# Vers une mesure simultanée du mélange et du transport turbulents en écoulement bicouche

Noé Clavier, Yvan Dossmann, Romain Volk, Mickaël Bourgoin

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Noé Clavier, Yvan Dossmann, Romain Volk, Mickaël Bourgoin. Vers une mesure simultanée du mélange et du transport turbulents en écoulement bicouche. 26ème congrès français de Mécanique, Aug 2025, Metz (France), France. 2025. [hal-05292922](https://hal-univ-lorraine.fr/hal-05292922v1)

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Submitted on 1 Oct 2025

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# Vers une mesure simultanée du mélange et du transport turbulents en écoulement bicouche

Noé Clavier<sup>a,*</sup>, Yvan Dossmann<sup>b,c,**</sup>, Romain Volk<sup>b</sup>, Mickaël Bourgoin<sup>b</sup>

a. Département de Physique, École Normale Supérieure PSL, 24, rue Lhomond, 75005 Paris

b. Laboratoire de Physique, École Normale Supérieure de Lyon, 46 allée d'Italie, 69007 Lyon

c. LEMTA, Université de Lorraine, 2 avenue de la Forêt de la Haye, 54500 Vendoeuvre-lès-Nancy

* noe.clavier@ens.psl.eu

** yvan.dossmann@ens-lyon.fr

## Résumé :

Les processus de mélange stratifié, et notamment la génération d'interfaces de densité jouent un rôle important dans la redistribution verticale de traceurs et particules dans l'océan. Nous étudions ici un fluide bicouche comme stratification modèle, dont l'interface de densité est soumise à une excitation turbulente autrement homogène et isotrope produite par un dispositif de forçage aléatoire disposé au fond de la cuve. Nous caractérisons certaines propriétés du mélange turbulent du champ de densité $ρ(r, t)$ grâce à des mesures d'absorbance par la technique d'atténuation lumineuse (LAT). Le phénomène de raidissement de l'interface (layering) est mis en évidence pour des nombres de Richardson $Ri = \Delta\rho gL/\rho_0 U^2 > Ri_c$ (où $\rho_0$ et $\rho_0 + \Delta\rho$ sont les densités des deux couches, L l'échelle de longueur typique et U la vitesse intégrale). Nous estimons $Ri_c \in [55, 90]$. De plus, les données confortent l'existence d'une loi de puissance d'exposant $-5/4$ entre la vitesse de déplacement de l'interface et le nombre de Rossby comme suggérée dans des études expérimentales (Rouse et Dodu, La Houille Blanche, 1955). Nous implémentons en parallèle la méthode de suivi lagrangienne de particules et mesurons de façon synchrone les trajectoires de particules inertielles initialement confinées à l'interface. Nous identifions leur fréquence typique d'oscillation. À mesure que le mélange progresse, les variations de flottabilité relatives entraînent une dynamique verticale accrue des particules inertielles. La mesure synchrone des trajectoires et du mélange dans des expériences contrôlées est une première étape vers l'étude du transport de particules, notamment plastiques et microplastiques, dans un environnement réaliste.

## Abstract :

Stratified mixing processes, and in particular the generation of density interfaces, play an important role in the vertical redistribution of tracers and particles in the ocean. Here we study a bilayer fluid as a model stratification, whose density interface is subjected to an otherwise homogeneous and isotropic turbulent excitation produced by a random forcing device located at the bottom of the tank. We characterise some properties of the turbulent mixing of the $ρ(r, t)$ density field using absorbance measurements by the light attenuation technique (LAT). The layering phenomenon is demonstrated for Richardson numbers $Ri = \Delta\rho gL/\rho_0 U^2 > Ri_c$ (where $\rho_0$ and $\rho_0 + \Delta\rho$ are the densities of the two layers, L the typical length scale and U the integral velocity). We estimate $Ri_c \in [55, 90]$. Furthermore, the data supports the existence of a power law of exponent $-5/4$ between the interface displacement velocity
and the Rossby number as suggested in experimental studies (Rouse and Dodu, La Houille Blanche, 1955). In parallel, we implement the Lagrangian particle tracking method and synchronously measure the trajectories of inertial particles initially confined to the interface. We identify their typical oscillation frequency. As mixing progresses, changes in relative buoyancy lead to increased vertical dynamics of the inertial particles. Synchronous measurement of trajectories and mixing in controlled experiments is a first step towards studying the transport of particles, particularly plastics and microplastics, in a realistic environment.

# Mots clefs : stratification ; turbulence ; mélange ; microplastiques ; light attenuation technique

## 1 Introduction

Oceanic pollution is a major global issue. Microplastics (< 5 mm) and nanoplastics (< 1 µm) considerably affect marine life due to their toxicity, but also because they are rafts which can transport potentially invasive micro-organism species from one ecosystem to another [1, 2]. The most recent studies indicate that 10⁷ tons of plastic entered the oceans for the sole year 2010 [3] and suggest that this figure could rise by 1 order of magnitude by 2025 [4]. Nonetheless, estimates of the amount of plastics at the surface of the oceans correspond to only a few percent of this global yearly input, in spite of their low density [1]. The vast majority of plastic waste has gone missing and consensus about their location is far from being reached [3]. The main hypotheses include, among others, beaching processes and vertical transport to the deeper layers of the oceans. In particular, the latter is supported by the little data available in which plastic is found in significant amount down to very large depths. It hence appears that particle transport is closely related to mixing processes of the different oceanic layers.

Oceanic flows obey the well-known Navier-Stokes equations describing momentum and mass conservation. They must be completed by transport equations of the scalar quantities on which density depends, i.e. temperature and salinity. For both the laboratory and the oceanic currents, density fluctuations are of order 10% or less and the so-called Boussinesq approximation is justified : density fluctuations can be neglected everywhere but in the gravitational term [5]. Since the equation of state relating the density $\rho$ to the salinity of saline water is linear when the relative variations of density considered are as large as 10⁻³ [6] the governing equations read :

$$ egin{cases} 
\nabla \mathbf{u} = 0 \\ 
\partial_t b + \mathbf{u} \nabla b = \nu_s \nabla^2 b \\ 
\partial_t \mathbf{u} + \mathbf{u} \nabla \mathbf{u} = -\frac{1}{\rho_0} \nabla p + \nu \nabla^2 \mathbf{u} + b \mathbf{e}_z 
\end{cases} 
\quad (1) $$

where $\mathbf{u}(\mathbf{r}, t)$ is the velocity of the fluid, $\nu_s$ the diffusivity of salt in water, $\rho_0$ the density of fresh water and $b \equiv -g(\rho - \rho_0)/\rho_0$ the buoyancy. Here double-diffusivity effects have been neglected and the temperature is assumed to be constant. Therefore, 3 main dimensionless parameters define the parameters space :

$$ Re \equiv \frac{UL}{\nu}, \quad Pr \equiv \frac{\nu}{\nu_s}, \quad Ri \equiv \frac{\Delta \rho g L}{\rho_0 U^2} $$

The Reynolds number $Re$ compares momentum advection with its diffusion and the Prandtl number $Pr$
momentum diffusion with salinity (i.e. buoyancy) diffusion. The Richardson number $Ri$ is the ratio of the potential energy stored in stratification to the turbulent kinetic energy. Following [7], we define it based on the integral length scale $L$ (correlation length) of turbulence and velocity at this scale $U$.

Modelling the properties of turbulence and mixing under external forcing and initial stratification has been the subject of considerable research since the 1960s [5]. However, technical challenges are many. Even a weak stratification breaks down isotropy. Turbulence in stratified flows is known to be spatially and temporarily intermittent with a potential key role of rare events. Among the large number of physical quantities, the definition of several of them are still debated, such as the seemingly straight-forward *mixing efficiency* [8]. Therefore, the question of whether idealised laboratory experiments or simulations are relevant to real geophysical flows is not even settled yet [5]. In simulations, the challenge lies in the computational cost the small-scale processes, which must be parametrised in global circulation models [3]. A common idealised experimental model is the two-layer fluid, i.e. a system made of a deep, initially homogeneous layer of density $\rho_l$ and a shallow, lighter layer at $\rho_u < \rho_l$ with a sharp transition between the two, and $Ri \gg 1$. Experiments are often further idealised by the use of very specific forcing such as oscillating grids crossing the interface [9, 10]. Measurements of density are often pointlike [9] or done by the light-attenuation technique (LAT). In this latter case, one fluid phase is initially dyed and the absorbance of the solution is a proxy of the density at each time and position. Since LAT only measures densities integrated along the optical path, it is common practice to have an experimental tank whose depth $L_y$ (along the optical path) is small compared to its width $L_x$, but to the cost of breaking isotropy.

In this paper, we report on LAT measurements of mixing in which a two-layer fluid in a tank with horizontal aspect ratio $1$ ($L_x = L_y$), under forcing from an array of randomly-actuated synthetic jets (RASJA). This type of forcing ensures a somewhat homogeneous, isotropic turbulent forcing coming from the bottom phase [11, 12]. It is more realistic than grid forcing for no solid object directly disturbs the density interface. We study the influence of the initial stratification and vary $Ri$ from 37 to 134. We also present a setup in which the trajectory of plastic particles trapped at the interface can be studied simultaneously with the mixing of the fluid. Such dual measurements could prove useful to understand the connection between microplastic transport and oceanic layer mixing.

We first describe the setup and the light-attenuation technique which we use to measure the fluid's density. Then we show how the density interface is displaced up under mixing and a power law is found that relate its velocity to the density jump. We observe that, counter-intuitively, the strong interfaces become even stronger under mixing and relate this observation to the theory of layering proposed by Phillips [13] and Posmentier [14]. The evolution of the potential energy of the fluid over time is investigated and an estimate of the mixing efficiency is proposed. Last, we present the dual particle tracking-LAT setup that allows for a simultaneous characterisation of particle transport and density mixing. Preliminary results are shown.

## 2 Experimental methods

Experiments are carried out using the water tank and turbulence-forcing system depicted in Figure 1. The tank is 160-cm high and has a square horizontal section of inner width $L_x = L_y = 30$ cm. The bottom of the tank is closed, whilst the top is opened. The tank is filled up with tap water which can be pumped in and out from two holes drilled through the bottom face. Density is varied through the concentration in sodium chloride. In order to create a two-layer fluid, we first fill the tank up to the desired level with tap water in which sodium chloride is dissolved. Then, the second layer is created
<figure><img src="image_1.png" /><figcaption>FIGURE 1 – **Left** : Schematif of the setup. **Right** : Schematic of the float used to create two-layer fluids.</figcaption></figure>

using a specially-designed float (Fig. 1). It is made of a 3D-printed structure in the shape of a square reservoir, surrounded by plastic foam to ensure buoyancy. The bottom face of the float, however, is not made of plastic but of compressed sponges. The reservoir is filled with fresh water which flows very slowly through the sponge. The resulting interface is typically 2 centimeters thick.

At the bottom of the tank, 16 bilge pumps are laid out in a 4×4 grid. The pumps are all independently controlled by 16 relays connected to a Raspberry-Pi controller and powered with a common adjustable voltage source. The exit pipes of the pumps are equipped with check valves. They are all parallel—pointing upwards—and the nearest-neighbour nozzles are 7.5 cm away. This choice of spacing ensures planar symmetry of the forcing with respect to the wall—the outer nozzles-to-wall distance is half the inter-nozzle distance—which tends to cancel out the stress tensor at the boundaries and suppress secondary mean flows [15]. The pumps are independently and randomly actuated. Suction occurs near the bottom surface of the tank and always simultaneously with the jets, which are therefore called “synthetic jets” (because of the zero net mass flux across any closed surface surrounding a pump). Such a forcing is commonly referred to as RASJA (for Randomly Actuated Synthetic Jet Array) and it proves to be able to produce relatively homogeneous and isotropic turbulence in *xy* planes [11]. We set *z* = 0 at the nozzles’ ends (40 cm above the bottom face) and *x* = *y* = 0 in the middle of the tank. The random actuation of the jets can be performed in various ways, but [11] found that the following scheme ensures the most homogeneous, isotropic turbulence : (i) for each pump, a random power-on duration is drawn for each pump according to a gaussian distribution of mean τon and width σon (ii) after this time, a power-off duration is similarly drawn with parameters τoff and σoff (iii) the process is repeated. [12] extensively characterised the flow this forcing produces in the present setup. We set accordingly τon = 4 s, τoff = 28 s and σon,off = τon,off/3 for all pumps. In this configuration, in the volume *x*, *y* ∈ [-4, 4] cm, *z* ∈ [75, 85] cm, the mean flow remains small compared to the fluctuations,
and the latter are satisfactorily isotropic :

$$ \begin{array}{c} \frac{\langle u_x \rangle}{\sqrt{\langle u_x^2 \rangle}} \\ 3.2\% \\ \\ \frac{|\langle u_z \rangle|}{\sqrt{\langle u_z^2 \rangle}} \\ 10\% \\ \\ \frac{\sqrt{\langle u_z^2 \rangle}}{\sqrt{\langle u_x^2 \rangle}} - 1 \\ 1.1\% \end{array} $$

The present work focuses on a significantly larger region in the xy plane ($x, y \in [-12, 12]$ cm), which may slightly impede isotropy due to boundary effects. However, here the interfaces are further away from the forcing ($z \in [90, 105]$ cm) and the typical forcing power is smaller as well. Since [12] showed that isotropy improves with height—and most likely with decreasing forcing, we can therefore be confident that turbulence remains satisfactorily isotropic in our region of interest. In addition, [12] observed that the turbulent kinetic energy decreases exponentially with $z$ with a typical attenuation length $L_d = 28.4$ cm, to be compared to the measurement area of height 25 cm $\sim L_d$. Turbulence will therefore decrease significantly in intensity as the interface travels up throughout each experiment. Most importantly, it must be stressed that the presence of a density interface is likely to affect significantly the flow by changing the boundary conditions, especially in the vertical direction. In the case of grid turbulence, [1] reported strengthened mean circulations because vertical scales had to be transferred to the horizontal ones at the interface. Nevertheless, it is not a flaw of the setup but an intrinsic feature of turbulence in stratified fluids and the above characterisation can be seen as that of the forcing device only, rather than that of the actual flow.

The flow is monitored using two Ximea MQ022MG-CM black and white HD cameras whose respective optical axes are perpendicular to the sides of the tank and to each other (see Fig. 1). Two LED panels are placed against the two faces opposite to the cameras. Cameras are synchronously triggered by a function generator. The salty-phase is acidified to pH $\sim 5.3$ to prevent calcium carbonate precipitation and ensure transparency. In tracking experiments, the refractive index of the two layers are adjusted to the same value by the addition of ethanol to the upper layer. We carry out light-attenuation measurements (LAT), a technique that has been tried and tested for 30 years for geophysical or stratified flows [16–18]. The dense layer is dyed with E133 before any turbulence is forced or dilution is made. $\nu_s \approx \nu_{dye} \ll \nu$, hence the dye concentration is proportional to the salt concentration throughout the experiment. The density of the dense, salty phase is itself a linear function of the salt concentration. In LAT, the absorbance $A_{2/1} \equiv \log_{10}(I_2/I_1)$ of the solution is measured, where $I_1$ and $I_2$ are the intensity of the light passing through the water tank at a given position in two different flow states 1 and 2, measured by a given pixel of one camera. They are related to the density via Beer-Lambert law extended to polychromatic light :

$$ A_{2/1} = \log_{10} \left( \frac{\kappa e^{\tilde{\beta}(\rho_1 - \rho_0)} + 1 - \kappa}{\kappa e^{\tilde{\beta}(\rho_2 - \rho_0)} + 1 - \kappa} \right) \quad (2) $$

where $\tilde{\beta}$ and $\kappa$ are fitting parameters, $\rho_0$ is the density of fresh water and $\rho_1$ and $\rho_2$ are the densities of states 1 and 2 integrated along the optical path. The origin of equation (2) is explained in Appendix A.
<figure><img src="image_2.png" /><figcaption>FIGURE 2 – Density maps seen by one camera for $\Delta\rho = 70 \text{ kg/m}^3$. Strong mixing event alternate with calm phases (marked with *) for which an interface position is defined. The upper layer is quiet, just nibbled.</figcaption></figure>

# 3 Results

## 3.1 Scaling of the interface displacement velocity

We perform 7 experiments with the upper layer density $\rho_u = 1003 \pm 1 \text{ kg/m}^3$ in all cases and the initial lower layer density $\rho_l$ ranging from $1042$ to $1110 \pm 2 \text{ kg/m}^3$. We measure the evolution of density with time, during circa 30 minutes or until the interface has travelled out of the measurement volume. The RASJA is actuated with the same parameters for all experiments. Therefore, the control parameter which we vary is directly the Richardson number. However, the values of $U$ and $L$ in the case of a two-layer fluid are unknown—and actually ill-defined due to the isotropy breaking. Using the reference values for the forcing from [12], we estimate that $Ri$ ranges from circa 37 to 134. Nonetheless, in the following we choose to express the results as a function of $\Delta\rho \propto Ri$ rather than $Ri$, since the former is the quantity which we accurately measure.

Typical density maps obtained are reported in Fig. 2. A general feature observed for all interfaces is that very strong mixing events (e.g. $t = 690$ s) alternate with much calmer states (marked with *). We also find that density remains uniform and constant in the upper layer, in agreement with the well-known observation that the interface acts as a barrier for turbulence in such a way that the upper layer is undisturbed, but nibbled from the bottom [9, 19]. When the interface looks quiet enough, it makes sense to average $\rho$ along $x$. Note that because the measurement volume is not parallepipedic (see Fig. 1), the average we take is somewhat weighted: the sides of the tank will count for a bit less than the middle, for instance. However, if the interface is quiet and uniform this shouldn't matter. The averaged density profiles and their respective gradient are shown in Fig. 3a,b ($\Delta\rho = 42 \text{ kg/m}^3$) and Fig. 4a,b ($\Delta\rho = 106 \text{ kg/m}^3$). Three features are observed in the evolution of these profiles. First, the progressive upward displacement of the interface is caused by the mixing induced by the random forcing events. Second, The displacement is associated with a slow yet measurable decrease of the density in the lower layer. Third, the density gradient exhibits two different behaviours in the two experiments. For $\Delta\rho = 42 \text{ kg/m}^3$, a progressive thickening of the interface and decrease of the maximum gradient is measured, while the interface clearly sharpens for $\Delta\rho = 106 \text{ kg/m}^3$. The observed layering mechanism at larger $\Delta\rho$ is discussed in the next section.
<figure><img src="image_3.png" /><figcaption>FIGURE 3 – Average density (a), density gradient (b), and BPE variation (c) profiles at selected times where the interface is relatively quiet, for $\Delta\rho = 42 \text{ kg/m}^3$. Blue : Initial state. From light to dark grey as mixing advances. The dashed and solid lines correspond to each of the two cameras.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIGURE 4 – Average density (a), density gradient (b), and BPE variation (c) profiles at selected times where the interface is relatively quiet, for $\Delta\rho = 106 \text{ kg/m}^3$. Blue : Initial state. From light to dark grey as mixing advances. The dashed and solid lines correspond to each of the two cameras.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>FIGURE 5 – (a) Displacement of the interface over time for several density jumps $\Delta\rho$ (in kg/m$^3$). (b) Time-averaged interface velocity for several density jumps $\Delta\rho$. It is consistent with a (-5/4)-power-law (solid line) which is also the best-fit exponent. For reference, the dotted lines represent exponents 10% larger or lesser.</figcaption></figure>

A useful measure to quantify mixing and be able to parameterise it in global circulation models is the so-called *mixing rate*. A first way to define that rate is to consider the velocity at which the interface is displaced up. The position of the quiet interface $z_i(t)$ is plotted against time for various $\Delta\rho$ in Fig. 5a. But our data shows two different regimes : for $\Delta\rho \le 50 \text{ kg/m}^3$ (filled symbols), the velocity significantly decreases with time/altitude, whereas is is relatively constant for $\Delta\rho \ge 60 \text{ kg/m}^3$ in spite of the decreasing forcing intensity. It is not possible to tell, based on $z_i$ alone, whether two different mechanisms are at stake or if the time scales over which the system evolves are just so large at large $\Delta\rho$ that the data shown in Fig. 5 only correspond to the linear start of the low-$\Delta\rho$ ones. In any case, the larger $\Delta\rho$ the slower the displacement. Fig. 5b shows the interface velocity $\dot{z}_i(t=0)$ estimated from the first half of each of the curves $z_i(t)$, as a function of $\Delta\rho/\rho_0$. We find a power-law $\overline{z_i} \propto (\Delta\rho/\rho_0)^{-5/4}$. Based on both experimental data and theoretical considerations, Rouse et al. [9] found a relation that implies the same exponent, although in a different configuration (oscillating grid turbulence, interface maintained at a constant height). [10] too found a similar scaling law with an exponent in the estimated range [-5/2, -1] (best-fit : -5/4), again in the case of grid turbulence but with a moving interface. The present results suggest that this scaling law could be valid in a more general framework, making a step towards the generalisation of experimental results to more realistic systems.

## 3.2 Layering

The examination of the mixing speed is not sufficient to describe the processes at stake for it could not state whether or not the mixing mechanisms are qualitatively the same, regardless of *Ri*. Therefore, we now focus on the evolution of the interface thickness with mixing. Fig. 6 shows that, for large *Ri*, the interface becomes thinner under mixing. This is the counter-intuitive *layering* phenomenon, already observed in the 1950s but for constant interface height and grid turbulence [9]. Again, our experiments suggest that mixing phenomenology may be quite insensitive to the type of forcing. It is challenging to
<figure><img src="image_6.png" /><figcaption>FIGURE 6 – Raw picture of an interface with $\Delta\rho = 103 \text{ kg/m}^3$ (a) before and (b) after 300 s of turbulent forcing and 60 s of rest. Layering is clearly visible. Parallax effects are small (the interface only moved by 1.5 cm = 1/15th of the total image) and at this $z_i$ they would tend to compensate layering anyway.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 7 – (a)-(g) $N^2$, normalised by its maximum throughout the given experiment, against altitude $z$ and the position $z_i$ of the interface at this time, for various $\Delta\rho$. (h) Same quantity throughout an experiment where a static interface is moved all the way down. Blue lines are guides only. (a)-(c) exhibit a thinning of the interface characteristic of layering. (f)-(g) don't. (d)-(e) are intermediate.</figcaption></figure>
define a scalar measure of layering in the present experiment, for the thickness of the interface is not
easily defined when strong mixing events create waves and non-monotonic and/or asymmetric density
profiles. Therefore, we study the full map of an intensive quantity instead : the buoyancy frequency
$N \equiv \sqrt{-g(\partial_z \rho)/\rho}$, the frequency at which a fluid particle initially moved away from its equilibrium
position will oscillate around it. The larger $N$, the stronger the interface. $N^2$ is plotted in Fig. 7a-g against
the altitude $z$ and the interface position at each time $z_i(t)$. Parallax problems bias these results : when
the interface is, say, close to the top of the measurement volume, then it is seen from the bottom with
an angle and appears thicker than it really is. To quantify the parallax effects in our complex geometry,
we conducted a simple experiment where a static interface is moved all the way down the z-axis (by
emptying the tank from the bottom). We plot $N^2$ also for this reference experiment in Fig. 7h.

For large $Ri$ ($\Delta\rho \ge 70$ kg/m$^3$), the large-gradient region narrows significantly as the interface travels up, in proportions that are significant regardless of parallax effects as the guiding lines of Fig. 7 show. Notably, parallax tends to enlarge the apparent large-gradient region when it is away from the optical axis ($z = 95$ cm), whereas graphs 7a-c show a constant thinning of that region throughout the experiment. On the other hand, no such layering is observed at $\Delta\rho \le 42$ kg/m$^3$ but the large-gradient area thickens instead, like one would intuitively expect. From the data presented here, density jumps $\Delta\rho \in [50, 60]$ kg/m$^3$ seem to belong to some intermediate, transitional state with neither thinning nor thickening of the interface. It is possible that they exhibit one of the two aforementioned behaviour but on time scales longer than that of our experiment. This picture—layering at large $Ri$, transient at medium $Ri$, no layering at small $Ri$—is consistent with the theory of layering proposed by [13] and [14]. This theory is based on a stability analysis under the assumption that the local mixing efficiency $\eta$—corresponding here to the amount of mixing under a given injection of energy—increases monotonically with $Ri$ from 0 up to a maximum at $Ri_c$ and then decreases for $Ri > Ri_c$. Such a relation is expected to be true because (i) it is hard to mix something almost homogeneous already ($Ri \to 0$) (ii) it is hard to mix something very stratified hence extremely stable ($Ri \to \infty$). However, it has not been demonstrated yet. Let us consider a strongly stratified fluid ($Ri > Ri_c$) under this assumption. Assume that a local perturbation increases the density gradient $|\partial_z \rho|$ at $z_1$. $Ri(z_1)$ increases and thus $\eta(z_1)$ decreases : the buoyancy flux towards the region just above diminishes. Because the total $\Delta\rho$ is unchanged, the perturbation of $\partial_z \rho$ at $z_1$ must be accompanied by a symmetrical opposite perturbation, say, at $z_2 > z_1$, through which the buoyancy flux increases. As a result, the region around $z_2$ gets more homogenised and less mixed with $z < z_1$ : a layer forms around at $z_2$ and a sharp interface at $z_1$. Evidently, the fluid is not “unmixed” by layering : the buoyancy potential energy does increase, the evolution is irreversible and the interface can sharpen only because it also moves upwards. The present results place $Ri_c$ approximately in the range 55–90.

## 3.3 Potential Energy budget

Another approach to the mixing speed is to consider how much power is involved in the mixing process.
Following [8], we introduce the total background potential energy

$$ \mathrm{BPE}_{\mathrm{tot}}(t) \equiv \int d^3\mathbf{r} \hat{\rho} gz \quad (3) $$

where $\hat{\rho}(z, t)$ is the sorted density profile obtained from $\rho(\mathbf{r}, t)$ by an adiabatic, incompressible rearrangement of fluid parcels such that $\hat{\rho}(z, t)$ is everywhere statically stable ($\partial_z \hat{\rho} \le 0$). It represents the energy that is not available to the system and increases as the fluid is irreversibly mixed. $BPE_{tot}$ de-
<figure><img src="image_8.png" /><figcaption>FIGURE 8 – Extended density, integrated BPE and Mass variation profiles for $\Delta\rho = 42 \text{ kg/m}^3$. Colours and lines as in Fig. 3.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 9 – Extended density, integrated BPE and Mass variation profiles for $\Delta\rho = 106 \text{ kg/m}^3$. Colours and lines as in Fig. 4.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>FIGURE 10 – Background Potential energy (a) and Dissipation evolution (b).</figcaption></figure>

pends on the entire tank volume rather than only on the mixing zone. The density profiles taken at rest permit to quantify the evolution of this background potential energy. In the following, we consider the partially integrated energy $BPE(t, z) \equiv \int_0^z d^3\mathbf{r}'\hat{\rho}(t, z')gz'$. The variation of BPE in the measurement field shown in Fig. 3c and Fig. 4c, with the origin taken at the bottom of the field shows an increase for both thickening and layering scenarios. In order to evaluate the evolution of the integrated $BPE(z, t)$ with respect to the energy input, density profiles are extrapolated to the base of the tank, assuming a constant density equal to the value at the bottom of the measurement window as shown in Fig. 8a and Fig. 9a for $\Delta\rho = 42 \text{ kg/m}^3$ and $\Delta\rho = 106 \text{ kg/m}^3$. This assumption is supported by the intense turbulent mixing forced by the pumps that homogenize the bottom layer, and the evaluation of the integrated mass $m(z, t) \equiv \int_0^z d^3\mathbf{r}'\rho(z', t)$ that varies by less than 0.1% over time (Fig 8c and Fig 9c). At each time, the integration of $\Delta BPE(t, z) = BPE(t, z) - BPE(0, z)$, in Fig 8b and Fig 9b exhibit a decreasing vertical trend in the lower layer owing to the density decrease by mixing with lighter fluid parcels. The rise of the interface overcompensates this decrease and leads to a strong positive variation of $\Delta BPE$ in the narrow region surrounding the interface. This behaviour confirms that mixing processes participate to the rise of the center of mass of the fluid for both the scouring and layering scenarios.

The evolution of the background potential energy of the fluid column is assessed using $\Delta BPE(t, H)$ ($H$ is the height of the tank), where measurement times are chosen when the fluid is in transient quiet states. The overall BPE variation with time is displayed in Fig. 10a. All experiments display similar trends, with a fast BPE growth rate when the interface is low at early times and a deceleration of the growth rate at later times. Note that while the initial density difference varies by a factor of circa 4 through the experiments, this variation hardly affects $\Delta BPE$ changes in the course of the experiment, particularly at large $\Delta\rho$. The decrease of the BPE growth rate is mirrored in the evolution of the energy dissipated around the interface $D(t) = \int_0^t dt'eL_x L_y \rho_0 \epsilon(z_i(t))$ shown in Fig. 10b. The latter is evaluated assuming a typical (fixed) interface thickness of $e = 2$ cm and relying on the profile of energy dissipation rate
<figure><img src="image_11.png" /><figcaption>FIGURE 11 – Mixing efficiency evolution for 7 experiments, with $\Delta\rho$ varying from 28 kg/m$^3$ to 106 kg/m$^3$;</figcaption></figure>

$\varepsilon(z)$ measured in a homogeneous column by [12]. While these assumptions lack information regarding the role of the interface on the dissipation profile, they allow a first estimate of the amount of energy contributing to the irreversible BPE increase.

The ratio $\Gamma(t) \equiv \Delta BPE(t, H)/D(t)$ evaluates the fraction of energy that has transferred to irreversibly mixing the fluid from the beginning of the process to time $t$. While the canonical value of $\Gamma = 1/6$ issued from grid turbulence experiments has been widely used in parameterisations of mixing processes, recent studies have shown the strong process-dependent behaviour of the mixing efficiency, with values ranging from 10% in lock-exchange gravity currents to more than 70% for topography induced nonlinear mixing. The present experiments display values in the range [6–8]%- under the present assumptions regarding $\epsilon$- which is close to the lower bounds for measured mixing processes.

# 4 Towards the simultaneous characterisation of particle transport

## 4.1 Lagrangian tracking

In the second part of this work, inertial particles are added to the fluid in order to characterise their transport properties and try to correlate them to the measures of mixing identified above. As a first step and to simplify their tracking, spherical particles of diameter $d_p = 5 \pm 0.1$ mm and density $\rho_p = 1024 \pm 2$ kg/m$^3$ are chosen. For a given experiment, the density of each layer and of the particles are chosen so that the stable position of the particles is initially at the interface. Light-attenuation measurements are still carried out. In addition, the individual 3D trajectories of a set of particles is recorded. This tracking is performed following the method developed by [20]. First, cameras are calibrated using $N$ pictures of a test pattern moved to well-known locations within the measurement area along its normal axis (Fig. 12a). The pixel-to-real-world relation is inferred for these $N$ planes. Then, the spherical particles of well-known radius are detected in experimental pictures using circular Hough transforms. The particle centres
<figure><img src="image_12.png" /><figcaption>FIGURE 12 – (a) N pictures of a test-pattern moved along *w* provide N pixel-to-real-world transformations. (b) The 2D position of a particle (grey disk) is translated to N real-world positions through which a ray is fitted. (c) The midpoint of the shortest segment between the two cameras' rays is the particle's actual position (solid lines, red cross). With a second detection (dotted lines, black cross), false positives are likely to appear (triangle).</figcaption></figure>

coordinates in a camera's world are translated to *N* real-world positions in the calibration planes (Fig. 12b). The refractive index of both phases have been matched and light hence propagates in a straight line within the measurement area. A line is fitted through the *N* points accordingly. For each frame, all the lines corresponding to one camera are compared to those of the other one. When 2 lines pass close enough to each other, a particle must be present there, say at the midpoint (Fig. 12c). The shortest distance between the two rays—which would ideally cross—is at most 500 µm and generally less than 200 µm. As a result, we get a set of the locations at which a (any) particle is present for each time $t_i = i \times dt$, where $f = 1/dt = 25$ Hz is the frame rate. The Lagrangian trajectory of an individual particle is reconstructed in the chronological order, by looking to whether a particle has been detected at $t_{n+1}$ at a distance less than $D_{max}$ of the predicted position based on the trajectory during $\{t_i\}_{i \le n}$. This prediction is made by a third-order polynomial extrapolation with $D_{max} = 2$ mm. Each time the bead is not detected or gets out of the measurement volume, the trajectory is interrupted.

## 4.2 Trajectories under mixing events

The method was tested on beads attached to wires and moved by hand before we tried to apply it to an interface populated with 10 to 20 beads. But because particles are confined at the interface which is seen from the side by the cameras, the light rays are almost all co-planar. Therefore, the number of false intersections (see Fig. 12c) gets dramatically large as the number of beads increases. A criterion on the bead apparent radii—which must been consistent between the two cameras—is implemented but does not improve the situation much because the radii are affected by the remaining refractive index gradients and thus not measured accurately. Although we the false positives are excluded at the trajectory reconstruction step, the trajectories we obtain are very short in particular during the strong mixing events and their statistical properties are most likely biased. In the preliminary results which follow, we circumven-
<figure><img src="image_13.png" /><figcaption>FIGURE 13 – Example trajectories of a particle during ~380 s. Each trajectory ends if the tracking fails or the particle goes out of the measurement volume. Note the aspect ratio *xy/z*.</figcaption></figure>

ted these issues by using at most two beads at a time. An example of the trajectories obtained is given in Fig. 13. A 23-minute experiment with a single bead yielded 58 trajectories of at least 25 frames (1 second), together representing 98.5% of the detected positions and covering 50% of the observation time (17500 frames). We checked that during the majority of the 50% remaining observation time, the bead was indeed not visible by one of the cameras. To study a larger number of beads and get better statistics, adding a third camera with a different angle and increasing the frame rate could prove very efficient.

The data presented here is only preliminary. Nevertheless, we can already highlight a few of the properties of the particles trajectories in the case of large-density interface ($\Delta\rho = 59 \text{ kg/m}^3$) and relatively heavy beads ($(\rho_l - \rho_p)/\Delta\rho = 0.22$). First, the probability density function (PDF) of the position of the bead during trajectories longer than 10 frames is plotted in Fig. 14 in coordinates $\{x, y, z - z_i(t)\}$. The particle positions seem evenly distributed along $x$ and $y$, which is reassuring because it suggests that boundary effects are still small and turbulence homogeneous in $xy$ planes. The drop of the PDF at the sides is simply due to the fact that it is quite unlikely for the bead to stay close to the edge of the measurement volume for a long time without ever exiting it. The PDF of the $z$-positions highlights the relative confinement of the particle close to the interface, but reveals that the particle can travel quite far away from it under the action of turbulence, typically 3 times the interface thickness. Fig. 14 shows that $\bar{z} = -30 \text{ mm}$, because the bead's density is closer to that of the lower fluid layer. Hence the beads stay rather below the middle of the interface.

We also observe that the beads mostly oscillate along $z$. They move along $x$ and $y$ at some significant velocity too, but without clear periodicity. To quantify these assertions and explain the oscillations, we compute the Fourier spectrum of each component of the bead's velocity $v(t)$. However, the longest trajectories are about 40 s long, which is only of the order of the forcing time scale $\tau_{\text{on}} + \tau_{\text{off}} = 32 \text{ s}$. As a result, a single trajectory is not necessarily representative of the overall dynamics. Therefore, we don't plot the spectrum of a single trajectory but the average spectrum of the $M = 21$ longest trajectories (Fig. 14). They all last 15 s at least and their cumulative duration reaches 521 s. The choice of $M$ is a trade-off between number of trajectories and low-frequency resolution. Spectra of $v_x$ and $v_y$ overlap quite well, suggesting that the transport dynamics is isotropic in $xy$ planes. Vertical velocities, however, are affected : low-frequency components are comparatively small, which is consistent with the idea that
<figure><img src="image_14.png" /><figcaption>FIGURE 14 – (a) PDF of the position of the beads, $\Delta\rho = 59 \text{ kg/m}^3$. $z = 0$ is the interface position. (b) Average of $M$ power spectrum of the velocity along each direction. The coloured surroundings represent the standard deviation over $\sqrt{M}$. The grey area depicts the range of $N^2$.</figcaption></figure>

the particles cannot keep a vertical velocity unchanged for too long without buoyancy bringing it back to its equilibrium position. It is, however, non-negligible for frequencies as small as $10^{-1} \text{ Hz}$. Although no sharp peak is found, the spectrum clearly reaches a maximum in the frequency range 0.2 – 0.6 Hz. It is comparable to the range of buoyancy frequency $N$ observed across the interface, although ~20% smaller. Correction of the expression to account for a linear drag tends to slightly decrease $N$ but by a negligible amount. At fast scales, the spectrum of $v_z$ collapses with the two others. It could indicate that buoyancy effects do not dominate the velocity dynamics at these scales.

## 5 Conclusion

In this work, we have investigated the mixing in two-layer fluids focusing on the influence of the overall Richardson number $Ri_0$, using random synthetic jets as a generator of isotropic, homogeneous turbulence. Mixing was studied from two perspectives. Firstly, we determined the density field using light attenuation measurements. First, a power law relation between the interface velocity and $\Delta\rho$ (i.e. $Ri_0$) was found with a $-5/4 \pm 1/8$ exponent also obtained for grid turbulence [9, 10]. Second, the layering phenomenon at large Richardson was clearly observed between the initial and final state and, using the buoyancy frequency evolution as a criterion, we estimate $Ri_c \in [55, 90]$. Third, we provide an estimate of the mixing efficiency in our system, of [6–8]%, close to the lower bound of previous measurements of mixing processes.

In addition, we presented a setup and methods to simultaneously investigate particle transport using the same setup and cameras. The preliminary results show that the particles behaviour remains isotropic in $xy$ planes even in the presence of stratification and 5 cm away from the walls only. The typical vertical oscillation frequency of the beads was found to be close to the buoyancy frequency. With larger data sets, cross-analysis of the mixing and transport measurements could be carried out. Finding relations between the two would mean that some information on local transport of plastics can be inferred from the larger-scale mixing properties, but the indicators to consider are not yet clear.
Promising developments include using linear stratifications or low-Richardson density jumps, using actual microplastics collected in situ and adjusting the forcing parameters to better match those of the oceans and estuaries, in order to attempt to bridge the relevance gap which still separates experiments and geophysical flows. Collaboration with specialists of microplastics in the oceans (chemists, biologists, geophysicists) should fuel further laboratory research.

# Appendices

## A Model for the extended Beer-Lambert law

In the large-density-gradients, large-salt-concentration regime we focus on here, the equation of state of saline water is linear. Its density $\rho$ reads $\rho = \rho_0 + \alpha c_{NaCl}$ where $\alpha$ is a known constant and $c_{dye}$ and $c_{NaCl}$ the dye and salt molar concentrations, respectively. We do not control accurately the salt-to-dye concentration ratio, therefore $\tilde{\alpha}$ is not known quantitatively. However, experiments with various $Ri$ are carried out in a row by successive dilutions, which ensures that $\tilde{\alpha}$ is a constant throughout the measurement campain. The usual absorbance-concentration relation is Beer-Lambert law, which states that the logarithm of the light intensity $I_2$ of each pixel normalised by the intensity $I_1$ of (any) other picture with varies linearly with $c$. Thus, $A_{2/1} \equiv -\log_{10}(I_2/I_1) = \tilde{\beta}(\rho_2 - \rho_1)$ where $\tilde{\beta}$ is a constant throughout the measurement campain. However, the LED panels used are polychromatic and Beer-Lambert law must be modified accordingly. Let us assume that the lighting intensity $I_0(\lambda)$ is constant in the wavelength range $[\Lambda_1, \Lambda_2]$ and is zero elsewhere, and the molar attenuation coefficient of the dye $\varepsilon(\lambda)$ to have a similar behaviour but in the range $[\lambda_1, \lambda_2]$. Then,

$$I_{1,2} = \int d\lambda \mathcal{I}_0(\lambda) 10^{-\varepsilon(\lambda)lc_{1,2}} \quad (4)$$

yields

$$A_{2/1} = \log_{10} \left( \frac{\kappa e^{\tilde{\beta}(\rho_1 - \rho_0)} + 1 - \kappa}{\kappa e^{\tilde{\beta}(\rho_2 - \rho_0)} + 1 - \kappa} \right) \quad (5)$$

with $\tilde{\beta}$ as above, $l$ a length, $\kappa = (\lambda_2 - \lambda_1)/(\Lambda_2 - \Lambda_1)$ and $\rho_0$ the density of fresh water ([21] used this relation too in a less general case).

Throughout the series of successive experiments, calibration pictures of the tank filled with a homogeneous, saline layer are taken, and the density of the fluid is measured with a densimeter. Then the fitting parameters $\kappa$ and $\tilde{\beta}$ are inferred. Fig. 15 diplays the results of this procedure. The thick lines show that the model fits very well the entire data set with adjusted values $\tilde{\beta} = 7.08\text{cm}^3/\text{g}$ and $\kappa = 0.73$ falling within 15% of estimates based on order of magnitudes $\tilde{\beta} \approx 6$ and $\kappa \approx 0.72$. For reference, dashed lines show that Beer-Lambert law is not sufficient. However, more thorough analysis of the data reveals that the very good agreement on average to the model above hides significant spatial inhomogeneity. In particular, due to the geometry of the measurement area (see Fig. 1), light rays coming from the sides, top and bottom cross the fluid on longer distances which influences $l$ and hence $\tilde{\beta}$. We therefore divide the pictures in 144 sections (each 100 pixels wide, i.e. $\approx 1.8$ cm) and adjust the model parameters for each of them. This process corrects for the majority of the heterogeneity observed.
<figure><img src="image_15.png" /><figcaption>FIGURE 15 – Absorbance to density relation. **Dashed lines**: linear fit (Beer-Lambert's law). **Solid lines**: linear-exponential model (2) with $\kappa = 0.73$. Each colour corresponds to a camera.</figcaption></figure>

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