Keywords: Taylor-Couette flow, Stratified rotating flows, Mixing, Pycnocline

Taylor-Couette flow in a two-layer stratified fluid: instabilities and mixing In this preliminary experimental study we investigate the mixing of a salt-stratified two-layer fluid in a Taylor-Couette flow. We focus on two flow regimes, one in which waves are present in the pycnocline and vortices in the homogeneous layers, and one with vortices also in the pycnocline. The transition between these two regimes differs significantly compared to the wave-vortex transition found in a uniform stratification, investigated by [START_REF] Caton | Stability and bifurcations in stratified Taylor-Couette flow[END_REF] Stability and bifurcations in stratified Taylor-Couette flow. J. Fluid Mech. 419,, and is accompanied by a larger jump in fundamental frequency of the density fluctuations.

Most density gradients show a strong tendency to sharpen with time, with layer formation in the case of an initially thick interface. On a long time-scale, the entrainment of a smooth pycnocline shows cascade-like bifurcations before its transition to chaos. In the latter case, a decrease of the initial density gradient is observed.

We discuss the mixing efficiency, measured by the Richardson flux number Ri f . Its value is a function of the Reynolds number ∼ Re k , with k = 3/2, and for Reynolds numbers O(10 3 ), the mixing efficiency is of order O(10 -2 ).

Introduction

Mixing in density stratified fluids plays an important role in numerous geophysical and engineering applications, and is responsible for the transport of heat, nutrients and pollutants in the oceans and atmosphere, as well as in various technical devices. The driving mechanisms for mixing in oceanographic processes have been thoroughly discussed in the literature over the past five decades (see [START_REF] Kato | On the penetration of the turbulent layer into a stratified fluid[END_REF][START_REF] Linden | Mixing in stratified fluids[END_REF][START_REF] Fernando | Turbulent mixing in stratified fluids[END_REF][START_REF] Staquet | Internal gravity waves: from instabilities to turbulence[END_REF]Sommeria, 2002, Caulfield andPeltier, 2002). Typical experiments on mixing include the entrainment and overturning motion in shear flows (e.g., [START_REF] Strang | Entrainment and mixing in stratified shear flows[END_REF], stirring by different mechanical devices such as oscillating grids, or entrainment of vortex rings [START_REF] Linden | The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment[END_REF], for a review see [START_REF] Fernando | Turbulent mixing in stratified fluids[END_REF].

In many geophysical flows, mixing is produced by anisotropic flows having a significant horizontal vorticity component. For example, Langmuir circulation, which is characterized by the occurrence of organized horizontal helical vortex-structures with their axis directed along the wind-induced main flow, is thought to play a significant role for the mixing of the ocean pycnocline [START_REF] Garrett | Processes in the surface mixed layer of the ocean[END_REF]. Motivated by this application, we investigate the evolution of an interface-or pycnocline-of a two-layer fluid in the presence of Taylor vortices. Taylor-Couette flows are extensively investigated for homogeneous fluids and give a controlled access to different flow regimes, ranging from wavy flows, wavy-vortex flows and turbulence (see [START_REF] Andereck | Flow regimes in a circular Couette system with independent rotating cylinders[END_REF]. The extension to Taylor-Couette flows with linear axial stratification have been investigated experimentally and theoretically by Hopfinger and co-workers (see [START_REF] Boubnov | Stratified circular Couette flow: instability and flow regimes[END_REF][START_REF] Boubnov | Experimental study of circular Couette flow in a stratified fluid[END_REF][START_REF] Caton | Stability and bifurcations in stratified Taylor-Couette flow[END_REF] and numerically by [START_REF] Hua | First transitions in circular Couette flow with axial stratification[END_REF] and new regimes are shown to occur as a consequence of the different dynamics. Stratification increases the threshold rotation-rate, Ω c , for the onset of the Taylor vortices, while for small rotation-stratification frequency ratio, Ω/N, the Taylor vortices are flattened. In the two-layer stratification, the dynamics are more complicated, since the buoyancy frequency, N, is not constant, and flow properties may vary locally. Waves generated inside the pycnocline find critical layers above and beneath, leading to reflection or absorption, while vortices outside the pycnocline perturb the stratified layer and the motion there in. These effects may alter the dynamics as well as the regimes found in linearly stratified TC-flows and have not been investigated before. In the next section we present our experimental setup, followed by the results on the different observed regimes in Section 3.1 and the mixing in Section 3.2. We briefly discuss the results in Section 4.

Experimental setup and procedure

Apparatus

The experimental apparatus consists of two Plexiglas coaxial cylinders with radii, respectively, a = 15 cm and b = 20 cm, and is an enlarged version of the stratified Taylor-Couette system described in [START_REF] Caton | Stability and bifurcations in stratified Taylor-Couette flow[END_REF] further referred to as CJH. The outer cylinder is fixed while the inner cylinder is driven by a DC-motor at a rotation rate Ω of a value between 0.08 and 0.56 s -1 . The total depth of the fluid in the experiments, H, is 76 cm yielding a moderate value of aspect ratio A = H/d = 15.2, where d = ba. The annular space between the cylinders was filled with a two-layer salt-stratified fluid and tap-water is used as working fluid. The vertical density distribution can be approximated as

ρ(z) = ρ 0 1 - ǫ 2 tanh 2(z -h) δ , where ρ 0 = ρ 1 + ρ 2 2 and ǫ = ρ 2 -ρ 1 ρ 0 ,
with z = 0 at the free surface and the z-axis directed vertically upwards; h is the depth of the upper fluid (in experiments h = H/2), δ the characteristic thickness of the pycnocline, and ρ 1 , ρ 2 and ρ 0 are the fluid densities in the upper and lower layers and mean density, respectively. Experiments with a smooth pycnocline were performed at large δ, obtained after having the interface diffused during several days. The density difference between the layers was varied within the range 0.0015 ≤ ǫ ≤ 0.02. In the otherwise quiescent fluid, parameter δ changed with time t as δ ∼ t 1/2 , owing to the molecular diffusion of salt. Both, sharp and smooth pycnoclines were studied with typical The Taylor number, Ta, is defined as 4Ω 2 Y 2 • d 4 /(b 2 /a 2 -1) ≈ 3.2 × 10 7 Ω 2 and the Grashof number Gr = N 2 d 4 /ν 2 = 6.25 × 10 6 N 2 with N = (gǫ/δ) 0.5 . values δ/d around 0.2 and 2, respectively. The Reynolds number Re = Ωad/ν (where ν is the kinematic viscosity) was varied between 600 and 4200. A list with the experimental parameters is given in Table 1, which includes the experimental values of the Taylor number and Grashof number.

For an identification of the observed flow patterns, we used the shadow-graph technique, which visualizes variations in the second spatial derivative of the instantaneous density profile by light and dark projections of a punctual light-source that shines through the fluid. The white and dark-gray band represents the interface (see Fig. 1a). These light intensity fluctuations were recorded with a B&W analog camera. From the images a fixed vertical line of pixels was selected in real time and added with a prescribed sampling rate to the stack of lines taken at previous time steps. This vertical line of pixels was taken at 1 cm distance from the wall of the outer cylinder (see arrows in Fig. 1a and Fig. 2a). For the measurement of mixing across the density interface we used laser-induced fluorescence.

Experimental results

Flow regimes and entrainment in a smooth pycnocline

We have noted two different flow regimes depending on the rotation rate while keeping the stratification of the interface constant. For low rotation values, Taylor vortices are present in the two homogeneous layers, while the flow is stabilized and axisymmetric in the centre of the pycnocline. Since the buoyancy frequency decreases to zero at the edges of the pycnocline, waves are generated at some level above the pycnocline centre. This regime is referred to as the wave-vortex regime. For higher rotation rates, vortices are present, both in the pycnocline and the two homogeneous layers. The latter is referred to as the vortex-vortex regime.

A typical example of the wave-vortex regime, is illustrated in Fig. 1a, showing highgradient interfaces on the upper and lower edges of the pycnocline, with alteration of dark and light zones inside the pycnocline. Fig. 1b shows the spatio-temporal image of the vertical line of pixels marked by the black arrow in Fig. 1a. Fig. 1b reveals the wave motion inside the pycnocline. Note that inside the pycnocline, the density varies with height. At the centre of the pycnocline where the buoyancy is maximal, waves are suppressed in agreement with results in a linearly stratified fluid by [START_REF] Boubnov | Experimental study of circular Couette flow in a stratified fluid[END_REF], while waves with increasing amplitude occur with distance from the pycnocline-centre and decreasing buoyancy. These waves are progressive and propagate towards the edges of the pycnocline. The time-history of the light fluctuations inside the pycnocline yields sine curves with a well defined frequency Ω f .

The saw-like patterns at the pycnocline edges represent the motion of the vortices, which interact with the waves that are periodically emitted in the pycnocline. The waves break above and below the pycnocline where the local buoyancy frequency is relatively small, and dense (light) fluid is entrained into the upper (lower) layer. Nevertheless little mixing takes place, and the initial density profile is quickly recovered after turning off the inner cylinder rotation. Note, that in contrast to the standing waves observed in the wavy regime by CJH in a linearly stratified fluid, we observe progressive wave patterns; their amplitude increases with distance from the pycnocline centre due to the decreasing buoyancy frequency, as can be inferred from the variation in wave-signature in the shadow-graph images (see Fig. 1).

For higher rotation rates the wave motion is unstable, leading to the formation of stratified vortices in the pycnocline region. The threshold rotation-rate for the occurrence of vortex motion corresponded well with that in a linear stratified fluid found by [START_REF] Boubnov | Experimental study of circular Couette flow in a stratified fluid[END_REF] in the same device for Ω c ≥ 0.075N + 0.075, with N the maximum stratification of the pycnocline. Fig. 2a shows an example of this vortex-vortex regime, consisting of a system of narrow layers separated by clearly visible interfaces. The spatiotemporal image of this flow is shown in Fig. 2b for the same δ/d and ǫ as in Fig. 1.I n the pycnocline we notice periodic oscillations with upward and downward propagation direction and a well-pronounced fundamental frequency Ω f . At the edges of the pycnocline, the noise increases and fluctuations at frequencies lower than the fundamental one appear (typically, at Ω f /2 and Ω f /3).

Five minutes after having turned off the rotation of the inner cylinder the shadow-graph images reveal a well-pronounced layered structure (see Fig. 2c), showing that the stratified vortices effectively mix the pycnocline. The ratio between the vertical and horizontal sizes of the layered structure in Fig. 2c is remarkably low. It reaches 1:6 in the middle of the pycnocline and slightly increases to 1:5 at the edges of the pycnocline, due to a decrease of the local buoyancy frequency. Note, however, that the vertical size of the structure does not increase in inverse proportion to the local buoyancy frequency. The vertical size of the layers at the edges of the pycnocline is 1.2 times greater than in the middle of the pycnocline, while the initial local buoyancy frequency at the horizons of these layers is 1.4 times smaller than in the middle of the pycnocline. Also, it should be noted that the aspect ratio of 1:6 is equal to the lowest aspect ratio observed in the same apparatus by [START_REF] Boubnov | Experimental study of circular Couette flow in a stratified fluid[END_REF] for a uniform stratification. In a smaller Taylor-Couette device, the onset of the stratified vortex regime in the uniformly stratified fluid leads to the formation of structures having an aspect ratio between vertical and horizontal size around 1:2 (see CJH).

The transition from the wave-vortex regime to the vortex-vortex regime is accompanied by a jump in the mean rotation frequency Ω f of the 3D flow-structure. This (fundamental) frequency can be measured from the light-intensity fluctuations shown in Fig. 1b, where one single revolution period of the flow structure around the cylinder axis is depicted by a horizontal bar. For a purely azimuthal flow the rotation frequency Ω 0 with which the flow structure drifts can be derived for the Taylor-Couette flow, and yields after averaging over the volume in the gap (see CJH):

Ω 0 = - a 2 b 2 -a 2 + 2a 2 b 2 (b 2 -a 2 ) 2 ln b a Ω.
For our setup this gives Ω 0 =0.411Ω. When the axisymmetry of the system is broken by an azimuthal mode, m ≥ 1, the flow-structure precesses relative to the mean rotation frequency with a frequency, α, given by the relation [START_REF] Knobloch | Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows[END_REF][START_REF] Caton | Primary and secondary Hopf bifurcations in stratified Taylor-Couette flow[END_REF]:

Ω f = m Ω 0 +α(m).
As N increases, the frequency jump occurs at higher Ω due to the stabilizing effect of the stratification (see Fig. 3). Both, in the wave-vortex regime and the vortex-vortex regime, Ω f increases linearly with Ω. The least square approximation yields for the wave-vortex regime

Ω f = 0.41Ω + 0.0014 (1)
and for the vortex-vortex regime

Ω f = 0.81Ω + 0.138. (2)
In the wave-vortex regime, the dependence of Ω f on Ω is remarkably close to the angular velocity for purely azimuthal flow, Ω 0 . Surprisingly, this result corresponds with the Ω f (Ω)-relation in the stratified vortex regime, characterized by counter-rotating vortex pairs with azimuthal mode m = 1 (see CJH), whereas the nature of the wave-vortex regime, with progressive waves in the pycnocline and vortical motion at the edges of the pycnocline, is altogether different. Also in both cases, the evolution of Ω f with Ω does not depend on stratification since the data obtained for different values of N fall on the same straight line. Though further research is required to give a sound explanation, possibly, the drift and precession of the vortical motions in the weakly stratified upper and lower part of the pycnocline govern the drift at the interface, suggesting the correspondence in fundamental frequency with the stratified-vortex regime and the precession value of exactly twice that in CJH.

In the vortex-vortex regime, the shadow-graph lines are propagating in up-and downward directions (see Fig. 4), while from ( 1) and ( 2) we note that in the vortex-vortex regime the coefficient of Ω is twice the value of the coefficient in the wave-vortex regime, and the precession about 100 times as large. This high value of the precession suggests the presence of higher modes (m = 2), most likely to be expected in the layers with absent, or relatively weak stratification. At the transition in a two-layer fluid from the wave-vortex regime to the vortex-vortex regime we note a clear jump in frequency Ω f . In contrast, in the stratified vortex regime in a linearly stratified fluid (see CJH) the measured frequency of the density fluctuations decreases with Ω in the wave regime and then grows linearly with Ω without any jump in frequency.

Because of the gradual erosion of the pycnocline by the Taylor vortices in the outer homogeneous regions, the parameter δ (and thus also N) slowly varies in time and different flow transitions are observed to occur. An example is shown in Fig. 4 for the vortex-vortex regime. Up to the time T 1 the process is essentially similar to the one depicted in Fig. 2b, with sinuous oscillations at a certain fundamental frequency. After T 2 , one clearly notices the varying oscillation and the subsequent gradual increase of stochastic noise, apparent from the noise in grey levels in Fig. 4. The described behavior reminds to a classic scenario of transition to chaos via period-doubling cascades [START_REF] Landau | Fluid Mechanics[END_REF].

Since the buoyancy frequency is maximum in the middle of the pycnocline and decreases to zero at its edges, the entrained fluid there has initially less potential energy. Therefore, with the decrease of the pycnocline thickness, the entrainment-rate gradually decreases. Indeed, when scrutinizing the pycnocline edges with time in Fig. 4 one can notice small curvatures.

Mixing in a two-fluid system

In this subsection, we focus our attention on the mixing of a two-fluid system with a sharp interface with δ/d around 0.2. To keep the duration of experimental runs within reasonable limits, experiments were conducted at Reynolds numbers of 2000 and higher. The flow evolution of which a typical spatio-temporal image of the process is shown in Fig. 5a , was visualized with the laser-induced-fluorescence (LIF) technique. The mixing process can be schematically divided into three stages. In the first stage, the initial density distribution is reduced to a sharp interface with nearly zero δ, as is apparent from the high contrast between light and dark zones in Fig. 5a. In the next stage, entrainment in the upper (and lower) layer of the fluid at the interface gradually decreases the density difference, ǫ(t), until reaching a critical value ǫ c at time T i , when the sharp interface starts to fade. In the subsequent stage the two density fluids mix to one homogeneous layer reached at time T s ; after time T s the light intensity in the image in Fig. 5a has turned uniform grey. These times, T i and T s are, respectively, referred to as the interface and stratification life time.

The time-evolution of the light intensity distribution over depth J(z/D) is shown in Fig. 5b. J is normalized such that at t = 0, its mean initial value in the upper and lower fluid correspond, respectively, to 0 and 100. Assuming that J is directly proportional to the local density, we can calculate the evolution of the potential energy P of the system with time, which is shown in Fig. 5c in normalized units, with the initial and final levels taken as 0 and 1. For our experimental setup, the variation of the potential energy between the initial two-fluid and the final homogeneous state corresponds in dimensional form to P = ǫρgπ (b 2 -a 2 )h 2 /2. We can define the flux Richardson number as the ratio Ri f = P/ E, where E is the mechanical energy input in the system. The total mechanical energy required to mix the two-layer fluid can be estimated as E = MΩT s , where M is the torque acting on the inner rotating cylinder. We estimated this torque from [START_REF] Dubrulle | Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection[END_REF] for a homogeneous fluid, assuming that it is the same as in a stratified fluid. According to [START_REF] Dubrulle | Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection[END_REF] the torque is given by M = GρνH, where G is calculated from the gap width and Re = 3800 according to their relation (20), and gives G = 2.51 × 10 6 . With ρ = 1000 kg/m 3 , T = 1.9 × 10 -3 N m and the work performed during the mixing time T s = 4000 s the mechanical energy is then E = 3.82 J. The potential energy variation for two ideally separated layers is P = ǫρ 0 gπ(b 2 -a 2 )h 2 /2 (where h = H/2, ǫ = 0.003), which in our case gives P = 0.117 J, yielding a flux Richardson number, Ri f = P/ E = 0.031.

For a fixed rotation rate, i.e., fixed Reynolds number, the variation of T i and T s is displayed as a function of the interface density difference, ǫ,i nFig. 6. Both, T i and T s are directly proportional to ǫ implying that the interface and stratification life-times are proportional to the initial potential energy; the mixing efficiency remains thus constant.

For a fixed stratification, ǫ, the dependence of T i and T s on Ω (∼ Re) are shown in Fig. 7 on a logarithmic scale. Within the parameter range under consideration, T i and T s are proportional to Ω -4 . For the range of Re-numbers studied in the present experiments the torque M ∼ Re n , with n = 3/2 (see [START_REF] Dubrulle | Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection[END_REF]. Since the total variation of potential energy P is the same for all experimental data shown in Fig. 7,w ec a n conclude that Ri f = P/ E = 1/(MiΩT s ) ∼ Re k , with k = 3/2.

Discussion

Uniform stratification is normally considered as a generic case for the studies on continuously stratified fluids. New dynamical effects occur when the spatial scale of the buoyancy frequency variation is comparable with the characteristic dimension of the flow. In the present study the vertical extension of the pycnocline is comparable with the gap width of the experimental system. We noticed a marked difference in flow evolution with that in a fluid with constant buoyancy frequency. The transition from the wave-vortex to the vortexvortex regime differs significantly from the one described for the uniform stratification in CJH and is accompanied by a jump of the fundamental frequency of the density fluctuations in the flow. Such a jump implies an increase of the azimuthal mode-number. Physically, the main difference of the present system with the linearly stratified case, is the perturbation of the pycnocline by the Taylor vortices in the relatively weakly stratified layers above and below the pycnocline centre. This explains the presence of higher azimuthal modes.

Generally, experiments on mixing in stratified fluids with smooth density profiles (see [START_REF] Fernando | Turbulent mixing in stratified fluids[END_REF], show a strong tendency toward sharpening of density gradients, with layer formation in case of a thick interface. Depending on the density difference ǫ, an increase of the initial density gradient as well as a decrease of the initial density gradient has been observed.

The mixing efficiency for Reynolds numbers O(10 3 ) is found as low as 0.031 compared to values of 0.1 and 0.2 in, respectively, numerical simulations on unstable shear flows [START_REF] Caulfield | The anatomy of the mixing transition in homogeneous and stratified free shear layers[END_REF] and experiments on turbulence in stratified fluids [START_REF] Park | Turbulent mixing in stratified fluids, layer formation and energetics[END_REF]. The reason must be that the main torque is on the primary azimuthal flow, while the Taylor vortices that mix the flow, are of secondary order. In addition, they are highly anisotropic in contrast to generally investigated turbulent flows. To make a proper comparison with former studies on mixing efficiency, we will need to know the exact strength of the the Taylor vortices; this will be presented in the near future. Nevertheless, these values may be interesting in view of the mixing due to other secondary flows such as for instance Langmuir circulation, which are presently of interest in view of the mixing of the ocean upper layer.

Fig. 1 .

 1 Fig. 1. Shadow-graph image of the gap between the cylinders (a) and (b) spatio-temporal image of the selected vertical line of pixels in the wave-vortex regime. The picture contains 674 vertical lines taken at the acquisition frequency 1 Hz, and time increases from left to right. Experimental parameters: ǫ = 0.024, δ/d = 1.56, Ω = 0.151 rad/s.

Fig. 2 .

 2 Fig. 2. Typical shadow-graph image of the gap between the cylinders (a), (b) spatio-temporal image of the selected vertical line of pixels for a total acquisition time of 443 s at the sampling frequency 1 Hz, and (c) shadow-graph image of the gap between the cylinders after the experimental run. Experimental parameters ǫ = 0.024, δ/d = 1.56, Ω = 0.268 rad/s.

Fig. 3 .

 3 Fig. 3. Wave-vortex transition (open symbols: ǫ = 0.006, δ/d = 1.7, N = 0.83 rad/s; black symbols: ǫ = 0.024, δ/d = 1.56, N = 1.74 rad/s).

Fig. 4 .

 4 Fig. 4. Spatio-temporal shadow-graph image of entrainment in a smooth pycnocline. The acquisition frequency is 0.05 Hz and the total acquisition time amounts 9.8 h. Experimental run C with Ω = 0.16 rad/s and times T 1 = 2940 s, T 2 = 7320 s after the onset of rotation.

Fig. 5 .

 5 Fig. 5. (a) Spatio-temporal image of the mixing of a two-fluid system with a sharp pycnocline. The image contains 496 vertical lines taken with acquisition frequency 0.1 Hz; the vertical size of the image is 10d. (b) Light-intensity profiles J(z/d) at different stages of the mixing process, with time-step t = 1000 s between the profiles and solid black lines representing the initial profile at t = 0. (c) Time-evolution of the normalized potential energy P. Experimental parameters: ǫ = 0.003, Ω = 0.5 rad/s, Re = 3800.

Fig. 6 .

 6 Fig. 6. Values of T i , (open circles) and T s , (black circles), vs. ǫ. The values T are obtained from spatio-temporal shadow-graph images obtained at acquisition frequency of 0.1 Hz. The rotation rate is kept constant at Ω = 0.5 rad/s (Re = 3800).

Fig. 7 .

 7 Fig. 7. Log-log plot of values of T i (open circles) and T s (black circles) vs. ǫ. The values T are obtained from spatiotemporal shadow-graph images obtained at acquisition frequency of 0.1 Hz. The stratification of the interface is ǫ = 0.003. The solid line has a slope of Ω -4 .

Table 1

 1 Governing parameters of the experimental runs, where series A and B and experiment C correspond to runs with an initially smooth interface whereas in series D, E the interface was initially sharp.

	Experiment series	ǫ	δ/d	Ω (s -1 )	N (s -1 )	Ta × 10 5	Gr × 10 6
	A	0.006	1.7	0.068	0.83	1.5	4.3
				0.084		2.3
				0.098		3.1
				0.111		3.9
				0.125		5.0
				0.139		6.2
				0.154		7.6
				0.165		10.6
	B	0.024	1.56	0.097	1.74	3.0	19
				0.123		4.8
				0.151		7.3
				0.178		10.1
				0.209		14.0
				0.268		23.0
				0.293		27.3
				0.33		34.9
	C	0.006	1.7	0.16	0.83	8.2	4.3
	D	0.0015	0.2	0.5	1.21	80	9.3
		0.003			1.72		18.5
		0.006			2.43		37.5
		0.009			2.97		55
		0.02			4.43		123
	E	0.003	0.2	0.29	1.72	27	18.5
				0.35		39
				0.41		54
				0.46		68
				0.53		90

Acknowledgements

The authors acknowledge the fruitful discussions with Beatrice Janiaud and Emil Hopfinger, and the technical assistance of Pierre Carecchio in setting up the experimental device. Special thanks are due to Bruno Voisin for interesting discussions and steady friendly help. EE was supported by the fellowship MENRT-246 in 2001 which made this research at the LEGI possible and partially by the fund of integration projects SB RAS under grant No. 3.13.1. The experimental work was financed by EPSHOM contact no. 0287 028004702925.


H Michallet
email: herve.michallet@hmg.inpg.fr.2legi.jean-paul.barbier-neyret@hmg.inpg.fr.

J.-P Barbier-Neyret

M Mory

I Piedra-Cueva

AN OPTICAL SYSTEM FOR MONITORING SEABED EVOLUTIONS AT A COASTAL STRUCTURE

An optical system has been designed for measuring the bed level, sand grain motions inside the soil and the presence of suspension at the toe of a coastal structure exposed to breaking waves. These measurements were conducted in the framework of the LIMAS project. We show that the soil is not strictly stable even at great depths (about 1 m). The generation of pressure differences in the soil is the main mechanism for this instability at the wall facing the waves. At the corner of the structure, the shear induced by large longitudinal velocities enhances the sediment transport and scour.

INTRODUCTION

The sediment transport at the toe of a coastal structure exposed to breaking waves is generally very difficult to observe. Due to suspension and aeration, it is often impossible to visualize the bed level. An optical system has been designed to estimate the behaviour of the soil under breaking wave activity.

Conductivity probes, as described by [START_REF] Babu | Measurement of scour in cohesive soils around a vertical pile-simplified instrumentation and regression analysis[END_REF], may be used to follow bed evolution and scour in laboratory experiments with fresh water. However, it appears that the discrepancies between conductivities in the saturated soil and in the water column are not so large in sea-water and this may complicate the interpretation of the measurements [START_REF] Cassen | Resistive measurement of seabed local evolution and in depth sediment response to the wave action[END_REF].

Optical backscatter systems (OBS) are often used to estimate sediment concentrations (e.g. [START_REF] Conley | Cross-shore sediment transport partitioning in the nearshore during a storm event[END_REF]. Other studies have also used this technique to estimate the bed level evolution (e.g. [START_REF] Lee | Sediment transport on macrotidal flats in Garolim Bay, west coast of Korea: significance of wind waves and asymmetry of tidal currents[END_REF]. The physical principle of our system is basically the same. Instead of trying to carefully estimate the sediment concentration in the water column, we focussed in the interpretation of bed level evolution and also the response inside the soil. We show in the present paper that we may qualify the soil stability by sampling at high frequencies (100 Hz).

This work was performed in the framework of field measurements of the LIMAS project [START_REF] Bonjean | Monitoring of the foundations of a coastal structure submitted to breaking waves: occurrence of momentary liquefaction[END_REF][START_REF] Mory | Momentary liquefaction caused by waves around a coastal structure[END_REF][START_REF] Piedra-Cueva | Wavelet analysis of pressure measurements near a coastal structure[END_REF]. The main goal of these experiments was to determine the occurrence of liquefaction through pressure measurements at various depths. A last world war bunker exposed to wave activity at high tide was instrumented. Two different configurations have been considered: the instruments located at the middle of the bunker's front facing the waves and the instruments located at the bunker's corner where significant scour has been observed (the latter configuration is shown in Fig. 1). Special attention has been paid to the estimation of the bed level for interpreting the pressure variations. The aim of the present paper is to describe the designed optical system and soil response in more detail. Moreover, the comparison between measurements in the two configurations is presented.

The paper is organized in the following manner. We first describe the experimental configuration and the deployed instruments with special emphasis on the optical system. The bed evolution is discussed for two configurations (bunker's front and corner) and a range of timescales, from small (wave) to large (tide). A general view of soil stability at the toe of a coastal structure exposed to breaking waves is thus given.

CONFIGURATION OF THE EXPERIMENTS

Instruments considered

The experimental set-up has been described in [START_REF] Mory | Momentary liquefaction caused by waves around a coastal structure[END_REF]. Five pressure and eight optical sensors are placed inside a steel beam, the tips of the sensors facing outward. The five pressure sensors are spaced every 30 cm, five optical sensors at the same level and three other placed inbetween the four upper pressure sensors. The steel beam is rigidly fixed to the bunker in order to (i) avoid any displacement of the beam relatively to the structure and (ii) have sensors in the water column as well as deep into the soil.

A graduated pole was also used to estimate the bed level during high tide (see Fig. 1). The pole would stand on the sand bed and fall down when erosion occurred. The pole could be lifted up and placed on the new bed level if accretion occurred. This system is very robust and relatively accurate with a precision of the order of 1 cm. Its main drawback is poor time resolution, which depends on the action of the operator.

Underwater video recordings were conducted under moderate wave activity. This was an additional check of the bed level estimate and suspension activity.

Principle of the optical system

A sensor is made up of two optical fibres set in parallel inside the steel beam, the fibre tips facing outward. One of the fibres is the emitter of light, the other one is the receptor. It is possible to determine what is in front of the sensor based on the reflected light level. In order to remove surrounding light effects, the incoming light is switched alternatively on and off (at 500 Hz). The level of backscattered light when the emission is off is suppressed from the measure when the emission is on. Fibre tips are 0.8 mm in diameter and 0.5 mm apart in order to reduce the measurement volume to a few mm 3 . In order to detect rapid variations of the soil properties as liquefaction events, the sampling frequency was set to 100 Hz.

Calibration

The response to a calibrating test is shown in Fig. 2. The beam was laid horizontally under water, sensors facing upward. Sand is poured on the sensors at time t = 9 s. The response of the system is a rapid increase. The beam is then turned upside down to let the sand fall at t = 45 s. We note that it takes a few seconds for the output to stabilize at the minimum level corresponding to clear water. The beam is then turned aside. At t = 120 s, a cloud of sand is dropped in front of the sensor. The threshold level has been set to 2.5 V.

Since the sediment colour and size vary widely in our field conditions (see for instance the pictures in Fig. 3), any calibration to precisely infer sediment concentration is hopeless. We therefore do not intend to translate output voltages into concentrations or soil compaction. We nevertheless deduce the following information from the measurements: (i) whether the sensor is in the soil; (ii) whether there is grain mobility in the soil; (iii) whether there is sediment suspension in the water column. The determination of these three aspects is justified in the next section.

OBSERVATIONS AND MEASUREMENTS Sand motions at the time scale of the waves

On September 29, the wave climate was calm enough to make videos of rather good quality. Video frames of this experiment are shown in Fig. 3. In that sequence, the beam is at the bunker's corner and only P1 (on the left of the beam) and o1 (on the right) are in the water column. The next sensors P2 and o1-2 are 20 cm and 5 cm below the soil/water interface respectively. The corresponding times series of pressure and optical outputs are shown in Fig. 4a andb. It is clear from these videos that below the wave troughs there is a strong offshore flow and shear that produces sediment transport (three top pictures in Fig. 3). Large sand grain motions are recorded at the optical sensor located 5 cm below the soil/water interface (sensor o1-2 in Fig. 4b). As the wave crest approaches, a vortex is generated at the bunker's corner and a large amount of sediment is put into suspension (t=2.6 s) reaching the area in front of the upper optical sensor (o1 in Fig. 4b at t=2.8 to 5.4 s). Again, shear produces significant motion of the upper part of the soil around t=10 s. From the three last pictures in Fig. 3 we note that the sediment concentration can vary very rapidly and that the optical sensor response frequency (see Fig. 4b) is good.

If we now look at the response of an optical sensor located deeper into the soil (o2-3 in Fig. 4d is located 35 cm below the soil/water interface), we observe smaller but significant variations that are in phase with the pressure differences shown in Fig. 4c (o2-3 is located in between P2 and P3). This is a general feature of the field campaign measurements. Optical outputs recorded small variations deep into the soil, which are in phase with the pressure differences.

The variations in output voltage are not directly proportional to sand grain displacements. Large sand grain motions may induce a small response of the system if there is a light change in sand grain colour, size and density. Besides, a large variation of the output voltage may be due to a small displacement of an air bubble trapped into the soil. Nevertheless, since all recorded motions are in phase with large variations of pressure differences, we are confident that the system is well designed for estimating soil stability at a coastal structure.

Mean sand response over a 10 mn period

Power density spectra have been computed over 10 mn time series and are plotted in Fig. 5. We first consider the wave energy as determined from pressure variations (top panels). We note the peak wave frequency around 0.1 Hz and also the harmonics. In the corner configuration (top-right panel) we observe low frequency waves due to the complex refraction and reflection in between the bunkers. Once inside the soil the energy clearly decreases with depth. From these pressure measurements we may infer that the soil/water interface is located between P2 and P3.

We now consider the response of the optical sensors (bottom panels). The energy level indicates the activity at the sensor's depth. In the corner configuration, sensors in the water column record large fluctuations. The high level of energy indicates that sediment is put into suspension with almost every wave. We note that almost all optical sensors present a peak of energy at the wave frequency (around 0.1 Hz). This shows that, even deep into the soil, pressure variations are recorded at the wave time scale. The energy level varies with depth, which may be linked to the gas content. We expect larger sand grain motions if air bubbles are located close the sensor. Two sensors require special attention: o2-3 (left) and o3 (right). They show a relatively high energy level at low frequencies (below 0.1 Hz). This is probably due to the fact that these sensors are the closest to the soil/water interface and thus experience bed level variations over the 10 mn period considered.

Optical response at the time scale of the tide: water depth evolution

In Fig. 6 we show the evolution of the optical outputs over a tide for the two different configurations.

The mean and maximum water levels inferred from the pressure sensors are also shown in the top panels of Fig. 6. A threshold value for the optical output has been determined from calibration. We have also shown that large fluctuations indicate suspension in the water column or large changes in the bed properties (from the video, see Figs 3 and4). We may thus estimate a threshold on the fluctuations of the response and plot Fig. 7. We note that the optical estimation of the bed level agrees with the pole measurement in the front configuration. In the corner configuration (panel on the right of Fig. 7), the agreement is rather poor. This is mainly due to local scour at the pole foot induced by the large flow velocities observed.

We now consider the bunker's front configuration in the left panels on Fig. 6. Clearly, waves are breaking against the bunker and significant sediment activity is noticed at the beginning of the tide. This induces an erosion at the structure's toe of about 27 cm as measured with the pole: o2, which was in the soil at the preceding low tide, is in the water column at 15:00 (output below the threshold value). Apparently, the soil/water interface is close to o2-3 since its output varies significantly between 15:10 and 15:30. At high tide (between 16:00 and 18:00), the waves break over the bunker and the soil does not change much. When the water level is lower (18:30 -19:00), the waves are breaking against the bunker once more: erosion is measured and even o3 output level is changing. At the end of the tide (after 19:00), accretion is observed. In the other configuration (panels on the right of Fig. 6) we observe the same general feature, erosion at the beginning of the tide and accretion at the end. Both processes are now stronger, more than 60 cm of erosion and a relatively weaker accretion of about 20 cm are verified. Suspension is observed all tide long, although the wave energy is a lot weaker compared to the other configuration. Here, in the corner of the bunker, shear induced by the strong longitudinal velocities (as observed in the video, see Fig. 3) is the key sediment transport mechanism.

Particular events

Figure 8 shows a 20 s time series with interesting features to understand the bed response to breaking waves. A wave is breaking on the structure at t=114 s. Optical variations at each depth are recorded in phase with such shock wave. Pressure difference P3-P2 is rapidly increasing to reach a value of about 50 cm water depth (at t=116, 119, 126 or 128.5); o3 and also o2-3 response is clearly visible. The pressure difference is close to the effective weight of the soil between the two pressure sensors (see [START_REF] Mory | Momentary liquefaction caused by waves around a coastal structure[END_REF]. This probably breaks the soil static equilibrium condition locally and thus induces sand mobility.

Two other time series are shown in Figs 9 and 10. The threshold value of pressure difference that would break the soil static equilibrium is plotted as a dotted line in the middle panels. We note that P3-P2 may slightly overshoot this value and a response of o2-3 is simultaneously recorded. This confirms the occurrence of momentary liquefaction. In the front configuration, this happens at the back of the wave a s the water level drops (Fig. 9) while, for the corner configuration, it appears later in the wave trough (Fig. 10). In the latter case, the backwash flow shear leads to large pressure differences that induce sediment mobility and probably enhance the sheet flow.

CONCLUSIONS

The soil response to breaking waves on a coastal structure was monitored for the first time with an optical system. We have shown its relevance for estimating soil stability. The designed optical system enables to estimate the position of the bed level at the toe of a coastal structure exposed to breaking waves. The configuration chosen has an uncertainty of the order of the spacing of the sensors (15 cm). Moreover, we have shown that the system determines the sediment activity in the water column. Strong fluctuations indicate that suspension is passing in front of the sensor. A more precise calibration of the system for determining the suspended sediment concentration will be subject of future work. In addition, sand grain mobility inside the soil has been determined. Due to large amounts of gas content [START_REF] Sandven | Development of Sampler for measurement of gas content[END_REF], pressure variations modify the trapped air bubbles size and therefore the sand grain arrangement at great depths.

A shock wave, due to a large wave breaking exactly on the structure, is measured at all instrumented depths (maximum depth: 1 m). When pressure differences break the soil static equilibrium, liquefaction occurs and soil mobility is measured by the optical system. Even if not new in its conception, the designed optical system is proven valuable for engineers and scientists for determining soil and sediment activity at coastal structures.

In the front configuration, pressure differences inside the soil are mainly due to rapid changes of water level due to breaking. In the corner configuration, pressure differences are due to the shear at the bed level. In both cases, these pressure differences may overcome the effective weight of the soil and break its static equilibrium.

Fig

  Fig. 1. Instrumented bunker (corner configuration).

Fig. 2 .

 2 Fig. 2. Typical response of an optical sensor to a calibrating test (see text).

Fig. 3 .

 3 Fig.3. Video frames showing soil response (corner configuration) that correspond to t=2.12, 2.20, 2.28, 2.60, 3.80, 9.24, 10.80 and 11.12 in Fig.4(from left to right and top to bottom).

Fig. 4 .

 4 Fig. 4. Time series of the three upper pressure sensors (a); two upper optical sensors (b); two upper pressure differences (c) and the forth upper optical sensor located between P2 and P3 (d).

Fig. 5 .

 5 Fig. 5. Power density spectra of 10 mn sequences of pressure (top panels) and optical (bottom panels) measurements for the two configurations: bunker's face (left panels: Sept. 25 15:42-15:52) and bunker's corner (right panels: Oct. 2 8:48-8:58).

Fig. 6 .

 6 Fig. 6. Optical response over a tide (bottom panels) for the two configurations: bunker's face (left) and bunker's corner (right). There is an offset of multiples of 10V for clarity. The threshold value for each sensor is plotted in a dotted line. The mean water level (solid line) and the maximum water level (dashed line) are shown in the top panels.

Fig. 7 .

 7 Fig. 7. Comparison of bed level and sediment stability estimations from optical responses with graduated pole measurements over one tide in the two configurations: bunker's face (left) and bunker's corner (right).

Fig. 8 .Fig. 9 .

 89 Fig. 8. Time series of pressure (a); pressure differences (b) and the optical outputs (8 bottom panels), front configuration, Sept. 27 16:58:54. The bed level is close to o2 and P2; o5 and P5 are about 90 cm deep.

ACKNOWLEDGEMENTS

This study was partially funded by the European Commission Research Directorate, FP5, specific program "Energy, Environment and Sustainable Development," Contract No EVK3-CT-2000-00038, Liquefaction Around Marine Structures LIMAS.