# Why Do Waves Get So Big? Waves-Storm Resonance Lab Canadian Hurricane

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Why Do Waves Get So Big? Waves-Storm Resonance Lab Canadian Hurricane
Centre Peter Bowyer Allan MacAfee

Between Oct. 91 and Sept. 95, EC’s East Coast
NOMAD buoys reported some extraordinary waves events, with the 100-year waves being greatly exceeded 3 times in those 4 years

“The Perfect Storm” - October 1991 “Storm of the Century” - March 1993
Hurricane Luis - September 1995 . . . each reported significant wave heights of 17+ metres, and maximum wave heights 30+ metres

Our work that has followed
has focussed on the problem of “what makes big waves?”

Basics of Waves l = cT

( ) SPEED OF WAVES IN WATER 2p c = gl tanh 2pd 2p l
1 2 ( ) g = gravitational constant c = gl tanh 2pd 2p l l = wavelength d = water depth In deep water, d / l gets very large, so tanh (big #) , therefore, c = gl 2p 1 2

SPEED OF WAVES IN DEEP WATER
Since l = cT c = gT 2p l = gT 2 Therefore, 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.

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 = gT 2p 2 2 2

COMPOSITION OF WAVES Any 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.

COMPOSITION OF WAVES In reality, the sea is a superposition of many wave sets.

WAVE GENERATION & DECAY

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 blowing Duration - the length of time the wind persists from a certain fetch

SIGNIFICANT WAVE HEIGHTS
Hsig a V2 tanh [a (F/V2)a] V = wind speed F = fetch length “Law of Diminishing Returns” is at work

FULLY DEVELOPED SEA SEA CHANGING TO SWELL
RIPPLES CHOP WIND WAVES WIND FETCH of the wind

THE WAVE SPECTRUM ENERGY For a simple sine wave, the energy is proportional to the square of the wave height However, the real sea is a combination of many sine waves 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

WAVE SPECTRA FOR DIFFERENT WIND SPEEDS
As wind speed is increased, not only is more energy available (higher wave heights), but longer waves (longer period or lower frequency) are also present. Also, the period of maximum energy shifts to longer period waves. 40 knots SPECTRAL ENERGY 30 knots 20 knots 60 20 10 5 PERIOD (sec)

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) 2 a 2 a 4 a Best Wave Model Uses “Good” Winds

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 H H = average height of (1/n)th highest waves H = 1.3 H H = 1.8 to 2.2 times the H SIG AV SIG 1/n 1/10 SIG SIG MAX

WAVE GROUPS Although individual crests advance at a speed corresponding to their wavelength (as a coherent unit), the group advances at its 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 than leading waves and faster than trailing waves.) The energy is equally split between KE and PE, however, the KE is associated with movement of particles in nearly closed orbits....therefore, the KE is not propagated. The PE is associated with net displacements of particles and this moves along with the wave at the wave speed.

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) 2 g

TYPICAL WAVE PERIODS 6m waves developed by a marginal gale
Period (sec) Group Speed (kts) 5 8 6 9 7 11 8 12 9 14 6m waves developed by a marginal gale 10 15 11 17 12 18 13 20 14 21 15 23 16 24 16m “fully-developed” seas in a big storm 17 26 18 27 19 29 Long swell well ahead of a storm system 20 30

Bretschneider Wave Calculator

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.

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.

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%.

Moving Fetches Most interesting wind systems are
not stationary. Determining fetches in a moving system can be complex and time consuming. As it turns out, the determination of the fetch is as critical as the determination of the wind speed

Cyclone Motion WIND 1 1 X O 3 3 4 4 WIND WIND 2 2 WIND

O . . . . . Winds Perpendicular to Fetch Motion 1 Time T0 X PC PD PB
PA

O . . . . . . . . . . Winds Perpendicular to Fetch Motion 1 Time T1 X
Waves from PA & PC & PE grew for less than 1 time-step before moving outside the fetch PB & PD have grown for X O 1 Time T1 . . . . PC PD . . PB . . . . PE PA

O . . . . . . . . . . . . . . . Winds Perpendicular to Fetch Motion 1
Waves from PB moved outside the fetch after 1 time-step Waves from PD grew for almost 2 time-steps Winds Perpendicular to Fetch Motion X O 1 Time T2 . . . . . . PC PD . . . PB . . . . . . PE PA

O . . . . . Winds Perpendicular to Fetch Motion 2 Time T0 X PC PD PB
PA

O . . . . . . . . . . Winds Perpendicular to Fetch Motion 2 Time T1 X
Waves from PA & PD & PE grew for less than 1 time-step before moving outside the fetch PC & PB have grown for X O 2 Time T1 . . . . PC PD . . PB . . PE . . PA

O . . . . . . . . . . . . . . . Winds Perpendicular to Fetch Motion 2
Waves from PB moved outside the fetch after 1 time-step Waves from PC have grown for 2 time-steps Winds Perpendicular to Fetch Motion X O 2 Time T2 . . . . . . PC PD . . . PB . . . PE . . . PA

Winds Perpendicular to Fetch Motion Waves from PC grew for more than 2 time-steps before moving outside the fetch X O 2 Time T3 . . . . . . . . PC PD . . . . PB . . . . PE . . . . PA

Winds Opposing Fetch Motion O 3 X . . Time T0 PC PD . PB . PE . PA

O . . . . . . . . . . . . Winds Opposing Fetch Motion 3 Time T1 X
Waves from PC & PD grew for 1 time-step Waves from PE & PB & PA grew for less than 1 time-step before moving outside the fetch O 3 X Time T1 . . PC . PD . . PB . . . PE . . PA . .

O . . . . . . . . . . . . . . . Winds Opposing Fetch Motion 3 Time T2
Waves from PC & PD moved outside the fetch before growing for even 2 time-steps O 3 X Time T2 . . PC . PD . . . PB . . . PE . . . PA . . .

Winds With Fetch Motion O X 4 . . Time T0 PC PD . PB . PE . PA

O . . . . . . . . . . Winds With Fetch Motion 4 Time T1 X
Waves from PA fell behind the fetch before even 1 time-step Winds With Fetch Motion O X 4 Time T1 . . . . PC . PD . . PB . . PE . PA

O . . . . . . . . . . . . . . . Winds With Fetch Motion 4 Time T2 X
Waves from PE grew for more than 1 time-step but were then outrun by the fetch O X 4 Time T2 . . . . . . . PC . . PD . . . PB . . PE . PA

O . . . . . . . . . . . . . . . . . . . . Winds With Fetch Motion 4
Waves from PC & PB & PD are still growing after 3 time-steps . . . although those from PB will soon be outrun by the fetch Winds With Fetch Motion O Time T3 X 4 . . . . . . . . . . . PC . . . PD . . . PB . . PE . PA

Waves from PC & PD are still growing after 4 time-steps Winds With Fetch Motion Time T4 O X 4 . . . . . . . . . . . . . . . . PC . . . PD . . . PB . . PE . PA

Winds With Fetch Motion Time T5 O . X 4 Waves from PD were finally outrun by the fetch after more than 4 time-steps PC are still growing after 5 time-steps . . . . . . . . . . . . . . . . . . . . PC . . . PD . . . PB . . PE . PA

“Fetch Reduction” always occurs in Quadrants 1-2-3 . . . as long as the cyclone is moving

“Fetch Enhancement” may occur in Quadrant 4 . . . depending on the speed of the cyclone

Waves Moving in Same Direction as Their Storm Potential for Resonance

A Fetch moving much faster than waves Mid-latitude systems

A B Fetch moving Waves moving much faster much faster than waves
Mid-latitude systems Waves moving much faster than fetch Tropical storms in the tropics

A B C Fetch moving Waves moving Some waves much faster much faster
than waves Mid-latitude systems Waves moving much faster than fetch Tropical storms in the tropics Some waves in harmony with fetch Strong wind systems in mid-latitudes

A B C D Fetch moving much faster than waves Mid-latitude systems
Waves moving much faster than fetch Tropical storms in the tropics Some waves in harmony with fetch Strong wind systems in mid-latitudes Perfect Waves-Storm Resonance The REAL “Perfect Storms”

Scanning Radar Altimetry data for Hurricane Bonnie (98) as
NASA Scanning Radar Altimetry data for Hurricane Bonnie (98) as it tracked northward off the US east coast.

Courtesy of Ed Walsh, NASA (runs automatically in SlideShow mode)

The dominant waves are in the front right quadrant where a degree of
resonance exists in a single spectral mode. This allows for ease of modelling and parametrization.

Fetches moving with constant speed

All of the waves quickly outrun the slow- moving storm system The steady-state solution is reached in 7+ hours.

Most of the waves still outrun the slower moving storm system The steady-state solution is reached in 10+ hours.

Waves leaving the trailing edge of the fetch are outrun by the wind system . . . however, all others move out ahead. The steady-state solution is reached in 17+ hours.

Most waves are soon outrun by the initially quicker moving wind system . . . however, waves starting at the leading edge “hold on long enough” until their speed catches up to the system speed . . . and eventually outruns the system. Steady state is reached in 30 hrs.

All waves are quickly outrun by the much quicker wind system and growth is very limited. The steady-state solution is reached in under 5 hours.

Maximum wave growth occurs for a constant system speed of 20.7 knots. All speeds greater or less than this will result in lower wave heights. The steady-state solution is not reached until 34 hours.

Even for speeds only slightly greater, there is a significant difference in the resonance of the storm- waves system.

Consider . . . - a fixed fetch length (eg. 50 nmi) - fixed wind speed throughout fetch (eg. 50 kts) What is the relationship between fetch-enhancement and storm speed?

Storm-Wave Resonance Calculator

Fetch Enhancement Fetch Reduction

Extreme Enhancement Significant Enhancement Enhancement Fetch Reduction

Enhancement Reduction

Observation: 1. The greater the wind speed,
the greater the enhancement

Maximum Possible Significant Wave Heights
35-knot winds

Maximum Possible Significant Wave Heights
50-knot winds

Maximum Possible Significant Wave Heights
65-knot winds

Maximum Possible Significant Wave Heights
80-knot winds

Maximum Possible Significant Wave Heights
95-knot winds

Observations: 2. The smaller the fetch, the greater the enhancement
3. The greater the wind speed, the greater the optimum storm speed

Unlikely durations

Observation: 4. Optimum fetch-enhancement is
more probable for high wind / small fetch events than low wind / high fetch events because the required durations are on the order of 1-day (ie: these events can actually occur)

Conclusion 1 Observations 1-4 select sub-synoptic scale storm systems for the greatest potential for optimum resonance (tropical cyclones and polar lows)

Optimum Enhancement

Optimum Enhancement

Average speed of TCs north of 40oN Average speed of TCs south of 30oN
50 nm Fetch 100 nm Fetch Average speed of TCs south of 30oN

Observation: 5. Fetch enhancement for tropical cyclones of TS or Hurricane strength is greater in mid latitudes than in the tropics.

Conclusion 2 The highest waves with tropical storms or hurricanes could be expected in mid latitudes . . . . . . a unique problem with ETs

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.

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

Accelerating Fetches If the storm system accelerates with a speed
matching that of the waves it generates, the conditions for perfect resonance exists and wave growth is more easily maximized.

Storms moving in a straight line, at the optimum speed, can result in the potential for phenomenally large waves to develop

Luis (95) Felix (95) Bonnie (98) Danielle (98)

HURRICANE LUIS Sept. 10-11, 1995 11 / 06Z 105 knots 39 11 / 00Z
38 10 / 18Z 85 knots 32 10 / 12Z 85 knots 25 10 / 06Z 85 knots

HURRICANE LUIS Sept. 10-11, 1995 * Wave Field at Sept.11 - 01Z 2
105 knots 2 39 Max Reported Sig. Waves 17+ metres 3 QEII 11 / 00Z * 95 knots 4 38 For 17 metres, 85 kts is required throughout this box for about 14 hours 5 10 / 18Z 10 85 knots 6 32 9 10 / 12Z 85 knots 7 8 25 10 / 06Z 85 knots

Storm speed 40+ knots and increasing Nearest point to storm

Since Luis was quickly outstripping the waves generated by its wind-field, the waves were not as large as they might of been, had Luis moved considerably slower.

TS FELIX Aug. 21-22, 1995 22 / 12Z 50 knots 43 22 / 06Z 50 knots 35
31 21 / 18Z 28 50 knots 21 / 12Z 50 knots 21 / 06Z 55 knots 21 / 00Z 60 knots

TS FELIX Aug. 21-22, 1995 * Wave Field at Aug.22 - 00Z Max Reported
50 knots 43 22 / 06Z 50 knots 35 22 / 00Z * 50 knots 31 21 / 18Z Max Reported Sig. Waves 13+ metres 2 7 28 50 knots 6 21 / 12Z 3 50 knots 5 For 13 metres, 50 kts is required throughout this box for about 36 hours 21 / 06Z 55 knots 4 21 / 00Z 60 knots

Storm speed 30+ knots and increasing Nearest point to storm

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.

TS BONNIE Aug , 1998 30 / 12Z 30 / 18Z 30 / 06Z 25 26 34 45 knots 45 knots 45 knots 30 / 00Z 28 45 knots 29 / 18Z 26 45 knots 29 / 12Z 18 45 knots 29 / 06Z 29 / 00Z 18 45 knots 45 knots

TS BONNIE Aug. 29-30, 1998 * Wave Field at Aug.30 - 00Z 7
25 26 34 45 knots 45 knots 45 knots 30 / 00Z * 28 45 knots 29 / 18Z 26 45 knots 29 / 12Z 18 45 knots 29 / 06Z 7 For 11 metres, 45 kts is required throughout this box for about 42 hours 29 / 00Z 18 45 knots 6 Max Reported Sig. Waves 11 metres 45 knots 2 5 3 4

Storm speed 25+ knots and increasing Nearest point to storm

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.

HURRICANE DANIELLE Sept. 2-3, 1998
04 / 00Z 23 03 / 18Z 65 knots 65 knots * 23 03 / 12Z 70 knots 24 03 / 06Z 70 knots 30 03 / 00Z 70 knots 32 02 / 18Z 70 knots

HURRICANE DANIELLE Sept. 2-3, 1998
Wave Field at Sept Z 04 / 00Z 23 03 / 18Z 65 knots 23 65 knots 03 / 12Z 70 knots 24 * 03 / 06Z 2 3 4 70 knots 5 30 6 For 16 metres, 70 kts is required throughout this box for about 16 hours 7 03 / 00Z 9 70 knots 32 8 02 / 18Z 70 knots Max Reported Sig. Waves 16 metres

Storm speed 25 knots and slowing Nearest point to storm

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.

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

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

Exercise Using the Wave Resonance Calculator on the
Workstation, what significant wave heights would you expect along the coastline of Atlantic Canada with different fetch scenarios A-N?

Modelling the Problem Specifically for TCs
Resonance is, uniquely, a unidirectional problem, so the spectral issues simplify considerably Since 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.

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

The output of the “trapped-fetch” model shows trajectories, maximum Hsig, and duration of wave growth from independent fetch areas within the tropical cyclone

For Bonnie, the CHC model predicts a maximum Hsig passing through southern Maritime waters of 10.3m. Compare this to the maximum Hsig of 10.8m reported by a weather buoy.

For Luis, the CHC model predicts a maximum Hsig of 18.9 m to the right-of-track. A weather buoy reported a maximum Hsig of 17.1 m.

For Danielle, the CHC model predicts a maximum Hsig of 15.5 m passing through southern Maritime waters. A weather buoy reported 15.8 m.