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Under consideration for publication inJ. Fluid Mech. 1 Stokes drift for inertial particles transported by water waves 2 1 1 2 0 G. BOFFETTA , M. MARTINS AFONSO , 2 A. MAZZINO3 M. ONORATO1, AND F. SANTAMARIA1 ct 1Dipartimento diFisica and INFN,Universitàdi Torino, O via Pietro Giuria 1, 10125 Torino, Italy 2Institutde Mathématiques et deModélisation de Montpellier, CNRS UMR 5149, 8 UniversitéMontpellier 2, c.c.051, 34095 Montpellier cedex 5, France 3Dipartimento diIngegneria Civile, Chimica e Ambientale, Universitàdi Genova, ] n via Montallegro 1, 16145 Genova, Italy y d (Received 9 October2012) - u We study the effect of surface gravity waves on the motion of inertial particles in an l f incompressible fluid. Using the multiple-scale technique, we perform an analytical cal- . s culation which allows us to predict the dynamics of such particles; results are shown c i for both the infinite- and finite-depth regimes. Numerical simulations based on the ve- s locity field resulting from the second-order Stokes theory for the surface elevation have y h been performed, and an excellent agreement with the analytical predictions is observed. p Suchanagreementseemsto holdevenbeyondthe formalapplicability ofthe theory.We [ find that the presence of inertia leads to a non-negligible correction to the well-known 1 horizontal Stokes drift; moreover, we find that the vertical velocity is also affected by a v drift.Thelatterresultmayhavesomerelevantconsequencesontherateofsedimentation 6 of particles of finite size. We underline that such a drift would also be observed in the 6 (hypothetical) absence of the gravitationalforce. 1 2 . 0 1. Introduction 1 2 The study of the Stokes drift is a problem of paramountimportance both from a fun- 1 damentalpointofview(Van den Broeck1999)andinconnectionwithapplications,espe- : v ciallyintheareaofsedimenttransport(Longuet-Higgins1953;Nielsen1992;Vittori & Blondeaux i 1996; Blondeaux, Brocchini & Vittori 2002; Blondeaux et al. 2012). As far as the first X point is concerned, the Stokes drift is for instance responsible of important fluid-mixing r a mechanisms suchasmass andmomentumtransportnearthe free-surface,aswellas vertical mixing enhancement owing to turbulent kinetic-energy production (Kantha et al. 2009).Intheocean,theStokesdriftisthoughttobeoneimportantingredientresponsible fortheLangmuircirculation(McWilliams, Sullivan & Moeng1997).Inrelationtoappli- cations, it is known that an accurate evaluation of the Stokes drift is important for the correct representation of surface physics in ocean general circulation models and ocean models at smaller scales. Other relevant effects on the ocean circulation are discussed, e.g., by Nielsen (1992). Since the seminal paper by Stokes (1847), Stokes drift has been recognized as an important example that illustrates the difference between the Eulerian and the La- grangian statistics (Longuet-Higgins 1986). It predicts that a fluid particle (i.e. a tracer of negligible inertia) experiences a mean drift in the direction of wave propagation proportional to U2/c, where U is the amplitude of the wave-induced velocity and c is 2 G. Boffetta, M. Martins Afonso, A. Mazzino, M. Onorato, and F. Santamaria the wave phase velocity. Because the Stokes drift originates from the difference between averages, it is relevant for all floating and suspended particles present in the water column, and not only for fluid particles considered in the original derivation. In- ertia of finite-size particles with density different from the fluid modifies Lagrangian averages with respect to Eulerian ones. This has important consequences on particle dispersion in both laminar and turbulent flows (see, e.g., Squires & Eaton 1991; Wang et al. 1992; Falkovich& Pumir 2004; Boffetta, De Lillo & Gamba 2004;Bec et al. 2006;Martins Afonso2008;Martins Afonso, Mazzino & Olla2009;Bec et al.2010;Martins Afonso & Mazzino 2010; Martins Afonso, Mazzino & Muratore-Ginanneschi 2012), and we expect that inertia might affect the Stokes drift experienced by inertial particles. Previous studies in thefieldhaveinvestigatedthecaseofparticlesclosetobeneutrallybuoyantinavelocity field generated by internal gravity waves (Grinshpun et al. 2000) and small particles of generic density in deep water in the presence of surface gravity waves (Eames 2008). Ourmainaimhereistopushforwardtheanalysesperformedbythesepreviousstudies and to investigate the role of inertia on the resulting Stokes drift for linear and second- order Stokes waves over arbitrary depth. For this purpose, a multiple-scale expansion has been exploited to re-sum secular terms, the cumulative effect of which gives rise to a generalized Stokes drift. As a result of our analysis, we show that inertia induces a correction to the horizontal Stokes drift which is second order in particle inertia, and generates a vertical drift (a first-order effect) which modifies the sedimentation velocity. Interestingly, this verticaldrift has a dynamical originas it is active even in the absence ofgravity,aremarkableresultnotpointedoutinpreviousstudies.The analyticalresults carriedoutbymeansofasymptoticmethodsarecorroboratedbyasetofdirectnumerical simulations which extend the range of validity of our results beyond the perturbative regime. Theremainingofthispaperisorganizedasfollow.InSection2wereporttheanalytical calculations and results based on the multiple-scale expansion. Section 3 is devoted to the numerical results and their comparison with analytical predictions. In Section 4 we extend our results to the case of finite depth, and Section 5 is devoted to conclusions. 2. Multiple-scale analysis of inertial-particle motion We consider the motion of small inertial particles transportedby Stokes waves propa- gating in arbitrary depth h. The reason for considering a second-order expansion in the wave motion (Stokes wave) comes from the fact that the Stokes drift is a second-order effect in the wave amplitude. The two-dimensional, irrotational and incompressible velocity field u = (u,w) induced by a Stokes wave is (Whitham 1974; Kundu & Cohen 1990) Ucosh[k(z+h)] 3U2cosh[2k(z+h)] u= cos(kx ωt)+ cos[2(kx ωt)]  sinh(kh) − 4c sinh4(kh) − (2.1)  w = Usinh[k(z+h)]sin(kx ωt)+ 3U2sinh[2k(z+h)]sin[2(kx ωt)], sinh(kh) − 4c sinh4(kh) − where x and z are the horizontal and the vertical coordinate, respectively; k is the wave number and ω is the angular frequency related to k via the dispersion relation, ω = gktanh(kh) (strictly speaking,one should include the nonlinear correctionto the dispeprsion relation; however,this turns out to be inessential in our analysis). The phase velocityisc=ω/k,andthe maximumvelocityU atthe surface(z =0)ofthe first-order solutionisrelatedtothewaveamplitudeAbyU =ωA.Theequations(2.1)areobtained underthehypothesisofsmallsteepnessǫ=kA,which,becauseoftherelationbetweenU Stokes drift for inertial particles 3 and A, is equivalentto the Froude number defined as Fr=U/c.Note that, in equations (2.1),the meanflowthat shouldappearatthe same orderasthe secondharmonicis not included becausewe aredealingwitha monochromaticwave,andnotwithwavepackets characterized by modulation length (see e.g. Whitham 1974, p. 474 for a discussion). The motion of a small inertial particle transported by the fluid flow u through the Stokes drag (with typicalresponse time τ) and subject to gravityaccelerationg is given by dx =V (2.2) dt dV u V du = − +(1 β)g+β , (2.3) dt τ − dt wherex(t)andV(t)representtheparticlepositionandvelocity.In(2.3)theadded-mass effecthasbeentakenintoaccountviathedimensionlessnumberβ =3ρ /(ρ +2ρ ),built f f p from the fluid, ρ , and particle, ρ , densities (Maxey & Riley 1983; Gatignol 1983). f p In order to have an explicit expression for the particle velocity, we expand (2.3) per- turbatively in τ (Balkovsky, Falkovich & Fouxon 2001), to obtain: du d2u V =u+τ(1 β) g +τ2(1 β) +O(τ3). (2.4) − (cid:18) − dt(cid:19) − dt2 Forsimplicity,inthefollowingwewillfirstconsiderthedeep-waterlimit, i.e.kh , →∞ for which the coefficients of the second order term in the velocity field vanish and (2.1) simplifies to: u(x,z,t)= Uekzcos(kx ωt),Uekzsin(kx ωt) . (2.5) − − In the case of deep-water grav(cid:0)ity waves, the dispersion relation redu(cid:1)ces to ω =√gk. Besides the steepness, another dimensionless number, Γ (1 β)gτc/U2, can be ≡ − introduced as the ratio of the bare gravitational sedimentation velocity, (1 β)gτ, over the drift velocity, U2/c. Note that this number can be written as Γ = (1− β)St/Fr2, − where we have introduced the Stokes number St ωτ. Indeed, since the velocity field ≡ (2.5) decays exponentially with z, the effect of the Stokes drift is modulated by the sedimentation process. Therefore, in the perturbation expansion, one has to decide at which order in ǫ the gravitationalterm has to be taken into account. We have analyzed different,physicallyrelevant,casesandwefoundthatthemostinterestingregimeiswhen Γ=O(1). Physically,this choice correspondsto a situationin which gravityis relatively small and particles experience a finite horizontal Stokes drift during sedimentation. In other words, the particle settling velocity in still fluids, (1 β)gτ, made dimensionless − via the wave velocity c, is assumed to be asymptotically U2/c2, i.e. O(ǫ2). ∝ Upon momentarily introducing the dimensionless variables x kx, t ωt and u 7→ 7→ 7→ u/U, we can expand u and its Lagrangian derivatives at the second order in ǫ, and substituting in (2.2)–(2.4) we obtain for the particle motion: x˙ =ǫ u St(1 β)w St2(1 β)u +ǫ2St2(1 β)e2z +O(St3)+O(ǫ3), (2.6) − − − − − (cid:2) (cid:3) z˙ =ǫ w+St(1 β)u St2(1 β)w ǫ2 Γ+St(1 β)e2z +O(St3)+O(ǫ3). (2.7) − − − − − A sta(cid:2)ndard expansion of the coord(cid:3)inate (cid:2)as x = x +ǫx (cid:3)+ǫ2x +... leads to sec- 0 1 2 ular terms in the equation at the second order. Therefore, we are led to analyze the problem within the multiple-scale formalism. To this aim, we introduce an independent slow time variable T ǫ2t — the specific choice of the power 2 as the exponent of ≡ ǫ is obviously dictated by the fact that the secularity appears in the equation at the 4 G. Boffetta, M. Martins Afonso, A. Mazzino, M. Onorato, and F. Santamaria secondorderoftheexpansioninǫ—,andweconsiderthedynamicalvariablesasafunc- tion of both times, x(t,T) = x (t,T)+ǫx (t,T)+ǫ2x (t,T) (Bender & Orszag 1978; 0 1 2 Bensoussan, Lions & Papanicolaou1978). At the secondorder, from the solvability con- dition which removes secular terms, we obtain an expression for the mean Lagrangian velocity which, going back to dimensional variables, gives the drift velocities: U2 u = 1 β(1 β)St2 e2k[z0−(1−β)gτt] , (2.8) d c − − (cid:2) (cid:3) U2 w = (1 β)gτ (1 β)St e2k[z0−(1−β)gτt] . (2.9) d − − − − c In the limit of fluid particles, St = 0 and β = 1, the above expressions recover the drift derived by Stokes: u = e2kz0U2/c, w = 0. Inertia induces a correction of order St2 d d to the horizontal drift velocity (2.8). The interesting result is that inertia produces a drift velocity also in the vertical direction, which corrects the sedimentation velocity. The vertical drift is present even in the absence of gravity (g =0), and is a consequence of the interplay between delayed particle dynamics (due to inertia) and asymmetric vertical dependence of the velocity field. The vertical drift (2.9) can be rewritten as w = (1 β)gτ 1+Fr2e2k[z0−(1−β)gτt] , from which we see that the correction to the d bare s−edim−entatio{n due to gravity is sim}ply proportional to Fr2. To our knowledge, the correction to sedimentation velocity induced by water waves has never been discussed before.Wealsonoticethat,hereandinwhatfollows,nolong-timedivergenceappearsfor light particles (β >1, i.e. Γ<0), because in that case the particle evolution is bounded by the water surface (z =0 plus wave corrections). Anothersimple,andphysicallyrelevant,predictiononecanderivefrom(2.8)isthenet displacement of particles from the point of release. For heavy particles (β < 1) released at the surface (z = 0), a time integration of (2.8) from 0 to gives for the total 0 ∞ displacement: U2 Fr21 β(1 β)St2 ∆x= 1 β(1 β)St2 = − − . (2.10) 2k(1 β)gτc − − 2k (1 β)St − (cid:2) (cid:3) − Of course, expression (2.10) is in principle valid only for small St, such that ∆x>0. 3. Numerical simulations In this Section we report the numerical results obtained from the integration of (2.2) and (2.3), with the aim of verifying the analytical predictions of the previous Section, and also to check the robustness of these predictions with respect to the expansion parameters. Figure1showstwotypicalexamplesoftrajectoriesofslightlyheavyandlightparticles inthedeep-waterlimit.Fromthetrajectoriesthedriftvelocityisobtainedbycomputing averagesoverthe Lagrangianperiod (whichis different fromthe Eulerian,waveperiod). An example of the resulting velocity components is shown in Fig. 2 for a wave with ǫ = 0.33. Despite the fact that the steepness is not so small, the agreement with the theoretical prediction (2.8) and (2.9) is very good. From the plot of the vertical velocity w weseethattherelativecorrectioninducedbywavestothebaresedimentationvelocity d at the initial time is Fr2 0.11. ≈ For steeper waves, the agreement between numerical simulations and perturbative predictions worsens, but (2.8) and (2.9) still capture qualitatively the fact that bigger waves induce larger corrections to the sedimentation velocity. One should also realize Stokes drift for inertial particles 5 0 -0.1 z -0.2 -0.3 0 0.05 0.1 0.15 0.2 0.25 x Figure 1. Two examples of trajectories of slightly heavy (β =0.99, continuous line) and light (β = 1.01, dotted line) particles transported by a linear wave (2.5) with ǫ = Fr = 0.33 and St = 2.22 in deep water. The initial position for particles is x(0) = 0, z(0) = 0 (heavy) and x(0)=0.13, z(0)=−0.5 (light). that for a larger steepness other effects such as wave breaking, clearly not included in the theory or simulation, can take place. The total horizontal displacement of heavy particles released at the surface of deep water for different values of the parameters is shown in Fig. 3, together with the predictions givenby (2.10). The agreementis very goodnotonly in the perturbative regimeof small St in which (2.10) is derived. Deviations are observable only for St=O(1). As mentioned in Section 2, the analytical result (2.9) implies that inertial particles have a vertical drift even in the absence of gravity. This drift acts in the same direction of sedimentation as it has the same sign of the gravitational term, independently on β. By comparing (2.8) and (2.9), we see that for g =0 the mean motion is along a straight line, with slope given by w /u = (1 β)St/[1 β(1 β)St2]. A simple consequence d d − − − − is that in this case the total displacement of a heavy particle over deep water diverges, in agreement with (2.10) with g 0. Figure 4 shows the slope of the mean motion as → a function of St for different values of β. Again, for small and moderate values of the parameter, the agreement with the analytical prediction is very good. 4. Predictions for arbitrary depth In this Section we generalize the calculations of Section 2 to the velocity field (2.1) inducedbyaStokeswaveinwaterofarbitrarydepthh.Thecalculationfollowsthesame steps of the deep-water limit, with the additional complication due to the presence of 6 G. Boffetta, M. Martins Afonso, A. Mazzino, M. Onorato, and F. Santamaria 0.04 0.03 d u 0.02 0.01 0 0 5 10 15 t -0.0195 d -0.0205 w -0.0215 0 5 10 15 t Figure2.Numerical(squares)vs.theoretical(solidline)driftvelocity:horizontal(upperpanel) and vertical (lower panel) components. The dashed line represents the settling velocity in the absence of wave motion wd =−(1−β)gτ.Parameters: ǫ=Fr=0.33, St=2.22, β=0.99. Stokes drift for inertial particles 7 1000 2 r F / 100 k x ∆ 10 0.01 0.1 1 St Figure 3. Total horizontal displacement of heavy particles with β = 0.9 (squares), β = 0.99 (circles) and β = 0.995 (triangles), settling beneath a linear wave with ǫ = Fr = 0.33, as a function of St. Lines represent theoretical predictions (2.10). new terms. The final result, which generalizes equations (2.8)–(2.9), reads: U21 β(1 β)St2 u = − − cosh 2k[z +h (1 β)gτt] , (4.1) d c 2sinh2(kh) { 0 − − } U2 (1 β)St w = (1 β)gτ − sinh 2k[z +h (1 β)gτt] . (4.2) d − − − c 2sinh2(kh) { 0 − − } Wefirstnoticethat,bytakingthelimitkh ,theaboveexpressionsrecoverthedeep- →∞ waterresultofSection2.Wealsoobservethat,remarkably,thesecond-orderterminthe Stokes wave expansion (i.e. the one proportional to U2 in (2.1)) does not contribute in the drift (4.1)–(4.2), as long as we stop the expansion (2.6)–(2.7) at the second order in ǫ. Its effect will eventually appear in higher-order terms in the expansion, which are not considered here. From this point of view, the perturbative analysis for finite depth has the same structure of the infinite-depth limit. We have performed numerical simulations of particle motion given by (2.2) and (2.3) inthevelocityfield(2.1).Alsointhiscasewehavefoundaverygoodagreementbetween the numerically-evaluated drift velocities and the analytical predictions (4.1) and (4.2), even for values of the expansion parameters relatively large, confirming the robustness of the multiple-scale technique. 5. Conclusions We have considered the problem of Stokes drift induced by water waves on small inertial particles with two complementary perspectives. On the one hand, our results give the correction to the horizontal Stokes drift induced by inertia. This correction is 8 G. Boffetta, M. Martins Afonso, A. Mazzino, M. Onorato, and F. Santamaria 0 -0.1 d u ) β - -0.2 1 ( / d w -0.3 -0.4 0 0.1 0.2 0.3 0.4 0.5 St Figure 4. Ratio of vertical to horizontal drift velocity wd/((1 − β)ud as a function of St for heavy particles in the absence of gravity (g = 0). Parameters: ǫ = 0.33, β = 0.9 (squares), β = 0.99 (circles), β = 0.995 (triangles). The line represents the prediction wd/(1−β)ud =−St/[1−β(1−β)St2]. found to be second order in the particle Stokes number, with a sign which depends on the particle density relative to water. On the other hand, we also obtain a vertical drift, which therefore represents a correction to the sedimentation velocity induced by wave motion on the surface. This effect, which results to be first order in the Stokes number, has never been discussedbefore andis ofpossible relevance,e.g.,in the field ofsediment transport in coastal regions. We conclude observing that, although the present analysis is performed in the ideal world of linear two-dimensionalwater waves and in the absence of any interactions with physical boundaries, we expect that our main findings will survive in more complex and realistic situations. Indeed, we have already shown that, in the more complex case of second-order waves on arbitrary depth, our results do not change qualitatively. There- fore, it would be extremely interesting to study the drift of inertial particles, and its effect on sedimentation, in more realistic simulations of wave motion and in laboratory experiments, where a precise determination of the mean velocity and falling velocity is possible. 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