Rapid population growth and major trends of world economy growth have led to significant energy needs in our country. Benin, Gulf of Guinea country, although with a significant coastal network powered by potential energy from breaking waves, has experienced a deficit and a critical energy instability, marked by recurrent power cuts and disruption of the national economy. To ensure the integration of this source of renewable energy in the Benin energy mix and sustainably reduce the energy deficit in progress, this work has aimed to study the dissipation of wave energy at the bathymetric breaking in the breakers zone of Cotonou coast. Sea conditions and the statistics parameters of the breaking waves under perturbation effect of the seabed were evaluated to predict the beginning of the breaking. The modeling is based on the Navier-Stokes equation in which the viscosity and the interactions between the molecules of the oceanic fluid are neglected. The nonlinear wave dispersion relation is also used. The results obtained for this purpose showed that water particles have an almost parabolic motion during their fall; their velocity is higher than those of the early breaking. In this area, the waves dissipate about 80% of their energy: it generates turbulence which leads to a strong setting in motion of sediments.
Waves are oscillations of the sea surface, generated by the wind energy and maintained by gravity [
Benin is a coastal state of the Gulf of Guinea. As such, it has access to the ocean which is at the core of perpetual wave motions [
The bathymetric map (
Benin is a country of the Gulf of Guinea located between the parallel 6˚15' and 12˚30' north latitude the one hand and meridians 1˚ and 3˚40' east longitude on the other hand (
and cut off in two places, namely, the Bouche du Roy and the channel mouth of Cotonou. Its coastal zone is between latitude 6˚15' and 6˚23' north [
Waves parameters, significant wave height ( H s ) and peak period ( T p ) are recorded every 30 min. The buoy data (
The gravity on the coast of Benin is approximately g = 9.80 N / kg and the density of sea water is ρ = 1025 kg / m 3 [
As part of the expansion of the port of Cotonou, the Millennium Challenge Account-Benin (MCA-Benin) performed a series of measures in terms of wave power. These measures are performed with two stations of wave recorder buoy called WCP1 and WCP2 whose geographical coordinates are respectively (N 6˚20.118'; E 2˚27.255') and (N 6˚20.373'; E 2˚26.140') [
Data analysis yielded the following diagrams which respectively reflect the period T (
From the analysis of these curves, it appears that the swells in the Gulf of Guinea in Cotonou have:
• A height which varies between 0.4 m and 2 m (0.4 m ≤ Ho ≤ 2 m) with a common average value of about 1.1 m (Hom ≈ 1.1 m);
• A period of peak that ranges between 8 s and 18 s (8 s ≤ T ≤ 18 s) and whose average value is about 12 s (Tm ≈ 12 s);
• A propagation direction that lies between the south (S) and southwest (SW) with a strong dominance of the direction from the south to the southwest (SSW).
As for the sea breeze or short swells generated by local winds, they have a period that varies between 3 s and 7 s.
When swells become nonlinear under the effect of shoaling (lifting of swell), refraction, reflection, diffraction... it is induced by the disturbing effect of the seabed, the wave dispersion relation proposed by Dalrympleetal., to improve angularity [
ω 2 = g k tanh ( k h ) (1)
with h = d + H 2
Group velocity C g = ∂ ω ∂ k and phase velocity C φ = ω k have been inferred from that relation of nonlinear dispersion [
Bathymetric breaking occurs in shallow waters where tanh ( k h ) ≈ k h ; thus we have.
C g = C φ = g ( d + H 2 ) (2)
Using the results obtained with the theory of Boussinesq and the breaking standard of Kaminsky and Krauss (1993) [
H o and L o = g T 2 2 π represent the off height and wavelength.
{ d b = H o 0.908 L o 0.092 1.33 ( tan β ) 0.216 H b = 0.903 H o ( L o H o ) 0.227 ( tan β ) 0.054 L b = 0.613 T 2 H o 0.454 L o − 0.459 π ( tan β ) 0.108
⇒ μ b = d b L o = H o 0.908 L o − 0.918 1 , 33 ( tan β ) 0.216 (3)
Thus at the breaking point, we have:
C g b = C φ b = g ( d b + H b 2 ) (4)
The breakers zone is the area where swells break for the first time. In this area, the vertical rise of the free surface behaves like a wave of expansion during the breaking. When the maximum height obtained begins to drop at the end of shoaling, water particles located on the surface follow the break and drop sharply.
The forces of pressure and gravity and that of Coriolis caused by viscosity make it possible to establish the Navier-Stokes equation below [
∂ u ∂ t + ( u ⋅ ∇ ) u = − 1 ρ ∇ p + g + μ ρ Δ u (5)
So, the acceleration a of dropping particles is:
a = d u d t = ∂ u ∂ t + ( u ⋅ ∇ ) u = 1 ρ ∇ p + g − μ ρ Δ u (6)
At the surface, the pressure is nearly constant and equal to the atmospheric pressure ( p = p a t m = C s t e ⇒ ∇ p → 0 ). As the viscosity μ is negligible compared to
the density ρ of the ocean ( μ ≪ ρ ⇒ μ ρ → 0 ), we obtain the following [
a = g (7)
By approximating the direction of swell spread to a straight line, the wave’s position r and its velocity v are specified on the mark ( O , i , k ) such as the origin O (
At the breaking point, the height of the wave is H b and its group velocity C g b . Therefore, the early breaking begins at a position r o with a velocity u 1 as.
r o { x o = 0 z o > 0 and u 1 { u 1 x = C g b u 1 z = 0 (8)
The position and velocity are respectively given by:
r ( x , z ) = − 1 2 a t 2 + u 1 t + r o
⇒ r ( x , z ) { x ( t ) = C g b t z ( t ) = η ( t ) = − 1 2 g t 2 + z o (9)
u = d r d t ⇒ u ( x , z ) { u x = C g b u z = − g t (10)
Thus, the path equation of the water particles located at the surface during the breaking in the breakers zone is:
η ( x ) = z = − 1 2 d b + H b x 2 + H b 2 with 0 ≤ x ≤ l r (11)
If at the end of the breakers zone the particles renew contact with the ocean at a point r C with a velocity u c and a horizontal component u 2 , then we get the following:
r C { x C = l r z C ≤ 0 and u c { u C x = u 2 = C g b u C z = − g t C with H b = z C − z o (12)
At the end of a time t C , the particle slopes on the free surface at rest and twist it towards the depth at a position r c ( l r , z c ) ; thus we obtain:
r c ( l r , z c ) { l r = C g b t C z c = − 1 2 g t C 2 + z o ⇒ { l r = H b ( 2 d b + H b ) t C = 2 H b g (13)
The local depth of water d c at the sloping point is as:
d c = d b − l r tan β = d b − H b ( 2 d b + H b ) tan β (14)
From the previous results, we have:
C g = u = C g b 2 + g 2 t 2 = C g b 1 + g 2 t 2 C g b 2 if 0 ≤ t ≤ t C (15)
Let δ v = C g C g b be the coefficient of variations of the group velocity in the zone
δ v = C g C g b = 1 + 2 g t 2 2 d b + H b with 0 ≤ t ≤ 2 H b g (16)
If C g c is the velocity of particles at the sloping point, then the ratio δ v c = C g c C g b
expressing the variations in velocity between the end and the beginning of the breakers zones is given by:
δ v c = C g c C g b = 1 + 2 g t c 2 2 d b + H b with t c = 2 H b g
⇒ δ v c = C g c C g b = 1 + 4 1 + 2 d b H b (17)
The total energy carried by a swell is given by:
E = 1 8 ρ g H 2 (18)
when the breaking ends (in Surf and Swash zones) where d ≤ d b , the height of wave drops if the local depth of water d falls; according to Bonneton P. 2002, this is expressed by the relation below:
H ( d ) = H b [ σ ( d d b ) − 1 2 + ( 1 − σ ) ( d d b ) 1 4 ] − 1 (19)
with σ = 2 H b T tan β g ( d b + H b 2 )
H b and d b are respectively the height of swell and the local depth of water at the bathymetric breaking point. Thus, we obtain the following at the first dropping point of enrolled water particles:
H c = H b [ σ ( d c d b ) − 1 2 + ( 1 − σ ) ( d c d b ) 1 4 ] − 1 (20)
with σ = 2 H b 2 T tan β g ( 2 d b + H b )
Let τ d = E w b − E w c E w b be the dissipated energy rate in the breakers zone during
the breaking where E w b and E w c represent respectively the total energy at the beginning and at the end of the breakers zone.
⇒ τ d = 1 − ( H c H b ) 2
⇒ τ d = 1 − [ σ ( 1 − l r tan β d b ) − 1 / 2 + ( 1 − σ ) ( 1 − l r tan β d b ) 1 / 4 ] − 2 (21)
Graphs of
The curves in
The graph below (
The curves of
- The horizontal length l r of the breakers zone varies between 2 m and 7 m approximately ( 2 m ≤ l r ≤ 7 m ).
- The fall duration of the water particles fluctuates between 1.5 s and 3.5 𝑠 approximately ( 2 s ≤ t c ≤ 2.5 s ).
- Variations of the curves in
● The length l r varies between 5.5 m and 4.3 m approximately ( 4.3 m ≤ l r ≤ 5.5 m ).
● The fall time t c corresponds to figures between 2 s and 2.5 s approximately ( 2 s ≤ t c ≤ 2.5 s ).
Variations in the coefficient δ v c (
- δ v c > 1 ∀ T , H o and β : therefore the velocity acquired by the water particles at the end of the breakers zone is always higher than their velocity at the beginning of this zone; the velocity of the particles increases in the breakers zone.
- δ v c is a monotonic function of the wave period T but a descending function of their off height H o : the longer the period is (the smaller the height of swell is), the more important the velocity of water particles is in the breakers zone.
The evolution of the curve of
slope of the seabed. But this growth tends a limit when tan β → 1 ⇒ 0 ≤ β ≤ π 4 .
The graph of
● The dissipation rate increases with the off height of the swells; it is therefore a monotonic function of the wave height.
● The rate decreases as the frequency of these waves increases: it is a decreasing function of the period.
● Each breaking wave loses between 70% and 90% of their energy in the breakers zone in Cotonou with an average slope tan β ≈ 0.05 when the period fluctuates between 8 s and 18 s and the height between 0.4 m and 2 m.
● Breaking is the major source of dissipation of wave energy on the study site.
As for the diagram of
- The dissipation of the breaking waves energy in the breakers zone increases with the slope of the seabed.
- On the study site, when T = 12 s and 0.4 m ≤ H o ≤ 2 m , the rate of dissipation varies between 40% and 95% as the slope of the seabed to the horizontal varies between 0.001 rad and 0.2 rad.
Dissipation of breaking wave energy is a shaft of renewable energy. It appears from this study that the breaking waves in the breakers zone at the coast of Benin lose about 80% of their total energy. This generates turbulence on this site, and leads to a strong set in motion of non-cohesive sediments: it heightens coastal erosion that one can observe on this coast. This dissipation increases with the off height of waves but decreases with their period; it is a monotonic function of the slope of the seabed. In addition, results show that the horizontal length lr of the breakers zone is an increasing function of the period and the wave height can vary between 2 m and 7 m. As for the fall time of the water particles in this zone, it fluctuates between 1.5 s and 3.5 s. Similarly, the velocity of the water particles increases in this zone depending on the period of the swells and the slope of the seabed but constitutes a decreasing function of their off height Ho.
In Benin context characterized by a large energy deficit, the development of this sea renewable energy would be a good perspective to address the crisis and improve the modern conveniences of the country in terms of energy.
The authors thank Benin Center for Scientific Research and Innovation (CBRSI) for accompanying this research work.
Authors have declared that no competing interests exist.
We (Author and Co-Authors) confirm that neither the manuscript nor any part of its contents is currently under review or published in any other journal. All authors (Author and Co-Authors) have approved the manuscript and are in agreement with its submission to JMP.
Acclassato, O.G., Tokpohozin, N.B., Akowanou, C.D., Houékpohéha, A.M., Houngue, G.H. and Kounouhéwa, B.B. (2022) Study of Dissipating of Wave Energy in the Breakers Zone of the Gulf of Guinea: Case of Autonomous Port of Cotonou in Benin Coastal Zone. Journal of Modern Physics, 13, 1272-1286. https://doi.org/10.4236/jmp.2022.139076
ρ : Density of sea water (kg/m3);
μ : Fluid dynamic viscosity (sea water);
r : Vector position of a point located on the free surface;
η : Vertical elevation of the water level relative to the reference (m);
β : Inclination of the seabed relative to the horizontal (rad);
g: Gravity acceleration (m/s2);
H: Wave crest-to-trough height (m);
H 0 : Offshore wave height peak to trough (m);
H b : Wave crest-to-trough height at breaking point (m);
H c : Peak-to-trough height of the swell in the breaker zone (m);
L: Wave length of the swell (m);
L 0 : Wavelength of the offshore swell (m);
L b : Wavelength of the swell at the breaking point (m);
L c : Wavelength of the swell in the breaker zone (m);
l r : Horizontal length of the breaker zone (m);
T = 2 π ω : Swell period (s);
d: Surface-seabed distance near in coastal areas (m);
d b : Local water depth at the breaking point (m);
d c : Local water depth in the breaker zone (m);
C g : Wave group speed (m/s);
C φ : wave phase speed (m/s);
C g b : Swell group speed at breaking point (m/s);
C φ b : Swell phase speed at breaking point (m/s);
C g c : Swell group speed in the breaker zone (m/s);
E: Total wave energy (J/m);
E w b : Total energy at the start of the breakers zone (J/m);
E w c : Total energy at the end of the breaker zone (J/m);
t c : Duration of fall of water particles in the breakers zone (s);
τ d = E w b − E w c E w b : Rate of energy dissipated in the breakers zone;
δ v = C g C g b : Coefficient of variations of group speed in the breaker zone;
δ v c = C g c C g b : Rate of the speeds of the water particles enlisted at the end and at the start of the breakers zone.