Inﬂuence of regular surface waves on the propagation of gravity 1 currents: experimental and numerical modeling

The propagation of gravity currents is analyzed in the presence of regular surface waves, 5 both experimentally and numerically, by using a full-depth lock-exchange conﬁguration. Full- 6 depth lock-exchange releases have been reproduced in a wave ﬂume, both in the absence 7 and in the presence of regular waves, considering two ﬂuids having densities ρ 0 and ρ 1 , 8 with ρ 0 < ρ 1 . Boussinesq gravity currents have been considered here ( ρ 0 /ρ 1 ∼ 1), with 9 values of the reduced gravity g (cid:48) in the range 0 . 01 ÷ 0 . 1 m / s 2 , while monochromatic waves 10 have been generated in intermediate water depth. The experimental results show that the 11 hydrodynamics of the density current is signiﬁcantly aﬀected by the presence of the wave 12 motion. In particular, the front shows a pulsating behavior, the shape of the front itself is 13 less steep than in the absence of waves, while turbulence at the interface between the two 14 ﬂuids is damped out. In the present test conditions, the average velocity of the advancing 15 front may be decreased in the presence of the combined ﬂow, as a function of the relative 16 importance of buoyancy compared to wave-induced Stokes-drift. Moreover, a new numerical 17 model is proposed, aiming at obtaining a simple, eﬃcient and accurate tool to simulate the 18 combined motion of gravity currents and surface waves. The model is derived by assuming 19 that surface waves are not aﬀected by gravity current propagation at leading order and 20 that the total velocity ﬁeld is the sum of velocities forced by the orbital motion and those forced by buoyancy. A Boussinesq-type wave model for nonstratiﬁed ﬂuids is solved, and 22 its results are used as input of a gravity current model for stratiﬁed ﬂows. Comparisons of 23 the numerical results with the present experimental data demonstrate the capability of the 24 model to predict the main features of the analyzed phenomena concerning propagation of 25 the density current (averaged velocities, front height, etc.), the increase of entrainment of 26 the ambient ﬂuid into the density current in the presence of the waves and the intra-wave 27 pulsating movement of the heavy front. 28


INTRODUCTION
gravity currents over both smooth and rough bottoms (La Rocca et al. 2008). Furthermore, 48 Theiler and Franca (2016) analyzed the influence of the released volume in full-depth lockexchange experiments, obtaining that density currents with high volume of release conserved In particular: (i) a laboratory study has been carried out in a wave flume to investigate the initial buoyancy velocity u b = √ g h, as well as by affecting the turbulence of the flow.

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Concerning the numerical model, it should be mentioned that in the past gravity currents dissipation, but they are computationally very expensive.

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In the above framework, trying to fill the gap between the two mentioned approaches, a 128 new computationally efficient two-dimensional numerical model is proposed for investigating 129 the combined wave-gravity current flow, including the non-homogenous density distribution a porous beach allows to minimize wave reflection.
ρ 0 < ρ 1 , at the onshore side (see Figure 1). This situation is an idealization of real coastal 161 environments where surface waves are generated offshore in a denser fluid and propagate 162 toward regions where lighter waters are present (e.g. estuaries or industrial discharges).

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To obtain different densities, tap water is mixed with sodium chloride (NaCl). Moreover, 164 a diluted green dust organic food dye, combination of E102, E131, E514, is used to highlight 165 the gravity current. The volume of salted water in the lock is about 0.5m 3 . The concentration 166 of the diluted dye is about 0.004%, thus differential diffusive effects between the brine and 167 the dye are negligible. 168 Before starting each experiment, samples of the colored salt water and of the fresh water 169 were gathered to measure the actual densities, ρ 1 and ρ 0 of the two fluids respectively. The was measured by using a high precision scale (0.0001 g accuracy). By following the above 175 procedure, the error in measuring the density is estimated to be smaller than 1 g/m 3 .

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Full-depth two-dimensional lock-exchange experiments have been carried out without and 177 with superimposed surface regular waves.

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In the first case, starting from an hydrostatic condition, the sluice gate is removed and In the second case, again starting from hydrostatic conditions, the wavemaker generates 183 a train of monochromatic waves which propagates in the onshore direction. As soon as the first wave crest is about to hit the gate, i.e. about 5.7 s after the wavemaker starts according 185 to the present wave characteristics, the gate itself is manually removed. The removal takes 186 about 0.3 s, while the wave period is about 1 s. It should be noted that the wave crest is the 187 point which can be determined with the highest precision along the wave profile O(4cm).

188
The above experimental procedure guarantees that the removal of the gate is performed in 189 such a way that wave reflection from the gate does not affect the wave train which interacts 190 with the gravity current.

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Thanks to the small scale apparatus, wave generation within the flume is highly repeat- where measurements of the front are gathered, has been carefully checked, as well as the 209 minimization of lens distortion. A 10 cm by 10 cm grid on the glass wall of the flume helped 210 the metric calibration of the images.

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A simple automatic procedure was implemented to analyze the recorded images, to re-212 cover the shape front of the gravity current and to calculate the front velocity and front 213 height.

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Preliminarily, it has been checked that the camera had a linear transfer function between 215 light absorbance and concentration of the colored salt water, as suggested by Kolar et al.

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For each experiment the recorded high resolution video sampled at 50 frames per second 218 are treated to grab single snapshots as color RGB images, which are thus converted into 219 gray scale images. Then, the measuring region between the sluice gate and a section about 220 40 cm downstream of the sluice gate itself is isolated into the images (see Fig. 2a).

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The initial frame is considered as a reference frame. The intensity I i 0 of i-th pixel of the 222 above grayscale reference frame is subtracted from the intensity of the corresponding pixel 223 of each subsequent n-th frame I i n in order to obtain an enhanced image with intensities (1) 225 with n = 1, 2, . . . , N , and N the total number of analyzed frames.

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By following such a procedure, the resulting image (see Fig. 2b) highlights the dynamics 227 of the front, since all the pixels of the enhanced image share the same reference level of light 228 intensity and background disturbances are automatically removed.

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To facilitate the measure of the front characteristics, the grayscale images were converted 230 into binary black and white images (see Fig. 2c), by using a threshold on the pixel intensity, 231 whose value has been determined through a sensitivity analysis. The front location was 232 determined simply by counting the number of black pixels at the bottom along the horizontal 233 direction, while the shape of the front is recovered as the interface between the black and 234 white regions. By following the above procedure, it is estimated that the errors on the 235 location of the front of the gravity current is smaller than 0.5 mm. Figure 2 shows an example of the outcome of the adopted image processing. The second column reports the water depth h, the third column shows the initial aspect ratio 242 R, defined as the ratio between the initial depth of the current h and the initial length of the give the densities of the light and of the heavy fluids, ρ 0 and ρ 1 respectively. The sixth 252 column indicates the dimensionless density γ = ρ 0 /ρ 1 , which is always close to 1, since only 253 Boussinesq currents have been considered here. The seventh column presents the reduced  of the front position before and after t 0s , by reporting also the value of t 0s and the linear 281 function used to estimate the average front velocity U . For some tests the duration of initial 282 transient may be significant, particularly for the tests where the reduced gravity is very small 283 (e.g. S001-S006), since inertia prevails on buoyancy effects. If the reduced gravity is larger, 284 as in tests S007-S009, the initial transient is much smaller.

285
Analogously, Figure 4 shows the results obtained for the tests W001-W009, with super-286 imposed regular surface waves. In this case, the front oscillates while propagating onshore,

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with the same frequency of the waves. Moreover, by following the procedure described above, 288 the estimate of U is not affected by the oscillation of the front position due to the orbital 289 motion, and in turn by the actual phase of the waves. Finally, it may be noticed that when 290 waves are superimposed to the gravity current, in general, t 0s is smaller than in the absence 291 of waves, since inertial effects are overridden by the orbital motion.
292 Table 3 summarizes the measured average front velocity U and the kinematic and dynamic  The small values of the wave steepness ka confirms that linear waves have been generated.

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Wave reflection in the flume has been measured by means of the two-gauge method proposed 314 by Goda and Suzuki (1976). In all the tests the reflection coefficient is smaller than 15%, 315 therefore almost purely progressive waves have been obtained, similar to the ones in the 316 nearshore region, in the presence of gravel or sandy beaches.
In order to preliminarily validate the tests carried out in quiescient ambient fluid, Figure 5 318 illustrate on a log-log scale the evolution of the dimensionless front position for the tests S001-

319
S009 . The comparison with the reference slope equal to one (see for example Marino et al. with the Froude number being equal to F = 0.539.

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The interface between the lighter and the denser fluid is unstable. Quasi two-dimensional h F ∼ 0.8m, and also the front steepness is reduced to about 50 • . Moreover, the relative rate 345 of advancing of the front F = U/u b in the onshore direction is sligltly slower compared to 346 the classical lock-exchange case, being F = 0.507.

347
The presence of the orbital wave motion interacts with the formation and evolution of 372 F = U/u b = 0.5 is also shown in the Figure. As expected, data from tests S001-S009 tends to 373 collapse on such a line. In general, the interaction with the wave motion induces a decrease 374 of the relative front velocity, i.e. of the Froude number F . In particular such a decrease is 375 larger when the buoyancy velocity is smaller, while in the case of u b > 0.1 m/s, i.e. for the 376 g larger than 1.00 m/s 2 , there is no significant difference between the two cases.

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It must be considered that in the case of the combined gravity current-wave motion, 378 the average speed of the gravity current should be influenced by the integral wave mass 379 transport. In general, such a mass transport is composed by the offshore directed Stokes where u and w are the horizontal and vertical components of the total velocity u ≡ (u, v); p 423 is the total pressure; τ xz and τ xz are the viscous and turbulent stresses respectively; ρ is the 424 local density; g is the gravity acceleration; κ is the diffusion coefficient of the species that constitutes the density variation, which is equal to zero in the case of immiscible fluids.

426
Modelling the surface wave motion in a lock-exchange release is a complex task, which, 427 thanks to the adopted Boussinesq approximation i.e. γ = ρ 0 /ρ 1 ∼ 1, is tackled here by 428 decoupling the homogeneous density wave motion and the gravity current propagation forced 429 by actual density gradients.

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The total velocity field u is obtained by linearly adding up the velocity field due to the 431 wave motion u B and the one due to the gravity current u d : The gravity current velocity field u d (x, z, t) ≡ (u d , w d ) is established due to the variable 438 density field ρ(x, z, t), which is comprised in the range ρ o ÷ ρ 1 , and it is evaluated as a 439 function of the pressure field induced both by buoyancy and by the waves.  Coupled with the gravity current model which will be described in the following Sec- where u p is the depth averaged potential velocity, which coincides with the depth-averaged 473 total orbital velocity u in the absence of breaking waves, since in this case the rotational 474 velocity u r is null. ∆ 1 and ∆ 2 are coefficients given by the following expressions: Once the horizontal component of orbital velocity u B is obtained over the entire domain, 477 the vertical component w B can be derived numerically on the basis of the continuity equation, 478 as follows: The pressure term related to density variation can be calculated by integrating: in which ∆ρ = ρ − ρ 0 is the local density variation with respect to the reference value ρ 0 , and 500 ν and ν t represent the kinematic and the eddy viscosities respectively. The eddy viscosity is 501 estimated on the basis of the formulation proposed by Smagorinsky (1964) for sub-grid scale 502 turbulence, as a function of the local derivatives of the velocity field and the local grid size: where C is a constant which has been considered equal to 0.004; ∆x and ∆z are the dimension The pressure p d is assumed equal to zero on the free surface ζ. Once p d is estimated 509 through integration of eq. (11), it can be inserted in the horizontal momentum: where a no slip boundary condition is used at the bottom. Since the wave motion has been  2) and (6): Moreover, in the adopted formulation the free surface elevation ζ should be the sum of 520 the two contributions ζ B and ζ d , due respectively to waves and buoyancy, i.e. ζ = ζ B + ζ d .

521
Here it is assumed that at leading order the wave motion dominates and the contribution due  11) and (14), 536 respectively.

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It is worth to specify that, in the absence of waves, the gravity current formulation  Figure 11 shows the comparison between the evolution of the gravity current in the 562 laboratory flume and that simulated by the proposed numerical model both for the tests 563 S007 and W007. The first test is a classical full depth lock-exchange, the second one is the 564 same test carried out in the presence of a superimposed regular surface wave field. In both 565 cases, the reduced gravity current is g = 0.047 m/s 2 .

566
The laboratory images allow to highlight the presence of an entrainment layer at the 567 interfaces of the two fluids. Therefore the capabilies of the numerical model can be tested 568 to describe the density variability both in time and space.

569
In particular, the dimensionless density differences ∆ρ * calculated by the model are 570 defined as with ρ being the actual value of the computed density. In Figure 12 the isolines of ∆ρ * are 573 superimposed to the snapshots of the front. Such a representation permits to compare the 574 measured and calculated propagation of the positive and negative fronts and the evolution 575 of their shape, also by considering the variability of density across such fronts.

576
A fairly good agreement between experimental and modeled heavy front is obtained at 577 the front, since the numerical model is able to catch the overall dynamics of the gravity 578 current propagation, both in the absence and in the presence of the waves.

579
The position in time of the two fronts is similar (see Figure 11). Both in the lab and in  The main discrepancies can be observed during the initial stages, after the removal of 593 the gate, whereas the simulated propagation of the front is quite similar to that observed 594 in the lab. One of the possible reasons for such discrepancies is the different mechanisms 595 of gate opening, which is manual in the lab and instantaneous in the model. Moreover, probably because of the adopted scaling and of the chosen simple turbulence closure, the 597 proposed model is more suited to reproduce the horizontal propagation of the front, rather 598 than the initially vertical dam-break dynamics. A simple falling body approach is adopted to 599 estimate the initial dam-break duration, at which the numerical model may not be accurate.  In the presence of surface waves superimposed to the gravity current, the comparison 633 between the experimental data and the numerical results is again generally fairly good, as 634 shown in Figure 13, and the model is generally able to catch the reduction of averaged 635 velocity of the front due to the orbital motion.

636
Only for test W004, which corresponds to the lowest value of reduced gravity g there is a 637 mismatch. The physical meaning of such a different behaviour is that the gravity current is 638 more influenced by external forces when the buoyancy is low. In particular, the presence of 639 orbital velocity induced by surface waves significantly reduces the gravity current velocity for 640 small g . Such an effect is not caught by the proposed decoupled numerical model, which may 641 be not suitable to treat cases with small initial density differences, as shown also in Figure 12. An important consequence of the presence of surface wave is that the instantaneous veloc-650 ity of the front may be significantly different from the average velocity U . Its maximum value 651 is to 4 times larger, in the investigated conditions. This strongly influences the transport 652 processess of materials triggered by gravity currents in nearshore regions.

653
Moreover, in both numerical and experimental data, shown in Figure 13, the front posi-654 tion oscillates with a period which matches that of surface waves. This occurs not only at 655 the forefront (∆ρ * = 1/8), but also in the region immediately upstream (∆ρ * = 1/2; 1/4),

656
indicating that the thickness of the entrainment layer periodically varies due to the waves.

657
More in details, the amplitude of the wave-generated front oscillations is larger for smaller 658 values of ∆ρ * .

659
In order to further analyze the above wave-induced gravity current front oscillations, 660 Figure 14 reports the measured and calculated normalized spectral components of the front 661 positions X/X peak , obtained by removing the trend related to the current propagation.

662
The comparison confirms that the model is able to predict the overall dynamics of the 663 combined wave-gravity current flow. Indeed, as expected, since the flow oscillations are due 664 to waves, the highest peak occurs at the surface wave frequency, f w = 1/T w both in the 665 model and in the experimental data. Small differences may be attributed to the different 666 spectral frequency discretization used when analyzing the two datasets. Furthermore, the 667 model is able to catch the secondary peak which appears at frequency lower than f w which 668 may be be due to a long wave induced by the initial opening of the lock gate. Moreover, the 669 width of the spectrum becomes larger as g increases. This is probably due to the exchange 670 of momentum at higher frequency induced by the larger gravity current velocities and to a 671 more unstable shear layer between the heavy and the light fluid.