6 ( ) SPEED OF WAVES IN WATER 2p c = gl tanh 2pd 2p l 12()g = gravitational constantc = gl tanh 2pd2p ll = wavelengthd = water depthIn deep water, d / l gets very large, sotanh (big #) , therefore,c = gl2p12
7 SPEED OF WAVES IN DEEP WATER Since l = cTc = gT2pl = gT2Therefore, speed (c) and wavelength (l)are only a function of period (T).Since the speed is a function of the period,deep water is a “dispersive” medium.
8 BASIC DEEP WATER WAVE FORMULAE c (m/s) = 1.56 T (sec)c (kts) = 3.03 T (sec)l (m) = 1.56 T (sec)l (ft) = 5.12 T (sec)S = H = H (m)l T (sec)c = gT2p222
9 COMPOSITION OF WAVESAny observed wave pattern on the ocean can be shown to be comprised of a number of simple waves, which can differ from each other in height, wavelength and direction.The above profile is the result of two waves differing in l only.
10 COMPOSITION OF WAVESIn reality, the sea is a superposition of many wave sets.
12 WIND-WAVES...depend on: Wind speed - the speed of the wind Fetch - the area of sea surface over which a wind of constant direction (within 30o) and steady speed is, or has been blowingDuration - the length of time the wind persists from a certain fetch
13 SIGNIFICANT WAVE HEIGHTS Hsig a V2 tanh [a (F/V2)a]V = wind speedF = fetch length“Law of Diminishing Returns”is at work
14 FULLY DEVELOPED SEA SEA CHANGING TO SWELL RIPPLESCHOPWIND WAVESWINDFETCH of the wind
15 THE WAVE SPECTRUMENERGYFor a simple sine wave, the energy is proportional to the square of the wave height However, the real sea is a combination of many sinewaves of varying l and T The simplest way of determining the “energy” of the sea is to examine the relative amounts of energy contained within different period ranges in the sea surface. The plot here of energy vs. frequency (1/T) is a typical energy (or wave) spectrum.PERIOD
16 WAVE SPECTRA FOR DIFFERENT WIND SPEEDS As wind speed isincreased, not onlyis more energyavailable (higherwave heights), butlonger waves(longer period orlower frequency) arealso present. Also,the period ofmaximum energyshifts to longerperiod waves.40 knotsSPECTRAL ENERGY30 knots20 knots6020105PERIOD (sec)
17 WAVE ENERGY vs. WIND SPEED Wave energy is very sensitive to wind speed:Wave Energy (Wave Height)Wave Height (Wind Speed)Wave Energy (Wind Speed)2a2a4aBest Wave Model Uses “Good” Winds
18 STATISTICAL DESCRIPTION OF WAVE HEIGHTS H = significant wave height= average height of highest 1/3 waves in record(corresponds roughly to visually observed heights)H = average of all height .625 HH = average height of (1/n)th highest wavesH = 1.3 HH = 1.8 to 2.2 times the HSIGAVSIG1/n1/10SIGSIGMAX
19 WAVE GROUPSAlthough individual crests advance at a speed correspondingto their wavelength (as a coherent unit), the group advances atits own speed, called the GROUP SPEED....Cg.The group speed is the speed at which the energy propagates(moves with the speed of the “middle” of the pack...slower thanleading waves and faster than trailing waves.)The energy is equally split between KE and PE, however, theKE is associated with movement of particles in nearly closedorbits....therefore, the KE is not propagated. The PE isassociated with net displacements of particles and this movesalong with the wave at the wave speed.
20 WAVE GROUPS C (kts) = C = 1.51 T (sec) 2 Therefore, only 1/2 of the energy (PE) is propagated at the wave speed which is the same as the total energy moving at 1/2 of the wave speed.C (kts) = C = T (sec)2g
21 TYPICAL WAVE PERIODS 6m waves developed by a marginal gale Period (sec)Group Speed (kts)58697118129146m waves developedby a marginal gale101511171218132014211523162416m “fully-developed”seas in a big storm172618271929Long swell well aheadof a storm system2030
24 STATIONARY FETCHES - Example 1 Coast-lines - The fetch at point B is the distance AB. The fetch at point D is the distance CD. Since AB > CD, a wind from the coast would generate greater waves at B than at D because of the proximity of the coastline.
25 STATIONARY FETCHES - Example 2 Curvature - The fetch at point B is limited by the curvature of the flow upwind. The fetch is now the distance upwind from B to the point where the wind direction becomes more than 30o different from that at B.
26 STATIONARY FETCHES - Example 3 Fanning out of flow The fetch at point B is limited by the fanning out of the isobars upstream, therefore giving decreasing wind speeds. In this case, the criterion that is recommended is for wind-speed reductions > 20%.
27 Moving Fetches Most interesting wind systems are not stationary. Determining fetches in amoving system can be complex and timeconsuming.As it turns out, the determination of thefetch is as critical as the determinationof the wind speed
29 O . . . . . Winds Perpendicular to Fetch Motion 1 Time T0 X PC PD PB PA
30 O . . . . . . . . . . Winds Perpendicular to Fetch Motion 1 Time T1 X Waves fromPA & PC & PEgrew for less than1 time-stepbefore movingoutside the fetchPB & PDhave grown forXO1Time T1....PCPD..PB....PEPA
31 O . . . . . . . . . . . . . . . Winds Perpendicular to Fetch Motion 1 Waves fromPB moved outsidethe fetch after1 time-stepWaves from PDgrew for almost2 time-stepsWinds Perpendicularto Fetch MotionXO1Time T2......PCPD...PB......PEPA
32 O . . . . . Winds Perpendicular to Fetch Motion 2 Time T0 X PC PD PB PA
33 O . . . . . . . . . . Winds Perpendicular to Fetch Motion 2 Time T1 X Waves fromPA & PD & PEgrew for less than1 time-stepbefore movingoutside the fetchPC & PBhave grown forXO2Time T1....PCPD..PB..PE..PA
34 O . . . . . . . . . . . . . . . Winds Perpendicular to Fetch Motion 2 Waves fromPB moved outsidethe fetch after1 time-stepWaves from PChave grown for2 time-stepsWinds Perpendicularto Fetch MotionXO2Time T2......PCPD...PB...PE...PA
35 Winds Perpendicularto Fetch MotionWaves from PCgrew for morethan 2 time-stepsbefore movingoutside the fetchXO2Time T3........PCPD....PB....PE....PA
37 O . . . . . . . . . . . . Winds Opposing Fetch Motion 3 Time T1 X Waves from PC & PDgrew for 1 time-stepWaves fromPE & PB & PAgrew for less than1 time-stepbefore movingoutside the fetchO3XTime T1..PC.PD..PB...PE..PA..
38 O . . . . . . . . . . . . . . . Winds Opposing Fetch Motion 3 Time T2 Waves from PC & PDmoved outside thefetch before growingfor even2 time-stepsO3XTime T2..PC.PD...PB...PE...PA...
49 A B Fetch moving Waves moving much faster much faster than waves Mid-latitudesystemsWaves movingmuch fasterthan fetchTropical stormsin the tropics
50 A B C Fetch moving Waves moving Some waves much faster much faster than wavesMid-latitudesystemsWaves movingmuch fasterthan fetchTropical stormsin the tropicsSome wavesin harmonywith fetchStrong windsystems inmid-latitudes
51 A B C D Fetch moving much faster than waves Mid-latitude systems Waves movingmuch fasterthan fetchTropical stormsin the tropicsSome wavesin harmonywith fetchStrong windsystems inmid-latitudesPerfectWaves-StormResonanceThe REAL“Perfect Storms”
52 Scanning Radar Altimetry data for Hurricane Bonnie (98) as NASAScanning Radar Altimetry datafor Hurricane Bonnie (98) asit tracked northward off theUS east coast.
53 Courtesy of Ed Walsh, NASA (runs automatically in SlideShow mode)
54 The dominant waves are in the front right quadrant where a degree of resonance exists in a single spectralmode.This allows for ease ofmodelling and parametrization.
56 All of thewaves quicklyoutrun the slow-moving stormsystemThe steady-statesolution isreached in7+ hours.
57 Most of thewaves stilloutrun theslower movingstorm systemThe steady-statesolution isreached in10+ hours.
58 Waves leavingthe trailing edgeof the fetch areoutrun by thewind system . . .however, allothers move outahead.The steady-statesolution isreached in17+ hours.
59 Most waves aresoon outrun by theinitially quickermoving windsystem . . .however, wavesstarting at theleading edge “holdon long enough”until their speedcatches up to thesystem speed . . .and eventuallyoutruns the system.Steady state isreached in 30 hrs.
60 All waves arequickly outrunby the muchquicker windsystem andgrowth is verylimited.The steady-statesolution isreached inunder 5 hours.
61 Maximum wavegrowth occursfor a constantsystem speed of20.7 knots. Allspeeds greateror less than thiswill result inlower waveheights.The steady-statesolution is notreached until34 hours.
62 Even for speedsonly slightlygreater, thereis a significantdifference inthe resonanceof the storm-waves system.
63 Consider . . .- a fixed fetch length (eg. 50 nmi)- fixed wind speed throughout fetch(eg. 50 kts)What is the relationship betweenfetch-enhancement and storm speed?
84 Observation: 4. Optimum fetch-enhancement is more probable for high wind / smallfetch events than low wind / highfetch events because therequired durations are on the orderof 1-day (ie: these events canactually occur)
85 Conclusion 1Observations 1-4 select sub-synoptic scale storm systems for the greatest potential for optimum resonance(tropical cyclones and polar lows)
90 Average speed of TCs north of 40oN Average speed of TCs south of 30oN 50 nm Fetch100 nm FetchAverage speed of TCssouth of 30oN
91 Observation:5. Fetch enhancement for tropical cyclones of TS or Hurricane strength is greater in mid latitudes than in the tropics.
92 Conclusion 2The highest waves with tropical storms or hurricanes could be expected in mid latitudes . . .. . . a unique problem with ETs
93 As tropical cyclones become ET, they are typically increasing in speed under the influence of a mid-latitude stream. The seas associated with these systems (and even minimal hurricanes that move “quickly”) can be greater than those associated with major hurricanes.
94 For example: A Hurricane in the Tropics Moving 5 kts For example: A Hurricane in the Tropics Moving 5 kts area of 65-kt winds over a fetch length of 100 nm can generate Hsig of almost 11 m. A Tropical Storm in Mid-Latitudes Moving 191/2 kts area of 50-kt winds over a fetch length of 100 nm can generate Hsig of almost 15 m.-27
95 Accelerating Fetches If the storm system accelerates with a speed matching that of the waves it generates, theconditions for perfect resonance exists andwave growth is more easily maximized.
106 The waves in Felix were quite large, considering the wind field The waves in Felix were quite large, considering the wind field. However, like Luis, Felix began outstripping the waves which might have eventually grown larger, had Felix’s translation remained below 30 knots.
110 The waves with Bonnie were close to being fully-developed for the wind field associated with the storm. The proximity of the wave max with the storm centre shows that the two moved in reasonable harmony. A slightly slower translation speed for the storm might have resulted in slighly larger waves.
114 Like Bonnie, the max in the wave-field with Danielle was very close to the storm centre showing that the two were moving in reasonable harmony. Danielle actually began slowing, allowing the wave-field to catch up and grow larger what might be expected.
115 The Worst-Case Scenario is, therefore, a storm centre The Worst-Case Scenario is, therefore, a storm centre moving in a straight line, - covering a large distance over open ocean, - increasing in speed, continually matching the speed of the waves that corresponds closely with the peak in the energy spectrum
116 The Pattern area of maximum waves to the right of track - very tight gradient in the wave field at the leading edge as the trapped-fetch arrives, it brings with it a “wall of water” (no forerunners to warn of storm) - waves can subside rather quickly in the wake of these storms, however, the trailing gradient is usually much weaker
117 Exercise Using the Wave Resonance Calculator on the Workstation, what significant wave heightswould you expect along the coastline of AtlanticCanada with different fetch scenarios A-N?
118 Modelling the Problem Specifically for TCs Resonance is, uniquely, a unidirectional problem, so the spectral issues simplify considerablySince wind is, by far, the most critical issue, even a simple first generation parametric approach will give superior results, if the wind field is treated accurately.
119 Currently, we are tackling this problem by coupling the CHC’s hurricane wind model with a single-purpose wave model that looks only at the maximum possible Hsig with a given storm