Evolution in the Balearic Islands.
PhD Thesis
Author:
Amaya ´ Alvarez Ellacur´ıa
Advisors:
Dr. Alejandro Orfila F¨ orster Prof. Ra´ ul Medina Santamar´ıa
IMEDEA
Universitat de les Illes Balears
Date: June 2010
Agradecimientos
Son muchas las personas que me han hecho f´acil y sobretodo interesante el camino hasta aqu´ı. Agradezco a Joaqu´ın Tintor´e el soporte que me ha ofrecido todos estos a˜nos, a mi codirector de Tesis, Ra´ul Medina, por hacer tan fascinante el movimiento de la arena en sus clases. A Alejandro Orfila le agradezco su determinaci´on y empe˜no en que realizara el doctorado y a Jano le quiero dar las gracias por todas esas horas de dedi- caci´on, por estar siempre disponible, por leer con tanto inter´es todo lo que escrib´ıa y por ense˜nareme tanto sobre cual debe ser el proceso de la investigaci´on (nunca un embudo tuvo tanta importancia). Pero, ante todo Jano, muchas gracias por tu amistad.
Muchas gracias a todos mis compa˜neros del TMOOS, que siempre han estado dis- puestos a ayudar. A Guille, por ayudarme tanto con Cala Millor, a Enrique por escuchar y comprender mis quejas, a Benja por ayudarme en todos mis muestreos y a Tolo por tener siempre soluci´on a mis dudas en general y de matlab en particular.
Quiero agradecer infinitamente a toda la gente que me ha visto todos los d´ıas en el IMEDEA, y muchos d´ıas tambi´en fuera. Hab´eis hecho todo mucho m´as interesante y divertido. Sois muchos y seguro que me dejar´ıa a alguno as´ı que espero que os deis por aludidos todos aquellos que: hab´eis compartido comidas en el IMEDEA, o una, dos o tres galletas, incluso rara vez chocolate. Los que hab´eis jugado a volley playa o a f´utbol, los que hab´eis estado en mi azotea o yo en la vuestra. A los que os sint´ais identificados en m´as de una situaci´on, much´ısimas gracias.
Aunque en la distancia, muchos amigos han estado pendientes de mis progresos: Mar y Carlota, la Pepis y los pantaneros. Otros desde cerquita han soportado charlas infinitas sobre si la tesis esto o lo otro. In´es, gracias por acompa˜narme tan bien, es una suerte saber que siempre est´as ah´ı. Tomeu, ”he pasado gusto de” fumar cigarritos al sol, mirar la pantalla ”a la sombra”, salir los jueves por la noche, tomar caf´es por la ma˜nana, y todo, siempre contigo. Por fin ha llegado... ”nuestra primera tesis”.
Por ´ultimo quiero agradecer a mi familia todo el apoyo que me ha dado antes y durante el doctorado. A Marga y Lu´ıs por estar tan pendientes de mis progresos y ani- marme siempre. A mis padres por llevarme a la playa tantos a˜nos, porque siempre est´an ah´ı y porque lo ´unico que esperan de mi es que sea feliz. A Pablo que, aunque en la distancia, siempre ha estado pendiente. A Itzi que hace mucho que sabe lo que me pasa s´olo con mirarme a la cara y siempre encuentra una manera de solucionarlo. A ´Alvaro, por hacer que sonr´ıa todos los d´ıas, nunca imagin´e que se pudiera tener tanta suerte en la vida.
palabra
Resumen
El estudio de la hidrodin´amica y morfodin´amica de playas es de gran inter´es para la gesti´on costera. El uso de las playas como espacios recreativos ha obligado a las au- toridades a dise˜nar protocolos de seguridad y a implantar vigilancia en las playas. Estos lugares adem´as de una gran importancia socio-econ´omica tienen un alto valor ecol´ogico y son el primer elemento de protecci´on de la costa frente a los temporales y subidas del nivel del mar. Las pol´ıticas de protecci´on de estas ´areas, as´ı como las decisiones tomadas para la mejora de la seguridad en playas deber´ıan estar basadas en los conocimientos cient´ıficos adquiridos en los ´ultimos a˜nos en los campos de la hidrodin´amica y morfodin´amica litoral.
En esta Tesis se presentan tres estudios realizados con el objetivo de desarrollar herramientas ´utiles para la gesti´on costera, principalmente en el aspecto de la seguri- dad en playas, que ofrezcan informaci´on ´util tanto a gestores como socorristas y usuarios.
Una de las principales causas de rescates en playas es la aparici´on de corrientes generadas por la rotura de las olas. Estas corrientes son conocidas como rip currents o corrientes de retorno. La predicci´on a corto plazo de estas corrientes es, desde hace 20 a˜nos, uno de los principales focos de estudio de la hidrodin´amica de playas. El inter´es del Govern Balear por mejorar la seguridad en las playas del archipi´elago hizo posible el desarrollo de un sistema de predicci´on de oleaje y corrientes de retorno en una playa de la Isla de Mallorca, Cala Millor. Este sistema utiliza como datos iniciales las predicciones de oleaje en aguas profundas frente a la playa que se propagan mediante un modelo hacia aguas someras para obtener las alturas de ola en la playa. El acoplamiento de un segundo modelo de generaci´on de corrientes, que utiliza el oleaje propagado como dato inicial, permite la obtenci´on de las corrientes generadas en la playa a causa de la rotura del oleaje. El sistema ha estado operativo durante 3 a˜nos, ofreciendo informaci´on de altura de ola, direcci´on e intensidad de corrientes, con un horizonte de 36 horas. Los datos han estado disponibles v´ıa web para los socorristas y gestores.
Los resultados del proyecto piloto de predicci´on de oleaje y corrientes en Cala Millor fueron la base de la ampliaci´on del estudio a otras playas del litoral Balear. La imple- mentaci´on del sistema en diferentes playas hizo necesarias algunas modificaciones para su correcto funcionamiento. Se desarroll´o un nuevo sistema de predicci´on de oleaje y el nivel de riesgo asociado dependiendo de las caracter´ısticas de cada playa. En este estudio ha sido fundamental la comunicaci´on con los socorristas, quienes aportaron datos visuales diarios de las condiciones de oleaje y del nivel de riesgo asociado. Estos datos han sido incorporados en la calibraci´on del sistema de predicci´on.
La falta de datos en las zonas de estudio, tanto batimetr´ıcos como de oleaje y corri- entes, es uno de los principales problemas para el desarrollo de herramientas operacionales con alto grado de fiabilidad. El uso de sistemas de observaci´on remotos permite paliar esta deficiencia y ofrece la posibilidad de obtener largas series de datos, con una alta resoluci´on espacial y temporal. El an´alisis de im´agenes de video se ha utilizado en el estudio de la variablidad de la l´ınea de costa a lo largo de un a˜no y medio en la playa de Cala Millor. El estudio se ha realizado con im´agenes digitales diarias que han sido procesadas para extraer la posici´on de la l´ınea de costa. La variabilidad de est´a l´ınea puede asociarse con los movimientos de crecimiento y erosi´on de una playa. El avance en este tipo de estudios mediante el uso de im´agenes digitales permitir´a en un futuro la obtenci´on de datos fiables de batimetr´ıa y oleaje.
Durante el desarrollo de esta Tesis, los conocimientos adquiridos tamb´en han servido para la realizaci´on de otros estudios publicados. Estos trabajos se han incluido al final del documento.
Esta Tesis se ha realizado gracias a la financiaci´on de la Direcci´o General d’Emerg`encies del Govern de les Illes Balears.
Agradecimientos i
Resumen iii
List of Figures vii
List of Tables x
1 Motivation 1
1.1 Outlines . . . 3
2 Coastal Zone 5 2.1 Coastal Zone Definition . . . 5
2.2 Coastal Zone Processes . . . 6
2.2.1 Nearshore Hydrodynamics . . . 6
2.2.2 Nearshore Morphodynamics . . . 10
2.2.3 Beach Morphodynamic Classification. Rip Currents . . . 11
2.3 Numerical Models for Nearshore Studies . . . 13
2.3.1 Wave Propagation . . . 13
2.3.2 Surf Zone Currents . . . 14
2.4 Nearshore Observations. Videomonitoring Systems . . . 15
2.5 Beach Safety Studies . . . 16
2.6 Processes and Models Studied in the Thesis . . . 17
3 A Nearshore Wave and Currents Forecasting System 19 3.1 Abstract . . . 19
3.2 Introduction . . . 20
3.3 Data and Methodology . . . 21
3.3.1 Deep Waters Forecast Data . . . 22
3.3.2 Waves Propagation . . . 22
3.3.3 Wave-breaking Current Generation . . . 24 v
3.3.4 Bathymetries . . . 27
3.4 Study Site . . . 28
3.4.1 Wave Climate . . . 29
3.5 Results . . . 31
3.5.1 Validation . . . 32
3.6 Discusion . . . 36
3.7 Conclusions . . . 37
4 An Alert System for Beach Hazard Management in the Balearic Islands 39 4.1 Abstract . . . 39
4.2 Introduction . . . 40
4.3 Study Areas . . . 42
4.4 Data and Methods . . . 46
4.4.1 Nearshore Wave Climate . . . 47
4.4.2 Forecast Wave Data . . . 48
4.4.3 Hazard Level . . . 48
4.5 Results and Discussion . . . 50
4.5.1 Lifeguards Data . . . 51
4.5.2 Validation . . . 51
4.5.3 Calibration . . . 53
4.6 Conclusions . . . 55
5 Short-Term Shoreline Evolution in a Low-Energy Beach 57 5.1 Abstract . . . 57
5.2 Introduction . . . 58
5.3 Data and Methodology . . . 60
5.3.1 Coastline Extraction . . . 60
5.3.2 ANN Development . . . 62
5.3.3 Shoreline Analysis . . . 64
5.3.4 Wave Data . . . 64
5.4 Results and Discussion . . . 65
5.5 Conclusions . . . 71
6 General Conclusions 73 7 Future Work 75 Bibliography 77 A Navier-Stokes Equations 83 A.1 Navier-Stokes Equations . . . 83
B Radiation Stress 85 B.1 Radiation Stress . . . 85
C Mild slope equation 89
C.1 Mild slope equation . . . 89 C.1.1 Parabolic approximation . . . 91
D Publications 95
2.1 Coastal zone . . . 6
2.2 Waves classification . . . 7
2.3 Morphodynamic scales . . . 10
2.4 Morphodynamic states . . . 12
3.1 Models scheme . . . 21
3.2 Grids picture . . . 23
3.3 Boundary conditions . . . 27
3.4 Bathymetries 2007 . . . 28
3.5 Geographic location of the pilot area . . . 29
3.6 Long term distribution . . . 30
3.7 Annual wave maximums distribution . . . 30
3.8 Wave roses . . . 31
3.9 Hs andTm at deep waters . . . 32
3.10 Buoy-prediction Hs correlation . . . 33
3.11 Buoy-prediction Tm correlation . . . 33
3.12 Significant wave height serie . . . 34
3.13 Aerial photography . . . 35
4.1 Study sites . . . 42
4.2 Wave roses . . . 44
4.3 Hazard alert system scheme . . . 46
4.4 Deep and shallow wave height series . . . 52
4.5 Surfzone circulation at IB5 . . . 54
5.1 Timex images and UTM image . . . 61
5.2 Cala Millor location . . . 61
5.3 ANN scheme . . . 62
5.4 Error between ANN detection and topographic data in Barcelona . . . 63 5.5 Spatial modes of the temporal variance from the EOF analysis,EOF1,2,3 . 66
ix
5.6 Temporal amplitude of theEOF1,2,3 . . . 66 5.7 Temporal amplitudes of 1st and 2ndmode . . . 68 5.8 Surfzone currents generated by Navier-Stokes model . . . 69 5.9 Mean differences inter shorelines, related standard deviation and associ-
atedHs,50SW4 . . . 71
4.1 Beaches characteristics . . . 45
4.2 Grids charactheristics . . . 47
4.3 Minimum and maximum incident directions for the hazard level definition in each beach . . . 49
4.4 Hazard level definition . . . 49
4.5 Results of lifeguards and forecasted hazard . . . 50
4.6 Differences between propagated and interpolated Hs . . . 51
4.7 Comparison between flags posted and hazard level forecast . . . 53
4.8 Calibration of hazard level . . . 54 5.1 Correlation between first and second amplitudes (a1, a2) and wave contidions 67
xi
Motivation
Beaches are highly dynamical systems conforming preferred mechanisms of coastal de- fence with an enormous ecological and economic value. The correct management of beaches should be based on a scientific knowledge on the natural processes occurring there, providing accurate answer to three main related topics: protection, recreation and the support of their natural values. Each of these requirements refer to a specific role of the beach surface: (i) absorbing/dissipating the incident wave energy during storms reducing its impact on the hinterland, (ii) offering an environment for leisure and (iii) supplying a physical substrate for the development of coastal ecosystems.
In the last three decades several areas of knowledge have been involved in the advance of coastal management policies as well as on the support of natural values of beaches, but only until recently little attention was paid over the beach safety as a scientific issue.
As a result scientific knowledge on beach dynamics has not been applied as a tool by coastal managers to improve safety policies in an operational sense.
The recreational use of beaches has introduced beach safety as an important concept to be taken into account for their management. The risk in beaches increases as the users do. Beach hazards are the elements of the beach-surf environment that expose the public to danger or harm.
From the hydrodynamical point of view there are three hazards common to all beaches: water depth, breaking waves and surf zone currents. All these hazards are present in the nearshore where waves break and find the shore. Therefore, improving the knowledge on the natural processes taking place in this area is important for safety management purposes.
1
Besides management interests, nearshore zone also embodies a significant research interest. There are variations of morphology at temporal scales of hours to centuries and spatial scales of meters to hundreds of kilometers related to hydrodynamic processes also varying between seconds to years. High-resolution measurement techniques are necessary to solve small-scale coastal processes at spatiotemporal scales in the order of (tens of) meters and hours to days, whereas the collection of long-term data sets is of decisive importance to study large-scale coastal behavior at spatial scales of 1 - 100 km and tem- poral scales of months to decades. Nowadays the advances in measuring technologies allows the acquisition of data required for the small and intermediate scale process and after twenty years of continuous monitoring of specific sites long scale process can be quantitatively studied.
In the year 2000 a group of coastal researchers developed a list of scientific priorities issues in nearshore research (Thornton et al., 2000). Among the major topics identified, they prioritized the study on breaking waves, bottom boundary layers and associated turbulence, breaking-wave induced currents and nearshore sediment transport. A com- plete understandig of the above issues is still not available and it is crucial to develop proper models to provide accurate answers to beach managers questions.
Even the nearshore processes are not completely understood, large advances have been made since the early eighties thanks in part to the development of new numerical and physical models. Several studies can be found in the literature trying to elucidate the nearshore evolution, although only a few of them were done in collaboration with the coastal managers.
In this Thesis a scientific approach to nearshore dynamics is presented in order to test a diagnostic and prognostic tool for the use by coastal managers. This is the last goal of all research, to contribute to a better development of the society and its environment.
Specifically in this Thesis three different approaches are presented, implemented and used to improve beach safety conditions as well as to define strategies for the beach management. This Thesis covers a broad area mostly unexplored which lies between science and management which I believe would at the end benefit both worlds.
1.1 Outlines
This Thesis is divided in 7 chapters and 4 appendixes with the following contents. Chap- ter 2 provides an introduction to the coastal zone and its hydrodynamics and morpho- dynamics. A brief introduction on hydrodynamic theory and beach morphodynamic is presented together with the importance of temporal and spacial scales in beach evolu- tion. Numerical models used in the nearshore are also introduced paying attention to those used in the Thesis. Finally, the video monitoring systems are briefly introduced as a technology for measuring beach variability.
Chapters 3, 4 and 5 present three different studies, based on published papers. These chapters are the core of the Thesis where the different techniques were applied to better understand the beach as a dynamical system. Chapter 3 is an edited version of the paper
”A Nearshore wave and currents forecasting system”published in the Journal of Coastal Research (Alvarez-Ellacuria et al., 2010). The study presents a real time nearshore fore- casting system to better manage beach safety and was carried out on a pilot beach in Mallorca Island, financed by the Balearic Government. The tools developed are nowa- days used by the lifeguards.
Chapter 4 is an edited version of the paper ”Beach Hazard Alert System associated to hydrodynamic conditions in the Balearic Islands” published in Coastal Management (Alvarez-Ellacuria et al., 2009). This work is the spin off of the previous study. In this case, a tool to forecast nearshore conditions was developed for 15 beaches of the Balearic Islands. Results of this work are also being used by the Balearic Government to inform lifeguards and beach users about the beach state. Finally Chapter 5 presents an edited version of the paper ”Shoreline short-term evolution in a low energy beach”, submitted to Marine Geology.
Finally, Chapter 6 summarizes the Thesis presenting the main conclusions of all the studies presented, and Chapter 7 introduces the future work to be done in the next years.
For completeness four appendixes are included in the Thesis since I understood that they were necessary for understanding the work. Appendix A, presents the hydrodynamic equations. In appendix B the radiation stresses for the surf zone currents generation are derived. Appendix C presents the mild slope equation and its parabolic approximation which is the basis for the wave propagation model used in the Thesis. Finally Appendix D, other publications produced during my PhD period are included.
Coastal Zone
2.1 Coastal Zone Definition
The coastal zone is defined throughout this Thesis as the zone between the continental shelf and the land where morphodynamic processes are driven by the marine dynamics . Its width depends on the coastal typology, the coastal shelf and the sea climate. In a sandy coast exposed to strong winds, the coastal zone includes the dunes, where dynamics depend on the capability on transporting sand from the beach. The offshore limit depends on the sea climate and is defined as the point where hydrodynamic does not affect the sediment transport at the bottom. A scheme showing the different parts of the coastal zone is shown in Figure (2.1). This area is divided into offshore region where there is no sediment movement; the nearshore zone, where the energy from waves is dissipated through wave breaking; the foreshore where the sea-land intersection is present and finally the backshore, the dry part of the beach.
In this Thesis, we will focus on the nearshore zone extending seaward from the shore- line to just beyond the region in which the waves break. In this zone part of the energy dissipated in the wave breaking processes is used to generate surf zone currents which are the main driving mechanisms for sediment transport along and across shore directions.
Here is where beach safety problems arise due to rapid depth variations, wave breaking and their associated currents.
5
Figure 2.1: Schematic view of the coastal zone. (Raudkivi, 1998)
2.2 Coastal Zone Processes
Nearshore hydrodynamics and beach morphodynamics can not be studied independently.
Sediment transport is the result of the hydrodynamic forces that at the same time are modified by the bottom which is dynamic in the nearshore. Understanding processes as wave breaking, bar formation and surfzone currents generation are far to be completely achieved. There are many unknowns in coastal dynamics and coastal morphology due to the high degree of non linearity in the process involved.
In the next sections an overview of the hydrodynamic processes governing this area will be described as well as the resulting morphodynamics and the models used to study it.
2.2.1 Nearshore Hydrodynamics
In the ocean there is always some kind of wave that propagates the mechanic energy through the interphase water-atmosphere. Waves are a manifestation of forces acting on the fluid tending to deform it against the action of gravity and surface tension. The energy input can be from different sources: wind, meteorological perturbations, earthquakes, planetary attraction, etc. As a consequence of the variability in the original forces, a large range of waves are present int he ocean (Figure (2.2)).
Four parameters are necessary to describe a wave at certain point: wave length (L), wave height (H), wave period (T) and the local water depth (h). A first classification for waves can be done depending on the relation between depth and wave length. For those waves whereh/L≥1, which corresponds to intermediate and deep waters, the regime is
the so-called Stokes regime. The long wave regime or shallow water waves corresponds to those waves with h/L≪1. Inside the Stokes regime, small-amplitude wavesH/h≪1, are studied with the Airy theory or Stokes first order theory.
Ultragravity
Figure 2.2: Distribution of ocean surface wave energy illustrating the classification of surface waves by wave band, primary disturbance force, and primary restoring force.
To obtain the governing equations for small amplitude waves we have to impose the mass conservation and the conservation of momentum which read respectively
∂ρ
∂t +ρ∇ ·⃗u= 0 −h < z <0, (2.1)
∑
i
F⃗i=m·⃗a −h < z <0, (2.2)
where ρis density of water,∇= (∂
∂x,∂y∂ )
is the nabla operator,⃗uis the fluid velocity, F⃗iare the forces acting on a volume of fluid, m is the mass in a volume of fluid and⃗athe acceleration. For an incompressible and newtonian fluid, Eq.(2.1) and Eq.(2.2) become,
∇ ·⃗u= 0 −h < z <0, (2.3) D⃗u
Dt =−1
ρ∇p+ν∇2⃗u+g⃗k −h < z <0. (2.4) where DtD is the total derivative (e.g. DtD = ∂t∂ +⃗u· ∇), pthe pressure,ν the dynamic viscosity and gthe gravity (see Appendix A for completeness).
Considering an ideal fluid (e.g. and inviscid fluid) Eq. (2.4) can be rewritten as the Euler equations:
Du Dt =−1
ρ
∂p
∂x, Dv
Dt =−1 ρ
∂p
∂y, (2.5)
Dw Dt =−1
ρ
∂p
∂z−g.
Assuming irrotational flow, a velocity potentialΦ= Φ(x, y, z, t) can be defined, such as,
⃗
u=−∇Φ, (2.6)
and therefore, conservation of mass Eq.(2.1) becomes the Laplace equation,
∇2Φ=∂2Φ
∂x2 +∂2Φ
∂y2 +∂2Φ
∂z2 = 0. (2.7)
This is an elliptic equation of the second kind and therefore requires boundary con- ditions in all boundaries. The problem is fully defined in Eq.(2.8)-Eq.(2.11).
∂2Φ
∂x2 +∂2Φ
∂z2 = 0 −h < z <0, (2.8)
−∂Φ
∂z =∂η
∂t +∇η· ∇Φ z= 0, (2.9)
−∂Φ
∂t +1 2
[ (∂Φ
∂x)2+ (∂Φ
∂z)2 ]
+p
ρ+gz=C(t) z= 0, (2.10)
−∂Φ
∂z +∇h· ∇Φ= 0 z=−h. (2.11) where η is the free surface andx=x(x, y) For the free surface, two conditions have to be specified; the free surface kinematic boundary condition, Eq.(2.9) which establishes that the free surface is a material surface. The dynamic free surface boundary condi- tion, Eq.(2.10) impose continuity of the stresses across this boundary. At the bottom the kinematic bottom boundary condition, Eq.(2.11) establish no flux across the bottom.
To solve Eq.(2.8)-(2.11) analytically, the method of separation variables is used where Φ is defined by the product of functions that only depend on one of the independent variables.
Φ(x, z, t) =X(x)Z(z)T(t). (2.12) The solution correspond to a wave propagating in positive (negative) x axis.
Φ(x, z, t) =−Ag ω
coshk(h+z)
coshkh sin (kx−ωt). (2.13)
From this solution, values of free surface, velocities acceleration and pressure are ob- tained:
η=1 g
(∂Φ
∂t )
z=0
=Acos(kx−ωt), (2.14) u=−∂Φ
∂x =Agk ω
coshk(h+z)
coshkh cos (kx−ωt), (2.15) w=−∂Φ
∂z =Agk ω
sinhk(h+z)
coshkh sin (kx−ωt), (2.16) ax=∂u
∂t =Agkcoshk(h+z)
coshkh sin (kx−ωt), (2.17) aw=∂w
∂t =−Agksinhk(h+z)
coshkh cos (kx−ωt), (2.18) p=ρ∂Φ
∂t −ρgz=ρgAcoshk(h+z)
coshkh cos(kx−ωt)−ρgz. (2.19) The energy associated to the wave motion is due to the moving water particles (kinetic) and to the displacement of a mass from a position of equilibrium against a gravitational field (potential). The sum of both energies is the total energy associated to a wave per unit surface area:
E=ρgH2
8 , (2.20)
whereρis water density and H is the wave height.
As waves break, some of this energy is dissipated trough turbulence and by generating surf zone currents. The explanation for the generation of surf zone currents circulation can be studied using the radiation stress concept, which represents the excess of momen- tum flux due to the presence of waves. Using linear wave theory, the Radiation Stress can be defined as,
Sxx=E (
2n−1 2
)
, (2.21)
Syy =E (
n−1 2
)
, (2.22)
where n is:
n= 1 2
(
1 + 2kh sinh 2kh
)
. (2.23)
If the progressive wave is propagating at some angle Θ to the x axis, the values of Sxx andSyy are modified,
Sxx=E [
n(cos2Θ + 1)−1 2 ]
, (2.24)
Syy =E [
n(sin2Θ + 1)−1 2 ]
. (2.25)
In this case there is an additional term representing the flux in the x direction of the y component of momentum, denotedSxy
Sxy= E
2nsin 2ω. (2.26)
The radiation stress will be later used in modeling the nearshore circulation. (See Ap- pendix B for further details).
2.2.2 Nearshore Morphodynamics
Coastal morphodynamics can be defined as the mutual adjustment of topography and fluid dynamics where sediment is in constant movement. This implies that the bottom topography of the nearshore will adjust to accommodate the fluid motions produced by waves, tides and other currents, which in turn will influence the wave and tide processes (Short, 1999). Field experiments and numerical models have shown that nearshore wave transformation, circulation, and bathymetric change involve coupled processes at many spatial and temporal scales. All the scales are totally related on a feedback system represented in Figure (2.3).
Figure 2.3: Coupling of the small-, intermediate-, and large-scale processes, (Thornton et al.,
2000) .
2.2.3 Beach Morphodynamic Classification. Rip Currents
Beach systems are comprehended in terms of three-dimensional morphodynamic models that feature quantitative parameters (wave breaking height, sediment fall velocity, wave period and beach slope) and boundary conditions for definable processes association (e.g.
presence or absence of bars as well as its type). Wright and Short (1984) made a clas- sification of beaches into three categories using data from the analysis of their evolution during 6 years in a number of Australian study sites. This classification relates beach state observations to forcing factors: dissipative, intermediate (from the intermediate- dissipative domain to the intermediate-reflective domain) and reflective modes. This classification is quantified by means of a non-dimensional fall velocity parameter, which is defined as,
Ω = Hb T ωs
(2.27) where Hb is the breaking wave height, T is the wave period and ωs is the sediment fall velocity. A scheme of the morphodynamic states defined is shown Figure (2.4).
Flux generated during wave breaking over the bar feeds the longshore currents that find the offshore way in a rip current. When the shore part of the crescentic bar reach the shoreface, the transition to the transverse bar and rip is made and highly dissipative transverse bars are generated, alternating with deeper zones highly reflective and with strong offshore currents.
These circulatory systems are formed by rip and longshore currents. The rip current per se consist of two converging feeder currents, the rip neck which occupies the rip channel across the bar and the rip head. The largest velocities are encountered in the rip neck while flows expand and decelerate in the rip head. Feeders of rip currents usually are the longshore currents, these are continuous shore-parallel flows within the surf zone.
The generation of these currents is directly associated to the radiation stress defined in the previous section. The excess of energy during wave breaking is transformed in longshore or offshore flows, this flows can reach velocities up to 1 m/s and become a hazard for swimmers who can easy being carried offshore.
Figure 2.4: Morphodynamic states described by Wright and Short (1984)
2.3 Numerical Models for Nearshore Studies
2.3.1 Wave Propagation
The numerical models developed for the study of the nearshore zone usually simplifies some processes adding as well some empirical parameters to avoid the resolution of tur- bulent processes. There are two main groups of numerical propagation models, i) those models where the wave phase is solved and ii) those where the wave phase is averaged.
The first type of models, are based on momentum and mass equations. The simplest deterministic (phase resolving) models are derived from the Euler equation for potential flows (Laplace equation + boundary conditions) under the hypothesis of weak non line- arity and in the limit of shallow water.
The second group is composed by those models where the phase is averaged and the governing equations are the spectral energy balance. These stochastic (phase-averaged) models are derived from deterministic equations by applying a turbulence-like closure hypothesis to the infinite set of coupled equations governing the evolution of the spectral moments.
The election of the type of model depends on what is going to be studied. The phase averaged models are used in large coastal areas (O∼10 km), where waves are generated, but diffraction and wave-wave interaction are the major concern. The resolving phase models need an specific spatial resolution associated to the wave length, and they are normally applied in specific coastal areas (O∼1 km).
Mild slope equation and its parabolic approximation.
Berkhoff (1972) noted that the important properties of linear progressive water waves could be predicted by a weighted vertically integrated model.The underlaying assumption of his theory is that evanescent modes are not important for waves propagating over a slowly varying bathymetry (Liu and Losada, 2002). His equation is known as the mild slope equation and is derived in terms of the surface displacement,η(x, y) =A(x, y)eiωt:
∇ ·(CCg∇η) +k2CCgη= 0, (2.28) where C is the wave celerity andCg the group velocity given respectively by:
C=√
(g/k) tanh(kh), (2.29)
Cg=C{1 + sinh 2kh2kh }
2 (2.30)
whereh(x, y) is the local water depth andgis the acceleration of gravity. The local wave number,k(x, y), is related to the angular frequency of the wavesω, and the water depth hby the linear dispersion relationship,
ω2=gktanh(kh). (2.31)
This equation can be applied to a wave system with multiple wave components as long as the system is linear and these components do not interact with each other. However one problem of the mild slope equation is specifying boundary conditions along the shoreline as the location of the breaker cannot be determined a priori. A solution to this problem is to apply the parabolic approximation. Radder (1979), developed a parabolic model, which has several advantages over the elliptic mild slope equation. First, the boundary condition at the downwave end of the model area is no longer necessary and secondly, very efficient solution techniques can be implemented for the finite difference solver of the model. Kirby and Dalrymple (1983b) included non linear effects in the parabolic equation which are represented by the last term of Eq.(2.32):
2ikccg
∂A
∂x + 2k(k−k0)ccgA+i∂kccg
∂x A+∂ccg
∂y
∂2A
∂y2 −kccgk3 c cg
D|A|2A= 0, (2.32) wherek0 is the reference wave number andD is part of the non linear term, given by
D=cosh(4kh) + 8−2 tanh2(kh)
8 sinh4(kh) , (2.33)
For further details on mild slope equation and its parabolic approximation see Appendix C.
2.3.2 Surf Zone Currents
The surf zone is a highly dynamic area where energy from waves is partially dissipated through turbulence in the boundary layer and transformed in short and long waves, mean sea level variations and currents. The current in the surf zone is composed of motions at many scales, forced by several processes. Schematically, the total current u can be expressed as a superposition of these interrelated components:
u=uw+ut+ua+u0+ui, (2.34) where uw is the steady current driven by breaking waves, ut is the tidal current, ua is the wind-driven current, andu0 andui are the oscillatory flows due to wind waves and infragravity waves (CEM, 2002) . Three types of surf zone currents exist : (i) bed return flows (undertow); (ii) rip currents flow (cell circulation); and (iii) longshore currents. All these current systems are due to cross- and/or longshore components of radiation stress gradients associated with wave breaking mainly.
The surfzone current models developed since the seventies are based in the solution of the momentum and continuity averaged equations. The 2-DH models solve the equations vertically integrated and as result the velocity components in the horizontal axis are obtained, Eq(2.35, 2.36, 2.37).
∂η
∂t +∂(U H)
∂x +∂(V H)
∂y = 0 (2.35)
∂U
∂t +U∂U
∂x +V∂U
∂y +g∂η
∂x + 1 ρH
∂
∂x(Sxx) + 1 ρH
∂
∂y(Sxy) (2.36) +gU
c2H(U2+V2)1/2−ϵ [(∂2U
∂x2 +∂2U
∂y2 )]
= 0,
∂V
∂t +U∂V
∂x +V∂V
∂y +g∂η
∂y + 1 ρH
∂
∂x(Sxy) + 1 ρH
∂
∂y(Syy) (2.37) +gV
c2H(U2+V2)1/2−ϵ [(∂2V
∂x2 +∂2V
∂y2 )]
= 0,
whereη is the free surface, U and V the depth averaged currents velocities in the x and y directions andSxx,Syy andSxy are the components of the radiation stress.
These models are normally developed by finite difference and averaged over a wave period. The final result are the mean flows of the nearshore circulation system. For more detail on these equations see section data and methodology on Chapter 3.
2.4 Nearshore Observations. Videomonitoring Sys- tems
Beach monitoring can be made at a variety of temporal scales ranging from fractions of a second to months or years and spatial scales ranging from a millimeters to tens of kilometers. High-resolution measurement techniques are necessary to resolve small-scale coastal processes at spatiotemporal scales in the order of (tens of) meters and hours to days, whereas the collection of long-term data sets is of decisive importance to study large-scale coastal behavior at spatial scales of 1 - 100 km and temporal scales of months to decades (DeVriend et al., 1993). The new technologies allow improvements for ex- amining nearshore processes by extending the measurements to both larger and smaller space-time scales with increased resolution and accuracy. The challenge is to assimilate the data into improved models to provide accurate predictions of nearshore processes.
Coastal monitoring can be made by two different approaches: 1) in situ observations and
2) remote sensors. The first approach provides a high temporal but low spatial resolu- tion. The remote sensors such as radar, satellites or video monitoring systems offer high spatial and temporal resolution but the information obtained is restricted only to the sea surface. For morphodynamic studies, video monitoring systems are a good alternative due to their low cost together with the high temporal and spatial resolution achieved and their large spatial coverage (Ojeda, 2009).
The first video monitoring system, was developed in 1992 by the Oregon State Uni- versity and it is know as the ARGUS system. Some other similar systems are to date available (IMEDEA and IH Cantabria developed the SIRENA and HOrus systems res- pectively) which are powerful and affordable tools for nearshore morphodynamic moni- toring. These video monitoring systems are composed of an on-field node and a central server which are remotely connected. The on-field node captures the images and broad- cast them to the central server using standard communication protocols. This structure makes the remote station a simple and autonomous system hereby allowing for the repli- cation and simultaneous operation of several remote stations from the same central node.
Four types of statistical products are usually defined in the video monitoring sys- tems: a mean image, a variance image, time stacks and snapshots. Mean images show the patterns of high frequency variability. Time stacks consist in cross-shore transects perpendicular to the coast (in the real world) where all pixel intensity is stored. Wave rays and breaking zones can be determined from these products. These images allow also the estimation of wave celerity and therefore the estimation of bathymetry assuming shallow water theory. The image variance is used to filter some postprocessing products indicating those areas where variability is higher. Finally during the image capture process, an hourly snapshot is recorded. This product can be the basis for beach and coastal zone management activities (e.g. beach uses, beach cleaning or identification of rip currents). From these statistical products several outputs are possible as detecting the shoreline position or the offshore bars location, beach profiles estimation of volume changes, statistics of wave run-up, etc. A broad review of applications can be found in the special issue of Coastal Engineering of June 2007, (54).
2.5 Beach Safety Studies
Beach safety is mainly conducted by lifeguards associations, which mark safety levels for beach usage. Statistics in US and Australia show that rip currents seem to be the major cause of rescues and drownings (MacMahan et al., 2006). These data are still not available in Spanish Mediterranean coasts, although it has already been reported the existence of hydrodynamic hazards defined by Short (1999).
Previous studies to redress beach safety problems tried to integrate or to establish appropriate warning system using morphodynamical models linked with some hydrody- namical information. Short and Hogan (1994), related wave heights with morphody- namical beach states to create a safety rating for 721 beaches of the Australian coast.
These authors, combined the non-dimensional parameter Ω, with sea conditions. For the definition of the safety rating they used the water depth, the size of breaking waves, the prevalence and intensity of rips and the existence of longshore currents. Benedet et al. (2004), created a storm hazard-risk category for some Florida beaches relating beach morphology, presence or absence of dunes and presence or absence of some coastal development. Scott et al. (2007), related beach rescue statistics to the nearshore mor- phology to identify specific hazards in the Southwest of England. They derived a risk coefficient using the average number of people estimated to be in the water per hour and the number of individuals assisted/rescued per hour at a specific location. Even if these studies give a description of different hazard level depending on physical beach charac- teristics and hydrodynamics, risk levels based on nearshore forecast of wave conditions in an operational way is still under development.
2.6 Processes and Models Studied in the Thesis
This Thesis focuses in the intermediate spatial and temporal scales, where the driving dynamics are from hours to weeks and the spatial scales from 1 m to 100 m. Within this range fall the wind waves, nearshore currents, bar movement and beach short term erosion or accretion among other processes. Even major waves presented in Figure (2.2) have a role on the mid term beach evolution, major changes are induced by gravity waves with periods between 1-30 s and normally generated by the wind.
From a morphodynamic point of view the work focuses on intermediate morphody- namic beach stages where beach circulatory systems appear, taking special care on the formation of rip channels and the associated rip currents. These rip currents will oc- cur mainly when the beach is on the intermediate states ”rhythmic bar and beach” and
”transverse bar and beach” as defined by Wright and Short (1984).
For the obtention of hydrodynamical data, punctual deployment of instruments were used in the areas of study. Besides in situ data two numerical models for propagating waves from offshore to the nearshore and breaking waves currents generations were im- plemented. The models here used are base on the hydrodynamic linear theory presented before.
A Nearshore Wave and
Currents Forecasting System
This chapter is based on the paper. ’A Nearshore Wave and Currents Forecasting System’, Journal of Coastal Research, 26(3), 503-509.
Authors:Amaya Alvarez-Ellacuria, Alejandro Orfila, Maitane Olabarrieta, Ra´ul Medina, Guillermo Vizoso and Joaqu´ın Tintor´e.
3.1 Abstract
An operational forecasting system for nearshore waves and wave-induced currents is pre- sented. The forecasting system (FS) has been built to provide real time information of nearshore conditions for beach safety purposes. The system has been built in a modular way with four different autonomous submodels providing, twice a day, a 36 hour wave and current forecast, with a temporal resolution of six hours. Making use of a mild slope parabolic model, hourly deep water wave spectra are propagated to the shore. The resulting radiation stresses are introduced in a depth integrated Navier Stokes model in order to derive the resulting current fields. The system has been implemented in a beach, located in the north-eastern part of Mallorca Island (Western Mediterranean), charac- terized by its high touristic pressure during summer season. The FS has been running for three years being a valuable tool for local authorities for beach safety management.
19
3.2 Introduction
The coastal zone is one of the most complex and variable marine systems since its dy- namics are subjected to the effects derived from a complex geometry where bathymetry plays a crucial role in wave propagation. Moreover coasts are under the forcing of waves, wind currents, tides, etc., at a wide range of temporal scales in all their boundaries (i.e., surface, bottom, lateral, and internally) which make them highly variable environments.
Despite the socio-economical relevance of coastal areas, modeling, observation and con- tinuous monitoring of coastal variability is to date scarce due to the intrinsic complexity of these systems. Besides the morphological importance of coastal areas, they are a ma- jor recreational resource around the world and human activities have been constantly growing in the last three decades. Coastal management has increasingly relied on the scientific results obtained from different research fields that have been transferred to new engineering methodologies and applications to environmental systems in search of new, more integrated and sustainable solutions to coastal problems. Beach erosion and coastal evolution are in a global change context, top scientific issues but also an increasing de- mand for accurate information in an operational sense of short term variability is also required by governments and end users. Continuous observation of coastal variability is expensive and sometimes impossible to obtain. Comprehensive information in coastal areas is nowadays required in order to establish efficient coastal zone monitoring as well as to develop management policies to effectively study these marine systems (Smit et al., 2007). The scarcity, and in most cases the lack, of information becomes a problem when scientists need to asses the current state (diagnostic) of specific coastal systems as well as to build predictive models (prognostic) of their evolution. Evolution of physical systems are expressed in terms of differential equations which need data to establish the initial conditions of the system. Moreover, due to their non linear nature, continuous update of data are required to correct the deviations of predictions from the real state.
Experiments to provide a complete overview of nearshore dynamics have been shown to be the most useful tool to improve the understanding of short term variations of coastal dynamics (Herbers et al., 2003; Reiners et al., 2004). Unfortunately maintaining such experiments which is required for any operational activity is logistically and economi- cally difficult. Only recently, with the development of new observing technologies has been possible to provide observations in a continuous way of some aspects of nearshore variability on all relevant time scales (Davidson et al., 2007). Besides, the increasing ca- pacity of computers have been made possible to solve complex numerical models in short time and operational systems for deep water wave conditions and currents are presently available (Hodur, 1997); (Bidlot et al., 2002).
This study presents the applicability of numerical models to an operational system in the nearshore region. An operational forecasting system (FS) for nearshore waves and wave-induced currents is developed to improve the beach management. The FS has been implemented in an intermediate barred beach in the western Mediterranean Sea. Waves and currents are predicted twice a day with a 36 hours horizon.
3.3 Data and Methodology
The Forecasting System (FS) has been built in a modular way with 4 independent sub- models, Figure (3.1). Hourly wind predictions are provided daily by the National Mete- orology Institute (AEMET) using the HIRLAM model over a mesh of 0.2 degrees for the 24 hours horizon and over a mesh of 0.5 degrees after 1 day. Wind fields at 10 m height are used as the meteorological input for the WAM cycle 4 wave model on the oceanic scale. These predictions are carried out operationally over the Mediterranean Sea by the Spanish Harbour Authority providing wave height and wave direction for the next 3 days.
2DH NAVIER-STOKES MODEL
SURFZONE CURRENTS GENERATION
HIRLAM MODEL WIND PREDICTION
WAM MODEL
DEEP-INTERMEDIATE WATERS WAVE PREDICTION
MILD SLOPE MODEL
ONSHORE WAVE PROPAGATION
Input: wind fields 10m above the sea
Input: Radiation Input: wave bidimensional spectra
2DH NAVIER-STOKES MODEL
SURFZONE CURRENTS GENERATION
HIRLAM MODEL WIND PREDICTION
WAM MODEL
DEEP-INTERMEDIATE WATERS WAVE PREDICTION
MILD SLOPE MODEL
ONSHORE WAVE PROPAGATION
Input: wind fields 10m above the sea
Input: Radiation Input: wave bidimensional spectra
HIRLAM MODEL WIND PREDICTION
WAM MODEL
DEEP-INTERMEDIATE WATERS WAVE PREDICTION
MILD SLOPE MODEL
ONSHORE WAVE PROPAGATION
Input: wind fields 10m above the sea
Input: Radiation Stress Input: wave bidimensional spectra
Figure 3.1: Scheme of the different models used in the forecasting system.
The deep water wave conditions at two nodes near Cala Millor, the studied beach, (39◦40′N, 3◦30′E; 39◦35′N, 3◦30′E) (see diamonds in Figure (3.5)), are propagated to shallow waters using a mild slope model. To proper simulate the different incoming di- rections two different meshes were implemented in the area with a resolution of 15m. The wave propagation model solves the mild slope equation with the parabolic approximation (Kirby and Dalrymple, 1983a).
3.3.1 Deep Waters Forecast Data
In 1988, the first implementation of the 3rd generation model WAM (Wave Analysis Model) was published. In Spain this model is called WAME and is operated by the Spanish Harbours Authority in collaboration with the AEMET which provides the me- teorological data necessary for generating waves from the HIRLAM (HIgh Resolution Limited Area Model).
HIRLAM is operational in the AEMET since 1995. Four runs are made daily (00, 06, 12, 18 UTC) with a forecast horizon of 72 hours. The model runs over a mesh of 0.2 degrees for the 24 hours horizon and over a mesh of 0.5 degrees after 1 day. Wind fields at 10m height are used as the meteorological input for the WAM.
As mentioned before the WAM is based on the energy transport equation. The mesh has a resolution of 0.125 degrees in the Mediterranean sea. The information generated is the energy directional spectra, from which the significant wave height Hs, the peak periodTp, mean direction and other wave parameters can be obtained.
3.3.2 Waves Propagation
The spectra obtained from the WAM model is discretized into 13 frequencies (0.074− 0.2323 Hz) and 16 directions (352−142◦) and then introduced as input for the propagation of waves to the beach front. The model used in this part of the processes, OLUCA- SP (OLUCA-SP, 2003);(Gonz´alez et al., 2007), solves the mild slope equation with the parabolic approximation (Kirby and Dalrymple, 1983a), e.g.,
D2Φ
D2t + (∇ ·U)DΦ
Dt − ∇(ccg∇Φ) + (ω2−k2ccg)Φ+ 2ω[k∇Φ2 (3.1)
− k2
2ωcosh2kh]Φ+ω2k2DA|A|2Φ+iωγ 2Φ= 0, whereD is a part of the non linear term:
D=cosh(4kh) + 8−2 tanh2(kh)
8 sinh4(kh) , (3.2)
In Eq.(3.1) Φis the velocity potential at the free surface and Φ2 is the velocity po- tential for a long wave. The non linear termω2k2DA|A|2Φrepresents the dispersion due to amplitude. The dissipation term iωγ2Φ, is used to model the energy dissipation due to bottom friction.
The non linear interactions (wave-current interaction and wave dispersion due to wave height) was incorporated empirically by modifying the dispersion equation. The model includes refraction-diffraction with wave-current interaction and predicts the energy lose due to wave breaking. Within the parabolic approximation one has not to define all the boundary conditions, but one initial condition at the offshore boundary and lateral con- ditions. In addition this approximation allows using implicit resolution schemes useful to reduce the time computation. It must be taken into account that this approximation has a limitation in the propagation wave angle that must be under ±55 from the main axis (x) and that the waves reflected effect is neglected. The angle limitation implies that to model all incoming waves arriving to the beach, two pair of meshes with different orientation have to be constructed, see Figure (3.2). For each orientation two meshes are used: an offshore mesh with a resolution of 150x150m and a nested mesh arriving to the beach with a resolution of 15x15m.
The principal outputs of this model are theHsat each grid point and the radiation stress, which is used as input for the current model.
Figure 3.2: Grids used by the system
3.3.3 Wave-breaking Current Generation
The COPLA (COPLA-SP, 2003), solves the flux equations in the surf zone. The input data are the output data from the wave model. This model is based in the solution of the movement and continuity averaged equations. To save computational time, vertically averaged equations are implemented. Finite differences with an implicit schemes are used to solve this equations.
The hypothesis applied to this model are:
• The fluid is homogeneous, incompressible and its density is constant.
• The bottom variation is small so the current velocities (u,v) are independent of the bottom (2DH).
• The associated movements to the beach currents are permanent and therefore the equations can be averaged in time.
• Viscosity effects are negligible except in the boundaries, so the flux is irrotational.
• The turbulent fluctuations generated by the waves are neglected.
• Coriolis is neglected since the domains are small.
• Currents are weak enough to neglect is iteration with the waves train.
Integrating the Navier Stokes equations in depth and averaging over a wave period in a coordinate system placed on the mean sea level (x= crosshore direction, y= longshore direction, z= vertical direction), using the hypothesis described before, we obtain the continuity and movement equations:
∂η
∂t +∂(U H)
∂x +∂(V H)
∂y = 0 (3.3)
∂U
∂t +U∂U
∂x +V∂U
∂y +g∂η
∂x+ 1 ρH
∂
∂x(Sxx) + 1 ρH
∂
∂y(Sxy) (3.4) +gU
c2H(U2+V2)1/2−ϵ [(∂2U
∂x2 +∂2U
∂y2 )]
= 0,
∂V
∂t +U∂V
∂x +V∂V
∂y +g∂η
∂y + 1 ρH
∂
∂x(Sxy) + 1 ρH
∂
∂y(Syy) (3.5) +gV
c2H(U2+V2)1/2−ϵ [(∂2V
∂x2 +∂2V
∂y2 )]
= 0,
where
H =η+h, (3.6)
Sxx= 1 T
∫ t+T t
∫ η
−h
(ρu2+p)dzdt− 1 T
∫ t+T t
∫ 0
−h
p0dzdt, (3.7) Syy= 1
T
∫ t+T t
∫ η
−h
(ρv2+p)dzdt− 1 T
∫ t+T t
∫ 0
−h
p0dzdt, (3.8) Sxy= 1
T
∫ t+T t
∫ η
−h
(ρuv)dzdt, (3.9)
V = 1 T
∫ t+T t
∫ η
−h
v(x, y, z, t)dzdt, (3.10) U = 1
T
∫ t+T t
∫ η
−h
u(x, y, z, t)dzdt, (3.11) η= 1
T
∫ t+T t
η(x, y, t)dt. (3.12) The radiation stress due to irregular waves depend on the radiation stress generated by each energy component. These components are propagated forming an angle θ respect to x axis, the linear addition of this components in each point of the domain give the next expressions:
Sxx(x, y) = 1 2ρg
Nf
∑
j=1 Nθ
∑
l=1
|Ajl|2 [
nj(1 +cos2θjl)−1 2 ]
, (3.13)
Syy(x, y) =1 2ρg
Nf
∑
j=1 Nθ
∑
l=1
|Ajl|2 [
nj(1 +sin2θjl)−1 2f
]
, (3.14)
Sxy(x, y) =1 4ρg
Nf
∑
j=1 Nθ
∑
l=1
|Ajl|2njsin(2θjl), (3.15)
nj= 1 2
(
1 + 2kjh sinh(2kkjh)
)
. (3.16)
The dependent variables of the model are η the free surface, U and V the averaged currents velocities in a time period in directions x and y.
In Eq.(3.3)-Eq.(3.16), the different variables are:
• Ajl(x,y): amplitude for the frequency and direction components (j,l).
• h: distance from bottom to mean sea level
• nj: relation between group velocityCgj and velocity o j componentCj.
• t: time ;T: wave period.
• Sxx: radiation stress over the x plane all over the x axis.
• Sxy: radiation stress over the y plane and all long the x axis,Sxy: Syx.
• Syy: radiation stress over the y plane all over the y axis.
• η(x,y,t): free surface elevation from mean sea level.
• u: instant velocity in x direction.
• v: instant velocity in y direction.
• H: wave height.
• kj: wave number associated to the frequency component j.
• θjl: angle of the wave number vector with the x axis for a frequency component j an direction component l.
• c: Chezy coefficient.
• ϵ: Eddy viscosity coefficient.
• p: total pressure.
• p0: static pressure starting at the reference mid level.
• g: gravity acceleration.
• ρ: flux density.
For the application of the model in a beach, the equations presented are solved by using finite differences on a rectangular mesh. Initial conditions areηandU,V att= 0.
As they are unknown they are assumed to be zero and the values of radiation stress obtained by the wave propagation are used inside the equations to obtain the velocities.
The boundary conditions are shown in Figure (3.3).
Offshore boundary condition (h=0,absorbent)
Lateral boundary open(U=0,V=0) absorbent
Lateral boundary closed(U=0,V=0) reflectant
Reflectant boundary condition at the shoreline (U=0, V=0 )
Figure 3.3: Boundary conditions used in COPLA
3.3.4 Bathymetries
This is a crucial point since bathymetry has to be done at least after every storm event to get the exact position of the sandbars and rip channels. Nowadays this is performed with a ship mounted Biosonics DE-4000 echo sounder equipped with a 200 KHz transducer.
The draught of the boat allows sampling to depths of about 0.5 m. Inshore-offshore and along coast echo sounding transects are sampled perpendicular to the bathymetric gradient with a separation of 50 m between transects. Acoustic pulse rate is set to 25s−1 and the sampling speed was set to 3 knots, which allows for a horizontal resolution of 1 m. This procedure provides accurate bottom mapping but is very time consuming. The importance of updating the bathymetry for the correct predictions of rip existence and location can be seen in Figure (3.4) where two bathymetries of 2007 can be compared.
Figure 3.4: Bathymetries of June (left panel) and September (right panel) 2007.
3.4 Study Site
The FS has been implemented in Cala Millor, located in the northeast coast of Mallorca Island, see Figure (3.5). The beach is an open bay with an area around 14km2, extending to depths up to 20-25 meters. Near the coast to 8m depth, the bathymetry presents a regular slope indented with sand bars near the shore that migrate from offshore to on- shore between mild wave conditions periods (Figure (3.5c)). Significatively changes in bathymetry are produced by the bars movement, between 0.5 and 4.5 m depth. The sediment in shallower waters (0-2 m) present rocky substrate, mainly in the center an south part of the beach. Cala Millor can be classified morphodynamically as an interme- diate beach, being the predominant states rhythmic bar and beach, and transverse bar and rip. These bars define the surf zone currents and the the response to incident waves.
At depths from 8 m to 35 m the seabed is covered by thePosidonia oceanicameadow, an endemic seagrass species of the Mediterranean Sea (Infantes et al., 2009). Cala Millor is a tourist resort with a population of 6,072 permanent inhabitants. However, during the summer this number can increase up to 17,046 inhabitants. The tourist occupancy takes values between 74.6% and 91.3% of the hotel bed places during summer season.
This means that the Cala Millor real population achieves the value of 20,263 inhabitants;
three times more people than the rest of the year. Mean daily number of beach users
Figure 3.5: Geographic location of the pilot area. Symbols in panel (b) correspond to deep water wave buoy (circle); hipocas node (triangle) and the deep water WAM pre- diction points (diamonds). Point A in panel (c) indicates location of the acoustic currentmeter moored during the 02/28/2006−03/08/2006 experiment.
fluctuates from 6,400 to 6,800 users, at the end of the tourist season these rates mean that at least 500,000 individuals have engaged in recreational activities at Cala Millor beach. The tidal regime is microtidal with a spring range of less than 0.25 m (G´omez- Pujol et al., 2007). Attending to the criteria of Wright and Short (1984), Cala Millor is an intermediate barred beach.
3.4.1 Wave Climate
Wave climate characterization has been performed using HIPOCAS data, consisting on hourly wave data from a 44 years wave hindcast (Soares et al., 2002). These hindcast models have become a powerful tool not only for engineering or prediction scales but for climate purposes involving large temporal periods (Ca˜nellas et al., 2007).
For the study area, the virtual wave gage is located 10 km offshore at a depth of 50m (triangle in Figure ( 3.5b)) showing significant wave heights (Hs) above 1 meter during 50% of time from the long term probability distribution (log normal) see Figure (3.6) and typical peak periods (T p) between 3 and 6 seconds, the methodology used for the description of wave regimes has been the same as the one used by (Ca˜nellas et al., 2007)
