basic wave theory review graham warren bureau of meteorology australia
TRANSCRIPT
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 3Workshop 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 4Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 5Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Properties of Waves
• Wavelength (metres)
• Height H (=2x amplitude) (metres)
• Period T (seconds)
• Phase velocity c = c T
17 June, 2003 6Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Total wave height
• Height of the wind waves = Hw
• Height of Swell waves = Hsw
• Total wave height = (Hw2 + Hsw
2)1/2
17 June, 2003 7Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Dispersion
• Dispersion is the variation of wave speed with wavelength
• Define
• Dispersion relation is– deep water:– shallow water:
/2;/2 kT)tanh(2 kdgk
gk2dgk 22
17 June, 2003 8Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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
kcg /
17 June, 2003 9Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 10Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
The Wave Spectrum
)sin( 01
0 j
n
jj tja
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 11Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Wave Growth
• Basic concepts
• Manual forecasting techniques
• Changing Wind
• Swell Forecasting
17 June, 2003 12Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 13Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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
22
u
gXh
g
u
u
gth
g
uH xtc
)()(2u
gXp
g
u
u
gtp
g
uT xtc
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
hg
uHc
)(pg
uTc
17 June, 2003 16Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 >80km2.8m
5.8sFetch=25km
1.8m 4s
17 June, 2003 18Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 19Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 20Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Wave Heights with Changing Wind 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 21Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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.
Wave Heights with Changing Wind 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 25Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 27Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 28Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 29Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 30Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 31Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Remote SensingWaves from ERS-2 Radar Altimeter
17 June, 2003 32Workshop on Wind Wave and Storm Surge Analysis and Forecasting for 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 33Workshop on Wind Wave and Storm Surge Analysis and Forecasting for Carribean Countries
Finally….
• The accuracy of any wave forecast is dependant on the accuracy of the wind forecast.