Geothermal gradients and heterogeneous permeability are commonly observed in natural geological formations for underground CO$_2$ sequestration. In this study, we conduct three-dimensional direct numerical simulations on the double-diffusive convection with both unstable temperature and concentration gradients in homogeneous and heterogeneous porous media. For homogeneous porous media, the root-mean-squared velocity increases linearly with density ratio defined as the buoyancy ratio by temperature and concentration differences. The flow structures show no remarkable changes when temperature Rayleigh number ${Ra}_T$ is less than its critical value, but alter from sheet-like to cellular structures as ${Ra}_T$ surpasses this threshold. The concentration wavenumber scales approximately as $k_{rS}\sim {Ra}_e^{0.47}$ with a defined effective Rayleigh number ${Ra}_e$. By using a scale analysis, the concentration flux exhibits a consistent linear relation with the total driving forces for all simulations. For heterogeneous porous media, where the Dykstra–Parsons coefficient $V_{DP}$ and correlation length $l_{r}$ determine the spatial distribution of the permeability field, the flow is strengthened in places with higher permeability. The velocity and concentration flux are less affected by $l_{r}$ than that by $V_{DP}$. For small correlation length, the flow structures coarsen and their characteristic width generally increases with increasing heterogeneity. For large correlation length, small structures emerge in the regions with large permeability, which can be attributed to the intensified local Rayleigh number triggering more vigorous convection there. The variations of concentration flux with $l_{r}$ and $V_{DP}$ can be explained by the portion of area covered by high concentration with large vertical velocity near the boundaries.

We investigate experimentally the effect of double diffusion in the salt-fingering configuration on entrainment in turbulent plumes. Plumes over a range of source buoyancy fluxes $B_{0}$ and source density ratios $R_{\unicode[STIX]{x1D70C}}$ are examined. When the plumes are double diffusive ($R_{\unicode[STIX]{x1D70C}}>0$) the entrainment coefficient $\unicode[STIX]{x1D6FC}$ is not constant, with an up to 20 % reduction from the value found for single-diffusive plumes, that is, plumes with $R_{\unicode[STIX]{x1D70C}}=0$. The scale of reduction is found to be in direct relation to the source density ratio and is inversely related to the distance travelled by the plume, indicating that double-diffusive effects decrease as the plume evolves. We propose an explanation for the observed reduction in the entrainment coefficient on the basis of differential diffusion hindering large-scale engulfment at the edge of the plume.

Wedge-shaped layers of ultramafic and mafic cumulates in Intrusion II of the Caledonian Honningsvåg Intrusive Suite suggest crystallization on an inclined magma chamber floor from a compositionally-zoned and density-stratified magma.

Cyclic unit 8 (140–100 m thick) consists of a distally-thinning olivine gabbro (denoted paoC) macrolayer overlain by a distally-thickening gabbronorite, pahC. New mineral data in four traverses across cyclic unit 8 show systematic compositional changes; the Mg# of the mafic phases decreases upwards through the unit and distally, both along the base and along the paoC/pahC interface.

A crystallization model based on an effectively continuously-zoned magma chamber with numerous, relatively thin, double-diffusive magma layers is proposed. Differential migration of horizontal isopleths (e.g. Mg# and aSiO2) in response to fractional crystallization and assimilation of country rock can explain the variations in the Mg# of the cumulates.

Radko’s recent article (J. Fluid Mech., vol. 805, 2016, pp. 147–170) entitled Thermohaline layering in dynamically and diffusively stable shear flows, is slated to become a seminal reference in the field of fluid dynamics. It proposes an elegant solution to the long-standing question of why thermohaline staircases form in the high-latitude oceans. Equally importantly, it provides a rare and interesting example of how two physical processes that are both strongly stable when considered individually, can trigger a linear instability when they interact.

We examine the dynamics of a binary mixture in a cubic cell subjected to a temperature differential and oscillatory forcing. The Soret effect, which is negative in the present study, provides a coupling mechanism by which a temperature gradient establishes a concentration gradient in a mixture. We present the results of experiments that were performed on the International Space Station (ISS) and compare the observations with the results of direct numerical simulations. The evolution of temperature and concentration fields is investigated by optical digital interferometry. One advantage of the experimental technique is the observation of the fields along two perpendicular directions of the cell, allowing us to restore the three-dimensional field. Experimental evidence disproves speculations that the ISS microgravity environment always affects diffusion-controlled processes. Furthermore, we demonstrate that imposed vibrations with constant frequency and amplitude create slow mean flows and that they do influence the diffusion kinetics. The perturbation of the diffusive fields scales as the square of the vibrational velocity. In addition to calculations of the full three-dimensional Navier–Stokes equations, a two-time-scale computational methodology is used for situations in which the forcing period is very small compared to the natural time scales of the problem. The simulations show excellent agreement with experimental observations.

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use three-dimensional direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state, and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Part 1 (Traxler et al., J. Fluid Mech. doi:10.1017/jfm.2011.98, 2011), a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic γ-modes are also excited and grow according to the γ-instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular γ-mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values and find that the progenitor γ-mode is expected to grow on a time scale of a few hours and leads to the formation of a thermohaline staircase in about one day with an initial spacing in the order of 1–2 m.

Double-diffusive instabilities are often invoked to explain enhanced transport in stably stratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean's thermocline and typically increases diapycnal mixing by 2 orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability and the γ-instability as limiting cases. We find that the fastest growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and γ instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase-formation theory.

This work is a parametric study of the fluxes of heat and salt across unsheared and sheared double-diffusive interfaces using one-dimensional-turbulence (ODT) simulations. It is motivated by the need to understand how these fluxes scale with parameters related to the fluid molecular properties and background shear. Comparisons are made throughout with previous models and available measurements. In unsheared interfaces, ODT simulations show that the dimensionless heat flux Nu scales with the stability parameter Rρ, Rayleigh number Ra and Prandtl number Pr as Nu ~ (Ra/Rρ)0.37±0.03 when Pr varies from 3 to 100 and as Nu ~ (Ra/Rρ)0.31Pr0.22±0.04 when Pr varies from 0.01 to 1. Here Ra/Rρ can be seen as the ratio of destabilizing and stabilizing effects. The simulation results also indicate that the ratio of salt and heat fluxes Rf is independent of Pr, scales with the Lewis number Le as Rf ~ Le0.41±0.04 when Rρ is large enough and deviates from this expression for low values of Rρ, when the interface becomes heavily eroded. In sheared interfaces, the simulations show three flow regimes. When the Richardson number Ri ≪ 1, shear-induced mixing dominates, the heat flux scales with the horizontal velocity difference across the interface and Rf = Rρ. Near Ri ~ 1 the heat and salt fluxes are seen to increase abruptly as the shear increases. The flow structure and scaling of the fluxes are similar to those of unsheared interfaces when Ri ≫ 1.

The formation and evolution of double-diffusive interleaving is experimentally investigated with the purpose of analysing the influence of the convective flow structures, at different scales, on the mean flow. Recently, Krishnamurti (J. Fluid Mech., vol. 558, 2006, p. 113) has shown that, in the case of a continuous stratification experiment, the Reynolds stresses, due to convective flow patches, are able to vertically transport horizontal momentum, maintaining the mean flow. This mechanism is similar to the turbulent wind observed in thermal convection. In this study, the interleaving is produced using the classical set-up of Ruddick & Turner (Deep-Sea Res., vol. 558, 1979, p. 903). The dam-break experiments better resemble the case of oceanic fronts, where interleaving is commonly observed. The flow structures are investigated by measuring the two-dimensional flow fields using the particle image velocimetry technique. The resulting two-dimensional vector fields reveal complex fine-scale flow structures, and convective patterns are observed inside the finger-favourable layers. Vortical structures at scales comparable with the layer thickness are embedded in these regions and seem to be responsible for sustaining the horizontal mean flow against the viscous dissipations, especially in a region close to the layer nose. A spectral analysis of the flow fields suggest that the energy balance is governed by an inverse energy cascade, which implies a transfer of energy from the smaller scales to the larger scales (mean flow).