Overturning circulation driven by breaking internal waves ...
Modeling Breaking Waves
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![Page 1: Modeling Breaking Waves](https://reader036.fdocuments.in/reader036/viewer/2022062422/568138d7550346895da08f53/html5/thumbnails/1.jpg)
Modeling Breaking Waves
Zoe Boekelheide
Scientific Computing
April 30, 2003
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Questions
Qualitative#1: Can I model breaking waves?#2: Can I make waves that look like actual
ocean waves?Quantitative
#3: How do the time scales of waves of different amplitudes compare?
#4: Can I calculate a variable to measure “surfability” of a wave?
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The Model
One-dimensional modelAssumes deep water with zero viscosityMethod borrowed from M.S. Longuet-
Higgins and E.D. Cokelet, “The deformation of steep surface waves on water”
Fourth-order Runge-Kutta solver
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The Waves
I picked a realistic wave profile to test, and tried it for 5 different ratios of amplitude to wavelength
Amplitude/Wavelength
.31
.41
.62
.83
1.25
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0 1 2 3 4 5 6 7-2
0
2
0 1 2 3 4 5 6 7-2
0
2
0 1 2 3 4 5 6 7-2
0
2
0 1 2 3 4 5 6 7-2
0
2
amplitude/wavelength = 0.41
step 0
step 5
step 10
step 15
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#1: Can I Model Breaking Waves?
YES!
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#2: Do they actually look like ocean waves?
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0 1 2 3 4 5 6 7
-4
-3
-2
-1
0
1
2
3
4
Zoe's model wave
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Successes:The waves steepen over time like breaking
ocean waves do.They break like ocean waves do.
Failures:The “curl” doesn’t actually curl, it kind of
just floats in the air.
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#3: How do the time scales of waves of different amplitudes
compare?
The 5 waves I tested all had the same behavior, but on very different time scales.
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Amplitude/ Wavelength
Time scale (Δt)
.31 .008
.41 .004
.62 .0015
.83 .0005
1.25 .00033
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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
1
2
3
4
5
6
7
8x 10
-3
amplitude/wavelength
timescale
Time scale of wave breaking versus amplitude
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#4: Can I calculate a variable to measure “surfability” of a
wave?
Simple rules for surfing:You can only surf on parts of a wave that
make an angle between 0° and 35° with the horizontal
You can only surf on a concave surface of a wave
Takeshi Sugimoto, ”How to Ride a Wave”
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Define variable surfability
Surfability = (distance along wave satisfying the simple surfing conditions) x (time scale)
The surfability variable indicates only how much the wave is able to be surfed—not the quality of the surfing
Now I can calculate a value for each of my five waves and compare.
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Amplitude/ Wavelength
Surfability
.31 .2944
.41 .1135
.62 .0310
.83 .0079
1.25 .0033
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0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
amplitude/wavelength
surf-ability
Surfability of a wave versus amplitude
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Conclusions
#1: I can model breaking waves!#2: But not perfectly…#3: Waves with different amplitudes have the
same behavior, but over different time scales. Larger amplitude waves break faster.
#4: Smaller amplitude waves are more surfable, probably because they last longer.