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Results in Applied Mathematics 12 (2021) 100191
Contents lists available at ScienceDirect
Results in Applied Mathematics
journal homepage: www.elsevier.com/locate/results-in-applied-mathematics
Quantification of wave attenuation in mangroves in Manila
Bay using nonlinear Shallow Water Equations
Ikha Magdalena a,b, Raynaldi La’lang a, Renier Mendoza c,∗
aFaculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Indonesia
bCenter for Coastal and Marine Development, Bandung Institute of Technology, Indonesia
cInstitute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines
article info
Article history:
Received 1 June 2021
Received in revised form 1 August 2021
Accepted 12 September 2021
Available online 22 September 2021
Keywords:
Wave attenuation
2D nonlinear shallow water equations
Mangrove
Tsunami
Manila Bay
abstract
In this paper, we investigate the wave attenuation by mangroves using modified 2D
Nonlinear Shallow Water Equations. Numerically, we solve the model using a staggered
finite volume method that is free from damping error. Further, several benchmark tests
are performed to show the robustness of our numerical model. Then, we implement
our numerical model to investigate the effects of mangroves in the wave attenuation
caused by tsunamis in Manila Bay. The bathymetry profile of Manila Bay used in
this study is obtained from the General Bathymetric Chart of the Oceans (GEBCO).
Numerical results show that the presence of mangroves can reduce ocean waves near
the coast. Simulations on varying density and width of mangrove forest quantify how
much the tsunami wave height can be reduced. The results of this work may guide the
policymakers of the Philippines in deciding the best strategy in rehabilitating Manila
Bay.
©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC
BY license (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Manila Bay is a body of water situated on the west part of Luzon and is bounded by four provinces and Metro Manila.
Because of its strategic location, Manila Bay has been the seat of socio-economic development since pre-Hispanic times.
It also has abundant natural resources, which have been a source of livelihood for residents living in the coastal areas
surrounding the bay [1].
Due to rapid urbanization, Manila Bay faces several threats: marine pollution [2,3], eutrophication and hypoxia [4,5],
overexploitation of fishery resources [6], invasive species [7], and erosion [1].
Manila Bay is also prone to tsunamis [8]. Tsunamis are oceanographic phenomena that represent a water wave or
series of waves caused by an impulsive vertical displacement of the body of water [9]. Tsunami scenarios in Manila Bay
can potentially originate from a giant rupture along the Manila trench [10] or eruption of the Corregidor caldera located
in the mouth of the bay [11].
The destruction caused by tsunamis can be massive [12]. Thus, mitigation strategies are studied [13,14]. One strategy
is by utilizing coastal mangrove forests. It was shown in [15] how coastal mangrove vegetations mitigated the 2004 Asian
tsunami in 18 coastal hamlets along the southeast coast of India. Furthermore, the existing coastal vegetation in front of
∗Corresponding author.
E-mail address: rmendoza@math.upd.edu.ph (R. Mendoza).
https://doi.org/10.1016/j.rinam.2021.100191
2590-0374/©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/
licenses/by/4.0/).
I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 1. The observation domain of 2D-Shallow Water Equations (2D-SWEs). The porous domain Ω2is where the mangroves are situated. The variables
of the 2D-SWEs are also illustrated.
settlements on the west coast of Aceh, Indonesia significantly reduced casualties caused by the 2004 Asian tsunami by an
average of 5% [16]. It was studied in [17–20] how mangrove forests can reduce wave height. Energy reduction for waves
propagating in mangroves on varying slopes has been shown in [21]. Wave attenuation in coastal mangrove forests in
Vietnam was studied in [22].
In this study, we investigate through numerical simulations, the tsunami wave attenuation in mangroves in Manila
Bay. We use a 2D nonlinear Shallow Water Equations (SWE) to model the tsunami wave propagation. SWE models are
popular in simulating tsunami propagation because of their efficient computation capability without sacrificing too much
accuracy [23]. SWE models have been used in several tsunami studies [24–30]. To solve the 2D nonlinear SWE, we use the
finite volume method (FVM), which is a discretization scheme for the numerical simulations of various types of partial
differential equations [31]. FVM has been extensively used in several applications [29,31–33]. In our model, we use FVM
in a staggered grid, with a momentum conservative scheme, as discussed in [34–40].
In the next section, we present the mathematical model that we use in this work. Then, the numerical method to solve
the model is discussed. Furthermore, we perform some benchmark tests and investigate the effects of adding mangroves
in Manila Bay in reducing the tsunami wave heights. Finally, we give concluding remarks and recommendations.
2. Mathematical model
In this section, we derive the mathematical model to investigate the propagation of waves passing through a mangrove
forest. For this purpose, the observation domain will be divided into three domains Ω1,Ω2, and Ω3. Domain Ω2is for
the porous domain and Ω1,Ω3are the free domain. Here we use 2D-Shallow Water Equations that have also been used
in [34,41]. The 2D-SWEs consist of three hyperbolic equations, mass conservation, and momentum balance in the x- and
y-directions:
ηt+(hu)x+(hv)y=0,(1)
ut+uux+vuy+gηx=0,(2)
vt+uvx+vvy+gηy=0.(3)
Here, hdenotes the total water depth and gis the acceleration due to gravity. The variables dand ηrefer to the water
depth at still condition and the free surface elevation, respectively (see Fig. 1). Thus, we have h=d+η. The wave velocity
along the x-direction is denoted by uand the wave velocity along the y-direction is denoted by vas shown in Fig. 1.
As the incoming wave enters the porous domain Ω2, the wave interacts with the vegetation which causes friction. We
modify the model by adding a friction factor into the momentum equations. Thus, Eqs. (2) and (3) become
ut+uux+vuy+gηx+cf
u√u2+v2
h=0,(4)
vt+uvx+vvy+gηy+cfv√u2+v2
h=0,(5)
2
I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 2. 2D staggered grid scheme (Arakawa C-Grid) scheme.
with a piecewise constant friction functions of cfdescribed as
cf=0,x∈Ω1,Ω3,
f,x∈Ω2.(6)
Hence, the governing equations for the whole domain are given by Eqs. (1),(4),(5), and (6).
3. Numerical method
We now solve the governing Eqs. (1),(4)–(6) numerically using the finite volume method on a staggered grid method.
First, we consider the 2D shallow water equations with the observation domain Ω:= [0,Lx]×[0,Ly]and the time interval
t:= [0,T]. The spatial domain is divided into a rectangular grid of Nx×Nyand the time interval tis divided into Nttime
steps. In this staggered arrangement, the surface elevation ηand water depth hare evaluated in the mass conservation
equations cell on the full grid xi,j,i=1,2,...,Nxand j=1,2,...,Ny. In contrast, the velocity in the x-direction uand
the velocity and y-direction vare evaluated on the half grid xi−1
2,jand xi,j−1
2, as shown in Fig. 2. This arrangement gives
us advantage in terms of the imposing boundary condition. If we use collocated grid, then we need to give a coupled
boundaries to fulfill the well-posedness of our problem. However, using a staggered approach, we only need to define the
incoming wave velocity from the left and moving boundary on the right.
The approximations of ηand hat point xi,jand time tnare represented by ηn
i,jand hn
i,j, respectively. The approximation
of uat point xi+1
2,jand time tnis denoted by un
i+1
2,j,n∈ {1,2,...,Nt}. Similarly, the approximation of vat xi,j+1
2and time
tnis denoted by vn
i,j+1
2
,n∈ {1,2,...,Nt}. Applying forward time centered space numerical scheme, the discretization of
the mass conservation equation of the 2D-SWEs (Eq. (1)) becomes
ηn+1
i,j−ηn
i,j
∆t+qun
i+1
2,j−qun
i−1
2,j
∆x+qvn
i,j+1
2−qvn
i,j−1
2
∆y=0,(7)
where qun
i+1
2,j=∗hn
i+1
2,jun
i+1
2,jand qvn
i,j+1
2=∗hn
i,j+1
2
vn
i,i+1
2
.
Note that the values of hon the half grid xi+1
2,jand xi,j+1
2are missing, then we approximate those terms using the
upwind approximation, for all (i,j)∈{1,...,Nx}× {1,...,Ny}:
∗hn
i+1
2,j=
hn
i,j,un
i+1
2,j≥0
hn
i+1,j,un
i+1
2,j<0(8)
∗hn
i,j+1
2=
hn
i,j, vn
i,j+1
2≥0
hn
i,j+1, vn
i,j+1
2
<0(9)
3
I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Table 1
The errors and rate of convergence on the standing wave simulation.
E∆x(t=10) E∆t(x=10)
∆x E Rate of convergence ∆t E Rate of convergence
1 0.362989 1 1.416436
0.5 0.187381 0.953948 0.5 0.599868 1.239549
0.25 0.094108 0.993583 0.25 0.336278 0.83499
0.125 0.047118 0.998042 0.125 0.189068 0.830746
0.0625 0.023561 0.999867 0.0625 0.091228 1.051362
0.03125 0.01178 1.000033 0.03125 0.047637 0.937402
Next, we discretize the momentum balance equations of the 2D linear SWEs (Eqs. (4)–(5)) and obtain
un+1
i+1
2,j−un
i+1
2,j
∆t+
uqx
i,j
hx
i+1
2,jun
i+1
2,j−un
i−1
2,j
∆x+
vqx
i,j−1
2
hx
i+1
2,jun
i+1
2,j−un
i+1
2,j−1
∆y
+gηn+1
i+1,j−ηn+1
i,j
∆x+cfun+1
i+1
2,j
un
i+1
2,j2
+vn
i+1
2,j2
hx
i+1
2,j
=0,(10)
vn+1
i,j+1
2−vn
i.j+1
2
∆t+
uqy
i−1
2,j
hy
i,j+1
2vn
i,j+1
2−vn
i−1,j+1
2
∆x+
vqy
i,j
hy
i,j+1
2vn
i,j+1
2−vn
i,j−1
2
∆y
+gηn+1
i,j+1−ηn+1
i,j
∆y+cfvn+1
i,j+1
2
un
i,j+1
22
+vn
i,j+1
22
hy
i,j+1
2
=0,(11)
where
uqx
i,j=ui,jhx
i,j,uqy
i,j=ui,jhy
i,j,
vqx
i,j=vi,jhx
i,j,vqy
i,j=vi,jhy
i,j,
hx
i+1
2,j=hi,j+hi+1,j
2,hy
i,j+1
2=hi,j+hi,j+1
2.
Notice that in both momentum equations, ηis approximated implicitly by the term ηn+1
i,j. This was applied to guarantee
the stability of the scheme as mentioned in [41]. The semi-implicit method was used to approximate the friction terms
in both equations to avoid changes in the stability condition of the scheme. Moreover, to simulate the wave propagation
over a dry bed, we add wet-dry procedure into our numerical scheme. We calculate the velocity when the water depth is
positive. This procedure is also included in the friction terms that can handle dry area correctly. Applying this procedure,
we can simulate the tsunami propagates on land accurately. In the next section, we implement the numerical scheme
based on Eqs. (7)–(11) to simulate several cases of wave propagation including wave that propagates over a dry area such
as island or shoreline.
3.1. Error and rate of convergence
We use a one-dimensional standing wave simulation to elaborate the convergence of the numerical scheme. This
simulation is produced on a domain of length L=25 m and constant depth d=4 m, initial condition of η(x,0)=
cos π
Lx, and boundary conditions of u(0,t)=u(L,t)=0. The analytical solution to this simulation is identical to that
of vibrating string of infinite length: η(x,t)=cos (ωt)cos (kx), where in this case we have ω=k√gd,k=π
L.
The error of the numerical simulation is calculated as E∆x(tn)=maxi=0,...,Nxη(xi,tn)−ηn
i, and E∆t(xi)=
maxn=0,...,Ntη(xi,tn)−ηn
iwhere ηn
iis the numerical solution at point (xi,tn). Then the rate of convergence is ap-
proximated as logE1
E2
log∆1
∆2. As seen in Table 1, the rate of convergence of this numerical scheme is 1 in both ∆xand
∆t.
4
I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 3. Laboratory experiment setup by Briggs et al. [42].
4. Simulations and discussion
In this section, the staggered scheme of 2D wave simulation is validated by conducting several benchmark test
simulations. First, we simulate wave propagation over a conical island. Second, we validate our numerical scheme for
the tsunami in Monai Valley Island in Japan. We show that successfully simulating these test cases means our numerical
scheme can handle nonlinear phenomena that involve wet-dry areas very well. After that, we implement our numerical
scheme for simulating the wave interaction with a mangrove forest in Manila Bay.
4.1. Solitary wave runup on a conical island
In this subsection, we run a numerical simulation to see the interaction of a solitary wave around a conical island.
This simulation is motivated by the 1992 Flores Island tsunami. Here, we compare the numerical surface waves with
the experimental data that was obtained from the laboratory experiment at the Coastal Engineering Research Center,
Vicksburg, Mississippi. The basin has 30-m-wide, 25-m-long, and 60-cm-deep dimensions with a conical island in the
center. The slope of the island is 1:4 and the water depth is 0.32 m. Details of the experiment can be found in [42].
For the numerical simulations, the water depth is set to be
d(x,y)=max (hc,min (hb,a+rb)) ,
where
hc= −0,305,hb=0,32,rb=3,6,rc=1,1,a=hb−hc
rb−rc
and
r=(x−12,96)2+(y−13,8)2,
for any x,yin the observation domain. The numerical simulations were performed using ∆x=0.1,∆y=0.5, and
∆t=∆x
4√2gH . The initial conditions are given by u(x,y,0)=v(x,y,0)=η(x,y,0)=0. The boundary conditions
are u(Lx,y,t)=v(x,0,t)=vx,Ly,t=0 (where Lx,Lyare the x,yboundaries) and u(0,y,t)is generated by the
wavemaker as seen in Fig. 4. Because mangroves are not present in this simulation and bed friction is neglected then the
value of cfin Eq. (6) is set to 0. The solitary wave has an amplitude 0.015 m coming from the left boundary propagates
to the right, then we record the surface elevations at four locations denoted by WG6, WG9, WG16, and WG22, as shown
in Fig. 3 and listed in Table 2. Three wave gauges WG9, WG16, WG22 are located at the initial shoreline surrounding
the island, whereas WG6 is located near the front face of the island. Fig. 5 shows the propagation and evolution of the
soliton. When the soliton reaches the island, the waves diffract and move around the conical island, then the waves move
forward leaving the island. From Fig. 6, our numerical simulations replicate the experimental data from the four gauges
well. The numerical waves profile has a similar trend with experimental data as well as the reflections. The computed
errors in all the gauges are also small as shown in Fig. 6.
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I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 4. Influx wave for the conical island experiment.
Fig. 5. Numerical simulation of the propagation of soliton on a conical island.
Table 2
Positions of wave gauges WG6, WG9, WG16, and WG22.
Wave Gauge ID x(m) y(m)
6 9.36 13.80
9 10.72 13.80
16 12.96 11.53
22 15.20 13.80
4.2. Wave runup onto a complex beach (Monai Valley)
This simulation is reproducing the 1993 Hokkaido–Nansei–Oki (HNO) tsunami. The tsunami that struck Okushiri Island,
Japan, reached an extreme height of 30 m in a small valley near the Monai village. The experiment was conducted using
a laboratory model scaled as 1:400 of the actual bathymetry of Monai Valley in Central Research Institute for Electric
Power Industry (CRIEPI), Abiko, Japan. This simulation is a well-known benchmark test for numerical models to investigate
tsunami phenomena, and all data are provided openly available at [43].
To validate our numerical model, we run a numerical simulation performed on a 5.4880 m ×3.4020 m computational
domain with spatial partition ∆x=∆y=0.014 and time increment ∆t=1.72 ×10−4. The initial conditions are given
by u(x,y,0)=v(x,y,0)=η(x,y,0)=0. The boundary conditions are u(Lx,y,t)=v(x,0,t)=vx,Ly,t=0 (where
Lx,Lyare the x,yboundaries) and u(0,y,t)is generated by the wavemaker as seen in Fig. 7. Like the previous example,
the value of cfis also set to 0. The challenging part in this benchmark test is the wave that propagates not only in wet
area but also several dry areas. Therefore, we apply the wet-dry procedure with a threshold of water thickness set to
10−5. The bathymetry data we got from [43] is shown in Fig. 8.
6
I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 6. The comparison between our numerical results and the experimental data of the Flores Island tsunami.
Fig. 7. Initial wave for Monai Valley experiment.
For wave influx, a leading-depression wave is used to represent the incident wave from offshore. The incoming wave,
as shown in Fig. 8, propagates towards the Monai beach. The wave approaches the shoreline at t=14 s. Moreover, the
maximum run-up happens at a recorded time t=16.3 s near Monai Valley. Fig. 9 shows that our numerical scheme has
estimated the wave run-up over the complex beach Monai Valley successfully with a mean average error of around 0.35%.
4.3. Wave passing through mangroves in Manila Bay
Finally, we simulate the wave attenuation in mangroves in Manila Bay. This study aims to show how the restoration
of mangrove forests in Manila Bay can protect the coasts against tsunamis.
In our simulations, we use a gridded bathymetry of Manila Bay provided by the General Bathymetric Chart of the Oceans
(GEBCO), https://www.gebco.net/. To be specific, the data capture the seafloor topography of a portion of Manila Bay, with
the longitude of 120.94 to 121.00, and the latitude of 14.51 to 14.65, which gives a domain of around 6 km by 15 km, with
grid sizes ∆x=46.25 m,∆y=231.25 m, and a time step of ∆t=1.87 s. Please refer to Fig. 10. The initial conditions are
given by u(x,y,0)=v(x,y,0)=η(x,y,0)=0. The boundary conditions are u(Lx,y,t)=v(x,0,t)=vx,Ly,t=0
(where Lx,Lyare the x,yboundaries); and on x=0 are east-traveling sinusoidal waves with initial amplitude of 2 m
and period of 60 s: u(0,y,t)=2 sin 2π
60 t. The presence of mangroves is modeled by applying a coefficient of friction cf
7
I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 8. Scaled bathymetry of Monai Valley. The bottom topography is shown on the left. The locations of the wave gauges are depicted by green
dots on the right.
Fig. 9. Comparison of our numerical simulation results against the experimental data of wave run-up over a Bathymetry of Monai Valley.
along the areas in front of the coastline. We vary the friction coefficient (cf=0,0.2,0.4,0.6,0.8) and the width of the
mangrove forest (100 m from the coast and 200 m from the coast). The values of cfare based on the values used in [39]
while the values for the width of the mangrove forest are based on the values used in [17]. Intuitively, the higher the
value of cfis, the denser the mangrove forest is. A wider mangrove forest will make the porous domain Ω2larger, which
in turn, should make the wave height reduction higher.
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I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 10. Bathymetry of a portion of Manila Bay and the placement of mangroves along the coast of the bay. The bottom topography of Manila Bay
is shown on the left. As shown on the right, the mangroves are depicted by orange dots that propagate from the beach towards the ocean. The
observation points (wave gauges) are depicted as red dots on the right.
Fig. 11. A 40-minute time series of surface elevation (η) at the three observation points. Simulations are done on varying coefficients of friction cf
and mangrove width.
The graphs in Fig. 11 show the evolution of surface elevation at the three different observation points: P-1(120.9565,
14.63), P-2(120.9555, 14.595), and P-3(120.989, 14.53). As expected, it can be seen how denser mangrove vegetation can
help reduce the wave height more. Similarly, the wave height reduction is greater for the case when the width of the
forest is higher. For the forest with a width of 200 m, wave height reduction is already significant even if cf=0.2.
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I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Fig. 12. A 40-minute time series of surface elevation (η) at the three observation points. Simulations are produced by limiting the length of the
mangrove zone to be within 1 km of the sensors.
Additionally, the tide is much more stable, especially as the zone width is increased. In the case when the width is 200
m and the coefficient of friction is 0.8, the presence of mangroves can reduce ocean waves near the coast, by as much as
45%.
Because Manila Bay is already a highly urbanized coastal area, mangroves cannot be planted on some of its portions.
Hence, it is interesting to investigate the effects of mangroves on reducing the wave height. In Fig. 12, we present the
simulations when the mangroves are only placed on three portions of the bay. The length of the mangrove forests is
within 1 kilometer of the observation point. It can be seen how wave height reductions are still observed in the sensors.
5. Insights, conclusion, and recommendations
In the early 1990s, Manila Bay has extensive mangrove plantations, which were subsequently replaced by fishponds,
settlements, and port infrastructure [44]. The reduction of mangrove areas has further compromised their ecological
function of support for fishery production and of serving as habitats for both marine and terrestrial biodiversity.
Furthermore, about 70 percent of Philippine mangroves have been lost or severely degraded during the past 50 years
[45].
In 2020, the Philippine Department of Environment and Natural Resources started installing crushed dolomite boulders
along a portion of the Manila Bay beach [46]. The Marine Science Institute, University of the Philippines Diliman issued a
statement that the use of crushed dolomite sand will not help solve the environmental problems in Manila Bay [47].
Moreover, the Institute of Biology, University of the Philippines Diliman recommended mangrove rehabilitation as a
cheaper and more cost-effective nature-based solution than the dolomite dumping project [48].
Mangroves are a source of valuable plant products. They serve as nesting grounds for hundreds of bird species, as
well as nurseries, and are home to a wide variety of animals [49]. The mangrove ecosystem also provides a livelihood for
residents living in the coastal community [50]. Mangrove also plays a role in climate change mitigation particularly in
carbon absorption [51]. Furthermore, mangroves provide flood protection [52].
In this paper, we were able to quantify the wave attenuation in mangroves near the coast of Manila Bay. Our
numerical simulations emphasize another importance of restoring coastal mangrove vegetation in Manila Bay. Restoration
of mangrove forests in Manila Bay aligns with the 2008 SC Mandamus directing 13 Philippine government agencies to
clean up, rehabilitate, and preserve the bay [53]. We hope that the results of this study can guide the government in
making policies for the improvement of Manila Bay.
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I. Magdalena, R. La’lang and R. Mendoza Results in Applied Mathematics 12 (2021) 100191
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgment
The authors acknowledge the ITB research grant 158/IT1.B07.1/TA.00/2021 and ministry of education, culture, research,
and technology 2/E1/KP.PTNBH/2021 for the funding support.
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