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]
