Where does river water go when it enters the ocean? Does it spread out everywhere? Does it do a loop-de-loop?
Does it flow along the coast? It sounds like it should be easy, but it's anything but.
It's a research problem, I did it for my PhD,
river water flowing down a river,
hits the mouth and it enters the ocean. But where does it actually go? We don't really know.
(Brady: I always thought it just, like, would blast out to sea like in a...)
(slowing plume.)
The Coriolis force is zero on the equator,
so the river Amazon does just blast out in a plume like you described.
But then when we move into the Northern hemisphere and the Southern hemisphere
the Earth's rotation creates this Coriolis force, which is super, super important.
So that is going to cause your river in the Northern hemisphere to turn to the right.
And you have to imagine it, when I say 'right,' you have to imagine travelling down the river and as you reach the coast turn right.
So that could be east or west depending on where your river's flowing. And then the southern hemisphere,
same thing, but you're gonna turn to the left.
And we can see this in satellite images. You get this by doing experiments, that's what I did: did some experiments in the lab.
You know, went clockwise/anti-clockwise you see right and left.
Did it without rotation and it looks like the Amazon in that big plume.
Ultimately, the Earth's rotation is the most important factor in determining where this river water goes.
Because the effect of the Coriolis force,
it actually scales exactly as sine of latitude, as in the sine function.
So as you move further away
it gets bigger and bigger and bigger and as you move further south it gets more and more negative.
We want to model a river entering the ocean. So here's our river. This is our coast. We're gonna be in the northern hemisphere,
being currently in the UK. River water comes in and then it's going to turn to the right.
This is what the Coriolis force does. So the water's coming in. It's turning to the side and it's flowing along the coast.
So what you actually get are two very different features.
You see these on satellite images, or you can do experiments - you can actually see these two things forming.
So there's this sort of large whirlpool that forms near the river mouth,
but there's this particular point here where it divides. So you're a particle coming in,
you can either
go like this for a while, or sort of stay there, or if you're near enough the edge you sort of go around
and then you enter this current. Now, the bit near the source
is interesting from a maths viewpoint, but it's not super big in terms of trying to understand this problem.
You just have a big clump of river water near your river mouth,
we would expect that. Perhaps. But then going along the coast what we want to know is, well, how far
along the coast does this river water go?
How deep is this river water? And
maybe how far away from the coast is it? Most of the pollution in our oceans comes from rivers.
So if we're going to start cleaning up our oceans
the sensible thing is to start where there is the most pollution, and this could of course
you know be either tiny plastic beads or chemical pollution that you can't see - so it's not a case of there's a giant blob of
garbage, let's clean it up.
You need to think about, you need to model this. You need to think about the physics, the maths of it and
you need to be clever. So what we want to know is, we want to know its width,
so I'm going to call 'w' naught. We want to know how fast it's going,
velocity, 'u' naught, and we want to know how deep it is and I'm gonna call that h naught. So if we're looking at it
from the side on you've got your river surface with the waves and you've got your sort of current here
which is going to be your fresh water, your river. And then below it here, you've got your sea, your saltwater.
So this is the view from above we want to know the width of it, how far off the coast,
wanna know how fast it's moving, and
then if we sort of flip it on its side, we want to know how far down in the ocean
does that current go? Because if we know these things, you know, we clean up that amount of distance from the coast.
We, we clean up the top layer of the ocean. It's incredibly important information. We're actually
you know, gonna clean up our oceans, fixing pollution and the maths is also awesome, which is an added bonus.
We're doing fluid mechanics. So the key thing really is to think about, what are the variables?
What are the parameters that are actually going to affect
this particular situation.
So first of all, I've hopefully convinced you that the Earth's rotation is
incredibly important. If we're on the equator with the Amazon, it behaves very differently in the northern hemisphere, the southern hemisphere.
So we're going to have our Coriolis parameter. Our rotation rate is going to be f.
It's going to play such a major role that we're also to assume that this guy, the rotation, dominates.
What else might affect the movement of this water? Perhaps how much water is leaving the river?
So what is the the volume flux, or the outflow rate from our river? So I'm going to call that one Q.
Q is our volume flux.
So I've got a giant river like the Amazon compared to a tiny little stream, different things will happen. And then it turns out
the third and final thing that we're going to need is going to be the density difference.
River water is going to float on top of the salt water. That's why we want to know the depth of that top layer.
So we're going to model that with what we call the reduced gravity, g prime, but for our purposes, that's just the density difference.
It's just a formula which involves the density of the river water, the density of the salt water (the ocean), and gravity.
So these are our three key parameters and importantly we've said rotation dominates.
We're then going to have to make some more assumptions.
So fluid mechanics means
Navier-Stokes equations. I've talked a lot about Navier-Stokes equations in the past
so I'm not going to do all of that again.
But in short, set of equations that model the flow of every fluid. They are super difficult,
super tricky, we don't really understand them. We can't really solve them. So we're going to make loads and loads of assumptions.
So, as I said, first assumption is rotation dominates.
We then also assume that in our particular setup
the key velocity of our river current is going to be in the direction along the coast.
There will be some velocity in this boundary current, in this current where, you know,
it might move a little bit this way, and it might move a little bit down or up,
but generally it's going along the coast. The rotation is pushing it in that direction.
So what you're doing by assuming that that velocity is dominant,
you're reducing the dimensions of the Navier-Stokes. Because as we know, we don't know if solutions exist in 3D.
So let's not think about 3D. Let's just consider 1D, let's make it easy for ourselves.
This is how we get around the difficulty of the Navier-Stokes in practice.
So we assume we have this dominant velocity along the coast. And again, we can look at satellite data,
we can look at experiments, and this is true. So that's gonna be our second assumption.
We then assume something called hydrostatic pressure,
which basically just says that your pressure only matters in the depth of the water.
We don't need to worry too much about what that means, but it helps us a lot.
So we have that. And then we're gonna have two final things. The next one is going to be that our ocean is infinitely deep.
Ridiculous, I know, but
bear with me because the idea here is, as we will see with our eventual answers,
the river current- that bit of fresh water on top of the ocean- it's pretty shallow, right?
So we're talking 10 to 20 metres compared to an ocean that's hundreds of metres deep. So
it seems, at least to me, hopefully to everybody else, that
the fact that your ocean is so much deeper than that top 10 metres, it's hundreds of metres down,
you don't expect the bottom of the ocean to interfere with the top layer of your river water.
So it may as well be infinitely deep. And by doing that it makes the maths a lot easier
which is always nice. And then our very final thing which is sort of the key to the whole model,
is that the potential vorticity is conserved.
So in short it says that your vorticity is sort of how swirly your - it's like the
whirlpool-iness, swirliness of your water. It just says that if that's at a particular value, it will stay at that value.
It's something we do all the time in fluid mechanics. So we assume that the PV, potential vorticity, is conserved.
We have these three important parameters:
rotation rate,
volume flux - the amount of water coming in - and the density difference. We make these five assumptions and as if by magic
we get these incredibly nice
formulae for the three things we're interested in. (Brady: Are the Navier-Stokes equations what give them to you?)
Yes, so it starts in Navier-Stokes,
but we're using the the classic applied mathematician trick of,
let's just make all of these assumptions and
worry about whether they were good or bad at the end when we get our result. And what we find,
we get this really neat formula to say that the maximum depth h naught is equal to 2 times the rotation rate,
times the volume flux,
divided by the density difference;
all to the half. Even though we started with the Navier-Stokes equations, these incredibly complicated
mathematical things we don't understand. It's amazing, it genuinely is amazing that this pops out from
such a complicated set of equations. And you can think a little bit about whether this makes sense.
This is saying that the river current gets deeper when your rotation rate is larger.
Says the river current gets deeper when your volume flux is larger.
You have more water leaving your river, you expect that the resulting current to be deeper.
And then finally it says that it's deeper when your density difference is smaller.
So the density of the two waters are really close together,
they mix together really well. Whereas when they're really far apart you've got super salty ocean water, really fresh river water -
they don't mix. You know, it's like mixing water and oil versus mixing salt water, fresh water. You get that difference.
So it all makes perfect sense and it came out of the equation and it involves the three things we thought were important.
(Brady: Tom, not being very familiar with these actual parameters, like what a flux density is and that sort of stuff,)
(what type of numbers do you normally get from typical Earth-sized rivers?)
So f is really small. f is one rotation of the Earth in 24 hours. So f is tiny, right, because the Earth's huge,
so it moves very, very slowly. Q, the volume flux,
so the River Rhine is the biggest river in the North Sea, one of the ones I studied with this work and that we
compared and got similar results to satellite data.
You're looking at like 2,000 metres Q per second. For the density difference if your fresh water is 1000,
then your ocean water is 1025. Sort of, it's one way of measuring it. So it's slightly,
slightly more dense, so the g prime value, it tends to be quite small.
So f is quite small, g prime is quite small, but Q can be quite big.
(Brady: So how deep does that give us, what kind of depths? ) - Sorry, so for the River Rhine
this guy comes out at about 10 to 20 metres.
Because of course it will vary depending on the season. So in the summer
there's a smaller volume flux, you know, the densities might be more similar. Whereas in the winter really large volume flux.
Different things can happen in different seasons, but you get a value between 10 and 20 metres,
you measure it with a ship, you measure it on the satellite. It's about 10 metres deep.
So it's sort of, it's giving us the right kind of values. So if we're going to clean up
the North Sea along the coast of the Netherlands, France and Germany, we only need to clean the top 10 metres.
That's the kind of knowledge you can get from these models. Then we also get the width.
So the width is important because we want to know how far offshore to clean up.
Thinking practically, we want to know do we clean up 10-20 kilometres out to sea or do we have to go all the way out
100 kilometres? And we get a really neat formula for the width. w naught, we've got an 8 this time, g prime Q
divided by f cubed, all to the 1/4. Like before let's think a little bit does this make sense? If we have a
larger volume flux
we expect it to be deeper and we expect it to be wider. There's more water. If we have a larger density difference
this is telling us that
it's going to be wider. So we saw that it was shallower and here this is telling us it will be wider.
Again, if you've got the same amount of water
but you change the density difference, if this is shallower, it's got to go somewhere
so it sort of goes wider, becomes further away.
And then with the rotation, if the rotation is larger, think about
one of those fairground rides where you're in that sort of circular thing and it spins really fast and you're pushed back against the wall.
So the faster it goes, the more you're forced outwards
so you expect it to be less wide, because you're forced more towards the coast the faster
it's moving. And this guy is gonna spit out- it's around 10 to 20 kilometres?
So it's much bigger than the top one because f is so small, because f is tiny
we've got it cubed on the bottom so you get a much bigger number.
So this is now kilometres, 10 to 20 kilometres for the River Rhine.
Again, it's around 20 kilometres on the satellites.
So we know just clean up top 10 metres, 20 kilometres out to sea. And then our final result we can get the velocity.
So the velocity is going to follow a similar pattern except the numbers this time are ridiculous.
So we have 3 over 2 to the nine-quarters -
maybe my favourite number. f, g prime, Q to the 1/4. Again, let's just think about, does this make sense?
We go faster if the Earth rotates faster, if our experiment rotates faster.
Okay, that seems sensible. If there's a larger density difference
they're not going to mix as much.
So if they're not using their energy to mix
they're probably going to use that energy to go faster. And then, again, if there's more water it's going to be pushed through faster.
So they all make complete physical sense. This gives us a value of around one metre per second?
So not super fast, but again the River Rhine, it's around 0.5 to 1 metres per second as well.
So all of them are within the ballpark and this kind of knowledge actually then,
you know, we can use this. We can go out to the River Rhine, go out to the Dutch coast and start
using this, putting this to practice and cleaning up
our oceans by just doing some maths, some Navier-Stokes, some assumptions and sort of getting these really neat formula
explaining what's happening. (Brady: This plume or this current that goes along the coast,)
(obviously it's constantly being fed by more and more water as the river just keeps flowing and flowing and flowing.)
(How long is this plume? How does it end, do the pollutants follow the whole coast of Europe forever and - ?)
Yeah, great question.
So the River Rhine plume flows along the Dutch coast and then along
Germany and then it hits the sort of the corner bit where Denmark then pops up.
And in that corner is the river Elbe, which is the second largest river in the North Sea.
So it kind of merges into the
outflow from the river Elbe and then the river Elbe flows up along the coast of Denmark and then goes into the
Baltic Sea and then sort of all gets trapped
in the Baltic Sea. And then you've got all kinds of different things happening in there because it's quite a
stagnant ocean and it only has one
inlet in and out and it's quite large so you get weird things going on in the Baltic Sea and,
entering a different sort of modelling regime, different physics happening there.
This video was the third in a kind of trilogy about Navier-Stokes equations. To see the other two
there are links on the screen and in the video description.
[Preview] The full Navier-Stokes equations,
as written down on our piece of paper. (Brady: Hang on which way around are we?) - If we're talking this - yeah