Yes, it worked.
So what I'm going to do is I see a number
of faces who have seen me talk about similar things before.
So for those, the first third of the lecture will be familiar.
But I see other faces who probably haven't seen this yet.
This is the basis on which things stand that I do.
And then I will go through four examples
of how these multi-scale ideas that I follow and methods
that I follow can lead to applications.
I will not go into much detail.
I would rather, in this talk, give you four examples
where, in the end, some, I think, interesting conclusions
can be drawn.
So the first part of the lecture,
I should start with listing a number of collaborators
that have contributed to this endeavor over the years.
And they come from many places in many countries.
And Deutsche Forschungske Meinchaft is
paying lots of my bills, which I appreciate very much.
And now the first part is on an asymptotic modeling framework
that I've developed starting in 97 at the Potsdam Institute
when I was an engineer still.
I hadn't looked into meteorology or climate research at all.
And first thing I did was I opened textbooks
of theoretical meteorology, like Petlosky's book or Gil's book.
And in each of the chapters, I found
that different scales, phenomena on different scales,
were presented and discussed.
And that made all sense to me because I
was familiar with multiple scales from combustion engineering.
And then I found behind each of the pictures,
I found a different set of equations
that were proposed to be the right equations to handle
these things in atmospheric contexts.
And I don't expect you in the back
to be able to read these equations I show here.
I just want to emphasize from a mathematical point of view,
each of these sets of equations is completely different
in terms of its mathematical structure.
On the small scales, we know we have
something that very much looks like three-dimensional gas
dynamics with a momentum balance and a mass balance
and an entropy transport.
On the intermediate scales where we
see the high and low pressure systems travel
in the mid-latitudes, we have the quasi-geostrophic system.
And of course, it's very different from this system
as it involves one scalar being advected.
And from the scalar, by stationary elliptic constraints
and equations, you can reconstruct everything.
So since I was a mathematics professor from that point on
and also was head of the Data and Computation Department
at Potsdam Institute, I had the challenge
of talking to the mathematicians and telling them
what the climate researchers were doing,
and at the same time, tried to export some mathematics
into the climate community.
And so I thought, how would I explain to a mathematician
where all these equations come from?
Without going in steps that combine physical intuition,
probably for the boundary layer processes,
the Boussinesq equations are enough.
So let's look at those and then derive the boundary layer
equations.
Mathematicians wouldn't buy that because they think
we have the compressible flow equations.
Why don't we start from those and always
do the same procedure of starting
from the common set of equations that we
believe are relevant for the atmosphere
and then derive these things?
That was what I wanted to achieve.
And so I started like an engineer would do.
I looked at some quantities of influence
that are universal for meteorological applications.
So they do not depend on the phenomenon I'm interested in,
like mid-latitudes and optic dynamics
is 1,000 kilometer characteristic scale,
but that's something that's specific to the phenomenon.
I was interested in universal quantities.
And there are, of course, a couple of them.
Radius of Earth, Earth rotation rate, gravitational constant,
the reference pressure, a reference temperature
that's basically due to the balance of radiation
between the sun and Earth and the universe.
And then the potential temperature variability
across the troposphere and between the equator and pole
is roughly the same order of magnitude
and it's also rather stable over long terms.
So I took that as a reference.
And then there is a couple of quantities
that come from the constitutive equations.
And if you take these eight numbers, one of them
is already dimensionless.
You have four physical fundamental units.
You get three dimensionless parameters out of that,
according to Buckingham's pi theorem.
And you can combine them in various ways,
but three independent choices are the ones I've listed here.
One, the first is intuitive.
It's the pressure scale height versus the radius of Earth.
That's, of course, very small because the atmospheric layer
is very thin.
The second is this potential temperature difference
across the troposphere or between equator and pole divided
by the reference temperature.
That's also a small number.
And then this funny quantity here
is the isothermal sound speed divided
by the velocity of points due to rotate and at the equator
due to the rotation of Earth.
So Earth rotates and therefore points on the equator rotate.
And the velocity ratio is about a half,
meaning that at the equator, when we are sitting there,
we are zooming around at Mach 2.
Although we don't feel it because the Earth
is rotating with it.
The air is rotating with us, but that's what it is.
Then I did a drastic step that I was familiar with,
but I met with some criticism when
I talked about this first.
Namely, I took a distinguished limit
coupling the three parameters to each other.
You would say normally, physically, they
don't have anything to do with each other in principle.
So how can you do that?
I went the other way around.
I said, I'm interested in exploiting
the smallness of these parameters
and doing theory and deriving limit equations.
And if you try to derive a general asymptotic limit where
three parameters are small, and you
try to keep them independent of each other.
So you claim that as you go to 0 with the parameters,
it doesn't matter what's the sequence of the limits
that you take.
Whether you first let this guy go to 0, and then this,
and then this, or the other way around, or something else.
If you claim this was independent,
you have to prove it.
Because in general, this is not the case.
And it even isn't the case for the linear oscillator
with small mass and small damping.
Let me go to the blackboard for the second
and explain that if we look at the linear oscillator,
and we think of epsilon and delta both being small,
then suppose we let the mass be small to begin with,
and then look at small delta.
It's clear what's going to happen.
When the mass is very small, the damping wins.
And in the limit, we get the damped equation whose solutions
do not show any oscillations.
If, however, we let the delta go to 0 first, the damping,
and look at small mass, then we get rapid oscillations.
So therefore, it's clear that depending
on which sequence you take in letting epsilon and delta go
to 0, you get completely different answers.
And the reason is that the function y, that depends on time,
and of course on the initial data,
which I neglect for the moment in the notation,
and it depends on epsilon and delta,
this beast, mathematically speaking,
doesn't have a Fréchet derivative at the origin.
It doesn't have a full gradient.
But it has directional derivatives.
So if you are in the epsilon delta space, epsilon and delta,
and you look at what happens at the origin,
and you ask, is there in the tangent direction
to any curve that moves into the origin,
is there a derivative with respect,
a directional derivative at the origin?
You find the answer is yes.
But along each curve into the origin,
the limit is different in principle.
And this is why in general, when you
have multiple small parameters, you
have to take distinguished limits
to find anything meaningful.
Of course, that also means that if you
take different distinguished limits,
you might get different answers.
But that's consistent with our observations
because the dynamics on Earth is, for example,
very different than the dynamics of Jupiter,
where the big red spot is a synoptic phenomenon.
It is on the scale of the planet,
whereas ours are synoptic dynamics.
In mid-latitudes, the synoptic scale
is much smaller than the planetary scale.
And that has to do with the fact that the distinguished
limit for Jupiter would be a different one
than you would have to take for Earth.
So this motivates why I looked consciously,
I looked for a distinguished limit.
And staring at the numbers, this is clearly small.
And the numbers suggest that this is epsilon cubed.
If I label this as epsilon, here, 1 half could also be unity.
But I'll show you later on that if you pick unity here
or square root of epsilon there, it
makes a difference in terms of the scales
that you see in the end.
I'll get to this.
So anyway, having done that, I'm
ready to basically try to ask, what
is the relation of my small epsilon
now to the standard dimensionless parameters
we know from the textbooks?
Like Mach numbers, Rossby numbers of various types,
internal wave fruit numbers, or the external wave fruit
numbers.
And it so happens that all this dimensionless numbers we
know in the textbooks, they have a certain relation
to epsilon now.
I can identify that.
What we typically call the synoptic scale
is related to the pressure scale height, for example,
by an epsilon squared power.
All of this is now nailed and fixed.
And we look at the dynamics in the limit of small epsilon.
We non-dimensionalize all the quantities
by length scales by the scale height times
by the advection time scale and thermodynamic quantities
by the reference quantities that I mentioned before.
By the way, a little twist.
Martin Clausen, is he here?
No, he's unfortunately not here.
But I remember very well, in 2001 or so, I talked to Martin
and asked him, in the textbooks, you always
get U reference is 10 meters per second.
Where the heck does that come from?
And Martin told me, it might be the thermal wind relation,
the thermal wind shear that you get from the equator
to pole difference.
And when I arrived at this point,
I liked it very much because the reference quantities I noted
here, they do not contain a flow velocity naturally.
You don't see it.
However, by this question, this scaling relationship
for the thermal wind, that's exactly what you get.
You can write down from the quantities
I noted as being universal.
I get a reference, a flow velocity.
It is the thermal wind, and it has
the right order of magnitude.
So I solved that puzzle on the way that puzzled me, at least,
to begin with, with Martin's head.
So now I can non-dimensionalize the equations.
And here they are, the full compressible flow equations,
horizontal momentum, vertical momentum, mass conservation,
and an entropy transport equation.
I keep the general terms here not resolved in detail
and written down because I will not
go into the details of these later on.
The inviscid equations will be sufficient for what
I will discuss.
And they are for most of the derivations
you find in the textbooks also.
Anyway, so now we have small epsilon
appearing in various places.
Epsilon cubed is 1 over Mach number squared.
That's what should appear here.
And this is the Rossby number based on the scale height.
So this is why it's small.
This is 1 over Rossby.
This is small.
So the Rossby number based on the scale height is large.
Of course, when I go to the synoptic scale,
it's going to be a small Rossby number.
So now the question is, how do I now
look at the question of what are scale-dependent phenomena?
Remember, I started off with the argument
that in the textbooks you are faced with certain phenomena
that are of interest and are to be studied mathematically.
And these live on characteristic scales.
And now I am only allowed to look at different scales
in the sense of asymptotics, meaning
I want to study, for example, what
happens when I look at a long length scale using
a rescaled coordinate, epsilon squared x, for example.
And remember, two slides ago, I said
that the synoptic scale, mid-latitude synoptic scale,
is 1 over epsilon squared times the scale height,
the pressure scale height.
So therefore, if I use a coordinate that's
the dimensional x divided by the synoptic scale, which
would be the natural scale to use
in trying to derive the mid-latitude dynamical
equations, this is x prime divided
by h scale over epsilon squared.
So I multiply by epsilon squared.
And this is epsilon squared times my dimensionless x.
So that coordinate, let me call it xi,
resolves naturally what is called the synoptic scale
traditionally.
So this is the natural coordinate
Petlosky would use in his derivation.
And then if I combine this with the appropriate law
advective time scale on this large horizontal scale,
then by standard single scale asymptotic arguments,
I can actually derive the quasi-geostrophic equations.
If I scale differently, for example, I
take a scalar xi, which depends on the large scale,
but it's rescaled itself by 1 over epsilon,
I can resolve an isotropic phenomena.
So that would mean I have a synoptic scale phenomenon,
but it has level sets of this coordinate xi of epsilon
squared x.
And if I do so and resolve a smaller characteristic scale
normal to the level sets, then I can, for example,
derive the semi-geostrophic equations,
which are made to resolve an isotropically scaled
phenomena in space.
So this is how you can now play with coordinate scalings
in powers of epsilon to derive a long list of reduced model
equations that you actually find in the textbooks.
And there wasn't any one of those
that I couldn't derive until Arakawa and Conroe came up
with their reduced model, where it's not obvious how to do that.
But everything else I've so far been
able to capture in this framework.
That was very nice, because now I
could tell the mathematicians, how
do the meteorologists arrive at the reduced models?
Well, they start with the compressible equations.
They use globally valid reference quantities
to make them everything dimensionless.
They end up with three small parameters.
There is a particular distinguished limit.
And this is a new insight.
I can derive all of these equations
without ever having to go back and change this.
So this distinguished limit is, in fact,
consistent with all the reduced models
meteorologists have been deriving
for the dynamics of the atmosphere before, which is very
nice and wasn't known before.
By the way, if you wanted to do the similar thing for the ocean,
this would have to change.
You get a different distinguished limit
that naturally would, for example,
make for much smaller synoptic scales, as we know.
So it can be done for the ocean as well,
but you have to go back to this place and do something about it.
Anyway, so then in the annual reviews paper in 2010,
this was basically the culmination and summary
of it all.
I was able to draw a diagram like this.
On the horizontal, I have length scales rescaled
by powers of epsilon.
On the vertical, I have time scales also rescaled
by powers of epsilon.
And now I can, in cross lines here, in these coordinates,
I can locate, for example, the hydrostatic primitive
equations.
I can locate Boussinesq on the small scales, the QG equations,
or the planetary geostrophic equations, which
is already very nice.
Now I have sorted this out.
That is, however, only a starting point
from my perspective, because now I have a justification
to ask the question, what happens if there is,
for example, processes being described
by the hydrostatic primitive equations present
at the same time where I have phenomena that are better
described in the near equatorial region, for example,
by the weak temperature gradient approximation.
They coexist.
It's an interaction across the scales, the length scales,
at the same time scale, for example.
Now I can use mathematical tools of multiple scales asymptotics
to study scale interactions.
And I'm justified to do so because I
know all of these models come from one
and the same distinguished limit.
There's one epsilon that characterizes them all.
And the only difference is coordinate scalings.
So that justifies using the methods of multiple scales
asymptotics that would mean going back to here.
And instead of, for example, using just x
as a horizontal space coordinate,
use x and epsilon squared x together in one ansatz,
and then work out how these functions depend
on the two scales.
And there's techniques for doing so,
which I spare you at this stage.
So that summarizes this is the first part of the lecture.
And many of you have seen me talk about this before,
probably with almost the same slides.
But I think it's a very powerful approach that
allows you to really zoom into the particular phenomena
and pull them out and see how they interact.
And now I'm going to give you four examples of how,
over time, we have used this approach
to make out new physical insights or hypotheses
for mechanisms or thought about numerics.
First example, motion and structure
of atmospheric vortices.
And it's all about birth of hurricanes.
When you look at the subtropical Atlantic,
typically you have a wave train coming from the Atlas
Mountains moving across.
And then if you zoom into one of those
and you move with a frame of reference
that follows the wave, you can then
draw streamlines, which Dunkerton and colleagues
have done in 2009.
And then what you observe is that under certain conditions,
when the storms get strong enough or the motions get
strong enough, you see the streamlines wrap into each
other and you form something that you could
validly call a vortex.
It's not just a meandering motion,
but it really is a circular motion
in the moving frame of reference.
And this is typically the beginning
of something that can or cannot become a strong storm
or even a hurricane.
Notice in this regime, the flow velocities and length scales
are still such that the Rosby number,
the characteristic Rosby number, is small, about a tenth.
Like you would have it in derivations
of the quasi-geostrophic equations.
Now, when you look at a full-fledged hurricane,
then the radius of maximum wind is about 52 to 100 kilometers.
The circumferential velocities around the center
are in the range of 30 to 60 meters per second.
And when you form the Rosby number, that's 10.
So going from this regime where the folding over
of a baroclinic instability is starting
to generate a circular motion all the way to a hurricane,
means you have to go from small to large, large Rosby numbers.
That cannot be done in one and the same limit process.
So you have to have, if you want to describe the growth starting
from one end going to the other, you
have to look into the middle regime, which
is Rosby numbers of order unity.
And it so happens that when we had submitted the paper that I'm
describing right now to JFM, Mike Montgomery
was one of the referees.
And we had submitted it, assuming it was set in most of the papers
we had read before.
If you want to do theory for hurricanes,
you look at the gradient wind regime.
So we did that.
We did it asymptotically.
And then we got a result, which was sort of interesting.
And then Mike Montgomery said, yeah, but wait a second.
Gradient wind regime, you explain to me.
This is what we explained in the paper.
You divide this term by that term,
and you get Rosby numbers of order unity.
Because in the gradient wind regime,
the centripetal forces, and the pressure gradient forces,
and the Coriolis are all of the same order of magnitude, which
means that in particular, this term and this term
are comparable.
And that means you have Rosby numbers of order unity.
And then Mike said, wait a second.
That's not a hurricane.
Hurricanes have large Rosby numbers,
as I showed you before.
So we learned, aha, we are not looking at hurricanes.
We are looking at the transition regime,
where we can actually bridge between what is not
a hurricane yet and what wants to become a hurricane.
And in that sense, the paper became a lot more interesting,
because mechanisms of transition are much less understood
than the structure of full fledged hurricanes.
So in that sense, we rewrote the introduction.
Mike was happy, and we all had learned something from it.
Just to the youngsters in the room,
so battling with referees is a very productive process.
I recommend it very strongly.
Anyway, so this is what the regime we looked at.
And now, I want to just give you an idea of what
we needed to do to set up the asymptotics in a way
that we could really address this initial regime
that Dankatan et al. had been looking at.
Well, they gave us another picture, which
is this one, where they plotted the velocity
potential of the horizontal motion
in this moving frame of reference.
Velocity potential is non-trivial,
because if you just focus on the horizontal,
you have a divergent or convergent flow coming
from motions near the boundary layer
and then the diverging motion at the top.
And these velocity potential contours,
they show very nicely at 12 kilometer
and about one kilometer height that the center of activity
is displaced between top and bottom
by something that's 200 kilometers in this case.
Think about it.
This is 200 kilometers.
So we are talking about a phenomena
that vertically stretches across the troposphere, which
is 10 kilometers.
And horizontally, the phenomenon covers 200 kilometers.
So it's a very strongly tilted pancake-like vortex.
It's nothing like a vertical vortex that's
slightly perturbed, nothing at all, not at all.
It's very tilted, and you have to be
able to account for it if you want
to catch this transition regime.
So what we did was to set up an asymptotic analysis where
we said that the large-scale flow is supposed
to be quasi-geostrophic.
We look at a center core flow that has a higher resolution.
And we allow the center line of the near
at leading order axisymmetric distributions of velocity.
We allow the center to be a curve that evolves in time
and can meander around and has horizontal displacements
comparable to the size of the vortex,
meaning about 200 kilometers.
And then some scaling arguments that basically
rely on the Rothby number being supposed
to be order unity in the core and order epsilon in the far
field.
This is consistent with QG here and gradient wind regime there.
And at the same time requiring that the total circulation
of motions in the far field and the total circulation
in the core should be comparable.
Because basically, what you think of
is you pull in mass from far distance,
comes in with a planetary circulation.
You pull it into a smaller radius
and you get the period effect from ice skating.
And that means that the circulation is
of the same order of magnitude.
These two rules, Rothby number scaling
and the same order of magnitude of the circulation
gives us uniquely what is the scale of the radius,
the meso scale for the typical radius of the vortex,
and then the velocity scale that comes with it
so that everything matches up.
So then we had a two-scale problem.
And the internal flow is gradient wind regime.
The far field is quasi-geostrophic.
And the rest is a lot of pages of detail
matched asymptotic expansions.
Let me just give you one result that came out in the end
and that we think is rather nice.
And we're trying to pinpoint it with 3D simulations
at this time.
And that result is a mechanism for the spinning up or spinning
down of a vortex of this type that's
related to asymmetric heating, for example,
from latent heat release of clouds.
And what happens is, if you write down
the dynamical evolution equation for the circumferential
velocity, these terms are basically
the circumferential momentum balance
as you would have it anyway.
So this is in an axisymmetric flow, nothing spectacular.
On the right-hand side, you get a term
in a vortex rotating with the rotation direction of Earth.
This term is positive.
And then there is a term, UR star,
that looks somehow dimensionally something
like a radial velocity, but just dimensionally so.
And it comes as an average of the vertical velocity,
a circumferential average of the vertical velocity times
the tilt of the vortex.
And basically, when you trace back where it comes from,
it's the net vertical transport of horizontal momentum.
That's due to the fact that when you tilt a vortex,
the parcels have to start meandering.
That's due to this interaction of the tilt of the vortex,
the pressured perturbations, and the hydrostatic balance
in the vertical.
And if the vertical velocity is not driven by heat release,
which is due to these two terms, if there is no heat release,
no external heating, then it so happens
that the vortex just meanders around,
and there's no strengthening or weakening
of the primary circulation.
Nothing happens.
However, if the notation is with the two indices here,
one is the first Fourier mode, sine or cosine,
and one and two of the second numbers
indicate cosine and sine structures.
So if you have a certain orientation
of the latent heat release or the diabatic terms,
relative to the tilt of the vortex,
you can spin it up or spin it down.
You can create a momentum source or a momentum sink.
And that mechanism is especially independent
of any vertical secondary circulation,
as it's typically assumed to be responsible for spinning up
a hurricane.
When things of convection taking place
and it becoming stronger and stronger,
and it sucks in air from far distances in the boundary layer,
and that spins up the vortex by the pirouette effect.
But there is a chicken and egg hen problem,
because how do you create the secondary circulation
in the first place if the vortex wasn't strong enough yet?
And that mechanism is independent of that.
And it so happens that in numerical simulations
by Dalin Sung, for example, who retraced real life hurricanes
in numerical simulations, he could
show that typically you see very asymmetric distributions
of clouds around the circumference of a vortex.
And that indicates that you have this latent heat release
of condensation and evaporation again while parses move around.
And that makes for, in fact, asymmetric distributions.
So in a project that we have now in our new collaborative research
center, one of the projects is addressing the questions
behind this.
And in the end, we hope that by describing these terms
with microphysics closures, self-consistently,
that we would be able to say something
about the sign of this U star that comes from the clouds.
So this was a theory that describes large displacements
and a nonlinear theory for the core dynamics and motion
of a concentrated asymptotic atmospheric vortex
in the transition regime between QG and not right
quite being a hurricane yet.
And we found a new spin up criterion.
And in this project, we want to follow up and pinpoint it
by data as well as simulations.
So that was one application.
Second application has to do with a longstanding question.
You find many papers in the literature that argue,
do these unelastic or pseudo incompressible equations,
do they make any sense?
Can you use them?
And why can you use them?
Or can you not use them?
They are attractive because if you use soundproof flow models,
you don't have to care about acoustic modes.
And you don't have to discretize them.
And that's an advantage in terms of time steps as well as
other funny phenomena.
But there is a disadvantage.
You truncate your equations.
And some of the physics might get lost.
And so the questions we were asking here
were, first of all, where do the soundproof models
live in this diagram on lengths and time scales?
And basically, they cover, interestingly, a blurred region.
You cannot pinpoint them in one point on the diagram.
And I want to explain to you why that is so.
Here, just for summary, this is the compressible flow
equations with, again, mass balance
and horizontal momentum, vertical momentum,
and entropy transport equation.
This capital P is just rho theta.
But it also happens to be just a function of the pressure.
And importantly, in the soundproof models,
you want to keep both advection.
That's, for example, responsible for vertical advection
of moisture and then condensation, et cetera.
And like here, waves, the trace of which
you can see here in a cloud deck that's
excited by an island over the ocean
where internal gravity waves get triggered.
And then you can see them in the cloud patterns.
So internal waves, as well as advection, should be kept.
But you want to get rid of acoustics.
You want to get rid of sound waves.
So in the argument that I'm going
to show you why the soundproof models live in a blurred area
and do not have a unique point in this diagram,
I need to go into looking at the dimensional analysis
of the characteristic time scales in these problems.
Well, if you just focus on the design regime of these models,
this is about 10 kilometers horizontally and vertically
and 20 minute time scale, then an inverse time
scale for advection is just u divided by the scale height
or this 10 kilometer scale.
Sound speed goes like the isothermal or isentropic
sound speed divided by the scale height.
That's the sound frequency.
And the ratio of the two in non-dimensional terms
is, of course, 1 over the Mach number.
That's the definition of it.
If you look at internal waves and you non-dimensionalize
by dividing by u ref over h scale,
you get a quantity that looks like, yes, that's
the same ratio as the sound frequency multiplied
by the dimensional stratification
of potential temperature.
And now, either this quantity is out of unity,
then the internal waves are as fast as the sound waves.
And if you want to do asymptotics,
they go away together and you end up
with a model that only has advection.
That, by the way, was done some time years ago.
And you end up with a model that only knows
horizontal motions in layers.
And at the time, it was dismissed
as being meaningless because there's no gravity waves.
But it so happens that that's exactly the weak temperature
gradient approximation that's all the time used nowadays
in tropical meteorology.
So that's somehow a limit process
where you remove internal waves as well as sound.
But that's not what we are talking about here.
We want a model where advection and internal waves are kept
and sound is removed.
You need that term to be small, the stratification
to be small in order to achieve that.
And that's actually reasonable.
Across the scale height of 10 kilometers,
the potential temperature changes by something like 30 Kelvin,
but not by 300 Kelvin, which is our reference temperature.
So therefore, small stratification is meaningful.
However, if you make it so small that it's actually
order epsilon squared so it can cancel the 1 over epsilon here,
you're in trouble.
This is what Ogura and Phillips did in 1962.
And they actually assumed the stratification
to be order epsilon squared.
And then they could do a single time scale analysis
because there was only one fast process and two slow ones.
But if you have an epsilon squared stratification,
what does it mean?
Epsilon is about 130 is the Mach number.
Epsilon squared is 1 over 1,000, roughly.
And so the stratification you are left with in that regime
is just 1 Kelvin potential temperature stratification
across the troposphere.
It's way too weak.
And this is why Ogura and Phillips, among other things,
were probably criticized at the time.
So what you need to do to be realistic,
you need to keep the stratification that's
in between being order 1, that's too strong,
and order epsilon squared, that's too weak.
So we introduced in a paper in 2010,
we introduced an intermediate stratification,
leaving this exponent mu that's supposed
to be somewhere between 0 and 2, leaving it open
and asking, how can we pick it so that we still
get consistency between the compressible and the pseudo
incompressible or the soundproof models?
So we're asking, what's the factor of mu?
How large can it be?
Or how small can it be to still have
consistency between the compressible equations
and the soundproof equations?
That's the key question.
And to go after this, what we did
is we rewrote the equations in such a fashion
that we decompose into a linear part that
contains all the fast terms that are multiplied
by 1 over some power of epsilon, and the non-linear and even
smaller terms on the right-hand side.
And then we focused on the linear equations, which
show terms proportional to 1 over epsilon.
These are responsible for sound waves.
And terms proportional to this epsilon to the power mu,
those are responsible for internal gravity waves.
And then we said, let's take that linear system
and do a spectral decomposition, vertical mode expansion,
or a separation of variables ansatz.
So horizontally, we just have traveling waves as usual.
And vertically, we try to find the structure functions
that are compatible with these horizontal traveling waves.
That's a standard approach.
You plug this in to your equations.
You eliminate a couple of the variables,
keeping only one, and you end up with the Taylor-Goldstein
equation.
That's standard.
So the Taylor-Goldstein equation is this thing here.
And interestingly, now, the only place
where the compressibility appears is in this point.
This is where the compressibility remains.
And if you switch this off this term,
then you have the pseudo incompressible linear modes.
If you leave the term, you have the compressible linear modes.
Now, there is a lot of theory about, no, I have to say first.
If you drop this term, then you can see immediately,
this is a second order differential equation
for this vertical velocity structure function.
And it's an eigenvalue problem, because if you
want to solve this equation with homogeneous boundary
conditions, we picked rigid lids on top and bottom.
And you want to, since it's a homogeneous equation that
has w as a factor everywhere, you
can only solve it with certain frequencies.
These are the eigenfrequencies.
So it's an eigenvalue problem.
And we know everything about Sturm-Liouville problems
of this type.
And we have a full spectrum that has nicely separated eigenvalues.
It has orthogonal eigenfunctions, a set of orthogonal eigenfunctions.
Everything is fine for the pseudo incompressible case.
But we also know that if you take a Sturm-Liouville problem
and you slightly perturb the coefficient function here
or there, then the reaction of the solution
to it of the eigenvalue solution is proportional to the perturbation
amplitude.
So it's a continuous perturbation problem.
The solutions depend smoothly on the coefficient functions.
And that we could use to argue why having a small epsilon to the power
mu here will induce only epsilon to the power mu perturbations
to the eigenvalues and eigenmodes of the compressible system.
So this is great, because now we know the vertical structure
functions for epsilon to the power mu beings anyhow small.
They are very close to the compressible structure functions.
The waves look the same.
They have the same spatial structure.
And they only depart from each other by this epsilon to the power mu.
Now came the next question.
We also, unfortunately, have a small difference
in the eigenvalues that means in the frequencies.
Or if you wish, in the phase velocities of these waves.
Now suppose you start a pseudo incompressible
and a compressible solution with the same initial data.
They will start moving.
And the phases differ slowly from each other
because of the small difference in the eigenvalues.
We are interested not in the fast modes.
We are interested in the slowest mode in the system, which is advection.
So we have to wait a long time until we really
reach the time scale that's of interest.
So the requirement is that if we let these two modes oscillate
over advective time scales.
So in our internal wave time scale, it means asymptotically long.
Epsilon to the power minus nu.
We still want phase consistency, asymptotic phase consistency,
between the compressible and the pseudo incompressible modes.
And that led us to conclude that mu has to be larger than 2 thirds.
And if it's larger than 2 thirds, you still have asymptotic consistency
over advective time scales.
Which is to conclude, well, if you run the pseudo incompressible model,
or the un-elastic, by the way, that's the same.
They are consistent with the compressible equations
over advective time scales, provided you didn't have acoustics to begin with.
That's an assumption.
If you have acoustics to begin with, of course,
the pseudo incompressible equations cannot capture them.
But if you have acoustically balanced initial data,
the solutions stay closed over advective time scales.
And epsilon to the power 2 thirds means, aha, this is now 1 third T.
To the power 2 thirds, and you plug in the numbers,
you come up with potential temperature differences
vertically across the troposphere that are allowed to be as large as 40 Kelvin.
And that's realistic.
So in fact, that's a justification for using,
at least in the design regime of 10 to 50, 100 kilometers,
and 20 minute time scales, advective time scales,
you can safely use the soundproof models instead of the compressible ones,
and you still should get reasonable results.
It doesn't mean yet that if you go to planetary scales, it's still true.
That's a totally different story.
But that's not what we claim.
So that was the second example.
Let me give you a third example that has to do with waves,
internal waves, and how they can interact possibly with clouds.
And the idea was to ask, suppose we were in the tropics.
This was work with Andy Mida, who was very much into tropical dynamics
and studying multiscale phenomena in the tropics.
We asked, suppose we have deep convective columns
that have saturated air in them.
And there is a gravity wave passing over it on a larger scale.
So the gravity wave length scale would be the 10 to 100 kilometer scale,
and the clouds would be an epsilon smaller in horizontal extent,
so about a kilometer or so.
And they would stretch possibly 10 kilometers in height,
so deep convection columns being passed by an internal wave.
Well, that's a classical multiple scales problem.
We introduced a fast time scale and a fast space scale
and did the asymptotics and worked it all out.
And what came out in joint work with Daniel Ruprecht,
my PhD student at the time and with Andy,
we found a system of equations that seemed not to have been written down before,
that has the DTs think of it as a partial derivative for a moment.
Then this is linear wave dynamics, unless you had this red term here.
But if everything was zero on the right hand side except for that term here,
then this is just linearized internal wave dynamics
on the design scale for the pseudo incompressible approximation.
However, there is a factor of 1 over sigma,
where sigma is the cloud fraction in a horizontal slice.
So if we have these columns, and we take a horizontal slice,
and then we compute the percentage of cloud cover.
So how much of the area in a horizontal slice is covered by saturated air?
That's the sigma parameter.
And you can see here that sigma parameter weakens the stratification.
N squared is the buoyancy frequency squared.
And so effectively, to be expected if you think about it,
at presence of a cloud where the restoring force
from the background stratification is overcome because of condensation effects,
it reduces the strength of the stratification effectively.
So it slows down internal waves.
But it's coupled all to the vertical dynamics within the columns.
So an internal wave passing over creates vertical motions in the towers.
And there is a separate set of equations that describes the conditional average
of vertical velocity and potential temperature conditioned on being in a cloud tower,
but averaged over the large scale.
And so this coupled system of equations gives new dynamics.
That's not linearized dynamics of internal waves.
But it's a new mode.
What's interesting about the mode, one phenomenon we could find out
is the following.
Normally, if you take a wind blowing over a mountain range, say,
and it's stably stratified, then you would get lee waves.
And it's very well known that there is a cutoff, a high wave number cutoff.
If the waves are shorter than this, the waves in the topography are shorter
than this, then the internal waves become evanescent.
They don't radiate anymore.
But it so happens that if you put in the cloud towers,
then you suddenly also get a large wave number, a low wave number cutoff,
so a long wavelengths cutoff, which isn't present if you don't have any clouds.
And if you implement the reduced model that we derived and run it,
and you have topography just as I showed in the previous graph,
the superposition of, for example, a sign of 2x topography,
and you have sigma equals 0, meaning no clouds,
then you can see the trace of both of the components radiating upwards.
It's both the sine x and the sine of 2x modes radiating.
If you crank up the sigma parameter and increase the cloud cover to 10%,
then you can see that the long wave component is basically gone,
and you end up with just a short wave component,
which means that this long wavelengths cutoff that comes from the clouds has kicked in.
And if you go to even larger sigma, then you can even kill the shorter wave cloud,
and you get evanescent waves only, even though in the dry case,
you would have both waves radiate upwards.
So in that sense, these interactions of waves and clouds
can actually modify the wave dynamics quite non-trivially on large length scales,
which seems to be a new effect.
And now let me come very quickly to the last point,
which very briefly cuts into questions about numerical methods related to these asymptotic limit regimes.
And what we did here is the following.
We had observed from the previous discussions that, at least on small scales,
a compressible flow solver that's made for atmospheric applications,
where acoustic modes are unimportant, should behave like a pseudo incompressible model.
So it should suppress sound and basically behave as if it was a pseudo incompressible code.
So we challenged ourselves in writing a compressible flow solver.
It's based on this way of noting the compressible flow equations.
As I showed before, mass balance, momentum balance,
and then this transport equation for potential temperature written in conservation form.
And we put an alpha parameter here.
If you drop this term, you have the pseudo incompressible model.
If you keep the term with alpha equals 1, you have a fully compressible code.
And then, however, you can play with it and have alpha between 0 and 1
and basically continuously move between the compressible and the pseudo incompressible modes.
By implementing a semi-implicit discretization, as is more or less familiar in many contexts in meteorology,
we were able to write it all in such a fashion that all the terms here are discretized exactly in the same way
for both of the limit cases.
And we can allow for an alpha parameter that's between 0 and 1, including both.
So one important application of this is you can now make a comparison between models,
the pseudo incompressible and the compressible model, based on really identical numerics.
There is a study by Pyotr Smolakiewicz and Andreas Dornbach in 2007,
where they did a similar study and showed, well, these models are very close to each other.
Even though in many publications before, it was said, well, if you take one and the same phenomenon
and you solve the an elastic or the compressible equations, you get very different answers.
And it so happens that most of these conclusions were due to the fact that both models were discretized in a different fashion.
They use different numerics.
And the difference in the numerics can make a larger difference than just the difference in these models.
And so with this kind of approach, we can actually compare models based on identical numerics,
ruling out that numerical differences make for fake differences, suggesting difference in models that is not there.
Another application that I'm discussing with the colleagues at the ECMWF now is the following application.
It has to do with data assimilation and the question of how can you generate balanced initial data
when you feed a model with new information.
So this is a very simple test case where a hot air bubble is supposed to rise in a neutral environment.
And after a while, when you started with a circular symmetric distribution of a potential temperature, positive perturbation,
a bubble rises and it forms a mushroom type vortex.
That's very much a test case everybody knows and you often use this as one of the first tests to check.
Now, what we did was to monitor at this point in the domain versus time the increment of pressure from one time to the time step to the next.
So basically we get a trace of pressure time derivatives, if you wish.
If you run it compressibly with large time steps that are supposed to resolve the convection,
advective time, advective current numbers of order unity, we see oscillations of pressure like this,
10 to the minus 4 in terms of pressure increments.
And the straight line down here, the dashed straight line, is basically the same result.
However, when you run a pseudo incompressible code.
So the pressure differences you see in a basically acoustically balanced code are nothing like 10 to the minus 4.
They are much, much smaller.
So all we see here in terms of oscillations are slowed down acoustics from our same implicit solver.
If we take the same problem, however, and we start with alpha equals zero in the first time step,
meaning the first time step is run pseudo incompressibly.
Then, obviously, given the initial imbalance that we have in here,
the code generates a consistent pressure distribution that matches that so that no sound is present.
And then you crank up alpha to go to 0.7 in the next and to fully compressible in the third time step,
and from then on you run it compressibly.
What you see is that the deviations or the time increments of pressure have come down by three orders of magnitude.
They now are comparable to the trace that you see in the pseudo incompressible model.
And there is only small oscillations left over.
The trends that you see that this curve goes up here and this goes up here is also actually meaningful and systematic for the following reason.
The compressible code has three energy compartments.
Internal energy, kinetic energy, and potential energy.
Whereas the pseudo incompressible model lacks the internal energy part.
It only has kinetic and potential.
And this difference is, in fact, consistent with what goes into internal energy changes due to the rising of the bubble.
So we checked that also in some detail.
You could decrease the oscillations here some more by making the transition smoother,
but I thought that we get this kind of result where you never can see any differences in the potential temperature controls, by the way, no matter what you do.
This is just discussing pressure.
That you could do this with just three steps is very attractive in the context of data assimilation, if you think about it.
Suppose you have an implicit model that is doing weather forecasts.
You import new data from a data assimilation procedure, then there's no guarantee that you know what is the pressure field that is consistent with the new velocity distributions that you have assimilated.
Typically, you have some methods of filtering the data and taking many time steps to assimilate the new information step by step, etc., in order to get rid of the imbalances that normally would assume.
If you just use that technique, take the code, instead of making the next time step compressibly, make it pseudo incompressibly, and from then on run compressibly again, you have to do nothing else.
It balances immediately, and you can run on.
Of course, at this stage, we haven't proven it in a real-life context.
It's a hypothesis and a proposal, but people at ECMWF are interested, and we're in the process of trying things out over there.
What I did was I introduced, in some lengths, my asymptotic modeling framework, which I'm still proud of and use in a lot of different contexts.
I think it's very powerful, and what I showed you is four different results or applications that came out of this line of thought and used together with multiple scales techniques,
structure and motion of atmospheric vortices, justification of soundproof models for realistic stratification regimes, interactions of clouds and waves that were unexpected somehow and not seen before on long wavelengths,
and then one application of using these ideas to create balanced numerical methods and using them possibly in data assimilation contexts.
And with that, I thank you for your attention.