The increased wave-induced drift of floating marine litter: A mechanism for the increased wave-induced drift of floating marine litter
R. Calvert, M.L. McAllister, C. Whittaker, A. Raby, A.G.L. Borthwick, T.S. van den Bremer
aa r X i v : . [ phy s i c s . f l u - dyn ] F e b Under consideration for publication in J. Fluid Mech. A mechanism for the increased wave-induceddrift of floating marine litter
R. Calvert a,b , M.L. McAllister a , C. Whittaker c , A. Raby d , A.G.L.Borthwick b and T.S. van den Bremer a,e a Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK b School of Engineering, The University of Edinburgh, Edinburgh, EH9 3FB, UK c Department of Civil and Environmental Engineering, University of Auckland, Auckland 1010,New Zealand d School of Engineering, University of Plymouth, Plymouth PL4 8AA, UK e Faculty of Civil Engineering and Geosciences, Delft University of Technology, 2628 CD, Delft,The Netherlands(Received February 22, 2021)
Periodic water waves generate Stokes drift as manifest from the orbits of Lagrangianparticles not fully closing. Stokes drift can contribute to the transport of floating ma-rine litter, including plastic. Previously, marine litter objects have been considered tobe perfect Lagrangian tracers, travelling with the Stokes drift of the waves. However,floating marine litter objects have large ranges of sizes and densities, which potentiallyresult in different rates of transport by waves due to the non-Lagrangian behaviour of theobjects. Through a combination of theory and experiments for idealised spherical objectsin deep-water waves, we show that different objects are transported at different ratesdepending on their size and density, and that larger buoyant objects can have increaseddrift compared with Lagrangian tracers. We show that the mechanism for the increaseddrift observed in our experiments comprises the variable submergence and the corre-sponding dynamic buoyancy force components in a direction perpendicular to the localwater surface. This leads to an amplification of the drift of these objects compared to theStokes drift when averaged over the wave cycle. Using an expansion in wave steepness,we derive a closed-form approximation for this increased drift, which can be included inocean-scale models of marine litter transport.
1. Introduction
In the last half century large concentrations of plastic have polluted the oceans, withharmful effects on marine wildlife and potentially on human health (Ostle et al. et al. et al. et al. et al.
Calvert et al. under the backward-moving trough results in orbits that do not close, i.e. a Lagrangian-mean drift, known as Stokes drift (Stokes 1847). Stokes drift in deep water is proportionalto the square of wave steepness and decays with depth at twice the rate of the oscilla-tory water particle velocity (see e.g. the review by van den Bremer & Breivik (2017)).Ocean surface gravity waves are driven by wind, and thus Stokes drift has often beenassumed to be locally proportional to the wind forcing (Weber 1983). However, wavesare slow to build and, once established as swell, waves can travel long distances withlittle dispersion (Ardhuin et al. et al. ® Development Group 2016), can be used to predict Stokes drift (Webb& Fox-Kemper 2011, 2015).Several authors have considered the effect of Stokes drift on the transport of floatingmarine litter. In an early study, Kubota (1994) found that Stokes drift derived from localwind fields did not make a significant contribution towards debris transport. However,more recent studies that included the entire wave field showed that Stokes drift could playan important role. For example, Iwasaki et al. (2017) found that Stokes drift transportedplastic towards the coast in the Sea of Japan during winter, and Delandmeter & Van Se-bille (2019) reported similar behaviour in the Norwegian Sea. Stokes drift could enabledebris to leak out of the Indian Ocean (Dobler et al. et al. et al. et al. (2014) modelled the plastic beaching process by including Stokes drift and sinkingvelocity and observed that larger plastic debris was selectively moved onshore. All theforegoing studies have simply assumed that floating marine litter objects are transportedwith the Stokes drift; in other words, that they are perfect Lagrangian tracers.If a particle is infinitesimally small and has the same density as water, it will behavepurely as a Lagrangian tracer and will be transported with the Stokes drift. This is notnecessarily true for an object of finite size or of a density different to that of water. Asthe inertia of such an object becomes important, the fluid will exert a drag on the objectowing to the relative velocity between the object and fluid. Furthermore, the objectmay rise, sink, or float depending on the density difference. The literature distinguishesbetween fully submerged and floating objects, discussed separately below.The motion of a fully submerged sphere in unsteady flow with viscous drag can be de-scribed by the Maxey–Riley equations (Maxey & Riley 1983). Based on this pioneeringwork, Eames (2008) and Santamaria et al. (2013) examined how far slightly positivelyor negatively buoyant objects would be transported by regular waves. They defined thedistance transported as either the horizontal distance transported whilst a negativelybuoyant object sinks from the free surface to the sea floor or the horizontal distancetransported whilst a positively buoyant object rises from the sea floor to the free sur-face. Eames (2008) and Santamaria et al. (2013) used an expansion in wave steepnessand Stokes number to arrive at analytical solutions for small objects. To leading orderand for negatively buoyant objects, Eames (2008) showed such small objects are trans-ported with a mean horizontal Stokes drift velocity and sediment with their terminal fallvelocity. Santamaria et al. (2013) predicted that positively buoyant objects would expe-rience an increase in drift owing to their inertia. Although Eames (2008) and Santamaria et al. (2013) considered the object’s inertia when examining transport by waves, bothconsidered completely submerged objects.Also considering fully submerged objects, DiBenedetto & Ouellette (2018) first showednon-spherical objects have a preferential orientation under waves, confirming this re-sult numerically (DiBenedetto & Ouellette 2018) and experimentally (DiBenedetto et al.he increased wave-induced drift of floating marine litter et al. et al. (1979). These authors considered the free surface to be an oscillating slope witha vertical force balance between gravity and buoyancy, whilst the horizontal part of thebuoyancy force induces object motion in what Rumer et al. (1979) termed the slope-sliding effect. Shen & Zhong (2001) further extended the slope-sliding model, proceedingto find analytical solutions of the object motion in limit of no added mass or no resis-tance. Huang et al. (2016) found that the drift of relatively large floating discs, used tomodel floating ice sheets, increased beyond the Stokes drift in physical experiments. Thiscould be explained by numerical solutions to an equation of motion based on a rotatingcoordinate system which aligned with the free surface, leaving the physical mechanismat work unclear.Although not focusing on waves, Beron-Vera et al. (2016) showed that the inertia ofan undrogued drifter is important for their accumulation in subtropic gyres. The studyintegrated a Maxey–Riley equation that modelled the variable submergence of surfacedrifters and included forcing from current and wind velocities, by varying the relativeeffect of each with the submerged volume of the drifter. The drag formulation assumedlinear dependence of force on the density ratio between the object and water, as hasbeen experimentally validated by Miron et al. (2020). The Maxey–Riley equation hasbeen extended to model floating Sargassum rafts (Beron-Vera & Miron 2020).Surface tension can be important in the response of small inertial particles under waveaction, as shown by Falkovich et al. (2005), who found that hydrophobic and hydrophilicparticles concentrate in antinodes and nodes of a standing wave, respectively. Denissenko et al. (2006) demonstrated the importance of surface tension when predicting time scalesof small particle clusters in standing waves. In this paper, we do not examine the effectof surface tension, which places a lower limit on the size of particles for which our modelis valid.This paper examines the transport of inertial, finite-size floating marine litter under theinfluence of non-breaking waves. Our derivation starts from Newton’s second law, withbuoyancy, gravity and drag force components. Using a transformed coordinate system,similar but not equivalent to Huang et al. (2016), that vertically translates and is orientedorthogonally to the time-varying free surface, we ensure that the dynamic buoyancyterm is directed normal to the free surface. In this model, the drag force changes withsubmergence of the object, and we formulate a drag coefficient that is valid across a rangeof Reynolds numbers. We use perturbation methods to derive a closed-form solutionfor the transport of inertial, finite-size floating spherical objects, which is then used tointerpret the physical mechanism for their enhanced transport compared to the Stokesdrift. Numerical and analytical solutions are compared for viscous drag. In order toobserve the predicted response, we perform experiments in a laboratory wave flume.This paper is laid out as follows. §2 presents the theoretical model. §3 describes solu-tions obtained using perturbation methods for viscous drag. §4 compares the analyticalsolutions thus obtained against numerical solutions of the model. The numerical solutionsare also used to compare model predictions of viscous and non-viscous drag. Conclusionsare drawn in §5.
Calvert et al.
Figure 1: Diagram of the two coordinate systems used to describe a floating object ofdiameter D : a stationary laboratory coordinate system ( x , z ) and a vertically translatingand rotating coordinate system ( τ , n ) with its origin at the vertical position of thefree surface z = η p and the τ -axis aligned tangential to the free surface. The vector x p locates the centre of the object relative to the origin of the stationary coordinate system, tan θ = ∂η/∂x is the slope of the free surface, and s is the (variable) submergence.
2. Mathematical model
Equation of motion of a floating object
The motion of a floating inertial object is described by Newton’s second law: m ˙ v = F ≡ B + M + G + R , (2.1)where m is the mass of the object and v its velocity with the dot denoting a derivativewith respect to time. The total force on the object F can be decomposed into a buoyancyforce B , an added-mass force M , a gravity force G and a resistance force R , which areformulated below. The buoyancy and added-mass forces arise from the integral of pressurearound the object. For simplicity, we will assume the object is spherical with diameter D . Throughout, it is assumed that the object is small relative to the wavelength, suchthat D/λ ≪ , with D the diameter of the object and λ the wavelength. This has fourimportant consequences. First, the wave field is unaffected by the presence of the object;in other words, there is no diffraction. Second, the free surface can be approximated asan (inclined) straight line on the scale of the object. Third, we can approximate the(relative) velocity field between the liquid and object, which determines the drag on theobject, as the velocity at a point. Fourth, the buoyancy force can be computed fromthe submergence measured relative to the free surface. Nevertheless, the model neglectssurface tension. This assumption is reasonable for floating objects provided the followingthreshold criterion (e.g Falkovich et al. (2005)) is met: D/ > p γ/ ( ρg ) , where γ issurface tension, ρ is density of water and g is gravitational acceleration. For water, thecriterion is satisfied for objects of diameter exceeding 5.4 mm, resulting in the findingsbeing invalid for microplastic. However, such small plastics are likely to behave as purelyLagrangian tracers.We first adopt a stationary two-dimensional laboratory coordinate system ( x , z ) withthe vertical coordinate z measured upwards from the undisturbed free surface. To definethe forces on the object, a second, moving coordinate system ( τ , n ) is established that he increased wave-induced drift of floating marine litter z = η ( x, t ) and aligns locally with the τ -axistangential to the free surface at the position of the object x p and the n -axis normalto it, as shown in figure 1. The coordinate transformation takes the form of a verticaltranslation followed by a clockwise rotation through the angle θ = arctan ( ∂η/∂x ) , bothat the horizontal position of the object x p : (cid:20) τn (cid:21) = (cid:20) ∂ x η ( x, t ) | x p − ∂ x η ( x, t ) | x p (cid:21) (cid:20) xz − η ( x p , t ) (cid:21) Ξ( x p , t ) with Ξ( x p , t ) ≡ (cid:16) (cid:0) ∂ x η ( x, t ) | x p (cid:1) (cid:17) − / , (2.2)where a small-angle approximation on θ has been used and Ξ is required for the deter-minant of the transformation matrix to be unity and thus conserve area. The quantities ∂ x η ( x, t ) , η ( x, t ) and Ξ( x, t ) are evaluated at the object position x p ( t ) and are thus solelyfunctions of time t . The coordinate system ( τ , n ) does not translate in the horizontaldirection, enabling direct estimation of the object’s horizontal drift v x = ˙ x p , where theoverbar denotes an average over the wave cycle. The time-dependent unit normal vectorsare e τ = (cid:2) , ∂ x η ( x, t ) | x p (cid:3) Ξ( x p , t ) and e n = (cid:2) − ∂ x η ( x, t ) | x p , (cid:3) Ξ( x p , t ) . (2.3a,b)It should be emphasized that ( τ , n ) is an accelerating coordinate system, both in termsof rotation and vertical translation. Inverting (2.3): e x = (cid:0) e τ ( t ) − ∂ x η ( x, t ) | x p e n ( t ) (cid:1) Ξ( x p , t ) and e z = (cid:0) ∂ x η ( x, t ) | x p e τ ( t ) + e n ( t ) (cid:1) Ξ( x p , t ) . (2.4a,b)For the time-dependent unit normal vectors e τ ( t ) and e n ( t ) :d e τ ( t ) d t = ˙ θ p e n ( t ) and d e n ( t ) d t = − ˙ θ p e τ ( t ) with ˙ θ p = d t ( ∂ x η ( x, t ) | x p )Ξ p , (2.5a,b)in which θ p ( t ) ≡ θ ( x p ( t ) , t ) , Ξ p ( t ) = Ξ( x p ( t ) , t ) , and d t ≡ d / d t .Denoting the position of the object as x p = x p e x + z p e z = η p e z + τ p e τ + n p e n with η p ( t ) ≡ η ( x p ( t ) , t ) , its velocity may be written as: v = v x e x + v z e z = (cid:16) ˙ τ p − ˙ θ p n p + ˙ η p ∂ x η | x p Ξ p (cid:17) e τ + (cid:16) ˙ n p + ˙ θ p τ p + ˙ η p Ξ p (cid:17) e n , (2.6)where we have used (2.5) for the time derivatives of the unit vectors e τ , and e n , and e z was substituted for from (2.4b). The velocity in the translating reference frame v ∗ isrelated to the velocity in the stationary reference frame v by v ∗ = v − ˙ η p e z , where bothvectors can be expressed in any arbitrary set of orthogonal components, such as e x and e z or e τ and e n . Accordingly, the acceleration of the object can be written as: ˙ v = ˙ v x e x + ˙ v z e z = (cid:16) ¨ τ p − ¨ θ p n p − θ p ˙ n p − ( ˙ θ p ) τ p + ¨ η p ∂ x η | x p Ξ p (cid:17) e τ + (cid:16) ¨ n p + ¨ θ p τ p + 2 ˙ θ p ˙ τ p − ( ˙ θ p ) n p + ¨ η p Ξ p (cid:17) e n . (2.7)To evaluate (2.7), the double time derivatives ¨ θ p and ¨ η p must be evaluated explicitly.The double time derivative ¨ θ p can be obtained by differentiating with respect to timetwice using the relationship θ p = arctan (cid:0) ∂η/∂x | x p (cid:1) , noting that x p is a function of timerequiring the chain rule, to obtain: ¨ θ p = (cid:16) ∂ ttx η | x p +2 ˙ x p ∂ txx η | x p + ( ˙ x p ) ∂ xxx η | x p +¨ x p ∂ xx η | x p (cid:17) Ξ p + (cid:0) ∂ tx η | x p + ˙ x p ∂ xx η | x p (cid:1) p ˙Ξ p . (2.8) Calvert et al.
Similarly, the double time derivative ¨ η p takes into account the dependence of the freesurface η p ( x p ( t ) , t ) on time t and the time-dependent horizontal position x p ( t ) , whichgives through the chain rule after differentiating twice: ¨ η p = ∂ tt η | x p +2 ˙ x p ∂ tx η | x p + ( ˙ x p ) ∂ xx η | x p +¨ x p ∂ x η | x p . (2.9)Substituting (2.8) and (2.9) into (2.7) and (2.7) thence into (2.1) results in two second-order differential equations in the ( n, τ ) coordinate system, which are explicitly givenby (A 1) and (A 2) in appendix A. These two equations contain three second-order timederivatives, and so a third (kinematic) equation relating the second-order derivatives isrequired to solve the system. Such an equation can for example be found by taking thedot product of (2.7) and e x (see (A 3) in appendix A).For convenience, we express the normal coordinate of the centre of the object n p interms of the submergence depth s (see figure 1). To do so, we assume that D/λ ≪ so that the free surface is a locally straight line with n -coordinate n s = − ∂ x η | x p x p Ξ p (using (2.2), setting x = x p and z = η p ). The submergence depth is then given by s = D/ − ( n p − n s ) = D/ − n p − x p ∂ x η | x p Ξ p , where D is the diameter of the object.From (2.6), the following expression is obtained for the horizontal velocity of the object: ˙ x p = (cid:16) ˙ τ p − ˙ θ p (cid:0) n p + τ p ∂ x η | x p (cid:1) − ˙ n p ∂ x η | x p (cid:17) Ξ p . (2.10)It should be noted that ˙ s = − ˙ n p − d t ( x p ∂ x η | x p Ξ p ) .2.1.1. Buoyancy and added mass
We decompose total pressure p into an undisturbed component p undisturbed and a dis-turbed component p disturbed owing to the presence of the object. Assuming an objectthat is small relative to the wavelength ( D/λ ≪ ), the undisturbed pressure varies as p undisturbed = ρ f g ( η ( x, t ) − z ) on the scale of the object with ρ f the density of the fluid,so that the dynamic free surface boundary condition p undisturbed ( z = η ) = 0 is satisfied,the variation with depth is hydrostatic, and any depth-dependent variation owing thewaves (cf. exp( k z ) with k the wavenumber) is ignored.The undisturbed pressure integrated around the wetted surface results in a buoyancyforce acting in the normal direction to the free surface, B n ( t ) = gmβ V s V Ξ − p = gmβ (cid:18) s ( t ) D (cid:19) − (cid:18) s ( t ) D (cid:19) ! Ξ − p , (2.11)where g is the gravitational constant, V s is the submerged and V the total volume of thesphere, and β ≡ ρ o /ρ f is the ratio of object to fluid density. By including ρ f gη ( x, t ) inthe undisturbed pressure, we have included the Froude–Krylov force resulting from thewaves.The disturbed component of pressure leads to added mass terms, as derived by Maxey& Riley (1983): M τ = C m,τ ( s ) mβ ( ˙ u τ ( ˜x p , t ) − ˙ v τ ) and M n = C m,n ( s ) mβ ( ˙ u n ( ˜x p , t ) − ˙ v n ) , (2.12a,b)where C m = ( C m,τ , C m,n ) is the added mass coefficient, which is deliberately left as anunspecified function of submergence s ( t ) at this stage of the derivation.The small-diameter assumption leaves the vertical location, where we should evaluatethe velocity of the surrounding fluid in (2.12), unspecified. We set this location to be atthe free surface, ˜ x p = ( x p , η p ) . he increased wave-induced drift of floating marine litter V s as a function of the variablesubmergence s ( t ) ; (centre) the projected area of a submerged sphere moving in the normaldirection ( e n ); and (right) the projected area of a submerged sphere moving in thetangential direction ( e τ ). All diagrams are shown in the ( τ , n ) coordinate system.2.1.2. Gravity forces
The gravity force acts in the vertical direction, and has the following components inthe moving coordinate system, G τ ( t ) = − mg∂ x η | x p Ξ p ( t ) and G n ( t ) = − mg Ξ p ( t ) . (2.13a,b)2.1.3. Resistance forces
The resistance terms are caused by drag on the object when it has a velocity relativeto that of the surrounding liquid. To begin, we assume viscous drag. We assume thisdrag depends on the submergence of the object and, specifically, we assume the drag isproportional to the submerged projected area of the sphere in the tangential and normaldirections (see figure 2). Other drag formulations are discussed and examined in §4. Theresistance force in the tangential direction is, R τ = 3 πρ f νD ˆ A s ,τ ( u ∗ τ − v ∗ τ ) , (2.14)where u ∗ τ and v ∗ τ are the velocity components in the τ -direction of the surrounding fluidand the object velocity respectively (in the moving reference frame). The normalised areain the tangential direction ˆ A s ,τ is the projected area of the submerged sphere, A s ,τ = D ζ − sin( ζ )) with ζ ≡ − (1 − s/D ) , (2.15)normalised by the maximum projected area A = πD / , so that ˆ A s ,τ = A s ,τ /A . Assumingthe drag is proportional to the submerged projected area following Beron-Vera et al. (2016), which has been validated for steady flows (Miron et al. et al. u ∗ τ at the free surface, ˜ x p = ( x p , η p ) .Similar to the τ -direction, we have for the n -direction, R n = 3 πρ f νD ˆ A s ,n ( u ∗ n − v ∗ n ) , (2.16)where we have evaluated the velocity of the surrounding fluid at the same location ˜ x p as for the tangential resistance force. The submerged projected area of a sphere in thenormal direction is given by (see figure 2): A s ,n = πs ( t ) ( D − s ( t )) , (2.17) Calvert et al. which again, is normalised by the maximum projected area of a sphere A = πD / , sothat ˆ A s ,n = A s ,n /A . Later, in §4, other drag formulations are considered to examine therobustness of the model’s predictions.2.2. Fluid velocity for surface gravity waves
We consider unidirectional deep-water surface gravity waves propagating over a hori-zontal bed in the ( x, z ) -coordinate system, with z measured vertically upwards fromstill water level, and the free surface located at z = η . For irrotational flow of inviscid,incompressible fluid, the governing (Laplace) equation is, ∇ φ = 0 for − d ≤ z ≤ η , (2.18)where φ is the velocity potential and d depth. Equation (2.18) is solved subject to theno-flow bottom boundary condition, ∂ z φ = 0 for z = − d , (2.19)and the kinematic and dynamic linear free surface boundary conditions, u z − ∂ t η − u∂ x η = 0 and gη + ∂ t φ + 12 ( ∇ φ ) = 0 at z = η , (2.20a,b)where the velocity components are u x = ∂ x φ and u z = ∂ z φ .
3. Perturbation theory for viscous drag
To interpret the physical mechanism behind the drift predicted by the model derivedin §2, we use perturbation theory to establish an analytical solution. We do so here forthe case of viscous drag, as this allows inclusion of drag at first order in our expansion.We will discuss limitations of viscous drag in §3.4 and consider numerical solutions ofour model in §4 in which the assumption of viscous drag is relaxed. We consider onlyperiodic, weakly nonlinear, deep-water surface gravity waves, so that k d ≫ with k the wavenumber. We perturb the object position x p in a Stokes-type expansion in wavesteepness ( α = k a , where a the wave amplitude), giving x p ( t ) = x (0) p + α x (1) p ( t ) (cid:12)(cid:12)(cid:12) x (0) p + α x (2) p ( t ) (cid:12)(cid:12)(cid:12) x (0) p + O ( α ) . (3.1)where the superscript corresponds to the order in α , and x (0) p is the object label and thusnot a function of time. As we are interested in wave-induced drift, which arises at secondorder, we only pursue those terms necessary to obtain this drift.Applying a perturbation expansion in the same small parameter α to the governingequation of the fluid (2.18) and its boundary conditions (2.19) and (2.20) allows the freesurface η and the velocity potential φ to be determined, and we do so up to second order.Although the perturbation theory solutions in this section are for regular waves, theexperiments introduced in appendix B make use of long (or narrow-bandwidth) wavepackets for practical reasons. We assume that inertial effects do not arise on the scaleof the packets, as justified in appendix C, so that we can correct for the presence of awave packet simply by accounting for its Eulerian mean flow. Table 1 lists the resultingsolutions, whose derivation and laboratory validation is given in more detail by Van denBremer et al. (2019) for deep water and Calvert et al. (2019) for intermediate depth. Weconsider only deep-water waves here ( k d ≫ ). The solutions for the Eulerian return flowand the second-order surface elevation are based on wave packets with envelope | A | . Itis assumed that the wave packets are narrow banded and that the Eulerian return flow is he increased wave-induced drift of floating marine litter Field Symbol SolutionFirst-order horizontal velocity u (1) x A ω exp( iϕ + k z ) First-order vertical velocity u (1) z − A ω i exp( iϕ + k z ) First-order free surface elevation η (1) A exp( iϕ ) Second-order horizontal Eulerian velocity u (2) x − ω d | A | corresponding time-integrated displacement ∆ x (2) E − ω d R t t | A | d t Second-order horizontal Stokes drift velocity u (2) S k ω | A | exp(2 k z ) corresponding time-integrated displacement ∆ x (2) S k ω R t t | A | d t Table 1: First and second-order solutions for the kinematic properties of deep-water sur-face gravity waves, with A = a ˆ A the wave amplitude envelope, a its amplitude, ˆ A a non-dimensional envelope, ω the carrier wave frequency, and k the carrier wavenum-ber. Where complex fields are given, the real part is understood, and ϕ = k x − ω t .The first three rows are first-order solutions, valid for regular waves or wave packets.The remaining rows comprise second-order solutions for the wave-averaged Eulerian andStokes velocities and the set-down. The second-order wave-averaged Eulerian velocityonly arises for wave packets, considered in the experiments in appendix B.shallow, corresponding to a depth that is small relative to the packet length (Calvert et al. (2019) establish the Eulerian return flow without the shallow return flow assumption).In practice, inclusion of the effect of the return flow merely leads to a small correction ofless than for our laboratory experiments.3.1. Zeroth-order in wave steepness: O ( α ) At zeroth-order in wave steepness, wave forcing evidently does not play a role. Only thenormal direction of (2.1) has any forcing at zeroth order, where the following leading-order static balance is achieved between buoyancy force and gravity, F (0) n = gmβ " (cid:18) s (0) D (cid:19) − (cid:18) s (0) D (cid:19) − gm = 0 . (3.2)We have used the fact that Ξ p = 1 at zeroth order and note that (3.2) is only valid fora floating sphere, i.e. | D/ − s (0) |≤ D/ . Equation (3.2) is a cubic equation, which canbe readily solved numerically for the depth of submergence of a floating sphere in theabsence of waves s (0) . 3.2. First-order in wave steepness: O ( α ) We begin by expressing the projected areas of the sphere required to calculate the tan-gential and normal resistance forces as series expansions around s (0) . The submergedprojected area of a sphere in the tangential direction (2.15) can be approximated by A s ,τ ( s ) = A s ,τ ( s (0) ) + 2 D s s (0) D − (cid:18) s (0) D (cid:19) s (1) + O ( α ) , (3.3)where we have obtained ∂ s ( A s ,τ ) from (2.15) by implicit differentiation. For the sub-merged projected area of a sphere in the normal direction, it is sufficient for our purposesto evaluate A s ,n ( s ) at zeroth order, i.e. A s ,n ( s ) = A s ,n ( s (0) ) + O ( α ) .0 Calvert et al.
The tangential direction
To first-order of approximation, the velocity and acceleration in the horizontal coor-dinate x and the tangential coordinate τ are equal, i.e. ˙ x (1) p = v (1) x = v (1) τ and ¨ x (1) p =˙ v (1) x = ˙ v (1) τ . The only forces that play a role are the tangential components of the addedmass, gravity and the resistance force. The first-order added-mass terms in the tangentialdirection are M (1) τ = C m mβ ( ˙ u (1) x − ¨ x (1) p ) , (3.4)where we now assume for simplicity that the added-mass coefficient C m is a constant andindependent of direction. Other added-mass formulations are discussed and examined in§4.In a potential flow, a fully submerged sphere has an added mass coefficient of / .Instead of deriving the complicated dependence of C m on the object’s density, we inter-polate linearly between the values for a sphere that is fully submerged ( β = 1 , C m = 1 / )and a sphere that is entirely out of the water ( β = 0 , C m = 0 ) and set C m = β/ . Therobustness of this assumption is investigated numerically in §4.The resistance force (2.14) can be approximated as: R (1) τ = Γ R mω ˆ A (0)s ,τ ( u (1) x | ˜ x (0) p − ˙ x (1) p ) with Γ R ≡ πνDβV ω , (3.5)where the non-dimensional coefficient Γ R measures the importance of the resistance force.From the object’s equation of motion (2.1) we thus obtain: (cid:18) C m β (cid:19) ¨ x (1) p = C m β ˙ u (1) x | ˜ x (0) p − g∂ x η (1) | x (0) p +Γ R ˆ A (0)s ,τ ω (cid:16) u (1) x | ˜ x (0) p − ˙ x (1) p (cid:17) . (3.6)We seek a solution to the forced second-order ordinary differential equation (3.6) of theform x (1) p = R ( iX (1) a exp( iϕ (0) p )) with ϕ (0) p = k x (0) p − ω t + ϕ and ϕ = arg( A ) ,ignoring initial transients. The complex coefficient X (1) represents the amplitude andphase change of the horizontal motion of the object relative to that of an idealized La-grangian object under the influence of waves at the same order, x (1) L = R ( ia exp( iϕ (0) p )) .We obtain X (1) = 1 , i.e. there is no horizontal motion amplification compared to that ofa Lagrangian particle.3.2.2. The normal direction
Expressing the submergence depth s in terms of the vertical coordinate z p , we havewithout approximation that s = D/ − ( z p − η p )Ξ p . Therefore, the velocity and acceler-ation in the vertical coordinate z and the normal coordinate n are related to first orderby: ˙ z (1) p = v (1) z = − ˙ s (1) + ˙ η (1) p and ¨ z (1) p = ˙ v (1) z = − ¨ s (1) + ¨ η (1) p . (3.7a,b)We first approximate the buoyancy force (2.11) by: B (1) n = Γ B mω s (1) with Γ B ≡ βk D s (0) D − (cid:18) s (0) D (cid:19) ! , (3.8)the added-mass terms by: M (1) n = C m mβ ¨ s (1) , (3.9)and the resistance force (2.16) by: R (1) n = Γ R mω ˆ A (0)s ,n ˙ s (1) , (3.10) he increased wave-induced drift of floating marine litter (a) (b) Figure 3: For viscous drag, magnitudes of the first-order horizontal motion amplification X (1) (a) and the variable submergence S (1) (b) as functions of dimensionless object size D/λ for different density ratios β = ρ o /ρ f , where the density ratio for each colour isshown in the legend. We have set C m = β/ . Numerical and analytical solutions fromperturbation theory are denoted by crosses and solid lines, respectively.where we have used u (1) z ( z = 0) = ˙ η (1) p from the linearised kinematic free surface boundarycondition and v (1) n = ˙ z (1) p . The new non-dimensional coefficient Γ B measures the strengthof dynamic buoyancy, and Γ R measures the strength of the resistance force, as for thetangential resistance force in (3.5). From the object’s equation of motion (2.1) we thusobtain: (cid:18) C m β (cid:19) (cid:16) ¨ η (1) p − ¨ s (1) (cid:17) = C m β ˙ u (1) z | ˜ x (0) p +Γ B ω s (1) + Γ R ˆ A (0)s ,n ω ˙ s (1) , (3.11)where we note gravity only enters at zeroth order. As for the tangential direction, weseek a solution to the forced second-order ordinary differential equation (3.11) of the form s (1) = R ( S (1) a exp( iϕ (0) p )) with ϕ (0) p = k x (0) p − ω t + ϕ and ϕ = arg( A ) , ignoringinitial transients. We find for the non-dimensional submergence at first order S (1) :(3.12) S (1) = 1 + C m β − Γ B − i Γ R ˆ A (0)s ,n (cid:18) C m β − Γ B (cid:19) + (cid:16) Γ R ˆ A (0)s ,n (cid:17) . Figures 3 and 4 respectively show the magnitudes and arguments of the first-order so-lutions for the horizontal motion amplification X (1) and the variable submergence S (1) .In these figures, the purely Lagrangian limit, in which the object is simply transportedwith the Stokes drift and floats on the moving surface, corresponds to X (1) = 1 , S (1) = 0 .This limit is obtained as the object size tends to zero. Note that the phase of variablesubmergence in this limit is non-zero, arg ( S (1) ) → π/ . This is because both imaginaryand real parts of the variable submergence tend to zero, with the imaginary part ap-proaching zero at a faster rate. As our model is only valid for objects that are smallrelative to the wave length, we truncate the x -axis at D/λ = 6% . Diffraction of thewave field typically only becomes important for D/λ > .As confirmed in figure 3a, the magnitude of the horizontal motion | X (1) | is equivalent tothat of a purely Lagrangian tracer. Turning to figure 4a, the argument of the horizontalmotion arg( X (1) ) is evidently also zero. As shown in figure 3b, the magnitude of the2 Calvert et al. (a) (b)
Figure 4: For viscous drag, arguments of the first-order horizontal motion amplification X (1) (a) and the variable submergence S (1) (b) as functions of dimensionless object size D/λ for viscous drag and for different density ratios β = ρ o /ρ f , as shown in the legend.We have set C m = β/ . Numerical and analytical solutions from perturbation theory aredenoted by crosses and solid lines, respectively.variable submergence |S (1) | increases monotonically with object size and does so at alarger rate for density ratios closer to unity. Variable submergence is driven by the freesurface elevation and governed by drag, dynamic buoyancy, and (added) mass, whichare respectively the resistance, spring, and inertia terms of a forced spring-mass-dampersystem (cf. (3.11)). The larger the object, the more dominant is the acceleration of thefree surface, which acts as an apparent force in the moving reference frame in whichthe variable submergence is defined, thus increasing the ‘bobbing’ of the object. Thelower the density ratio, the stronger the buoyancy force and the stiffer the ‘spring’. Theresponse in variable submergence for a stiffer ‘spring’ is smaller. The argument of variablesubmergence arg( S (1) ) decreases monotonically with object size and growing importanceof inertia but is dependent on the density ratio, as shown in figure 4b.At first order in steepness the tangential and normal directions are independent, andso it is possible for there to be a significant change in first-order variable submergencewhilst the first-order horizontal motion remains unchanged. As can be seen in the nextsection, a change in first-order variable submergence results in a change in horizontalmotion at second order.3.3. Second-order in wave steepness: O ( α ) The equation of motion (2.1) resolved in the horizontal direction and at second order ofapproximation gives: ¨ x (2) p = 1 m (cid:18) F (2) τ − ∂ x η (1) (cid:12)(cid:12)(cid:12) x (0) p F (1) n (cid:19) . (3.13)In order to examine the wave-induced drift of a floating object in periodic waves, weconsider the steady wave-averaged transport and set ¨ x (2) p = 0 , so that the resultant forcemust be zero. We will now consider the tangential and normal force contributions to(3.13) in turn.3.3.1. Tangential and normal directions
In the tangential direction, the added-mass terms at second order can be obtainedfrom the combination of an expansion in the horizontal and vertical displacements of the he increased wave-induced drift of floating marine litter M (2) τ = C m mβ (cid:16) ˙ u (2) x + x (1) p ∂ x ˙ u (1) x | ˜ x (0) p + η (1) p ∂ z ˙ u (1) x | ˜ x (0) p + ˙ u (1) z | ˜ x (0) p ∂ x η (1) | x (0) p + ˙ x (1) p ∂ x u (1) x | ˜ x (0) p + ˙ η (1) p ∂ z u (1) x | ˜ x (0) p − ˙ v (2) τ (cid:17) . (3.14)In addition to the added-mass terms, the tangential force consists of a correction to thetangential component of gravity due to the object’s horizontal displacement, G (2) τ = − mg∂ xx η (1) (cid:12)(cid:12)(cid:12) x (0) p x (1) p , (3.15)and a tangential resistance force, R (2) τ = 3 πρ f νD (cid:16) ˆ A (1)s ,τ (cid:16) u (1) τ,p − v (1) τ (cid:17) + ˆ A (0)s ,τ (cid:16) u (2) τ,p − v (2) τ (cid:17)(cid:17) . (3.16)For the first-order velocity components, we have u (1) τ,p = u (1) x | ˜ x (0) p and v (1) τ = ˙ x (1) p . Notingfrom the coordinate transformation that u τ = u x + ∂ x η | x p u z + O ( α ) , we obtain for thesecond-order accurate horizontal fluid velocity component at the object position: u (2) τ,p = u (2) x | ˜ x (0) p + ∂ x u (1) x | ˜ x (0) p x (1) p + ∂ z u (1) x | ˜ x (0) p ˜ z (1) p + ∂ x η (1) | x (0) p u (1) z | ˜ x (0) p . (3.17)We set the second-order Eulerian wave-induced velocity u (2) x to zero for the regular wavesconsidered here. The object’s horizontal velocity component at second order is: v (2) τ = ˙ x (2) p + ∂ x η (1) | x (0) p ˙ z (1) p , (3.18)where ˙ x (2) p is the quantity that is ultimately of interest. Combining (3.17) and (3.18) andsubstituting into (3.16) gives: (3.19) R (2) τ = 3 πρ f νD ˆ A (1)s ,τ (cid:16) u (1) x | ˜ x (0) p − ˙ x (1) p (cid:17) + ˆ A (0)s ,τ (cid:16) ∂ x u (1) x | ˜ x (0) p x (1) p + ∂ z u (1) x | ˜ x (0) p η (1) p − ˙ x (2) p + ∂ x η (1) | x (0) p ˙ s (1) (cid:17) ! , where we have substituted u (2) x = 0 , ˙ z (1) p = ˙ η (1) p − ˙ s (1) and u (1) z | ˜ x (0) p = ˙ η (1) p from the lin-earised kinematic free surface boundary condition. We use the notation ˆ A (1)s ,τ = ˆ A ′ (0)s ,τ ( s (1) /D ) with ˆ A ′ (0)s ,τ ≡ ∂ ˆ s ˆ A s ,τ (ˆ s ) | ˆ s (0) and ˆ s ≡ s/D according to (3.3).In the normal direction, the total force at first order consists of a buoyancy force, anadded mass and a resistance force already evaluated in (3.8), (3.9) and (3.10), respectively.3.3.2. The wave-induced drift
Substituting the first-order solutions for x (1) p (i.e. X (1) = 1 ) and for s (1) from (3.12)and for the wave quantities from table 1 and averaging over the waves, we obtain the4 Calvert et al.
Figure 5: For viscous drag, wave-induced drift amplification X (2) as a function of dimen-sionless object size D/λ for different density ratios β = ρ o /ρ f (see legend). We have set C m = β/ . Numerical and analytical solutions from perturbation theory are denoted bycrosses and solid lines, respectively.following expression from (3.13) for the wave-induced drift of the object v x = ˙ x (2) p : v x = u S Adjusted Stokes drift z }| { − R ( S (1) ) | {z } Increasesdrift + 1ˆ A (0)s ,τ Γ R Buoyancyresolved intothe x -direction z }| { − Γ B I ( S (1) ) | {z } Increases drift + Added mass z }| { C m I ( S (1) ) β | {z } Negligible effect + Normal drag z }| { ˆ A (0)s ,n ˆ A (0)s ,τ R ( S (1) ) | {z } Reduces drift , (3.20)where u S = k ω a is the Stokes drift. We define the drift amplification factor X (2) ≡ v x /u S , so that X (2) corresponds to the terms inside the square brackets in (3.20) dividedby . Equation (3.20) is the main result of this paper, and we will interpret it below. Thetext above the terms explains their physical origins, and the text below their effect onthe wave-induced drift of the object compared to the Stokes drift.We begin by examining the wave-induced drift amplification factor X (2) as a functionof object size and for different density ratios in figure 5. It is evident that the drift isenhanced and increasingly so for larger and heavier objects. Figure 6 examines the con-tributions to X (2) of the four components in (3.20): the adjusted Stokes drift, buoyancyresolved in the x -direction, normal drag, and added mass, which we will discuss in turn.In (3.20) and figure 6, X (2) = 1 corresponds to objects that do not experience an increasein drift and are simply transported with the Stokes drift (i.e. v x = u S ).3.3.3. Adjusted Stokes drift
The adjusted Stokes drift terms in (3.20) reflect change in linear object trajectory.For unmodified horizontal motion ( X (1) = 1 ) and zero variable submergence ( S (1) = 0 ),we obtain X (2) = 1 from the adjusted Stokes drift terms alone. For larger objects, theincrease in the vertical motion due to ‘bobbing’ of the object effectively enhances the he increased wave-induced drift of floating marine litter (a)(b) Figure 6: For viscous drag, contributions to the wave-induced drift amplification X (2) from the five components in (3.20) as a function of non-dimensional object size D/λ fordensity ratio β = 0 . and C m = β/ .Stokes drift, as shown in figure 6. This mechanism occurs because the linear variablesubmergence changes the object’s orbit and hence its velocity and time spent undertrough and crest. Integration of the linear velocity component along the linear orbitresults in Stokes drift. Hence, changes to velocity and orbit result in an adjusted Stokesdrift.3.3.4. Buoyancy resolved in the x -direction The mechanism through which buoyancy, when resolved in the x -direction and aver-aged over the wave cycle, can increase the drift of an object is illustrated in figure 7.Without variable submergence (left column), the dynamic buoyancy force is simply zero.With variable submergence but without drag in the normal direction (middle column),the first-order buoyancy force resolved in the x -direction does not result in a net forceon the object, as the first-order buoyancy force and the first-order slope required to re-solve this force into the x -direction are out of phase. It is only in the presence of a dragcomponent in the normal direction (right column) that a phase lag in the submergencedepth arises and a net force results. As shown in figure 6, the buoyancy force thus makesa relatively large contribution to the object’s drift.3.3.5. Normal drag
Although normal drag is required to create the phase difference that leads to the netbuoyancy force resolved in the x -direction, normal drag also acts to reduce the magnitudeof the ‘bobbing’ mechanism and thus reduces the drift motion, as shown in figure 3.The horizontal direction component of normal drag opposes the horizontal directioncomponent of buoyancy force, with the balance resulting in a drift that is greater than theadjusted Stokes drift discussed above. Tangential drag, through the inverse dependenceof X (2) on the projected area ˆ A (0) s,τ and the effective drag coefficient Γ R in (3.20), acts to6 Calvert et al.
No variable submergence Variable submergence Variable submergenceNo normal drag With normal drag S (1) = 0 S (1) is real S (1) is complex s (1) ∂ x η (1) = 0 s (1) ∂ x η (1) = 0 s (1) ∂ x η (1) = A k I ( S (1) )2 s (1) is out of phase with ∂ x η (1) . The in-phase component of s (1) with ∂ x η (1) has a meancomponent in the x -direction.No enhanced drift. No mean component andno enhanced drift. This mean componentcauses an enhanced drift. Figure 7: Schematics of the object trajectory (red) and free surface (blue) for threecases: no variable submergence, variable submergence with no normal drag, and variablesubmergence with normal drag. The schematics illustrate the physical mechanism forincreased drift arising from variable submergence s (1) , where variable submergence anddrag are in the n -direction, and a mean motion in the x -direction is created due tothe slope of the free surface ∂ x η (1) . For this illustration, we have chosen a density ratio β = 1 / .reduce the increase in object drift, by effectively ‘anchoring’ the object to the fluid andits Stokes drift.3.3.6. Added mass
At first order, the object accelerates in the normal direction, experiencing an inertiaforce in addition to the buoyancy force and the normal drag discussed above, and so anadded mass term has to be take into account. As shown in figure 3, the contribution byadded mass is relatively small and acts to reduce drift.3.4.
Limitation on validity of viscous drag
Although the preceding analysis has demonstrated how enhanced drift of non-infinitesimalobjects may arise, the underlying assumption of viscous drag places an upper limit onobject size. The maximum Reynolds number that arises from the linear motion in thenormal direction is estimated from: Re max = a ω |S (1) | Dν ≤ , (3.21)where we take to be the maximum Reynolds number for drag to be considered viscous.Noting that S (1) ( D/λ , β ) and taking β = 0 . , we obtain from (3.21) for the maximumdiameter that: S (1) ( D max /λ , β = 0 .
8) (
D/λ ) = k ναω π (3.22) he increased wave-induced drift of floating marine litter α = 0 . and frequency f = 1 . Hz, the right-hand side of (3.22) becomes equal to . × − . Fitting a linear curve S (1) = 5 . D/λ to figure 3b, we can solve the quadratic (3.22) in D/λ and obtain amaximum diameter to wavelength ratio of . corresponding to Re max = 2 . Examiningfigure 5, we can conclude that drift enhancement is negligible for such small objects. Wewill therefore have to use a realistic, non-viscous drag formulation, as discussed in thenext section.
4. Numerical solutions
To validate the perturbation theory for viscous drag in §3 and to explore the predictionsof our model for realistic, non-viscous drag, we set out to obtain numerical solutions ofour model. Specifically, we solved the set of differential equations (A 1-A 3) with theforces described in detail in §2 using a numerical ordinary differential equation solver.The fluid velocity and free surface elevation from table 1 were used as input. We firstconsider viscous drag in §4.1 and then non-viscous drag in §4.2, distinguishing conditions(notably Reynolds numbers) that are representative of laboratory (§4.2.1; see appendixB for further details) and field scale (§4.2.2). Appendix D discusses the small-object limitof the numerical solutions. Alternative drag and added-mass formulations are examinedin appendix EThe numerical solutions commenced from an initial condition in the absence of waveswith the object depth set at the static submergence given by numerical solution of (3.2).Numerical integration in time was carried out using an explicit Runge-Kutta methodwith variable time step based on Dormand & Prince’s (1980) formulation which is fifthorder in time and fourth order in accuracy. Avoiding initial transients, wave forcingwas ramped up using half of a Gaussian envelope to steady state. A convergence studyshowed that a Gaussian half width set to 20 wavelengths was sufficient to avoid initialtransients, whilst the spatial and temporal convergence were in part resolved by the vari-able time step method and checked explicitly for the largest objects. Once the objectmotion reached steady state, its motion components in the x and z directions were effec-tively linearised using a band-pass filter between . f and . f . The linear phase wasdetermined using the cross-correlation of the linearised object motion and the linearisedEulerian velocity evaluated at the object position in both directions. The object driftvelocity was calculated as the gradient of a straight line fitted to the sub-harmonic x ( t ) motion obtained by low-pass filtering at . f .4.1. Viscous drag
The crosses in figures 3, 4 and 5 display the numerical solutions of the model with aviscous drag formulation for a (small) steepness α = 0 . . Near perfect agreement isevident with the perturbation theory solutions shown as continuous lines for both thefirst-order amplitudes (figure 3) and phases (figures 4) and the second-order drift (figure5). Tiny discrepancies between perturbation theory and numerical simulations in thesefigures are due to the inherent inclusion of higher-order terms (beyond second-order)in steepness in the numerical simulations. The comparison verifies both the numericalmodel and the second-order perturbation theory.8 Calvert et al.
Non-viscous drag
To overcome the maximum Reynolds-number limit of the viscous drag formulation (ofRe ≡ | u − v | D/ν = 2 . × ), we also consider the following non-viscous drag formulation:(4.1) R j ( t ) = 12 C d ( Re ) ρ f A s ,j (cid:12)(cid:12) u ∗ j (˜ x p , t ) − v ∗ j ( t ) (cid:12)(cid:12) (cid:0) u ∗ j (˜ x p , t ) − v ∗ j ( t ) (cid:1) , where the indices j = n, τ represent the tangential and normal directions; and drag isdetermined using an experimentally-fitted, non-viscous drag coefficient C d . We choosea formulation of the drag coefficient C d ( Re ) that captures both viscous drag at smallReynolds number, which is linear in velocity difference, and form drag at high Reynoldsnumber. Specifically, we use the fit to experimental data for drag on a sphere obtainedby Morrison (2013, page 625), which is accurate for Re < × : C d ( Re ) = 24 Re +2 . Re / Re / . +0 .
411 ( Re / (2 . × )) − . (1 + Re / (2 . × )) − +0 . Re / (1 × )1 + Re / (1 × ) , (4.2)where (4.2) is the same in both directions because the Reynolds number is independentof direction (Re ≡ | u − v | D/ν ). Taking the small-object and thus the small-Reynolds-number limit of the drag force in (4.1) we can recover the viscous drag on a partiallysubmerged sphere (2.14) and (2.16).4.2.1.
Laboratory scale results
At laboratory scale, we set f = 1 . Hz, corresponding to λ = 1 . m and α = 0 . .With object diameters up to D = 60 mm, we obtain D/λ = 6% , where the limit of valid-ity for viscous drag is D/λ = 0 . (see §3.4). At laboratory scale, figure 8 compares theanalytically predicted linear motion using viscous drag with the corresponding numer-ical results using non-viscous drag. The response in the normal direction is unchangedbecause the forcing is inertial with little effect from drag. As the object size increases,inertia increasingly dominates over drag. A small decrease in horizontal linear motion isevident reaching a few percent for larger objects. The results for small objects are thesame because the non-viscous drag recovers viscous drag in the small object limit.The drift amplification increases slightly when using non-viscous drag for larger objects,as seen in figure 9. This is because the (tangential) drag force for larger objects is lower fornon-viscous drag than for viscous drag, resulting in reduced resistance to increased driftcompared to the Stokes drift. The maximum Reynolds number reached in the numericalsolutions at laboratory scale was Re max = 3 . × .4.2.2. Field scale results
We set a wave frequency of f = 0 . Hz and a steepness of α = 0 . to represent atypical wind wave at field scale. The frequency of . Hz corresponds to the peak in thespectrum with α = 0 . at the upper end of the steepness range for wind waves in theocean (Toffoli & Bitner-Gregersen 2017). This steepness corresponds to a dimensionalwave amplitude of a = 0 . m. The difference between viscous and non-viscous dragresults will be larger at field scale owing to the higher value of Reynolds numbers, whichreached a maximum of Re max = 7 . × in the numerical simulations.Figure 10a shows the linear horizontal motion, which is mostly unchanged from theperturbation theory result. The magnitude of variable submergence is inertia-driven andthus very similar to the viscous analytical result shown in figure 10b.The drift amplification for field scale simulations using non-viscous drag shown in figure11 is greater than the perturbation theory result based on viscous drag, and even more so he increased wave-induced drift of floating marine litter (a) (b) Figure 8: Laboratory scale numerical simulation results using non-viscous drag for mag-nitudes of the first-order horizontal motion amplification X (1) (a) and the variable sub-mergence S (1) (b) as functions of dimensionless object size D/λ for different densityratios β = ρ o /ρ f , where the density ratio corresponding to each colour is listed in thelegend. Here, C m = β/ . Numerical and analytical solutions from perturbation theoryare denoted by crosses and solid lines, respectively. Figure 9: Laboratory scale numerical simulation results using non-viscous drag for wave-induced drift amplification X (2) as a function of dimensionless object size D/λ and fordifferent density ratios β = ρ o /ρ f (see legend). Here, C m = β/ . Analytical solutionsusing viscous drag from perturbation theory are denoted by solid lines.than at laboratory scale. This is because the non-viscous drag force is now considerablysmaller than its viscous equivalent (taken outside the range of Reynolds numbers forwhich it is valid). The (tangential) drag force obtained for larger objects is lower fornon-viscous drag than for a viscous drag formulation, resulting in reduced resistance toincreased drift compared to the Stokes drift.Using the results from field-scale numerical simulations for non-viscous drag, a mdiameter object of density ρ p = 0 . g/cm leads to a increase in drift ( X (2) = 1 . ).0 Calvert et al. (a) (b)
Figure 10: Field scale numerical simulation results using non-viscous drag for magnitudesof the first-order horizontal motion amplification X (1) (a) and variable submergence S (1) (b) as functions of dimensionless object size D/λ for different density ratios β = ρ o /ρ f ,where the density ratio corresponding to each colour is shown in the legend. Field scalehere denotes a . Hz wave with a steepness of α = 0 . . Here, C m = β/ . Numericaland analytical solutions from perturbation theory are denoted by crosses and solid lines,respectively. Figure 11: Field scale numerical simulation results using non-viscous drag for the wave-induced drift amplification X (2) as a function of dimensionless object size D/λ fordifferent density ratios β = ρ o /ρ f (see legend). Field scale is modelled by a . Hz wavewith a steepness of α = 0 . . Here, C m = β/ . Analytical solutions using viscous dragfrom perturbation theory are denoted by solid lines.This is a significant increase compared to the Stokes drift infinitesimal objects wouldexperience. By comparison, a . m diameter object in the same wave field does notexperience any drift amplification ( X (2) = 1 ) and behaves as a perfectly Lagrangiantracer. he increased wave-induced drift of floating marine litter
5. Conclusions
In this paper, we have developed a model for the transport of spherical, finite-size,floating marine debris by deep-water waves. Using a Stokes-like expansion in wave steep-ness, we have derived closed-form solutions for the linear response and the wave-induceddrift of an object forced by regular waves and experiencing viscous drag. These closed-form solutions match numerical solutions of our model in the case of viscous drag. Ourmodel recovers the Lagrangian limit as object size tends to zero, meaning that smallobjects are simply transported with the Stokes drift of surface gravity waves.Through our perturbation solutions, we have identified two mechanisms for increaseddrift. The first arises from the change in magnitude of the linear orbits, especially itsvertical component. The second arises when an out-of-phase variable submergence isresolved in the horizontal direction by the slope of the free surface. The second mechanismrequires buoyancy and drag to be acting normal to the free surface, where the drag isrequired to create the phase difference that gives rise to the drift when averaged overthe wave cycle. In any realistic oceanographic scenario, an non-viscous drag is requiredin order for the drift amplification to be significant. To observe the predicted effect,we have carried out laboratory wave flume experiments for a range of object sizes anddensities (see appendix B). The experiments show that an increase in wave-induced driftoccurs. However, due to large experimental error, the present results have not been usedto validate the theoretical model or choice of physics contained within.The main driver for an increased drift is predicted to be an object’s size relative to thewavelength. Thus, in the real ocean, where wavelengths range from - m, increaseddrift will likely only be observed where shorter wavelengths are present, such as in gulfsor smaller seas. Modelling an object with a diameter of m and density of . g / cm floating on a wave with a s period and a steepness of α ≡ k a = 0 . , typical of amoderately steep wind wave, results in a % increase in wave-induced drift comparedto the Stokes drift for such a wave. In the same wave field, an object with a diameterof . m would not experience an increase in drift at all. High-quality experiments arerecommended at larger scale, covering a wider range of object sizes and considering theeffect of object shape. Insights from the present work should be useful in the developmentof more sophisticated models for tracking floating marine litter. Acknowledgement
TSvdB acknowledges a Royal Academy of Engineering Research Fellowship.
Declaration of interests
The authors report no conflict of interest.2
Calvert et al.
Appendix A. Equations of motion
Substituting (2.8) and (2.9) into (2.7), and (2.7) into (2.1) results in two second-orderdifferential equations in the ( n, τ ) coordinate system: ¨ τ p − (cid:0) − ( ∂ x η | x p ) Ξ p + ∂ xx η | x p Ξ p n p (cid:1) ¨ x p = 1 m (1 + C m,τ β ) F τ + n θ p ˙ n p + ( ˙ θ p ) τ p − ∂ x η | x p Ξ p (cid:0) ∂ tt η | x p + 2 ˙ x p ∂ tx η | x p + ( ˙ x p ) ∂ xx η | x p (cid:1) + n p h(cid:0) ∂ tx η | x p + ˙ x p ∂ xx η | x p (cid:1) p ˙Ξ p + (cid:0) ∂ ttx η | x p + 2 ˙ x p ∂ txx η | x p + ( ˙ x p ) ∂ xxx η | x p (cid:1) Ξ p io , (A 1)(A 2) ¨ n p + (cid:0) ∂ x η | x p Ξ p + ∂ xx η | x p Ξ p τ p (cid:1) ¨ x p = 1 m (1 + C m,n β ) F n − n θ p ˙ τ p − ( ˙ θ p ) n p + Ξ p (cid:0) ∂ tt η | x p + 2 ˙ x p ∂ tx η | x p + ( ˙ x p ) ∂ xx η | x p (cid:1) + τ p h(cid:0) ∂ tx η | x p + ˙ x p ∂ xx η | x p (cid:1) p ˙Ξ p − (cid:0) ∂ ttx η | x p + 2 ˙ x p ∂ txx η | x p + ( ˙ x p ) ∂ xxx η | x p (cid:1) Ξ p io ,where we have kept all the second-order time derivatives on the left-hand side. We nowhave two equations in terms of three second-order time derivatives, namely ¨ τ p , ¨ n p and ¨ x p , and require a third equation to solve the system. We obtain this third (kinematic)equation by taking the dot product of (2.7), in which we have substituted for ¨ θ p and ¨ η p from (2.8) and (2.9), and e x , giving: ¨ x p (cid:2) ∂ xx η | x p Ξ p (cid:0) n p + ∂ x η | x p τ p (cid:1)(cid:3) − ¨ τ p Ξ p + ¨ n p ∂ x η | x p Ξ p =Ξ p n − n p h(cid:0) ∂ tx η | x p + ˙ x p ∂ xx η | x p (cid:1) p ˙Ξ p + (cid:0) ∂ ttx η | x p + 2 ˙ x p ∂ txx η | x p + ( ˙ x p ) ∂ xxx η | x p (cid:1) Ξ p i − θ p ˙ n p − ( ˙ θ p ) τ p − ∂ x η | x p h τ p h(cid:0) ∂ tx η | x p + ˙ x p ∂ xx η | x p (cid:1) p ˙Ξ p + (cid:0) ∂ ttx η | x p + 2 ˙ x p ∂ txx η | x p + ( ˙ x p ) ∂ xxx η | x p (cid:1) Ξ p i + 2 ˙ θ p ˙ τ p − ( ˙ θ p ) n p io . (A 3) Appendix B. Wave flume experiments
B.1.
Set-up and data acquisition
A series of object tracking experiments were conducted in the Sediment Wave Flume inthe Coastal, Ocean and Sediment Transport (COAST) Laboratory at the University ofPlymouth, UK. The flume has length m, width . m, and was filled with water to . m depth, as shown in figure 12. A double-element piston-type wavemaker suppliedby Edinburgh Designs Ltd (EDL) was used to generate a wave packet with a spectralshape that linearly focuses to a Gaussian packet, A = a exp (cid:0) − ( x f − c g, t ) / σ (cid:1) , ata measurement zone centred x f = 9 . m from the rest position of the wavemaker.The wave packet was made as long as possible to make it quasi-monochromatic whilstavoiding reflection ( ǫ = 1 / ( k σ ) = 0 . ) with a steepness α = a k = 0 . and peakfrequency f = 1 . Hz.Despite our perturbation theory solutions being for periodic waves, we used quasi-monochromatic wave packets in our laboratory experiments because wave-induced trans-port is much easier to measure experimentally for wave packets (see van den Bremer et al. (2019) and Calvert et al. (2019) and the discussion in Monismith (2020)). In appendixC, we confirm that the slow modulation associated with the wave packet does not result he increased wave-induced drift of floating marine litter Piston-typewavemaker 9.75mxyz x Measurement zonePhotron CameraGauges Absorption zone
Figure 12: Experimental set-up used to track the motion of floating objects under wavemotion generated by a double-element piston-type wave maker at the COAST Labora-tory, University of Plymouth, UK.in any additional non-inertial behaviour of the object. As a result, our model predictionsfor periodic waves and the wave packets considered in our experiments are equivalent.We controlled the wavemaker using linear wave theory. Although sub-harmonic errorwaves at second order generated for wave packets (e.g. Nielsen & Baldock (2010); Orsza-ghova et al. (2014)) can lead to spurious wave-induced displacements (Calvert et al. et al.
Hz free surface elevation measure-ments. Five gauges were located close to the focus location at cm intervals, as shownin figure 12 . Two gauges were located significant distances before and after the focus loca-tion. After propagating through the measurement zone, the dispersed wave packets wereabsorbed by mesh-filled wedges within an absorption zone located at the downstream endof the wave flume. To ensure near-quiescent initial conditions for each experiment, thewater surface was allowed to settle for minutes between experiments. A Photron SA4high-speed camera captured the object motions at frames/s, resolution of by pixels, and shutter speed of / s. Optical distortion was removed using mmchequerboard images and MATLAB’s inbuilt image processing package.B.2. Matrix of experiments
In the experiments, we selected a peak frequency of f = 1 . Hz, corresponding toa wavelength of λ = 1 . m and non-dimensional water depth k d = 3 . . We thenvaried systematically the diameter D and the density ρ o of the spherical floating object,with values for the 16 experiments listed in table 2. Object size was limited by cameraresolution and the MATLAB tracking algorithm. Density was varied by filling hollowspheres with different ratios of epoxy to glass micro-ball filler. Each experiment wasrepeated five times. B.3. Data processing
B.3.1.
Free surface elevation
Wave packets were created from narrow-banded spectra to allow frequency filtering toseparate the linear and second-order sub-harmonic components in the wave gauge signal.A band-pass filter between . f and . f was used to extract the linear free surface4 Calvert et al.
Experiment D [m] ρ o [kgm − ] D/λ [%] β [-]1 0.051 508 5.1 0.512 0.051 551 5.1 0.553 0.051 620 5.1 0.624 0.051 703 5.1 0705 0.038 597 3.8 0.606 0.038 637 3.8 0.637 0.038 678 3.8 0.688 0.038 750 3.8 0.759 0.025 649 2.5 0.6510 0.025 678 2.5 0.6811 0.025 700 2.5 0.7012 0.025 809 2.5 0.8113 0.019 647 1.9 0.6514 0.019 679 1.9 0.6815 0.019 654 1.9 0.6516 0.019 807 1.9 0.81 Table 2: Matrix of experiments listing dimensional object diameter D , object density ρ o ,non-dimensional object diameter D/λ , and density ratio β = ρ o /ρ f .Figure 13: Time histories of object horizontal position for each experiment. Each panelshows the five repeated experiments in different colours. he increased wave-induced drift of floating marine litter A was calculated using the Hilbert transform of thelinear free surface elevation. Use of the measured envelope at the location where thetrajectories were measured, to calculate purely Lagrangian displacement, accounted forany dissipation or non-linear dispersion between the wavemaker and the zone of interest.B.3.2. Object tracking
Profile images of the floating white spheres were illuminated from various angles andcaptured by the Photron camera. The trajectories of the floating objects were trackedby identifying their position in each frame using a circle finding algorithm. The apparentsize of the circle in the image was used to calibrate the pixel scale against the knownsize of the sphere. This also reduced any errors from out-of-plane motion not capturedby the single camera. The horizontal components of the raw trajectories, repeated fivetimes, are shown in figure 13.Every effort was made to settle the sphere at the start of each experiment in order togive it a zero initial velocity. This was not completely possible due to air flows over thewater surface and slight disturbance from human touch. A linear fit in the time domain,assuming a constant pre-existing drift velocity, was used to remove motion before thearrival wave packet from the raw orbits in figure 13. The focus location was determinedas coinciding with the position of the maximum of the linearised vertical motion envelopeof the object. The difference in object location and exact focus location in the flume hadnegligible effect because of the very long wave packets used.The magnitudes of the linear response were determined by filtering the horizontal andvertical motion components with a band-pass filter of . - . f , followed by a Hilberttransform to obtain the envelope A . Note that frequency filtering was only applied tovelocities, and numerical integration was used to calculate displacements. The maximummagnitude of the envelope was then normalised by wave amplitude a to obtain X (1) and unity subtracted from the normalised vertical motion to give S (1) (the normal andvertical directions equivalent up to first-order accuracy). We were not able to extractthe linear phase from the experiments because exact spatial and temporal matching ofEulerian wave-gauge data and Lagrangian object positions could not be achieved. A low-pass filter at . f was used to extract the sub-harmonic horizontal velocity component.The drift value X (2) was then determined by subtracting the Eulerian return flow fromthe maximum value of the sub-harmonic horizontal velocity component flow and dividingby the Stokes drift.B.4. Comparison between theory and experiments
B.4.1.
First-order in wave steepness: O ( α ) Figure 14 presents the first-order magnitudes | X (1) | and |S (1) | as functions of dimen-sionless diameter ( D/λ ) for each experiment, with colour corresponding to density ratio.Comparison is made with numerical solutions of our model for non-viscous drag and an-alytical solutions using viscous drag. Overall, the horizontal motion in figure 14a is ofsimilar magnitude to what is theoretically predicted ( X (1) ) with some variability, asquantified by the error bars. We note that a decrease of a few percent in the numericalsimulation solutions to | X (1) | is equivalent to a (small) dimensional decrease in the hor-izontal motion less than mm. The first-order variable submergence |S (1) | in figure 14bincreases monotonically with dimensionless diameter ( D/λ ), as predicted by theory.The experiments do not show a consistent trend with density for either linear motioncomponent. We note that the densities are not equally spaced or the same for each sizesphere owing to practical constraints on filling the spheres with different ratios of epoxyto glass micro-ball filler (see table 2 for the experimental matrix). The error bars shown6 Calvert et al. (a) (b)
Figure 14: Magnitude of the first-order motion as a function of non-dimensional objectsize
D/λ for different density ratios (see legend): analytical solution with viscous drag(solid lines) and experiments (circles). The density ratios for the numerical solutions arelisted in the legend; density ratios for the experiments are labelled using the same colourscale. The error bars are obtained from repeated experiments and correspond to twostandard deviations.for each experiment, which are twice the standard deviation of the five repeats, are largeenough to mask any trend in density. Although we could measure the overall density ofthe spheres accurately, we emphasize that we were not able to measure its uniformitywithin the sphere.Errors could have arisen from various physical sources that can account for the rela-tively large standard deviations. The initial motion of the object was hard to eliminate.Air conditioning was switched off, but there were occasional air flows over the flume.The method of taking the value of sub-harmonic velocity at the peak of the wave packethas been shown numerically to match regular waves in appendix C. However, inertia atpacket scale can be seen in figure 15 as the velocity does not go to zero after the packetpasses. Although a 10-minute delay was prescribed between experiments to allow waterin the flume to settle, there may have been residual currents still present. The theoreticalmodel also has uncertainty, as can be seen in the sensitivity analysis in appendix E, whicharises from the choice of drag and added mass formulations, and the exclusion of certainphysics from the model, such as surface tension.B.4.2. Second-order in wave steepness: O ( α ) Figure 15 presents time histories of the normalised sub-harmonic horizontal objectvelocity component for all 16 experiments, having first removed motion ahead of thewave packet and the Eulerian mean flow associated with the wave packet. In all cases,the non-dimensional sub-harmonic horizontal object velocity exceeds or is very close tounity near focus, and has a Gaussian-like profile, reducing close to zero within about25 s either side of focus. The distributions are slightly skewed, with a faster rising limbthan falling. There is more variability after focus than before. Using the peak values fromfigure 15, figure 16 shows the dimensionless drift factor X (2) for each experiment as afunction of dimensionless diameter, with colour indicating density ratio. Drift increaseswith non-dimensional diameter and, as for the first-order results, the trend with densityis unclear from the experiments and masked by substantial variability. We note that thedensity of floating plastic in the ocean typically has a small range between - kg / m and may thus be a less important variable than object size. The trend with object he increased wave-induced drift of floating marine litter X (2) exp = ( v (2) | t =0 x − u (2) x | t =0 ) / ( u s | t =0 ) where u s | t =0 = ω k a . The mean of the five repeated experiments isshown as a continuous red line, and the confidence band corresponding to two standarddeviations is shaded in grey, with five lines overlaid for each individual experiment.size is consistent between experiments and theory, both presenting a similar increase withsize.The experiments show that sufficiently large floating objects experience an increasein wave-induced drift. However, the experimental results are not sufficiently accurate tovalidate the theoretical model. In future work, it is therefore intended to carry out moreexperiments aimed at validating the model. Appendix C. Wavepackets vs. periodic waves
We use numerical solutions (see §4) to the model developed in §2 to examine thedifference in predictions for objects subject to the quasi-monochromatic wave packets weuse in our experiments and periodic waves. The processing of the trajectory data from thenumerical simulations using wave packets was the same as for the experiments describedin appendix B. Figure 17 shows the almost identical first-order response as a function ofnon-dimensional object diameter at different density ratios for periodic waves (crosses)versus wave packets of the same bandwidth as in experiments (circles). Figure 18 showsthe corresponding second-order drift amplification factors. Very slight differences are onlypredicted for larger object sizes for which the role of inertia is more dominant. For wave8
Calvert et al.
Figure 16: Second order drift amplification factor X (2) as a function of non-dimensionalobject size for different density ratios (see legend): analytical solution with viscous drag(solid lines) and experiments (circles). The density ratios for the numerical solutions arelisted in the legend; density ratios for the experiments are labelled using the same colourscale. The error bars are obtained from repeated experiments and correspond to twostandard deviations. (a) (b) Figure 17: Numerical predictions of the magnitude of the first-order horizontal motionamplification X (1) (a) and the variable submergence S (1) (b) as functions of dimensionlessobject size D/λ for non-viscous drag and for different density ratios β = ρ o /ρ f (seelegend). In the figure, periodic waves are denoted by crosses and wave packets of thesame bandwidth as in the experiments by circles.packets, a slightly smaller drift motion is predicted, because the time required for inertialobjects to reach steady state is longer for larger objects. he increased wave-induced drift of floating marine litter Figure 18: Numerical predictions of the magnitude of the second-order horizontal motionamplification X (2) as functions of dimensionless object size D/λ for non-viscous dragand for different density ratios β = ρ o /ρ f (see legend). In the figure, periodic waves aredenoted by crosses and wave packets of the same bandwidth as in the experiments bycircles. Appendix D. Limiting behaviour of the numerical solutions
To confirm the model developed in §2 is correct, including its cumbersome coordinatetransforms, we examine the perfectly Lagrangian limit (§D.1) and the small-object limit(§D.2) of its numerical solutions obtained using MATLAB’s ODE15s solver.D.1.
The Lagrangian limit
To obtain the Lagrangian limit, we replace the forces on the object by the accelerationsa Lagrangian particle would experience under linear periodic waves: ¨ x p = a ω sin( ϕ ) exp ( k z p ) , ¨ z p = − a ω cos( ϕ ) exp ( k z p ) , (D 1a,b)where ϕ = k x p − ω t + ϕ . The accelerations are then mapped to the translating coordi-nate system and expressed in the ( n, τ )-directions. The system is then solved numericallyin ( n, τ )-coordinates and the results mapped back onto ( x, z )-coordinates, providing con-firmation our transformations are correct. As shown in figure 19, we obtain the correctamplitude of the vertical and horizontal linear motion and the correct Stokes drift.D.2. Small-object limit
As object size tends to zero, D → , the solution should recover the behaviour of aperfectly Lagrangian tracer. This has been explicitly checked by numerically solving foran object of non-dimensional diameter D/λ = 1 × − , which results in X (1) = 1 . , S (1) = 0 . and X (2) = 1 . .0 Calvert et al. (a) (b)(c) (d)
Figure 19: Trajectory of a perfectly Lagrangian tracer obtained using a numerical solutionof the present model with forcing provided by (D 1). The top two panels (a, b) displaythe horizontal and vertical motions x p ( t ) and z p ( t ) , with the blue dashed line showing thetheoretical Stokes drift displacement and the red lines the superimposed wave amplitudes.The bottom two panels (c, d) show the tracer particle positions in the ( n, τ )-coordinatesystem. Appendix E. Alternative drag and added-mass formulations
This appendix examines several alternative approaches to modelling the drag (§E.1)and added-mass (§E.2) forces on a floating object. Results are obtained from numericalsolutions at laboratory scale conditions as in §4.E.1.
Drag
Although drag on a fully submerged sphere away from a free surface and in steady flowis well defined across a large range of Reynolds numbers (e.g. Morrison (2013)), thedrag force on a partially submerged, floating object in the unsteady flow field arisingfrom surface waves is not. To understand the implications for our model’s predictions,we consider the following drag formulations: viscous drag with C d = 24 / Re , non-viscousdrag with C d = C d (Re) based on Morrison (2013), and turbulent drag with C d = 1 / . he increased wave-induced drift of floating marine litter Viscous drag: C d = 24 / Re For the viscous drag coefficient C d = 24 / Re , we consider three cases: a case based onsubmergence-dependent and thus time-varying projected area A PA ( t ) = ( A s ,n ( t ) , A s ,τ ( t )) ,as in the paper, a case that ignores the time-dependence and sets A PA = A (0)PA ≡ A PA ( s (0) ) , and a case that is based on the time-varying, direction-independent sub-merged surface area A SA ( t ) . To compute the drag force, we use (2.14) and (2.16). For asphere, the submerged surface area A SA ( t ) = πDs ( t ) . We normalize this by the surfacearea of a sphere A FS = πD , so that ˆ A SA ( t ) = s ( t ) /D and replace both ˆ A s,τ in (2.14)and ˆ A s,n in (2.16) by ˆ A SA . As a result of this normalization, the drag forces on a fullysubmerged sphere based on projected area and based on submerged area are equal.The first-order horizontal motion remains unchanged and so is not presented here.Variable submergence and second-order drift solutions are shown in figure 20. It is evidentthat inclusion of time-varying submergence in the projected area and replacing projectedby submerged area has a negligible effect on the first-order submergence and only a veryminor effect on the drift.E.1.2. Non-viscous drag: C d = C d (Re) For the non-viscous drag coefficient, which is based on a fit to experimental data for afully submerged sphere (4.2) (from Morrison (2013)), we consider two cases. First, we setthe drag to be proportional to the submergence-dependent, time-varying projected area A PA ( t ) , which is the approach used in the paper. Second, we ignore the time dependenceand use the projected area of the sphere without waves A PA = A (0)PA ≡ A PA ( s (0) ) .Again, the first-order horizontal motion is unchanged and not presented here. Themagnitude of the variable submergence and the drift are presented in figure 20. Thevariable submergence responses in these two cases are very similar to each other and tothe viscous drag cases discussed above. The solutions for drift are similar to the viscoussolution for small objects, diverging as the object size increases. For larger objects, thedrift is significantly larger than when modelled with viscous drag. This is caused by therelative reduction in the drag force. There is a slight increase in drift when the projectedarea is time dependent.E.1.3. Turbulent drag: C d = 1 / We capture the turbulent-drag limit by setting C d = 1 / , which we consider to be thepractical large-object limit of (4.2). We consider two cases; similar to non-viscous drag, wehave used the time-dependent projected areas A PA and also consider time-independentprojected areas of a sphere in the absence of waves A (0)PA .Again, the linear horizontal motion is unchanged and so not presented. The variablesubmergence is slightly decreased when compared with the viscous and non-viscous casesfor larger object sizes, which results in a smaller adjusted Stokes drift. The increasein drift is larger than the viscous cases because of the relative reduction in drag, butsmaller than the non-viscous cases. The comparative increase observed when using time-dependent submerged projected area, seen for non-viscous drag, can also be observedwith turbulent drag. E.2. Added mass
Maxey & Riley (1983) derived the added mass for a fully submerged sphere in a low-Reynolds regime and found the added-mass coefficient to be C m = 1 / . Hulme (1982)studied a floating hemisphere under wave forcing and derived independent surge andheave added-mass coefficients as functions of non-dimensional object size k D/ . The2 Calvert et al. (a) (b)
Figure 20: The effect of alternative drag formulations on the numerical predictions of first-order variable submergence S (1) (left) and second-order drift X (2) (right) as a functionof non-dimensional object size D/λ for a density ratio β = 0 . at laboratory scaleconditions. The lines correspond to different drag formulations, labelled in the legend,using either viscous drag ( C d = 24 / Re, solid lines), non-viscous drag ( C d = C d ( Re ) ,dashed lines) or turbulent drag ( C d = 1 / , dotted lines), which either vary with the time-varying projected area in the respective directions ( A ( t ) = A PA ( t ) ), with the constantprojected area in the respective directions ( A = A (0) PA ), or with the submerged surfacearea ( A ( t ) = A SA ( t ) ).range of non-dimensional object sizes in the present study is < k D/ ≤ . , whichcorresponds to added-mass coefficients in the range . ≤ C m,n ≤ . in heave and . ≤ C m,τ ≤ . in surge (Hulme 1982).We consider two categories of added-mass formulations: direction independent anddependent. In the first category, we consider C m = 0 , C m = 0 . representative of a sub-merged sphere in a low-Reynolds regime, and C m = 0 . β for an added mass that increaseslinearly with depth of submergence in the absence of waves but remains time independent.In the second category, we consider constant C m = (0 . , . representative of a hemi-sphere (Hulme (1982)), C m = 2 β (0 . , . so that the added mass recovers Hulme’s(1982) result for a hemisphere and is zero for an entirely unsubmerged sphere. Finally, weextend this to a submergence and time-dependent added mass: C m = 2(0 . , . s ( t ) /D .As for the different drag formulations, the first-order horizontal motion is insensitiveto our choice of added-mass formulation. Figure 21 shows the first-order variable submer-gence and drift responses obtained for the different added-mass formulations considered.The left panel of figure 21 shows the relative insensitivity of the variable-submergenceresponse to the different added-mass formulations. The variable submergence exhibits aslight increase when the added mass is directionally dependent and a function of sub-mergence. Drift, shown in the right panel of figure 21, is more sensitive to the choiceof added-mass formulation. Direction-independent formulations result in a smaller in-crease in drift compared to their direction-dependent counterparts. The smallest increasein wave-induced transport (excluding the special case of zero added mass C m = 0 ) is C m = 0 . β which is used to generate the analytical and numerical solutions presented inthe paper. REFERENCESArdhuin, F., Brandt, P., Gaultier, L., Donlon, C., Battaglia, A., Boy, F., Casal, he increased wave-induced drift of floating marine litter (a) (b) Figure 21: The effect of alternative added-mass formulations on the numerical predictionsof first-order variable submergence S (1) (left) and drift X (2) (right) as a function of non-dimensional object size D/λ for a density ratio β = 0 . at laboratory scale conditions fornon-viscous drag. The lines correspond to different added-mass formulations, describedin the legend, with solid lines for directionally independent added-mass formulations,and dashed lines for added-mass formulations decomposed into normal and tangentialdirections. T., Chapron, B., Collard, F., Cravatte, S. & others
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