# Langmuir turbulence in suspended kelp farms

Tong Bo1<EMAIL_ADDRESS>James C. McWilliams1  Chao Yan2  Marcelo
Chamecki1  1Department of Atmospheric and Oceanic Sciences, University of
California Los Angeles, Los Angeles, CA 90095-1565, USA. 2Institute of Urban
Meteorology, China Meteorological Administration, Beijing 100089, China.

###### Abstract

This study investigates the influence of suspended kelp farms on ocean mixed
layer hydrodynamics in the presence of currents and waves. We use the large
eddy simulation method, where the wave effect is incorporated by solving the
wave-averaged equations. Distinct Langmuir circulation patterns are generated
within various suspended farm configurations, including horizontally uniform
kelp blocks and spaced kelp rows. Intensified turbulence arises from the farm-
generated Langmuir circulation, as opposed to the standard Langmuir turbulence
observed without a farm. The creation of Langmuir circulation within the farm
is attributed to two primary factors depending on farm configuration: (1)
enhanced vertical shear due to kelp frond area density variability, and (2)
enhanced lateral shear due to canopy discontinuity at lateral edges of spaced
rows. Both enhanced vertical and lateral shear of streamwise velocity,
representing the lateral and vertical vorticity components respectively, can
be tilted into downstream vorticity to create Langmuir circulation. This
vorticity tilting is driven by the Craik-Leibovich vortex force associated
with the Stokes drift of surface gravity waves. In addition to the farm-
generated Langmuir turbulence, canopy shear layer turbulence is created at the
farm bottom edge due to drag discontinuity. The intensity of different types
of turbulence depends on both kelp frond area density and the geometric
configuration of the farm. The farm-generated turbulence has substantial
consequences for nutrient supply and kelp growth. These findings also
underscore the significance of the presence of obstacle structures in
modifying ocean mixed layer characteristics.

##  1 Introduction

Marine macroalgae, such as kelp, provide essential habitats, shelter, and food
sources for a diverse range of marine species, with immense importance for
biodiversity preservation and ecosystem health (e.g., Dayton, 1985; Teagle et
al., 2017). The cultivation and harvest of macroalgae also has the potential
to become a sustainable strategy for biofuel production, food supply, and
carbon sequestration (Ghadiryanfar et al., 2016; Ferdouse et al., 2018). Given
the constraints posed by the ecological carrying capacity of existing
nearshore aquaculture, recent interest has thus arisen in expanding macroalgal
farming offshore (Troell et al., 2009; Yan et al., 2021; Frieder et al.,
2022). These offshore macroalgal farms are usually attached to suspended
structures near the ocean surface, typically within the ocean mixed layer
(OML).

An essential factor affecting the performance of suspended macroalgal farms is
their interaction with the hydrodynamic processes in the OML (Yan et al.,
2021; Frieder et al., 2022). Kelp exerts a drag force on the flow (e.g., Thom,
1971; Jackson, 1997), resulting in attenuation in current velocity and wave
motions (Rosman et al., 2007; Monismith et al., 2022). Discontinuities in drag
can also lead to development of shear layers and eddies at the boundaries of
the canopy (Plew, 2011), which may cause significant modifications in OML
turbulence (Yan et al., 2021). These altered hydrodynamic conditions due to
the presence of kelp can determine nutrient availability, chemical transport,
and salinity and temperature conditions in the farms, thereby affecting kelp
growth.

Moreover, the variability of farm configurations, e.g., farm geometry and
orientation with respect to currents and waves, can introduce added complexity
into the interaction between kelp farms and OML turbulence. In addition, kelp
growth and harvesting can effectively alter the frond surface area density
(Frieder et al., 2022), consequently influencing the drag force and canopy
flow profiles. Comprehensive understanding of the complex hydrodynamic
processes in the OML with the presence of suspended farms is therefore crucial
for optimally designing farm configurations and tactically managing harvesting
practices.

The investigation of suspended farm hydrodynamics, beyond its direct
implications for farm performance, also contributes to our broader
understanding of how obstacle structures modify the OML. Various obstacle
structures located near the ocean surface boundary, e.g., aquatic vegetation,
engineered offshore platforms, ships, buoys, and sea ice, have the potential
to influence the hydrodynamic interactions among winds, waves, and currents.
These modifications to OML hydrodynamics may consequently result in
alterations to other OML characteristics, e.g., heat transport and salinity
mixing.

Suspended kelp farms, hydrodynamically classified as suspended canopies, share
similarities with submerged canopies that are located on the bottom boundary
(Plew, 2011; Tseung et al., 2016). The emergence of shear layer turbulence at
the top of the canopy has received considerable attention in the context of
submerged canopy flow (e.g., Finnigan, 2000; Nepf, 2012). Likewise, for a
suspended canopy, a shear layer can develop at the bottom of the canopy,
leading to generation of turbulence and the exchange of momentum and scalars
between the canopy and the underlying flow (Plew, 2011). Additionally, for
suspended canopies of finite dimensions, an adjustment region typically
develops within the canopy starting from the leading edge (Tseung et al.,
2016), where the flow adapts to the drag imposed by the canopy, similar to
that of finite-length submerged canopies (Belcher et al., 2003; Rominger &
Nepf, 2011). Subsequent to this adjustment region, a fully developed canopy
flow region emerges, followed by a wake region downstream from the canopy.
While the suspended canopy in general behaves like an inverted submerged
canopy (Plew, 2011), distinct hydrodynamic conditions can arise, as a result
of the different boundary conditions at the surface and the prevalent presence
of surface gravity waves in the OML.

A prominent turbulent process in the OML is the Langmuir circulation driven by
wind and waves (e.g., Leibovich, 1983; McWilliams et al., 1997; Thorpe, 2004).
It is typically visible on the ocean surface as streaks of foam or debris,
i.e., surface convergence lines, and are usually roughly parallel to the
directions of wind and waves (Langmuir, 1938). The generation of Langmuir
circulation depends critically on the interaction between the sheared wind-
driven current and the Stokes drift of surface gravity waves (e.g., Craik &
Leibovich, 1976; Craik, 1977; Leibovich, 1977). Vorticity tilting due to the
vertically sheared Stokes drift can cause instability to generate coherent
Langmuir cell structures aligned with the downwind direction, and this is
known as the Craik–Leibovich (CL) 2 mechanism. The intensified turbulence
associated with Langmuir circulation can enhance the vertical transport and
mixing in the upper ocean (McWilliams et al., 1997; McWilliams & Sullivan,
2000) and could even be important for producing and maintaining the uniform
OML (Li & Garrett, 1997; Thorpe, 2004).

Nevertheless, the interplay between Langmuir turbulence and aquatic vegetation
remains largely unexplored. The significance of Langmuir circulation extends
beyond hydrodynamics, also playing a crucial role in biogeochemical transport
in the OML and influencing the distribution of macroalgae (Evans & Taylor,
1980; Qiao et al., 2009; Dierssen et al., 2009). On the other hand, the
presence of vegetation can modify wind-driven currents and waves in the OML,
and is thus expected to affect the generation of Langmuir turbulence. Yan et
al. (2021) investigated the creation of attached Langmuir circulation in a
suspended macroalgal farm with a specific configuration, where spaced rows of
kelp are aligned parallel to the flow and waves. In addition to the generation
of canopy shear layer turbulence below the farm, Langmuir-type turbulence was
found to occur within the farm, with a stronger magnitude than the standard
Langmmuir turbulence generated without a farm. These various types of farm-
generated turbulence can significantly affect nutrient transport in the OML,
potentially leading to feedback on farm performance. Therefore, a more
comprehensive examination of the physical mechanisms behind turbulence
generation by suspended farms, e.g., the interaction between the Stokes drift
and farm-modulated ocean currents, is necessary for an improved understanding
of OML hydrodynamics. The dependence of canopy flow properties on different
farm configurations also merits further investigation, as a crucial aspect of
offshore farm planning and nutrient management.

In this study, we employ large eddy simulation (LES) to understand the
influence of suspended kelp farms on OML hydrodynamics, with a particular
focus on the mechanisms of turbulence generation. We also explore how varying
farm configurations and frond density distributions affect these turbulence
generation mechanisms. Section 2 describes the numerical method and the
various farm configurations investigated in this study. Section 3 presents the
statistics of mean flow, secondary flow, and turbulence in the farm. In
particular, we focus on three representative farm configurations to highlight
the various flow patterns arising from different horizontal arrangements and
vertical frond density profiles. Section 4 examines the energy budget to
understand sources of farm-generated turbulence. Section 5 investigates the
vorticity dynamics in the farm to illustrate the generation mechanisms of
Langmuir circulations. In § 6, we explore other farm parameters that affect
turbulence generation, including the effective frond area density, farm
orientation, and farm length. Section 7 presents the conclusion.

##  2 Methods

###  2.1 Model description

We use LES to investigate turbulence generation associated with suspended kelp
farms. LES is a widely used tool in studying OML turbulence and more detailed
discussion can be found in Chamecki et al. (2019). The code used in this study
has been validated against Langmuir turbulence simulations in McWilliams et
al. (1997) and applied to previous research in the macroalgal farm and
boundary layer flow (Yan et al., 2021, 2022). The present LES framework is
based on the wave-averaged and grid-filtered equations for mass, momentum, and
heat:

$\nabla\cdot\tilde{\boldsymbol{u}}=0,$ (1)
$\displaystyle\frac{\partial{\tilde{\boldsymbol{u}}}}{\partial{t}}+\tilde{\boldsymbol{u}}\cdot\nabla\tilde{\boldsymbol{u}}$
$\displaystyle=$
$\displaystyle-\nabla\Pi-f\boldsymbol{e}_{z}\times\left(\tilde{\boldsymbol{u}}+{\boldsymbol{u}}_{s}-{\boldsymbol{u}}_{g}\right)+{\boldsymbol{u}}_{s}\times\tilde{\boldsymbol{\zeta}}$
(2)
$\displaystyle+\left(1-\frac{\tilde{\rho}}{\rho_{0}}\right)g\boldsymbol{e}_{z}-\nabla\cdot\boldsymbol{\tau}^{d}-\boldsymbol{F}_{D},$
$\frac{\partial{\tilde{\theta}}}{\partial{t}}+\left(\tilde{\boldsymbol{u}}+{\boldsymbol{u}}_{s}\right)\cdot\nabla\tilde{\theta}=-\nabla\cdot\boldsymbol{\pi}_{\theta}.$
(3)

This mathematical model was initially introduced in McWilliams et al. (1997)
by extending the original CL equations (Craik & Leibovich, 1976), with the
effects of planetary rotation and advection of scalars by Stokes drift
incorporated.

The tilde in (1), (2), and (3) represent the grid-filtered variables. In the
Cartesian coordinate system $\boldsymbol{x}=(x,y,z)$, the velocity vector is
$\tilde{\boldsymbol{u}}=(\tilde{u},\tilde{v},\tilde{w})$, i.e., the
streamwise, lateral (cross-stream), and vertical components, respectively. In
(2), $\Pi$ is the modified pressure (e.g., Chamecki et al., 2019), $f$ is the
Coriolis frequency, $g$ is the gravitational acceleration,
$\boldsymbol{e}_{z}$ is the unit vector in the vertical direction, and
$\tilde{\boldsymbol{\zeta}}=\nabla\times\tilde{\boldsymbol{u}}$ is the
filtered vorticity. Here $\boldsymbol{u}_{s}$ is the Stokes drift associated
with surface gravity waves, and $\boldsymbol{u}_{g}$ is a geostrophic current
that represents the effect of mesoscale ocean flows. The geostrophic current
is driven by an external pressure gradient
$f\boldsymbol{e}_{z}\times{\boldsymbol{u}}_{g}$. The term $\boldsymbol{F}_{D}$
represents the drag force imposed by the canopy onto the flow, and the
detailed treatment of canopy drag will be described later.

In (2), $\tilde{\rho}$ is the filtered density, and $\rho_{0}$ is the
reference density. We assume that variations in density are only caused by the
potential temperature $\tilde{\theta}$ via an linear relationship
$\rho=\rho_{0}[1-\alpha(\theta-\theta_{0})]$, where $\alpha=2\times 10^{-4}$
K-1 is the thermal exapansion coefficient and $\theta_{0}$ is the reference
potential temperature corresponding to $\rho_{0}$. The term
$\boldsymbol{\tau}^{d}$ is the deviatoric part of the subgrid-scale (SGS)
stress tensor
$\boldsymbol{\tau}=\widetilde{\boldsymbol{u}\boldsymbol{u}}-\tilde{\boldsymbol{u}}\tilde{\boldsymbol{u}}$,
and
$\boldsymbol{\pi}_{\theta}=\widetilde{\boldsymbol{u}\theta}-\tilde{\boldsymbol{u}}\tilde{\theta}$
is the SGS heat flux. The SGS stress is modeled using the Lagrangian scale-
dependent dynamic Smagorinsky model (Bou-Zeid et al., 2005). The SGS heat flux
is modeled using an eddy diffusivity closure, with diffusivity obtained from
SGS viscosity and a prescribed value of turbulent Prandtl number $Pr_{t}=0.4$.
Molecular viscosity and diffusivity are assumed to be negligible for high
Reynolds number flows examined in this study.

The surface waves are not explicitly resolved in the model, and the time-
averaged influences of waves are incorporated by imposing the Stokes drift
$\boldsymbol{u}_{s}$. We consider a simple case of monochromatic deep water
wave propagating in $x$ direction, with amplitude $a_{w}$ and frequency
$\omega=\sqrt{gk}$, where $k$ is the wave number. The Stokes drift velocity
thus reduces to $\boldsymbol{u}_{s}=(u_{s},0,0)$, and

$u_{s}=U_{s}\textrm{e}^{2kz},$ (4)

where $U_{s}=\omega ka_{w}^{2}$ is the Stokes drift velocity at the surface.
The effects of waves on OML turbulence is represented by the CL vortex force
${\boldsymbol{u}}_{s}\times\tilde{\boldsymbol{\zeta}}=(0,-u_{s}\tilde{\zeta}_{z},u_{s}\tilde{\zeta}_{y})$,
i.e., the third term on the right side of (2). While the presence of the
canopy may additionally influence waves, Stokes drift, and the wave-current
interaction (e.g., Rosman et al., 2013; Luhar et al., 2010, 2013), these
effects are estimated to be reasonably small for the macroalgal farm
simulations considered here and thus have been neglected (Yan et al., 2021).

The parameterization of the canopy drag force $F_{D}$ in (2) is expressed as
(Shaw & Schumann, 1992; Pan et al., 2014; Yan et al., 2021)

$\boldsymbol{F}_{D}=\frac{1}{2}C_{D}a\boldsymbol{P}\cdot\left(|\tilde{\boldsymbol{u}}|\tilde{\boldsymbol{u}}\right).$
(5)

Here $C_{D}$ is the drag coefficient, and $|\tilde{\boldsymbol{u}}|$ is the
magnitude of the filtered velocity. Henceforth, for simplicity we will drop
the tilde symbols that denote grid-filtered variables. We use $C_{D}=0.0148$
based on the experimental study of Utter & Denny (1996), and more detailed
discussion on this choice of $C_{D}$ can be found in Yan et al. (2021). In
(5), $a$ is the frond surface area density (or foliage area density, area per
volume, m-1). The frond surface area of macroalgae is obtained by conversion
of the algal biomass (Frieder et al., 2022). Rather than directly resolving
the geometry of the macroalgae fronds and stipes, their overall drag on the
flow is characterized through this quadratic formula in the model, with a
representative frond area density $a$ for each grid cell. The coefficient
tensor $\boldsymbol{P}$ stands for the projection of frond surface area into
each direction, and in the present study we use
$\boldsymbol{P}=\frac{1}{2}\boldsymbol{I}$, where $\boldsymbol{I}$ is the
identity matrix (Yan et al., 2021).

In the realistic farm setup, macroalgae are attached to subsurface structures
at the bottom of the canopy (Charrier et al., 2018; Yan et al., 2021), and
buoyancy typically keeps them upright in the water column (e.g., Koehl &
Wainwright, 1977). We therefore assume macroalgae fronds and stipes to
maintain an approximately fixed position in the flow, except for exhibiting
small-amplitude oscillations passively following the wave orbital motion. This
assumption is supported by the analyses in Yan et al. (2021), which examined a
set of dimensionless parameters including the Cauchy number and buoyancy
number. It is also consistent with the omission of the wave orbital velocity
in (5), neglecting the interaction between wave and canopy drag (Yan et al.,
2021; McWilliams, 2023). A more detailed description of macroalgal farm
configurations will be presented in § 2.3.

The present LES framework resolves the flow and temperature fields using a
grid structure with horizontally collocated points and a vertically staggered
arrangement. A pseudospectral method is used in the horizontal direction, and
vertical derivatives are discretized with a second-order central finite-
difference method. Aliasing errors arising from the nonlinear terms are
removed through padding on the basis of the 3/2 rule. The equations are
advanced in time using the fully explicit second-order Adams-Bashforth scheme.

###  2.2 Simulation setup

The present study aims to investigate the mechanisms of turbulence generation
in the OML in the presence of suspended kelp farms, with a focus on examining
various farm configurations and frond area density distributions. Therefore,
instead of considering a range of oceanic factors like wind stress, waves,
currents, and mixed layer depth, we only select one set of representative
oceanic conditions. The simulation parameters used here are generally the same
as those in McWilliams et al. (1997). The flow is driven by a constant wind
stress $\tau_{w}=0.037$ N m-2 at the surface boundary, corresponding to a wind
speed at 10-m height above the surface of $U_{10}=5$ m s-1 and a friction
velocity of $u_{*}=0.0061$ m s-1. The deep water waves have an amplitude of
$a_{w}=0.8$ m, and the wavelength is $\lambda=60$ m, corresponding to a wave
period of 6.2 s. This leads to a surface Stokes velocity of $U_{s}=0.068$ m
s-1 and a turbulent Langmuir number of $La_{t}=\sqrt{u_{*}/U_{s}}=0.3$.

In addition to the wind-driven current, a geostrophic flow
$\boldsymbol{u}_{g}=(u_{g},0,0)$ is imposed in $x$-direction, i.e., same as
the direction of wind and waves. The geostrophic flow is driven by an external
pressure gradient $fu_{g}$ in $y$-direction, with a constant value of
$u_{g}=0.2$ m s-1, assuming that variations of mesoscale flow are small within
the temporal and spatial scales relevant to the present study. The Coriolis
frequency $f=10^{-4}$ s-1 corresponds to 45∘ N latitude. The initial mixed
layer depth of the upstream inflow is 25 m, and a stably stratified layer is
beneath it, with a uniform temperature gradient
$\text{d}\theta/\text{d}z=0.01$ K m-1 (buoyancy frequency $N=0.0044$ s-1). We
assume no heat flux at the surface boundary.

Figure 1: Model domain and three types of farm configurations. $(a)$ and
$(b)$: The farm block configuration (top view and side view). $(c)$ and $(d)$:
The configuration with kelp rows aligned with the current direction. $(e)$ and
$(f)$: The configuration with kelp rows oriented perpendicular to the current
direction.

Kelp farm simulations are conducted on a $L_{x}\times L_{y}\times
L_{z}=800\times 208\times 120$ m3 domain, with $N_{x}\times N_{y}\times
N_{z}=400\times 104\times 240$ grid cells. The mesh is uniformly distributed,
with a horizontal resolution of 2 m and a vertical resolution of 0.5 m. A
sensitivity test on grid size was conducted in Yan et al. (2021), and doubling
the resolution in all the three dimensions yielded consistent results. The
simulations are run for 15000 s to allow for the adjustment of the OML to the
suspended canopy, and for another 9000 s after the fully developed turbulence
state is reached to analyze turbulence statistics. A quasi-equilibrium state
with converged turbulence statistics is established in the 9000 s period used
for analysis, as examined in Yan et al. (2021).

The farm is located in the middle of the domain from $x=0$ to $x=L_{MF}$, with
a farm length of $L_{MF}=400$ m (figure 1). The upstream boundary is at
$x=-L_{u}=-150$ m, and the downstream boundary is at a distance of $L_{d}=250$
m from the farm trailing edge. In the $y$-direction the farm extends across
the entire domain with a periodic boundary, i.e., effectively assuming an
infinite farm width to eliminate the complexities arising from the lateral
farm edges. In the vertical direction the farm is between the sea surface and
$h_{b}=20$ m (the farm base), i.e., the depth at which the suspended structure
is deployed. While most simulations are set as $L_{x}=800$ m and $L_{MF}=400$
m, an additional simulation is conducted with an extended domain length of
$L_{x}=1200$ m (and $N_{x}=600$) and farm length of $L_{MF}=800$ m to examine
the effect of a longer farm. More detailed descriptions of farm parameters
will be presented in § 2.3.

A precursor inflow method is used to simulate the spatially evolving flow in
the kelp farm simulations (Churchfield et al., 2012; Stevens et al., 2014; Yan
et al., 2021). In this method the turbulent velocity and temperature fields at
the upstream boundary of the domain are obtained from a precursor simulation.
The precursor simulation is separately conducted with identical conditions
without the farm, until the turbulent flow reaches a quasi-equilibrium state.
A fringe region (length $L_{fr}=100$ m) is used at the downstream end of the
domain of the farm simulations. In this fringe region the flow field is
smoothly forced toward the inflow conditions provided by the precursor
simulation at the end of every time step. The precursor inflow method allows
the turbulence produced by the precursor simulation to enter the domain of the
farm simulations, while permitting the farm wakes to exit without cycling back
through the periodic boundary conditions.

###  2.3 Farm configuration

The cultivation of macroalgae in open ocean environments involves a diverse
range of aquaculture structures. A representative farm configuration is
considered here and consists of a series of organized longlines spaced
horizontally (Yan et al., 2021; Frieder et al., 2022). Each longline is
anchored at both ends and is also connected to surface buoys. Growth ropes,
where kelp is seeded, are attached perpendicular to the longline. Kelp is
cultivated at $h_{b}=20$ m, i.e., the longline deployment depth, and grows
upright due to their buoyancy. The frond surface area density is assumed to be
horizontally uniform within each canopy row (each longline set) for
simplicity. In the present study, we explore various farm configurations by
varying the canopy row spacing and orientation and by comparing different
vertical profiles of frond surface area density within the row.

Two vertical profiles of frond surface area density are considered, which
represent two different growth stages of kelp (Frieder et al., 2022): (1) an
intermediate growth stage with kelp extending from the farm base to around 2 m
below the sea surface; (2) a fully grown stage with kelp extending from the
farm base to the sea surface, with notably high frond area density at the top
due to a large portion of the fronds floating at the sea surface. The frond
surface area density profiles of the two stages are obtained by conversion of
the algal biomass (Frieder et al., 2022) and are plotted in figure 2. Harvest
practices typically concentrate on the uppermost 1-2 m part of kelp, resulting
in the reduction of frond density to zero near the surface. This causes the
frond density profile to revert to the earlier growth stage from the fully
grown stage. Henceforth, we refer to the earlier growth stage – with a low
frond density near the surface – as the ‘harvested’ profile, and the fully
grown stage – with a high frond density near the surface – is referred to as
the ‘ripe’ profile.

Figure 2: Vertical profiles of frond surface area density $a$, normalized by
the farm base depth $h_{b}$. The solid line represents the harvested profile,
and the dash-dotted line represents the ripe profile. The depth average value
$\langle a\rangle_{z}=1.14$ m-1 ($\langle a\rangle_{z}h_{b}=23$) for the
harvested profile, and $\langle a\rangle_{z}=2.20$ m-1 ($\langle
a\rangle_{z}h_{b}=44$) for the ripe profile.

In addition, we examine a variety of kelp row arrangements for each vertical
profile of frond area density. The farm parameters for all the simulations are
summarized in table 1 in Appendix A. The first set of arrangements (cases ‘S’)
has the longlines aligned parallel to the $x$-direction (the direction of
waves and geostrophic flow), extending the length of the farm. The longlines
are repeated at a fixed distance $S_{MF}$ in the $y$-direction (figure 1$c$
and $d$). We conduct a range of farm simulations with varying $S_{MF}$ and
kelp row width (growth rope length) $W_{MF}$ (details provided in Appendix A).

The effective frond area density over the farm is defined as

$\left\langle
a\right\rangle_{xyz}=\frac{1}{L_{MF}L_{y}h_{b}}\int_{0}^{L_{MF}}\int_{-L_{y}/2}^{L_{y}/2}\int_{-h_{b}}^{0}a\
\text{d}x\text{d}y\text{d}z.$ (6)

The frond area density $a$ takes the form depicted in figure 2 within the kelp
rows and is 0 in the gaps between rows. The effective frond density thus
decreases as the spacing between kelp rows increases, while keeping the row
width unchanged.

Another set of farm arrangements (cases ‘B’) assumes a scenario where the kelp
rows are positioned closely enough so that there is no gap in between, i.e.,
essentially forming a uniform kelp farm block (figure 1$a$ and $b$). The two
profiles in figure 2 are employed in simulations involving this farm block
configuration. Additionally, we conduct farm block simulations with varying
effective density by introducing a multiplication factor to each frond density
profile.

In the third set of farm arrangements (cases ‘PS’), the longlines are rotated
by 90 degree, so that kelp rows are oriented parallel to the $y$-direction,
perpendicular to the geostrophic flow (figure 1$e$ and $f$). This set of
arrangements also includes a range of values of $S_{MF}$ and $W_{MF}$,
mirroring the above-mentioned simulations with longlines aligned parallel to
the $x$-direction.

Specifically, we have selected three simulations as representatives for an in-
depth examination of the turbulence generation mechanisms. These selected
cases are S26H (spaced farm rows parallel to the $x$-direction with a
harvested profile, $S_{MF}=26$ m and $W_{MF}=8$ m; refer to table 1) and B1H
and B1R (i.e., farm block simulations with harvested and ripe profiles,
respectively). By comparing these three simulations, we aim to elucidate the
impacts of cross-stream spacing and the vertical frond area density profile on
OML turbulence generation. The effects of effective density and kelp row
orientation will be discussed subsequently, following the analysis of
turbulence generation mechanisms.

##  3 Farm hydrodynamics

In this section we present the adjustment of mean flow to the kelp farm as
well as the turbulence generated by the farm. We first introduce a flow
decomposition to isolate distinct flow components. The instantaneous flow
field can be split into the time-averaged and fluctuating components, i.e.,

$\boldsymbol{u}(x,y,z,t)=\overline{\boldsymbol{u}}(x,y,z)+\boldsymbol{u}^{\prime}(x,y,z,t).$
(7)

The overline represents the time average, and the prime represents temporal
fluctuations about the time average. Further, the time-averaged flow field is
decomposed into a cross-stream average and a steady cross-stream deviation,

$\overline{\boldsymbol{u}}(x,y,z)=\left\langle\overline{\boldsymbol{u}}\right\rangle_{y}(x,z)+\overline{\boldsymbol{u}}^{c}(x,y,z),$
(8)

where $\left\langle\cdot\right\rangle_{y}$ denotes the spatial averaging in
$y$-direction.

The temporal and cross-stream average
$\left\langle\overline{\boldsymbol{u}}\right\rangle_{y}$ is defined as the
mean flow. The steady cross-stream deviation $\overline{\boldsymbol{u}}^{c}$
is referred to as the secondary flow component, representing the stationary
circulation structure generated by lateral variations in farm geometry. Note
that $\overline{\boldsymbol{u}}^{c}$ only exists in farm configurations with
laterally spaced farm rows and is negligible in the farm block or in kelp rows
that are perpendicular to the geostrophic flow (details shown in following
sections). The transient fluctuation $\boldsymbol{u}^{\prime}$ represents the
turbulence component.

Similarly, the covariance between velocity and any field $\phi$ can be
decomposed as

$\left\langle\overline{\boldsymbol{u}\phi}\right\rangle_{y}=\left\langle\overline{\boldsymbol{u}}\right\rangle_{y}\left\langle\overline{\phi}\right\rangle_{y}+\left\langle\overline{\boldsymbol{u}}^{c}\overline{\phi}^{c}\right\rangle_{y}+\left\langle\overline{\boldsymbol{u}^{\prime}\phi^{\prime}}\right\rangle_{y}.$
(9)

The first term on the right side stands for the contribution from the mean
flow; the second term represents the effect of the secondary flow, akin to a
dispersive flux (Finnigan, 2000); the third term represents the turbulent
flux.

Specifically, we focus on three simulations, S26H, B1H, and B1R, as
representatives to illustrate the influences of kelp row spacing in the cross-
stream direction and the vertical profiles of frond area density. Section 3.1
presents the adjustment of the time-mean flow field in the presence of kelp
farms. Section 3.2 provides an overview of the hydrodynamic characteristics of
all the different farm configurations. Section 3.3 compares the Langmuir
circulation patterns across the three representative cases. Subsequently, we
quantify turbulence and steady secondary flows in the three cases in § 3.4,
and calculate the vertical velocity skewness in § 3.5 to characterize the
turbulence generated by the farm.

###  3.1 Mean flow

The mean flow structure is substantially altered by the presence of kelp
farms. Figure 3 shows the adjustment of mean flow in case S26H (spaced rows
aligned with $x$-direction) as an example. As flow enters the canopy region,
the streamwise velocity decreases due to the drag force exerted by the kelp
(figure 3$a$). A shear layer develops beneath the farm as a result of
discontinuity in kelp frond density. In addition, vertical shear is also
increased within the canopy due to the vertical variability of frond area
density (figure 2). The shear within the farm is most pronounced near the
leading edge, and gradually diminishes downstream as turbulence generated in
the farm leads to vertical mixing of momentum.

Figure 3: Side views of mean flow in case S26H (spaced rows aligned with the
current, harvested profile). $(a)$: Normalized streamwise velocity
$\left\langle\overline{u}\right\rangle_{y}/u_{g}$. $(b)$: Normalized vertical
velocity $\left\langle\overline{w}\right\rangle_{y}/u_{*}$. The mean flow is
averaged in time and in the cross-stream direction. Dotted rectangles show the
extent of the farm, and the solid gray line in $(b)$ represents the mixed
layer depth. Note that $\left\langle\overline{u}\right\rangle_{y}$ is
normalized by $u_{g}$ and $\left\langle\overline{w}\right\rangle_{y}$ is
normalized by $u_{*}$, and $\left\langle\overline{u}\right\rangle_{y}$ is
generally much larger than $\left\langle\overline{w}\right\rangle_{y}$.

At the canopy leading edge, as the streamwise flow is decelerated by the
pressure gradient set up by the canopy drag, the mean downward vertical
velocity develops as a result of mass conservation (figure 3$b$). Likewise,
the pressure decrease at the trailing edge induces an upward velocity in the
farm’s wake region. Note that a fringe region is located between the wake
region and the downstream end of the domain (not shown), where the flow field
is forced toward the precursor inflow conditions to avoid the wake effect on
periodic horizontal boundary conditions. The farm-induced mean flow adjustment
in case S26H closely aligns with the previous study by Yan et al. (2021);
other cases involving different farm configurations will be explored further.

We define the mixed layer depth (MLD) $z_{i}$ as the depth at which the
laterally averaged potential temperature
$\left\langle\overline{\theta}\right\rangle_{y}$ first deviates from its
surface value by $\Delta\theta$ (e.g., Kara et al., 2000), here using
$\Delta\theta=0.01$ K. The MLD is around 25 m at the upstream boundary where
the inflow originates. The mixed layer deepens to nearly 40 m due to the
downwelling created by the farm, which then recovers downstream from the farm.
The farm is always situated above the pycnocline, and the buoyancy effect is
thus considered to have negligible influence on turbulence generation within
the farm.

Figure 4: Vertical profiles of streamwise velocity
$\left\langle\overline{u}\right\rangle_{y}$ $(a)$ and vertical velocity
$\left\langle\overline{w}\right\rangle_{y}$ $(b)$. The velocities are time-
averaged and cross-stream averaged, at $x/h_{b}=2.5$. Note that the
geostrophic current $u_{g}=0.2$ m/s has been subtracted in $(a)$. The
streamwise velocity in $(a)$ is normalized by $u_{g}$, and the vertical
velocity in $(b)$ is normalized by the friction velocity $u_{*}=0.0061$ m s-1.
Black lines represent case S26H, spaced rows aligned with the geostrophic
current, with the harvested profile; red lines represent case B1H, farm block
with the harvested profile; blue lines represent case B1R, farm block with the
ripe profile. Additionally, the dash-dotted line in $(a)$ shows the streamwise
velocity profile at the upstream boundary (inflow condition at
$x/h_{b}=-7.5$). The dashed line in $(a)$ represents the vertical profile of
Stokes drift $u_{s}$. The dotted horizontal lines mark the farm bottom, and
the thin solid horizontal lines represent the inflow mixed layer depth.

The canopy drag length $L_{c}$ is defined as (Belcher et al., 2003; Rominger &
Nepf, 2011; Belcher et al., 2012)

$L_{c}=\frac{2}{C_{D}\left\langle a\right\rangle_{xyz}},$ (10)

where $C_{D}$ is the drag coefficient, and $\left\langle a\right\rangle_{xyz}$
is the effective frond density in (6). The factor of 2 in (10) accounts for
the projection of frond surface area, as denoted by the term $\boldsymbol{P}$
in (5). The canopy drag length is $L_{c}=347$ m for case S26H. Note that
$L_{c}$ varies with the effective density and thus the farm configuration, and
case B1R (farm block, ripe profile) has the smallest $L_{c}$ of 61 m. The
length of the adjustment region scales in proportion to $L_{c}$, typically by
a factor of $4.5-6$ (Belcher et al., 2012). Therefore, the adjustment region
length for the farm configurations in this study is either comparable to or
larger than the farm length, suggesting that the canopy flow does not reach a
fully developed state prior to exiting the farm. This explains the streamwise
variations found in the mean flow field within the farm (figure 3) and also
agrees with the analysis of the turbulence field presented in the following
sections.

The mean flow properties in the other simulations B1H and B1R are compared
with S26H (figure 4). The vertical profiles in figure 4 are averaged in the
cross-stream direction, and also averaged in the streamwise direction from the
leading edge to $5h_{b}$ into the farm, a distance comparable to the canopy
drag length $L_{c}$ in cases B1H and B1R. Generally cases B1H and B1R are
similar to case S26H shown above, except that B1H and B1R demonstrate more
pronounced downwelling at the leading edge and stronger vertical shear below
the canopy, due to their higher effective frond area density and greater drag.
In addition, the vertical shear of the mean streamwise flow within the farm is
also stronger in B1H compared to S26H, because of the higher effective density
in B1H. Moreover, while the streamwise mean velocity monotonically increases
from the canopy bottom to the sea surface in cases S26H and B1H, the vertical
shear reverses near the surface in B1R, because the presence of the uppermost
dense layer in B1R locally enhances drag and decelerates flow.

Note that the selected cases have a relatively high effective density
$\left\langle a\right\rangle_{xyz}$. Other simulations that are not presented
here, e.g., with larger spacing between kelp rows or with a lower frond area
density, generally have weaker downwelling at the farm leading edge and weaker
shear in the mean streamwsie flow.

###  3.2 Overview

Figure 5: Mean flow ($\overline{u}$, $\overline{w}$, and shear
$\partial\overline{u}/\partial z$), secondary flow ($\overline{w}^{c}$), and
turbulence ($w^{\prime}$) statistics within the farm ($a$-$c$) and below the
farm ($d$-$f$). Each point represents a simulation (see the legends for
details). Note that row 0∘ and row 90∘ denote kelp rows aligned with and
perpendicular to the current, respectively. The horizontal dashed line in
($a$) represents the intensity of standard Langmuir turbulence in the absence
of a farm. The thick gray lines in ($a$) are fitting curves for the three
types of farm configurations. Averaging within the farm is conducted between
$z=0$ and $-h_{b}$, and averaging below the farm is between $-h_{b}$ and
$-2h_{b}$. The horizontal axis in ($a$, $c$-$f$) is the effective density
$\left\langle a\right\rangle_{xyz}$ averaged within the farm. Note that the
shear and vertical velocity are negative below the farm in ($e$) and ($f$).
The mean streamwise velocity is normalized by the the geostrophic velocity
$u_{g}=0.2$ m/s, the mean vertical velocity is normalized by the friction
velocity $u_{*}=0.0061$ m s-1, and the variance terms are normalized by
$u_{*}^{2}$.

Along with modifications to the mean flow, the presence of the farm also
affects Langmuir circulation patterns and turbulence intensity. In this
section, we present an overview of the distinct flow patterns associated with
various farm configurations, before moving on to detailed analysis of Langmuir
circulation and turbulence. The magnitudes of mean flow, secondary flow, and
turbulence components within and below the canopy are summarized in figure 5
for all the farm configurations in table 1.

The mean streamwise flow within the farm decreases with the increased
effective density $\left\langle a\right\rangle_{xyz}$ due to the greater kelp
drag (figure 5$c$), leading to stronger downwelling below the farm as a result
of mass conservation (figure 5$f$). In the shear layer below the canopy, the
vertical shear of streamwise mean flow generally increases in magnitude with
the increased effective frond density (figure 5$e$). Correspondingly, the
turbulence intensity in the shear layer – quantified by the variance of the
temporal fluctuating component of vertical velocity
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{xyz}$ – increases
with the increased $\left\langle a\right\rangle_{xyz}$ (figure 5$d$). This is
consistent with that expected for classical canopy flow, where the canopy
effect positively depends on $\left\langle a\right\rangle_{xyz}$ (e.g., Poggi
et al., 2004; Bailey & Stoll, 2013). For shear layer statistics, the average
is calculated within a vertical distance equal to $h_{b}$ beneath the canopy
bottom, i.e.,

$\left\langle\cdot\right\rangle_{xyz}=\frac{1}{L_{MF}L_{y}h_{b}}\int_{0}^{L_{MF}}\int_{-L_{y}/2}^{L_{y}/2}\int_{-2h_{b}}^{-h_{b}}\cdot\
\text{d}x\text{d}y\text{d}z$ (11)

for averaging below the farm. Here we select the vertical component
$w^{\prime}$ because of its direct relevance to vertical transport in kelp
farms. More discussions about the other components $u^{\prime}$ and
$v^{\prime}$ and turbulence anisotropy will be presented later. Note that the
scatter in figure 5$(d)$ and $(e)$ primarily results from the two different
vertical profiles of frond area density. The dense layer near the surface in
the ripe profile has a small influence on the shear layer dynamics below the
canopy, and excluding this dense surface layer from the effective density
calculation can lead to improved alignment in figure 5$(d)$ and $(e)$.

As a contrast to the shear layer turbulence generated below the canopy,
turbulence intensity within the farm displays a more complex dependence on the
effective frond density (figure 5$a$). Note that within this depth range,
e.g., between the sea surface and $z=-h_{b}$, standard Langmuir turbulence is
expected to occur in the absence of the canopy (McWilliams et al., 1997). In
the presence of the canopy, for kelp frond density with the harvested vertical
profile, $\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$ is
enhanced compared to that of the standard Langmuir turbulence. Moreover, the
intensity of enhanced turbulence within the farm positively depends on the
effective density $\left\langle a\right\rangle_{xyz}$. However, for farm
configurations with the ripe vertical profile, turbulence is inhibited in farm
blocks and kelp rows that are perpendicular to the geostrophic current. For
farms with kelp rows aligned with the current,
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$ is enhanced
compared to the standard Langmuir turbulence only for cases with an
intermediate effective density $\left\langle a\right\rangle_{xyz}$. This
enhancement diminishes as $\left\langle a\right\rangle_{xyz}$ increases, i.e.,
with closely spaced rows that asymptotically resemble a farm block, or, as
$\left\langle a\right\rangle_{xyz}$ decreases, asymptoting toward a scenario
without a farm.

For canopy flow without the influence of waves and Langmuir circulation, the
turbulence intensity within the canopy is usually anticipated to decrease with
increased effective density, due to the weaker penetration of shear layer
eddies (Poggi et al., 2004; Bailey & Stoll, 2013). Nevertheless, our
simulations show that the turbulence intensity in the farm either positively
depends on the effective density or exhibits a more complex dependence. This
contrasting dependence on effective density underscores the distinct type of
turbulence that arises from the interaction between the canopy and the OML,
which is called the farm-generated Langmuir turbulence and will be presented
in the following sections.

Moreover, secondary flow
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$ (stationary
circulation) can be generated within the farm exclusively in cases with kelp
rows aligned with the current (figure 5$b$). The occurrence of stationary
secondary circulation associated with spaced kelp rows is consistent with the
previous study by Yan et al. (2021), which referred to these flow patterns as
‘attached Langmuir circulation’. The intensity of
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$ is always
stronger in cases with the harvested profile compared to the ripe profile. Our
simulations do not allow for a clear identification of the maximum intensity
with respect to the lateral spacing $S_{MF}$, and the peak of
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$ appears to
fall within the range of $S_{MF}=52$ m and 208 m. Note that secondary flow
never occurs in the shear layer below the canopy.

Overall, these various flow statistics highlight the distinction between the
hydrodynamics within the farm (within the OML where Langmuir turbulence is
expected) and the classical shear layer turbulence beneath the farm. In
addition, the contrast between different farm simulations suggests that these
hydrodynamic processes depend on both the vertical frond density profile and
horizontal farm arrangement, and these factors will be analyzed in detail
below.

###  3.3 Langmuir circulations

Three representative simulations are selected to investigate the distinct flow
patterns associated different farm configurations, i.e., case S26H (spaced
rows aligned with $x$-direction, harvested profile), case B1H (farm block,
harvested profile), and case B1R (farm block, ripe profile). Snapshots of
instantaneous vertical velocity of the three simulations are compared in
figure 6, on a horizontal plane at $z=-0.25h_{b}$.

Figure 6: Snapshots of normalized vertical velocity $w/u_{*}$ on a horizontal
plane at $z=-0.25h_{b}$, for cases S26H $(a)$, B1H $(b)$, and B1R $(c)$.
Dotted rectangles show the extent of the farm block or rows.

Upstream of the farm, the elongated streaks of downward vertical velocity are
indicative of standard Langmuir circulation (e.g., McWilliams et al., 1997).
These patterns are typically featured by stronger downward motions within
narrower regions compared to the broader and weaker upward motions. In
addition, the streaks are generally rotated to the right of the direction of
the wind and waves ($x$-direction) as a result of Coriolis. The standard
Langmuir circulation patterns are transient, characterized by their continuous
cycles of formation, evolution, and dissipation.

For the case with laterally spaced kelp rows (case S26H), Langmuir circulation
within the farm area shows a notable increase in magnitude compared to the
standard Langmuir circulation in the upstream region (figure 6$a$). Moreover,
these Langmuir circulation patterns are locked in space in the cross-stream
direction, generally with upward motions within the kelp rows and downward
motions within the gaps in between. The Langmuir patterns in the farm are
generally stationary in time, although the smaller-scale coherent structures
are still transient. These patterns are termed as the ‘attached Langmuir
circulation’ (Yan et al., 2021) because of their locked-in-space
characteristics.

For the farm block case with the harvested profile (case B1H), Langmuir
circulation is also enhanced within the farm compared to the upstream region
(figure 6$b$). However, these farm-enhanced Langmuir circulation patterns are
completely transient and not locked in space due to the absence of repeated
kelp rows, and we thus refer to these patterns as ‘unattached Langmuir
circulation’. In the vertical direction, both the attached and unattached
Langmuir circulation patterns have comparable dimentions to the farm height,
with their maximum intensity found at a depth of around 10 m (quantitative
results shown in § 3.4).

Furthermore, in contrast to the Langmuir patterns found in case B1H, Langmuir
circulation notably vanishes in the farm in case B1R (farm block with the ripe
profile, see figure 6$c$). This implies an absence of the Langmuir circulation
generation mechanism or an increase in dissipation in case B1R, as will be
investigated below. It is also worthwhile noting that the vertical velocity
exhibits patterns aligned with the lateral direction in case B1R, in contrast
to the streaks aligned with the streamwise direction in other cases. These
distinct patterns in case B1R (e.g., for $x/h_{b}$ from 5 to 10) corresponds
to shear-generated turbulence within the farm, as the canopy drag force
significantly decelerates flow near the sea surface and enhances the vertical
shear (figure 4$a$). Moreover, shear layer turbulence generated at the farm
bottom edge could penetrate into the farm in cases where Langmuir turbulence
is inhibited (e.g., for $x/h_{b}>15$ in figure 6$c$). The distinct Langmuir
patterns as well as the penetration of shear layer eddies will be further
analyzed in subsequent sections.

Additionally, a simulation with laterally spaced kelp rows with the ripe
profile (case S26R) is examined. In this case, attached Langmuir circulation
is generated in the farm (not shown) that resembles the patterns found in case
S26H, while the magnitude of vertical velocity is smaller in case S26R
compared to S26H.

###  3.4 Langmuir turbulence and bottom shear layer turbulence

The temporally and laterally averaged vertical velocity variance associated
with transient eddies
($\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$) is calculated
to quantify the farm-generated turbulence (figure 7). The vertical component
$w^{\prime}$ is selected because Langmuir turbulence is usually characterized
by its large vertical velocity variance (McWilliams et al., 1997; Yan et al.,
2021), and the vertical component also directly influences vertical transport.

Figure 7: Side views of the transient component of vertical velocity variance
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}/u_{*}^{2}$ for
cases S26H $(a)$, B1H $(b)$, and B1R $(c)$. The results are temporally and
laterally averaged. Dotted rectangles show the extent of the farm.

Standard Langmuir turbulence occurs upstream of the farm, with the maximum
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$ found slightly
below the sea surface, e.g., around 5-10 m, consistent with McWilliams et al.
(1997). Within the farm, similar Langmuir-type turbulence is generated in the
two cases with the harvested profile (B1H and S26H, figure 7$a$ and $b$). The
farm-generated Langmuir turbulence has a stronger magnitude compared to the
standard Langmuir turbulence in the upstream region, while its vertical
variance $\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$ also
peaks at a similar depth of around 5-10 m. In the streamwise direction, the
intensity of farm-generated Langmuir turbulence increases from the leading
edge of the farm as flow adjusts to the canopy drag, with a maximum at around
$x/h_{b}=10$ (also see figure 9). Turbulence intensity then decreases toward
the farm trailing edge, due to the decrease in the production mechanisms and
the damping by kelp drag and viscosity (details examined in the energy budget
calculation in § 4). In addition, the farm-generated Langmuir turbulence is
stronger in the farm block case B1H than the spaced rows case S26H (both with
a harvested profile), and this aligns with the positive dependence on
effective frond area density in figure 5$(a)$. By contrast, no intensified
turbulence is found in the farm block with a ripe profile (case B1R) (figure
7$c$), consistent with the absence of Langmuir patterns in the map view plot
of vertical velocity (figure 6$c$).

The intensified turbulence in the kelp farm is referred to as Langmuir-type
because its generation relates to the Stokes drift (see § 4), akin to the
standard Langmuir turbulence. Furthermore, the intensified Langmuir-type
turbulence in the farm completely disappears in another set of test cases
(S26H-NW, B1H-NW, B1R-NW, not presented here), where the surface wave forcing
is excluded. This provides corroborating evidence that the Langmuir-type
turbulence results from the interaction between waves and canopy flow.

In addition to the Langmuir-type turbulence generated within the farm, shear
layer turbulence occurs at the bottom edge of the canopy. The shear layer
turbulence, as a characteristic of classical canopy flow, is consistently
found across all the simulations regardless of the presence or absence of
Langmuir-type turbulence within the farm. The intensity of shear layer
turbulence typically increases with the increased effective density (e.g.,
Bailey & Stoll, 2013), so that the farm block cases (B1H and B1R) exhibit
stronger turbulence below the farm compared to the case with spaced kelp rows
(S26H).

While the shear layer turbulence mostly occurs beneath the canopy, it also
penetrates into the canopy, in particular in the downstream part of the farm.
Nevertheless, the penetrated shear layer turbulence is overall much weaker
than the farm-generated Langmuir turbulence (as will be substantiated by the
skewness analysis in § 3.5). Therefore, in cases with enhanced Langmuir
turbulence, shear layer turbulence has a minimal influence on the turbulence
statistics calculated within the farm. This point is also supported by the
dependence of turbulence intensity on the effective density $\left\langle
a\right\rangle_{xyz}$ in figure 5$(a)$. Turbulence intensity within the canopy
is expected to decrease with increased $\left\langle a\right\rangle_{xyz}$,
assuming penetration of shear layer is the dominant source of turbulence
(Poggi et al., 2004; Bailey & Stoll, 2013). Nevertheless, our simulations show
a more complex dependence of turbulence intensity on $\left\langle
a\right\rangle_{xyz}$, due to the dominance of Langmuir-type turbulence within
the farm.

We have primarily focused on the vertical component $w^{\prime}$, and the
horizontal components $u^{\prime}$ and $v^{\prime}$ generally display similar
distributions to $w^{\prime}$. However, it is worthwhile noting that the
transient eddies are not isotropic, and this turbulence anisotropy also
provides insights into distinguishing between different types of farm-
generated turbulence. The shear layer turbulence typically has a stronger
streamwise component $u^{\prime}$ compared to the other components, while the
Langmuir-type turbulence is featured by a relatively stronger vertical
component $w^{\prime}$.

Figure 8: Side view of the secondary flow component of vertical velocity
variance
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}/u_{*}^{2}$ for
cases S26H. The results are temporally and laterally averaged. The dotted
rectangle shows the extent of the farm.

In figure 8, we plot the temporally and laterally averaged vertical velocity
variance associated with the steady secondary flow, denoted as
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$. Note that
secondary flow includes both the lateral variations in streamwise velocity
($\overline{u}^{c}$) and the stationary lateral circulation
($\overline{v}^{c}$ and $\overline{w}^{c}$). Secondary flow exclusively occurs
in the case with laterally spaced kelp rows (case S26H), where attached
Langmuir circulation occurs. In farm block simulations where transient
Langmuir circulation has been found,
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$ is generally
negligible. The region of high
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$ roughly
coincides with that of high
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$ for case S26H,
while the peak of
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}$ is slightly
upstream of $\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}$
(figure 9).

Figure 9: Streamwise variations of
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}/u_{*}^{2}$
(dashed lines) and
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}/u_{*}^{2}$
(solid lines) for cases S26H (black), B1H (red), and B1R (blue). The results
are temporally, laterally, and vertically ($z=0$ to $-h_{b}$) averaged.
Vertical dotted lines show the extent of the farm.

Figure 10: Spectra of the vertical velocity versus the lateral-direction
wavenumber $k_{y}$. $(a)$: Cases S26H (rows aligned with the $x$-direction, 26
m spacing, solid black line) and S52H (52 m spacing, solid yellow line), at
$x/h_{b}=10$ and averaged between $z=0$ and $z=-h_{b}$. The dash-dotted black
line represents the inflow condition at the upstream boundary
($x/h_{b}=-7.5$). Two vertical gray lines indicate the corresponding spacing
between kelp rows, 26 m and 52 m, respectively. $(b)$: Cases B1H (solid blue
line) and B1R (solid red line).

We calculated the lateral-direction wavenumber spectra of the vertical
velocity to investigate the characteristic lateral spacing of Langmuir
circulations. In case S26H (rows with a spacing of 26 m), the spectral peak in
the farm aligns with the lateral row spacing $S_{MF}=26$ m (figure 10$a$).
Similarly, for case S52H with rows spaced at 52 m, the peak wavelength
corresponds to $S_{MF}=52$ m. Note that these spectra are calculated from the
total vertical velocity $w$, and the energy peaks in the above two cases are
predominantly contributed by the secondary flow component $\overline{w}^{c}$.
The peaks would be considerably less distinct if calculating the spectra using
the turbulent component $w^{\prime}$ alone. In case B1H (farm block with the
harvested profile), the spectral peak corresponds to a wavelength of around 30
m (figure 10$b$). This peak wavelength is slightly smaller than that of the
inflow condition with standard Langmuir turbulence. In contrast to the spaced
rows, the farm block exhibits a less distinct peak, due to the absence of
secondary flow structures caused by row spacing. For Case B1R (farm block with
the ripe profile), the spectrum exhibits no apparent peak because of the
inhibition of Langmuir turbulence in the farm, with a weakened overall
magnitude compared to that upstream of the farm.

###  3.5 Vertical velocity skewness

The skewness of the vertical turbulence component, defined as
$\left\langle\overline{{w^{\prime}}^{3}}\right\rangle_{y}/\left\langle\overline{{w^{\prime}}^{2}}\right\rangle_{y}^{3/2}$,
is plotted in figure 11 to illustrate the distinct attributes of various types
of turbulence. The standard Langmuir turbulence in the upstream region is
characterized by its negative skewness of vertical velocity (McWilliams et
al., 1997). The negative skewness indicates stronger downward motions confined
within narrower regions in comparison to broader and weaker upward motions,
consistent with the Langmuir circulation patterns shown in figure 6.

Figure 11: Side views of the skewness of $w^{\prime}$ (transient component of
vertical velocity) for cases S26H $(a)$, B1H $(b)$, and B1R $(c)$. The results
are temporally and laterally averaged. Dotted rectangles show the extent of
the farm.

In the shear layer generated at the canopy bottom, the vertical velocity
skewness is generally negative beneath the canopy and positive near the canopy
edge (figure 11). The positive skewness of vertical velocity near the canopy
edge indicates the dominance of sweep events that bring high-momentum fluid
into the canopy, consistent with those found in classical canopy flow without
waves (Raupach & Thom, 1981; Katul et al., 1997; Poggi et al., 2004). Note
that the skewness sign in the suspended canopy is opposite to that of
submerged benthic canopies due to the reversed geometry of the problem setups
(i.e., here sweep events are characterized by intensified positive vertical
velocities into the canopy). Conversely, the negative skewness of vertical
velocity below the canopy is indicative of the prevalence of ejection events.

The vertical velocity skewness of the farm-generated Langmuir turbulence
appears less clear, displaying mostly negative values. However, the boundary
between shear layer turbulence and farm-generated Langmuir turbulence remains
discernible through their difference in the skewness sign inside the canopy.
Specifically, in case B1H in figure 11$(b)$, the positive vertical velocity
skewness associated with shear layer turbulence occupies less than one third
of the canopy height. Furthermore, the region with the highest turbulence
intensity is located above the region with positive skewness (figure 7$b$),
underscoring that the majority of energy within the farm is attributed to
Langmuir-type turbulence rather than the penetrated shear layer turbulence.

For case B1R (farm block with the ripe profile), the concept of skewness
within the farm lacks significance due to the absence of Langmuir-type
turbulence. However, indications of shear-generated turbulence can be found
within the farm, e.g., in the range of $x/h_{b}$ from 0 to 5, where the mean
flow is adjusting to the canopy drag (figure 11$c$). This shear-generated
turbulence within the farm exhibits similar characteristics to the shear layer
turbulence below the canopy bottom edge, although with lower intensity. It
results from the vertical variability in frond area density, and then rapidly
dissipates downstream in the farm.

Figure 12: Side view of the skewness of $\overline{w}^{c}$ (steady secondary
flow component of vertical velocity) for case S26H. The results are temporally
and laterally averaged. The dotted rectangle shows the extent of the farm.

Similarly, we define a skewness for the steady secondary flow as
$\left\langle{\overline{w}^{c}}^{3}\right\rangle_{y}/\left\langle{\overline{w}^{c}}^{2}\right\rangle_{y}^{3/2}$.
The steady secondary velocity is generally upward in the kelp rows and
downward in the gaps between kelp rows (figure 6). In cases where the kelp
rows are narrower than the gaps, e.g., case S26H in figure 12, the upward
secondary flow is typically stronger in magnitude than the downward flow as a
result of mass balance. Therefore, a positive skewness of the steady secondary
vertical velocity is evident in the upstream part of the farm (figure 12).
Toward the downstream region of the farm, the vertical secondary velocity
skewness becomes less distinct. This can be attributed to the lateral
expansion of Langmuir wakes linked to kelp rows, so that regions of upwelling
may extend beyond the constraints of kelp rows (figure 6$a$, also see Appendix
B for a clearer presentation of the Langmuir wake pattern in case S208H, the
single-row simulation). As one may anticipate, the skewness of secondary flow
varies with changes in farm width and lateral spacing across different farm
configurations. As an example, the skewness
$\left\langle{\overline{w}^{c}}^{3}\right\rangle_{y}/\left\langle{\overline{w}^{c}}^{2}\right\rangle_{y}^{3/2}$
in case S208H can reach up to a value of 3, where the upwelling of secondary
flow is confined within a single kelp row before wake expansion occurs.

The skewness of turbulence and secondary flow investigated above also
influences diffusive and dispersive fluxes in the farm, potentially leading to
asymmetric vertical nutrient transport (Wyngaard & Brost, 1984; Wyngaard &
Weil, 1991; Chor et al., 2020, 2021). Note that spurious oscillations
displaying alternating positive and negative skewness can be found in the
secondary flow and turbulence components, e.g., around $z/h_{b}=-2$ and
$x/h_{b}=15$ in figure 12 and around $z/h_{b}=-2$ and $x/h_{b}=20$ in figure
11$(a)$. These oscillations occur in regions where $\overline{w}^{c}$ or
$w^{\prime}$ is minimal and thus do not have practical significance.

##  4 Kinetic energy balance

In this section we investigate the kinetic energy equations to understand the
sources of Langmuir circulation and turbulence. Following the convention in
(8) and (9), we decompose the total kinetic energy $K_{T}$ into the mean flow
energy $K_{M}$, secondary flow energy $K_{SE}$, and turbulence kinetic energy
$K_{TE}$, i.e.,

$K_{T}=\frac{1}{2}\left\langle\overline{u_{i}u_{i}}\right\rangle_{y}=\underbrace{\frac{1}{2}\left\langle\overline{u_{i}}\right\rangle_{y}\left\langle\overline{u_{i}}\right\rangle_{y}\vphantom{\frac{1}{2}\left\langle\overline{u^{\prime}_{i}u^{\prime}_{i}}\right\rangle_{y}}}_{K_{M}}+\underbrace{\frac{1}{2}\left\langle\overline{u_{i}}^{c}\overline{u_{i}}^{c}\right\rangle_{y}\vphantom{\frac{1}{2}\left\langle\overline{u^{\prime}_{i}u^{\prime}_{i}}\right\rangle_{y}}}_{K_{SE}}+\underbrace{\frac{1}{2}\left\langle\overline{u^{\prime}_{i}u^{\prime}_{i}}\right\rangle_{y}}_{K_{TE}}.$
(12)

We can derive the transport equations for $K_{M}$, $K_{SE}$, and $K_{TE}$ from
the governing equations (1) and (2) (Yan et al., 2021): eq:tke-budget-KM DKMDt
&= - C_M-SE - C_M-TE + S_M \+ D_M + T_M + R_M,
eq:tke-budget-KSE DKSEDt = C_M-SE - C_SE-TE + S_SE \+ D_SE + T_SE,
eq:tke-budget-KTE DKTEDt = C_M-TE + C_SE-TE + S_TE + ϵ_TE + D_TE + T_TE. Here
the material derivative is $\text{D}/\text{D}t=\partial/\partial
t+\left\langle\overline{u_{j}}\right\rangle_{y}\partial/\partial
x_{j}+u_{s}\partial/\partial x$. Note that we have neglected the buoyancy
production terms, as the farm is located within the OML. Terms $C_{M-SE}$,
$C_{M-TE}$, and $C_{SE-TE}$ represent the energy conversion between the mean
flow $K_{M}$, secondary flow $K_{SE}$, and turbulence $K_{TE}$ (also referred
to as shear production), written as C_M-SE &= - ⟨¯u_i^c ¯u_j^c⟩_y ∂⟨¯ui⟩y∂xj,
C_M-TE = - ⟨¯u_i’ u_j’⟩_y ∂⟨¯ui⟩y∂xj,
C_SE-TE = - ⟨¯u_i’ u_j’ ∂¯uic∂xj⟩_y. Terms $S_{M}$, $S_{SE}$, and $S_{TE}$
stand for Stokes production that transfers energy between waves and the three
flow components, and are given by

$\displaystyle
S_{M}=-\left\langle\overline{u}\right\rangle_{y}\left\langle\overline{w}\right\rangle_{y}\frac{\partial{u_{s}}}{\partial{z}},\quad
S_{SE}=-\left\langle\overline{u}^{c}\overline{w}^{c}\right\rangle_{y}\frac{\partial{u_{s}}}{\partial{z}},\quad
S_{TE}=-\left\langle\overline{u^{\prime}w^{\prime}}\right\rangle_{y}\frac{\partial{u_{s}}}{\partial{z}}.$
(13a-c)

The SGS dissipation term is

$\epsilon_{TE}=-\left\langle\overline{\tau^{\prime}_{ij}\frac{\partial{u^{\prime}_{i}}}{\partial{x_{j}}}}\right\rangle_{y}.$
(14)

The majority of energy dissipation is expected to occur in the small-scale
transient eddies for turbulent flows with a large Reynolds number (e.g., Pope,
2000). Therefore, the energy loss of the larger-scale mean flow and secondary
flow due to direct SGS dissipation is considered negligible, and we only focus
on the dissipation of turbulence kinetic energy. The canopy drag dissipation
terms are

$\displaystyle
D_{M}=-\left\langle\overline{u_{i}}\right\rangle_{y}\left\langle\overline{F_{D,i}}\right\rangle_{y},\quad
D_{SE}=-\left\langle\overline{u_{i}}^{c}\overline{F_{D,i}}^{c}\right\rangle_{y},\quad
D_{TE}=-\left\langle\overline{u^{\prime}_{i}F^{\prime}_{D,i}}\right\rangle_{y},$
(15a-c)

which represent the energy loss or gain associated with canopy drag. In
addition, several flux terms representing kinetic energy transport associated
with the resolved stress, SGS stress, and pressure are collected into the
transport terms $T_{M}$, $T_{SE}$, and $T_{TE}$. The last term
$R_{M}=f\left\langle\overline{v}\right\rangle_{y}(u_{g}-u_{s})$ in the mean
flow energy budget represents the energy transfer between the mean flow,
surface waves and the background geostrophic current, resembling the concept
of Stokes-Coriolis work (Suzuki & Fox-Kemper, 2016; Yan et al., 2021).

Figure 13: Side views of terms in the $K_{TE}$ budget in (LABEL:eq:tke-budget-
KTE) for case B1H (farm block, harvested profile). The results are temporally
and laterally averaged. Dotted rectangles show the extent of the farm.

We calculate the turbulent kinetic energy budget for case B1H (farm block with
the harvested profile), and the major source and sink terms are shown in
figure 13. The Stokes production is the dominant source of turbulence upstream
of the farm, where standard Langmuir turbulence is found. This agrees with the
widely recognized mechanism wherein the generation of Langmuir circulation
results from vorticity tilting by the Stokes drift (Craik, 1977; Leibovich,
1977, 1983). Within the farm, the Stokes production becomes significantly
stronger, which, together with the shear production, contributes to the
enhanced Langmuir-type turbulence (figures 13 and 7). Below the farm, shear
production provides the dominant source of shear layer turbulence, and Stokes
production is weak because the Stokes drift diminishes at greater depth. Both
dissipation and canopy drag dissipation terms tend to destroy the turbulence
generated within the farm, while in the shear layer below the canopy, the
dissipation term is the only sink of turbulence kinetic energy. Note that the
secondary flow energy is negligible in the farm block, and there is no energy
conversion associated with secondary flow.

Figure 14: Side views of terms in the $K_{TE}$ budget in (LABEL:eq:tke-budget-
KTE) for case B1R (farm block, ripe profile). The results are temporally and
laterally averaged. Dotted rectangles show the extent of the farm. Note that
differences between case B1R and case B1H (figure 13) primarily lie in the
energy production mechanisms. The dissipation and canopy drag dissipation
terms are not shown here, because the energy loss mechanisms in case B1R are
similar to case B1H.

In case B1R (farm block with the ripe profile), the shear production term is
the major source of turbulence kinetic energy in the shear layer below the
canopy (figure 14), consistent with case B1H discussed above (farm block with
the harvested profile). However, when focusing within the farm, both the
Stokes production and shear production diminish due to the presence of a dense
layer near the surface (characteristic of the ripe profile). This explains the
absence of Langmuir-type turbulence in figure 7$(c)$. In particular, negative
values of Stokes production are even found near the sea surface, indicating
that the shear of the Stokes drift velocity contributes to disrupting vortex
formation rather than stretching vortex tubes (detailed mechanisms are
explained in § 5). Negative vertical shear of the Eulerian velocity arises due
to the presence of a dense kelp layer near the surface, which dominates over
the positive vertical shear of the Stokes drift (figure 4). This results in a
negative sign for the Reynolds stress $\overline{u^{\prime}w^{\prime}}$, and
consequently, the Stokes production term acts as a sink for turbulence kinetic
energy. The dissipation and kelp drag dissipation terms are not shown for case
B1R. The distribution of dissipation generally correlates with that of
turbulence intensity, and strong kelp drag dissipation occurs where areas with
high kelp frond density intersect with intense turbulence.

It is worthwhile noting that the originally farm-generated vertical shear has
a comparable magnitude in cases B1R and B1H, e.g., near the farm leading edge
in the range of $x/h_{b}$ between 0 and 5 (figure 4). In fact, case B1R has
even more pronounced sheared flow near the surface due to the existence of the
dense layer. Nonetheless, the rapid downstream increase in shear production in
case B1H compared to B1R, e.g., for $x/h_{b}$ from 5 to 15, indicates that its
strength does not solely depend on the vertical shear induced by variations in
frond area density. Instead, the increased Stokes production in case B1H plays
a critical role in initiating the growth of turbulence and thus contributing
to the enhancement of shear production. Further investigation of the distinct
Stokes mechanisms in the two cases will be presented in § 5.

Figure 15: Side views of terms in the $K_{TE}$ budget in (LABEL:eq:tke-budget-
KTE) for case S26H (rows aligned with the current, harvested profile). The
results are temporally and laterally averaged. Dotted rectangles show the
extent of the farm.

The turbulence kinetic energy budget is also evaluated for the case with
laterally spaced rows (S26H, harvested profile). The distributions of Stokes
production (figure 15), dissipation, and canopy drag dissipation (not shown)
are generally similar to those in the farm block case (B1H, harvested profile,
figure 13), although with smaller magnitudes due to the overall lower
effective density of the spaced rows. However, there is a distinction in shear
production in the spaced rows as compared to the farm block, owing to the
existence of secondary flow. Unlike the farm block case, where shear
production directly converts mean flow energy to turbulence kinetic energy, a
significant portion of turbulence kinetic energy in spaced rows is generated
through the conversion of secondary flow energy.

Figure 16: Side views of terms in the $K_{SE}$ budget in (LABEL:eq:tke-budget-
KSE) for case S26H (rows aligned with the current, harvested profile). The
results are temporally and laterally averaged. Dotted rectangles show the
extent of the farm.

We subsequently calculate the secondary flow energy budget (figure 16). Note
that the secondary flow energy encompasses both cross-stream variations in
streamwise velocity ($\overline{u}^{c}$) and the steady lateral circulation
($\overline{v}^{c}$ and $\overline{w}^{c}$). Canopy dissipation $D_{SE}$ is a
significant source term in the secondary flow energy budget. Although the
canopy drag leads to energy loss for both the mean flow and turbulence (hence
named as a dissipation term), it can lead to a gain in secondary flow energy.
This energy gain through $D_{SE}$ is evident as cross-stream variations in the
drag force, enhancing the cross-stream variability of streamwise flow. In
addition, the Stokes production term is another major source of secondary flow
energy, contributing to the generation of attached Langmuir circulation.

##  5 Vorticity dynamics of Langmuir circulations

The vorticity dynamics is investigated to understand the generation mechanisms
of Langmuir circulation associated with various farm configurations. By
manipulating the governing equations (1) and (2), we derive the transport
equation for vorticity $\boldsymbol{\zeta}$ (e.g., Fujiwara et al., 2018):

$\displaystyle\frac{\partial{\boldsymbol{\zeta}}}{\partial{t}}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{\zeta}+$
$\displaystyle(\boldsymbol{u}_{s}\cdot\nabla)\boldsymbol{\zeta}=(\boldsymbol{\zeta}\cdot\nabla)\boldsymbol{u}+(\boldsymbol{\zeta}\cdot\nabla)\boldsymbol{u}_{s}-\nabla\times\boldsymbol{F}_{D}$
(16)
$\displaystyle+\nabla\times(\nabla\cdot\boldsymbol{\tau}^{d})-\nabla\times\frac{\rho}{\rho_{0}}g\boldsymbol{e}_{z}+f\boldsymbol{e}_{z}\cdot\nabla(\boldsymbol{u}+\boldsymbol{u}_{s}).$

The time derivative and advection terms are on the left side. The first two
terms on the right side represent vorticity tilting and Stokes drift-vorticity
tilting, respectively. The third term corresponds to the vorticity generation
associated with canopy drag. The fourth to sixth terms represent the
contributions of SGS stress, baroclinicity, and rotation. For the sake of
clarity, we express the equations for each vorticity component separately
eq:vort-x DζxDt &= ζ_x∂u∂x + ζ_y∂u∂y \+ ζ_z∂u∂z + ζ_z∂us∂z + R_ζ,x,
eq:vort-y DζyDt = ζ_x∂v∂x + ζ_y∂v∂y \+ ζ_z∂v∂z - ∂FD,x∂z+ R_ζ,y,
eq:vort-z DζzDt = ζ_x∂w∂x + ζ_y∂w∂y \+ ζ_z∂w∂z + ∂FD,x∂y+ R_ζ,z. The material
derivative is defined as $\text{D}/\text{D}t=\partial/\partial
t+\left\langle\overline{u_{j}}\right\rangle_{y}\partial/\partial
x_{j}+u_{s}\partial/\partial x$, the same as (4). Note that we have assumed
$F_{D,x}\gg F_{D,y},F_{D,z}$, given that the streamwise velocity is much
greater than lateral circulation, and we thus only focus on the influence of
$F_{D,x}$ on vorticity. The term related to SGS stress diffuses vorticity
rather than generating it; the baroclinic effect can be neglected because the
farm is within the mixed layer; the effect of Earth rotation is minimal. We
thus group these factors into the residual terms, denoted as $R_{\zeta,x}$,
$R_{\zeta,y}$ and $R_{\zeta,z}$.

The vorticity is calculated for case S26H (spaced kelp rows aligned with the
current, harvested profile). Here we introduce the phase average in the cross-
stream direction, denoted as $\langle\cdot\rangle_{p}$, to average over
equivalent positions of the periodically repeated rows. For any field $\phi$,
the cross-stream phase average is defined as

$\left\langle\phi\right\rangle_{p}(x,y,z)=\frac{1}{N}\sum_{n=0}^{N-1}\phi\left(x,y+nS_{MF}+\frac{1}{2}W_{MF},z\right),$
(17)

where $N$ is the total number of rows.

Figure 17: Top views of vorticity components $\zeta_{x}$ ($a$), $\zeta_{y}$
($c$), $\zeta_{z}$ ($e$), and their primary forcing terms ($b$, $d$, and $f$)
in (5) for case S26H (rows aligned with the current, harvested profile). The
results are time-averaged, depth-averaged ($z=0$ to $-h_{b}$), and cross-
phase-averaged. Dotted rectangles show the extent of the kelp row.

The dominant phase-averaged terms in the vorticity equation are shown in
figure 17. The persistent downstream vorticity $\zeta_{x}$ corresponds to the
attached Langmuir circulation generated by kelp rows (recall figure 6$a$). The
generation of attached Langmuir circulation is attributed to the tilting of
vertical vorticity $\zeta_{z}$ by the Stokes drift into the downstream
direction, i.e., the term $\zeta_{z}(\partial u_{s}/\partial z)$ in
(LABEL:eq:vort-x). This tilting process aligns with the classical mechanism
that gives rise to standard Langmuir circulation (e.g., Leibovich, 1983). In
the farm the vertical vorticity $\zeta_{z}$ is predominantly generated due to
the drag discontinuity at the lateral edges of kelp rows, i.e., $\partial
F_{D,x}/\partial y$ in (LABEL:eq:vort-z). The enhanced lateral shear in
streamwise velocity ($\zeta_{z}$) due to spaced kelp rows can thus lead to
more intensified Langmuir circulation ($\zeta_{x}$) within the farm (figures
17 and 18$a$), compared to the standard Langmuir circulation upstream of the
farm (Yan et al., 2021). Note that the vorticity field in figure 17 was
averaged over time to show the persistent forcing and stationary Langmuir
circulation associated with spaced kelp rows.

Figure 18: Three-dimensional representation of vortex structures, as revealed
by isosurfaces of $\lambda_{2}$. $(a)$: Cross-phase-averaged and time-averaged
streamwise vorticity $\left\langle\overline{\zeta}_{x}\right\rangle_{p}$ for a
kelp row in case S26H (rows aligned with the current, harvested profile).
$(b)$: A zoom-in view of the instantaneous streamwise vorticity $\zeta_{x}$
for case B1H (farm block, harvested profile). Note the differences in colormap
range and axis range between the two plots.

Subsequently, we investigate the vorticity dynamics in case B1H (farm block,
harvested profile). Here we analyze a representative snapshot as opposed to
the time-averaged field, because the Langmuir circulation patterns in the farm
block are transient and will be smeared out through time-averaging. The
strongest vorticity is generally aligned with the streamwise direction, i.e.,
the $\zeta_{x}$ component (figure 19$a$). This is particularly evident in
regions of intensified Langmuir circulation, e.g., around $x/h_{b}$ from 10 to
15 within the farm (comparing figure 19$a$ with figure 6$b$). The pronounced
$\zeta_{x}$ component is consistent with the feature of standard Langmuir
circulation (McWilliams et al., 1997). The generation of $\zeta_{x}$ is
primarily due to the vorticity tilting by the Stokes drift, i.e.,
$\zeta_{z}(\partial u_{s}/\partial z)$, similar to that in the kelp rows.

Figure 19: Top views of vorticity components $\zeta_{x}$ ($a$), $\zeta_{y}$
($c$), $\zeta_{z}$ ($e$), and their primary forcing terms ($b$, $d$, and $f$)
in (5) for case B1H (farm block, harvested profile). The results are snapshots
of depth-averaged ($z=0$ to $-h_{b}$) results. Dotted rectangles show the
extent of the farm.

However, in contrast to the vorticity dynamics in the kelp rows, the influence
of $\partial F_{D,x}/\partial y$ is diminished in the farm block due to the
absence of lateral variations in kelp frond density. Instead, the vorticity
generation is initiated from the leading edge of the farm block, where
$\partial F_{D,x}/\partial z$, resulting from the vertical frond density
variability, produces cross-stream vorticity $\zeta_{y}$. The drag force is
typically smaller near the surface for the harvested profile, which thus leads
to a positive vertical shear with higher streamwise velocity on top of lower
velocity, i.e., positive $\zeta_{y}$ (figure 19$c$). The higher streamwise
velocity near the surface tends to concentrate toward regions with surface-
convergence (indicative of downwelling), as a result of lateral and vertical
advection by Langmuir circulation. Similarly, lower streamwise velocity is
transported to the upwelling regions in the cross-section. The associated
lateral variability in streamwise velocity enhances vertical vorticity
$\zeta_{z}$ (figure 19$e$), which can subsequently be tilted into $\zeta_{x}$
by the Stokes drift (figure 19$a$). The increased $\zeta_{x}$ represents
stronger Langmuir circulation that can furthur interact with the other
vorticity components $\zeta_{y}$ and $\zeta_{z}$. This interaction leads to
positive feedback, thereby intensifying Langmuir circulation.

The processes described above are consistent with the CL2 mechanism (Craik,
1977; Leibovich, 1977, 1983), which is widely recognized as a key driving
factor for standard Langmuir circulation. According to the CL2 mechanism, the
vorticity tilting due to Stokes drift can cause instability and produce
Langmuir circulation cells when the vertical shear of the Eulerian wind-driven
current is in the same direction as the vertically sheared Stokes drift. In
our study, the presence of the kelp farm further enhances the vertical shear
in the OML through the vertical variability of canopy drag, thus promoting the
growth of instability and leading to more intensified Langmuir circulation.

Note that the patterns of $\zeta_{y}$ and $\zeta_{z}$ are generally consistent
with the forcing terms $\partial F_{D,x}/\partial z$ and $\zeta_{y}(\partial
w/\partial y)$ in the upstream region of the farm, where these vorticity
components are initially generated (figure 19). Other vorticity tilting terms
become more prominent as Langmuir circulation is rapidly enhanced downstream.
This leads to the more complex vorticity patterns found in the downstream part
of the farm.

Figure 20: Top views of vorticity components $\zeta_{x}$ ($a$), $\zeta_{y}$
($c$), $\zeta_{z}$ ($e$), and forcing terms ($b$, $d$, and $f$) in (5) for
case B1R (farm block, ripe profile). These are snapshots of depth-averaged
($z=0$ to $-h_{b}$) results. Dotted rectangles show the extent of the farm.

In contrast, in the farm block with the ripe profile (case B1R), the presence
of the dense layer near the surface tends to decelerate the streamwise
current. Consequently, negative vertical shear arises with lower streamwise
velocity on top of higher velocity, in particular near the sea surface where
the Stokes drift shear is most pronounced (figure 4$a$). This leads to the
negative depth-averaged $\zeta_{y}$ in figure 20$(c)$, in contrast to the
positive $\zeta_{y}$ in figure 19$(c)$ associated with the harvested profile
(case B1H). The negative vertical shear due to the ripe profile in case B1R
results in a scenario that suppresses the growth of CL2 instability. It thus
explains the weak vorticity and absence of Langmuir circulation patterns in
the farm block with the ripe profile (figure 20 and figure 6$c$).

Furthermore, we examine the streamwise evolution of different vorticity
components. In the farm block with the ripe profile (case B1R), cross-stream
vorticity $\zeta_{y}$ promptly increases at the leading edge driven by the
canopy drag gradient $\partial F_{D,x}/\partial z$. Downstream of the leading
edge, all the vorticity components tend to diminish in magnitude through the
farm due to the lack of Langmuir circulation generation. In the farm block
with the harvested profile (case B1H), the rapid growth of $\zeta_{y}$ at the
leading edge subsequently causes the increase of $\zeta_{z}$ and $\zeta_{x}$,
as a result of the tilting mechanism. In comparison, within the spaced kelp
rows (case S26H), the vertical vorticity $\zeta_{z}$ is the vorticity
component that initially increases, driven by $\partial F_{D,x}/\partial y$ at
the lateral edges of kelp rows, and this increase in $\zeta_{z}$ is followed
by the intensification of $\zeta_{x}$ and $\zeta_{y}$. This contrast again
demonstrates the distinction in vorticity generation mechanisms between the
two farm configurations. It also elucidates why Langmuir turbulence still
occurs in kelp rows aligned with the current even with the ripe profile (e.g.,
case S26R): $\zeta_{z}$ can be persistently generated by the spaced kelp rows
and then tilted by the Stokes drift to produce $\zeta_{x}$, even if a vertical
shear that favors the CL2 mechanism is not created.

Figure 21: Top views of vorticity components $\zeta_{x}$ ($a$), $\zeta_{y}$
($b$), and $\zeta_{z}$ ($c$) for case B1H (farm block, harvested profile).
These are snapshots of depth-averaged ($z=-h_{b}$ to $-2h_{b}$) results below
the farm. Dotted rectangles show the extent of the farm.

Differing from the Langmuir patterns found within the farm, the vorticity
patterns beneath the farm generally resemble the classical shear layer
turbulence (Finnigan et al., 2009; Bailey & Stoll, 2016) (figure 21). Negative
cross-stream vorticity $\zeta_{y}$ is generated due to the drag discontinuity
($\partial F_{D,x}/\partial z$) at the canopy bottom edge. Other vorticity
components $\zeta_{x}$ and $\zeta_{z}$ subsequently increase as a result of
vorticity tilting. The influence of Stokes drift rapidly diminishes with
depth, and the Stokes drift - vorticity tilting term is thus negligible below
the farm.

##  6 Sensitivity to other farm parameters

###  6.1 Effective density

In the previous sections, we focused on distinctions between farm blocks and
spaced rows and between different vertical frond density profiles. In
addition, the farm effective density is also an key characteristic affecting
the hydrodynamics of suspended farms (Poggi et al., 2004; Belcher et al.,
2012; Bailey & Stoll, 2013). Figure 22$a$ shows the turbulence kinetic energy
in the farm block with a lower effective density $\left\langle
a\right\rangle_{xyz}$ (case B0.3H, harvested profile, with a density lower
than case B1H by a factor of 0.3). Overall, the shear layer turbulence below
the canopy is weaker in case B0.3H compared to B1H, consistent with the
expected dependence on effective density for classical canopy flow (Poggi et
al., 2004; Bailey & Stoll, 2013).

Turbulence intensity within the canopy typically decreases with the increased
effective density $\left\langle a\right\rangle_{xyz}$ for classical canopy
flow without waves. This is because a higher effective density reduces the
penetration of shear layer turbulence into the canopy and also enhances the
dissipation of turbulence by canopy drag. However, in our study turbulence is
weaker in the lower-density farm block compared to the higher-density case
(figure 22$a$ and figure 7$b$). This is because the lower density leads to
weaker shear (or vorticity) in the canopy, and thus a weaker Stokes drift -
vorticity tilting mechanism and less Langmuir turbulence production (§§ 4 and
5).

Moreover, the region with maximum Langmuir turbulence intensity occurs at a
larger downstream distance from the farm leading edge in the lower effective
density case compared to the higher density case. This is because it takes a
longer distance for flow to adjust to canopy drag and for shear to develop
with a lower effective density. This longer development distance is consistent
with (10), where the adjustment length is expected to be inversely
proportional to the effective density.

Figure 22: Side views of the transient component of vertical velocity variance
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}/u_{*}^{2}$.
$(a)$: Case B0.3H, farm block with a lower frond area density, harvested
profile. $(b)$: Case PS26H, spaced rows oriented perpendicular to the
geostrophic current, harvested profile. The results are temporally and
laterally averaged, similar to figure 7. Dotted rectangles show the extent of
the farm block or rows. Note that cases B0.3H and PS26H have an equal
effective density $\left\langle a\right\rangle_{xyz}$.

###  6.2 Farm orientation

The dependence of turbulence generation on farm orientation is also examined
through a set of simulations with kelp rows oriented perpendicular to the
direction of current and waves (cases ‘PS’). This is particularly relevant to
the farm design problem in realistic ocean environments, given the variability
of submesoscale processes. Farms with kelp rows perpendicular to the flow
generally behave similarly to farm blocks that have an identical effective
density $\left\langle a\right\rangle_{xyz}$, e.g., case PS26H in figure
22$(b)$ and case B0.3H in figure 22$(a)$, both of which have a same effective
density of 0.35 m-1. Note that the configuration with spaced rows could lead
to additional stationary streamwise deviations in canopy flow, typically
corresponding to a dispersive flux (Bailey & Stoll, 2013; Li & Bou-Zeid,
2019). Nevertheless, the influences of dispersive processes appear to be
minimal in the examined cases, and the statistics of kelp rows perpendicular
to the flow are broadly similar to uniform farm blocks (figure 5).

###  6.3 Synthesis

We have presented cases B0.3H and PS26H (refer to table 1) above as a
representative to illustrate the impacts of effective density and farm
orientation. Other simulations with various density or farm configurations
generally show a consistent dependence on effective density $\left\langle
a\right\rangle_{xyz}$. As has been summarized in figure 5$(d)$, the intensity
of shear layer turbulence below the farm positively depends on the effective
density for all the simulations. In addition, for cases with the harvested
profile, which favors the generation of Langmuir-type turbulence, turbulence
intensity within the farm increases as the effective density is increased
(figure 5$a$).

By contrast, for the ripe profile, no obvious dependence of within-farm
turbulence intensity on the effective density is found for farm blocks and
kelp rows oriented perpendicular to the flow (figure 5$a$), because of the
absence of Langmuir circulation. For kelp rows aligned with the geostrophic
flow with the ripe profile, turbulence intensity asymtotes toward a farm block
scenario or no farm scenario with either increasing or decreasing effective
density, and the maximum intensity is achieved with an effective density of
around 0.5 m-1.

The distance for farm-generated Langmuir turbulence to peak generally
increases with the decreased effective density $\left\langle
a\right\rangle_{xyz}$, for configurations with a farm block or kelp rows
perpendicular to flow. Nevertheless, in kelp rows aligned with flow, the
distance for Langmuir turbulence to develop is much smaller than a farm block
with an identical effective density, e.g., comparing figure 7$(a)$ with figure
22$(a)$. This is because kelp rows aligned with the current can lead to rapid
growth of Langmuir turbulence through the creation of vorticity at lateral
edges of rows by the drag force gradient $\partial F_{D,x}/\partial y$. It
further implies that this type of configuration may result in a shorter
adjustment region length compared to that expected with the effective density
in (10).

###  6.4 Extended farm length

Figure 23: Turbulence and secondary flow statistics in the long farm
simulation (case S26LH, spaced rows aligned with the current, harvested
profile). $(a)$: Streamwise variations of vertical velocity variance
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}/u_{*}^{2}$
(turbulence component, dashed line) and
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}/u_{*}^{2}$
(secondary flow component, solid line). The results are temporally, laterally,
and vertically ($z=0$ to $-h_{b}$) averaged. Vertical dotted lines show the
extent of the farm. $(b)$ and $(c)$: Side views of
$\left\langle\overline{w^{\prime}w^{\prime}}\right\rangle_{y}/u_{*}^{2}$ and
$\left\langle\overline{w}^{c}\overline{w}^{c}\right\rangle_{y}/u_{*}^{2}$. The
results are temporally and laterally averaged. Dotted rectangles show the
extent of the farm. $(d)$: Side view of the skewness of $w^{\prime}$.

We ran an additional simulation (case S26LH, refer to table 1) with an
extended farm length of $L_{MF}=800$ m to further investigate the downstream
development of farm-generated turbulence. The long farm case S26LH has the
same configuration as case S26H (spaced kelp rows aligned with the flow with
$S_{MF}=26$ m, harvested profile). The turbulence and secondary flow
characteristics in the upstream part of the long farm are generally similar to
that of the short farm, e.g., comparing figure 23 with figures 7$(a)$, 8, and
11$(a)$.

In the upstream part of the long farm, secondary flow diminishes after around
$x/h_{b}=15$, and turbulence intensity also remains at a stable value after a
decay from its peak value (figure 23$a$ and $b$). Turbulence statistics near
the trailing edge of the long farm are broadly similar to those of another
simulation with an infinite farm length (not shown). The canopy flow is thus
considered as having reached an equilibrium state toward the long farm
trailing edge.

We note that the farm-generated Langmuir-type turbulence and secondary flow
investigated in this study primarily occurs in the upstream part of the farm
where flow is adapting to the canopy drag, before the establishment of an
equilibrium canopy flow. Nevertheless, our investigation is relevant to the
farm design problem, given that the farm length is comparable to or less than
the canopy adjustment length in a range of realistic farm design
considerations.

The turbulence intensity near the long farm trailing edge, in spite of the
disappearance of secondary flow there, is still higher than that of the
standard Langmuir turbulence observed upstream of the farm (figure 23$a$).
Moreover, it is worthwhile noting that positive skewness of $w^{\prime}$ only
occurs in the lower one third of the canopy in the equilibrium state at the
trailing edge (figure 23$d$). The positive skewness of $w^{\prime}$ is a
typical feature of the shear layer turbulence that is penetrated into the
canopy, corresponding to the prevalent presence of sweep events (Raupach &
Thom, 1981; Katul et al., 1997; Poggi et al., 2004). The negative skewness of
$w^{\prime}$ above the penetrated shear layer is indicative of the Langmuir-
type turbulence (McWilliams et al., 1997). This indicates that the shear layer
turbulence growing from the farm bottom does not penetrate through the entire
canopy in the equilibrium state.

We calculate the penetration length scale $\delta_{e}$ based on the empirical
relationship (Nepf et al., 2007; Nepf & Ghisalberti, 2008)

$\delta_{e}=\frac{0.23}{\frac{1}{2}C_{D}\left\langle a\right\rangle_{xyz}}.$
(18)

Here the factor of $\frac{1}{2}$ accounts for the projection of frond surface
area (term $\boldsymbol{P}$ in (5)). The calculated $\delta_{e}$ is four times
larger than the canopy height, suggesting that shear layer eddies would
penetrate to the sea surface in fully developed canopy flow. As a comparison,
in the simulation without waves (case S26H-NW), positive skewness of
$w^{\prime}$, which represents shear layer turbulence, occupies the entire
farm height after around $x/h_{b}=15$, and this is consistent with the
prediction from (18). Nevertheless, the coexistence of both positive and
negative skewness of $w^{\prime}$ in figure 23$(d)$ implies that wave effects
can impede the penetration of shear layer eddies. An adjusted OML is
established in the equilibrium state, characterized by a combination of
Langmuir-type turbulence and penetrated shear layer turbulence.

##  7 Conclusion

In this study, we investigate the influence of various suspended farm
configurations on the OML hydrodynamics. The drag force induced by kelp can
alter the vertical profile of mean flow and leads to generation of shear layer
turbulence beneath the farm. Moreover, Langmuir circulation can be generated
within the farm, differing from the standard Langmuir circulation in the
absence of a farm. These modifications to the OML depend on both the farm
horizontal arrangement and the vertical profile of frond area density.

In farm blocks or spaced kelp rows that are perpendicular to the current,
distinctions in Langmuir circulation patterns emerge between the harvested
profile and ripe profile (without and with the dense layer near the surface,
respectively). Enhanced Langmuir circulation occurs in the farm with the
harvested profile, characterized by transient patterns that have a larger
magnitude compared to the standard Langmuir circulation upstream of the farm.
The average turbulence intensity in the farm, dominantly contributed by
Langmuir turbulence, has a positive dependence on the effective density
$\left\langle a\right\rangle_{xyz}$. The generation of Langmuir turbulence is
attributed to the establishment of a vertically sheared current profile that
favors CL2 instability, as a result of the vertical variability in canopy
drag. The lateral vorticity (vertical shear of streamwise velocity) gives rise
to vertical vorticity, which is subsequently tilted into streamwise vorticity
by the Stokes drift to create Langmuir circulation. By contrast, for the ripe
profile, a reversed vertical shear occurs near the sea surface due to the
presence of the dense kelp layer, and Langmuir turbulence is absent in the
farm due to the suppression of CL2 instability.

The farm configuration involving spaced kelp rows aligned with the flow leads
to the generation of attached Langmuir circulation for both the harvested and
ripe profiles. The attached Langmuir circulation is spatially locked in the
lateral direction, comprised of both a turbulence component and a stationary
secondary flow component. Strong lateral shear is generated at the lateral
edges of kelp rows, and this vertical vorticity is subsequently tilted into
the streamwise direction by the Stokes drift, producing attached Langmuir
circulation. The intensity of Langmuir turbulence monotonically increases with
the effective density $\left\langle a\right\rangle_{xyz}$ for the harvested
profile. However, for the ripe profile, the enhancement of Langmuir turbulence
diminishes at either high $\left\langle a\right\rangle_{xyz}$, resembling a
farm block, or low $\left\langle a\right\rangle_{xyz}$, resembling a scenario
without a farm. The peak value occurs at an intermediate effective density,
which is around $\left\langle a\right\rangle_{xyz}=0.5$m-1 in our simulations.
Additionally, the strength of the secondary flow increases from the kelp row
lateral spacing $S_{MF}$ of 13 m to 52 m, and decays at $S_{MF}=208$ m. The
simulations do not allow for an accurate determination of the maximum
secondary flow intensity with respect to $S_{MF}$, and it appears to fall
within the range of $S_{MF}=52$ m and 208 m.

The vertical shear of mean flow is enhanced below the farm, leading to
development of shear layer turbulence. The intensity of shear layer turbulence
generally increases with the effective density $\left\langle
a\right\rangle_{xyz}$, while displaying a minimal dependence on the specific
farm configuration.

We have focused on a specific set of representative oceanic conditions in the
present study, and the implications of turbulence generation can be extended
to various wave and current conditions. Stronger winds and waves typically
increase the Stokes drift and the CL vortex force, therefore further
intensifying Langmuir turbulence within the farm configurations that favor
turbulence generation. Weaker ocean currents would lead to a reduced level of
shear layer turbulence below the farm, as its generation depends on the
magnitude of shear (figure 5$d$ and $e$). Moreover, the enhancement of
Langmuir turbulence in the farm relies on the interaction between the Stokes
drift and farm-modulated currents. The canopy drag effect would diminish under
conditions of weaker ocean currents, which is thus expected to result in less
deviation from the standard Langmuir turbulence.

The different mean flow, secondary flow, and turbulence characteristics
associated with various farm configurations have significant implications for
nutrient transport and farm performance, as will be examined in an upcoming
companion study. Moreover, the presence of suspended farms notably alters
turbulence intensity within the OML. These findings underscore the potential
of floating obstacle structures in reshaping characteristics of the ocean
surface boundary layer. The presence of such floating structures may enhance
the shear in ocean currents, giving rise to shear layer turbulence.
Additionally, the modified velocity profile caused by floating structures can
interact with surface gravity waves, thereby influencing the generation of
Langmuir turbulence in the OML.

[Acknowledgements]The authors thank Kristen Davis for helpful discussions.

[Funding]The research leading to these results was funded by the ARPA-E
MARINER Program DE-AR0000920. We would like to acknowledge high-performance
computing support from Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s
Computational and Information Systems Laboratory, sponsored by the National
Science Foundation. Chao Yan is supported by the youth innovation team of
China Meteorological Administration (No. CMA2024QN12).

[Declaration of interests]The authors report no conflict of interest.

##  Appendix A Detailed farm parameters

Case | Wave |  $L_{MF}$ (m) |  $S_{MF}$ (m) |  $W_{MF}$ (m) | Frond vert. prof. | Orientation |  $\left\langle a\right\rangle_{xyz}$ (m-1)
---|---|---|---|---|---|---|---
B1H $\star$ | Yes | 400 | - | 208 | Harvested | - | 1.14
B0.3H | Yes | 400 | - | 208 | Harvested | - | 0.35
B0.16H | Yes | 400 | - | 208 | Harvested | - | 0.18
B0.08H | Yes | 400 | - | 208 | Harvested | - | 0.09
B1R $\star$ | Yes | 400 | - | 208 | Ripe | - | 2.20
B0.3R | Yes | 400 | - | 208 | Ripe | - | 0.68
B0.16R | Yes | 400 | - | 208 | Ripe | - | 0.35
B0.08R | Yes | 400 | - | 208 | Ripe | - | 0.18
S26H $\star$ | Yes | 400 | 26 | 8 | Harvested | Aligned | 0.35
S13H | Yes | 400 | 12 | 8 | Harvested | Aligned | 0.70
S16H | Yes | 400 | 16 | 8 | Harvested | Aligned | 0.57
S52H | Yes | 400 | 52 | 8 | Harvested | Aligned | 0.18
S52W20H | Yes | 400 | 52 | 20 | Harvested | Aligned | 0.44
S208H | Yes | 400 | 208 | 8 | Harvested | Aligned | 0.04
S26R | Yes | 400 | 26 | 8 | Ripe | Aligned | 0.67
S13R | Yes | 400 | 13 | 8 | Ripe | Aligned | 1.35
S16R | Yes | 400 | 16 | 8 | Ripe | Aligned | 1.10
S52R | Yes | 400 | 52 | 8 | Ripe | Aligned | 0.34
S52W20R | Yes | 400 | 52 | 20 | Ripe | Aligned | 0.85
S208R | Yes | 400 | 208 | 8 | Ripe | Aligned | 0.08
S26LH | Yes | 800 | 26 | 8 | Harvested | Aligned | 0.35
PS26H | Yes | 400 | 26 | 8 | Harvested | Perpendicular | 0.35
PS52H | Yes | 400 | 52 | 8 | Harvested | Perpendicular | 0.18
PS52W20H | Yes | 400 | 52 | 20 | Harvested | Perpendicular | 0.44
PS26R | Yes | 400 | 26 | 8 | Ripe | Perpendicular | 0.67
PS52R | Yes | 400 | 52 | 8 | Ripe | Perpendicular | 0.34
PS52W20R | Yes | 400 | 52 | 20 | Ripe | Perpendicular | 0.85
PRE | Yes | - | - | - | - | - | -
B1H-NW | No | 400 | - | 208 | Harvested | - | 1.14
B1R-NW | No | 400 | - | 208 | Ripe | - | 2.20
S26H-NW | No | 400 | 26 | 8 | Harvested | Aligned | 0.35
PRE-NW | No | - | - | - | - | - | -
Table 1: Farm parameters. The cases selected for detailed analysis are marked
with stars. The letters ‘H’ and ‘R’ denote harvested and ripe profiles,
respectively. The letter ‘B’ represents farm block cases, and 1, 0.3, 0.16,
and 0.08 are the frond density multiplication factors, influencing the
effective density $\left\langle a\right\rangle_{xyz}$. The letter ‘S’
represents spaced kelp rows aligned with $x$-direction, and ‘PS’ represents
spaced kelp rows oriented perpendicular to $x$-direction. Numeric values 13,
16, 26, 52, and 208 denote the row spacing parameter ($S_{MF}$, in meters).
Most configurations have a row width $W_{MF}$ of 8 m, and several cases have a
row width of 20 m, denoted by ‘W20’. Additionally, a long farm simulation is
named as S26LH, characterized by an extended farm length $L_{MF}$ of 800 m.
The precursor simulation in absence of a farm is refered to as PRE. The term
‘-NW’ is used to denote the simulations conducted without any surface wave
forcing, meaning the Stokes drift velocity is zero.

A range of farm simulations with spaced rows are conducted (table 1), where
the spacing parameter $S_{MF}$ varies from 13 m to 208 m. The kelp row width
is $W_{MF}=8$ m for different $S_{MF}$. There is also an additional case with
an increased row width of $W_{MF}=20$ m for $S_{MF}=52$ m. We primarily focus
on cases with the farm length of $L_{MF}=400$ m, and a long farm simulation
with $L_{MF}=800$ m is also investigated for $S_{MF}=26$ m and $W_{MF}=8$ m.

For farm block simulations, each frond density profile is additionally
multiplied by a factor of 0.3, 0.16, and 0.08, to investigate the influence of
decreased effective density (table 1). Note that the kelp farm block
simulations with a multiplication factor of 0.3 and 0.16 result in an
identical effective density $\left\langle a\right\rangle_{xyz}$ to the spaced
kelp row simulations with $S_{MF}=26$ m and $S_{MF}=52$ m, respectively, for a
row width of $W_{MF}=8$ m.

##  Appendix B A single kelp row

Figure 24: Top views of vertical velocity $w$ $(a)$ and vorticity components
$\zeta_{x}$, $\zeta_{y}$, and $\zeta_{z}$ (b-d) for case S208H (a single row
aligned with the current, harvested profile). These results are time-averaged
and depth-averaged ($z=0$ to $-h_{b}$). Dotted rectangles show the extent of
the kelp row.

Figure 24 shows the simulation of a single kelp row aligned with the current
and waves (case S208H, harvested profile, corresponding to a lateral spacing
$S_{MF}$ of 208 m under periodic boundary conditions in $y$-direction). The
regions with strong farm-generated vertical vorticity $\zeta_{z}$ remain close
to the lateral edges of the kelp row. The vertical velocity $w$ and streamwise
vorticity $\zeta_{x}$, which correspond to the attached Langmuir circulation,
exhibit a downstream expansion in width. This lateral expansion of Langmuir
circulation, i.e., the Langmuir wake pattern associated with the farm, can
thus influence the horizontal distribution of farm-generated turbulence and
secondary flow. Moreover, in kelp rows with smaller lateral spacing $S_{MF}$,
the expansion of the wake may encounter interference due to the presence of
other neighboring rows.

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