Tesis Amaya Alvarez Ellacuria by Alejandro Orfila

Nearshore Hydrodynamics and Shoreline 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 fácil y sobretodo interesante el camino hasta aqu´ı. Agradezco a Joaqu´ın Tintoré 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 determinación y empe˜ no en que realizara el doctorado y a Jano le quiero dar las gracias por todas esas horas de dedicaci´ on, por estar siempre disponible, por leer con tanto interés todo lo que escrib´ıa y por ense˜ nareme tanto sobre cual debe ser el proceso de la investigación (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 dispuestos 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 solución 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 también fuera. Habéis hecho todo mucho más 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: habéis compartido comidas en el IMEDEA, o una, dos o tres galletas, incluso rara vez chocolate. Los que habéis jugado a volley playa o a f´ utbol, los que habéis estado en mi azotea o yo en la vuestra. A los que os sintáis identificados en más de una situación, 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. Inés, gracias por acompa˜ narme tan bien, es una suerte saber que siempre estás ah´ı. Tomeu, ”he pasado gusto de” fumar cigarritos al sol, mirar la pantalla ”a la sombra”, salir los jueves por la noche, tomar cafés por la ma˜ nana, y todo, siempre contigo. Por fin ha llegado... ”nuestra primera tesis”. Por u ´ltimo 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 animarme siempre. A mis padres por llevarme a la playa tantos a˜ nos, porque siempre están ah´ı y porque lo u ńico 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 ´ sólo con mirarme a la cara y siempre encuentra una manera de solucionarlo. A Alvaro, por hacer que sonr´ıa todos los d´ıas, nunca imaginé que se pudiera tener tanta suerte en la vida.

ii palabra

iii

Resumen El estudio de la hidrodinámica y morfodinámica de playas es de gran interés para la gestión costera. El uso de las playas como espacios recreativos ha obligado a las autoridades a dise˜ nar protocolos de seguridad y a implantar vigilancia en las playas. Estos lugares además de una gran importancia socio-económica tienen un alto valor ecológico y son el primer elemento de protección de la costa frente a los temporales y subidas del nivel del mar. Las pol´ıticas de protección de estas áreas, 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 u ´ltimos a˜ nos en los campos de la hidrodinámica y morfodinámica litoral. En esta Tesis se presentan tres estudios realizados con el objetivo de desarrollar herramientas u ´tiles para la gestión costera, principalmente en el aspecto de la seguridad en playas, que ofrezcan información u ´til tanto a gestores como socorristas y usuarios. Una de las principales causas de rescates en playas es la aparición de corrientes generadas por la rotura de las olas. Estas corrientes son conocidas como rip currents o corrientes de retorno. La predicción a corto plazo de estas corrientes es, desde hace 20 a˜ nos, uno de los principales focos de estudio de la hidrodinámica de playas. El interés del Govern Balear por mejorar la seguridad en las playas del archipiélago hizo posible el desarrollo de un sistema de predicción 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 generación de corrientes, que utiliza el oleaje propagado como dato inicial, permite la obtención de las corrientes generadas en la playa a causa de la rotura del oleaje. El sistema ha estado operativo durante 3 a˜ nos, ofreciendo información de altura de ola, dirección 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 predicción de oleaje y corrientes en Cala Millor fueron la base de la ampliación del estudio a otras playas del litoral Balear. La implementaci´ on del sistema en diferentes playas hizo necesarias algunas modificaciones para su correcto funcionamiento. Se desarrolló un nuevo sistema de predicción 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 calibración del sistema de predicción.

iv La falta de datos en las zonas de estudio, tanto batimetr´ıcos como de oleaje y corrientes, es uno de los principales problemas para el desarrollo de herramientas operacionales con alto grado de fiabilidad. El uso de sistemas de observación remotos permite paliar esta deficiencia y ofrece la posibilidad de obtener largas series de datos, con una alta resolución espacial y temporal. El análisis de imágenes 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 imágenes digitales diarias que han sido procesadas para extraer la posición de la l´ınea de costa. La variabilidad de está l´ınea puede asociarse con los movimientos de crecimiento y erosión de una playa. El avance en este tipo de estudios mediante el uso de imágenes digitales permitirá en un futuro la obtención de datos fiables de batimetr´ıa y oleaje. Durante el desarrollo de esta Tesis, los conocimientos adquiridos tambén han servido para la realización 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’Emergències del Govern de les Illes Balears.

Contents Agradecimientos

Resumen

iii

List of Figures

vii

List of Tables

1 Motivation 1.1 Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2 Coastal Zone 2.1 Coastal Zone DeďŹ nition . . . . . . . . . . . . . . . . . . . . 2.2 Coastal Zone Processes . . . . . . . . . . . . . . . . . . . . . 2.2.1 Nearshore Hydrodynamics . . . . . . . . . . . . . . . 2.2.2 Nearshore Morphodynamics . . . . . . . . . . . . . . 2.2.3 Beach Morphodynamic ClassiďŹ cation. Rip Currents 2.3 Numerical Models for Nearshore Studies . . . . . . . . . . . 2.3.1 Wave Propagation . . . . . . . . . . . . . . . . . . . 2.3.2 Surf Zone Currents . . . . . . . . . . . . . . . . . . . 2.4 Nearshore Observations. Videomonitoring Systems . . . . . 2.5 Beach Safety Studies . . . . . . . . . . . . . . . . . . . . . . 2.6 Processes and Models Studied in the Thesis . . . . . . . . . 3 A Nearshore Wave and Currents Forecasting System 3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Data and Methodology . . . . . . . . . . . . . . . . . . . 3.3.1 Deep Waters Forecast Data . . . . . . . . . . . . 3.3.2 Waves Propagation . . . . . . . . . . . . . . . . . 3.3.3 Wave-breaking Current Generation . . . . . . . . v

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5 5 6 6 10 11 13 13 14 15 16 17

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19 19 20 21 22 22 24

CONTENTS

3.4 3.5 3.6 3.7

3.3.4 Bathymetries Study Site . . . . . . 3.4.1 Wave Climate Results . . . . . . . . 3.5.1 Validation . . Discusion . . . . . . Conclusions . . . . .

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27 28 29 31 32 36 37

4 An Alert System for Beach Hazard Management in the Balearic Islands 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Study Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Data and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Nearshore Wave Climate . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Forecast Wave Data . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Hazard Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Lifeguards Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 40 42 46 47 48 48 50 51 51 53 55

5 Short-Term Shoreline Evolution 5.1 Abstract . . . . . . . . . . . . . 5.2 Introduction . . . . . . . . . . . 5.3 Data and Methodology . . . . . 5.3.1 Coastline Extraction . . 5.3.2 ANN Development . . . 5.3.3 Shoreline Analysis . . . 5.3.4 Wave Data . . . . . . . 5.4 Results and Discussion . . . . . 5.5 Conclusions . . . . . . . . . . .

57 57 58 60 60 62 64 64 65 71

in a Low-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Beach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 General Conclusions

7 Future Work

Bibliography

A Navier-Stokes Equations 83 A.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

CONTENTS

vii

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

List of Figures 2.1 2.2 2.3 2.4

Coastal zone . . . . . Waves classiďŹ cation . . Morphodynamic scales Morphodynamic states

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6 7 10 12

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

Models scheme . . . . . . . . . . . . Grids picture . . . . . . . . . . . . . Boundary conditions . . . . . . . . . Bathymetries 2007 . . . . . . . . . . Geographic location of the pilot area Long term distribution . . . . . . . . Annual wave maximums distribution Wave roses . . . . . . . . . . . . . . Hs and Tm at deep waters . . . . . . Buoy-prediction Hs correlation . . . Buoy-prediction Tm correlation . . . SigniďŹ cant wave height serie . . . . . Aerial photography . . . . . . . . . .

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21 23 27 28 29 30 30 31 32 33 33 34 35

4.1 4.2 4.3 4.4 4.5

Study sites . . . . . . . . . . Wave roses . . . . . . . . . . Hazard alert system scheme . Deep and shallow wave height Surfzone circulation at IB5 .

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42 44 46 52 54

5.1 5.2 5.3 5.4 5.5

Timex images and UTM image . . . . . . . . . . . . . . . . . . . . . . . Cala Millor location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ANN scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error between ANN detection and topographic data in Barcelona . . . . Spatial modes of the temporal variance from the EOF analysis,EOF1,2,3

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61 61 62 63 66

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LIST OF FIGURES 5.6 5.7 5.8 5.9

Temporal amplitude of the EOF1,2,3 . . . . . . . . . . . . . . . . Temporal amplitudes of 1s t and 2n d mode . . . . . . . . . . . . . Surfzone currents generated by Navier-Stokes model . . . . . . . Mean diďŹ&#x20AC;erences inter shorelines, related standard deviation and SW 4 ated Hs,50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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66 68 69 71

List of Tables 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Beaches characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grids charactheristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum and maximum incident directions for the hazard level definition in each beach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hazard level definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of lifeguards and forecasted hazard . . . . . . . . . . . . . . . . . Differences between propagated and interpolated Hs . . . . . . . . . . . . Comparison between flags posted and hazard level forecast . . . . . . . . Calibration of hazard level . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

Correlation between ﬁrst and second amplitudes (a1, a2) and wave contidions 67

45 47 49 49 50 51 53 54

Chapter 1

Motivation Beaches are highly dynamical systems conforming preferred mechanisms of coastal defence 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. Motivation

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 temporal 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 complete 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. Motivation

1.1

Outlines

This Thesis is divided in 7 chapters and 4 appendixes with the following contents. Chapter 2 provides an introduction to the coastal zone and its hydrodynamics and morphodynamics. A brief introduction on hydrodynamic theory and beach morphodynamic is presented together with the importance of temporal and spacial scales in beach evolution. 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 forecasting 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 nowadays 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.

Chapter 2

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 shoreline 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.

2. Coastal Zone

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 where h/L ≥ 1, which corresponds to intermediate and deep waters, the regime is

2. Coastal Zone

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 waves H/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 + ρ∇ · ⃗u = 0 ∂t∑ F⃗i = m · ⃗a

−h < z < 0,

(2.1)

−h < z < 0,

(2.2)

( ) ∂ ∂ where ρ is density of water, ∇ = ∂x , ∂y is the nabla operator, ⃗u is the fluid velocity, F⃗i are the forces acting on a volume of fluid, m is the mass in a volume of fluid and ⃗a the acceleration. For an incompressible and newtonian fluid, Eq.(2.1) and Eq.(2.2) become, ∇ · ⃗u = 0 1 D⃗u = − ∇p + ν∇2 ⃗u + g⃗k Dt ρ

−h < z < 0,

(2.3)

−h < z < 0.

(2.4)

D ∂ D is the total derivative (e.g. Dt = ∂t + ⃗u · ∇), p the pressure, ν the dynamic where Dt viscosity and g the gravity (see Appendix A for completeness).

2. Coastal Zone

Considering an ideal ﬂuid (e.g. and inviscid ﬂuid) Eq. (2.4) can be rewritten as the Euler equations: Du 1 ∂p =− , Dt ρ ∂x Dv 1 ∂p =− , Dt ρ ∂y Dw 1 ∂p =− − g. Dt ρ ∂z

(2.5)

Assuming irrotational ﬂow, a velocity potential Φ = 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, ∇2 Φ =

∂2Φ ∂2Φ ∂2Φ + + = 0. ∂x2 ∂y 2 ∂z 2

(2.7)

This is an elliptic equation of the second kind and therefore requires boundary conditions in all boundaries. The problem is fully deﬁned in Eq.(2.8)-Eq.(2.11). ∂2Φ ∂2Φ + =0 −h < z < 0, ∂x2 ∂z 2 ∂Φ ∂η − = + ∇η · ∇Φ z = 0, ∂z] ∂t [ ∂Φ 1 ∂Φ 2 ∂Φ 2 p − + ( ) +( ) + + gz = C(t) z = 0, ∂t 2 ∂x ∂z ρ ∂Φ − + ∇h · ∇Φ = 0 z = −h. ∂z

(2.8) (2.9) (2.10) (2.11)

where η is the free surface and x = 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 condition, 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 cosh k(h + z) sin (kx − ωt). ω cosh kh

(2.13)

2. Coastal Zone

From this solution, values of free surface, velocities acceleration and pressure are obtained: ( ) 1 ∂Φ η= = A cos(kx − ωt), (2.14) g ∂t z=0 ∂Φ Agk cosh k(h + z) u=− = cos (kx − ωt), (2.15) ∂x ω cosh kh ∂Φ Agk sinh k(h + z) w=− = sin (kx − ωt), (2.16) ∂z ω cosh kh ∂u cosh k(h + z) ax = = Agk sin (kx − ωt), (2.17) ∂t cosh kh ∂w sinh k(h + z) aw = = −Agk cos (kx − ωt), (2.18) ∂t cosh kh ∂Φ cosh k(h + z) p=ρ − ρgz = ρgA cos(kx − ωt) − ρgz. (2.19) ∂t cosh kh 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: H2 E = ρg , (2.20) 8 where ρ is 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 momentum flux due to the presence of waves. Using linear wave theory, the Radiation Stress can be defined as, ) ( 1 , Sxx = E 2n − (2.21) 2 ( ) 1 Syy = E n − , 2 where n is: n=

1 2

( 1+

2kh sinh 2kh

(2.22)

) .

(2.23)

If the progressive wave is propagating at some angle Θ to the x axis, the values of Sxx and Syy are modiﬁed, ] [ 1 2 , Sxx = E n(cos Θ + 1) − (2.24) 2 [ ] 1 Syy = E n(sin2 Θ + 1) − . (2.25) 2

2. Coastal Zone

In this case there is an additional term representing the ﬂux in the x direction of the y component of momentum, denoted Sxy Sxy =

E n sin 2ω. 2

(2.26)

The radiation stress will be later used in modeling the nearshore circulation. (See Appendix 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. Coastal Zone

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 classification 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 intermediatedissipative 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.

2. Coastal Zone

Figure 2.4: Morphodynamic states described by Wright and Short (1984)

2. Coastal Zone

2.3 2.3.1

Numerical Models for Nearshore Studies 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 turbulent 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 linearity 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) + k 2 CCg η = 0,

(2.28)

where C is the wave celerity and Cg the group velocity given respectively by: C=

√ (g/k) tanh(kh),

Cg = C

{1 +

2kh sinh 2kh }

(2.29) (2.30)

2. Coastal Zone

where h(x, y) is the local water depth and g is the acceleration of gravity. The local wave number, k(x, y), is related to the angular frequency of the waves ω, and the water depth h by the linear dispersion relationship, ω 2 = gk tanh(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 ∂kccg ∂ccg ∂ 2 A c + 2k(k − k0 )ccg A + i A+ − kccg k 3 D|A|2 A = 0, 2 ∂x ∂x ∂y ∂y cg

(2.32)

where k0 is the reference wave number and D 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 + u a + u 0 + u i ,

(2.34)

where uw is the steady current driven by breaking waves, ut is the tidal current, ua is the wind-driven current, and u0 and ui 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.

2. Coastal Zone

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 ∂(U H) ∂(V H) + + =0 ∂t ∂x ∂y ∂U ∂U ∂η 1 ∂ 1 ∂ ∂U +U +V +g + (Sxx ) + (Sxy ) ∂t ∂x ∂y ∂x ρH ∂x ρH ∂y [( 2 )] gU ∂ U ∂2U 2 2 1/2 + 2 (U + V ) − ϵ + = 0, c H ∂x2 ∂y 2 ∂V ∂V ∂V ∂η 1 ∂ 1 ∂ +U +V +g + (Sxy ) + (Syy ) ∂t ∂x ∂y ∂y ρH ∂x ρH ∂y [( 2 )] gV ∂ V ∂2V + 2 (U 2 + V 2 )1/2 − ϵ + = 0, c H ∂x2 ∂y 2

(2.35)

(2.36)

(2.37)

where η is the free surface, U and V the depth averaged currents velocities in the x and y directions and Sxx , Syy and Sxy 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 examining 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. Coastal Zone

2) remote sensors. The first approach provides a high temporal but low spatial resolution. 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 University 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 respectively) which are powerful and affordable tools for nearshore morphodynamic monitoring. 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 broadcast them to the central server using standard communication protocols. This structure makes the remote station a simple and autonomous system hereby allowing for the replication and simultaneous operation of several remote stations from the same central node. Four types of statistical products are usually defined in the video monitoring systems: 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).

2. Coastal Zone

Previous studies to redress beach safety problems tried to integrate or to establish appropriate warning system using morphodynamical models linked with some hydrodynamical information. Short and Hogan (1994), related wave heights with morphodynamical 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 morphology 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 characteristics 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 morphodynamic 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 occur 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 implemented. The models here used are base on the hydrodynamic linear theory presented before.

Chapter 3

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 presented. 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), characterized 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.

3.2

3. A Nearshore Wave and Currents Forecasting System

Introduction

The coastal zone is one of the most complex and variable marine systems since its dynamics 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 continuous 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 major 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 demand 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 economically 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 capacity 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).

3. A Nearshore Wave and Currents Forecasting System

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 submodels, Figure (3.1). Hourly wind predictions are provided daily by the National Meteorology 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.

HIRLAM MODEL WIND PREDICTION Input: wind fields 10m above the sea

WAM MODEL DEEP-INTERMEDIATE WATERS WAVE PREDICTION Input: wave bidimensional spectra

MILD SLOPE MODEL ONSHORE WAVE PROPAGATION Input: Radiation Stress

2DH NAVIER-STOKES MODEL SURFZONE CURRENTS GENERATION Figure 3.1: Scheme of the diﬀerent models used in the forecasting system.

3. A Nearshore Wave and Currents 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 directions 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 meteorological 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 period Tp , 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, OLUCASP (OLUCA-SP, 2003);(Gonz´ alez et al., 2007), solves the mild slope equation with the parabolic approximation (Kirby and Dalrymple, 1983a), e.g., D2 Φ DΦ + (∇ · U) − ∇(ccg ∇Φ) + (ω 2 − k 2 ccg )Φ + 2ω[k∇Φ2 2 D t Dt k2 γ − ]Φ + ω 2 k 2 DA|A|2 Φ + iω Φ = 0, 2 2ω cosh2 kh

(3.1)

where D is a part of the non linear term: D=

cosh(4kh) + 8 − 2 tanh2 (kh) , 8 sinh4 (kh)

(3.2)

3. A Nearshore Wave and Currents Forecasting System

In Eq.(3.1) Φ is the velocity potential at the free surface and Φ2 is the velocity potential for a long wave. The non linear term ω 2 k 2 DA|A|2 Φ represents the dispersion due to amplitude. The dissipation term iω γ2 Φ, 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 conditions. 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 the Hs at 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. A Nearshore Wave and Currents Forecasting 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 ∂(U H) ∂(V H) + + =0 ∂t ∂x ∂y ∂U ∂U ∂U ∂η 1 ∂ 1 ∂ +U +V +g + (Sxx ) + (Sxy ) ∂t ∂x ∂y ∂x ρH ∂x ρH ∂y [( 2 )] gU ∂ U ∂2U 2 2 1/2 + 2 (U + V ) − ϵ + = 0, c H ∂x2 ∂y 2 ∂V ∂V ∂η 1 ∂ 1 ∂ ∂V +U +V +g + (Sxy ) + (Syy ) ∂t ∂x ∂y ∂y ρH ∂x ρH ∂y [( 2 )] gV ∂ V ∂2V + 2 (U 2 + V 2 )1/2 − ϵ + = 0, c H ∂x2 ∂y 2

(3.3)

(3.4)

(3.5)

3. A Nearshore Wave and Currents Forecasting System

where

Sxx = Syy =

1 T 1 T

∫

t+T t

∫

−h t+T ∫ η

(ρu2 + p)dzdt −

−h

(ρv 2 + p)dzdt − Sxy = V = U=

1 T 1 T

∫

1 T

(3.6)

p0 dzdt,

(3.8)

(ρuv)dzdt,

(3.9)

v(x, y, z, t)dzdt,

(3.10)

u(x, y, z, t)dzdt,

(3.11)

t+T

∫ 1 T t ∫ t+T ∫

t+T

−h t+T ∫ η

−h

1 η= T

−h

∫

(3.7)

−h t+T ∫ 0

∫

H = η + h, ∫ 0 p0 dzdt,

∫

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: [ ] Nf Nθ 1 ∑∑ 1 2 2 Sxx (x, y) = ρg |Ajl | nj (1 + cos θjl ) − , 2 j=1 2

(3.13)

[ ] Nf Nθ 1 ∑∑ 1 ρg |Ajl |2 nj (1 + sin2 θjl ) − f , 2 j=1 2

(3.14)

l=1

Syy (x, y) =

l=1

Nf Nθ 1 ∑∑ Sxy (x, y) = ρg |Ajl |2 nj sin(2θj l), 4 j=1 l=1 ( ) 1 2kj h nj = 1+ . 2 sinh(2kkj h)

(3.15) (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 velocity Cgj and velocity o j component Cj .

3. A Nearshore Wave and Currents Forecasting System • 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.

3. A Nearshore Wave and Currents Forecasting System

For the application of the model in a beach, the equations presented are solved by using ﬁnite diﬀerences on a rectangular mesh. Initial conditions are η and U , V at t = 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 25 s−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.

3. A Nearshore Wave and Currents Forecasting System

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 onshore 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 intermediate 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 the Posidonia 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

3. A Nearshore Wave and Currents Forecasting System

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 prediction 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´omezPujol 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)

3. A Nearshore Wave and Currents Forecasting System

As seen in Figure (3.7), maximum waves around 6m height occur each two years.

Figure 3.6: Annual long-term distribution for HIPOCAS node 1433

Figure 3.7: Wave Extremal regime for HIPOCAS node 1433

The annual and summer wave conditions are shown in Figure (3.8). The most energetic waves are usually from the north and northeast, which are the most common directions, with wave heights between 1-4m.

3. A Nearshore Wave and Currents Forecasting System

The information obtained in this characterization is used to deďŹ ne the grids orientations, and the range values for the bidimensional input spectra of the system.

Figure 3.8: Annual wave regime (upon panel) and summer wave regime (bottom panel) for the deep water point (triangle in Figure (3.5b)).

3.5

Results

The FS system is operational since April 2005, providing 36 hours horizon daily forecast of the wave height, direction and surf zone currents for the Search and Rescue authorities who are responsible for the beach safety. Maps with beach conditions (waves and currents) are updated automatically via internal ftp and used to manage the risk at the beach. Besides, wave conditions are stored for statistical purposes.

3. A Nearshore Wave and Currents Forecasting System

The highly variable environment with sandbars moving continuously onshore (oﬀshore) depending on mild (storm) conditions, force us to perform bathymetric surveys every two months on average. New bathymetries are used to generate the new numerical meshes were models are run.

3.5.1

Validation

To test the performance of the FS, a ﬁeld experiment was carried out from February 28th to March 8th 2006. For this period, time series of wave height and wave period measured with a buoy located at 45 m depth (circle in Figure ( 2.1b)) are compared with the time series of the predicted wave conditions (northern diamond in Figure (3.5b)) provided by the Spanish Harbours Authority (Figure (3.9)).

Figure 3.9: Signiﬁcant wave height (top panel) and mean period (bottom panel) at 45m depth. Solid lines correspond to the forecast and dashed lines to observations.

The correlation between predicted and measured data for the signiﬁcant wave height during 2005-2007 is around 86% (Figure (3.10)) with a RMS 0.37; for the mean period around 78 % (Figure (3.11)) with a RMS 1.32 and for the mean direction 91 % with a RMS 0.56.

3. A Nearshore Wave and Currents Forecasting System

buoy−prediction Hs correlation 6

buoy data

2 y = 0.87*x + 0.14 1

3 prediction data

Figure 3.10: Correlation between measured and predicted Hs .

buoy−prediction Tm correlation 10 9 8

buoy data

7 6 5 4 y = 0.71*x + 1.8 3 2

6 7 prediction data

Figure 3.11: Correlation between measured and predicted Tm .

During the experiment, an acoustic wave and currentmeter AWAC, was moored at 10m depth, approximately 700 m offshore (point A, Figure (3.5c). This instrument provided the incident wave height, period and direction during the first 20 minutes of each hour, at a sample rate of 2Hz. Time series comparing measured significant wave height and the predicted by the system are shown in Figure (3.12).The high values registered during the first hours of March 1st can be confusing as buoy data for the same hours in deep waters, see Figure (3.9), do not show any important change in wave height for

3. A Nearshore Wave and Currents Forecasting System

those hours. The difference between data of last hours of February 28th and first hours of March 1st is the incoming wave directions which for the 28th are around 11N and for the 1st are around 22N. This difference implies a big energy lost for the northern waves during the refraction process. As seen, in March 6th a storm over the area generated

Figure 3.12: Signiﬁcant wave height at 10 meters (location A in Figure (3.5c)) during the ﬁeld experiment. Solid line correspond to measurements and triangles to the hourly prediction provided by the FS.During the period marked in gray there were no available forecasting data since incoming waves were from southwest.

significant wave heights up to 1,5 m and maximum peak periods of 10 s, which are reasonable well predicted by the system. During the period marked in gray there was no available forecast since the incoming waves were from the southwest. As explained before, the directions propagated into our study zone go from 352(NNW) to 142(SE), and so if incoming waves are outside this range no propagation is made. Although the predictions are well correlated with observations (78.4% of agreement), the observed discrepancies could be related with WAM period predictions which do not correlate exactly with the buoy data during the first five days of march and with the fact that local winds are not included in the system. Additionally, a current meter was moored at a water depth of 1.5 m to measure the mean velocities every minute. However, measured currents did not show any indication of seaward outgoing currents. During the first 3 days directions of measured currents correlate well with the predictions made by the model, unfortunately during the storm event the currentmeter was buried and therefore there is no current data available for this period. In order to have an overall view of the current system in Cala Millor beach, the Spanish Air Force performed a photographic survey all over the beach. This kind of data is very useful since it provides visual signals of surf zone currents.

3. A Nearshore Wave and Currents Forecasting System

Figure (3.13), shows the aerial photography of May 30th 2005 (upper right side) where the accumulated leaves of the endemic specie P. oceanica serves as a passive tracer for the identification of the wave-induced currents. Rip location can be visually identify by a line of foam, seaweed, or debris moving steadily seaward (Komar, 1998). In the same picture the predicted currents the same day over the whole study area are shown. Comparing the aerial photography with the forecasted current field, seems that the system is able to predict the outgoing flows. This visual estimation of a rip current is made by several lifeguards associations as well as by the NOOA, in which web explains how to identify rip currents where ”a line of foam, seaweed, or debris moving steadily seaward”.

Figure 3.13: Current ﬁeld for May 30th 2005 at 18.00. The aerial photography at that time is represented at the upper right side of the picture, while the predicted rip current is represented in the bottom right side. The location of the predicted rip- current coincides with the area in which the P. oceanica leaves.

3.6

3. A Nearshore Wave and Currents Forecasting System

Discusion

The validation shows a good agreement between forecast data and field data, but the field experiment only provided punctual data, which could be wrongly extrapolated to all the beach. An intensive experiment with several instruments in different profiles of the beach would provide a better 2D validation. In our experiment wave data were surveyed at 10 m depth where most waves have been affected by the refraction and shoaling, even the data collected show agreement with forecast it would be a better comparison to survey data at 5-3 m depth, just before the bar, where non linearity is higher and surf zone currents are generated. The settlement of only one currentmeter in the zone where rip currents should appear makes extremely difficult to validate the existence of offshore currents. As said by MacMahan et al. (2006) complete rip current experiments require three types of measurements, 1) comprehensive velocity measurements within the rip channel and neighboring shoal, 2) accurate measure of the bathymetry, and 3) offshore directional wave measurements. Considerable difficulties are encountered in simultaneously obtaining all three of these measurements. Bathymetric data were took before the experiment but nearshore waves and currents are certainly strongly affected by the migrating sand bars which were not updated during the experiment, in this period there was sediment movement as the burying of the currentmeter proves. This lack of bathymetric data seems to be one of the main problems in having an operational system in the nearshore region some examples are (Allard et al., 2008) and (Neilla et al., 2007). The election of wave propagation model can be discussed as there are new version of models like SWAN (Booij, 1996) able to reproduce almost all the wave transformation in shallow waters and with some advantages over the parabolic models, as the inclusion of wind generation waves. The parabolic model has been tested in several field experiments giving good results in wave modeling nearshore, but for the area here studied the inclusion of local winds, specially during summer season when seabreeze is important, would improve the predictions during summer time when beach risk increases due to the high number of beach users. An improvement would be to implement a model like SWAN in the area of deep and intermediate water, and to nest a mesh of a parabolic model in the shallow waters, as this type of models can be run over meshes with bigger spatial resolution. Even some implementations could be done, and an intense validation should be repeated, the qualitative comparisons made with aerial photography and lifeguard data

3. A Nearshore Wave and Currents Forecasting System

oﬀer good results of the system all long the summers of 2005, 2006 and 2007. It must be take into account that the system here presented is the ﬁrst approach to better understanding the nearshore hydrodynamics in Balearic Islands beaches and future work will be directed to develop better versions of the FS.

3.7

Conclusions

Surf zone hydrodynamics control all the processes in the region where waves break and generate currents. The study of wave transformation and current generation needs of some assumptions to simplify this highly non linear processes. Modeling the processes of the surf zone currents requires a good definition of the boundaries, specially the bottom boundary which will influence in wave transformation. Numerical modeling facilitates advancements in our understanding of coastal changes and can provide predictive capabilities for resource managers. This work presents an approach to improve beach safety management by the development of an operational forecasting system for the prediction of waves and wave-breaking currents in surf zone waters. In the Forecasting System developed, nearshore waves are obtained by the propagation of deep water wave conditions to the area of interest. Breaking currents are obtained after the inclusion of the radiation stresses in a depth integrated Navier-Stokes model. The FS has been working for three years providing the nearshore hydrodynamics in an autonomous and operational way. Some interruptions occurred during this period mainly due to problems related with connections between the local server and the server located in the Spanish Harbours Authority. Field experiments were carried out to test the forecasting capabilities of the system showing that predictions of wave height and direction are well correlated with observations. Directions of currents in the surf zone are also well predicted. Wave breaking currents were also qualitatively validated using aerial photography. A constant communication with beach lifeguards existed during 2005, 2006 and 2007, information took from their reports showed good agreement with the FS. Main differences are obtained those days with strong breeze and small waves in deep water. It was noticed that sea breeze generate an amount of water onshore that generate longshore currents which were not predicted by the system. Although the FS developed provides good estimations of the occurrence, location and intensity of wave breaking currents, several improvements have to be implemented. First, since accurate bathymetry is continuously required it is necessary to develop low cost techniques to obtain bathymetry. In this sense matching video-based techniques with the presented approach has to be explored. Besides the inclusion of local wind forcing such as the breeze would improve the input wave conditions at the outer beach boundary.

Chapter 4

An Alert System for Beach Hazard Management in the Balearic Islands This chapter is an edited version of the paper ’An Alert System for Beach Hazard Management in the Balearic Islands’, Coastal Management, 37:6, 569-584. Authors: Amaya Alvarez-Ellacuria, Alejandro Orfila, Maitane Olabarrieta, Llu´ıs G´ omez-Pujol, Ra´ ul Medina and Joaqu´ın Tintor´e .

4.1

Abstract

A real time beach hazard level associated with nearshore hydrodynamics is presented in this paper. The suitability of the discussed alert system is illustrated via its application to ﬁfteen beaches in the Balearic Islands (Western Mediterranean Sea) providing nearshore safety conditions for beach management. The system provides daily forecasts of nearshore wave conditions using the deep water wave forecasts. The shallow water wave data (wave height, period and direction) together with the morphology of the site (presence of bars, capes, beach type, etc.) are used to deﬁne a hazard level (low, medium and high) associated with local conditions. The resulting hazard level is transmitted via SMS to lifeguards and local authorities for real time beach management. The low computational cost of this system after the implementation and calibration, results in a very suitable approach for beach management in order to mitigate hydrodynamic risks. 39

4.2

4. Alert System for Beach Hazard Management

Introduction

Coastal areas are among the most complex and variable marine systems. The dynamic of shallow seas is the result of complex interaction of physical forcings acting over a complex geometry. Moreover, a detailed knowledge of beach hydrodynamics is crucial for three primary reasons: beaches are coastal protection features; they are the basis for substantial economic activities and finally, they are the ecosystem for many marine and terrestrial species. In addition, coastal tourism, especially beach recreational uses increased in the second half of the last century with sea bathing becoming the most prominent activity worldwide. Beach hazard rating and safety have become key issues in coastal management and consequently, the development of tools for accurate estimates of nearshore conditions in order to minimize hazards related to beach hydrodynamics is nowadays a public demand. Safety in terms of beach management refers to nearshore conditions which may change into dangerous for users by increasing the risk of drowning or injury. Both risks are clearly related to two main groups of factors; the first one includes those aspects related to user-behavior and the spatiotemporal distribution of users on the beach (Hartmann, 2006; Jiménez et al., 2007; Manolios and Mackie, 1988) and the second group takes into account those aspects associated with the physical features of the beach such as beach and surf topography, water depth, breaking waves, currents and the presence of reef and rocks (Klein et al., 2003; Short, 1999). While there is a vast literature on rip currents (see MacMahan et al. (2006)), there are only few works referring to beach hazard parameters. Previous studies to redress beach safety problems integrated beach morphodynamical models and hydrodynamical information in order to assess the beach hazard level concept. Short and Hogan (1994), linked wave heights with morphodynamical beach states to create a safety rating for 721 beaches in Australia These authors, combined the non-dimensional parameter , defined as Ω = Hb /T ωs , where Hb is the breaking wave height, T the wave period and ωs the sediment fall velocity, linking hydrodynamic and sediment conditions with morphological configurations or scenarios. For the definition of the safety rating they further used the water depth, the size of breaking waves, the prevalence and intensity of rip currents and the existence of longshore currents. Benedet et al. (2004), created a storm hazard-risk category for several Florida beaches relating beach morphology to the presence or absence of some coastal development. Recently, Scott et al. (2007) added beach rescue statistics to the nearshore morphology to identify specific hazards in Southwest 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. In spite of these studies describe different hazard levels depending on physical

4. Alert System for Beach Hazard Management

beach characteristics and hydrodynamics to our knowledge, operational systems based on nearshore wave forecasts are still under development.

A nearshore operational system has to be able to provide reliable estimations of wave and current conditions within a reasonable time period. Nowadays, the increasing capacity of computers allows to solve complex wave propagation numerical models in short time, making possible the forecast of waves and currents in the frame of operational systems (Hodur, 1997; Bidlot et al., 2002). Nearshore waves are more difficult to predict than open ocean waves as there are many factors modifying their propagation (Kobayashi and Yasuda, 2004). Nevertheless, when the area of interest is too large numerical models for operational purposes might be computationally expensive. However, there are some alternatives to the above forecasting systems based on first physical principles since forecasts can be made as well empirically with a learning process directly from observations. In this sense, (Browne et al., 2007) presented a nearshore swell estimation for 17 beaches of the Australian coast by using artificial neural networks together with surf reporters which showed better results than swell propagation obtained from numerical models. In the same way, NOAA has been using lifeguards’ reports to create a predictive risk index due to rip currents in US beaches. This predictive index was first developed by Lascody (1998) and later modified by Engle et al. (2002).

In 2004, environmental management authorities of the Government of the Balearic Islands requested a relocatable forecasting system (FS) to predict in real time waves and rip currents. The system developed, fall within those based on first physical principles and was tested in a pilot area (Cala Millor) in the north-eastern coast of Mallorca Island (Western Mediterranean). The FS was formed by four physical independent submodels providing wave and current forecasts twice a day with a 36 hours forecasting horizon. Even though results were very accurate for the forecast of waves and rip currents (see validation in (Alvarez-Ellacuria et al., 2009)), some problems arose when authorities tried to apply the FS to different littoral beaches. Firstly, forecast was made in real time with a wave propagation model being computationally very expensive since a dedicated computer was required at each location. Secondly, updated bathymetries were needed for accurate forecasts in the surf zone in order to know the exact position of the submerged sandbars, with the subsequent economic cost. Within this framework, in this study we present an alternative methodology for the development of a Hazard Alert System (HAS) for beach safety management based on the definition of hazard levels associated with nearshore waves. The hazard level is obtained using a database of nearshore waves, daily deep water wave forecast and lifeguards’ observations. At a first stage, a database of different hydrodynamic conditions at each beach is constructed (150 cases are generated for each site) using a numerical model which propagates combinations of deep water

4. Alert System for Beach Hazard Management

directions, wave heights and peak period. Then the hazard level for each location is deﬁned using the wave height and direction of incoming waves at 10 m depth. Finally the system is calibrated using in situ observations provided by lifeguards.

4.3

Study Areas

The Balearic archipelago, is located in the western Mediterranean, about 100 km eastward to the Iberian Peninsula coastline. The four major islands are Mallorca, Menorca, Eivissa and Formentera (Figure (4.1)).

Figure 4.1: Geographic location of the Balearic Islands (a); Mallorca island (b); Ibiza and Formentera islands (c), and Menorca island (d). In b, c and d the dot show study sites and the triangle the deep water wave data used for wave climate characterization (HIPOCAS nodes).

Although cliﬀ coasts characterize a large part of the littoral of the Balearic Islands, there are 869 beaches with a total of 120 km of sandy coastline. Regional studies of sandy beaches and shelf sediments highlight that bioclasts are the main constituents of os and Ahr, 1997; G´omez-Pujol et al., 2007). beach and dune sediments (Forn´

4. Alert System for Beach Hazard Management

Maximum wave height at deep waters rarely exceed 8 m in the Balearic Sea. These extreme waves are the result of storms driven by heavy Mistral winds (up to 40 m/s) with a large associated fetch going from Ligurian Sea to the Balearic channel.The north western and central part of the Balearic Sea are forced by northerly winds (Mistral) during the main part of the year, while the eastern part is generally modulated by a seasonal variability (Ca˜ nellas et al., 2007). HIPOCAS dataset (Soares et al., 2002), consisting on a 44 year hindcast of sea conditions, shows that significant wave height (Hs ) achieves close to 1m the 50% of time and peak periods (Tp ) are around 6 s ranging between 2 and 12 s. Wave climate analysis indicates a gradient in sea wave height and direction among the islands. Location near the northern side of Menorca Island (Figure (4.2a)) shows northern waves and highest values for Hs than locations in the west coast of Formentera (Figure (4.2b)), where smaller values in Hs are recorded with a major influence of southern winds. Balearic Islands have a recent history of intense tourist development and exploitation Kent et al. (2002). About 13 million of tourists visited the Balearic Islands in 2007, mainly between May and September. During these months beaches were overcrowded; hence users’ safety was a major concern for local authorities. This interest lead to the selection of 17 beaches in the Balearic Islands to develop a HAS for beach safety. All these beaches are classified as medium to high risk by the authorities owing to the existence of surfzone currents. Lifeguard services are provided five months per year, especially during the summer season when beach usage is at its maximum. Five of these beaches are located in Mallorca and range between 334 m and 2.5 km length and 42 to 68 m width (Figure (4.1b)). Six beaches are sited in Menorca (Figure (4.1d)), ranging between 335 m to 2.6 km length and 14 to 110 m width. Two more beaches were selected for both, Ibiza and Formentera Islands (Figure (4.1c)) with lengths between 85 m to 2 km and mean width between 20 and 47 m. At least in 13 beaches, beachrock or reef crops up induce rip currents. According to Short’s classification (Short, 2006) , two types of rips can be identified at Balearic beaches: (a) accretion rips, which are related to the presence and displacement of sandbars onshore/offshore and (b) topographic rips associated with deep channels that can migrate fast (days or weeks). The main features of each beach are summarized in Table (4.1).

4. Alert System for Beach Hazard Management

Figure 4.2: Wave climate for the HIPOCAS nodes a) H6, b) H13.

Ses casetes des Capellans Cala Mesquida Cala Millor Sa Coma Es Trenc Platja des Tancats S’Arenal de Tirant Punta Prima Son Bou Sant Toms Platja de Son Xoriguer Cala Tarida Cala des Comte Cala Llenya Cala Nova Platja de Llevant Platja de Migjorn

MA1 MA2 MA3 MA4 MA5 ME1 ME2 ME3 ME4 ME5 ME5 IB1 IB2 IB3 IB4 IB5 IB6

405.40 334.12 1682.10 793.72 2455.97 336.82 417.82 274.69 2571.40 530.29 81.67 168.17 85.18 193.19 118.37 808.04 2030.15

Length (m) 60.46 68.40 60.40 57.54 42.15 34.49 37.82 110.58 69.37 14.95 72.97 23.48 20.85 33.41 47.13 41.45 22.78

Mean width(m) Fine sands Coarse sands Medium / fine sands Medium / fine sands Fine / very fine sands Medium / fine sands Coarse / medium sands Medium sands Fine sands Medium / fine sands Medium sands n.a. n.a. n.a. n.a. n.a. n.a.

Sediment Size

x x

x x x x x x x x x

Rips Reef related

x x

x x x x x

Bar related

θ 330-90 330-90 0-180 0-180 105-255 330-60 285-60 330-60 90-270 90-270 120-270 190-360 190-360 0-180 0-180 0-140 140-270

H1-1314 H2-1359 H3-1433 H3-1433 H4-1611 H5-1192 H6-1194 H7-1365 H8-1319 H8-1319 H9-1317 H10-1319 H10-1319 H11-1735 H11-1735 H12-1865 H13-1949

HIPOCAS node

Table 4.1: Identiﬁcation and principal characteristics of studied beaches: length, mean with, sediment size, rip type, range of wave directions arriving to the beach and HIPOCAS node identiﬁcation

Beaches

4. Alert System for Beach Hazard Management 45

4.4

4. Alert System for Beach Hazard Management

Data and Methods

In this section the core of the system is presented. Previous work was devoted to characterize the wave climate at each location and to generate a large database of cases by propagation of different incoming waves. Long term deep water characteristics for each study site are firstly obtained from the HIPOCAS database. Afterwards, a large number of selected combinations of deep water conditions (Hs , Tp and θ) are propagated to the shore using a mild slope parabolic model generating a look up table (interpolation matrix). In this matrix, the deep water waves are related to nearshore waves by means of a propagation coefficient for each location. For the operational part, the wave forecast system of the Spanish Harbour Authority (Puertos del Estado) provides wave forecast at deep waters (between 50 and 150 meters depth) near each beach twice a day. From these forecasts, nearshore wave conditions are obtained using the look up table for the next 36 hours. An initial assessment of the hazard level is produced based on wave height and incoming direction for each site at 10 m depth. The potential hazard levels are in a second step validated and calibrated with observations provided by lifeguards during 2007. An explanatory scheme of the system is shown in Figure (4.3). PREPROCESSING SELECTION OF SEA STATES

NEARSHORE

ONSHORE PROPAGATION

DATABASE

HAZARD INDEX DEFINITION LOW HAZARD LEVEL

DEFINITION OF HAZARD LEVEL

SELECTION OF MEDIUM HAZARD LEVEL

HS & q

LIMITS

HIGH HAZARD LEVEL

HAZARD ALERT SYSTEM DEEP WATER FORECAST DATA RECEPTION

INTERPOLATION OF FORECAST DATA. HS & q AT 10m

LIFEGUARD DATA REPORT

DEPTH

HAZARD LEVEL FORECAST

CALIBRATION OF HAZARD LEVEL DEFINITION

Figure 4.3: Hazard alert system scheme.

4. Alert System for Beach Hazard Management

4.4.1

Nearshore Wave Climate

From the long term probability distribution obtained from the entire HIPOCAS database (385.000 hourly data) selected combinations of the long term (between 5% and 95% probability distribution) of significant wave height (0.5 < Hs < 5m) and wave period (3 < Tp < 12s) have been propagated to the beach by a mild slope parabolic alez et al., 2007; ?). To model the different incoming wave directions, four model (Gonz´ different grids have been implemented implemented at each beach: a coarse grid for deep and intermediate waters with a resolution varying between 60 m and 100 m and a fine grid with a resolution between 15 m and 25 m (Table (4.2) shows grid resolutions and orientation to the north). The cases propagated are a combination of Hs (0.5, 1, 2, 3, 5 m), and Tp (3, 5, 7, 9, 12 s), and directions depending on each beach exposition (see Table (4.1)). More than 2000 cases were propagated with a mean computation time of about 15 minutes for each case. The interpolation matrix provides the wave data at 10 m depth ( θ10 , Tp and Hs10 ).

angle

Mesh1 δx,δy (m)

MA1 MA2 MA3 MA4 MA5 ME1 ME2 ME3 ME4 ME5 ME6 IB1 IB2 IB3 IB4 IB5 IB6

20 0 50 50 210 320 10 40 220 220 220 345 345 60 60 60 210

80,80 100,100 100,100 80,80 100,100 80,80 60,60 60,60 100,100 100,100 60,60 80,80 60,60 100,100 100,100 100,100 60,60

Mesh2 δx,δy (m) 20,20 20,20 20,20 20,20 20,20 20,20 20,20 20,20 25,25 25,25 20,20 20,20 20,20 20,20 20,20 20,20 15,15

angle

Mesh1 δx,δy (m)

Mesh2 δx,δy (m)

60 75 130 130 150 40 60 130 180 180 160 235 235 130 130 90 150

100,100 100,100 100,100 100,100 80,80 80,80 60,60 60,60 100,100 100,100 60,60 80,80 80,80 100,100 100,100 100,100 100,100

20,20 20,20 20,20 20,20 20,20 20,20 20,20 20,20 25,25 25,25 20,20 20,20 20,20 20,20 20,20 20,20 25,25

Table 4.2: The grid orientation in degrees respect to the North and model grids resolution in meters (δx,δy).

4. Alert System for Beach Hazard Management

Once the database is created, the daily forecast of deep water waves was introduced in the matrix to get the wave parameters at each beach. For forecasted deep water conditions slightly diﬀerent from those of propagated cases, a linear interpolation was used within the matrix. Finally two beaches, IB1 and IB2, were removed from the study since propagation was not satisfactory due to inaccurate bathymetry.

4.4.2

Forecast Wave Data

Wave forecast is provided over the western Mediterranean Sea by the Spanish Harbour Authority 3 days in advance. The model runs over a mesh of 0.083 at deep and intermediate waters. This model is run every 12 hours giving Hs , Tp and mean direction (θ) at deep waters. For each beach, the forecast point at deep water is as close as possible to the HIPOCAS node employed in the climate characterization. These forecast points are located between 50m and 150 m depth and between 5 km and 14 km oﬀshore.

4.4.3

Hazard Level

Three different hazard levels have been defined depending on nearshore wave conditions. Level one indicates low hazard due to wave breaking and current generation; level two implies a moderate hazard and level three indicates high hazard due to hydrodynamic conditions. For the definition of the hazard level the angle of exposition respect the open sea was obtained (see Table (4.1)). Between these angles, the directions perpendicular to the shoreline correspond to the hazardous incoming wave directions and thus a new set of angles and have been defined for these directions (Table (4.3)). Moreover for the definition of these angles, the morphodynamic beach states were also taken into account by using the lifeguards’ description of each location as well as aerial photographs and sediment size where available. The definition of the different hazard levels as a function of incoming wave height, and wave direction (θ ) is presented in Table (4.4). The low hazard level is defined in two cases: i) for wave heights at the beach under 0.5 m and all incoming wave directions, which is the average wave height in the Balearic Islands generated by a mild fetch and ii) wave height is between 0.5 and 1 m and the incoming directions lie outside the angles and Medium level is set in two scenarios: i) when incoming directions are inside α1 and α2 and wave heights are between 0.5 and 0.8 m and ii) when wave heights are between 1 and 2 m and incoming directions are outside α1 and α2 .

4. Alert System for Beach Hazard Management Beach

α1

α2

MA1 MA2 MA3 MA4 MA5 ME1 ME2 ME3 ME4 ME5 ME6 IB3 IB4 IB5 IB6

350 5 70 80 170 350 350 80 170 170 170 60 140 80 170

10 60 110 120 200 20 10 220 200 200 200 110 160 110 220

Table 4.3: Minimum and maximum incident directions for the hazard level deﬁnition in each beach.

A high hazard level is defined when i) wave height is higher than 0.8 m and the incoming directions lie inside α1 and α2 and ii) for all directions and wave heights greater than 2 m. It is clear that since wave energy is proportional to the square of wave height, hazard levels will increase as wave height does. The levels here defined are subject to redefinition in specific areas based on the analysis of lifeguard data.

Hs(m)

θmin

θmax

Hazard index

< 0.5 ≥ 0.5 & ≤ 1 ≥ 0.5 & ≤ 0.8 > 0.8 >1&<2 ≥2

0 < α1 ≥ α1 ≥ α1 < α1 0

360 > α2 ≤ α2 ≤ α2 > α2 360

1 1 2 3 2 3

Table 4.4: Hazard level deﬁnition. θmin & θmax describe the range of incoming directions used for the hazard levels deﬁnition.

4.5

4. Alert System for Beach Hazard Management

Results and Discussion

The system herein presented for the Balearic Islands has been operating since summer 2007 generating automatically the hazard level at the selected beaches. Once the hazard levels are produced for the beaches, a SMS is sent to the on-site lifeguard. Additionally a daily report describing deep water conditions, nearshore waves and the hazard level for bathing is sent via mail to management authorities, who then display the information on their public web server. Results for the period comprised between July and September 2007 are shown in Table (4.5). All locations were catalogued as low hazard during 60%of the implementation time. Among them, seven sites, MA1, MA3, MA5, ME3, ME4 ME6 and IB6, obtained the referred level for more than 90% of the time. Beaches in the north coast present more energetic wave conditions and therefore hazard levels rise to 2 or 3. Moreover, in some of the studied beaches the presence of beachrocks and reefs also increased the hazard level, (i.e. IB5). Beach

observations

low (%)

medium (%)

high(%)

MA1∗ MA2 MA3 MA4 MA5 ME1 ME2 ME3 ME4∗ ME5 ME6 IB3∗ IB4 IB5 IB6

∗

94.02 60.98 90.28 67.09 100 76.44 67.92 92.73 99.67 100 99.07 81.77 98.8 70.45 100

0 13.41 9.72 18.99 0 8.05 7.55 3.64 0.33 0 0.93 14.29 0 17.05 0

5.98 25.61 0 13.92 0 15.52 24.53 3.64 0 0 0 3.94 1.2 12.5 0

78 68 139 38 48 106 96 ∗

48 98 ∗

166 81 50

Table 4.5: Number of lifeguards observations and forecasted hazard level statistics using the same dates of the observations.∗ Refers to beaches where there is no lifeguard data, the statistics here are made with data from July to September 2007.

Since particular conditions at each site are crucial for the deﬁnition of its hazard level, calibration with in situ data is required both, to test the HAS and to redress the results.

4. Alert System for Beach Hazard Management

Hence some aspects that the system does not take into account (i.e. local wind, fixed topography in bathymetry under 10 m depth) should be incorporated. Moreover there is a significant variation in beach hazards and their severity depending on the nature of the hydrodynamic conditions and the beach type state (Scott et al., 2007). This beach state is only taken into account in the presented approach at the first stage of the HAS; seasonal variations are not included in the system, which might vary the results.

4.5.1

Lifeguards Data

Daily reports on beach conditions are produced by lifeguards during summertime, containing information about the local weather, sea state and number of rescues. To avoid irrelevant data for the validation ( e.g. people injured in the beach by jellyfish, sunstroke or other facts), lifeguards report twice a day on wave height and direction, wind direction, presence of submerged sandbars and the flag placed every day on the beach (red indicating some danger, yellow for unstable water conditions and green for good swimming conditions). The number of reports was different for each beach. For beaches ME4, MA1 and IB3, data was not available and therefore validation was not possible. The number of reports varied between 38 and 166 (see Table (4.5) showing number of samples for each beach). Beach

Hs max dif (m)

θmax max dif

Hs mean dif (m)

θ mean dif

MA3 MA5 ME2 ME6 IB5

0.109 0.142 0.703 0.25387 0.350

4.857 15.01 17.694 8.0918 4.16

0.02904 0.031 0.07410 0.060217 0.0439

1.458 4.974 2.855 4.022 1.453

Table 4.6: Maximum and mean diﬀerences between results of Hs and θ propagated and interpolated

4.5.2

Validation

Deep water conditions from a buoy located at 45 m depth were compared to those from the forecasting deep water model at the location MA3. The correlation between predicted and measured data during 2005-2007 was around 86% for the signiﬁcant wave height with a RMS 0.37; around 78% for the mean period with a RMS 1.32 and around 91% for the mean direction whit a RMS 0.56. Deep water wave height and the corresponding propagation to 10m depth are shown in Figure (4.4) at MA4, IB5 and, ME6. Gaps in shallow water wave height correspond to those directions outside the angles of exposure.

4. Alert System for Beach Hazard Management

Figure 4.4: Temporal series deep water wave height (solid line), and corresponding propagation at 10m depth (dashed line) at MA4 (top), IB5 (center) and ME6 (bottom).

To test the validity of the look-up table, 24 sea states were propagated at five beaches and the resulting wave parameters compared with the interpolated data. From the results shown in Table (4.6), mean differences in wave height are under 1 cm in all beaches and maximum differences range between 10 and 70 cm mainly occurring when Tp is higher than 10 s, which is an unusual value for the Balearic coasts. Mean differences in direction can be considered to be small (between 1.5 and 5). The use of alternative to numerical models for nearshore propagation have been already presented and validated in other studies (Browne et al., 2007). Results for the hazard level (low, medium and high) provided by the system during July, August and September 2007 and flags posted on each beach (green, yellow and red) are displayed in Table (4.7). Results are only available for twelve beaches, since lifeguard’s data were not available at MA1, ME4 and IB3. Flag colours do not depend only on the hydrodynamic conditions but also on other factors, such as presence of jellyfish, water contamination or wind conditions. To solve for this a filter was developed to transform red and yellow flags into green flags when the wave height observed by the lifeguard was less than 0.5 m and the predicted wave height was less than 0.3 m. As seen in Table (4.7), agreement between hazard level and its corresponding flag were between

4. Alert System for Beach Hazard Management

56% and 92%. Higher accuracy was achieved in beaches located in eastern and southern coasts, where green flags were set over 60% of the time. If comparison is made for sea states, the agreement between green flag and low hazard level is around 90%, 18% between yellow flags and medium hazard level and 49% between red flags and high hazard level. .In three sites, ME1, ME3 and IB5 the agreement were under 60% . In these locations the main problem was the conditions sets for the definition of the hazard levels mostly for the medium and high hazard levels. Beach

observations

green

low

yellow

medium

red

high

accuracy (%)

MA2 MA3 MA4 MA5 ME1 ME2 ME3 ME5 ME6 IB4 IB5 IB6

78 68 139 38 48 106 96 48 98 166 81 50

44 66 112 30 26 65 59 38 84 106 49 46

37 63 80 30 21 61 53 38 83 104 38 46

12 0 17 8 15 29 35 9 14 54 6 4

4 0 6 0 4 5 1 0 0 0 0 0

22 2 10 0 7 12 2 1 0 6 26 0

15 0 5 0 3 12 0 0 0 0 9 0

71.8 92.6 65.46 78.9 58.3 73.6 56.2 79.1 84 62 58 92

Table 4.7: Number of green, yellow and red flags posted and the number of coincidences between low hazard level and green flag, medium hazard level and yellow flag and high hazard level and red flag. The last column is the accuracy of the hazard level prediction compared with flags posted.

4.5.3

Calibration

Data from lifeguards is crucial to address the different ranges for the definition of hazard levels. Calibration was done in those beaches where agreement between the defined hazard levels and flags posted by lifeguards was under 84% where a higher range of incoming directions was set and values for Hs10 adapted to each specific beach. This calibration was based on the information provided by the lifeguards. For instance, in IB5 a large rock was sticking out in the surfzone, thus modifying the hydrodynamics by generating strong rip currents with relatively small wave heights (Figure (4.5)). Correlations between flags and the new redefined HAS levels are shown in Table (4.8).

4. Alert System for Beach Hazard Management 1º26’35’’E

38º44’20’’N

Outgoing flow Incoming waves

38º44’15’’N

Breaking zone

1º26’35’’E

Figure 4.5: Aerial view of IB5. Submerged reef is contoured and surfzone circulation is marked with arrows.

beach

observations

green

low

yellow

medium

red

high

accuracy (%)

MA2 MA5 ME1 ME2 ME5 IB4 IB5

78 38 48 106 48 166 81

44 30 26 65 38 106 49

38 30 20 60 38 85 33

12 8 15 29 9 54 6

6 3 6 13 1 27 1

22 0 7 12 1 6 26

16 0 5 11 1 1 18

76.9 86.8 64.5 79.2 83.3 68 64.2

Table 4.8: New results obtained with the calibration of hazard levels in each beach.

4. Alert System for Beach Hazard Management

4.6

Conclusions

In this paper a Hazard Alert System (HAS) for beach safety management is developed and applied to 15 beaches in the Balearic Islands. The combination of numerical modeling and available historical information from lifeguards allows the development of this system. HAS is computed twice a day providing hazard level information for the next 36 hours to coastal managers. The low computational cost of the presented approach makes it very suitable for its implementation to large areas of the littoral. The agreement between the observations provided by lifeguards and forecasted beach conditions is higher in low and high hazard levels, whereas differences increase in intermediate conditions (yellow flag and medium level). The calibration with lifeguard’s data introduces new wave height and direction ranges for the definition of hazard levels, improving the performance of the Balearic HAS. After calibration new definitions have been developed for each beach. Since observation is necessary to validate as well as to calibrate the system, new improvements should be focused on the application of autonomous observing platforms. For instance, remote video images would provide hydrodynamic conditions and beach occupation in real time. The implementation of the HAS system is enhanced by local knowledge of beach dynamics that is positively increased by lifeguards’ data.

Chapter 5

Short-Term Shoreline Evolution in a Low-Energy Beach This chapter is based on a paper submitted to Marine Geology. Authors:Amaya Alvarez-Ellacuria, Alejandro Orfila, Llu´ıs G´ omez-Pujol, Gonzalo Simarro and Nelson Obreg´ on.

5.1

Abstract

In this study it is analyzed the short term variability in a microtidal low energetic beach by using weekly shorelines obtained after applying an ANN to daily video images. The spatial and temporal variability of the beach is decomposed by temporal EOFs, obtaining that the first EOF explains the shoreline changes due to cross-shore sediment transport and the second and third EOF the changes induced by longshore sediment transport. The first EOF is significantly correlated with cross-shore component of the significant wave height at shallow waters averaged for the previous four days before shoreline detection. The second EOF is significantly correlated with the averaged deep water waves for the previous 7 days.

5.2

5. Short-Term Shoreline Evolution in a Low-Energy Beach

Introduction

Beaches are among the most challenging environments to study in part due to the incomplete understanding of the physical processes acting at several spatial and temporal scales, but also because large data sets are needed to describe completely nearshore processes. Beach morphology is the result of complex hydrodynamics (waves, currents, tides as well as their nonlinear interaction) which are forced in a wide range of spatial and temporal scales. Therefore, morphological response of beaches is a highly complicated phenomenon that can be studied at different temporal scales: very long term (centuries to millennia), long term (decades to centuries), middle term (years to decades) and short term (hours to years) (Stive et al., 2002). These features make the observation, modeling and continuous monitoring of beach variability cumbersome, expensive, and sometimes impossible to obtain. The scarcity of information becomes a problem when users and governments need to asses the current state and possible scenarios after intervention in coastal zones. Comprehensive information in coastal areas is required in order to study these marine systems and to establish efficient coastal management programs. In the short term, beach variability depends on waves and tides principally, which are responsible of the sediment transport all along and across the beach. The cross-shore and longshore variability of beaches, while usually assumed orthogonal, are closely connected. The cross-shore transport is linked to the beach profile adjustment to the wave energy. It has been observed that in a dynamically stable beach configuration, sandbars usually move offshore during storm episodes, when strong seawards currents (undertows) dominate the sediment transport phenomena (Gallagher et al., 1998). Onshore sandbar migration occurs between storm events when wave energy is lower (Hoefel and Elgar, 2003). When waves come to the shore with a certain angle it results in sediment transport along the beach. This longshore transport results from the combined effect of breaking waves, which place sand in motion, and the presence of a longshore current in the surf zone. The morphodynamic response of beaches to low energy conditions is slower than under the presence of storm events, being the morphology in the later a sequence of past conditions rather than the actual (Smit et al., 2007). Storms lead to transient changes to beach morphology that can be seen in a long perspective as noise superimposed to the equilibrium state (Reeve and Spivack, 2004). Understanding the beach erosion and accretion processes is important for coastal management and engineering works such as the design of coastal protection, developing

5. Short-Term Shoreline Evolution in a Low-Energy Beach

hazard levels or assesing sea-level rise. During the last decades, much of the research in beach morphology has been focussed in the development of simplified models of beach state indicators in order to overcome the problem of solving the hydrodynamic equations coupled with sediment transport models. Several studies have defined the relation between wave and current conditions to sand bars or rip channels formation and development (Lippmann and Holman, 1990; Hoefel and Elgar, 2003; Plant et al., 2006; MacMahan et al., 2006). Measuring hydrodynamic and morphological features on beach is not free of problems. Although many efforts have been made to provide information on coastal zone processes, only recently, with the development of new technologies, it became possible to obtain information with appropriate temporal and spatial resolution. Several experiments and coastal monitoring schemes have been developed and carried out in the last decade (Senechal et al., 2009; Rogers and Ravens, 2008; Almar et al., 2008). The most complex ones are field based coastal and oceanographic facilities. These facilities have a broad capacity to produce spatial and temporal measurements of physical and environmental variables (Proctor et al., 2004; Petit et al., 2001). Unfortunately, they require a large economical investment and have a relatively long installation and set-up times before producing meaningful data sets. Besides, long term maintenance and sustainability become a liability if funds are not readily available. Another alternative to measure coastal processes is based on remote sensors. In this way, information can be acquired automatically, continuously, and periodically from high resolution digital images. The quality of remotely sensed information depends on the image resolution and quality; images are affected by adverse weather conditions and the accuracy of measurements can be lower than traditional techniques. However, it is an alternative that utilizes a significantly lower amount of human, economic, and computational resources hence, allowing a better continuity and frequency in data acquisition. Among optical remote sensors, fixed digital video cameras is an attractive alternative for coastal monitoring. Video based coastal surf zone monitoring systems are low cost systems, that can be implemented in coastal areas and permit the estimation of several littoral processes from surface signatures on the image. Shoreline position has been used as a coastal state indicator to quantify coastline retreat and beach evolution (Ojeda and Guillén, 2008; Miller and Dean, 2007; Pearre and Puelo, 2009). For the study of shoreline variability at short and medium term a database with high spatial and temporal resolution is needed. In this sense, video systems have proved to be an efficient tool to monitor the shoreline with a high temporal and spatial resolution (Kroon et al., 2007).

5. Short-Term Shoreline Evolution in a Low-Energy Beach

In this paper we analyze the high frequency variability of shoreline in a barred low energy microtidal beach. Weekly shorelines were extracted from a video based coastal observing system and they are analyzed using Empirical Orthogonal Function (EOF) analysis. Deep water wave conditions were propagated for the period studied and wave height used to elucidate the temporal response of beach variability. The Chapater is structured as follows. Section 5.3 presents the observing system, the algorithm for shoreline extraction and the wave data used. Section 5.4 presents the dynamical aspects of the ﬁrst two EOF modes and ﬁnally in Section 5.5 the main conclusions are outlined.

5.3 5.3.1

Data and Methodology Coastline Extraction

Coastline data were obtained from a video monitoring system (Nieto et al., 2010). The use of digital images allows an objective analysis as well as repeatability of the detection technique. Three types of images were obtained from the video system: i) snap shot images, ii) variance images which contains the standard deviation of the image intensity and iii) time exposure images consisting of the result of averaging 4500 images taken in 10 minutes, (Figure (5.1 a,b,c)). Time exposure images were used for shoreline detection since they remove visual features related to individual waves or moving objects such as people present on the beach being suitable to detect the shoreline or patterns of breaking waves (Plant et al., 2007). Images were obtained every hour of daylight. The ﬁrst step to detect shorelines was to transform RGB images into HSV images (Hue Saturation Value). Previous works on shoreline detection used histogram of hue and saturation values or grayscale intensity to distinguish between wet and dry pixels (Aarninkhof et al., 2003). As already noted by Ojeda and Guill´en (2008), some problems appear when using this methodology. We found that using the image resulting of multiplying the red channel and the saturation channel gave better results for shoreline detection in the area of interest. Images were obtained in Cala Millor Beach, from February 2007 till July 2008. Cala Millor is located in northeast coast of Mallorca Island (Northwestern Mediterranean) (Figure (5.2)). It is 1700 m long with a variable width going from 15 m to 30 m.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

NO Shoreline pattern

Shoreline pattern

Figure 5.1: Timex images (a,b,c), black lines limit the useless parts of the images; in image b there are plotted examples of shoreline patterns. Shorline over an utm converted image, d.

40oN

50’

40’

30’

20’

15’

30’

45’

3oE

15’

30’

Figure 5.2: Geographic location of the pilot area, left. Aerial photo of Cala Millor Beach, black lines deﬁne the area captured by the cameras.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

Typical significant wave heigths Hs , are around 1m and the associated periods Tp around 3-6 s. Energetic waves come from the north and northeast with wave heights between 1-4 m, (Alvarez-Ellacuria et al., 2010). The beach has a configuration of transverse bars placed between 0.5-4.5 m depth (Tintoré et al., 2009). In this location three cameras were installed on the top of building of 45 m height, which provides a perfect platform.

5.3.2

ANN Development

A methodology for automatic detection of shoreline based on an artificial neural network (ANN) was developed. We used a feed-forward ANN, which is equivalent to a multivariate multiple non-linear regression model (a scheme of the ANN is shown in Figure (5.3)). The network consists on a layer of input nodes, {xi }m i=1 (pixel intensity values) and a bias value 1, defining the output signal which is the shoreline location. The ANN learns the definition of shoreline patterns by a large number of training examples that consist of input/output pairs (xi /d). The connecting weights Wi , are chosen at random initially and a non linear logistic function is used to adapt the weight values to reduce the difference between simulated output y, and output signals d, reducing the error ε. For each camera a neural network was trained for 200 epoch (an epoch is one iteration

Figure 5.3: ANN scheme

of evaluating a network output and updating the network weights with the chosen algorithm) (Kingston et al., 2000). During the training process around 10.000 patterns were introduced in the ANN from which around 5.000 consisting in shoreline patterns. Shoreline position was defined by the surrounding pixel values. Each pattern contains 82 variables, 81 pixel intensity values and one value to define if it is a shoreline pattern or not (0 or 1) (see Figure (5.1b) for an example of training patterns). The existence of shoreline position is defined by the central pixel of the pattern. When the objective function FO, which depends on the square error ε is minimum, the iteration process finished and the weights obtained Wi are used over new images to detect shoreline patterns.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

Training patterns were obtained from those images where shoreline was correctly detected by using an edge detector over the result of the red channel multiplied by the saturation channel. The training processes was done only over images taken between 11:00 and 15:00 pm, to avoid differences due to sunrise or sunset light. As a result of the ANN three matrix of 81 weights, one for each camera, were obtained. These weights were multiplied as a moving filter over all the images to find the shoreline patterns and then the central pixel was defined as shoreline. Some filters were applied over the images to avoid wrong shoreline patterns due to shadows, umbrellas or threes. Finally the three images were joined and converted to UTM coordinates and a smoothing filter applied over the complete shoreline (Figure (5.1d)). To test the new methodology for the shoreline detection a validation was done by using the ANN for an ARGUS system installed in Barcelona beaches (Ojeda and Guillén, 2008). These images were provided by the Department of Marine Geology at CMIMA. Shoreline detected was compared with topographical measurements of shoreline. The mean differences between the two methods did not exceed the 2.3 m which is under the 4.7 m obtained when comparing the detection made by using the ARGUS software (Ojeda and Guillén, 2008). The error between the shoreline detected with the ANN methodology and shoreline topographic measurements for the same date is shown in Figure (5.4).

7 6

error (m)

5 4 3 2 1 0 0

100 150 longshore (m)

200

Figure 5.4: Error between ANN detected shoreline and topographic measurements in Somorrostro Beach, solid line marks the mean error.

5.3.3

5. Short-Term Shoreline Evolution in a Low-Energy Beach

Shoreline Analysis

To study shoreline variability a coordinate conversion was used to transform the oblique coordinates for the images into real coordinates (Holland et al., 1997). An example of converted image is shown in Figure (5.1d) where the beach was divided into three parts, north, central and south. This latter is a small portion around 80 m length. This irregular division is due to the fact that the south part of the beach was not completely captured by the Camera 3 (Figure (5.1c)). Pixel resolution range between 10 cm and 4 m in the northern part of Camera 1 (Figure (5.1a)). Video images are analyzed across 630 m longshore and 70 m cross-shore. Due to lens distortion not all the image pixels were used, and so the farthest parts of the beach were not analyzed. Some of the shorelines were wrongly detected due to weather conditions as rain, fog or cloudy sky. Moreover there were two gaps in the data set due to electronic problems and to dust coverage on the lens that were interpolated from the next and past shorelines in time. We obtain 71 shorelines weekly separated for the period of February 2007 to July 2008. An analysis was done over different days, to study sensitivity in front of different wave regime and tides. Mean differences between hourly shorelines were between 0.2-1 m with a maximum standard deviation of 5 m. These values are inside the pixel resolution and therefore we have not averaged the daily shorelines using the best shoreline detected each day. All the shorelines where pass through a five point running average filter to remove outlayers. These shorelines were interpolated to obtain equidistant points in longshore obtaining a set of 71 weekly shorelines. Shoreline spatial and temporal variability was analyzed using the Empirical Orthogonal Function (EOF) technique (Miller and Dean, 2007; Medell´ın et al., 2008). This analysis was used to identify dominant patterns of variability within uniformly distributed data set. EOF’s are the eigenfunctions of a covariance matrix, which when ordered by eigenvalue, represent the dominant pattern of the variance. The amplitudes of the EOF which represents the temporal variation of each spatial mode were finally related to the wave conditions.

5.3.4

Wave Data

Wave climate was obtained by a reanalysis of WAM model. In the western Mediterranean Sea, these data are provided by the Spanish Harbors Authority every three hours over a mesh of 0.125. We used for the period of study the signiﬁcant wave height (Hs ), peak period (Tp ) and direction from deep waters (ca. 50 m). Wave climate at the shore was

5. Short-Term Shoreline Evolution in a Low-Energy Beach

done by using an interpolation matrix (Alvarez-Ellacuria et al., 2009). This matrix was created using diﬀerent combinations of Hs , Tp and directions that were propagated using a mild slope parabolic model (Kirby and Ozkan, 1994; Liu and Losada, 2002). The use of this method enables a fast propagation of whole deep data set which otherwise would be computationally too expensive. Deep and shallow water waves were correlated with temporal EOF amplitudes. Correlations were done with the mean value of Hs obtained averaging the 7 days before each shoreline extraction. Moreover, correlation was also computed for the Hs averaged over the last 6, 5, 4, 3, 2 and 1 previous days.

5.4

Results and Discussion

The first three EOF explain the 78% of the total variability. The spatial patterns and their corresponding amplitudes for these EOF are presented in Figure (5.5) and Figure (5.6) respectively. EOF1 , which represents the 50.7% of the total variance is negative in its spatial mode (see Figure (5.5), top panel). This, indicates that EOF1 represents acrretion/erosion of the beach depending on negative/positive values of its amplitude (see Figure (5.6), top panel). Moreover, EOF1 can be related with the sediment supply provided by the bar system. It is widely accepted that in a dynamically stable beach configuration, sandbars usually move offshore during storm episodes, when strong seawards currents (undertows) dominate the sediment transport phenomena and onshore sandbar migration occurs between storm events when wave energy is lower. Thereby, negative values of the temporal amplitude of EOF1 correspond to accretion processes associated to mild wave conditions and positive values to erosion after storm events with offshore sandbar migration. EOF2 , which represents the 17.4% of the total variance explains the beach variability induced by longshore sediment transport as a response of the different incoming wave direction. The spatial mode of EOF2 shows a central area where the mode is negative being positive in the northern and southern part (see Figure (5.5), middle panel). This different configuration produces a modal variation in this EOF depending on the sign of its corresponding amplitude (Figure (5.6), middle panel). Finally, EOF3 explains a 10.1% of the total variability. The spatial pattern of this EOF turns around a central point with positive values in the north part of the beach and negative values in the south (Figure (5.5), bottom panel). Therefore, positive amplitudes of EOF3 explains a clockwise rotation of the beach and negative values of this temporal

5. Short-Term Shoreline Evolution in a Low-Energy Beach

amplitude a counterclockwise rotation (Figure (5.6), bottom panel).

Figure 5.5: EOF1,2,3 spatial modes of the temporal variance from the EOF analysis.

Figure 5.6: Temporal amplitude of the EOF1,2,3 and wave data related to shorelines.

To better understand the physical meaning given to the ďŹ rst two EOF modes, the temporal amplitudes a1 and a2 were correlated with the deep and shallow water wave

5. Short-Term Shoreline Evolution in a Low-Energy Beach

conditions. Moreover, shallow water waves were decomposed in a longshore and crossshore (to the beach) components. As expected, no significant correlation was found between a1 and the longshore wave components for shallow water waves. We did not either obtain significant correlation between a2 and the shallow water components of wave data but significant correlations were obtained for this amplitude and deep water wave conditions . Table (5.1) displays significant (at 95% of confidence level), correlation coefficients between the amplitude a1 and the cross-shore wave component in shallow waters and between a2 and deep water wave conditions.

SW a1-Hs50 ⊥ DW a2-Hs50

1 day

1-2 days

1-3 days

1-4 days

1-5 days

1-6 days

1-7 days

0,31 -

0,37 0,28

0,38 0,36

0,43 0,48

0,36 0,55

0,31 0,55

0,29 0,57

Table 5.1: Correlation between ﬁrst amplitude (a1) and cross-shore (⊥) component of shallow (SW ) wave data and correlation between second amplitude (a2) and deep waters (DW )wave data. Correlations are given for wave data corresponding to periods ranging from 1 to 7 days before shoreline extraction

Even if a1 is also significantly correlated with deep water waves, correlation coefficients are much lower and therefore not shown in the table. The wave conditions have been evaluated for the average of the last 7, 6, 5, 4, 3, 2 and 1 previous days of the shoreline extraction. First temporal amplitude and cross shore wave component in shallow SW ⊥) presents the highest correlation for the significant wave height averwaters (Hs,50 SW,4 ). For the second aged for the four days before the shoreline detection (hereafter, Hs,50 amplitude, a2 , highest correlations are obtained for the averaged deep water wave conditions of the previous 7 days. Correlation between the net shoreline differences and wave conditions at the nearshore were significatively found for all wave averaged periods but SW ⊥. Since the net differences are again the highest value, -0.58, was obtained for the Hs,50 only related with the accretion or retrieve of the shoreline this suggests that cross-shore movements are correlated with the mean wave climate during the last four days. To further test the physical explanation of EOF1 and EOF2 we represent the two amplitudes of these EOFs in an Euclidean space (Figure (5.7)) where four different areas have been defined by the intersection of the axes. Areas A1 and A2 correspond to positive values of the temporal amplitude of EOF1 and positive and negative values for the temporal amplitude of EOF2 respectively. Those shorelines whose first two EOFs lie near the ordinate axes correspond to beach states where the longshore sediment dominates the dynamics. The positive ordinate axes indicates a beach eroding in the central part and a sediment supply in the southern and northern part while negative ordinate indicates an accretion in the central part with sediment transported form the two extremes.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

Figure 5.7: Temporal amplitudes of 1s t and 2n d mode. Inside the graphic there are displayed the week number and the signiďŹ cant wave height and direction of the 50% of the four days before

Large values in ordinate axis can be associated with changes in the beach shape without changes in the total dry area (e.g., shorelines 5 and 71). Those shorelines near the abscissa axes indicate modal states where the cross-shore sediment dominates. Therefore positive abscissa correspond mainly to erosion induced by severe storms while negative abscissa to accretion under relatively mild conditions. In fact, points near the abscissa axes correspond to that changes in these shorelines have been produced in the total beach area (e.g., shorelines 59 and 12). Within A1 area, states are a mixture of both processes where the shoreline is eroding and the central part provides sediment mostly for the southern part. It has to be noted that wave directions in A1 are mostly from ENE and in A2 are mainly ESE or E. The dynamics induced by these waves explain the variation of EOF2 since, as explained, is related to longshore movements. Relevant values of Hs as well as their directions are indicated in Figure (5.7). Most of the shorelines measured in summer (crosses) fall near the origin indicating almost null variability during this period.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

To test the morphodynamical explanation given to the EOF analysis, a numerical alez et al., 2007). The model is a 2D depth integrated model was implemented (Gonz´ Navier-Stokes solver with the radiation stresses as a forcing term. The stresses are obtained by propagating the water wave conditions using a mild slope parabolic model (Alvarez-Ellacuria et al., 2010). Water wave conditions are the mean significant wave Hs,50 of the four days before the shoreline detection. This data is obtained from the time series of offshore wave conditions. Three different cases were selected to study specific shorelines. The first case correspond to the period between the weeks 4 and 5 where mean wave conditions at the beach were Hs,50 = 0.48m, Tp = 7.6s and θ = EN E(74) (Figure (5.8), top panel).

Figure 5.8: Surfzone currents generated by mean shallow water conditions between shorelines 4-5, top panel, 65-66, center panel and 68-69 bottom panel. Solid lines represent the resulting shoreline after the currents, dashed line represent the shoreline before the currents action

As seen, prevailing breaking currents for this period induce a longshore sediment transport to the southern part. Sediment from the central part is transported to the south producing a shoreline retrieve in the central part and an accretion in the south (see grey line for the shoreline corresponding to week 4 and black line for week 5). We want to remark here that the shoreline for the week 4 is near negative abscissa axes in Figure (5.7) while shoreline for the week 5 is in the positive ordinate axes which explains the sediment supply from the central part to the south.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

The second case corresponds to the period between shorelines 65 and 66 where mean wave conditions were Hs,50 = 1.7m, Tp = 6.4s and θ = E(109) . Under these conditions sediment is transported cross-shore moving back the shoreline all along the beach from week 65 (grey) to week 66 (black) (Figure (5.8), middle panel). Note that shoreline 65 is over the ordinate axes in Figure (5.7) while shoreline 66 is near the positive abscissa axes explaining not only the cross shore erosion occurred between these two weeks but also the pattern followed by the shoreline. Finally we explain the prevailing ESE conditions which occur between shorelines 68 and 69. The nearshore wave conditions for this period is Hs,50 = 0.33m, Tp = 6.6s and θ = ESE(121). A longshore transport drives the sediment from the southern part of the beach to the central part (Figure (5.8), bottom panel). In this case both shorelines remain in the same abscissas value but increase in negative values in the ordinate axes (Figure (5.7). Therefore, we can explain changes between these two shorelines as the result of a longshore sediment transport which produces changes in the shape of the beach but almost no differences in the total dry area. A further description of the dynamical sense of EOF modes can be obtained by computing the average shoreline difference between two consecutive measurements and its standard deviation (Figure (5.9), top and middle panel). The shoreline mean difference is related with changes in the beach area while large values of its standard deviation with changes in shape that are not necessarily related with an accretion or erosion of the dry beach. For instance, between week 4 and 5 there is almost no change in mean shoreline position (Figure (5.9), top) but it presents a large value for its standard deviation (Figure ( 5.9)). This situation correspond to mild wave conditions (see Figure (5.8), top). In fact, this situation correspond in Figure (5.7) to large differences in the ordinate axis but small differences in the abscissa axis which correspond to longshore sediment transport. A similar situation is obtained for the difference between week 68-69 with wave conditions inducing sediment transport in the opposite direction. SW 4 Large accretion corresponds to low energy conditions of Hs50 (Figure (5.9), bottom), while erosion corresponds to high energetic conditions (e.g. between weeks 5-6,7-8, 14-15, 45-46 and 65-66).

The above results could be understood as the beach responding to wave climate by using two mechanism: ﬁrst, the beach defense in front wave climate is to retrieve its position by adding sand to the submerged sandbar. When this mechanism is not suﬃcient, the beach redistribute the sand alongshore to better dissipate the energy from waves. This redistribution is from the central part to the two lateral sides and vice-versa.

5. Short-Term Shoreline Evolution in a Low-Energy Beach

Figure 5.9: Mean diﬀerences inter shorelines, top panel. Related standard deviation, middle SW 4 panel and associated Hs,50 , bottom panel.

5.5

Conclusions

In this work we have analyzed the short term variability of a microtidal low energetic beach by the analysis its shoreline variability. By applying an ANN to raw images obtained from a video monitoring system, daily shorelines where obtained for the period of February 2007 to July 2008. The best detected image for each day was used to obtain a set of 71 weekly data. To study the spatial and temporal variability of the beach a temporal EOF decomposition was applied to these data. From this analysis we can conclude that the first EOF correspond to the shoreline change due to cross shore sediment transport while the second and third EOF to the change induced by longshore sediment transport. The first two EOF which represent the 68% of the beach shoreline variability have been correlated with the significant mean wave height for different periods. We found that the temporal amplitude of EOF1 , showed the highest correlation with the perpendicular component of the significant wave height at shallow waters averaged for SW 4 the previous four days after shoreline detection (Hs,50 ). For the temporal amplitude of EOF2 correlation with deep water data were obtained which is larger for the mean

5. Short-Term Shoreline Evolution in a Low-Energy Beach

signiďŹ cant wave height averaged for the previous 7 days. No correlation was found for the second EOF and the shallow water wave data (parallel and perpendicular to the beach) due to wave refraction to the shore. The beach studied which is characterized as a microtidal low energy wave dominated shows two temporal scales of interrelation between wave forcing and morphological change in the short-term. We found that these scales are 4 days for the cross-shore sediment transport and 7 days for the longshore sediment transport.

Chapter 6

General Conclusions The aim of this Thesis has been to generate tools for beach management based on scientific knowledge of nearshore hydrodynamics and morphodynamics and to develop systems capable of offering useful information to coastal managers. Specifically, the tools developed, have been applied to build operational systems for beach safety as well as morphodynamic evolution. In this Thesis an operational system for the prediction of nearshore waves and associated wave breaking currents has been successfully implemented. This system provided good estimations of the occurrence, location and intensity of wave breaking currents. Results were disseminated via web and coastal managers and lifeguards obtained useful information for beach security issues. Extension of the above system for large areas of the littoral presents some drawbacks. First, to propagate accurately wave conditions from deep to shallow waters, a detailed bathymetry is needed. In shallow and sandy areas, bottom evolves constantly as waves do and therefore bathymetry has to be updated since it plays a crucial role in wave propagation and in the generation of breaking currents. This fact, makes the above system unrealistic since the large economical cost derived of generating constantly bathymetric surveys. Secondly, the numerical models used in the operational system require high resolution domains to solve all kind of incoming waves. This implies the use of powerful computational resources that not always are available in the coastal management administration. In this Thesis, these problems have been solved by modifying the propagation system, generating a large catalog of propagations and linking deep water wave conditions with the different databases. This new forecasting system requires a constant feedback between beach managers and lifeguards.

6. General Conclusions

Daily reports have been included in the new system to calibrate as well as to validate the forecasted risk level at each location. Finally 15 beaches around the Balearic Islands have available the risk forecasting system for beach conditions. Although the forecasting systems developed provide good results, during this Thesis arise other problems associated with the lack of having in real time autonomous data of wave and beach morphology. These problems were solved using a video remote sensing platform which provides the possibility of having at a relatively low cost data of wave conditions, shoreline position, areas of breaking waves, etc. In the Thesis the beach morphodynamic response to wave climate in the short and medium term has been achieved by the analysis of the digital images of the video system. A new methodology has been developed to detect automatically shorelines using Artificial Neural Networks. The use of Empirical Orthogonal Functions together with numerical models has provided the general overview of the dynamics of a microtidal low energetic beach. By the combination of these novel techniques the dynamical pattern of the study site has been solved obtaining that sediment transport responds fundamentally at two different temporal scales. The cross-shore movements is related to the first EOF which turns to be correlated with the wave regime of the previous four days prior shoreline extraction. The longshore movements are related to the second EOF which depends on deep water wave climate seven days before shoreline extraction. This results are the basis for the hypothesis that the beach responds to wave climate by using two mechanism: first, the beach defense itself in front wave climate retrieving its position by adding sand to the submerged sandbar. When this mechanism is not sufficient, the beach redistribute the sand alongshore to better dissipate the energy from waves. This hypothesis should be demonstrated in a future work by studying the sediment transport processes between sandbar and dry beach. The studies presented in this Thesis are the first attempt to include the hydrodynamic and morphodynamic knowledge as an additional tool for beach management in the Balearic Island. As outlined in this Thesis, scientific improvements related to nearshore hydrodynamics and morphodynamics have a high interest for the costal management and should be spread in a useful form for being easily comprehensible by society.

Chapter 7

Future Work During my PhD research I have found large gaps in the explanation of some hydrodynamic and morphodynamic processes occurring in the nearshore. In the area of morphodynamics these gaps are specially important in the explanation of the sediment transport processes between bars and dry beach. Beach erosion and coastal evolution are in a global change context, top scientific issues. 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. For instance, large gaps remain in our knowledge on the mechanisms involved in the sediment transport processes. Masselink and Puleo (2006) concluded that one of principal factors in the lack of understanding of sediment transport processes is due to the fact that the surfzone and swash zone are treated as independent areas. van Maanen et al. (2008) highlighted as a priority the development of new coupled models for the surf and swash zone processes. Development of new methods of observation that will provide reliable, well-validated and practical models for longshore and crossshore sediment transport is necessary. The causes of shoreward sediment transport and sandbar migration are not known, and therefore models for beach evolution are not accurate . Over the years, a number of processes have been held responsible for onshore bar migration including wave skewness (Ruessink et al., 2007), wave asymmetry Hoefel and Elgar (2003), near-bed streaming, Stokes drift and some others (van Maanen et al., 2008). It is widely acknowledged that the majority of coastal sediment transport occurs in the nearshore zone, including the wave-breaking zone (the zone where the predominant 75

7. Future Work

processes are the breaking waves) and the swash zone (Masselink and Russell, 2006). The improvement of the current theories and models makes necessary the quantification of sediment transport both in the surf and swash zones under a global point of view. Efforts should be now focussed on investigating which processes are responsible for the onshore/offshore sediment transport from the bar to the dry beach under an experimental and theoretical approach. Future research will be focused in describing morphodynamically the accretion/erosion processes of natural barred sandy beaches by analyzing experimental data measured over the surf and swash zones. A vast body of knowledge of individual processes is available from the literature but, to date, none of the published studies deal with an integrated point of view the sediment transport in the surf and the swash zone and between them. Many of the studies point in their conclusions the need of new coupled models for sediment transport that describe the surf and swash zone processes. Next step in the sediment transport studies should be to advance in the knowledge of the processes involved in the onshore/offshore sediment transport by merging on field experiments with numerical models. The interaction between hydrodynamics and morphodynamics demands a multidisciplinary approach integrating field experiment techniques, fundaments in nearshore hydrodynamics, numerical modeling, sediment transport and morphodynamics.

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Appendix A

Navier-Stokes Equations A.1

Navier-Stokes Equations

The Navier-Stokes equations describe the motion of viscous ﬂuid substances such as liquids and gases. These equations arise from applying Newton’s second law to ﬂuid motion. Newtons second law ∑ F⃗ = m · ⃗a. (A.1) Forces acting over a mass can be separate in Volumetric forces + Surface forces = Inertia forces Volumetric forces act all over the mass without physical contact, normally this forces are represented by gravity. F⃗ = Fx + Fy + Fz .

(A.2)

Surface forces are formed by shear and normal stresses, next equations represent the surfaces forces acting on x direction. ( ) ( ) ∂σxx ∆x ∂σxx ∆x fx = σxx + ∆y∆z − σxx − ∆y∆z ∂x 2 ∂x 2 ( ) ( ) ∂τyx ∆y ∂τyx ∆y + τyx + ∆x∆z − τyx − ∆x∆z ∂y 2 ∂y 2 ( ) ( ) ∂τzx ∆z ∂τzx ∆z + τzx + ∆x∆y − τzx − ∆x∆y, ∂z 2 ∂z 2 ( fx =

∂σxx ∂τyx ∂τzx + + ∂x ∂y ∂z 83

) dV.

(A.3)

A. Navier-Stokes Equations

Inertia forces are represented by the m⃗a and the acceleration can be described as Du ∂u ∂u dx ∂u dy ∂u dz = + + + , Dt ∂t ∂x dt ∂y dt ∂z dt ∂u ∂u ∂u ∂u Du = +u +v +w . Dt ∂t ∂x ∂y ∂z

(A.4)

Newtons 2nd law, for the forces acting in x axis can be write as Fx dm + fx = ax dm, ) ∂σxx ∂τyx ∂τzx Du Fx ρdV + + + dV = ρdV , ∂x ∂y ∂z Dt ( ) ∂τyx ∂τzx ∂σxx Du + + =ρ − Fx . ∂x ∂y ∂z Dt (

where

( ) ∂σxx ∂p ∂ ∂u 2 ∂ =− + 2µ − µ (∇ · ⃗u), ∂x ∂x ∂x ∂x 3 ∂x ( ) ∂τyx ∂ ∂u ∂v =µ + , ∂y ∂y ∂y ∂x ( ) ∂τyx ∂ ∂u ∂w =µ + , ∂z ∂w ∂w ∂x

(A.5) (A.6) (A.7)

(A.8) (A.9) (A.10)

substituting the terms in the Eq.(A.7) 1 − ∇p + µ∇2 u + µ∇(∇ · ⃗u) = ρ(ax − Fx ). 3

(A.11)

Oredering the terms we arrive to the Navier-Stokes equation showed in the introduction. D⃗u 1 1 = − ∇p + ν∇2 ⃗u + ν∇(∇ · ⃗u) + ⃗g Dt ρ 3 where ν = µρ , is the kinematic viscosity and the volumetric force Fx = ⃗g , is the gravity.

(A.12)

Appendix B

Radiation Stress B.1

Radiation Stress

There is a net mass transport in the wave due to the orbital velocities asymmetry. This mass transport, M, is deﬁned as, ∫

ρudz.

(B.1)

−h

This mass transport is integred through all the water column and on the wave period. The overline means, f=

1 T

∫

t+T

f dt.

(B.2)

However, in the region between the trough and the wave crest, the horizontal velocity must be obtained by the Taylor series, as the problem in linear theory is deﬁned between -h and 0.

∂u

u(x, η) = u(x, 0) + η , (B.3) ∂z z=0

gAk cosh k(h + z)

gA2 k 2 tanh kh cos2 (kx − ωt) cos(kx − ωt) +

ω cosh kh ω z=0 gAk = cos(kx − ωt) + A2 kω cos2 (kx − ωt). ω

The surface velocity is periodic, yet faster at the wave crest than at the wave trough, as the second term is always positive at these two phase positions. This asymmetry of velocity indicates that more ﬂuid moves in the wave direction under the wave crest than 85

B. Radiation Stress

in the trough region. This is , in fact, true. If we average u(x,η) over a wave period there is a mean transport of water ∫ 1 T A2 kω (kA)2 C u(x, η) = u(x, η)dt = = , (B.4) T 0 2 2 where C is the phase speed of the wave. To obtain the mass transport M, ∫

∫

M= −h

∫

(ρu)dz =

ρ¯ udz +

∫

−h η

ρudz

(B.5)

E . C

(B.6)

ρudz = ρηu = 0

There is no mass ﬂow except due to the contribution of the region bounded vertically by η. This mass transport has a momentum associated with it, which means that forces will be generated whenever this momentum changes magnitude or direction by Newton’s second law. ∫ η (B.7) (ρu)udz = M Cg , −h

where u2 =

gA2 k 1 + cosh 2k(h + z), sinh 2kh

(B.8)

where M is the mass flux, and Cg is the group celerity, the speed at which the wave energy propagates, defined as, ( ) 1 2kh Cg = C (B.9) 1+ = nC, 2 sinh2kh The flux of momentum in the direction of the wave past a section and the pressure force per unit width is defined as ∫ Ix = M Cg +

p(z)dz,

(B.10)

−h

second term represents the induced pressure forces, these forces can be represented as ∫ η¯ 1 Fp = ρg(¯ η − z)dz = ρg (h + η¯)2 . (B.11) 2 −h Eq(B.10) can be rewritten as

1 Ix = ρg (h + η¯)2 + {M Cg + 2

∫

1 p(z)dz − ρg (h + η¯)2 }. 2 −h

(B.12)

B. Radiation Stress

The second term represents the excess of the momentum ﬂux respect to the hydrostatic forces, is known as the Radiation Stress, Sxx . Using linear theory the Radiation Stress can be deﬁned as ( ) 1 Sxx = E 2n − (B.13) , 2 (B.14) where, 1 ρgH 2 , 8 ( ) 1 2kh n= 1+ . 2 sinh 2kh E=

(B.15) (B.16)

The same term can be obtained for the y axis 1 Syy = E(n − ). 2

(B.17) (B.18)

If the progressive wave is propagating at some angle Θ to the x axis, the values of Sxx and Syy are modiﬁed, [ ] 1 2 Sxx = E n(cos Θ + 1) − (B.19) , 2 [ ] 1 Syy = E n(sin2 Θ + 1) − , (B.20) 2 (B.21) for this case there is an additional term representing the ﬂux in the x direction of the y component of momentum, denoted Sxy Sxy =

E n sin 2ω. 2

(B.22)

Appendix C

Mild slope equation C.1

Mild slope equation

The Mild Slope Equation describes the propagation of the wave in time over the horizontal x,y-domain over a slowly varying depth. For the description of motion we use the velocity potential Φ which satisﬁes the Laplace equation ∇2 Φ = 0.

(C.1)

The velocity potential using separation of variables, can be written as, Φ = ϕs (x, y, 0, t) · f(z),

(C.2)

where ϕs is the velocity potential at the MWL (mean water level) and f (z) is the depth variation: cosh k(z + h) f (z) = , (C.3) cosh kh The solution of a traveling wave in the x-direction is: Φ(x, z, t) = −

Ag cosh k(h + z) sin (kx − ωt). ω cosh kh

(C.4)

Substituting in C.4 cosh k(h + z) by, cosh k(z + h) = cosh kz cosh kh + sinh kz sinh kh,

(C.5)

and deriving twice in z axis we obtain ∂2Φ = k2 Φ, ∂z 2 89

(C.6)

C. Mild slope equation

and the Laplace equation Eq.(C.1) can be rewritten as ∇2 Φ + k2 Φ = 0.

(C.7)

To incorporate the eﬀect of the depth variation, we use the Green’s 2nd theorem ∫

(ϕ1 ∇2 ϕ2 − ϕ2 ∇2 ϕ1 )dx = [ϕ1 ∇ϕ2 − ϕ2 ∇ϕ1 ]ba ,

(C.8)

∫

(

−h

∂ 2 f(z) ∂2Φ f(z) − Φ ∂z 2 ∂z 2

)

( dz =

∂f(z) ∂Φ f(z) − Φ ∂z ∂z

)

( − 0

∂f(z) ∂Φ f(z) − Φ ∂z ∂z

) , −h

(C.9) we substitute variables of the Eq.(C.9) by using the linearized boundary conditions for the initial equation Eq.(C.10), Eq.(C.11) and the equalities found for the f(z) in Eq.(C.12) and Eq.(C.14) ∂Φ 1 ∂ 2 Φ + =0 ∂z g ∂z 2 ∂Φ + ∇h · ∇Φ = 0 ∂z (

∂f(z) ∂z

z = 0,

(C.10)

z = −h,

(C.11)

) = 0,

(C.12)

−h

f(0) = 1, (

∂f(z) ∂z

) = k tanh kh = 0

(C.13) ω2 . g

(C.14)

The equation Eq.(C.9) can be rewritten as ∫ −

0 −h

(f(z)∇2 Φ + Φk 2 f(z))dz = −

1 ∂ 2 ϕs ω2 f(0) − ϕ − (∇h · ∇Φf)−h . s g ∂t2 g

(C.15)

From Eq.(C.2), we obtain ∇Φ = f ∇ϕs + ϕs ∇f ,

(C.16)

∇2 Φ = f ∇2 ϕs + 2∇ϕs ∇f + ϕs ∇2 f ,

(C.17)

substituted into Eq.(C.15) gives ∫

{f 2 ∇2 ϕs + ∇ϕs ∇f 2 + ϕs f ∇2 f + ϕs f 2 k 2 }dz ( ) 1 ∂ 2 ϕs 2 = + ϕs ω − ∇h · (f 2 ∇ϕs + ϕs f ∇f )−h . g ∂t2 −h

(C.18)

C. Mild slope equation

Rearranging terms Eq(C.18) may be written ∫ ∫

∫

−h 0

∇2h (f 2 ∇ϕs )dz + [∇hf 2 ∇ϕs ]−h + ϕs k 2

1 =− ϕf ∇ f dz − ϕ∇h · [f ∇f ]−h + g −h

(

f 2 dz −h

) ∂ 2 ϕs 2 + ϕs ω . ∂t2

(C.19)

We apply Leibniz’rule to the ﬁrst two terms ∫ ∇(∇ϕs ) ∫ =−

∫

0 2

−h

f dz) + ϕs k

0 −h

ϕs f ∇2h f dz − ϕs ∇h · [f ∇f ]−h +

From Eq.(C.3)

∫

f(z) dz = −h

1 g

(

f 2 dz −h

) ∂ 2 ϕs 2 + ϕ ω . s ∂t2

ccg , g

(C.20)

(C.21)

substituting into Eq.(C.20) ∂ 2 ϕs − ∇ · (ccg ∇ϕs ) + (ω 2 − k 2 ccg )ϕs ∂t2 {∫

}

f ∇ dz + f + ∇h[f ∇f ]−h gϕs . 2

= −h

(C.22)

If we introduce the mild-slope assumption that ∇ ≪ kh we can avoid the terms of the right hand side of the equation having a result the time varying version of the Mild slope Equation ∂ 2 ϕs − ∇ · (ccg ∇ϕs ) + (ω 2 − k 2 ccg )ϕs = 0, ∂t2

(C.23)

if a time harmonic motion is introduced we get the dynamic free surface as ζ(x, y, t) = a(x, y)eiwt ,

(C.24)

the mild slope equation can be explained in terms of surface variation ∇ · (ccg ∇a) + k 2 ccg a = 0.

C.1.1

(C.25)

Parabolic approximation

An elliptic equation implies that the solution requires that it is solved in a closed domain with boundary conditions speciﬁed along the entire boundary, other disadvantage of this equation is that is numerically time-consuming to solve. The parabolic approximation

C. Mild slope equation

transforms the elliptic equation into an equation which is of ﬁrst order in the main direction of wave propagation(x)and of second order in the y-direction. The equation(C.23) can be rewrite in the expanded form as ccg

∂ccg ∂ϕs ∂ccg ∂ 2 ϕs ∂ 2 ϕs + + + k 2 ccg ϕs = 0. ∂x2 ∂x ∂x ∂y ∂x2

(C.26)

To correct derive the formula it is assume: The wave motion propagates in a direction close to the x-axis. We use the form of ϕs ϕs (x, y, t) = −

∫ ig ′ A (x, y)ei( kdx−ωt) + c.c., ω

(C.27)

where c.c. stands for the complex conjugate component and A’(x,y) is the (slowly varying)amplitude of the free surface. The imaginary part of A(x,y) represents the phase diﬀerences between the actual wave motion and the motion described by the factor exponential. A(x,y) will include the slow amplitude variations along the wave front. The variation of the amplitude A is assumed to have diﬀerent length scales in the x and y directions. LX = O(ϵ2 Lx ),

(C.28)

where Lx is of the order of the wave length and ϵ is a small parameter. LY = O(ϵLx ),

(C.29)

the variation of the amplitude in the y-direction is an order of magnitude slower than in the x-direction. The equation C.27 becomes ϕs = −

∫ ig ′ A(X,Y ) ei( k(X,Y ) dx−ωt) . ω

(C.30)

The depth variation is assumed of the same magnitude in the x and y directions with ∇h = O(ϵ2 ) so that ∂k ∂k , = O(ϵ2 k), (C.31) ∂x ∂y and similar for ccg which is a function of k. Now the surface velocity potential can be write as ) ∫ ( ′ ig 2 ∂A ′ ϕs = − ϵ + ikA ei( k(X,Y ) dx−ωt) . 2ω ∂X ϕ2s = −

ig 2ω

( ) ∫ ∂ 2 A′ ∂A ′ ∂k ′ 2 2 ′ 2 ϵ4 + ϵ 2ik A − k A ei( k(X,Y ) dx−ωt) . + ϵ 2i ∂X 2 ∂X ∂X

ig Substituting into the equation(C.26) and dividing by − 2w .

(C.32)

(C.33)

C. Mild slope equation

( ) ∂(ccg ) 2 ∂A′ ∂ 2 A′ ∂A′ ϵ + ikA′ + ccg (ϵ4 + ϵ2 2ik 2 ∂x ∂x ∂x ∂x ( ) 2 ′ ∂k ∂ccg ∂ A +ϵ2 i A′ − k 2 A′ ) + ϵ2 + k 2 ccg A′ = 0 ∂x ∂y ∂y 2

ϵ2

We neglect the O(1) and O(ϵ4 ) and consider only O(ϵ2 ) and we get ( ) ∂A′ ∂kccg ′ ∂ccg ∂ 2 A′ 2ikccg +i A + =0 ∂x ∂x ∂y ∂y 2

(C.34)

(C.35)

The amplitude is redeﬁned including a constant reference number to k0 A = A′ ei(k0 x−

∫

kdx)

(C.36)

Substituting the new amplitude in the equation(C.34) we get the parabolic equation on the form ∂A ∂kccg ∂ccg ∂ 2 A 2ikccg + 2k(k − k0 )ccg A + i A+ (C.37) ∂x ∂x ∂y ∂y 2 Kirby and Dalrymple (1983b) included in this equation a term representing the ﬁrst approximation to nonlinear eﬀects, the last term of equation(C.38). 2ikccg

∂A ∂kccg ∂ccg ∂ 2 A c + 2k(k − k0 )ccg A + i A+ − kccg k 3 D|A|2 A = 0 ∂x ∂x ∂y ∂y 2 cg

(C.38)

Appendix D

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