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Wind Wave and Storm Surge workshop-Halifax 16-20/06/03

WAVES IN SHALLOW WATER Jean-Michel Lefèvre Meteo-France Marine and Oceanography Jean-Michel.Lefevre@meteo.fr. Aknowledgement to Jaak Monbaliu from UKL for having provided some of the material. Wind Wave and Storm Surge workshop-Halifax 16-20/06/03. INTRODUCTION

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Wind Wave and Storm Surge workshop-Halifax 16-20/06/03

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  1. WAVES IN SHALLOW WATERJean-Michel LefèvreMeteo-FranceMarine and OceanographyJean-Michel.Lefevre@meteo.fr Aknowledgement to Jaak Monbaliu from UKL for having provided some of the material Wind Wave and Storm Surge workshop-Halifax 16-20/06/03 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  2. INTRODUCTION Swell propagation is a « quasi » linear process outside the surf zone water particle trajectories in progressive waves of different relative depth (Dean & Dalrymple, 1991; their fig 4.3) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  3. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  4. For practical purposes, one considers that a wave is moving in deep water when its wave length is less than twice the depth and that a wave is moving in shallow water when its wave length is more than 1/20 of the depth. In those two cases the dispersion relations can be approximated (with a error less than a few%) with the following relations : in deep water : in shallow water : =1OOm d<5 m T=/sqrt(gd)=14s T=10s, =156m d>78 m d</20 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  5. So long waves (large period) fell the bottom at larger depth than short waves, and the longer the swell is, the more important the wave transformation is. water particle velocities (Dean & Dalrymple, 1991; their fig 4.1) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  6. water particle trajectory (Dean & Dalrymple, 1991; their fig 4.2) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  7. Particle kinematics progressive waves Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  8. horizontal and vertical velocity Water particles in a wave over shallow water move in an almost closed circular path near the surface. The orbits become progressively flattened with depth as shown on the figure below. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  9. STANDING WAVE Can be written as the superposition of two progressive waves propagating in the opposite directions In shalow water, the wave period T necessary to produce a seiche in a closed basin of dimension L is T=2xL/sqrt(gd) or with wave lenght of two times L. For an open basin on one edge, T=4xL/sqrt(gd) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  10. distribution of water particle velocities in a standing wave (Dean & Dalrymple, 1991; their fig 4.6) There is no crest propagation. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  11. particle displacement in a standing wave mean position (x1,z1) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  12. particle velocities in a standing wave Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  13. Engineering wave properties Starting from a flat surface, it is the energy necessary to move the water in order to get a wavy surface, because of the gravity force. potential energy waves (Dean & Dalrymple, 1991; their fig 4.11) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  14. potential energy Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  15. kinetic energy mean energy, over one wave lenght or one wave period, due to the orbital velocities of the water particles Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  16. total energy Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  17. wave power power transmitted to a vertical surface of 1m width from the surface to the bottom. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  18. energy flux work done by the dynamic pressure averaged over one wave period Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  19. hydrostatic dynamic pressure response factor Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  20. group velocity Cg group velocity Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  21. wave group characteristics (Dean & Dalrymple, 1991; their fig 4.12) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  22. integrate up to mean water surface  neglect 2-nd order terms in wave height Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  23. waves entering shallow water Conservation of waves scalar phase function define temporal variation in the wave number vector is balanced by spatial changes in the wave angular frequency Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  24. wave number vector (Dean & Dalrymple, 1991; their fig 4.14) conservation of waves (Dean & Dalrymple, 1991; their fig 4.15) bottom beach Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  25. refraction no alongshore variations : (Snell’s law) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  26. Snell’s law states that the component of the wave number vector parallel to the isobath is conserved along trajectories. Also since =kc, and because  is conserved along trajectories the Snell’s law reads: The component of the phase speed parallel to the isobath is also conserved along trajectories Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  27. In shallow water this expression simplifies to: Let us consider the case of a wave propagating perpendicular to the isobaths. From Snell’s law we have: Initially and the direction of wave propagation is not modified. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  28. wave rays idealized bathymetry (Dean & Dalrymple, 1991; their fig 4.17) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  29. Conservation of energy in case or regular isobaths refraction coeff. shoaling coeff. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  30. REFRACTION g b c a h d e f Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  31. These equations can be numerically integrated in order to determine ray paths. Manual methods have also been developed. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  32. h=8m T=10s 0=40° h/gT2 =0.0082 =20° KR=0.905 refraction parallel contours (Dean & Dalrymple, 1991; their fig 4.18) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  33. refraction parallel contours (Dean & Dalrymple, 1991; their fig 4.19) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  34. calculated rays beach with rip channels (Dean & Dalrymple, 1991; their fig 4.21) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  35. Spectral approach •  Numerical spectral wave model can deals with depth refraction • Modern wave prediction is based on the solution of the so called energy balance equation. This equation is derived from the principle of the conservation of wave action.  In this theory, it is assumed that: • Amplitudes, frequencies and wave lengths of individual waves vary slowly with respect to the intrinsic space scale (the wave length) and time scale (the wave period), and that linear theory is valid. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  36. This means that in regions where characteristics of waves are varying rapidly, the theory presented here is not valid and others formulations are required. In the surf zone where waves are breaking, one cannot apply this theory, in « theory ».However the breaking effect can be taken into account in spectral models. • The balance equation for the action density spectrum N reads: • (1) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  37. Where: =( ) denotes the propagation velocity of the wave group in the four dimensional phase space of (spatial coordinates) and (wave number vector), isthe density action, F is the energy spectrum,  is the intrinsic angular frequency, S is the source term. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  38. The velocity of a wave group is given by the propagation equations: Where: isthe absolute frequency, (in the fixe frame) W denotes the dispersion relation,U(x,t) is the current velocity Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  39. The intrinsic frequency  depends on k(x,t) and the water depth d(x,t) through the dispersion relation: which is the wave frequency in a frame moving with the current Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  40. Spectral approach Equation (1) holds in any coordinate system (x,l,t) where x denotes spatial co-ordinates (spherical or Cartesian) and l denotes the two independent spectral variables (chosen from, k, , , and ). In Cartesian coordinates, without current and source terms the energy balance equation reduces to the energy spectrum propagation equation: Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  41. where, denotes the group velocity and , denotes the rate of change of direction of a wave component following the path of the wave component. The second term of the equation: represents the change in the energy spectrum due to the divergence of energy density fluxand is termed the shoaling contribution. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  42. The last term: describes the redistribution of spectral energy density over the spectrum. It corresponds to a change of direction of a spectral component and is termed the angular refraction. In the general case of a two dimensional bathymetry, the distinction between shoaling and angular refraction is not useful for quantitative studies since most of the time it is not possible to separate the two processes: The consideration of simple cases can help us understanding these processes. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  43. Shoaling In the simple situation of waves initially propagating perpendicular (for instance in the x direction) to parallel isobaths we have seen that the direction of the rays remains constant Equation the propagation equation becomes: The energy flux normal to shore remains constant. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  44. As waves propagate from deep water into shallow water, their velocity initially increases slightly and then decreases. Therefore, energy increasesafter a slight decrease. Energy growth is however limited by wave breaking. In the surf zone, the linear approximation is not longer valid and other approximate equations are needed. While, wave length decreases, wavesare becoming more and more steep until they break. What happens to the absolute frequency (or period) in the stationary case?. We have: Stationarity implies: Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  45. and since we have: So the absolute absoluteperiod remains constant along the path. In terms of energy E instead of spectral energy density F(k), the balance equation in stationary conditions reads: Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  46. This means that the energy fluxremains constant between two ray paths. If denotes the length of a crest (section) between two ray paths, then: Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  47. In the situation where waves approach a straight beach (with straight isobaths), this is equivalent to: where  is the local angle of incidence of the waves. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  48. h (mètres) k (m-1)  (mètres) H/H0 100 50 25 15 10 5 2 0,040 0,041 0,046 0,055 0,065 0,090 0,142 157 153 137 114 97 70 44 1 0,95 0,91 0,94 1 1,12 1,36 APPLICATION For a wave of 10 s period, and k0 = 0.04 m-1 with a height of H0 off shore, one can compute the wave transformation, λ et H/H0 at several water depths: H/H0=(1+2kh/sinh 2kh) k~ k0/(tanh k0 h) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  49. CURRENT REFRACTION Strong currents are normally present in shallow water areas. Modelling waves in such places often requires taking into account wave-current interactions. This can be done with the use of a coupled wave-current-surge model but this is beyond the scope of this workshop. We will only discuss the effect of slowly varying and limited current speed on waves. Wave kinematics along rays is described through the following relations: Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

  50. For A steady medium with a constant depth these equations simplify to: Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

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