Breeze-forced oscillations and strongly nonlinear tide-generated internal solitons

ANEXO II - PROGRAMA DE DOCTORADO EN OCEANOGRAF´ IA - Departamento de F´ısica Bienio 2006 – 2008 Breeze-forced oscillations and strongly nonlinear tide-generated internal solitons Oscilaciones forzadas por las brisas y ondas internas solitarias de origen mareal fuertemente no lineales Tesis Doctoral presentada por D. Borja Aguiar Gonz´alez para obtener el grado de Doctor por la Universidad de Las Palmas de Gran Canaria. Dirigida por el Dr. ´ Angel Rodr´ıguez Santana, el Dr. Theo Gerkema y el Dr. Jes´us Cisneros Aguirre. El Director El Director El Director El Doctorando Las Palmas de Gran Canaria a de Abril de 2013 v ANEXO I D. Salvador Galv´an Herrera Secretario del Departamento de F´ısica de la Universidad de Las Palmas de Gran Canaria, CERTIFICA, Que el Consejo de Doctores del Departamento en sesi´on extraordinaria tom´o el acuerdo de dar el consetimiento para su tramitaci´on a la tesis doctoral titulada ‘Breeze-forced oscillations and strongly nonlinear tide-generated internal solitons’ presentada por el doctorando D. Borja Aguiar Gonz´alez y dirigida por los Doctores ´ Angel Rodr´ıguez Santana, Theo Gerkema y Jes´us Cisneros Aguirre. Y para que as´ı conste, y a efectos de lo previsto en el Art. N◦6 del Reglamento para la Elaboraci´on, Defensa, Tribunal y Evaluaci´on de Tesis Doctorales de la Universidad de Las Palmas de Gran Canaria, firmo la presente en Las Palmas de Gran Canaria, a de Abril de 2013. iii A mis padres Mappi e Ignacio A mi hermano Jose Thesis Preview The present Thesis entitled ‘Breeze-forced oscillations and strongly nonlinear tide-generated internal solitons’ was developed at the Universidad de Las Palmas de Gran Canaria and the Royal Netherlands Institute for Sea Research. Financial support for this PhD study was provided by the Ministry of Science and Innovation (Spanish Government) through a FPU grant awarded to D. Borja Aguiar Gonz´alez, and the research project PROMECA (CTM2008-04057/MAR), whose Chief Investigator is Dr. ´ Angel Rodr´ıguez Santana. This PhD research has been co-supervised by Dr. ´ Angel Rodr´ıguez Santana and Dr. Jes´us Cisneros Aguirre from the Department of Physics at the Universidad de Las Palmas de Gran Canaria, and Dr. Theo Gerkema from the Department of Physical Oceanography at the Royal Netherlands Institute for Sea Research. The structure of the Thesis is organized as follows. The first half of the document deals with breeze-forced oscillations. Chapter 1 introduces the ‘state of the art’ on the study of this phenomenon. Next we present an observational study on breeze-forced oscillations poleward the critical latitude for diurnalinertial resonance along Chapters 2–3. The second half of the document is devoted to modeling strongly nonlinear tide-generated internal solitons. Subsequently, Chapter 4 provides the empirical and theoretical background on the phenomenon of study. Chapter 5 presents the derivation of a theoretical model on strongly nonlinear tide-generated internal solitons; and, Chapter 6 discusses on numerical experiments performed with the derived model. Conclusions and future research are presented in Chapter 7. Finally, a summary in Spanish is included in Chapter 8 as required by the PhD Thesis Regulations from the Universidad de Las Palmas de Gran Canaria (BOULPGC. Art. 2, iii Chap. 2, November 5th 2008). The references are listed at the end of the document. iv Presentaci´on de la Tesis La presente Tesis Doctoral se titula ‘Oscilaciones forzadas por las brisas y ondas internas solitarias de origen mareal fuertemente no lineales’ y se ha desarrollado entre la Universidad de Las Palmas de Gran Canaria y el Royal Netherlands Institute for Sea Research. La financiaci´on ha procedido del Ministerio de Ciencia e Innovaci´on (Gobierno de Espa˜na) a trav´es de la concesi´on de una beca del Programa de Formaci´on de Profesorado Universitario (FPU) al doctorando D. Borja Aguiar Gonz´alez, y del proyecto de investigaci´on PROMECA (CTM2008-04057/MAR), del cual el Dr. ´ Angel Rodr´ıguez Santana es Investigador Principal. Esta Tesis Doctoral ha sido codirigida por el Dr. ´ Angel Rodr´ıguez Santana y el Dr. Jes´us Cisneros Aguirre, ambos del Departamento de F´ısica de la Universidad de Las Palmas de Gran Canaria, y por el Dr. Theo Gerkema del Departamento de Oceanograf´ıa F´ısica del Royal Netherlands Institute for Sea Research. La estructura del documento se ha organizado de la siguiente manera. La primera parte de la Tesis Doctoral (Cap´ıtulos 1-3) investiga la respuesta del oc´eano al forzamiento de las brisas. El Cap´ıtulo 1 introduce el ‘estado del arte’ en el estudio de este fen´omeno f´ısico. Los Cap´ıtulos 2-3 forman el n´ucleo de un estudio observacional sobre oscilaciones forzadas por las brisas en latitudes cr´ıticas para la resonancia diurno-inercial. La segunda parte de la Tesis Doctoral (Cap´ıtulos 4-6) desarrolla un modelo te´orico para la simulaci´on de ondas internas solitarias de origen mareal fuertemente no lineales. El Cap´ıtulo 4 introduce el ‘estado del arte’ en el estudio de este fen´omeno f´ısico. El Cap´ıtulo 5 describe el desarrollo del modelo te´orico. El Cap´ıtulo 6 presenta la discusi´on de diferentes simulaciones num´ericas obtenidas con el modelo. Las conclusiones y trabajo futuro abarcan el contenido del Cap´ıtulo 7. Por ´ultimo se incluye v un resumen en espa˜nol de la Tesis Doctoral en el Cap´ıtulo 8 para cumplir con los requisitos establecidos por el Reglamento para la Elaboraci´on, Tribunal, Defensa y Evaluaci´on de Tesis Doctorales de la Universidad de Las Palmas de Gran Canaria (BOULPGC. Art. 2, Cap. 2, 5 de Noviembre de 2008). Las referencias citadas a lo largo del documento figuran al final del mismo. vi CONTENTS 7 Conclusions and Future Research 141 8 Summary of the Thesis in Spanish 143 8.1 Un Escenario Forzado por las Brisas . . . . . . . . . . . . . . . 145 8.1.1 Brisas Tierra-Mar . . . . . . . . . . . . . . . . . . . . . 145 8.1.2 Oscilaciones Forzadas por las Brisas . . . . . . . . . . . 152 8.1.3 Marco del Estudio, Objetivos y Metodolog´ıa . . . . . . 170 8.2 Ondas Internas Solitarias en el Oc´eano . . . . . . . . . . . . . . 174 8.2.1 Estudios Observacionales . . . . . . . . . . . . . . . . . 174 8.2.2 Descripci´on Matem´atica de los Solitones . . . . . . . . . 183 8.2.3 Marco del estudio, Objetivos y Metodolog´ıa . . . . . . . 191 8.3 Conclusiones y Trabajo Futuro . . . . . . . . . . . . . . . . . . 195 A Rotary Wavelet Method 197 A.1 Rotary Wavelet Power Spectrum . . . . . . . . . . . . . . . . . 197 A.2 Rotary Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.3 Statistical Significance Test . . . . . . . . . . . . . . . . . . . . 200 A.4 Cross-Rotary Wavelet Power Spectrum . . . . . . . . . . . . . . 201 A.5 Rotary Wavelet Coherence Squared Spectrum . . . . . . . . . . 201 A.6 Rotary Wavelet Phase Spectrum . . . . . . . . . . . . . . . . . 202 B Theorem for Derivatives of Integrals with Variable Boundaries 205 C Derivation of Function Gi(x, t)207 C.1 Upper Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 C.2 Lower Layer Over an Oscillating Topography . . . . . . . . . . 209 D Derivation of ∂p′ 2(1) ∂x 213 E Substituting ¯u2in terms of ¯u1215 References 219 Data Acknowledgements 233 Agradecimientos 235 Curriculum Vitae 239 xiii CONTENTS xiv List of Figures 1.1 Diagram of a Lake Breeze Regime . . . . . . . . . . . . . . . . 2 1.2 Diagram of a Land Breeze Regime . . . . . . . . . . . . . . . . 3 1.3 Seasonal Variability of Sea-land Breezes on the Texas-Louisiana shelf ................................. 5 1.4 The Effect of Coastlines on Sea Breezes . . . . . . . . . . . . . 7 1.5 Seasonal Variability of Breeze-Forced Oscillations on the TexasLouisiana shelf . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Breeze-Forced Oscillations on the Namibian shelf . . . . . . . . 10 1.7 Phase Evolution of Breeze-Forced and Free Near-Inertial Oscillations................................ 12 1.8 Slab Model of Forced Oscillations . . . . . . . . . . . . . . . . . 14 1.9 A Continuos Model with Friction of Forced Oscillations . . . . 18 1.10 Effects of Breeze-Forced Oscillations on the Vertical Mixing . . 21 1.11 Regions for Diurnal/Semidiurnal Resonance . . . . . . . . . . . 23 2.1 Oceanographic Buoys - Puertos del Estado . . . . . . . . . . . 27 2.2 Rotary Wavelet Power Spectrum of Wind at the Buoy Gulf of C´adiz ................................ 31 2.3 Rotary Wavelet Power Spectrum of Wind at the Buoy Valencia II 32 2.4 Rotary Wavelet Power Spectrum of Wind at the Buoy Cape Pe˜nas ................................ 33 2.5 Rotary Wavelet Power Spectrum of Currents at the Buoy Gulf ofC´adiz............................... 34 2.6 Rotary Wavelet Power Spectrum of Currents at the Buoy ValenciaII............................... 35 2.7 Rotary Wavelet Power Spectrum of Currents at the Buoy Cape Pe˜nas ................................ 36 xv LIST OF FIGURES 2.8 Averaged Wavelet Variance of Wind and Currents from the Buoy at the Gulf of C´adiz . . . . . . . . . . . . . . . . . . . . . 39 2.9 Averaged Wavelet Variance of Wind and Currents from the Buoy at the Gulf of Valencia . . . . . . . . . . . . . . . . . . . 40 2.10 Averaged Wavelet Variance of Wind and Currents from the Buoy at the Cape Pe˜nas area . . . . . . . . . . . . . . . . . . . 41 2.11 Squared Wavelet Coherency and Phase Spectra between Wind and Currents (Clockwise Components) measured at the Buoy Gulf of C´adiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.12 Squared Wavelet Coherency and Phase Spectra between Wind and Current (Clockwise Components) measured at the Buoy Cape Pe˜nas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 MREA04 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Wind Time series for the Bay of Set´ubal . . . . . . . . . . . . . 55 3.3 Breeze-Forced Oscillations on the Portuguese shelf . . . . . . . 57 3.4 Surface Contours for the Bay of Set´ubal . . . . . . . . . . . . . 60 3.5 Diapycnal Mixing Analysis . . . . . . . . . . . . . . . . . . . . 62 3.6 Rotary Wavelet Power Spectrum of Wind from a Meteorological Station at Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1 Schematic of Canonical Soliton Packets . . . . . . . . . . . . . 72 4.2 Solitary Waves over the Oregon shelf . . . . . . . . . . . . . . . 74 4.3 Evolution of a Solitons Packet in the Sulu Sea . . . . . . . . . . 75 4.4 Remote Sensing SAR (Synthetic Aperture Radar) images of Solitons in the Sulu Sea . . . . . . . . . . . . . . . . . . . . . . 77 4.5 Solitary wave solutions of the eKdV equation . . . . . . . . . . 82 4.6 Comparison of KdV and MCC theories . . . . . . . . . . . . . . 83 6.1 Configuration of the Numerical Experiments . . . . . . . . . . . 125 6.2 Moving Topography vs. Tidal Motion . . . . . . . . . . . . . . 128 6.3 Generation of the Linear Hydrostatic Internal Tide (Rotationless case, µ=0)...........................131 6.4 Generation of the Linear Hydrostatic Internal Tide (at Mid Latitudes: φ= 45◦,µ= 4.61)...................132 6.5 Generation of the Linear Hydrostatic Internal Tide (at High Latitudes: φ= 90◦,µ= 6.52)...................133 6.6 Generation of ‘Table-Top’ Solitons . . . . . . . . . . . . . . . . 135 6.7 Time evolution of ‘Table-Top’ Solitons . . . . . . . . . . . . . . 136 xvi LIST OF FIGURES 6.8 Generation of Solitons between Two Layers of Equal Thickness 137 6.9 Time evolution of Solitons between Two Layers of Equal Thickness .................................137 6.10 Time evolution of Strongly Nonlinear Solitons dispersed by the Effects of the Earth’s Rotation (at Mid Latitudes: φ= 45◦, µ= 4.61) ..............................138 8.1 Diagrama de un r´egimen de brisas de lagos . . . . . . . . . . . 146 8.2 Diagrama de un r´egimen de brisa de tierra . . . . . . . . . . . . 147 8.3 Variabilidad estacional de las brisas mar-tierra en la plataforma de Texas-Louisiana . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.4 El efecto de la costa sobre las brisas . . . . . . . . . . . . . . . 151 8.5 Variabilidad estacional de las oscilaciones forzadas por la brisa en la plataforma de Texas-Louisiana . . . . . . . . . . . . . . . 154 8.6 Oscilaciones forzadas por la brisa en la plataforma de Namibia 156 8.7 Evoluci´on de la fase de oscilaciones forzadas por brisa y casiinercial ‘libres’ . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.8 Modelo de oscilaciones forzadas . . . . . . . . . . . . . . . . . . 160 8.9 Modelo Continuo con fricci´on de oscilaciones forzadas . . . . . 166 8.10 Efectos de las oscilaciones forzadas por brisa en la mezcla vertical169 8.11 Regiones para resonancia diurna/semidiurna . . . . . . . . . . . 171 8.12 Esquema de paquetes de solit´on can´onico . . . . . . . . . . . . 176 8.13 Ondas solitarias sobre la plataforma de Oregon . . . . . . . . . 178 8.14 Evoluci´on de un paquete de solitones en el Mar de Sulu . . . . 180 8.15 Im´agenes de teledetecci´on SAR (Synthetic Aperture Radar) de solitones en el Mar de Sulu . . . . . . . . . . . . . . . . . . . . 181 8.16 Soluciones de ondas solitarias de la ecuaci´on eKdV . . . . . . . 188 8.17 Comparaci´on de las teor´ıas KdV y MCC . . . . . . . . . . . . . 189 xvii LIST OF FIGURES xviii Chapter 1 Introduction to a Breeze-Forced Scenario This chapter presents a review on observations, theoretical and modeling works from the existing literature on the topic ‘breeze-forced oscillations within the critical latitudes for diurnal-inertial resonance’, which for short we may refer in future chapters as ‘resonant breeze-forced oscillations’. In Section 1.1 we start the introduction to a breeze-forced scenario with a brief description of the fundamental aspects of sea-land breezes as a forcing to the coastal ocean. Therein, we focus on the physics of resonant breeze-forced oscillations in Section 1.2. We discuss the scope of this PhD study within the described context in Section 1.3. 1.1 Sea-Land Breezes Sea-land breezes are thermally-induced winds caused by the differential heating and cooling of sea and land at diurnal cycles. This phenomenon is supported due to the ocean has a higher specific heat capacity than land, what entails that during daytime land warms up more than the ocean and so the air above it. The process reverses at night time, when land cools more than the ocean. This results in land-water air pressure differences driving systems of breezes, generally onshore-offshore, along about two thirds of the earth’s coasts (Simpson,1994). Thus we have the scenario where sea-land breezes can 1 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO occur, i.e. along coastlines adjacent to large bodies of water: oceans, seas, lakes, rivers,... Figure 1.1: Diagram of a Lake Breeze Regime - Idealized illustration of a typical lake breeze circulation and its associated front. The dashed line represents the outer boundary of the inflow layer. The frontal zone is not shown to scale. [From Sills (1998)]. A general description of a sea-land breeze event may start with calm weather conditions where surface pressures over land and sea are initially the same along a straight coastline. After daytime insolation, air over land is heated and starts to expand. Then, a fall of the pressure at low levels in the atmosphere inland leads to a pressure gradient between air over land and air over sea. Once the pressure gradient is acting, one can say the onset of the breeze system has taken place. This pressure gradient working during daytime is the driving force of the onshore surface winds known as ‘sea breezes’. Figures 1.1 and 1.2 show two idealized schemes for a breeze system. These schemes detail lake-land breezes acting in the Great Lakes (North America) 2 1.1 Sea-Land Breezes based on previous literature values (Moroz,1967;Lyons,1972;Lyons and Olsson,1973;Keen and Lyons,1978) and allow us to explain the main dynamics of sea-land breezes. However, though thickness layer and velocity values maintain rather similar numbers, it should be noticed that horizontal scales have been observed to be longer in oceanic areas in comparison with those in lakeland breeze systems (Simpson,1994;Hyder et al.,2002;Rippeth et al.,2002; Simpson et al.,2002;Gille et al.,2003;Zhang et al.,2009;Hyder et al.,2011). Oceanic horizontal scales may extend from several tens to a few hundreds of kilometres alongshore, and more than 100 km offshore from the coast (Simpson,1994;Gille et al.,2003;Aparna et al.,2005;Gille et al.,2005;Zhang et al., 2009). Figure 1.2: Diagram of a Land Breeze Regime - Idealized illustration of a typical land breeze circulation and its associated front. The dashed line represents the outer boundary of the outflow layer. The frontal zone is not shown to scale. [From Sills (1998)]. The thickness of the sea breeze layer may reach a height from the surface between 300 −1000 m, depending on the site, with typical wind speeds about 3 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO 4−7 m s−1(Figure 1.1). Nonetheless, it has been also found that it can even exceed 10 m s−1in some cases (Pattiaratchi et al.,1997). This cool and wet air flowing inland converges to the warm and dry air ascending from the heated land. The result is a frontal zone (1-2 km wide) parallel to the coast where a line of cumulus clouds tends to arise. This cloud formation is an evidence of the so called ‘sea-breeze front’. Above the sea breezes flowing inland, a weaker return flow of 2 −4 m s−1balances the system at a height of ∼500 −2000 m, and sinks over inshore waters closing the sea breeze cell. During the night time and the early morning, land becomes cooler than the ocean and the process reverses (Figure 1.2). Thereof, the air contracts inland and descends creating a zone of relatively high pressure. The consequent pressure gradient between land and sea drives the onset of the seaward winds known as ‘land breezes’ (Figure 1.2). The cooling of the atmosphere over land is restricted to a shallower layer at night than the layer of heating of the air during the day time. This makes land breezes to be tipically shallower and weaker than sea breezes. Therefore, the thickness of the land breeze layer presents a lower height varying between 50 −400 m with typical wind speeds <4 m s−1. These offshore breezes are balanced by a return flow (<1 m s−1) over sea moving landward at a height of 100 −1000 m and descending over the coast. In this case, the convergence zone tends to form cumulus clouds over the sea, where the air flows upward. This boundary is known as the ‘land-breeze front’. As expected from thermally-induced winds, the strength of sea-land breeze systems is directly proportional to the temperature gradient between air over land and air over the ocean (Pielke and Segal,1986). Hence, frequency and intensity of sea-land breeze events are higher during periods of strong daytime heating and night time cooling in the absence of large scale wind systems. In tropical and subtropical climates, where the atmospheric pressure gradient between land and sea tends to be steady throughout the year, sea-land breeze events are expected to occur at any season. On the contrary, in temperate climates, sea-land breezes have often to overcome winds from different directions driven by the passage of cyclones and anticyclones which affects its appearance (Simpson,1994). At these latitudes, the breeze system needs a temperature difference between land and sea which must be large enough to promote breezes getting over large scale wind systems. Therefore, they are most frequently observed during spring and summer months when large sea-land temperature differences are accompanied by favourable synoptic conditions. A nice exam4 1.2 Breeze-Forced Oscillations (2002) show an illustrative example for a breeze forcing which is interrupted (after day 31 in Figure 1.7) and the resultant motion switches to the inertial frequency so that, when viewed as a diurnal-forced motion, there is a regular increase in the phase lag with time (Figure 1.7). Thus the alternation of diurnal-forced and transient-inertial oscillations is an expected feaure of windforced motions. Additionally, some authors Hyder et al. (2002); Rippeth et al. (2002); Sobarzo et al. (2007); Hyder et al. (2011) have also observed in time series data that there is a beat period of ∼(2π/(ω−f) in which the strength of the periodic diurnal current components (ω) oscillates in combinaton with free phase inertial current components (f). Geographically, near-inertial motions are a widespread feature in deep and coastal oceans which brings into play an important source of energy available for mixing (Webster,1968;Millot and Crepon,1981;Salat et al.,1992;Font et al.,1995;van Haren et al.,1999;Knight et al.,2002;van Aken et al., 2005;Sobarzo et al.,2007;Chaigneau et al.,2008). However, near-inertial energy level is not uniform worlwide and is more variable than the rest of the spectrum (Fu,1981;Garret,2001;Gerkema and Shrira,2005). Breeze-forced oscillations contribute (among other phenomena not addressed here) to those geographical differences on the near-inertial energy levels. This is based on the fact that depending on the latitude in which the breeze forcing is acting, the ocean response may be resonant. Thus, near 30◦N/S, inertial and diurnal periods are close to each other and the transfer of momentum and energy from breeze forcing to inertial motions can be significantly increased (Hyder et al., 2002;Rippeth et al.,2002;Simpson et al.,2002;Sobarzo et al.,2007). On the basis of all mentioned differences between breeze-forced oscillations and ‘free’ near-inertial oscillations, valuable efforts have been done in the last decades to develop theory and model breeze-forced oscillations (Craig,1989b; Chen and Xie,1997;Rippeth et al.,2002;Simpson et al.,2002;Hyder et al., 2002,2011). In the following, we review theoretical and modeling works which focus on the observed behaviour of the ocean to breeze forcing. Additionally, we point out those modeling results which predict interesting features not yet confirmed with observations and thus needed of further research. This review provides a deeper insight into the physics behind the generation and evolution of breeze-forced oscillations. 11 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO Figure 1.7: Phase Evolution of Breeze-Forced and Free Near-Inertial Oscillations - Amplitude and phase of the diurnal motion over the full recording period. (a) Current at 34-m anticlockwise (*) and clockwise (+), (b) current at 126 m anticlockwise (x) and clockwise (•), (c) phase of anticlockwise component at 34 m (*) and 126 m (x), (d) wind amplitude as anticlockwise (*) and clockwise (+) components, and (e) phase of the wind components (*) anticlockwise and clockwise (+). The straight sloping line in (c) indicates the rate of phase change of a pure inertial oscillation (15.5◦d−1). [From Simpson et al. (2002) - Fig. 7]. 12 1.2 Breeze-Forced Oscillations 1.2.1 Periodic Forcing and Near-Inertial Resonance Theoretical and modeling advances on the ocean response produced by periodic forcing (tidal and wind forcing) at a particular frequency have been greatly benefited from the studies of Battisti and Clarke (1982) and Craig (1989a,b). In the following we describe the dynamics of breeze-forced oscillations by introducing some recent illuminating works. Critical Latitudes for Resonance The essential mechanism and latitudinal variation of diurnal wind-forced motions are clearly illustrated in Simpson et al. (2002) with a simple slab model for a water column of depth Hvertically uniform in velocity forced by an oscillating wind stress τswith no horizontal pressure gradients (Figure 1.8). Frictional damping is introduced linearly via (−ru, −rv), leaving the dynamical equations for the complex velocity w=u+iv as ∂w ∂t +ifω =−rw +τs ρH (1.1) Then, the complex amplitudes for clockwise (W−) and anticlowise (W+) forcing at frequency ωare W−=A− i(f−ω) + r′;W+=A+ i(f+ω) + r′(1.2) where τs/(ρH) = A±e±ωt and r′=r/(ρH). As it can be seen in Figure 1.8 (top), the ocean response is resonant (W→A/r′) for the clockwise case at the critical latitud of 30◦N where f=ω, and for the anticlockwise case at the critical latitud of 30◦S where f=−ω. In other words, the ocean response is resonant when diurnal and inertial periods are close to each other. Under these circumstances, current and forcing are in phase and the transfer of momentum and energy is significantly enhanced. It is also shown that the phase of the current changes rapidly between limiting values of +π/2 and −π/2 farther than 30◦N/S, and that this effect is sharper when using a higher coefficient for frictional damping. 13 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO Figure 1.8: Slab Model of Forced Oscillations - Variation of the response function Z=W/A with latitude for a simple slab model of forced oscillations with r′= 10−5,2 10−5,410−5s−1. The amplitude (a) and phase (b) are shown for clockwise (solid line) and anticlockwise (dotted line). [From Simpson et al. (2002) - Figure 1]. Dynamics of Rotary Breeze-Forced Oscillations If we now move a step forward, we find the two-layer model developed by Simpson et al. (2002), which allows us to introduce the physics behind breeze-forced oscillations in coastal areas. They hypothesised an oscillatory diurnal wind stress acting directly at the ocean surface and in the presence of a coast. This condition produces a coast-normal surface slope variation which drives the transfer of momentum to the whole water column though a pressure gradient of comparable magnitude but opposite in phase to the surface forcing. From this starting point, Simpson et al. (2002) use a frictionless two-layer 14 1.2 Breeze-Forced Oscillations analytical model with the upper layer being forced by a diurnal wind stress and the opposing surface slope resulting from this wind forcing; and, the lower layer being forced solely by the surface slope. This external pressure gradient forces the lower layer leading to phase-shifted motions with relatively the same amplitude as in the upper layer. The model successes on reproducing the key features of reported breeze-forced oscillations in the literature: anticyclonic currents rotating ∼180◦out of phase above and below the pycnocline with energetic amplitudes of similar magnitud and almost constant phase within every layer (Chen et al.,1996;Rippeth et al.,2002;Simpson et al.,2002;Zhang et al.,2009;Hyder et al.,2011). In contrast to what is observed with free near-inertial motions (Millot and Crepon,1981;Orli´c,1987), which exhibit an increasing phase delay and less energetic currents in deeper waters due to momentum is transferred downward through the shear stress described by classical Ekman theory. Simpson et al. (2002) set their simple two-layer model to account the diurnal wind-forced ocean response as follows. They consider the flow in a shelf region bounded by a coastline extending in the y direction at x= 0 and with a depth profile H(x). The layers are assumed to be uniform in density (ρ) and velocity (u, v) but decoupled from each other by a frictionless interface. Depth of each layer is given by h1and h2, the upper and lower layers, respectively. The motion is forced by an oscillatory wind stress (τx, tauy) at the diurnal frequency ω, which acts directly only at the surface layer. Hence, they derive the following momentum equations for the upper (subscript 1) and lower (subscript 2) layers ∂u1 ∂t −fv1=τx ρh1−g∂η ∂x,(1.3) ∂v1 ∂t +fu1=τy ρh1 ; ∂u2 ∂t −fv2=−g∂η ∂x,(1.4) ∂v2 ∂t +fu2= 0; where ηis the surface elevation, gis gravitational accelaration, and fis the Corilis frequency. The key component of this model enters into play through 15 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO the ‘Craig approximation’ (Craig,1989b) (eq. 1.5), which sets up the role of the coastal boundary inducing a pressure gradient in response to surface stress forcing. The approximation derives from the lowest order vertically integrated solution, after assuming that the ratio of the shelf width Lto the barotropic wavelength 2π(gH)1/2/ω is small. This requirement of the ‘Craig approximation’ implies that the time of transit of a barotropic wave accross the shelf is small in comparison to the inertial period. Then, the surface slope induced by the applied wind stress can be defined as ∂η ∂x =τx+i(f/ω)τy ρgH .(1.5) Because of the no-normal-flow condition at the coast, the diurnal crossslope wind stress τxgenerates an opposing surface slope component. In accordance, the diurnal alongshore wind stress τyinduces a coast-parallel, depth uniform current V=iπ/(ωgH) that, in geostrophic balance, requires a surface slope component of amplitude (f/ω)τy/(ρgH). As the system is derived, it is clear that the wind stress forcing and the opposing surface slope govern the upper layer, while in the lower layer only operates the induced surface slope variation (the pressure gradient associated with the surface slope acts throughout the entire water column). Defining the complex velocity as w=u+iv =W eiωt, and introducing the ‘Craig approximation’ into the momentum equations, eq. (1.3) and eq. (1.4), Simpson et al. (2002) describe the diurnal-inertial resonance in the Southern Hemisphere through the regular oscillatory solutions for anticlockwise motion W1=γTx+i(γ+f/ω + 1)Ty iρH(f+ω); (1.6) W2=−(Tx−if/ωTy) iρH(f+ω).(1.7) with γ=h2/h1, positive ωcorresponding to anticlockwise motion; Tx and Tybeing the complex amplitudes of the wind stress; and W1and W2 16 1.2 Breeze-Forced Oscillations the complex amplitudes of the motion in upper and lower layer, respectively. Thus, the ocean response, eq. (1.6) and eq. (1.7), for the anticyclonic motion will be enhanced at latitudes close to 30◦S where it enters in resonance whith f→ −ωso that f/ω → −1 leading to W1≈γTx+i(γ+iTy) ρH(f+ω)≈W2.(1.8) This particular solution for the steady state response of the ocean near critical latitudes reproduces the main features of observed wind-forced diurnalinertial oscillations that may be influenced by land boundaries (Chen et al. (1996); Rippeth et al. (2002); Simpson et al. (2002); Zhang et al. (2009); Hyder et al. (2011)). The phase of the forced-motion oscillating within the diurnal-inertial band is shifted by 180◦between upper and lower layers, and the velocity amplitude is of comparable magnitude in the two layers when the factor γ, which is a function of the depth of the pycnocline, is of order unity as it occurrs in the study area of Simpson et al. (2002). Analogous effects are obtained from the regular oscillatory solutions for clockwise motion in the Northern Hemisphere (see Rippeth et al. (2002)). The ocean response is then enhanced at latitudes close to 30◦N where f→ω(Northern Hemisphere) and the motion is resonant. In both hemispheres, the ocean response to cyclonic forcing at the diurnal frequency is remarkably weaker in the ratio |(f+ω)| /|(f−ω|). It should be noted that the previous model (Simpson et al.,2002) ignores internal and frictional effects in the mechanism. However, these effects of frictional coupling between layers have been further explored by Rippeth et al. (2002) using a continuos model with friction based on the same dynamics as the model of Simpson et al. (2002). Hence, the new model includes many more layers and shear stresses specified in terms of an eddy viscosity. This modeling work was tested for four different cases (Figure 1.9): run A including a pycnocline and the coastal boundary condition; run B including a pycnocline but not a coastal boundary condition; run C not including a pycnocline, but the coastal boundary condition; and, run D not including neither a pycnocline nor the coastal boundary condition. The aim was to evaluate whether the coastal boundary condition or an existing pycnocline determines the vertical structure of wind-forced motions. 17 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO Figure 1.9: A Continuos Model with Friction of Forced Oscillations - Results from the continuous model. The predicted near surface (solid line) and near bed (dotted line) along-shore current components. (a) Run A: results from the model run which includes a pycnocline and the coastal boundary condition. (b) Run B: results from the model run which includes a pycnocline but does not include a coastal boundary. (c) Run C: results from the model run which includes the coastal boundary but no pycnocline. (d) Run D: results from the model run with no coastal boundary or pycnocline. [From Rippeth et al. (2002) - Fig. 9]. Run results only reproduced the characteristic phase shift and the penetration of energy to near the bed in the cases where the coastal boundary condition was included (run A and run C in Figure 1.9), whereas in its absence (run B and run D) forced motions did not exhibit these features. The authors also found that internal shear stresses did not greatly modify the frictionless 18 1.2 Breeze-Forced Oscillations response to forcing (run A and run C), and confirmed that the reverse flow in the lower layers primarily results from a barotropic pressure gradient (‘Craig approximation’) set up by the applied wind stress and the no-normal-flow condition at the coastal boundary, which is the basis of the model. Consequently, these results point out that the upper and lower layers of breeze-forced motions behave differently than those in purely inertial motions, where the out of phase layers are not driven by a constant forcing but by phase propagation effects and the pycnocline has as well a main role in the propagation. Modeling results from Rippeth et al. (2002) also find a beating cycle between diurnal and inertial periods, despite using a diurnal forcing. In the case of runs without a pycnocline (runs C-D), the beating dies after 4 days as the transient signal is damped out, leaving a steady state diurnal period solution. This behaviour appear to be consistent with previously mentioned beat periods found in observational data (Hyder et al.,2002;Sobarzo et al., 2007;Hyder et al.,2011). 1.2.2 Stratification and Vertical Mixing Near-inertial oscillations close to the critical latitude for diurnal-inertial resonance play an important role on vertical mixing processes, specially in coastal stratifed areas with low tidal energy (Rippeth et al.,2002;Simpson et al.,2002; Hyder et al.,2011). As highlighted in Hyder et al. (2011), earlier suggestions that NIOs brings into play around half the kinetic energy in the worlds oceans (Pollard and Millard,1970;Pollard,1980,1970) are now confirmed with new observations. Thus, an estimated energy flux of ∼0.5−0.7 TW can be found in the literature, a value comparable with that from the internal tides of 0.9TW (Park et al.,2005;Watanabe and Hibiya,2002;Alford,2003a,b;Munk and Wunsch,1998). The first baroclinic mode of NIOs, with antiphase motions between layers above and below the pycnocline, carries along high vertical shears which promote dissipation and vertical mixing in stratified areas. Previous works about strong inertial oscillations (Knight et al.,2002;van Haren,2000) estimated the gradient Richardson number, Ri, to assess mixing conditions promoted by NIOs. This nondimensional number is calculated by Ri =N2 S2,(1.9) 19 1. INTRODUCTION TO A BREEZE-FORCED SCENARIO where N2is the squared Brunt-V¨ais¨al¨a frequency (squared buoyancy frequency), and S2is the squared vertical current shear applied to a fluid parcel. The squared Brunt-V¨ais¨al¨a frequency is given by N2=−g ρ0 ∆ρ ∆z,(1.10) where ρis the potential density calculated from temperature and conductivity data; gis the gravity constant; ρ0is the averaged potential density; and ∆zis the vertical distance between the top and the bottom of the fluid parcel. Finally, the vertical squared shear is given by S2=∆u ∆z2+∆u ∆z2,(1.11) where ∆uand ∆vare the east-west and north-south current differences between the top and bottom of the fluid parcel, respectively. Then, typical values for Ri < 1 indicate the generation of Kelvin-Helmholtz instabilities caused by the vertical shear that triggers mixing in a stratified fluid (Miles, 1986;Van Gastel and Pelegr´ı,2004). On the contrary, if Ri >> 1, buoyancy is dominant since there is insufficient kinetic energy to homogenize the water column. Knight et al. (2002) estimated Ri using ADCP and CTD data in the North Sea and found values less than 1 when strong inertial currents carried large vertical shears across the thermocline, although never approached to critical values of 0.25. Dissipation measurements were also made and indicated that mixing within the thermocline layer was more intense and was associated to high mean thermocline diffusion coefficients when large inertial current shears were acting. Modeling results and observations in the North Sea (van Haren, 2000) also support the importance of tidal and inertial shear across stratification for vertical exchange likely due to internal shear-driven turbulence. If we now consider inertial motions acting around the critical latitudes for diurnal-inertial resonance, one finds that the near-resonant response to diurnal wind forcing involves an efficient transfer of momentum and energy to the ocean with diurnal-inertial motions dominating the kinetic energy spectrum (Simpson et al.,2002). Under these conditions, it is reasonable to expect that breeze-forced oscillations represent the major source of turbulent kinetic energy driving vertical mixing, specially in the absence of friction resulting 20 2.2 Data and Methodology forced mechanism evolves in both situations. The locations of the three buoys are represented in Figure 2.1 and their corresponding inertial periods, offshore distances and depths are listed in Table 2.1. −5000 −5000 −5000 −5000 −4000 −4000 −4000 −4000 −4000 −4000 −4000 −3000 −3000 −3000 −3000 −3000 −3000 −3000 −3000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −2000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −1000 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −700 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −500 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 −100 Cape Peñas (2242) Bay of Setúbal (Chapter 3 in This Work) Gulf of Cádiz (2342) Valencia II (2630) 12oW 9oW 6oW 3oW 0o 3oE 36oN 38oN 40oN 42oN 44oN Longitude Latitude Figure 2.1: Oceanographic Buoys - Puertos del Estado - Map of the Iberian Peninsula showing the oceanographic buoys used in this study and provided by Puertos del Estado from the REDEXT network (Red Exterior de Boyas). Isobath of 200 m is highlighted in black. In the Gulf of Cadiz, Gulf of Valencia and Cape Pe˜nas area, time series of surface current velocity are collected from a UCM-60 current meter settled in Seawatch REDEXT buoys, which have been providing oceanographic and meteorological data since August 1996, September 2005 and March 1998, respectively. The current meter measures velocity at 3 m depth (range: 0 to 3 m s−11% accuracy and current direction within ±1◦). Meteorological sensors (Aandera 2740 and Aandera 3590) measure wind velocity (up to 70 m s−1) and direction at 3 m over the surface with an accuracy of 1.5% and 1%, respectively (Criado-Aldeanueva et al.,2009). Both surface current and 27 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT wind velocity data are obtained by averaging the measurements of the first 10 min of each hour, providing hourly data to the user. All data marked as of dubitous quality by Puertos del Estado during control tests were discarded from our analysis. Gaps in time series, usually due to maintenance or failure of the instruments, have been linearly interpolated when its duration was less than 8 hours. Longer gaps have been left blank. Years with high recurrence of long period gaps were not considered in our study for clarity on the data representation over annual cycles. Additionally, prior to any further analysis, principal tidal constituents were removed from the current measurements using the T Tide Harmonic Analysis Toolbox of Pawlowicz et al. (2002) to eliminate the influence of principal tidal energy (Zhang et al.,2009). Buoy Position Offshore Dist. (km)/ Inertial (Code) (Lat./Long.) Depth (m) Period (hr) Gulf of C´adiz (2342) 36.48◦/-6.96◦80 / 450 20.13 Valencia II (2630) 39.52◦/0.21◦59 / 260 18.81 Cape Pe˜nas (2242) 43.74◦/-6.17◦19 / 450 17.31 Table 2.1: Oceanographic Buoys from Puertos del Estado - Oceanographic buoys used in this study and provided by Puertos del Estado from the REDEXT network (Red Exterior de Boyas). This chapter attempts to provide a basis for the analysis of resonant breezeforced oscillations where rotary wavelet method (Torrence and Compo,1998; Hormaz´abal et al.,2002) is applied to vector time-series of current and wind velocity (see Appendix A). This method separates vector time-series into clockwise (CW) and counterclockwise (CCW) components within time-frequency space (rotary wavelet power spectrum). Further application of the rotary wavelet method on two complex time-series leads to cross-rotary wavelet power spectrum, which provides an estimate of the joint power content of two timeseries for rotary components rotating in the same sense. And finally, the rotary wavelet coherency and phase spectrum which, respectively, find significant time-frequency regions where the two time series covary and detects any lag between corotating components of both series, although not necessarily have high power content in common (Hormaz´abal et al.,2002). To conclude it is important to note that none time series data were filtered in order to preserve lowand high-frequency variability in our wavelet results. The entire picture provided by rotary wavelet analysis results allow us to explore within the same figure all frequency bands interacting simultaneously in 28 2.3 Results and Discussion time and frequency domains for both the atmosphere and the ocean. However, characterization of the general atmosphere/ocean circulation in the areas of study is beyond the scope of this research1and, hence, dominant long-period processes will be addressed here only when they play a role on the dynamics of the analyzed breeze-forced scenario. 2.3 Results and Discussion 2.3.1 Characterization of Three Wind-Forced Scenarios We present a wavelet rotary analysis of meteorological and oceanographic time series data from three different regions around the Iberian Peninsula to characterize the temporal evolution adn variability of sea-land breezes and the ocean response to this wind forcing. Hence, we explore qualitatively simultaneous and co-located measurements of wind at 3 m above the sea surface and ocean currents at 3 m depth. In the next section we use the rotary wavelet method to quantify current variance in response to breeze forcing as well as to investigate the effects of the phase correlation between the forcing and the ocean response in the enhancement of breeze-forced oscillations. Geographically, the Gulf of C´adiz and the Gulf of Valencia are framed within the critical latitudes for diurnal-inertial resonance (30◦±10◦N/S). The Cape Pe˜nas area bounds the poleward limit though being out of this range of latitudes (Figure 2.1; see also Table 2.1 for details on local inertial periods). The three areas present a wide continental shelf for the development of the coastal breeze-forced oscillations subject of this study. A) Temporal Evolution of Sea-land Breezes Sea-land breezes are thermally-induced winds and so their strength is directly proportional to the temperature gradient between air over land and air over the ocean (Pielke and Segal,1986). Consequently, one may expect to find higher recurrence and intensity of sea-land breeze events during periods 1Details on the main currents and tidal components for REDEXT buoys can be found in http://www.puertos.es. 29 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT of strong daytime heating and night time cooling in the absence of large scale wind systems. These conditions can be found specially during summer months when large sea-land temperature differences are accompanied by favourable synoptic conditions and sea-land breezes do not need to overcome winds from different directions driven by the passage of cyclones and anticyclones. On the contrary, during winter months breezes need a larger temperature difference between land and sea than in summer to promote breezes getting over large scale wind systems1. As a consequence, sea-land breezes are less probable to occur under this scenario. Spring and autumn months represent the transitional periods between both situations. This pattern responds to the climatology of a sea-land breeze system (Hunter et al.,2007;Zhang et al., 2009). Figures 2.2-2.4 show the rotary wavelet power spectra of wind at the buoys located in the Gulf of C´adiz, the Gulf of Valencia and the Cape Pe˜nas area, respectively. Firstly, in Figure 2.2, we observe that the temporal evolution of the wind variance at the Gulf of C´adiz shows a clear seasonal pattern within the diurnal band, the weather band (2-8 days) and the intraseasonal band (16-64 days), for both the clockwise and counterclockwise senses. The diurnal wind variance appears as a well-defined packet which is visibly enhanced from the early spring to the early autumn (April-October) in comparison to the lately autumn and winter months (November-March). On the contrary, the weather band and the intraseasonal band involve broader phenomena in the frequency domain and are specially enhanced during winter months, when large scale wind systems are more active. The weakening of these low-frequency bands occur from the early spring to the early autumn, what greatly benefits the onset and enhacement of diurnal-sea land breezes which do not have to overcome winds from large scale scale systems. 1It has been tested with time series data of atmospheric pressure (not shown here) from the three REDEXT buoys that the three areas of study present regularly calm weather conditions during summer months and a more intense large scale activity during winter months. 30 2.3 Results and Discussion Figure 2.2: Rotary Wavelet Power Spectrum of Wind at the Buoy Gulf of C´adiz - Rotary wavelet power spectrum of surface winds (3 m above sea surface) measured at the Buoy Gulf of C´adiz. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel) Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. The pattern described above is consistent with that indicated for the climatology of a sea-land breeze system (Hunter et al.,2007;Zhang et al.,2009). The diurnal wind variance observed during nonsummer seasons may be forced by mechanisms other than sea-land breezes, such as synoptic wind events. The same cause has been pointed out for diurnal wind variance observed during non-breezes periods on the Texas-Lousiana shelf (Chen et al.,1996;Zhang et al.,2009). For this reason in Section 2.3.2 we will use only data from spring-summer months for our analysis on the effects of the phase correlation between the breeze forcing and the subsequent ocean response giving rise to resonant breeze-forced oscillations. 31 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT Figure 2.3: Rotary Wavelet Power Spectrum of Wind at the Buoy Valencia II - Rotary wavelet power spectrum of surface winds (3 m above sea surface) measured at the Buoy Valencia II. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel) Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. Figure 2.3 presents the rotary wavelet power spectrum of wind for the buoy deployed at the Gulf of Valencia. The temporal evolution of the wind variance exhibits in this area a slightly different pattern to that observed in the Gulf of C´adiz. The weather and intraseasonal bands show also an enhacement of variance during the winter months; however, in general terms, large scale wind systems are weaker than those observed in the Gulf of C´adiz for the same months. This may explain that the diurnal wind variance appears as a more continued packet of activity throughout the year since there may exist more days with favourable synoptic conditions for the development of sea-land breezes even during winter months. 32 2.3 Results and Discussion As it occurs at the buoy of the Gulf of C´adiz, both senses of rotation (Figure 2.3 - intermediate, lower pannels) are noticeable in the signal of the sea-land breeze system. Figure 2.4: Rotary Wavelet Power Spectrum of Wind at the Buoy Cape Pe˜nas - Rotary wavelet power spectrum of surface winds (3 m above sea surface) measured at the Buoy Cape Pe˜nas. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel) Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. Finally, Figure 2.4 shows the rotary wavelet power spectrum of wind measured from the buoy deployed at the Cape Pe˜nas area. In this case, the diurnal wind variance does not present a clear seasonal pattern, neither the continued presence of well-packaged signal. On the contrary, sporadic and continuous events dominate the diurnal band, specially in the clockwise sense of rotation (Figure 2.4 - intermediate pannel). This might be the response to a more in33 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT tense weather and intraseasonal bands in comparison with the other two areas of study, what difficults here the development of a sea-land breeze system. It is also worth mentioning that the diurnal wind variance vanishes for the counterclockwise sense during summer months, while it is relatively more active during winter months (Figure 2.4 - lower pannel). B) Temporal Evolution of Wind-Forced Oscillations Now, surface ocean currents measured from the three REDEXT buoys subject of this research are similarly analyzed with the rotary wavelet method. Thus, Figures 2.5-2.7 present the rotary wavelet power spectra of currents at the Gulf of C´adiz, the Gulf of Valencia and the Cape Pe˜nas, respectively. Figure 2.5: Rotary Wavelet Power Spectrum of Currents at the Buoy Gulf of C´adiz - Rotary wavelet power spectrum of surface currents (at 3 m depth) measured at the Buoy Gulf of C´adiz. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel) Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. 34 2.3 Results and Discussion At this point we focus our attention on the temporal evolution of the diurnal-inertial variance, which is the band of the main concern for the study of breeze-forced oscillations in the Northern Hemisphere. Thus, from Figures 2.52.6 it becomes evident that the diurnal-inertial signal exhibits a well-defined packet of enhanced variance in the clockwise sense of rotation (Figure 2.5 and 2.6, intermediate pannels) from the early spring to late autumn months. These observations, made from measurements taken in the Gulf of C´adiz and the Gulf of Valencia, are in good agreement with the seasonal pattern described for sea-land breeze systems in other regions (e.g. the Texas-Louisiana shelf (Zhang et al.,2009)). Figure 2.6: Rotary Wavelet Power Spectrum of Currents at the Buoy Valencia II - Rotary wavelet power spectrum of surface currents (at 3 m depth) measured at the Buoy Valencia II. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel). Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. In this regard, it is interesting to highlight that, in spite of a noticeable 35 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT signal of diurnal wind forcing all along the year in the Gulf of Valencia (Figure 2.3), breeze-forced oscillation variance is enchanced only from the early spring to the late autumn months (Figure 2.6 - intermediate pannel). This scenario responds to previous description given in Chapter 1, where the vertical structure of breeze-forced oscillations was shown to be partially supported by a first baroclinic mode marked by the depth of the pycnocline in stratified waters (Simpson et al.,2002;Rippeth et al.,2002;Zhang et al.,2009;Hyder et al.,2011). This explains that during winter months in the Gulf of Valencia, even in presence of breeze-forcing, the ocean response does not develop an effective transfer of momentum from wind to the interior of the water column. Figure 2.7: Rotary Wavelet Power Spectrum of Currents at the Buoy Cape Pe˜nas - Rotary wavelet power spectrum of surface currents (at 3 m depth) measured at the Buoy Cape Pe˜nas. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel) Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. The previous two areas of study, the Guf of C´adiz and the Gulf of Valen36 2.3 Results and Discussion and accumulation of diurnal-inertial current energy via breeze-forced oscillations as we describe in the following. Figure 2.11: Squared Wavelet Coherency and Phase Spectra between Wind and Current (Clockwise Components) measured at the Buoy Gulf of C´adiz - (top pannel) Rotary wavelet power spectrum of surface winds. (intermediate pannel) Rotary wavelet power spectrum of surface currents. (lower pannel) Squared Wavelet Coherency (γ2) and Phase Spectra between Wind and Current (Clockwise Components). The vectors indicate the phase difference between the wind and current at different frequencies. Positive phase angles between wind and current indicates that current leads wind. Accordingly, vectors are in phase pointing straight up, out of phase pointing straight down, and current leads wind by 90◦pointing right. For clarity only one vector is plotted for each day. Only phase angles (vectors) for γ2>0.7 are plotted. The four patches of very high coherency within the diurnal-inertial band (bottom pannel) are of special interest owing to its associated high diurnal wind (top pannel) and current (intermediate pannel) variance. These patches 43 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT are representative of breeze-forced events and are observed during 8-13 April, 19-23 April, 28 April - 2 May and 3-17 May. All these events start with the same pattern, diurnal wind (breezes) variance arises preceding the increase of diurnal-inertial current variance (breeze-forced oscillations), which is in phase with vectors pointing straight up. At times after the onset of the phenomenon, the phase difference between wind forcing and forced-currents slightly increases to positive phase angles (vectors deviate pointing to the right), and therefore currents leads wind forcing. This pattern may be consistent with an alternation of cycles of forced oscillations, when current motions are in phase with wind forcing, and cycles of decreased forcing when the currents tend to revert to ‘free’ inertial oscillations (Simpson et al.,2002). This would explain that in Figure 2.11, whenever the forcing decreases, the observed phase difference between wind and currents increases and tends to show inertial current phase leading diurnal wind phase since at latitudes poleward of 30◦N/S ‘free’ inertial motions are of higher frequency than diurnal motions. If the wind forcing vanishes completely after several days being active, the situation could be similar to that observed for the breeze-forced event starting on 3 May (Figure 2.11 - bottom pannel). After more than 10 days of constant wind forcing, diurnal wind variance drops on 17 May; however diurnal-inertial current variance is still high and from then ‘free’ inertial oscillations dominate (see in intermediate pannel the decreasing tendency from diurnal to inertial period). As expected, low coherency values between wind and currents are found for days after 17 May, when the breeze forcing has disappeared (top pannel). The enhancement of diurnal-inertial current variance observed in Figure 2.11, and related to a high coherency between forcing and forced-motions, suggests that energetic breeze-forced events are promoted when the relationship between wind and currents lasts for several cycles. These results are in good agreement with previous observations reported by Zhang et al. (2009) on the Texas-Louisiana shelf (equatorward of 30◦N). They suggested that the longer this in phase relationship persists, the larger the magnitude of the diurnalinertial current becomes until a steady state is reached. 44 2.3 Results and Discussion Figure 2.12: Squared Wavelet Coherency and Phase Spectra between Wind and Current (Clockwise Components) measured at the Buoy Cape Pe˜nas - top pannel) Rotary wavelet power spectrum of surface winds. (intermediate pannel) Rotary wavelet power spectrum of surface currents. (lower pannel) Squared Wavelet Coherency (γ2) and Phase Spectra between Wind and Current (Clockwise Components). The vectors indicate the phase difference between the wind and current at different frequencies. Positive phase angles between wind and current indicates that current leads wind. Accordingly, vectors are in phase pointing straight up, out of phase pointing straight down, and current leads wind by 90◦pointing right. For clarity only one vector is plotted for each day. Only phase angles (vectors) for γ2>0.7 are plotted. Finally, Figure 2.12 (Cape Pe˜nas) shows an analogous analysis to that developed in Figure 2.11 for the Gulf of C´adiz. However, as we previously advanced, the Cape pe˜nas area is out of the range of diurnal-inertial resonance and serve us here as a ‘blank test’ of a situation in which resonant diurnalinertial oscillations are not expected to occur. Thus, in comparison with results from the Gulf of C´adiz time series data, here the wind and current show very 45 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT low coherency and the diurnal-inertial phase difference (bottom pannel) is highly variable for most of the analysed period (July-August 2005)1. It is clear that there exists diurnal wind forcing acting during this period (top pannel); however, there is no evidence of this wind forces periodic diurnalinertial currents. The inertial current variance (intermediate pannel) seems to respond to impulsive injections of momentum from diurnal wind (top pannel) although beyond that initial pulse, near-inertial currents may oscillate ‘freely’. Therefore, the dominant ocean response appears to be in the form of transient oscillations which do not accumulate efficiently near-inertial energy in time as it occurs in regions of diurnal-inertial resonance where forced currents may last for several cycles. 2.4 Conclusions We use time series data of surface current velocity and wind velocity over annual cycles from three different regions around the Iberian Peninsula: the Gulf of C´adiz and the Gulf of Valencia (both regions within the critical latitudes for resonance); and, the Cape Pe˜nas area (out of the critical latitudes for resonance). Rotary wavelet analysis results provide evidence that sea-land breeze forcing enhances clockwise rotary current motions (breeze-forced oscillations) which greatly contribute to near-inertial energy budget and its variability in the Gulf of C´adiz and the Gulf of Valencia (Northern Hemisphere). These results also support that the near-resonant condition in these areas promotes the generation and accumulation of diurnal-inertial current energy via breeze-forced oscillations. Squared wavelet coherency and phase spectra suggest that energetic breeze-forced events are enhanced when the relationship between wind and currents lasts for several cycles and present a high coherency between the forcing and forced-motions being in phase. We find that the diurnal-inertial current variance attributed to breezeforced oscillations exhibits a clear seasonal pattern which peaks during springsummer months, presumably due to observed strong breeze events take place 1This analysis has been performed with other periods of time in the area of study and analogous results were found. 46 2.4 Conclusions when coastal waters are sufficiently stratified1. This enhancement of diurnalinertial current variance due to breeze-forced oscillations in the Gulf of C´adiz and the Gulf of Valencia has been estimated to be as large as on order of magnitude higher during summer months (reaching values of up to 225 cm2s−2) than in winter months (values are less than 10 cm2s−2). Despite the fact that this research focuses on time series data from the Gulf of C´adiz and the Gulf of Valencia, we find that these results and methodology may have applications to the study of breeze-forced oscillations over other coastal areas poleward of the critical latitude for diurnal-inertial resonance (30◦N/S). The inertial current variance in the Cape Pe˜nas area also exhibits a seasonal pattern and increases its values from winter months (less than 10 cm2s−2) to summer months (up to 75 cm2s−2). However, this enhacenment is not as large as in the Gulf of C´adiz and the Gulf of Valencia, where resonance between diurnal wind forcing and inertial motions from the ocean response may occur. The inertial current variance in the Cape Pe˜nas area seems to respond to impulsive injections of momentum from diurnal wind. The dominant ocean response appears to be in the form of transient ‘free’ oscillations which do not efficiently accumulate near-inertial energy in time as it occurs in regions of diurnal-inertial resonance where forced currents may last for several cycles. These results support the role of resonant breeze-forced oscillations on the generation and accumulation of near-inertial energy which may be available for promoting vertical mixing with important consequences on coastal dynamics as we will explore in the following chapter. 1We explore the effects of breeze-forced oscillations in stratified waters, and associated diapycnal mixing proccess, in Chapter 3. 47 2. BREEZE-FORCED OSC. AROUND THE POLEWARD LIMIT 48 Chapter 3 Breeze-Forced Oscillations and Diapycnal Mixing1 3.1 Outline This chapter presents an analysis of diapycnal mixing processes triggered by breeze-forced oscillations (BFOs) with observational data. Our interest focused on the relative importance of this phenomenon overcoming the vertical stability of the water column. The goal is to highlight breeze-forced oscillation dynamics in areas where the coast exhibits a well-developed sea-land breeze regime; and, hence, the ocean response is strongly controlled by breeze-forced dynamics. Our research is based on previous literature and new observations of BFOs taken during the Maritime Rapid Environmental Assessment 2004 (MREA04) sea trial off the west coast of the Iberian Peninsula, within the critical latitudes for diurnal-inertial resonance. Additionally, we also count with wind data from a meteorological land station provided by the Instituto Hidrogr´afico - Portugal Analysis of the measurements shows evidence of dominant circular anticyclonic oscillations forced by sea-land breezes acting during the MREA exercise into the Bay of Set´ubal (Portugal). Furthermore, currents were distributed 1Aguiar-Gonz´alez B., Rodr´ıguez-Santana A., Cisneros-Aguirre J., Mart´ınez-Marrero. 2011. Diurnal-inertial motions and diapycnal mixing on the Portuguese shelf. Cont. Shelf. Res. 31, 1193–1201. 49 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING into two main layers with highly stratified and vertically sheared flows. Surface stratification entered into the system through the Sado Estuary (freshwater input). As it will be discussed, breeze-forced currents were found to promote mixing in stratified shallow waters of the shelf area over the scale of a few days. The period of observation counts with days of breeze and relaxed-wind events, what allowed us to track the evolution of the water column at different stages. Wind conditions were determined with an Aanderaa meteorological buoy placed in the Bay of Set´ubal during the MREA04 experiment. Currents data were recorded from two Barny ADCP (Acoustic Doppler Current Profiler) deployments located on the north and south sides of the bay, respectively. Both ADCP’s covered almost the full depth of water (∼112 m) with 24 bins of 4 m size at both sites. Simultaneous CTD (Conductivity, Temperature, Depth) and data were used to analyse the strength of observed breeze-forced oscillations in promoting diapycnal mixing through the squared vertical shear (S2), squared buoyancy frequency (N2) and gradient Richardson Number (Ri) of the water column. The potential role that breeze-forced oscillations may play in shelf areas with low tidal energy has been recurrently highlighted in the literature. Vertically sheared flows in stratified waters may lead to diapycnal mixing and, correspondingly, one could expect that breeze-forced oscillations in stratified waters may promote mixing throghout the water column. Nevertheless, to the best of our knowledge only one paper previous to this study (Zhang et al., 2009) has addressed an analysis which explicitely supports observational evidence of this role. The present research provides a new data set of breeze-forced oscillations in stratified waters near the critical latitudes (30◦±10◦N/S) for diurnal-inertial resonance, where they can greatly contribute to triggering diapycnal mixing. The results of our analysis show that the ocean response to breeze forcing may be enhancing vertical mixing between surface and deeper waters in the Bay of Set´ubal through breeze-forced diunal-inertial currents. A detailed description of the data and methodology used in this chapter is given in Section 3.2. In Section 3.3 we combine data from two ADCP deployments, available CTD casts, a meteorological buoy and a meteorological land station to draw the scenario where the evolution of the water column is analyzed and discussed in terms of breeze-forced oscillation events promoting diapycnal mixing. The chapter ends with conclusions in Section 3.4. 50 3.2 Data and Methodology 3.2 Data and Methodology The MREA04 experiment was performed by the NATO Undersea Research Centre (NURC) from 29 March-19 April 2004 to test the reliability and quality of several real-time coastal oceanic prediction systems (e.g., Rixen and Ferreira-Coelho (2007); Allard et al. (2008); Ko et al. (2008); Leslie et al. (2008); Lam et al. (2009)). The operational area extended off the west coast of Portugal from 40◦N - 36◦N and from 11◦W - 8◦W. This study uses observational data taken over the continental shelf off Portugal, into the Bay of Set´ubal (Figure 3.1). Longitude Latitude −100 −500 −1000 −800 −200 55 56 57 58 59 60 6162 6364 65 66 67 68 69 70 71 72 73 74 75 76 77 79 80 81 82 83 84 85 86 87 88 89 78 T 10 T 11 T 12 T 13 T 14 T 15 T 16 10’ 9oW 50’ 40’ 38oN 6’ 12’ 18’ 24’ 30’ Bay of Setubal Sines Sado Estuary ADCP: 4−14 April Anderaa Meteorological Buoy: 4−13 April CTD Station A: 4 April CTD Station B: 7,8 April CTD Station C: 9−10 April CTD Station D: 12 April CTD Station E: 13 April Figure 3.1: MREA04 Data Set - Operational area over the continental shelf off Portugal (Bay of Set´ubal) during the MREA04 sea trial. CTD stations were sampled several times, represented by coloured circles for each sampling day. Locations of Barny North and Barny South ADCP deployments (close to CTD station 62 and CTD station 78, respectively), an Anderaa meteorological buoy and bottom topography are also shown. The Sado estuary influences the area from the north side of the bay. 51 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING This region is characterised by a steep bathymetry because of the Set´ubal Canyon, and by fluvial inputs coming from the Sado estuary on the northern side of the bay. Related to tides, the Bay of Set´ubal is mainly dominated by lunar (M2) and solar (S2) semidiurnal components (Martins et al.,2002). Vertical temperature and salinity profiles were obtained using Seabird 911 Plus CTD real-time pumped systems. The CTD stations were sampled several times to get incoming temperature and salinity data in real-time to run, correct and compare the forecasts made by the coastal oceanic prediction models tested during the MREA04 trial. A total of 195 CTD casts were distributed among 95 different sites over the continental shelf and open ocean area from 31 March-16 April 2004. In this chapter, we present the 35 different sites that were analised to get an overview of the coastal ocean conditions involving the observed breeze-forced oscillations, and study the associated diapycnal mixing processes into the Bay of Set´ubal. These sites spanned a regular CTD grid with a spatial resolution of 5 km between CTD stations and transects (Figure 3.1). Currents data were obtained from two RDI Workhorse ADCP’s deployed on Barny-type trawl safe platforms and situated on the north (Barny North ADCP) and south (Barny South ADCP) sides of the bay at 106,6 m and 118 m depth, respectively (Figure 3.1). Both deployments of Barny Sentinel ADCP’s recorded vertical profiles of horizontal velocity (u,v) with a broadband of 307.2 kHz and 24 bins of 4 m size. The ADCP heads were about 6 m from the bottom. Both profilers sampled data once every minute for the period of 4−14 April 2004. The rotary spectral method (Hayashi,1979;van Aken et al.,2005) was applied to examine the variance in the current oscillations for each frequency band as clockwise and counterclockwise rotating variances. Because breezeforced oscillations involve currents rotating in a clockwise (counterclockwise) direction in the Northern (Southern) Hemisphere, this technique contributes to determining its presence. Next the diurnal-inertial band, in which breeze-forced oscillation peak, was isolated from the rest of the frequencies to characterise its properties. Previously, to filter this band, the original current records were decimated to 1 datum per hour after lowpass filtering. Then frequency-band of interest was isolated via band-pass filtering with a fourth-order Butterworth filter with zero-phase response and quarter-power points at {c−1ωc, c ωc}around 52 3.3 Results and Discussion (as also shown in Rippeth et al. (2002)). In modeling these energetic oscillations for MREA exercises, it is important to highlight that these are essentially wind-forced motions. Therefore, accurate meteorological forcing represents a critical target to get realistic predictions in real-time, similar to previous suggestions for near-inertial motions observed during the Maritime Rapid Environmental Assessment 2003 (MREA03) exercise, which was performed off of the west coast of Italy near the Elba Island (Coelho and Robinson,2003). It has been attempted (research not shown here) to forecast the observed breeze-forced oscillations with a coastal oceanic prediction model around the ADCP’s sites during the MREA04 sea trial. Tests were capable of reproducing the essential physics of these oscillations, although the forecasted oscillations are weaker and phase lagged in comparison with the observations. Recent efforts and improvements have been done in modeling for MREA exercises with adaptive sampling on short-time scales (Wang et al., 2009;Haley and Lermusiaux,2010;Lermusiaux et al.,2010;Ueckermann and Lermusiaux,2010). 3.3.3 Diapycnal Mixing and Breeze-Forced Events The role that observed breeze-forced oscillations could be playing in the interior of the Bay of Set´ubal responds to its associated high vertical shear within stratified layers (Pelegr´ı and Sangr`a,1998;Rodr´ıguez-Santana et al.,1999, 2001) and is discussed here in terms of diapycnal mixing processes. The entire Bay of Set´ubal exhibited different stages of stratification during the MRE04 sea trial that involved breeze-forced oscillations and intrusions of fluvial waters coming from the Sado estuary. As is shown by the surface salinity and potential density fields in Figure 3.4a,d, this estuarine signal strongly affected the northern and western sides of the bay on 7-8 April, enlarging the area influenced on 9-10 April (Figure 3.4b,e). After a few days, this lighter and freshwater signal had already disappeared (12 April; Figure 3.4c,f). In the following we assess breeze-forced currents triggering diapycnal vertical mixing between oceanic and estuarine waters in the interior of the bay. We focus on that scenario accounting for the interaction between diurnal-inertial oscillation current events and stratification due to freshwater inputs. 59 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING Longitude Latitude 12’ 6’ 9oW 54’ 48’ 42’ 5’ 10’ 38oN 15.00’ 20’ 25’ 30’ 7−8 April T 10 T 11 T 12 T 13 T 14 T 15 T 16 (a) Longitude Latitude 12’ 6’ 9oW 54’ 48’ 42’ 5’ 10’ 38oN 15.00’ 20’ 25’ 30’ 7−8 April T 10 T 11 T 12 T 13 T 14 T 15 T 16 (d) Longitude 12’ 6’ 9oW 54’ 48’ 42’ 5’ 10’ 38oN 15.00’ 20’ 25’ 30’ 9−10 April T 13 T 14 T 15 T 16 (b) Longitude 12’ 6’ 9oW 54’ 48’ 42’ 5’ 10’ 38oN 15.00’ 20’ 25’ 30’ 9−10 April T 13 T 14 T 15 T 16 (e) Longitude 12’ 6’ 9oW 54’ 48’ 42’ 5’ 10’ 38oN 15.00’ 20’ 25’ 30’ 12 April T 12 T 13 T 14 T 15 (c) 35.7 35.75 35.8 35.85 35.9 35.95 36 36.05 Longitude 12’ 6’ 9oW 54’ 48’ 42’ 5’ 10’ 38oN 15.00’ 20’ 25’ 30’ 12 April T 12 T 13 T 14 T 15 (f) kg·m−3 26.4 26.5 26.6 26.7 26.8 Figure 3.4: Surface Contours for the Bay of Set´ubal - (a-c) Surface salinity and (d-f) surface potential density contours computed from CTD stations B, CTD stations C and CTD stations D, respectively. Blue stars correspond to the locations of Barny North and Barny South ADCP deployments. An input of fluvial waters (less salty and less dense than oceanic waters) coming from the Sado estuary is clear in surface contours (a) and (d). To this end, the squared vertical shear (S2), squared buoyancy frequency (N2) and the gradient Richardson Number (Ri) were studied for the Barny North ADCP and CTD station 62 (Figure 3.5a-d), and the Barny South ADCP and CTD station 78 (Figure 3.5e-h). Critical values of N < 0.7 cph were not considered in this analysis to make a reliable estimate of Ri (Muench et al., 2002). Barny North site Figure 3.5a shows the vertical distribution of potential density at CTD station 62 on three different days: 4 April (grey line) and 13 April (brown line), when breeze-forced events were detected (Figure 3.3e); and 7 April (black line), corresponding to a relaxation of current oscillations. These captured ‘scenes’ allowed us to analyse the strength of breeze-forced diurnal-inertial motions in triggering diapycnal mixing processes according to its evolution over time. 60 3.3 Results and Discussion Based on these profiles, the water column was found to be highly stratified on 7 April from 10 −20 m depth, when breeze-forced currents vanished, reaching a maximum N2of approximately 2×10−4s−2(Figure 3.5b). This maximum responded to the intrusion of estuarine waters (warmer and lighter than oceanic waters) into the bay (Figure 3.4a,d). On the contrary, profiles on 4 and 13 April exhibited lower levels of stratification near the surface and all along the water column until a depth of approximately 60 m, where N2started to increase for next 20 m, approaching values of 2×10−5s−2. For these two sampled days in which breeze-forced oscillations were stronger (Figure 3.3e), the squared vertical shear S2(Figure 3.5c) showed two pairs of maximum values. These two peaks concern the depths where the maximum currents were acting within the described out-ofphase layer configuration for diurnal-inertial oscillations. The largest peaks were both over 3×10−5s−2at 80 −90 m on 4 April and occupied a broader range of depths on 13 April (60 −80 m). The secondary peaks placed around 30 −60 m on 4 April, and 20 −50 m on 13 April, with values of 1.3×10−5s−2 and 2.1×10−5s−2, respectively. With regards to the profile sampled when there was a period of relaxedwinds (7 April in Figure 3.5c), two maxima of S2were also observed at similar depths to the previous ones, but with inverted positions. The largest peak was located between 30 −40 m with S2∼3.3×10−5s−2, and the second largest peak was located over 80 m with S2∼1.6×10−5s−2. These peaks might be the response to attenuated diurnal-inertial currents during 7 −8 April, which decayed in a transitional state, disturbed by the intrusion of estuarine waters into the bay. The break in the vertical structure of the water column caused by this intrusion is evident by comparison of the potential density profiles taken on 4 April and 7 April (Figure 3.5a). From the combination of previous data (N2and S2), the vertical distribution of the gradient Richardson number was estimated (Figure 3.5d) to locate the depths at which the generation of Kelvin-Helmholtz instabilities caused by vertical shears in a stratified fluid could be suitable to trigger mixing and account for the observed changes in the vertical distribution of the potential density. 61 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING 26.5 26.6 26.7 26.8 26.9 27 10 20 30 40 50 60 70 80 90 100 σθ (kg·m−3) Depth (m) 04−Apr−2004 15:30:00 07−Apr−2004 20:55:00 13−Apr−2004 17:37:00 (a) 0 0.5 1 1.5 2 x 10−4 10 20 30 40 50 60 70 80 90 100 N2 (s−2) Depth (m) 04−Apr−2004 15:30:00 07−Apr−2004 20:55:00 13−Apr−2004 17:37:00 (b) 0 1 2 3 4 5 x 10−5 10 20 30 40 50 60 70 80 90 100 S2 (s−2) Depth (m) 04−Apr−2004 16:00:00 07−Apr−2004 21:00:00 13−Apr−2004 18:00:00 (c) −2 −1 0 1 2 3 10 20 30 40 50 60 70 80 90 100 Log10Ri Depth (m) 04−Apr−2004 16:00:00 07−Apr−2004 21:00:00 13−Apr−2004 18:00:00 (d) 26.5 26.6 26.7 26.8 26.9 27 10 20 30 40 50 60 70 80 90 100 110 σθ (kg·m−3) Depth (m) 08−Apr−2004 04:10:00 10−Apr−2004 01:34:00 12−Apr−2004 15:10:00 (e) 0 0.5 1 1.5 2 2.5 3 x 10−4 10 20 30 40 50 60 70 80 90 100 110 N2 (s−2) Depth (m) 08−Apr−2004 04:10:00 10−Apr−2004 01:34:00 12−Apr−2004 15:10:00 (f) 0 0.5 1 1.5 2 2.5 x 10−4 10 20 30 40 50 60 70 80 90 100 110 S2 (s−2) Depth (m) 08−Apr−2004 04:00:00 10−Apr−2004 02:00:00 12−Apr−2004 15:00:00 (g) −2 −1 0 1 2 3 10 20 30 40 50 60 70 80 90 100 110 Log10Ri Depth (m) (h) Figure 3.5: Diapycnal Mixing Analysis - Vertical time-profiles of (a,e) potential density, σθ; (b,f) squared buoyancy frequency, N2; (c,g) squared vertical shear, S2and (d,h) logarithm of the gradient Richardson number, Ri. These plots combine data from CTD station 62 together with Barny North ADCP (a-d) and data from CTD station 78 together with Barny South ADCP (e-h) after low-pass filtering oscillations shorter than 9 hours. Horizontal dashed lines represent the range of depths sampled with the Barny North (South) ADCP. Subcritical values (Ri < 0.25) were found at several depths during breezeforced events. On 4 April, the two maximum peaks of S2corresponded with values lower and close to subcritical Ri, respectively. The first peak, Ri ∼0.04, was found at 50 m and was associated with high vertical diapycnal diffusivities coefficients, kv, of approximately 3×10−3m2s−1. The second peak, Ri ∼0.27, was also associated with high values, kv∼4×10−4m2s−1, around 90 m depth. Analogously, the two peaks of high S2on 13 April coincided approximately with low Ri values of 0.02 over 30 m and 0.23 at 90 m, which were associ62 3.3 Results and Discussion ated with an enhanced kvof ∼4×10−3and ∼5×10−4m2s−1, respectively. Overall, on 13 April, the whole profile supported values of Ri < 1, giving rise to high diapycnal diffusivities, especially in the range of the upper layer (20-50 m depth) observed for breeze-forced oscillations during 10 −14 April (Figure 3.3e). It is worth noting that, during the breeze-relaxed period on 7 −8 April, Ri values larger than 1 were obtained with high levels of high stratification near the surface and attributable to strong inputs of fluvial waters on these days. Nevertheless, at deeper levels (around 80 m), Ri values close to 0.1 were also found with kv∼2×10−3m2s−1. The common pattern observed on 4 and 13 April during breeze-forced current events was a two-layer vertical structure with high values of S2embedding stratified waters. As a result, the pycnocline that partially contributed to supporting this two-layer configuration was significantly eroded. In such a way, this triggered a vertical displacement of the interface with high stratification, now smoothed, towards deeper waters. These results suggest that the displacement was promoted by the high vertical shear of the upper and lower layers acting over the depths of the highest stratification. If we now recall the theoretical works of Rippeth et al. (2002) and Simpson et al. (2002), one can understand why in presence of a noticeable breezeforcing over 8-13 April (Figure 3.2d) there is, on the contrary, a weakening of the breeze-forced current system. These studies predict that breeze-forced currents at the lower layer, in relation to those in the upper layer, vanish by a factor γwhich is dependent of the depth of the pycnocline following γ=h2/h1 (being h1(h2) the thickness of the upper (lower layer)). Thus, the deepening and smoothing of the interface observed along 8-13 April (Figure 3.5a-d) could have driven the self erosion of the baroclinic mode which supported the breezeforced current system (Figure 3.3e) owing to its associated vertical diapycnal mixing. A similar finding was reported by Zhang et al. (2009) in the Gulf of Mexico, where they observed that the deepening of the mixed layer depth driven by freshwater inputs and vertical mixing weakened the diurnal-inertial ocean response to breeze-forcing. Barny South site Regarding the Barny South ADCP deployment and CTD station 78, Figure 3.5e presents the vertical profiles of potential density on three different 63 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING days: 8 April (black line), characterised by the absence of breeze-forced events; and 10 April (red line) and 12 April (green line), which involved increasing breeze-forced oscillations (Figure 3.3e). As can be observed in Figure 3.5f, the maximum values of N2that partially supported the baroclinic mode of these diurnal-inertial motions increased their depth from 8 April (10 m) to 12 April (45 m) while decreasing their values from approximately 3×10−4s−2to 1×10−4s−2. This deepening of the pycnocline likely occurred according to the mechanism previously explained for breezeforced currents oscillating around the Barny North ADCP site. On 10 April, the distribution of vertical shear (Figure 3.5g) exhibited a maximum at 20 m, which was above the interface of high stratification, with an S2of approximately 2.7×10−4s−2. A secondary peak of S2∼3×10−5s−2 was found on the same day for the lower layers, which is similar to the situation described for the Barny North ADCP. Likewise, on 12 April, a maximum of S2was also found above the interface of high stratification with values of ∼4.3×10−5s−2at approximately 35 m. For days when breeze-forced diurnal-inertial oscillations were acting, the main peaks of S2coincided with low Ri values of 0.05 at 20 m depth on 10 April and 0.17 at 35 m depth on 12 April (Figure 3.5h). The values of kvassociated with these Ri values were also relatively high, approximately 2.6×10−3m2s−1on 10 April and 8×10−4m2s−1on 12 April. As a consequence, the combination of strong vertical shears and surface inputs of stratification from the estuary produced the same effects as illustrated for the area sampled by the Barny North ADCP and CTD station 62. Thus, a noticeable modification of the density vertical distribution at relatively short-time scales was also found in response to diapycnal mixing processes enhanced by breeze-forced motions. Additionally, it was also found that again the welldefined baroclinic mode structure which supported the breeze-forced currents (Figure 3.5e-h on 10 April) broke up due to vertical mixing (Figure 3.5e-h on 12 April), hence weakening the diurnal-inertial ocean response even in presence of breeze-forcing (Figure 3.3f). CTD and ADCP observations reported here, although they were not planned during the MREA04 sea trial for the present research, successfully capture the critical moments and consequences of the interactions between breeze-forced current events and stratified waters in the interior of the bay. These results contribute to explaining the main vertical changes observed in the CTD casts 64 3.3 Results and Discussion next to the ADCP deployments during observed breeze-forced events and support that these events may play an important role in promoting vertical mixing. This mechanism is of special interest in areas where the water column is stratified and strong vertical shears, as shown in this study, can be responsible for the diapycnal transport of nutrients. In these cases, breeze-forced oscillations may be able to promote regions of high biological productivity, as it has been observed for strong inertial currents in the North Sea (van Haren et al., 1999). 3.3.4 Impact of BFOs in the Bay of Set´ubal One may expect that breeze-forced oscillations reported here are not an isolated feature of the area of study but be part of the hydrodynamics of the bay. Counting with shelf conditions next to an estuary which supplies freshwater to the surface ocean, one has half of the required scenario for breeze-forced oscillations. However, the forcing to the system cannot be assumed to be a regular pattern of the area without analysing the local sea-land breeze system. In this section we use wind time series data from a land meteorological station at Sines (Figure 3.1), which bounds the southern side of the Bay of Set´ubal, in order to determine whether the area may be a suitable candidate for further research on the impact of breeze-forced oscillations in stratified waters. With this aim the rotary wavelet spectrum is shown in Figure 3.6 for wind data during 2004. As it can be observed, the result of the wavelet analysis gives evidence of the same seasonal pattern reported in Chapter 2for diurnal wind over three different locations placed off the Iberian Peninsula coast. Thus, the variability for this frequency band exhibits, as expected, a stronger and more recurrent presence of sea-land breezes during the summer months. These results, together with previous analysis, suggest that the Bay of Set´ubal, which is framed within the critical latitudes for diurnal-inertial resonance, seems to be a suitable candidate area for the enhancement of diurnalinertial motions when sea-land breezes are acting and, hence, to promote diapycnal vertical mixing with important consequences for the shelf dynamics in the interior of the bay. 65 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING Figure 3.6: Rotary Wavelet Power Spectrum of Wind from a Meteorological Station at Sines - Rotary wavelet power spectrum of winds (at 10 m above the ground) measured at Sines from April 2004 to December 2004. (upper pannel) Total rotary wavelet power spectrum. (intermediate pannel) Rotary wavelet power spectrum of clockwise component. (lower pannel) Rotary wavelet power spectrum of counterclockwise component. The two black lines on either end indicate the ‘cone of influence’ where edge effects become important. 3.4 Conclusions The Bay of Set´ubal is located within the critical latitudes (30◦±10◦N/S) where inertial period approaches diurnal period and a near-resonance response between free inertial oscillations and diurnal wind forcing may occurr. Under these conditiond, we show evidence of breeze-forced oscillations (BFOs) enhancing diurnal-inertial energy in the interior of the bay and promoting vertical diapycnal mixing. The squared vertical shear (S2), squared buoyancy frequency (N2) and the gradient Richardson Number (Ri) were estimated to analyse the scenario in 66 3.4 Conclusions which enhanced vertical shears developed during breeze-forced events led to stratified and vertically sheared flows with subcritical values of Ri. These zones were found to promote vertical mixing across the pycnocline due to strong squared vertical shears about 10−5s−2. In the end, these mixing processes drove the self-erosion of the pycnocline that partially supported the interface between the two-layers, thereby homogenising the water column in their wake. At the Barny South ADCP site, the vertical variation in the highest peaks of stratification was especially clear, shifting approximately 15 m in two and a half days and involving an efficient mechanism to transport properties vertically throughout the water column. The above results point out that there may exist an internal control on the strength of the ocean response to breeze forcing that has not been described before to the best of our knowledge. This follows from the previously discussed role that the depth of the mixed layer plays on the vertical structure of BFOs supporting its baroclinic mode and determining the intensity of the forced currents above and below the pycnocline (Rippeth et al.,2002;Simpson et al.,2002). Starting from this basis, we observe that vertical diapycnal mixing promoted by breeze-forced oscillations led to the deepening and smoothing of the pycnocline which supported them and, hence, weakened the ocean response to breeze-forcing following a self-erosion mechanism. Hence, we suggest that the effect of the vertical mixing must be included in theoretical models to reallistically predict the evolution of the vertical structure of breeze-forced oscillations; otherwise, initialising a model with a prescribed pycnocline which is fixed along the simulation will not allow the decay of breeze-forced oscillations due to the deepening of the mixed layer driven by the associated vertical mixing. From this basis, we consider that further research of time series data (currents, temperature and salinity) covering the total water depth in combination with co-located simultaneous wind data would benefit greatly the understanding of how breeze-forced oscillations evolve in time and space, with special attention to the role that associated diapycnal mixing may be playing on the vertical structure of of these forced-motions. Remarks on this Chapter The rotary wavelet spectrum of wind data shown in this chapter represents an extension to the publication of Aguiar-Gonz´alez et al. (2011). 67 3. BREEZE-FORCED OSC. AND DIAPYCNAL MIXING The analysis of diapycnal mixing processes associated to breeze-forced oscillations and the discussion of the role that enhanced mixing might be playing on the vertical structure of these forced motions Aguiar-Gonz´alez et al. (2011) represents the main contribution of Chapter 3to this topic. To this regard we find of interest to discuss a later paper of Hyder et al. (2011) in which the authors report observations and simulations of wind-forced oscillations on the Namibian shelf, i. e. equatorward the critical latitude for diurnal-inertial resonance. They found that the General Ocean Turbulence Model (GOTM) 1-D simulations of diurnal forcing, including the first order coast-normal surface slope response to diurnal forcing, represented the principal features of the observed diurnal anticyclonic current. However, they also found that the vertical structure of simulated diurnal anti-cyclonic amplitudes was significantly different to those observed. The problem was mainly based on the upper layer depth was too shallow and the uper layer ampitude was too strong. They suggest that differences could arise because the simulated downward transfer of wind forced momentum is too small or the relative magnitude of diurnal wind and slope forcing for the model is not correct, i. e. the Craig approximation is invalid. As the downward transfer of wind momentum is controlled by mixing, they point out that in their 1-D simulations unresolved 3-D processes such as internal waves may be missrepresenting an important source of mixing. Also, the sensivity of the upper layer thickness and amplitude to latitude in their simulations was discussed and tested as possible explanation for the disagreement between observations and model results. The most interesting aspect to our concern is the series of experiments that the authors undertook to elucidate the origin of this disagreement. With this aim they modified the standard configuration of their runs. Among those experiments, they included a simulation employing an unrealistic minimum TKE of 1 x 10−3m2s−2to represent a large unknown unresolved mixing source1. For standad model runs the authors had employed a minimum turbulence kinetic energy (TKE) kmin of 1 x 10−6m2s−2. With this modification, the simulations improved and provided a much better representation of vertical phase variation, deepening the upper layer depth and reducing the error in the lag between the wind and the upper layer current. Broadly, we find those modeling results of great importance as they support our earlier suggestion, based on observational analysis, that there may exist an 1Literally reproduced from Hyder et al. (2011). 68 4.1 Solitary Waves in the Ocean As long as soliton generation is linked to internal tide activiy, solitary wave packets will exhibit the same variability as tides, and hence the properties of generated solitons will vary on semididurnal, diurnal, fortnightly, seasonal and semiannual cycles. Therefore one can expect, for instance, more and larger solitons during spring tides than during neap tides (Pingree and Mardell, 1985). Depending on the site and time, solitary wave generation may be even possible only during spring tides, being absent at neap tides (Apel et al.,1985; Osborne and Burch,1980). Figure 4.3: Evolution of a Solitons Packet in the Sulu Sea - Composite of two figures reported in Apel et al. (1985). Left-hand side: Line drawing renditions of DMSP (Defense Meteorolological Satellite Program) images, interpreting striations as surface signatures of packets of internal solitary waves. Several solitons occur in each group, and up to five packets are visible in images. Interpacket distance range from 56 to 198 km, depending on azimuth and forthnightly tidal phase. Intersoliton distances, which defines wavelengths, range from 5 to 16 km. Two different DMSP images at times yield direct time-of-flight speeds of approximately 2.4 m s−1. Right-hand side: Schematic diagram summarizing the evolution of a soliton packet in the Sulu Sea: (a) In the sill region an internal hydraulic jump is produced by strong ebb (southward) flow over the Pearl Bank sill. (b) As the tidal flow changes to a weak flow, a broad thermocline depression moves over the sill, propagating northward. (c) Eventually solitary waves begin to form on this depression as nonlinear and dispersive effects begin to balance. (d) After traveling a distance approximately 200 km the soliton packet is welldeveloped. Scales shown are typical. [From Apel et al. (1985): Right-hand side of Fig. 13; and Fig. 25]. 75 4. INTRODUCTION TO A SOLITONS SCENARIO An illustrative case of large-amplitude solitons generated from a sill occur in the Sulu Sea (Apel et al.,1985;Liu et al.,1985). The study area is shown in the left-hand side of Figure 4.3. The mechanism consists on the release of a low-mode internal tide from the Pearl Bank sill (Figure 4.3 - Right-hand side: a - b). Subsequenty, as the internal tide propagates nonlinear steepening and dispersive effects come into play leading to the appearance of solitary waves distributed into packets separated by tidal cycles (Figure 4.3 - Right-hand side: c - d). The generation mechanism over a shelf break and over a shallow promontory (sills, banks, seamounts, ...) is similar, however solitons propagating oceanward evolves differently than those traveling shoreward since the former are generally out of the area controlled by bathymetry. Hence, solitary waves propagating on the shelf, where the pycnocline depth shoals and the water depth diminishes, are conditioned by bottom friction effects. This leads to refraction of the waves, driving the packet crests along isobaths and decreasing their amplitudes, speeds and wavelengths, independently from wherever solitons were generated. A clear example of wave refraction of solitons when water depth shoals can be found in Zeng and Alpers (2004) for solitary waves traveling to shallow waters, afresh in the Sulu Sea (Figure 4.4). Nevertheless, solitary wave packets have been also observed traveling oceanward far away from shelf breaks and promotory areas (e.g. solitons in the Bay of Biscay (Gerkema,2001;New and da Silva,2002). These solitons respond to a different generation mechanism. Then, we need to consider that internal tidal energy may radiate away either horizontally as interfacial waves in the thermocline (mechanism explained above and termed ‘direct generation’) and/or vertically as ‘beams’ into the stratified continuum below (mechanism known as ‘local generation’) (New,1988,1990;New and Pingree,1992;Gerkema,2001; Akylas et al.,2007;da Silva et al.,2007;Da Silva and New,2009). These beams emanate from critical topographies where bottom slope matches the slope of the beam paths and travel at an angle to the vertical into the deep ocean. After reflection from the seafloor, scattering of these beams at the base of the thermocline induces the generation of internal solitary waves propagating at long distances from the shelf break or promontory which caused the initial beam of internal tidal energy (New,1990;Gerkema,2001;Da Silva and New, 2009). Other possible mechanism of soliton generation include shear-flow instability just up-current of the break/sill (Farmer and Armi,1999), upstream blocking (Lee and Beardsley,1974), transcritical generation (Grimshaw,1986; 76 4.1 Solitary Waves in the Ocean Melville and Helfrich,1987). Figure 4.4: Remote Sensing SAR (Synthetic Aperture Radar) images of Solitons in the Sulu Sea - (a) ERS-2 SAR image acquired during orbit 14529 on 30 Januray, 1998, at 02:24 UTC showing sea surface manifestations of three internal wave packets generated at ( three different locations; (b) ERS-2 SAR image of the same area, but aquired during orbit 09018 on 10 January, 1997, at 02:24 UTC which was at a slightly later phase of the tidal cycle; (c) ERS-2 SAR acquired during orbit 06284 on 3 July, 1996, at 02:27 UTC showing strong refraction of internal solitary waves by bottom topography. [From Zeng and Alpers (2004) - Figure 1]. As one can imagine, a better understanding of the dominating mechanisms by which solitons are generated will benefit from the assessment of the candidate areas for soliton activity. Additionally, the interest on oceanic solitons is supported by the impact these waves have on various aspects: ocean mixing 77 4. INTRODUCTION TO A SOLITONS SCENARIO and biological productivity (Sandstrom and Elliott,1984;Pinkel,2000;Sangr`a et al.,2001;Moum et al.,2003;Macias et al.,2006), water acoustics (Tiemann et al.,2001), sediment transport (Sandstrom and Elliott,1984;Bogucki et al., 1997;Butman et al.,2006), oil platforms (Osborne and Burch,1980), etc. The uncertainty about the relative importance of each generating mechanism worlwide leads to the topic of oceanic solitons still represents a challenging fieldwork. Fine reviews of the existing empirical and theoretical background on solitons can be found in Ostrovsky and Stepanyants (1989); Grimshaw et al. (1998); Apel (2002); Apel et al. (2006); Helfrich and Melville (2006). The seed for next section is linked to the earliest scientific report describing the surface manifestation of a solitary wave, which was made by J. Scott Russell in 1838 on a propagating and unchanging single hump in a Scottish canal (Russell,1838b,a). Half a century later, Korteweg and de Vries provided for the first time a theoretical explanation to this important phenomenon in the field of mathematical physics (Korteweg and de Vries,1895). That work opened an entire research line on the study of soliton properties from a theoretical point of view whose interest persists in time. In the following we focus on introducing the solitary wave theory which is of main concern for the background of this PhD research. 4.2 A Mathematical Approach to Solitons To set the scope of the present thesis, we first discuss soliton wave theory based on the applications of the quadratic Korteweg-de Vries equation (KdV) and its extended cubic version (eKdV), which describe the dominant wave characteristics and evolution of solitons under weakly (KdV) and more strongly nonlinear conditions (eKdV), respectively, given an initial profile. Theoretical models are traditionally built up starting from the most simplified situation possible, which gets more comprehensive by including new elements in modeling the process as long as a previous scenario is understood. On solitary wave theory, it is not surprising that KdV-type models have become the classical approach to describe nonlinear dispersive waves. They have been widely used in the literature and show good aggreement with dominant relationships between phase speed, amplitude and wavelength scales in comparison with reported observations of oceanic solitons (Osborne and Burch,1980;Pingree and Mardell, 1985;Ostrovsky and Stepanyants,1989;Stanton and Ostrovsky,1998;Xu and Yin,2012). Additionally, they permit modeling the phenomenon of interest 78 4.2 A Mathematical Approach to Solitons with a reduced set of equations. However, not all quantitative features are fully well captured. The Korteweg-de Vries (KdV) equation is a quadratic nonlinear model which admits form-preserving solitary wave solutions travelling in one dimension and based on a balance between nonlinearity and dispersive effects. For simplicity, here we discuss the KdV equation in terms of a two-layer fluid with a rigid lid at the surface, even bottom and no mean flow (Djordjevic and Redekopp,1978;Kakutani and Yamasaki,1978), following ∂η ∂t +c0 ∂η ∂x +αη ∂η ∂x +β∂3η ∂x3= 0 (4.1) where tis time; xis the spatial variable in the direction of wave propagation; h1(h2) is the thickness of the upper (lower) layer; gis the gravitational acceleration; η(x, t) is the interfacial displacement from its rest level; and α, βand c0are the so-called environmental coefficients of the medium describing nonlinearity, dispersion and long-wavelength phase speed which read c2 0= Λρg h1h2 h1+h2 , α =3 2 h1−h2 h1h2 c0, β =1 6h1h2c0,(4.2) with Λρ= (ρ2−ρ1)/ρ2. Additionally, relative differences in density between the two layers (Λρ) are assumed to be small (Boussinesq approximation), what holds in the ocean with typical values of the order 10−3. This KdV equation is applicable in shallow waters to stratified fluids where two main assumptions are taken: 1) The interfacial displacement is much smaller (but of finite amplitude) than the depth of either layer (i. e., weak nonlinearity); and, 2) waves are long (but of finite length) in comparison with the fluid depth (i. e., weak nonhydrostatic dispersion). Thus, the KdV equation is scaled by the small parameters α=a/H and β= (H/l)2in such a form that both are small and of comparable order (β=O(α)≪1); taking aas a measure of the wave amplitude, las the length scale of the wave, and Has the intrinsic vertical scale. The limit in which waves are infinitesimal (α→0 leads to linearity) and of infinite length (β→0 leads to hydrostacy) reduces 4.1 to ∂η ∂t +c0∂η ∂x, which simply describes rightward travelling waves. Wherever waves are assumed to be small but of finite amplitude and long 79 4. INTRODUCTION TO A SOLITONS SCENARIO but of finite wavelength, parameters αand βtake values different from zero to include, respectively, nonlinear and nonhydrostatic dispersive effects in the KdV evolution equation. The prototypical analytical solution to the KdV equation for a single soliton is the hyperbolic secant profile, which reads η(x, t) = η0sech2x−ct λ(4.3) where cis the nonlinear phase speed derived from KdV theory and related to the linear phase speed c0and amplitude; and λis the characteristic width of the soliton, being related to the amplitude of the displacement η0and the environmental cofficients: c=c0+αη0 3, λ2=12β αη0 (4.4) As λ2must be positive, for a thin upper layer with h1< h2we have that α < 0, what results in downgoing displacements of the interface between the layers (η < 0). On the other hand, if the lower layer shoals in such a manner that h2< h1, then solitons reverse to upgoing displacements (α < 0, η < 0). At the configuration in which h1=h2, the quadratic nonlinear coefficient α vanishes and a higher-order (cubic) nonlinear coefficient must be included in the KdV equation (Djordjevic and Redekopp,1978;Kakutani and Yamasaki, 1978), leading to its extended version (eKdV): ∂η ∂t + (c0+αη +α1η2)∂η ∂x +β∂3η ∂x3= 0 (4.5) where the coefficient α1, absent in the Kdv equation (4.1), is the above mentioned cubic nonlinear coefficient α1=3c0 (h1h2)2h7 8(h1−h2)2−h3 2+h3 1 h1+h2i (4.6) 80 4.2 A Mathematical Approach to Solitons It should be noticed that while the sign of αdepends on the layer depths, it turns out that α1is always negative. As the cubic term α1is O(α2), the eKdV should formally include also additional dispersive and nonlinear dispersive terms of order O(β2) and O(αβ), respectively, to balance nonlinear and dispersive effects at a higher-order of the KdV equation (Koop and Butler,1981;Lamb and Yan,1996;GrimShaw et al.,2002). Nevertheless, if the quadratic nonlinear coefficient αis of order O(α) (i. e., nonlinear quadratic effects are small), as occurs when |h1−h2|/(h1h2)≪1, then the eKdV is assymptotically consistent, but requires the balance β=O(α2) (Helfrich and Melville,2006). The modified KdV (mKdV) equation appears when the interface is located at the critical critical thickness ratio (h1=h2) and the quadratic nonlinear coefficient αis zero. Then the equation (4.5) reduces to ∂η ∂t + (c0+α1η2)∂η ∂x +β∂3η ∂x3= 0 (4.7) Kakutani and Yamasaki (1978) found that cubic nonlinearity of the mKdV equation governs nonlinear long gravity waves at the critical thickness ratio, whereas near this configuration nonlinearity is governed by an equation of a combined form of the KdV and modified KdV equation. Solitary wave behaviour of eKdV equation (Kakutani and Yamasaki,1978; Ostrovsky and Stepanyants,1989;Gerkema and Zimmerman,2008) can be described by η(x, t) = η0 b+ (1 −b)cosh2γ(x−ct),(4.8) where c=c0+η0 3α+1 2α1η0, γ2=η0(α+1 2α1η0) 12β,(4.9) b=−η0α1 2α+α1η0 , Lw=η−1 0Z∞ −∞ ηdx (4.10) 81 4. INTRODUCTION TO A SOLITONS SCENARIO with η0being a measure of the maximum amplitude of the soliton; Lw the characteristic wave-width; c, the nonlinear phase speed derived from eKdV theory; and, band γparameters defining the system. Figure 4.5 shows eKdV solitary waves approaching its maximum amplitude and becoming flat-topped, a radical departure from the classical KdV soliton which has no formal mathematical limit to its amplitude. Figure 4.5: Solitary wave solutions of the eKdV equation - Examples of solitary wave solutions of the eKdV equation (Equation 6) for the arbitrary choice of the parameters β=α=α1. As the maximum wave amplitude increases, the waves eventually broaden and develop a flat crest at the maximum amplitude η0max = 1. [From Helfrich and Melville (2006) - Figure 4]. As previously pointed out, within the framework of the two-layer model, α1is always negative (4.6). However, for more general stratifications and background shear flows the cubic nonlinear coefficient may be either negative or positive. In the latter case, soliton solutions of both positive and negative polarities may exist regardless of the sign of α(Grimshaw et al.,1997,2004). Cubic KdV, though including higher nonlinearities, is still based on an expansion of a small parameter (α=a/H) and so is still subject to the assumption of weak nonlinearity. A useful and well-known radical departure from the weakly nonlinear two-layer KdV model was derived by Miyata (1985, 1988) and Choi and Camassa (1999). The result of their work was an equiva82 4.2 A Mathematical Approach to Solitons lent set of bi-directional wave equations with full nonlinearity, α=O(1) and first-order weakly dispersive effects, β≪1. Figure 4.6: Comparison of KdV and MCC theories - Comparison of solitary wave properties from the two-layer KdV (red), eKdV (blue), and MCC (orange) theories. The top row shows the wave speed cvs. amplitude η0; and the bottom row shows the wavelength (here read characteristic wave-width of the soliton)Lw vs. η0. The comparison is done for the two stratifications h1/h2= 1/4 (left column) and h1/h2= 2/3 (right column). For both the eKdV and MCC waves the maximum wave amplitude corresponds to the end of the speed curves. [From Helfrich and Melville (2006) - Figure 5]. Helfrich and Melville (2006) highlighted with Figure 4.6 the deviation among the weakly nonlinear KdV, the more strongly nonlinear eKdV and the fully nonlinear Miyata-Choi-Camassa (MCC) models. The amplitude depen83 4. INTRODUCTION TO A SOLITONS SCENARIO dence of the wave speed cand characteristic wave-width Lw1from the three models is shown for two different layer-thickness (h1/h2= 1/4 and h1/h2= 2/3) in a two-fluid layer system. Remarkable differences arise when comparing results from the weakly nonlinear KdV and the strongly nonlinear eKdV and MCC models, for both cand Lw, even for relatively small amplitudes. The solitary wave solutions of the eKdV and MCC equations increase its phase speed cwith growing amplitude until they start to slow as they approach its maximum η0max (Figure 4.6 - top row). On the contrary, KdV solutions show the phase speed increasing linearly with amplitude. Additionally, it also worth noting that the maximum wave amplitude for both the eKdV and MCC waves corresponds to the end of the speed curves (what it does not occur with KdV solitons). As previously shown in Figure 4.5 for eKdV solitary waves, its characteristic wave-width becomes narrower as they grow (increasing η0) to lately broaden when the upper limit is approached (η0max). This behaviour is also exhibited by the MCC theory (Figure 4.6 - bottom row); and, unlike classical KdV solitary waves whose wave-width gets narrower as they become larger until its maximum amplitude. On a whole, comparison of the wave shapes and properties between the eKdV and MCC theories agree quite well for 0.4< h1/(h1+h2)<0.6, where the scaling requirements of eKdV are reasonably met (Helfrich and Melville, 2006). Outside this range, differences among the two theories start to grow (see Figure 4.6 - left column, bottom row). As we have described, the Korteweg-de Vries (KdV) equation and its extended version (eKdV) with no forcing terms included, provide an appropiate mathematical tool for studying the evolution of a given initial profile under weakly nonlinear (KdV and eKdV) conditions. To go to strongly nonlinear settings, MCC-type of equations need to be used. But for these, no model exists that includes a forcing mechanism. It is the principal goal of the following chapters to extend MCC to include the mechanism of forcing by barotropic tides over topography. Throughout the next section, the basis and scope of the present thesis are discussed. 1For KdV solitary waves Lw = 2γ−1. 84 5.4 Vertically Integrated Equations 5.4 Vertically Integrated Equations From here on we follow a procedure analagous to that of Choi & Camassa (1999), i.e. we vertically integrate the equations over their layers, and then consider subsequently the orders δ0and δ1in order to obtain a closed set for the weakly nonhydrostatic equations. The layer-mean ¯ f1of a function f1(x, z, t) for the upper layer is defined as ¯ f1(x, t) = 1 η1Zh1 ζ dz f1(x, z, t), η1=h1−ζ(5.17) and for the lower layer as ¯ f2(x, t) = 1 η2Zζ −h2+h dz f2(x, z, t), η2=h2−h+ζ . (5.18) Here, ηirepresents the thickness of the layer (depending on the interfacial displacement ζ). Notice that the boundaries contain a dependence on tand x via ζ(t, x) and h(t, x). 5.4.1 Upper Fluid-Layer Vertical integration of the continuity equation (5.9), from the interface (ζ) to the surface (h1) yields ∂η1 ∂t +∂(η1¯u1) ∂x = 0 .(5.19) Here we used the theorem for derivatives of integrals with variable boundaries (see Appendix B): ∂ ∂x ZB(x) A(x) f(x, z)dz =ZB(x) A(x) ∂f(x, z) ∂x dz+f(x, B(x))dB(x) dx −f(x, A(x))dA(x) dx . The vertical integration of the Euler equation (5.10) involves the following steps. For the acceleration term we get Zh1 ζ dz ∂u1 ∂t =∂ ∂t Zh1 ζ dz u1+u1|z=ζ ∂ζ ∂t =∂(η1¯u1) ∂t +u1|z=ζ ∂ζ ∂t . 91 5. A GENERATION MODEL OF SOLITONS The vertical integration of the nonlinear terms, which can be rewritten (u2 1)x+ (w1u1)z, yields Zh1 ζ dz [(u2 1)x+ (w1u1)z] = ∂ ∂x Zh1 ζ dz u2 1+u2 1z=ζ ∂ζ ∂x + (w1u1)|h1 ζ =∂η1u2 1 ∂x −∂ζ ∂t u1|z=ζ The vertically integrated first horizontal momentum equation becomes ∂(η1¯u1) ∂t +∂(η1u1u1) ∂x −µη1¯v1=−η1∂p′ 1 ∂x .(5.20) The other horizontal momentum equation is treated similarly. The nonlinear terms can be rewritten as (uv)x+ (wv)z, and become after vertical integration Zh1 ζ dz (u1v1)x+ (w1v1)z=∂ ∂x Zh1 ζ dz u1v1+ (u1v1)|z=ζ ∂ζ ∂x −(w1v1)|z=ζ =∂ ∂x(η1u1v1)−∂ζ ∂t v1|z=ζ Then, the vertically integrated second horizontal momentum equation gives ∂(η1¯v1) ∂t +∂(η1u1v1) ∂x +µη1¯u1= 0 .(5.21) 5.4.2 Lower Fluid-Layer For the lower layer one proceeds similarly, except that now both boundaries are variable when one integrates vertically from the interface (ζ) to the bottom (−h2 + h). First, the vertical integration of the continuity equation yields Zζ −h2+h dz ∂u2 ∂x =∂ ∂x Zζ −h2+h dz u2−u2|z=ζ ∂ζ ∂x +u2|z=−h2+h ∂h ∂x 92 5.4 Vertically Integrated Equations and w2|ζ −h2+h=∂ζ ∂t +u2|z=ζ ∂ζ ∂x −∂h ∂t −u2|z=−h2+h ∂h ∂x Combined, this yields ∂η2 ∂t +∂(η2¯u2) ∂x = 0 .(5.22) The vertical integration of the Euler equation (5.10) involves the following steps. For the acceleration term we get Zζ −h2+h dz ∂u2 ∂t =∂ ∂t Zζ −h2+h dz u2−u2|z=ζ ∂ζ ∂t +u2|z=−h2+h ∂h ∂t =∂(η2¯u2) ∂t −u2|z=ζ ∂ζ ∂t +u2|z=−h2+h ∂h ∂t . The vertical integration of the nonlinear terms, which can be rewritten (u2 2)x+ (w2u2)z, yields Zζ −h2+h dz [(u2 2)x+ (w2u2)z] =∂ ∂x Zζ −h2+h dz u2 2−u2 2z=ζ ∂ζ ∂x +u2 2z=−h2+h ∂h ∂x + (w2u2)|ζ −h2+h =∂η2u2 2 ∂x −u2 2z=ζ ∂ζ ∂x +u2 2z=−h2+h ∂h ∂x +∂ζ ∂t +u2 ∂ζ ∂xu2|z=ζ −∂h ∂t +u2 ∂h ∂xu2|z=−h2+h =∂η2u2 2 ∂x +∂ζ ∂t u2|z=ζ−∂h ∂t u2|z=−h2+h The vertically integrated first horizontal momentum equation becomes ∂(η2¯u2) ∂t +∂(η2u2u2) ∂x −µη2¯v2=−η2∂p′ 2 ∂x .(5.23) 93 5. A GENERATION MODEL OF SOLITONS The other horizontal momentum equation is treated similarly. The acceleration term gives Zζ −h2+h dz ∂v2 ∂t =∂(η2¯v2) ∂t −v2|z=ζ ∂ζ ∂t +v2|z=−h2+h ∂h ∂t . The nonlinear terms can be rewritten as (uv)x+ (wv)z, and yield after vertical integration Zζ −h2+h dz (u2v2)x+ (w2v2)z =∂ ∂x Zh1 ζ dz u2v2−(u2v2)|z=ζ ∂ζ ∂x + (u2v2)|z=−h2+h ∂h ∂x +∂ζ ∂t +u2 ∂ζ ∂xv2|z=ζ−∂h ∂t +u2 ∂h ∂xv2|z=−h2+h =∂ ∂x(η2u2v2) + ∂ζ ∂t v2|z=ζ−∂h ∂t v2|z=−h2+h Thus, the vertically integrated second horizontal momentum equation gives ∂(η2¯v2) ∂t +∂(η2u2v2) ∂x +µη2¯u2= 0 .(5.24) 5.5 Expansion in δ The six integrated equations (5.19 -5.24) derived so far, are exact but do not form a closed set. The variables η1,η2and ζcount as one unknown, but we have also ¯ui, ¯vi,p′ i,x.uiuiand uivi. The last two expressions will be cast in terms of ¯uiand ¯viby using the vertical momentum equation, expanded in terms of the small parameter δ. Moreover, continuity of pressure at the interface is used to connect the pressure in the lower and upper layer. All in all, the six equations are thus modified to contain only six unknowns. We make a formal expansion of the unknowns as, for example, ¯ui= ¯ui(0) +δ¯ui(1) +··· 94 5.5 Expansion in δ 5.5.1 Lowest Order The vertical momentum equation reduces to ∂p′ i(0)/∂z = 0 neglecting terms of order δ. At lowest order, (perturbation) pressure is vertically constant in each layer. For convenience, we introduce P=p′ 2(0), being a function of tand x. It then follows from continuity of pressure at the interface, that p′ 1(0) =P−ζ. Thus, ∂p′ 1 ∂x =∂P ∂x −∂ζ ∂x +O(δ),∂p′ 2 ∂x =∂P ∂x +O(δ). Returning to the original horizontal momentum equations, it is now natural to assume that the horizontal velocities, too, are independent of zwithin each layer, given the z-independence of pressure. Thus, uiui= ¯u2 i+O(δ), uivi= ¯ui¯vi+O(δ). At lowest order, then, the set of integrated equations is closed, in terms of the six variables ¯ui, ¯vi,ζand P: ∂(η1¯u1) ∂t +∂(η1¯u2 1) ∂x −µη1¯v1=−η1∂P ∂x −∂ζ ∂x+O(δ) (5.25) ∂(η2¯u2) ∂t +∂(η2¯u2 2) ∂x −µη2¯v2=−η2 ∂P ∂x +O(δ) (5.26) ∂(ηi¯vi) ∂t +∂(ηi¯ui¯vi) ∂x +µηi¯ui=O(δ) (5.27) ∂ηi ∂t +∂(ηi¯ui) ∂x = 0 .(5.28) (Notice that the last equation, representing the vertically integrated continuity equations, is exact and involves no approximation in terms of δ.) We can obtain the lowest-order expressions wi(0) from the continuity equation wi(0) =−z∂¯ui(0) ∂x +ci(t, x) where ciare constants of integration, which are determined by using the boundary conditions at the surface (5.13) and bottom (5.14). For these we get: w1(0) = (h1−z)∂¯u1(0) ∂x (5.29) w2(0) = (h−h2−z)∂¯u2(0) ∂x +D2h(5.30) 95 5. A GENERATION MODEL OF SOLITONS where the operator Diis defined as ∂/∂t + ¯ui(0)∂/∂x 5.5.2 Next Order At this order, we include terms of order δ. The key problem is, again, to close the set of six vertically integrated equations by deriving closed expressions for the horizontal pressure gradients p′ i,x as well as for the contributions of uiui and uiviin the nonlinear terms. The latter problem is particularly simple. At order δ, the products contain one lowest-order field, which is independent of z, hence uiui=1 ηiZdz u2 i=1 ηiZdz (ui(0) +δui(1) +···)2 =1 ηiZdz (ui(0)2+ 2δui(0)ui(1) +···) =ui(0)2+ 2δui(0) 1 ηiZdz ui(1) +··· = ¯ui(0)2+ 2δ¯ui(0) ¯ui(1) +··· = (¯ui(0) +δ¯ui(1) +···)2 = ¯u2 i+O(δ2) what leads to uiui= ¯u2 i+O(δ2), uivi= ¯ui¯vi+O(δ2). This means that we can write the horizontal momentum equations as ∂(ηi¯ui) ∂t +∂(ηi¯u2 i) ∂x −µηi¯vi=−ηi∂p′ i(0) +δ∂p′ i(1) ∂x +O(δ2) (5.31) ∂(ηi¯vi) ∂t +∂(ηi¯ui¯vi) ∂x +µηi¯ui=O(δ2) (5.32) ∂ηi ∂t +∂(ηi¯ui) ∂x = 0 .(5.33) 96 5.5 Expansion in δ The remaining problem is to find an expression for p′ i(1). Before we enter that problem, we first simplify the above equations by combining them, using ∂(ηi¯ui) ∂t +∂(ηi¯u2 i) ∂x =ηi ∂¯ui ∂t + ¯ui ∂ηi ∂t +ηi¯ui ∂¯ui ∂x + ¯ui ∂(ηi¯ui) ∂x =ηi ∂¯ui ∂t +ηi¯ui ∂¯ui ∂x (5.34) and ∂(ηi¯vi) ∂t +∂(ηi¯ui¯vi) ∂x =ηi ∂¯vi ∂t + ¯vi ∂ηi ∂t +ηi¯ui ∂¯vi ∂x + ¯vi ∂(ηi¯ui) ∂x =ηi ∂¯vi ∂t +ηi¯ui ∂¯vi ∂x .(5.35) Hence, ∂¯ui ∂t + ¯ui ∂¯ui ∂x −µ¯vi=−∂p′ i(0) +δ∂p′ i(1) ∂x +O(δ2) (5.36) ∂¯vi ∂t + ¯ui ∂¯vi ∂x +µ¯ui=O(δ2) (5.37) ∂ηi ∂t +∂(ηi¯ui) ∂x = 0.(5.38) At order δ, the vertical momentum equations reads ∂wi(0) ∂t +ui(0) ∂wi(0) ∂x +wi(0) ∂wi(0) ∂z =−∂p′ i(1) ∂z .(5.39) We now use the lowest order expressions for the velocity components to find an expression for p′ i(1). 5.5.2.1 Pressure in Upper Layer With the lowest-order expression for the vertical velocity, w1(0) = (h1−z)∂¯u1(0) ∂x , 97 5. A GENERATION MODEL OF SOLITONS the left-hand side of (5.39) becomes (h1−z)∂2¯u1(0) ∂x∂t + ¯u1(0)(h1−z)∂2¯u1(0) ∂x2−(h1−z)∂¯u1(0) ∂x 2 = (h1−z)h∂2¯u1(0) ∂x∂t + ¯u1(0) ∂2¯u1(0) ∂x2−∂¯u1(0) ∂x 2i ≡(h1−z)G1,(5.40) where we introduced Gi=∂2¯ui(0) ∂x∂t + ¯ui(0) ∂2¯ui(0) ∂x2−∂¯ui(0) ∂x 2(5.41) Hence, ∂p′ 1(1) ∂z = (z−h1)G1.(5.42) Integration now gives p′ 1(1) = (z2/2−h1z)G1−(ζ2/2−h1ζ)G1 =1 2hz(z−2h1)−ζ(ζ−2h1)iG1(5.43) In the first equality, we used the freedom to add a constant of integration (in fact, a function of xand t), to add terms such that pressure is zero at the interface1. Taking the derivative to x, ∂p′ 1(1) ∂x = (h1−ζ)∂ζ ∂xG1+1 2hz(z−2h1)−ζ(ζ−2h1)i∂G1 ∂x ,(5.44) and then the mean over the upper layer, ∂p′ 1(1) ∂x = (h1−ζ)∂ζ ∂xG1−1 2ζ(ζ−2h1)∂G1 ∂x +1 2 ∂G1 ∂x 1 η1Zh1 ζ dz z(z−2h1) =η1 ∂ζ ∂xG1−1 2ζ(ζ−2h1)∂G1 ∂x +1 2 ∂G1 ∂x 1 η1z3/3−h1z2 h1 ζ =η1 ∂ζ ∂xG1−1 2ζ(ζ−2h1)∂G1 ∂x −1 2 ∂G1 ∂x 1 η1h2h3 1/3 + ζ2(ζ/3−h1)i =η1 ∂ζ ∂xG1−η2 1 3 ∂G1 ∂x =−η1 ∂η1 ∂x G1−η2 1 3 ∂G1 ∂x =−1 3η1 ∂ ∂x(η3 1G1).(5.45) 1The same condition must later be imposed on p′ 2(1), since at first order the requirement of continuity of pressure becomes p′ 1(1) =p′ 2(1) at z=ζ. 98 5.5 Expansion in δ The last equality allow us to write ∂p′ 1 ∂x =∂p′ 1(0) ∂x +δ∂p′ 1(1) ∂x +O(δ2) as ∂p′ 1 ∂x =∂P ∂x −∂ζ ∂x −δh1 3η1 ∂(η3 1G1) ∂x i+O(δ2).(5.46) 5.5.2.2 Pressure in Lower Layer With the lowest-order expression for the vertical velocity, w2(0) = (h−h2−z)∂¯u2(0) ∂x +D2h the left-hand side of (5.39) becomes ∂h ∂t ∂¯u2(0) ∂x + (h−h2−z)∂2¯u2(0) ∂x∂t +∂2h ∂t2+∂¯u2(0) ∂t ∂h ∂x + ¯u2(0) ∂2h ∂x∂t +¯u2(0)h∂h ∂x ∂¯u2(0) ∂x + (h−h2−z)∂2¯u2(0) ∂x2+∂2h ∂x∂t +∂¯u2(0) ∂x ∂h ∂x + ¯u2(0) ∂2h ∂x2i −h(h−h2−z)∂¯u2(0) ∂x +∂h ∂t + ¯u2(0) ∂h ∂xi∂¯u2(0) ∂x = (h−h2−z)G2+∂2h ∂t2+∂¯u2(0) ∂t ∂h ∂x + 2¯u2(0) ∂2h ∂x∂t + ¯u2(0) ∂¯u2(0) ∂x ∂h ∂x +¯u2(0)2∂2h ∂x2 = (h−h2−z)G2+D2 2h , (5.47) Hence ∂p′ 2(1) ∂z = (h2−h+z)G2−D2 2h . (5.48) 99 5. A GENERATION MODEL OF SOLITONS Integration now gives p′ 2(1) = ([h2−h]z+z2/2)G2−zD2 2h−([h2−h]ζ+ζ2/2)G2+ζD2 2h where the constant’ of integration has been chosen such that p′ 2(1) vanishes at the interface (in accordance with the requirement stated above). Next we take the derivative to x: ∂p′ 2(1) ∂x =−z∂h ∂xG2+ ([h2−h]z+z2/2)∂G2 ∂x −z∂ ∂x(D2 2h) +∂h ∂xζG2−(h2−h)∂ζ ∂xG2−ζ∂ζ ∂xG2−([h2−h]ζ+ζ2/2)∂G2 ∂x +∂ζ ∂xD2 2h+ζ∂ ∂x(D2 2h) =zh(h2−h)∂G2 ∂x −∂h ∂xG2−∂ ∂x(D2 2h)i+z2 2 ∂G2 ∂x +∂h ∂xζG2−η2 ∂ζ ∂xG2−([h2−h]ζ+ζ2/2)∂G2 ∂x +∂ζ ∂xD2 2h+ζ∂ ∂x(D2 2h). Finally, taking the depth-average we have ∂p′ 2(1) ∂x =h(h2−h)∂G2 ∂x −∂h ∂xG2−∂ ∂x(D2 2h)i1 η2Zζ −h2+h dz z +1 2 ∂G2 ∂x 1 η2Zζ −h2+h dz z2+∂h ∂xζG2−η2 ∂ζ ∂xG2 −([h2−h]ζ+ζ2/2)∂G2 ∂x +∂ζ ∂xD2 2h+ζ∂ ∂x(D2 2h) (5.49) This expression, following the derivation in Appendix D, evolves to ∂p′ 2(1) ∂x =−1 3η2 ∂ ∂x(η3 2G2)−1 2η2G2 ∂h ∂x +η2 2 ∂ ∂x(D2 2h) + ∂ζ ∂xD2 2h(5.50) 100 5.7 Numerical Modeling where Y(tn, xj) represents a collection of known variables whose values may be dependent on time and/or space. If we now discretize the derivative in the expression above, we get yn+1 j−yn j ∆t=Y(tn, xj) (5.73) where yn+1 jis an unknown quantity which can be expressed for all jpositions in terms of the known quantities of the previous time-levels of y(tn, xj) and Y(tn, xj). This operation is solved numerically using third-order AdamsBashforth approximation (Durran,1999) as yn+1 j=yn j+∆t 12 23Yn j−16Yn−1 j+ 5Yn−2 j(5.74) This is the idea we keep in mind, we calculate values at the new time-steps by using known values at the previous time-steps for all spatial positions. Hence, we need an scheme which coherently solve our equations in a way that we always count with the required known values to move our system to the new time-step over the space domain. To this end, equations for v1(5.65), v2(5.66) and ζ(5.38) are easily reorganized to be solved numerically with the time derivative in the left hand-side following ∂¯v1 ∂t =−¯u1 ∂¯v1 ∂x −µ¯u1+O(δ2) (5.65) ∂¯v2 ∂t =−µ¯u2−¯u2 ∂¯v2 ∂x +O(δ2) (5.66) ∂ζ ∂t = (h1−ζ)∂¯u1 ∂x −¯u1 ∂ζ ∂x (5.38) as required for applying Adams-Bashforth appromixation. 107 5. A GENERATION MODEL OF SOLITONS However, solving ¯u1numerically is not straightforward. After collecting time derivatives in the left hand-side and remaining terms in the right handside, the horizontal momemtum equation of ¯u1(5.64) evolves to an expression 1 in the form of a∂¯u1 ∂t +b∂2¯u1 ∂x∂t +c∂3¯u1 ∂x2∂t =Y(tn, xj) (5.75) where a,band ccollect spatial derivatives of ζ(x, t) and h(x, t) (space-time dependent variables); and Y(tn, xj) represents remaining terms, as previously. If now we operate the time derivative as a common factor in the left-hand side, the result leads to ∂ ∂ta¯u1+b∂¯u1 ∂x +c∂2¯u1 ∂x2=Y(tn, xj) + ∂a ∂t ¯u1+∂b ∂t ∂¯u1 ∂x +∂c ∂t ∂2¯u1 ∂x2(5.76) what helps us to introduce a new variable, ¯ U1, to turn our problem into a numerically solvable expression following ∂¯ U1 ∂t =Y(tn, xj) + ∂a ∂t ¯u1+∂b ∂t ∂¯u1 ∂x +∂c ∂t ∂2¯u1 ∂x2(5.77) In the following, for clarity purposes, we refer to the right-hand side above as R(tn, xj) ∂¯ U1 ∂t =R(tn, xj) (5.78) It is important to recall here that R(tn, xj) is a known quantity since both Y(tn, xj) and the spatial derivatives of ¯u1are both evaluated at the current time-step (n); and, the time derivatives of a,band c, which involve values of ζat the current (n) and new time-step (n+ 1), have been previously achivied with (5.38) via Adams-Bashforth approximation (5.74). 1For simplicity, we deal at this point with the generic expression (5.75) which represents the the horizontal momemtum equation of ¯u1(5.64) to be solved numerically. The procedure to that end is the matter of Section 5.7.2. Here we leave that work aside, as now we focused on explaining the numerical strategy we applied to solve ¯u1once we get an expression as (5.75). 108 5.7 Numerical Modeling Next, we proceed to discretize the spatial derivatives in ¯ U1using (5.68 and 5.69), what results in ¯ U1=aj¯u1j+bj ¯u1j+1 2∆x−bj ¯u1j−1 2∆x−cj ¯u1j+1 (∆x)2−2cj ¯u1j (∆x)2+cj ¯u1j−1 (∆x)2 (5.79) and grouping terms, it yields ¯ U1=aj−2cj 2∆x¯u1j+−bj 2∆x+cj (∆x)2¯u1j−1+bj 2∆x−cj (∆x)2¯u1j+1 (5.80) which we rewrite by introducing factors d,eand fas follows U1j=dj¯u1j+ej¯u1j−1+fj¯u1j+1 (5.81) Subsequently, when we discretize the time derivative of ¯ U1 ∂U1 ∂t =U1n+1 j−U1n j ∆t and apply Adams-Bashftorth (5.74), we obtain ¯ U1n+1 j=¯ U1n j+∆t 12 23Rn j−16Rn−1 j+ 5Rn−2 j(5.82) where ¯ U1n+1 jactually includes ¯ U1n+1 j=dn+1 j¯u1n+1 j+en+1 j¯u1n+1 j−1+fn+1 j¯u1n+1 j+1 (5.83) To close our system we still need to obtain ¯u1n+1 jfor all jterms. To that end, the equation above is more complicated to solve and gives rise to 109 5. A GENERATION MODEL OF SOLITONS implicit equations as we have not only the unknown ¯u1n+1 j, but also ¯u1n+1 j−1and ¯u1n+1 j+1 , which come from the mixed second and third derivatives of u1in (5.75). However, this is a well-known problem (Logan,1987) which can be solved following a procedure designed to deal with a numerically efficient resolution (Thomas algorithm). Firstly, we consider our space grid with j= 1...J and choose the boundaries of the interval sufficiently far away from the generation area to ensure that the internal tides will not reach them within the period of time under consideration. Consequently we may set ¯u1n=ζn J= 0 (5.84) for all n, and similarly for ¯u2, ¯viand ζ. Then we solve ¯u1n+1 for known quantities of ¯ U1n+1 and factorsdn+1 j,en+1 jand fn+1 jwithin the space grid j= 2...J −1. Consequently we can write equation (5.83) in matrix form as a tridiagonal system:           dn+1 2fn+1 2 en+1 3dn+1 3fn+1 3 . . . . . . . . . en+1 J−2dn+1 J−2fn+1 J−2 en+1 J−1dn+1 J−1                     ¯u1n+1 2 ¯u1n+1 3 . . . ¯u1n+1 J−2 ¯u1n+1 J−1           =            ¯ U1n+1 1 ¯ U1n+1 2 . . . ¯ U1n+1 J−2 ¯ U1n+1 J−1            (5.85) (5.86) The absence of d2and fJ−1is due to we previously set ¯u1= 0 onto the boundaries, what we introduce in the system as d2= 0 fJ−2= 0 (5.87) Then, equation (5.85) can be solved for ¯u1atthe next time step n+1. The procedure requires to calculate firstly 110 5.7 Numerical Modeling ¯ dj=dj−ejfj−1(5.88) ¯ fj=fj/¯ dj(5.89) ¯gj= ( ¯ U1j−ejgj−1)/¯ dj(5.90) for j= 2...J −1 (notice that fJ−1= 0), and subsequently ¯u1n+1 J−j=gJ−j−¯ fJ−j¯u1n+1 J−j+1 (5.91) for j= 1...J −2. Following this, we accomplish ¯un+1 1for all jterms and, hence, ¯u2n+1 in a straightforward manner using the expression (5.60), what closes the model resolution for every new time level n+ 1. With this we have drawn a numerical strategy which can be used to solve successfully the model object of this thesis with numerical techniques. 5.7.2 Preliminaries to Solve ¯u1 The mathematical work presented throughout this subsection is optional under the reader’s convenience and may be skipped without loss of continuity for the understanding of the numerical strategy we use to solve the model. The main goal here is to detail the procedure we follow to obtain a numerically solvable expression for ¯u1in the form of (5.75) taking its horizontal momentum equation (5.64) as starting point. For simplicity in future manipulations we work by collecting terms with analogous physical effects: linear, non-linear and non-hydrostatic dispersive effects: ∂¯u1 ∂t =µ¯v1−¯u1 ∂¯u1 ∂x +∂ζ ∂x +1 (1 −h)∂(Uh) ∂t +∂ ∂x(η1¯u2 1+η2¯u2 2) −µ(η1¯v1+η2¯v2)−η1 ∂ζ ∂x+δ1−η1 (1 −h)hη1G1 ∂η1 ∂x +η2 1 3 ∂G1 ∂x i +δη2 (1 −h)h−η2G2 ∂ζ ∂x −η2 2 3 ∂G2 ∂x +η2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi(5.92) 111 5. A GENERATION MODEL OF SOLITONS where we introduce for convenience the factors δ1and δ2in front of dispersive terms from the upper layer, and dispersive terms from the lower layer and the topography, respectively, δ1=δ1−η1 (1 −h);δ2=δη2 (1 −h)(5.93) leading to ∂¯u1 ∂t =µ¯v1−¯u1 ∂¯u1 ∂x +∂ζ ∂x +1 (1 −h)∂(Uh) ∂t +∂ ∂x(η1¯u2 1+η2¯u2 2) −µ(η1¯v1+η2¯v2)−η1 ∂ζ ∂x+δ1hη1G1 ∂η1 ∂x +η2 1 3 ∂G1 ∂x i +δ2h−η2G2 ∂ζ ∂x −η2 2 3 ∂G2 ∂x +η2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi(5.94) We proceed working out linear and non-linear terms of equation above, what leads to deal with the following µ¯v1−¯u1 ∂¯u1 ∂x +∂ζ ∂x +1 (1 −h)∂(Uh) ∂t +∂ ∂xη1¯u2 1+η2¯u2 2−µη1¯v1+η2¯v2−η1 ∂ζ ∂x where we introduce equation (5.57) to substitute the time-derivative of the oscillating topography, h, by its space-derivative µ¯v1−¯u1 ∂¯u1 ∂x +∂ζ ∂x +1 (1 −h)h∂U ∂t +U2∂h ∂x + ¯u1 ∂η1¯u1 ∂x + ¯u1η1 ∂¯u1 ∂x +¯u2 ∂η2¯u2 ∂x + ¯u2η2 ∂¯u2 ∂x −µη1¯v1+η2¯v2−η1 ∂ζ ∂x Next, we introduce the continuity equation (5.38) 112 5.7 Numerical Modeling µ¯v1−¯u1 ∂¯u1 ∂x +∂ζ ∂x +1 (1 −h)h∂U ∂t +U2∂h ∂x −¯u1 ∂η1 ∂t + ¯u1η1 ∂¯u1 ∂x −¯u2 ∂η2 ∂t + ¯u2η2 ∂¯u2 ∂x −µη1¯v1+η2¯v2−η1 ∂ζ ∂x as well as the definition of η1and η2for involved derivatives µ¯v1−¯u1 ∂¯u1 ∂x +∂ζ ∂x +1 (1 −h)h∂U ∂t +U2∂h ∂x + (¯u1−¯u2)∂ζ ∂t + ¯u1η1 ∂¯u1 ∂x +¯u2 ∂h ∂t + ¯u2η2 ∂¯u2 ∂x −µη1¯v1+η2¯v2−η1 ∂ζ ∂x At this point, we can clearly separate linear and non-linear terms grouped as linear =µ¯v1+∂ζ ∂x +1 (1 −h)h∂U ∂t +U2∂h ∂x + ¯u2 ∂h ∂t −µη1¯v1+η2¯v2−η1 ∂ζ ∂x(5.95) and nonlinear =−¯u1 ∂¯u1 ∂x +1 (1 −h)(¯u1−¯u2)∂ζ ∂t + ¯u1η1 ∂¯u1 ∂x + ¯u2η2 ∂¯u2 ∂x (5.96) Accordingly, equation (5.94) can be rewritten as ∂¯u1 ∂t =linear +nonlinear +δ1hη1G1 ∂η1 ∂x +η2 1 3 ∂G1 ∂x i +δ2h−η2G2 ∂ζ ∂x −η2 2 3 ∂G2 ∂x +η2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi(5.97) 113 5. A GENERATION MODEL OF SOLITONS We continue working out dispersive terms from the upper layer in equation (5.97), what leads to deal with δ1hη1G1 ∂η1 ∂x +η2 1 3 ∂G1 ∂x i(5.98) taking special care of terms with ∂¯u1 ∂t involved in G1(5.41). Therefore, it yields δ1h−η1 ∂ζ ∂xh∂2¯u1 ∂x∂t + ¯u1 ∂2¯u1 ∂x2−∂¯u1 ∂x 2i+η2 1 3 ∂ ∂x∂2¯u1 ∂x∂t + ¯u1 ∂2¯u1 ∂x2−∂¯u1 ∂x 2i δ1h−η1 ∂ζ ∂xh∂2¯u1 ∂x∂t + ¯u1 ∂2¯u1 ∂x2−∂¯u1 ∂x 2i+η2 1 3 ∂3¯u1 ∂x2∂t +η2 1 3h¯u1 ∂3¯u1 ∂x3−∂¯u1 ∂x ∂2¯u1 ∂x2ii Leaving on the left-hand side terms with time-derivatives for u-velocity components, equation (5.97) develops as ∂¯u1 ∂t +δ1hη1 ∂ζ ∂x ∂2¯u1 ∂x∂t −η2 1 3 ∂3¯u1 ∂x2∂ti=linear +nonlinear +δ1h−η1 ∂ζ ∂xh¯u1 ∂2¯u1 ∂x2−∂¯u1 ∂x 2i+η2 1 3h¯u1 ∂3¯u1 ∂x3−∂¯u1 ∂x ∂2¯u1 ∂x2ii +δ2h−η2G2 ∂ζ ∂x −η2 2 3 ∂G2 ∂x +η2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi(5.99) where we can group the non-hydrostatic dispersive terms related to upper layer on the right-hand side under the variable dispersive1, following dispersive1=δ1h−η1 ∂ζ ∂xh¯u1 ∂2¯u1 ∂x2−∂¯u1 ∂x 2i+η2 1 3h¯u1 ∂3¯u1 ∂x3−∂¯u1 ∂x ∂2¯u1 ∂x2ii (5.100) and the equation (5.99) can be rewritten as 114 5.7 Numerical Modeling ∂¯u1 ∂t +δ1hη1 ∂ζ ∂x ∂2¯u1 ∂x∂t −η2 1 3 ∂3¯u1 ∂x2∂ti=linear +nonlinear +dispersive1 +δ2h−η2G2 ∂ζ ∂x −η2 2 3 ∂G2 ∂x +η2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi (5.101) Next, we continue working out dispersive terms from the lower layer in equation (5.101), what leads to deal with δ2h−η2G2 ∂ζ ∂x −η2 2 3 ∂G2 ∂x i taking special care of terms with ∂¯u2 ∂t involved in G2(5.41). Then, we obtain δ2h−η2 ∂ζ ∂xh∂2¯u2 ∂x∂t + ¯u2 ∂2¯u2 ∂x2−∂¯u2 ∂x 2i−η2 2 3h∂3¯u2 ∂x2∂t + ¯u2 ∂3¯u2 ∂x3−∂¯u2 ∂x ∂2¯u2 ∂x2ii Leaving on the left-hand side terms with time-derivatives for u-velocity components, equation (5.101) develops as ∂¯u1 ∂t +δ1hη1 ∂ζ ∂x ∂2¯u1 ∂x∂t −η2 1 3 ∂3¯u1 ∂x2∂ti+δ2hη2 ∂ζ ∂x ∂2¯u2 ∂x∂t +η2 2 3 ∂3¯u2 ∂x2∂ti= linear +nonlinear +dispersive1+δ2h−η2 ∂ζ ∂xh¯u2 ∂2¯u2 ∂x2−∂¯u2 ∂x 2i −η2 2 3h¯u2 ∂3¯u2 ∂x3−∂¯u2 ∂x ∂2¯u2 ∂x2i+η2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi(5.102) where we can group the non-hydrostatic dispersive terms related to upper layer on the right-hand side under the variable dispersive2, following dispersive2=δ2h−η2 ∂ζ ∂xh¯u2 ∂2¯u2 ∂x2−∂¯u2 ∂x 2i−η2 2 3h¯u2 ∂3¯u2 ∂x3−∂¯u2 ∂x ∂2¯u2 ∂x2ii (5.103) 115 5. A GENERATION MODEL OF SOLITONS and the equation (5.102) can be rewritten as ∂¯u1 ∂t +δ1hη1 ∂ζ ∂x ∂2¯u1 ∂x∂t −η2 1 3 ∂3¯u1 ∂x2∂ti+δ2hη2 ∂ζ ∂x ∂2¯u2 ∂x∂t +η2 2 3 ∂3¯u2 ∂x2∂ti= linear +nonlinear +dispersive1+dispersive2 +δ2hη2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi(5.104) At this stage it is important to notice that we have obtained in equation (5.104) terms with time derivatives of ¯u2on the left-hand side. Once we finish to turn out all dispersive terms from topography, we will deal with the complete left-hand side, grouping terms and substituting ¯u2in terms of ¯u1 following Appendix E. Now we continue working out dispersive terms from the topography in equation (5.104), what leads to deal with δ2hη2G2 2 ∂h ∂x +η2 2 ∂(D2 2h) ∂x +∂ζ ∂xD2 2hi taking special care of terms with ∂¯u2 ∂t involved in G2(5.41). Then we get δ2hη2 2 ∂h ∂xh∂2¯u2 ∂x∂t + ¯u2 ∂2¯u2 ∂x2−∂¯u2 ∂x 2i +η2 2 ∂ ∂x∂2h ∂t2+∂¯u2 ∂t ∂h ∂x + 2¯u2 ∂2h ∂x∂t + ¯u2 ∂¯u2 ∂x ∂h ∂x + ¯u2 2 ∂2h ∂x2 +∂ζ ∂xh∂2h ∂t2+∂¯u2 ∂t ∂h ∂x + 2¯u2 ∂2h ∂x∂t + ¯u2 ∂¯u2 ∂x ∂h ∂x + ¯u2 2 ∂2h ∂x2ii (5.105) where making some simplifications we obtain δ2hη2 ∂2¯u2 ∂x∂t ∂h ∂x +η2¯u2 ∂2¯u2 ∂x2 ∂h ∂x +η2 2h∂3h ∂x∂t2+∂¯u2 ∂t ∂2h ∂x2+ 2∂¯u2 ∂x ∂2h ∂x∂t +2¯u2 ∂3h ∂x2∂t + 3¯u2 ∂¯u2 ∂x ∂2h ∂x2+ ¯u2 2 ∂3h ∂x3i+∂ζ ∂xh∂2h ∂t2+∂¯u2 ∂t ∂h ∂x + 2¯u2 ∂2h ∂x∂t +¯u2 ∂¯u2 ∂x ∂h ∂x + ¯u2 2 ∂2h ∂x2ii(5.106) 116 Chapter 6 Numerical Experiments on Solitons1 6.1 Outline The main goal of this chapter is to present various numerical experiments which demonstrate the capability of the model we have derived on reproducing two of the more distinguishable properties attributed to a fully nonlinear set of equations: the generation of ‘table-top’ solitons when approaching the theoretical maximum amplitude given appropriate initial conditions; and, 2) the appearance of nonlinearitiess under a configuration in which the two-fluid system consists of two layers of equal thickness. In Section6.2 we deal with key aspects about the configuration of the numerical experiments before starting the presentation of the main results. In Subsection 6.2.1 we recall to the reader the main components of the twofuid layer system which configure the model. Discussion on the criteria to choose the resolution of the space-time grid needed for solving numerically the modeld is addressed in Subsection 6.2.2. As we have already described, the ‘tide-topography’ interaction is introduced in the model through a moving topography and a fluid initally at rest; unlike the original problem in which 1Aguiar-Gonz´alez B., Gerkema T. A Model for the Generation of Strongly Nonlinear, Weakly Nonhydrostatic Interfacial Waves. In Preparation to be Submitted to an International refereed Journal 123 6. NUMERICAL EXPERIMENTS ON SOLITONS a barotropic tidal flow over topography generates internal tides. This situation leads to the question of whether the model can be realistically considered equivalent to oceanic conditions. Consequently, we use in Subsection 6.2.3 the generation model of weakly nonlinear weakly nonhydrostatic solitons derived in Gerkema (1996), which works with tidal motion, as a benchmark for testing our fully nonlinear weakly nonhydrostatic generation model of solitons, which works with moving topography. Next, we summarize in Subsection 6.2.4 the main parameters we set for the numerical experiments shown throughout the chapter. Firstly we present some simulations on the generation of the linear hydrostatic internal tides to test the model in absence of nonlinearities and nonhydrostatic effects (Section 6.3). Subsequently, we move forward and include the latter effects in the simulations focusing on two special cases. On one hand, we show in Section 6.4.1 the generation of strongly nonlinear solitons approaching its theoretical limiting amplitude and hence evolving to large-amplitude flat ‘table-top’ solitons. On the other hand, we deal with two layers of equal thickness and nonlinearities in Section 6.4.2. Conclusions are addressed in Section 6.5. 6.2 Configuration of the Numerical Experiments 6.2.1 The Two-Fluid Layer System A general diagram of the two-fluid layer system is shown in Figure 6.1. We assume that the upper and lower layer consists of incompressible, inviscid, homogeneous fluids of density ρ1and ρ2, and have a thickness of h1and h2, respectively. The rest level of the interface is located at z= 0, the upward direction of the z-axis is positive. Here, ηirepresents the thickness of the layer (depending on the interfacial displacement ζ). Furthermore, the upper surface is covered by a rigid lid. The system is supposed to be uniform in the y(i. e. transverse) direction; the topography is consequently assumed to be infinite in that direction. We mimick the ‘tide-topography’interaction which generates the internal tide in the system with a moving bottom which oscillates horizontally. However, in order to facilitate the interpretation of the simulations, we transformed the 124 6.2 Configuration of the Numerical Experiments results with moving topography to the frame in which the topography is at rest and located symmetrically with respect to the center of the x-axis. In all experiments fluid starts moving to the right at t= 0 (i. e. topography moving to the left). The waves are generated near the origin in x-axis due to the ‘tide-topography’ interaction; on the negative (positive) x-axis, waves travel to the left (right). As the forcing enters in the simulation asymmetrically with fluid at rest moving to the right, it is expected that wave packets in the front appear rather different when comparing both sides (negative vs. positive x-domain). These fronts are the transients, which are influenced by the way the experiment is started. Consequently, the steady solution at both sides of the x-axis is reached after several tidal periods have passed away. For this reason, we start the observation of all our results when the signal has become periodic (at least after 6 tidal periods) to avoid transient effects. Figure 6.1: Configuration of the Numerical Experiments - A general diagram of the two-fluid layer system. For the numerical experiments we present throughout this chapter we have defined the topography1as h(x, t) = b1 1 + (x/b2)2(6.1) 1The topography is set analytically to ensure an unstaggered domain for second and third derivatives of h(x, t). Nevertheless, other analytical functions may be also used depending on the desired topography. 125 6. NUMERICAL EXPERIMENTS ON SOLITONS with xbeing the grid positions in space; and, b1and b2being the parameters which set the height and width of a symmetric sill, respectively. Also, the topographic obstacle (ridge, sill, ...) is always centred in the x-axis and the length of the x-domain is chosen to be large enough to prevent that generated waves reach the boundaries. It is worth while to recall that the oscillation of the topography is included within the model (on page 102) as h=h(X) with X(x, t) = x−acos twith a being an arbitrary constant. 6.2.2 Setting the Space-Time Grid The choice of the steps Λt, Λxintroduced in Chapter 5(Section 5.7) was based on two main requirements. Firstly, the resolution in x(Λx) must be sufficiently fine to resolve third-derivative terms and ensure that any short, solitary-like waves are properly resolved. Nevertheless, dealing with equivalent equations to Miyata (1988) and Choi and Camassa (1999), as we do in our model, Kelvin-Helmholtz instabilities are not filtered out. In this regard, Jo and Choi (2002) found that solitary waves of sufficient amplitude could be unstable at high wave numbers to Kelvin-Helmholtz instability. Thus, if the grid resolution is too fine, unstable short waves will emerge near the wave crest and ultimately overwhelm the calculations and explode numerically (Jo and Choi,2002;Helfrich and Melville,2006;Helfrich and Grimshaw,2008). In some cases, the instability can be controlled by filtering out wavenumbers above a threshold (W. Choi 2007, personal communication cited in Helfrich and Grimshaw (2008)). For our numerical experiments we consider a Λx course enough to prevent the problem. A second condition follows from the requirement of stability. Then, for a given spatial step one may take the Courant-Friedrichs-Lewy condition for the linearized equations as an indication of the required time step. The criterion implies that Λx/Λtshould be larger than the phase speed of the wave; taking special care where the advection by the barotropic tidal flow (here mimicked with the moving topography) should be added to the phase speed to apply the criterion properly (Gerkema, 1994). For the simulations we present, it was not needed to filter out wavenumbers above a threshold to control Kelvin-Helmholtz instabilities as we designed the space-time grid to avoid this problem following previous conditions. However, in some cases, specially in the simulations where the forcing was fairly strong, 126 6.2 Configuration of the Numerical Experiments an additional trick was needed to retain stability around the generation area1 (Gerkema,1994). In those cases averages were taken in the vicinity of the top of the ridge (around the steepest part of the topography), where the instabilities arised. At one particular point (xj,tn) in space-time, new values of ¯ui, ¯viand ζwere calculated by taking the average of the old values at xj−1, xjand xj+1, and subsequently in time between tnand tn−1. The disturbance provoked by this procedure was tested and found to be a minor effect only, as it was only applied over the closest region to the top of the topography. 6.2.3 Basic Tests A Galilean (or inertial) reference frame is a frame with constant and rectilinear motion with respect to one another. Consequently, an inertial frame of reference describes time and space homogenously and isotropically with no time dependance. Hence, observations made in one inertial frame can be converted to observations in another inertial frame by a simple transformation (the socalled Galilean transformation), as physical laws follows the same behaviour in all inertial frames. Our moving topography is not an inertial frame as it does not move with constant motion but accelerated. In theses cases, observations made in a non-inertial frame cannot be transformed directly to observations made in an inertial frame. Physics will vary depending on the acceleration of the noninertial frame with respect to the inertial frame and, consequently, regular physics forces will need to be supplemented by ficticious forces. In our case, this situation leads to the question of whether a moving topography can be realistically considered equivalent to a barotropic tidal flow over topography. This comparison is not straightforward and requires further research before the model can be applied to the simulation of oceanic observations. The main area of attention is the top of the topography from where the waves emerge and where our non-inertial frame (the moving topography) is accelerated. Far away from this point, nonlinear and nonhydrostatic effects acting over the internal tide are presumably equivalent as the bottom there does not move and is flat. 1This was needed to be applied only in the case for the generation of ‘table-top’ solitons in Section 6.4.1. 127 6. NUMERICAL EXPERIMENTS ON SOLITONS −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 Distance (km) −20 −15 −10 −5 0 5 10 15 20 −5 −2.5 0 2.5 5 Distance (km) −20 −15 −10 −5 0 5 10 15 20 −10 −5 0 5 10 Interfacial Displacement, ζ (m) Distance (km) Figure 6.2: Moving Topography vs. Tidal Motion - Black lines is the model derived in this thesis and grey line is themodel derived in Gerkema (1996). The upper and lower layer thickness are h1= 30 m and h2= 70 m. respectively. The tidal flow is 20 cm s−1in (a); 40 cm s−1in (b); and, 80 cm s−1in (c). Here we use the generation model of weakly nonlinear weakly nonhydrostatic solitons derived in Gerkema (1996), which works with tidal motion, as 128 6.2 Configuration of the Numerical Experiments a benchmark for testing our fully nonlinear weakly nonhydrostatic generation model of solitons, which works with moving topography. The comparison is obviosuly not straightforward as both models are of different nature, weakly and strongly nonlinear effects already make them different. For this reason we focus only on comparing the response of the model over the top of the topography, where the ficticious forces may become important and the internal tide is generated. Any differences which may appear far away from the generation area are indeed expected due to different nonlinear effects acting in every model. Also for this reason we present the results centred only over this area and under weakly nonlinear settings in order to make them more comparable (the matter here is to test that the height of the topography is not a problem). For clarity on the results we use for this comparison the same configuration1 we use for the numerical experiments that we will present throughout the chapter. Thus, Figure 6.2 suggests that an internal tide generated with a relatively small moving topography is equivalent to that generated via a tidal flow over topography. 6.2.4 Set of Experiments The model operates with equations in nondimensinal form; however, we present results from the numerical experiments after dimensionalisation. To organize the presentation of the numerical experiments, we chose a configuration which we keep fixed for most of the simulations. For simplicity on the magnitudes, we set a total water depth of 100 m (h= 100) with the parameters for the thickness of the upper and lower layers being h1= 30 and h2= 70 (the upper layer being thinner than the lower layer). Regarding the topography, the height of the sill was set at 40 m with decresing amplitude covering mainly the interval from -10 km to 10 km over the x-axis, thus b1= 40 and b2= 10000. Furthermore, the horizontal oscillation of the moving topography (forcing) is always of semidiurnal fequency. The strength of the stratification at the interface was set by a value of g′= 0.01 m s−2. For this configuration, the small parameter δretaining weak nonhydrostatic effects keeps the same value of δ= 1.67e-06. 1See following subsection for details. 129 6. NUMERICAL EXPERIMENTS ON SOLITONS The space-time grid is solved in all cases with Λx= 20 m and Λt=∼4.5 s. Nevertheless, for numerical experiments in Section 6.3 (linear hydrostatic simulations) and Section 6.4.2 (simulation with two layers of equal thickness), we use the same Λx, but a longer time step, Λt=∼10 s. In case any of the above mentioned values change in a simulation for a specific purpose, it will be mentioned in the text. 6.3 Generation of the Linear Hydrostatic Internal Tide We start with the generation of the linear hydrostatic internal tide. With this aim, we linearize model equations derived throughout Chapter 5(and summarized in Section 5.7.3), which then read ∂¯u1 ∂t =µ¯v1+∂ζ ∂x +1 (1 −h)h∂U ∂t +U2∂h ∂x + ¯u2 ∂h ∂t −µη1¯v1+η2¯v2 −h1 ∂ζ ∂x,(6.2) ¯u2=Uh −η1¯u1 η2 ,(5.60) ∂¯v1 ∂t =−µ¯u1,(6.3) ∂¯v2 ∂t =−µ¯u2,(6.4) ∂ζ ∂t =h1 ∂¯u1 ∂x .(6.5) 130 6.3 Generation of the Linear Hydrostatic Internal Tide −13 −12 −11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Interfacial Displacement, ζ (m) −250 −225 −200 −175 −150 −125 −100 −75 −50 −25 0 25 50 75 100 125 150 175 200 225 250 −130 −120 −110 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 Interfacial Displacement, ζ (m) Distance (km) Figure 6.3: Generation of the Linear Hydrostatic Internal Tide (Rotationless case, µ= 0)- Time evolution of the linear, hydrostatic internal tide during the sixth tidal period. Influence of the tidal advection is shown in (top) with weak ‘tidal flow’ (5 cm s−1); and (bottom) with fairly strong ‘tidal flow’ (50 cm s−1), where the wave is no longer sinusoidal (‘quasi-nonlinear’ case) because of the higher harmonics. Time evolves from the highest profiles downward. First we consider the rotationless case (µ= 0), so that there is no transverse velocity component either. Consequently, Eq. (6.3) and (6.4 drop as well as terms wich involve Coriolis effects. Figure 6.3 presents two cases of a linear, hydrostatic internal tide after 6 tidal periods have passed away. The waves are generated over the topography, centred in the origin of the x-axis, and subsequently travel away to the left (negative x-axis) and to the right (x-axis) from the topography. The upper part of the figure shows ‘tidal flow’ moving to the left (topography moving to the right); the lower half shows ‘tidal flow’ moving to the right (topography moving to the left). The lapse of time between 131 6. NUMERICAL EXPERIMENTS ON SOLITONS sucessive representations is 1 12 of tidal period. The ‘tidal flow’ in Figure 6.3 (top) is weak, 5 cm s−1, and consequently internal tides arise as sinusoidal waves. On the contrary, in Figure 6.3 (bottom) the ‘tidal flow’ is strong, 50 cm s−1; hence, the barotropic tidal advection becomes significant as observed and higher harmonics give rise to ‘quasinonlinear’ waves (Maas and Zimmerman,1989). If we now include the effects of the Earth’s rotation on the generation of the linear hydrostatic internal tide, we need to solve the complete set of equations above. We present two cases: at mid (φ= 45◦,µ= 4.61) and high latitudes (φ= 90◦,µ= 6.52) for testing weak and strong Earth’s rotation effects, respectively. −0.75 −0.5 −0.25 0 0.25 0.5 0.75 Interfacial Displacement, ζ (m) −250 −225 −200 −175 −150 −125 −100 −75 −50 −25 0 25 50 75 100 125 150 175 200 225 250 −10 −7.5 −5 −2.5 0 2.5 5 7.5 10 Interfacial Displacement, ζ (m) Distance (km) Figure 6.4: Generation of the Linear Hydrostatic Internal Tide (at Mid Latitudes: φ=45◦,µ= 4.61)- The linear, hydrostatic internal tide during the sixth tidal period. Influence of the Earth’s rotation is shown in (top) over a weak ‘tidal flow’ (5 cm s−1); and (bottom) over a strong ‘tidal flow’ (50 cm s−1). 132 Agradecimientos Este trabajo ha crecido bajo la mirada c´omplice de personas muy especiales. Es el momento de darles las gracias. Algunos nombres se repetir´an, diferentes son los motivos y los lugares, e incluso los idiomas. Intentar´e mantener un orden, pero lo que ahora se presenta me voy a permitir el capricho de dejar que fluya y oscile con su per´ıodo particular, el m´ıo. Estoy convencido que lo entender´an... La presente Tesis Doctoral ha sido dirigida por el Dr. ´ Angel Rodr´ıguez Santana (Universidad de Las Palmas de Gran Canaria - ULPGC), el Dr. Theo Gerkema (Royal Netherlands Institute for Sea Research - NIOZ) y el Dr. Jes´us Cisneros Aguirre (Universidad de Las Palmas de Gran Canaria - ULPGC), a quienes agradezco la confianza y dedicaci´on que me han brindado durante estos a˜nos de mi etapa de formaci´on como investigador. Aprecio enormemente el tiempo que hab´eis dedicado a guiar este trabajo con vuestra valiosa experiencia. Me encuentro tambi´en agradecido al Dr. Samuel Hormaz´abal de la Universidad Cat´olica de Valpara´ıso, cuyo contribuci´on a mi trabajo de investigaci´on ha sido muy importante. Gracias por tu dedicaci´on y generosa hospitalidad. Este trabajo se ha nutrido tambi´en de la experiencia del Dr. Antonio Mart´ınez Marrero, quien siempre mostr´o inter´es y encontr´o tiempo para atender mis dudas. Dentro del Grupo de Investigaci´on OFYGA, del cual he formado parte durante mi etapa como doctorando en la ULPGC, quiero tambi´en agradecer a la Dra. ´ Angeles Marrero D´ıaz su dedicaci´on y atenci´on conmigo, siempre disponible para ayudar. No pod´ıa ser de otro modo, dentro del Grupo OFYGA, agradezco muy especialmente a ´ Angel y a Jes´us su continuado inter´es en mi formaci´on, en mi presente y en mi futuro, siempre dispuestos a ofrecerme nuevas oportunidades, participaci´on en campa˜nas oceanogr´aficas, contactos con otros centros de investigaci´on, etc ... ¡Gracias! Al Dr. Josep Llu´ıs Pelegr´ı, al Dr. Pablo Sangr`a Inciarte y al Dr. Miguel Bruno Mej´ıas quisiera agradecer su confianza al permitirme la oportunidad de participar en varias campa˜nas oceanogr´aficas a su cargo. En todas ellas la experiencia vivida fue siempre enriquecedora. Durante mis a˜nos de doctorado tuve la generosa oportunidad de colaborar con la Dra. May G´omez y el Dr. Ted Packard, gracias por contar conmigo en 235 Agradecimientos vuestra investigaci´on. Ted, investigar a tu lado es un placer, tu vitalidad es inspiradora, mil gracias por contagiarme tu entusiasmo por la ciencia. Estos a˜nos de laboratorio no han podido contar con mejores compa˜neros de trabajo y amigos: M´onica, Francis, Carolina, Mireya, Sheila, Charles, Bego,... Compartir tiempo y espacio con ustedes ha sido un placer. No se me ocurre mejor escenario para trabajar y compartir vivencias durante el d´ıa. Mis inicios en la programaci´on con Matlab y escritura en L A T EX fueron mucho m´as amenos de lo que pudiera ser habitual al profano. Y eso fue gracias al tiempo que generosamente me dedic´o el Dr. Francisco Mach´ın. Esta tesis tiene mucho de lo que he aprendido de ti. Guardo especialmente en el recuerdo el tiempo compartido juntos en Barcelona. Idaira, Aridane, Isis, Inma, Rosa,... est´an vinculados de manera especial a mis primeros a˜nos de curiosidad oceanogr´afica durante la carrera. En lo que a problemas inform´aticos se refiere, gracias por tu ayuda Jorge, m´as de un disco duro se ha salvado gracias a ti. Mis inicios en la oceanograf´ıa con Mac tuvieron alg´un que otro contratiempo que r´apidamente se resolvi´o gracias a la buena disposici´on y ayuda de David. Desde que inici´e la carrera en Ciencias del Mar (CCM) hasta el d´ıa de hoy he compartido mis vivencias oceanogr´aficas con muchos amigos que vienen ahora a mi memoria ... es imposible nombrarlos a todos, pero no por ello dejo de decirles desde aqu´ı: gracias por estar ah´ı. Desde hace m´as de una d´ecada que cuento con vuestra amistad, cari˜no y apoyo... ¡menuda suerte!... Gracias M´onica, ´ Angela, Fede. En el Centre Mediterrani D’Investigacions Marines i Ambientals (CMIMACSIC) en Barcelona realic´e mis cursos de M´aster y Doctorado en Oceanograf´ıa a trav´es del programa organizado por la ULPGC. All`a el Dr. Josep Llu´ıs Pelegr´ı em va rebre amablement durant la meva estada, oferint-me totes les facilitats d’acc´es i ubicaci´o al Departament d’Oceanografia F´ısica per superar amb `exit la meva primera etapa investigadora. Moltes Gr`acies Josep Llu´ıs! After my stay at the CMIMA-CSIC, I travelled to The Hague (The Netherlands) to collaborate with Dr. Frans-Peter A. Lam and Dr. Mathijs Schouten at the Netherlands Organisation for Applied Scientific Research (TNO). During this period I was introduced to HOPS numerical modeling. The time I spent at TNO is unforgettable, working with both of you is one of the gifts 236 I will always remember from my PhD time. There are not enough words to thank all I received from you. I appreciated very much your advices and careful guide. I also thank Prof. Dr. P. F. J. Lermusiaux (Massachusetts Institute of Technology, MIT) for his help, inputs and comments in the distance on modeling work for MREA exercises. From my period at TNO I cannot forget about Benoit and Jeroen for all their help at my arrival and for the joyful time we shared, as well as with Mathieu, Antonio, Jacqueline, Michael,... My stays at NIOZ were possible thanks to Dr. Frans-Peter A. Lam, who firstly introduced me to Prof. Dr. L. R. M. Maas and Dr. T. Gerkema. The Netherlands always brings to me nice memories. My time there was special, and that was thanks to the people I met on my way. They made my stay to be great: joyful and fruitful at the same time. I already mentioned people from The Hague at TNO, but I also met wonderful people I want to remember in these lines at Texel during my stays at NIOZ, specially Jenny, Femke, Carola, Anna and Andrea. I also want to thank the Physical Oceanography Department at NIOZ, which kindly hosted me during my PhD time there. In this regard, I am grateful to Prof. Dr. Leo Maas, who always shared his interest on my work with refreshing and inspiring questions. At this point, it may be obvious to the reader that I deeply feel this thesis is half spanish, half dutch. And the dutch side of this research was developed thanks to Dr. Theo Gerkema. I appreciate a lot your hospitality and all the time you have spent working with me and supervising my PhD study. I am also grateful for your continuous interest on helping me with future steps:.¡Muchas Gracias por todo Theo! He aprendido y disfrutado much´ısimo trabajando a tu lado. ´ Angela, gracias por estar ah´ı siempre, por los muchos a˜nos de amistad desde que nos conocemos, y por todos los que vendr´an. Muchas gracias por tu apoyo en lo personal y en lo laboral. Nuestras conferencias Gran Canaria - Texel, y luego al rev´es, son inolvidables... el destino es curioso, pero contigo al lado adem´as de curioso es siempre agradable. M´onica, muchas son las horas que hemos compartido dentro y fuera del lab, dentro y fuera de ‘la isla’. Muchos son los grandes momentos vividos en estos a˜nos de amistad (y los que vendr´an): viajes sorpresa, excursiones aventura, videoconferencias en la distancia,... Los he disfrutado y disfruto much´ısimo. Muchas gracias por tu apoyo en lo personal y en lo laboral, por tu compa˜n´ıa, por estar ah´ı siempre. 237 Agradecimientos Goonie, gracias por tirar de mi pantal´on y guiar cada uno de mis pasos aquella noche de Febrero, tuviste que tirar fuerte, y en varias ocasiones, pero nunca te falt´o fuerza a pesar de tu peque˜no tama˜no. Estoy seguro que est´as en el cielo de las pizzas y los churros. Las primeras palabras de esta tesis van dedicadas a ellos, y de ellos es el honor de cerrarla: mis padres Mappi e Ignacio, y mi hermano Jose. Ustedes han hecho posible que haya llegado hasta aqu´ı. Gracias por vuestra compa˜n´ıa y cari˜no. No tengo palabras para agradecerles que siempre pueda contar con ustedes. Gracias por todas vuestras visitas cuando estoy fuera de ‘la isla’, me hacen sentir pieza importante en vuestra vida y siempre suponen una inyecci´on de alegr´ıa. Mappi, gracias por explicarme ‘la teor´ıa del sandwich’, aplicable a todos los ´ambitos de la vida. Si he terminado esta tesis es gracias a que la he aplicado una vez m´as. Gracias por ayudarme siempre a ver lo esencial y a encontrar mi camino. Ignacio, en momentos de estr´es ‘tus teor´ıas de las ondas’ me hicieron re´ır y ver alguna que otra luz, gracias por ambos efectos, llegaban siempre que hac´ıan falta. Jose, gracias por tus ´animos, por alegrarte e interesarte siempre por lo que hago, por tu tiempo conmigo y por cuidar siempre de tu ‘hermanito peque˜no’ (aunque ya no sea tan peque˜nito a´un te necesito). ¡Gracias mis peque˜nos, gracias por apoyarme, millones de gracias! 238 Curriculum Vitae Borja Aguiar Gonz´alez was born in Las Palmas de Gran Canaria (Canary Islands, Spain) on September 29th1982. From his childhood, summertime and vacations were always linked to life in the coast, specially in Lanzarote and Fuerteventura, where he used to share maritime activities with his family and friends. Thus, he became interested in all phenomena related to the sea, and decided to study the Bachelor of Marine Sciences in the fall of 2000 at the Universidad de Las Palmas de Gran Canaria (ULPGC). From the very beginning he was involved in numerous projects as a trainee, most of them related to coastal management. He spent the last years of his bachelor collaborating with the research group OFYGA (in the Deparment of Physics), which introduced him to physical oceanography. After finishing the bachelor, he decided to continue his education as a researcher with this research group. In January 2007, he is awarded with the PhD grant which supports the realization of this thesis. He developed his research partially at ULPGC and the Royal Netherlands Institute for Sea Research (NIOZ). During his years as a PhD student, he had also the opportunity to work with outstanding professionals at the Centre Mediterrani D’Investigacions Marines i Ambientals (CMIMA-CSIC), the Netherlands Organisation for Applied Scientific Research (TNO) and the Universidad de Concepci´on (UdeC). Working at sea he learns and enjoys the most the job as oceanographer. Onboard the ships B/O Garc´ıa del Cid,B/O Sarmiento de Gamboa,Hidrogr´afico Malaspina and BIO Hesp´erides he has sailed along the north-western coast of Africa, from the Strait of Gribraltar to the Cape Verde Islands; and, from the Southern South America to the Antartic Peninsula across the Drake Passage. The focus of his PhD research activities have been the observational study of resonant breeze-forced oscillations1,2and modeling of strongly nonlinear 1Aguiar-Gonz´alez B., Rodr´ıguez-Santana A., Cisneros-Aguirre J., Mart´ınez-Marrero. 2011. Diurnal-inertial motions and diapycnal mixing on the Portuguese shelf. Cont. Shelf. Res. 31, 1193–1201. 2Aguiar-Gonz´alez B., Hormaz´abal S., Rodr´ıguez-Santana A., Cisneros-Aguirre J., Mart´ınez-Marrero. Breeze-Forced Oscillations around The Poleward Limit for DiurnalInertial Resonance. In Preparation to be Submitted to an International refereed Journal 239 Curriculum Vitae tide-generated internal solitons1. During the PhD time he also worked on HOPS (Harvard Ocean Prediction System) modeling for the Maritime Rapid Environmental Assessment 2004 (MREA04). In the recent years, he has been working on the contribution of mesoscale activity to near-inertial energy variability with drifters, remote sensing images and altimetry data (research in progress); as well as on the dominant processes which control oceanic mixing around the South Shetland Islands in Antartica (research in progress). As a result of his multidisciplinary education and the interaction with other research groups, he has recently co-authored a paper on theoretical and experimental biochemistry2. 1Aguiar-Gonz´alez B., Gerkema T. A Model for the Generation of Strongly Nonlinear, Weakly Nonhydrostatic Interfacial Waves. In Preparation to be Submitted to an International refereed Journal 2Aguiar-Gonz´alez B., Packard T., Berdalet E., Roy S., G´omez M. 2012. Respiration predicted from an Enzyme Kinetic Model and the Metabolic Theory of Ecology in two species of marine bacteria. J. Exp. Mar. Biol. Ecol. 412, 1–12 240