Applicability and failure of the flux-gradient laws in double-diffusive convection

2014 ◽  
Vol 750 ◽  
pp. 33-72 ◽  
Author(s):  
Timour Radko
Keyword(s):  
Large Scale ◽  
Numerical Models ◽  
Vertical Transport ◽  
Double Diffusive Convection ◽  
Flux Gradient ◽  
Salinity Gradients ◽  
Diffusive Convection ◽  
Small Scales ◽  
New Formulation ◽  
Double Diffusive

AbstractDouble-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 ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H_{pof}$) 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.

Oscillatory double-diffusive instabilities in a vertical slot

2001 ◽  
Vol 426 ◽  
pp. 347-354 ◽  
Author(s):  
OLIVER S. KERR
Keyword(s):  
Prandtl Number ◽  
Linear Stability ◽  
Salinity Gradient ◽  
Double Diffusive Convection ◽  
Salinity Gradients ◽  
Vertical Slot ◽  
Diffusive Convection ◽  
Prandtl Numbers ◽  
Double Diffusive ◽  
Oscillatory Instabilities

Recent linear stability analyses of double-diffusive convection in a laterally heated vertical slot containing water have shown that for very weak (or no) vertical salinity gradient the initial instabilities are steady, but as the salinity gradient is increased there is a transition to oscillatory instabilities. For higher Prandtl number fluids the initial instabilities in a slot with no stratification can be oscillatory or wave-like. We show that the oscillatory instabilities in water are linked to these higher Prandtl number oscillatory instabilities. The salinity gradient has a destabilizing effect on these oscillations, making them appear for Prandtl numbers where oscillatory instabilities are not possible in the absence of salinity gradients. We derive an asymptotic description for this mode of instability.

Layered double-diffusive convection in porous media

1981 ◽  
Vol 102 ◽  
pp. 221-248 ◽  
Author(s):  
R. W. Griffiths
Keyword(s):  
Porous Medium ◽  
Flux Ratio ◽  
Geothermal System ◽  
Double Diffusive Convection ◽  
Salinity Gradients ◽  
Diffusive Convection ◽  
Diffusive Interface ◽  
Convection In Porous Media ◽  
And Diffusion ◽  
Double Diffusive

In this paper it is shown that layered double-diffusive convection of a fluid within a porous medium is possible. A thin ‘diffusive’ interface was observed in a Hele Shaw cell and in a laboratory porous medium, with salt and sugar or heat and salt as the diffusing components. Heat–salt and salt–sugar fluxes through two-layer convection systems were measured and are compared with predictions of a model. For the thermohaline system the salt and heat buoyancy fluxes are approximately in the ratio r ≃ ετm½, where ε is the porosity and Tm is the appropriate ratio of diffusivities. The behaviour of the heat flux is explained in terms of a coupling between purely thermal convection within each convecting layer and diffusion through the density interface. Salinity gradients are important only within the interface. The presence of a ‘diffusive’ interface in the Wairakei geothermal system is postulated. The ratio of heat and salt fluxes (that can be estimated from existing observations) through this convection system is consistent with the laboratory flux ratio.

Large-scale Langmuir circulation and double-diffusive convection: evolution equations and flow transitions

1994 ◽  
Vol 276 ◽  
pp. 189-210 ◽  
Author(s):  
Stephen M. Cox ◽  
Sidney Leibovich
Keyword(s):  
Boundary Conditions ◽  
Large Scale ◽  
Evolution Equations ◽  
Mixed Boundary ◽  
Travelling Waves ◽  
Temperature Fields ◽  
Double Diffusive Convection ◽  
Langmuir Circulation ◽  
Diffusive Convection ◽  
Double Diffusive

Two-dimensional Langmuir circulation in a layer of stably stratified water and the mathematically analogous problem of double-diffusive convection are studied with mixed boundary conditions. When the Biot numbers that occur in the mechanical boundary conditions are small and the destabilizing factors are large enough, the system will be unstable to perturbations of large horizontal length. The instability may be either direct or oscillatory depending on the control parameters. Evolution equations are derived here to account for both cases and for the transition between them. These evolution equations are not limited to small disturbances of the nonconvective basic velocity and temperature fields, provided the spatial variations in the horizontal remain small. The direct bifurcation may be supercritical or subcritical, while in the case of oscillatory motions, stable finite-amplitude travelling waves emerge. At the transition, travelling waves, standing waves, and modulated travelling waves all are stable in sub-regimes.

scholarly journals Settling-driven large-scale instabilities in double-diffusive convection

2020 ◽  
Vol 901 ◽  
Author(s):  
Raphael Ouillon ◽  
Philip Edel ◽  
Pascale Garaud ◽  
Eckart Meiburg
Keyword(s):  
Large Scale ◽  
Double Diffusive Convection ◽  
Diffusive Convection ◽  
Image Position ◽  
Double Diffusive

Abstract

New Method for Estimating Double-Diffusive Dissipation Rates

2021 ◽  
Author(s):  
Leo Middleton ◽  
Elizabeth Fine ◽  
Jennifer MacKinnon ◽  
Matthew Alford ◽  
John Taylor
Keyword(s):  
Heat Fluxes ◽  
New Method ◽  
The Arctic ◽  
Small Scale ◽  
Double Diffusive Convection ◽  
Dissipation Rates ◽  
Diffusive Convection ◽  
Small Scales ◽  
Microstructure Measurements ◽  
Double Diffusive

<p>Understanding the transport of heat in the Arctic ocean will be vital for predicting the fate of sea-ice in the decades to come. Small-scale turbulence is an important driver of heat transport and one of the major forms of this turbulence is known as `double-diffusive convection'. Double diffusion refers to a variety of turbulent processes in which potential energy is released into kinetic energy, made possible in the ocean by the difference in molecular diffusivities between salinity and temperature.  The most direct measurements of ocean mixing require sampling velocity or temperature gradients on scales <1mm, so-called microstructure measurements. Here we present a new method for estimating the energy dissipated by double-diffusive convection using temperature and salinity measurements on larger scales (100s to 1000s of metres). The method estimates the up-gradient diapycnal buoyancy flux, which is hypothesised to balance the dissipation rate. To calculate the temperature and salinity gradients on small scales we apply a canonical scaling for compensated thermohaline variance (or `spice') and project the gradients down to small scales. We apply the method to a high-resolution survey of temperature and salinity through a subsurface Arctic eddy (Fine et al. 2018) and compare the results with simultaneous microstructure measurements. The new technique can reproduce up to 70% of the observed dissipation rates to within a factor of 3. This suggests the method could be used to estimate the dissipation and heat fluxes associated with double-diffusive convection in regions without microstructure measurements. Finally, we show the method maintains predictive skill when applied to a sub-sampling of the CTD data at lower resolutions.</p>

scholarly journals Rotating double-diffusive convection in stably stratified planetary cores

2019 ◽  
Vol 219 (Supplement_1) ◽  
pp. S195-S218 ◽  
Author(s):  
R Monville ◽  
J Vidal ◽  
D Cébron ◽  
N Schaeffer
Keyword(s):  
Large Scale ◽  
Critical Rayleigh Number ◽  
Marginal Stability ◽  
Early Earth ◽  
Double Diffusive Convection ◽  
Earth Core ◽  
Diffusive Convection ◽  
Non Linear ◽  
Rayleigh Numbers ◽  
Double Diffusive

SUMMARY In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection (DDC). We investigate rotating DDC (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal thermal and compositional source terms. We focus on the finger regime, in which the thermal gradient is stabilizing whereas the compositional one is destabilizing. First, we perform a global linear stability analysis in spheres. The critical Rayleigh numbers drastically drop for stably stratified fluids, yielding large-scale convective motions where local analyses predict stability. We evidence the inviscid nature of this large-scale double-diffusive instability, enabling the determination of the marginal stability curve at realistic planetary regimes. In particular, we show that in stably stratified spheres, the Rayleigh numbers Ra at the onset evolve like Ra ∼ Ek−1, where Ek is the Ekman number. This differs from rotating convection in unstably stratified spheres, for which Ra ∼ Ek−4/3. The domain of existence of inviscid convection thus increases as Ek−1/3. Secondly, we perform non-linear simulations. We find a transition between two regimes of RDDC, controlled by the strength of the stratification. Furthermore, far from the RDDC onset, we find a dominating equatorially antisymmetric, large-scale zonal flow slightly above the associated linear onset. Unexpectedly, a purely linear mechanism can explain this phenomenon, even far from the instability onset, yielding a symmetry breaking of the non-linear flow at saturation. For even stronger stable stratification, the flow becomes mainly equatorially symmetric and intense zonal jets develop. Finally, we apply our results to the early Earth core. Double diffusion can reduce the critical Rayleigh number by four decades for realistic core conditions. We suggest that the early Earth core was prone to turbulent RDDC, with large-scale zonal flows.

scholarly journals Layering and vertical transport in sheared double-diffusive convection in the diffusive regime

2021 ◽  
Vol 933 ◽  
Author(s):  
Yantao Yang ◽  
Roberto Verzicco ◽  
Detlef Lohse ◽  
C.P. Caulfield
Keyword(s):  
Initial Perturbation ◽  
Flux Ratio ◽  
Vertical Transport ◽  
Salt Flux ◽  
Double Diffusive Convection ◽  
Control Parameters ◽  
Diffusive Regime ◽  
Diffusive Convection ◽  
Wide Range ◽  
Double Diffusive

A sequence of two- and three-dimensional simulations are conducted for the double-diffusive convection (DDC) flows in the diffusive regime subjected to an imposed shear. For a wide range of control parameters, and for sufficiently strong perturbation of the conductive initial state, staircase-like structures spontaneously develop, with relatively well-mixed layers separated by sharp interfaces of enhanced scalar gradient. Such staircases appear to be robust even in the presence of strong shear over very long times, with early-time coarsening of the observed layers. For the same set of control parameters, different asymptotic layered states, with markedly different vertical scalar fluxes, can arise for different initial perturbation structures. The imposed shear significantly spatio-temporally modifies the vertical transport of the various scalars. The flux ratio $\gamma ^*$ (i.e. the ratio between the density fluxes due to the total salt flux and the total heat flux) is found, at steady state, to be essentially equal to the square root of the ratio of the salt diffusivity to the thermal diffusivity, consistent with the physical model proposed by Linden & Shirtcliffe (J. Fluid Mech., vol. 87, 1978, pp. 417–432) and the variational arguments presented by Stern (J. Fluid Mech., vol. 114, 1982, pp. 105–121) for unsheared double-diffusive convection.

Chaotic dynamics of large-scale double-diffusive convection in a porous medium

2018 ◽  
Vol 364 ◽  
pp. 1-7 ◽  
Author(s):  
Shutaro Kondo ◽  
Hiroshi Gotoda ◽  
Takaya Miyano ◽  
Isao T. Tokuda
Keyword(s):  
Porous Medium ◽  
Chaotic Dynamics ◽  
Large Scale ◽  
Double Diffusive Convection ◽  
Diffusive Convection ◽  
Double Diffusive
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