## GCMs and the Navier-Stokes Equations

Kaufmann and Stern contained a reference to the provocatively titled Govindan et al. [2002], Global Climate Models Violate Scaling of the Observed Atmospheric Variability, Phys. Rev. Lett. 89, available here . I’ll comment at some time on the scaling issues, but it contained the following concise description of GCMs which I liked:

The models [coupled atmosphere-ocean general circulation models (AOGCMs) ] provide numerical solutions of the Navier Stokes equations devised for simulating mesoscale to large-scale atmospheric and oceanic dynamics. In addition to the explicitly resolved scales of motions, the models also contain parametrization schemes representing the so-called subgrid-scale processes, such as radiative transfer, turbulent mixing, boundary layer processes, cumulus convection, precipitation, and gravity wave drag. A radiative transfer scheme, for example, is necessary for simulating the role of various greenhouse gases such as CO2 and the effect of aerosol particles. The differences among the models usually lie in the selection of the numerical methods employed, the choice of the spatial resolution, and the subgrid-scale.

Note carefully the focus on Navier-Stokes equations, which are notoriously intractable with very difficult mathematics. In fact, they are one of the Clay Insitute’s seven "Hilbert" problems for the 21st century – a \$1 million prize is attached to any progress on the mathematics of the Navier-Stokes equations. So one should not assume that brute force numerical methods necessarily evade difficult and subtle mathematical problems. The Clay Institute’s snapshot of the problem is:

Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.

Charles Fefferman writes in the Clay Institute problem statement paper, Existence and Smoothness of the Navier-Stokes equation here :

Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behaviour of solutions of the Navier-Stokes equations. Since we don’t even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.

A discussion in Physics World is here. Ray Girvan writes here:

In May 2002, the Clay Mathematics Institute (CMI) of Cambridge, Massachusetts, in an initiative to further the study of mathematics, allocated a \$7m prize fund for the solution of seven Millennium Problems, ‘focusing on important classic questions that have resisted solution over the years’. One of the \$1m problems stands out for its massive practical importance: the solution of the Navier-Stokes equations (NSEs) for fluid flow.

Although there are many named variants and special cases, the fundamental equations are the incompressible Navier-Stokes for Newtonian fluids. In their most compact form, they comprise a pair of vector partial differential equations (PDEs): one expresses the forces acting (pressure, viscosity and body forces); the other is the continuity equation, which says that divergence of the velocity field is zero for an incompressible fluid (that is, ‘what comes in, goes out’).

The NSEs are among the most-studied partial differential systems, the subject of around 15-20 published papers a week. Nevertheless, they’re among the least understood at a theoretical level.

Figure from Girvan article.

In short, A GCM "control run", is essentially one numerical run from a hugely complicated Navier-Stokes equation, the deep mathematical properties of which mathematicians say they know very little. Climatologists on the other hand appear to know the results to high degres of certainty – remarkable.

Reference: R. B. Govindan, Dmitry Vyushin, Armin Bunde, Stephen Brenner,Shlomo Havlin,1 and Hans-Joachim Schellnhuber, Global Climate Models Violate Scaling of the Observed Atmospheric Variability. 2002, Phys Rev Letters, 89 http://www.atmosp.physics.utoronto.ca/people/vyushin/Papers/Govindan_Vyushin_PRL_2002.pdf

1. Peter Hartley
Posted Dec 22, 2005 at 10:20 AM | Permalink

Clearly the climatologists get so much money from AGW research that the \$1 million prize for having solved the Navier-Stokes equations is of such trivial value by comparison that they haven’t bothered to claim the prize.

2. TCO
Posted Dec 22, 2005 at 10:30 AM | Permalink

Why do they use Navier Stokes at all? I would think some sort of heat balance (with feedback mechanisms, etc.) would be better, no? You would still have physical insights, etc. but would not be using a fluid flow model to model a system that is more of a heat balance type problem.

3. TCO
Posted Dec 22, 2005 at 10:31 AM | Permalink

I don’t need to use turbulent flow models to understand a pressure vessel that has nucleate boiling.

BTW, this is really said in defense of them! In defense of being able to do modeling and get useful insights!

4. Steve McIntyre
Posted Dec 22, 2005 at 10:59 AM | Permalink

#2 – if your interest is primarily in global temperature, it’s hard to see why you need the terabytes of operations – maybe you do, I haven’t delved into the matter. I notice that, trom time to time, they model GCM outputs by simpler models and use these simpler models to project GCM outputs. That’s what they did in IPCC TAR – see Wigley and Raper. Also, Stone et al, cited by Gavin, uses Energy Balance Models (EBMs) to estimate GCM results. Kaufmann and Stern argue that the GCM estimates of global temperature are actually worse than linear estimates using only forcing inputs. Maybe using GCMs adds ritual to the process. They are fancy and expensive and take eons to run. They look impressive and widen the gulf between clerics and laity. I don’t see a big effort at de-mystification by Gavin and his crowd. They prefer papal bulls and councils of bishops.

5. Peter Hartley
Posted Dec 22, 2005 at 11:06 AM | Permalink

TCO: Ross or Chris would be better people to discuss this issue than me, but if I understand them correctly your comment is closely related to the point they make about AGW differing fundamentally from a real greenhouse. They point out that a real greenhouse is an enclosed system so we do not need to worry about the turbulent flow. The inside of the greenhouse has to heat up enough until the outgoing radiation is sufficient to balance the incoming radiation. It really is a radiation balance problem. On the other hand, the so-called “greenhouse gas” issue is quite different in that a large part of the energy transfer between the ground (and near ground air) and the troposphere, and between low latitudes and higher latitudes at ground level, takes the form of turbulent flows — cloud and cyclone formation, wind flows, ocean currents etc. There is no “glass ceiling” to contain the turbulent flow as in a real greenhouse. Your pressure vessel seems more like a real greenhouse and unlike the real atmosphere and oceans.

6. TCO
Posted Dec 22, 2005 at 12:03 PM | Permalink

I still don’t quite buy it. I can do useful thermodynamic analysis of “open systems” like the Brayton cycle for instance.

7. Fredrik
Posted Dec 22, 2005 at 3:06 PM | Permalink

Incompressable version of Navier-Stokes equation?
The pressure goes down as you go up in the atmosphere.
So as apart from a very local, solution I think you have to use the version with varying pressures.
These equations beeing very sensitive to initial conditions – as you put it here chaotic and turbulent. Our as in the book (Essex and Mcitrick – Taken by storm) everytime the thunder strikes you have a new problem to solve. So no.1 The NS-equation is not solved… or?

The point about chaotic systems spills over to the ‘problem about climate and global warming’ – the current physics describing processes underlying the climatic system are turbulent and chaotic. So we have to stay with statistics – the process of describing the effect of the unknown or the in principle unknowable. So everybody unless you have really on your feet – be agnosics!

Yours Truely
Fredrik

8. Reid B
Posted Dec 22, 2005 at 5:37 PM | Permalink

GCM’s can never be accurate as time progresses in the model. Like a broken clock, they can be accurate by coincidence but not because the underlying partial differential equations are predicting the future.

When equations can predict size and intensity of solar changes, volcanic eruptions and quadrillions of random events, then we may have truly skillful GCM’s. That isn’t going to happen.

The future cannot be predicted by projecting the past.

9. Juan Rivero
Posted Dec 22, 2005 at 6:00 PM | Permalink

The Navier-Stokes equations are usually undestood to mean the equations of fluid flow with a particular kind of stress tensor. The GCMs do NOT use Naver-Stokes. The base of an atmospheric GCM is a set of equations called the “primitive equations”. These represent conservation of momentum, the continuity equation (conservation of mass), the first law of thermodynamics (conservation of thermal energy) and lastly, an equation of state. If we track water vapor we need another continuity equation. The atmosphere is assumed to be hydrostatic at the scales modeled (~10-100 Km). The ocean part is similar, but continuty amounts to zero divergence, because the ocean is taken to be incompressible.

All that said, we then come to the forcings. These all occur at scales far below the “grid scale”. These include condensation and precip, radiation, and friction. This is where the problems occur, not in the numerics (there are some problems with numerics but I do not think they are insoluble).
They are modeled by “parameterizations”, i.e. functions of the variables you actually track.
The parameterizations cannot follow physical laws, because either these laws are intractable or because we don’t know how to model them in a finite-resolution model.

One can fling many tomatoes at the GCMs but one should understand something about them before flinging.

10. Steve McIntyre
Posted Dec 22, 2005 at 6:53 PM | Permalink

The attribution of Navier-Stokes to GCMs was by Govindan et al in Phys Rev Letters, so you might want to sent a note to Phys Rev Letters on the topic. I haven’t seen them issue a Corrigendum. A question: presumably Navier-Stokes applies. So if they don’t use Navier-Stokes, what is the effect of not using it if it’s a Navier-Stokes system? I don’t see how you avoid the Navier-Stokes problems by ignoring it, if that’s what they do.

I’m discussing GCMs because people want to and have made it clear that I’m not familiar with the literature. So if I mis-state something, I apologize.

11. Steve McIntyre
Posted Dec 22, 2005 at 10:46 PM | Permalink

#9, 10: Juan, the following appears to contradict what you said: "The 3-D primitive equations neglect vertical acceleration and are derived from the Navier-Stokes equations by invoking scaling arguments for the atmospheric general" here

So if the "primitive equations" are a simplified version of Navier-Stokes, it looks to me like all of my comments stand.

12. Bob K
Posted Dec 22, 2005 at 11:41 PM | Permalink

Steve,

Your link in #11 doesn’t work. Think you have to remove the characters before http://

13. Posted Dec 23, 2005 at 6:45 AM | Permalink

Dear Steve,

what you write is morally right, but on the other hand, be sure that the Clay institute people are slightly more demanding and an approximate numerical solution of the NS equations in one situation – even if it were pretty accurate – certainly won’t earn you a million of bucks.

All the best
Lubos

14. Ross McKitrick
Posted Dec 23, 2005 at 4:44 PM | Permalink

Peter (#5) re-states part of the TBS argument exactly right. The “Greenhouse Effect” does not work like a greenhouse, and using the metaphor assumes away everything that makes the problem complicated. Greenhouses warm because the glass dampens the fluid motions and the radiation has to intensify to maintain energy balance, which guarantees warming inside the greenhouse. The theory is unambiguous–Temperature enters the equation of radiative transfer as a level. In the atmosphere it’s the opposite. CO2 dampens the radiation drain (a little bit), and the unconstrained fluid motions have to change to maintain energy balance. The theory does not exist to enable predictions about how temperature levels will change. Even if the NSE could be solved and computed, temperature enters as a gradient rather than a level, so you’ll only be computing how differentials change, not averages across levels. GCMs may be admirable for many reasons, but they are not a replacement for theoretical solutions to the physical problem. If theoretical solutions are currently impossible, that’s life.
#9: As I understand it an “equation of state” exists only in equilibrium. How can one be defined for the Earth’s climate?

15. John A
Posted Dec 23, 2005 at 6:08 PM | Permalink

Can I ask a dumb question: Do GCMs model the fact that the Earth is not an inertial system, ie it rotates?

16. Terry
Posted Dec 24, 2005 at 12:08 AM | Permalink

Re: #15.

I sure hope so. The Coriolus acceleration is critical to understanding weather patterns.

17. Willis Eschenbach
Posted Dec 24, 2005 at 12:49 AM | Permalink

Re 15: I’d like an answer to that as well, because

1. winds are driven by rotation of the earth, and would exist even if the earth were not heated by the sun.

2. winds are central to the global heat balance, because other things being equal, evaporation is directly proportional to wind speed. They are also important in that the number of airborne particles (sea spray, dust, natural aerosols, etc.) are wind dependent.

3. rotation plus sun gives us the trade winds and the operation of the Hadley cells, which is central to the global temperature balance.

4. by means of winds, the atmosphere exchanges momentum with the ocean, leading to natural cyclical temperature phenomena, as shown in the COADS data.

In short, if the GCMs can’t replicate the winds, they’re in real trouble.

w.

18. Pat Frank
Posted Dec 24, 2005 at 12:58 AM | Permalink

Regarding GCMs and predictions, Tad Murty’s August 30 article in the Toronto Star specifically discusses predictions and outcomes:

"Tomorrow’s forecast: Hysterical; Data fail to back up claims weather is getting worse, says Tad Murty"

The link is long, sorry: http://tinyurl.com/7pdnm

Perhaps his most telling comment: "[A]fter being associated with such [GCM] simulations for the past 45 years, I have little faith in their predictions. With a very slight tweaking of one single parameter (low cloud amount) in the model, forecasts can change abruptly from global warming to an ice age."

Maybe we should ban cloud-seeding experiments.

John adds: Use tinyurl.com and save on electrons!

19. Hans Erren
Posted Dec 24, 2005 at 2:17 AM | Permalink

re 17:

I remember from last year: A climate model created two Intertropical Convergence Zones (ITCZ), this was considered a minor point.

20. epica
Posted Dec 24, 2005 at 4:40 AM | Permalink

#15
“Can I ask a dumb question: Do GCMs model the fact that the Earth is not an inertial system, ie it rotates?”
It’s hard to believe that someone who is asking such type of questions on the other hand has a strong opinion about it. It’s like someone not knowing what a nuclear reaction is but is completely convinced that nuclear power plants will safe the world’s energy problem. Try a bit more Wittgenstein:Whereof one cannot speak, thereof one must be silent.

21. John A
Posted Dec 24, 2005 at 4:55 AM | Permalink

Re: #20

A simple “Yes” or “No” would suffice.

22. Willis Eschenbach
Posted Dec 24, 2005 at 5:21 AM | Permalink

Re #21: Excellent, John. Crisp, to the point, no extra words. I have to learn to resist the urge to make … um … let me call them “intemperate comments”.

And like you, I’m waiting for the yes or no …

w.

23. epica
Posted Dec 24, 2005 at 5:42 AM | Permalink

#21 and 22
The primitive equations are formulated on a rotating frame. As an exercise you learn that 1st year in physics. I just thought that someone who found out that the parametrisation of the water cycle in GCMs is manipulated to get certain sensitivities to greenhouse gases as John A should have looked into many lines of GCM code and read many many papers. He then certainly will know such kind of basics, at least much more basics than the parametrisation of the water cycle. But I was wrong.

24. fFreddy
Posted Dec 24, 2005 at 6:06 AM | Permalink

Bugger Wittgenstein. Whereof one cannot speak, one should ask questions until one can.
A Merry Christmas to one and all.

25. Doug L
Posted Dec 24, 2005 at 6:32 AM | Permalink

Re #20

I have played with a computer model, so I guess I’m entitled to speak. The model is called “Railroad Tycoon II” it has an editor which allows you to do experiments. I noticed two things. It’s incredibly time consuming to generate a single data point and it’s easy to think you’ve figured out some “truth” about how the model works only to find out later, you didn’t take something into account.

26. Hans Erren
Posted Dec 24, 2005 at 8:12 AM | Permalink

Paleoclimatology Reconstructing Climates of the Quaternary. Bradley, R.S., 1999. Second Edition. Academic Press International Geophysics Series, Volume 64. ISBN 012124010X

— quote from page 475-476 —

Convective thunderstorms, for example, play a critical role on a global scale in latent and sensible heat transfer from the surface to the atmosphere, but individually they are too small to be represented by even a 2 degree by 2 degree grid spacing. In such cases the process is represented in a sinmplified manner as a function of of other variables, a procedure known as as parametrisation (parametric representation).

— end quote —

Likewise, hurricanes are also parametrically represented.

27. John A
Posted Dec 24, 2005 at 9:18 AM | Permalink

Re #26

That’s relief. Otherwise you’d think they were guessing.

If hurricanes are parametrized in a simplifed manner as a function of other variables, then how do climate modellers know how many hurricances there are supposed to be?

28. Douglas Hoyt
Posted Dec 24, 2005 at 9:32 AM | Permalink

Any parameterization can be fit by several different equations that to the eye look identical over the range where the fit is made, but, nonetheless, have quite different variations outside the range of fit. So if the climate drifts outside of the range of fit, how can we tell if the climate model is still be sensible?

In a more general sense, it seems to me that these parameterization equations would interact with each other and the corresponding output may not be anything like the output if an exact solution were available using the laws of physics.

29. Dave Dardinger
Posted Dec 24, 2005 at 9:34 AM | Permalink

Hans,

Exactly how do these parameters get entered in? For instance, there’s bound to be a relationship between cloudiness and thunderstorms, or between average wind velocity and cloud formation (via dust for water droplet formation). Don’t these sort of interactions require treating things as variables, not parameters, or is that part of the parameterization process? I’ve always thought parameterization as a look-up table sort of thing, rather than a hard-coded value sort of thing, but in either case can the value be changed during the course of a run or not? If it can’t then I’d think a lot of the value of a GCM would be lost.

30. Willis Eschenbach
Posted Dec 24, 2005 at 3:02 PM | Permalink

Re 21: Epica, you say:

The primitive equations are formulated on a rotating frame.

Unfortunately, you then continue with a bunch of insults, which detract heavily from the point you are making.

Setting that bit of unprofessional and unpleasant behaviour aside, what do you mean by the “primitive equations”? Do you mean:

1. The equations in the GCMs are all primitive (which is quite believable, given their often ridiculous results), and are all formulated on a rotating frame

or

2. Some of the equations in the GCMs are “primitive” (meaning what? which equations are they?) and some are not, and only the “primitive” equations are done on a “rotating frame”.

Finally, is the rotating frame spherical or cylindrical?

Please leave out the insults in your reply, as they only make you look foolish. As my science professor used to say, “The only stupid questions are the ones you don’t ask” …

Thanks,

w.

PS — I note that Trenberth et. al. state that the size of the Hadley Cell circulation (one consequence of a rotating earth) is underestimated in most GCMs by ~20%. This does not inspire a lot of confidence in your “primitive equations” and how they are handled in GCMs.

31. Juan Rivero
Posted Dec 27, 2005 at 4:10 PM | Permalink

```The attribution of Navier-Stokes to GCMs was by Govindan et al in Phys Rev Letters, so you might ant to sent a note to Phys Rev Letters on the topic. I haven't seen them issue a Corrigendum.A question: presumably Navier-Stokes applies. So if they don't use Navier-Stokes, what is the effect of not using it if it's a Navier-Stokes system? I don't see how you avoid the Navier-Stokes problems >by ignoring it, if that's what they do. ```
In most classical fluid mechanics texts, “Navier-Stokes” refers to a particular stress tensor for the fluid. It gives rise to a frictional dissipation term — I can’t write equations easily here — proportional to the Laplacian of u and v (horizontal components of velocity). The proportionality constant is the molecular viscosity. Now, in the atmosphere at the scales we are interested in, the Reynolds number is so high that the viscosity is negligible and one has to use one of the various (and unsatisfactory) eddy dissipation parameterizations. Some of these look like Navier-Stokes with an “eddy viscosity” coefficient. In that sense they look like N-S. I suppose that is why the PRL contribution used the term. Terminology is a bear. I think one should use “Navier-Stokes” for the original equations, and use “primitive equations” for those used by GCMs.

In aircraft design, similar equations (numerical models) are used with great success, to the point where wind tunnels are dying out. So a priori one can’t say that numerical models are useless. But any numerical solution is a particular solution of a problem. In order to collect the megabuck, one would have to come up with general solutions of the problem.

32. beng
Posted Feb 15, 2006 at 9:18 AM | Permalink

There’s some interesting discussion on GCMs at Pielke Sr’s site here:

33. John Creighton
Posted Dec 12, 2007 at 10:39 PM | Permalink

I’m not sure if this has been discussed before but how valid are the Navier-Stokes stokes equations for the climate assuming we could solve them? Most fluid equations are only valid over distances where the mean free path is small relative to the state properties of the fluid.

What happens in the upper portions of the atmosphere? Does the mean free path exceed the the distance at which the pressure and temperature of the fluid change? Then of course photons although not considered part of the fluid have very long mean free paths.

34. Geoff Sherrington
Posted Dec 13, 2007 at 1:28 AM | Permalink

Navier-Stokes.

I suspect these are used to frighten people who question the dogma. I questioned a scientist in Australia’s eminent Government research body, CSIRO, about a cold upwelling of ocean off NSW and received this response:

Sent: Wednesday, May 02, 2007 7:06 PM
Subject: Cold upwelling off Sydney

Geoff,
I’m sorry but there’s a lot more to it than just the eqn of state.
The dynamical equations are called the Navier-Stokes equations but to get a feel for the dynamics its best to consider various simplifications. Principle (sic) of these is the ‘Thermal wind’ equation. Simpler still is the concept of geostrophic balance – a near-equilibrium that says that if sea level is forms a round depression, the current velocity (in the S Hem) is clockwise around it. The pressure gradient then balances the Coriolis force. The vertical gradient of the velocity field depends on the horizontal gradient of the density field. Some eddies are only shallow. Some reach way down. Some are strong at middepth, zero at the surface. The outflow of the Mediteranean forms these. Homework: Google all those terms.

Scientists do not react well to suggestions to Google terms, especially when they have worked with related concepts, so I replied in part:

Ref: The Stern Review: A Dual Critique (authors acknowledged)
Quote from reference: “The governing equations of fluid dynamics (Navier-Stokes) have resisted solution for over 100 years; indeed the Clay Institute is offering a \$1 million prize to anyone who can merely prove a solution exists.”

So, are CSIRO going to claim the Clay Institute reward, or are they going to be correctly scientific as the climate change debate continues with its component of oceanography?”

After receiving no answer, I concluded that bully boy tactics include the fluent throwing around of expressions like Navier-Stokes, Reynold’s Numbers, Coriolis Forces etc.

The exchange of emails, well-intentioned on my part, resulted in NIL progress. I never did discover if their models were finally checked for compliance with equations of state. I now suspect bully-boy work each time I see reference to Navier-Stokes. It’s a dominant male thingo.

35. Philip Mulholland
Posted Dec 13, 2007 at 2:37 AM | Permalink

In my book there is no such thing as a dumb question, but I am sure that there are plenty of dumb answers.

36. Geoff Sherrington
Posted Dec 13, 2007 at 3:41 AM | Permalink

Re # 1 Peter Hartley

Don’t answer if you don’t wish to, but are you familiar with the Tasman Institute and Michael Porter and Texas?

37. Tom Vonk
Posted Dec 13, 2007 at 7:41 AM | Permalink

Im not sure if this has been discussed before but how valid are the Navier-Stokes stokes equations for the climate assuming we could solve them? Most fluid equations are only valid over distances where the mean free path is small relative to the state properties of the fluid.

The Navier Stokes being a local continuous conservation equation , it is valid as long as the continuity requirement stays valid .
That’s true for climate purposes or otherwise .
There are situations where the mean distance between molecules is of the same order of magnitude as the flow characteristics and then Navier Stokes can no more be used – f.ex the space shuttle reentrance in the atmosphere can’t be calculated with N-S .
However as has been seen , the climate models do NOT solve N-S .
They don’t do so because they use a hundred kilometer grid and with such ridiculously low resolution they wouldn’t even capture
the coarsest characteristics of the relevant solutions .

Can one therefore say that N-S is without relevance for climate models ?
Of course not because N-S contains a vital information about energy balance , namely energy dissipation .
If that was neglected altogether , the results would be completely irrealistic .
But as energy dissipation , be it by turbulence or by friction with the surface happens at scales far smaller than the resolution of the models , it must be replaced by different coefficients that are supposed to be consistent with what N-S would say at the appropriate scale .
How the models handle that and does it give realistic figures is the true and serious question .
Only an example relating to the friction :
– you can take some average constant value (w/m²) of dissipation for the whole planet
– you can consider different constant coefficients depending on the location only (obviously wind friction over Antarctica
will be much lower than wind friction over the roaring fifties where the ocean is always exceptionnaly rough)
– you can consider that the coefficient is some function of the horizontal wind speed component at surface

All of the above are approximations with unknown impacts on the solutions because energy dissipation varies both in time and in space and explaining that it would all “average out” over several years simply doesn’t compute .
I assume that if the modellers do correctly their work , they try different formulas and look at the sensibility (aka what did it do to the results of the model) .
Now knowing how extremely complicated energy dissipation problems are , I doubt that they spend much time looking in it and would rather bet that if certains realistic formulas pose problems to the model behaviour , they’d rather tune it out .
Of course not being familiar with teh details of GCMs , I don’t know what is exactly the code and how they use/tune it for the above mentioned issues .

38. John Creighton
Posted Dec 14, 2007 at 12:16 AM | Permalink

I tried to write out a little more clearly my question about weather the equations are valid:

The laws of Fluid mechanics are valid when the intrinsic properties of the fluid change slowly relative to the mean free path of the particles in the fluid.

In the case of the climate all energy must inevitableness leave the earth through radiation. Photons effect the intrinsic properties of a fluid but our ability to measure fluid properties is on much smaller scales then the distance that light travels before interacting with the fluid. While we could try to treat the two systems as separate, photons behave like particles in the sense that they can collide with molecules and exchange energy trough collisions.

Because the mean free path of photons is much larger then molecules the distribution of photons my suggest a different temperature then the distribution of particles. Moreover when the particles absorb select frequencies of photons the distribution of photons is no longer a black body distribution and the distribution of particle is likely no longer a perfect Boltzmann distribution. If the particle distribution is altered from an idealized distribution by the photons then what consequence will this have on thermodynamic properties like, conductivity, viscosity and entropy.

http://fluid-entropy.blogspot.com/2007/12/though-disorder-blog-was-born.html

My bigger question though is how do the climate models deal with radiation? Do they deal with each frequency separately? How many different frequency bins do they use to describe the state of the system?

39. John Creighton
Posted Jan 4, 2008 at 10:13 PM | Permalink

Someone mentioned stephan Boltzmann’s law doesn’t apply to T_earth. There is a law of probability f(E[x]) approaches E[f(x)] as the variance as the variance of x approaches zero. Basically it works because for small variance the function is roughly linear for most values of the random variable.

The variance in the temperature from one place on the earth to another as a percentage of the temperature in Kalvin is quite small.

40. John Creighton
Posted Jan 4, 2008 at 10:17 PM | Permalink

ops I replied in the wrong thread. lol

41. Phil.
Posted Jan 4, 2008 at 11:57 PM | Permalink

Difficult as the N-S and Euler equations are the wings of all the Boeings you’ve flown on were designed using a solver written by a friend of mine, difficult doesn’t mean impossible.

42. Jan Pompe
Posted Jan 5, 2008 at 4:40 AM | Permalink

Phil might I remind you that one of the issues mentioned in the head post is scaling.

While and approximate solution might be possible for a small area over a short time period using numerical methods it doesn’t mean it is or will be in the foreseeable future over the scale of continents over a century.

43. Posted Jan 5, 2008 at 8:20 AM | Permalink

Phil–
But how is the success of Boeing relevant? Predicting high Re flow over airfoils prior to stalling has always been more tractable than fully three dimensional flows with mixing due to natural convection. This isn’t a secret.

As your friend to use his model to predict drag on the airplane if it’s put in a wind tunnel sideways.

44. Arthur Smith
Posted Jan 5, 2008 at 10:19 AM | Permalink

Oh, no the dreaded Navier-Stokes equations! The ones that govern all fluid flow, the ones that are used, for instance to help design aircraft wing shapes, race car geometries, the ones we rely on for computing weather forecasts, the ones that are at the basis of all mathematical analysis of fluids such as computing the properties of bearings and so much more…

In short, if GCM’s were *not* based on Navier-Stokes, they would be doing something very fundamentally wrong. Just because there are no analytic solutions does not mean they cannot be addressed computationally, as others have observed here – without computational tractability, all those other applications of Navier-Stokes would be worthless and computational fluid dynamics would be an impossible field.

A GCM “control run”, is essentially one numerical run from a hugely complicated Navier-Stokes equation, the deep mathematical properties of which mathematicians say they know very little. Climatologists on the other hand appear to know the results to high degres of certainty – remarkable.

can most charitably be described as a display of deep ignorance. The most fundamental wrong thing here is your claim that “climatologists […] appear to know the results to [a] high degree of certainty” – they do not: GCM results are compared to one another to look for uncertainties and they indeed exhibit uncertainties to a high degree – the IPCC reports are full of uncertainty estimates. But that does not mean they are not useful – far from it, these approaches are the best we have.

Furthermore, I fail to see how your claim of certainty even addresses the sort of issues that seem to keep coming up on this site, that touting the results of the GCM’s is alarmist etc. If we had certainty that the outcome of anthropogenic climate change would be bad, that would be one thing. If we had certainty it would not be so bad, that would be another. But we don’t have either situation, we have a situation where, almost fundamentally from the basic equations governing the problem, we have a high degree of uncertainty.

Given that high degree of uncertainty, what would the reasonable engineer do? Surely recommend action in the way that provides a comfortable safety margin to even the worst case of the uncertain range in question. The more uncertain we are about the GCM results, the *greater* the need for action, since surely a more certain GCM number for climate sensitivity, for instance, would be well within the worst case we see now.

45. Jan Pompe
Posted Jan 5, 2008 at 12:39 PM | Permalink

Given that high degree of uncertainty, what would the reasonable engineer do?

Look for a better method before going off half cocked with solutions, that could cost lives or livelihoods, for somthing that might not even be a problem or soluble in the way they suggest in the event it is a problem.

46. Robert Pangilinan
Posted Jan 5, 2008 at 12:52 PM | Permalink

Re 44:

Given that high degree of uncertainty, what would the reasonable
engineer do? Surely recommend action in the way that provides a
comfortable safety margin to even the worst case of the uncertain
range in question. The more uncertain we are about the GCM results,
the *greater* the need for action, since surely a more certain GCM
number for climate sensitivity, for instance, would be well within the
worst case we see now.

Huh?

So, which worst case scenario should we prepare for now?
An impending tropicalization of Antarctica or the
Antarctization of the tropics? Both?

I think you are wrong in your belief. It is the people
who are used to spend ranpantly other peoples money who would
think in the way you describe.
Engineers are trained to think in
economic terms and I doubt any engineer would
recommend preparing for the worst case scenario if the
economic costs are too high. Otherwise, all vehicles in
roads today would be built like tanks.

Proactive AGW proponents are alarmists because it is their
job to sensationalize. Imagine how much media mileage
you are going to get if you say something along the
lines of “the results of computer models have a lot of
uncertainty but the worst case scenario is that
the temperature will rise 2 degrees this century.
I suggest that we deliberately cripple our economy, spend
trillions in carbon mitigation, and commit untold sacrifices
just to be on the safe side.”

47. Posted Jan 5, 2008 at 4:13 PM | Permalink

Arthur– I believe your area is not fluid dynamics?
First, GCM’s don’t solve the Navier Stokes equations. They can’t– it’s too computationally intensive. They solved highly parameterized versions.

Second, there are tractable fluid dynamics problems and intractable ones. Which are which is understood by those who have taken at least two courses in fluid mechanics, because this is explained.

Lubrication limit involves low Reynolds numbers, laminar flows. In fact, they deal with Stokes Equations, not the full Navier Stokes. The non-linear terms (aka “the hard part”) vanish in this limit. Those flows, and solutions are non turbulent, and highly amenable to analytical and numerical solutions.

Wings are external flows, and when we build aircraft, we only care about performance prior to stall, when the boundary layer separates. We only care about the wing facing into the free stream velocity because airplanes don’t fly sideways.

Aircraft may be super cool, and designing them is not a snap, but the fact is, these features lead to tremendous simplifications that permit quite accurate approximate solutoin methods in the range where aircraft can fly. (It’s more difficult to solve the flow field in situtaion where they don’t fly, but we can tell when the solution methods break down, and that corresponds with “plane won’t fly”. So, who cares if we can’t predict that.)

And as to weather forecasts, who can predict the weather more than a month in advance? Heck, around here (Chicago area), I don’t trust forecasts more than one week in advance. Heck, often 1 day in advance! If it weren’t for sattelites, we’d have practically no predictive ability at all.

FWIW: Intercomparison of a variety of highly parameterized models as a method of validation is a rather unusual method of testing for accuracy.

In other fields, what a reasonable engineer does is take more experimental data.

We do no also have the option of running real numerical experiments using the types of codes that solve the Navier Stokes directly. Unfortunately, these are too computationally intensive to use to test GCMs.

FWIW– I think AGW is probable. I think we should ramp up our nuclear power generation capacity. But why spew non-sense about fluid mechanics codes to persuade people? Lots of engineers took fluid mechanics, and they know why the “Boeing uses code”, and “we can design bearings” tells us very little about the accuracy GCM’s.

GCM’s solve different, more complicated class of flows: fully 3-d with heat transfer, phase change, density variations due to temperature changes and turbulence, all wrapped into one nice neat package. And we want to predict changes in global average temperature within 1C?

Maybe it can be done. Maybe it can’t.

I think AGW is probable, but I think so based on zero-order models and the general trend in temperature.

48. Phil.
Posted Jan 5, 2008 at 5:13 PM | Permalink

Re #43

Phil
But how is the success of Boeing relevant? Predicting high Re flow over airfoils prior to stalling has always been more tractable than fully three dimensional flows with mixing due to natural convection. This isnt a secret.

His solvers are for full 3-d over whole aircraft, the whole point was that when he started doing it the idea of so, even for a airfoil was thought to be almost impossible and originally required super computers. After a while it could be done on a laptop! The suggestion made in the opening piece is that the N-S equations are totally impossible, I was trying to point out that is not the case. Also more complex situations can also be modelled, another friend of mind participated in the development of the Fluent code which models flows which include chemical reaction, radiation, multiphase and variable geometry

49. Posted Jan 5, 2008 at 5:44 PM | Permalink

Phil– flow over airfoils don’t have the features of other flows. I’m familiar with Fluent. Unfortunately, if you read the journal article describing the parameterizations in GCM’s they sound qualitatively like Fluent circa 1980, not Fluent circa 2007. (The reason for the difference is the size of the problem. You can kick in Fluent’s Smagorinski model for small-ish problems, but you can’t do it when predicting flow for the entire planet.)

On your friends code: you name your friend’s solver so we can read up, see what type general class of model it uses, and the range of problems it applied to solve?

Once again: Ask your friend if he can predict drag or forces on components of the aircraft turned sideways?

There are codes and there are codes. There are flows and there are flows. It is true the sideways aircraft problem is not relevant to designing aircraft. But I ask this because that problem is relevant to explaining what different types of highly parameterized codes do well and what they don’t do well.

50. Phil.
Posted Jan 5, 2008 at 6:00 PM | Permalink

Re #49

Phil flow over airfoils dont have the features of other flows. Im familiar with Fluent. Unfortunately, if you read the journal article describing the parameterizations in GCMs they sound qualitatively like Fluent circa 1980, not Fluent circa 2007. (The reason for the difference is the size of the problem. You can kick in Fluents Smagorinski model for small-ish problems, but you cant do it when predicting flow for the entire planet.)

On your friends code: you name your friends solver so we can read up, see what type general class of model it uses, and the range of problems it applied to solve?

Once again: Ask your friend if he can predict drag or forces on components of the aircraft turned sideways?

There are codes and there are codes. There are flows and there are flows. It is true the sideways aircraft problem is not relevant to designing aircraft. But I ask this because that problem is relevant to explaining what different types of highly parameterized codes do well and what they dont do well.

I know all the above, I’m drawing attention to the overkill in the intro., if the intro had been more like what you said above I wouldn’t have mentioned it. I remember in the 80’s all the comments on the limits of cfd and the use of supercomputers, nowadays those problems can be done on desktops.

If you want to check out Tony Jameson’s codes feel free.

51. Posted Jan 5, 2008 at 6:50 PM | Permalink

@Phil– Thanks. Now that I have a name, I can read what types of problems it does.

Yes– I see your point about Steve going a bit over the top. I give him some slack because he’s a geologist.

OTOH, I’ve read similarly over blow rhetoric on the other side from climate scientists running code claiming outrageous amounts believability. (Of course, I heard similar things from engineers in the 80s. People who run codes believe codes, period.)

I’ll have to Google a bit to find it. But there was something to the effect that GCM’s are nothing more than stuff we’ve know for 200 years. That would mean GCM’s predict stuff based on knowladge available before Prandtl!

Unfortunately, it’s all over the top on both sides.

Hey, I believed airfoils generated lift even when Launder, and Whitelaw were putting out those early CFD RANS codes. We knew that without CFD RANS codes. I knew spheres had drag, and roughly how much, and how to use methods discussed in Schlicting when CFD codes still predicted the separation point poorly.

The fact the codes were poor didn’t make me conclude airfoils had no lift. I think AGW is probable– but not because of GCM models.

But, I didn’t feel any need to say codes were better than they were just because I believed airfoils generated lift. Nor because the codes held promise.

At this point, I’m not going to say GCM models have the internal workings to make them a priori predictive just because I think AGW is probable.

52. Phil.
Posted Jan 5, 2008 at 7:14 PM | Permalink

Re #51

Yeah I remember giving Jim a hard time about those models too! I was very sorry to learn of his death last year, unfortunately too late to make the funeral.
However, poor though those codes were on airfoils, about 10 years later Tony was optimizing wing designs for Boeing, McD-D, et al.

53. Larry
Posted Jan 5, 2008 at 7:27 PM | Permalink

Uhh….Boeing and Mickey D are the same company. IIRC, they merged about 10 years ago.

54. Phil.
Posted Jan 5, 2008 at 7:33 PM | Permalink

Re #53

Concentrate Larry, they weren’t then!

55. Larry
Posted Jan 5, 2008 at 7:38 PM | Permalink

I assumed you were 22, and your friends were of a similar age.

56. Posted Jan 5, 2008 at 7:47 PM | Permalink

However, poor though those codes were on airfoils, about 10 years later Tony was optimizing wing designs for Boeing, McD-D, et al.

Agreed. Used judiciously, poor codes were better than no codes. Improvements come quickly. Even detractors agreed with this. Also, at the time, a lot of data was collected, and people knew not to feel confident applying models to flows that were much too different from real flows.

Plus…. airfoils that aren’t stalled are easier than bluff bodies, or fully 3-D terrains.

With respect to GCM’s, in addition to the circa-1980s Fluent (with some circa 60s parameterizations), there are other things that I find unsettling.

For example, I sure wish the peer reviewed papers I’ve gotten a hold of would at least mention the baseline tempeturature for the model predicted temperature anomolies in honest to goodness Kelvins. (They give traceable values for the actual empirical values.)

After all, the maximum absolute humidity of water in air of humidity in the air is a non-linear function of temperature. The melting point of ice depends on temperature.

Plus, logical consistency would require that if we really should only care about temperature anomolies, then they could still jolly well mention the reference temperature in the journal articles. It would require 1 sentence in the 30 page articles. Does it match the baseline temperature for the empirical data within 0.1C? Is it 10C off? 50C? I can find no answer in the few papers I’ve read. (I haven’t read many, but you’d think it would be in at least half of the papers!)

Oh well….

57. Posted Jan 5, 2008 at 8:24 PM | Permalink

@larry–
“Jim” in 52 was James Whitelaw, and Phil said he regretted missing the funeral. So I kinda figured Phil was more than 22! (Though the picture I found by Googling doesn’t look that much older. )

To give you guesses on ages: When I began my Ph.D. Nick Vlachos was my advisor. Nick worked with Whitelaw. But Nick didn’t get tenure, and I finished with Shao Lee Soo. Nick and Shao Lee were interested in measuring turbulence properties in particulate flows, I used one of Bill Bachalo’s PDPA devices to make measurements. Which would be entirely uselss information to you, Larry. I will only say that,based on dates of a papers Phil published, I’d now guess Phil is my age plus or minus 10 years. But women don’t reveal their ages, right? Still…my age plus or minus 10 would not make him 22. 🙂

58. Phil.
Posted Jan 5, 2008 at 9:11 PM | Permalink

Re #57

There’s a few names from the past, unfortunately regarding age it’s almost certainly on the plus side of the ledger! 😦
I remember Nick when he was a grad student, one of the Greek contingent at IC back then (you probably remember Dinos Arcoumanis too).

59. Geoff Sherrington
Posted Jan 5, 2008 at 9:22 PM | Permalink

Re # 327 Neal J King

We can agree on some things, no probs. Beer’s law measurments at lab scale typically use a near-parallel beam through a homogenous medium. This is different to the isotropic radiation from a point in an atmosphere, where the vector can involve other factors such as temperature and density gradients.

Re management structures, the free enterprise structure has been the engine of global economic advancement. I’m simply saying that a management structure for global climatology based on the corporate structure would be rather more effective than the present model. How do you choose a head? How does the World Bank do it? On merit.

One strength of the corporate model is accountability. We miss a lot of that at present in climate work. Another strength is the setting of objectives and structuring to achieve them. Have a read the IPCC charter up above somewhere and you’ll see it is charged with recommending remedies to problems. How many pages have you read from the IPCC devoted to remedies? It’s not even obeying its one-paragraph mission. People in industry who disobey like that can be out the door by sunset.

Within my country, the essence of the work we did in industry was required by law to be reported to the appropriate government department for assessment and archiving. The archives, over the years, formed a wonderful reference asset. They usually included swags of raw data and we did not avoid our duty to complete them diligently. They were also windows on the way famous predecessors thought and problem-solved. Better than FAQs.

60. Posted Jan 5, 2008 at 9:49 PM | Permalink

@Phil,
I don’t think I met Dinos. I was at Urbana, not Imperial College.

61. Arthur Smith
Posted Jan 5, 2008 at 10:19 PM | Permalink

Geoff Sherrington (#59) – The entire Working-Group 3 report from IPCC is devoted to looking at “remedies”, costs of mitigation, possible solutions, etc. It’s well worth reading – only 807 pages.

62. John Creighton
Posted Feb 15, 2008 at 12:13 AM | Permalink

I mentioned fractals and turbulence in another thread:

I found some papers via Google:
http://www.me.jhu.edu/~meneveau/pubs-fractals.html

63. Agile Aspect
Posted Jun 23, 2010 at 5:56 PM | Permalink

I’m surprised to hear the models are attempting to solve the Navier-Stokes equations – even the linearized form.

How does one distill the non-physical parameters of “sensitivity” and “forcing” from the Navier-Stokes equations?

Has the simplified version of the Navier-Stokes equations, namely, the Heat Equation, been solved for a rotating 3 dimensional sphere where half of the sphere is being periodically cooled while the other half is being heated?