## Eddy Who?

### Or: A very short introduction to turbulence

guest post by Tim van Beek

Have a look at this picture:

Then look at this one:

Do they look similar?

They should! They are both examples of a Kelvin-Helmoltz instability.

The first graphic is a picture of billow clouds (the fancier name is altostratus undulatus clouds):

The picture is taken from:

• C. Donald Ahrens: Meteorology Today, 9th edition, Brooks/Cole, Kentucky, 2009.

The second graphic:

shows a lab experiment and is taken from:

• G.L. Brown and A. Roshko, online available Density effects and large structure in turbulent mixing layers, Journal of Fluid Mechanics 64 (1974), 775-816.

Isn’t it strange that clouds in the sky would show the same pattern as some gases in a small laboratory experiment? The reason for this is not quite understood today. In this post, I would like to talk a little bit about what is known.

#### Matter that tries to get out of its own way

Fluids like water and air can be well described by Newton’s laws of classical mechanics. When you start learning about classical mechanics, you consider discrete masses, most of the time. Billiard balls, for example. But it is possible to formulate Newton’s laws of motion for fluids by treating them as ‘infinitely many infinitesimally small’ billiard balls, all pushing and rubbing against each other and therefore trying to get out of the way of each other.

If we do this, we get some equations describing fluid flow: the Navier-Stokes equations.

The Navier-Stokes equations are a complicated set of nonlinear partial differential equations. A lot of mathematical questions about them are still unanswered, like: under what conditions is there a smooth solution to these equations? If you can answer that question, you will win one of the a million dollars from the Clay Mathematics Institute.

If you completed the standard curriculum of physics as I did, chances are that you never attended a class on fluid dynamics. At least I never did. When you take a first look at the field, you will notice: the literature about the Navier-Stokes equations alone is huge! Not to mention all the special aspects of numerical simulations, special aspects of the climate and so on.

So it is nice to find a pedagogical introduction to the subject for people who have some background knowledge in partial differential equations, for the mathematical theory:

• C. Foias, R. Rosa, O. Manley and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge U. Press, Cambridge, 2001.

So, there is a lot of fun to be had for the mathematically inclined. But today I would like to talk about an aspect of fluid flows that also has a tremendous practical importance, especially for the climate of the Earth: turbulence!

There is no precise definition of turbulence, but people know it when they see it. A fluid can flow in layers, with one layer above the other, maybe slightly slower or faster. Material of one layer does hardly mix with material of another layer. These flows are called laminar flows. When a laminar flow gets faster and faster, it turns into a turbulent flow at some point:

This is a fluid flow inside a circular pipe, with a layer of some darker fluid in the middle.

As a first guess we could say that a characteristic property of turbulent flow is the presence of circular flows, commonly called eddies.

#### Tempests in Teapots

A funny aspect of the Navier-Stokes equations is that they don’t come with any recommended length scale. Properties of the fluid flow like velocity and pressure are modelled as smooth functions of continuous time and space. Of course we know that this model does not work on a atomic length scale, where we have to consider individual atoms. Pressure and velocity of a fluid flow don’t make any sense on a length scale that is smaller than the average distance between electrons and the atom nucleus.

We know this, but this is a fact that is not present in the model comprised by the Navier-Stokes equations!

But let us look at bigger length scales. An interesting feature of the solutions of the Navier-Stokes equations is that there are fluid flows that stretch over hundreds of meters that look like fluid flows that stretch over centimeters only. And it is really astonishing that this phenomenon can be observed in nature. This is another example of the unreasonable effectiveness of mathematical models.

You have seen an example of this in the introduction already. That was a boundary layer instability. Here is a full blown turbulent example:

The last two pictures are from the book:

• Arkady Tsinober: An Informal Conceptual Introduction to Turbulence, 2nd edition, Springer, Fluid Mechanics and Its Applications Volume 92, Berlin, 2009.

This is a nice introduction to the subject, especially if you are more interested in phenomenology than mathematical details.

Maybe you noticed the “Reynolds number” in the label text of the last picture. What is that?

People in business administration like management ratios; they throw all the confusing information they have about a company into a big mixer and extract one or two numbers that tell them where they stand, like business volume and earnings. People in hydrodynamics are somewhat similar; they define all kinds of “numbers” that condense a lot of information about fluid flows.

A CEO would want to know if the earnings of his company are positive or negative. We would like to know a number that tells us if a fluid flow is laminar or turbulent. Luckily, such a number already exists. It is the Reynolds number! A low number indicates a laminar flow, a high number a turbulent flow. Like the calculation of the revenue of a company, the calculation of the Reynolds number of a given fluid flow is not an exact science. Instead there is some measure of estimation necessary. The definition involves, for example, a “characteristic length scale”. This is a fuzzy concept that usually involves some object that interacts with – in our case – the fluid flow. The characteristic length scale in this case is the physical dimension of the object. While there is usually no objectively correct way to assing a “characteristic length” to a three dimensional object, this concept allows us nevertheless to distinguish the scattering of water waves on a ocean liner (length scale ≈ 103 meter) from their scattering on a peanut (length scale ≈ 10-2 meter).

The following graphic shows laminar and turbulent flows and their characteristic Reynolds numbers:

This graphic is from the book

• Thomas Bohr, Mogens H. Jensen, Giovanni Paladin and Angelo Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge U. Press, Cambridge, 1998

But let us leave the Reynolds number for now and turn to one of its ingredients: viscosity. Understanding viscosity is important for understanding how eddies in a fluid flow are connected to energy dissipation.

#### Eddies dissipate kinetic energy

“Eddies,” said Ford, “in the space-time continuum.”

“Ah,” nodded Arthur, “is he? Is he?” He pushed his hands into the pocket of his dressing gown and looked knowledgeably into the distance.

“What?” said Ford.

“Er, who,” said Arthur, “is Eddy, then, exactly, then?”

– from Douglas Adams: Life, the Universe and Everything

A fluid flow can be pictured as consisting of a lot of small fluid packages that move alongside each other. In many situations, there will be some friction between these packages. In the case of fluids, this friction is called viscosity.

It is an empirical fact that at small velocities fluid flows are laminar: there are layers upon layers, with one layer moving at a constant speed, and almost no mixing. At the boundaries, the fluid will attach to the surrounding material, and the relative fluid flow will be zero. If you picture such a flow between a plate that is at rest, and a plate that is moving forward, you will see that due to friction between the layers a force needs to be exerted to keep the moving plate moving:

In the simplest approximation, you will have to exert some force $F$ per unit area $A$, in order to sustain a linear increase of the velocity of the upper plate along the y-axis, $\partial u / \partial y.$ The constant of proportionality is called the viscosity $\mu$:

$\displaystyle{ \frac{F}{A} = \mu \frac{\partial u}{\partial y}}$

More friction means a bigger viscosity: honey has a bigger viscosity than water.

If you stir honey, the fluid flow will come to a halt rather fast. The energy that you put in to start the fluid flow is turned to heat by dissipation. This mechanism is of course related to friction and therefore to viscosity.

It is possible to formulate an exact formula for this dissipation process using the Navier-Stokes equations. It is not hard to prove it, but I will only explain the involved gadgets.

A fluid flow in three dimensions can be described by stating the velocity of the fluid flow at a certain time $t$ and $\vec{x} \in \mathbb{R}^3$ (I’m not specifying the region of the fluid flow or any boundary or initial conditions). Let’s call the velocity $\vec{u}(t, \vec{x})$.

Let’s assume that the fluid has a constant density $\rho$. Such a fluid is called incompressible. For convenience let us assume that the density is 1: $\rho = 1$. Then the kinetic energy $E(\vec{u})$ of a fluid flow at a fixed time $t$ is given by

$\displaystyle{ E(\vec{u}) = \int \| \vec{u}(t, \vec{x}) \|^2 \; d^3 x }$

Let’s just assume that this integral is finite for the moment. This is the first gadget we need.

The second gadget is called enstrophy $\epsilon$ of the fluid flow. This is a measure of how much eddies there are. It is the integral

$\displaystyle{\epsilon = \int \| \nabla \times \vec{u} \|^2 \; d^3 x }$

where $\nabla \times$ denotes the curl of the fluid velocity. The faster the fluid rotates, the bigger the curl is.

(The math geeks will notice that the vector fields $\vec{u}$ that have a finite kinetic energy and a finite enstrophy are precisely the elements of the Sobolev space $H^1(\mathbb{R}^3)$)

Here is the relationship of the decay of the kinetic energy and the enstrophy, which is a consequence of the Navier-Stokes equations (and suitable boundary conditions):

$\displaystyle{\frac{d}{d t} E = - \mu \epsilon}$

This equation says that the energy decays with time, and it decays faster if there is a higher viscosity, and if there are more and stronger eddies.

If you are interested in the mathematically precise derivation of this equation, you can look it up in the book I already mentioned:

• C. Foias, R. Rosa, O. Manley and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge U. Press, Cambridge, 2001.

This connection of eddies and dissipation could indicate that there is also a connection of eddies and some maximum entropy principle. Since eddies maximize dissipation, natural fluid flows should somehow tend towards the production of eddies. It would be interesting to know more about this!

In this post we have seen eddies at different length scales. There are buzzwords in meteorology for this:

You have seen eddies at the ‘microscale’ (left) and at the ‘mesoscale’ (middle). A blog post about eddies should of course mention the most famous eddy of the last decade, which formed at the ‘synoptic scale’:

Do you recognize it? That was Hurricane Katrina.

It is obviously important to understand disasters like this one on the synoptic scale. This is an active topic of ongoing research, both in meteorology and in climate science.

### 46 Responses to Eddy Who?

1. silkenpaw says:

I’m afraid the math is beyond be but it is amazing that these phenomena look so similar at such hugely differing scales.

The photo of the clouds is awesome. Had no idea such things existed; I thought it was a Photoshop trick when I first saw the picture.

2. Bruce Blackwell says:

How very interesting. I am a retired physicist. Like you said, physicists are rarely exposed to the intricacies of fluid dynamics in their coursework. I think this is because those that design physics curricula are afraid to delve into those many fields of “classical” physics that are messy, carrying the baggage of many unsolved problems. All the classical problems are supposed to be solved, right? We have moved on to quantum physics, QED, string theory, and left the classical world behind. One needs to be reminded of the challenge and interest that many classical problems still present. In my dotage I have been looking into lightning, another classical field with many remaining mysteries.

• John Baez says:

Indeed, in my early dotage I have belatedly realized that the classical universe at macroscopic scales is a really exciting place for physics, both in terms of theory and in terms of ‘saving the planet’.

3. a missing ingredient in your description is that turbulence is intrinsically 3D. 2D guys like the hurricane Katrina are somewhat different.

nice post, though

• Tim van Beek says:

I’m not sure I understand the difference, although classical vector analysis concepts like rotation do make sense in three dimensions only, of course.
Is there a simple description of how more or less “two dimensional” phenomena like Katrina are different from 3 dimensional turbulence?

• Frederik De Roo says:

2D turbulence: “energy and enstrophy conserved, and no vortex stretching”
3D turbulence: “enstrophy not conserved, and vortex stretching present”

There is also quasi-geostrophic turbulence (Katrina?) which conserves enstrophy but does allow vortex stretching.

4. John Baez says:

Nice post, Tim! Someday I’ll write a blog entry on the Navier-Stokes equation, but just so people can see it and start getting less scared of it, here it is:

$\displaystyle{ \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f} }$

This is for a Newtonian incompressible fluid. We’ll see what ‘Newtonian’ means in a minute.

The left-hand side here is the rate of change of momentum per unit volume of a little ‘piece’ of the fluid. $\rho$ is the density and

$\displaystyle{ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}}$

is the acceleration of a little piece of fluid. This acceleration involves the rate of change of velocity as you just sit in one place:

$\displaystyle{ \frac{\partial \mathbf{v}}{\partial t} }$

but also the rate of change of velocity caused by moving to a new place where the velocity is different:

$\displaystyle{ \mathbf{v} \cdot \nabla \mathbf{v} }$

The sum

$\displaystyle{ \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}}$

is called the convective acceleration: click the link if you feel you want to understand this concept better!

In short, the left-hand side is a close relative of $m a$ from Newton’s law. Simiarly, the right-hand side is a close relative of $F$! It’s the force per unit volume on a little piece of fluid:

$\displaystyle{ -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f} }$

$p$ is the pressure, and $-\nabla p$, minus the gradient of the pressure, is the force due to pressure.

$\mu$ is the viscosity, and $\mu \nabla^2 \mathbf{v}$ is the force per unit volume due to viscosity. For example, if a little piece of fluid is moving slower in the $x$ direction than the average of its neighbors, the $x$ component of $\mu \nabla^2 \mathbf{v}$ will be positive, meaning that its neighbors will be pushing on it, trying to make it go faster in that direction! This simple linear formula is only valid for Newtonian fluids. Blood is a notably non-Newtonian fluid, because red blood cells push up against their neighbors in a much more complicated way. So, if someone uses the old saying “blood is thicker than water” on you, now you can annoy them by adding: “and it’s also non-Newtonian!”

Finally, $\mathbf{f}$ is any additional force per unit volume that we might be imposing on the liquid.

5. John Baez says:

The coolest concept – turbulence is a strange attractor embedded into a finite-dimensional submanifold of the solution space of stationary Navier-Stokes, produced from the Hopf bifurcation and infinite sequence of period-doubling bifurcations.

I find it a bit hard to imagine turbulence as sitting inside the space of stationary solutions of the Navier-Stokes equations. I have, however, seen some interesting things about finite-dimensional submanifolds of the space of solutions of Navier-Stokes.

Hmm, here’s one:

We construct fi nite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on $\mathbb{R}^2$ and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways.

And here’s another:

By imposing regularity properties on the terms of the Navier-Stokes Equations (with periodic boundary conditions), we can fin d in $H$ a maximal compact invariant set $A$ that is also the minimal set that attracts all bounded sets $X \subset X$, and we call this set the ‘global attractor’. On the global attractor, we can extend our semidynamical system to a true dynamical system. Finally, I will show that $A$ is finite-dimensional by constructing an explicit bound on its fractal dimension, and I will discuss in what sense this implies that the dynamics of the attractor are determined by a finite number of degrees of freedom, by showing that we can parametrize the attractor with a fi nite set of coordinates. The reader should have some familiarity with the languages of Banach, Hilbert and Sobolev spaces, as well as with the basic notations of PDEs and dynamical systems. For conciseness, some well known inequalities and estimates will be utilized without proof.

This is again the 2-dimensional Navier-Stokes equations. I’m not aware of similar results for the 3d case. I also don’t understand the qualitative nature of this ‘global attractor’. What are the solutions in there like?

6. Roger Witte says:

Presumably, temperature differentials enter into the equation indirectly via Boyle’s law and, therefore, turn up in the pressure term?

• Tim van Beek says:

AFAIK thermodynamical concepts like temperature are not necessary in order to derive the Navier-Stokes equations. That is, Newton could have formulated them himself without inventing thermodynamics.

If you have a fluid flow where a gas behaves like an ideal gas, you can use the pressure to calculate the temperature in a volume using the ideal gas law, of course.

• Yrogirg says:

AFAIK thermodynamical concepts like temperature are not necessary in order to derive the Navier-Stokes equations. That is, Newton could have formulated them himself without inventing thermodynamics.

btw, I’ve remembered a nice way to derive NS equations, I like it very much. One actually need only to write down the equation for the energy $\rho u + \frac{1}{2} \rho v^2$. This equation should remain the same under the Galileo transform ($+ \boldsymbol v_0$ to velocities and some changes to derivatives). Since we have $v^2$ thus we’ll get the sum of 3 terms:

multiplied by $\boldsymbol v_0^2$
multiplied by $\boldsymbol v_0$
no $\boldsymbol v_0$

Since only the last one should survive for the equation to preserve its form under the Galileo transformation, the other two should equal zero for all $\boldsymbol v_0$ — and we’ll get equations for $\rho$ and $\rho \boldsymbol v$. I learned that trick from the book “Extended Irreversible Thermodynamics” by David Jou, José Casas-Vázquez, Georgy Lebon. Once I’ve even conducted the derivation, though with the help of Maxima.

• John Baez says:

I like that derivation method, Ygorirg! Thanks for telling us about it. Symmetry arguments always appeal to me, and the Galilei group is sadly underappreciated compared to the Poincaré group, which is famously useful in special relativity.

You’ll note that the authors of Extended Irreversible Thermodynamics are a superset of the authors of Understanding Non-equilibrium Thermodynamics, which I’ve been discussing over in our chat about the principle of minimal entropy production. I should read both these books very carefully!

• Frederik De Roo says:

To borrow a few lines from Wikipedia:

The stress terms […] are yet unknown, so the general form of the equations of motion is not usable to solve problems.

(The stress terms include the pressure and a generalized form of the laplacian of the velocity, see John’s comment above)

The Navier–Stokes equations are strictly a statement of the conservation of momentum. In order to fully describe fluid flow, more information is needed (how much depends on the assumptions made).

If I remember well, the stress terms can be derived from the assumption of conservation of energy, but this introduces other unknowns, so in the end one has to resort to an equation of state or an approximation…

PS do you really mean Boyle’s law?

7. Yrogirg says:

The usual (Newtonian) fluid can be described by fields of density, temperature and velocity. Thus you need a set of equations for $\rho$, $T$, $\boldsymbol v$. With the equation of state you can switch $\rho$ or $T$ to $p$.

However, maybe it is better to think of 5 or even 7 equations:

1 for $\rho$ — conservation of mass
2 for $\rho \boldsymbol v$ — balance of momentum
3 for energy $\rho \varepsilon = \rho u + \frac{1}{2} \rho v^2$ — conservation of energy

— equilibrium properties of the substance
4 equation of state $p = p(\rho, T)$
5 equation for internal energy $u = u(\rho, T)$

— non-equilibrium properties of the substance
6 viscosity $\mu(\rho, T)$ (let us assume there are no other viscosities)
7 heat conductivity $\kappa(\rho, T)$

It is a slightly vague question, what set of equations to call the Navier-Stokes equation. I think the common variant is the first 3. But it doesn’t really matter — let them be just equations governing motion of Newtonian fluids.

8. Robert Smart says:

Did you mean “connection of eddies and some maximum entropy production principle”?

• As opposed to the maximum entropy principle?

My deep math hobby is analyzing the Fokker-Planck equation in the context of slowly varying disorder. The equation becomes much simpler due to some integrating identities and it is fun to see what kinds of behaviors it matches up with. So far it works well for explaining anomalous transport in PV semiconductors and dispersive transport through porous media. I use the Maximum Entropy Principle to vary the diffusion and drift coefficients around their mean values.

9. Yrogirg says:

Just to make things completely clear

I use the Maximum Entropy Principle to vary the diffusion and drift coefficients around their mean values.

So you mean “Maximum Entropy Principle”, not “Maximum Entropy Production Principle”, don’t you? In that case I really want to know what do you think about the Maximum Entropy Production Principle — I’m collecting people’s opinions on it.

• John Baez says:

I’m collecting references on the maximum entropy production principle and also Prigogine’s minimum entropy production principle. I just put them here:

Extremal principles in non-equilibrium thermodynamics, Azimuth Library.

The principle of maximum entropy production has been applied to climate science—see the references above. Unfortunately, I don’t have anything intelligent to say on this subject. I need more time to think about it.

10. I think the distinction between MEP and MEPP is that with MEP we use variational methods to estimate stochastic variates with minimal information and assumed constraints. This is really an information entropy calculation at its heart. In contrast, with MEPP, we actually try to predict which way reactions will proceed depending on thermodynamic calculations of the possible entropy growth gradient.

I think it was Prigogine that pushed the idea of minimum entropy production. This was for a system in equilibrium, which I can also rationalize. Yet the fact that we have both a Maximum Entropy Production Principle and a Minimum Entropy Production Principle makes it kind of confusing.

• John Baez says:

WebHubTel wrote:

I think it was Prigogine that pushed the idea of minimum entropy production.

Right. It was part of the work he won the Nobel prize for.

This was for a system in equilibrium, which I can also rationalize.

Actually Prigogine’s mininum entropy production principle applies to a certain special class of systems in steady state, not equilibrium. I’m trying to give more precise details here:

Minimum entropy production, Azimuth Wiki.

For example, it applies to a pot of water with a low flame under it, such that the water forms convection cells whose velocity is low, and constant in time at each point in the water:

When you turn up the flame higher, the flow become turbulent and Prigogine’s principle of least entropy production—or at least the usual proof of it!—breaks down.

Of course, this is precisely when things get interesting!

Yet the fact that we have both a Maximum Entropy Production Principle and a Minimum Entropy Production Principle makes it kind of confusing.

You darn well bet it’s confusing!

This is what I want to understand. There are some potentially very helpful comments here. However, I have not digested them yet.

• John, Thanks for straightening out the equilibrium vs steady-state premise (and this is just after your excellent discussion relative to the earth and the incoming and outgoing radiation).

• Suppose that we are in a steady-state with an ensemble of particles; and have established that the probability density function is in a maximum entropy configuration given the constraints.

In this case wouldn’t this be the case of minimum entropy production ala Prigogine, because the entropy production gradient is zero at this maximum entropy PDF?

• Yrogirg says:

Suppose that we are in a steady-state with an ensemble of particles; and have established that the probability density function is in a maximum entropy configuration given the constraints.

I think it is worth emphasizing that maximization of the entropy given certain constraints does not necessary lead to the true probability density function. Although this result can be used to construct the true one.

I’d like to know is there any cases when MaxEnt gives the true probability density function for a steady non-equilibrium state?

• John Baez says:

WebHubTel writes:

In this case wouldn’t this be the case of minimum entropy production ala Prigogine, because the entropy production gradient is zero at this maximum entropy PDF?

Yes, that’s true given the extremely restrictive assumptions needed to prove Prigonine’s principle. Unfortunately, those assumptions exclude most interesting situations.

These authors:

• Georgy Lebon and David Jou, Understanding Non-equilibrium Thermodynamics, Springer, Berlin, 2008.

give the following list of assumptions which they claim are needed to prove Prigogine’s minimum entropy production principle:

• Time-independent boundary conditions,

• Linear phenomenological laws,

• Constant phenomenological coefficients,

• Symmetry of the phenomenological coefficients.

The real killer is “linear phenomenological laws”: it means your system has to respond in a linear way to any change in boundary conditions.

For a pot of water as shown above, this linearity assumption is approximately true if the heat on the stove is low: double the heat and the convection cells swirl around twice as fast, remaining otherwise unchanged. But if you turn up the heat past a certain point, the flow changes qualitatively and Prigogine’s result no longer applies!

So, if I have a completely sealed pressure cooker with a high flame under it, I just don’t know if what it’s doing will minimize entropy production. Ditto for the Earth with the Sun shining on it.

It would be great to extend Prigogine’s result beyond the linear regime, but I suspect that it’s not true without some carefully chosen assumptions.

Right now I’m doing something simpler: I’m struggling to turn Prigogine’s result into an actual mathematical theorem of the sort I can understand! Lebon and Jou’s book is very helpful, but their argument makes at least one leap of logic.

I’ve started trying to straighten things out here:

Minimum entropy production, Azimuth Library.

but I realized yesterday that my attempt so far is deeply flawed. Luckily, I think I know how to fix it using ideas from another project I’m engaged in, on electrical circuits. So, I’m planning to write a paper about all this stuff. Once I get the linear case done, I’ll be in a position to contemplate the nonlinear generalizations!

• Yrogirg says:

The real killer is “linear phenomenological laws”: it means your system has to respond in a linear way to any change in boundary conditions.

I thought “linear phenomenological laws” means that thermodynamical fluxes depend linearly on thermodynamical forces. Thus it is really crucial what to call fluxes and forces. Ultimately one can say that Newtonian fluid is always governed by linear phenomenological laws — Fourier’s law for heat conduction and Newton’s model for molecular stresses. In the usual formulation of the minimum entropy production it is said it is valid for steady states and that’s why it fails when Bernard’s cells collapse, though the laws remain linear.

But anyway, let’s assume $1/T_h - 1/T_c$ to be the force and heat flux $Q$ to be the flux. If you state that Bernard’s cells correspond to a linear regime, than it seems there are two linear regimes! The first one when the heat is transported due to heat conduction alone rather than by any type of convection. I’m not an expert to claim that the regime with pure heat conductivity is possible, but Prigogine himself in his books claims many times that it is true. It is a canonical example of self-organization — pure heat conduction -> Bernard’s cells.

Does minimal entropy production theorem implies that at some point heat conduction produces more entropy then Bernard’s convection?

• Yrogirg says:

Oh, according to Wikipedia (Rayleigh–Bénard convection) heat transport via heat conduction is unstable and it spontaneously turns into convection. It still would be interesting to compare entropy production for the unstable (conduction) mode and the stable (convection) mode and to figure out what principle can be used to get rid of the unstable solution.

Huh, I misspelled “Bénard” in my previous comment.

11. John Baez says:

Speaking of eddies and hurricanes, here are some images of Hurricane Irene from the Cooperative Institute for Meteorological Satellite Studies at the University of Wisconsin-Madison:

I haven’t found any information on the second image, so I don’t know exactly what it represents. The colors describe “total precipitable water” in millimeters, which sounds like the amount of water in clouds—but if so, why is it only measured over the ocean?

• Frederik De Roo says:

Some googling:

Total precipitable water (TPW) in the atmosphere is the amount of water that can be obtained from the surface to the “top” of the atmosphere if all of the water and water vapor were condensed to a liquid phase.

but I’m quite sure it is measured across the whole (visible part of) the globe:

Microwave sounders are able to measure very low levels of microwave radiation naturally emitted by the Earth at different frequencies. Even water vapor emits microwave radiation that can be measured by microwave sounders.

I think that the land surface is later subtracted, maybe for visibility (but then drawing borders would be sufficient, I guess) or maybe also because the measurements are less reliable over land (it’s a guess, I’m not sure which other factors could contribute to the microwave channels that measure water vapor) or maybe because it is less important, as a hurricane is “fueled” over sea and decays over land.

• John Baez says:

Thanks, Frederik! Could their method involve bouncing microwaves from a satellite off the ocean surface and measuring the intensity of the microwaves that come back? That might explain why they don’t do these measurements on land.

Someone is doing a lot of measurements, to produce hourly scans of such a huge area!

• Frederik De Roo says:

Not really “bouncing microwaves” ;-)

As far as I know one measures the thermal spectrum of the earth in response to the insolation, and then tries to identify the signature of water vapour in the atmosphere. I’m quite sure water vapour above land will also give a response.

With respect to measuring the thermal spectrum: of course one can only measure a limited number of spectral channels but these are well chosen to discriminate as much as possible between the most interesting emission spectra. But there may be contamination from atmospheric effects (aerosols etc).

If you want more details, e.g. in this Technical Document on TPW from MODIS is written:

The retrieval algorithm relies on observations of water vapor attenuation of near-IR solar radiation reflected by surfaces and clouds. The product is produced only over areas that have reflective surfaces in the near-IR. Techniques employing ratios of water vapor absorbing channels centered near 0.905, 0.936, and 0.94 mm with atmospheric window channels at 0.865 and 1.24 mm are used. The ratios partially remove the effects of variation of surface reflectance with wavelengths and result in the atmospheric water vapor transmittances. The column water vapor amounts are derived from the transmittances based on theoretical radiative transfer calculations and using look-up table procedures.

There are a few thousand satellites in orbit.

(PS: despite the discussion about black bodies and planets in another thread, one can measure the spectrum emitted by the earth, and this spectrum is compared with the theoretical Planck spectrum to identify attenuation by e.g. water vapour in the atmosphere – of course for the shorter wavelengths the reflected sunlight also plays a role)

• Frederik De Roo says:

I’ve been sloppy when I wrote:

the thermal spectrum of the Earth,

mixing a general explanation and some particulars about TPW determination. In general there are two important contributions:

* the real thermal spectrum of the Earth, which exists because of the insolation;
* the reflected sunlight spectrum (including near-infrared) which is a more direct response to the solar radiation;

but for TPW the latter of the two is important.

For those who have read Tim van Beek’s post A Quantum of Warmth this should come as no surprize.

12. Boris Borcic says:

I react tangentially to the inserted figure 1.12 from Tsinober’s book of a dyed drop in clear water, upside-down next to a Nevada nuclear test. While you don’t exactly make the claim for the picture itself, you do claim that the picture illustrates at least a component of “the unreasonable effectiveness of mathematical models”. I have an issue with this. While I agree there is relevance, I can’t track any sense of clearly supportive illustration. Quite the contrary, in fact, since what you display is a remarkable naked-eye math-free example of a similarity that speaks more for analog models than mathematical models.

This might actually pass as appropriate humor – after all, it’s not as if you were hiding anything – but it feels something of a pity akin to wasting the data from a super-exceptional LHC collision.

• Tim van Beek says:

Since the post is mostly a “tell me what turbulence is all about in two minutes while I wait for my coffee”, one shouldn’t take everything to heart that is said there.

Anyway, high energy physicists are proud to say that the standard model (or quantum electrodynamics, to be more precise) is the most successful physical theory known today, with the most convincing agreement of theory and experiment. I was half-jokingly thinking about the possibility that – depending on your reference frame – the Navier-Stokes equations may be more worthy of this praise, because they successfully predict a lot more phenomena on a wider range of length scales, and have a lot of applications in engineering, too.

• Boris Borcic says:

I see – and I shouldn’t in any case forget to thank you for that pretty morsel. It makes me regret another image of parallel flows that I’ve lost for decades, that of a chaotic short time-scale aside a perfectly regular long time-scale photograph of the same turbulent flow out of a submerged nozzle – like an ad for perturbative computations. I had it on my wall for years.

I don’t really mean to deprive the Navier-Stokes equations of the glory that flows display, but at the same time I believe the focus put on the equations blinds to… what similarly paired images can in a sense directly teach about flows.

13. Robert Smart says:

The opposite of analog is discrete/digital. Mathematics can be analog, indeed that’s what calculus is about. The opposite of mathematical is hand-wavy. Not that there’s anything wrong with that…

• Boris Borcic says:

I wonder if perhaps I’ve been betrayed by my browser autocorrector that pushed me to “analog” rather than “analogical”.

Anyway, 1) isn’t there an ingrained sense to “mathematical modeling” that implies predictions get read off mathematical computations 2) isn’t there an acknowledged and contrasting sense to “analogical modeling” that implies prediction follows from reading answers directly off another physical system, with little or no use of mathematical computations ?

14. Robert Smart says:

We all make hand-wavy (including analogical) arguments most of the time. Mathematicians think about how to make it more precise, i.e. mathematical. The two reasons for not doing that are: (a) you don’t know how, which brings the whole argument into question; or (b) you don’t want to be that transparent and make life easy for your opponents. Well mathbabe has something to say about when mathematical modelling misleads people who don’t understand or don’t have access to the model being used: http://mathbabe.org/2011/08/28/what-is-the-mission-statement-of-the-mathematician/.

• Boris Borcic says:

Making hand-wavy arguments shouldn’t be confused with talking over the other person’s head, although it’s of course possible to do both at once. The problem with your confusion of the mathematical with “making things precise” is that it makes it too easy to dismiss precision and nuance in ordinary language that’s not obviously backed by mathematics. Most notably, it short shrifts the idea that a physical equivalence can replace a mathematical equivalence.

15. Phil Henshaw says:

I studied the forms of air currents as energy transport systems within buildings for years, and found both their stable and unstable complex organization truly remarkable.

There are a variety of examples, like the new form of high speed surface layer convection I patented, which was then tossed out because the second appeals board wanted me to have an equation for it. I didn’t have an equation, just a way to produce it. The system’s not supposed to work that way, but that’s what happened.
fyi http://synapse9.com/airwork.htm#fluwall

The other one I first thought to mention is the one we all see demonstrated fairly frequently at the end of a nice dinner. Someone blows out some candles. If you just watch you often see smoke rising from the wick in a very thin column and then billowing out at a certain distance above. the same thing can also be seen using a stick of incense in a very quiet place, which produces smoke that is a little more visible.

I think the mechanism of turbulence is a little easier to observe in that extremely simple case, as it’s even possible to watch the self correction mechanism of the columnar flow, and see it begin to “fishtail” and go out of control just before the disintegration of the laminar flow into a complex tumbling occurs.

16. very interesting discussion but seems to be a far-fetch. what about somewhat simpler problems – could we know the amount of energy dissipated by turbulence in the bulk of turbulent flow? could we estimate the rate of energy transfer from large eddies to small ones and vice versa?

• John Baez says:

Alex wrote:

could we know the amount of energy dissipated by turbulence in the bulk of turbulent flow? could we estimate the rate of energy transfer from large eddies to small ones and vice versa?

Kolmogorov came up with a theory to answer them in 1941, and this theory seems at least approximately true, but it’s a complicated business which I don’t understand nearly as well as I’d like to.

17. […] This is called viscosity. I already explained it back in the post Eddy Who? High viscosity means that there is a lot of friction […]

18. nodota says:

Reblogged this on nodota.

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