simulation of wave climate in the nazare area using ... · wavewatch iii is a wave generation and...

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Simulation of Wave Climate in the Nazare area using

WAVEWATCH III

Nuno Ricardo Cascarejo Caeiro

Thesis to obtain the Master in Science Degree in

Mestrado Integrado em Engenharia Mecânica

Supervisor: Prof.Dr. Ramiro Joaquim de Jesus Neves

Co-supervisor: Francisco Javier Campuzano Guillen

Examination Committee

Chairperson: Prof.Dr. Edgar Caetano Fernandes

Supervisor: Prof.Dr. Ramiro Joaquim de Jesus Neves

Members of the Committee: Prof.Dr. António José Nunes de Almeida Sarmento

May 2017

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Acknowledgments

It has been an enriching journey that now ends with the feeling of accomplishment. I would like to

express my gratitude to my colleagues that have traveled this path along with me and with whom I’ve

shared a lot of memorable moments.

I’m thankful to all the teachers with whom I came across, for exceeding yourselves so that each class

could be an inspiring learning experience. I would like to leave a word of remembrance to Prof. José

Miguel C. Mendes Lopes, for his touching guidance since the beginning of this journey either as

professor and tutor

I am deeply grateful to Prof. Ramiro Neves for the ambitious challenge proposed and for his generous

advice throughout the development of this work.

I am thankful to MARETEC, for friendly welcome me into this research group. A special word is given

to Guilherme Franz and Francisco Campuzano for being always available to patiently share their

expertise with me.

I would like to thank my family for all the support, particularly to my sister, my lovely little nieces and to

my parents for providing and supporting the best education I could have.

Finally I would like to thank Sofia, for always being comprehensive and giving me the encouragement

that made this work possible.

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Resumo

As ondas oceânicas constituem um dos mais desafiantes fenómenos naturais para modelação visto

serem um escoamento com superfície livre e pressão não hidrostática, numa geometria natural

complexa e com transferência de energia ao longo do escoamento. A complexidade aumenta ainda

por a uma taxa de transferência de energia depender da própria onda e porque, por ser lenta, exige a

simulação de domínios oceânicos, muito extensos. A quantificação destes processos é ainda

dificultada pelas incertezas associadas aos resultados dos modelos meteorológicos necessários à

geração e propagação das ondas.

O Modelo WAVEWATCH III é um modelo de geração e de propagação de ondas, aplicável à escala

do oceano, também capaz de representar a propagação de ondas em zonas próximas da costa. Este

modelo foi configurado para a região da Nazaré ao longo de um período de 61 dias sendo que a

validação do modelo foi obtida pela comparação dos resultados com os dados reais da bóia

oceanográfica Monican02, colocada pelo Instituto Hidrográfico de Portugal junto à linha costeira da

Nazaré, revelando uma correlação superior a 90% no que diz respeito à altura significativa.

Um total de 8 diferentes simulações foram consideradas para análise, alterando as definições do

modelo de forma a poder avaliar o impacto desses parâmetros na qualidade dos resultados. Desta

análise foi possível concluir que a melhoria do passo temporal conduz a resultados mais precisos; um

aumento da resolução do modelo de vento é também um factor de melhoria nas características

simuladas da onda; o valor sugerido para o coeficiente de atrito do fundo para condições de swell é,

na realidade, o mais adequado para todas as condições marítimas; e o modelo tende a produzir

melhores resultados para simulações efectuadas num período de Inverno, em comparação com um

período de Primavera. Para a resolução da malha computacional os resultados não comprovaram a

hipótese de que uma maior resolução exerce uma influência positiva nas características de onda

simuladas.

Palavras-chave: Modelo de ondas, WAVEWATCH III, resolução da malha, passo temporal, atrito do

fundo, modelo de vento.

ii

Abstract

Ocean waves are one of the most challenging natural processes to model since it is a free surface

flow with no hydrostatic pressure, in a natural complex geometry with energy transfer throughout the

flow. The complexity is even increased because the energy transfer rate is depending on the wave

itself and since it is a slow process it requires the simulation of very large oceanic domains. The

quantification of such processes is even more difficult because of the uncertainties related to the

meteorological models results, necessary for the wave generation and propagation.

WAVEWATCH III is a wave generation and propagation model developed for oceanic applications but

also capable of representing waves in near shore areas. The model was configured in the Nazare

area for a 61 days period time, and the validation was performed by confronting the obtained results

with data from the oceanographic buoy Monican02, placed close to Nazare shoreline by Instituto

Hidrográfico de Portugal, which showed correlations above 90% for significant wave height.

A total of 8 different simulation runs were performed, with varying definition settings in order to

evaluate the impact of those parameters in the quality of results. From this analysis it was possible to

conclude that an increase in temporal resolution could lead to more accurate results, a wind model

with better resolution is also a factor of improvement in the wave’s simulated characteristics, the

suggested value for the bottom friction coefficient in swell conditions is the more appropriate for all

ocean state conditions and the model tends to provide a better correlation for simulations conducted

during a winter period rather than a summer period. Regarding the grid resolution of the model, results

didn’t corroborate the hypothesis that a more detailed grid would exert positive influence in the

obtained wave characteristics.

Key Words: Wave model, WAVEWATCH III, grid resolution, time step, bottom friction, wind model.

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Content

1 Introduction ....................................................................................................................................... 1

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

1.1.1 Waves in Nazare ................................................................................................................. 1

1.1.2 Wave modeling .................................................................................................................... 3

1.2 Methodology ............................................................................................................................ 4

1.3 Objectives ................................................................................................................................ 6

2 Literature Review ............................................................................................................................. 7

2.1 Introduction .............................................................................................................................. 7

2.2 Linear Wave Theory ................................................................................................................ 8

2.2.1 Random-phase/amplitude model ....................................................................................... 10

2.3 Generation and Propagation ................................................................................................. 12

2.3.1 Mechanism of Wave’s generation ..................................................................................... 12

2.3.2 Propagation of Waves ....................................................................................................... 12

2.4 Nearshore Processes ............................................................................................................ 13

2.4.1 Linear wave theory for coastal waters ............................................................................... 13

2.4.2 Transformation processes ................................................................................................. 15

2.5 Wave Model - WAVEWATCH III............................................................................................ 19

2.5.1 Governing Equations ......................................................................................................... 19

2.5.2 Wave physics parameterization ......................................................................................... 20

3 Wave Model ................................................................................................................................... 24

3.1 Model Definition ..................................................................................................................... 24

3.1.1 Model ................................................................................................................................. 24

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3.1.2 Grid Resolution .................................................................................................................. 27

3.1.3 Spectral Information .......................................................................................................... 27

3.1.4 Time Steps ......................................................................................................................... 27

3.1.5 Validation Period................................................................................................................ 29

3.1.6 Wind Model ........................................................................................................................ 29

3.1.7 Bottom Friction Coefficient ................................................................................................ 29

3.1.8 Test Scenarios ................................................................................................................... 30

3.2 Model Validation .................................................................................................................... 31

3.2.1 Real Data ........................................................................................................................... 31

3.2.2 Statistical parameters ........................................................................................................ 32

4 Results ........................................................................................................................................... 34

4.1 Results Presentation ............................................................................................................. 34

4.2 Results Analysis .................................................................................................................... 47

4.2.1 Grid Resolution .................................................................................................................. 48

4.2.2 Time Step .......................................................................................................................... 48

4.2.3 Wind Model ........................................................................................................................ 49

4.2.4 Bottom Friction ................................................................................................................... 50

4.2.5 Seasonal Comparison ....................................................................................................... 50

5 Conclusions and Future Developments ......................................................................................... 51

5.1 Conclusions ........................................................................................................................... 51

5.2 Future Developments ............................................................................................................ 52

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List of Tables

Table 2.1- Approximations for wave characteristics depending on water depth ................................... 14

Table 2.2 - The relative importance of the various processes affecting the evolution of waves in

oceanic and coastal waters (after Batjjes, 1994)................................................................................... 15

Table 2.3–WW III Parameterizations configured for this work .............................................................. 23

Table 3.1 - Computational grids ............................................................................................................ 24

Table 3.2 - Time step components for the conducted model runs ........................................................ 29

Table 3.3 – Parameters definition for all the conducted simulation runs .............................................. 30

Table 4.1 - Statistical Parameters for Significant Wave Height simulated results ................................ 34

Table 4.2 - Statistical Parameters for Mean Period simulated results .................................................. 34

Table 4.3 - Statistical Parameters for Mean Direction simulated results .............................................. 35

Table 4.4 - Computational cost vs Correlation for different time steps (R1, R3, R4) ............................ 49

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List of figures

Figure 1.1 – Nazare submarine Canyon ................................................................................................. 1

Figure 1.2 - Frequencies and periods of the vertical motions of the ocean surface (after Munk, 1950) . 4

Figure 1.3 - Spatial Grids ......................................................................................................................... 5

Figure 2.1 - Basic characteristics of waves ............................................................................................. 7

Figure 2.2 - Boundary Conditions for the linear wave theory .................................................................. 8

Figure 2.3 - Sum of a large number of harmonic wave components, travelling across the ocean with

different periods, directions, amplitudes and phases (after Pierson et al., 1955) ................................. 11

Figure 2.4 - Definition of distinct depth zones ....................................................................................... 13

Figure 2.5–Graphical representation of the hyperbolic tangent ............................................................ 14

Figure 3.1 - D1 North Atlantic spatial grid ............................................................................................. 25

Figure 3.2 - D2 Southwest Europe spatial grid...................................................................................... 25

Figure 3.3 - D3 Portugal Spatialgrid ...................................................................................................... 26

Figure 3.4 - D4 Nazare Spatial grid ....................................................................................................... 26

Figure 3.5 - Geographic representation of Monican01 and Monican02 (from IHP) .............................. 31

Figure 4.1 - R1 Hs time series and scatter plot ...................................................................................... 35

Figure 4.2 - R1 Tm time series and scatter plot ..................................................................................... 36

Figure 4.3 - R1 Dir time series and scatter plot ..................................................................................... 36


Figure 4.5 - R2 Tm time series and scatter plot ..................................................................................... 37

Figure 4.6–R2 Dir time series and scatter plot ...................................................................................... 38


Figure 4.8– R3 Tm time series and scatter plot ..................................................................................... 39

Figure 4.9 - R3 Dir time series and scatter plot ..................................................................................... 39

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Figure 4.10 – R4 Hs time series and scatter plot ................................................................................... 40

Figure 4.11 - R4 Tm time series and scatter plot ................................................................................... 40

Figure 4.12 - R4 Dir time series and scatter plot ................................................................................... 41

Figure 4.13 - R5 Hs time series and scatter plot .................................................................................... 41







Figure 4.20 – R7Tm time series and scatter plot ................................................................................... 45

Figure 4.21 – R7Dir time series and scatter plot ................................................................................... 45




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1 Introduction

1.1 Motivation

1.1.1 Waves in Nazare

In the west of Portugal there is an old seaside town that provided the settlement of fishing

communities since early ages due to the richness and abundance of its waters.

This abundance of nutrients in the Nazare waters was proven to be strongly related with the

hydrodynamic processes that took place in a large submarine canyon that enabled the existence of

great depths at short distances from the shoreline.

The Nazare canyon is the largest submarine canyon in Europe, with a total length greater than 200km,

that spreads from the Iberian abyssal plain with depths over 5000m to the Nazare harbour where the

depth is around 50m. Its large dimensions, small longitudinal slope and proximity to the shore line

categorize it like a Gouf type, the most rare of the submarine canyons.

This canyon is also responsible for focusing surface wave energy and contribute for the formation of

huge waves in Praia do Norte, that had uneasy fishermen for many years, and more recently are

claiming world’s attention with the surfing performances of adventurous athletes like Garret McNamara

who broke the World Record by surfing the largest wave ever, at approximately 23,8m, in November

2011.

Figure 1.1 – Nazare submarine Canyon

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The very specific characteristics of this area and the importance of getting the adequate information

regarding the wave climate for all the maritime activities that explore the ocean in the Nazare area,

gave context to this effort of understanding and replicate the sea conditions in Nazare.

Since during my master’s classes I had only a brief contact with wave’s physics as a subject from the

renewable energies course it meant that I would have to approach wave modelling from the start.

I was however deeply driven by the curiosity of understanding an unfamiliar process, model it and

replicate its conditions as close as possible, and was fortunate enough to develop this work in

MARETEC (Marine Environment and Technology Centre), an IST research group with high academic

standards that supported me along this journey.

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1.1.2 Wave modeling

In wave modelling there are two major groups of models: (a) the so called phase resolving models that

solve the momentum, and mass conservation equations and compute the velocity and the free surface

as a function of time and space and (b) the spectral models that solve the propagation of the wave

spectrum providing the amplitude of the waves and their wavelength. Using the later models the

position of the free surface is not known, but the energy carried by the waves and their height are. The

former models are much more complex and are useful mostly at local scale (e.g. ports and

breakwaters, where diffraction and refraction are very strong, while the others can be used in large

spatial scales, and thus top simulate wave generation, but are less convenient to address the

interaction between waves and structures. This thesis is based on the spectral model WAVEWATCH

III.

The first attempt to model waves so that their behaviour could be predicted has occurred for the

planning and preparation of DDay (June 1944).

In the late 1950’s, many researchers started to develop parameterizations that could represent the

behaviour of wave processes in order to include them in phase averaged models. Philips (1957) and

Miles (1957) theories for wind-wave interaction and Hasselman (1962) concepts for wave-wave

interactions represented an important step in wave modelling.

In this so called first generation wave models, there was no concentration of effort in solving full

energy conservation equation, not only because nonlinear wave interactions were underestimated at

the time but also due to lack of computational resources.

Second generation wave models, available by the early 1980’s started to approach nonlinear wave

interactions by the use of a simple parametric formulation.

The first third generation wave model was WAM (WAMDI group, 1988) and is still the base of all

current wave models. The main differences from the predecessors were the appliance of a realistic

relationship for nonlinear wave interactions and the use of a wave dissipation term instead of a

saturate spectrum level assumption.

Third generation wave models were originally designed for oceanic scale simulations but, as

formulations of near shore processes started to be developed, these parameterizations were gradually

introduced, increasing the versatility of these models.

Regarding the currently most used wave models we can consider the already mentioned WAM, the

SWAN (Booij et al.1999) which was developed for computation of waves in coastal regions and inland

waters, and WAVEWATCH (Tolman, 1991). The WAVEWATCH is perhaps the most used model,

adopted by several entities as the Instituto Hidrográfico de Portugal and the website Wind Guru.

Although it was primarily developed for deepwater processes, in the last years new versions have

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been released that had included nearshore processes parameterizations in order to make it more

adequate to replicate all the process of generation, propagation and breaking of waves.

All the simulations performed in the scope of this thesis were made in MARETEC, in the Alameda

Campus of Instituto Superior Técnico (IST).

1.2 Methodology

Waves, as vertical motions of the ocean surface, can be grouped into different types according to their

characteristics and source of energy. For instance, tides are waves with a very long period generated

by the interaction between the oceans and the moon or the sun, tsunamis are waves generated by a

submarine landslide or an earthquake, and wind-generated waves are waves formed by the

interaction between the wind and the ocean.

Figure 1.2 - Frequencies and periods of the vertical motions of the ocean surface (after Munk, 1950)

The waves we are interested to address in this thesis are the surface gravity waves, which are waves

generated by the wind and dominated by gravity (period larger than ¼ s). This category embraces two

types of waves:

Swell – Regular and long crested waves generated in a storm that had travelled a great

distance.

Wind Sea – Irregular and short crested waves generated by the local wind.

Considering that the Portuguese coast is frequently reached by waves formed in storms with centre in

the middle of the Atlantic Ocean as well as waves created in the nearby area by the local wind, a

downscaling approach has been followed in order to consider the swell into the regional and high

resolution model.

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A set of different grids were configured in WAVEWACTH III, with the largest one including the North

Atlantic and thus the oceanic deep water processes of wave generation, and three progressively

smaller grids with increasing resolution in order to properly represent the bottom topography and land

boundaries of the shoreline.

The first grid to be used for simulation is the North Atlantic Ocean. After the simulation of wave

conditions in this larger domain has been conducted for the period of interest the output from this

simulation is used as input boundary conditions for the next grid, Southwest Europe, which output will

be used for Portugal domain which, at last, will provide input boundary conditions for the high

resolution Nazare grid.

The nested scheme of the 4 different domains is presented below:

D1 –North Atlantic Ocean

D2 – Southwest Europe

D3 – Portugal

D4 - Nazare

Figure 1.3 - Spatial Grids

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1.3 Objectives

The present work has the intention to acknowledge the effort that has been done through the years to

understand and replicate the ocean conditions, providing an overview of the background and state of

art of wave modelling.

It also intends to enable the configuration of a third generation wave model in order to produce a

hindcast for a coastal area so that considerations can be made about WAVEWATCH III ability to

model the wave climate in nearshore areas.

It further aims to analyse the impact of varying definition settings in order to ascertain about the

influence of these parameters in the model results.

From that parameter analysis the following hypothesis are intended to be confirmed:

An improvement of the grid resolution leads to better results.

Reduction of time step conducts to more accurate results.

Wind forcing from a wind model with better resolution is more appropriate for wave simulation

in nearshore areas.

Bottom friction coefficient for swell wind states is more adequate than the value suggested for

wind sea states.

Simulations conducted during the winter period have a better correlation than simulations

conducted during the summer period.

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2 Literature Review

In the present chapter a presentation of the basic concepts of oceanic wind waves is made, including

the Linear Theory which characterizes the wave behaviour, the processes associated to generation

and propagation of waves and also the specific processes that occur in coastal areas. At last, a

presentation of the WAVEWATCH III is made, describing the equations in which it is based and the

formulations that are available and were adopted during this project in order to properly represent the

natural processes that affect waves.

2.1 Introduction

Despite the misleading perception that the movement of waves could give us as traveling particles of

water, waves are in fact energy being transfer through the vibration of water particles.

In the following figure some basic wave characteristics are presented

Figure 2.1 - Basic characteristics of waves

Considering the zero as the mean of surface elevation, the amplitude of a wave, a, it is defined as

the surface elevation from zero to its maximum, the wave’s crest, and from zero to the lowest

value, the wave’s trough. Wave height is measured from the crest to the trough which means that

for a symmetric wave is equivalent to the double of amplitude.

The period of a wave, T, is the amount of time that is necessary to perform a full cycle and the

distance between two points with the same phase is the wavelength, λ.

The phase velocity of a wave is the rate at which the phase of the wave propagates in space, and

can be defined as the wavelength over the period.

https://en.wikipedia.org/wiki/Wave

https://en.wikipedia.org/wiki/Phase_(waves)

https://en.wikipedia.org/wiki/Wave_propagation

8

𝒄 =

𝛌

𝑻

(2.1.1)

2.2 Linear Wave Theory

Linear Wave Theory was first derived by G.B.Airy (1845) and withstands the basis of our

understanding about how individual waves can be mathematically modelled and how we can predict

with reasonable accuracy its behaviour through space and time regarding interactions with other

waves and bottom bathymetry.

In order to derive the linear theory it is necessary to make some simplifications, so let us consider a

2D vertical plane (x,z) as shown in the figure presented below:

Figure 2.2 - Boundary Conditions for the linear wave theory

The following assumptions are made:

The bottom is plane and at a constant depth, (𝑧 = −𝑑)

The wave period and wavelength are constant (periodic wave)

The fluid is homogeneous, incompressible

Wave height much smaller than wavelength, (𝐻/𝜆 ≪ 1)

Viscous and turbulent stresses are neglected

9

Conducting a mass balance in the considered control volume we derive the continuity

equation:

𝜕𝑢𝑥

𝜕𝑥

+𝜕𝑢𝑧

𝜕𝑧

= 0

(2.2.1)

In order to simplify the analytical solution we then introduce the concept of velocity potential, such that:

𝑢𝑥 =𝜕𝜑

𝜕𝑥, 𝑢𝑦 =

𝜕𝜑

𝜕𝑦, 𝑢𝑧 =

𝜕𝜑

𝜕𝑧

(2.2.2)

We can write the continuity equation (2.2) in terms of the velocity potential, by replacing the spatial

derivatives, obtaining the Laplace Equation:

∇2φ =∂2φ

∂2x+

∂2φ

∂2z= 0

(2.2.3)

The following initial boundary conditions are used to solve the Laplace equation for the velocity

potential:

𝜕𝜑

𝜕𝑧= 0 𝑎𝑡 𝑧 = −𝑑 (𝑘𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑏. 𝑐. )

(2.2.4)

𝜕𝜑

𝜕𝑧−

𝜕𝜂

𝜕𝑧= 0 𝑎𝑡 𝑧 = 𝜂 (𝑘𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑏. 𝑐. )

(2.2.5)

𝜕𝜑

𝜕𝑡+ 𝑔𝜂 = 0 𝑎𝑡 𝑧 = 0 (𝑑𝑦𝑛𝑎𝑚𝑖𝑐 𝑏. 𝑐. )

(2.2.6)

The solution yields the velocity potential as:

𝜑(𝑥, 𝑧, 𝑡) =𝜔𝑎

𝑘

cosh[𝑘(𝑑 + 𝑧)]

sinh(𝑘𝑑)cos(𝜔𝑡 − 𝑘𝑥)

(2.2.7)

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Where 𝑘 is the wave number defined as:

𝑘 =2𝜋

𝜆

(2.2.8)

And 𝜔 is the wave frequency, defined as:

𝜔 =2𝜋

𝑇

(2.2.9)

Developing the dynamic boundary condition at 𝑧 = 0 we then obtain the surface elevation:

𝜂(𝑥, 𝑡) = 𝑎 sin(𝜔𝑡 − 𝑘𝑥)

(2.2.10)

And combining both boundary conditions at 𝑧 = 0 we obtain the dispersion relation, which states a

unique relation between frequency, wavenumber and water depth.

𝜔2 = 𝑔𝑘 tanh (𝑘𝑑)

(2.2.11)

The propagation speed or wave celerity previously defined can now be rearranged so that:

𝑐 =𝜆

𝑇=

𝜔

𝑘= √

𝑔

𝑘tanh(𝑘𝑑)

(2.2.12)

That allows us to conclude that wave celerity depends on the water depth in which the wave is

propagating.

2.2.1 Random-phase/amplitude model

It may not be easy to find any resemblance between the irregular wave patterns observed in the

ocean and the regular wave behaviour described by the linear theory, however, we can look at the

ocean as a sum of a large number of harmonic, regular waves, each with its own frequency and

direction, a constant amplitude and a randomly chosen phase.

11

Figure 2.3 - Sum of a large number of harmonic wave components, travelling across the ocean with different periods, directions, amplitudes and phases (after Pierson et al., 1955)

𝜂(𝑥, 𝑡) = ∑ 𝑎𝑛sin (𝜔𝑛𝑡 − 𝑘𝑛𝑥 + 𝜙𝑛)

𝑁

𝑛=1

(2.2.1.1)

Having in mind that an ocean wave is the summation of a large number of waves, each one with its

own phase velocity, it is convenient to define group velocity as the velocity at which the overall shape

of the waves' amplitudes, known as wave envelope, propagates through space.

𝑐𝑔 =𝜕𝜔

𝜕𝑘= 𝑛𝑐

(2.2.1.2)

Where c in the phase speed of the wave and n is (from the dispersion relationship, eq.2.2.11):

𝑛 =1

2(1 +

2𝑘𝑑

sinh (2𝑘𝑑))

(2.2.1.3)

12

2.3 Generation and Propagation

2.3.1 Mechanism of Wave’s generation

The most accepted theory regarding generation of waves by wind is the Miles-Philips theory. In

this theory, waves are the result of a resonant interaction between the surface of the water and

turbulent fluctuations in the air (pressure). At the beginning wind induced pressure waves interact

with an entirely flat sea causing some irregularities in the sea surface, and, as a result of this

interaction, small capillary waves first begin to grow.

Once the sea contains capillary waves, there is an increase in the surface roughness, and the

action of the wind over these small waves will produce some pressure variations along the wave

profile, which will enhance the waves grow. This is an exponential development, since as the

waves start to grow the pressure variation along their profile will increase which will intensify the

energy transfer between the wind and the wave and consequently the wave’s growth.

2.3.2 Propagation of Waves

There are three factors that influence the level of energy transferred to waves: Wind velocity; wind

fetch which is the length of water over which a given wind has blown; and duration, as the

amount of time those winds blow over the same part of the ocean.

Once waves are generated they start to travel across the ocean. As we’ve seen from the

dispersion relationship, low-frequency waves travel faster than high frequency waves. The initially

random field of waves created in a storm will therefore disintegrate into more regular fields in the

direction of propagation, with the low frequency waves in the lead and the high frequency waves

on the trailing edge, in a process called frequency-dispersion. Similarly the waves will disintegrate

in a range of directions in a process called direction-dispersion. Due to these processes, waves

that have travelled across the ocean will change from short-crested to long-crested and grow more

and more regular, being denominated as Swell.

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2.4 Nearshore Processes

2.4.1 Linear wave theory for coastal waters

When the waves enter coastal waters their amplitude, direction and celerity are affected. This was an

expected behaviour considering that most of the parameters obtained from the linear wave theory are

dependent on the water depth. Accordingly it’s important to remark the differences and additional

processes regarding propagation, wave-wave interactions and dissipation.

Figure 2.4 - Definition of distinct depth zones

Recalling the dispersion relation (eq.2.2.11) it is possible to conclude that if the period remains

constant and the depth, d, decreases, then the wave speed and wavelength will also decrease,

confirming that in deep waters waves travel faster than in shallow waters.

The dispersion relationship is dependent on a trigonometric function, the hyperbolic tangent, which

has the following graphical representation, typically called “the S curve”.

14

Figure 2.5–Graphical representation of the hyperbolic tangent

In the ocean the water depth is typically much greater than the wavelength, 𝑑 >> , which means that

𝑘𝑑 ≫ 1. From figure 2.5 we can see that for deepwater the following simplification can be made:

tanh (𝑘𝑑) ≈ 1

(2.4.1.1)

The same logic can be applied for shallow waters, considering that the wavelength is much greater

than the water depth, >> 𝑑:

tanh (𝑘𝑑) ≈ 𝑘𝑑

(2.4.1.2)

From these simplifications we can obtain approximations for the wave parameters, depending on the

water depth:

Deepwater Shallow water

Phase velocity 𝑐 =𝑔

𝜔 𝑐 = √𝑔𝑑

Group velocity 𝑐𝑔 =𝑐

2=

𝑔

2𝜔 𝑐𝑔 = 𝑐 = √𝑔𝑑

Dispersion relationship 𝜔2 = 𝑔𝑘 𝜔2 = 𝑔𝑘2𝑑

Table 2.1- Approximations for wave characteristics depending on water depth

15

2.4.2 Transformation processes

Not only the propagation of waves is affected, but also generation, nonlinear wave-wave interactions

and dissipation processes will be influenced by the limitation of depth. The processes that already

occurred in deep waters will behave differently and there are also new processes that need to be

taken into account when modelling waves in coastal waters.

Process

Coastal waters

Oceanic waters Shelf seas Nearshore Harbour

Wind generation XXX XXX X 0

Quadruplet wave-wave interactions XXX XXX X 0

White-capping XXX XXX X 0

Bottom friction 0 XX XX 0

Bottom refraction/shoaling 0 XX XXX XX

Breaking 0 X XXX 0

Triad wave-wave interactions 0 0 XX XX

Reflection 0 0 X/XX XXX

Diffraction 0 0 X XXX

Table 2.2 - The relative importance of the various processes affecting the evolution of waves in oceanic and coastal waters (after Batjjes, 1994)

In the above table it is structured the importance that each process plays depending on the water

depth: XXX – Dominant, XX –Significant but not dominant, X – Of minor importance, 0 – Negligible.

2.4.2.1 Wind generation, quadruplets and whitecapping

Regarding wind generation, the amount of transferred energy will increase, since as we’ve seen, the

waves slow down when entering shallow waters, thus increasing the difference between the wind

velocity and phase speed, increasing consequently the amount of transferred energy.

As the waves become steeper, the quadruplet wave-wave interactions and white capping processes

will also be enhanced.

16

2.4.2.2 Bottom friction

When propagating in coastal waters, waves dissipate energy in a thin, turbulent boundary layer near

the bottom that is created by the wave-induce motion. This process consists of an energy and

momentum transfer between the orbital wave’s motion immediately above the boundary layer and the

turbulent movement of water particles in that layer. The intensity of this dissipation depends on the

wave conditions and the characteristics of the bottom.

2.4.2.3 Surf Breaking

The depth induced breaking is a poorly understood non linear process responsible for the breaking of

waves as they reach a certain minimum depth.

The maximum height that a wave can reach before it breaks can be described as directly related to

the water depth.

If the shore is a flat beach, the Iribarren (1949) parameter allows the classification of the wave

according to the type of breaking: spilling, plunging, collapsing and surging.

𝐼𝑟 = 𝑡𝑎𝑛𝛽/(𝐻/)1

2

(2.4.2.3.1)

Where 𝛽 represents the beach slope.

2.4.2.4 Shoaling

In the absence of any generation or dissipation of wave energy and assuming that the wave

propagates through a constant direction, the energetic flux between two consecutive wave crests will

remain constant. So, when the wavelength decreases due to depth effects the wave amplitude will

increase in order to assure energy conservation:

(𝐸𝑎𝑐𝑔𝑎). 𝑏𝑎 = (𝐸𝑏𝑐𝑔𝑏

). 𝑏𝑏

(2.4.2.4.1)

E represents energy per unit crest length,𝑐𝑔 is the wave group velocity and 𝑏 the distance between two

lateral sides.

17

Energy per unit of length can be computed as:

�� =1

8𝜌𝑔𝐻2

(2.4.2.4.2)

Rearranging eqs.2.4.2.4.1 and 2.4.2.4.2 we obtain the following relationship between wave heights at

different depths:

𝐻𝑏 = 𝐻𝑎√𝑐𝑔𝑎

𝑐𝑔𝑏

√𝑏𝑎

𝑏𝑏

(2.4.2.4.3)

Where index b is associated to shallow waters and a to deep waters.

2.4.2.5 Refraction

As a harmonic wave approaches the coast its crests tend to become perpendicular to the shore, this is

caused by the change of the phase speed induced by depth variation.

The wave crest propagates faster in deep water than it does in shallow water, therefore if the wave

approaches a straight shoreline with an oblique incidence, 𝜃, the part of the wave crest closer to shore

is in shallower water and moving slower than the part away from the shore in deeper water. As the

wave crest in deeper water moves faster the wave crest tends to become parallel to the shore.

This process can be represented by Snell’s Law:

sin 𝜃

𝑐=

sin 𝜃0

𝑐0

(2.4.2.5.1)

In which 𝑐 and 𝑐0 correspond to phase speed in shallow and deep waters, respectively, and 𝜃 is the

incidence wave angle.

18

2.4.2.6 Diffraction

If a wave finds an obstacle like an island or a breakwater, energy is laterally transmitted through the

crest into this so-called 'shadow zone.'

The intensity of the diffraction effect is related with the proportion of the obstacle in relation with the

wavelength.

2.4.2.7 Reflection

When a wave reach an obstacle, a portion of its energy will be reflected in a new direction, so that the

nearby area wave motion will be characterized by the interference of the incoming wave with one or

more reflected waves.

This phenomenon depends strongly on the nature of the obstacle, as it dictates the amount of

reflected energy and also the returning directions, for instance for a gentle beach, reflection does not

usually play an important part in the waves behavior as its energy it’s barely reflected.

All these processes can be accommodated by the linear wave theory as long as the waves do not

enter very shallow water, where the nonlinear effects have to be accounted for. When the waters are

too shallow there is the need of adopting a nonlinear theory.

19

2.5 Wave Model - WAVEWATCH III

Wavewatch III

WAVEWATCHIII™ (Tolman, 2002b, 2008) is a 3rd

generation wind wave spectral modeling software

developed at the Marine Modelling and Analysis Branch (MMAB) of the Center of Environmental

Prediction (NOAA/NCEP).

Although it is based on WAVEWATCH I and WAVEWATCH II which were developed at Delft

University of Technology and NASA Goddard Space Flight Center, respectively, it differs from its

predecessors in all major aspects as governing equations, program structure, numerical and physical

approaches.

All the model simulations conducted during this thesis were run on WAVEWATCH III (WW III) version

4.18, available since March 2014.

2.5.1 Governing Equations

In the absence of currents, the energy of a wave package is a conserved quantity but the same does

not apply when wave-currentinteractions are considered. Since WW III allows the inclusion of such

interactions, it uses an explicit third order finite difference scheme for solving spectral action density

balance equation rather than the energy balance equation in order to assure conservation.

𝐷𝑁

𝐷𝑡=

𝑆

𝜎

(2.5.1.1)

∂𝑁

∂𝑡+ 𝛻𝑥 ∙ ��𝑁 +

∂

∂𝑡��𝑁 +

∂

∂𝜃��𝑁 =

𝑆

𝜎

(2.5.1.2)

where

𝑥 = 𝑐𝑔 + 𝑈

(2.5.1.3)

k = −

∂σ

∂d

∂d

∂s− k ∙

∂U

∂s

(2.5.1.4)

20

𝜃 = −

1

𝑘[𝜕𝜎

𝜕𝑑

𝜕𝑑

𝜕𝑚− 𝑘 ∙

𝜕𝑈

𝜕𝑚]

(2.5.1.5)

On the left hand side of the equation terms relative to local rate of change of action density are

presented as well as propagation effects. The right hand side of the equation is reserved for source

and sink functions.

𝑆 = 𝑆𝑖𝑛 + 𝑆𝑛𝑙 + 𝑆𝑑𝑠 + 𝑆𝑙𝑛 + 𝑆𝑏𝑜𝑡 + 𝑆𝑑𝑏 + 𝑆𝑡𝑟 + 𝑆𝑠𝑐 + 𝑆𝑖𝑐𝑒 + 𝑆𝑟𝑒𝑓 + 𝑆𝑥𝑥

(2.5.1.6)

The first three source terms are associated with deep water processes, namely 𝑆𝑖𝑛, which represents

wind-wave interaction, 𝑆𝑛𝑙 which characterizes nonlinear wave-wave interactions and𝑆𝑑𝑠, which relates

to dissipation as whitecapping. A linear input term is also included in order to represent more

realistically the initial wave growth process, 𝑆𝑙𝑛.

In shallow waters, it is necessary to take into account additional processes like the wave-bottom

interaction, 𝑆𝑏𝑜𝑡, and, in extremely shallow waters depth-induced breaking and triad wave-wave

interactions,𝑆𝑑𝑏and 𝑆𝑡𝑟 respectively.

There are other terms also available in WWIII such as 𝑆𝑠𝑐 which accounts for the scattering effects that

may occur due to the physical conditions of the sea bottom, 𝑆𝑖𝑐𝑒that represent wave-ice

interactions,and 𝑆𝑟𝑒𝑓which stands for reflection by a shore line or other floating objects. At last, 𝑆𝑥𝑥 is

reserved in the model to be user defined.

2.5.2 Wave physics parameterization

Non-linear interactions, Snl

Although WWIII provides the option to choose between three different parametrizations for this term

representation, the one elected was the DIA (Discrete Interaction Approximation), which is the more

widespread, mostly due to its computational economy.

In this parameterization, a simplification of the wave vectors is made regarding the 4 wave

components that promote resonant nonlinear interactions.

21

Linear Input, Sln

This term is important for the start of the model from quiescent conditions and for enhancing the initial

wave growth behavior. The used parameterization was the one by Cavaleri and Malanotte-Rizzoli

(1981) with a low-frequency filter, introduced by Tolman (1992).

𝑆𝑙𝑛(𝑘, 𝜃) = 80 (𝜌𝑎

𝜌𝑤

)2

𝑔−2𝑘−1 max[0, 𝑢∗ 𝑐𝑜𝑠(𝜃 − 𝜃𝑤)]4𝐺

(2.5.2.1)

ρa and ρw are the air and water density, respectively, and G represents the introduced filter.

G = exp [− (f

ffilt

)−4

]

(2.5.2.2)

Energy input and dissipation, Sin and Sds

The source term package of Tolman and Chalikov (1996) contains both the input source term of

Chalikov and Belevich (1993) and Chalikov (1995) and two dissipation constituents.

The input source is given as

𝑆𝑖𝑛(𝑘, 𝜃) = 𝜎𝛽𝑁(𝑘, 𝜃)

(2.5.2.3)

The low frequency dissipation term which is based on an analogy with dissipation due to turbulence, is

presented as follow,

𝑆𝑑𝑠,𝑙(𝑘, 𝜃) = −2𝑢∗ℎ𝑘2𝜙𝑁(𝑘, 𝜃)

(2.5.2.4)

And finally, the empirical high frequency dissipation:

𝑆𝑑𝑠,ℎ(𝑘, 𝜃) = −𝑎0 (𝑢∗

𝑔)

2

𝑓3𝛼𝑛𝐵𝑁(𝑘, 𝜃)

(2.5.2.5)

22

Bottom friction, Sbot

The formulations that were developed to model bottom friction dissipation can be generalized into an

equation (Weber, 1991) which contains a dissipation coefficient, 𝐶𝑓, that depends on hydrodynamic

and sediment properties. There are two main approaches to estimate the shear stress: the drag-law

models which determine empirically a constant coefficient for every wave condition and bottom

properties and the eddy-viscosity models that describe the dissipation as a function of the bottom

characteristics.

Alternatively, this frictional turbulent process can be represented by the empirical JONSWAP model

(Hasselman et al. 1973), described as:

𝑆𝑏𝑜𝑡(𝑘, 𝜃) = 2𝑛 − 0,5

𝑔𝑑𝑁(𝑘, 𝜃)

(2.5.2.6)

is an empirical constant which is estimated as = −0,038𝑚2𝑠−3 for swell (Hasselman et al., 1973),

and as = −0,067𝑚2𝑠−3 (Bows and Komen, 1983) for wind sea states.

Surf Breaking

In order to enhance WW III performance in shallow waters, a formulation to account for depth-induced

breaking was added. The included approach was derived byBattjes and Jansen (1978) based on the

assumption that all the waves in a wave field that exceed a certain threshold height, dependent on

bottom topography parameters, will break.

The relation between maximum height and water depth is defined through a McCowan-type criterion,

which consists of a simple constant ratio:

𝐻𝑚𝑎𝑥 = 𝛾𝑑

(2.5.2.7)

Where d is the local water depth and 𝛾 a constant defined through laboratory and field tests. The

average value found by Battjes and Jansen (1978) was 𝛾 = 0,73, which is set as default value in the

model.

The following depth-induced breaking dissipation source function is obtained:

23

𝑆𝑑𝑏(𝑘, 𝜃) = −𝛼𝛿

𝐸𝐹(𝑘, 𝜃)

(2.5.2.8)

E is the total spectral energy, 𝛼 is a tunable parameter and 𝛿 is the bulk rate of spectral energy density

dissipation of the fraction of breaking waves.

In the next table, a schematization of the source terms parametrizations that were configured is

presented:

Source Term Parameterization

Linear Input, Sln Cavaleri and Malanotte-Rizzoli

Wind Input, Sin Tolman and Chalikov

Dissipation, Sds Tolman and Chalikov

Nonlinear interactions, Snl DIA

Bottom Friction, Sbot JONSWAP

Depth Induced breaking, Sdb Battjes-Jansen

Table 2.3–WW III Parameterizations configured for this work

24

3 Wave Model

3.1 Model Definition

3.1.1 Model

With the intention of obtaining the wave characteristics in the Nazare area, it is necessary to create a

model capable of representing the processes of generation and propagation that take place in an

oceanic scale as well as the wave’s transformation mechanism as it approaches the coast line.

In order to achieve this, a downscaling approach was followed, where 4 different grids with

progressively decreasing dimensions and gradually increasing resolutions were used. The ocean

conditions were first simulated at a greater scale, in a grid that contains the North Atlantic. The results

from this simulation provided input boundary conditions for the nested grid – Southwest Europe, which

is a grid that involves a smaller area and has the double of the resolution of the previous one. After the

model simulated the ocean conditions for this grid, the output will be again used as input for the next

nested grid – Portugal, and similarly from Portugal to Nazare, which is the smallest grid with the best

resolution form the set.

A total set of 4 grids were used, the first two –D1 North Atlantic and D2 Southwest Europe – with a

coarse grid and the other 2, considered for analysis – D3 Portugal and D4 Nazare with a more

detailed grid, as displayed in the following table:

Domain Latitude Longitude Resolution Grid size

D1 North Atlantic 15°𝑁: 75°𝑁 90°𝑊: 5°𝐸 0,5° × 0,5° 191 × 121

D2 Southwest Europe 33°𝑁: 48°𝑁 24°𝑊: 0° 0,25° × 0,25° 97 × 61

D3 Portugal 35,53°𝑁: 42,97°𝑁 11,77°𝑊: 7,12°𝑊 0,05° × 0,05° 92 × 149

D4 Nazare 39,02°𝑁: 40,08°𝑁 10,38°𝑊: 8,86°𝑊 0,02° × 0,02° 76 × 53

Table 3.1 - Computational grids

25

Figure 3.1 - D1 North Atlantic spatial grid

Figure 3.2 - D2 Southwest Europe spatial grid

26

Figure 3.3 - D3 Portugal Spatialgrid

Figure 3.4 - D4 Nazare Spatial grid

With the aim of understanding and ascertain the influence of different model parameters in the results

quality, several model runs were performed and in each one of them a single parameter was changed

so that the results could be compared with the original scenario.

27

3.1.2 Grid Resolution

At an oceanic scale it is recommendable to use a coarse grid in order to save some time in

computational calculations. Also the bathymetry resolution is not a factor so determinant for deep

waters as it is for shallow waters since at very large depths the effects caused by the bottom of the

ocean are barely noticed. But in a nearshore area the bathymetry is very important to recreate all the

processes associates to depth changes. So, recalling that the number of grid points is dependent on

both the resolution and dimensions of the grid, the definition of the 4 tested grids was careful to

consider a total number of grid points reasonably low in order to have acceptable computational costs,

but sufficiently large to meet a certain level of precision required for the proper representation of the

bottom layout.

With this in mind, none of the computational grids used had more than 200 points in a single direction.

In order to avoid numerical instabilities, the resolution between any grid and the corresponding nested

grid – a relationship commonly known as father and son- was improved no more than 5 times.

3.1.3 Spectral Information

In WW III there are 5 different parameters concerning spectral information: minimum frequency,

increment, number of frequencies, number of directions and directional offset.

Minimum frequency was set to 0,04118 Hz, with a 10% increment, which implies an upper limit of

0,4056Hz. The reason for this is that the energy distributions outside this interval is not meaningful.

The integration was made along 24 different directions, or, in other words, considering intervals of 15

degrees and the relative offset of the directional increment was set as null.

All the above definitions were maintained constant throughout all the tested scenarios.

3.1.4 Time Steps

One of the most important parameters to be defined in each grid it’s the time step, as it very important

to find a commitment between numerical precision and computational economy.

In WW III the time step comprises 4 different components: global time step ∆𝑡𝑔, spatial time step ∆𝑡𝑥𝑦,

directional time step∆𝑡𝑘 and source time step ∆𝑡𝑆.

The first component to be determined is the spatial time step, as it shall respect the Courant-

Friederichs-Levy (CFL) criterion which postulates that the speed of fastest waves in the model must

be less than or equal to the grid spacing divided by the time step:

∆𝑡𝑥𝑦 = ∆𝑥

𝐶𝑔

(3.1.4.1)

28

=

40×106

360× ∆𝑥 × cos(𝑚𝑎𝑥𝑙𝑎𝑡)

1,15×𝑔

4𝜋×

1

𝑓

(3.1.4.2)

=40 × 106 × ∆𝑥 × cos(𝑚𝑎𝑥𝑙𝑎𝑡) × 4𝜋 × 𝑓

360 × 1,15 × 𝑔

(3.1.4.3)

= 123766 × ∆𝑥 × cos(𝑚𝑎𝑥𝑙𝑎𝑡) × 𝑓

(3.1.4.4)

According to WW III authors suggestions, the global time step, ∆𝑡𝑔, is then set as 2 or 3 times the

spatial time step, ∆𝑡𝑥𝑦 , in order to ensure this commitment between numerical precision and

computational economy.

Once the global time step is defined, the directional time step can be defined as well as half of the

global time step in order to take into account correctly refraction effects.

For the source time step it’s assigned the minimum possible value,∆𝑡𝑆 = 15𝑠.

In all the simulation runs conducted the time step was defined as the maximum possible value,

respecting the above rules. These were considered as standard conditions. Exception was made in

specific test scenarios where the objective was to ascertain the time step impact (R3 and R4).

The defined time steps (in seconds) for the conducted model runs are presented in the table below:

Global Time Step

∆𝒕𝒈

Spatial Time Step

∆𝒕𝒙𝒚

Direc.Timestep

∆𝒕𝒌

Source Time step

∆𝑡𝑆

R1 400 180 200 15

R2 200 90 100 15

R3 200 90 100 15

R4 100 45 50 15

R5 400 180 200 15

R6 200 90 100 15

R7 200 90 100 15

R8 400 180 200 15

29

Table 3.2 - Time step components for the conducted model runs

3.1.5 Validation Period

Most of the simulations for the presented test scenarios were relative to the period from the 31st

of

April of 2015 to the 31st of June of 2015 (61 days).

There were additionally two simulations conducted on the period from the 4th of September of 2011 to

the 3rd

of November 2011 so that results could be analyzed in a perspective of a seasonal

comparison.

3.1.6 Wind Model

In the large scale grids North Atlantic and Southeast Europe the adopted wind model was the Global

Forecast System (GFS) produced by the National Centre for Environmental Prediction (NCEP), which

has a 50km x 50km resolution. In the Portugal grid there was the intention of comparing GFS with a

model with better resolution so the Weather Research & Forecasting Mode (WRF) was also used with

a resolution of 9km x 9km. All the remaining simulation runs were forced by the WRF.

3.1.7 Bottom Friction Coefficient

It is suggested by the authors the use of the default value 𝛤 = −0,067𝑚2𝑠−3 in the parameterization

that accounts for the friction between the wave and the bottom surface of the water. This value is

associated to wind seas states. Additionally it is proposed the value of 𝛤 = −0,038𝑚2𝑠−3 which

represents sea conditions where swell plays a dominant role.

Despite that, works have emerged (Holthuijsen et al, 2010) stating that the value = −0,038𝑚2𝑠−3 is

a better approximation for both sea states.

All model runs were defined according to the swell value, = −0,038𝑚2𝑠−3, with the exception of one

– R6 - which had the bottom friction factor assigned to the wind sea state value 𝛤 = −0,067𝑚2𝑠−3 so

that considerations could be made about the impact that each of the values promoted.

No currents interference was included which means that refraction is due only to spatial variations of

water depths.

30

3.1.8 Test Scenarios

A total of 8 simulations were conducted, changing always one parameter in each run in order to

analyze the influence in the results. The tested scenarios are presented in the table below:

Grid Resolution Time Step Wind Model Test Period Bottom

Friction

R1 Portugal

Portugal 0.05° × 0.05° Standard WRF Summer -0,038

R2 Nazaré

Nazare 0.02° × 0.02° Standard WRF Summer -0,038

R3 Portugal

½ Tstep Portugal 0.05° × 0.05° ½ Standard WRF Summer -0,038

R4 Portugal

¼ Tstep Portugal 0.05° × 0.05° ¼ Standard WRF Summer -0,038

R5 Portugal

GFS

Portugal 0.05° × 0.05° Standard GFS Summer -0,038

R6 Nazare

Bot.Friction Nazare 0.02° × 0.02° Standard WRF Summer -0,067

R7 Nazare

Winter

Nazare 0.02° × 0.02° Standard WRF Summer -0,038

R8 Portugal

Winter

Portugal 0.05° × 0.05° Standard WRF Winter -0,038

Table 3.3 – Parameters definition for all the conducted simulation runs

31

3.2 Model Validation

3.2.1 Real Data

In order to evaluate the quality of the obtained results it’s necessary to establish a comparison with

real data. The source of the data used for these comparisons is a SEWATCH wavescan buoy,

identified as Monican02. This is one of two buoys placed at different parts of the Nazare Canyon by

Instituto Hidrográfico de Portugal in order to retrieve information than enable the study of maritime

conditions and processes in this area.

These buoys are equipped with a number of sensors that allow the acquisition of several

meteorological and oceanographic parameters such as significant wave height, mean period, wave

direction, temperature, current direction and velocity, air pressure, etc.

In the following figure both buoys are represented, Monican01 (39º 30.9'N 09º 38.2' W) is located at

an approximated distance of 50Km from the coastline at a depth of 2000m and Monican02

(39º33.661'N 09º12.632'W) is placed at a distance of 10Km where the water depth is just a few 70m.

Due to data availability only information from Monican02 was used in this study.

Figure 3.5 - Geographic representation of Monican01 and Monican02 (from IHP)

32

The wave characteristics that are subjected to comparison in this study are next described:

Significant wave height, Hs, defined as the mean of the highest one-third of waves in the wave record.

Experiments show that this value is close to the visually estimated wave height, the respective unit is

meters (m).

𝐻𝑠 =1

𝑁/3∑ 𝐻𝑗

𝑁/3

𝑗=1

(3.2.1.1)

Mean zero-crossing wave period,Tm,as the time interval between the start and end of the wave,

measured in seconds (s)

𝑇𝑚 =1

𝑁∑ 𝑇𝑖

𝑁

𝑖=1

(3.2.1.2)

And mean wave direction defined as the direction from which the waves are coming.

Since there is never just a single direction for the ocean waves, the mean wave direction is defined as

the most common direction, in a record, from which the waves are coming. The units are degrees, with

North as zero and a clockwise counting.

3.2.2 Statistical parameters

By visual examination of the graphs it’s possible to make a qualitative analysis of the results and to

take some general conclusions. However, for a more detailed analysis in order to attain quantitative

elements that allow more precise conclusions regarding the models validation it’s necessary to use

some statistical values.

The following parameters were used:

Real data mean value,

�� =∑ 𝑋𝑖

𝑛𝑖=1

𝑛

(3.2.2.1)

33

Model data mean value,

�� =∑ 𝑌𝑖

𝑛𝑖=1

𝑛

(3.2.2.2)

Bias,

𝐵𝑖𝑎𝑠 =∑ (𝑋𝑖 −𝑛

𝑖=1 𝑌𝑖)

𝑛

(3.2.2.3)

Root mean squared error,

𝑅𝑀𝑆𝐸 = √∑ (𝑋𝑖 −𝑛

𝑖=1 𝑌𝑖)2

𝑛

(3.2.2.4)

Scatter index,

𝑆𝐼 =𝑅𝑀𝑆𝐸

��

(3.2.2.5)

Pearson correlation coefficient,

𝑅 =∑ (𝑋𝑖 − ��)(𝑌𝑖 − ��)𝑛

𝑖=1

√∑ (𝑋𝑖 − ��)2𝑛

𝑖=1 ∑ (𝑌𝑖 − ��)2𝑛

𝑖=1

(3.2.2.6)

34

4 Results

4.1 Results Presentation

The results obtained in the performed simulations will be presented next as a group of tables and

charts. Firstly a set of tables containing the calculated statistical parameters is presented so that an

overall perception of the results is provided. The presented parameters are the correlation coefficient –

R, the slope –S, the Bias, the root mean square error – RMSE and the scatter index –SI. The

identification of each simulation is the same presented in table 3.3.

Time series diagrams are also added for each particular run, enabling an evaluation of the significant

wave height, mean period and mean direction tendency to follow the real data curves, during the 61

days of simulation. Scatter plotsare presented displaying the dispersion of the obtained results by

opposing the real data in the x axis with the simulation values in the y axis.

Hs R1 Portugal

R2 Nazaré

R3 Portugal ½ Tstep

R4 Portugal ¼ Tstep

R5 Portugal

GFS

R6 Nazare

Bot.Friction

R7 Nazare Winter

R8 Portugal Winter

R 0,9213 0,9102 0,9222 0,9225 0,9157 0,8603 0,9566 0,9567

S 1,005 0,9654 1,0042 1,0029 0,9344 0,9986 0,8514 0,9381

Bias -0,01 -0,10 -0,01 -0,01 -0,14 -0,02 -0,22 -0,23

RMSE 0,2766 0,311 0,2753 0,2748 0,3206 0,3619 0,41908 0,4333

SI 0,1442 0,1622 0,1435 0,1424 0,1672 0,1887 0,2042 0,2111

Table 4.1 - Statistical Parameters for Significant Wave Height simulated results

Tm R1

Portugal

R2 Nazaré



R5 Portugal

GFS

R6 Nazare

Bot.Friction

R7 Nazare Winter

R8 Portugal Winter

R 0,8470 0,8490 0,8545 0,8560 0,8726 0,7981 0,8836 0,8836

S 0,6560 0,6399 0,6705 0,6738 0,6632 0,6548 0,7681 0,7681

Bias -0,61 -0,75 -0,60 -0,60 -0,84 -1,06 -0,97 -0,96

RMSE 1,13707 1,23782 1,1014 1,0973 1,244 1,4664 1,2976 1,2907

SI 0,1577 0,1717 0,1528 0,1407 0,1726 0,2034 0,1683 0,1674

Table 4.2 - Statistical Parameters for Mean Period simulated results

35

Dir R1 Portugal

R2 Nazaré



R5 Portugal

GFS

R6 Nazare

Bot.Friction

R7 Nazare Winter

R8 Portugal Winter

R 0,8755 0,9059 0,9020 0,8997 0,8703 0,9407 0,8129 0,8643

S 0,7990 0,8274 0,8486 0,8427 0,7629 0,9714 0,7014 0,8112

Bias 1,17 3,57 1,10 1,16 4,23 4,75 3,40 3,33

RMSE 14,1707 13,4515 12,9580 13,0788 15,307 12,108 16,1873 16,0618

SI 0,0453 0,0430 0,0414 0,0420 0,0489 0,0387 0,0518 0,0514

Table 4.3 - Statistical Parameters for Mean Direction simulated results

4.1.1.1 R1 - Portugal

Significant Wave Height, Hs

Figure 4.1 - R1 Hs time series and scatter plot

36

Mean Period, Tm

Figure 4.2 - R1 Tm time series and scatter plot

Mean Direction, Dir

Figure 4.3 - R1 Dir time series and scatter plot

37

4.1.1.2 R2 - Nazare



Mean Period, Tm


38

Mean Direction, Dir

Figure 4.6–R2 Dir time series and scatter plot

4.1.1.3 R3- Portugal ½ Time Step



39

Mean Period, Tm

Figure 4.8– R3 Tm time series and scatter plot

Mean Direction, Dir


40

4.1.1.4 R4- Portugal ¼ Time Step


Figure 4.10 – R4 Hs time series and scatter plot

Mean Period, Tm


41

Mean Direction, Dir


4.1.1.5 R5- Portugal GFS



42

Mean Period, Tm


Mean Direction, Dir


43

4.1.1.6 R6 - Nazare Bottom Friction



Mean Period, Tm


44

Mean Direction, Dir


4.1.1.7 R7- Nazare Winter



45

Mean Period, Tm

Figure 4.20 – R7Tm time series and scatter plot

Mean Direction, Dir

Figure 4.21 – R7Dir time series and scatter plot

46

4.1.1.8 R8- Portugal Winter



Mean Period, Tm


47

Mean Direction, Dir


4.2 Results Analysis

As we look at the graphs that represent the simulation curves behaviour against the real data it’s

possible to conclude that in general the results are fairly reasonable good, in particular for the

significant wave height.

The statistical parameters also validate this conclusion, the results for all the simulations don’t contain

major differences, with the exception of R6 – Nazare Bottom Friction, which is the only run with a

correlation parameter below 90% for the significant wave height.

It’s difficult to conduct an overall analysis of the results for the 3 statistical parameters since they don’t

evolve in the same direction from run to run, i.e. when for instance the quality of significant wave

height results increase, the quality of period or/ and direction could be worse. The exceptions are the

R3 and R4, where all the wave parameters were enhanced as the time step was reduced.

By analysing the mean direction time series it’s also possible to confirm that the Portuguese nearshore

wave pattern is most of the time regular with the incoming wave direction usually from North/West.

In the direction time series there’s always one value out of the tendency. In winter runs it’s explainable

by the type of chosen representation, since a value slightly higher than 0 degrees it’s close to 360

degrees but it’s not possible to represent them that way in a time series. For the summer runs it’s

more difficult to explain it, but it is probably related to the input wind.

48

4.2.1 Grid Resolution

The results comparisons between the Portugal and Nazare grids didn’t provide the expected results

since Nazare, the grid with best resolution, 2𝑘𝑚 𝑥 2𝑘𝑚, didn’t increase the accuracy of the significant

wave height simulation results relatively to the Portugal grid, which as a more coarse grid, 5𝑘𝑚 𝑥 5𝑘𝑚.

In the winter period the results for Hsare practically the same between Father and son (R8 and R7) but

in the summer period (R1 and R2) the results correlation slightly decreases.

This wasn’t the expected results, since they do not corroborate the hypothesis that a better resolution

in the grid corroborated with a higher quality bathymetry should lead to the improvement of the results

regarding significant wave height and period.

4.2.2 Time Step

The comparison between R1, R3 and R4 allows verifying the improvements in the results when the 4

components of the time step are reduced to half and to a quarter, respectively.

As expected the quality of all the wave parameters simulated has improved when the time step is

reduced since the ocean conditions are calculated in more frequently.

For R1 we recall that the global time step was defined as 200s, which means that for the period of

analysis considered calculations were made every 200s (3:20min). In R3 the global time step was set

as half of 200s,100s (1:40min) which means that the interval of calculations were reduced, or, in other

words, the number of calculations was increased. At last, in R4 the global time step considered was

50s.

This approach did indeed improve the quality of every wave parameters as the time step was reduced,

however, it had very high computational costs since that for every time step interval the total number

of equation is equivalent to the total number of grid points multiplied by the frequency intervals along

the number of defined directions.

In the table below a comparison is made between the improvements in the correlation factor and the

computation time required by each case.

49

R1

Portugal

R3

Portugal ½Tstep

R4

Portugal ¼ Tstep

Duration 9h50 16h27 48h25

R Hs 0,9213 0,9222 0,9225

R Tm 0,8470 0,8545 0,8560

R Dir 0,8755 0,9020 0,8997

Table 4.4 - Computational cost vs Correlation for different time steps (R1, R3, R4)

As we can see a slight increase in the correlation implies a great computational cost. This is a

substantial limitation considering that with such timing requirements the ability of the model to be used

for forecasting purposes becomes seriously affected.

4.2.3 Wind Model

As the wind is the main driving force of wave’s generation, the quality of the wave model results are as

predicted directly related with the quality of the input wind.

In this analysis R1 and R5 were compared, evidencing the differences between wind forced by a low

resolution model , GFS, and a more detailed model, WRF9, respectively.

Results show a better correspondence between the significant wave height from the WRF9, given that

the GFS model (R5) results tend to be overestimated.

For the GFS, with a resolution much less detailed than the grid resolution, it’s necessary to interpolate

the wind velocity for more points than for the WRF9. If we consider that this interpolation could assign

higher values of velocity this overestimation of wind velocity would imply an overestimation of energy

transfer and correspondingly larger significant wave heights than the real data indicates.

When simulating large areas such as North Atlantic Ocean or Southwest Europe a large scale wind

model such as GFS with 0,5° resolution seems to be appropriate, but for smaller confined areas a

more detailed wind model is necessary in order to represent more accurately the wind changes

caused by temperature gradients in shore areas.

50

4.2.4 Bottom Friction

In order to evaluate the bottom friction coefficient, R2 and R6 are compared. Both are configured in

the Nazare grid, but in R2 the bottom friction coefficient was defined with the swell value =

−0,038𝑚2𝑠−3 and R6 with the default value proposed for wind sea states = −0,067𝑚2𝑠−3.

The results for significant wave height and period are the less accurate from the total set of runs,

which supports the argument that the value recommended for wind sea states overestimate the

dissipation of energy. It’s possible to see in the wave height time series that the results from this run

do not follow as close as R2 the monican02 wave height tendency, which is more visible in the highest

values.

4.2.5 Seasonal Comparison

The two model runs conducted in the winter period, R7 and R8, respectively using the Nazare and

Portugal grid, were the ones which delivered the best correlation for significant wave height and

period. Despite that, regarding the wave height their values of bias are the most negative from the set

which indicates that the results are also the most overestimated.

One possible explanation for this is that the WW III is more suited for the oceanic process modelling,

and during winter time, when swell from storms in the Atlantic Ocean are more frequent than in the

summer time, its ability to describe the energy created in a storm propagating across the ocean leads

to better results comparing to summer time, when local wind generated waves is predominant.

The time series shows wave heights exceeding quite oftenly 3m, which is characteristic from storm

situations in this time of the year, and also highest values for period comparing with summer time

which is usually associated with waves that travelled a longer distance.

51

5 Conclusions and Future Developments

5.1 Conclusions

The main objective of this thesis was achieved since wave conditions in a nearshore area were

successful simulated, providing fairly good results from all the tested wave models.

For all the runs, the bias is slightly negative which denotes a tendency in the model to overestimate

the results.

Furthermore an analysis to different parameters and input data was performed and some conclusions

were obtained.

For grid resolution the results weren’t as expected since an improvement in grid resolution didn’t led to

better results.

As for time step, it was confirmed that a decrease in the overall time step, followed by a proportional

reduction of the other time step components, produces results with better quality. Despite that the

increase in the computational costs in doing so is too high for the marginally accuracy improvements.

The wind comparison between GFS and WRF9 also supported that a wind model with better

resolution is more appropriate to be used in coastal applications since its greater level of detail

represents with greater precision the wind changes due to the shoreline characteristics.

Regarding the bottom friction coefficient, it was confirmed that the lowest indicated value is more

appropriate to be included in the parameterization regardless of the ocean state characteristics.

The seasonal analysis enabled the evaluation of the contrast between a summer and a winter period

of simulation. The best correlated results for the winter period seems to indicate a better ability of WW

III to reproduce deepwater processes which are more dominant during the winter time.

With the nearshore parameterizations that have been introduced in the last years, WAVEWATCH III

has now the capability of being applied to shallow water areas. However, as the model authors admit,

surf-zone physics implemented so far are still fairly rudimentary, which means that there’s still room for

improvement in order to make this wave model adequate for coastal areas as it is for deep waters.

If we wanted to simulate the extreme wave climate that reaches Praia do Norte, it would be necessary

to use a phase-resolving model, such as the Boussinesq (1872) Model that takes into account the

vertical structure of the horizontal and vertical flow velocity.

52

5.2 Future Developments

There’s a set of future developments that can be performed so that further conclusions can be taken

regarding WW III ability to model nearshore areas, more specifically the Nazare area.

A larger period of simulation (>1year) could also be used in order to have a more general set of results

that embrace both winter and summer periods.

The simulation period could also be defined according to the availability of both Monican01 and

Monican02 so that it could analyzed the differences in the waves as they pass through these two

points.

SWAN model could be used to simulate wave conditions in Nazare and then the results could be

compared with the ones from this study, in order to quantify the WW III difficulty to perform in

nearshore areas.

53

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Internet references:

http://polar.ncep.noaa.gov

http://monican.hidrografico.pt/

http://pt.magicseaweed.com/

http://polar.ncep.noaa.gov/

http://monican.hidrografico.pt/

http://pt.magicseaweed.com/

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