TRAPPED-FETCH WAVES The Theory from an Observational Perspective Peter J Bowyer Allan W MacAfee 50 50 45 45 40 40 35 35 System Speed (kt) System Speed (kt) 30 30 25 25 20 20 15 15 10 10 5 5 Fetch Enhancements Wind Speed vs System Fetch Speed Enhancements for 250 nmi Fetch Length Wind Speed vs System Speed for 250 nmi Fetch Length Fetch Reduction Fetch Reduction Extreme Enhancement Extreme Enhancement Significant Enhancement Significant Enhancement Fetch Enhancement Fetch Enhancement 0 0 20 20 27 27 34 34 41 41 48 48 55 55 62 62 69 69 76 83 90 97 76 83 90 97 Wind Speed (kt) Wind Speed (kt) 104 104 111 111 118 118 125 125 132 132 139 139 146 146
BACKGROUND Since 1990 a significant number of TCs have moved through eastern Canadian waters, accompanied by extreme waves (H SIG 13-17 m and/or H MAX 25-30 m) generated by a dynamic ( trapped ) fetch (Bowyer & MacAfee) 1995 Interest in trapped-fetch waves spawned 2000 Basic theory & model developed 2003 Theory & model significantly refined 2004 Papers submitted to WAF (awaiting 2 nd review)
Cyclone Motion WIND 1 WIND XO 3 4 WIND 2 WIND
Winds Perpendicular to Fetch Motion 1 XO Which waves will remain in in the the fetch the the longest? P C P E P B P D P A Time T 0
Winds Perpendicular to Fetch Motion 1 XO Waves from P A & P C & P E grew for less than 1 time-step before moving outside the fetch Waves from P B & P D have grown for 1 time-step P C P E P B P D P A Time T 1
Winds Perpendicular to Fetch Motion 1 XO Waves from P B moved outside the fetch after 1 time-step Waves from P D grew for almost 2 time-steps Time T 2 P C P E P B P D P A
Winds Perpendicular to Fetch Motion XO Which waves will remain in in the the fetch the the longest? 2 P C P D P E P B Time T 0 P A
Winds Perpendicular to Fetch Motion XO 2 Waves from P A & P D & P E grew for less than 1 time-step before moving outside the fetch Waves from P C & P B have grown for 1 time-step P D P B P C P E P A Time T 1
Winds Perpendicular to Fetch Motion XO Waves from P B moved outside the fetch after 1 time-step Waves from P C have grown for 2 time-steps 2 P D P B P C P E P A Time T 2
Winds Perpendicular to Fetch Motion Waves from P C grew for more than 2 time-steps before moving outside the fetch XO 2 Time T 3 P C P D P B P E P A
Winds Opposing Fetch Motion 3 XO Which waves will remain in in the the fetch the the longest? P C P D P B P E P A Time T 0
3 Winds Opposing Fetch Motion XO Waves from P C & P D grew for 1 time-step Waves from P E & P B & P A grew for less than 1 time-step before moving outside the fetch P C P D Time T 1 P E P B P A
Winds Opposing Fetch Motion Waves from P C & P D moved outside the fetch before growing for even 2 time-steps 3 XO P C P E P B P D P A Time T 2
Winds With Fetch Motion XO 4 Which waves will remain in in the the fetch the the longest? P C P D P E P B Time T 0 P A
Winds With Fetch Motion Waves from P A fell behind the fetch before even 1 time-step XO 4 P C P B P D Time T 1 P E P A
Winds With Fetch Motion Waves from P E grew for more than 1 time-step but were then outrun by the fetch XO 4 P C P B P D Time T 2 P E P A
Winds With Fetch Motion XO 4 Waves from P C & P B & P D are still growing after 3 time-steps although those from P B will soon be outrun by the fetch Time T 3 P C P B P E P D P A
Winds With Fetch Motion Waves from P C & P D are still growing after 4 time-steps XO 4 Time T 4 P C P B P E P D P A
Winds With Fetch Motion Time T 5 XO 4 P C P B P E P D P A Waves from P D were finally outrun by the fetch after more than 4 time-steps Waves from P C are still growing after 5 time-steps
Which fetch generates the the smallest waves? Cyclone Motion Fetch Reduction always occurs in Quadrants 1-2-3 as long as the cyclone is moving WIND 1 WIND XO 3 4 WIND 2 WIND
Cyclone Motion Fetch Enhancement may occur in Quadrant 4 depending on the speed of the cyclone WIND 1 WIND XO 3 4 WIND 2 WIND
Waves Moving in Same Direction as Their Storm Potential for Resonance
A Fetch moving much faster than waves Mid-latitude systems
A Fetch moving much faster than waves Mid-latitude systems B Waves moving much faster than fetch Tropical storms in the tropics
A Fetch moving much faster than waves Mid-latitude systems B Waves moving much faster than fetch Tropical storms in the tropics C Some waves in harmony with fetch Strong wind systems in mid-latitudes
Fetches moving with constant speed
All of the waves quickly outrun the slow- moving storm system The steady-state state solution is reached in 7+ hours
Most of the waves still outrun the slower moving storm system The steady-state 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 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 state solution is reached in under 5 hours
Maximum wave growth occurs for a constant system speed of 207 knots All speeds greater or less than this will result in lower wave heights The steady-state 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
Maximum Enhancement Ratio 25 2 15 1 05 0 Fetch Enhancement Fetch Reduction 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 m s -1 Storm Speed (kt / m s -1 )
250 20 kt Maximum Wave Growth (h) 200 150 100 50 40 kt 60 kt 80 kt 100 kt 120 kt 140 kt 0 0 10 20 30 40 50 0 5 10 15 20 25 m s -1 Storm Speed (kt / m s -1 )
35 Maximum Enhancement Ratio 3 25 2 15 1 05 20 kt 40 kt 60 kt 80 kt 100 kt 120 kt 140 kt Observation The greater the wind speed, the greater the enhancement 0 0 10 20 30 40 50 0 5 10 15 20 25 m s -1 Storm Speed (kt / m s -1 )
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 Observation The smaller the fetch, the greater the enhancement
3 Optimum Enhancement 25 2 15 50 Fetch 150 Fetch 250 Fetch 1 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10 20 30 40 50 60 70 m s -1 Wind Speed (kt / m s -1 )
25 Wind Speed vs Storm Speed for Maximum Fetch Enhancement 50 nm Fetch Area 20 Optimum Storm Speed (kt) 15 10 5 0 Observation The greater the wind speed, the greater the optimum storm speed y = -0001x 2 + 03131x + 365 R 2 = 09986 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 10 0 Wind Speed (kt)
45 Fetch Enhancements Wind Speed vs System Speed for 50 nmi Fetch Length System Speed (kt) 40 35 30 25 20 15 10 5 0 20 26 Fetch Reduction Extreme Enhancement Significant Enhancement Fetch Enhancement 32 38 44 50 56 62 68 74 80 86 92 98 104 110 116 122 128 134 140 Wind Speed (kt) 146 Fetch Enhancements Wind Speed vs System Speed for 250 nmi Fetch Length System Speed (kt) 50 45 40 35 30 25 20 15 10 5 Fetch Reduction Extreme Enhancement Significant Enhancement Fetch Enhancement 0 20 27 34 41 48 55 62 69 76 83 90 97 Wind Speed (kt) 104 111 118 125 132 139 146
Maximum Wave Growth (h) 500 450 400 350 300 250 200 150 100 250 Fetch 150 Fetch Observation 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) 50 50 Fetch 0 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10 20 30 40 50 60 70 m s - 1 Wind Speed (kt / m s -1 )
Conclusion 1 Sub-synoptic scale storm systems, like tropical cyclones and polar lows, have the greatest potential for optimum resonance
50 45 25 VCRIT (kt) 40 35 30 25 20 Average speed of TCs north of 40 N 50 Fetch 150 Fetch 250 Fetch 20 15 10 15 10 5 Average speed of TCs south of 30 N 0 0 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10 20 30 40 50 60 70 m s -1 Wind Speed (kt / m s -1 ) Observation Fetch enhancement for TS or Hurricanes is greater in mid latitudes than in the tropics 5
As tropical cyclones undergo extratropical transition (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
Conclusion 2 Very high waves with TCs should be expected in mid latitudes as these systems become ET
Example: Three 50-kt tropical storms identical in in all all ways except translation speed 30kt 15kt 10kt
Maximum H sig generated is 53 m
Maximum H sig generated is 91 m
Maximum H sig generated is 63 m
30kt 15kt 10kt Storm speed is is very important 53m 91m 63m
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 50 45 100-kt wind 25 Wave Group Speed (kt) 40 35 30 25 20 15 10 5 80-kt wind 60-kt wind 40-kt wind 20-kt wind 20 15 10 5 0 0 0 10 20 30 40 50 60 70 80 90 100 110 120 Duration of Wave Growth (h)
Storms moving in a straight line, at the optimum speed, can result in the potential for phenomenally large waves to develop
The Worst-Case Scenario is 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 (little or 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