Modeling Breaking Waves

Zoe Boekelheide

Scientific Computing

April 30, 2003

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?

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

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

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

#1: Can I Model Breaking Waves?

YES!

#2: Do they actually look like ocean waves?

0 1 2 3 4 5 6 7

-4

-3

-2

-1

0

1

2

3

4

Zoe's model wave

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.

#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.

Amplitude/ Wavelength

Time scale (Δt)

.31 .008

.41 .004

.62 .0015

.83 .0005

1.25 .00033

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

#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”

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.

Amplitude/ Wavelength

Surfability

.31 .2944

.41 .1135

.62 .0310

.83 .0079

1.25 .0033

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

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.