So our next speaker is Michael Brenner.
Michael has worked on a variety of problems
in non-linear mechanics, in fields ranging from physics
to biological evolution.
Much was mentioned this morning about the interactions
of the applied math department with EAPs
and other departments at MIT back
in the days of Lorenz and Charney.
I can say that many of us have quite fond memories of when
Michael was in our applied math department in the late '90s.
And he's now at Harvard.
And I give you Michael.
Thank you, Dan.
Thank you for having me.
You're nice.
You've always been much too nice to me.
Anyway, so I'm going to give a talk that
relates to this meeting in various ways
but also doesn't relate to this meeting in various ways.
So we've been on this quest--
I've basically been on this quest
since I was a graduate student.
I didn't even know when I was a graduate student
that I was on this quest, but apparently, I was on it.
Which was the question is, is could one
actually see in real time what the turbulent cascade looks
like?
And, actually, my thesis advisor told me to work on this problem
and I couldn't do it at the time.
And 20 years later, for some reason--
it was like a crisis or something--
I started working on it.
And I'm here to report on what we've
been thinking about with this.
And I mean at the end of the day,
this talk will have two things.
It will have some ideas.
It will have a really spectacular experiment
by a grandson of Harry.
An academic grandson of Harry, not a real grandson of Harry.
And I guess it relates to what we heard in the morning,
because--
at least what I think is-- this is
sort of a siren's warning song about just how under-resolved
your simulations are likely to be.
So that's if you'd like to think about that.
So this is a much simpler problem.
So OK, so can I get this thing--
OK, so these are the people who help-- uh oh.
So Harry's academic grandson--
I think that's right-- is Shmuel.
Shmuel?
Shmuel.
And Shmuel and his graduate student,
Ryan, did what I think is really a stunning experiment.
I'll show you at the end of the talk.
But then at the beginning of the talk,
I'm going to give you nonsense theoretical arguments,
and that's everyone else's fault.
So this-- like Harry-- so we sort of start at the beginning,
namely with GI Taylor.
So to my knowledge, the first person
who asked this question in a seriously quantitative way
was GI Taylor, who wrote this paper in 1936 in which he
asked, basically, for a solution to the Navier-Stokes equation
that amplified vorticity.
And at the time, computers were even more underdeveloped
than they were when Lorenz did his work.
And so Taylor resorted to getting a graduate student
named Green to be his computer and to compute a Taylor series
expansion of the Navier-Stokes equation to fifth order in time
so that he could plot--
with tables of it-- so he could plot
the dissipation rate or the vorticity
as a function of time.
And he saw that the vorticity went up,
it was amplified from the initial solution
by about a factor of 10, depending
on the Reynolds number.
And then it started to go down.
So the question, I think, that underlied all of this
is that we all know-- and basically at that time,
he also knew, even though it wasn't written down--
about the turbulent cascade.
That on average, energy goes from large-scale
to small-scale.
But the question is to please identify
the actual events that caused that transfer to be mediated.
So just in case you think that this isn't interesting,
I'm going to show you an experiment that
got me interested in this a couple of years ago.
This was an old experiment by Lim and Nickels,
in which what they did-- here's a movie--
they took two vortex rings and they
let them collide against each other.
There's a red one and a blue one,
and it formed an explosion.
So it's sort of beautiful.
And so here are frames from their movie.
And what you see is, you take a red ring and a blue ring
and you collide them.
And you get little rings, which are half red and have blue.
And the paper was published in Nature
because the editors were excited that the thing was
half red and half blue.
So now, of course, anyone who knows
anything about fluid mechanics knows that that's
completely ridiculous.
I mean, of course they're half red and half blue,
because there's an instability, which
is well-known in this experiment that
leads to vortices which reconnect and make
half red and half blue.
So the interesting figure in this paper,
however, was slightly later--
you had to keep reading--
in which the authors increased the Reynolds number
to about 3,500.
Look at this picture.
And when they collide, they produced smoke.
This is a blow up.
And it's really amazing, actually, because the time
that this took, this was 2.74 seconds.
The thing went from something coherent to smoke.
And so the question is-- so this is presumably
happening constantly in a turbulent flow--
this experiment was cleverly designed
so that the phenomena was stationary in the laboratory
frame.
And so even at this point, they could take a picture of it.
And the question is to please identify the dynamics that
leads to this smoke.
OK, so this talk has two parts.
And Dan, please, if I'm four minutes before,
tell me, because I just want to make sure I show you these.
Oh, that's the time.
Wow, you guys are--
I didn't know that.
The speakers have a thing.
Huh.
OK.
So anyway, this talk has two parts.
The first part is theory.
And I'm going to just give you a sketch of theory.
And what I'm going to basically tell
you about is a mechanism that we invented
for some strange reason, which essentially
involves iterative cascades occurring during this process.
And then, I'm going to show you Shmuel and Ryan's experiment,
in which they managed to visualize
this in the experiment that I just showed you.
So in mathematics, the question that I'm talking about
has been very popular in the last 20 years.
20 years?
20 years, because it's a famous problem.
It's one of these clay problems.
And the mathematics people talk about this
as the question of smoothness of the Euler and Navier-Stokes
equations, which I must admit, I always
thought was sort of boring because it's
posed to be quite mathematical.
I mean, on dimensional grounds, if u as the velocity
field, the gradient of u has a scale which is 1 over a time,
and so if it's timed to a singularity where there's
actual blow up of vorticity, then the scaling law
should look like this.
And the math community has spent a lot of time
studying whether or not this formula is correct.
And basically, no one knows at this moment.
Practically, whether or not there's
a singularity is essentially irrelevant in practice.
On the other hand--
but what matters, and what I think
is really an important problem, is
to decide, to identify what the mechanism is
that's leading to this process.
Whether it's singular or not, it just doesn't really matter.
And so the interesting thing about the experiment
that I showed you is it shows you that something happens.
And one would just like to be able to describe
in some way what it is.
And the notion is is that because there is a clear scale
separation-- that is, you go from a big thing
to like smoke something-- then there
should be some dynamics that one could characterize
that governs that transition.
So OK, so I'm to just sketch calculations
and then I'll start talking more quickly so that I
can get to the experiment.
So basically, we did a calculation
in the simplest way.
We started out with two rings, a red ring and a blue ring.
And if you assume mathematically that the radius
of the ring, that the core radius
is much smaller than the radius of curvature,
then there's a very nice, simple description
than one can write down and solve
for the dynamics of the rings, which
is the basically the Biot-Savart law from electrostatics.
And this law is not uniformly accurate, but it's intuitive
and it's accurate, as long as these assumptions hold.
And so we basically are going to start
by just showing you solutions of that for this ring problem.
So there are two pieces of physics
that are involved in these equations for two
colliding rings.
One is that there's the self interaction of the filament.
There's the fact that the filament interacts with itself.
And that gives what I call the smoke ring law.
It's because it's curved.
You know, smoke rings translate.
The other is is that the two rings interact with each other.
And if you look at the rings closely,
they look like 2 point vortices that
are sort of next to each other.
And that causes the ring to expand.
So those are the two basic bits of physics
that are in this equation.
And so in order to sort of close this and think about it
properly, you have to say something about what's
happening to the core.
And in the simplest model, right-- because their core
contains the vorticity-- and in the simplest model
as the thing expands, then the core should shrink,
because the total amount of vorticity is conserved.
And so one can basically sort of just
write down a phenomenological law, which
also has been studied in the literature--
and it's not so bad-- that says that the area of the core,
basically, decreases like 1 over the stretching rate
of the thing.
And so that means that the vorticity is actually
growing like 1 over the area, or it's
sort of growing like the stretching rate of the core.
So OK, so this is a well-posed math problem that you
can study if you're bored.
It's correct as long as the core radius is small.
And so we spent some time studying this.
Let me just show you a simulation of this quickly.
So you see, these are two things that are coming together.
There's a red one and a blue one.
And the curvature is actually diverging.
So if you look, the curvature is diverging in the solution.
So it's actually a singularity of the Biot-Savart equations.
The problem is is that it's not a singularity that is--
the singularity of base scaling laws,
that basically says that any scale goes
like the square root of time.
And one can as one does-- if you're a physicist or whatever
I am--
sort of write down similarity solutions
and characterize the dynamics.
And you can do that in--
I don't know, there's math.
OK.
And what you find is that the similarity solution that comes
out looks like a double tent.
So what it looks like is there's two tents and the tents
meet at a point.
And the point is where all the action is.
And basically, we spent a lot of time characterizing all
the solutions of these tents.
I don't know why.
So double tents-- if you look in the literature--
have long been observed.
So this is a paper from the '80s, where there are
two rings that are colliding.
And you see, they make double tents.
Vortex reconnection often has double tents.
And we, just to check, did simulations of this.
This is Rodolfo, who was a post-doc at Harvard.
And this is the Biot-Savart equations.
And you'll see that as they collide,
they make lots of double tents.
Double tents.
So there are tents.
So the action is happening in the tents.
So what happens at the tents?
The curvature blows up at the tents.
So the thing is is that if you look at the solutions, what
you discover very quickly is that the core radius doesn't
shrink quickly enough for the approximation
that I just stated to be uniformly accurate.
And so at some point, you lose double tents.
And so what happens in practice-- and everyone who
studies fluid mechanics knows this--
is that the tents flatten and you have two flattened things.
So this is hidden math.
Basically, we're able to calculate
how much flattening there is.
That is, when these tents [INAUDIBLE]
and how much does it flatten.
And it turns out there's a formula, which
says that the aspect ratio, a is the--
on this picture-- a is this dimension
and b is this small thickness.
This aspect ratio-- this to this--
basically goes like the radius of curvature
of the perturbation divided by the core radius to a power.
And the power is about two.
And this aspect ratio, if you put in actual numbers,
this is a very, very large aspect ratio.
So what this says, actually, just from the point of view
of theory, is that you will-- just by colliding these
things--
make very small land scales very quickly.
You'll make very thin sheets.
And so you can go and look at the literature
and there actually aren't thin sheets
in the simulational literature.
The reason is because nobody's been able to resolve them.
In fact, all of simulate-- because they become so thin so
quickly when you simulate the damn thing that you basically
run out of resolution, given the resolution goes
like the cube of the box and all that.
And I mean, these are pretty ubiquitous anyway.
So we then were sort of wondering.
Now, of course, sheets are not smoke.
And so there was a paper of Terry Tao
that I don't have time to describe
that was sort of interesting.
And because of what we were sort of wondering, well,
what happens after you make the sheets, right?
And that would make small scales.
And so the following picture emerged.
So what happens to sheets of vorticity?
And so if you read old literature like Lord Rayleigh
he tells you that sheets are unstable.
And so sheets are unstable.
And there's a long literature about the instability
of sheets.
And so we thought, well maybe the filaments
will create these sheets and then
the sheets will create more filaments.
And then, the filaments will again collide and make
more sheets.
And you see, it could just go around and around.
You know what I mean?
Again and again.
And then at this point, you come to the point
where mathematics is impossible.
You can't calculate in this regime.
I can sort of describe to you--
but what we did-- and this is sort of inspired by Lorenz,
I think, at heart-- was we did the only thing you could do,
which is we made a map.
We made a map.
Like, Lorenz had maps, well we had maps.
So we basically derived a map from our similarity solution,
assuming that this is what happens.
Please say how much vorticity, how much circulation there
is in every filament and what the thickness is.
And could this actually continue forever?
And what we discovered is that it is not
inconsistent with the equation of motion
that this would happen.
But, of course, that's far from a proof.
OK, so now we come to Shmuel.
And so this experiment-- so Shmuel--
these guys are just great experimentalists.
Anyway, I'm just going to show you.
So instead of talking about it, they made a movie.
And I'm just going to show you their movie.
This was a movie of what they did,
which I think summarizes the whole thing better than I
could do.
Oh, shoot.
It has to-- as long as the movie plays.
Oh no.
No, no.
The movie has to play.
This is the highlight of the talk.
This is like one of these things.
You know, it's the only part of the talk which is reasonable.
OK, so here's their movie.
So when vortex rings collide, they rapidly
break in-- so this is repeating the Lim and Nickels experiment.
This is in a lab at the basement of a building at Harvard.
And actually, if you can turn off the lights in front,
actually.
If somebody could turn off the screen lights because actually,
this movie gets better.
Sorry.
I can brag about it because I didn't make it.
It's really nice.
If you could get the lights right there.
If somebody could turn them off.
Is it possible, you guys, to turn off the light?
OK.
So you can't visualize from that.
So what Shmuel did was to basically
put a laser sheet in the center and to scan the laser sheet
at very high field and then do three-dimensional
reconstructions of the thing.
So these are the tents.
Which they'd break down.
So these are scale bars, this is time.
But now, you can sort of see.
OK, so now this is 3D high-speed scanning light microscopy.
So there's a laser sheet and it scans at a rapid rate
and then there's a three-dimensional
reconstruction.
It's amazing what you can do with modern software.
This is experiment, I just wanted to be--
So the technical details in this experiment
were many, as you can imagine.
So this is one core dyed, there's
another core at the bottom which you can't see.
So now I want you to watch.
Look.
Do you see this?
It makes a sheet.
Look how thin the sheet is.
And now, the sheet breaks and makes a hole.
There's a hole in the sheet.
Now, there are two more filaments.
The hole, you can only make a hole
in the sheet with viscosity.
So for those of you who are interested in fluid mechanics.
So viscosity has just come into the problem,
but it's now gone because it's now very, very inertial.
So do you see the threshold?
There's a dye threshold, but you can change in the software.
OK, now the tertiary filaments will go.
And this is a single picture actuary.
So this is actually a snapshot, there's no movie.
You see one sheet to two filaments
to tertiary filaments.
OK, so that's the experiment.
Like I said, that was by far the best part of this talk.
So we also did numerical simulations.
I only have 17 seconds left.
Oh, I'm OK?
I have four minutes?
Oh, wow.
That's the good way to do this.
Including questions.
OK, well I need to make sure there are no questions,
so I should talk a lot.
So we did simulations.
Actually in simulations, it was a real challenge
to basically get.
Our goal was very modest in the simulations,
we just wanted to see the seed of one iteration.
That was it.
And Rodolfo, who is a post-doc, managed
to get this to work with a code.
This is sort of part of it.
Unfortunately, there are symmetry-- this movie
could be made better.
In fact, what we ended up doing was calculating the dye as well
as the vorticity.
So one problem with the experiment, of course,
is that you're measuring dye, you're not measuring vorticity.
And you might worry whether vorticity and dye
are the same thing.
And you'll learn from these simulations
that they are the same thing.
And actually, if you go through this, you can see.
Unfortunately, there's a symmetry,
this thing is periodic.
And so it's this way, but if you stare at it,
you can see that it actually-- if you look at vorticity there,
the dye-- you see it broke down into a thing.
See this?
Look.
You see it?
Oh.
See, there it goes.
It went from one to two.
OK, so that's the thing.
I could talk about it more, but I only have a minute left.
So let's see.
So in summary, I guess I was really surprised
by this whole little endeavor.
I sort of thought, first of all, that the question
of whether there are singularities in the Euler
equation was stupid.
I also thought that--
this is probably recorded actually, oops--
that is, I also thought that the question
of whether the viscosity regularized it
was also stupid, because of course it does.
That was my opinion.
And I also thought that--
I thought lots of things.
I also thought that the fact that no one had ever seen one
in a simulation must mean that they couldn't exist,
because everybody's been trying.
Like, there are all these people who are trying.
And I guess now what I think is we're not
even close to the computational power
that it requires to do this.
The truth is, the only way to resolve what I just showed you
at the moment is with experiment.
There is no way to do this without massive remeshing
in a simulation.
But if you do remeshing-- as many of you probably
know-- you introduce extra factors
in the simulation, which makes them
much harder to control and to be sure that they're accurate.
In terms of the role of viscosity,
you will notice that in the experiment,
you saw that the way that the cascade went
was different than what we imagined in a certain sense.
That the sheet formed and then the sheet actually popped.
The popping of the sheet, there's a theorem.
The popping of the sheet can only
happen because of viscosity.
So that meant it made such a small scale
that viscosity came in then.
But then, it was not viscous anymore.
Right?
Because it was then continuing as these filaments,
which were massively going around very quickly.
In fact for a long time, Shmuel and Ryan
were not able to see this phenomena in the experiments,
because what actually happens is is when you pop the hole,
then you have these vortex filaments, which
are highly curved.
And they start rotating very, very quickly.
And if you don't time resolve fast enough,
they rotate so quickly that it smears out.
And it's exactly when the hole pops,
so it's exactly when you'd like to see what is going on.
But the fact that they start rotating
quickly after the hole pops shows
that this is a topological change that basically
leads to the phenomenon.
So I must admit, I don't really know what to do next.
You know what I mean?
Actually, Shmuel and Ryan are continuing
to increase the Reynolds number.
That was at a Reynolds number that was about 8,000
based on the instability wavelength.
They can go up in principle to 25,000.
It is clear that things get more complicated as the Reynolds
number goes up, although there seem to be
remnants of this phenomena.
For those of you who have thought about this before,
there are other instabilities that
occur with colliding filaments.
And in particular, there's the famous elliptical instability,
which was actually partly developed here
in the math department with Willem
Malkus and his collaborators.
And also, in fact, in aeronautics
by Sheila Widnall and her collaborators.
So it's at MIT, the instability.
But it's an instability, which happens,
which basically no one's ever really
been able to understand the non-linear features of.
And it sets in.
But I think this is very interesting.
And if nothing else, it should provide a cautionary tale
about how little you're resolving, even if you try.
I'm done now.
[APPLAUSE]
Oh, I should put up these people.
These guys.
We have time for some questions.
And there's one right here in the front
and then there'll be one in the back.
Just a question to understand you better.
You had two rings interacting-- one of the figures--
that ended up what we call islands.
Could you-- then we have two rings of currents.
There were two rings.
The rings collide.
They're separated by vacuum that is infinite res--
No, they're in water, actually.
The experiment's done in water.
No, but I'm referring to in theory.
Oh, the theory.
I see.
Well no, the theory, I mean, they're separated by a fluid.
They're rings that are in a fluid.
Their sources are--
I don't know what you mean by vacuum.
No, because if you have two rings
and they're conducting currents--
Oh no, no, these aren't--
there's no electricity.
We're solving the Navier-Stokes equations or the Euler
equations.
All right, all right.
By the way, if you interested in rings, the [INAUDIBLE],,
there are plenty of jobs for you.
[LAUGHTER]
Rings?
Laboratory class, [INAUDIBLE] astrophysics.
What?
Rings, I--
Have you ever heard of the smoke ring
model of jets in astrophysics?
Oh, yeah.
And that's also under-resolved.
We've got one more question.
Yeah, I just wanted to remark that elliptical instability was
basically my thesis with Sheila.
But I also wanted to just mention
some really beautiful work by Bruce Bailey that
was done after some years later, which
gave a really nice analytical interpretation of how
the elliptical instability works.
But you're completely right, we don't
know what the non-linear fate of the elliptical instability is
and what role, if any, it actually plays
in the generation of smoke.
Right.
And actually at some point, I would love to show you
the movies of what happens when you go higher,
because the interplay between what I showed--
which is basically the crow instability and the elliptical
instability as you go through this,
as the Reynolds number gets higher--
the experiments are fascinating, and it's sort of
very hard to-- but I apologize for getting
the references wrong.
That's what happens when things just
start coming out of my mouth.
I did get to Sheila, though, eventually.
Yeah, you got Sheila.
Yeah I did.
I got to Sheila.
So it wasn't--
Thank you, Michael.
OK.
[APPLAUSE]
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