nearshore sticky waters

Ocean Modelling 80 (2014) 49–58

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Ocean Modelling

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Nearshore sticky waters

http://dx.doi.org/10.1016/j.ocemod.2014.06.0031463-5003/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Mathematics, University of Arizona,Tucson, AZ 85721 USA. Tel.: +1 520 621 4367.

E-mail address: [email protected] (J.M. Restrepo).

Juan M. Restrepo a,b,c,⇑, Shankar C. Venkataramani a, Clint Dawson d,e

a Department of Mathematics, University of Arizona, Tucson, AZ 85721 USAb Department of Atmospheric Sciences, University of Arizona, Tucson, AZ 85721 USAc Department of Physics, University of Arizona, Tucson, AZ 85721 USAd Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX 78712, USAe Institute of Computational and Engineering Sciences, University of Texas, Austin, TX 78712, USA

a r t i c l e i n f o

Article history:Received 24 January 2014Received in revised form 21 April 2014Accepted 16 June 2014Available online 2 July 2014

Keywords:Oil slickPollutant transportShallow water flowsMixing and dispersionWaves and currents

a b s t r a c t

Wind- and current-driven flotsam, oil spills, pollutants, and nutrients, approaching the nearshore will fre-quently appear to slow down/park just beyond the break zone, where waves break. Moreover, the portionof these tracers that beach will do so only after a long time. Explaining why these tracers park and at whatrate they reach the shore has important implications on a variety of different nearshore environmentalissues, including the determination of what subscale processes are essential in computer models forthe simulation of pollutant transport in the nearshore. Using a simple model we provide an explanationfor the underlying mechanism responsible for the parking of tracers, not subject to inertial effects, therole played by the bottom topography, and the non-uniform dispersion which leads, in some circum-stances, to the eventual landing of all or a portion of the tracers. We refer to the parking phenomenonin this environment as nearshore sticky waters.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Oil from spills, red tides, flotsam and other suspended and sur-face tracers approach the nearshore, carried by winds and currents.It is not uncommon, however, that these debris and tracers slowdown and park themselves, somewhere beyond the break zone(See Fig. 1); eventually, a portion of these reach the beach zoneby the action of turbulence and tidal effects, in combination withinertial effects on the debris. The tendency of tracers to park them-selves in certain areas of the Great Barrier Reef has been noted.Wolanski and Spagnol (2000), who reported the phenomenon,and denoted it as ‘‘sticky waters’’. Though we will not be discussingestuarine environments and the mechanism at play in the GreatBarrier Reef situation may be different from the nearshore case,we will borrow this terminology and refer to the phenomenonwe investigate in this paper as ‘‘nearshore sticky waters’’.

Of obvious environmental, economic, and social importance,understanding why nearshore sticky waters occur is also funda-mental to improved environmental assessments of coastal settings.Moreover, as part of a larger research agenda aimed at improvingmodels for pollutant transport in ocean general circulation models,nearshore sticky waters offers a field-verifiable problem with

which to test contaminant advection reaction and dispersionmodels.

The focus is on tracer transport phenomena, with length scalesseveral times larger than the depth and temporal scales of hours,weeks. That is, we are mostly concerned with large-scale pollution‘‘disasters’’, such as large-scale red tides, significant oil spills, etc.Although we consider long time and space scales, we cannot ignoredepth dependent features of the flow and the transport of tracers.The tracer may be buoyant but not necessarily entirely residing onthe surface of the ocean; We therefore consider a layered (insteadof simply depth averaged) model for the tracer and account for thevertical structure of the advective velocity. We defer considerationof tracers with non-trivial inertial effects to a separate study. Obvi-ously, tracers advect and diffuse in the alongshore direction as wellas in the cross-shore direction. In fact, advection/diffusion in thelongshore direction is usually more intense in many non-estuarineenvironments. However, if we consider a situation where the long-shore variations of the tracer concentrations are small, the diver-gence of the flux in the longshore direction is negligible, and it isappropriate to consider a one-dimensional problem in the cross-shore direction.

Nearshore sticky waters will refer to the slowing down or theparking of the tracer approaching the shore. In a sticky water situ-ation the center of mass of the incoming tracer that is approachingthe shore at advective speeds will experience a partial or totalslowing down. Whatever tracer amounts reach the shore will do

Fig. 1. A red tide event, off the coast of Florida. The event occurs nearly annuallyalong the state’s Gulf Coast. Image courtesy of P. Schmidt, Charlotte Sun. For anexample that has more surfzone wave action, see Fig. 1 of Grant et al. (2005).

50 J.M. Restrepo et al. / Ocean Modelling 80 (2014) 49–58

so by the action of dispersive effects, usually higher inside thebreakzone than in deeper waters. In the cross-shore direction, largescale currents are typically weak, close to the shore. In a wave-dominated nearshore setting, the typical advection velocity wouldbe the residual flow due to the waves, the Stokes drift velocity. Thelength scales are those of the long waves, i.e., waves which havewavelengths that are large when compared to the depth. The diffu-sive length scale is typified by large-scale eddies; if the break zoneis a significant source of mixing, the length scale would be the dis-tance between the start of the breaking of the waves and the shore.When advective and diffusive effects are those in balance, the dif-fusive time scale is large, in the order of hours. There is consensusthat in wave-dominated beaches the dissipation of waves is differ-ent inside and outside of the breakzone, the latter being consider-ably smaller than the former. Mei (1989), Chapter 10, describes thetheoretical development of a model for wave action dissipation,based upon dimensional analysis and homogeneous turbulenceconcepts (see also Svendsen and Putrevu, 1994, for further devel-opments). The model used in Uchiyama et al. (2009) is that ofThornton and Guza (1983), which is one of several based uponhydraulic jump parametrizations. Analysis of field data of the dis-persion of tracers in the nearshore suggest that the diffusivity ismuch higher in the break zone than outside. Dispersion estimatesbased upon the dimensional analysis model of Svendsen andPutrevu (1994) are off by orders of magnitude, when comparedto field data (see Feddersen, 2012a). A possible explanation forthe discrepancy might lie in the fact that the dimensional parame-trization is based upon homogeneous turbulence conditions and ismore typical of the smaller scale vertical diffusion, rather than thelarger eddy-scale transverse diffusion (Feddersen, privatecommunication).

The basic depth-averaged hydrodynamics, appropriate to thesescales, are captured by the similarly scaled vortex force model inMcWilliams et al. (2004) (see McWilliams and Restrepo, 1999;Restrepo, 2001 for background and Lane et al., 2007 for a compar-ison of this ‘‘vortex force’’ model and the ‘‘radiation stress’’ alterna-tive. See also Smith (2006)). In the following form, the model hasbeen used to study nearshore problems, such as longshore currents(in Uchiyama et al., 2009), and rip currents (in Weir et al., 2011).The depth-averaged momentum balance reads:

Dvc

Dt¼ �grf� v½uSt�? þ N; ð1Þ

where v is the vorticity of the depth-averaged velocityvcðx; y; tÞ :¼ ðuc; vcÞ, and N encompasses bottom drag, wind forcing,and dissipation; it also encompasses momentum transfers from

wave breaking to the current momentum (see Restrepo et al.,2011). The vortex force is the second term on the right hand side,which couples the residual flow due to the waves to the rotationin the current vc . The depth-averaged Stokes drift velocity isdenoted by uSt :¼ ðuSt ;vStÞ; the operator ? is used to obtain½uSt�? ¼ ð�vSt;uStÞ.

The continuity equation reads

@f@t¼ � @f

c

@t�r � ½Hðvc þ uStÞ�; ð2Þ

where H ¼ fc þ Hðx; tÞ is the local water column depth andfc ¼ fþ f is the composite sea elevation; f is the quasi-static seaelevation. The waves are found via conservation equations for thewave action A, and wavenumber k. For the wave action, the equa-tion is

@A@tþr � ðCGAÞ ¼ NA; ð3Þ

where NA is the loss term and CG is the absolute group velocity,

CG ¼ vc þ R

2k2 1þ 2kHsinh 2kH

� �k: ð4Þ

The relative frequency is x ¼ vc � kþ R, where the frequency satis-fies the dispersion relation R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigk tanhðkHÞ

p. The wave action, the

Stokes drift velocity and the quasi-static sea elevation response aregiven by

A :¼ 12R

qgA2; uSt :¼ 1

qHAk; f ¼ � A2k2 sinhð2kHÞ ; ð5Þ

respectively. A is the wave amplitude and k is the magnitude of thewavenumber k. The wavenumber conservation equations are

@k@tþrðRþ vc � kÞ ¼ 0: ð6Þ

The evolution equation for a tracer h, (see McWilliams et al., 2004),is

@h@tþ ðvc þ uStÞ � rh ¼ Nh; ð7Þ

where Nh is the tracer dispersion term.The simplest situation we consider is that of a flow with mean

shoreward-directed velocity, transporting the pollutant towardland, flowing over a sloped and featureless bathymetry. We con-sider a nearshore domain that has only transverse extent x anddepth z; the water column increases in depth, away from the shore.Consideration of the actual mechanism that is generating the cur-rent field makes the basic story presented here richer, but isbeyond the scope of this paper. Instead we focus on the basic kine-matics of the tracers.

The advecting mean current, with a shore-directed component,might consist entirely or partially of a wave-induced flow, theStokes drift velocity (see Mei, 1989). For specificity we will assume,in fact, that the advective mean current is exclusively composed ofthe Stokes drift and that these are generated by shore-directedwaves. (As the reader will eventually surmise we could haveassumed instead the presence of currents not associated withwaves, or even considered the case where both wave-inducedflows and currents are present; nearshore sticky waters conditionsdo not require the presence of wave-generated currents). Accord-ing to (1), however, this Stokes drift will not generate a vortexforce. If the velocity at the shore end, at x ¼ 0, is zero, the cross-shore component of the depth-averaged current ucðx; tÞ must beequal and opposite to the cross-shore component of the depth-averaged Stokes drift velocity (in Uchiyama et al., 2009 we recog-nized it as the anti-Stokes current, but more generally it is theundertow current. See Lentz and Fewings, 2012 for more details

J.M. Restrepo et al. / Ocean Modelling 80 (2014) 49–58 51

concerning the anti-Stokes current. This is a very readable intro-duction to the nearshore flow environment). The Stokes drift veloc-ity depends on the wave action and the wavenumber by (5) and(6). In Uchiyama et al. (2009), Fig. 1, are shown the somewhat typ-ical slight increases in the wave action, as the waves approach thebreakzone, followed by their partial or full dissipation due to wavebreaking, with an ensuing transfer of momentum to the currents.The resulting cross-shore component of the Stokes drift velocityincreases as the waves shoal, and then diminishes drastically orbecomes insignificant. If a separate current can be identified inthe flow, and waves are present, an ensuing transfer of momentumensues when the waves break. The currents are also subjected tobottom/form drag, sea elevation gradients, etc.

Momentum transfers from the breaking wave field to the cur-rents via N in (1), can generate currents, however, we will assumein what follows that velocities thus so generated are inconsequen-tial. The depth-dependence of the current will prove essential: TheStokes drift drops off exponentially as a function of the depth. Thevelocity at z ¼ �HðxÞ, must satisfy W ¼ 0;U :¼ ðU;VÞ ¼ 0, where Wand U :¼ ðU;VÞ are the vertical and transverse three-dimensionaland time dependent Eulerian velocity components, respectively.(The boundary condition W ¼ �U � rH, applies to an inviscid flow).We assume that the wavenumber k ¼ �kx, where x is the unittransverse vector, pointing away from the beach. With thisassumption, and (4) and (5) it is understood that reflections fromthe shore are insignificant, thanks to dissipative processes embod-ied by NA.

Since vc þ uSt ¼ 0 near the beach, the evolution of the depth-averaged tracer there is purely diffusive, by (7). It is generallyagreed that diffusivity is small outside of the breakzone. If the dif-fusivity near the shore were insignificant, there would be littlechange of the depth-averaged tracer distribution. However, theflow in the breakzone is complex. At the larger scales these turbu-lent and boundary layer effects manifest themselves as enhanceddiffusivity and viscosity.

2. The model

In this paper, we propose a kinematic model for oil transport inthe nearshore, and in the model we include the following effects(1) A mean advective flow that is depth dependent and is shoredirected on the surface, (2) A dispersion model that models thetransport due to the fluctuating component of the velocity, andaccounts for the enhanced diffusion in the break zone, (3) A simpli-fied oil model which includes the effects of buoyant stratificationof the oil into a surface slick and a bulk suspension, and (4) Anexchange interaction which allows the bulk suspended oil to resur-face and turbulence to entrain the surface oil into the bulk. Themodel is purely kinematic, and applies for various ‘‘physical’’ mod-els of the underlying mechanisms with appropriate parameteriza-tions of the advective velocity, dispersion, mixing layer depth andexchange rate.

Fig. 2. Schematic cross-section of the model domain. A light, thin oil slick sits atop the ocas a concentration. The distance from the shore, at x ¼ 0, is denoted by x. The break zonedescribed by z ¼ �HðxÞ.

Fig. 2 depicts the physical domain. A list of symbols usedappears in Table 1. The quiescent ocean level is at z ¼ 0, the basinis bounded below, at z ¼ �HðxÞ. The domain extends from x ¼ 0,the shore end, where the depth is H0 P 0, to x ¼ X where the depthis H1 P H0. The bathymetry HðxÞ will be sloped and featureless:

HðxÞ ¼ H0 þmx; 0 6 x 6 X;

where H0 is the depth in the nearshore, and m P 0 is the slope. Wedistinguish two oceanic regimes in our problem: the high mixingsurf zone, corresponding to 0 6 x 6 L, and the deep ocean zone,from L < x 6 X. L is typically tens to hundreds of meters. The pollu-tant (for example, oil), or the tracer (for example, an algal bloom) issubject to buoyancy effects. Oil in the surface slick may beentrained by the action of wave breaking and turbulent mixing.The oil may also resurface, at a rate dependent on the size of thedroplets. We will assume that, in the most general case, there is avery thin layer of pure oil, riding on the ocean surface. This layer,which we will denote as the oil slick, has thickness sðx; tÞ, typicallymicrometric. Immediately below is a layer of ocean in which thebulk of the oil is found, in suspension. As depicted in Fig. 2, the layercontaining the suspended oil is assumed to have a maximum thick-ness P. (It is possible to estimate P in terms of the flow, but doing sois beyond the scope of this study. Here we treat P as a parameter).We will denote this oil in suspension as the interior oil. bðx; tÞ isthickness of an ‘‘equivalent’’ pure oil layer containing the sameamount of oil as the interior. Assuming that the interior oil is uni-formly distributed within the mixed layer, we have the equationof state

bðx; tÞ ¼ Bðx; tÞnðxÞ; ð8Þ

where B denotes the (dimensionless) volume fraction of the oil insuspension, and nðxÞ is the local depth of the mixed layer. Weapproximate nðxÞ as a smooth approximation to min ðHðxÞ; PÞ.

We now discuss the various mechanisms that transport the tra-cer (oil) and build a mathematical model for this process. Becausethe oil slick moves with the surface, the cross-shore component ofthe oil slick velocity uS � UStðx;0; tÞ :¼ USt , the Stokes drift velocity,

evaluated at the surface, USt � �A2kR. (The mean Eulerian velocityhas been set to zero). More realistically Stokes drift velocitydepends on x since the waves obey a dispersion relation that is afunction of x, via its dependence on the water column depth; how-ever, the results presented in the analysis that follows would onlychange quantitatively. (Idealized descriptions for simple shoregeometries and waves can be found in Lentz and Fewings(2012)). The (time-averaged) advective flux in the slick is thusgiven by uSsðx; tÞ. We will be using a simple proxy in place of theactual velocity Uðx; z; tÞ, consistent with the essential kinematicsdetailed in Section 1: Assuming a parabolic profile for the Lagrang-

ian mean velocity (See Fig. 3a), we get U ¼ USt 1þ 4zHðxÞ þ 3z2

HðxÞ2

� �by

requiring that U equals USt at z ¼ 0, equals zero at z ¼ �HðxÞ and

has zero depth average i.e.,R 0�H U dz ¼ 0. Undoubtedly, this is a

ean. The ocean’s mixed layer of thickness P is laden with oil droplets, accounted forextends to x ¼ L. The ocean surface is at z ¼ 0 and bottom topography is fixed and

Fig. 3. For the examples considered, (a) parabolic velocity profile uðx; zÞ, at H ¼ 20 m; (b) dispersion DðxÞ, over the whole domain. It increases substantially close to the shore,where the flow is more turbulent. (c) The normalized effective velocity vðxÞ=jUSt j in (12), for different mixing depths P. See Table 1 for parameters.


crude approximation of the real flow, consisting of a flow, withmean shoreward-directed velocity, and an undertow, but it is qual-itatively consistent with Pearson et al., 2009, Fig. 8, (ignoring massfluxes that shift the mean sea level away from z ¼ 0, cf., (5)). Aver-aging U over the mixed layer �nðxÞ 6 z 6 0, we get the bulk velocity

uBðxÞ ¼ USt HðxÞ � nðxÞ½ �2=HðxÞ2: ð9Þ

The (time-averaged) advective flux in the bulk is thus given byuBðxÞbðx; tÞ.

The velocity fluctuations about the mean contribute to aneffective dispersion (diffusivity) of the tracer. Field measurementsindicate that the vertical and horizontal diffusivities have differ-ent dominant mechanics and different magnitudes (seeFeddersen, 2012a,b and references cited in these). At scales ofthe surfzone itself, which are much larger, typically, than thewater column depth, the horizontal diffusivity is dominated byeddies at scales larger than the depth. Measurement, character-ization and parametrizations of the diffusivity tensor in the near-shore and the surfzone has come a long way in the last 20 years.Recent field measurements of large scale cross shore and longshore diffusivities are reported in Clark et al. (2010). Theseappear to be poorly captured in every respect by older estimatesderived by dimensional analysis arguments in Svendsen andPutrevu (1994) (see also Pearson et al., 2009) with regard tocross-shore diffusivities, for example. This is clear in Figure 13of Clark et al. (2010).

For the purpose of formulating a simple model for the stickywater phenomenon we will adopt a very crude cross shore disper-sion model. From field campaigns, Spydell et al. (2009) give a roughestimate of the long-term cross shore diffusivity in the bulk, of0:05 m2/s, away from the break zone, and 0:5� 1:75 m2/s in thenearshore, under mild sea conditions. (Noted in their report, how-ever, is that the diffusivity changes over time and it is generally lar-ger in the longshore direction). We assume that the interior oil andthe slick viscosities are equal, and given by the simple model

DðxÞ ¼ Deddy þ SðxÞDL; ð10Þ

where SðxÞ ¼ ð1þ exp½ðx� LÞ=w�Þ�1. This crude model is similar tothe one proposed in Rippy et al. (2013), which is inspired by fieldmeasurements. In the examples that follow L ¼ 200 m andw ¼ 20 m is the width of the transition of the sigmoid.Deddy ¼ 0:05 m2/s is the background eddy diffusivity, andDL ¼ 1:6 m2/s the enhanced diffusivity in the nearshore due to tur-bulence and wave breaking (Fig. 3b). Fick’s law gives a diffusive flux�DðxÞ @

@x s on the surface. In the subsurface region, the diffusive fluxis driven by gradients of the bulk concentration B but its effect on bis found via the equation of state, (8). Fick’s law along with integra-

tion in the depth of the mixed layer gives a diffusive flux�nðxÞDðxÞ @

@x B in the bulk.The last process we model is the exchange of material between

the surface and the bulk due to wave-mixing and buoyancy. Onlong time scales corresponding to averaging over many waves, asimple model for the net flux from the slick into the suspensionis the linear expression 1

sðxÞ ðð1� cÞs� cPBÞ, where c 2 ð0;1Þ is aparameter which sets the relative proportions of the oil in the slickand in suspension for vertical equilibrium, viz.,

sPB¼ c

1� c:

sðxÞ is the time scale for the vertical mixing. It is a measure of thetotal kinetic energy (TKE) in the fluctuating velocity field. Conse-quently dimensional analysis suggests that sðxÞ � P2=DðxÞ, the dif-fusion time scale over the typical depth of the mixed layer. (SeeTkalich and Chan, 2002, and references contained therein, for analternative formulation of this exchange flux and for more detailson the mixing/buoyancy physics).

Neglecting sources and sinks of oil, the conservation laws for sand b, obtained from the above fluxes are:

@s@tþ @½uSðxÞs�

@x¼ �ð1� cÞs� cPB

sðxÞ þ @

@xDðxÞ @s

@x

� �; ð11Þ

@b@tþ @½uBðxÞb�

@x¼ ð1� cÞs� cPB

sðxÞ þ @

@xnðxÞDðxÞ @

@xB

� �;

) @b@tþ @½vðxÞb�

@x¼ ð1� cÞs� cPB

sðxÞ þ @

@xDðxÞ @b

@x

� �; ð12Þ

where we have used (8) and

vðxÞ :¼ uBðxÞ þ DðxÞ 1nðxÞ

dnðxÞdx

;

(see Fig. 3c). Note that the ‘‘effective’’ velocity v of the oil in suspen-sion is not the depth-averaged velocity of the flow uB; rather it has acontribution that depends on the eddy diffusivity DðxÞ and also thebottom topography HðxÞ if HðxÞ < P, the maximum depth of themixed layer.

To conserve total mass we specify zero flux conditions at x ¼ 0and x ¼ L:

uSðxÞs� DðxÞ @@x

s ¼ 0 at x ¼ 0 and L;

vðxÞb� DðxÞ @@x

b ¼ 0 at x ¼ 0 and L: ð13Þ

Eqs. (11)–(13), together with specified initial conditions sðx; t ¼ 0Þand bðx; t ¼ 0Þ give a complete mathematical model which can besolved numerically. We denote this model as the PDE model.


3. Finite dimensional reduction of the model

By specifying appropriate functional forms for the diffusivityDðxÞ and the vertical mixing time sðxÞ in the PDE model Eqs.(11)–(13), we can describe a variety of scenarios. Conversely, a nat-ural question is the extent to which simple choices for the func-tions, DðxÞ and sðxÞ, as motivated in the previous section, yieldresults that are valid for real physical flows. In this section wedemonstrate that the results we obtain from the PDE model arerobust, and depend only on gross features of DðxÞ and sðxÞ, butnot on fine details of these functions.

To this end we first derive a reduced finite dimensional modelthat describes the evolution of Gaussian pulse initial conditions.The ratio of the vertical and horizontal time scales is given byðP2=DÞðUSt=XÞK 0:01 for P � 1 to 6 m, even in the deep oceanwhere D is small. Thus, the vertical tracer distribution equilibratesvery rapidly. We can determine s and b in terms of the total oilq ¼ sþ b by setting the exchange flux to zero yielding

s � cPcP þ ð1� cÞn q; b � ð1� cÞn

cP þ ð1� cÞn q:

Substituting in the equations for s and b, we obtain

@q@t¼ @

@xDðxÞ @q

@x� ueðxÞq

� �; ueðxÞ ¼

cPuS þ ð1� cÞnvðxÞcP þ ð1� cÞn : ð14Þ

Observe that these equations no longer depend on sðxÞ, showingthat as long as the mixing in the vertical direction is rapid on thescale of the horizontal transport, the precise choice of the mixingtime is irrelevant.

3.1. Initial asymptotics: pulse solutions

Setting ‘q � 1ffiffiffiffiffiffiffiffiffiffiffiffi2pr2ðtÞp exp � ðx�lðtÞÞ2

2r2ðtÞ

h idescribes an evolving unit

mass Gaussian pulse with mean (peak location) lðtÞ and variance

(width) r2ðtÞ. We expect this Gaussian pulse ansatz to be valid as

long as the peak of the pulse is ‘‘far’’ from the shore on the scale

of its width, lðtÞ � rðtÞ. Using this ansatz in (14) and computing

the first and second moments yields the ODE model,

@

@tl ¼ ueðlÞ þ

@D@xðlÞ; @

@tr2 ¼ 2DðlÞ; ð15Þ

where we have assumed that D and ue are slowly varying on thescale of the width rðtÞ. This model works best at early times, whenrðtÞ=lðtÞ � 1, so the pulse does not ‘‘feel’’ the boundary conditionat the shore x ¼ 0.

3.2. Steady states

Solving (14) for long times t !1, we obtain the steady state

q! q1 ¼ C expZ

ueðxÞDðxÞ dx

� �; C is a normalizing constant:

ð16Þ

q1 has a maximum at the x values where ue changes sign. Outsidethe break zone (DðxÞ is small), or in sufficiently deep water (nðxÞ ¼ Pso n0ðxÞ ¼ 0), we have vðxÞ ¼ uBðxÞ < 0. Thus ue is negative(shoreward).

If ue from (14) is negative for all x the maximum of the steadystate distribution of q is at x ¼ 0, and this is the case if H0 is suffi-ciently large (see Fig. 3c). However if P > H0, then the depth aver-aged bulk velocity uB is zero at x ¼ 0. In this case,vð0Þ ¼ Dð0Þm=H0 > 0 so the effective velocity ue > 0 for suffi-ciently small c (see (14)). In this scenario ue will change sign awayfrom x ¼ 0 (see Fig. 3c).

Of course, in a physical situation, we do not get to choose c. Anatural question is in what circumstances is the maximum of thesteady state distributions ‘‘significantly’’ away from the shore, i.e.the location of the maximum is on the scale of L, the width ofthe break zone. As we will see below, this corresponds to a signif-icant slowing of the tracers as they approach the shore. The follow-ing argument gives estimates of the relevant parameter regime.

From (9), we see that uB ¼ 0 for all x < C ¼ ðP � H0Þ=m. Thismotivates the definition of the non dimensional parameter

b ¼ CL¼ P � H0

H1 � H0

� �XL: ð17Þ

If b < 0 then P < HðxÞ for all x. If 0 6 b 6 1, the point where P ¼ HðxÞis in the break zone, and if b > 1, this point is outside the breakzone.

For x < C, i.e. the region where HðxÞ < P;uB ¼ 0 and the defini-tion of v in (14) yields

ueðxÞ > 0() ð1� cÞDðxÞm P �cPUSt ¼ cPjUStj;

where we note that the stokes drift USt is negative as it is directedshoreward. This condition can be rearranged to give

DðxÞ > c1� c

� �PjUStj

m Dthreshold: ð18Þ

Using typical values for P;USt and m, we get PjUSt jm � ð0:25� 5Þm2/s

so unless c is incredibly small, Dthreshold � Deddy the eddy diffusivityoutside the break zone, justifying our claim above that ue < 0(shoreward) outside the break zone. To get that ueðxÞ > 0 for x onthe scale of L, we thus need two conditions to hold. From (17), weneed b J Oð1Þ, and from (18) and (10), we need thatDL P Dthreshold. This motivates the definition of a second dimension-less parameter

d ¼ DL

Dthreshold¼ ð1� cÞDLðH1 � H0Þ

cPjUStjX: ð19Þ

3.3. Approach to the steady state

In terms of the steady state distribution q1, the long timeasymptotics of (14) are

qðx; tÞ � q1ðxÞ þ f ðxÞe�k1t ;@

@xDðxÞ @f

@x� ueðxÞf

� �¼ �k1f ð20Þ

�k1 is the largest negative eigenvalue of the above operator withno-flux boundary conditions at x ¼ 0 and L, and f is a correspondingeigenfunction. Eqs. (15) and (20) give finite dimensional approxima-tions to the short and long time behavior of the solutions. We canobtain a uniformly valid approximation by matching the twodescriptions.

We need to switch from the short time to the long time descrip-tion once the pulse (Eq. (15)) is close enough to the shore that it feelsthe effect of the no-flux boundary condition. We thus distinguishshort and long times by comparing l2ðtÞ=r2ðtÞ with a fixed thresh-old a, which we set at 4 in the rest of the paper (See discussionbelow). This corresponds to changing from the initial to the finalasymptotics when 2:5% of the total oil has ‘‘beached’’ and 84% ofthe oil is within 3rðtÞ from the shore. At the time te determined bylðteÞ ¼

ffiffiffiap

rðteÞ we switch from using (15)–(20) to describe qðx; tÞ,i.e., for t > te

qðx; tÞ � q1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pr2ðteÞ

s1

1þErfðffiffiffiffiffiffiffiffia=2

pÞ

exp �ðx�lðteÞÞ2

2r2ðteÞ

" #�q1

" #e�k1ðt�teÞ;

ð21Þ

Table 1Table of symbols.

Name Symbol Units

Transverse coordinate x ¼ ðx; yÞ mDepth coordinate z mCross-shore coordinate x mAlongshore coordinate y mTime t sUpward unit vector z –Cross-shore unit vector x –Gravity g m/s2

Exchange relaxation time s sMinimum depth H0 mMaximum depth H1 mTopographic extent X mSurf zone extent L mMixing layer thickness P mTransition width w mBottom gradient m m/mBottom topography HðxÞ mMean (current) sea elevation fc ¼ fþ f m

Quasi-static sea elevation correction f m

Sea elevation f mTotal water column H ¼ Hðx; tÞ þ fc m3D velocity Uðx; z; tÞ ¼ ðU;V ;WÞ m/sDepth-averaged Eulerian velocity vcðx; tÞ ¼ ðuc ; vcÞ m/sDepth-averaged Stokes drift velocity uStðx; tÞ ¼ ðuSt ;vStÞ m/sCross-shore Stokes drift, at the

surfaceUSt m/s

Slick advective velocity uS m/sBulk advective velocity uBðxÞ m/sEffective bulk oil velocity vðxÞ ¼ uBðxÞ þ DðxÞ 1

nðxÞdnðxÞ

dxm/s

Oil mixed depth n �minðHðxÞ; PÞ mAverage oil velocity (ODE) ue m/sHorizontal oil dispersion D ¼ Deddy þ SðxÞDL m2/sOil dispersion transition sigmoid S ¼ 1þ expðx�L

w Þ �1 –

Horizontal dispersion, surf DL m2/sHorizontal dispersion, far-field Deddy m2/sHorizontal threshold dispersion Dthreshold m2/sSlick thickness sðx; tÞ mBulk oil thickness bðx; tÞ mBulk oil volume fraction Bðx; tÞ –Vertical vorticity v s�1

Wave action A kg/sLoss term in the action equation NA kg/s2

Wave height A mRelative group velocity CG m/sWavenumber vector k m�1

Wavenumber magnitude k m�1

Wave frequency R rad/sRelative wave frequency x rad/sTracer h tracer

unitsTracer diffusion Nh tracer

units/sMomentum source/sinks/dissipation N m4/s2/kgAverage oil distribution (ODE) q mLongtime oil distribution (ODE) q1 mSmallest (in magnitude) non-zero

eigenvalue (PDE)k1 s�1

Eigenfunction associated with k1

(PDE)f ðxÞ m

Switching time (ODE) te sProxy for peak of oil distribution

(ODE)lðtÞ m

Variance of oil distribution (ODE) r2ðtÞ mRatio of mean-square to variance at

te

a –

Ratio of surface oil to total oil c –Stalling parameter b –Ratio of dispersion d ¼ DL=Dthreshold –Proportionality constant C m

Table 2Parameter values for the example depicted in Figs. 3–7.

Units Value Units Value Units Value

H0 m 1.2 H1 m 20 X m 1000L m 200 P m 1 – 6 w m 20Deddy m2/s 0.05 DL m2/s 1.6 USt m/s 0:0203


where we have normalized the Gaussian to have unit mass in ð0;1Þand k1 is determined numerically by discretizing the operator in(20).

3.4. Patching asymptotic solutions

Eqs. (15) and (21) together give a (piecewise defined) compositesolution which we denote as the reduced or ODE model to contrastwith the PDE model in Section 2. A natural question is the depen-dence of the matched solution (21) on the (somewhat arbitrary)choice of threshold a. Insofar as there is an overlap region whereboth (15) and (20) give good approximations to the true solution,the composite solution is insensitive to the precise choice of aand gives a good approximation to the true solution from thePDE model for all time (see Bender and Orszag, 1999). We havealso verified this numerically by taking a ¼ 1 and a ¼ 9, which givesimilar results to taking a ¼ 4.

We get the following predictions for the dynamics of pulse solu-tions from the ODE model. For small times, the Gaussian pulsesolution has an maximum away from the shore x ¼ 0 for q; s andb. For long times, the maximum for q (respectively s and b)depends on the location of the maximum of the steady state forq (respectively s and b), which in turn is determined by the signof the effective velocity ue in (14) (and c and n). If the steady solu-tion has a maximum away from the shore, then the maximum of q(resp. s and b) is away from the shore for all time. This indicatesnearshore stickiness. Conversely if the steady solution has maxi-mum at x ¼ 0, the matched solution will have two local maxima,one away from the shore for the Gaussian pulse, one at the bound-ary from the steady state. Further the global maximum will jumpinstantaneously to the shore at a critical time when the values atthe two local maxima are equal.

Finally, the steady solution q1 has a maximum that is signifi-cantly away from the shore, leading to nearshore stickiness, pro-vided that b J Oð1Þ (as shown in (17)) and DðxÞ > Dthreshold in thebreakzone (as shown in (18)). These criteria are robust and dependon the mixing layer depth P and the bottom topography H0;m andthe enhanced diffusivity DL in the breakzone, but are insensitive tothe details of the vertical mixing and in particular to the specificchoice for the mixing time sðxÞ.

4. Model outcomes

In this section we present, compare, and discuss results fromnumerically solving the PDE model (Section 2) and the ODE model(Section 3). The PDE model was numerically integrated using aCrank–Nicholson method with centered differencing for the diffu-sion terms and upwind differencing for the advection terms, whilethe ODE model was solved using ode45 of matlab. Table 2 summa-rizes the values of the parameters used in the illustrative examplesthat follow. Before embarking on results, we can use b and d to pre-dict, approximately, what conditions are required for stalling out-side of the break zone, if we accept the parameter values in Table 2.From (17), we see that for stalling to occur near the edge of thebreakzone, we require a P J 3:76 m. From (19), and forP � 3:76 m, we can ask whether DL ¼ 1:6 m2/s is above theDthreshold required to see stalling. We find, approximately, thatDthreshold ¼ 4 c

1�c. Hence, cK 0:29, for Dthreshold K 1:6 m2/s.

Fig. 3a displays the velocity uðx; tÞ at x ¼ X. Superimposed is adashed line indicating P ¼ 1. Fig. 3b depicts the diffusivity DðxÞ,used in the calculations. The effective velocity vðxÞ in (12) is shownin Fig. 3c, for several P. For various choices of P and c, we describe


the evolution from an initial condition corresponding to a symmet-ric oil slick on the surface, namely

sðx; t ¼ 0Þ ¼ expð�0:001ðx� 500Þ2Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1000pp

; bðx; t ¼ 0Þ ¼ 0:

Case I: P ¼ 1; c ¼ 0:9. Since c is high most of the oil is in theslick, rather than the interior. The space–time evolution of thesðx; tÞ field is shown in Fig. 4a and b depicts the contours ofsðx; tÞ. The dynamics of the oil slick and the interior oil arephase-locked because the vertical mixing is very rapid on the scaleof the horizontal transport. The bðx; tÞ field takes up some of the oiland travels at the same rate as the surface oil sðx; tÞ.

Fig. 5a illustrates the slick s, at four different times. For shorttimes, the pulse is symmetric. For longer times, the pulse losesits symmetry, broadening more toward the beach due to a combi-nation of the effective advection velocity ue, the enhanced turbu-lent diffusion D in the near shore and the no-flux boundarycondition. Fig. 5 also includes the ‘‘ballistic’’ prediction, which isdefined as the distance/time relationship given by the advectivespeed USt .

Fig. 5b shows the track traced out by the maximum of sðx; tÞ(mx) and the center of mass (com). For comparison we superim-pose a hypothetical ballistic trajectory, for oil traveling at speedUSt . A slowdown of the pulse is evident. It begins to happen before

Fig. 4. Evolution of the (a) oil slick sðx; tÞ; (b) contours of s

Fig. 5. (a) Evolution of oil slick and interior oil, at four different times. P ¼ 1; c ¼ 0:9. Thedramatically as it approaches the shore. Dashed, b at t ¼ 0:21 hrs is shown, for scale. Subswell as its center of mass (com) slow down slightly outside of the nearshore, compartrajectories.

the oil reaches the edge of the turbulent nearshore, which herespans x ¼ ½0;200�m. Since P < HðxÞ;uBðxÞ;vðxÞ and ueðxÞ are neverzero (cf., Fig. 3c). The steady state solutions for s and b have theirmaxima at x ¼ 0, and as one would expect the maxima for both sand b jump instantaneously from the interior to the boundary.The center of mass of the oil slick, should approach the center ofmass of the steady distribution, and thus stays away from theshore. We also plot the ODE model trajectories of the maximumand the center of mass. The agreement between the PDE andODE trajectories is excellent.

Case II: P ¼ 1; c ¼ 0:1. As c is decreased ue gets closer to uB andv, which are smaller in magnitude than USt . This causes a slow-down of the center of mass of q compared to the ballistic trajectory.(See Fig. 6a.) The oil, which started in the slick, transitions to theinterior and the bulk of the oil in short order. We note the v andue for this case and the previous case are strictly negative, becauseP ¼ 1 (See Fig. 3c), so the maximum for s and b in the steady state isat x ¼ 0 (See Fig. 7a.) Thus we expect the maxima to jump to zero,and this is borne out by simulations from the full model and alsothe reduced model, as depicted in Fig. 6a.

Cases III and IV: c ¼ 0:1; P ¼ 3 and P ¼ 6 respectively. In thesecases, P ¼ HðxÞ at x ¼ 95 m (inside the break zone) and x ¼ 254 m(outside the break zone), respectively. The effective velocity v is

ðx; tÞ; P ¼ 1; c ¼ 0:9. For this case, d ¼ 0:17; b ¼ �0:05.

oil slick (solid), which is initially symmetric about its center of mass, changes shapeequent bðx; tÞ are similar in shape to sðx; tÞ. (b) The maximum (mx) of the oil slick ased to the ballistic trajectory. The maxima of b and s are tracing nearly identical

Fig. 6. Trajectories of maxima (mx) and centers of mass (com), c ¼ 0:1, (a) P ¼ 1, (b) P ¼ 3, (c) P ¼ 6. As is evident as well in Fig. 5, the ODE model captures the dynamics ofthe full model very well.

Fig. 7. Steady state of s and b. c ¼ 0:1. (a) P ¼ 1; d ¼ 13:34;b ¼ �0:05; (b) P ¼ 3; d ¼ 4:45; b ¼ 0:48; (c) P ¼ 6; d ¼ 7:22; b ¼ 1:28. Note that as P is increased to 3 and beyond,the location of the maximum moves away from the shore to the edge of the surf zone.

Fig. 8. Contours of bðx; tÞ; c ¼ 0:1, (a) P ¼ 3, (b) P ¼ 6.


positive on the shallow side of P ¼ H, as shown in Fig. 3c and theeffective velocity ue changes sign. The maximum for the steadystates of b are now in the interior of the domain, while the maxi-mum for the steady states of s remain at x ¼ 0 (See Fig. 7b and cWe thus predict that the maxima for s will jump instantaneously,but the maxima for b will remain away from the shore. This isindeed the case, for both the full and the reduced models as shownin Fig. 6b and c.

Since c ¼ 0:1, the initial pulse of oil in the slick transitions quicklyto the interior, and remains there. Also, the maximum of b stays awayfrom the shore, so this is an example of nearshore sticky waters. Thespace–time evolution of bðx; tÞ is shown in Fig. 8, showing the slow-ing and parking of the tracers in the bulk away from the shore. Wewish to draw attention to the agreement between the PDE and theODE models in Fig. 6b and c, even in situations where the ballisticand actual trajectories are very different.


5. Conclusions

The model we propose describes, in a simple way, how buoyantcontaminants that are impervious to inertial effects, may slowdown as they approach nearshore environments. Evidence ofunusually long residence times for tracers, measured in breakzonefield experiments, is reported from time to time (see for example,Reniers et al. (2009)). The mechanism depends on vertical varia-tions of the tracer density and the cross-shore component of thevelocity, and thus on buoyant stratification and topographic fea-tures, rather than on blocking or stationary structures in theLagrangian trajectories induced by the velocity field (see Haller,2001, for details on Lagrangian Coherent Structures). It is also dif-ferent from the surface-wave deceleration mechanism outside ofthe surf zone, that is explored in Ohlmann et al. (2012).

The phenomenon and the dynamics detailed here should haverelevance to the long-term transport of other contaminants andbiological material in which inertial effects are negligible. In envi-ronmental mitigation problems the timing of landfall of pollutantsand the manner in which these make it to land are crucial. Wehope this simple model motivates efforts to improve forecastingcirculation models for nearshore flows and transport, with betterthe modeling of breakzone boundary conditions, and nearshoresubscale transport of pollutants.

The proposed model (Section 2) is a set of coupled PDEs and,despite its relative simplicity, it has multiple, potentially relevantparameters (See Table 1). In order to gain further insight into thedynamics of the PDE model, we derived a finite dimensional reduc-tion in Section 3. We gain significant insight from studying thereduced model, namely we can explain the evolution of the verticalstratification of the oil density, the evolution of the oil density fromsymmetric to shoreward skewed distributions, the eventual fate ofthe maximum and the center of mass of the oil distribution, both inthe surface and in the interior including an explanation of sticki-ness phenomenon, and a prediction for the time and space scalesfor the complex dynamics of the oil distribution.

Most importantly, the reduced model demonstrates that near-shore stickiness is a robust phenomenon, and helps identify thekey parameters (or combinations thereof) which govern the grossbehavior of the system. The determining factors are the twodimensionless quantities

b ¼ P � H0

H1 � H0

� �XL; d ¼ ð1� cÞDLðH1 � H0Þ

cPjUStjX:

These quantities indicate the outcomes would change if the topo-graphical parameters or the magnitude of the USt ; P; c, and/or DL

are changed. A general conclusion from our analysis is that increas-ing b and/or increasing d (equivalently decreasing c) leads to morestickiness, i.e., the oil approaches the shore more slowly and stallsnear the break zone for much longer.

The parameter b depends on the topography of the nearshore,the sea conditions and the nature of the pollutant, as all of theseinfluence P. Locations where the topography drops steeply evennear the shore, such as in the Southern California Bight, will havesmall b and are thus less ‘‘sticky’’. Conversely a typical situationin the Gulf of Mexico or on the mid and southern portions of theAtlantic US coast would have P large when compared to H, evenat considerable distances from the shore, i.e. b large, so the near-shore is considerably more ‘‘sticky’’. The parameter d depends onthe enhanced turbulent diffusivity DL in the breakzone as well asthe parameter c which governs the fraction of the oil which is inthe slick. Increasing the turbulent eddy diffusivity or decreasingthe fraction of the total oil in the slick both lead to increasedstickiness.

The models we have developed have the virtue of simplicity andbeing analytically tractable, but they lack many of the ingredientsof more realistic physical models. In particular we do not includeinertial effects, and the transient setup and setdown process (seeLentz and Fewings, 2012). One expects the mixed layer to have adiffuse boundary unlike our simple model with a sharply discon-tinuous density at a constant layer depth P. The mixed layer candepend on the sea state and the oil or pollutant constituents. Forinstance, the mixed layer depth and the concentration of the oilin suspension will tend to increase with Langmuir turbulenceand wave breaking activity. We use a kinematically specified flowwith a parabolic profile, but a more realistic model should includea dynamical model for the ocean with a realistic vertical meanLagrangian velocity structure, at least close to the shore, as wellas a careful treatment of the form-stress for the coupling of windeffects, which are recognized as very important in the case of sur-face oil slick dynamics (see Elliot and Hurford, 1989). Finally, wehave a simplified description of buoyancy and vertical mixingeffects. These are important issues which we consider in a sequelto this paper. While these effects are relevant, we believe they willnot change the basic conclusions we have drawn from the simpli-fied model, and it gives a robust description of the mechanismswhich can lead to the observed apparent stalling of incoming pol-lutants approaching a shore.

Acknowledgements

We received funding from GoMRI/BP and from NSF DMS grant1109856. JMR wishes to thank the Statistics and Applied Mathe-matical Sciences Institute, an NSF funded institute, in which someof this research was done. JMR also thanks the J. Tinsley Oden Fel-lowship program at the University of Texas for its support. SV wasalso supported in part by the NSF DMS grant 0807501. Prof. J. C.McWilliams is acknowledged for bringing to our attention theproblem of sticky waters in the Great Barrier Reef, which lead usto ask whether sticky waters occur in the nearshore setting. Weare also very grateful to Prof. F. Feddersen for sharing with us hisexpertise on dispersive processes in the surfzone. His suggestionsconsiderably improved this paper.

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