Mechanisms for the Formation of Rhythmic Topography in the Nearshore Region by Giovanni Coco A thesis submitted to the University of Plymouth in partial fulfilment for the degree of DOCTOR OF PHILOSOPHY Institute of Marine Studies Faculty of Science October 1999 LIBRARY STORE REFERENCE ONLY 2000 Class NO umim 90 0406131 9 Mechanisms for the Formation of Rhythmic Topography in the Nearshore Region Giovanni Coco Abstract The possibility that the periodic features observed in the nearshore region are the result of self- organisational processes is investigated in this work. The behaviour of two numerical models, based on different techniques, has been analysed in order to describe the formation of periodic features in the surf and swash zone respectively. The appearance of periodic patterns in the nearshore region has been traditionally linked to the presence of standing edge waves with the topographic changes passively driven by the flow patterns. A more recent approach indicates the possibility that periodic patterns appear because of feedback processes between beach morphology and flow. In the first model, the coupling between topographic irregularities and wave driven mean water motion in the surf zone is examined. This coupling occurs due to the fact that the topographic perturbations produce excess gradients in the wave radiation stress that cause a steady circulation. To investigate this mechanism, the linearised stability problem in the case of an originally plane sloping beach and normal wave incidence is solved. It is shown that the basic topography can be unstable with respect to two different modes: a giant cusp pattern with shore attached transverse bars that extend across the whole surf zone and a crescentic pattern with alternate shoals and pools at both sides of the breaking line showing a mirroring effect. For the swash zone, the formation of beach cusps has been investigated. The several theories proposed in the past have been analysed and all the field and laboratory measurements available in the literature collected in order to test such theories. It is suggested that, with the available measurements it is not possible to distinguish between the standing edge wave model and the self-organisation approach. A numerical model based on self-organisation has been here developed and tested in order to understand the processes occurring during beach cusp formation and development, to evaluate the sensitivity towards the parameters used and to look at how the model might relate to field observations. Results obtained confirm the validity of the self-organisation approach and its capacity to predict beach cusp spacing with values in fair agreement with the available field measurements and with most of the input parameters primarily affecting the rate of the process rather than the final spacing. However, changes in the random seed and runs for large numbers of swash cycles reveal a dynamical system with significant unpredictable behaviour. A qualitative comparison between the model results and field measurements collected by Masselink et al. (1997) during beach cusp formation and development has also been performed on the basis of a non-linear fractal technique. Results indicate beach locations and time-scales where non-linearities are more important and self- organisation can play a fundamental role. Ill TABLE OF CONTENTS Abstract List of Figures List of Plates List of Tables Acknowledgements Author's Declaration 1. INTRODUCTION 1 2. MORPHODYNAMIC MODELLING 4 2.1 Introduction 4 2.2 Complex dynamics 5 2.3 Complexity in coastal processes 6 2.4 Stability theory 7 2.4.1 Linear stability analysis 7 2.4.2 Non-linear stability and bifurcation theory 9 2.4.3 On stability models in coastal morphodynamics 10 2.4.4 Stability models applied to coastal morphodynamics: a review 12 2.4.4.1 Ripples 12 2.4.4.2 Cuspate features 14 2.4.4.3 Transverse and welded bars 16 2.4.4.4 Crescentic bars 18 2.5 Self-organisation and cellular automata models 21 2.5.1 Basic theory of cellular automata 21 2.5.2 Self-organisation models applied to morphodynamics: a review 22 2.5.2.1 River dynamics 23 2.5.2.2 Eolian ripples 25 2.6 Comparison between stability theory and self-organisation models 26 3. A STABILITY MODEL FOR THE FORMATION OF RHYTHMIC PATTERNS IN THE SURF ZONE 28 3.1 Introduction 28 3.2 Governing equations and stability analysis 29 3.2.1 Governing equations 29 iv 3.2.2 Basic state 31 3.2.3 Linear stability equation 31 3.2.4 Sediment transport parameterisation 35 3.3 Analysis of the instability mechanism 36 3.3.1 Bottom evolution equation 37 3.3.2 Flow over topography (FOT) problem 38 3.3.3 The instability mechanism in a simple case 40 3.4 Numerical simulation 42 3.5 Discussion 52 4. BEACH CUSPS: A COMPARISON OF DATA AND THEORIES FOR THEIR FORMATION 56 4.1 Introduction 56 4.2 Field and laboratory observations 57 4.2.1 Beach cusp occurrence and nature 63 4.2.2 The sediment properties of beach cusps 65 4.2.3 Current patterns 67 4.3 Beach cusp formation theories 68 4.3.1 Instabilities in the breaking waves or in the swash 68 4.3.2 Instability of the littoral drift 69 4.3.3 Rip currents 70 4.3.4 Intersecting wave trains 70 4.3.5 Edge wave theory 71 4.3.6 Theories related to swash dynamics 78 4.4 Compatibility between the edge wave and the self-organisational approach 79 4.5 Discussion 82 5. A SELF-ORGANISATION MODEL FOR THE SWASH ZONE 83 5.1 Introduction 83 5.2 Model description 84 5.2.1 Hydrodynamics 84 5.2.2 Sediment dynamics 85 5.2.2.1 Morphological smoothing 86 5.2.2.2 Angle of repose 87 5.2.3 Boundary conditions 87 5.2.4 Random features of the model 89 5.2.5 Model simulation 89 5.2.6 Physical mechanism 92 5.3 Sensitivity to model parameterisation 92 5.4 Tests of model behaviour 96 5.5 Long-term model behaviour 101 5.6 Results over a non-planar topography 104 5.7 Numerical simulations of edge wave and self-organisation compatibility 108 5.8 Discussion 113 6. COMPARISON BETWEEN MODELS AND FIELD MEASUREMENTS: THE FRACTAL APPROACH 115 6.1 Introduction 115 6.2 Analysis of time series through non-linear techniques 116 6.3 The fractal approach 118 6.4 Data available 121 6.5 Results 125 6.6 Discussion 133 7. CONCLUSIONS AND FUTURE DIRECTIONS 135 REFERENCES 139 VI LIST OF FIGURES CHAPTER 2 Figure 2.1 Sketch of morphodynamic instability CHAPTER 3 Figure 3.1 Sketch of the geometry and co-ordinate system Figure 3.2 Sketch of the bed-surf instability mechanism in the idealised case on a monotonically increasing a/Do fiinction Figure 3.3 The sum of the squares of the cross-shore velocities for the first seven edge wave modes on a linear slope (arbitrary units) Figure 3.4 Instability curves for different values of r, N=0.01, y=0.02, a(0)=0.1 Figure 3.5 Instability curves for different values of y, r=0.5, N=0.01, a(0)=0.1 Figure 3.6 Instability curves for different values of N, r=0.5, Y=0.02, a(0)=0.1 Figure 3.7 Contour lines of the topographic perturbation for k=3.0, r=0.5, N=0.01, y=0.02, a(0)=0.1. For these plots the alongshore direction is on the horizontal axis while the vertical axis indicates the cross-shore direction; darker areas correspond to greater depths Figure 3.8 Topographic perturbation and relative flow pattern for k=3.0, r=0.5, N=0.01,7=0.02, a(0)=O.I Figure 3.9 3d-view of the topographic perturbation (basic slope and perturbation amplitude have been chosen arbitrarily) for k=3.0, r=0.5, N=O.OI, y=0.02, a(0)=0.1 Figure 3.10 Instability curves for different values of N=0.01, r=0.5, y=0.02, a(0)=0.0 Figure 3.11 Contour lines of the topographic perturbation for mode n=2 and k=l (r=0.5, N=0.01,y=0.02, a(0)=0.0) Figure 3.12 Topographic perturbation and relative flow pattern for mode n=2 and k=l (r=0.5, N=0.01, Y=0.02, a(0)=0.0) Figure 3.13 3d-view of the topographic perturbation (basic slope and perturbation amplitude have been chosen arbitrarily) for mode n=2 and k=l (r=0.5, N=0.01,Y=0.02, a(0)=0.0) Figure 3.14 Instability curves for different values of e, r=0.5, N=0.01, y=0.02, a(0)=0.1 Figure 3.15 Instability curves for different values of r, N=0.01, y=0.02, a(x)=constant Figure 3.16 Instability curves for different values of y, r=0.5, N=0.01, a(x)=constant Figure 3.17 Instability curves for different values of N, r=0.5, y=0.02, a(x)=constant Figure 3.18 Contour lines of the topographic perturbation for k=4.0 (r=0.5, N=0.01, vu y=0.02, a(x)=constant) Figure 3.19 Topographic perturbation and relative flow pattern for mode for k=4.0 (r=0.5, N=0.01, Y=0.02, a(x)=constant) Figure 3.20 3d-view of the topographic perturbation (basic slope and perturbation amplitude have been chosen arbitrarily) for k=4.0 (r=0.5, N=0.01, y=0.02, a(x)=constant) Figure 3.21 Contour lines of the topographic perturbation for k=4.0 (r=0.5, N=0,01, 7=0.02, a(0)=0.3) Figure 3.22 Contour lines of the topographic perturbation for k=4.0 (r=0.5, N=0.01, 7=0.02, a(0)=0.5) Figure 3.23 Contour lines of the topographic perturbation for k=4.0 (r=0.5, N=0.01, 7=0.02, a(0)=0.7) CHAPTER 4 Figure 4.1 Planar and profile view of a cuspate system Figure 4.2 Variation of measured cusp spacing and mean diameter Figure 4.3 Swash circulation pattern over a cuspate beach Figure 4.4 Variation of measured cusp spacing and wave height Figure 4.5 Cross-shore variations in the amplitude of edge waves (n = 0-3) Figure 4.6 3D view of and edge wave of mode n = 2 Figure 4.7 Maximum runup and cusp spacing for subharmonic edge waves Figure 4.8 Maximum runup and cusp spacing for synchronous edge waves Figure 4.9 Comparison of measured cusp spacing with subharmonic and synchronous mode zero edge wavelength.