Does it spread out everywhere?

Does it do look dealing?

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 the river hits the mouth, enters the ocean.

But where does it actually go?

We don't really know.

I always thought just what would blast out to sea like in a slowing plume.

The Coriolis force is zero on the equator to the rear.

Amazon does just blast out in a plume like you describe.

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 traveling down the river.

And as you reach the coast, turn right that could be east or west of heading on where your rivers flow, and then in the Southern Hemisphere, same thing.

But you're gonna turn to the left on.

We can see this.

The satellite images.

You get this by doing experiments.

That's what I did.

Do some experiments in the lab when clockwise anti clockwise you see right and left did it without rotation.

And it looks like the Amazon in that big blue.

Ultimately, the Earth's rotation is the most important factor in determining where this river water goes.

Because the effect off the Coriolis force, he actually scales exactly as sign off 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 against more, 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 for me.

So there's this sort of large whirlpool that forms near the river mouth.

But there's this particular point here where it divides that you're a particle coming in.

You can either go like this for a while.

We'll sort of stay there.

Or, if you are near enough the edge, you sort of go around and then you enter this current, now the bit near the source.

It's interesting for a mass 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 pops, 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 on?

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 operations, 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 to model this.

You'd think about physics the matter of it, and you need to be clever.

So what we want to know is we want to know it's with so called w not.

We want to know how fast it's going velocity.

You know what?

We want to know how deep it is.

I'm gonna call that hate note.

So 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 gonna be your fresh water, your river and then below it.

Here, you've got your C your salt water.

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 clean up the top layer of the ocean.

It's incredibly important information, actually clean operations.

Fixing pollution on the math is also awesome.

We're doing fluid mechanics.

So the key thing really is to think about what are the variables?

What are the parameters?

They're actually gonna effect 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 North Hemisphere, the Southern Hemisphere.

So we're going to have our Coriolis parameter.

Our rotation rate is going to be F to play such a major role that we're also going 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 volume flux out flow rate from a river?

So I'm gonna call that one.

Q Q.

Is our value flux.

So I've got a giant river like the Amazon fed to a tiny little stream.

Different things will happen.

And then it turns out the third vile thing we're going to need is going to be the density difference.

River water is going to float on top off 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 off the salt water, the ocean on dhe gravity.

So these are 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 Naevia stakes equations.

I've talked a lot of the estate's equations in the past, so I'm not gonna 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 gonna make loads and loads of assumptions.

So I said, first assumption is, rotation dominates.

We then also issue in our particular set up the key velocity off 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 vocation is pushing it in that direction.

So what you're doing by by assuming that that velocity is dominant, you're reducing the dimensions.

Often obvious takes because, as we know, we don't know if solutions exist in three days.

So let's not think about three D.

Let's just consider one day.

Let's make it easy for ourselves.

This is how we get around the difficulty of an obvious 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.

That's gonna be our second assumption.

We then assume something called hydrostatic pressure, which basically just say is 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 to final things.

The next one is going to be that our ocean is infinitely deep, ridiculous, but bear with me because the idea here is as as we will see with with our eventual answers, the river current, that bit of fresh water on top of the ocean.

It's pretty shallow, so we're talking 10 to 20 meters compared to an ocean that's hundreds of meters deep.

So it seems at least to me, hopefully everybody else that the fact that your ocean is so much deeper than that top 10 meters it's hundreds of meters down.

You don't expect the bottom of the ocean to interfere with the top layer off your river water, so it may as well be infinitely deep on.

By doing that, it makes the math 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 say's that your vorticity is sort of how swirly your it's like the will pulling us swirling mess of your water.

It just say that if that's what a particular value, it will stay at that about you.

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, amount of water coming in on the density difference.

We make these five assumptions on as if by magic, we get these incredibly nice formula for the three things were interested in having Levy of Stokes equations.

What gives them, too?

Yes, So it starts to Naevia Stokes.

But we're using the classic applied mathematician trick off.

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 on what we find, we get this really neat formula to say that the maximum depth hates naught is equal to two times the rotation rate times the volume flux divided by the density difference.

Oh, to the half.

Even though we started with the Naevia Stokes equations thes incredibly complicated mathematical things we don't understand.

It's amazingly, Daniel is amazing that this pops out from from such a complicated set of equations on.

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, it's larger.

You have more water leaving your river.

You expect the resulting current to be deep.

And then finally, it say's that it's deeper when you're density difference.

It's smaller, so the density of the two waters are really close together.

They mix together really well.

Where is when they're really far apart?

You 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 on it.

Came out of the equation uninvolved.

The three things we thought were important, 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 size rivers?

So F is really small.

F is one rotation of the earth in 24 hours, so f his tiny because the earth's huge, so it moves very, very slight.

Cue the volume flux, so the River Rhine is the biggest river in the North Sea, while the ones I studied with this work and we compared and got similar results to satellite data you're looking at like 2000 meters Q per second for the density difference.

If your fresh water is is 1000 then your ocean waters.

1000 25 sort of is one way of measuring it, so it's slightly slightly more dense.

So the G prime value is Jones to be quite small, so F is quite small.

G prime is quite small, but you could be quite big.

So how deep does that give us?

What kind of dips story?

So for the River Rhine, this guy comes out at about 10 to 20 meters because, of course, it will vary depending on the season.

So in the summer there's a smaller volume.

Flux density is much more similar, whereas in the winter really large volume flexed.

Everything's gonna happen in different seasons.

But you you get a vibe between 10 20 meters.

You measure it with a ship you measure on the satellite.

It's about 10 meters deep, so it's sort of it's giving us the right kind of values.

So we're gonna clean up the North Sea along the coast of the Netherlands, France and Germany.

We only need to clean the top 10 meters.

That's the kind of knowledge you can get from 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 kilometers out to sea, or do we have to go away out 100 kilometers?

We get really neat formula for the width.

W not forgotten.

Eight This time G Prime Q Divided by F Cube or to the 1/4 like before, Let's think a little bit.

Does this make sense?

If we have a larger volume flux, we expected to be deeper on, We expect it to be wide, that 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.