Generation of attached Langmuir circulations by a suspended macroalgal farm
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Generation of attached Langmuircirculations by a suspended macroalgal farm
Chao Yan, James C. McWilliams and Marcelo Chamecki † Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA90025, USA(Received xx; revised xx; accepted xx)
In this study, we focus on Langmuir turbulence in the deep ocean with the presenceof a large macroalgal farm using a Large Eddy Simulation method. The wave-currentinteractions are modelled by solving the wave-averaged equations. The hydrodynamicprocess over the farm is found to drive a persistent flow pattern similar to Langmuircirculations but is locked in space across the farm. These secondary circulations aregenerated because the cross-stream shear produced by the rows of canopy elements leadsto a steady vertical vorticity field, which is then rotated to the downstream directionunder the effect of vortex force. Since the driving mechanism is similar to the Craik-Leibovich type 2 instability theory, these secondary circulations are also termed asattached Langmuir circulations. We then apply a triple decomposition on the flow fieldto unveil the underlying kinematics and energy transfer between the mean flow, thesecondary flow resulting from the farm drag, and the transient eddies. Flow visualizationsand statistics suggest that the attached Langmuir circulations result from the adjustmentof the upper ocean mixed layer to the macroalgal farm, and they will weaken (if notdisappear) when the flow reaches an equilibrium state within the farm. The triple-decomposed energy budgets reveal that the energy of the secondary flow is transferredfrom the mean flow under the action of canopy drag, while the transient eddies feed onwave energy transferred by the Stokes drift and energy conversion from the secondaryflow.
1. Introduction
Macroalgae, also known as seaweeds, are an important component in temperate marineecosystems (Dayton 1985; Schiel & Forster 2015). Providing shelter, food and protectionfor many species of marine living creatures, macroalgae play a paramount role in pre-serving biodiversity and promoting sustainable aquaculture production. Macroalgal forestharvesting also contributes enormously to various applications, such as remediation of eu-trophication pollution, biofuel production, food and pharmaceutical processing, etc. Thedesire to increase the productivity of aquaculture spurs the growing need for aquafarmdevelopment in the ocean, where the canopy grows near the surface and is supported by afloating structure (Troell et al. et al. et al. b ). Therefore, understanding and quantifying the diverse hydrodynamicprocesses that occur in the presence of macroalgal farms is essential in evaluating anddesigning optimal farm configurations, as well as assessing their environmental impacts. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b C. Yan, J. C. McWilliams and M. Chamecki
From a hydrodynamics perspective, aquatic vegetation can be classified as submerged,emergent, or suspended based on its growth form. Submerged and emergent vegetationare attached to the bottom floor, and occupy a fraction or all of the water depth.The flow structures and mass transport over such canopies have been well documented(Nepf 2012 a , b ; Yan et al. et al. a ).Through laboratory experiments of suspended canopies in shallow waters, Plew (2011 a )concluded that the additional bottom boundary layer (BBL) associated with the oceanfloor affects the penetration of the shear layer into the suspended canopy. Based on themeasurements of Plew (2011 a ), Huai et al. (2012) proposed a simple analytical modelfor the vertical profile of streamwise velocity. While these studies focus on flow overuniform canopies (i.e. essentially infinite size), where the flow has been fully adjustedto the canopy, common aquaculture structures are of finite size and the correspondingcanopy flow displays distinct spatial distribution patterns.The finite dimensions and spatial arrangement of the suspended canopy lead to flowpatterns different from the fully developed scenario (Tseung et al. et al. (2016) and Zhao et al. (2017), the flow over a suspended canopy of finitesize is similar to the terrestrial flow over forest patches (Belcher et al. et al. et al. et al. et al. (2007) explored the effects of giant kelp forestson ocean flows through a field experiment at the coast of Santa Cruz, California. Theyhighlighted the importance of the Stokes drift in cross-shore transport within the kelpcanopy. Rosman et al. (2013) conducted experiments at a scaled laboratory flume toexamine the interaction of surface waves and currents with kelp forests, and concludedthat these interactions must be taken into account when modeling flow and transportwithin kelp forests.One of the distinct features widely observed in the upper ocean is the presence of eneration of attached Langmuir circulations et al. et al. et al. et al. et al.
2. Methods
Mathematical model
For the past three decades, the LES technique has been widely adopted to studyturbulence in the OML. Detailed discussion of the LES framework and assumptionsunderpinning its applicability can be found in the review paper by Chamecki et al. (2019). In the present work, the dynamics of Langmuir turbulence in the presence of amacroalgae canopy are captured using the LES method by solving the wave-averagedequations described by McWiliams et al. (1997). This mathematical model is built uponthe original Craik-Leibovich equations (Craik & Leibovich 1976) with the inclusion ofplanetary rotation and Stokes drift advection of scalar fields, ∇ · (cid:101) u = 0 , (2.1) ∂ (cid:101) u ∂t + (cid:101) u ·∇ (cid:101) u = −∇ Π − f e z × ( (cid:101) u + u s − u g )+ u s × (cid:101) ζ + (cid:18) − (cid:101) ρρ (cid:19) g e z + ∇· τ d − F D , (2.2) ∂ (cid:101) ρ∂t + ( (cid:101) u + u s ) · ∇ (cid:101) ρ = ∇ · τ ρ , (2.3)Here, the tilde indicates grid-filtered variables, (cid:101) ρ is the filtered seawater density, ρ isthe reference density, Π is the generalized pressure, f is the Coriolis frequency, g =9 .
81 m s − is the gravitational acceleration, e z is the unit vector in the vertical direction,and (cid:101) u = ( (cid:101) u, (cid:101) v, (cid:101) w ) is the velocity vector represented in the Cartesian coordinate system x = ( x, y, z ), with x , y , and z being the downstream, cross-stream, and vertical directions,respectively. The vertical coordinate is defined positive upward with z = 0 at the ocean C. Yan, J. C. McWilliams and M. Chamecki surface. The geostrophic current u g = ( u g , ,
0) is driven by an external mean pressuregradient force with magnitude f u g applied in the y -direction. The canopy is treated asa source of flow resistance, and its effect is accounted for by adding a drag force F D tothe momentum equation.In (2.2) and (2.3), τ d is the deviatoric part of the subgrid-scale (SGS) stress tensor τ = (cid:101) u (cid:101) u − (cid:102) uu , and τ ρ = (cid:101) u (cid:101) ρ − (cid:102) u ρ is the SGS buoyancy flux. We assume that the changesin the seawater density ρ are caused by the varying potential temperature θ , and thesetwo variables are linearly related by ρ = ρ [1 − α ( θ − θ )], where α = 2 × − K − isthe thermal expansion coefficient, and θ is the reference potential temperature at which ρ is measured. The SGS stress tensor is modeled using the Lagrangian scale-dependentdynamic Smagorinsky SGS model (Bou-Zeid et al. Pr t = 0 .
4. The viscous force is assumed to be negligible for the high-Reynoldsnumber flows considered in the present study.The Stokes drift u s induced by surface gravity waves is imposed in the governingequations to reflect the time-averaged effects of the wave field on the oceanic turbulence,since the surface wave motions are not explicitly resolved in our simulations. The thirdterm on the right-hand-side (RHS) of (2.2) is the CL vortex force u s × (cid:101) ζ (here (cid:101) ζ = ∇ × (cid:101) u is the vorticity field), which represents the interaction of wind-driven turbulence andsurface gravity waves. For simplicity, we only consider a steady monochromatic wave.Assuming that the surface gravity wave propagates along the mean wind direction (i.e.the x -direction), the Stoke drift velocity reduces to u s = ( u s ( z ) , , u s is givenby: u s = U s e kz (2.4)in which k is the wavenumber and U s is the wave-induced Stokes drift at the surface.Then, the vortex force u s × (cid:101) ζ reduces to (0 , − u s (cid:101) ζ z , u s (cid:101) ζ y ). Note that the presence of thecanopy can attenuate the waves and impact the Stokes drift profile (Rosman et al. et al. (1984), we have estimated theeffects of canopy drag on the surface waves for the specific canopy and wave parametersused in this study and found only a small attenuation of about 3% in wave amplitude and6% in the magnitude of the Stokes drift (see appendix B). These estimates are consistentwith those obtained in flume measurements by Rosman et al. (2013). For the sake ofsimplicity, we neglect wave attenuation in this study.Finally, there is evidence suggesting that surface waves can induce a mean current inthe direction of the wave propagation within aquatic canopies (Luhar et al. et al. et al. et al. et al. et al. u g imposed in our simulations. Thus, we expect that theoverall effects of this wave-induced current to be small, and we neglect them in adoptinga wave-averaged approach.2.2. Numerical representation of macroalgal farm
For the cultivation of macroalgae, the aquaculture structures being deployed in theopen ocean are varied, but common practice is to suspend seeded materials from surfacebuoys and mooring structures (Charrier et al. eneration of attached Langmuir circulations Figure 1.
Schematic of the spatial morphology of the suspended macroalgae farm: ( a ) spatialarrangement of the macroalgal farm; ( b ) frond area density profile for each macroalgae row, a ( z ), normalized by the canopy height h MF . cultivation strategy for the macroalgal of interest (giant kelp) is shown in figure 1 a .The macroalgal farm comprises parallel lines of seeded growing ropes with a lengthof W MF = 8 m coiled around a backbone (or longline). Each backbone line, with alength L MF , is anchored at each end and connected to surface buoys (not shown). Eachmacroalgae consists of 8 fronds with an average length h MF = 19m, which are assumed tobe in an upright posture by virtue of the buoyancy provided by the gas-filled floats (calledpneumatocysts). The lateral spacing between two adjacent rows of canopy elements is S MF = 26m.The frond surface area of the cultivated macroalgae species is obtained by conversionof vertically-resolved algal biomass generated from a macroalgal growth model (ChristinaFrieder, personal communication) using allometric relationships (Fram et al. a ( z ), which is shown in figure 1 b .FAD is the total (one-sided) frond surface area per unit volume of space (m − ), withoutexplicit differentiation among blades, fronds, and stipes, etc. Since our main focus hereis to examine the adjustment of OML as it flows over the farm, canopy parameters suchas a ( z ), h MF , S MF are kept constant (the only exception being the length L MF ) and asensitivity study to farm design is beyond the scope of this study.The drag per unit mass F D in (2.2) represents the effect of the canopy as a momentumsink for the flow field, and it is parameterized as (Shaw & Schumann 1992; Pan et al. F D = 12 C D a ( z ) P · | (cid:101) u | (cid:101) u (2.5)in which C D is the drag coefficient and | (cid:101) u | is the magnitude of the resolved velocityvector. For the sake of simplicity, the tilde symbols used to denote resolved variables areomitted hereafter. The coefficient tensor P = P x e x e x + P x e y e y + P z e z e z is employedhere to account for the projection of total foliage area onto the orthogonal planes withnormal in each one of the Cartesian directions. Note that the expression for P involvesthe dyadic products of the standard basis vectors e x , e y , and e z , so that P is also asecond-order tensor. This projection operation is commonly used for terrestrial canopies(Legg & Powell 1979; Aylor & Flesch 2001; Pan et al. P x , P y , and P z depend on the geometry of the canopy and thus on the specific details ofeach plant species (Aylor & Flesch 2001). In the absence of observational data to specify C. Yan, J. C. McWilliams and M. Chamecki these coefficients, we make the assumption of isotropic distribution of FAD (e.g. thefraction of FAD projected towards each direction is always the same), which correspondsto P x = P y = P z = 1 / C D is a key input parameter in the drag model (2.5) that canaffect the accuracy for the prediction of turbulence statistics (Pinard & Wilson 2001).Generally, C D is estimated from the reduced momentum balance based on experimentalmeasurements, where large uncertainty exists depending on the formulations of themomentum equation being used (Cescatti & Marcolla 2004; Pan et al. et al. C D of constant value forterretrial canopies (Shaw & Schumann 1992; Dupont & Brunet 2008; Finnigan et al. et al. (2014) introduceda velocity-dependent C D in their LES study to account for the reconfiguration of theflexible cornfield in response to the surrounding flow (Vogel 1989). However, giant kelpelements do not bend with the flowing water in the same way as many terrestrial plantsor seagrasses do, because they possess many gas-filled floats that can keep the frondsupward to the surface via buoyancy forces (Koehl & Wainwright 1977; Henderson 2019).In our LES cases, we use the value of C D = 0 . Numerical scheme
The present LES framework employs a Cartesian grid using a vertically staggered ar-rangement, with the horizontal velocity components, pressure and potential temperature( u, v, p, θ ) defined at the cell center, while the vertical velocity component ( w ) is storedat the cell face. Spatial derivatives in the horizontal directions are treated with pseudo-spectral differentiation, while the derivatives in the vertical direction are discretized usinga second-order centered-difference scheme. Aliasing errors associated with the non-linearterms are removed via padding based on the 3/2 rule. Time advancement is performedusing the fully explicit second-order accurate Adams-Bashforth scheme. The numericalcode has been validated against the LES study of McWiliams et al. (1997) for Langmuirturbulence in deep ocean by Yang et al. (2015).The LES domain with dimensions of L x × L y × L z is shown in figure 2. For clarity, theorigin of the coordinate system is defined at the leading edge of the farm in the centrallongitudinal plane, and the z -axis is pointing upward. The top boundary is specified as eneration of attached Langmuir circulations Figure 2.
Sketch of the LES computational model for Langmuir turbulence with the presenceof suspended macroalgae farm in deep ocean: ( a ) side view and ( b ) plan view. A fringe region oflength L fr towards the end of the domain is used to force the velocity and potential temperatureback to the inflow, so periodic conditions are satisfied in the horizontal plane. a non-deforming surface exposed to wind shear stress. A sponge layer is imposed withinthe bottom 20% of the domain to damp out fluctuations of velocity and temperature,thus avoiding the reflection of the internal gravity waves.The backbone line is at a depth h b = 20m below the surface while the canopy height is h MF = 19m, leaving a canopy-free layer at the top 1m near the ocean surface to representtypical harvest practices. A domain depth of L z = 6 h b is chosen to avoid the interferencewith the bottom boundary condition as the flow is deflected below the canopy. Thecross-stream domain size L y = 8 S MF is tailored to encompass N = 8 parallel rowsof macroalgae elements, the longitudinal axes of which are aligned in the downstreamdirection. Periodic boundary conditions are imposed in the horizontal directions, whichwill enable us to exclude the complexities brought by the limited width of the farm. Theinlet is positioned L u = 7 . h b upwind from the farm leading edge, and the outlet is ata distance L d = 12 . h b downstream of the farm trailing edge. Thus, the domain size inthe downstream direction is L x = L MF + 20 h b .A fringe region of length L fr = 5 h b is used at the end of the domain (see figure 2)to enable simulations of spatially evolving boundary layer flows in a periodic domainusing pseudo-spectral numerics (Stevens et al. L fr is duplicated from the precursorsimulation on the fringe region of the actual simulation at the end of every time step.Then, any variable φ (i.e. velocity and potential temperature) in the fringe region isdetermined as a weighted average of fields in the precursor and actual simulations (alsosee Stevens et al. φ ( x, y, z, t ) = f ( x ) · φ pre ( x, y, z, t ) + [1 − f ( x )] · φ act ( x, y, z, t ) , (2.6)in which φ pre and φ act are, respectively, the field in the precursor and actual domains, C. Yan, J. C. McWilliams and M. Chamecki and f ( x ) is the weighting function expressed as, f ( x ) = (cid:40) (cid:104) − cos (cid:16) π x − x s x e − x s (cid:17)(cid:105) , x s (cid:54) x (cid:54) x e , x > x e . (2.7)Here, x represents the downstream position, x s = L x − L u − L fr is the starting pointof the fringe region, x e = L x − L u − L fr is the position beyond which φ = φ pre . Thelength of the fringe region must be large enough to enable a smooth transition of thefield φ from the farm wake flow to the inflow condition. To avoid any possible upstreaminfluence from the fringe region, only solutions up to x = x s − h b are analyzed.2.4. Simulation parameters
Our major goal is to report new flow features that develop around suspended aquafarmsunder realistic oceanic conditions. Therefore, instead of exploring the vast parameterspace of possible ocean states (e.g. varying degrees of wind, waves, currents, and surfacebuoyancy forcing, etc.), we only focus on one set of very typical conditions encounteredin the deep ocean. The flow is driven by two main forcings, i.e. the overlying atmosphericflow and a geostrophic current, in a uniformly rotating environment with the Coriolisfrequency f = 1 . × − s − (corresponding to a latitude of 45 ◦ N). The simulationparameters are chosen to be the same as those used in McWiliams et al. (1997), whichserves as benchmark case in the literature on Langmuir turbulence (Polton et al. et al. τ w = 0 .
37 N m − is applied at the air-seainterface and aligned with the wave field in the downstream direction. The correspondingwind speed at 10-m height is U = 5 m s − , and the friction velocity at the oceansurface is u ∗ = 6 . × − m s − . The wave field consists of monochromatic waves withwavelength λ = 60 m (corresponding to a wave period T w = 6 . a w = 0 . U s = 0 .
068 m s − . The resulting turbulent Langmuirnumber La t = (cid:112) u ∗ /U s = 0 .
3, which is typical for wind-wave equilibrium conditions inthe open ocean (Belcher 2012).A geostrophic current u g = 0 . − in the downstream direction is superimposed onthe flow field to represent the effect of mesoscale flow features, which are considered tobehave as a constant flow on the time and spatial scales of interest here (5 hrs and a fewkilometers). The upper mixed layer is bounded by a stably stratified layer below witha constant temperature gradient d θ/ d z = 0 .
01 K m − . Since surface heating or coolingwould add another layer of complexity associated with buoyancy effects on turbulence,we assume zero buoyancy flux at the ocean surface for the simulations considered here.Table 1 summarizes the simulation parameters and resolution of six different casesconsidered here. In the table, N x , N y , and N z are the number of grid points in the x , y , and z directions, respectively. Simulation cases CLT/LT and CST/ST represent the modellingof Langmuir turbulence and pure shear-driven turbulence in the presence/absence ofmacroalgal farm, respectively. These four cases are carried out to evaluate the effects ofmacroalgae canopy and the role of surface gravity waves on the flow features. The shear-driven cases CST and ST are conducted in the absence of any surface wave forcing, i.e.the wave-induced Stokes drift velocity is zero. For a boundary layer flow within andunder a suspended canopy of finite size, whether or not the boundary layer can reach afully developed stage depends on the length of the canopy (Tseung et al. L MF = 800 m), referred to as caseCLTL, is performed to explore the limit of fully developed flow. We focus mostly on theresults of the CLT simulation and use CLTL only when investigating the downstreamflow development. The mesh is uniformly distributed, with a horizontal resolution ∆ h = 2 eneration of attached Langmuir circulations Case Canopy Wave La t L MF (m) L x (m) × L y (m) × L z (m) N x × N y × N z CLT Yes Yes 0.3 400 800 × ×
120 400 × × × ×
120 800 × × × ×
120 600 × × × ×
120 400 × × × ×
120 200 × × × ×
120 200 × × Table 1.
Parameters of the LES runs m and vertical resolution of ∆ z = 0 . u ∗ /La / T in Langmuirturbulence (Grant & Belcher 2009), we use the surface friction velocity u ∗ as the scalingvelocity throughout the paper to facilitate the comparison between Langmuir and shear-driven turbulence.A snapshot of the vertical velocity w/u ∗ on a horizontal plane at z = − . h b for caseCLTF is shown in figure 3. The elongated streaks of downward vertical velocity readilyobserved upstream from the farm leading edge are signatures of Langmuir circulations.They are oriented to the right of the wind direction (i.e. x -direction), and are transientstructures that are continuously generated and dissipated. As the OML flows into thefarm, however, a persistent pattern with stronger downward and upward velocitiesalternating laterally is clearly seen, roughly parallel to the canopy rows. The magnitude of w/u ∗ within the farm region can be as large as 8.0 (the colorbar has been saturated), whilethe typical values for Langmuir and shear turbulence in the absence of the farm for thesame ocean conditions are 1.6 and 0.75, respectively (e.g. see McWiliams et al. Flow decomposition
The statistics for cases LT and ST are obtained by averaging both temporally andhorizontally, indicated by (cid:104) · (cid:105) . Note that the time average and spatial average are indi-cated by an overbar and a pair of angled brackets, respectively. The physical quantitiesfor CLT and CST are first averaged in the temporal dimension. Because of the three-dimensional spatial heterogeneity of the flow, these time-averaged statistics are subjectto larger random errors than the spatial-temporal averaging used for cases LT and ST.Thus, either a spatial or phase averaging operation in the cross-stream y direction isalso used, indicated respectively by (cid:104) (cid:105) y or (cid:104) (cid:105) p . Given the idealized cross-stream canopy0 C. Yan, J. C. McWilliams and M. Chamecki
Figure 3.
Snapshot of the normalized vertical velocity w/u ∗ on a horizontal plane ( z = − . h b )for case CLTF. The black dashed rectangles represent the region occupied by macroalgae canopy.The blue and red color indicate downwelling and upwelling regions. heterogeneity, the cross-phase average defined here, different from the wave-phase averageintroduced in deriving equation (2.2), corresponds to averaging over equivalent positionsin cross-stream phases. For any time-averaged field φ , the cross-phase averaging can beexpressed as, (cid:10) φ (cid:11) p ( x, y, z ) = 1 N N − (cid:88) n =0 φ ( x, y + nS MF , z ) , (2.8)where N = 8 is the number of macroalgae rows.Hereafter, we use the cross-stream average to define the (primary) mean field (cid:104) φ (cid:105) y ( x, z ),and the deviations from the mean field are decomposed into a secondary-flow componentand a transient component. Thus, instantaneous flow quantities, such as the velocity field u , can be represented by, u = u + u (cid:48) = (cid:104) u (cid:105) y + u c + u (cid:48) , (2.9)Here, u (cid:48) denotes the transient fluctuation from u , while the secondary-flow disturbance u c = u − (cid:104) u (cid:105) y is stationary in time and represents the lateral structure of the time-averaged velocity field induced by the farm geometry. As the transient fluctuationand secondary-flow disturbance are uncorrelated, the covariance between the velocitycomponent u i and any field φ can be written as, (cid:10) u i φ (cid:11) y = (cid:104) u i (cid:105) y (cid:10) φ (cid:11) y + (cid:68) u ic φ c (cid:69) y + (cid:68) u (cid:48) i φ (cid:48) (cid:69) y , (2.10)The three terms on the RHS represent the of contributions from the mean flow, thesecondary-flow part, and the transient part, respectively.Finally, in some cases we further average results in the vertical direction (depth-averaged), from the free surface z = 0 to a fixed depth z = z t with z t = − h b , which arethen represented by (cid:104) u (cid:105) yz = 1 | z t | (cid:90) z t (cid:104) u (cid:105) y d z. (2.11)
3. Langmuir turbulence in the presence of canopy
Adjustment of the mean flow
The OML undergoes significant changes as it approaches and flows over the farm.Here, we present the mean flow for case CLTL to offer a more complete picture of the eneration of attached Langmuir circulations Figure 4. ( a ) Hodographs of the mean velocity vector ( (cid:104) u (cid:105) y , (cid:104) v (cid:105) y ) in the vertical at four differentdownstream positions as noted in the legend are also included, and downstream variation of thedepth-averaged mean velocity vector ( (cid:104) u (cid:105) yz , (cid:104) v (cid:105) yz ) (black line); ( b ) profiles of the resolvedmomentum stress (cid:10) u (cid:48) w (cid:48) (cid:11) y at these selected downstream locations. Circles indicate values at thesurface z/h b = 0, and asterisks indicate the canopy bottom z/h b = − spatial development of the upper OML. Figure 4 a shows the hodographs of the meanhorizontal velocity vector ( (cid:104) u (cid:105) y , (cid:104) v (cid:105) y ) at four different downstream positions. Upstreamfrom the canopy leading edge ( x/h b = −
5, purple line), the hodograph follows a typicalStokes-Ekman spiral in Langmuir turbulence, with the cross-stream velocity pointingto the right of the wind stress (i.e., (cid:104) v (cid:105) y <
0) and most of the shear located near thesurface (the horizontal velocity is nearly uniform within most of the OML depth dueto strong vertical mixing). As the flow moves into the farm ( x/h b = 10 , , , x/h b = 20(blue line), the cross-stream component of the flow switches direction within the OML,and at x/h b = 30 (red line), the cross-stream flow is completely reversed (i.e., to the leftof the wind direction within the entire depth of the OML). Also included in the figure isthe downstream variation of the depth-averaged horizontal velocity vector ( (cid:104) u (cid:105) yz , (cid:104) v (cid:105) yz )(black line). Downstream from the leading edge, we can see that the depth-averagedmean flow direction changes sign at x/h b ≈
18, indicating a change in the direction ofcross-stream advection within the farm.The overall change in the direction of the cross-stream flow can be understood basedon the differences of surface and bottom boundary layers in the presence of the rotation.In the northern hemisphere, the horizontal transport is oriented to the right of the windstress in surface Ekman layers, and to the left of the main current in bottom Ekman layers(McWilliams 2006). In the present case, the canopy introduces a vertically distributeddrag that is more pronounced near the bottom of the farm (where the LAD and themean velocities are larger). Therefore, the sign of (cid:104) v (cid:105) y depends critically on the relativeimportance of shear stresses at the top and bottom of the farm. Specifically, if the stressnear the ocean surface dominates over the stress around the canopy bottom, then (cid:104) v (cid:105) y is aligned to the right of the wind as in the wind-stress driven mixed layer (McWiliams et al. (cid:104) v (cid:105) y is directed to the left of thewind (right of the bottom stress) (Taylor & Sarkar 2008). Because the former scales with u ∗ and the latter with u g , we expect the flow behavior for a fixed canopy configurationto depend on the ratio u g /u ∗ . Figure 4 b shows the vertical profiles of (cid:10) u (cid:48) w (cid:48) (cid:11) y at the2 C. Yan, J. C. McWilliams and M. Chamecki
Figure 5.
The time- and cross-stream-averaged vertical velocity (cid:104) w (cid:105) y , nomalized by u ∗ , forcase CLTL along the x − z plane. The black dashed rectangle represents the location where themacroalgae is planted. The black solid line marks the mixed layer depth, which is defined as theposition where the temperature exceeds a certain percentage of the mixed layer value. selected four downstream locations. It clearly shows that the turbulence within the farmhas not reached a fully developed state in the downstream direction, and the complexityof hodographs from figure 4 a also reflects this fact. Along the x − direction, the flowtransitions from a surface-stress-dominated regime to a bottom-stress-dominated flow,which explains the switch in mean cross-stream flow direction shown in figure 4 a .Figure 5 displays the mean vertical velocity (cid:104) w (cid:105) y /u ∗ along the x − z plane for caseCLTL. The region occupied by the macroalgae canopy is highlighted in a dashed rectan-gle. The (cid:104) w (cid:105) y /u ∗ exhibits a small value near the inlet, which implies that the macroalgaefarm poses a minor impact on the inflow. As the flow approaches the macroalgae farm,the canopy drag obstructs the fluid. The associated pressure gradient across the leadingedge decelerates the flow within a region upstream of the canopy (termed the “impactregion” in Belcher et al. z i . As it develops downstream, the shear turbulence near the bottomof the macroalgae canopy gradually erodes the stratification by entraining denser waterinto the upper mixed layer. Here, we define MLD as the location at which the potentialtemperature first exceeds a certain percentage of the mixed layer temperature θ ML . Thus z i = { z : (cid:104) θ (cid:105) y ( x, z ) − θ ML = χθ ML } (3.1)where χ is a predefined constant. This definition is adapted from the potential temper-ature contour method in Sullivan et al. (1998). The downstream evolution of the MLDindicates that, for the present configuration in which the MLD is comparable to thedepth of the backbone line, the shear layer at the bottom of the farm creates a localperturbation in the depth of the OML, which seems to recover downstream from thefarm. 3.2. Attached Langmuir circulations
Figure 6 shows the contours of the secondary-flow part of the vertical velocity (cid:104) w c (cid:105) p /u ∗ for case CLT in the cross-sections noted in the caption. In the figure, we can observe aregular pattern of (cid:104) w c (cid:105) p alternating between positive and negative values along the cross-stream direction, indicating the steady upwelling and downwelling motions induced by thepresence of the canopy. This organized pattern is the signature of pairs of steady counter-rotating circulatory flows with axis approximately aligned in the streamwise direction.These upweling and downweling regions extend to the bottom of the OML. We infer that eneration of attached Langmuir circulations Figure 6.
The normalized secondary-flow part of vertical velocity (cid:104) w c (cid:105) p /u ∗ , averaged over timeand cross-phase, for case CLT on a x − y plane at z = 0 . h b ( a ), and y − z plane (facingupstream) at x = 2 . h b ( b ), x = 7 . h b ( c ), x = 12 . h b ( d ), and x = 17 . h b ( e ). The black solidline marks the mixed layer depth. The black dashed rectangles represent the location where themacroalgae is planted. The velocity has been cross-phase-averaged and remapped to the entireplane. The extreme colors of the colorbar are saturated to highlight the spatial variation of thestrength of the cell pattern. these flows are primarily driven by the wave-current interaction since these features arenot observed in the shear-driven case CST (not shown). We refer to these flow structuresas attached Langmuir circulations because (i) their position is determined by the spatialstructure of the canopy, and (ii) their formation depends critically on the wave-inducedStokes drift via a mechanism that resembles the CL type 2 instability, which will bedescribed in section 4.While the standard Langmuir circulations appear as unsteady structures that movearound in the flow (see figure 3), the attached Langmuir cells are more steady andregularly spaced. For the present canopy configuration, the separation between neigh-boring pairs of attached Langmuir cells is determined by the lateral spacing betweenconsecutive rows of macroalgae elements, but test runs suggest that this could change ifthe distance between canopy rows is significantly larger (not shown). As the flow movesdownstream, the strength of the canopy-induced Langmuir circulations exhibits a non-monotonic variation. The downwelling velocity reaches its maximum value at x ≈ . h b with a magnitude of approximately 8 u ∗ (figure 6 b ). The cell pattern then graduallydecays until x ≈ . h b (figure 6 c ), and recovers at a lower level further downstreamtowards the trailing edge of the farm (figure 6d). The orientation of Langmuir cells can beidentified by the elongated downwelling streaks. Owing to the non-zero component in themean cross-stream velocity (see figure 4 a ), the canopy-attached Langmuir circulationsare oblique to the downstream direction. The upwelling and downwelling bands are mildlydeflected to the right of the wind for x/h b < .
0, and then aligned somewhat to theleft of the wind for x/h > .
0, in agreement with the change in cross-stream velocitydiscussed in the previous section. This complex pattern is discussed further in section 4,where results for the long farm case (case CLTL) are presented.4
C. Yan, J. C. McWilliams and M. Chamecki
Figure 7.
The cross-stream- and depth-averaged secondary-flow part of velocity variances forCLT (solid lines) and CST (dashed lines), together with the results from CLTF (dash-dottedline). The vertical dotted lines mark the leading and trailing edge of the farm.
As clearly seen in figures 3-6, the flow field has not reached a fully developedstate at the trailing edge of the farm (true for both the short and long farms).For canopy flows, the canopy-drag length is defined as L c = (cid:0) C D a (cid:1) − where a = W MF / ( S MF h b ) (cid:82) − h b a ( z ) P x d z is the effective FAD. This length scale neglectsthe vertical and horizontal structure of the canopy, and characterizes the distanceover which the flow adjusts to the mean drag of canopy elements (Belcher et al. ah b ≈ . L c ≈ . h b . Note that the short and long farms have lengths of approximately equalto L c and 2 L c , suggesting that the upper mixed layer flow does not fully adjust to thecanopy in these two cases.To quantify the strength of the attached Langmuir circulations, we focus on the threecomponents of velocity variances due to the contribution from the secondary flow. Figure7 shows the downstream variation of the depth-averaged mean velocity variances for casesCLT and CST. The results from CLTF are also included to examine the sensitivity togrid resolution. The comparison shows that the finer resolution simulation (CLTF) yieldsrelatively larger variances than CLT in all three velocity components, but the overallvariations observed in CLTF conform qualitatively to those in CLT. Thus, we considerthe simulations with moderate resolution (CLT and CST, etc.) to be a good startingpoint to explore Langmuir turbulence in the presence of marine plants. It is interestingto note that (cid:104) u c u c (cid:105) yz shows negligible differences within the farm between CLT and CST,suggesting that the canopy effect on the streamwise velocity component of the secondaryflow is not impacted by the surface waves. This also indicates that (cid:104) u c u c (cid:105) yz is dominatedby the lateral variation in mean velocity due to the spatially varying drag. For case CST,the magnitudes of (cid:104) v c v c (cid:105) yz and (cid:104) w c w c (cid:105) yz within the canopy exceed their upstream levelsby roughly an order of magnitude, suggesting that the presence of canopy rows leads tosome secondary circulations driven by adjustment to the canopy drag, which may alsobe impacted by spatial variation in the turbulent stresses (i.e. Prandtl’s secondary flowof the second kind) (Bradshaw 1987). In the simulation with the Stokes drift (case CLT),however, (cid:104) v c v c (cid:105) yz and (cid:104) w c w c (cid:105) yz are about two orders of magnitude greater than that inthe Stokesless simulation (case CST). The downstream enhancement and reduction of (cid:104) v c v c (cid:105) yz and (cid:104) w c w c (cid:105) yz within the canopy for case CLT are consistent with the pattern ofthe vertical velocity in figure 6. Therefore, we conclude that, for the present configuration, eneration of attached Langmuir circulations Figure 8.
The transient part of the vertical velocity variance (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ for case CLT in the x − y plane at z = 0 . h b ( a ), and y − z plane at x = 2 . h b ( b ), x = 7 . h b ( c ), x = 12 . h b ( d ),and x = 17 . h b ( e ). the presence of Stokes drift is a key factor enabling the mean streamwise flow structureinduced by the farm drag to develop into strong secondary circulations. As discussedabove, these eddies are roughly two-dimensional with centerline approximately aligned inthe downstream direction, justifying the nomenclature “attached Langmuir circulations”.Based on these results, hereafter we interpret the streamwise component of the secondaryflow as a product of the spatial structure of the canopy drag, and the crosswise andvertical components of the secondary flow in simulations with Stokes drift as attachedLangmuir circulations. 3.3. Langmuir turbulence intensity
Langmuir turbulence intensity is often characterized by large vertical velocity variance.Our interest is centered on how the macroalgae farm alters the spatial evolution ofturbulence levels and associated turbulent mixing efficiency. In the figure 8, we plotthe time- and cross-phase-averaged vertical velocity variance due to transient eddies (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ for case CLT. Similar to that in standard Langmuir turbulence, the verticalintensity (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ peaks at a subsurface level, even in the presence of a shear layernear the surface due to canopy discontinuity (top 1m). In the nearfield downstreamfrom the leading edge (0 < x/h b < (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ is decreased within the canopyand increased near the canopy bottom (figure 8a and b). This is because the canopydrag dampens the vertical kinetic energy within the canopy, but the shear layer at thecanopy bottom can inject additional energy from the mean flow via shear production (seesection 5). Further downstream, (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ first increases, with the maximum valueoccurring at 9 < x/h b <
11 (figure 8a), and then decreases towards the trailing edge.The energetics of the upper mixed layer, which will be covered in section 5, suggest thatthe enhancement and reduction of (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ are mainly determined by two processes:(i) the energy exchanges with the attached Langmuir circulations and (ii) the shearproduction associated with the lateral/vertical shear in streamwise velocity caused by the6 C. Yan, J. C. McWilliams and M. Chamecki
Figure 9.
The transient part of the vertical velocity standard deviation (cid:104) w (cid:48) w (cid:48) (cid:105) / y /u ∗ for CLT( a ) and CST ( b ) in the x − z plane. The black solid line marks the mixed layer depth. canopy structure. In the downstream cross-section (figure 8 c-d ), a clear pattern emergeswith increased (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ at the bottom and outer edge of the canopy rows and reducedintensity in the lower half of the canopy row where the leaf area density is high (figure1b).Figure 9 shows the comparison of the RMS of the transient vertical velocity fluctu-ation (cid:104) w (cid:48) w (cid:48) (cid:105) / y /u ∗ between Langmuir (case CLT, upper panel) and shear turbulence(case CST, lower panel). Upstream from the leading edge, (cid:104) w (cid:48) w (cid:48) (cid:105) / y from case CLTis about twice as large as that from case CST. This is because Langmuir turbulenceyields significantly higher vertical velocity intensity compared to the pure shear-driventurbulence scenario (McWiliams et al. (cid:104) w (cid:48) w (cid:48) (cid:105) / y /u ∗ is similar towhat is expected for open-channel flow over a suspended canopy (see figure 16e in Tseung et al. (cid:104) w (cid:48) w (cid:48) (cid:105) / y withinthe growing shear layer. Towards the end of the farm, the shear layer penetrates overthe entire canopy depth, a phenomenon that usually occurs for sparse canopies (Nepf2012 a ). Interestingly, in the simulation that includes the wave-induced Stokes drift (figure9a), the shear layer turbulence seems to merge with Langmuir turbulence at around x/h b ≈
4, and the turbulence levels near the ocean surface are further enhanced withinthe canopy (for 6 < x/h b <
12) as compared to the Stokesless counterpart (figure 9b).This can be attributed to the presence of attached Langmuir circulations described abovein section 3.2. This difference between the two cases also confirms that the enhancementof (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ in figure 8 is due to the turbulence modulation by the attached Langmuircirculations.Since transient eddies and attached Langmuir circulations coexist as the fluid impingesupon and flows over the farm (figure 3), it is desirable to compare the energy associatedwith transient eddies from that of attached Langmuir circulations. In figure 10, weplot the vertical velocity variances due to the contribution from the transient eddiesand attached Langmuir circulations as noted in the caption. Again, only some minordifferences exist between CLT and CLTF within the farm region, building confidence onthe use of the coarser simulations to analyze the flow. To evaluate if the flow has fully eneration of attached Langmuir circulations Figure 10.
Downstream variations of (cid:104) w (cid:48) w (cid:48) (cid:105) yz /u ∗ (dashed line) and (cid:104) w c w c (cid:105) yz /u ∗ (solid line)for CLT (red), CST (blue), CLTF (green), and CLTL (black). The end of the farm is locatedat x/h b = 20 for CLT/CST/CLTF, and x/h b = 40 for CLTL. The grey dotted lines mark thevalues of the depth-averaged vertical velocity variance for normal Langmuir turbulence (caseLT, upper line) and pure shear-driven turbulence (case ST, lower line). adjusted to the canopy towards the end of the farm in cases CLT, the results from CLTLare also shown. The discrepancies between cases CLT and CLTL (black and red lines) aremainly located near the end of the farm in CLT ( x/h b = 20) due to the trailing edge effect.As the farm extends further downstream (case CLTL, L MF = 40 h b ), (cid:104) w c w c (cid:105) CLT yz does notbecome uniform but still evolves in the streamwise direction within the farm (blacksolid line). It is observed that the attached Langmuir circulations gradually attenuatein strength from x/h b ≈
20 and eventually fade away at x/h b ≈
32 (black solid line).This suggests that their existence is a result of flow adjustment to the suspended farm offinite size rather than a fully developed state. While the attached Langmuir circulationsdisappear, the vertical velocity variance of transient eddies for case CLTL (cid:104) w (cid:48) w (cid:48) (cid:105) CLTL yz isincreasing from x/h b ≈
30 towards the end of the farm (black dashed line). The enhanced (cid:104) w (cid:48) w (cid:48) (cid:105) CLTL yz of transient eddies is mainly attributed to the canopy shear in the horizontaldirection, which no longer assists the generation of attached Langmuir circulations as theflow has reached an equilibrium state. Except in the nearfield downstream of the leadingedge, (cid:104) w (cid:48) w (cid:48) (cid:105) CLT yz is much larger than (cid:104) w c w c (cid:105) CLT yz throughout the remaining part of thefarm. 3.4. Comparison against standard Langmuir circulations
To compare the attached Langmuir cells with the traditional Langmuir cells thatappear in the absence of the farm, we employ a conditional sampling approach for theLES solutions to educe the coherent structure of both fields (also see McWiliams et al. et al. et al. E is defined as all ( x s , y s , t )instances that satisfy w ( x s , y s , z ∗ , t ) (cid:54) − σ w (cid:12)(cid:12) max , where σ w is the RMS of transientvertical velocity and z ∗ is the depth at which σ w attains its maximum value, denoted as σ w (cid:12)(cid:12) max . The ordered pair ( x s , y s ) represents a set of grid points in the horizontal space.For case LT, σ w = (cid:104) w (cid:48) w (cid:48) (cid:105) / and ( x s , y s ) enumerates the entire horizontal domain; whilefor case CLT, σ w = (cid:104) w (cid:48) w (cid:48) (cid:105) / y is a function of x s , and ( x s , y s ) only contains grid pointsat the center of the canopy spacing along the x -direction. Thus, the conditional averagefor any quantity, denoted as (cid:98) φ , is obtained with, (cid:98) φ ( x s , y s , x (cid:48) , y (cid:48) , z, t ) = (cid:10) φ ( x s + x (cid:48) , y s + y (cid:48) , z, t ) (cid:12)(cid:12) E (cid:11) , (3.2)8 C. Yan, J. C. McWilliams and M. Chamecki
Figure 11.
Contour plots of the conditional-averaged transient vertical velocity (cid:99) w (cid:48) /u ∗ in y − z planes for case LT ( a ), and case CLT at different downstream locations ( b ) x s = 2 . h b , ( c ) x s = 7 . h b , ( d ) x s = 12 . h b , and (e) x s = 17 . h b . The black solid line marks the mixed layerdepth. It should be noted that ( x, y ) is the absolute coordinate in the horizontal plane based onthe Cartesian system defined in figure 2, while ( x s , y s ) denotes the reference point with( x (cid:48) , y (cid:48) ) being the distance from ( x s , y s ) in the horizontal direction. Only when the flowis horizontally inhomogeneous should ( x s , y s ) be equal to ( x, y ). To reduce the samplingerror, the sampled flow field for case CLT is then further smoothed by moving averagewith window size in the streamwise direction given by x s − h b / < x < x s + h b / (cid:98) w/u ∗ in y (cid:48) − z planes for cases LT and CLT as notedin the caption. Note that the mean vertical velocity (cid:104) w (cid:105) y has been removed for the caseCLT before conditional averaging operations to better compare the distinct attachedLangmuir circulations against standard Langmuir circulations (e.g. (cid:104) w (cid:105) y is identicallyzero for LT but not for CLT). In both cases (LT and CLT), the Langmuir cells extenddown to the bottom of the OML. The Langmuir cell pattern for LT (figure 11 a ) appearsasymmetrical about the longitudinal plane because of the Ekman shear. The row spacinghappens to be very close in width to the natural lateral size of the downwelling region instandard Langmuir circulations, and this may be related to the geometric characteristicsof the attached Langmuir cells presented here. This canopy row spacing also plays a role indetermining the separation between neighboring attached circulations as described above,and the effects of varying row spacing should be explored in the future. The downwellingvelocity is greater than the upwelling velocity for both cases, but the upwelling motionsincrease by an order of magnitude in the presence of the canopy. This is partly caused bythe fact that the obstruction of farm rows constrains the lateral extension of upwelling eneration of attached Langmuir circulations Figure 12.
Sketch illustrating the mechanism for attached Langmuir circulations generated dueto the presence of a farm in the upper ocean. The cross-varying current excited by the farm isrotated by the Stoke drift, producing the attached Langmuir circulations (black solid curves)that persist across the farm. regions compared to standard Langmuir turbulence regime, producing stronger upwellingto conserve mass.
4. Mechanism for attached Langmuir circulations
The standard Langmuir cells in a horizontally uniform OML (e.g. case LT) aregenerated through the CL2 instability, which is triggered by the wave-induced Stokes driftvelocity acting upon a cross-stream perturbation in an otherwise horizontally uniformcurrent (Craik 1977; Leibovich 1983; Suzuki & Fox-Kemper 2016). The instability arisesfrom the torques produced by the variations of vortex force u s × (cid:101) ζ that appears in (2.2),which leads to overturning cellular motions with downstream vorticity (Leibovich 1977,1983). This flow pattern drives the well-known Langmuir circulations that are transientin nature in the sense that they can survive for long periods of time but they alsooccasionally merge and disappear (McWiliams et al. ζ z thatinteracts with the wave-induced Stokes drift in a way similar to the CL2 instability.Specifically, the vertical component of vorticity ζ z associated with the cross-streamanomaly introduces a cross-stream vortex force − u s ζ z that carries fluid parcels towardsthe planes of local maximum u where fluid sinks due to continuity (Leibovich 1983;Thorpe 2004). Because the horizontal shear is persistent within the farm and the Stokedrift associated with the waves is horizontally uniform, such interaction gives rise to theformation of attached Langmuir circulations that are stationary and stable within the0 C. Yan, J. C. McWilliams and M. Chamecki
Figure 13.
The time- and cross-phase-averaged vortex force: ( a ) cross-stream component − u s (cid:104) ζ z (cid:105) p · h b /u ∗ and ( b ) vertical component u s (cid:104) ζ y (cid:105) p · h b /u ∗ ; and ( c ) secondary-flow part ofvertical velocity (cid:104) w c (cid:105) p /u ∗ for case CLTL at z = − . h b . farm. This leads to downwelling regions in the high velocity regions between the canopyrows and upwelling regions within the rows of canopy elements. A schematic diagramillustrating the generation of such circulations is shown in figure 12. The black closedcurves provide an illustration of the swirling streamlines in the plane perpendicular tothe canopy axis.Figures 13 a and b show the cross-stream and vertical components of vortex force, i.e. − u s (cid:104) ζ z (cid:105) p and u s (cid:104) ζ y (cid:105) p respectively, in the x − y plane at z = − . h b for the CLTL case. Interms of magnitude, the cross-stream component − u s (cid:104) ζ z (cid:105) p dominates over the verticalcomponent u s (cid:104) ζ y (cid:105) p down to about x/h b (cid:54)
30, while they are both negligibly smalltowards the end of the longer farm. Consistent with the pattern of the coherent partof vertical velocity (cid:104) w c (cid:105) p /u ∗ (figure 13c), the vortex force alternates in sign periodicallyacross the farm, forming pairs of equal magnitude, oppositely directed forces in the cross-stream direction. Very close to the leading edge (0 < x/h b < − u s (cid:104) ζ z (cid:105) p is positive (pointing in the positive y − direction) and negative (pointingin the negative y − direction) near the left and right edges of the canopy rows, respectively.In consequence, the action of − u s (cid:104) ζ z (cid:105) p drives upwelling motions within the farm rowsand downwelling motions in the spacing (see figure 13 c ), as illustrated in figure 12. Thispattern is clearly disrupted downstream from the leading edge, as discussed below.To further characterize the flow structure associated with the attached Langmuircirculation, we look at the streamwise vorticity ζ x . Figure 14 plots the contours of (cid:104) ζ x (cid:105) p for case CLTL at several cross-sections as noted in the caption. As described above, it isthe Stokes drift rotation of vertical vorticity ζ z that produces downstream vorticity ζ x of alternating signs in the cross-stream direction. Although the heterogeneous canopy inthe absence of the Stokes drift (case CST) also generates turbulence-driven secondaryflows (because of spatial variability of the turbulent stresses), it fails to yield any regularpatterns in the streamwise vorticity as those shown in Figure 14 (not shown).To better visualize the overturning circulations, we plot streamlines on y − z cross-sections in figure 14 b - d (Akselsen & Ellingsen 2019, 2020). We determine the streamlines eneration of attached Langmuir circulations ψ computed from ∂ ψ∂y + ∂ ψ∂z = − ζ x , (4.1)The streamlines in figure 14 portray pairs of counter-rotating vortices, with the axesaligned to the right of the wind for 0 < x/h b <
10 and tilted to the left of the windafter x/h b ≈
10. Since the attached Langmuir cells are not strictly aligned with the x − direction, the use of (cid:104) ζ x (cid:105) p only captures the largest downstream component of thethree-dimensional vortices, and thus documents weaker overturning motions relativeto the full form of coherent circulations. The variations of (cid:104) ζ x (cid:105) p resemble that of thesecondary-flow part of vertical velocity in figure 6 a , with the maximum magnitudeappearing in the nearfield downstream from the leading edge ( x = 2 . h b ∼ . h b ).Towards the end of farm, the negative downstream vortices vanish and only the weakpositive vortices are left. This is mainly because the cross-stream vortex force therewithis not strong enough (figure 13 a ) to sustain a downstream counter-rotating vortex pair.In an idealized configuration in which the incoming mean flow is perfectly parallel tothe farm rows, we would expect an organized flow structure similar to that shown infigure 12. However, as it is clearly seen in figures 6, 13, and 14, the patterns that emergefrom the simulation are far more complex. The attached Langmuir cells meander in thecross-stream direction and their amplitude changes in a non-monotonic way as a functionof distance from the leading edge of the farm. These departures from the idealizedscenario are mostly caused by the cross-stream advection, as seen by the superpositionof horizontal velocity vectors onto the streamwise vorticity in figure 14a. In particular,the shift in cross-stream velocity from negative to positive around x/h b ≈
15 discussedin section 3.2 produces a similar change in the effect of advection, causing the upwellingmotions to be displaced to the right of the farm row in the region near the leading edge(i.e., up to x/h b ≈
10) and to the left of the row for x/h b > (cid:104) ζ z (cid:105) and (cid:104) ζ y (cid:105) ). This is mostly because the canopy drag continues to generatelateral shear at the canopy edges, strongly influencing the position of (cid:104) ζ z (cid:105) and (cid:104) ζ y (cid:105) . As aconsequence, in the region between 10 < x/h b <
15, the upwelling/downwelling branchesof the attached Langmuir cells no longer coincide with the divergence/convergence of thecross-stream vortex force (compare figures 13 a and 13 c ), leading to the weakening ofthe attached Langmuir cells around x/h b = 12 followed by a restrengthening at themore favorable position with the upwelling within the canopy row. This process appearsmostly as an abrupt left shift of the flow structure at x/h b ≈
12. Towards the end of thefarm, − u s (cid:104) ζ z (cid:105) p is significantly reduced, and is no longer capable of driving clear attachedLangmuir circulations (see figures 13 c and 14), which is also consistent with the decayof the vertical variance for the secondary-flow component of the flow seen in figure 10.
5. Mixed layer energetics
In this section, we examine the budget of the kinetic energy in the mixed layer, whichwill reveal the energy source for the secondary flow in our LES solutions. Following thedecomposition strategy described in section 2.5, the total kinetic energy ( K = (cid:104) u i u i (cid:105) y / C. Yan, J. C. McWilliams and M. Chamecki
Figure 14.
The time- and cross-phase-averaged downstream vorticity (cid:104) ζ x (cid:105) p h b /u ∗ with overlaidhorizontal velocity vector (a scale factor of 1/5 is applied to (cid:104) u (cid:105) p for better visualization) forcase CLTL at z = − . h b ( a ), and in the y − z plane at four different downstream locations ( b ) x = 2 . h b , ( c ) x = 7 . h b , ( d ) x = 12 . h b , and ( e ) x = 17 . h b , overlying the two-dimensionalstreamfunction (cid:104) ψ (cid:105) p (grey lines) computed from ζ x . as, K = 12 (cid:104) u i u i (cid:105) y = 12 (cid:104) u i (cid:105) y (cid:104) u i (cid:105) y (cid:124) (cid:123)(cid:122) (cid:125) K M + 12 (cid:104) u ic u ic (cid:105) y (cid:124) (cid:123)(cid:122) (cid:125) K SE + 12 (cid:68) u (cid:48) i u (cid:48) i (cid:69) y (cid:124) (cid:123)(cid:122) (cid:125) K TE (5.1)Here, K M represents the mean kinetic energy, K SE is the kinetic energy of the secondarymean flow (which includes lateral variations in the flow produced by the spatial structureof the farm and the attached Langmuir circulations), and K TE is the turbulent kineticenergy. By manipulating the governing equations (2.1) and (2.2), the transport equationsfor K M , K SE and K TE can be obtained as follows,D K M D t = − C M − SE − C M − TE + S M + B M + ε M + D M + T M + R M , (5.2 a )D K SE D t = C M − SE − C SE − TE + S SE + B SE + ε SE + D SE + T SE , (5.2 b )D K TE D t = C M − TE + C SE − TE + S TE + B TE + ε TE + D TE + T SE . (5.2 c )in which the material derivative D / D t = ∂/∂t + (cid:104) u j (cid:105) y ∂/∂x j + u s ∂/∂x . Note that theprescribed wave and current conditions, namely u g = ( u g , ,
0) and u s = ( u s ( z ) , , K M , K SE , and K TE as implied in eneration of attached Langmuir circulations C M − SE = − (cid:104) u ic u jc (cid:105) y ∂ (cid:104) u i (cid:105) y ∂x j ,C M − TE = − (cid:68) u (cid:48) i u (cid:48) j (cid:69) y ∂ (cid:104) u i (cid:105) y ∂x j ,C SE − TE = − (cid:28) u (cid:48) i u (cid:48) j ∂u ic ∂x j (cid:29) y . (5.3)Note that the Einstein summations convention is used. As an example, C M − SE > K SE at the expense of K M , as this term appears asa source in the equation for K SE (5.2 b ) and a sink in the equation for K M (5.2 a ). Thus,it represents the energy transfer rate from the mean flow to the secondary flow.The third terms on the RHS of equations (5.2) are the Stokes production terms thatreflect the energy conversion between the waves and the decomposed field, S M = − (cid:104) u (cid:105) y (cid:104) w (cid:105) y ∂u s /∂z, S SE = − (cid:104) u c w c (cid:105) y ∂u s /∂z, S TE = − (cid:10) u (cid:48) w (cid:48) (cid:11) y ∂u s /∂z (5.4)Interestingly, the Stokes production, which only makes contribution to the turbulentkinetic energy in a horizontally homogeneous OML, now also appears in the budgetequation of mean kinetic energy in our LES experiments because of a non-zero andspatially evolving mean vertical velocity (cid:104) w (cid:105) y field. The fourth term is the buoyancyproduction term, B M = αg (cid:104) w (cid:105) y (cid:16)(cid:10) θ (cid:11) y − θ (cid:17) , B SE = αg (cid:68) w c θ c (cid:69) y , B TE = αg (cid:10) w (cid:48) θ (cid:48) (cid:11) y (5.5)Here, B M represents an exchange of mean kinetic energy K M with the potential energy.The fifth term in (5.2) is the SGS dissipation term, ε M = − (cid:104) τ ij (cid:105) y ∂ (cid:104) u i (cid:105) y /∂x j , ε SE = − (cid:104) τ ijc ∂u ic /∂x j (cid:105) y , ε TE = − (cid:68) τ (cid:48) ij ∂u (cid:48) i /∂x j (cid:69) y (5.6)In light of the energy cascade phenomenology (Pope 2000), we expect most of the energydissipation occurs at the small-scale transient eddies, while the energy loss of the large-scale mean flow and secondary flow to direct SGS dissipation effects is negligible, i.e. ε M , ε SE (cid:28) ε TE . Thus, we will assume ε ≈ ε TE in interpreting the LES solutions, anddo not partition the total dissipation ε into 3 components as in (5.6). The sixth term in(5.2) is the canopy destruction term, D M = − (cid:104) u i (cid:105) y (cid:10) F D,i (cid:11) y , D SE = − (cid:68) u ic F D,ic (cid:69) y , D TE = − (cid:68) u (cid:48) i F (cid:48) D,i (cid:69) y (5.7)which represents the energy gain/loss of each component of the flow field (i.e. mean flow,secondary flow, and transient eddies) due to the action of canopy drag. The terms in flux4 C. Yan, J. C. McWilliams and M. Chamecki form are grouped together as a transport term in (5.2), T M = ∂∂x j (cid:104) (cid:104) u i (cid:105) y (cid:104) τ ij (cid:105) y + (cid:104) u j (cid:105) y (cid:104) u (cid:105) y u s − (cid:104) u j (cid:105) y (cid:10) Π (cid:11) y − (cid:104) u i (cid:105) y (cid:104) u ic u jc (cid:105) y − (cid:104) u i (cid:105) y (cid:68) u (cid:48) i u (cid:48) j (cid:69) y (cid:21) , (5.8 a ) T SE = ∂∂x j (cid:28) u ic τ ijc − u ic u ic u jc − u jc Π c − u (cid:48) i u (cid:48) j u ic + u jc u c u s (cid:29) y , (5.8 b ) T TE = ∂∂x j (cid:28) u (cid:48) i τ (cid:48) ij − u (cid:48) i u (cid:48) i u (cid:48) j / − u (cid:48) j Π (cid:48) + u (cid:48) j u (cid:48) u s − u jc u (cid:48) i u (cid:48) i (cid:29) y . (5.8 c )which represents the transport of kinetic energy ( K M , K SE , or K TE ) through resolvedmomentum stresses, SGS stresses, and pressure. The last term on the RHS of (5.2 a )represents the effect of Coriolis force associated with the Stokes drift and geostrophiccurrent, R M = f (cid:104) v (cid:105) y ( u g − u s ) (5.9)which transfers energy from surface waves and external larger-scale field to the meanflow (Suzuki & Fox-Kemper 2016).Figure 15 a shows the downstream variation of the depth-averaged kinetic energy forthe triply-decomposed field. Within the canopy region (0 < x/h b < K M decreasesbecause the farm drains the mean kinetic energy by decelerating the time-mean flow. Asthe OML flow impinges upon the farm, both K SE and K TE increase in the near-fielddownstream of the leading edge. While K TE maintains at a high level after that, K SE gradually decreases towards the end of the farm. This suggests that, in the presenceof a suspended farm, the flow within the canopy region is in a highly turbulent statebut the organized secondary circulations become less intense as the fluid moves furtherdownstream. The downstream variations of the various production and destruction termsin the kinetic energy budget equation (5.2) for the mean flow, secondary flow, andtransient eddies are depicted in figure 15 b - d , respectively, using u ∗ and h b as the scalingparameters (transport terms are not shown). To facilitate interpretation, the curves arecolor coded according to the diagram depicting energy exchanges shown in figure 16,which provides a summary of the energy budget for the three components of the flowintegrated over the entire farm.The Stokes production S M is the main source for K M (figure 15 b ), except it is negativeafter x/h b ≈
32, mainly because of the upward deflection near the trailing edge, i.e. (cid:104) w (cid:105) y > S M = − (cid:104) u (cid:105) y (cid:104) w (cid:105) y ∂u s /∂z <
0. Contrary to expectations, theenergy conversion term − C M − TE is mostly positive along the farm (green dashed linein figure 15 b and d ), indicating that the transient eddies lose kinetic energy to themean flow. The canopy destruction term D M is the primary sink term for K M as thehydrodynamic drag imparted by the farm consistently removes the momentum from theflow (e.g. ∂ (cid:104) u (cid:105) y /∂x < C M − SE (red solid line in figure15 b and c ) constitutes the secondary energy sink for K M , i.e. energy is transferred fromthe mean flow to the secondary flow. This is mainly because the leading order term of C M − SE in (5.3) is − (cid:104) u c u c (cid:105) y ∂ (cid:104) u (cid:105) y /∂x >
0. Since the geostrophic current and Stokesdrift velocity are prescribed, the sign of Coriolis-related term R M in (5.9) is directlydetermined by the cross-stream velocity (cid:104) v (cid:105) y , which goes to the right of the wind (i.e. (cid:104) v (cid:105) y <
0) as in standard Langmuir turbulence before x/h b ≈
18 and then turn to the leftof the wind (i.e. (cid:104) v (cid:105) y >
0) after that (not shown). The flow veering is largely caused by eneration of attached Langmuir circulations Figure 15.
Budget terms of the depth-averaged kinetic energy in the upper surface layer forcase CLTL: (a) downstream variation of the triply-decomposed kinetic energy; and partitionof conversion, Stokes production, buoyancy production and canopy destruction for (b) K m , (c) K SE , and (d) K TE . The terms are normalized by h b /u ∗ . the modification of the suspended farm on the vertical momentum transfer, given that f (cid:104) v (cid:105) y ∼ ∂ (cid:10) u (cid:48) w (cid:48) (cid:11) y /∂z as yielded from a reduced form of (2.2).In terms of the secondary flow, the canopy-related term D SE is a major source term for K SE (black dotted line in figure 15 c ), mainly because it is the spatial arrangement of thefarm that leads to persistent variations in the streamwise flow across the farm. Apart fromthe energy conversion from the mean flow C M − SE , another important source term for thesecondary mean flow is the Stokes production S SE , which is the main source of energyto the attached Langmuir circulations. This is true everywhere except for the region9 < x/h b <
12 where S SE is negative. In this local range, S SE serves as an sink of K SE and the energy transferred from the mean flow C M − SE is also decreasing (red solid line),which to some extent explains the local attenuation of attached Langmuir circulations at x/h b = 12 . x/h b > S SE is approximately zero because the coherent6 C. Yan, J. C. McWilliams and M. Chamecki
Figure 16.
Schematic diagram of the depth-averaged energy budget for the mean flow,secondary flow, and transient eddies. The arrow-lines represent the transfer of energy integratedover the entire farm length, with the direction of net energy flow indicated by heavy arrowheads.The number alongside each arrow-line is the farm-averaged value of the corresponding term,normalized by h b /u ∗ . Note that the transport terms are not included here, thus the energybudget for each component is not closed. vertical velocity w c almost vanishes (figure 10) and hence the momentum stress due tothe secondary flow (cid:104) u c w c (cid:105) y in (5.4) is negligibly small. These three source terms ( D SE , S SE , and C M − SE ) are responsible for the maintenance of secondary flow (including theattached Langmuir circulations) in the adjustment region downstream of the leadingedge, whereas the exchange with the transient eddies C SE − TE constantly extracts energyfrom the secondary flow to support the turbulence level (purple dashed line in figure15c).As shown in figure 15 d , the transient eddies feed on wave energy transferred by theStokes drift shear (blue dash-dotted line) and energy conversion from the secondary flow.The transient eddies lose energy mostly via three processes: (i) energy transfer to themean flow; (ii) energy removal due to the canopy drag; and (iii) energy dissipation atthe small scales (represented by the SGS dissipation). As Langmuir turbulence in thepresence of canopy features strong shear layers and wave forcing, and we assume noincoming or outgoing buoyancy flux at the surface, the buoyant production terms for thesecondary flow and transient eddies ( B SE and B TE ) are negligibly small in comparison.
6. Conclusions
In this study, a fine-scale LES model is used to explore how Langmuir turbulencein deep ocean evolves as it flows over and through a row-structured macroalgae farm.The ocean flow is driven by a constant wind stress and a geostrophic current, under theinfluences of surface gravity waves, planetary rotation, and stable interior stratification.The effects of Langmuir turbulence are accounted for by adding the CL vortex forceto the momentum equation without explicitly resolving the surface waves. For the casestudied here, the drag force at the bottom of the farm becomes dominant over the windforcing at the surface with increasing distance downstream from the leading edge. As aresult, the mean horizontal flow switches from the canonical surface forced Ekman layerto a regime that resembles a bottom Ekman layer. This transition is evident in the changeof direction of the mean current perpendicular to the wind. eneration of attached Langmuir circulations (cid:104) v (cid:48) v (cid:48) (cid:105) / y and (cid:104) w (cid:48) w (cid:48) (cid:105) / y ), is much larger under the effect of Stokes driftassociated with the surface waves (case CLT) compared to the pure shear-driven scenario(case CST), which is also consistent with previous studies in the absence of the canopy(McWiliams et al. (cid:104) w (cid:48) w (cid:48) (cid:105) / p /u ∗ for 0 < x/h b <
4. Further downstream, the attached Langmuircirculations promote strong enhancement of turbulence. This enhancement slowly fadesas the flow adjusts to the canopy and the strength in the secondary flow decays (figure 10).The presence of the canopy leads to the formation of the attached Langmuir circulationsand to local enhancement of the turbulence. Both flow modifications are expected toenhance vertical mixing within the OML and possibly help the entrainment of nutrientsfrom the pycnocline.Analysis of kinetic energy budget shows that, as the flow moves downstream of thecanopy leading edge, the canopy drag acts as an energy sink for the mean flow andtransient fluctuations, while serving as a major source for the kinetic energy of thesecondary mean flow. If the canopy is long enough, the secondary flow pattern vanisheswhen the oceanic turbulence is fully adjusted to the macroalgal farm. Therefore, thisflow feature arises from the adjustment of the upper mixed layer to the aquafarm.The conclusions drawn here are valid for conditions in which the effect of Stokesdrift dominates over that of wind stress and external pressure gradient forcing (i.e. thesolutions are posed in the Langmuir turbulence regime). Despite the simplification madehere (e.g. plant reconfiguration, monochromatic waves, etc.), we are optimistic that thefindings presented above are relevant to realistic practice, and could serve as guidance forthe design of large scale macroalgae systems. Still, the attached Langmuir circulations8
C. Yan, J. C. McWilliams and M. Chamecki from our LES solutions and their potential implication on nutrient uptake by aquaculturefarms await field observations to confirm their veracity.From a fluid dynamics perspective, the physical flow presented here encompassesa variety of processes (stratification, Coriolis acceleration, wave-driven transport, anda canopy, etc). One of our main goals is to make it clear that these flow featuresare important in practice, in conditions under which macroalgal farms are deployed.As it turns out, most of the complexity involved in our setup is essential for theattached Langmuir eddies to develop (waves, mean current, non-uniform canopy, anddownstream flow development). There are some possible simplifications that wouldallow one to reduce the parameter space and simplify the problem, bringing it to amore manageable fundamental configuration (e.g., removing the effects of planetaryrotation and stratification). The results in this paper warrant further investigation of amore fundamental nature in simplified conditions, which could help reconcile a bit thecomplexity of the flow features we discovered with a more traditional fluid dynamicalinvestigation of the parameter space.
Acknowledgements
This work is supported by the ARPA-E MARINER Program (DE-AR0000920). Wethank the three anonymous reviewers for their constructive comments which led toimprovements of the manuscript.
Declaration of Interests
The authors report no conflict of interest.
Appendix A. Motion of buoyant, flexible macroalgae in upper OML
The stipe reconfiguration in response to the flowing water depends on the oceanparameters (wave amplitude, wave period, and current) and the mechanical properties ofmacroalgae (stipe length, Young’s modulus, density, and buoyancy). We decompose theupper OML flow into two parts, i.e. the steady flow (geostrophic current) and oscillatoryflow (wave orbital velocity), and analyze the motion of buoyant, flexible macroalgae withrespect to flow components separately. For each plant, the stipe bundles are simplifiedto have a circular cross-section, with length l s = 20 m, radius r s = 0 . I = π r s / E = 1 × Pa, and density ρ s = 595 kg m − (properties taken from Utter & Denny 1996; Henderson 2019).In a unidirectional steady current (e.g. u g = 0 . − ), the key parameters deter-mining the form of macroalgae elements in sustained flow conditions are the dimension-less Cauchy number Ca (fluid drag/elastic force) and buoyancy number B (buoyancyforce/elastic force) defined as (Luhar & Nepf 2011), Ca = 12 ρC D r s l s u g EI/l s , (A 1) B = ( ρ − ρ s ) g π r s l s EI/l s , (A 2)in which ρ = 1010 kg m − is the density of water. The ratio Ca/B measures therelative importance of fluid drag and buoyancy force. Note that the flexibility of blades eneration of attached Langmuir circulations
Ca/B = 2 . × − (cid:28) Ca =304 . , B = 1 . × ), and the bending angle of the stipe bundles ξ = 0 . ◦ (estimatedby equation (12) in Luhar & Nepf 2011), suggesting the buoyancy force dominates overthe fluid drag and the stipe bundles deform very little relative to its vertical position.For wave-induced oscillatory flows, such as a sinusoidal wave with surface elevation η = a w cos ( kx − σt w ), Henderson (2019) introduced a new dimensionless buoyancy number β and stiffness number S , β = ( ρ − ρ s ) gr s t w ρC D l s u w , (A 3) S = EIt w ρC D r s l s u w , (A 4)Here, σ = 2 π /a w is the angular frequency, t w = 2 π / √ gk and u w = σa w are thewave period and orbital velocity scale, respectively. Based on the monochromatic waveparameters reported in section 2.4 ( t w = 6 . u w = 0 .
81 m s − ), the resulting β = 1 . S = 3 . × − . The relative magnitude of buoyancy and elasticity scales with γ = β/S / = 184 (cid:29)
1, which suggests that the elasticity plays a negligible role here.As β is of order unity, the stipe displacement and the wave-induced water motion arecomparable, i.e. the stipe bend with the waves. Note that our estimates of S and β aredifferent from those in Henderson (2019) because different values of wave (e.g. periodand amplitude) and canopy parameters (e.g. length and drag coefficient) are used here. Appendix B. Deep-water wave attenuation by suspended canopies
Surface waves propagating through marine plants lose energy due to the drag exertedby the canopy, leading to attenuation in wave heights (Dalrymple et al. et al. et al. (1984) for dampingby rigid cylinders in coastal regions, and considers suspended macroalgal farms in deepwater (described previously in the main text).Assuming that energy dissipation is dominated by the canopy drag force, the conser-vation of wave energy equation is, ∂ ( E w c g ) ∂x = − α D ε D , (B 1)in which E w = ρga w is the energy density per unit area of sea surface waves, a w is thewave amplitude, and c g = (cid:112) g/k is the wave group velocity. The prefactor α D accountsfor the reduction in dissipation arising from the motion of buoyant, flexible macroalgae,and it is a function of β and S defined in appendix A expressed as (Henderson 2019), α D = (cid:20) C S S + C β β C S S + C β β (cid:21) / , (B 2)in which C S = 1 / C β = 1 /
16. For the highly flexible macroalgae ( β = 1 .
06 and S = 3 . × − in appendix A), the value of α D is 0.51. ε D is the mean depth-integrated0 C. Yan, J. C. McWilliams and M. Chamecki wave dissipation due to canopy drag force, ε D = (cid:90) − h b D x u x d z, (B 3)in which the overline denotes averaging over a complete wave period, u x = σae kz cos ( kx − σt ) is the horizontal velocity due to wave orbital motions, σ = √ gk is theangular frequency, and D x = ρC D (cid:104) a (cid:105) y P x | u x | u x is the wave drag force on the canopywith (cid:104) a (cid:105) y being the lateral-averaged FAD. Substituting equation (B 3) into equation(B 1) yields, 12 gc g ∂a w ∂x = − Ga w , (B 4)in which G = 43 α D C D P x σ (cid:90) − h b ae kz d z, (B 5)The solution of (B 4) is a w a w = (cid:18) Ga w gc g x (cid:19) − , (B 6)in which a w (=0.8 m here) is the incident wave amplitude before entering the macroalagecanopy. From the values of parameters reported above, the wave height decay over a 800-m (400-m) long farm is about 2.9% (1.4%), and the corresponding decay in Stokes driftvelocity is 5.7% (2.8%). REFERENCESAbdolahpour, M., Hambleton, M. & Ghisalberti, M.
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