H M De La

Rosa Zambrano

G Verhille

P Le Gal

Fragmentation of magnetic particle aggregates in turbulence

Particle aggregation is an ubiquitous phenomenon encountered in many natural and industrial environments. We describe here the equilibrium state of the aggregation/fragmentation process of inertial scale particles clustering in turbulence i.e. when the particles and the aggregates are larger than the Kolmogorov dissipative scale η. For this purpose, we seeded a high Reynolds number turbulent von Kármán flow with millimetric nearly neutrally buoyant magnetic particles. Each magnetic dipole imposes a torque and a force on the other magnets at the origin of the cohesion of the aggregates. On the contrary, turbulent fluctuations impose external stresses which tend to fragment the aggregates. Using video image analyses, we have performed the three-dimensional reconstruction of the aggregates and measured their characteristic sizes. The average number of particles inside each aggregate as a function of the intensity of turbulence can then be deduced. Assuming a Kolmogorov inertial scaling law for the turbulent velocity increments, we predict theoretically the aggregate mean size which is in agreement with our experimental results.

I. INTRODUCTION

The study of aggregation and fragmentation of flocs in turbulent flows is of great importance in engineering processes such as in the paper or textile making industry [START_REF] Fredrik Lundell | Fluid Mechanics of Papermaking[END_REF] where aggregation of particles or fibers is harmfully controlled. Aggregation is also very frequently observed in natural systems where it concerns a huge range of scales from nano or micro organic particles that can be transported by winds or marine currents [START_REF] Miller | Biological Oceanography[END_REF] to dust grains and even rocks that aggregate in accretion disks during planet formation [START_REF] Blum | The Growth Mechanisms of Macroscopic Bodies in Protoplanetary Disks[END_REF]. If the effects of turbulence on aggregation and fragmentation have been widely studied for particles whose size is within the viscous range of turbulence [START_REF] Matthäus | Modelling the breakup of solid aggregates in turbulent flows[END_REF][START_REF] Pumir | Collisional Aggregation Due to Turbulence[END_REF], its consequences within the inertial range are poorly documented and not yet understood [START_REF] Oyegbile | Flocculation kinetics and hydrodynamic interactions in natural and engineered flow systems: A review[END_REF] despite their tremendous importance.

In the present study, we analyse the effect of a turbulent flow acting on aggregates of magnetic beads whose sizes are within the inertial range of turbulence. We performed a three-dimensional video image analysis to detect the aggregates and measure their characteristic sizes. While low intensity turbulence allows large aggregates, strong turbulence favors small aggregates: an equilibrium average size being reached when fragmentation by turbulence is balanced by the aggregation of individual particles on existing clusters or by the merging of smaller clusters [START_REF]Drei Vortrage uber Diffusion. Brownsche Bewegung und Koagulation von Kolloidteilchen[END_REF]. Using Kolmogorov scaling arguments we propose a theoretical model whose predictions are validated by our experiments that are performed in a fully turbulent von Kármán flow. We choose millimetric magnetic particles because the separation force between two adjacent magnets is well characterized and relatively uniform through the thousand of particles we use. Then we will use a fractal description of the aggregate shapes to measure their dimensions in order to predict their average size as a function of the turbulence intensity.

II. EXPERIMENTAL SET-UP AND MEASUREMENT TECHNIQUES.

The turbulence is generated in the so called von Kármán flow by the counter-rotation of two impellers of radius R = 8.6 cm, fitted with straight blades (h = 5 mm), that can rotate at a controlled frequency f ranging from 8.5 to 30 Hz. FIG. 1 shows a drawing of our set-up that differs from classical installations [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnetohydrodynamiques dans un ecoulement de von Karman turbulent[END_REF][START_REF] Machicoane | Large sphere motion in a nonhomogeneous turbulent flow[END_REF] by the presence of two grids parallel

r [m] f [Hz] . 1/r (D LL /C K ) 3/2 [m 2 /s 3 ] [m 2 /s 3 ] b) a)
FIG. 2. a) Normalized longitudinal second-order velocity structure function for the different impeller rotation frequencies f . b) Energy dissipation rate as a function of the impellers rotation frequency f . to the disks. The purpose of these grids is to avoid the aggregates to get close to and break on the rotating disks. The transparent acrylic cubic container is filled with 8 liters of water and the visualization is performed through two of its perpendicular sides. With the help of LED light sources, diffused by a translucient screen, an homogeneous backlight is obtained and used to image the aggregates allowing good contrast and depth of the visual field.

In order to characterize the turbulence, we have measured its properties by Particle Image Velocimetry (PIV) [START_REF] Thielicke | PIVlab -Towards User-friendly, Affordable and Accurate Digital Particle Image Velocimetry in MATLAB[END_REF] in a 11 cm x 17 cm section of a meridional plane between the grids. Let us note that the turbulent kinetic energy u 2 is at least 5 times the mean flow energy u 2 . From the 2D PIV velocity fields, we can also measure the energy dissipation rate which is estimated using the longitudinal second-order velocity structure function D LL (r) ≡ (u(x + r, t) -u(x, t)) 2 where r is the spatial variable [START_REF] Stephen | Turbulent Flows[END_REF][START_REF] Kolmogorov | Dissipation of energy in locally isotropic turbulence[END_REF][START_REF] Xu | Accurate estimate of turbulent dissipation rate using PIV data[END_REF]. FIG. 2a) shows the variation of D LL (r) 3/2 /r (normalized by the Kolmogorov constant C K ) as a function of r for different disk frequencies. In each case, the inertial regime corresponds to the plateaus extending between ∼ 1 mm (larger than η because of the limited resolution of the PIV measure) and the integral scale L i ∼ 3 cm. The typical length of an aggregate L ag is smaller than the integral scale (L ag ∼ 1 cm). At this scale the pressure force induced by the mean flow is much smaller than the one due to turbulence. Hence, the aggregate will be mainly sensitive to the turbulent fluctuations. The energy dissipation rate is given by the amplitude of these plateaus using a Kolmogorov constant C K = 2.12 [START_REF] Katepalli | On the universality of the Kolmogorov constant[END_REF]. The variation of with the rotation frequency f is plotted in FIG. 2b). As expected, varies with the third power of f : = (2πα) 3 R 2 f 3 , in agreement with classical results in turbulent von Kármán flow [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnetohydrodynamiques dans un ecoulement de von Karman turbulent[END_REF][START_REF] Labbé | Power Fluctuations in Turbulent Swirling Flows[END_REF], where here α = 3.24 × 10 -2 . TABLE I summarizes the turbulence parameters.

The one thousand beads used for the experiments were made from 1 mm in diameter spherical neodymium magnets. In order to characterize the magnetic interaction, the separation force of a sample of magnetic bead pairs is measured when the pairs are separated by a stepper motor. At contact, the separation force is equal to 0.135 N. FIG. 3a) shows the evolution of the attractive force between two magnets with parallel magnetic momentum m. As expected, this force decreases as r -4 b , where r b is the distance between the two dipoles [START_REF] Landau | Electrodynamics of Continuous Media[END_REF]. Let us remark two important features that prevent the direct use of these magnetic beads: first, their density being around 7, they would sink in water to the bottom of the container, and second, the separation force at contact would be too large compared to the expected hydrodynamic force thus preventing any fragmentation of the clusters by turbulence. Because of these two difficulties, we coated the magnets with a layer of wax which has two benefic effects: first, it reduces the apparent density of the particles and second, by increasing the bead diameter, it decreases the cohesion force at the contact. The characteristic of the final wax coated beads are presented in FIG. 3b), c) and d), where it can be seen that their shape is ovoidal because of the manufacturing process dipping them into melted wax. They have an aspect ratio around 1.5, an average main axis equal to 3.7 mm, a mean density of 1.14. Furthermore, the attractive force F B is decreased and equal to 4 10 -4 N for a separation distance of 3.7 mm corresponding to the average major axis along which the magnetic dipole is aligned. To test the eventual role of the bouyancy during the aggregation/fragmentation process, some of our experimental runs were performed with salty water having a density of 1.14. As we will see later, very limited effects of sedimentation were detected, and most of our experimental runs were finally realized with fresh water.

IV. EXPERIMENTAL PROCEDURES AND MEASUREMENTS.

In order to relate the size of the aggregates to the intensity of turbulence, different experimental runs were performed at different rotation frequencies. At the beginning of each experiment, all the magnetic beads are placed inside the tank packed in a single large cluster, then the disks are set into rotation. Under the action of turbulence, the initial large aggregate breaks into smaller ones of different sizes. After some time, an equilibrium state is reached where the accretion of single particles or of small clusters versus the fragmentation of aggregates equilibrates. For each frequency, when the stationary regime is achieved (and checked by visual inspection), we record with two synchronized CCD cameras, a 255 or 500 frame video sequence of the flow at a rate of 5 images per second. We choose on purpose a low video rate in order to perform statistics of the aggregate size from uncorrelated images. Three different regimes are detected and illustrated in FIG. 4. As expected, at a low frequency (8.5 Hz on FIG. 4a) the aggregates are made of a relatively large number of magnets; while at higher frequency, the size of the aggregates becomes smaller (see FIG. 4b) for 14.5 Hz) until the flow is seeded by merely only individual particles (see FIG. 4c) for 19 Hz).

To measure the average volumes of the aggregates, we have performed a three-dimensional digital reconstruction of each of them using the convex hull volume reconstruction method [START_REF] Kong-Man (german) Cheung | Shape-From-Silhouette Across Time Part I: Theory and Algorithms[END_REF]. As shown in [START_REF] Adhikari | Visual hull method for tomographic PIV measurement of flow around moving objects[END_REF], the precision of the volume reconstruction increases with the number of cameras. Here with only two cameras, the volumes are always overestimated by this measurement. However, most of the time the shape of the aggregates is not concave, then this error should not impact our results. Moreover, the packing coefficients, introduced later in the next section, will consider the concavity of the aggregates. The principle of the reconstruction is the following: an initial cubic volume is determined at the center of the tank, a region seen by both cameras. This volume is then divided in 8 cubes with equal side which are projected on the two images. If the projection of a cube is empty in at least one image, this cube is discarded. Otherwise, this cube is divided into 8 smaller cubes and the process is repeated until the current cube is completely filled and then saved. Finally, the volume of each aggregate is determined using the positions and the sizes of each saved cube.

V. A MODEL FOR THE AGGREGATE EQUILIBRIUM SIZE.

On average, aggregates are constituted by a number N of magnetic particles of diameter d. We can then express their average volume V , cross section area S and length L as a function of N and d:

V = c 3 N 3/d 3 d 3 , S = c 2 N 2/d 2 d 2 , L = c 1 N 1/d 1 d (1)
where 1/d 1 , 2/d 2 and 3/d 3 are fractal dimensions characterizing the shapes of the aggregates and c 1 , c 2 and c 3 three parameters representing their level of packing. From the inspection of the aggregates as can be seen on FIG. 4, they appear to be quite compact without any arborescence. As a consequence and in order to simplify our analysis we will assume in the following that d 3 = 3, and thus will keep the possible fractal arrangements of clusters only in the expressions of S and L. Following the seminal idea of Kolmogorov [START_REF] Kolmogorov | On the breakage of drops in a turbulent flow[END_REF] concerning drop breakage in turbulence, we show now that the mean size of the aggregates is given by the balance between the hydrodynamic force F H acting on their cross section and the magnetic force F B that tends to keep two particles clustered together. Indeed, when an aggregate breaks into smaller ones, the last link between the initial aggregate and a particle (belonging or not to another cluster) is controlled by a unique dipolar interaction i.e. by the force F B . This assumption differs strongly from the classical hypothesis where the number of broken bonds scales with the number of aggregated particles [START_REF] Bache | Floc rupture and turbulence: a framework for analysis[END_REF]. Here the aggregates have a mean size L within the inertial range of turbulence. Then the hydrodynamic force F H applied on the aggregate is driven by the pressure [START_REF] Nauman | Turbulent Transport of Material Particles: An Experimental Study of Finite Size Effects[END_REF]:

F H = 1 2 ρ(δu L ) 2 S ( 2
)
where δu L is the typical velocity increment at scale L and ρ the fluid density. Taking into account that δu L = ( L) 1/3 , that is the classical law of homogeneous and isotropic turbulence [START_REF] Kolmogorov | Dissipation of energy in locally isotropic turbulence[END_REF] together with our measurement of the dissipation rate , we can express the equilibrium regime of the fragmentation/aggregation process:

F H = 2ρ(απf ) 2 R 2 d 4 N β 2/3 c 2 c 2/3 1 = F B ( 3
)
with β = (3d 1 + d 2 )/d 1 d 2 .
For high rotation rates of the impellers, we have observed that aggregates are completely fragmented and that only individual particles are present in the flow (see FIG. 4c)). Therefore aggregates reach their smallest size when they are constituted by only N c = 2 beads i.e. chain-like structures implying that c 1 = 1, c 2 = π/4, c 3 = π/6, d 1 = 1 and d 2 = 2 giving β c = 5/2. A threshold frequency f c for the aggregation onset is thus obtained from equation (3). For our set-up, we calculate a value f c ≈ 25 Hz. This value corresponds to a dissipation rate c ∼ 1 m 2 /s 3 . Using this threshold, we can recast equation (3) to predict the average number of particles per aggregate:

N = f c f 3/β N c a βc/β or N = c 1/β N c a βc/β (4)
with a = (4/π c

2/3 1 c 2 ) 3/2βc . As β c = 5/2, a = (4/π c 2/3 1 c 2 ) 3/5
, a value that remains close to 1 for the different arrangements that can be encountered for packing beads [START_REF] Conway | Sphere Packings, Lattices, and Groups[END_REF].

As can be observed on FIG. 4a), in the low frequency or weakest turbulence regime, the aggregates seem quite smooth and compact. In this case d 1 = d 2 = 3 and β = 4/3. As a consequence, equation (4) leads to a scaling law valid far from the aggregation onset for the average number N of particles per aggregate as a function of the rotation frequency f of the disks or of the dissipation rate of turbulence:

N ∝ f f c -9/4 or N ∝ c -3/4
(5) A similar scaling law is predicted in [START_REF] Bache | Floc rupture and turbulence: a framework for analysis[END_REF] but the assumption on the number of bonds broken at fragmentation leads to a different exponent. These results are analogous to the pioneer work of Kolmogorov on the drop fragmentation in turbulence where the size of the dropplet D drop is related to the turbulent intensity: D drop ∼ -K with K = 2/5 [START_REF] Kolmogorov | On the breakage of drops in a turbulent flow[END_REF].

VI. COMPARISON BETWEEN EXPERIMENTAL RESULTS AND THEORY.

After having processed the synchronized video images issued from the two cameras, a statistical study of the aggregate size and shape distribution is realized. Note that as our model applies only for N 2, the individual particles are not taken into account in the statistical analysis. First, the measurement for each aggregate of its surface S and length L versus its volume V leads to the determination of the average d 1 and d 2 and β wich is plotted on FIG. 5a) versus / c together with its fit. FIG. 5a) presents also the evolution of the measured average number N exp of particles per aggregate for the two different densities of fresh and salty water. N exp is calculated from the volume of each aggregate divided by the volume πd 3 /6 of a single particle. It is then an upper bound of the real number of particles N that depends on the packing number c 3 : N exp = 6c 3 /π N with c 3 between π/6 for chains and 1/ √ 2 for the maximum admissible packing of spheres [START_REF] Conway | Sphere Packings, Lattices, and Groups[END_REF]. We can conclude that the value of c 3 is quite constraint and stays in any case close to 0.5 thus justifying our choice to keep c 3 = π/6. Using a smooth sigmoid function β f it = β 0 /(1 + exp((b )/σ)) + β ∞ that fits the experimental values of β together with the threshold value found theoretically and a fitting coefficient a = 0.9 and 6c 3 /π = 1, we see that our theoretical model nicely reproduces the experimental results on the entire range of exploration (solid lines on FIG. 5a)). Between the large and smooth aggregate asymptotic regime given by the -3/4 scaling law (5) and the aggregation threshold, there exists a smooth transition where the shape of the aggregates becomes less regular, acquiring non trivial fractal dimensions. Finally, under the threshold, the average measured number N T of aggregated or not particles, becomes close to the single particle limit as can be observed in FIG. 5b). We also observe that the results of the experiments using fresh and salty water are very close one to the other, justifying our assumption to neglect the settling.

VII. CONCLUSIONS.

In classical studies on aggregation of nano or micro particles in turbulent flows, those are usually smaller than the Kolmogorov's scale of turbulence and thus are essentially sensitive to viscous effects. In this study, we have deliberately chosen to study the fragmentation of particle aggregates whose sizes are within the inertial range of turbulence. Experiments were performed using in house manufactured millimetric magnetic particles that aggregate thanks to their magnetic field. The consideration of a classical model of turbulence based on Kolmogorov similarity hypothesis leads to the prediction of the average size of the aggregates as a function of the turbulence intensity. Experimental measurements -using an original three dimensional aggregate detection and reconstruction software -and theoretical predictions agree very well. Therefore, we claim that knowing the intensity of turbulence, and knowing the cohesive force and shapes of the particles, our model can predict the average size of the aggregates. We expect that our result can be used in different engineering situations but we claim also that it can be used to probe turbulence itself: knowing the force field between particles, the sole observation of the aggregate sizes and shapes should lead to the estimation of the level of turbulence.

This work has been carried out in the framework of the Labex MEC Project (No. ANR-10-LABX-0092) and of the A*MIDEX Project (No. ANR-11-IDEX-0001-02), funded by the

2 FIG. 1 .

 21 FIG. 1. Sketch of the turbulent von Kármán flow setup.

  -2 m 2 /s 3 Taylor microscale, λ = 15νu 2 rms / 1.6 -3.2 mm Taylor-Reynolds number, Re λ = 15/ν u 2 rms 600 -1000 Kolmogorov length scale, η = (ν 3 / ) 1/4 28 -65 µm Integral scale, L i 3 cm TABLE I. Principal characteristics of turbulence.

  FIG. a) Magnetic force F B between the magnets as a function of their separation distance r. b) Photograph of some beads coated by wax. c) Main axis length probability distribution function (PDF) and d) density PDF, with their Gaussian fit.

FIG. 4 .

 4 FIG. 4. Observation of the three different regimes as a function of the turbulence intensity. a) f = 8.5 Hz, b) f = 14.5 Hz, c) f = 19 Hz.

1 FIG. 5

 15 FIG.5. a) Experimental results for fresh water: red filled circles (N exp ) and blue filled squares (β) and salt water results: red empty squares (N exp ) and blue empty triangles (β) and theoretical predictions (solid lines) of the average number of particles N per aggregate, versus the dissipation rate of turbulence. A smooth sigmoid evolution for β is fitted through the fresh water data and is used together with the measured threshold f c = 25 and fitting parameter a = 0.9 and 6c 3 /π = 1 to predict N exp . b) Measure of the average number N T of particles per aggregate taking into account individual particles in the statistics.

'Investissements d'Avenir' French Government program managed by the French National Research Agency (ANR). HDLR would like to thank CONACYT for his international PhD grant.


Kanako Yano

Yuji Tasaka

Yuichi Murai

Yasushi Takeda

Patrice Le Gal

Patrice Le

Investigation of Magnetic Permeability and Effective-kinematic Viscosity of Liquid Metal in Uniform Magnetic Field

Keywords: MHD, Liquid metal, Magnetic permeability, Effective-kinematic viscosity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

MHD experiments in a large setup have been done to clarify the mechanism of the geo-dynamo. In order to simulate the phenomenon, huge Reynolds number and also huge magnetic Reynolds number should be realized in the setup. The magnetic Reynolds number is defined as Rem = µmσUD, where µm, σ, U and D are respectively, magnetic permeability, electric conductivity, characteristic velocity and characteristic length of the setup. Magnetic diffusivity is also defined as λ = 1/µmσ, and thus the magnetic Reynolds number becomes Rem = UD /λ. Even a huge experimental setup with several meters in the characteristic length, its magnetic Reynolds number is incomparably smaller than that of the earth. The possibility to increase the magnetic Reynolds number is in decreasing magnetic diffusivity, e.g. by using liquid with high magnetic permeability. Usual liquid metals using for the MHD experiment are sodium (Na), mercury (Hg) and gallium (Ga). The magnetic permeability becomes small by Na > Hg > Ga, however, order of these values are almost same. Moreover, using sodium, which has the largest magnetic permeability in these liquid metals, involves risk to react with water.

To solve this problem, we attempt to realize high magnetic permeability by using ferromagnetic particles. By mixing the ferromagnetic particle into the mercury, we may be able to increase the magnetic permeability dramatically. But we have to estimate this value to use it for the MHD experiment. In this study, we aim to establish the way to determine the magnetic permeability of unknown fluids by investigating spin-up of the fluid, and to verify its performance using well-known fluid. When the ferromagnetic fluid put into a uniform magnetic field, its kinematic viscosity may change. We also estimate the effective-kinematic viscosity of the ferromagnetic fluid under the uniform magnetic field.

Experimental setup and experiment

The illustration of the experimental setup is shown in Figure 1. The setup consists of a rotating open cylinder put in uniform magnetic field generated by a pair of Helmholtz coil, and measurement equipment. The cylinder is made of vinyl chloride and is rotated by a stepping motor via a belt. The inner diameter of the cylinder is 30 mm. The inner diameter of coil is 260 mm and the outer is 275 mm.

The distance of a pair of coils is 220 mm. Rotating speed is determined by a rev counter mounted on the axis of the rotation. Testing fluid, Galinstan (GaInSn), is poured in the cylinder with covering by water to prevent oxidation. A probe for a gauss meter is mounted at outside of the rotating cylinder perpendicular to the direction of the magnetic field. The rotating cylinder and the Helmholtz coils are covered by a magnetic shield made of stainless steel to reduce noise.

When the cylinder starts rotation, advection of the fluid changes the orientation of the uniform magnetic field B0. Different component of the magnetic field ∆B perpendicular to the uniform magnetic field appears and it depends on magnetic permeability µm, electric conductivity σ and rotating speed Ω, namely ∆B ~ µmσ Β0Ω. Magnetic diffusivity λ = 1/(µmσ ) can be determined by investigating the variation of the magnetic field for changing the rotation speed. The variation of the magnetic field detected by the gauss meter was recorded by an oscilloscope via an amplifier and a low-pass filter.

Results and discussions

The magnetic field ∆B, which is induced by advection of the fluid in the rotating cylinder, gradually increases with spin-up of the fluid, and it has the maximum value when the fluid is fully spin-up. Forward rotation and reverse rotation of the cylinder were repeated and the maximum value of ∆B is obtained as a mean value of the maximum values obtained at the each rotation; to reduce a long period fluctuation of the magnetic field. Figure 2 shows the maximum value at each rotating speed. The magnetic filed linearly increases with respect to the rotating speed.

We attempt to determinate magnetic permeability by using the slope of the relational expression between the magnetic field and the rotating speed explained in the last section. This slope is expressed as Β0 /λ, B0 is uniform magnetic field. In this study B0 is 1.85 mG, so λ is calculated as 0.389 m 2 /s. The magnetic diffusivity of Galinstan is known as 0.361 m 2 /s. The magnetic diffusivity calculated by the experiment agrees with the exact value, and magnetic permeability of Galinstan is determined as 0.80 × 10 -6 H/m for σ = 3.46 × 10 6 S/m.

( ) [1 exp( )] t B t B τ ∆ = ∆ - - . (1)
This equation can be transformed into logarithmic expression, max ( ) log 1

B t t B τ   ∆ - - =   ∆   , ( 2
)
where ∆Bmax is the maximal value of the magnetic field. The left-side part of the equation is defined as F(t). By fitting the obtained dataset, which is the temporal variation of the magnetic field just after the reverse rotation of the cylinder, to the equation (2), we obtains variations of F(t) at each rotating speed as shown in Figure 3; the horizontal axis is measurement time and the vertical axis is F(t). The black lines are linear approximated function. For the 0 10 20 30 40

1.0 2.0 3.0 t [s] F(t) = 0.0901t F(t) = 0.0539t + 0.5 (
F(t) = 0.0941t (b) Ω = 247.3 [rpm]
Fig. 3 Dataset of the temporal variation of the magnetic field at each rotating speed and its fitting function lower rotating speed, two kinds of the slope exist. This is because turbulent flow due to the strong shear flow induced by reverse rotation of the cylinder and/or the momentum propagation from the bottom plate of the cylinder.

From the equation (2) τ is determined as the slope of F(t).

When two kinds of the slope exist, we selected the later one to calculate τ. Double logarithmic plot of τ (Ω) is shown in figure 4, where the black line represents the linear approximated function. Spin-up time in the present configuration is determined by the following equation (1)(2) ;

1 2 2 . L τ νΩ   =     (3)
In the logarithmic expression, it becomes

1 log( ) log( ) , 2 C τ Ω = - +
where C is constant number. According to this expression, the slope of log(Ω ) is -0.5 and the slope of experiment result is almost the same value. The effective-kinematic viscosity is determinate for each rotating speed, and the mean value of them becomes 2.21 mm 2 /s. This value is twice larger than the actual value, around 1 mm 2 /s. This difference may be caused by effects of Eckman layer and Stewartson layer, which are neglected in the equation (3).

Conclusions

We attempted to establish the method to measure the magnetic permeability and the effective-kinematic viscosity of unknown fluids by investigating spin-up of the fluid. Verification of the method has been done for Galinstan. Obtained results are reasonable and the applicability of the method has been confirmed. It is omitted here but the properties of mercury were also measured as reasonable values. Such other results will be presented at the conference. In the next step, we apply the established method for ferromagnetic fluid with changing amount of ferromagnetic particles.

Fig. 1

 1 Fig.1 Experimental setup

Fig. 2

 2 Fig.2 Maximum value of the perpendicular component of the magnetic field versus rotating speed

  a) Ω = 66.7[rpm]

Fig. 4

 4 Fig.4 Spin-up time τ versus rotating speed Ω for Galinstan in the logarithmic expression


Patrice Le Gal
email: legal@irphe.univ-mrs.fr

Miklos Vincze

Uwe Harlander

The Barostrat Instability: the baroclinic instability in a rotating stratified fluid

Our project aims to describe the baroclinic instability that destabilizes an initially stratified layer of fluid. Classically, this instability is studied using pure fluid. Here, the originality of our experiment comes from the use of a layer of water initially stratified with salt. Before rotation is started, double convection sets in within the stratified layer with a strongly nonhomogeneous pattern consisting of a double diffusive staircase at the bottom of the container in the very dense water layer and a shallow convective cell in the top surface layer. These two layers are separated by a motionless stably stratified zone. As radial motions take place due to the presence of these convective cells, the action of the Coriolis force generates strong zonal flows as soon as rotation is started. Then, above a rotation rate threshold, the baroclinic instability destabilizes the flow in the top shallow convective layer, generating a ring of pancake vortices sitting above the stably stratified layer. Therefore, this laboratory arrangement mimics the presence of a stratosphere above a baroclinic unstable troposphere. Infrared camera images measure the temperature distributions at the water surface and PIV velocity maps describe the wavy flow pattern at different altitudes. In some regimes, some wave trains have been detected. These waves might be traces of Internal Gravity Waves generated by the fluid motions in the baroclinic unstable layer.

Introduction

The differentially heated rotating annulus is a widely studied experimental apparatus for modeling large-scale features of the mid-latitude atmosphere (planetary waves, cyclogenesis). In the classic set-up, a rotating cylindrical tank is divided into three coaxial sections: the innermost domain is kept at a lower temperature, whereas the outer rim of the tank is heated. The working fluid in the annular gap in between thus experiences a radial temperature difference. These boundary conditions imitate the meridional temperature gradient of the terrestrial (or more generally planetary) atmosphere between the poles and the equator. The Coriolis effect arising due to the rotation of the tank modifies this convective pattern and yields a strong zonal flow with directions alternating with depth. Then at large enough values of the rotation rate, this azimuthaly symmetric basic flow becomes unstable and leads to the well known baroclinic instability with the formation of cyclonic and anticyclonic eddies in the full water depth. In the present work, we study a modified version of this experiment, in which besides the aforementioned radial temperature difference a vertical salinity stratification characterized by a buoyancy frequency N is also present. Whereas the baroclinic instability experiments are classical models of mid-latitude atmospheric dynamics, our aim here is to create a juxtaposition of convective and motionless stratified layers in the laboratory. We expect therefore that our original experimental arrangement can mimic the convective and radiative zones of stars, the tropospheres and stratospheres of planetary atmospheres or the surface turbulent sea layers above deeper stratified waters. Let us stress in particular that the exchange of momentum and energy between these layers (by the propagation of internal gravity waves for instance) is a major issue in astrophysics and geophysical sciences, in particular for the parametrization of subgrid hydrodynamics scales of forecast models. Therefore, we claim that our present set-up may lead to new insights in geo-and astrophysical applications.

Experimental set-up and methods

The experiments of the present study were conducted in the laboratory set-up sketched in Fig. 1. The working fluid consists of sodium-chloride/de-ionized water solution, with Lewis number Le ≡ κ T /κ S ≈ 100, where κ T and κ S are the diffusion coefficients for heat and salt, respectively. The radii of the inner (cooling) and outer (heating) cylinders were a = 4.5 cm and b = 12 cm, yielding a d = b -a = 7.5 cm wide annular cavity that was filled up to heights ranging from D = 10 cm to D = 13 cm in the different runs. In the experimental runs discussed here, the lateral temperature difference ∆T was set between 4 K and 10 K, and a continuously stratified salinity profile with a buoyancy frequency 1 rad/s < N < 5 rad/s, was initially prepared in the annular cavity with the standard double-bucket technique. Large enough stable vertical salinity gradients may inhibit the formation of full-depth (unicellular) overturning flow, even in such laterally heated systems as the rotating annulus experiments. The vertical extent of a convective cell is naturally limited by the condition that the initial (saline) density difference between the top and bottom of the cell cannot exceed the (thermal) horizontal density difference between the lateral sidewalls, i.e.:

ρ α∆T ≈ λ ∂ρ ∂z , ( 1
)
where ρ is the average density within the cell, α the coefficient of thermal expansion and λ the characteristic vertical (direction z) extent of the convective cell. This scale can be expressed using the buoyancy frequency N (z) = g ρ(z) -1 |dρ(z)/dz|, in the form:

λ(z) = gα∆T N 2 (z) . (2)
This characteristic vertical scale of double-diffusive convection is referred to as the Chen scale, since the pioneering experimental analysis of [START_REF] Chen | Stability of thermal convection in a salinity gradient due to lateral heating[END_REF]. Within each cell, the convective flow yields mixing and the initial stratification vanishes. Thus, a so-called "double-diffusive staircase" can develop with jump-wise density changes between these locally homogenized cells. Even before starting rotation, the onset of double diffusive convection can be observed visually in experiments. Generally, most of our prepared vertical profiles yielded conditions where convection in the bulk is mostly inhibited. However, the no-flux conditions for salinity at the bottom and the top of the water body translate to N ≈ 0 conditions at these regions in terms of buoyancy frequency. Thus in thin layers at the vicinity of these boundaries where N < N crit holds, localized convective cells could still invade the water layer on a depth ≈ λ, as seen in Fig. 2. Note that the stratified zone will be gently eroded in time, but it is worth noting that we can keep the confinement of convection for several days before complete mixing of salt water. Rotation rate Ω was increased gradually before reaching its target value (in increments of ∆Ω ≈ 0.2 rpm in every 5 minutes). At this point the system was left undisturbed for 30-40 minutes, thus the early transients of the rotating double-diffusive cell formation were not observed. Then, PIV measurements were conducted using horizontally illuminated laser sheets at different water depths. Two slightly different arrangements were used. For the first one, a camera together with a laser were fixed on an arm from the sidewall of the container. Close ups of the velocity fields could be acquired in this manner. In the second set-up, complete views of the velocity fields in the full annulus could be observed from a co-rotating camera platform situated above the tank (as depicted in Fig. 1). Alongside these PIV cameras, an infrared thermographic camera was also mounted on the platform to enable the simultaneous observation of the temperature field at the free water surface. At the end of the PIV measurements, rotation rate Ω was lowered gradually (in the same manner as in the spin-up phase), and after the flow reached a full stop, a second salinity profile was also recorded.

The Barostrat Instability

As expected, above a rotation rate threshold (see [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF]), the baroclinic instability destabilizes the flow in a shallow layer, generating a ring of vertically confined vortices.

The azimuthal wavenumber m of the instability is a function of the thickness of the convective layer (itself depending on the local density stratification and on the temperature difference) and on the rotation rate Ω. Aside to infrared temperature measurements of the free surface fluid layer, Particle Image Velocimetry (PIV) measurements were also conducted using horizontally illuminated laser sheets at different water depths, which were observed from a co-rotating camera. Figure 3 shows an example of the infrared image of an azimuthal wavenumber m=4 unstable pattern superimposed with its corresponding velocity field measured closed to the surface. To our knowledge, these are the first experimental observations of these combined convective and baroclinic instabilities in a rotating stratified layer and we coin this phenomenon the "Barostrat Instability". We performed different experiments at the following values of global parameters: ∆T = 10 K and Ω = 2rpm (denoted as Exp. 4), ∆T = 9.6K and Ω = 3 rpm (Exp. 5), ∆T = 9 K and Ω = 4rpm (Exp. 6), ∆T = 9 K and Ω = 5 rpm (Exp. 7). Fig. 4 presents the vertical mean azimuthal velocity profile as measured in the experimental runs denoted Exp.4 to Exp.7, as mentioned above. For each point we calculated the mean velocity over the whole investigated horizontal plane, which corresponds to approximatively one sixth of the cylindrical gap in the azimuthal direction. Since the presence of baroclinic instability breaks the axial symmetry of the flow, temporal averaging was performed upon a sufficient timespan to permit that the flow from different segments can contribute to the final azimuthal mean (between one and 2.5 minutes). Although the vertical resolution is rather poor (9 data points in the full water layer) one can easily recognize on Fig. 4 the opposite signs of the zonal flows at the borders of the convective cells. In the small region between them only weak flow was measured. It is to be emphasized, that despite the differences in rotation rate Ω and temperature difference ∆T , all four experiments yielded very similar results. Moreover, it is possible to estimate in each case the vertical velocity gradients that should be proportional to the radial temperature gradients divided by the rotation rate Ω as predicted by the thermal wind equation. This equation expresses the geostrophic balance within the flow and our velocity and temperature gradients measures are in fairly good agreement with this equilibrium as can be seen in Fig. 5).

Perspectives

The next question is then to explore the possible ways to escape from the geostrophic equilibrium with the generation of Internal Gravity Waves (IGWs) by the Barostrat Instability as it can be the case for instance in unbalanced dynamics of atmospheres; see for instance O' Sullivan and Dunkerton (1995) or [START_REF] Plougonven | Inertia-Gravity Waves Spontaneously Generated by Jets and Fronts. Part I: Different Baroclinic Life Cycles[END_REF]. For the first time Jacoby et al. [START_REF] Jacoby | Generation of inertia-gravity waves in the rotating thermal annulus by a localized boundary layer instability[END_REF] looked for IGWs in a classical continuously stratified, differentially heated rotating annulus. They found signatures for high-frequency waves but surprisingly the waves were related to boundary layer instabilities and not to the fronts of baroclinic waves. Recently, [START_REF] Borchert | Gravity wave emission in an atmospherelike configuration of the differentially heated rotating annulus experiment[END_REF] and [START_REF] Randriamampianina | Inertiagravity waves in a liquid-filled, differentially heated, rotating annulus[END_REF] numerically investigated the occurrence of IGWs by the baroclinic instability. Although we cannot prove yet that the Barostrat Instability can generically generate IGWs, Figure 6 displays at snapshot of our PIV velocity fields that exhibits a wave train whose characteristics are compatible with IGWs. The comparison between these PIV measurements of the velocity field with numerical simulations of the baroclinic instability in the atmosphere by O'Sullivan and Dunkerton (1995) is striking.

Figure 1 :

 1 Figure 1: A schematic drawing of the experimental set-up, with parameters a = 4.5 cm, b = 12 cm, D = 10 -13 cm, Ω = 1.7 -2.7 rpm, ∆T = 6K. The direction of the tank's rotation is also indicated.

Figure 2 :

 2 Figure 2: A snapshot of the double diffusive cell at the top of the experimental tank during a non-rotating control experiment; visualization is performed with rhodamine dye. Note, that convection is confined in a region of depth ≈ λ and does not invade the region under the boundary where N is larger.

Figure 3 :

 3 Figure 3: Infrared image superimposed with the PIV velocity field near the surface. ∆T = 6 K, Ω = 2.7 rpm.

Figure 4 :

 4 Figure 4: Vertical azimuthal velocity profile as calculated from the four experiments mentioned above. Both convective cells sustaining strong zonal flows of opposite signs at top and bottom of the whole layer are visible.

Figure 5 :

 5 Figure 5: Verification of the geostrophic balance for the four experiments mentioned above. The geostrophic balance within the flow, as given by the thermal wind equation, is illustrated by the -1 slope line in this log-log representation.

Figure 6 :

 6 Figure 6: Comparison between experimental (∆T = 10 K, N = 3.3 rad/s and Ω = 2 rpm) measurement of the velocity field near the surface (divergence of the field is in color) (left) and the numerical simulation of the emission of an internal wave train by the baroclinic instability in the atmosphere (from O'Sullivan and Dunkerton (1995)) (right).

VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 -Sept. 1, 2016


Miklos Vincze
email: vincze.m@lecso.elte.hu

Uwe Harlander

Patrice Le Gal

Double-diffusive convection and baroclinic instability in a differentially heated and initially stratified rotating system: the barostrat instability

A water-filled differentially heated rotating annulus with initially prepared stable vertical salinity profiles is studied in the laboratory. Based on twodimensional horizontal particle image velocimetry (PIV) data, and infrared camera visualizations, we describe the appearance and the characteristics of the baroclinic instability in this original configuration. First, we show that when the salinity profile is linear and confined between two non stratified layers at top and bottom, only two separate shallow fluid layers can be destabilized. These unstable layers appear nearby the top and the bottom of the tank with a stratified motionless zone between them. This laboratory arrangement is thus particularly interesting to model geophysical or astrophysical situations where stratified regions are often juxtaposed to convective ones. Then, for more general but stable initial density profiles, statistical measures are introduced to quantify the extent of the baroclinic instability at given depths and to analyze the connections between this depth-dependence and the vertical salinity profiles. We find that, although the presence of stable stratification generally hinders full-depth overturning, double-diffusive convection can yield development of multicellular sideways convection in shallow layers and subsequently to a multilayered baroclinic instability. Therefore we conclude that by decreasing the characteristic vertical scale of the flow, stratification may even enhance the formation of cyclonic and anticyclonic eddies (and thus, mixing) in a local sense.

Introduction

The differentially heated rotating fluid annulus is a widely studied experimental apparatus for modeling large-scale features of the mid-latitude atmosphere (planetary waves, cyclogenesis, etc.). In the classic set-up, first studied by the group of [START_REF] Fultz | Studies of thermal convection in a rotating cylinder with implications for atmospheric motions Meteor[END_REF] and, in parallel by [START_REF] Fowlis | Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes[END_REF], a rotating cylindrical tank is divided into three coaxial sections: the innermost domain is kept at a lower temperature, whereas the outer rim of the tank is heated. The working fluid in the annular gap in between thus experiences lateral temperature difference ∆T in the radial direction, which initiates a 'sideways-convective' circulation. These boundary conditions imitate the meridional temperature gradient of the terrestrial atmosphere between the poles and the equator. The Coriolis effect arising due to the rotation of the tank modifies this axisymmetric basic flow markedly, and leads first to the formation of an azimuthal flow and then to baroclinic instability with cyclonic and anticyclonic eddies in the full water depth. In the present work, we study a modified version of this experiment, in which -besides the aforementioned radial temperature difference -vertical salinity stratification is also present.

Whereas the baroclinic instability experiments are classical models of mid-latitude atmospheric dynamics, our aim is to create a juxtaposition of convective and motionless stratified layers in the laboratory. We expect therefore that the lessons learned from our original experimental set-up may well be useful for future laboratory arrangements (in somewhat different geometries and material properties) that could mimic the convective and radiative zones of stars, the tropospheres and stratospheres of planetary atmospheres or the surface turbulent sea layers above deeper stratified waters. Let us stress in particular that the exchange of momentum and energy between these layers (in particular by the propagation of internal gravity waves) is a major issue in astrophysics, in atmospheric sciences as well as in oceanography. Therefore, we claim that our present set-up may lead to new insights in geo-and astrophysical applications [START_REF] Plougonven | Inertia-Gravity Waves Spontaneously Generated by Jets and Fronts. Part I: Different Baroclinic Life Cycles[END_REF][START_REF] Nettelmann | An exploration of double diffusive convection in Jupiter as a result of hydrogen-helium phase separation[END_REF].

Small variations in the density of a fluid parcel (δρ) are connected to those of salinity (δS) and temperature (δT ) in the form of δρ = -αδT + βδS, where α and β denote the coefficients of thermal expansion and saline contraction, respectively. The global thermohaline ocean circulation is driven by the endless competition between the thermal and saline contributions to the buoyancy of each and every parcel of seawater [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics -Fundamentals and Large-Scale Circulation[END_REF]. Better understanding the nature of this complex interplay has been the key motivation of the present study.

Large enough (stable) vertical salinity gradients may inhibit the formation of fulldepth (unicellular) overturning flow, even in such laterally heated systems as the rotating annulus experiments. The vertical extent λ of a convective cell is naturally limited by the condition that the initial (saline) density difference between the top and bottom of the cell cannot exceed the (thermal) horizontal density difference between the lateral sidewalls, i.e.:

ρ α∆T ≈ λ ∂ρ ∂z , (1)
where ρ is the average density within the cell. This scale can be expressed with buoyancy frequency N(z) = g ρ(z) -1 |dρ(z)/dz|, in the form

λ(z) = gα∆T N 2 (z) . (2)
This characteristic vertical scale of (non-rotating) double-diffusive convection is referred to as the Chen scale, since the pioneering experimental analysis of Chen et al (1971).

Within each cell the convective flow yields mixing and the initial stratification vanishes. Thus, a so-called 'double-diffusive staircase' develops with jump-wise density changes between these locally homogenized cells. Double-diffusive staircases are formed in natural water bodies, ranging from saline lakes to oceans [START_REF] Boehrer | Double-Diffusive Convection in Lakes Encyclopedia of Lakes and Reservoirs[END_REF][START_REF] Schmitt | Double diffusion in oceanography[END_REF]. However, due to the inevitable mixing between the cells, such configurations are mostly transient. Hopefully, as the spatial scales involved in the baroclinic as well as in the convective instabilities are quite large, the transient characteristic times calculated on these lengths are long enough to permit a full investigation. Some of the experimental runs that we will describe later lasted several days. The paper is organized as follows: after having recalled the basic principles of double-diffusive convection (and its response to rotation) and then of the baroclinic instability, in sections 2 and 3 respectively, in section 4 we describe our experimental arrangement and measurement methods. Section 5 is devoted to the presentation of the results and their analysis that will concern two sets of experimental runs. In the first set (subsection 5.2), the salinity profiles were mainly linear (with a single buoyancy frequency in the core of the fluid) with two non-stratified regions at the top and bottom of the full water layer. The corresponding results permit to test the possibility to confine the baroclinic instability in two shallow layers separated by a motionless zone. In a second part of this result section (subsection 5.3), more complex salinity profile have been realized with vertical variations of the buoyancy frequency. In this case, multilayered convection sets in. The analysis is thus much more complex and necessitates a particular method to discriminate and quantify the dynamics inside these independent layers. Section 7 summarizes our main results and opens discussions on the future possibilities of our original experimental set-up.

Basics of double-diffusive dynamics and its response to rotation

For the sake of qualitative demonstration, in Fig. 1a-c we present typical vertical profiles of total density, temperature, and horizontal velocity u, from two-dimensional solutions of the non-rotating Boussinesq equations, acquired via numerical simulation using the open-source software package Advanced Ocean Modeling of [START_REF] Kämpf | Advanced Ocean Modelling Using Open-Source Software[END_REF]. No-flux conditions were prescribed at the top and bottom boundaries of the rectangular solution domain, and differential heating was applied at its lateral walls: the temperature at the vertical sidewall at the left (right) hand side of the domain was prescribed to be ∆T /2 = 4 K colder (warmer) that the initially uniform temperature T 0 of the fluid. The fluid had a stable and continuous vertical density gradient and was at rest at time zero. Each profile of Fig. 1 was extracted from the vertical column at the middle of the domain. Since these graphs serve demonstration purposes only, no units are given. In panel (a) it is clearly visible that the initial density stratification (dashed line) has developed into the aforementioned double-diffusive staircase (solid line, red online); see also the corresponding velocity profiles in panel c). In qualitative agreement with the formula (2), the largest cells -both in terms of vertical size λ and of water fluxappear at mid-depths, where the initial salinity profile exhibited the smallest gradient. Note, however, that in the regions characterized by the largest density gradients, the flow is basically blocked (the horizontal velocities are practically zero): in the spirit of (2), the large local N would yield very shallow convective layers, but their formation is inhibited by viscous effects. This "viscous cut-off" bounds the cell thickness from below: it cannot be smaller then a certain λ crit , or, in terms of the buoyancy frequency, there is a maximum N crit , above which convection is locally blocked.

In their experiments, Chen et al (1971) investigated the development of convective rolls in a rectangular vertical tank filled up with saline water of initially uniform vertical stratification with differential heating at two opposing lateral walls. His results are expressed in terms of a certain version of Rayleigh number Ra Chen , where the vertical scale is given by λ -as derived from (2) -and the characteristic temperature difference is measured between the lateral walls; thus Ra Chen incorporates the effects of both the horizontal and the vertical density gradients:

Ra Chen ≡ gα∆T νκ T λ 3 = (gα∆T ) 4 νκ T N 6 , (3)
where κ T is the thermal diffusivity and ν is the kinematic viscosity of the working fluid.

Chen et al (1971) found that the critical value of λ crit or, equivalently, N crit is encountered around Ra crit Chen ≈ 15000, below which convection is inhibited. If Ra Chen exceeds this threshold, simultaneously or successively formed multicellular doublediffusive convection rolls can develop [START_REF] Kranenborg | The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid[END_REF].

Introducing rotation around the vertical axis modifies the dynamics of doublediffusive convection markedly, as demonstrated in the theoretical work of [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF]. His conceptual set-up involves vertical salinity gradient and differentially heated lateral sidewalls with gapwidth d, which rotate at angular velocity Ω (i.e. Coriolis parameter f ≡ 2Ω). Unlike Chen, Kerr defines separate Rayleigh numbers for the vertical and horizontal directions (H and R, respectively), both taken with the same length scale d, which is, in fact, the only geometrical scale here, since the vertical extent of the domain is assumed to be infinite in the calculations. The horizontal Rayleigh number H reads as H ≡ gα∆T d 3 /(νκ T ), whereas its vertical counterpart R is given by R ≡ N 2 d 4 /(νκ T ). Note, that based on these formulae, the Chen scale λ (defined in (2)), can be expressed as λ = (H/R)d.

Kerr studied the f -dependence of the minimum H crit , below which the flow is blocked, at different fixed values of R and found two types of solutions in all cases. A schematic sketch of a typical stability diagram is shown in Fig. 2. One branch (green curve) is only slowly changing with f : this result is consistent with the findings of Chen et al (1971) in the f = 0 limit case.

This branch vanishes at a certain threshold f ′ , whose value depends on R: f ′ (R) is an increasing function of R. In the f > f ′ regime the other branch of solutions describes the appearance of a new regime (orange curve in Fig. 2). This branch decays in an inversely proportional manner with f -thus it is non-existent in the f = 0 limit -and its parameter A depends on R as well:

H crit (f ) = A(R) f -1 .
For a qualitative understanding of the effect of rotation on the double-diffusive background flow let us consider an experiment with counter-clockwise, i.e. 'Northern hemisphere'-like, rotation. As observed from the co-rotating frame of reference, the Coriolis force tends to push the horizontal flow branches to their right: at large enough values of rotation rate Ω and/or horizontal velocity u this action can yield strong zonal flows in the system. Their directions are alternating with depth, corresponding to the convective branches of double-diffusive convective cells; the Coriolis effect yields zonal propagation in the presence of radial buoyancy gradient. This is demonstrated in the composite photo of Fig. 3 taken during one of our experimental runs corresponding to the multi-layered cases that will be detailed in section 5.3. The figure presents the side view (through the transparent glass walls of the rotating tank) of dye patterns propagating away from a painted wooden stick (black) in 'eastward' (rightward) and For slow rotations, the slowly changing branch -here, referred to as 'Chen branch' -is stable, but above a certain threshold f ′ this branch vanishes, and only the sharply decreasing H crit ∼ f -1 Kerr's "rotating branch" survives. 'westward' (leftward) zonal directions at rotation rate Ω = 2.5 rpm, and average buoyancy frequency N = 3.26 rad/s.

It is important to emphasize, however, that the presence of baroclinic instability -resulting in zonal wave propagation in the annular tank -may largely add to the complexity of the emerging flow fields. Note also that even if [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF] never mentioned it, the solutions for f > f ′ seem very close in shape and wavelength to the classical baroclinic eddies.

Baroclinic instability

The most important non-dimensional parameter, traditionally used to describe the flow regimes in a rotating system with a finite kinematic viscosity ν is the Taylor number T a, which reads as follows:

T a = Ω 2 h 4d 5 ν 2 , ( 4
)
where h is the vertical scale and d is the 'gapwidth' of the apparatus (or more generally, the horizontal scale), as in the earlier formulae.

Another relevant parameter, the thermal Rossby number Ro T is defined as the ratio of the characteristic velocity of the flow -driven by the relative density change in the lateral direction, α∆T -to the rotation rate, in the form of

Ro T = h Ω 2 gα∆T d 2 , ( 5
)
where g is the acceleration due to gravity. Comparing the above formulae it is obvious that for a fixed ∆T (as in the experiments to be discussed here), T a and Ro T are inversely proportional to each other and therefore the combination Ω 2 /h can sufficiently parametrize the flow in the given experimental setup.

In a certain region of the Ro T -T a parameter space, the slope of the isopycnal surfaces in the water (which are not horizontal in this rotating system, due to thermal wind balance) surpasses a critical threshold and baroclinic instability emerges [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics -Fundamentals and Large-Scale Circulation[END_REF]. The extra potential energy stored in these tilted layers is then released via the excitation of complex wavy patterns, consisting of meandering zonal jets, accompanied by cyclonic and anticyclonic eddies. For ∆T = 6 K (that is the typical case for this study, as will be discussed later) the Taylor number corresponding to the onset of baroclinic instability is T a onset ≈ 6.33 × 10 6 (or equivalently, Ro onset T ≈ 3.1) ; above (in terms of Ro T , below) this threshold wave propagation is expected in the zonal direction [START_REF] Von Larcher | Experiments on transitions of baroclinic waves in a differentially heated rotating annulus[END_REF].

In the classic parametrization of homogeneous (i.e. thermally driven) baroclinic annulus experiments [START_REF] Vincze | An experimental study of regime transitions in a differentially heated baroclinic annulus with flat and sloping bottom topographies[END_REF][START_REF] Vincze | Benchmarking in a rotating annulus[END_REF] the vertical scale h that appears in the definitions of non-dimensional numbers (4) and ( 5) corresponds to the total water level in the tank, since the sideways convective unicellular flow penetrates through the whole depth D of the domain (h = D). The presence of vertical salinity stratification however, yields a system of shallow-layer convective cells (cf. Fig. 3), whose thicknesses may be determined by Chen's formula (2), and by the cut-off value λ crit . Therefore it is useful to introduce local versions of the Taylor and thermal Rossby numbers by taking h = λ(z) as vertical scale, corresponding to the local N. Hence, the local thermal Rossby number takes the form

Ro T (z) = gα∆T ΩN(z)d 2 , (6)
and the local T a(z) is written as

T a(z) = 4Ω 2 d 5 N 2 (z) ν 2 gα∆T . ( 7
)
It is reasonable to assume that such parameter combinations exist, where the global Taylor number calculated with the full water depth D does not exceed T a onset , but the local Taylor number T a(z), taken with the thickness of a given convective cell (h = λ(z) < D) is already above the threshold. The multicellular state can therefore enhance baroclinic instability within the cells, and thus open the way for wave propagation in such configurations, which (given their values of ∆T and Ω) would be stable in the homogeneous case. Also, due to differences in cell thicknesses λ(z), the 'baroclinicity' (i.e. the deviation of the flow field from the wave-free, axially symmetric state) in the set-up may vary from cell to cell, thus different flow states may be observed at different depths.

Experimental set-up and methods

The experiments of the present study were conducted in the laboratory set-up shown in Fig. 4. The working fluid consisted of sodium-chloride/de-ionized water solution, with Lewis number Le ≡ κ T /κ S ≈ 101, where κ T and κ S are the diffusion coefficients for heat and salt, respectively. The radii of the inner (cooling) and outer (heating) cylinders were a = 4.5 cm and b = 12 cm, yielding a d ≡ b -a = 7.5 cm wide annular cavity that was filled up to heights ranging from D = 10 cm to D = 13 cm in the different runs. The lateral temperature difference ∆T was set between 6 and 10 K in all cases. A continuously stratified salinity profile was prepared in the cavity before each measurement with the standard two-bucket technique [START_REF] Oster | Density gradient techniques[END_REF] two vessels containing saline and fresh water are connected with a pipe so that the freshwater inflow to the saline bucket yields a mixture of ever decreasing salinity in time. By raising or lowering one of the buckets during this filling-up procedure, we were able to adjust the water fluxes into the mixing compartment and, thus, create arbitrarily non-uniform stable salinity profiles.

Our experiments investigated variations of the flow field with height for different types of initial salinity stratification. Thus, the fluid was seeded with 100µm -diameter tracers for particle image velocimetry (PIV). After filling up the tank, the initial salinity profiles were measured using a hand-held conductivity sensor. Then the differential heating was switched on.

Even before starting rotation, the onset of double diffusive convection could be observed visually. Generally, most of the N-values corresponding to the prepared vertical profiles yielded smaller Ra Chen 's than the aforementioned critical threshold of convection in a non-rotating system, Ra crit Chen ≈ 15000. Therefore, the convective flow in the bulk was mostly inhibited. However, the no-flux conditions for salinity at the bottom and the top of the water body translate to N ≈ 0 conditions at these regions in terms of buoyancy frequency. Thus in thin layers at the vicinity of these boundaries where N < N crit holds, localized convective cells could still partially invade the water column, as seen in Fig. 5, visualized with fluorescein dye. Rotation rate Ω was increased gradually before reaching its target value (in increments of ∆Ω ≈ 0.2 rpm in every 5 minutes). At this point the system was left undisturbed for 30-40 minutes, thus the early transients of the rotating double-diffusive cell formation were not observed. Then, PIV measurements were conducted using horizontally illuminated laser sheets at different water depths. Two slightly different arrangements were used for the two sets of experiments. For the first set, a camera together with a laser were fixed on an arm from the sidewall of the container ('narrow field' observations). Close-ups of the velocity fields could be acquired in this manner. In the second set of experiments, complete 'wide field' views of the velocity fields in the full annulus could be observed from a co-rotating camera platform situated above the tank (as depicted in Fig. 4). Alongside these PIV cameras, for both sets of experiments an infrared thermographic camera was also mounted on the platform to enable the simultaneous observation of the temperature field at the free water surface. At the end of the PIV measurements, rotation rate Ω was lowered gradually (in the same manner as in the spin-up phase), and after the flow reached a full stop, a second salinity profile was also recorded.

Results

General description of the unstable flow

The observation of surface temperature patterns in the tank using infrared camera revealed that -contrarily to the initial naïve expectations of the authors, but in full agreement with our hypothesis of baroclinic destabilization via the scale selection of λ, discussed subsection 2 -in experiments with salinity stratification the layer at the water surface becomes baroclinic unstable at much lower rotation rates Ω than in the homogeneous system. This phenomenon is demonstrated in Fig. 6. In panel a) the surface temperature field of a homogeneous control experiment is shown, where the vertical scale is set by the total water depth D = 12 cm. The standard temperature difference ∆T = 6 K and the applied rotation rate Ω = 2.5 rpm yields T a = 5.3 × 10 6 , that is below the aforementioned T a onset ≈ 6.33 × 10 6 , or equivalently, the corresponding Ro T is above Ro onset T ≈ 3.1, as determined by von [START_REF] Von Larcher | Experiments on transitions of baroclinic waves in a differentially heated rotating annulus[END_REF]. Consistently, the temperature distribution is axisymmetric indeed, implying that the fluid stays in the stable regime.

Panel b) shows the temperature pattern emerging in the stratified case for the same setting of parameters D, ∆T and Ω, resulting in a fourfold symmetric baroclinic wave, generally referred to as m = 4 in the literature. von Larcher and Egbers (2005) foundusing the same experimental tank and ∆T as here, but with homogeneous working fluid -that the lower limit of the m = 4 regime in terms of Taylor number is around T a ≈ 10 8 (or somewhat below, at T a ≈ 7 ×10 7 if 'spin-down' initial conditions are applied). From the formula (4) one can therefore calculate that the maximum characteristic vertical scale of the convection has to be approximately h ≈ 1 cm to explain the observed wave number. Taking this λ = h value as the Chen scale of (2) yields a local buoyancy frequency N ≈ 1 rad/s. This is consistent with the measured salinity profile, althoughdue to the finite size of used the hand-held conductivity sensor -there are no measured data points from the uppermost layer of the initial profile. Yet, based on the fact that 2 cm below the surface N = 2.5 rad/s was measured, and on the aforementioned noflux condition for salinity prescribing N = 0 at the top, it is reasonable to interpolate N ≈ 1 rad/s for the considered region. Moreover, λ ≈ 1 cm also matches the observed thickness of the spontaneously forming convection cells at the boundaries, cf. Fig. 5.

In order to obtain a more quantitative relationship between N and the 'baroclinicity' of the flow, one needs to investigate the bulk of the working fluid, where the buoyancy frequency profiles can be properly determined. This domain, however, is not accessible for infrared thermography (since the penetration depth of the applied infrared wavelength range in water is of the order of a millimeter). Thus, the buoyancy frequencies directly could only be compared with PIV data.

Two separated convective layers

In this section we will present experiments which we started from particularly simple density profiles as illustrated in figure 7a. The buoyancy frequency N is plotted versus the vertical coordinate in figure 7b. We performed different experiments at the following values of global parameters: ∆T = 10 K and Ω = 2 rpm (denoted as Exp. 4), ∆T = 9.6 K and Ω = 3 rpm (Exp. 5), ∆T = 9 K and Ω = 4 rpm (Exp. 6), ∆T = 9 K and Ω = 5 rpm (Exp. 7). In the most detail the first of these runs (i.e. 'Exp. 4') will be discussed. One can see in fig. 7 that the vertical density profile is almost linear. At the upper and lower boundaries the no-flux boundary condition imposes constant concentration and therefore constant density. Thus, close to these boundaries N approaches zero.

Contrarily to the non-stratified case, as explained in section 2, the imposed horizontal temperature gradient has to reach a certain threshold before sideways convection sets in. Figure 8 shows the top and bottom convective layers, where the fluid has been colored by the gentle injection of dye at the water surface. As visible and To obtain velocity vectors of the flow, the PIV method is suitable. As mentioned in section 4, in this series of experiments the narrow field camera (GoPro, 1080p, 30fps) and the co-rotating laser sheet were utilized to acquire video information. For calculating the velocities we used successive frames taken at equally distributed time intervals with increment of 0.5 s and we processed them with the public license MatPIV toolbox. The advantages of using a co-rotating measurement system are that the camera can be positioned near the free surface and thus the resolution will increase, and we don't need to correct the calculated velocities by subtracting the tank's rotation speed. As soon as rotation is started, a zonal flow is created by the action of the Coriolis force on the convective currents.

PIV maps have been acquired for 9 height levels. Depending on the rotation rate, and also on the intensity of the convective velocities, these zonal flows can be either stable or unstable in terms of baroclinic instability. Fig. 9 illustrates two horizontal close-ups of the velocity fields. The first one (panel a) is taken near the surface, inside the layer where convection is allowed to set in. Here a baroclinic wave can be observed.

At the convective region close to the bottom of the tank a convective cell appears too, but in contrast with the one near the surface it is baroclinic stable, possibly because the zonal flow is weakened by the bottom boundary layer. The next question is then to explore the possible generation of Inertial-Gravity Waves (IGWs) by the instability as it can be the case for instance in unbalanced dynamics of atmospheres; see for instance O' Sullivan and Dunkerton (1995) or [START_REF] Plougonven | Inertia-Gravity Waves Spontaneously Generated by Jets and Fronts. Part I: Different Baroclinic Life Cycles[END_REF]. For the first time Jacoby et al (2010) looked for IGWs in a classical continuously stratified, differentially heated rotating annulus. They found signatures for high-frequency waves but surprisingly the waves were related to boundary layer instabilities and not to the fronts of baroclinic waves. Recently, [START_REF] Borchert | Gravity wave emission in an atmosphere-like configuration of the differentially heated rotating annulus experiment[END_REF] and [START_REF] Randriamampianina | Inertiagravity waves in a liquid-filled, differentially heated, rotating annulus[END_REF] numerically investigated the occurrence of IGWs by the baroclinic instability. The former claimed to have found IGWs generated spontaneously in accordance with the waves detected in a two-layer baroclinic annulus by [START_REF] Lovegrove | Generation of inertia-gravity waves in a baroclinically unstable fluid Quart[END_REF] and [START_REF] Williams | Spontaneous generation and impact of inertia-gravity waves in a stratified, two-layer shear flow[END_REF]. The latter showed that in the continuously stratified case with small Prandtl number, IGWs are generated by Kelvin-Helmholtz instability and not via boundary layer instability. Although we cannot prove yet that the present instability can generically generate IGWs, the velocity field of Fig. 9a exhibits a wave train whose characteristics are compatible with IGWs.

The second close up (panel b) corresponds to the calm region in the center of the tank, where convection is blocked due to the effect of the strong initial stratification. Driven by the strong flow in the upper and lower convective layers there is still some azimuthal symmetric flow, but the velocities are an order of magnitude lower than the ones from the upper and lower convective layers.

This flow structure demonstrates the features that was illustrated by our numerical observation presented in section 2. Fig. 10 presents the vertical azimuthal velocity profile as measured in the experimental runs denoted Exp.4 to Exp.7, as mentioned above. For each point we calculated the mean velocity over the whole investigated plane, which corresponds to approximatively one sixth of the cylindrical gap in the azimuthal direction. Since the presence of baroclinic instability breaks the axial symmetry of the flow, temporal averaging was performed upon a sufficient timespan to permit that the flow from different segments can contribute to the final azimuthal mean (between one and 2.5 minutes). Although the vertical resolution is rather poor (9 data points in the full water layer) one can easily recognize the opposite signs of the zonal flows at the borders of the convective cells. In the small region between them only weak flow was measured. It is to be emphasized, that despite the differences in rotation rate Ω and temperature difference ∆T , all four experiments yielded very similar results. It is also worth mentioning here that this motionless layer can possibly sustain inertia-gravity Figure 9. PIV measurements of the flow at two different heights (near the surface at z = 12.5 cm and in the middle region at z = 9 cm) showing the azimuthal flow with or without the baroclinic instability. The small arrows represents the velocity vectors in the horizontal plane, the colored maps are the horizontal divergence fields. The stream lines are plotted in red. Close to the surface the velocities are an order of magnitude higher (v max ≈ 8 mm/s) than in the center (v max ≈ 1.1 mm/s).

waves (IGWs) that would be launched from the top or the bottom unstable zones.

Multilayered convection

In this section we present our findings from the second set of experiments, where more complex initial salinity profiles were studied, two of which are presented in panels a) and d) of Fig. 11. In panel a) both the initial (dashed line) and final (solid line) profiles are shown. Similarly to the simulated final profile in Fig. 1, the presence of double-diffusive staircase is apparent. Panels b) and e) show the values of buoyancy frequency N, as calculated from polynomial fits of the density profiles, and evaluated at the height levels where the 'wide field' PIV measurements were conducted. As an example of the PIV fields evaluated in this series of experiments Fig. 12 shows the acquired velocity vectors of a fully developed fourfold symmetric wave pattern obtained from ca. 1 cm below the water surface, superimposed onto the simultaneously obtained surface temperature field.

At each of these levels, the horizontal velocity field was calculated on a lattice of 49 × 66 grid points, yielding a spatial resolution of around 0.6 cm. In order to obtain a proper measure of 'baroclinicity' in such a slice of the flow, firstly, the angle ϕ i ∈ [0; 2π) between each velocity vector and the respective position vector r i -directed from the center of the tank, i.e. the axis of rotation -was computed (see the explanatory sketch in the inset at the bottom right corner of Fig. 13). Our approach is based on the distribution of these ϕ i -values. In the absence of baroclinic instability, the temperature pattern is axially symmetric and the flow is zonal (see Fig. 6a or state A in Fig. 13), thus the velocity field is highly anisotropic in the sense that the vast majority of the velocity vectors are perpendicular to their position vectors (ϕ i ≈ π/2, ∀i). At larger values of T a a meandering jet appears (state B in Fig. 13): the values of ϕ i scatter in a broader range around π/2. In a fully developed baroclinic unstable case (states C and D) the flow field is dominated by vortices, making the ϕ i density distribution close to uniform within the interval [0; 2π), i.e., in a certain sense, isotropic.

The right panel shows the cumulative density functions (CDF) of the ϕ i values of four measured vector fields, whose streamlines (states B through D) were sketched, as discussed above. Each count of the CDFs was calculated with a certain weight factor, assigned to each velocity vector proportionally to their relative magnitudes compared to the mean u (after the careful removal of outliers), in order to compensate for the inevitable uncertainties in the ϕ i values when measuring small vectors. In a perfectly isotropic case, the CDF curve would follow the CDF(ϕ i ) = ϕ i /(2π) linear function. The more anisotropic the velocity field, the larger the deviation of the CDF(ϕ i ) graph from this straight line. We quantify the anisotropy of the flow by defining a parameter γ as

γ = 1 n i CDF(ϕ i ) - ϕ i 2π , (8)
where n is the total number of the obtained 'useful' velocity vectors of the PIV data, which lie within the measurement domain (the annular cavity covers only a fraction of the rectangular image) and remain after the removal of clear outliers. This process typically left us with n ≈ 850 values of ϕ i for a given horizontal PIV slice. γ serves as the desired measure (order parameter) of baroclinic instability: generally, the smaller γ is, the more baroclinic unstable the system appears to be in the considered horizontal section. A perfectly axisymmetric flow, with all the velocity vectors aligned perpendicularly to their position vectors would yield γ = 0.75 (independently of whether the flow is clockwise or counterclockwise), whereas in a fully developed geostrophic turbulent state, where all directions ϕ are equally probable γ ≈ 0 would hold. The vertical profiles of anisotropy parameter γ(z) obtained for the two measurements of Fig. 11 are shown there in panels c) and f). We emphasize that the determination of these γ-values is completely independent from that of the density profiles, the former being obtained directly from the PIV data, and the latter via a hand-held conductivity probe. Yet, there is an apparent anticorrelation between γ(z) and N(z) in both shown cases, as seen when comparing panels b) to c), and e) to f) in Fig. 11.

The acquired γ(N) data points from four experiments with different vertical density profiles and slightly different rotation rates (ranging from Ω = 1.7 rpm to 2.7 rpm) are combined in Fig. 14a, with colors accounting for the corresponding values of local thermal Rossby number Ro T (z), as in ( 6): here each point is calculated from a complete horizontal PIV velocity slice. The overall trends are apparent: firstly, it is reassuring that anisotropy parameter γ seems to behave consistently with Ro T . Generally, the smaller Ro T is, the closer the system gets to geostrophic turbulence, yielding smaller γ's, as expected. Secondly, γ markedly anticorrelates with N in itself too, despite of the differences in the rotation rates, underlining the robustness of the N-dependence: the steeper the stratification, the smaller the vertical convection scale becomes, supporting the development of shallow layer geostrophic turbulence, i.e. smaller γ. Fig. 14b shows the same data set, now colored with respect to the corresponding relative average absolute horizontal velocities | u| , based on the PIV fields. These values have been normalized so that | u| = 1 corresponds to the largest observed average velocity. We found that two of the three weakest observed flow layers (the two blue and green data points on the right of the plot) appear at the largest buoyancy frequencies (N > 4.2 rad/s), which is in sharp contrast with the large velocities measured at only slightly smaller N's. This may well imply that here the blocking threshold is approached, and above N crit ≈ 4.2 rad/s (marked by a solid vertical line) the convection is inhibited; thus the corresponding parameters γ do not have actual physical meaning here, being results of the angle statistics of noise-dominated PIV fields. The consistency of this finding with the theoretical background is addressed in the next section.

Discussion

Chen et al (1971) and [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF] have analyzed the conditions under which double diffusive convection can set on in laterally heated, initially stratified set-ups. For a non-rotating case the criterion (as reported in the experimental study of Chen et al (1971)) was found to be that the 'combined' Rayleigh number, which incorporates the effects of both the salinity and the temperature gradients as defined in (3), must exceed a certain threshold Ra crit Chen ≈ 15000. From here, one can calculate the critical buoyancy frequency N crit Chen above which the viscous effects inhibit the formation of convective cells in our experimental tank. This estimation yields N crit Chen ≈ 1.7 rad/s, which appears to be consistent with the observation that in the non-rotating control experiments, cell formation was limited to the vicinity of the top and bottom boundaries, where N is small (see Fig. 5), and is also in fairly good agreement with the results of the first set of our rotating experiments, descussed in Section 5.2.

Based on this, we would barely see any flow in the bulk of the fluid at all, because most of our actual N-values scatter in the interval N ∈ (1.7; 4) rad/s, as shown in Figs. 11 and 14. Yet, in our second set of experiments, where more complex initial salinity profiles were investigated (section 5.3), once rotation was turned on, an alternating zonal flow emerged at mid-depths as well (see Fig. 3), implying convection. Therefore it is reasonable to assume that in the rotating case the system can also exhibit another type of dynamics, which we referred to as the Kerr branch. The states of this branch only exist in the presence of rotation, as seen in Fig. 2.

Kerr's analysis is based on a horizontal Rayleigh number H ≡ gα∆T d 3 /(νκ T ) and a vertical one R ≡ N 2 d 4 /(νκ T ), as discussed in subsection I.B. The average N of our multilayered experiments yields R ∼ 10 8 , and their typical rotation rate Ω ∼ 2 rpm translates to f ∼ 2 × 10 4 in the dimensionless units applied in the analysis (using d 2 /κ T as time units, cf. [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF]). From here, using the data presented by [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF], we can also estimate that H crit ∼ 10 6 is the corresponding critical horizontal Rayleigh number. Knowing now the ratio H crit /R ∼ 10 -2 , we can directly obtain the actual critical Chen scale as λ crit = (H crit /R)d, and finally, applying (2), we arrive at the theoretical prediction of N crit in the form of:

N Kerr crit = gα∆T (H crit /R)d ≈ 4 rad/s, (9)
that is in perfect agreement with the observed N crit ≈ 4.2 rad/s, implying that our multilayered experiments followed the 'Kerr (rotating) branch' of solutions, depicted in Fig. 2. Note, however, that -as mentioned above -our first set of experiments, where only two separate convective layers were observed at the small-N regions of the top and bottom of the tank, convection was blocked already at N ≈ 3.3 rad/s, and remained blocked, even when higher values of ∆T and f were applied. This observation implies that the dynamics in these experiments stayed on the 'Chen branch' of Fig. 2. The present study lacks an empirical determination of the breaking point f ′ , where -according to Kerr's linear stability analysis -the Chen branch is expected to vanish and the Kerr to remain the only possible solution. This transition between the two types of solutions needs to be clarified by future experiments.

Conclusion

To the best of our knowledge, we have reported here the very first experimental results on the combined effect of double convective and baroclinic instabilities in a rotating stratified layer and we decided to coin this phenomenon the 'barostrat instability'. Following the previous analyses of Chen et al (1971) and [START_REF] Kerr | The effect of rotation on double-diffusive convection in a laterally heated vertical slot[END_REF] we have observed the formation of patterns with different azimuthal wave numbers. As expected, these structures are confined in layers where the initial vertical density stratification was smaller. The thresholds for convective cell formation in terms of buoyancy frequency N and the observed typical vertical length scales were found to be in agreement with the theoretical predictions. We hope that this study will open new routes in the study of atmospheric baroclinic instabilities: in particular let us remark that the stable stratified zone that lies under the unstable shallow layer in the experiment mimics the stratosphere above the baroclinic unstable layers of the troposphere. The next question is then to explore the possible generation of inertial-gravity waves by the barostrat instability as it can be the case for instance in the unbalanced dynamics of atmospheric fronts.

As closing remarks, we emphasize that it was somewhat counter-intuitive for the authors to find that the presence of moderately strong vertical salinity stratification can increase isotropy, i.e. enhance the development of baroclinic instability in certain shallow layers via the formation of double-diffusive cells. This is seemingly in contrast with reasoning that stable stratification (large N) generally blocks baroclinic instability. Our findings imply that -although, indeed, the unicellular full-depth overturning may be inhibited -the steeper the stratification is, the shallower the double-diffusive cells may be, which then decreases the local value of thermal Rossby number Ro T , and by doing so, pushes the system (i.e. the given sideways-convective layer) towards quasigeostrophic turbulence, as long as the stratification does not exceed a critical value N crit above which the flow is indeed completely blocked.

With presenting the new findings of the present pilot study (which clearly needs to be followed by further experimental and numerical investigations) the authors hope to increase awareness within the community of the fact that the velocity field, and thus, horizontal mixing can vary in such a nontrivial way with N. This may be of interest in ocean dynamics, where horizontal (e.g. meridional) temperature difference and vertical salinity gradients co-exist.

Figure 1 .

 1 Figure 1. Initial (black dotted) and final (red) vertical profiles of total density (a), temperature (b) and horizontal velocity (c) from a 2D, non-rotating simulation of double-diffusive convection, demonstrating the development of a double-diffusive 'staircase'. Note the blocking at the regions of steeper density gradient. Units are arbitrary.

Figure 2 .

 2 Figure2. A sketch of the stability diagram H crit as a function of Coriolis parameter f , for a given R. For slow rotations, the slowly changing branch -here, referred to as 'Chen branch' -is stable, but above a certain threshold f ′ this branch vanishes, and only the sharply decreasing H crit ∼ f -1 Kerr's "rotating branch" survives.

Figure 3 .

 3 Figure 3. Dye tracing in a rotating experiment. The stick from where the dye was released is the thick vertical silhouette. The propagating dye fronts are marked with white dots for better visibility.

Figure 4 .

 4 Figure 4. A schematic drawing of the experimental set-up, with parameters a = 4.5 cm, b = 12 cm, D = 10 -13 cm, Ω = 1.7 -2.7 rpm, ∆T = 6 K. The direction of the tank's rotation is also indicated.

Figure 5 .

 5 Figure 5.A snapshot of 'classic' double diffusive cells at the bottom of the experimental tank during a non-rotating control experiment, visualized with rhodamine. Note, that the cells do not invade the regions farther above the boundary, where N is larger.

Figure 6 .

 6 Figure 6. Thermographic images of the surface temperature patterns of a homogeneous (a) and a salinity stratified (b) experimental run (in the latter case: N = 2.5 rad/s) with the same 'global' parameters: ∆T = 6 K, D = 12 cm, and Ω = 2.5 rpm.

Figure 7 .

 7 Figure 7. Saline water density (a) and the corresponding vertical buoyancy frequency profile (b) before starting the series of experiments. Visibly, except for the top and bottom, where the no-flux conditio yields weakly stratified layers, N is fairly constant around a value of N = 3.3 rad/s

Figure 8 .

 8 Figure 8. Visualization of the convective cell confined in the upper (a) and lower (b) layers of the fluid. In the middle region no convection occours.

Figure 10 .

 10 Figure 10. Vertical azimuthal velocity profile as calculated from four experiments corresponding to almost the same ∆T of about 10 K and different values of Ω showing three distinct zones: two unstable convective cells at bottom ant top of the full layer and a motionless layer that separate them in the middle plane.

Figure 11 .

 11 Figure 11. Vertical profiles of density (a and d), buoyancy frequency (b and e) and anisotropy parameter γ, defined in (8), for two experimets. The applied rotation rates were Ω = 2.24 rpm for the upper, and Ω = 1.7 rpm for the bottom row (∆T = 6 K).

Figure 12 .

 12 Figure 12. Surface temperature (color) and simultaneous horizontal velocity fields (arrows) from the uppermost 1 cm layer of one of the multilayered experimental runs (∆T = 6 K, Ω = 2.7 rpm).

Figure 13 .

 13 Figure 13. The development of baroclinic instability towards larger values of T a (smaller values of Ro T ). Sketches A-D (left) are cartoons of the streamlines in the flows to which the CDFs of the main panel belong. The right hand side shows the corresponding CDFs of velocity angle ϕ as obtained from actual PIV measurements. The inset demonstrates the geometric interpretation of angles ϕ i .

Figure 14 .

 14 Figure 14. Scatter plots of local anisotropy parameter γ vs. buoyancy frequency N , obtained from four experimental runs with different density profiles and rotation rates. The color coding is based on the local thermal Rossby number Ro T (a) and the normalized average horizontal velocity | u| (b). In panel b) the vertical solid line indicates N crit ≈ 4.2.

Acknowledgments

This work was supported by the European High-performance Infrastructures in Turbulence (EuHIT) program and by OTKA Grant No. NK100296. The authors are

grateful for Christian Maaß, Yongtai Wang, and Torsten Seelig for the technical support during the measurement campaigns. The fruitful discussions with Tamas Tel and Yossi Ashkenazy are also highly acknowledged.