Waves in the Ocean
Finally we deal with a phenomenon that we see all the time! Waves little waves and big waves Waves that rock you to sleep, waves that make you sick and waves that can kill you.
NOAA Photo Library
Period
Celerity = C = Wavelength/Period = T
2 T
C =
Wave Frequency
Wave Number
Wave Celerity
! s e v a
w ll a s e v a
W f o
s ei t r e p o r P
ci s a B
Note that these definitions work also for radio waves, gamma rays and light waves. In this case the interface is between the ocean and the atmosphere or between layers of different densities
or ripples
or seas
Types of Waves in the Ocean
Name Ripples Sea Swell Internal Planetary and Topographic Tsunamis Tides Typical Periods < 0.2 s 0.2 - 9 s 9 - 30 s min to several hrs Wave lengths 10-2 m 10s of m 100s of m 1 - 300 m Forcing Mechanism wind on sea surface " " current shear on stratification
hours to days 15 min to 1 hr several hrs
100-1000s km few 100s km 100s -1000s km
bathymetry/atmospheric pressure seismic, landslide, meteorite impact gravitational (moon and sun mainly)
Waves preserve their period The wavelength is a function of the period The celerity is a function of the wavelength Long waves travel faster than short waves
C=
Wave Period
seconds (From Carl Davies, Southampton Univ)
Small Amplitude Waves H < 1 20 In reality, wave steepness of small amplitude waves < 1 / 50
Period T
Celerity = C = Wavelength/Period = T
= A cos
2 t = A cos[ x t ] T
u = g t x
momentum balance
1 u = H t x
= A cos[ x t ]
2 C = g tanh h 2
1 2
g = tanh( h )
1 2
Obtained from equations of motion assuming the wave is progressive, incompressible and irrotational
cosh (20) = sinh(20) = 2.4 x 108
g C = tanh( h )
1 2
Deep water wave: < 2h ( h is depth ) or Short wave h h This means that is small and h is large
tanh( h ) 1
g C=
1 2
= g
g CT T = = g =g 2 2 2
Shallow water wave: > 20h ( h is depth ) or Long wave h This means that is large and h is small tanh( h ) h
g C = ( h )
1 2
= gh
Shallow Water Waves
Intermediate Waves Deep Water Waves
C = gh
g C = = tanh( h )
1 2
C = gT 2
2 = g tanh( h )
Dispersion relation
(relation between frequency and wave number)
Deep waves are dispersive Shallow waves are non-dispersive
Dispersive
C = g 2
1/ 2
= gT
Non-dispersive
C = gh
Waves in the area of the fetch Group Velocity Cg = Velocity of the wave envelope
Cg = C 2 h 1+ = 2 sinh(2 h )
In deep waters: Cg = 0.5 C In shallow waters: Cg = C
(green wave moving to the right; blue wave to the left)
Group Velocity = Velocity of the wave envelope
Rogue Wave generated by Constructive Wave Interference More information on Rogue Waves
Motion of the Wave Form
u=
2z
T 2 z H w= e sin(x t ) T
cos(x t )
Circular Motion
H = e sin(x t ) 2 2z H = e cos(x t ) 2
Short Deep Water Waves wind waves and swell offshore
2z
u=
Elliptical Motion
H g cos(x t ) 2 h
w=
z 1 + sin(x t ) T h
= =
Long Shallow Water Waves Tides, Tsunamis, Seiches, waves in shallow water
TH g sin(x t ) 4 h
H z 1 + cos(x t ) 2 h
What about INTERMEDIATE WATER WAVES?
Motion of water particles beneath waves -energy travels across the surface, not water particles
Orbits of water particles are not quite closed -- net displacement = STOKES DRIFT
Four factors controlling height and shape of wind waves
Wind Speed: Proportional Wind duration: e.g. synoptic vs. sea-breeze Fetch: Distance over which the wind blows. Original Sea State: begin from rough or smooth surface
Increased Wind Speed causes waves with increased height, wavelength and period
Fully developed sea -waves can no longer grow
Average and Significant Wave Height 25 20 15 10 5 0 20 30 40 50 60 70 80 90 Wind Speed
Ave. Hgt Sig. Hgt.
Wave Height
Significant Wave Height = average of highest 1/3 of the waves.
Small amplitude waves: Steepness < 1/20 (1/50 in real ocean) = A cos(x - t); A=H/2, = 2/, =2/T
deep waves (short) < 2h Phase speed or wave celerity C wavelength intermediate waves 2h < < 20h 1 g tanh h 2 2 = g tanh( h ) C= ( ) shallow waves (long) > 20h
C=
gT g g = = 2
C = [g h ]
1 2
gT 2 = 2
gT 2 tanh( h ) = 2
Cg = 2 h C 1+ 2 sinh(2 h )
u=
= CT
Cg = C ( h << 1)
H g cos(x t ) 2 h
Group velocity Cg
Cg = 0.5 C ( h >> 1)
u=
Wave particle horizontal velocity u vertical velocity w Horizontal displacements Vertical displacements Subsurface pressure
H
T
2z
cos(x t )
u=
H gT cosh(2 [h + z ] ) cos(x t ) 2 cosh(2h )
H gT sinh(2 [h + z ] ) sin(x t ) 2 cosh(2h ) H cosh(2 [h + z ] ) sin(x t ) 2 cosh(2h )
w=
H
T
2z
sin(x t )
w=
w=
z 1 + sin(x t ) T h TH g sin(x t ) 4 h
H = e 2
2 z
sin(x t ) cos(x t )
2z
= =
H e 2
2 z
H sinh(2 [h + z ] ) cos(x t ) 2 cosh(2h ) cosh(2 [h + z ] ) gz cosh(2h )
H z 1 + cos(x t ) 2 h
p = g ( z )
p = g e
gz
p = g
In shallow waters, waves refract, diffract, reflect and/or break Refraction Change of wave celerity (bending of rays) due to changes in bathymetry
Diffraction Change of wave celerity (bending of rays) due to the presence of an obstacle
Reflection
Waves break when steepness (H / ) < 1 / 7 or H / d ~ 0.8
Breaking
most common
Most desired by surfers
Progressive vs. Standing Waves Wind -generated waves are progressive
maximum motion associated with crests and troughs
= A cos( x t )
C u = A cos( x t ) H H
u =
0 7 - . 1 6 8
5 .
0 5 .
0 2 .
. 2
. 2
. 2
. 2
. 2
. 2
C
0 5 - . 1 -
time
This indicates that the flow is in phase with the elevation u
1 -
5 .
5 .
a C/H
0 7 - . 1 6 8
0 5 .
0 2 .
. 2
. 2
. 2
. 2
. 2
. 2
time
0 5 - .
1 -
1 -
5 .
= A cos x sin t
C u = A sin x cos t H
- . 0 7 1 6 8
. 5
0 . 5
0 . 2
. 2
. 2
. 2
. 2
. 2
. 2
This indicates that the flow is out of phase with the elevation by 90 degrees
1
time
- . 0 5
. 5
. 5
0. 5
u
0 - . 07 1 6 8 0. 2 8 3 2 1 . 2 8 3 2 2 . 2 8 3 2 3 . 2 8 3 2 4 . 2 8 3 2 5 . 2 8 3 2 6 . 2 8 3 2
a C/H
time
- . 05 1
. 5
Internal Waves
T = min to several hrs Wave length = 1 - 300 m Current shear on stratification
Convergences at troughs
They propagate roughly like a shallow water wave but the gravity restoring force is reduced by the difference in density.
2 1 d C = g 2 2 1 = 1 C = 10 0.001 100 = 1 m/s
Internal Waves in a closed basin
How about C of a surface (wind-induced) wave?
Standing Waves
Half-wave oscillator
Quarter-wave oscillator
Natural standing wave (lake, harbor, estuary) ---- seiche
Seiches
First Order Seiche Second Order Seiche
Example in the Balearic Islands
And 5 minutes later
Tsunamis
Created by earthquakes, underwater landslides, meteorites -cause a series of waves or wave train Harbor wave definitely not a tidal wave T = 15 min to 1 hour Wave length = 100s of km Speed of shallow water surface wave
C = g d = 10 4000 = 200m / s
Tsunami in Papua New Guinea 1960 Chile Tsunami
Tsunami in the Indian Ocean, Dec 26 2004
Countries affected
2004 Tsunami in the Indian Ocean
http://en.wikipedia.org/wiki/2004_Indian_Ocean_earthquake http://www.digitalglobe.com/tsunami_gallery.html
PLANETARY WAVES
POINCARE WAVES KELVIN WAVES ROSSBY WAVES Coriolis accelerations become important
Poincar Waves
= A cos( x t )
1 2
s=f
g u= A cos( x t ) 2 h (1 s )
g v = s A sin( x t ) y 2 h (1 s )
1 2
s 1 >f
gh C= 2 1 s
Cg = (1 s
1 2
) g h]
1 2
Cg < C dispersive waves
Poincar Wave >f Superinertial
= Ae
f y
cos( x t )
Kelvin Wave
g 2 f y C u = A e cos( x t ) h
v =0
C / f = Ro Rossby Radius
Ro
Needs a boundary or a guide on the right (northern hemisphere)
ROSSBY WAVE
Produced by variations in f -- Beta effect
+f
h
= const
