dynamical similarity – Bean Thinking

coffee at Watch House — What links a coffee to a guitar amplifier?

What links a coffee to music by the likes of Eric Clapton and Jimi Hendrix?

As we sit back and enjoy the aroma from our coffee, we may rue the fact that our precious brew is evaporating away. We know from experience that hot coffee evaporates faster than cold coffee and we may dimly remember the physics that explains why this is. But have you ever stopped to consider that it is this bit of your coffee that forms a link between your drink and those famous guitarists?

The link concerns the mechanism behind the evaporation. To evaporate out of the coffee, a water molecule needs to overcome a certain energy barrier, let’s call it W, in order to escape. Given that W is constant, the more energy a water molecule has, the greater its likelihood of escape. So we could say that the probability of a water molecule escaping the coffee goes as exp{-W/kT} which means, the higher the temperature, T, the smaller the ratio W/kT and hence the greater the probability (because the exponential is raised to a negative power and hence is a dividing factor). The k is a constant known as the Boltzmann constant.

thermometer in a nun mug — Hot coffee evaporates more. Something that Halley had noticed in his experiments at the Royal Society

Now think about how the amplifiers used by many musicians work. It seems that many guitarists favour valve amplifiers owing to the type of sound they produce. Certainly Clapton and Hendrix were well known for their use of valve amps. A valve amp works by a process of thermionic emission in which electrons are ‘evaporated’ from a hot metal wire before being accelerated to a positively charged plate. This bit is the ‘valve’. In order to escape the metal wire, the electrons have to overcome a certain energy barrier, let’s call it Ω. Just as with W and the coffee, this barrier is a property of the metal that the electron evaporates from. The more energy an electron has (the higher its temperature), the greater the likelihood of it escaping the metal filament and fulfilling its role in the valve amplifier. Hence the mathematics describing thermionic emission is the same as the mathematics describing the evaporation in your coffee cup¹ and the probability of thermionic emission goes as exp{-Ω/kT}.

Now the size of the barrier is of course different in the two cases (Ω is much larger than W) which is why you have to plug in your amplifier to the electricity supply rather than just let it sit on the table top. But this is a difference of size rather than of kind. It is another of those connections between your coffee cup and the world that can be stranger than you may at first think.

If you think of a connection between your coffee and an interesting bit of physics, why not share it in the comments section below.

¹This discussion originally appeared in (and was adapted from) the Feynmann Lectures on Physics, Vol. 1

vortices in coffee — A vortex … (Dragging a spoon through a cup of coffee)

Science involves designing experiments to test theories. I do not want to get distracted here by how a theory is defined or the precise ways in which a theory is tested by experiment. The point of this week’s Daily Grind is to look at the role of experiments in physics, where they can be used, where it is more difficult to use experiments to test hypotheses and, how this can be connected with coffee. Some physics can be relatively easily tested by observation or experiment: we can for example take photographs of distant no-longer-planets to test theories about the evolution of the solar system or measure the viscosity of a liquid as we add something to it. Yet there are some areas of physics where it is not immediately obvious how you would test any theory that you develop. One such area is atmospheric physics where the limitations of living on one planet with one atmosphere where many different things all happen at once, could potentially be a bit of a problem for doing experiments on the theories of atmospheric physics.

vortices, turbulence, coffee cup physics, coffee cup science — … is a vortex… (What happens if you put a coffee on a record player?)

Fortunately, there is a way in which atmospheric physicists can test their theories with experiment and, perhaps unsurprisingly for the Daily Grind, that way involves a cup of coffee (or tea). The route out is called “dynamical similarity” and it is a consequence of the fact that the same mathematics describes much of that which happens in a cup of tea as it does the atmosphere. It is true that a tea cup is a lot smaller than the atmosphere but a vortex in a tea cup is the same as a vortex in the atmosphere even if one is only a centimetre across while the other has a core size of many kilometres. The mathematics will be the same. This allows people to test hypotheses formed about the atmosphere in an environment that they can control and repeat.

A vortex in the atmosphere — … is a vortex.
(Typhoon Nangka, Image Credit: NASA image courtesy Jeff Schmaltz, LANCE/EOSDIS MODIS Rapid Response Team at NASA GSFC. Caption by Kathryn Hansen)

A couple of months ago, I wrote an article in Physics World about the connections between coffee and physics. Shortly after it came out, I got an email from Paul Williams alerting me to an article that he had written in the journal Weather called “Storm in a tea cup“. It turns out that the subject of his research had been to study the impact on the weather of the interaction of two types of atmospheric waves: Rossby Waves and Inertia-gravity waves. The method that he had used to test this was, if not quite a tea cup, a bucket which he could rotate. Rossby waves and inertia-gravity waves are both present in the atmosphere and can be induced, albeit on a smaller scale, in a bucket. He was using the concept of dynamical similarity to explore what happens in our atmosphere. And the experiment was important. Before his experiments, it had been thought that the effect of the interaction of these two sorts of waves was minimal. His experiments revealed that this may not be the case, the inertia-gravity waves can significantly affect the Rossby waves. Given that Rossby waves are responsible for cold/warm fronts and weather phenomena in mid-latitude regions of the world (such as the UK) his results, and his cup of tea, were potentially very important.

I’m always very happy to hear about what others are doing with science in a tea cup or a coffee mug. Please share any thoughts in the comments section below.

Paul Williams “Storm in a tea cup” can be found in Weather, 59, (4), p.96 (2004)

With apologies to Gertrude Stein.