What creates foam on stormy oceans?

[TriKri]

Hi! I'm going to simulate stormy oceans and the foam that is created on wave crests, and need to know what process it is that creates the foam. So, more specifically, do you have any idea what the conditions are for foam to be created?

 

[Greg Bernhardt]

Here is http://en.wikipedia.org/wiki/Sea_foam

Not sure what stormy sea foam is. Just churned up water and oxygen?

 

[geologeek]

Have a look at protein skimming or foam fractionation...whilst not a direct facsimile - it should give you the gist of why the foam is produced.

 

[TriKri]

Thanks for the answers. I found a few different research papers about this (I'm surprised there aren't more). First, I found the paper Realistic Animation of Fluid with Splash and Foam which (by judging from the pictures) seems to handle splash very well, but not oceanic whitecaps (also called just whitecaps, or white horses) which is foam created when the crests break because of strong wind, which is what I want to simulate. It manages this by converting water into "splash particles" on those locations where the curvature of the water exceeds a certain threshold. However, this methods requires a grid with fine resolution in order for the curvature to be able to reach that threshold, so for a coarse grid simulation like mine, this method would not work, unless the threshold was lowered drastically (which I fear would introduce new problems).

In turn, it concludes that there seems to be "few papers on handling of effects of splashes and foam with fluid", even though it mentions the paper Rendering Natural Waters (this link is to the version published in COMPUTER GRAPHICS forum) as one of them, which "makes crude approximations". However, this paper includes an empirical formula for the "fractional area of the wind-blown water surface that is covered by foam" $f$ (i.e. the average area of the foam/the total area). The definition of this fractional area looks like

$$f = 1.59 * 10^{-5}U^{2.55}\exp[0.0861(T_w - T_a)],$$
where $U$ is the wind speed, $T_w$ is the water temperature and $T_a$ is the air temperature. It then mentiones that "as a crude approximation to the true distribution, one can put whitecaps at positions on the surface where the amplitude of the waves is the largest". As a suggestion, the water surface would preferably first be high-pass filtered before determining at which parts of the surface "the amplitude of the waves is the largest", to prevent that some (large) regions of the surface that would happen to be a bit higher elevated than other regions would get much more whitecaps area. Or alternatively, maybe as a better approximation, one can put whitecaps at positions on the surface where the curvature of the waves is the largest. This obviously turns out to resemble the method used in the previously mentioned paper (my gosh, the method I feared would introduce new problems if it was used is back!).

Anyway, in turn, this paper mentions that the formula comes from the book Oceanic Whitecaps: Their Role in Air-Sea Exchange Processes, edited by E. C. Monahan and G. MacNiocaill, which I haven't read, so I really can't judge of which quality this empirical formula is. Would there for example be any whitecaps area at all under a certain wind speed threshold, say at 2 or 3 m/s? I doubt there would. According to the Beaufort scale, crests don't begin to break until there is at least a gentle breeze, which starts at about 3.4 m/s. According to this formula thogh, there would always be some whitecaps area when there is just a little amount of wind. As a remedy, one could subtract a small value $f_0 = f(U=U_0, T_w-T_a=\Delta T_0)$ (where, tentatively, $U_0 = 3.4\text{ m/s}$ and $\Delta T_0 = 0$) and create a corrected estimate $f^*$ of the fractional area, defined as

$$f^* = max(f-f_0,\,0).$$
Of course, the parameters would have to be readjusted, since we now have a new model for how to calculate the fraction between foam area and total area.

----------

If you want to use the splash particle model anyway, there are great news for you: It looks great. For example, the following really neat video seems to use a very similar model:

https://www.youtube.com/watch?v=c3JSbhgsoGg

By looking at the next video, I managed to get another clue about a trick that is probably used to make the foam that is formed on top of the surface look better. It looks like the foam particles are attracting each other, in order to prevent them from becoming just a homogenous grey mass on the surface of the water (for example, thin lines of foam will be formed where there were previously weak broad bands). This corresponds to some inherent property of sea foam bubbles to bind to each other since it minimizes the surface energy stored in the bubble walls. Anyway, here's the video:

https://www.youtube.com/watch?v=isXNkTiiAYQ

 

[Studiot]

I suggest you look also at the chemistry of colloidal or disperse systems.
Disperse systems are long term stable mixtures of more than one phase (state) and include colloids and gels (solid in liquid), air particulates (solid in gas), and foams (gas in solids or liquids).

This is a well developed science.

 

[TriKri]

How can I use that to simulate whitecaps? What is the connection between whitecaps and colloidal or dispersive systems?

 

[Studiot]

Well I did tell you that foam is a disperse system of air entrained in water.

 

[TriKri]

Right, that's the connection. Do you mean I will find out how whitecaps are formed by studying disperse systems? Thanks.

 

[Studiot]

Not only will the Physical chemistry tell you how, but how long they will last and why and many other things.

There is chemistry section here.

 

[TriKri]

I believe that the chemistry section may give me a lot of useful information about sea foam, but when it comes to whitecaps, I still need a mechanical model for breaking of the wave crests, which is when whitecaps are formed. I will take your suggestion though and study disperse systems to see what I may find out. Thanks again.

 

[Studiot]

Wave energy progresses at the group velocity of the waves.
When they enter shallow water their group speed decreases so for the same amount of energy flux their height must increase.
The frequency of arrival remains constant so the wavelength must decrease as speed decreases.
As their wavelength decreases and height increases the wave profile must therefore become steeper and the particle velocities must increase.

At the point where the max particle velocity exceeds the wave velocity the profile becomes unstable, spills over and the wave breaks.

The wave speed in shallow water is given by √(gd) where d is the depth.

Is this what you are looking for?

 

[TriKri]

That is not what I am looking for, sorry, although it was a nice explanation for wave breaking caused by shoaling. The whitecaps I'm talking about are the white parts of the surface on stormy oceans such as can be seen for example in this picture

I'm not exactly sure what it is that causes this foam (which I assume it is), but I've read it is formed because the wave crest breaks. This is probably just an extremely small part of the wave that breaks, maybe only a few centimeters. I believe the process is quite complex but maybe there is a bit written about it in the book Oceanic Whitecaps: Their Role in Air-Sea Exchange Processes, although it seems to be pretty difficult to create a good mathematical model for it, since the formula that was presented in the book that was later used to simulate whitecaps in Rendering Natural Waters was an empirical one.

Thanks anyway.

 

[TriKri]

I also ound the paper Real time modelling of multispectral ocean scenes, which also covers the rendering of white caps and which references a couple of older articles. This paper uses the vertical downward acceleration to determine where the whitecaps is located, and dynamically calculates a threshold, also this time in order to control the surface fraction.

So to calculate a surface fraction for the whitecaps and then use some relatively simple property of the surface as sort key to sort out where the whitecaps should be seems to be a quite popular method.

I guess this method, using the (Lagrangian) downward vertical acceleration, can be generalized by rewriting it as a function of the pressure gradient and the gravity. What one will find is that the pressure gradient is lower where the downward acceleration is larger. Under gravity, or in accelerated systems, a pressure gradient is formed. This is an indication that water and air are separated by some external force. If no pressure gradient is present, there is no force that actively keeps air and water separated, and so they will mix much more easily. Having a very low or no pressure gradient would in this case mean that the downward acceleration is very high so I guess that's a quite good indicator for when whitecaps should be created. A downward acceleration larger than the gravitational acceleration should definitely not be allowed at all, and therefore I guess when the two accelerations start to get close to each other, and when the wind strength is high, whitecaps are formed.

Then it's another question whether you will actually see these large downward accelerations in a simulation, especially when you are using a coarse grid. I do believe that the wind plays a fairly large roll in forming the tip of the waves, and unless you're doing a really detailed simulated, you may miss out on these sharp wave tips that are formed by the wind. But maybe it can be allowed to lower the threshold for coarser grids to get around this problem.

I guess one can also combine these methods with some theory for dispersive systems, so that whitecaps area doesn't just go away immediately when the threshold value isn't reached, but more like fades away. Maybe just a plain half time for the foam that doesn't meet the threshold requirements will do.

These are just some notes to myself. And if anyone else who would be interested in solving the same problem would find this thread. :)

 

[Studiot]

Are you aware of the work of

Sverdrup, Munk and Bretschneider

Darbyshire

Pierson, Neuman and James?

 

[TriKri]

No, I wasn't, but I will look them up! Thanks!