When the density of a fluid is a function of two scalars with different diffusivities, one of which is unstably stratified, an instability can set in even if the density initially decreases in the upward direction. This double-diffusive instability was first identified nearly seven decades ago in the seminal papers by Stommel, Arons & Blanchard (Reference Stommel, Arons and Blanchard1956) and Stern (Reference Stern1960), who showed that it is governed by the diffusivity ratio and the relative strength of the respective density gradients. It was subsequently shown to be important, for example, in parts of the tropical and subtropical ocean, where it contributes to the mixing of warmer, saltier surface water with colder, less salty deeper layers (Schmitt et al. Reference Schmitt, Perkins, Boyd and Stalcup1987; Schmitt Reference Schmitt1994, Reference Schmitt2003). Furthermore oceanographic measurements revealed the existence of so-called thermohaline staircases (Tait & Howe Reference Tait and Howe1968, Reference Tait and Howe1971), i.e. step-like structures of the temperature and salinity profiles whose existence was initially somewhat counterintuitive, as we expect diffusion-dominated process to smooth out gradients rather than enhance them, which implies the up-gradient transport of heat and salt. These staircases were hypothesised to result from a secondary instability of the double-diffusive process, which Radko (Reference Radko2003) confirmed when he demonstrated their origin to lie in the so-called $\gamma$ -instability of the mean field of the double-diffusive fingering. Subsequent direct numerical simulations (Stellmach et al. Reference Stellmach, Traxler, Garaud, Brummell and Radko2011; Traxler et al. Reference Traxler, Stellmach, Garaud, Radko and Brummell2011) proved consistent with this picture, cf. the comprehensive review provided in the excellent textbook by Radko (Reference Radko2013).

The recent paper by Pružina et al. (Reference Pružina, Zhou, Middleton and Taylor2025) significantly advances the field by providing a deeper understanding, along with a physically more intuitive explanation, of the transport processes responsible for the up-gradient transport of heat and salinity in terms of advective and diffusive processes. Toward this end, it builds on the earlier work of Winters et al. (Reference Winters, Lombard, Riley and D’Asaro1995), which established a novel framework for analysing mixing processes in stratified flows by means of quantifying the background potential energy (BPE) and available potential energy (APE) via the sorted density field. As Pružina et al. (Reference Pružina, Zhou, Middleton and Taylor2025) point out, in a stratified flow whose density depends on a singular scalar, APE can be converted into BPE but not vice versa. Double-diffusive flows, on the other hand, can convert BPE into APE and give rise to a negative eddy viscosity, thereby creating an energy reservoir that can drive fluid motion. By analysing Direct Numerical Simulation results for the evolution of the sorted density field, the authors are indeed able to demonstrate the existence of a negative eddy diffusivity everywhere in the flow, including deep inside the well-mixed layers, where one might expect the turbulent flow to generate a positive eddy diffusivity. In fact, the numerical simulation results by Pružina et al. (Reference Pružina, Zhou, Middleton and Taylor2025) show that the negative eddy viscosity is strongest in the interior of these well-mixed layers. This uniformly negative eddy viscosity is responsible for the up-gradient diapycnal flux of heat and salinity, and for the steepening of the temperature and salinity gradients and the eventual emergence of the thermohaline staircases. The authors furthermore discuss the conservation equation for the square of the buoyancy frequency, $N$ , in sorted density coordinates. The individual mathematical expressions in this equation lend themselves well to an interpretation in terms of physical transport processes in the form of advection of $N^2$ with a velocity given by the gradient of the eddy diffusivity in sorted density coordinates, diffusion of $N^2$ and a generalised source/sink term. Taken together, these observations modify our earlier picture of thermohaline staircases, as they suggest that double-diffusive effects are dominant not just near the steep interfaces that separate the well-mixed layers, but everywhere throughout these well-mixed layers. As Pružina et al. (Reference Pružina, Zhou, Middleton and Taylor2025) point out, it will be interesting to extend the analysis of the buoyancy frequency budget to systems with both double-diffusive and shear instabilities, in order to gain insight into which of these processes will dominate under a given set of conditions.

Figure 1.

Layer formation in sedimentary fingering convection. (a) Horizontally averaged density profiles at different times (horizontally staggered for clarity). Note how four initial layers gradually merge into one, while the interfaces separating them move upward with time. (b) Snapshots of the sediment concentration field at the times shown by the arrows (from Reali et al. (Reference Reali, Garaud, Alsinan and Meiburg2017)).

The work by Pružina et al. (Reference Pružina, Zhou, Middleton and Taylor2025) also opens the door to obtaining a deeper physical understanding of other, closely related systems, for example those found in Lake Kivu (Newman Reference Newman1976) and Lake Nyos (Schmid et al. Reference Schmid, Lorke, Dinkel, Tanyileke and Wuest2004), where heat is supplied to the lake bottom by geothermal springs. Further interesting occurrences of staircases have been observed in environments where one of the scalar fields influencing the effective density is a particulate phase with an associated Stokes settling velocity. The existence of layer formation in such sedimentary fingering systems, akin to the formation of thermohaline staircases, was recently demonstrated in the work of Alsinan, Meiburg & Garaud (Reference Alsinan, Meiburg and Garaud2017) and Reali et al. (Reference Reali, Garaud, Alsinan and Meiburg2017), and further explored by Ouillon et al. (Reference Ouillon, Edel, Garaud and Meiburg2020), cf. figure 1. As Reali et al. (Reference Reali, Garaud, Alsinan and Meiburg2017) point out, the presence of a settling velocity can extend the parameter space in which layer formation is observed, as it broadens the range susceptible to the $\gamma$ -instability. In addition, the settling velocity can result in a novel layering instability for parameter combinations that are stable with regard to the $\gamma$ -instability.

Another related example concerns double-diffusive instabilities in systems with phase transition. For example, it was recently discovered that thermohaline staircases can form in the Dead Sea, where the level of salinity is close to saturation, so that the descending salt fingers, as they undergo cooling, become oversaturated and precipitate halite crystals that sediment on the lake floor at a rate of approximately 10 $\rm cm\,yr^{-1}$ (Sirota et al. Reference Sirota, Ouillon, Mor, Meiburg, Enzel and Lensky2020). Field observations by these authors demonstrate that the thermohaline staircase formation is less pronounced in the somewhat diluted regions of the northern Dead Sea close to the inflow of the Jordan river, as opposed to the even saltier regions near its southern end. In addition, the field measurements show the seasonal intensification of the staircases throughout the summer, as the surface layer heats up and increases in salinity due to enhanced evaporation. All of these features render the Dead Sea the only place in the world today that enables us to study the processes that shape the formation of so-called ‘salt giants’, km thick salt deposits that are found in Earth’s crust at numerous locations around the world (Meijer & Krijgsman Reference Meijer and Krijgsman2005; Warren Reference Warren2010; Meiburg & Lensky Reference Meiburg and Lensky2025). It is reasonable to assume that the existence of thermohaline staircases in such halite-precipitating systems will influence the vertical salt flux and hence the effective sedimentation rate of salt crystals and its spatial variation, which in turn controls the architecture of the large-scale salt giants. These processes are of great interest to geologists as well as to the oil and gas industry, since the nearly impenetrable salt deposits can form effective traps for hydrocarbons. A fascinating example of the widespread formation of salt giants concerns the Mediterranean Sea, which approximately 6 million years ago underwent a process analogous to what is occurring in the Dead Sea today. The Strait of Gibraltar closed, cutting off the inflow from the North Atlantic, so that the sea level of the Mediterranean gradually dropped as a result of evaporation. As a result, its salinity level increased to the point where massive quantities of salt precipitated on the seafloor. This is known as the Messinian salinity crisis, and the resulting salt deposits are well documented in the geological record.

Taken together, the above findings and their implications for different research areas represent a beautiful example of how a fundamental discovery in one field (double-diffusive instabilities and the emergence of thermohaline staircases in fluid dynamics) can have wide-ranging implications for our understanding in seemingly unrelated areas (formation of ‘salt giants’ in geology), and for technological applications that flow from these (hydrocarbon recovery). This confirms the essential role curiosity-driven research plays within the broader realm of science and technology.