well described by the Miles–Howard criterion for all the stratified experiments considered, even though this criterion was originally developed for shear layers.

The Role of Sand in Wave Boundary Layers Over Primarily Muddy Seabeds: Implications for Wave‐Supported Gravity Flows

AbstractWe performed laboratory experiments to investigate the influence of sand content on the dynamics of wave‐supported gravity flows in mud‐dominant environments. The experiments were carried out in an oscillatory water tunnel with a sediment bed of either 1% or 13% sand. Low and high energy regimes are differentiated based on a Stokes Reynolds number ReΔ ≈ 500. In the low energy regime, the sand fraction influences flow dynamics primarily through ripple formation; no ripples form in the 1% sand experiments, whereas ripples form in the 13% experiments that increase turbulence and the wave boundary layer thickness, δm. In the high energy regime, small ripples form in both the 1% and 13% sand experiments and we observe high near‐bed suspended sediment concentrations. The influence of stratification on the boundary layer flow is characterized in terms of the gradient Richardson number Rig. The flow is weakly stratified inside the boundary layer for all runs and critically stratified at or above the top of the boundary layer. In the lower energy regime, the sand content reduces the relative influence of stratification in the boundary layer, shifting the elevation of critical stratification, LB, from approximately 1.3δm to 2.5δm in the 1% and 13% experiments, respectively. In both sets of experiments LB ≈ δm at the strongest wave energy, indicating a transition to strongly stratified dynamics.

Vertical dispersion of Lagrangian tracers in fully developed stably stratified turbulence

We study the effect of different forcing functions and of the local gradient Richardson number Rig on the vertical mixing of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation, and present a wave and continuous-time random walk model for single- and two- particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude based on the observed Lagrangian spectrum of vertical velocity, and a random walk process to capture overturning that depends on the statistics of Rig among other Eulerian quantities. The model is in good agreement with direct numerical simulations of stratified turbulence, where single- and two-particle dispersion differs from the homogeneous and isotropic case. Moreover, the model gives insight into the mixture of linear and non-linear physics in the problem, as well as on the different processes responsible for vertical turbulent dispersion.

Measurements of mixing parameters in atmospheric stably stratified parallel shear flow

The mixing coefficient Г = B/e, defined as the ratio of the magnitude of buoyancy flux B to the rate of turbulent kinetic energy (TKE) dissipation e, plays a key role in modeling atmospheric and oceanic flows. Г is sometimes estimated using yet another fundamental quantity, the flux Richardson number (or mixing efficiency) Rif = B/P, where P is the rate of production of TKE. In practice, Г and Rif are commonly assumed as constants, but studies show that they depend on multiple parameters determined by the type of flow, for example, the gradient Richardson number Rig for stratified shear flows. During the MATERHORN field program, direct measurements of velocity and temperature profiles as well as B, e, and P were made over an extended period using a densely instrumented flux tower. A ~ 90 min period of stratified parallel shear flow was identified in the data record, including recurrent intervals of nominally stationary flow. Measurements during this period supported the study of mixing parameters as reported in this paper. Even for this case of nature resembling an ideal flow, Г was found to be dependent on multiple parameters. Nevertheless, for periods robustly identified as shear flow in equilibrium with embedded turbulence, the measurements were in agreement with those of past stratified shear flow experiments in the laboratory. This result points to the challenges of parameterizing turbulent mixing in environmental flow models.

On the mixing models for stratified flows subjected to concomitant stable and unstable stratifications

ABSTRACTThe concurrence of stable and unstable stratification in stratified flows leads to dramatically different features of turbulent mixing. This unique flow is experimentally studied by introducing a horizontal jet of dense fluid into a pool of light fluid. The buoyancy flux from simultaneous velocity–density measurements is an indicator for competition between a stabilising mechanism and another destabilising mechanism. The difference of mixing efficiency, quantified by flux Richardson number Rif, between the (un)stable stratification is mainly contributed by the large-scale mixing. The behaviour of Rif can be modelled by the gradient Richardson number Rig linearly in the low-Ri case and nonlinearly in the high-Ri case (especially in a region where the counter-gradient flux emerges). The turbulent diapycnal diffusivity, quantifying the combined effect of reversible and irreversible mixing processes, increases as the buoyancy Reynolds number Reb increases only when Reb is large. The irreversible mixing diffusivity, which quantifies the sole irreversible mixing process, increases linearly as the turbulent Péclet number with the data points from the (un)stable stratification overlapped. The turbulent Prandtl number approaches 0.75 as Rig approaches zero, but does not show clear dependence on Rig in the examined regime.

Stratified Turbulence and Mixing Efficiency in a Salt Wedge Estuary

AbstractHigh-resolution observations of velocity, salinity, and turbulence quantities were collected in a salt wedge estuary to quantify the efficiency of stratified mixing in a high-energy environment. During the ebb tide, a midwater column layer of strong shear and stratification developed, exhibiting near-critical gradient Richardson numbers and turbulent kinetic energy (TKE) dissipation rates greater than 10−4 m2 s−3, based on inertial subrange spectra. Collocated estimates of scalar variance dissipation from microconductivity sensors were used to estimate buoyancy flux and the flux Richardson number Rif. The majority of the samples were outside the boundary layer, based on the ratio of Ozmidov and boundary length scales, and had a mean Rif = 0.23 ± 0.01 (dissipation flux coefficient Γ = 0.30 ± 0.02) and a median gradient Richardson number Rig = 0.25. The boundary-influenced subset of the data had decreased efficiency, with Rif = 0.17 ± 0.02 (Γ = 0.20 ± 0.03) and median Rig = 0.16. The relationship between Rif and Rig was consistent with a turbulent Prandtl number of 1. Acoustic backscatter imagery revealed coherent braids in the mixing layer during the early ebb and a transition to more homogeneous turbulence in the midebb. A temporal trend in efficiency was also visible, with higher efficiency in the early ebb and lower efficiency in the late ebb when the bottom boundary layer had greater influence on the flow. These findings show that mixing efficiency of turbulence in a continuously forced, energetic, free shear layer can be significantly greater than the broadly cited upper bound from Osborn of 0.15–0.17.

Dynamics of dense gravity currents and mixing in an up-sloping and converging vee-shaped channel

Detailed velocity and density measurements are used to investigate dense water dynamics in an inclined, silled channel of triangular cross-section with varying side slope α and adverse bed slope φ. For the steeper channel configuration considered (φ=3.6°), the dense-water bottom current is shown to be frictionally-controlled, with an internal flow structure characterized by a sharp pycnocline and decreasing isopycnal separation in the along-channel direction. For the milder up-sloping channel (φ=1.7°), the dense water outflow is shown to be hydraulically-controlled as the channel sill section is approached, with internal flow dynamics characterized by increasing isopycnal separation in the along-channel direction. Analysis of the gradient Richardson number Rig of the flow confirms that hydraulically-controlled flows dilute the active bottom water due to interfacial mixing. A gradually-varying internal flow model and a two-layer hydraulic modelling approach are shown, respectively, to represent adequately the outflow behaviour for these two bed slope conditions.

Large-Eddy Simulation of Deep-Cycle Turbulence in an Equatorial Undercurrent Model

Abstract Dynamical processes leading to deep-cycle turbulence in the Equatorial Undercurrent (EUC) are investigated using a high-resolution large-eddy simulation (LES) model. Components of the model include a background flow similar to the observed EUC, a steady westward wind stress, and a diurnal surface buoyancy flux. An LES of a 3-night period shows the presence of narrowband isopycnal oscillations near the local buoyancy frequency N as well as nightly bursts of deep-cycle turbulence at depths well below the surface mixed layer, the two phenomena that have been widely noted in observations. The deep cycle of turbulence is initiated when the surface heating in the evening relaxes, allowing a region with enhanced shear and a gradient Richardson number Rig less than 0.2 to form below the surface mixed layer. The region with enhanced shear moves downward into the EUC and is accompanied by shear instabilities and bursts of turbulence. The dissipation rate during the turbulence bursts is elevated by up to three orders of magnitude. Each burst is preceded by westward-propagating oscillations having a frequency of 0.004–0.005 Hz and a wavelength of 314–960 m. The Rig that was marginally stable in this region decreases to less than 0.2 prior to the bursts. A downward turbulent flux of momentum increases the shear at depth and reduces Rig. Evolution of the deep-cycle turbulence includes Kelvin–Helmholtz-like billows as well as vortices that penetrate downward and are stretched by the EUC shear.

Near-N Oscillations and Deep-Cycle Turbulence in an Upper-Equatorial Undercurrent Model

Abstract Direct numerical simulation (DNS) is used to investigate the role of shear instabilities in turbulent mixing in a model of the upper Equatorial Undercurrent (EUC). The background flow consists of a westward-moving surface mixed layer above a stably stratified EUC flowing to the east. An important characteristic of the eastward current is that the gradient Richardson number Rig is larger than ¼. Nevertheless, the overall flow is unstable and DNS is used to investigate the generation of intermittent bursts of turbulent motions within the EUC region where Rig > ¼. In this model, an asymmetric Holmboe instability emerges at the base of the mixed layer, moves at the speed of the local velocity, and ejects wisps of fluid from the EUC upward. At the crests of the Holmboe waves, secondary Kelvin–Helmholtz instabilities develop, leading to three-dimensional turbulent motions. Vortices formed by the Kelvin–Helmholtz instability are occasionally ejected downward and stretched by the EUC into a horseshoe configuration creating intermittent bursts of turbulence at depth. Vertically coherent oscillations, with wavelength and frequency matching those of the Holmboe waves, propagate horizontally in the EUC where the turbulent mixing by the horseshoe vortices occurs. The oscillations are able to transport momentum and energy from the mixed layer downward into the EUC. They do not overturn the isopycnals, however, and, though correlated in space and time with the turbulent bursts, are not directly responsible for their generation. These wavelike features and intermittent turbulent bursts are qualitatively similar to the near-N oscillations and the deep-cycle turbulence observed at the upper flank of the Pacific Equatorial Undercurrent.

Parameterizing individual effects of shear and stratification on mixing in stably stratified shear flows

This study proposes new parameterizations of diapycnal mixing by reanalyzing the results of previous laboratory and numerical experiments on homogeneous stably stratified shear flows. Unlike previous studies that use either the turbulent Froude number Fr or gradient Richardson number Rig, this study parameterizes nondimensional momentum and buoyancy fluxes as functions of Fr and a turbulent shear number Sh, in order to quantify individual effects of shear and stratification. Turbulent momentum flux is found to depend linearly on Sh and to decrease monotonically with decreasing Fr. Turbulent buoyancy flux has a peak at moderate Fr. With increasing Sh, it decreases and increases at high and low Fr, respectively. The increase of Sh also cause relatively small but significant decreases of nondimensional turbulent properties, such as the nondimensional conversion rate of turbulent potential energy to background potential energy. The proposed parameterizations lie within the scatter of limited available field data. The parameterizations may be reduced to Rig‐based ones by incorporating the relationship between Rigand turbulence intensity observed in the field. Existing stability functions for two‐equation turbulent closure schemes are found to over‐predict mixing efficiency at low Fr.

Efficiency of eddy mixing in a stable stratified atmospheric boundary layer

Based on a mesoscale RANS model of turbulence, the behavior of turbulent eddy mixing parameters is found to agree with the latest data of laboratory and atmospheric measurements. Some problems of the description of turbulent eddy mixing in the atmospheric boundary layer are studied. When the flow transforms to an extremely stable state, in particular, it is found the flux Richardson number Ri f can change nonmonotonically: it increases with increasing gradient Richardson number Rig until the state of saturation is reached at Ri g ≃ 1 and then decreases. The behavior of the coefficients of eddy diffusion of momentum and heat agrees with the concept of momentum (but not heat) transfer by internal waves propagating in an extremely stable atmospheric boundary layer.

The breaking of interfacial waves at a submerged bathymetric ridge

The breaking of periodic progressive two-layer interfacial waves at a Gaussian ridge is investigated through laboratory experiments. Length scales of the incident wave and topography are used to parameterize when and how breaking occurs. Qualitative observations suggest both shear and convection play a role in the instability of waves breaking at the ridge. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields from which the local gradient Richardson number Rig is calculated. The transition to breaking occurred when 0.2 ≤ Rig ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Rig than for similar waves breaking through shear instability in deep water (Troy & Koseff, J. Fluid Mech., vol. 543, 2005b, p. 107). Due to the effects of unsteadiness, nonlinear shoaling and flow separation, the canonical Rig > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Rig in waves between finite depth layers using scales of the incident wave scale and topography. The observed breaking transition corresponds with a constant estimated value of Rig from the model, suggesting that interfacial shear plays an important role in initial wave instability. For wave amplitudes above the initial breaking transition, convective breaking events are also observed.

Effects of ambient velocity shear on nonlinear internal wave associated mixing at the Columbia River plume front

Large‐amplitude nonlinear internal waves (NLIWs) are frequently observed propagating away from Columbia River tidal plume fronts. They are generated because of the deceleration of the frontal bulge. During the River Influences on Shelf Ecosystems project cruises, the velocity, density structure and acoustic backscatter of the plume fronts and frontal NLIWs were observed using a towed vehicle, vessel‐mounted instrumentation, and a vessel X band radar. These observations indicate that in the presence of strong ambient velocity shear, the NLIWs with maximum amplitudes occur well below the density interface and at a depth deeper than in the absence of shear. This deepening of the NLIW shear is associated with intensified vertical mixing below the interface. This is consistent with a dynamic analysis of NLIWs under the influence of sheared ambient flow, obtained from a high‐order KdV model. There are two mechanisms responsible for the vertical turbulent mixing intensification. The nonlinear interaction between ambient shear and the NLIWs increases total velocity shear and causes the gradient Richardson number Rig to decrease below a critical value in certain depth ranges. Also, because of the presence of the ambient shear, the maximum NLIW velocity shear occurs at a depth below the density interface where there is less stratification.

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.

Entrainment and mixing in stratified shear flows

The results of a laboratory experiment designed to study turbulent entrainment at sheared density interfaces are described. A stratified shear layer, across which a velocity difference ΔU and buoyancy difference Δb is imposed, separates a lighter upper turbulent layer of depth D from a quiescent, deep lower layer which is either homogeneous (two-layer case) or linearly stratified with a buoyancy frequency N (linearly stratified case). In the parameter ranges investigated the flow is mainly determined by two parameters: the bulk Richardson number RiB = ΔbD/ΔU2 and the frequency ratio fN = ND=ΔU.When RiB > 1.5, there is a growing significance of buoyancy effects upon the entrainment process; it is observed that interfacial instabilities locally mix heavy and light fluid layers, and thus facilitate the less energetic mixed-layer turbulent eddies in scouring the interface and lifting partially mixed fluid. The nature of the instability is dependent on RiB, or a related parameter, the local gradient Richardson number Rig = N2L/ (∂u/∂z)2, where NL is the local buoyancy frequency, u is the local streamwise velocity and z is the vertical coordinate. The transition from the Kelvin–Helmholtz (K-H) instability dominated regime to a second shear instability, namely growing Hölmböe waves, occurs through a transitional regime 3.2 < RiB < 5.8. The K-H activity completely subsided beyond RiB ∼ 5 or Rig ∼ 1. The transition period 3.2 < RiB < 5 was characterized by the presence of both K-H billows and wave-like features, interacting with each other while breaking and causing intense mixing. The flux Richardson number Rif or the mixing efficiency peaked during this transition period, with a maximum of Rif ∼ 0.4 at RiB ∼ 5 or Rig ∼ 1. The interface at 5 < RiB < 5.8 was dominated by ‘asymmetric’ interfacial waves, which gradually transitioned to (symmetric) Hölmböe waves at RiB > 5:8.Laser-induced fluorescence measurements of both the interfacial buoyancy flux and the entrainment rate showed a large disparity (as large as 50%) between the two-layer and the linearly stratified cases in the range 1.5 < RiB < 5. In particular, the buoyancy flux (and the entrainment rate) was higher when internal waves were not permitted to propagate into the deep layer, in which case more energy was available for interfacial mixing. When the lower layer was linearly stratified, the internal waves appeared to be excited by an ‘interfacial swelling’ phenomenon, characterized by the recurrence of groups or packets of K-H billows, their degeneration into turbulence and subsequent mixing, interfacial thickening and scouring of the thickened interface by turbulent eddies.Estimation of the turbulent kinetic energy (TKE) budget in the interfacial zone for the two-layer case based on the parameter α, where α = (−B + ε)/P, indicated an approximate balance (α ∼ 1) between the shear production P, buoyancy flux B and the dissipation rate ε, except in the range RiB < 5 where K-H driven mixing was active.