WAVES IN SHALLOW WATER Jean-Michel Lefèvre Meteo-France Marine and Oceanography Jean-Michel.Lefevre@meteo.fr Aknowledgement to Jaak Monbaliu.

WAVES IN SHALLOW WATER Jean-Michel Lefèvre Meteo-France Marine and Oceanography 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

Swell propagation is a « quasi » linear process outside the surf zone
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

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

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

water particle velocities (Dean & Dalrymple, 1991; their fig 4.1)
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

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

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

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

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

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

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

particle velocities in a standing wave

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

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

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

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

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

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

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

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

wave group characteristics (Dean & Dalrymple, 1991; their fig 4.12)

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

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

(Dean & Dalrymple, 1991; their fig 4.14) conservation of waves
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

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

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

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

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

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

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

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

refraction parallel contours (Dean & Dalrymple, 1991; their fig 4.18)
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

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

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

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

Where: =( ) denotes the propagation velocity of the wave group in the four dimensional phase space of (spatial coordinates) and (wave number vector), is the 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

The velocity of a wave group is given by the propagation equations: Where: is the 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

which is the wave frequency in a frame moving with the current
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

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

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 flux and is termed the shoaling contribution. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

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

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

As waves propagate from deep water into shallow water, their velocity initially increases slightly and then decreases. Therefore, energy increases after 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, waves are 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

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

This means that the energy flux remains 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

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

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) 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 1,12 1,36 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

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

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

The absolute frequency  (which is wave frequency in a fixed frame) remains constant along ray paths in a steady medium, not the intrinsinc frequency  (which is wave frequency in a frame moving with the current) unless the current is uniform. Consider the example of a long shore current decreasing towards the shore and waves initially propagating in the direction of the current. Equations implies that waves will turn towards the shore, similarly to what would happen without current but with depth refraction. Another simple example is a wave propagating in the direction of a current which is variable in the direction of propagation, in a steady medium. Since the absolute frequency remains constant along the ray path, we have: =const. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

where 0 is wave frequency at the location where there is no current. According to the dispersion relation in deep water, one can show that the above relation reduces to the quadratic equation in , where c denotes the phase velocity (c=/k) namely; which has one realistic solution if is positive. The critical situation appears when this term is equal to zero. That is if the current is equal and opposite to the local group velocity, but waves break shortly before this kinematic limits is reached. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

WAVE GROWTH INTO SHALLOW WATER In the theory of the equilibrium in the high frequency range of the wind wave k-spectrum, it was found according to dimensional arguments that its shape is universal and is k-3. Since in deep water the dispersion relation reduces to: 2=gk, we have k-3 dk=2-5 /g d The frequency spectrum F() which is related to the wave number spectrum F(k) through the relation: F(k)dk=F()d, has therefore a -5 tail. A new universal spectrum similar to the JOWSWAP spectrum and called the TMA spectrum was proposed (Bows et al. 1985): Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

FTMA(,d) = r(,d) FJONSWAP () where r(,d) is an empirically determined function The growth is limited by the depth in shallow water. Formula have been derived in order to compute the wave height as a function of depth, wind speed and fetch. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

In the case of an idealized flat bottom, a formulation for the maximum significant wave height (Hm) have been established by CERC (1973) based on various measurements. Once simplified, the formulation becomes for shallow water: d (m) U10(m/s) Hm(m) 1 10 0.3 5 20 1.5 30 2.7 50 6 0.5 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

BOTTOM DISSIPATION In shallow water, the ocean bottom is responsible for energy dissipation by: - bottom friction - percolation - wave induced bottom motion Other wave bottom interactions are also important in certain circumstances such as: - bottom scattering - bottom elasticity. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

The first process is conservative, the second is not. Outside the surf zone, bottom friction is usually the most relevant process. A version commonly used for the bottom friction source term comes from the JONSWAP experiment (Hasselmann et al. 1973): where  is an empirically determined coefficient. Several values have been proposed The typical value for  is 0.05 m2s-3. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

For the above relation reduces to : The typical time  scale for decreasing the energy by a factor 2.7 and therefore the wave height by a factor 1.65 is 200 times the depth d. During this time, the wave has travelling over a distance Examples: d (m)  (hours)  (Km) 10 0.5 17. 20 1. 50. 40 2. 150. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Depending on the nature of bottom sediment, percolation can contribute significantly or even dominates bottom dissipation (in cases of porous bottom) and the corresponding source term can be parametrized as: where a is also an empirically determined coefficient (a=0.01 m/s for coarse sand with a 1mm diameter and a= m/s for fine sand with a .25 mm diameter). For the above relation reduces to : Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

The typical time  scale for decreasing the energy by a factor 2.7 and therefore the wave height by a factor 1.65 depends on the size of the sand and on the wave lenght : for coarse sand: for fine sand:  (m)  (hours) 10 1. 100 10.  (m)  (hours) 10 0.05 100 0.5 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

WAVE BREAKING In deep water wave breaking is controlled by wave steepness. A theoretical upper limit for wave steepness is 0.14 or 1/7. In the surf zone, because of bottom friction, the upper part of the wave tends to increase its speed relative to the lower part. At some time the crest overtakes the preceding trough and the wave breaks. Several kinds of breaking can occur, depending on the steepness of the beach and on the steepness of the waves before entering shallow water areas; Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Spilling breaking The water from the crest falls along the forward face of the wave. Plunging breaking The water from the crest rapidly overtakes the wave base and plunges foward into the wave trough. Surging breaking The base of the wave surges up the beach before the crest can plunge forward Collapsing breaking This is between plunging and surging breaking Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

types of wave breaking (Dean & Dalrymple, 1991; their fig 4.22)

Wave Breaking Spilling when S<0.4, le plus fréquent : H/d=3/5 (gently and regular slopes) Plunging when 0.4<S<2 H/d>3/5 (steep slopes, obtacles) : beach slope Surging when S>2 le plus rare: pentes très fortes H0/ 0: steepness in deep water Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Wave breaking where possibly  depends on a number of parameters (e.g. Weggel, 1972) To make things simples, =0.6=3/5 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Steepness=kH off shore wave breaking dimensionless parameters
Breaking depth Steepness=kH off shore wave breaking dimensionless parameters (Dean & Dalrymple, 1991; their fig 4.23) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

The evolution of the waves in the surf zone is strongly nonlinear. However, it has been shown (Battjes and Janssen 1978), that the source term for wave breaking in the spectral energy balance can be parametrized with where is the mean frequency, Q is the fraction of breaking waves determined with Hm is the maximum possible wave height : Etot is the total energy. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

DIFFRACTION When waves propagate towards an obstacle, waves are diffracted behind the obstacle, towards the sheltered area so that the energy gradient does not become infinite. Diffraction cannot be described in the framework of the linear wave theory we presented because rapid changes in the wave characteristics occur when diffraction arises. This phenomenon is essentially relevant for coastal engineering applications. However, the cumulative effects of refraction in areas with irregular bathymetry may also cause locally large variations so that diffraction may arise, preventing energy from increasing unrealistically Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

diffraction (Dean & Dalrymple, 1991; their fig 4.24)

diffraction Laplace equation becomes Helmholtz equation Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

diffraction pattern - waves normal to breakwater
full line is an exact solution; dashed line is an approximate solution (Dean & Dalrymple, 1991; their fig 4.26) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

diffraction pattern behind a breakwater for normal incident waves
(Dean & Dalrymple, 1991; their fig 4.29) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Wave set up Variations of the wave energy (significant wave height) due to shallow water transformation (refraction, shoaling, breaking) cause a horizontal variation of the wave momentum  Gradients of the mean sea level and wave-driven currents Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Incident wave propagating towards de beach (x direction)
: mean surface elevation above still water -h: water depth Mxx=(2-0.5)E shoreward transport =0.5+kh/sinh(2kh) In deep water =0.5 and Mxx=E/2 In shalow water =1 and Mxx=E+E/2 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

The wave set up is about 15-20% Hrms (about Hs/1.4) Replace H by Hrms, not by H1/3, when applyng a formula with a single sinusoidal wave of height H to the actual sea state (also for wave power, wave energy…) Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Currents generated by breaking waves Long shore currents When waves approach the coastline at an oblique angle, a longshore current is established which flows parallel to the shoreline in the nearshore region with speed between 0.5 and 2 nm. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

Rip currents Rip current are strong, narrow currents with speeds up to 4 nm which flow seawards from the surf zone. Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

1. Seiche (an application from Steeve W. Lyons)
L=1m, d=0.5, assume shalow water (!) to make it simple Compute resonant wave period and fish speed to create such wave, so the fish on the left can get out of the tank where his fish mate is located Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

2.Wave power Compute the power of a wave of 20 m height, 100m crest lenght, and 20 second period. Same with respectively 2m, 100m, 7s And 2m, 100m, 14 s Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

3. Wave lenght computation
T=14 s Compute  for depth of 5, 3, 2 m Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

4. Tsunami 2m height (H), 200 km wave lenght, 5 km water depth (d) Is it deep or shalow water! Compute wave height just before breaking, depth where breaking starts (assuming Hb=0.6d and no angular refraction ). Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

5 Shoaling Compute wave height just before breaking, depth where breaking starts, and no angular refraction (assuming Hb=0.6d and no angular refraction ) for incoming deep water swell of: 5 m height (H), 7s wave period 5 m height (H), 14s wave period S1 S2 crête P1.s1 = P2.s2 E1.c1. s1 = E2.c2. S2 Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

6. Bottom refraction 1=45° d1=5m d2=2.5m 2=? Wind Wave and Storm Surge Workshop Halifax 16-20/06/03

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

7. Dissipation due to bottom friction
d (m)  (hours)  (Km) 10 20 40 . 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.

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WAVES IN SHALLOW WATER Jean-Michel Lefèvre Meteo-France Marine and Oceanography Jean-Michel.Lefevre@meteo.fr Aknowledgement to Jaak Monbaliu.

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Presentation on theme: "WAVES IN SHALLOW WATER Jean-Michel Lefèvre Meteo-France Marine and Oceanography Jean-Michel.Lefevre@meteo.fr Aknowledgement to Jaak Monbaliu."— Presentation transcript:

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