1

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

5

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

7

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.

9

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

10

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:

11

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

12

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

13

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.

16

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

17

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

19

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

20

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.

21

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

23

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

24

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

25

„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

26

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.

27

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

29

„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

30

31

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

32

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

33

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

34

Canyon

Island

Ridge

concentration dispersal

of wave energy

isodepth

wave front

wave ray

Refraction through depth change at irregular coasts

35

Erosion

Erosion

... finally leads to rectification of coasts

Refraction at irregular coast lines

Huth

Deposition