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

2. 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

3. “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

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

5. Basics of Waves
l = cT

6. ( ) SPEED OF WAVES IN WATER 2p c = gl tanh 2pd 2p l1 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

7. 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.

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 = gT 2p 2 2 2

9. 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.

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

11. WAVE GENERATION & DECAY

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 blowing
Duration - 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 speed
F = fetch length
“Law of Diminishing Returns”
is at work

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

15. 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

16. 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)

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) 2 a 2 a 4 a
Best 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 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

19. 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.

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) 2 g

21. 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

22. Bretschneider Wave Calculator

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 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

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

29. O . . . . . Winds Perpendicular to Fetch Motion 1 Time T0 X PC PD PBPA

30. 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

31. 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

32. O . . . . . Winds Perpendicular to Fetch Motion 2 Time T0 X PC PD PBPA

33. 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

34. 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

35. 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

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

37. 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 . .

38. 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 . . .

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

40. 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

41. 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

42. 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

43. 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

44. 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

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

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

47. Waves Moving in Same Direction as Their Storm Potential for Resonance

48. A
Fetch moving
much faster
than waves
Mid-latitude
systems

49. 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

50. 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

51. 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”

52. 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.

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 spectral
mode.
This allows for ease of
modelling and parametrization.

55. Fetches moving with
constant speed

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

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

58. 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.

59. 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.

60. 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.

61. 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.

62. Even for speeds
only slightly
greater, there
is a significant
difference in
the resonance
of 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 between
fetch-enhancement and storm speed?

64. Storm-Wave Resonance Calculator

68. Fetch Enhancement
Fetch Reduction

69. Extreme Enhancement
Significant Enhancement
Enhancement
Fetch Reduction

72. Enhancement
Reduction

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

75. Maximum Possible Significant Wave Heights
35-knot winds

76. Maximum Possible Significant Wave Heights
50-knot winds

77. Maximum Possible Significant Wave Heights
65-knot winds

78. Maximum Possible Significant Wave Heights
80-knot winds

79. Maximum Possible Significant Wave Heights
95-knot winds

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

82. Unlikely durations

84. 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)

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

87. Optimum Enhancement

88. Optimum Enhancement

90. 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

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

92. Conclusion 2
The 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, the
conditions for perfect resonance exists and
wave growth is more easily maximized.

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

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

99. HURRICANE LUIS Sept. 10-11, 1995 11 / 06Z 105 knots 39 11 / 00Z38
10 / 18Z
85 knots 32
10 / 12Z
85 knots 25
10 / 06Z
85 knots

100. 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

101. Storm speed
40+ knots
and
increasing
Nearest point
to storm

102. 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.

103. TS FELIX Aug. 21-22, 1995 22 / 12Z 50 knots 43 22 / 06Z 50 knots 3531
21 / 18Z 28
50 knots
21 / 12Z
50 knots
21 / 06Z
55 knots
21 / 00Z
60 knots

104. 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

105. Storm speed
30+ knots
and
increasing
Nearest point
to storm

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.

107. 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

108. TS BONNIE Aug. 29-30, 1998 * Wave Field at Aug.30 - 00Z 725 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

109. Storm speed
25+ knots
and
increasing
Nearest point
to storm

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.

111. 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

112. 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

113. Storm speed
25 knots
and
slowing
Nearest point
to storm

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 heights
would you expect along the coastline of Atlantic
Canada with different fetch scenarios A-N?

118. 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.

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

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

124. 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.

125. 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.

126. 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.