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Equilibration of salt fingers in the numerical experiment with Pr=7, τ = 0.01, R ρ = 1.9. Three-dimensional instantaneous temperature fields are shown for a) the early stage of linear growth at t=20 and b) the fully equilibrated state at t=50. Red/green color corresponds to high values of T and low values are shown in blue.
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This study examines mixing characteristics of double-diffusive convection for a wide range of fluids. Our approach involves Direct Numerical Simulation (DNS) utilizing de-aliased pseudo-spectral method. To expedite these simulations the numerical algorithm was parallelized using Message Passing Interface (MPI) calculations in both two and three dim...
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This paper presents a cooperative w ork between the French National Establishment for Aerospase Re-search ONERA and the Japanese National Aerospace Laboratory NAL, to investigate potentiality o f distributed environments for high performance computing in CFD. For this purpose the PEGASE Navier-Stokes solver, developed at ONERA, was implemented on d...
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... Thus, the fundamental difference between observations and Kunze's (1987) model of salt finger step structure is that the observations indicate thicker steps of several meters while the model assumes thin steps of order 0.5 m with very large gradients over which relatively small diffusivities can affect the vertical salt and heat transports. Radko and Smith (2012) have recently put forward a model where the linear growth of salt fingers to finite amplitude is balanced by secondary instabilities in the salt fingers. The layer of active salt finger mixing is thicker than the salt finger scale of maximum growth rate (about 0.5 m), in agreement with our observations for an order 7 m thickness for the sharp gradients steps, though they did not explicitly give the vertical scale of the salt finger zone. ...
... The layer of active salt finger mixing is thicker than the salt finger scale of maximum growth rate (about 0.5 m), in agreement with our observations for an order 7 m thickness for the sharp gradients steps, though they did not explicitly give the vertical scale of the salt finger zone. Radko and Smith (2012) made predictions for the heat and salt fluxes (and buoyancy flux ratio) as a function of R ρ . For the observed background stratification in our observations of the western Mediterranean, R ρ equals 1.28, for which their model produces mixing coefficients of k S = 3.5 × 10 −5 m 2 s −1 and k T = 1.7 × 10 −5 m 2 s −1 , and a buoyancy flux ratio of 0.62. ...
... Mixing coefficients based on the vertical gradients in the steps are of order 2 to 4 × 10 −5 m 2 s −1 , still an order of magnitude greater than those traditional theory for salt finger growth would suggest (eg. Kunze, 1987), but in apparent agreement with recent model results where salt finger growth is arrested by secondary instabilities (Radko and Smith, 2012). ...
Deep wintertime convection in the western Mediterranean in 2005-06
created a thick layer of warm, salty new deep water beneath the fresher,
colder, older deep water. Repeat surveys over the years 2006, 2008 and
2010 reveal that the transition region between the old and new deep
waters becomes progressively warmer and saltier. The increases in
temperature and salinity of this transition region represent a
convergence of heat and salt that is quantifiable from the repeat
surveys. These increases in temperature an salinity are likely due to
salt finger mixing processes that transport heat and salt downward
through the halocline-thermocline that connects the Levantine
Intermediate Water at about 400 m depth with the deep waters below 1500
m. Here the observed layer and step structure characteristic of salt
finger mixing in the halocline-thermocline is described. The estimated
downward heat and salt fluxes are used to explore how these fluxes
relate to the vertical gradients within the layer and step structures
and to the overall structure of the halocline-thermocline.
An attempt is made to quantify the impact of stochastic wave-induced shears on salt fingers associated with internal waves in the ocean. The wave environment is represented by the superposition of Fourier components conforming to the Garrett-Munk (GM) spectrum with random initial phase distribution. The resulting time series of vertical shear are incorporated into a finger-resolving numerical model, and the latter is used to evaluate the equilibrium diapycnal fluxes of heat and salt. The proposed procedure makes it possible to simulate salt fingers in shears that are representative of typical oceanic conditions. This study finds that the shear-induced modification of salt fingers is largely caused by near-inertial motions. These relatively slow waves act to align salt fingers in the direction of shear, thereby rendering the double-diffusive dynamics effectively two-dimensional. Internal waves reduce the equilibrium vertical fluxes of heat and salt by a factor of 2 relative to those in the unsheared three-dimensional environment, bringing them close to the values suggested by corresponding two-dimensional simulations.
The double diffusive convection between two parallel plates is numerically studied for a series of parameters. The flow is driven by the salinity difference and stabilised by the thermal field. Our simulations are directly compared with experiments by Hage & Tilgner (
Phys. Fluids
, vol. 22, 2010, 076603) for several sets of parameters and reasonable agreement is found. This, in particular, holds for the salinity flux and its dependence on the salinity Rayleigh number. Salt fingers are present in all simulations and extend through the entire height. The thermal Rayleigh number seems to have a minor influence on the salinity flux but affects the Reynolds number and the morphology of the flow. In addition to the numerical calculation, we apply the Grossmann–Lohse theory for Rayleigh–Bénard flow to the present problem without introducing any new coefficients. The theory successfully predicts the salinity flux both with respect to the scaling and even with respect to the absolute value for the numerical and experimental results.
Double-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers – thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the 'point-of-failure' scale (H pof) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that H pof can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport – the flux-gradient-aberrancy law – is proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.
This study examines the interaction of diffusive convection and shear through a series of 2D and 3D direct numerical simulations (DNS). The model employed is based on the Boussinesq equations of motion with oscillating shear represented by a forcing term in the momentum equation. This study calculates thermal diffusivities for a wide range of Froude numbers and density ratios and compares the results with those from the analysis of observational data gathered during a 2005 expedition to the eastern Weddell Sea. The patterns of layering and the strong dependence of thermal diffusivity on the density ratio described here are in agreement with observations. Additionally, the authors evaluate salinity fluxes that are inaccessible from field data and formulate a parameterization of buoyancy transport. The relative significance of double diffusion and shear is quantified through comparison of density fluxes, efficiency factor, and dissipation ratio for the regimes with/without diffusive convection. This study assesses the accuracy of the thermal production dissipation and turbulent kinetic energy balances, commonly used in microstructure-based observational studies, and quantifies the length of the averaging period required for reliable statistics and the spatial variability of heat flux.
A theoretical model is developed, which attempts to predict the lateral transport by mesoscale variability, generated and maintained by baroclinic instability of large-scale flows. The authors are particularly concerned by the role of secondary instabilities of primary baroclinically unstable modes in the saturation of their linear growth. Theory assumes that the fully developed equilibrium state is characterized by the comparable growth rates of primary and secondary instabilities. This assumption makes it possible to formulate an efficient algorithm for evaluating the equilibrium magnitude of mesoscale eddies as a function of the background parameters: vertical shear, stratification, beta effect, and bottom drag. The proposed technique is applied to two classical models of baroclinic instability—the Phillips two-layer model and the linearly stratified Eady model. Theory predicts that the eddy-driven lateral mixing rapidly intensifies with increasing shear and weakens when the beta effect is increased. The eddy transport is also sensitive to the stratification pattern, decreasing as the ratio of upper/lower layer depths in the Phillips model is decreased below unity. Theory is successfully tested by a series of direct numerical simulations that span a wide parameter range relevant for typical large-scale currents in the ocean. The spontaneous emergence of large-scale patterns induced by mesoscale variability, and their role in the cross-flow eddy transport, is examined using a suite of numerical simulations.
This study explores the dynamics of thermohaline staircases: well-defined stepped structures in temperature and salinity profiles, commonly observed in regions of active double diffusion. The evolution of staircases in time is frequently characterized by spontaneous layer-merging events. These phenomena, the authors argue, are essential in regulating the equilibrium layer thickness in fully developed staircases. The pattern and mechanics of merging events are explained using a combination of analytical considerations, direct numerical simulations, and data analysis. The theoretical merger model is based on the stability analysis for a series of identical steps and pertains to both forms of double diffusion: diffusive convection and salt fingering. The conceptual significance of the proposed model lies in its ability to describe merging events without assuming from the outset specific power laws for the vertical transport of heat and salt-the approach adopted by earlier merging models. The analysis of direct numerical simulations indicates that merging models based on the four-thirds flux laws offer adequate qualitative description of the evolutionary patterns but are less accurate than models that do not rely on such laws. Specific examples considered in this paper include the evolution of layers in the diffusive staircase in the Beaufort Gyre of the Arctic Ocean.
Three-dimensional dynamics of thermohaline staircases are investigated using a series of basin-scale staircase-resolving numerical simulations. The computational domain and forcing fields are chosen to reflect the size and structure of the North Atlantic subtropical thermocline. Salt-finger transport is parameterized using the flux-gradient formulation based on a suite of recent direct numerical simulations. Analysis of the spontaneous generation of thermohaline staircases suggests that thermohaline layering is a product of the gamma instability, associated with the variation of the flux ratio with the density ratio . After their formation, numerical staircases undergo a series of merging events, which systematically increase the size of layers. Ultimately, the system evolves into a steady equilibrium state with pronounced layers 20-50 m thick. The size of the region occupied by thermohaline staircases is controlled by the competition between turbulent mixing and double diffusion. Assuming, in accordance with observations, that staircases form when the density ratio is less than the critical value of , the authors arrive at an indirect estimate of the characteristic turbulent diffusivity in the subtropical thermocline.
Much progress has recently been made in understanding and quantifying
vertical mixing induced by double-diffusive instabilities such as fingering
convection (usually called thermohaline convection) and oscillatory
double-diffusive convection (a process closely related to semiconvection). This
was prompted in parts by advances in supercomputing, which allow us to run
Direct Numerical Simulations of these processes at parameter values approaching
those relevant in stellar interiors, and in parts by recent theoretical
developments in oceanography where such instabilities also occur. In this paper
I summarize these recent findings, and propose new mixing parametrizations for
both processes that can easily be implemented in stellar evolution codes.
Convection in a porous medium at high Rayleigh number Ra exhibits a striking quasisteady columnar structure with a well-defined and Ra-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ’heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber k and amplitude A ^ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter A=A ^Ra/k. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of A. The growth rate of the most unstable mode scales with A 4/9 for A≫1, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh-Bénard (Rayleigh-Darcy) cell at high Ra, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers k greater than k∼Ra 5/14 as Ra→∞. This stability criterion is consistent with hitherto unexplained numerical measurements of k in a Rayleigh-Darcy cell.
