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As the frequency and intensity of storms increase, a growing need exists for resilient shore protection techniques that have both environmental and economic benefits. In addition to producing seafood, aquaculture farms may also provide coastal protection benefits either alone or with other nature-based structures. In this paper, a generalized three-layer frequency dependent theoretical model is derived for random wave attenuation due to presence of biomass within the water column. The biomass can be characterized as submerged, emerged, suspended and floating canopies that can consist of natural aquatic vegetation with potential aquaculture systems of kelp or mussels. The present analytical solutions can reduce to the solutions by Mendez and Losada (2004), Chen and Zhao (2012) and Jacobsen et al. (2019) for submerged rigid aquatic vegetation. The present theoretical model incorporates the motion of these canopies using a cantilever-beam model for slender components and a buoy-on-rope model for elements with concentrated mass and buoyancy. Analytical results are compared with existing laboratory and field datasets for submerged and suspended canopies. The theoretical model was then used (in a case study at a field site in Northeastern US) to investigate the capacity of suspended mussel farms with submerged aquatic vegetation (SAV) to dissipate wave energy during a recent storm event. Compared to a dense SAV meadow in shallower water, the suspended aquaculture farms more effectively attenuate random waves with a smaller peak period and the higher frequency components of wave spectrum. The performance of suspended aquaculture farms is less affected by water level changes due to tides, surge and sea level rise, while the wave attenuation performance of SAV decreases with increasing water level due to decreased wave motion near the sea bed. Incorporating suspended aquaculture farms offshore significantly enhance the coastal protection effectiveness of SAV-based living shorelines and extend the wave attenuation capacity over a wider wave period and water level range. The combination of suspended aquaculture farms and traditional living shorelines provides a more effective nature-based coastal defense strategy than the traditional living shorelines alone.
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Aquaculture farms as nature-based coastal protection:
Random wave attenuation by suspended and
submerged canopies
Longhuan Zhua,, Kimberly Huguenarda, Qing-Ping Zoub, David W.
Fredrikssonc, Dongmei Xied
aDepartment of Civil and Environmental Engineering, University of Maine, Orono, ME,
04469-5711, USA
bThe Lyell Centre for Earth and Marine Science and Technology, Institute for
Infrastructure and Environment, Heriot-Watt University, Edinburgh, UK
cDepartment of Naval Architecture and Ocean Engineering, U.S. Naval Academy,
Annapolis, MD, 21402, USA
dCollege of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing, 210098,
China
Abstract
As the frequency and intensity of storms increase, a growing need exists for re-
silient shore protection techniques that have both environmental and economic
benefits. In addition to producing seafood, aquaculture farms may also provide
coastal protection benefits either alone or with other nature-based structures.
In this paper, a generalized three-layer frequency dependent theoretical model is
derived for random wave attenuation due to presence of biomass within the water
column. The biomass can be characterized as submerged, emerged, suspended
and floating canopies that can consist of natural aquatic vegetation with po-
tential aquaculture systems of kelp or mussels. The present analytical solutions
can reduce to the solutions by Mendez & Losada (2004), Chen & Zhao (2012)
and Jacobsen et al. (2019) for submerged rigid aquatic vegetation. The present
theoretical model incorporates the motion of these canopies using a cantilever-
beam model for slender components and a buoy-on-rope model for elements
with concentrated mass and buoyancy. Analytical results are compared with
existing laboratory and field datasets for submerged and suspended canopies.
Corresponding author
Email address: longhuan.zhu@maine.edu (Longhuan Zhu)
Preprint submitted to Coastal Engineering May 27, 2020
The theoretical model was then used (in a case study at a field site in Northeast-
ern US) to investigate the capacity of suspended mussel farms with submerged
aquatic vegetation (SAV) to dissipate wave energy during a recent storm event.
Compared to a dense SAV meadow in shallower water, the suspended aquacul-
ture farms more effectively attenuate random waves with a smaller peak period
and the higher frequency components of wave spectrum. The performance of
suspended aquaculture farms is less affected by water level changes due to tides,
surge and sea level rise, while the wave attenuation performance of SAV de-
creases with increasing water level due to decreased wave motion near the sea
bed. Incorporating suspended aquaculture farms offshore significantly enhance
the coastal protection effectiveness of SAV-based living shorelines and extend
the wave attenuation capacity over a wider wave period and water level range.
The combination of suspended aquaculture farms and traditional living shore-
lines provides a more effective nature-based coastal defense strategy than the
traditional living shorelines alone.
Keywords: Wave attenuation, Random waves, Suspended canopy, Vegetation,
Aquaculture farm, Natural coastal defense
1. Introduction1
Approximately 40% of the world’s population lives within 100 kilometers of2
the coast (MEA, 2005; Ferrario et al., 2014), and 71% of the coastal population3
lives within 50 kilometers of an estuary (UNEP, 2006). While coastal commu-4
nities benefit from proximity to seascapes, they are more vulnerable to natural5
coastal hazards and extreme events from the sea. For example, from 1900 to6
2017, 197 hurricanes with 206 landfalls in the USA caused about 2 trillion USD7
damage (normalized to 2018 value by considering the effects of inflation, wealth,8
and population), or annually about 17 billion USD (Weinkle et al., 2018). Due9
to climate change, more frequent and severe storms and rising sea level are likely10
to occur (Izaguirre et al., 2011; Tebaldi et al., 2012; Ondiviela et al., 2014).11
To mitigate storm damage, hard structures such as seawalls, breakwaters,12
2
and bulkheads have been used as coastal defenses. These structures, however,13
may aggravate land subsidence due to soil drainage, inhibit natural accumula-14
tion of sediments by tides and waves, adversely impact water quality, and cause15
coastal habitat loss (Syvitski et al., 2009; Currin et al., 2010; Pace, 2011; Tem-16
merman et al., 2013; Sutton-Grier et al., 2015). Additionally, these conventional17
hard engineering defenses are also seriously challenged due to their continual and18
costly maintenance, as well as their reconstruction and reinforcement to keep19
up with increasing flood risk are becoming unsustainable (Temmerman et al.,20
2013). Natural and nature-based infrastructure may be a viable alternative to21
hardened shoreline protection system with added economic and ecological bene-22
fits and ability to adapt to sea level rise and climate change (Borsje et al., 2011;23
Gedan et al., 2011; Temmerman et al., 2013).24
As an example of nature-based infrastructure, living shorelines including a25
variety of wetland plants, aquatic vegetation, kelp beds and oyster reefs have be-26
come a complement to hardened shoreline stabilization. Unlike many hardened27
coastal protection techniques, living shorelines can mitigate storm damage and28
erosion while enhancing productive habitat, improving water quality, producing29
food and adapting to rising sea level (Currin et al., 2010; Scyphers et al., 2011;30
Davis et al., 2015; Bilkovic et al., 2016; Gittman et al., 2016; Saleh & Weinstein,31
2016; Vuik et al., 2016; Moosavi, 2017; Leonardi et al., 2018; M¨oller, 2019). The32
protection of coastal ecosystems by wave attenuation is more effective in areas33
with relatively small tidal ranges (Bouma et al., 2014). Living shorelines at34
exposed, high-energy sites require structure such as breakwater or sill offshore35
to damp incident wave energy to sustain health growth of the living organisms36
(McGehee, 2016).37
Aquaculture systems may also act as nature-based infrastructure to attenu-38
ate wave energy and produce food at the same time. For example, Plew et al.39
(2005) observed that a 650 m ×2450 m mussel farm reduced wave energy by40
approximately 5%, 10%, and 17% at wave frequencies of 0.1, 0.2, and 0.25 Hz,41
respectively at low sea state. It was found that densely grown kelp may have ad-42
vantageous wave attenuation characteristics (Mork, 1996). For instance, Mork43
3
(1996) observed a 70% to 85% wave energy reduction across a 258 m long kelp44
bed (dominated by Laminaria hyperborea ) with the highest wave attenuation45
observed during low tide. Unlike the natural kelp beds rooted at the seabed,46
cultivated kelp is suspended near the surface from a longline (Peteiro & Freire,47
2013; Peteiro et al., 2016; Walls et al., 2017; Campbell et al., 2019; Grebe et al.,48
2019; Zhu et al., 2019), as shown on Fig. 1. Near surface cultivated kelp may
Figure 1: Canopy classification (from left to right): suspended aquaculture farms, floating
wetlands, submerged plants, and emergent plants (figure credit: Yu-Ying Chen).
49
damp more wave energy than bottom-rooted kelp since the wave motion de-50
creases towards the bottom. Kelp can also absorb carbon to mitigate climate51
change impacts and reduce nutrients to improve water quality, therefore, in-52
crease the growth rate of marine species (Duarte et al., 2017; Campbell et al.,53
2019). Other environmental benefits of kelp and seaweed farming include recy-54
cling inorganic nutrients and preventing eutrophication conditions (Yang et al.,55
2015; St´evant et al., 2017; Xiao et al., 2017; Campbell et al., 2019).56
Both mussels and kelp are often farmed near the surface on a horizontal type57
mooring system (Fig. 1). The wave attenuation characteristics of these aquacul-58
ture farms can be modeled in a similar way as natural, bottom-rooted submerged59
and emergent canopies such as kelp forests, seagrasses and salt marshes. In this60
4
study, the aquaculture structures are treated as suspended canopies according61
to the classification shown on Fig. 1. The classification is based on the vertical62
position in the water column and the “plant” height relative to the water depth63
(e.g., Plew, 2011; Huai et al., 2012; Chen et al., 2016; Zhu & Zou, 2017). The64
horizontal mussel and kelp farms shown on Fig. 1 are placed at an elevation with65
optimum light, temperature and nutrient conditions within the water column66
to achieve maximum growth. Fig. 1 also shows a row of nature-based floating67
wetlands and natural submerged and emergent plants.68
Extensive studies have been dedicated to better understanding and predict-69
ing wave attenuation by submerged and emergent vegetation as reviewed later in70
section 2.1. To model the wave attenuation by suspended canopies, Plew et al.71
(2005) developed a two-layer analytical solution for a floating longline mussel72
farm based on energy conservation equation with linear wave theory (Dalrymple73
et al., 1984). They represented random wave conditions using root-mean square74
wave height and peak wave period. Zhu & Zou (2017) extended the two-layer75
solution by Kobayashi et al. (1993) for submerged vegetation to a generalized76
three-layer theoretical solution for suspended and submerged vegetation. Zhu &77
Zou (2017) found that the wave attenuation by a submerged canopy decreases78
while the wave attenuation by a floating canopy increases with increasing wave79
frequency. The wave attenuation by a suspended canopy first increases and then80
decreases with increasing wave frequency. Combining an OpenFOAM (Higuera81
et al., 2013) hydrodynamics model with an immersed element vegetation model,82
Chen & Zou (2019) observed a strong jet formed at the top of a submerged flex-83
ible canopy in the opposite direction as the wave. Using a SWASH (Simulating84
WAves till SHore, Zijlema et al., 2011) model, Chen et al. (2019) investigated85
the wave-driven circulation cell induced by suspended canopies and found that86
the vertical position of the canopy also has significant effects on the wave-driven87
current in the canopy. Recently, SWASH was improved by Suzuki et al. (2019)88
to consider the drag of horizontal vegetation stems, vegetation canopy porosity89
and vegetation inertia, which can influence the wave dissipation. The effects of90
vegetation porosity on wave dissipation is of importance for dense vegetation. As91
5
time domain numerical models, OpenFOAM and SWASH are able to simulate92
waves with arbitrary frequency shape. However, the existing analytical models93
developed for suspended canopies in single characteristic frequency waves still94
need to be extended to frequency dependent models for random waves, which95
are a better representation of field conditions.96
The objective of this study is to develop a generalized three-layer frequency97
dependent theoretical model for random wave attenuation by submerged and98
suspended canopies. The analytical wave attenuation model is coupled with99
cantilever-beam and buoy-on-rope vegetation models to consider the motion100
of canopies with different type components. The coupled flow and vegeta-101
tion model is validated with laboratory experimental datasets for submerged102
canopies (Jacobsen et al., 2019) and laboratory and field datasets for suspended103
canopies (Seymour & Hanes, 1979). The validated coupled model is then ap-104
plied in the field near Saco, Maine in the Northeastern USA to investigate the105
potential of a mussel farm to damp storm during the January 2015 North Amer-106
ican blizzard. The effectiveness of using a suspended aquaculture farm alone107
and in combination with submerged aquatic vegetation (SAV) close to shore for108
wave attenuation is also investigated.109
2. Theory110
2.1. Background on analytical wave attenuation models111
Theoretical models have been developed to study the wave attenuation char-112
acteristics of submerged and emergent canopies by Dalrymple et al. (1984) and113
Kobayashi et al. (1993). Both studies represented the canopy as arrays of rigid,114
homogeneous cylinders subject to monochromatic wave action. Assuming the115
wave energy loss as the work performed by the drag of vegetation, Dalrymple116
et al. (1984) obtained the wave decay coefficient by solving the energy conserva-117
tion equation using linear wave theory. By solving the linearized incompressible118
Euler equations with assumptions of exponentially decayed wave height along119
the canopy and linearized drag, Kobayashi et al. (1993) obtained the same wave120
6
decay coefficient as Dalrymple et al. (1984). The wave decay coefficient is an121
explicit function of hydrodynamic conditions and canopy characteristics includ-122
ing blade length, width, and canopy density (defined as blade number per unit123
area).124
Both these analytical solutions have been widely used for calculating wave125
attenuation by submerged vegetation. The solution by Dalrymple et al. (1984)126
was modified by Mendez & Losada (2004) to consider random non-breaking and127
breaking waves propagating over a mildly sloped vegetation seabed by using the128
unmodified Raleigh distribution method and assuming a narrow-banded wave129
spectrum. The modification developed by Mendez & Losada (2004) has been im-130
plemented in the SWAN (Simulating WAves Nearshore) model by Suzuki et al.131
(2012), and the MDO (Mellor-Donelan-Oey) wave model for wind-generated132
waves and swells in deep and shallow waters by Marsooli et al. (2017). Recently,133
Losada et al. (2016) extended Mendez & Losada (2004) solution for combined134
wave and currents. The solution by Mendez & Losada (2004) was also used by135
Garzon et al. (2019) to analyze the wave attenuation by Spartina Saltmarshes136
in the Chesapeake Bay under storm surge conditions. These models based137
on the Mendez & Losada (2004) approach are limited to ideal narrow-banded138
waves. If applied to wide-banded waves, the Mendez & Losada (2004) based139
models would overestimate the dissipation for the wave components with higher140
frequency than the characteristic peak frequency and underestimate the dissi-141
pation for the wave components with lower frequency than the characteristic142
peak frequency (Jacobsen et al., 2019).143
To investigate the spectral distribution of energy dissipation, Chen & Zhao144
(2012) developed two analytical frequency dependent wave attenuation models145
for random waves and rigid vegetation by implementing the energy dissipation of146
random waves in Hasselmann & Collins (1968) and the joint distribution of wave147
heights and wave periods proposed by Longuet-Higgins (1983). To derive the148
wave attenuation solution based on the random waves in Hasselmann & Collins149
(1968), Chen & Zhao (2012) used the root mean square velocity to linearize150
the drag force following Madsen et al. (1988) such that |u| ≈ 2σu, where uis151
7
the horizontal wave velocity and σuis the standard deviation of u. Recently,152
Jacobsen et al. (2019) obtained a frequency distributed wave dissipation model153
by linearizing the drag force such that |u| ≈ p8/πσuestimated from 270,000154
numerical cases under JONSWAP spectrum so the linearization-induced mean155
error |u|/p8/πσu1is less than 0.01%. Borgman (1967) obtained the156
same result |u| ≈ p8/πσuby minimizing the mean square of the difference157
between the nonlinear drag and linearized drag for uin normal distribution.158
The Borgman (1967) method for the drag linearization can also be used for159
other probability distributions of u.160
Since most vegetation are flexible, the wave-induced motion of vegetation161
would reduce the relative velocity between wave-induced flow and vegetation162
and therefore the drag force, yielding less wave attenuation than rigid vegeta-163
tion (Mullarney & Henderson, 2010; van Veelen et al., 2020). To consider the164
effects of vegetation motion, one common practice is using a reduced bulk drag165
coefficient (e.g., Paul & Amos, 2011; Jadhav et al., 2013; Pinsky et al., 2013;166
Anderson & Smith, 2014; Hu et al., 2014; Zeller et al., 2014; M¨oller et al., 2014;167
Losada et al., 2016; Wu et al., 2016; Marsooli et al., 2017; Nowacki et al., 2017;168
Garzon et al., 2019; van Veelen et al., 2020). The bulk drag coefficient should be169
dependent on the Cauchy number (C a) incorporating the blade flexural rigidity170
related to vegetation motion. However, most of the empirical formulae in liter-171
ature for the bulk drag coefficient of flexible vegetation are expressed as a func-172
tion of Reynolds number (Re) or Keulegan–Carpenter number (KC) without173
incorporating the blade flexural rigidity. Consequently, the empirical formulae174
of bulk drag coefficient have different expressions for vegetation with different175
flexural rigidities for the same set of Re and KC numbers. This introduces176
uncertainty in modelling wave attenuation by flexible vegetation. To apply the177
original (unreduced) drag coefficient as previous studies and incorporate the ef-178
fects of blade motion at the same time, Luhar et al. (2017) proposed a reduced,179
effective blade length instead of a reduced drag coefficient to incorporate the180
effects of blade motion. The empirical formula for the effective blade length is181
dependent on blade flexural rigidity, therefore, can be readily applied to vege-182
8
tation of various flexural rigidities. The formula for the effective blade length183
was recently modified by considering the effects of rigid sheath of seagrass (Lei184
& Nepf, 2019b) and applied to combined waves and currents conditions (Lei &185
Nepf, 2019a). These empirical approaches do not need to resolve blade motion186
and therefore improve the computational efficiency by reducing the iterative187
computation for coupling wave and vegetation motion. These approaches, how-188
ever, require numerous datasets to derive the formulae for bulk drag coefficient189
and effective blade length. If the blade motion is directly resolved by the model,190
then the original unreduced drag coefficient and blade length can be used di-191
rectly without modification. Therefore, the number of experiments and model192
runs to calibrate the bulk drag coefficient and the uncertainty associate with193
the bulk drag coefficient are reduced.194
To resolve the blade motion, Asano et al. (1992) simplified the blade mo-195
tion as an oscillator with one degree of freedom by assuming blade deflection196
is linearly distributed along the length and also averaging deflection along the197
length. This method was then extended to consider irregular waves, wave reflec-198
tion, and evanescent modes by M´endez et al. (1999) for submerged vegetation.199
To analyze the depth dependence of the blade deflection as well as its effects on200
wave dissipation, Mullarney & Henderson (2010) modeled the blade as a contin-201
uous beam with Euler-Bernoulli techniques, where the governing equation for202
the blade motion is simplified as a balance between the flexural rigidity-induced203
restoring force and the drag force, assuming the inertia force and buoyancy204
are negligible. Recently, Henderson (2019) extended this model by including205
buoyancy but still neglected the inertia force, therefore the model is valid only206
for blades with small cross sectional area. In addition, the mass of vegeta-207
tion influences the natural frequency of the vegetation and further impacts the208
blade motion as well as the resonant conditions. To fully consider the grav-209
ity, buoyancy, structural damping, bending stiffness, virtual buoyancy, friction,210
drag and inertia forces, numerical models are often used to simulate the blade211
dynamics (e.g., Zeller et al., 2014; Zhu & Chen, 2015; Luhar & Nepf, 2016;212
Leclercq & de Langre, 2018; Zhu et al., 2018; Chen & Zou, 2019; Zhu et al.,213
9
2020). Recently, Zhu et al. (2020) used a cable model to capture the asym-214
metric “whip-like” blade motion and proposed mechanisms for the asymmetric215
blade motion in symmetric waves.216
To derive a generalized wave attenuation model for suspended and sub-217
merged canopies, the water column is divided into 3 layers with model set-up218
in section 2.2. The effects of canopy motion are incorporated by resolving the219
motion of individual canopy component using a cantilever-beam model or a220
buoy-on-rope model based on the type of the canopy component in section 2.3.221
These two structural dynamics models consider inertia force and are therefore222
applicable for large diameter structure such as mussel droppers. The frequency223
dependent theoretical wave attenuation model incorporating canopy motion is224
developed in section 2.4.225
2.2. Model set-up226
The mathematical approach is based on the three-layer model set-up shown227
on Fig. 2. As shown on Fig. 2, the horizontal coordinate, x, is positive in228
the direction of wave propagation (assumed to be perpendicular to the coast),229
with x= 0 at the leading edge of the canopy. The horizontal length of the230
canopy is defined as Lvsuch that x=Lvat the end of the canopy. The vertical231
coordinate, z, is positive upward with z= 0 at the still water level (SWL).232
The water column is divided into three layers with Layer 1 above the canopy,233
Layer 2 within the canopy, and Layer 3 below the canopy. The initial static234
thicknesses for each layer are denoted by d1,d2,d3, respectively. The thickness235
of Layer 2 (d2) also named the canopy height, is defined as the average sub-236
merged length of the canopy components. The water depth from the SWL is237
defined as h=d1+d2+d3, where the seafloor is located at z=hand assumed238
to be horizontal. This generalized three-layer model can be used to analyze the239
wave attenuation characteristics of the following four types of canopy configu-240
rations: (1) submerged (d16= 0 and d3= 0), (2) emergent (d1= 0 and d3= 0),241
(3) suspended in the water column (d16= 0 and d36= 0), and (4) floating on the242
surface (d1= 0 and d36= 0).243
10
Figure 2: Definition sketch of variables and coordinate system for the three-layer theoretical
model of waves propagating over a canopy. The coordinate system (x, z) with the origin at
the leading edge of the canopy (x= 0) and the still water level (SWL, z= 0), where xis
positive in the wave propagation direction from left to right and zis positive upward. The
water column is divided into three layers by the canopy. The thicknesses of layer 1, 2, and 3
are denoted as d1,d2, and d3, respectively. The canopy length is Lv. The water depth from
the SWL is defined as h=d1+d2+d3.
At many sites, sea surface profiles are better represented by random waves,244
which can be formulated as a superposition of monochromatic waves with a set245
of random phases. Thus, the water elevation can be expressed as246
η=
X
i=1
aicos (kixωit+ψi),(1)
where tis time, aiis the wave amplitude, kiis the wave number, ωiis the247
angular frequency and ψiis the random phase of the ith monochromatic wave248
component. As a sum of infinite independent random variables, the water eleva-249
tion tends toward a normal distribution according to the central limit theorem.250
Assuming that the random phase is distributed uniformly on (0,2π), the water251
elevation is normally distributed with a zero mean (hηi= 0, where h i indicates252
expected value) and a variance of σ2
η=η2=P
i=1 a2
i/2 = R
0Sηη (ω, x),253
where Sηη (ω, x) is the wave spectrum. For the convenience of expression, the254
index of summation iis omitted and ωis used to indicate the summation such255
11
that256
η=X
ω
a(kx ωt +ψ).(2)
According to linear wave theory (Dean & Dalrymple, 1991), the wave number257
and angular frequency satisfy the dispersion relation, ω2=gk tanh kh, where g258
is the gravitational acceleration. The wave orbital velocity (u) at a given level259
zis then written as260
u=X
ω
Γ cos(kx ωt +ψ),(3)
where Γ = cosh k(h+z)/sinh kh when zηand Γ = 0 when z > η.261
2.3. Models for the motion of canopy components262
The wave-induced motion of a canopy component is simulated by different263
models depending on the morphology and physical properties of the species.264
In this paper, we introduce cantilever-beam and buoy-on-rope models. The265
cantilever-beam model is applicable for slender species such as vegetation blades,266
kelp blades, and mussel droppers (Fig. 3). The buoy-on-rope model is applicable
Figure 3: Sketch for the cantilever-beam model and buoy-on-rope model for different species.
267
for species with concentrated mass and buoyancy supported by a tethered stipe268
whose mass and stiffness can be ignored, e.g., the bull kelp, Nereocystis luetkeana269
(Fig. 3).270
12
2.3.1. Cantilever-beam model271
The individual component of the canopies such as seagrass meadow, kelp for-272
est and mussel farms is modeled as a slender cantilever beam (Fig. 3), referred273
as a blade hereinafter. A typical blade having the averaged geometrical and274
physical properties of the canopy components is used to represent the canopy275
components. To simulate the large-amplitude deflection of a flexible blade, Zhu276
et al. (2020) introduced a cable model that can capture the asymmetric “whip277
like” motion of a flexible blade (Luhar & Nepf, 2016). To obtain the analytical278
solution for the horizontal displacement (ξ) of the blade, the governing equa-279
tions in Zhu et al. (2020) are linearized by assuming a small-amplitude motion280
such that the vertical displacement of the blade is negligible. The horizontal281
displacement, ξ(s, t) is a function of time tand the distance salong the blade282
length from the fixed end. The relation between the local coordinate sand the283
global coordinate zis given by284
z=
d1d2+s, blade fixed at the bottom end,
d1s, blade fixed at the tip end.
(4)
Neglecting tension and buoyancy, the linearized governing equation is given by285
ρvAc¨
ξ+EI ξ0000 =ρwAc˙u+1
2Cdρwb|u˙
ξ|(u˙
ξ) + CmρwAc( ˙u¨
ξ),(5)
where the dot ( ˙) indicates derivative with respect to t, the prime (0) indicates286
derivative with respect to s,ρwis the water density, ρvis the blade mass density,287
bis the projected blade width, Acis the blade cross sectional area, Eis the288
Young’s modulus of the blade, Iis second moment of the blade cross sectional289
area, Cdis the drag coefficient and Cmis the added mass coefficient. The terms290
on the right-hand side of (5) are virtual buoyancy, drag and added mass force291
per unit length modified from the Morison formula (Morison et al., 1950). To292
obtain an analytical solution to (5), the nonlinear drag 1/2Cdρwb|u˙
ξ|(u˙
ξ) is293
linearized as c(u˙
ξ), where the linearization coefficient (c) is calculated using294
the Borgman (1967) method. Substituting (3) into (5) yields295
m¨
ξ+c˙
ξ+EI ξ0000 =X
ω
Γ [ccos(kx ωt +ψ) + ωmIsin(kx ωt +ψ)] ,(6)
13
where m= (ρv+Cmρw)Acand mI= (1 + Cm)ρwAc. The boundary condi-296
tions for a cantilever beam are given by ξ(0, t) = 0, ξ0(0, t)=0, ξ00(l, t) = 0 and297
ξ000(l, t) = 0. Using a normal mode approach (Rao, 2007), the solution for the298
blade displacement is obtained in Appendix A as299
ξ=X
ω
aΓ [γssin(kx ωt +ψ) + γccos(kx ωt +ψ)] ,(7)
where γsand γcare the transfer functions given by300
γs=ω
Γ
X
n=1
φn
ωInλ2
nω2Dn2ζnλnω
(λ2
nω2)2+ (2ζnλnω)2(8)
and301
γc=ω
Γ
X
n=1
φn
Dnλ2
nω2+ωIn2ζnλnω
(λ2
nω2)2+ (2ζnλnω)2,(9)
where φn= (cos µnl+ cosh µnl) (sin µnssinh µns)+(sin µnl+ sinh µnl) (cosh µnscos µns)302
is the nth normal mode of the cantilever beam with µnbeing the nth solution303
of 1 + cos µl cosh µl = 0, λn=µ2
nqRl
0EI φ2
nds/ Rl
02
nds is the nth natural fre-304
quency of the blade, 2ζnλn=Rl
02
nds/ Rl
02
nds,Dn=Rl
0cΓφnds/ Rl
02
nds305
and In=Rl
0mIΓφnds/ Rl
02
nds. Since Γ is expressed in terms of zand φn
306
is expressed in terms of s, the relation between sand zin (4) is required to307
calculate the integral Rl
0Γφnds.308
The relative velocity of flow to blade ur=uξis given by309
ur=X
ω
Γ [(1 + γs) cos(kx ωt +ψ) + γcsin(kx ωt +ψ)] .(10)
According to the central limit theorem, the relative velocity also asymptotically310
approaches a normal distribution with zero mean (huri= 0) and the variance311
σ2
ur=u2
r=Z
0
ω2Γ2h(1 + γs)2+γ2
ciSηη (ω, x). (11)
Hence, the probability density function of uris given by312
p(ur) = 1
σur2π
e
u2
r
2σ2
ur .(12)
Using the Borgman (1967) method, the linearization coefficient (c) is obtained313
by minimizing the mean square difference between the nonlinear and linearized314
14
drag so that R
−∞ (1/2Cdρwb|ur|urcur)2p(ur)dur/∂c = 0, yielding315
c=1
2CdρwbR
−∞ |ur|u2
rp(ur)dur
R
−∞ u2
rp(ur)dur
=1
2Cdρwbr8
πσur.(13)
The linearization coefficient can be obtained iteratively through the following316
procedure. Starting from a static blade, an initial cis calculated from equation317
(13) with (11) by assuming γs= 0 and γc= 0. Once the blade displacement318
is obtained, ccan be recalculated from (13) and (11) with (8)and (9). Using319
the new value of c, the blade displacement can be updated. The procedure is320
repeated until a convergent solution is achieved.321
2.3.2. Buoy-on-rope model322
The bull kelp (Nereocystis luetkeana) is used as an example to describe the323
buoy-on-rope model (Denny et al., 1997), which is also used for other species324
such as Macrocystis pyrifera (Utter & Denny, 1996). The pneumatocyst (the325
ball-shape “float” structure) of Nereocystis luetkeana is modeled as a buoy and326
the stipe is modeled as a rope (Fig. 3). Therefore, the canopy component is327
modeled as a buoy attached to seabed by a thin, straight, non-buoyant rope.328
The inertia, drag and buoyancy act at the buoy center, zc=d1d2/2, where329
the canopy height d2is the diameter of the buoy. The horizontal displacement330
of the buoy and the fluid velocity at the buoy center is used to calculate the331
forces. The governing equation for buoy-on-rope model is given by332
ρvV¨
ξ+(ρwρv)V g
Rξ=ρwV˙u(zc)+ 1
2CdρwApu(zc)˙
ξhu(zc)˙
ξi+CmρwVh˙u(zc)¨
ξi,
(14)
where Ris the length of the tethered rope, Vis the volume of the buoy with pro-333
jected area of Ap. Similarly, the nonlinear drag force 1/2CdρwApu(zc)˙
ξhu(zc)˙
ξi
334
is linearized as Chu(zc)˙
ξi, where Cis obtained using the Borgman (1967)335
method. Substituting (3)into (14)yields336
M¨
ξ+C˙
ξ+=X
ω
Γ (zc) [Ccos(kx ωt +ψ) + ωMIsin(kx ωt +ψ)] ,
(15)
15
where M= (ρv+Cmρw)V,K= (ρwρv)V g/R, and MI= (1 + Cm)ρwV. The337
solution for (15) is338
ξ=X
ω
aΓ [γssin(kx ωt +ψ) + γccos(kx ωt +ψ)] ,(16)
where γsand γcare the transfer functions given by339
γs=Γ (zc)ω
ΓM
ωMIλ2ω2C(2ζλω)
(λ2ω2)2+ (2ζλω)2(17)
and340
γc=Γ (zc)ω
ΓM
Cλ2ω2+ωMI(2ζλω)
(λ2ω2)2+ (2ζλω)2,(18)
where λ=pK/M and 2ζλ =C/M. Similarly, the relative velocity ur=341
u˙
ξasymptotically approaches a normal distribution with zero mean and the342
variance σ2
urin a similar expression as (11) except for the transfer functions343
γsand γc, which are calculated using (17) and (18). Thus, the linearization344
coefficient (C) is given by345
C=1
2CdρwApr8
πσur,(19)
which is obtained iteratively using the same procedure for the cantilever-beam346
model.347
2.4. Solutions for random wave attenuation348
Following Dalrymple et al. (1984), Kobayashi et al. (1993), and Mendez &349
Losada (2004), the wave attenuation is assumed to come from the work of the350
canopy-induced drag force. The inertia force has a negligible contribution to351
wave attenuation since the mathematical expectation of the work due to the352
inertia force is zero because the relative acceleration and the relative velocity353
are out of phase in linear waves. The vertical frictional force is assumed negli-354
gible when compared with the horizontal drag force. The wave reflection from355
the canopy is also assumed negligible since the wave reflection has limited con-356
tributions to the wave attenuation for both submerged vegetation (Mendez &357
Losada, 2004) and suspended canopies (Seymour & Hanes, 1979). Some wave358
16
energy is converted into the kinematic and potential energy of the canopy at359
the beginning. However, once the canopy motion becomes steady, the energy360
needed to maintain the steady motion can be assumed negligible because the361
structural damping of the canopy components is negligible (Asano et al., 1992;362
endez et al., 1999). Lacking data, the velocity reduction in the canopy (Lowe,363
2005), the sheltering effects (Raupach & Thom, 1981; Abdelrhman, 2007; Et-364
minan et al., 2019), and the porosity effects (Mei et al., 2011; Nepf, 2011; Liu365
et al., 2015; Arnaud et al., 2017; Suzuki et al., 2019) are not considered. Using366
the linearized drag force, the energy conservation equation can be written as367
∂x Z
0
ρwgSηη (ω, x)cg=Zd1
d1d2*N1
2Cdρwbr8
πσuru2
r+dz, (20)
where cg= (ω/k)(1+2kh/ sinh 2kh)/2 is the group velocity and Nis the number368
of canopy components per unit horizontal area (also referred to as the canopy369
density). Substituting (11) into (20) yields the transmitted wave spectrum at370
distance xin relation to the incident wave spectrum at x= 0,371
Sηη (ω, x) = Sηη(ω , 0)e2β(ω)x,(21)
where the frequency dependent decay coefficient (β) is given by372
β(ω) = 22Nk2sinh2kh
πω(2kh + sinh 2kh)Zd1
d1d2
CdurΓ2h(1 + γs)2+γ2
cidz. (22)
The transfer functions γsand γcare selected based on the structural dynamics373
model used for the canopy motion. To evaluate the effect of the canopies on374
wave attenuation, the wave spectral dissipation ratio (SDR) and wave energy375
dissipation ratio (EDR) are used and defined as376
SDR = 1 Sηη (ω, Lv)
Sηη (ω, 0) (23)
and377
EDR = 1 R
0Sηη (ω, Lv)
R
0Sηη (ω, 0),(24)
respectively.378
17
3. Model-Data comparison379
3.1. Submerged canopy380
The model results were first compared with the laboratory experiments by381
Jacobsen et al. (2019) for a submerged canopy consisting of artificial vegetation.382
The wave conditions were based on a single peaked JONSWAP spectrum with383
a peak enhancement factor γ= 3.3 and peak wave period Tp= 1.15 s. The384
incident significant wave height at the leading edge of the canopy was Hs0= 3.7385
cm. The water depth was h= 0.685 m.386
The artificial vegetation was made of 4 mm-wide polypropylene blades with387
ρv920 kg/m3and E0.3 GPa (Ghisalberti & Nepf, 2002). Four blades were388
taped to a 6 mm-diameter PVC dowel and 60 mm above the bed. The canopy389
was 7.5 m long with a density of 566 dowels/m2therefore 2264 blades/m2. The390
blade length was 20, 40, and 60 cm such that d2/h ={0.38,0.67,0.96}. The391
blade thickness was 0.12, 0.2, 0.5 and 1.0 mm for the 20 cm-long blade and392
0.5 mm for the other blades. More details of the experiments can be found in393
Jacobsen et al. (2019).394
Based on the datasets for rigid flat plates in oscillatory flows (Keulegan &395
Carpenter, 1958; Sarpkaya & O’Keefe, 1996) with 1.7KC 118.2, Luhar &396
Nepf (2016) derived the drag coefficient and added mass coefficient,397
Cd= max(10KC 1/3,1.95) (25)
and398
Cm= min(Cm1, Cm2),(26)
where Cm1=
1+0.35KC2/3, K C < 20,
1+0.15KC2/3, K C 20
and Cm2= 1 + (K C 18)2/49 as399
described in Luhar (2012). Equations (25) and (26) are robust in calculating400
the hydrodynamic forces acting on flexible blades in regular waves (Zhu et al.,401
2020). To apply (25) and (26) to random waves, the KC number is calculated402
using the significant relative velocity (2σur) as KC = 2σurTp/b.403
The vegetation blade is modeled as a cantilever beam so that the wave404
attenuation model incorporating the cantilever beam model is used to calculate405
18
the wave decay coefficient, β. The model results using frequency dependent406
Cdand Cmin (25) and (26) as well as constant Cd= 1.95 and Cm= 1 are407
compared with the datasets of Jacobsen et al. (2019) on Fig. 4. It is noted that
Figure 4: Comparisons of calculated frequency (f) dependent wave decay coefficient (β) by the
present model and the data (black dotted lines) from Jacobsen et al. (2019). The model results
using frequency dependent and constant drag coefficient (Cd) and added mass coefficient (Cm)
are denoted by red solid and blue dashed lines, respectively. The submerged canopies with
blade lengths (l) of 20, 40 and 60 cm and thicknesses (d) of 0.12, 0.20, 0.5 and 1.0 mm are
subjected to random waves of JONSWAP spectrum with peak enhancement factor γ= 3.3,
peak wave period Tp= 1.15 s (vertical dashed black line) and incident significant wave height
of 3.7 cm at a water depth h= 0.685 m with normalized blade length (l/h) of 0.38 (a-d), 0.67
(e) and 0.96 (f). The canopy density is 566 shoots/m2(2264 blades/m2).
408
19
Cd= 1.95 is the minimum drag coefficient for (25).409
The model results are in a good agreement with the data with the root-mean-410
square-error (RMSE) of about 0.002 for the 20 cm-long blades (l/h = 0.38)411
with thickness d0.5 mm (Fig. 4a-c). For the thickest blades with d= 1412
mm, the decay coefficient is slightly overestimated for the lower frequency wave413
components (f < 0.8 Hz) resulting in a larger RMSE of 0.0032 (Fig. 4d). One414
possible reason is that the drag coefficient calculated using equation (25) might415
be overestimated for the thicker blades whose thickness-width ratio has reached416
0.25 and much larger than the thickness-width ratio (<0.1) of the experimental417
plates for the formula (25). The thicker blades are expected to have a smaller418
drag coefficient due to increased Reynolds number. Thus, the model results can419
be improved by using a smaller drag coefficient. For instance, the RMSE for420
the 1 mm-thick blades is reduced to 0.0016 by using Cd= 1.95 (Fig. 4d).421
For the longer blades that are nearly emergent (l/h 0.67), the model results422
calculated with frequency dependent hydrodynamic coefficients underestimate423
the observation with RMSE=0.0046 and 0.0073 for l= 40 cm (l/h = 0.67) and424
60 cm (l/h = 0.96), respectively, as shown on Fig. 4(e and f ) possibly due to425
the simplification of the cantilever beam model. Neglecting the large deflection-426
induced geometrical non-linearity, net buoyancy, and the net buoyancy-induced427
tension would underestimate the restoring capacity of the blades. Thus, the428
simplified model may overestimate the blade motion resulting in a smaller wave429
attenuation. This underestimation of wave attenuation is more obvious for430
longer blades because the effects of the large deflection-induced geometrical431
non-linearity, the net buoyancy and the net buoyancy-induced tension are more432
significant for longer blades. Compared to the shorter blade (l= 20 cm), the433
longer blade (l40 cm) is more flexible, therefore, the blade motion follows434
the flow more closely so that the relative velocity between the longer blade and435
flow is smaller, resulting in a larger Cd. Therefore, using a smaller Cd= 1.95436
enhances the underestimation as indicated by RMSE=0.0055 and 0.0177 for437
the l= 40 cm (l/h = 0.67) and 60 cm (l/h = 0.96), respectively on Fig. 4(e438
and f). A more precise formula for the hydrodynamic coefficients in random439
20
waves is desired for the nearly emergent canopies with l/h 0.67. However, the440
hydrodynamic coefficients in (25) and (26) as well as the constant hydrodynamic441
coefficients work well for the submerged vegetation (l/h 0.38) with a small442
RMSE of about 0.002.443
3.2. Suspended canopy444
The model results were also compared with the laboratory and field ex-445
periments by Seymour & Hanes (1979) for a suspended canopy consisting of446
spherical buoys. The field experiments for a suspended canopy consisting of447
arrays of tethered sphere buoys (Fig. 5) were conducted in San Diego Bay,
Figure 5: Sketch of the suspended canopy consisting of sphere components according to the
description of Seymour & Hanes (1979).
448
California, USA. The half-scale model tests for the field experiments were con-449
ducted in the 40-m long Wind Wave Channel at the Hydraulics Laboratory of450
Scripps Institution of Oceanography (Seymour & Hanes, 1979). The properties451
of the canopies in the laboratory and field experiments are shown in Table 1.452
For the laboratory experiments, the incident significant wave height was453
0.069 0.176 m and the peak frequency was 0.19 0.883 Hz. For the field454
experiments, two storms were observed on Jan 22, 1976 and Feb 9, 1976. The455
measured significant wave height was 0.17 0.44 m. The drag coefficient and456
21
Table 1: Properties for the suspended canopies consisting of sphere components in the labo-
ratory and field experiments (Seymour & Hanes, 1979).
In the lab In the field
(half scale) (full scale)
Sphere mass density [kg/m3] 40 85
Sphere diameter [cm] 15.8 29.2
Depth of sphere center [cm] 7.36 15.24 21.9
Effective tether length [cm] 83.8 168.0
Sphere spacing (along canopy length) [cm] 31.6 58.41
Sphere spacing (along canopy width) [cm] 31.6 58.41
Canopy width (perpendicular to wave direction) [m] 2.39 46
Canopy length (along wave direction) [m] 23 6
Water depth [m] 1.78 8
added mass coefficient for the tethered spheres are assumed as Cd= 0.5 and457
Cm= 0.5, respectively.458
The calculated transmitted wave spectrum and spectral dissipation ratio459
(SDR) are shown on Fig. 6. The calculated transmitted wave spectrum follows460
the shape of the incident wave spectrum. The SDR for the suspended canopy461
first increases and then decreases with increasing wave spectrum as expected.462
The comparison between the calculated and measured energy dissipation463
ratio (EDR) is shown on Fig. 7. Good agreement (RMSE=0.073) between464
model and data indicates that the present generalized analytical solutions are465
also applicable to suspended aquaculture farms with simple structures in other466
forms, such as cylinders, as long as the appropriate hydrodynamic coefficients467
are available.468
4. Case study at the field site469
The present frequency dependent theoretical model is now applied to analyze470
the wave attenuation capacity of suspended aquaculture farms at a field site and471
compared with that of submerged aquatic vegetation, as well as a combination472
of these two nature-based shore protection schemes.473
22
Figure 6: Comparisons between calculated and measured transmitted wave spectrum (Sηη )
as well as spectral dissipation ratio (SDR) versus wave frequency (f) for suspended canopies
with spheres in (a) laboratory and (b) field experiments by Seymour & Hanes (1979). The
incident significant wave height is Hs0and the peak period is Tp.
Figure 7: Comparisons between the calculated and measured wave energy dissipation ratio
(EDR) for laboratory (blue +) and field experiments (red ×) by Seymour & Hanes (1979).
23
The study site (42280200N,70210200W) is located at Saco Bay, Maine, USA474
as shown on Fig. 8 with a water depth of about 10.6 m. The January 2015
Figure 8: The study site at Saco Bay, Maine, USA (Sources: Esri, GEBCO, NOAA, National
Geographic, DeLorme, HERE, Geonames.org, and other contributors).
475
North American blizzard was a powerful and destructive extratropical storm476
that swept across the Saco Bay and along the coast of the Northeastern United477
States in January of 2015. To assess coastal flood risk and sea level rise effects478
during this storm event, Xie et al. (2019) constructed an integrated atmosphere-479
ocean-coast and overtopping-drainage modeling framework based on the coupled480
tide, surge and wave model, SWAN+ADCIRC. The wave spectrum and water481
level conditions output from the SWAN+ADCIRC model (Xie et al., 2019) are482
used to drive the present theoretical model. The canopies are oriented to be483
parallel to the dominant wave direction so that the present 1-D solutions can484
be applied.485
4.1. Properties for the mussel farm and submerged aquatic vegetation486
The mussel farm is simplified as arrays of cylinders which represent the487
mussel droppers. The cylinders have similar mechanical and hydrodynamic488
performances to the actual droppers. In this study, the geometric and physical489
properties of the cylinders are based on the measurements of the live droppers490
at the University of New Hampshire nearshore multi-trophic aquaculture site in491
the Gulf of Maine, USA (Knysh et al., 2020). The measured dropper diameter492
24
was 0.13 m, mass per unit length was 7.53 kg/m, and flexural rigidity was 0.79493
N/m2on July 3 (Knysh et al., 2020), which was about 160 days (years are not494
considered) after the January 2015 North American blizzard arrived at the study495
site. The geometrical properties of the droppers in January were estimated by496
assuming a growth rate of 7.5% per 40 days (Lauzon-Guay et al., 2006; Gagnon497
& Bergeron, 2017), which indicates that the diameter of the mussel dropper in498
January was about 75% of that in July. Therefore, the cylinder diameter was499
taken as b= 0.10 m, the mass per unit length was assumed as ρvAc= 4.46 kg/m,500
and the flexural rigidity was assumed as EI = 0.28 N/m2. The cylinders are501
assumed to be submerged half meter below the water surface so that d1= 0.5502
m. The length of the mussel dropper is assumed as l= 8 m following Plew503
et al. (2005) and Stevens et al. (2007) for a similar water depth. A sparse504
configuration with 0.06 droppers/m2(Plew et al., 2005; Gagnon & Bergeron,505
2017) and a dense configuration with 0.125 droppers/m2(e.g., mussel droppers506
are 0.5 m apart and the longline interval is about 16 m) are compared. Following507
Plew et al. (2009), Dewhurst (2016) and Knysh et al. (2020), the drag coefficient508
and added mass coefficient are assumed as Cd= 1.3 and Cm= 1, respectively,509
which are also comparable to the values in Raman-Nair & Colbourne (2003),510
Raman-Nair et al. (2008), Stevens et al. (2008),Plew et al. (2009), Gagnon &511
Bergeron (2017) and Landmann et al. (2019).512
The SAV is modeled as a rectangular plate based on the properties of Zostera513
marina, which is a common SAV in the Gulf of Maine, USA (Mattila et al., 1999;514
Gaeckle & Short, 2002; Beal et al., 2004; Neckles et al., 2005; Beem & Short,515
2009; Newell et al., 2010). The length of Zostera marina ranges from 10 to 150516
cm and the shoot density is about 50 1100 shoots/m2with 3 7 blades per517
shoot (Abdelrhman, 2007; Beem & Short, 2009; Bostr¨om & Bonsdorff, 1997;518
Gaeckle & Short, 2002; Mattila et al., 1999; Ondiviela et al., 2014). In January,519
however, the averaged blade length of Zostera marina is about 16 cm (Gaeckle520
& Short, 2002; Ondiviela et al., 2014). Thus, the SAV blade length is assumed as521
16 cm. The corresponding blade width and thickness as well as the sheath length522
and width are estimated using the empirical formula provided by Abdelrhman523
25
(2007), which yields a blade width of 3.7 mm, blade thickness of 0.11 mm, sheath524
length of 8 cm and sheath width of 3.4 mm. Following Abdelrhman (2007), the525
mass density is assumed ρv= 700 kg/m3. The Young’s modulus is assumed526
E= 0.26 GPa for the blades based on the measurements by Fonseca et al.527
(2007). The sheath is considered rigid following Lei & Nepf (2019b). A sparse528
SAV meadow with 200 shoots/m2and a dense meadow with 400 shoots/m2
529
are used in the study to investigate the variation of wave attenuation with530
vegetation density. The number of blades per shoot is assumed to be 5 so that531
there are 1000 blades/m2for the sparse configuration and 2000 blades/m2for532
the dense configuration. Due to small blade width and large significant wave533
height and peak period in a storm event, the calculated KC number is greater534
than 135, yielding a constant drag coefficient of 1.95 based on equation (25).535
The data comparison in section 3.1 has shown that Cd= 1.95 and Cm= 1536
work well for submerged vegetation with l/h < 0.38 (Fig. 4). Therefore, the537
drag coefficient and added mass coefficient are assumed Cd= 1.95 and Cm= 1,538
respectively, for this case study. The properties of the mussel farm and SAV539
meadow are summarized in Table 2. In this study, both mussel droppers and
Table 2: Properties of the mussel farm and submerged aquatic vegetation (SAV) meadow.
Mussel farm SAV
Canopy component Mussel dropper Zostera marina
Component properties Length: 8 m Blade length: 0.16 m
Diameter: 0.10 m Blade width: 3.7 mm
EI = 0.28 N/m2Blade thickness: 0.11 mm
ρvAc= 4.46 kg/m Young’s modulus: 0.26 GPa
Mass density: 700 kg/m3
Sheath length: 8 cm
Sheath width: 3.4 mm
Canopy density Sparse: 0.060 droppers/m2Sparse: 200 shoots/m2(1000 blades/m2)
Dense: 0.125 droppers/m2Sparse: 400 shoots/m2(2000 blades/m2)
Drag coefficient (Cd) 1.3 1.95
Added mass coefficient (Cm) 1 1
540
SAV are modeled as cantilever beams.541
26
4.2. Mussel farm and SAV at the same water depth542
The time evolution of tide, storm tide, storm surge at the study site during543
the January 2015 North American blizzard is given by the SWAN+ADCIRC544
model (Xie et al., 2019) and shown on Fig. 9(a). The tidal range is around 3.3
Figure 9: Time evolution of (a) tide, storm tide and storm surge during the January 2015
North American blizzard, (b) significant wave height Hs0and the corresponding peak wave
period Tp, (c) and (d) calculated wave energy dissipation ratio (EDR) by the suspended
mussel farm (blue lines) and submerged aquatic vegetation (SAV, red lines) using the wave
spectrum data. The canopy lengths are Lv= 100 m in (c) and Lv= 200 m in (d). The
canopy densities are shown in the legend.
545
m and the largest storm surge is about 0.8 m at the study site. The incident546
significant wave height (Hs0) and corresponding peak wave period (Tp) for every547
27
30 minutes are shown on Fig. 9(b). At the study site during the storm, the548
significant wave height reached 3.6 m with peak wave periods ranging from 5.2549
s to 13.5 s.550
The wave attenuation by the mussel farm and SAV at the same still water551
depth of 10.6 m during the storm is calculated with the wave spectral data552
from the SWAN+ADCIRC model. The calculated wave energy dissipation ratio553
(EDR) is shown on Fig. 9(c and d). The EDR of both SAV and mussels554
increases with incident significant wave height. However, the EDR decreases555
with water level resulting in an oscillating wave attenuation with the same period556
of the tidal cycle. This periodic behavior is more obvious for SAV because the557
mussels are less influenced by the tidal change since the mussels can move up558
and down with the buoys. The largest wave attenuation value occurs at the559
highest wave height during low tide. The larger (Lv= 200 m) and denser (0.125560
droppers/m2) mussel farm provides a more pronounced wave attenuation with561
EDR up to 0.32 (Fig. 9d), which is a bit more than that of the same size562
(Lv= 200 m) but sparse (200 shoots/m2) SAV with E DR up to 0.26. However,563
for the denser (400 shoots/m2) SAV with the same size (Lv= 200 m), the EDR564
can reach to 0.45. For the shorter period waves with Tp<9 s as shown on Fig.565
9(c and d), the mussel farm can damp more wave energy than the same size566
SAV since SAV at the ocean bottom has little effect on wave attenuation for567
short period waves whose energy is concentrated near the ocean surface.568
The comparisons for the selected wave spectrum as well as the associated569
spectral dissipation ratio (SDR) at 10:00 UTC (high tide with Hs0= 2.9 m570
and Tp= 9.2 s), 16:00 UTC (low tide with Hs0= 3.5 m and Tp= 13.5 s),571
and 22:00 UTC (high tide with Hs0= 3.5 m and Tp= 13.5 s) on Jan 27 are572
shown on Fig. 10. The SDR of the suspended mussel farm increases with wave573
frequency until reaching the maximum value, while the SDR of SAV decreases574
with wave frequency. As a result, the suspended mussel farm shows the ad-575
vantage of reducing higher frequency (shorter period) wave components over576
SAV. For example on Fig. 10(a2) with smaller Hs0and Tp, the SDR of the577
dense suspended mussel farm (0.125 droppers/m2) is larger than that of dense578
28
Figure 10: Comparisons of wave spectrum (Sηη ) and wave spectral dissipation ratio (SDR)
versus wave frequency (f) between the suspended mussel farm and submerged aquatic vege-
tation (SAV) with different canopy densities (shown in legend) at 10:00 UTC (a), 16:00 UTC
(b), and 22:00 UTC (c) on Jan 27. The canopy length is 200 m for both canopies. The
incident significant wave height and peak period are denoted by Hs0, and Tp, respectively.
SAV (400 shoots/m2) for f > 0.12 Hz (wave period T < 8.3 s) and sparse SAV579
(200 shoots/m2) for wave frequency f > 0.055 Hz (T < 18 s). The SDR of580
the sparse suspended mussel farm (0.06 droppers/m2) is larger than that of the581
dense SAV for wave frequency f > 0.16 Hz (T < 6.25 s) and sparse SAV for582
f > 0.12 Hz (T < 8.3s). As Hs0and Tpincreases, the threshold value of the583
wave frequency where the S DR of suspended mussel farm is larger than that584
29
of SAV increases to f > 0.167 Hz (T < 6 s) for the dense mussel farm and the585
dense SAV, f > 0.1 Hz (T < 10 s) for the dense mussel farm and the sparse586
SAV, f > 0.21 Hz (T < 4.8 s) for the sparse mussel farm and the dense SAV,587
and f > 0.17 Hz (T < 5.9 s) for the sparse mussel farm and the sparse SAV588
as shown on Fig. 10(b2). For the same Hs0and Tpat high tide, the SDR of589
both the mussel farm and SAV decreases due to the increase of water level. The590
threshold value of the wave frequency where the SDR of suspended mussel farm591
is larger than that of SAV decreases to f > 0.147 Hz (T < 6.8 s) for the dense592
mussel farm and the dense SAV, f > 0.095 Hz (T < 10.5 s) for the dense mussel593
farm and the sparse SAV, f > 0.18 Hz (T < 5.5 s) for the sparse mussel farm594
and the dense SAV, and f > 0.15 Hz (T < 6.7 s) for the sparse mussel farm and595
the sparse SAV as shown on Fig. 10(c2).596
4.3. Mussel farm and SAV at different water depths597
The previous section shows the advantages of suspended mussel farms on598
damping high frequency wave energy over SAV at the same water depth. Usu-599
ally, SAV colonizes in shallower water as shown on Fig. 1. To compare the per-600
formances of the suspended mussel farm and the shallow water SAV meadow,601
the water depth for SAV is set at 6 m so that maximum Hs0/h = 0.79 to602
avoid wave breaking. The water depth for the suspended mussel farm keeps603
the same at 10.6 m. The wave shoaling is incorporated using shoaling coeffi-604
cient Ks(ω) = pcgd(ω)/cgs (ω) (Dean & Dalrymple, 1991), where cgd(ω) and605
cgs(ω) are the wave group speed at deeper and shallower water depths, respec-606
tively. Correspondingly, EDR and SDR are calculated using the shoaled wave607
energy and wave spectrum. The canopy density is set as 200 shoots/m2(1000608
blades/m2) for SAV meadow and 0.125 droppers/m2for the mussel farm. The609
canopy length for SAV meadow is set as Lv= 100 m. For the mussel farm, two610
canopy lengths of 100 m and 200 m are designed for comparison.611
The wave attenuations of SAV and mussel farms as well as their combinations612
are shown on Fig. 11. The wave attenuation by SAV decreases dramatically613
with increasing water level while the suspended mussel farm is less affected by614
30
Figure 11: (a) Comparisons of wave energy dissipation ratio (E DR) between the sus-
pended mussel farm and submerged aquatic vegetation (SAV). The canopy densities are 0.125
droppers/m2for the suspended mussel farm and 200 shoots/m2(1000 blades/m2) for the SAV
meadow, respectively. The canopy length is 100 m for the SAV meadow. The canopy length
for the mussel farm is shown in the legend. (b1, c1, d1) The incident wave spectrum (Sηη )
and (b2, c2, d2) Comparisons of wave spectral dissipation ratio (SDR) versus wave frequency
(f) at 10:00 UTC (A), 16:00 UTC (B) and 22:00 UTC (C) on Jan 27. The incident significant
wave height and peak wave period are denoted by Hs0, and Tp, respectively.
the water level change. For example (Fig. 11a), the EDR of SAV decreases615
by 49% from 0.51 at low tide (Jan 27 16:00 UTC) to 0.26 at high tide (Jan 27616
22:00 UTC) with water level increment of 2.5 m while the E DR of the suspended617
mussel farm decreases by 29%. The combination of the suspended mussel farm618
31
and SAV provides a larger wave attenuation, especially for smaller significant619
wave period and low tide (Fig. 11a). For example at 10:00 UTC on Jan 27620
with Tp= 9.2 s and Hs0= 2.9 m, the EDR of SAV and a large mussel farm621
(Lv= 200 m) is 0.37, which is 1.5 times of the EDR = 0.15 of SAV. Adding a622
small mussel farm (Lv= 100 m) to SAV can also favorably improve the EDR623
of SAV to 0.27 by 80%. As Hs0and Tpincreases to Hs0= 3.5 m and Tp= 13.5624
s, the improvements of the wave attenuation of SAV by adding mussel farms are625
reduced because the mussel farm is more effective for reducing shorter period626
waves. However, the improvements still can reach up to 31% by adding a small627
mussel farm and 54% by adding a large mussel farm. The improvements of628
combined SAV and mussels hold for wave energy at all frequencies by taking629
the advantage of the canopy density of SAV and the vertical position of the630
suspended mussel as shown on Fig. 11(b2, c2 and d2).631
5. Discussion632
5.1. Wave attenuation characteristics of suspended aquaculture farms and SAV633
Wave attenuation occurs through the drag force which is determined by634
the horizontal wave orbital velocity. In shallow water waves, the amplitude of635
the horizontal wave orbital velocity is almost uniform with depth. Thus, the636
vertical position of the canopy has little effect on attenuating shallow water637
waves. Taking the advantages of canopy density, SAV can dissipate more wave638
energy than suspended mussel farms for long period waves. However, the wave639
attenuation of SAV is influenced by changes of water level. In shallow water, the640
wave attenuation of SAV decreases dramatically during high tide, storm surge,641
or storm tide (tide plus storm surge), which highlights the weakness of SAV in642
protecting coastlines during large storm tide conditions. This implies that severe643
erosion by storms may occur during high storm tide levels (in addition, higher644
waves may arrive at the shore without breaking during high tide). Therefore,645
living shorelines represented by SAV would be less effective during extreme646
events.647
32
Suspended aquaculture farms can work as living breakwaters to protect the648
coast due to their capacity for wave attenuation. The wave attenuation capacity649
of suspended aquaculture farms is mainly dictated by the canopy density and650
the size (length) in the wave direction. Unlike SAV, which is limited by water651
depth due to light and nutrients, the vertical locations of suspended aquaculture652
farms can be adjusted to optimize their growth. Consequently, there is no depth653
restriction for suspended farms and they can be quite large, e.g., the suspended654
mussel aquaculture farm off Gouqi Island in East China Sea has an area of about655
8 km2(Lin et al., 2016). In theory, the size of suspended aquaculture farms can656
be designed to achieve optimal wave attenuation. For example, for the incident657
significant wave height (Hs0) to be reduced to the transmitted significant wave658
height (HsT ) that will allow the living shorelines to thrive and mitigate coastal659
erosion, the size of the aquaculture farms can be designed as660
Lv>1
β(ωc)ln Hs0
HsT
,(27)
where ωcis the critical angular frequency such that R
0Sηη (ω, 0)e2β(ω)Lv=661
e2β(ωc)LvR
0Sηη (ω, 0). The existence of ωcis guaranteed according to the662
mean value theorem for definite integrals. For narrow-banded waves, ωccan be663
approximated using the peak wave angular frequency, ωcωp= 2π/Tp. The664
external factors such as the water depth and the vertical position of aquaculture665
farms should also be considered during the design. For places that are not suit-666
able to establish living shorelines, such as low-nutrient seabeds, the suspended667
aquaculture farms offer a viable alternative to SAV for nature-based coastal668
defense.669
This work has shown that suspended aquaculture farms can supplement SAV670
in wave attenuation. Suspended aquaculture farms attenuate shorter peak pe-671
riod waves and high frequency wave components more than SAV. Hence, adding672
suspended aquaculture farms to SAV-based living shorelines can compensate for673
the limitations of SAV for attenuating shorter period waves (such as boat wake)674
and enhance the wave attenuation capacity of SAV-based living shorelines for a675
wider range of wave frequency. The wave attenuation by SAV decreases during676
33
high tide or storm surge due to the increase in water level. The water level,677
however, has fewer influences on suspended aquaculture farms since they are678
located near the surface and move up and down with water level. Therefore,679
suspended aquaculture farms can enhance the wave attenuation capacity of SAV-680
based living shorelines during extreme events. The combination of suspended681
aquaculture farms and traditional living shorelines (such as SAV) is therefore a682
desirable nature-based coastal defense strategy.683
5.2. Simplified analytical solutions684
The generalized three-layer frequency dependent theoretical wave attenua-685
tion model developed in this paper is applicable to analyze the wave attenuation686
capacity of submerged, emerged, suspended, and floating canopies for random687
waves including narrow-banded and wide-banded wave conditions. The present688
analytical model provided a more precise consideration of the blade motion by689
incorporating the effects of inertia (neglected in Mullarney & Henderson, 2010;690
Henderson, 2019) and the mode shape (not considered in Asano et al., 1992;691
endez et al., 1999).692
The present model can reduce to previous models for submerged rigid vegeta-693
tion without motion by setting the transfer functions γsand γcas 0. Therefore,694
the decay coefficient βin (22) reduces to the solution for rigid blades and given695
by696
βR(ω) = 22Nk2
πω(2kh + sinh 2kh)Zd1
d1d2
Cdu[cosh k(h+z)]2dz, (28)
where σ2
u=R
0[ωcosh k(h+z)/sinh kh]2Sηη (ω, 0). If d3= 0, the solution697
in (28) reduces to the solution by Jacobsen et al. (2019) for submerged rigid698
vegetation. In this model, the nonlinear drag is linearized using the Borgman699
(1967) method such that |u| ≈ p8/πσu. If using the root mean square velocity700
to linearize the drag force following Madsen et al. (1988) such that |u| ≈ 2σu,701
the solution reduces to the Hasselmann & Collins (1968) based solution in Chen702
& Zhao (2012) for submerged rigid vegetation. For idealized narrow-banded703
waves such that Sηη 0 when ω6=ωp, the damping coefficient in (28) can be704
34
further simplified as705
βRN =1
12πCdbNkpHrms0
9 sinh kp(d2+d3)9 sinh kpd3+ sinh 3kp(d2+d3)sinh 3kpd3
(sinh 2kph+ 2kh) sinh kph,
(29)
where Hrms0=q8R
0Sηη (ω, 0)is the root mean square incident wave height706
and kpis the peak wave number calculated by solving ω2
p=gkptanh kph. For707
bottom-rooted rigid vegetation such that d3= 0, βRN in (29) reduces to the708
solution of Mendez & Losada (2004). The relationship between the present and709
previous models are shown on Fig. 12.
Figure 12: Relationship between the present solutions (22), (28), (29) and previous solutions
by Chen & Zhao (2012), Jacobsen et al. (2019), and Mendez & Losada (2004), where γsand
γcare the transfer function for the motion of canopy component, d3is the thickness of the
gap between the canopy and sea bed, uis the horizontal wave velocity and σuis the stand
deviation of u.
710
6. Conclusions711
A generalized three-layer frequency-dependent theoretical model for the wave712
attenuation by submerged and suspended canopies subjected to random waves713
was derived and validated with laboratory and field data. This model incorpo-714
rates the motion of canopies using a cantilever-beam model for slender canopy715
35
components and a buoy-on-rope model for canopy components with concen-716
trated mass and buoyancy. This frequency-dependent solution was used to717
demonstrate the shoreline protection capability of suspended mussel farms alone718
and in combination with submerged aquatic vegetation (SAV) to damp wave en-719
ergy at a field site in Saco Bay, Maine, USA during a January 2015 Blizzard.720
The results showed that both suspended mussel farms and SAV have the poten-721
tial to damp wave energy considerably during storm events. Suspended mussel722
farms are more effective at damping shorter waves and high frequency wave723
components of the wave spectrum while dense SAV colonized in shallower water724
have the advantages of damping longer waves and lower frequency wave compo-725
nents more effectively. However, the wave attenuation of SAV in shallow water726
decreases dramatically at the peak of storm tide due to increased water level,727
which decreases the wave motion reaching the ocean bottom. In contrast, sus-728
pended aquaculture farms can move up and down with water level change and729
are less affected by water level change. As a consequence, the combination of730
suspended aquaculture farms and traditional SAV-based living shorelines pro-731
vide an optimized nature-based shore protection scheme that can damp more732
wave energy for a wider wave frequency and water level range.733
The research of wave attenuation by suspended and floating canopies is still734
in its infancy. More laboratory and field experiments data for the hydrodynamic735
properties of suspended aquaculture farms (e.g., mussels and kelp) as well as736
wave attenuation are desirable. The present theoretical model assumed the737
blade motion as a linear vibration with small-amplitude. However, as long as738
the nonlinear effects of large-amplitude blade motion is negligible, the present739
theoretical model remains valid. In addition, the bottom friction, bedforms,740
bottom slope as well as wave-driven currents, wave and current conditions may741
also be significant for certain types of bottom rooted vegetation (e.g., Jensen742
et al., 1989; Myrhaug, 1995; Zou, 2004; Zou & Hay, 2003; Smyth & Hay, 2002;743
Maza et al., 2019; Abdolahpour et al., 2017; van Rooijen et al., 2020). Therefore,744
it is worthwhile to investigate the nonlinear effects of large-amplitude blade745
motion on the wave damping capacity of suspended canopies as well as the746
36
effects of bottom properties and wave-current conditions in the future work.747
Acknowledgments748
This work was completed as part of the PhD research of Longhuan Zhu who749
is supported by National Science Foundation award #IIA-1355457 to Maine EP-750
SCoR at the University of Maine. The authors benefited from discussions with751
Zhilong Liu and Haifei Chen. The authors wish to thank Stephen Cousins and752
Shane Moeykens at the University of Maine Advanced Research Computing for753
the computational support. The authors would like to thank Niels G. Jacobsen754
for sharing the data used on Fig. 4. The authors would also like to thank the755
anonymous reviewers for constructive comments that helped to improve this756
manuscript greatly.757
Appendix A. Normal mode solutions for blade displacements in ran-758
dom waves759
The governing equation (6) for the blade displacement in random waves is760
given by761
m¨
ξ+c˙
ξ+EI ξ0000 =X
ω
Γ [ccos(kx ωt +ψ) + ωmIsin(kx ωt +ψ)] (A.1)
with the boundary conditions ξ(0, t) = 0, ξ0(0, t) = 0, ξ00(l, t) = 0, and ξ000(l, t) =762
0 for a cantilever beam. The solution of (A.1) can be written as the linear763
superposition of components of different frequencies764
ξ= Σωξω,(A.2)
where ξωis the solution of765
m¨
ξω+c˙
ξω+EI ξ0000
ω=Γ [ccos(kx ωt +ψ) + ωmIsin(kx ωt +ψ)] .(A.3)
According to normal mode approach (Rao, 2007), the solution of (A.3) can be766
assumed as a linear superposition of the normal modes of the cantilever beam767
37
as768
ξω=X
n
φn(s)qn(t),(A.4)
where φn(s) is the nth normal mode and qnis the nth generalized coordinate769
or modal participation coefficient. The normal modes for a cantilever beam are770
found from the equation771
φ0000 µ4φ= 0 (A.5)
with boundary conditions φ(0) = 0, φ0(0) = 0, φ00(l) = 0, and φ000(l) = 0.772
Solving (A.5) yields the nth normal mode,773
φn= (cos µnl+ cosh µnl) (sin µnssinh µns)+(sin µnl+ sinh µnl) (cosh µnscos µns),
(A.6)
where µnis the nth solution of774
1 + cos µl cosh µl = 0.(A.7)
Using (A.5) associated with the boundary conditions, the normal modes are775
proved to satisfy the orthogonality conditions,776
Zl
0
G(s)φnφmds =
Rl
0G(s)φ2
nds, n =m,
0, n 6=m,
(A.8)
where G(s) is an arbitrary function. Substituting (A.4) into (A.3) yields777
mX
n
φn¨qn+cX
n
φn˙qn+EI X
n
φ0000
nqn
=Γ [ccos(kx ωt +ψ) + ωmIsin(kx ωt +ψ)] .
(A.9)
Multiplying (A.9) by φmand integrating from 0 to lresult in778
X
n Zl
0
nφmds¨qn+Zl
0
nφmds ˙qn+Zl
0
EI φ0000
nφmdsqn!
="Zl
0
cΓφmds cos(kx ωt +ψ) + ωZl
0
mIΓφmds sin(kx ωt +ψ)#.
(A.10)
38
Substituting (A.5) into (A.10) and using the orthogonality conditions (A.8) yield779
780
¨qn+ 2ζnλn˙qn+λ2
nqn=[Dncos(kx ωt +ψ) + ωInsin(kx ωt +ψ)] ,
(A.11)
where 2ζnλn=Rl
02
nds/ Rl
02
nds,λ2
n=µ4
nRl
0EI φ2
nds/ Rl
02
nds,Dn=781
Rl
0cΓφnds/ Rl
02
nds, and In=Rl
0mIΓφnds/ Rl
02
nds. The steady state solu-782
tion for (A.11) is783
qn=aωQssin(kx ωt +ψ) + aωQccos(kx ωt +ψ),(A.12)
where784
Qs=ωInλ2
nω2Dn2ζnλnω
(λ2
nω2)2+ (2ζnλnω)2(A.13)
and785
Qc=Dnλ2
nω2+ωIn2ζnλnω
(λ2
nω2)2+ (2ζnλnω)2.(A.14)
Substituting (A.6) and (A.12) into (A.4) and the result into (A.2) yields the786
blade displacement,787
ξ=X
ω
aΓ [γssin(kx ωt +ψ) + γccos(kx ωt +ψ)] ,(A.15)
where the transfer functions γsand γcare given by788
γs=ω
Γ
X
n=1
φn
ωInλ2
nω2Dn2ζnλnω
(λ2
nω2)2+ (2ζnλnω)2(A.16)
and789
γc=ω
Γ
X
n=1
φn
Dnλ2
nω2+ωIn2ζnλnω
(λ2
nω2)2+ (2ζnλnω)2.(A.17)
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