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Wind waves and swell
Gravity waves excited by the wind.
2
Capillary waves with
wave lengths smaller
than 17 mm.
Very fast dissipation.
Capillary waves
Ripples
Energ
y
Period
105 s 102 s104 s 10-1 s103 s
(28 h)
1 s10 s 10-2 s
(17 min)
Wind wavesSwellTsunamisTides
WindSun, moon Sea
quakes
gravity surface tension
(Frequency)
Restoring:
Forcing:
Wave
type:
Energy distribution of ocean waves
Latitudinal change
ordepthchange
Rossby
Kelvin waves
Coriolis
3
Forcing of wind waves
Friction &
pressure difference
Seung Joon Yang
Open University, 2008
4
Wind waves are determined by:
• the wind speed
• the duration of time the wind has been blowing
• the fetch (the distance over which the wind excites the waves)
Additional impact:
• Waves imported from remote wind regions.
• Interaction between waves
Typical periods: 1- O(100) s
Swell: Waves that have propagated away from the
region where they have been generated.
Often ending breaking at very calm coasts.
Wind waves:
Generated at the very same
location by the wind
Wind waves are typically „deep water waves“.
Waves generated by wind
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Waves are generated at wind speeds W exceeding W=0.7 m/s.
Beyond 0.9 m/s the steepness (h/l) increases with wind speed.
Effect of wind speed
Neumann, 1949
Gravity waves
Capillary waves
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Development of sea state with time
Start with short steep waves,
Growing of waves until most have c = 0.3*W (W= wind speed).
Steepness increases.
Steepness within a wave field decreases with age.
Fully developed sea state at constant wind:
The wave have developed towards an equilibrium between
Critical value for steepness s=h/l: 1/7,
or a= 120° for the cusp angle.
Finally long flat waves survive with c 1.4*W - swell which leaves the wind field behind.
Breaking of steep short waves; damping of short waves,
• wave generation • wave destruction by dissipation (or breaking) and
• waves leaving the fetch area.
Neither fetch nor duration are the limit for the wave field development.
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Wave energy of fully developed sea state as function of frequency
at different wind speeds.
With growing wind speed
• energy density (i.e. wave height) increases
• period of dominant wave gets longer (gray line)
• spectrum gets narrower
Moskowitz (1964)
Spectrum of fully developed sea state
E ∼ h2
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For some applied problems (shipping, design of oil platforms etc) information of the
total energy is requested:
Hmax = 1.6 H1/3
Characteristic number
„significant wave height“ H1/3 , mean height of the highest third.
Characteristic numbers of wave fields for practical purposes
Wave profile as a function of distance.
Neumann and Pierson,
1966
2
3/12
1HE
Or information on maximum wave height:
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Accidental superposition of waves at
a very special phase combination
can cause Rogue waves.
Observed up to 35 m height in
Atlantic.
Irregular superposition - sometimes also phase matters: Rogue Waves
Wave profile as a function of distance.
Neumann and Pierson,
1966
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Estimated 50-year maxima in the North Sea:
wave height (m)
and associated
wave period (s)
Maximum wave heights in the North Sea
OSPAR-Nordseereport
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Waves with residual transport: Stokes drift
z
Aer l
2
l
Simpson and Sharples, 2012Particle orbit:
Return flow - at some depth – slightly
smaller than flow in wave direction
return flow has smaller velocity
orbit is unclosed, net foreward motion
z
particle 0u U e sin( x t)
z
2 2 2 z
St 0u (z) U c e Stokes drift
at depth z
2
St 0
cU U
2
Transport (per
wave width) due
to Stokes drift
Natural waves might not be linear.
14
Non-linear waves – Stokes drift
Deep water waves
Shallow water waves
Current position of particle
Position of particle after
each period.
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2. With distance from wind field
because of radial spreading
Open University courses
b2n/l2
Wave damping b depends on wave
length l:
1. With time because of damping
through friction:
Decrease of wave energy density
Wind field
Small waves are damped faster
than long waves.
n: viscosity
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Geometric decrease of energy areal density
Length of wave front increases,
but total energy remains constant
(except decrease due to dissipation).
Decrease of energy flux per width
proportional to the distance
r2r
Spherical spreading loss Cylindrical spreading loss
18
12
l
HHdeep water waves
l
2
gc Phase velocity
„Deep-water“ waves:
Longer waves propagate faster than shorter waves.
Deep-water waves are dispersive.
tanh( H) 1
Dispersion relation of „deep-water“ waves
)tanh( Hg
c
l
2
Dispersion relation for gravity waves at
water depth H
2
g2
l
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l
2
gc
Propagation of swell over great
distances along great circles
Dispersion of wind waves - separation between short wind waves and longer swell
Dietrich, 1975
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Wavecrest buoy / wave rider buoy
Summerhayes and Thorpe, 1996
Mind the „Aliasing“!
Measuring wind waves or swell
http://www.bafg.de/DE/08_Ref/M1/04
_Aktuelles/Archiv/seegangsmessung
_radar_bfg
teil.pdf?__blob=publicationFile
The buoy follows the sea surface and
sensors measure the components of
acceleration.
The buoy is usually moored to the seabed
using a compliant tether.
Integrated twice, the acceleration data give
wave height variation. Twofold integration
emphasizes low frequencies.
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Since 1975
Radar principle: frequency-modulated pulse in
narrow angle
Carrier frequency ca. 14 GHz
Reflected signal recorded
• travel time distance to ocean surface
• Increase of echo curve measure of
wave height
- Genaue Kenntnis des Geoids notwendig (für große Skalen mittlerweile im mm-Bereich bekannt)
Measurement of wave height as side effect of satellite remote altimetry
22http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16
Distribution of the signal reflected by the sea surface
between wave crest and wave trough
Measuring wave height by satellite altimeter
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Santander, Spain, February 2014
Wave power plants at the coast?
Total power per width of wave front (W/m) h: wave height (m)
: wave period (s)
=1
32 ∙ 𝜋∙ 𝜌𝑔2ℎ2𝜏
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Power of wind waves and swell
(W/m)
Energy flux across a line of unit length; e.g. a shore line
𝑃𝑊 =1
32 ∙ 𝜋∙ 𝜌𝑔2ℎ2𝜏
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„Change“ from deep-water wave to shallow-water wave –
or
How does a short wave become a long wave
while even getting shorter?
• Orbit shape changes from circles to
ellipses.
• Phase speed c changes from
to
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Water depth H decreases.
l
2
gc gHc
l
Eventually the deep-water waves approach the coast. Particulary long swell arrives.
„Change“ from deep-water wave to shallow-water wave
z
erzr l
2
0)(
When H < l/2, „deep-water“ waves become „shallow-water“ waves.
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l
2
gc gHc
„Deep water waves “ „Shallow water waves “
Wind waves Swell
5 s 15 s
l 39 m 350 m
c 7.8 m/s 23 m/s
Water
depth H
18 m 5,5 m 4000 m
Tsunami
15 s > 15 s 17 min
l 200 m > 110 m 200 km
c 13 m/s 7.3 m/s 200 m/s
Dispersive:
Long waves move faster than
short ones.
Non-dispersive:
Phase velocity independent of
wave length.
Same wave, different water depth
gHl 2g
2l
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„Short / deep-water waves“ „Long / shallow-water waves“
wave length smaller =
wave crests get closer to each other
height increases (to meet the energy
budget)
steepness increases
waves become unstable and break
Wind
waves
swell
5 s 15 s
l 39 m 350 m
c 7.8 m/s 23 m/s
Water
depth H18 m 5,5 m
15 s 15 s
l 200 m 110 m
c 13 m/s 7.3 m/s
Transformation of waves approaching the shore
Total energy of the wave group propagating at the respective group velocity!
EH1*cH1 =EH2*cH2
𝑐 = 𝑐𝑔𝑟 = 𝑔𝐻
𝜆 = 𝜏 ∙ 𝑔𝐻
E1*c1= E4*c4Energy flux is constant c is decreasing
E must increase
h2 must increase
Transformation of shallow-water waves when approaching the shore
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Orientation of wave fronts
parallel to the beach
Refraction when approaching the shore
… because of refraction:
A change in phase speed leads
to a change on the direction of
wave propagation
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air
water
csin
sin c
a
b
phase
speed
higher
lower
Air air
water
Snellius law: refraction in optics
… refraction:
A change in phase speed leads
to a change on the direction of
wave propagation
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Transition to a medium with a different
phase speed results in change of wave
direction.
shw shw
dp dp
sin c
sin c
gHc
Orientation of wave fronts parallel to the beach
Phase speed
smaller
Refraction when approaching the shore
larger
Water depth Hi
decreasing towards
coast
Wave ray
Wave crest
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Canyon
Island
Ridge
concentration dispersal
of wave energy
isodepth
wave front
wave ray
Refraction through depth change at irregular coasts
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Erosion
Erosion
... finally leads to rectification of coasts
Refraction at irregular coast lines
Huth
Deposition