Mohammad A Al-Shudeifat
email: mohd.shudeifat@kustar.ac.ae

Highly efficient nonlinear energy sink

Keywords: Nonlinear oscillation, Nonlinear energy sink, Targeted energy transfer, Nonlinear stiffness, Negative stiffness, Passive energy pumping

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List of symbols

NES

Shock mitigation in small-and large-scale dynamic structures using nonlinear energy sinks (NESs) has

1
gained considerable interest in the past decade. Accordingly, an extensive research work has been done for investigating the targeted energy transfer through different designs of NESs. The NES is single or twodegree-of-freedom light-weighted device coupled to small-or large-scale dynamic structure through essentially nonlinear elements [START_REF] Vakakis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, I and II[END_REF][START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying Hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part II-resonance capture[END_REF][START_REF] Lee | Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results[END_REF][START_REF] Quinn | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance capture: part I[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part II, analytical study[END_REF][START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Al-Shudeifat | Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators[END_REF][START_REF] Sapsis | Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments[END_REF][START_REF] Vakakis | Shock isolation through the use of nonlinear energy sinks[END_REF][START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially coupled rotational nonlinear energy sink[END_REF][START_REF] Sigalov | Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling[END_REF][START_REF] Gendelman | Dynamics of an eccentric rotational nonlinear energy sink[END_REF]. It performs rapid and passive energy transfer and dissipation for significant portion of the vibration energy induced by impulse into the linear primary structure (due to shock, blast, earthquakes, etc.). Purely cubic springs have been employed in Type I, Type II, and Type III NESs where no negative or positive linear stiffness components were considered in these designs [START_REF] Vakakis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, I and II[END_REF][START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying Hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part II-resonance capture[END_REF][START_REF] Lee | Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results[END_REF][START_REF] Quinn | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance capture: part I[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part II, analytical study[END_REF][START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Al-Shudeifat | Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators[END_REF][START_REF] Sapsis | Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments[END_REF][START_REF] Vakakis | Shock isolation through the use of nonlinear energy sinks[END_REF]. Type I NES is composed of a small mass coupled to the linear structure through linear damping and essentially nonlinear stiffness elements where the Type II NES is coupled to the linear structure through both nonlinear stiffness and nonlinear damping elements. The Type III NES is composed of two masses where the upper NES mass is coupled to the bottom one through essentially nonlinear stiffness element, whereas the bottom NES mass is also coupled to the linear structure through essentially nonlinear stiffness element [START_REF] Sapsis | Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments[END_REF].

The strongly nonlinear inertia coupling in the rotating NES device which couples the rotating NES mass to the linear structure has been investigated in [START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially coupled rotational nonlinear energy sink[END_REF][START_REF] Sigalov | Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling[END_REF][START_REF] Gendelman | Dynamics of an eccentric rotational nonlinear energy sink[END_REF] where good performance of this NES for shock mitigation was observed. The strongly nonlinear vibro-impact coupling between the NES mass and the linear structure at the degree of freedom that exhibits high vibration amplitudes has also been investigated in [START_REF] Nucera | Targeted energy transfers in vibroimpact oscillators for seismic mitigation[END_REF][START_REF] Karayannis | Vibro-impact attachments as shock absorbers[END_REF][START_REF] Nucera | Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: I. Computational results[END_REF][START_REF] Lee | Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments[END_REF][START_REF] Georgiadis | Shock isolation through passive energy pumping caused by non-smooth nonlinearities[END_REF][START_REF] Al-Shudeifat | Numerical and experimental investigation of a highly effective single-sided vibro-impact nonlinear energy sink for shock mitigation[END_REF] as an efficient NES design. Recently, the most efficient single-sided vibro-impact (SSVI) NES was numerically and experimentally studied in [START_REF] Al-Shudeifat | Numerical and experimental investigation of a highly effective single-sided vibro-impact nonlinear energy sink for shock mitigation[END_REF]. Unlike other designs, The SSVI design was found to be efficient in shock mitigation even at severe initial input energies induced into the linear structure. Moreover, a set of SSVI NESs have been employed via synergy with Type I NESs with a 9-story test structure excited by a blast where the SSVI NESs have been found to have the main contribution for energy scattering to high modes which yields rapid and strong mitigation of the blast shock [START_REF] Wierschem | Experimental testing of a large 9-story structure equipped with multiple nonlinear energy sinks subjected to an impulsive loading[END_REF].

In all NESs, the basic mechanism for the TET takes place through single mode or cascade of modes resonance captures [START_REF] Vakakis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, I and II[END_REF][START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying Hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part II-resonance capture[END_REF][START_REF] Lee | Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results[END_REF][START_REF] Quinn | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance capture: part I[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part II, analytical study[END_REF][START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Sigalov | Resonance captures and targeted energy transfers in an inertially coupled rotational nonlinear energy sink[END_REF][START_REF] Sigalov | Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling[END_REF][START_REF] Gendelman | Dynamics of an eccentric rotational nonlinear energy sink[END_REF][START_REF] Al-Shudeifat | Numerical and experimental investigation of a highly effective single-sided vibro-impact nonlinear energy sink for shock mitigation[END_REF][START_REF] Wierschem | Experimental testing of a large 9-story structure equipped with multiple nonlinear energy sinks subjected to an impulsive loading[END_REF][START_REF] Quinn | Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments[END_REF][START_REF] Wierschem | Passive damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment[END_REF]. The NES performance for rapid passive energy transfer from the linear structure to the NES itself can be quantified through the percentage of the initial energy induced into the linear structure that is transferred and rapidly dissipated by the NES. Furthermore, effective measures for damping and stiffness have been introduced for studying the effect of the NES attachment on enhancing the damping and the stiffness of the linear structure [START_REF] Al-Shudeifat | Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators[END_REF][START_REF] Al-Shudeifat | Numerical and experimental investigation of a highly effective single-sided vibro-impact nonlinear energy sink for shock mitigation[END_REF][START_REF] Quinn | Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments[END_REF][START_REF] Wierschem | Passive damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment[END_REF]. As a result, the performance of the NES attachment was also accurately quantified through these measures.

In all above NESs, the NES mass is coupled to the linear structure through essentially nonlinear element. However, a completely different configuration of another NES design has been investigated in [START_REF] Jiang | Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results[END_REF][START_REF] Malatkar | Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator[END_REF] for studying the steady-state dynamics of a linear structure subjected to harmonic excitation. In this design, one side of the NES mass was weakly coupled to the linear structure by a linear spring, and the other side was grounded to the wall by an essentially nonlinear stiffness. Even though the authors in [START_REF] Jiang | Steady state passive nonlinear energy pumping in coupled oscillators: theoretical and experimental results[END_REF] have claimed that passive energy pumping has been achieved, the authors in [START_REF] Malatkar | Steady-state dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator[END_REF] have claimed that no evidence of energy pumping was observed by considering similar system of similar physical parameters.

The physical configuration of Type I NES used in the literature in [START_REF] Vakakis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, I and II[END_REF][START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying Hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part II-resonance capture[END_REF][START_REF] Lee | Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results[END_REF][START_REF] Quinn | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance capture: part I[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part II, analytical study[END_REF][START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Al-Shudeifat | Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators[END_REF][START_REF] Sapsis | Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments[END_REF][START_REF] Vakakis | Shock isolation through the use of nonlinear energy sinks[END_REF][START_REF] Wierschem | Experimental testing of a large 9-story structure equipped with multiple nonlinear energy sinks subjected to an impulsive loading[END_REF][START_REF] Quinn | Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments[END_REF][START_REF] Wierschem | Passive damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment[END_REF] can be modified to generate negative linear and nonlinear stiffness components. This modification is the main focus in this paper where the significant effect of the negative stiffness components in enhancing the TET in the NES is clearly addressed here. This bistable NES with the negative stiffness components has two stable equilibrium positions. Selecting one of these equilibrium positions to coincide with the equilibrium position of the coupling mass in the linear structure is the key behind the considerable enhancement in the NES performance for rapid passive energy transfer and dissipation in broadband fashion of input energies. In addition, significant finding is observed when the bistable NES attachment with negative linear and nonlinear stiffness components has a nonzero relative displacement relative to the linear primary structure away from the NES equilibrium positions. This leads to a strong suppression of the vibration amplitudes of the linear structure in the first cycle of oscillation for wide range of initial input energies induced into the linear structure. The enhanced NES design here through including small negative linear and nonlinear stiffness components is highly efficient design compared with all existing NESs investigated up to date in the related literature.

Realizing negative linear and nonlinear stiffness components in the NES

The configuration in Fig. 1 has been used in [START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF] for realizing the essentially nonlinear stiffness based on the geometric nonlinearity obtained by the transverse linear springs when they are not compressed or elongated at their vertical position. Hence, the resulting essentially nonlinear oscillator has no linear stiffness components. Here we assume that these transverse springs are compressed when they are aligned in their vertical position where an amount of potential energy is stored. This can be achieved by considering the bistable geometry in Fig. 1a which has two stable equilibrium positions at which both springs are not stretched or compressed. This configuration of the unloaded springs generates negative linear and nonlinear stiffness components during the motion of the mass. The two stable equilibrium positions of the NES mass (no tension or compression in the springs at their physical length L)a r ea tz = 0 and z = 2z c for z c > 0 relative to the selected reference of the displacement as shown in the figure. Consequently, the configuration of the NES is unstable at z = z c . For the following analysis, the initial position of the mass m is selected to coincide with the left equilibrium position of the system (i.e., z = 0). Consequently, the net horizontal force F nl of the springs becomes zero when the mass at the positions z = 0, z = z c , and z = 2z c . However, the net vertical force of the springs is always equal to zero due to the symmetry.

The realization to the negative linear and nonlinear stiffness components in the system is verified by deriving the relation of the net horizontal force F nl of the springs based on the geometry in Fig. 1bafterthemass moves to the right by the displacement z. Hence, for the deflection =-L + (L 2 0 + (-z c ) 2 ) 1/2 ) of each spring, the net nonlinear force of the springs acting on the mass is derived as

F nl =-2k (z -z c ) L 2 0 + (z -z c ) 2 1/2 -L L 2 0 + (z -z c ) 2 1/2 =-2k (z -z c ) 1-L L 2 0 +(z -z c ) 2 -1/2 , (1)
where L = (L 2 0 + z 2 c ) 1/2 which is the unloaded spring physical length. The Taylor-expansion about z = z c for the quantity

(L 2 0 + (z -z c ) 2 ) -1/2 yields L 2 0 + (z -z c ) 2 -1/2 = 1 L 0 - (z -z c ) 2 2L 3 0 + 3 (z -z c ) 4 8L 5 0 - 5 (z -z c ) 6 16L 7 0 + O (z -z c ) 8 . (2)
By substituting Eq. ( 2)into(1)forz = zz c we obtain where

F nl ∼ = Fnl =-2k 1 - L L 0 z - kL L 3 0 z3 + 3kL 4L 5 0 z5 - 5kL 8L 7 0 z7 + O z9 = k 1 z -k 2 z3 + k 3 z5 -k 4 z7 + O z9 , L 0 < L , (3)
k 1 = 2k L L 0 -1 , k 2 = kL L 3 0 k 3 = 3kL 4L 5 0 and k 4 = 5kL 8L 7 0 for L 0 < L.
For the arbitrary parameters of k = 0.3N/m,L 0 = 0.97 m, and L = 1 m, the results in Fig. 2 for the exact nonlinear force F nl in Eq. ( 1) and its corresponding approximated nonlinear force Fnl in Eq. ( 3)ofthe springs in Fig. 1 show the significance of including the seventh order nonlinear stiffness coefficients, because later we will see that the mass m is expected to exhibit high amplitudes of oscillation when the system in Fig. 1 is used as a nonlinear energy sink for passive energy transfer and dissipation.

Application for passive energy pumping

The nonlinear attachment in Fig. 1ao fm a s sm is now attached to the single-degree-of freedom linear dynamic oscillator of mass M(m ≪ M) as shown in Fig. 3. The purpose of that is to investigate the passive energy transfer from this linear structure to the modified NES attachment. The equilibrium position of the linear structure is selected to coincide with the left stable equilibrium position of the attachment at which springs are neither compressed nor elongated. This configuration shows that an amount of potential energy will be stored For the relative displacement between the NES mass and the linear structure mass z = x 1x 2 and z c > 0 the equations of motion are given as

m ẍ1 + λ ( ẋ1 -ẋ 2 ) -k 1 (z -z c ) + k 2 (z -z c ) 3 -k 3 (z -z c ) 5 + k 4 (z -z c ) 7 = 0, M ẍ2 + λ p ẋ2 -λ ( ẋ1 -ẋ 2 ) + k p x 2 + k 1 (z -z c ) -k 2 (z -z c ) 3 +k 3 (z -z c ) 5 -k 4 (z -z c ) 7 = 0, (4)
where x 1 is the NES mass displacement, x 2 is linear structure mass displacement, λ is the damping of the NES attachment, λ P is the damping of the linear structure, and k p is the stiffness of the linear structure.

Initially unloaded springs

Series of simulations for the system in Fig. 3 are performed here to compare the performance of the modified NES in Fig. 3 The Type I NES is now modified by the configuration in Fig. 3 for which L > L 0 and z c > 0 and optimized at the same optimized damping λ = 0.01 N s/m obtained previously for Type I NES. Hence, at ẋ2 (0) = 0.1 m/s we obtain two sets of the optimized values of k and L 0 for which η>99 % as shown in Fig. 4b. These sets are shown in this figure at η = 99.14 % to be k = 0.14 N/m and L 0 = 0.882L and at η = 99.09 % to be k = 2.9 N/m and L 0 = 0.993L. The performance of the modified NES at these two sets of the optimized parameters is compared with that of the optimized Type I NES for wide range of initial input energies induced into the linear structure mass, which are represented by their corresponding initial velocities as shown in Fig. 5. It is clear from the figure that for k = 0.14 N/m and L 0 = 0.882L of the modified NES the performance is within the range 97 % <η<99 % for wide range of input energies. This performance is considerably enhanced here compared with that of Type I NES as shown in the same figure. For 100 s of simulation the modified NES at k = 0.14 N/m and L 0 = 0.882L keeps its high performance as shown in Fig. 6 while 9 is strongly dominated by the 1:1 resonance capture. However, the response of the modified NES in Fig. 10 is dominated by a cascade of immediate resonance captures at the 1:1 NNM branch besides to a set of NNM branches of low frequencies as shown in the corresponding wavelet transform of the modified NES relative displacement. This energy scattering by an immediate cascade of resonance captures makes the modified NES to be capable to keep its high performance in shock mitigation at high initial input energy.

Furthermore, to verify the observed cascade of resonance captures by the modified NES, the time histories of the displacements and their corresponding wavelet transforms are obtained at an intermediate input energy of ẋ2 (0) = 0.25 m/s in Fig. 11 and at high input energy of ẋ2 (0) = 0.7 m/s in Fig. 12. It is clear in these figures that the modified NES keeps its strong and immediate cascade of resonance capture at 1:1 NNM branch and the NNM branches of low frequencies. However, at the high input energy in Fig. 12, the modified NES has an additional strong resonance capture by higher NNM frequency branch.

The actual nonlinear forces in the NESs are plott e di nF i g .13 based on the simulation results for the Type I NES and the modified one at the initial velocities of the primary mass of ẋ2 (0) = 0.1m/s,ẋ 2 (0) = 0.25 m/s, ẋ2 (0) = 0.4 m/s, and ẋ2 (0) = 0.7m/s.For low initial energy corresponding to ẋ2 (0) = 0.1m/s, the modified NES oscillates about its left equilibrium position as shown in Fig. 13a where the strong nonlinearity in the left branch of the force is clearly shown. For higher initial energies corresponding to the initial velocities in Fig. 13b,c,d The effect of the change in the modified NES damping and stiffness values is investigated in Fig. 14afor ẋ2 (0) = 0.1 m/s and in Fig. 14bfor ẋ2 (0) = 0.4m/s. It is found that, unlike Type I NES in Fig. 4a, the perfor- mance of the modified NES is not significantly affected by any significant change in its damping and stiffness. The figure shows that the modified NES is more robust to the change in damping and stiffness than the Type I NES. This is an important advantage to the modified NES for real life applications.

NES performance under multi input energies

The effect of multi input energies on the performance of the optimized NESs is investigated here. To enrich the comparison, the optimized linear absorber of which η = 99 % at the damping λ = 0.01 N s/m and the linear coupling stiffness k = 0.04 N/m is also considered. The mass of the linear structure is given an amount of initial input energy at time t = 0, and then another amount of external energy is induced at later time t = t 1 . The ratio of the maximum amplitude in the displacement of the primary mass when the NES (or the linear absorber) is unlocked to the maximum amplitude when the NES (or the linear absorber) is locked is plotted in Fig. 15 for 100 s of simulation at multi input energies. This ratio is shown in Fig. 15a for the modified NES, Fig. 15b for the Type I NES, and Fig. 15cfor the linear absorber at the corresponding velocities to the external input energies induced into the linear structure at t = 0 and t = 5 s. For Fig. 15d-f the external energies are induced into the linear structure at t = 0 and t = 10 s. This ratio for Type I NES, the modified NES and the linear absorber is found to exceed a unity for very small lesions in the figure. This common dynamic behavior between the considered NESs and the linear absorber can be considered insignificant to affect the overall performance in energy dissipation. Furthermore, the comparison between Fig. 15d of the modified NES and Fig. 15f of the linear absorber shows that the amplitude ratio is less for the system with the modified NES than that with the linear absorber as shown in the colored bar. Hence, the modified NES performs better than the linear absorber under multiple input energies.

Initially loaded springs

The case for initially loaded springs (compressed or elongated) is investigated here. The system is simulated for the parameters L = 1m, L 0 = 0.97L , k = 0.3N/m, m = 0.05 kg,λ = 0.016 N s/m, M = 1kg,λ p = 0.005 N s/m, and k p = 1 N/m. At these parameters, we obtain z c = 0.243 m, k 1 = 0.019, k 2 = 0.329, k 3 = 0.262, and k 4 = 0.232. The initial velocities and displacements for the simulation are set to, x 2 (0) = 0, ẋ2 (0) = v 0 for v 0 > 0, ẋ1 (0) = 0, and x 1 (0) =nz c for n = 0, 1, 2, 3, and 4. For n = 0 the NES attachment is now positioned at its right stable equilibrium position which coincides with the primary mass equilibrium position, for n = 1t h e springs of the NES attachment are in their vertical compressed position, for n = 2 the NES attachment is positioned at its left stable equilibrium position, and for n = 3 and n = 4 the NES attachment is positioned away from its left stable equilibrium position where the NES springs are now in tension. The energy pumping Fig. [START_REF] Nucera | Targeted energy transfers in vibroimpact oscillators for seismic mitigation[END_REF] Amplitude ratio (colored bar) under the effect of multi initial energies induced to the linear structures at different times in a and d for the modified NES; b, e for the Type I NES; and c, f for the linear absorber Fig. [START_REF] Karayannis | Vibro-impact attachments as shock absorbers[END_REF] Percentage of the initial energy induced into primary mass dissipated by the NES for various initial velocities of the primary mass and initial positions of the NES mass from the primary linear structure to the NES is again quantified by η. The simulation results for various initial velocities of the primary mass and different initial positions of the NES mass are shown in Fig. 16.I ti s observed in the figure that the modified NES with negative stiffness components still performs much better than Type I NES when its initial position is now at the right equilibrium position (x 1 (0) = 0) which coincides with primary mass initial position (x 2 (0) = 0). However, further enhancement in this NES performance is observed when x 1 =-4z c . This interesting finding shows that 99.5 % of the input energy induced into the linear structure is immediately dissipated by the NES when ẋ2 (0) = 0.07 m/s. To verify this observation, the time histories for the primary mass displacement and the NES mass relative displacement are plotted in Fig. 17a andb. The significant reduction in the vibration amplitudes of the primary linear structure is clear in the first cycle of oscillation. This reduction is due to fact that the NES force is dominated by the negative stiffness components which make this force to act in the opposite direction of the motion of the primary mass, even though both of the structure and NES masses are moving in the same direction. Similar observations are obtained for higher velocity of the linear structure as shown in Fig. 17c andd. These considerable enhancements in the NES performance are achieved here by the negative stiffness components. The NES relative displacements in Fig. 17 coincide with the range of the well-approximation of the NES force that previously plotted in Fig. 2b.

Two-story physical structure

The modified bistable NES with negative linear and nonlinear stiffness components is investigated here with the physical parameters of the two-degree-offreedom physical structure in [START_REF] Al-Shudeifat | Numerical and experimental investigation of a highly effective single-sided vibro-impact nonlinear energy sink for shock mitigation[END_REF]. The physical para- 19b. This jump between the equilibrium positions gives the NES mass long stroke of motion in which significant and rapid energy dissipation is achieved. The wavelet transform in Fig. 19c shows the strong immediate resonance capture between the NES mass and the top floor at 1:1 NNM branch near f 1 = 1.69 Hz. In addition, the modified NES keeps its strong cascade of resonance captures at 1:1 NNM branch near f 1 = 1.69 Hz and the NNM branches of low NNM frequencies as shown in Fig. 19d. This immediate cascade of resonance captures scatters a significant portion of the input energy in a very short time which leads to strong vibration suppression in the structure. Similar observations are obtained for the results in Fig. 20 where the time histories are obtained at high input energy corresponding to initial velocity of 0.82 m/sat each floor of the structure. These observations also agree well with those obtained for the single-floor linear structure considered in Fig. 3.

Concluded remarks

Realization of the negative linear and nonlinear stiffness components and their effect on the targeted energy transfer (TET) in the nonlinear energy sinks (NESs) have been addressed here. The linear and nonlinear negative stiffness components in the nonlinear bistable energy sink which are realized here through the modified configuration of the geometric nonlinearity have been found to considerably enhance the NES performance for passive energy pumping and local rapid energy dissipation. Furthermore, by considering the negative stiffness components in the NES with a proper initial position of the NES mass relative to the primary structure mass, the modified NES is found to be able to dissipate 97-99 % of the input energy induced into the linear structure mass for broadband initial energies. This ability for rapid energy transfer and dissipation makes the NES strongly suppress the vibration amplitudes of the linear structure even at the first cycle of oscillation. The high performance of the modified NES with negative stiffness components can be maintained for broadband of initial input energies by an immediate cascade of resonance captures at low and high NNM frequencies. In addition, the modified NES is found to maintain its high performance in shock mitigation for multi external energy inputs at different impulse times.

The findings in this paper are expected to enrich the fields of applications of this modified bistable NES. Finally, this NES can be named as Type VII NES to be distinguished from the other six types of NESs in the literature.

Fig. 1 Fig. 2

 12 Fig. 1 Physical configuration of the nonlinear damped oscillator

Fig. 3

 3 Fig. 3 Single-degree-offreedom linear structure coupled to the modified NES

  to the existing Type I NES in the literature[START_REF] Vakakis | Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, I and II[END_REF][START_REF] Gendelman | Energy pumping in nonlinear mechanical oscillators: part I-dynamics of the underlying Hamiltonian systems[END_REF][START_REF] Vakakis | Energy pumping in nonlinear mechanical oscillators: part II-resonance capture[END_REF][START_REF] Lee | Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment[END_REF][START_REF] Gourdon | Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results[END_REF][START_REF] Quinn | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance capture: part I[END_REF][START_REF] Sapsis | Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part II, analytical study[END_REF][START_REF] Mcfarland | Experimental study of non-linear energy pumping occurring at a single fast frequency[END_REF][START_REF] Al-Shudeifat | Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators[END_REF][START_REF] Sapsis | Effective stiffening and damping enhancement of structures with strongly nonlinear local attachments[END_REF][START_REF] Vakakis | Shock isolation through the use of nonlinear energy sinks[END_REF][START_REF] Wierschem | Experimental testing of a large 9-story structure equipped with multiple nonlinear energy sinks subjected to an impulsive loading[END_REF][START_REF] Quinn | Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments[END_REF][START_REF] Wierschem | Passive damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment[END_REF] obtained by setting L = L 0 . The primary linear structure in Fig.3is assumed to have the parameters M = 1kg, k p = 1 N/m, and λ p = 0.001 N s/m. Both Type I and the modified NESs are assumed to have the same mass of m = 0.05 kg and the unloaded spring length of L = 1m. The NES performance is quantified by the percentage of the initial input energy induced into the lin-ear structure that is passively transferred and dissipated by the NES where this percentage is represented by η. Based on η, the damping and the nonlinear stiffness coefficients of Type I NES (L = L 0 , z c = 0) are optimized in Fig. 4aa t ẋ2 (0) = 0.1 m/s to be λ = 0.01 N s/m and k nl = 1.63 N/m 3 at η = 98.7%.

Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10

 5678910 Fig. 5 Percentage of the initial energy induced into the primary mass dissipated by the Type I NES and the modified NESs for various initial velocities of the primary mass

Fig. 11

 11 Fig. 11 Time histories of the displacement of the primary mass and the relative displacement of the optimized modified NES mass and their corresponding wavelet transforms for ẋ2 (0) = 0.25 m/s

  , the modified NES exchanges oscillation between its left and right equilibrium positions. The oscillation through the two equilibrium positions gives the NES mass larger strokes (peak-to-peak amplitudes) of oscillation by which much more energy is dissipated by the modified NES than Type I NES. Unlike the modified NES, it is shown in the figure that the motion of the Type I NES is restricted by narrow strokes (peak-to-peak amplitudes) of oscillations at the intermediate and the high loads which restrict the amount of the energy transferred and dissipated by the NES.

Fig. 12 Fig. 13 Fig. 14

 121314 Fig. 12 Time histories of the displacement of the primary mass and the relative displacement of the optimized modified NES mass and their corresponding wavelet transforms for ẋ2 (0) = 0.7m/s

Fig. 17

 17 Fig. 17 Time histories of the displacement of the primary mass and the corresponding relative displacement of the NES mass for ẋ2 (0) = 0.07 m/sina and b and for ẋ2 (0) = 0.2 m/s in c and d

  meters of this two-story test structure are considered here where they are given in matrix form as M is the mass matrix, K is the stiffness matrix, and C is the damping matrix. This structure has the natural frequencies f 1 = 1.69 Hz and f 2 = 4.63 Hz. The Type I NES of mass m = 2 kg is coupled to top floor of the structure that has the mass M 2 = 24.2 kg. The damping λ and the nonlinear stiffness k nl of the Type I NES are optimized for this structure at equal initial velocities of 0.2 m/s of each floor. Consequently, the optimized physical parameters of Type I NES are λ = 3 N s/m and k nl = 3.2×10 5 N/m 3 . Using the same optimized damping λ = 3 N s/m of Type I NES, same NES mass of m = 2 kg, and the spring unloaded physical length L = 0.25 m, the modified NES with negative linear and nonlinear stiffness components is now optimized for L 0 and the linear transverse stiffnesskas shown in Fig. 18. It is observed that a wide zone of

Fig. 18 Fig. 19

 1819 Fig. 18 Percentage of the initial energy induced into the primary structure dissipated by the NES (colored bar) for various values of the transverse spring stiffness k and the transverse vertical

Fig. 20

 20 Fig. 20 Time histories of the displacement of the top floor and the relative displacement of the optimized modified NES mass and their corresponding wavelet transforms for ẋ2 (0) = 0.82 m/s


E Alekseenko
email: e.alekseenko@gmail.com

B Roux

Wind effect on bottom shear stress, erosion and redeposition on Zostera noltei restoration in a coastal lagoon; part 2

Keywords: Mediterranean lagoon, Etang de Berre, hydrodynamics, numerical modelling, bottom shear stress, erosion, sediment transport

This paper concerns wind effect on bottom shear stress (𝐵𝑆𝑆), resuspension and redeposition of bottom sediments in the nearshore areas of the Etang de Berre (EB), a semienclosed lagoon, in the context of Zostera noltei (Z.n.) restoration. As in our previous paper, the first step is to compare 𝐵𝑆𝑆 with its threshold, 𝐵𝑆𝑆 𝑐𝑟 , for a wind speed of 21 m/s. But, here, a new simulation is performed for 16 wind directions regularly spaced. It permits to analyze the combined effect, over one year, of these winds on the erosion risk. For that, a mean value of |𝐵𝑆𝑆|, i.e., a weighted average based on the frequency of each wind directions, has been determined and compared to 𝐵𝑆𝑆 𝑐𝑟 for a class of roughness parameters. A similar averaging of 𝐵𝑆𝑆 → has been also evaluated to obtain the direction of an average bottom current.

Then, the most important contribution of the present paper concerns the modelling of the wind-induced sediment erosion, transport and redeposition. The specific module SEDIM/MARS3D is used to determine how the thickness of a given mud-sand sedimentary layer can be changed by the wind-induced current during typical periods of 3 days of constant wind. This numerical study is performed for three wind speeds in the two main wind directions: N-NW and S-SE. The time evolution of this initial sedimentary layer permits to determine where and when the erosion would be large enough to be a stressor for the Z.n. restoration.

For an extreme erosion rate of 10 -3 kg/m 2 /s, for a strong wind speed of 21 m/s and an initial sedimentary layer of 15cm, all this layer would be eroded in several nearshore areas after 3 days. The erosion risk is maximum along the eastern shore, where an additive effect of the two opposite winds can be expected. Indeed, S-SE and N-NW winds create coastal jets of opposite direction and of about the same intensity exerting alongshore erosion and deposition at about the same places.

The present results concerning the time evolution of erosion depth induced by the wind improve our knowledge of the erosion risk on the Z.n. restoration in EB. They will be useful to appropriately orient the action of politics and managers for a future restoration program.

Introduction

The present work is an extension of a recent numerical study (Alekseenko et al., 2017) devoted to the restoration of Zostera noltei (Z.n.) -an intertidal dwarf-eelgrass -in the Etang de Berre (EB) which is one of the largest Mediterranean lagoon in South of France. The massive loss of submerged aquatic vegetation (SAV), all along the coastal ocean, is a common worldwide ecological problem. The conservation and restoration of seagrass meadows provide a strategy to mitigate climate change while conserving these important ecosystems [START_REF] Duarte | Assessing the CO2 capture potential of seagrass restoration projects[END_REF]. Concerning Z.n. meadows in EB, they are specially protected as they also provide habitats and feeding grounds for migratory birds (Natura 2000). Concerning Z.n. meadows in EB, it is mentioned by [START_REF] Warner | Environmental impacts of hydroelectric power and other anthropogenic developments on the hydro morphology and ecology of the durance channel and the Etang de Berre, southeast France[END_REF] that they were severely destroyed in the last fifty years due to anthropogenic water pollution and huge amount of silt and freshwater outflow from a hydroelectric power plant built in 1966. But, during the last decade, the hydropower outflow and silt amount were reduced and regulated, and resilience of Z.n. meadows were observed in situ during the last four years by divers of a non-profit association [START_REF] Bazile | Le rapide développement des herbiers de zostères de l'étang de Berre. 27 mai[END_REF][START_REF] Bazile | 1ère campagne de Survey des zostères de l'étang de Berre par drone (9-10 juillet[END_REF], in several areas very close to the shoreline. These observations are very promising as the divers confirm that these meadows are now increasing year after year [START_REF] Bazile | 1ère campagne de Survey des zostères de l'étang de Berre par drone (9-10 juillet[END_REF].

Z.n. -to be restored in EB -are marine flowering plants having a root and rhizome. They generally colonize soft substrates (mud-sand) which can be easily eroded by the bottom shear stress. The critical level of burial or erosion tolerated by this species is extremely low (between 4 and 8 cm) due to the small size of the species and the lack of vertical rhizomes [START_REF] Cabaço | Effects of burial and erosion on the seagrass Zostera noltii[END_REF]. These limits concerning resilience of Z.n. to burial or erosion disturbances are experimentally confirmed by a more recent work by [START_REF] Han | Resilience of Zostera noltii to burial or erosion disturbances[END_REF].

In our previous numerical study, the MARS3D numerical model was used to analyze the 3D current, salinity and temperature distribution induced by three meteorological, oceanic and anthropogenic forcings in this lagoon (Alekseenko et al., 2017). We have also analyzed the complex hydrodynamics due to the tide effect in the Caronte channel connecting EB to the sea (Alekseenko & Roux, 2017).

In the previous study, which was realized for large values of the dominant wind in N-NW direction, the main contribution concerned the bottom shear stress (𝐵𝑆𝑆) fields in the nearshore areas, for different values of the bottom roughness parameter. Then, the 𝐵𝑆𝑆 values were compared with the critical 𝐵𝑆𝑆 at which the mobility of such bottom sediments would occur.

As it is mentioned by [START_REF] Berlamont | Modelling (partly) cohesive sediment transport in sewer systems[END_REF], experimental research on the erosion of mixed sediments [START_REF] Torfs | Shape and scale effects on secondary currents and shear stress distributions in laboratory flumes[END_REF], Williamson & Torfs,1996) showed an increasing erosion resistance with increasing content of cohesive material.

• When the mixture only contains a very small amount of cohesive material the fines (i.e., the fraction smaller than 63 𝜇m) are washed out of the surface layer at very low values of the bed shear stress, lower than the critical shear stress needed to move the sand particles.

•

Adding more cohesive material, the fines fill the pore spaces in between the sand grains and make the mixture smoother and thus more difficult to erode. Such a reduction of the erodibility of mud-sand mixtures is also reported by [START_REF] Panagiotopoulos | The influence of clay on the threshold of movement of fine sandy beds[END_REF] when mud is added up to 50 % to two sorts of sand: 150 microns and 215 microns; their experiments showed that 𝐵𝑆𝑆 𝑐𝑟 would be increased of about 50 % and 100 %, respectively. Panagiotopoulos et al. (1996) recommend, when the shear strength (i.e. threshold of movement) of natural marine sand-mud deposits has to be investigated, to consider the modelling with caution. Indeed, it would be needed to consider the type of clay minerals, pH and temperature of the eroding (ambient) fluid, the chemical properties of the pore and eroding fluids, internal sediment structure, degree of sediment saturation and biological activity. But, such properties are very specific of any coastal hydro-system (lagoon or estuary); and we do not have such a detailed knowledge for EB.

A detailed overview of previous sediment transport modelling is given by Dufois & Le Hir (2015):

•

Chesher & Ockenden (1997) solve transport equations for sand and mud concentrations, with contrasted settling velocities. Interactions are accounted for by considering the same critical erosion shear stress for sand and mud, which depends on the mud content in the bed.

•

Van Ledden (2002) also consider two sediment fractions (sand and mud), both transported in suspension, and distinguish two regimes (non-cohesive and cohesive) in the formulation of erosion, depending on the mud content in the surficial sediment.

•

Le [START_REF] Hir | Dynamics of sand and mud mixtures: a multiprocess-based modelling strategy[END_REF] propose a modelling strategy applied to the suspended transport of sand and mud mixtures, and which accounts for consolidation of mixed sediments. The model state variables are the different classes of particles, generally classified according to their size, and grouped into categories that are either transported as bedload or in suspension. The bed is described as thin layers characterized by a distribution of these classes.

•

A more recent and very exhaustive review concerning modelling and measurement of critical bed-shear stress for sand-mud mixture has been presented by van Rijn (2016). It confirms the extreme diversity of the behavior of cohesive sediment (due to biogenic effect in particular), and the extreme variety of the critical bed-shear stress data obtained in-situ and/or in flumes. That means that it is extremely difficulty to derive some general law to express 𝐵𝑆𝑆 𝑐𝑟 as function of particle size for example. [START_REF] Amos | The erosion rates of cohesive sediments in Venice lagoon[END_REF] in their paper devoted to erosion rate of cohesive sediments in Venice lagoon pointed out the importance of the erosion rate per unit area of bed flux, E, which is an essential precursor to the prediction of eroded depth and the suspended sediment concentration for a given applied bed shear stress. Whilst solution of the erosion threshold is often reported, the evaluation of the variation of this threshold through the process of erosion (eroded depth) is usually omitted or not estimated. The field experiments of [START_REF] Amos | The erosion rates of cohesive sediments in Venice lagoon[END_REF] showed that E is changing throughout the period of a given applied shear stress. The maximum rate increases with stress; but the minimum rate appears roughly constant (about 10 -4 kg/m 2 /s) for the mudsand sediment in several stations in Venetian northern and southern lagoons: (13% sands, 61% silts, 26% clays) and (16% sands, 54% silts and 30% clays), respectively (after Tab.2 of [START_REF] Amos | The erosion rates of cohesive sediments in Venice lagoon[END_REF]. Note that this sediment composition is not far from that measured by [START_REF] Rigaud | Dynamique et biodisponibilité des éléments traces métalliques dans les sédiments de l'étang de Berre[END_REF] in the central part of EB.

In areas with high wave exposure and strong currents, seagrass may be damaged due to excessive sediment transport, which does not allow seeds to become established, or eroding/burying existing seagrass beds. As a result, wave-or current-exposed areas tend to have patchy seagrasses or are unvegetated [START_REF] Fonseca | Influence of physical setting on seagrass landscapes near Beaufort, North Carolina, USA[END_REF]). An intermediate flow regime may be optimal for seagrass growth and development.

Interactions between seagrasses and water flow in their natural environment when exposed to wind-driven currents, tides and waves have been intensively studied by [START_REF] Koch | Waves in Seagrass Systems : Review and Technical Recommendations[END_REF]. [START_REF] Kombiadou | Modelling the effects of Zostera noltei meadows on sediment dynamics : application to the Arcachon lagoon[END_REF] have also investigated the effects of an abundant vegetation to the near-bed hydrodynamics and to the sediment suspension in the Arcachon lagoon. They show that Z.n. meadows play an important role to velocity attenuation and to sediment stabilization, with flow and suspended sediment concentrations damping, compared to an unvegetated state. In the today state, the Z.n. meadows in EB are so sparse that our present modelling does not consider velocity attenuation by the vegetation.

The present study contains two main parts. The first part is still devoted to the wind effect on 𝐵𝑆𝑆 fields, but here we consider 16 wind directions; the frequency of which are given from a weather data base shown in Fig. 2 (after SOGREAH, 2003). In the second part, a specific sediment module of MARS3D (called SEDIM) is used to simulate the erosion and the transport of fine bottom sediments (mud and sand) which are initially uniformly distributed. The goal is to evaluate the risks to get of profound erosion and to get some scouring of lighter sediments that would be redeposited further afield; which would be deleterious for Z.n. restoration. SEDIM module is described in Subsection 3.4 and is used in Section 5.

Study site

EB has been under intense anthropogenic pressure for several decades [START_REF] Warner | Environmental impacts of hydroelectric power and other anthropogenic developments on the hydro morphology and ecology of the durance channel and the Etang de Berre, southeast France[END_REF]. At the middle of the 20 th century, Zostera meadows occupied over 6000 ha. But subsequently, the lagoon was disturbed by urban and industrial pollution, and since 1966 by a hydropower station leading to a very large increase of freshwater and silt inflow. [START_REF] Bernard | Long term changes in Zostera meadows in the Berre lagoon (Provence, Mediterranean Sea)[END_REF] confirmed the near extinction of Zostera in EB and mentioned that it was probably resulting from several causes, namely, pollution (including nutrients), low salinity and turbidity resulting from huge amounts of water and silt from hydropower station.

To alleviate the hydropower disturbance to Z.n., the annual hydropower runoff is reduced to 1.2 x 10 9 m 3 , since 2005. In addition, the maximal amount of silt associated with this annual runoff is limited to 60 000 tons.

In Fig. 1, we show four control areas (PA, PB, PF, PM) in the Grand Etang which correspond to the places where benthic vegetation existed 50 years ago and then disappeared completely (PF and PM, at the western shore) or partially (PA and PB, at the eastern shore). They also correspond to the main places which were initially considered for a replanting program, after this final hydropower regulation, as mentioned by [START_REF] Bernard | Long term changes in Zostera meadows in the Berre lagoon (Provence, Mediterranean Sea)[END_REF]. This replanting attempt was not successful; this is the main reason for which our numerical studies were focused on understanding the possible role of the bottom shear stress, which depends on the bottom granulometry, and salinity, in these four places.

A large part of EB bottom is composed of mud and fine sand carried for several decades from rivers and the sea. The granulometric distribution of sediments in two places in EB have been measured by [START_REF] Rigaud | Dynamique et biodisponibilité des éléments traces métalliques dans les sédiments de l'étang de Berre[END_REF]. In both cores, sediments are constituted of silts (51-81%) and clays (13-28%). Fine sands were also found but in proportions always less than 20%. The fraction larger than 2 mm which corresponds to shell residues exists only in some parts but always less than 15% in mass. The granulometry in the fraction in which the analysis has been performed is constant along the profiles in both cores (silts: about 81% in Grand Etang and about 85% in Etang de Vaïne).

Salinity and temperature measurements are reported by [START_REF] Delpy | Man-induced hydrological changes, metazooplankton communities and invasive species in the Berre Lagoon (Mediterranean Sea, France)[END_REF] from three permanent moorings in the Grand Etang (SA1 in the north, SA2 in the middle and SA3 in the southern part) equipped with five multiparametric probes, CTD SBE37 (Seabird), placed at five depths (see Tab.1 of [START_REF] Delpy | Man-induced hydrological changes, metazooplankton communities and invasive species in the Berre Lagoon (Mediterranean Sea, France)[END_REF]. These experimental results exhibit an almost uniform salinity in the vertical direction (i.e. no haline stratification), but with a periodic seasonal evolution: around 25 PSU in the summer time and less than 15 PSU in the winter time where the largest hydropower runoffs are used. These vertical profiles of salinity and temperature have been used to carefully validate our numerical model in a real situation, for a one-year period, from May 2009 to May 2010 (Alekseenko et al., 2017).

One conclusion of our previous work is that the salinity conditions in EB after the last operational regulation of the hydropower runoff (in 2005) is not a stressor for the SAV replanting and resilience (Alekseenko et al., 2017). So, the present work is mainly devoted to complement the previous study about wind effect on 𝐵𝑆𝑆 (in Section 4) and to study the time evolution of erosion and deposition induced by strong winds (in Section 5).

Model description

The numerical model concerns the 3D currents due to the three dynamical forcings: sea tide and freshwater runoff (with the consequent baroclinic pressure gradient) and strong wind.

The present study is realized with MARS3D developed by IFREMER [START_REF] Lazure | An external-internal mode coupling for a 3D hydrodynamical model for applications at regional scale (MARS)[END_REF][START_REF] Lazure | Development of a hydrodynamic model of the Bay of Biscay. Validation of hydrology[END_REF]. MARS3D contains a hydrodynamical model based on the governing equations proposed by [START_REF] Blumberg | A description of a Three-Dimensional Coastal Ocean Circulation Model[END_REF]. It is based on the system of incompressible Reynolds Averaged Navier-Stokes equations in the classical Boussinesq approximation with the hydrostatic assumption.

The surface elevation, 𝜂, is represented by 𝑧 = 𝜂(𝑥, 𝑦, 𝑡), and the bottom relief, 𝐻, by 𝑧 = -𝐻(𝑥, 𝑦). MARS3D uses a coordinate transformation "sigma" to permit a better resolution near the two boundary layers (at the surface and at the bottom). The water column is divided into layers in the transformed vertical "sigma" coordinate: 𝜎 = (𝑧 -𝜂)/(𝐻 + 𝜂); varying from -1 at the bottom to 0 at the free surface.

EB bathymetry obtained from the marine map NAVICARTE 505 of SHOM "Port-Saint-Louis-du-Rhone; Marseille/étang de Berre" using image recognition algorithm have been considered in the present numerical modelling. The numerical aspects of EB configuration and its hydrodynamic simulation were described in Alekseenko et al. (2013a,b) where the effects of refinement of the sigma-grid on the computed velocity fields were examined in order to choose an optimal grid for appropriate accuracy and acceptable computational time.

Boundary conditions

For the dynamical governing equations, the boundary conditions are the ones used by Alekseenko et al. (2013 a). The horizontal components of surface stress from wind, and of bottom friction are imposed as follows:

• for the surface stress components(𝜏 𝑠,𝑥 , 𝜏 𝑠,𝑦 ) (𝜏 𝑠,𝑥 , 𝜏 𝑠,𝑦 ) = 𝜌 𝑎 𝐶𝑑 𝑠 ‖𝑉 10 ‖(𝑈 10 , 𝑉 10 ),

where 𝑉 10 = (𝑈 10 , 𝑉 10 ) is the wind velocity 10 m above the lagoon surface, 𝜌 𝑎 = 1.25 kg/m 3 is the air density, 𝐶𝑑 𝑠 is the surface drag coefficient.

• for the bottom stress components (𝜏 𝑏,𝑥 , 𝜏 𝑏,𝑦 ) (𝜏 𝑏,𝑥 , 𝜏 𝑏,𝑦 ) = 𝜌 0 𝐶𝑑 𝑏 ‖𝑉‖(𝑈, 𝑉)

(2) with

𝐶𝑑 𝑏 = (𝜅/ ln ( 𝑧 + 𝐻 + 𝑧 0 𝑧 0 )) 2 , (3)
where 𝑉 is the velocity vector with horizontal components (𝑈, 𝑉), 𝐶𝑑 𝑏 is the bottom drag coefficient, 𝜅 = 0.4 is the Von Karman constant, 𝜌 0 is the reference density (𝜌 0 = 1015.27 kg/m 3 ; corresponding to a salinity of 20 PSU and a temperature of 10 𝑜 C), 𝑧 0 = 0.0001 m is the bottom roughness (corresponding to coarse sand). At surface (𝜎 = 0) and at bottom (𝜎 = -1), boundary conditions for temperature and salinity are:

𝜌 0 𝜈 𝐻 ( 𝜕𝑇 𝜕𝜎 , 𝜕𝑆 𝜕𝜎 ) = (0,0), (4)
where 𝜈 𝐻 is the vertical eddy viscosity.

For more details about the boundary conditions, and the meshing strategy the reader can see our last paper (Alekseenko et al., 2017).

Equation of state

As described in [START_REF] Lazure | An external-internal mode coupling for a 3D hydrodynamical model for applications at regional scale (MARS)[END_REF], in the MARS3D model, the dynamic equations are solved together with transport equations for salinity (S) and temperature (T). These transport equations are coupled with the dynamic equation through buoyancy and are solved in a way consistent with the mass conservation solver in order to be mass preserving, not only for water but also for any tracer.

The density 𝜌 is a mathematical function of temperature, salinity, and pressure. MARS3D includes the 1980 Equation of state, abbreviated EOS80.

Sediment transport

Exhaustive review of several works devoted to transport and fate of fine, cohesive sediments, and the exchange of fine sediment between the sea bed and the water column above have been given by [START_REF] Winterwerp | A conceptual framework for shear flow-induced erosion of soft cohesive sediment beds[END_REF] and by Van Rijn (2016). It concerns in particular the shear flow-induced erosion of soft cohesive sediments beds, for which they suggest to distinguish three types of erosion of muddy beds, as follows:

• floc erosion, which is the pick-up of individual particles and small-scale flocs of the top layer (millimeters) of the bed by the turbulent vortices of the fluid flow just above the bed, and which occurs for 0.5𝐵𝑆𝑆 𝑐𝑟 <𝐵𝑆𝑆<1.5𝐵𝑆𝑆 𝑐𝑟 with 𝐵𝑆𝑆 𝑐𝑟 -erosion threshold stress and 𝐵𝑆𝑆 -applied time-averaged bed-shear stress; some erosion may already occur for 𝐵𝑆𝑆<𝐵𝑆𝑆 𝑐𝑟 due to higher turbulent stresses; • surface erosion for 1.5𝐵𝑆𝑆 𝑐𝑟 <𝐵𝑆𝑆<3𝐵𝑆𝑆 𝑐𝑟 , which is the simultaneous mobilization of several layers of particles and flocs.

• mass erosion 𝐵𝑆𝑆>3𝐵𝑆𝑆 𝑐𝑟 , which is the erosion of lumps of bed material when the applied fluid stresses are larger than the undrained soil strength of the bed.

Concerning the modeling of transport sediment in semi-enclosed areas, such as EB, where it is generally necessary to consider both sandy and muddy sediments, we will use the sediment module coupled with MARS3D model (called SEDIM) provided by IFREMER (Le Hir et al, 2011; Cayocca et al, 2014).

MARS3D model manages particle advection in the water column, according to a depthaveraged formulation or to a 3D framework, whereas SEDIM module manages particle settlement in the water column as well as exchanges with the bed by erosion, deposition, and consolidation processes in the bed. In SEDIM module, processes such as deposition on bed sediment and erosion are modelled for each horizontal cell (dx*dy) separately (1D model, see Fig. 5 in Cayocca et al, 2014), with a variable number of layers whose thickness, porosity and concentration vary with time by sedimentation and resuspension events.

The modelling techniques in sediment transport simulations are split into two classes (Le Hir et al., 2011):

• "for non-cohesive sediment (sand and gravel), sediment evolution is computed by solving a continuity equation (Exner equation) where erosion or deposition result from the divergence of a transport capacity related to the hydrodynamic regime; this method is suitable when equilibrium is reached rapidly, which is the case for sediment with high settling velocity; • for fine sand and mud that are mostly transported in suspension, an advection/diffusion equation is solved (either depth-integrated or not) and the sediment evolution is straightforwardly deduced from erosion and deposition rates".

Le [START_REF] Hir | Dynamics of sand and mud mixtures: a multiprocess-based modelling strategy[END_REF] described in detail the sediment transport modelling strategy which offers the possibility to account: simultaneous bed load and suspended sediment transport, mixing of several sediment classes in the water column as well as in the sediment, consolidation of muddy and mixed sediments with possible segregation of sand particles by adapting Gibson theory, management of erosion fluxes depending on the cohesive or non-cohesive nature of the superficial sediment. Special attention was paid to the way the sediment layers content is updated after deposition, in order to simulate possible pore filling up between large particles in superficial sediment before creating a new layer".

According to Le Hir et al. ( 2011), the suspended transport is computed by solving an advection-diffusion equation ( 6) for the concentration of different particle classes which are grouped into three types defined by their behavior:

• "coarse type, generic for non-cohesive particles that are transported as bedload only: it is normally affected to coarse sand, and can be extended to gravels, cobbles and pebbles;

• sand type, generic for non-cohesive particles that are transported in suspension: it refers to fine sand, but also to medium sand, since the possibility to simulate transport capacities through an advection-diffusion process has been demonstrated [START_REF] Waeles | Modelling sand/mud transport and morphodynamics in the Seine river mouth (France) : an attempt using a process-based approach[END_REF]; • mud type, generic for cohesive particles; these are naturally transported in suspension but are also likely to flocculate (inducing a variation of settling velocity) and to undergo consolidation in the bed; the mud type is usually affected to silt and clay but could also be used for organic matter".

𝜕𝑐 𝜕𝑡 = - 𝜕𝑐𝑢 𝑖 𝜕𝑥 𝑖 + 𝜕𝑐𝑊 𝑠 𝜕𝑧 + 𝜕 𝜕𝑧 (𝐾 𝑧 𝜕𝑐 𝜕𝑧 ) + 𝐸 -𝐷 (5)
Where 𝑐 -sediment concentration in water, 𝑊 𝑠 -particle settling velocity, 𝐾 𝑧coefficient of vertical turbulent exchange, 𝐸 -erosion flux and 𝐷 -deposition flux.

The erosion flux for mixed sediment is taken from Le Hir et al. (2011). It is proportional to the erosion rate 𝐸 0 and it involves critical erosion bottom shear stresses: 𝜏 𝑒𝑠 for sand, and 𝜏 𝑒𝑚 for mud (see their equations 1 and 2). It is recalculated from Shields approach technique at each time step from taking into consideration fractions of a mixed sediment (see their Fig. 3).

For the deposition flux the formulation given by Le Hir et al. (2011, see their equation 3) is used. It depends on particle settling velocity 𝑊 𝑠 and on critical deposition bottom shear stress which, following [START_REF] Winterwerp | On the sedimentation rate of cohesive sediment[END_REF], is taken as 𝐵𝑆𝑆 𝑐𝑟 .

A major difficulty for implementing the model is the choice of the state variables, since natural size-spectra are often continuous and the computation cost increases with the number of state variables. The values of all the needed parameters for our numerical experiments (𝑊 𝑠 , 𝐸 0 , 𝐵𝑆𝑆 𝑐𝑟 , etc) are given in Section 5.1.

Description of model scenarios

As in our previous numerical study (Alekseenko et al., 2017), different scenarios are considered. In the first scenario (scen.#1), in absence of wind and starting from an initial homogeneous distribution of salinity (S=20 PSU) and a flow at rest, we obtain a stratification process (after 14 days) under the influence of two dynamical forcings: a seawater influx of high salinity (38 PSU) and a freshwater inflow produced by the hydropower station and two rivers. The aim is mainly to provide a plausible initial salinity stratification to start scen.#2, for which we consider a wind forcing.

Scen.#3 considers sediment transport and takes into account wind forcing starting from an initial homogeneous distribution of salinity (S=20 PSU) and a flow at rest.

For scen.#3, a thin bottom layer of mixed mud-sand sediments is considered with an initial thickness constant for the entire lagoon. The goal is to determine the temporal changes of this thickness as well as the changes in sediment composition induced bythe currents generated by a strong wind. This scenario is discussed in Section 5.1.

Seawater inflow

The Caronte channel by which the Mediterranean seawater penetrates into the lagoon is narrow (100-250 m wide) and long (6.5 km). From our previous studies (Alekseeko et al., 2013a,b), we expect that the tidal contribution of the seawater inflow does not create a substantial current in the nearshore replanting zones. So, the sea tide, semi-diurnal, is still simply represented by a harmonic law for the surface elevation at the open boundary of the Caronte channel:

𝜂(𝑡) = 0.3 cos ( 2𝜋(𝑡 -𝑡 0 )/𝑇), (6)
where 𝑡 0 denotes the initial time of computation, T is the tidal period (about 44714 s or 12h25 min). The tidal amplitude of 0.3 m was chosen corresponding to local tide-gauge measurements in the Golf of Fos at the entrance of the Caronte channel. It corresponds to a maximum amplitude (see REFMAR (2015).

The sea salinity at the entrance of the Caronte channel is of 38 PSU, according to the salinity measurements in Caronte channel (at the surface and at the bottom) shown in the Fig. 14 of the report by SOGREAH (2009).

Freshwater inflow

Hydropower runoff is fixed at its weekly average value, i.e., 80 m 3 /s. The two rivers, Arc and Touloubre, are considered with the mean runoffs of 5 m 3 /s and 3 m 3 /s, respectively.

Wind effect on 𝑩𝑺𝑺 for various wind directions; weighted average effect

The present section is aimed to extend the previous numerical study (Alekseenko et al. 2017) in the case of different wind directions, and a moderate hydropower runoff of 80 m 3 /s.

The weather database (Average daily wind speed at 10 m) provided by SOGREAH (2003) for the ten-years period 1992-2001 will be used (Fig. 2). It gives the mean annual frequency of wind speeds of 21 m/s and 11 m/s, in 16 directions. The dominant winds direction is N-NW, and the wind speed exceeds 11 m/s for over 102 days per year (Alekseenko et al, 2017, see their Table 2).

The numerical simulation has been organized in the way described in Section 3.5, for a wind speed of 21 m/s. First, the calculations are performed for the scen.#1 for 14 days, with the initial salinity S=20 PSU, and a sea salinity at the entrance of Caronte channel of 38 PSU. Concerning the bottom drag coefficient, the bottom roughness parameter 𝑧 0 =0.0001 m has been used in eq.( 3). It would correspond, according to Dufois & Le Hir (2015), to a particle diameter of d=1200 𝜇m (coarse sand).

Then, the time evolution of the 3D hydrodynamics has been calculated for the scen.#2 starting from the stratified field at the end of scen.#1. The same simulation strategy has been realized for each of the 16 wind directions of the wind rose. Concerning the surface drag, in eq.( 1), the value of 𝐶𝑑 𝑠 = 0.0032 given by Young (1999) has been used for the wind speed of 21 m/s.

The presentation of the results is limited to the bottom shear stress (𝐵𝑆𝑆) which is defined by 𝐵𝑆𝑆 = √𝜏 𝑏,𝑥 2 + 𝜏 𝑏,𝑦 2 . The 𝐵𝑆𝑆 maps for six directions, around the two main directions: N-NW and S-SE are given in Fig. 3. These maps are very similar. They all indicate the presence of downwind coastal jets along the eastern and western shores. In such a semienclosed lagoon, the hydrodynamics is characterized by a windward surface current, a downwelling at the opposite shore and return flow occurring over the deeper regions [START_REF] Csanady | Baroclinic boundary currents and long edge-waves in basins with sloping shores[END_REF]. For the wind directions of the N-NW sector, this downwelling mechanism increases 𝐵𝑆𝑆 along the Southern shore. Reversely, for the wind directions of the S-SE sector, the downwelling mechanism increases 𝐵𝑆𝑆 along the Northern shore (Fig. 3).

Then we tried to determine if some additive effect of all these wind contributions could generate a larger effect on 𝐵𝑆𝑆 or not, compared to the N-NW wind. For that purpose, a weighted summing of the 𝐵𝑆𝑆 values of each of the 16 wind directions has been realized by using the frequency of wind occurrence in each direction, given in Fig. 2. The corresponding 2D maps for the entire EB lagoon are given in Figs. 4a and4b. These two figures give the possibility, by comparing 𝐵𝑆𝑆 to 𝐵𝑆𝑆 𝑐𝑟 for different particle diameters, to give a rough estimation of the places in EB lagoon where the risk of erosion is higher. It is not in the scope of the present paper to determine the exact place where the erosion risk is higher. Our Figs. 4a and4b already indicate that the risk exists in several coastal areas. To determine more exactly the places of higher risk, it would be needed to measure the bottom sediment properties in the relevant places and, then, to repeat the numerical experiments with these more accurate properties.

Then, by summing the effect of the different wind directions on the two 𝐵𝑆𝑆 components (𝜏 𝑏,𝑥 , 𝜏 𝑏,𝑦 ), we can get the mean direction of the bottom current (Figs. 4c and4d), and we can expect that it will indicate the preferable direction for the Z.n. propagation (dynamic resilience) year after year.

Transport of bottom sediments: erosion and burial

The sediment transport is often a stressor for SAV restoration or resilience. According to Cabaco & Santos (2007) and [START_REF] Han | Resilience of Zostera noltii to burial or erosion disturbances[END_REF] the critical level of burial or erosion tolerated by Z.n. is extremely low (between 4 and 8 cm) due to the small size of the species and the lack of vertical rhizomes.

In the present work, a thin bottom layer of mixed mud-sand sediments is considered with an initial thickness constant for the entire lagoon. The goal is to determine the temporal changes of this thickness as well as the changes in sediment composition induced by the currents generated by a strong wind.

Model description

In this part, we use the sediment transport model SEDIM described in Section 3.4. The choice of particles has been inspired by the experimental data of Rigaud (2011), mentioned above, concerning mud-sand sediment in two deep places in EB. We consider an initial sediment composition with 25 % of fine sand of 160µm, 25% of very fine sand of 80µm, 25% of coarse silt of 40µm and 25% of medium silt of 20µm (Table 1).

SEDIM model involves a space filling procedure described by Le Hir et al (2011) and clearly illustrated on their Figs. 4a-b. This space filling procedure to obtain mixed mud-sand sediment is applied to an initial sediment column of thickness 120cm. Typical ranges of porosity (void fraction -a percentage between 0 and 100%) have to be respected whatever the combination of particle types. We took a porosity equal to 0.45 for pure sand and of 0.55 for mixed silt/sand (Hopmans & Rolson, 2000, see their Table 2.6). At the beginning of simulation, we use 5 sediment sublayers containing four particle types in equal concentration (100 kg/m 3 ).

After the space filling procedure, the thickness of the mixed sediment column is reduced to 15 cm, and the concentration of each particle type in each sublayer reaches 200 kg/m 3 . The thickness of each sublayer of mixed sediment became equal to 3 cm. SEDIM model involves a sedimentation process, for which we need to provide settling velocity 𝑊 𝑠 of the different suspended sediments and critical shear stress of deposition which, following [START_REF] Winterwerp | On the sedimentation rate of cohesive sediment[END_REF], is taken as 𝐵𝑆𝑆 𝑐𝑟 (Table 1). For the two cohesive substances (silts), the settling velocity is considered to be constant (i.e. we do not take into account a flocculation process) and has been estimated using the Stokes law. For non-cohesive particles (sands), the settling velocity is taken from Soulsby formulation (after Le Hir et al.,2011 or Dufois & Le [START_REF] Dufois | Formulating Fine to Medium Sand Erosion for Suspended Sediment Transport Models[END_REF].

For the deposition process, when all 5 sublayers are filled up, SEDIM model provides 5 additional sublayers to be filled.

The bottom roughness z0 which appears in eq.3 is assumed to only depend on the sediment size. After Dufois & Le [START_REF] Dufois | Formulating Fine to Medium Sand Erosion for Suspended Sediment Transport Models[END_REF], the following relationship is used: z0=2.5d/30 where d is the particle diameter. The diameter of the coarser modeled particle -fine sand (160µm) -has been chosen; leading to z0=13.3µm. Concerning the erosion rate E0 needed to calculate the erosion fluxes, most of the calculations have been performed for E0=0.001 kg/m 2 /s. According to the experimental data provided by Amos et al. ( 2010) for a semi-enclosed lagoon (Venice), this value corresponds to a maximum effect on the erosion flux. The influence of this important parameter is analyzed in Subsection 5.6.

Erosion and redeposition results for a N-NW wind of 21 m/s

Fig. 5 shows the time-evolution, every 6h, of sediment thickness for the N-NW wind of 21 m/s. To better identify the zones of erosion and deposition, we present the thickness relative to the initial level (0 m -at the sediment-water interface). The sediment is totally eroded when the thickness reaches -15 cm. Reversely, when the relative thickness is positive, it corresponds to deposition. The sediment is rapidly eroded on eastern nearshore of EB and in the Etang de Vaine. In these zones, almost all sands and silts have disappeared after the first 24 h. Eroded and deposition zones enlarge with time. In most of the central part of EB, the initial thickness increases in the range of 3-6 cm, but in the Etang de Vaine it increases up to 10 cm. So, according to the criteria given by [START_REF] Cabaço | Effects of burial and erosion on the seagrass Zostera noltii[END_REF], there is a risk of burial in the Etang de Vaine. Deposition zones could be seen also as zones which stock not only the eroded sediments but also Z.n. spores settling for the next reproduction. Fig. 6a shows the time-evolution of sediment thickness at the four nearshore control places (PA, PB, PF and PM; initially selected for Z.n. replanting). For this wind of 21 m/s, the initial sediment layer of 15 cm is totally eroded during the 40 hours. Erosion is faster at PA and PM: the erodible sediment thickness is reduced to almost zero after the first 4 and 10 hours, respectively. At PF, remarkably, sediment thickness slightly re-increases of 1-2 cm, cyclically, at the hours 42, 54 and 66. This can be related to the sea tide influence since this point is closer to the sea, and deeper than the others. Figs. 6a also show the time evolution of the sediment thickness at PBd, PAd and PVd, inside the deposition zones exhibited in Fig. 5. These points are selected near PB, PA and PV, respectively. The maximal sediment accumulation is found at PVd where the increase of sediment thickness became larger than 15 cm. This thickness evolution with time is particularly well correlated with the silt fraction changes shown in Figs. 6b-h. Silt fraction increases in all deposition points and decreases in all erosion points; this could be related to the fact that silt is firstly resuspended in the high energy currents due to its low 𝐵𝑆𝑆 𝑐𝑟 . Sand fraction increases in the nearshore control points while sediment thickness decreases; inducing an increase of sand concentration. Such a behavior of very high concentration vs very small residual sediment is mentioned by Le [START_REF] Hir | Dynamics of sand and mud mixtures: a multiprocess-based modelling strategy[END_REF]; they call it as a fast segregation of sand (see, their Fig. 7c). The nearshore control points selected for the replanting program are vulnerable for erosion and have tendency to contain more sands than silts in their sediment substrates, while the other points, where deposition mainly occurs, should contain more silts.

Erosion risk in EB is simply determined by comparing 𝐵𝑆𝑆 with the critical value, 𝐵𝑆𝑆 𝑐𝑟 , at which the sediment mobility occurs as a function of particle sizes. Due to the lack of available data about cohesiveness of EB sediments, 𝐵𝑆𝑆 𝑐𝑟 has been evaluated according to the work of [START_REF] Berenbrock | Simulation of Flow, Sediment Transport, and Sediment Mobility of the Lower Coeur d[END_REF] which follows the classical Shields approach for cohesionless sediment. For the four sediment types used in our model, 𝐵𝑆𝑆 𝑐𝑟 varies in the range of 0.070 -0.160 N/m 2 . In Figs.7, we construct 𝐵𝑆𝑆 classification maps (in N/m 2 ) for each sediment class based on their corresponding values of 𝐵𝑆𝑆 𝑐𝑟 (Table 1). These maps are useful to analyze areas subjected to different types of erosion (floc, surface and mass erosion) defined in Section 3.4. White areas correspond to the places without erosion (𝐵𝑆𝑆 < 0.5𝐵𝑆𝑆 𝑐𝑟 ). Most wide areas are impacted by floc erosion (0.5𝐵𝑆𝑆 𝑐𝑟 < 𝐵𝑆𝑆 < 1.5𝐵𝑆𝑆 𝑐𝑟 ; resuspention of individual particles of the surficial sediment layer). Surface erosion (1.5𝐵𝑆𝑆 𝑐𝑟 < 𝐵𝑆𝑆< 3𝐵𝑆𝑆 𝑐𝑟 ), appears in the central part for both types of sand, and in the northern part of EB for the very fine sand (Figs. 7a,b). For silts (Figs. 7c,d), these two areas are connecting and creating a pattern all along the central part of EB -from North to South. Surface erosion area is the largest for the smallest sediment class: medium silt (Fig. 7d).

Mass erosion (𝐵𝑆𝑆 > 3𝐵𝑆𝑆 𝑐𝑟 ) occurs for all sediment classes along most of the coastal zones. In addition, it occurs in the central part of EB, except for fine sand. Mass erosion areas enlarge when the grain size decreases. Mass erosion mostly impacts the smallest particles.

Erosion risk for different winds

We first analyze how 𝐵𝑆𝑆 maps depend on the wind speeds [START_REF] Hopmans | SSC 107 -INTRODUCTORY SOIL PHYSICS[END_REF][START_REF] Blumberg | A description of a Three-Dimensional Coastal Ocean Circulation Model[END_REF], and 6 m/s) of the N-NW wind after 3 days (Figs. 8). The goal is to evaluate erosion risk; i.e., the areas where 𝐵𝑆𝑆 exceeds 𝐵𝑆𝑆 𝑐𝑟 of each of the four particle types (Table 1). As it was expected, the stronger wind induces higher 𝐵𝑆𝑆 in the entire EB; highest 𝐵𝑆𝑆 is observed along the eastern shore. Along the eastern coastal zone, sediment bed could be strongly eroded, since their 𝐵𝑆𝑆 (yellow and orange colors) largely exceed 𝐵𝑆𝑆 𝑐𝑟 (Figs.8a).

Fig. 8c shows that even for N-NW wind of 6 m/s, 𝐵𝑆𝑆 reaches 0.1 N/m 2 (green colors) which is larger than 𝐵𝑆𝑆 𝑐𝑟 for silt in different places. It occurs in particular in mouth of Arc River, Etang de Vaine, near the entrance of the Caronte channel and inside this channel. So, sediment mobility and resuspension of fine sediments (silts) into water column could occur in these places (which are indicated in Fig. 1).

For moderate N-NW wind of 11 m/s, silt erosion (green colors) could occur in the major part of EB coastal zone (Fig. 8b). Some areas represented by magenta and blue colors would be not subject to erosion (their 𝐵𝑆𝑆 being lower than 𝐵𝑆𝑆 𝑐𝑟 ); so, these areas could be favorable for Z.n. resilience. For a stronger N-NW wind of 21 m/s, these areas reduce their surface, but still exist (Fig. 8a).

It is also interesting to analyze how the erosion risk would depend on wind direction. So, two wind speeds (21 m/s and 11 m/s) in S-SE direction have been selected, for the same initial sediment thickness of 15 cm. 𝐵𝑆𝑆 maps (Figs. 9a-b) reveal a risk of erosion over the eastern and western shores, and in some zones of the northern part of EB, marked by red colors. In addition, thickness maps (Figs. 9c-d) show that the highest erosion is obtained in areas where 𝐵𝑆𝑆 exceeds 1 N/m 2 for the wind speed of 21 m/s, and 0.1 N/m 2 for the wind speed of 11 m/s. This erosion risk along eastern shore in EB is more important than for wind of N-NW direction at the same wind speed of 21 m/s. This tendency is confirmed in Figs.10 by comparing the time-evolution of the sediment thickness. Erosion at the two eastern sites (PA and PB) is much larger for the S-SE wind; 15 cm are eroded at PA after 3 h and at PB after 8 h. For the two western sites (PM and PF), the behavior is different: the initial sediment thickness of 15 cm is totally eroded at PM after 28 h, while it increases at PF up to 30 cm due to redeposition process.

Concerning the three deposition points selected for N-NW wind, the initial sediment thickness of 15 cm is eroded at PAd during the first 12 h, while in PVd and PBd the sediment thickness increases (Fig. 10d). This means that at least for these two points, N-NW and S-SE winds contribute to increase deposition. On the contrary, a different behavior between S-SE and N-NW winds is found in PF (Fig. 10a) and in PAd (Fig. 10b). But this is without consequence for our analysis for erosion and burial risks.

For wind speed of 11 m/s, the time evolution of the sediment thickness is compared at the four control places for the two opposite directions N-NW (Fig. 11a) and S-SE (Fig. 11b). At PA, erosion is very fast for both wind directions. It is faster for S-SE wind (less frequent). At PB, the erosion appears also to be faster for S-SE wind. At PF, like in the case of larger wind speed, there is a deposition for S-SE wind, instead of erosion for N-NW wind. But, the way in which the comparison is made (only at four points) presents some drawback.

A global comparison between Figs.8a and 9a for the wind of 21 m/s, and Figs.8b and 9b for the wind of 11 m/s, show that both wind directions give an additive contribution on 𝐵𝑆𝑆 and would lead to the same conclusion about additive effect on erosion at least along the eastern and western shores.

It is interesting to point out that Paquier et al. ( 2014) presented seasonal bathymetric and beach topographic in a small shallow bay very close to PB in EB. The beach is subject, alternatively, to sedimentation and erosion, with changes attaining up to about 0.5m over a period of several months. Such large changes can be due not only to the bottom current generated by the strong winds during such a long period, but also by the wave dynamics which are not yet taken into account in our model.

Sensibility tests

As the exact characteristics of bottom sediments, such as grain-size proportion and erosion rate, are not yet available, and anyway are not constant in all EB, additional numerical experiments are performed to evaluate the impact of such uncertainty on the prediction of sediment thickness. Two tests have been performed for N-NW wind of 21 m/s. Two additional compositions have been tested (see Table 2). One with 90% of mud and 10% sand, called DF (deep fraction) which imitates the sedimentary layer in deep part of EB, and another one, called CF (coastal fraction), which is supposed to better represent coastal sediments with 50% of mud and 50% of sand.

Figs.12a show the time evolution of sediment thickness at PB and PA identified as erosion sites. At PA, the erosion is rapid: 4 hours for DF composition and 8 hours for CF composition. This is related to the fact that DF composition contains more mud which is rapidly suspended due to its lower 𝐵𝑆𝑆 𝑐𝑟 . At PB, the initial sedimentary layer is totally eroded for DF composition during the first 16h; while for CF composition the full erosion requires 72 hours.

Figs.12b show the time evolution of sediment thickness at PBd and PAd identified as redeposition sites. The redeposition is higher for DF composition which contains more silt. This is expected since the erosion was larger in this case.

Solid red curves representing CF composition in Figs. 12a-b for erosion points PB, PA could be compared to the curves plotted in Fig. 6a for the corresponding points, where the erosion was more rapid. This faster erosion can be explained by the fact that CF composition corresponds to a larger amount of sand of larger size (50% of sand of size of 200µm) than in the case of Fig. 6a (50% of sands of sizes of 80µm and 160µm, in equal proportion).

For the CF composition, we check the effect of erosion rate 𝐸 0 (second sensibility test), in the range of 10 -4 to 10 -3 kg/m 2 /s inspired by the work of [START_REF] Amos | The erosion rates of cohesive sediments in Venice lagoon[END_REF] for Venetian lagoons (Figs. [START_REF] Cabaço | Effects of burial and erosion on the seagrass Zostera noltii[END_REF].

The time evolution of the sediment thickness vs 𝐸 0 is presented at the four nearshore control points subjected to erosion in EB (Figs. 13a-d). As it was expected, erosion is less pronounced for lower 𝐸 0 . For 𝐸 0 = 10 -3 kg/m 2 /s: the initial sedimentary layer of 15 cm was totally eroded after 72 hours in all the control points (PA, PB, PF and PM).

For 𝐸 0 ten times lower, after 72 hours, the initial sediment layer is totally eroded only at PA. At the other erosion points, the erosion process is strongly decelerated: the decrease of sediment thickness is only of 1.5 cm at PB, and of 3 cm at PF.

The time evolution of the sediment thickness vs 𝐸 0 is also presented for the three deposition points PVd, PBd and PAd (Figs. 13e-g). Sediment thickness is increasing only in the range of 0.5 -1.5 cm during 72 hours (Figs. 13e-g). It is to remark that for the conditions discussed in this subsection, contrary to those used for Fig. 6a, the increases of bottom sediment thickness never reaches the critical level of deposition.

Discussion and conclusion

In Section 4, the 𝐵𝑆𝑆 maps have been determined for the 16 directions (given from a wind rose) of a wind speed of 21 m/s, in order to analyze the potential additional effect of these winds on the SAV roots, a mean value of |𝐵𝑆𝑆|. A weighted average based on the frequency of each wind directions, has been determined and compared to 𝐵𝑆𝑆 𝑐𝑟 corresponding to a class of roughness parameters. In addition, an averaging of the two components of 𝐵𝑆𝑆 has been evaluated, similarly, in order to obtain the direction of an average bottom current which could be compared with the preferential direction of propagation of resilient Z.n. observed year after year [START_REF] Bazile | 1ère campagne de Survey des zostères de l'étang de Berre par drone (9-10 juillet[END_REF].

In Section 5, we have analyzed how the thickness of a given layer of mud-sand bottom sediments is changed (resuspension and/or redeposition) due to wind effect in EB, for the two main wind directions (N-NW and S-SE) and for three wind speeds. Although bottom sediments in EB contain a large fraction of mud, their cohesiveness is not considered in the present work due to a lack of specific experimental data. The present modelling of sedimentary dynamics take mainly into account theoretical thresholds for sand and for mud, and surficial erosion flux which is proportional to an erosion rate. For this erosion rate, we used values in the range 10 -4 to 10 -3 kg/m 2 /s determined experimentally for the Venetian ones which have similar sediment composition.

The time evolution of the initial sedimentary layer, for a period of 3 days, permits to determine the area where the erosion would be large enough in EB to be a stressor for the SAV restoration (resilience or replanting). The main results are presented as 𝐵𝑆𝑆 maps for the entire EB. Then, time evolution of sediment thickness is presented at 4 places subjected mainly to erosion (PA and PB, on the eastern shore) and (PM and PF, on the western shore). It is also presented in three places subjected mainly to deposition (PAd and PBd, near PA and PB in EB) and PVd (in Etang de Vaïne).

Erosion is maximum in the areas where 𝐵𝑆𝑆 is maximum (e.g., at PA and PB) due to the coastal jets. It is larger for the S-SE wind of the same speed (but less frequent). Deposition is found in the nearshore areas where 𝐵𝑆𝑆 is close to zero due to coastal jets. The time evolution of the sedimentary thickness given in Fig. 5 permits to determine when and where the risk to overpass the critical levels of erosion and burial tolerated by Z.n. has to be expected. For an extreme erosion rate of 10 -3 kg/m 2 /s and a strong wind speed of 21 m/s, all the initial sediment layer of 15cm would be eroded in several nearshore areas. The erosion risk is maximum along the eastern shore, where an additive effect of the two opposite winds can be expected. Indeed, S-SE and N-NW winds create coastal jets of opposite direction and of about the same intensity (for the same speed) exerting alongshore erosion and deposition at about the same places.

𝐵𝑆𝑆 classification maps have been presented in Fig. 7 for the 3 types of erosion (floc, surface and mass) described by several authors. In Figs. [START_REF] Cabaço | Effects of burial and erosion on the seagrass Zostera noltii[END_REF], the temporal evolution of the sediment thickness for a lower erosion rate of 10 -4 kg/m 2 /s shows that, even in this case, the risk of an erosion overpassing the limit for Z.n. survival (between 4 and 8 cm; according to Cabaco & Santos 2007) would have to be expected in some places.

Of course, the accuracy of the present model to determine erosion and deposition is limited by the lack of adequacy of the input data that would be needed to calibrate it. In particular, the expression of 𝐵𝑆𝑆 𝑐𝑟 is limited to a cohesiveless sediment. The present results concerning the time evolution of erosion depth induced by the wind improve our knowledge of the erosion risk on the Z.n. restoration in EB. They will be useful to appropriately orient the action of politics and managers for a future restoration program.

Our results give insight on what future work should focus to improve our results. It will be not only needed to get reliable in situ data (at least in the possible restoration areas), but we plan also to consider in the present model the suspended matter (silts and clays involved in the hydropower inflow). In addition, we plan to take into account the effect of wave dynamics through the boundary conditions ( 1) and (2). -d) may be compared with those shown in Fig. 6a for E0 = 10 -3 kg/m2/s and the initial composition (25% fine sand, 25% very fine sand, 25% coarse silt, 25% medium silt).

Fig. 1 .

 1 Fig. 1. Etang de Berre and its tributaries (hydropower channel, Touloubre and Arc Rivers), and bathymetry. Black points correspond to three mooring stations, red stars -to four nearshore control points, black diamonds -to three sediment deposit points.

Fig. 2 .

 2 Fig. 2. Distribution of the average wind over 10 min, at the station of Port de Bouc, 10 m of height, for the period 1992-2001, 24,793 observations (Sogreah, 2003).

Fig. 3 .

 3 Fig. 3. Maps of BSS modulus for three wind directions in the N-NW and S-SE quadrants; for a wind speed of 21 m/s.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. BSS and vector field of the bottom stress. (a) BSS for N-NW of 21 m/s, (b) BSS for combined winds of 21 m/s, (c) bottom stress vector field for N-NW of 21 m/s, (d) bottom stress vector field for combined winds of 21 m/s. Contours on the vector field correspond to BSS = 0.5 N/m 2 .

Fig. 7 .

 7 Fig. 7. BSS classification maps (in N/m 2 ) of three types of erosion (see definition in Section 3.4): (a) fine sand of 160 µm, (b) very fine sand of 80 µm, (c) coarse silt of 40 µm, and (d) medium silt of 20 µm. Each sediment type has its own BSScr presented in the Table 1. Results were obtained for a N-NW wind of 21 m/s, after 3 days.

Fig. 8 .

 8 Fig. 8. BSS maps for three speeds of N-NW wind after 3 days; (a) 21 m/s, (b) 11 m/s, and (c) 6 m/s.

Fig. 9 .

 9 Fig. 9. BSS after 3 days for (a) S-SE wind of 21 m/s and (b) S-SE wind of 11 m/s; sediment thickness after 3 days for (c) S-SE wind of 21 m/s and (d) S-SE wind of 11 m/s.

Fig. 10 .Fig. 11 .Fig. 12 .

 101112 Fig. 10. Time-evolution of sediment thickness for S-SE wind of 21 m/s; (a) erosion at PB, PA, PM and PF (black horizontal line corresponds to the erosion limit of 8cm acceptable for Z.n. restoration); (b) redeposition at PVd,PBd and PAd.

Fig. 13 .

 13 Fig. 13. Sensibility test on the erosion rate E0 for N-NW wind of 21 m/s, for CF (coastal fraction case: 50% mud -50% sand); (a-d) at PB, PA, PM and PF (black horizontal line corresponds to the erosion limit of 8cm acceptable for Z.n. restoration); (e-g) at PVd, PBd and PAd. Four E0 values have been tested: 10 -4 ,2*10 -4 ,4*10 -4 ,10 -3 kg/m 2 /s. Results corresponding to E0= 10 -3 kg/m 2 /s (red curves in Figs.13a-d) may be compared with those shown in Fig.6afor E0 = 10 -3 kg/m2/s and the initial composition (25% fine sand, 25% very fine sand, 25% coarse silt, 25% medium silt).

Table 1 .

 1 Sediment composition and characteristics.

	Particle diameter (µm) BSScr (N/m 2 ) Particle density (kg/m 3 ) Initial fraction in sediment (%) Settling velocity Ws (m/s) E0 (kg/m 2 /s)	Medium silt Coarse silt Very fine sand 20 40 80 0.070 0.088 0.120 2650 2650 2650 25 25 25 constant estimated from Stokes law: 0.000928 constant estimated from Stokes law: Soulsby formulation (1997): 0.017063 0.000232 10 -3 10 -3 10 -3	Fine sand 160 0.160 2650 25 Soulsby formulation (1997): 0.004741 10 -3

Table 2 .

 2 Parameters of the sensibility tests.

	DF CF BSScr (N/m 2 )	1µm clay fraction (%) 20 10 0.005	20 µm medium silt fraction (%) 30 15 0.046	50µm coarse silt fraction (%) 40 25 0.1	200µm fine sand fraction (%) 10 50 0.175

Acknowledgements

We acknowledge financial supports by the French Water Agency (Agence de l'Eau-RMC -convention N 2010-0042, and the French Ministry of Foreign Affairs (ARCUS-Russia program N 2005_1513/DCSU). This study was done in the framework of the stateassignment of FASO Russia (theme № 0149-2018-0009). Part of this research is also a contribution to the grant of Russian Foundation for Basic Research (RFBR) N16-35-00526. We would like to thank our colleagues Christian Kharif, Richard Kotarba and also Alexander Sukhinov for their advises and discussions devoted to this work. We specially thank Pascal Bazile, diver of the non-profit Association l'Etang Nouveau, for his remarkable observation of zostera resilience in many places in EB.