01_Warren (1).ppt

Basic Wave Theory Review
Graham Warren
Bureau of Meteorology
Australia

17 June, 2003 2
Why Forecast Waves?
•SOLAS
•Shore
Protection
•Surf
•Oil and gas
exploration

17 June, 2003 3
Workshop on Wind Wave and Storm
Surge Analysis and Forecasting for
Carribean Countries
Wave Characteristics
• Some simple definitions
• Dispersion relation
• Deep water waves
• Wave Spectrum

17 June, 2003 4
Carribean Countries
Definitions
• Wind (or sea) waves - generated by the local
prevailing wind
• Swell waves - the regular longer period waves that
were generated by the winds of distant weather
systems. There may be several sets of swell waves
travelling in different directions, causing a
confused sea state.
• Sea state is the combination of wind waves and
swell.

17 June, 2003 5
Carribean Countries
Properties of Waves
• Wavelength  (metres)
• Height H (=2x amplitude) (metres)
• Period T (seconds)
• Phase velocity c
 = c T

17 June, 2003 6
Carribean Countries
Total wave height
• Height of the wind waves = Hw
• Height of Swell waves = Hsw
• Total wave height = (Hw
2 + Hsw
2)1/2

17 June, 2003 7
Carribean Countries
Dispersion
• Dispersion is the variation of wave speed
with wavelength
• Define
• Dispersion relation is
– deep water:
– shallow water:



 /
2
;
/
2 
 k
T
)
tanh(
2
kd
gk


gk

2

d
gk2
2



17 June, 2003 8
Carribean Countries
Group Velocity
• Phase velocity is the speed at which a particular
phase of the wave propagates
• Group velocity
– Velocity at which a group of waves travel
– Velocity of propagation of wave energy
k
cg 

 /


17 June, 2003 9
Carribean Countries
Deep Water Waves
• Applies when depth of water >  /4
• c2k2=gk
• Phase Velocity : c = g/=gT /2
•  = cT = gT2/2 = 1.56T2 m (T in secs)
• c=1.56T (m/sec)
• Group velocity: cg= gT /4 = c/2 = 0.78 T m/sec
• Thus: Longer waves travel faster

17 June, 2003 10
Carribean Countries
The Wave Spectrum
)
sin( 0
1
0 j
n
j
j t
j
a 


 

 

Fourier Analysis of wave trains:
Variance of the wave record is
obtained by averaging the
squares of the deviations of
each of the wave components
from the mean - gives wave
spectrum (Energy spectrum) Frequency
E

17 June, 2003 11
Carribean Countries
Wave Growth
• Basic concepts
• Manual forecasting techniques
• Changing Wind
• Swell Forecasting

17 June, 2003 12
Carribean Countries
Wave Heights, Wind and Fetch
• Energy from the wind is transferred to
waves
• Waves lose energy
– Whitecapping
– Interaction with sea floor etc
• The greater the wind speed, the higher
the waves
• The longer the duration of the wind, the
higher the waves
• The greater the distance over which the
wind blows (the FETCH) the higher the
waves.
Wave height
depends on a
balance
between
energy in and
energy out

17 June, 2003 13
Carribean Countries
Wind Wave Growth
• Growth usually explained by shear flow
instability
– Airflow sucks at crests and pushes on troughs
• Rate of growth is exponential as it depends
on the existing sea state and wave age
• Empirical formulae have been derived from
large data set
– Curves developed for manual forecasting

14
Characteristic Height and Period of Deep
Water Waves
• Empirical Studies show:
)
(
)
( 2
2
2
u
gX
h
g
u
u
gt
h
g
u
H x
t
c 

)
(
)
( 2
u
gX
p
g
u
u
gt
p
g
u
T x
t
c 

ht, hx, pt and px are dimensionless functions.
They all tend to a limit as the parameter (gt/u or
gX/u2) increases to ~ 105
t = duration of
wind
X=fetch
u = wind speed
g = 9.8m/s2
Duration limited Fetch limited

15
Wave Height and Period
hx()
ht()
px() pt()

= gt/u
or gX/u2
)
(
2

h
g
u
Hc 
)
(
p
g
u
Tc 

17 June, 2003 16
Carribean Countries
Wave Height and Period for General
Conditions
• Need to take the fetch and duration (time for
which the wind is blowing) into account
• Can use the general curves based on non-
dimensional parameters
– simple diagram, “complicated” calculation
• OR use a more complicated set of curves
– Complicated diagram, no calculation
• May need to take into account varying wind
conditions (changes in direction and/or speed)

17
Manual Wave Forecasting Diagram
(Gröen and Dorrestein, 1976
Need fetch
>80km
2.8m
5.8s
Fetch=25km
1.8m 4s

17 June, 2003 18
Carribean Countries
Range of Wave Heights and
Periods
• Wave heights can range from 0 to 2Hc
– The factor of 2 relates to the maximum wave
likely to be observed in a period of a few hours,
not the absolute maximum possible. The value
depends only weakly on the length of time.
• Most waves have periods in the range 0.5Tc
to 1.5Tc
– Important when forecasting swell

17 June, 2003 19
Carribean Countries
Wave Heights with Changing Wind
Conditions
1. Change in wind direction
• If wind direction changes by < 30°, calculate
waves conditions as if no change in direction has
occurred
• If wind direction changes by > 30°, treat existing
waves as swell waves, and start calculation for
new wind direction from scratch.
• As a rule of thumb, swell will decrease in height
by 25% over period of 12 hours

17 June, 2003 20
Carribean Countries
Conditions
2. Increasing wind speed (direction change <30°)
• New wind speed is V2
• Take wave height at time of increase = H1
• Calculate the duration required to achieve H1
given the new wind speed (=T1)
• If the new speed lasts for time T2, calculate wave
conditions assuming duration = T1 + T2 and speed
= V2.

17 June, 2003 21
Carribean Countries
Example of Increasing Wind
• An 8 m/s wind has blown for 6 hours, fetch
100km
• The wind gradually increases to 16m/s over
a 6 hour period.
• Estimate Hc and Tc at the end of the period
– For a quick calculation, when wind speed
increase is gradual from v1 to v2 over a period,
use speed = v2 – (v2-v1)/4 as the speed in the
calculation.

Conditions
3. Slackening wind speed
• When wind drops below speed needed to
maintain height of existing waves*, the
waves turn into swell.
• As a first approximation, swell height may
be reduced by 25% every 12 hours.
* The minimum wind speed that will produce the
existing wave height at the specified fetch

23
Swell Forecasting
• For distant storms, regard the source of the
swell as a point
– For nearby storms the situation is more
complicated
• Questions:
– When will the swell arrive?
– Which wavelengths are involved?
– What is the height of the swell?
Here we develop some simple, first
approximations

24
Swell Length and Arrival Time
• Longest wavelengths travel fastest, so they arrive
first
• Range of periods is T~ 0.5Tc to 1.5Tc
– Other periods exist, but the energy in them is small
–  = 1.56T2 m (T in secs)
• Speed is 1.515 T knots (T in secs)
– Longest waves arrive after time:
Time ~ distance (NM)/(1.5*1.5Tc) hrs
– Shortest waves take 3 times as long to arrive.
•Eg: Tc=6secs, distance = 600 nm, min time = 44 hours
maximum swell length = 126m

17 June, 2003 25
Carribean Countries
Swell Height
• Height of swell depends on
– Height of waves in source region, and extent of source
region
– Speed dispersion (longer waves and shorter waves have
different speeds – don’t arrive together)
– Angular spreading of the waves (height decreases with
distance as wave energy spreads over larger areas)
– Angle between wind direction and direction to storm

26
Angular Spreading of Swell from a Storm
Extent of storm
Wind direction in storm
Swell calculated here
Factor =0.15
Distance to storm/extent of storm
% spreading
factor for
energy
Take square
root for
swell height
Eg: Swell =
0.15 * Hc

17 June, 2003 27
Carribean Countries
Wave Measurements
• Visual observations
• Instruments for measuring waves
– Buoys
– Sub-surface pressure sensors
– Laser
• Remote sensing
– Radar Altimeter
– Synthetic Aperture Radar

17 June, 2003 28
Carribean Countries
Visual Observations
• Guide only as visual observations are not
generally reliable
• Observations of height tend to approximate
to the significant wave height

17 June, 2003 29
Carribean Countries
Instruments
• Wave buoys
– Vertical acceleration measured – can be
converted to wave height
• Wave staff
– Attached to platforms – wave height measured
by change in resistance or capacitance of the
wave staff

17 June, 2003 30
Carribean Countries
Instruments (2)
• Pressure sensors
– Mounted from platforms below surface –
change in pressure is measure of wave height
• Laser
– Attached to platforms – pointing downward

17 June, 2003 31
Carribean Countries
Remote Sensing
Waves from ERS-2 Radar Altimeter

17 June, 2003 32
Carribean Countries
Remote Sensing (2)
• Synthetic Aperture Radar
– Successive radar observations made along
satellite track
– Optical or digital processing produces high
grade imaging of the longer waves
– Wave directional spectrum (with 180o
ambiguity) obtained by analysis of image

17 June, 2003 33
Carribean Countries
Finally….
• The accuracy of any wave forecast is
dependant on the accuracy of the
wind forecast.

01_Warren (1).ppt

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