Stable Density Gradient Research Articles

Two regimes of dilute turbulent settling suspensions under shear

When turbulent flow is laden with negatively buoyant particles, their mean distribution in the direction of gravity can induce stable density gradients that penalize turbulent fluctuations. This effect is studied numerically for shear-driven flow with dilute noninertial sediment. The turbulent dynamics and sediment transport depend critically on particle settling velocity vs, splitting into two regimes: homogeneous weakly stratified turbulence and flow with developed turbulence atop an intermittent boundary layer. At intermediate vs, neither state can be sustained and the flow laminarizes. Published by the American Physical Society 2024

Convection in conditionally unstable seawater

Buoyancy-driven convection in a conditionally unstable ocean is studied theoretically. Conditionally unstable conditions are related to supercooled seawater. The freezing point is depressed due to increasing pressure, and upward motion (reduced pressure) leads to the formation of ice crystals in the form of frazil ice and hence a reduced bulk density of the rising mass parcel. For downward motion (increasing pressure), freezing does not occur. To model this one-way process, we take that rising parcels become lighter as they follow the adiabatic density lapse rate due to freezing. Through the action of a unit step function, we can model the fact that frazil ice is added only in the upward motion. Supercooled seawater is observed in the proximity of ice shelf fronts. We consider an idealized ice shelf and take the front to be vertical. For this geometry, we present linear analytical solutions as well as numerical results for nonlinear two-dimensional steady conditional convection in the presence of a stable environmental density gradient. With a parallel to moist convection in the atmosphere, we find convection cells near the ice front with rising fluid in a narrow region and sinking fluid over a much broader region.

Stably stratified exact coherent structures in shear flow: the effect of Prandtl number

We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number $Pr$, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, $Re=400$. In the two asymptotic limits $Pr \to 0$ and $Pr \to \infty$, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the $Pr \to 0$ limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number $Ri_b \lesssim O(Re^{-2}Pr^{-1})$. Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as $Pr\to\infty$. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing $Ri_b$. An asymptotic analysis is presented in the passive scalar regime $Ri_b \lesssim O(Re^{-2})$, which reveals $O(Pr^{-1/3})$-thick stratified boundary layers with $O(Pr^{-2/9})$-wide eruptions, giving rise to density fingers of $O(Pr^{-1/9})$ length and $O(Pr^{-4/9})$ width that invade an otherwise homogeneous interior. Finally, increasing $Re$ to $10^5$ in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.

Mass transport induced asymmetry in charge/discharge behavior of liquid metal batteries

Mass and charge transport in liquid metal batteries are closely intertwined because of the fully liquid interior of the cells. We found that charging and discharging cycles may show pronounced asymmetries. They are caused by the presence (charge) or absence (discharge) of solutal convection. While the direction of thermal gradients in the positive electrode of a liquid metal battery depends on boundary conditions and thermodynamics in a non-trivial manner, the solutal gradient predictably changes direction from charge to discharge. The unstable density distribution during charge drives a flow strong enough to prevent any concentration polarization. In contrast, during discharge, the stable density gradient suppresses convection and leads to a substantial mass transport overvoltage. We illustrate this scenario by experimental data, numerical simulations and a physical model.

Lee wave breaking region: the map of instability development scenarios

Numerical study of a stably stratified flow above the two-dimensional cosine-shaped obstacle has been performed by DNS and LES. These methods were implemented to solve the three-dimensional Navier–Stokes equations in the Boussinesq approximation, together with by the scalar diffusion equation. The results of scanning in the wide ranges of physical parameters (Reynolds and Prandtl/Schmidt numbers relating to laboratory experiment cases and atmospheric or oceanic situations) are presented for instability and turbulence development scenarios in the overturning internal lee waves. The latter is generated by the obstacle in a flow with the constant inflow values of velocity and stable density gradient. Evolution of lee-wave breaking is explored by visualization of velocity and scalar (density) fields, and the analysis of spectra. Based on the numerical simulation results, the power-law dependence on Reynolds number is demonstrated for the wavelength of the most unstable perturbation.

The turbulent wake of a towed grid in a stratified fluid

In a stable background density gradient, initially turbulent flows eventually evolve into a state dominated by low-Froude-number dynamics and frequently also contain persistent pattern information. Much empirical evidence has been gathered on these latter stages, but less on how they first got that way, and how information on the turbulence generator may potentially be encoded into the flow in a robust and long-lasting fashion. Here an experiment is described that examines the initial stages of evolution in the vertical plane of a turbulent grid-generated wake in a stratified ambient. Refractive-index-matched fluids allow optically based measurement of early ($Nt<2$) stages of the flow, even when there are strong variations in the local density gradient field. Suitably averaged flow measures show the interplay between internal wave motions and Kelvin–Helmholtz-generated vortical modes. The vertical shear is dominant at the wake edge, and the decay of horizontal vorticity is observed to be independent of $\mathit{Fr}$. Stratified turbulence, originating from Kelvin–Helmholtz instabilities, develops up to non-dimensional time $Nt\approx 10$, and the scale separation between Ozmidov and Kolmogorov scales is independent of $\mathit{Fr}$ at higher $Nt$. The detailed measurements in the near wake, with independent variation of both Reynolds and Froude numbers, while limited to one particular case, are sufficient to show that the initial turbulence in a stratified fluid is neither three-dimensional nor universal. The search for appropriately generalizable initial conditions may be more involved than hoped for.

Can a fractionally crystallized magma ocean explain the thermo-chemical evolution of Mars?

The impact heat accumulated during the late stage of planetary accretion can melt a significant part or even the entire mantle of a terrestrial body, giving rise to a global magma ocean. The subsequent cooling of the interior causes the magma ocean to freeze from the core–mantle boundary (CMB) to the surface due to the steeper slope of the mantle adiabat compared to the slope of the solidus.Assuming fractional crystallization of the magma ocean, dense cumulates are produced close to the surface, largely due to iron enrichment in the evolving magma ocean liquid. A gravitationally unstable mantle thus forms, which is prone to overturn. We investigate the cumulate overturn and its influence on the thermal evolution of Mars using mantle convection simulations in 2D cylindrical geometry. We present a suite of simulations using different initial conditions and a strongly temperature-dependent viscosity. We assume that all radiogenic heat sources have been enriched during the freezing-phase of the magma ocean in the uppermost 50 km and that the initial steam-atmosphere created by the degassing of the freezing magma ocean was rapidly lost, implying that the surface temperature is set to present-day values. In this case, a stagnant lid quickly forms on top of the convective interior preventing the uppermost dense cumulates to sink, even when allowing for a plastic yielding mechanism. Below this dense stagnant lid, the mantle chemical gradient settles to a stable configuration. The convection pattern is dominated by small-scale structures, which are difficult to reconcile with the large-scale volcanic features observed over Mars' surface and partial melting ceases in less than 900 Ma. Assuming that the stagnant lid can break because of additional mechanisms and allowing the uppermost dense layer to overturn, a stable density gradient is obtained, with the densest material and the entire amount of heat sources lying above the CMB. This stratification leads to a strong overheating of the lowermost mantle, whose temperature increases to values that exceed the liquidus. The iron-rich melt would most likely remain trapped in the lower part of the mantle. The upper mantle in that scenario cools rapidly and only shows partial melting during the first billion year of evolution. Therefore a fractionated global and deep magma ocean is difficult to reconcile with observations. Different scenarios assuming, for instance, a hemispherical or shallow magma ocean, or a crystallization sequence resulting in a lower density gradient than that implied by pure fractional crystallization will have to be considered.

The stratified layer at the core–mantle boundary caused by barodiffusion of oxygen, sulphur and silicon

Barodiffusion is the tendency of light elements to migrate down a pressure gradient. In the Earth’s outer core, this effect can lead to the development of a chemically stable layer beneath the core–mantle boundary (CMB). Barodiffusion has so far been considered unimportant relative to other effects, but here we show that it dominates at the CMB and leads to an order-100km-thick layer that is rich in light elements and stably stratified. Barodiffusion changes not only the equations governing molecular diffusion of light elements in the core but also the boundary condition at the CMB to a non-zero compositional gradient, a point missed by previous studies. The mathematical problem has the same form as the recently-proposed migration of light elements passing from the mantle into the core; the effect of barodiffusion is comparable provided all light elements in the outer core are included, not just the light element driving the convection as in previous studies. We therefore conclude that a substantial stable layer can exist at the top of the core independent of any mass flux across the CMB. We solve the relevant diffusion equations in a thin layer beneath the CMB for barodiffusion of oxygen, sulphur and silicon over the whole history of the core using diffusion constants obtained from first principles calculations. The lower boundary of the layer is defined to be the neutrally stable level where the stabilising barodiffusive gradient is equal and opposite to the destabilising gradients associated with buoyancy sources in the well-mixed bulk of the core. We assume no mass flux across the CMB, and find the compositional gradient imposed by barodiffusion to be so large that its stable density gradient could not be overcome by any destabilising gradient at any time. The light layer therefore develops at the top of the core immediately after core formation; solving the diffusion equations shows it to grow to a thickness of order 100km. The final thickness is remarkably insensitive to the model of core cooling used to specify the destabilising gradients in the well-mixed region of the core. We consider a variety of instability mechanisms and argue that the stratification is strong enough to inhibit virtually all radial motion within the layer, although conclusive evidence for the existence of stratification can only come from observations. The variation in composition is sufficiently strong to produce geomagnetic effects and seismic velocity anomalies of a fraction of a percent that could be, and may already have been, detected. Differences in the diffusion parameters for the three light elements cause differences in their relative concentrations in the layer, leaving the layer oxygen-rich relative to sulphur or silicon.

Stratified wake of a tilted cylinder. Part 2. Lee internal waves

AbstractThis experimental, numerical and theoretical study considers the lee internal waves generated by the wake of a circular cylinder, whose axis is tilted with respect to a stable density gradient. The main difference with the case of a horizontal cylinder is that the lee waves contain a large axial velocity, which are located in a row of lobes extending downstream from the cylinder. At small tilt angles, the wavelength is equal to $2\lrm{\pi} U/ N$, $U$ being the velocity of the cylinder and $N$ the Brunt–Väisälä frequency, which can be explained by the fact that the group velocity of the waves is small. The amplitude of the waves can be predicted using the Lighthill theory for dispersive waves applied to the case of a tilted bluff body. The flow around the cylinder is modelled empirically in order to reach a quantitative prediction in good agreement with the experimental and numerical results. The spatial structure of the predicted internal waves is qualitatively correct although some discrepancies arise because the advection by the flow around the cylinder is neglected.

Stratified wake of a tilted cylinder. Part 1. Suppression of a von Kármán vortex street

AbstractThis experimental and numerical study considers the two-dimensional stability of a circular cylinder wake, whose axis is tilted with respect to a stable density gradient. When the Reynolds number increases, the wake transitions from a steady flow to a periodic von Kármán vortex street as in a homogeneous fluid. However, the presence of a moderate stratification delays the appearance of the von Kármán vortex street, in agreement with the stabilization of shear flows by a density gradient. This stabilization, which does not occur for a vertical cylinder, increases with the tilt angle of the cylinder and is maximum for a horizontal cylinder. The critical Reynolds number increases when the stratification increases and diverges at a Froude number of order one for a horizontal cylinder. This critical Reynolds number can be predicted using the Richardson number based on the projection of the gravity and the density gradient in the direction of the shear, as was proposed by Candelier (J. Fluid Mech., vol. 685, pp. 191–201) for a tilted stratified jet. This picture is completely different for a strongly stratified wake since a new unstable mode appears, creating a von Kármán vortex street with a smaller Strouhal number. This surprising result is due to the presence of tilted vortices with no vertical velocity, i.e. with horizontal elliptic streamlines. This mode occurs in a band of Froude numbers which becomes smaller and smaller when the tilt angle increases, and eventually disappears for a horizontal cylinder. The presence of the tilt has thus a large impact on the structure of the wake at small Froude numbers and might need to be taken into account in geophysical flows.

Particle Image Velocimetry Measurements and Numerical Modeling of a Saline Density Current

Particle image velocimetry scalar measurements were carried out on the body of a stably stratified density current with an inlet Reynolds number of 2,300 and bulk Richardson number of 0.1. These measurements allowed the mass and momentum transport between the current and the less dense ambient fluid to be investigated. Reynolds stress, Reynolds flux, and shear production of turbulent kinetic-energy profiles revealed local maxima at the bed, as well as at the interface with the ambient fluid. Profiles of excess density variance and buoyancy production of turbulent kinetic energy revealed only local maxima at the interface with the ambient. These maxima decreased downstream as the stable density gradient reduced the turbulent intensities, until turbulence collapsed. A two-dimensional, unsteady, Reynolds-averaged Navier-Stokes (2DV-URANS) simulation was also performed on this density current. Good agreement was found between the modeled and measured normalized mean flow profiles. A comparison was also made between the measured and modeled outer flow scales of the density current. DOI: 10.1061/(ASCE)HY.1943-7900.0000304. © 2011 American Society of Civil Engineers. CE Database subject headings: Density currents; Flow measurement; Turbulence; Reynolds stress; Diffusion; Numerical models. Author keywords: Density current; Flow measurement; Navier-Stokes equations; Turbulence; Reynolds stress; Turbulent diffusion.

Instability and breakdown of a vertical vortex pair in a strongly stratified fluid

The dynamics of a counter-rotating pair of columnar vortices aligned parallel to a stable density gradient are investigated. By means of numerical simulation, we extend the linear analyses and laboratory experiments of Billant & Chomaz (J. Fluid Mech. vol. 418, p. 167; vol. 419, pp. 29, 65 (2000)) to the fully nonlinear, large-Reynolds-number regime. A range of stratifications and vertical length scales is considered, withFrh< 0.2 and 0.1 <Frz< 10. HereFrh≡U/(NR) andFrz≡Ukz/Nare the horizontal and vertical Froude numbers,UandRare the horizontal velocity and length scales of the vortices,Nis the Brunt–Väisälä frequency, and 2π/kzis the vertical wavelength of a small initial perturbation. At early times withFrz< 1, linear predictions for the zigzag instability are reproduced. Short-wavelength perturbations withFrz> 1 are found to be unstable as well, with growth rates only slightly less than those of the zigzag instability but with very different structure. At later times, the large-Reynolds-number evolution diverges profoundly from the moderate-Reynolds-number laboratory experiments as the instabilities transition to turbulence. For the zigzag instability, this transition occurs when density perturbations generated by the vortex bending become gravitationally unstable. The resulting turbulence rapidly destroys the vortex pair. We derive the criterion η/R≈ 0.2/Frzfor the onset of gravitational instability, where η is the maximum horizontal displacement of the bent vortices, and refine it to account for a finite twisting disturbance. Our simulations agree for the fastest growing wavelengths 0.3 <Frz< 0.8. Short perturbations withFrz> 1 saturate at low amplitude, preserving the columnar structure of the vortices well after the generation of turbulence. Viscosity is shown to suppress the transition to turbulence for Reynolds numberRe≲ 80/Frh, yielding laminar dynamics and, under certain conditions, pancake vortices like those observed in the laboratory.

Internal-wave radiation and optical measurements in stratified fluids

In many problems of considerable geophysical and industrial interest, fluid flows are characterized by the presence of a stable vertical density gradient. Under suitable conditions such fluids can support internal waves. In many aspects internal waves are similar to surface waves. Owing to small density difference across the fluid depth the effective gravity governing the dynamics of internal waves is typically by two or three orders of magnitude smaller than the terrestrial gravity. In a continuously stratified fluid the strength of gravity and viscous effects depends on the vertical density gradient, fluid viscosity, and characteristics of a wave-emitting disturbance. In the present report we focus our attention on experiments with internal waves generated by circular and rectilinear oscillations of cylinders in a uniformly stratified fluid. Quantitative measurements of internal waves are performed with the help of ‘synthetic’ schlieren technique based on computer evaluation of optical distortions in stratified fluids. The effects of the type and amplitude of oscillations on the generation of linear and nonlinear density-gradient disturbances are discussed.

Sinking velocities of phytoplankton measured on a stable density gradient by laser scanning.

Two particular difficulties in measuring the sinking velocities of phytoplankton cells are preventing convection within the sedimenting medium and determining the changing depth of the cells. These problems are overcome by using a density-stabilized sedimentation column scanned by a laser. For freshwater species, a suspension of phytoplankton is layered over a vertical density gradient of Percoll solution; as the cells sink down the column their relative concentration is measured by the forward scattering of light from a laser beam that repeatedly scans up and down the column. The Percoll gradient stabilizes the column, preventing vertical mixing by convection, radiation or perturbation of density by the descending cells. Measurements were made on suspensions of 15 microm polystyrene microspheres with a density of 1050 kg m-3; the mean velocity was 6.28 microm s-1, within 1.5% of that calculated by the Stokes equation, 6.36 microms-1. Measurements made on the filamentous cyanobacterium Planktothrix rubescens gave mean velocities within the theoretical range of values based on the range of size, shape, orientation and density of the particles in a modified Stokes equation. Measurements on marine phytoplankton may require density gradients prepared with other substances.

Destabilising A Fluid Saturated Gravity Modulated Porous Layer Heated From Above

The linear stability theory is used to investigate analytically the possibility of the motionless basic state in a porous layer heated from above and subjected to vibration. The linear stability results presented for the specific case of low amplitude vibration shows that there exists a bandwidth of frequencies for which the convection in a porous layer with a stable density gradient can be destabilized. In addition the scaled Darcy–Prandtl number is shown to influence the onset of the subharmonic and synchronous solutions.

Superior mirage effect in supercritical CO 2: Experiment and model

At temperatures near T c, supercritical CO 2 is highly compressible and it follows that a small temperature gradient will create a large density gradient. We established stable density gradients by maintaining the bottom of a sc-CO 2-cell cooler than its other sides. That created a large refractive index gradient that bent a laser beam through an angle that is roughly proportional to the density gradient. Such light bending causes, for example, the mirage of water on a hot road. A He–Ne laser beam traversed a 7.62 cm long cell and was deflected downward as much as 5.9°. The magnitude of the beam deflections agreed within 3% with a model that used only fundamental geometrical optics and known PVT data, but that employed no adjustable parameters.

Vertical density currents: a review of their potential role in the deposition and interpretation of deep-sea ash layers

Marine tephra layers form important chronostratigraphic markers and may also provide information on eruption dynamics and chronology. Recent experiments, and field data from the 1991 Pinatubo eruption, challenge tacit assumptions that, following atmospheric fallout on the ocean surface, marine sedimentation of ash-grade material occurs by Stokes Law settling of individual grains. Laboratory experiments have demonstrated that vertical density currents can be initiated at representative mass flux rates: further experiments, described herein, involving sustained particle fluxes entering stratified environments, demonstrate that ash particles can decouple from the transporting fluid, whose vertical motions are limited by the stable density gradient. In the ocean, vertical density currents generated by ash-loading and gravitational destabilization of the water column can overcome the strong stable density gradients in the ocean and transport ash particles vertically 1–3 orders of magnitude faster than possible by Stokesian settling, greatly reducing their residence time. Sea-bed sedimentation is inferred to involve settling of individual coarse or dense grains through the entire water column, plus suspension fallout from the polydisperse turbid layer delivered by convective plumes, yielding normal distribution grading. The turbid layer may continue laterally as a (nepheloid) current under the influence of bottom slopes and/or geostrophic forces, either mechanism possibly modifying the grain-size distribution of the suspension and hence the resulting ash layer by transporting selected fractions into different depositional locations. Consequently, and regardless of any subsequent reworking, deep-sea ash isopachs may not directly reflect an eruptive signal, and the ash layers may not directly record the order, rate, or location of arrival of ash particles at the ocean surface.

Intermittency of vertical density gradients at finescale and link with mixing processes

Nonlinear interactions among internal gravity waves lead to intermittent breaking events with density overturning and vertical turbulent mixing. We analyze the statistics of density fluctuations generated in this process, using direct numerical simulations. We then check the relevance of these results for in situ density profiles measured during the ARCANE experiment on the continental slope west of the Iberian peninsula, a region with strong internal tides. We focus the analysis in the depth range [400-1000 m] for which the temperature is uniform, and the density stratification is due mainly to salinity. We verify in the numerical simulations that the histograms of vertical density gradients become strongly skewed at small vertical scales. This intermittency of density fluctuations can be attributed to the formation of sheets with strong stable density gradients. These sheets result from convergent motion produced by surrounding breaking events. We check that the vertical density gradients at finescale can be represented by a model of layers and sheets distributed with a Poisson statistics, with a mean number of sheets per unit length, $\mu$. We next show that the mean sheet spacing $\mu^{-1}$ is proportional to the Ozmidov scale (proportional to the square root of the energy dissipation). Using the classical link between energy dissipation and vertical eddy diffusivity $K_d$, we deduce a link between the statistics of vertical density gradients and the vertical mixing, namely $K_d$ is proportional to $N \mu^{-2}$. For in situ measurements, we find a similar statistics of vertical density gradients, confirming earlier results obtained by Hayes et al. and Pinkel and Anderson in the open ocean thermocline. This allows us to estimate a rate of total energy dissipation and a vertical eddy diffusivity . Comparisons with other estimates of energy dissipation or vertical mixing would be required to validate this method.

Drifter Behavior on the Oregon–Washington Shelf during Downwelling-Favorable Winds*

Abstract Drifters released offshore of Oregon during predominantly downwelling favorable alongshore winds during three different deployments (October 1994, January 1998, and September 1998) display similar behavior: after being advected around in the offshore eddy field, they move onshore to a particular isobath and are advected poleward alongshore, without coming ashore. Numerical modeling results suggest that this may be due to downwelling circulation creating a marginally stable density gradient on the shelf inshore of the downwelling front, thereby increasing the vertical eddy diffusivity, which reduces the effective cross-shelf Ekman transport to nearly zero. The downwelling front itself is accompanied by a poleward jet, which carries drifters rapidly to the north. This behavior is consistent with previous modeling results.

Anisotropy in turbulence profiles of stratified wakes

At sufficiently high values of the Reynolds number (Re⩾4.5×103) and internal Froude number (F⩾4), initially turbulent bluff body wakes evolve in the presence of a stable background density gradient with wake-averaged mean and turbulence length and velocity scales that are independent of Re and F for at least two orders of magnitude extension in both parameters. The way in which the initially three-dimensional motions transition to the characteristic (and Re- and F-independent) late wakes (where vertical velocities, w≪u,v) is both of great practical interest, and complex, hence somewhat unclear. Here, digital particle imaging velocimetry type measurements on towed-sphere wakes are described, so that the development of anisotropy can be measured by the time development of turbulence profiles in horizontal and vertical centerplanes. The observed anisotropies can be associated with energy transfer to internal wave modes, and suppression of other vertical displacements, that contrasts with sphere wakes at similar Re in a homogeneous fluid. Maximum Reynolds stresses occur at the boundary of a sinuous undulation of the wake, which increases in amplitude up to Nt≈60 (N is the buoyancy frequency that characterizes the strength of the stratification). Although an intrinsic wake profile instability cannot be excluded, the observed wake element spacings can be accounted for by known spiral and Kelvin–Helmholtz instabilities in the near wake.