Maths, but not as you know it…
My PhD thesis on modelling the spread of river water in the ocean in its entirety – not for the faint hearted! Unless you are a researcher in fluid mechanics, I strongly recommend reading the summary articles here before tackling the beast below. If you have any questions/comments please do get in touch via the contact form.
Video of my ‘Teddy Talk’ at the 2019 St Edmund Hall open day.
Rivers are the major source of pollution in the oceans and if we are to clean them up, we first need to know where the majority of the pollution is concentrated. By creating a mathematical model for river outflows – verified by laboratory experiments and fieldwork – the goal is to be able to predict which areas are most susceptible to pollution from rivers and thus coordinate clean-up operations as effectively as possible.
If you’ve been following so far, we know why it’s useful to know where river water goes when it enters the ocean, why we can build a model of the situation in the lab, what the most important variables in the problem are and what the experimental setup looks like. I suppose you’re probably itching to know what actually happens when we open the floodgates and release the freshwater from the model river into the spinning saltwater tank that is our ocean. In which case, let me point you in the direction of the video below.
It probably doesn’t make much sense without some added explanation so here goes… This is a false colour image of an experiment viewed from above. The freshwater from the river is dyed red with food colouring which means that we can convert the colour intensity into a depth measurement. The more intense the red food colouring, i.e. the more of it there is, the deeper the current must be. The scale starts with black to represent no current (as is the case for the saltwater ocean), then increases with the current depth through red, yellow, green and finally blue for the deepest parts of the current.
If you look at the very beginning of the video you will see that as the river water is released from the source it travels forwards and then is immediately forced to turn to the right. This is due to the Coriolis force arising from the rotating tank (see article 3). As it turns back on itself it eventually collides with the tank wall where it then propagates as an anticlockwise boundary current. The anticlockwise direction is set by the direction of the rotation of the tank (also anticlockwise). The boundary current continues to travel around the edge of the tank, eventually filling the whole perimeter and returning back to the source by the end of the experiment.
As well as the propagating current, a second persistent feature can also be seen in the video: the outflow vortex. This is the large whirlpool-like feature that forms next to the source of freshwater. As the initial current turns to the right and back on itself to collide with the tank wall, the flow divides. One part moves anticlockwise and forms the boundary current that moves around the tank edge, whilst the other part continues in a clockwise direction and re-joins the initial jet of freshwater from the source. The result of this is to form a whirlpool next to the source which grows in size as the experiment progresses. Both features are labelled in the image below so that you can recognise them in the video.
Now that we’ve identified the two main features of the flow – the boundary current and the outflow vortex – the next step is to try to understand them in more detail. This is exactly what the first two sections of my thesis are about, beginning with the boundary current. For example, we might want to know how fast the current moves around the tank, how its depth changes as it does so and whether or not it has a constant width. For the outflow vortex, we are interested in similar properties, such as how deep the vortex is at its centre, what shape it forms at the surface and how it grows in size during an experiment. By looking at the experimental video you can begin to get a grasp on some of these questions, but to really understand them in detail you need two key ingredients: measurements and a mathematical theory.
In my thesis, I begin by discussing the mathematical models used to describe the flow and then compare this theory with the data collected from the experiments, with the hope that they will agree. The theory can only be correct if the measurements support it – which is pretty much my thesis in a nutshell. Do the predictions from the mathematical equations agree with the observations in the experiments? If we’re going to compare the two, we’d better start by forming some equations, which brings me nicely onto the next topic…
You can read the rest of the articles explaining my PhD thesis here.
Yes, you read that correctly. I really did spend a good chunk of the 4 years of my PhD spinning a giant fish tank. This isn’t just any old spinning fish tank though, as you may have guessed, it was specially designed to represent the real-world scenario of a river discharging into the ocean, but on a manageable scale in the lab. So what does such a setup look like? Well, below is a diagram of the tank, taken from my thesis (please excuse my crude drawing in Microsoft Word – it’s harder than you might think).
Let’s start with the tank itself. It’s made from acrylic (basically plastic) and is 1m x 1m with a depth of around 60cm, though it was only filled to 40cm with salt water. As you can see in the diagram, the tank is actually divided up into two sections by an internal barrier. The reason for this was to allow the source structure to be attached to the outside of the tank – I wasn’t allowed to drill holes into the actual tank despite all of my protests… It had to be attached in this way to allow the freshwater from the model river to flow into the main tank which contained the salt water in the model ocean.
The freshwater is stored in the reservoir (a plastic box) attached to the top of the metal structure surrounding the whole setup. This is to provide stability once the whole thing starts rotating, and also I assume to prevent things from flying off and hitting me. The river water flows out of the reservoir down a pipe and enters into the source structure. The amount of water released is controlled by a flow meter and an electronic switch. Once in the source structure, the water begins to accumulate until it resembles a flowing river and then it is released into the ocean (saltwater ambient). The water continues to flow throughout an experiment, much like a real river.
This all happens of course whilst the whole thing is spinning on a giant turntable. Turntable is an appropriate name because it basically just looks like a giant set of DJ decks (with only one disc a metre across). The speed is controlled by a computer and it can go pretty fast, trust me. Before the electronic switch was installed to start and stop the flow of freshwater from the reservoir, I used to have to climb up a ladder and flick the switch manually as it went spinning past. This was fine for the low speeds, but once things started speeding up I couldn’t flick the switch without knocking the entire structure which disturbed all of the measurements I had made. The only answer was to basically climb onto the structure myself and hang off the side whilst it went spinning round at high speed. If anyone ever saw me doing that I’m pretty sure I would have been thrown out of the lab, but hey when science calls ‘you gotta do what you gotta do’.
Now the source structure (shown in the diagram and picture above) was quite the piece of engineering… by which I mean an absolute nightmare to design. Most of the first year of my PhD was spent trying out different designs until finally we found one that gave a discharge that looked like an actual river. If the stream comes out too strong then it looks like a jet – think about squirting a hose in your garden. If it’s not strong enough then it’s just a point source – like a sponge slowly leaking out water. Neither of which represent a river. The trick was to feed the pipe carrying the freshwater into an l-shaped box filled with foam. The combination of the shape, plus the presence of the foam, meant that the box would become sufficiently filled with water before any of it exited into the ocean. It was key that the box filled up above the depth of the opening into the saltwater, so that the depth of the water when it left the source was known (and equal to the depth of the opening). We need to know this initial depth because the depth of the freshwater as it enters into the ocean is incredibly important in determining the properties of the current that forms, but that ladies and gentleman is another story for another day.
All of the articles explaining my PhD thesis can be found here.
My thesis is based on experiments. A weird thing for a mathematician to say you might think, but that’s the truth. It was always planned to be experimental in nature – it even says so in the title – and that’s because it isn’t practical to go to the nearest large scale river outflow (for my work that would be the Rhine in the Netherlands) and start trying to measure things. Fieldwork works well on a Geography trip: you put your wellies on and start splashing around in a stream, measuring the depth with a metre ruler and the speed of the stream by timing a little paper boat as it sails downstream… But things aren’t so easy when you’re talking about a river several kilometres wide and tens of metres deep. The bigger rivers are harder to measure, but the big rivers are precisely the ones that we need to look at, because they’re the only ones big enough to be affected by the earth’s rotation. But we’ll come to that. First up, how do we recreate a big river in the lab?
The trick is to scale things down, as you may have guessed, but there’s a little more to it than just building a scale model of a river. Those little wooden models of a city or building that architects use to help see how their plans will come to life are scale models of the real thing: they are built the same, just with all measurements at a ratio of 1:500 say of the real distances. For example, if a real-life football pitch is 100m in length and 75m wide, then for a 1:500 scale model it would be 20cm long and 15cm wide. A scale model is a good idea in principal, but when working with rivers that are 1km wide and 10m deep, once you scale it down to lab-size, your depth is about as thin as a piece of paper, which isn’t practical. We have to be a little cleverer as mathematicians and think about what properties of the river are the most important and then only include those in our lab model.
I’ll give you an example. Let’s suppose we are trying to work out how fast Usain Bolt travels when he runs the 100m. For ease of the numbers, we can say he runs 100m in 10.0 seconds (a slow day for Usain – he’d had a few too many chicken nuggets before the race). There are many other factors that will affect his speed:
We know that all of these things will affect Usain’s speed, but which do we actually think are important enough to include them in our model? If we ignore them all then at a first guess, we can just use the speed = distance/time triangle from school which gives 100/10 = 10m/s. This would be a first order estimate using just two properties: time and distance. If we want to include more information and get a more accurate answer, then maybe we can include the wind speed: 1m/s against him means he must run at 11m/s to cover 100m in 10 seconds. This is an increase of 10% on our first estimate of his speed, and so probably quite important. Because of the direction of the wind the rain will act against him too, though probably by only a very small amount. The coat will increase the air resistance slowing him down, but again probably quite small.
The key point here is that there are many factors that will affect the speed of Usain Bolt as he’s running the 100m, but as mathematicians our job is to figure out which ones are the most important and to only consider those. If we tried to model every small effect things would get very complicated very quickly and we don’t want that (trust me I’ve tried). Simple is good – so long as you don’t ignore the important bits…
For Usain’s speed we can probably keep the distance, time and the wind speed and that’s about it. Even if we included everything else I doubt the value would change very much from 11m/s, certainly by less than 10% which is a nice acceptable error that we can live with. For scaling rivers down to work with them in the lab we have to do the same thing: pick the important parts of the problem and ignore the rest. The key thing is picking the right bits – which we’ll come onto next time.
All of the articles explaining my PhD thesis can be found here.
It might seem like a simple question, but just think about it for a second… Water falls from the sky as rain, it flows over and under the ground and enters into a river. The river flows downstream, maybe passing through a few lakes along the way, until it reaches the ocean. Now what happens? The water has to enter the sea and will eventually be evaporated by heating from the sun and end up back in the atmosphere to form rain again. A lovely full-circle route called the water cycle – you probably learnt about it in Geography class. But, what if we could track a raindrop from the sky, into a river and then into the ocean. What would happen? Does it just flow into the ocean and then get mixed up and thrown around in the wind and waves? Or do tides drag it out further to sea? And what about the fact that the Earth is rotating? Perhaps not so simple after all…
Image credit: https://pmm.nasa.gov/education/water-cycle
This is essentially what my thesis is about. When water from a river enters the ocean, where does it go? It took me almost 4 years to figure it out – or more accurately to understand more about what’s actually happening… I certainly do not have all of the answers (as you’ll see). My plan is this series of articles is to try and explain 4 years’ worth of laboratory experiments, fieldwork, computer simulations and of course maths, so that anyone can understand what I’ve done and why it is important.
That’s probably as good a place to start as any: why is it important to understand more about where river water goes? Also known as the classic question faced by all researchers: why should we care? The grand big-picture answer is of course that by understanding more about the world around us, then and only then, can we begin to answer the fundamental questions about the meaning of life, the universe and our very being. That all sounds a little too Brain Cox to me so let’s try a simpler reason… pollution.
From waste outflows leaving factories to fertilisers used by farmers, it all flows into our rivers. And we want to know where this pollution will end up so that we can try to stop it causing too much damage. If we know where river water goes, then we know where pollution goes – easy (or so we hope). Pollution from fertilisers is a particular problem because it’s difficult to stop. If a factory is pumping out pollution into a river we can tell them to stop and to dispose of the waste properly by some other means. If we tell farmers to stop using fertilisers then they produce less crops, which means less food for us – hopefully you can see the issue.
The fertilisers used on crops seep into the soil and enter the underground water supply. This then flows into rivers, which flow into the sea. Fertilisers contain lots of nitrogen and this is great for growing plants – they love the stuff. The problem with having lots of nitrogen in the ocean is that it causes huge plankton blooms. Plankton are little plant-like things floating everywhere in the ocean, basically tiny sea plants. If you get lots of plankton blooming at the surface of the ocean, it blocks the sunlight from reaching the plankton beneath the surface and so they can’t photosynthesise to produce food, which means that they die. Lots of dead material means lots of bacteria. These little critters break down the dead plant material and when doing so use up oxygen in the water until eventually there’s none left. This is very bad news for fish – they need oxygen to breathe – and so they end up dying too. It’s basically a big circle of death which we scientists call eutrophication.
The good news is that if we know where the river water ends up and therefore where the fertiliser ends up, we can put measures in place to stop eutrophication from happening. So no more dead fish – hooray! At least not until they are caught in giant nets, but that’s a whole other kettle of fish… (pun very much intended).
So there you have it: knowing where river water goes means we can control pollution and stop fish from suffocating to death. And of course by understanding more about the world around us we can begin to answer the fundamental questions… nope I can’t do it. I’ll leave the star-gazing to Brain Cox.
All of the articles explaining my PhD thesis can be found here.
