The Constructal Law of Flow Systems

Guest Post by Willis Eschenbach

One of the most fundamental and far-reaching discoveries in modern thermodynamics is the Constructal Law (see the wiki entry as well). It was first formulated by Adrian Bejan in 1996. In one of his descriptions, the Constructal Law is:

For a finite-size (flow) system to persist in time (to live), its configuration must evolve such that it provides easier access to the imposed currents that flow through it.

Figure 1. An example of the Constructal Law processes at work in a river system. Formation of meanders, followed by cutting through a meander to form an oxbow lake. Image Source.

The Constructal Law has been described as being as important as the first two Laws of Thermodynamics, but most people have never heard of it. What does the Constructal Law mean in plain English, and what does it have to do with the climate?

Here is a different statement (pdf) of the Constructal Law, again from Bejan:

In 1996, the constructal law was formulated and proposed to expand thermodynamics in a fundamental way.

First was the proposal to recognize that there is a universal phenomenon not covered by the first law and the second law. That phenomenon is the generation of configuration, or the generation of ‘design’ in nature.

All thermodynamic systems in nature are flow systems (i.e. live, non-equilibrium systems), and they all have configuration. If they do not have it, then they acquire it, in time. The generation of configuration is ubiquitous, like other phenomena covered by other ‘laws’ in physics. Biological systems are configured. Geophysical systems are configured. Engineering and societal systems are configured. The configuration phenomenon unites the animate with the inanimate. All the other phenomena of physics (i.e. of ‘everything’) have this unifying power. Falling rocks, like falling animals, have weight, conserve energy, generate entropy, etc.

Second was the statement that this universal phenomenon should be covered by the constructal law. This law accounts for a natural tendency in time (from existing flow configurations, to easier flowing configurations). This tendency is distinct from the natural tendency summarized as the second law.

Again not necessarily the clearest statement, but the general idea of the Constructal Law is that flow systems continually evolve, within the physical constraints of the particular system, in order to maximize some variable(s).

A meandering river in bottomland is a good physical example to understand what this means. In the case of a river, what is being maximized by the flow system is the length of the river. However, this ideal condition is never achieved. Instead, the river length oscillates above and below a certain value.

As shown in Fig. 1, in an “S” shaped river, the moving water erodes the outside of the bends and deposits silt on the inside of the bends. Of course, this inevitably makes the river longer and longer. But when the river does this for a while, it gets too stretched out for the land to bear. At some point, the river cuts through and leaves an island and what will become an oxbow lake.

That leaves the river shorter. Again the lengthening process continues, until the river cuts through some other bend and shortens again. And as a result, the length of the river oscillates around some fixed value. It is constantly evolving to maximize the length, an ideal which it never attains.

Now, here’s the point of this whole example. Suppose I didn’t know about this active, evolutionary, homeostatic characteristic of rivers. If someone asked me if a river could be shortened, I’d say “Sure. Just cut through a meander.”. And if I cut through the bend I could physically measure the river length and prove that indeed, the river was shorter.

But would that really make the river shorter?

Of course not. Soon the relentless forces of flow would once again increase the length of the river until the next cutoff forms another oxbow lake, and the cycle repeats.

Net effect of my cut on the length of the river? None. The length of the river continues to oscillate around the same fixed value.

The key to understanding flow systems is that they are always “running as fast as they can”. They are not just idling along. They are not at some random speed. They are constantly evolving to maximize something. The Constructal Law ensures that they are up against the stops, so to speak, always going flat out.

What does all of this have to do with climate? The Earth’s climate is a huge flow system. It circulates air and water from the tropics to the poles and back. As a result the climate, like the river, is subject to the Constructal Law. This means that climate is constantly evolving to maximize something. Climate, like the river, is also “running as fast as it can”.

What does the climate flow system maximize? Because it is a heat engine (converting sunlight into the physical work of the planetary circulation), Bejan says (pdf) that it is doing a dual maximization. It maximizes the sum of the work done driving the planetary circulation, and the heat rejected back to space at the cold end of the heat engine. Again in Bejan’s words:

The earth surface model with natural convection loops allows us to estimate several quantities that characterize the global performance of atmospheric and oceanic circulation. We pursue this from the constructal point of view, which is that the circulation itself represents a flow geometry that is the result of the maximization of global performance subject to global constraints.

The first quantity is the mechanical power that could be generated by a power plant operating between Th and Tl, and driven by the heat input q. The power output (w) is dissipated by friction in fluid flow (a fluid brake system), and added fully to the heat current (qL) that the power plant rejects to Tl.

where Th and Tl are the temperatures of the hot and cold ends of the system. The system is maximizing the sum of work done and heat rejected.

There is a most fascinating interplay between those two. When the speed of the planetary circulation is low, so are the turbulent losses. So as speed increases, up to a certain point the sum of work done (circulation speed) and heat rejected is also increasing.

But as the speed increases further, the turbulence rapidly starts to interfere with the circulation. Soon, a condition exists where further speed increases actually decrease the total of work done and heat rejected. That is the point at which the system will naturally run. This is why nature has been described in the past as running at “the edge of turbulence”.

What does that mean for understanding the climate? This is a new area of scientific investigation. So I don’t know what all of that means, there’s lots of ramifications, some of which I may discuss in a future post. However, one thing I am sure of.

If we want to understand the climate, or to model the climate, we have to explicitly take the Constructal Law into account.

We are not modeling a simple system with some linear function relating forcing and response. That kind of simplistic understanding and modeling is not valid in the type of system where, for example, cutting a river shorter doesn’t make it any shorter. We are modeling a dynamic, evolving system which may not be affected by a given forcing. The modelers claim (falsely, but we’ll let that be) that their models are based on “physical principles”.

However, they have left one central, vital, physical principle out of the mix, the Construcal Law. And at the end of the day that means that all of their modelling is for naught. Sure, they can tweak the model so that the output resembles the actual climate. But the actual system does not change over time in a random way. It is not driven here and there by forcing fluctuations. It changes in accordance with the Constructal Law. The future evolution of the climate, what Bejan calls the “generation of configuration”, is ruled by the Constructal Law. It cannot be understood without it.

PS – For those that think that the Constructal Law is some crackpot theory, it is not. Bejan is one of the 100 most cited engineering authors of our time, and the results of the Constructal Law have been verified in a host of disciplines. It is indeed a new fundamental law of thermodynamics, one which we cannot ignore.

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jack mosevich

Mark Twain proposed a version of the Costructal law:
The Mississippi between Cairo and New Orleans was twelve hundred and fifteen miles long one hundred and seventy-six years ago. It was eleven hundred and eighty after the cut-off of 1722. It was one thousand and forty after the American Bend cut-off. It has lost sixty-seven miles since. Consequently its length is only nine hundred and seventy-three miles at present.
Now, if I wanted to be one of those ponderous scientific people, and `let on’ to prove what had occurred in the remote past by what had occurred in a given time in the recent past, or what will occur in the far future by what has occurred in late years, what an opportunity is here! Geology never had such a chance, nor such exact data to argue from! Nor `development of species’, either! Glacial epochs are great things, but they are vague–vague. Please observe. In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. This is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upward of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

George E. Smith

Very interesting post willis.
I can see an immediate area of application. Ocean currents are well known to meander just as if they were actual rivers cutting through land; so I suppose the Constructal law would apply to them too.
This is important when it comes to makign temperature measurements from surface waters; as was the historic case for the 70% of the earth surface that is ocean.
Because of meanders, a survey ship could go back to the exact same geographical co-ordinates; and end up in a totally different water structure from what was there six month ago.
So all of that hokey surface water proxies for lower tropospeheric Temperature, that is the bulk of the 150 years of global temperature data, must have a variance due to the constructal law, and ocean current meanders.
In Jan 2001 Dr John Christy et al also reported that the water surface Temperatures, and the lower troposphere temperatures were not the same; and were not correlated; but the standard assumption was that the water bucket reading was as good as meauring the air temperature.
Why would anyone expect them to be the same or correlated, when the air over Hawaii, could end up over California next week; so air and water don’t remain in thermal contact long enough to equilibrate.
So willis; what do YOU know about ocean current meanders and the Constructal law.

Dave Dardinger

How does the Constructal Law differ from maximizing entropy?

James Macdonald

As a meteorologist I can readily see this happening to the Jet stream. It oscillates between almost straight line flow and one of deep troughs and ridges, winding up as a cutoff low. Then the straighter line flow resumes. Almost identical to a river.
I am one of 15 scientists—climate realists- in the Hartford, CT area. Prof. Larry Gould is one of us as well as Art Horn.

J.Hansford

Constructal Law would be describing equilibrium in a dynamic system then…..?

etudiant

Wonderful. A major law that makes eminent sense hiding in plain sight.
Does make one worry what else we might be missing.
Meanwhile, it would seem that law should be applicable to the overall atmospheric flow system.
Unfortunately, the law gives zero guidance as to the configuration that is adopted and is even silent as to the claim that the system seeks to maximize the work done and the heat rejected by the atmosphere, even though Bejan makes that claim. Still, it gives another way to drive/evaluate the models, which is a real step forward. We may look for interesting new insights in the near future.

Curious George

Could you please formulate this “law” a little more mathematically? Unless you do, I propose an equivalent formulation of (a possilbly diferent) Constructal law:
The good should be rewarded, the bad should be punished.

Rhoda R

“It is indeed a new fundamental law of thermodynamics, one which we cannot ignore. ” I love it – that we are still finding fundamental laws of anything is great.

Douglas Dc

Here’s a good example of what Willis is talking about- Grande Ronde River Drainage.
I’ve rafted, fished that river, hunted, and dropped fire retardant on those steep ridges.
http://www.djensenphotography.com/images/ne_ore/mountains/winding_waters.htm
worth cruising Jensen’s website just for the pretty pictures and geology…..
Great post, Willis, BTW.

Gary Palmgren

I have an alternate description that comes directly from the 2nd law. The increase in entropy in any closed system can always be seen as the conversion of potential energy to low level uniform heat. Even temperature differences can be rigorously proven to be potential energy. The second law never allows any local uniform temperature to be turned back into local potential energy.
A dynamical system will evolve to maximize the generation of entropy at every location. In the river model we have two forms of potential energy that can be used to do work. One is the pressure head or total elevation from source to basin. The other is the kinetic energy of the moving water. (Yes kinetic energy can be looked at as a form of potential energy as the energy of motion can be used to do work.) At every location the system will try to minimize both. A short river will have the least amount of friction so the kinetic energy of the flowing water will be maximized. Erosion will occur rapidly and the meanders will increase. Friction increases increasing the generation of entropy.
When the meanders get to big, the friction is too high, the water backs up, and the pressure head increases. An increased pressure head results in the water crossing the meander to make a shorter path. Thus the river length oscillates around some constant as described by Willis Eschenbach.
I may be combining the structural law with the maximization of entropy, but I suspect they are both saying the same thing.

Willis:
“That leaves the river shorter. Again the lengthening process continues, until the river cuts through some other bend and shortens again. And as a result, the length of the river oscillates around some fixed value. It is constantly evolving to maximize the length, an ideal which it never attains.”
That’s not really what I’m seeing. I’m seeing a constant attempt to MINIMIZE the length of the river, which fits in with thermodynamics quite nicely. The original elongation was an aberation. Once in the lowest energy state possible between two points (straight line) there’s absolutely nothing to cause it to create additional corners.
The “generation of configuration” in this case is the lowest-energy path between two points. Once it has the optimal configuration there’s no reason for it to change.
I think your use of this example with climate isn’t really very useful. As conditions change, the optimal configuration may change, and the climate systems may drift in that direction, but that relies on variables changing, not an intrinsic part of the system to always seek change.

Willis Eschenbach

Curious George says:
November 15, 2010 at 6:19 pm

Could you please formulate this “law” a little more mathematically? …

Sure … if you can first give me a mathematical formulation of the Second Law of Thermodynamics.
The problem is that both these Laws only give the direction of the change. The Second Law says only says that net heat only flows from warm to cold. It doesn’t give numbers.
Similarly, the Constructal Law says that systems evolve to give easier access to the currents. And like the Second Law, it doesn’t give numbers.
However, there are a whole host of mathematical results made possible by the Constructal Law. See the references given in the head post for examples.

Alex Heyworth

As an interesting aside, the ratio of a river’s length to the length of a straight line between its source and mouth is approximately Pi.

Pamela Gray

Every single Wallowa County river fishing enthusiast understands this. I saw this law in action this past summer. That it can be applied to weather makes perfect sense. We are about to be inundated with an over the top windy pressure gradient in and around the NE portion of the Washington/Oregon state line. So I get to watch this law in action. You can see it in pressure gradients. Way cool.
The jet stream is another place to see this law in action.

Willis Eschenbach

Dave Dardinger says:
November 15, 2010 at 5:57 pm

How does the Constructal Law differ from maximizing entropy?

Thanks for the question, Dave. Bejan goes over this in great detail here (pdf).

D. Patterson

See:
The Constructal Law is about the time direction of the “movie” of design generation and evolution. It is not about optimality (min, max), end design, destiny or entropy.
http://www.constructal.org/en/theory/presentation.html

Alex Heyworth

PS, a reference for Pi being the ratio length/source to mouth distance:
Stølum, H.-H. “River Meandering as a Self-Organization Process.” Science 271, 1710-1713, 1996.

John Day

@CuriousGeorge:
> Could you please formulate this “law” a little more mathematically?
Ditto. I also can’t seem to get my head around this, sounds more like philosophy or hand-waving. It needs some theorems that can be tested or implemented constructively.
All bun, no meat.

Carl Chapman

Wouldn’t that imply that the earth’s climate would exhibit strong negative feedback?
My guess is that the feedback reduces temperature changes by forcings such as CO2 to 1/3. The IPCCs guess is that feedback increases changes to 3 times.
No warming since 1998, while CO2 goes up and up, makes the 3 times theory extremely hard to believe.

AusieDan

Hi Willis
Is this anything different from chaos theory?
Hurst wrote a paper on his 847 year study of the River Nile in 1951.
He was interested in river heights as he was planning the River Dam Project.
The language used was quite different but there seems to be some similarity.
I hope to contact you soon with a climate paper using this concept.
Regards Dan

Bruce

Meanwhile in CRU headquarters: “Ah these pesky models would work so much better if the Earth just would stop rotating. Hmm. Hey, I have an idea….”

Robert of Ottawa

A river “running as fast as it can” can be restated as losing energy as rapidly as possible; in the case of a river flow, this energy loss is restrained by the river sides and bottom. The maximum rate of energy loss is at a waterfall, such as the Rideau Falls here in Ottawa. http://www.google.ca/imgres?imgurl=http://farm4.static.flickr.com/3108/2918568397_8b972f3a76.jpg&imgrefurl=http://www.flickr.com/photos/41455313%40N08/galleries/72157622637220789&h=405&w=500&sz=104&tbnid=B7mGNPpDV5ItXM:&tbnh=105&tbnw=130&prev=/images%3Fq%3Drideau%2Bfalls%2Bottawa&zoom=1&q=rideau+falls+ottawa&hl=en&usg=__6fIaQPpQRHkyRjK6dT_mCVCm-bo=&sa=X&ei=OurhTIGmFYnLnAerq93-Dw&ved=0CCoQ9QEwAg
A flow cannot lose energy at a greater rate than this. So, interestingly this is a case where the increase in entropy actually has a maximum rate.

Jim

I’m wondering if the curves are carved during periods of high flow. The river breaches a bank and the turbulent waters carve a new curve, following generally the land. Then during periods of low flow, the river straightens itself out again. So it would be an interaction of water volume, the shape of the bank, and the lie of the land.

jimmi

This looks like one of those observations that are qualitative rather than quantitative, a bit like Le Chatelier’s Principle (A system when perturbed reacts to minimise the perturbation). It may even be a variant of Le Chatelier.
I think you can express the 2nd law mathematically if you start from statistical thermodynamics.

guidoLaMoto

This is really just another way of stating the Law of Survival of the Fittest: things survive when they don’t do anything that would make themselves go extinct. The Constructionist Law would only seem to hold for chaotic systems while they are exhibiting cyclicity or pseudocyclicity. The problem is, chaotic systems can reach bifurcation points where they take off in unexpected directions. We Warming Skeptics cannot necessarily take heart in this Law. Remember that the Amazon once flowed east to west.

Dave Springer

The path of least resistance is a very basic concept but not fundamental thermodynamic law as it’s derived from the first three.
Constructal law, at least as described by wiki, gets a little weird when it starts talking about nature as a designer. Basically it’s saying nature keeps improving on means of raising entropy. Not just following paths of least resistance but rather designing paths of least resistance. In that view we can make an argument that humans are an improved way of getting the low entropy in fossil fuels into a higher entropy configuration.
Let’s just hope nature didn’t design atom bombs for that purpose too.

David L. Hagen

Bejan published a very useful book:
Shape and Structure, from Engineering to Nature. Cambridge University Press, 2000, ISBN 0-521079388 2
Recently: Design with Constructal Theory
Ferenc Miskolczi has applied entropy maximization in his climate formulations.
Greenhouse effect in semi-transparent planetary atmospheres

We believe that the β parameter is governed by the maximum entropy principle, the system tries to convert as much SW radiation to LW radiation as possible, while obeying the 2OLR/(3 f ) = F0 + P0 condition. The cloud altitude, where the clear-sky A
OLR = OLR = ED depends only on the SW characteristics of the system (surface and cloud albedo, SW solar input) and alone, is a very important climate parameter.

Willis Eschenbach

AusieDan says:
November 15, 2010 at 7:11 pm

Hi Willis
Is this anything different from chaos theory?
Hurst wrote a paper on his 847 year study of the River Nile in 1951.
He was interested in river heights as he was planning the River Dam Project.
The language used was quite different but there seems to be some similarity.
I hope to contact you soon with a climate paper using this concept.
Regards Dan

The Constructal Law is different and distinct from both chaos theory and Hurst’s work on long-term persistance. Let me know how your paper goes.

Grey Lensman

Simple statement of the law
Symmetry is, Non symmetry works.
Thats why it is bad JU Ju to canalise and straighten rivers. That causes the flood disasters.

higley7

Wow! That is so cool!!!
Do not forget that the floods that occur carve out the various flood plains for the 10, 50, 100, 500, and 1000-year floods, helping the system dig out the next big bend.

Willis Eschenbach

John Day says:
November 15, 2010 at 7:06 pm

@CuriousGeorge:
> Could you please formulate this “law” a little more mathematically?
Ditto. I also can’t seem to get my head around this, sounds more like philosophy or hand-waving. It needs some theorems that can be tested or implemented constructively.
All bun, no meat.

In that case the Second Law of Thermodynamics is also “all bun, no meat”. It contains no more math than the Constructal Law.
In any case, here’s enough math and practical applications to give you a headache …

Willis,
The Second Law is quite quantitative (and falsifiable). It says, in one form, that entropy can’t decrease. Now entropy is a well-defined quantity, derived from other measureable quantities.
It’s true that it is somewhat unusual in being one-sided – it says entropy can’t decrease but makes no prediction about how much increase. But the statement of no decrease is precise.
The Constructal Law seems to lack that precision. How would you falsify it?

Willis Eschenbach

guidoLaMoto says:
November 15, 2010 at 7:28 pm

This is really just another way of stating the Law of Survival of the Fittest: things survive when they don’t do anything that would make themselves go extinct. The Constructionist Law would only seem to hold for chaotic systems while they are exhibiting cyclicity or pseudocyclicity.

No, no, and no. Please, folks. I have given you good references to a host of both very general and very detailed discussions of the Constructal Law. Read them first before you decide what the Constructal Law is and isn’t.
Bejan didn’t get to be one of the top cited scientific authors on the planet by writing foolishness, by being “all bun, no meat”, or by “handwaving”. C’mon, do your homework before uncapping your electronic pens …

James Sexton

Huh? I thought this particular line of thought fairly well understood. Maybe its because I’m old enough to have witnessed the making of an elbow lake, and “islands” that are no longer surrounded by water. If I live long enough, I’ll see this occur again. I hope not, the “Big Island” holds a special place in my heart. A place where the Neosho splits and then rejoins. Isolating ~ 10 sq. miles. One part of the river is lively, the other, dying. I digress.
Equilibrium is always sought by nature, yet never attained. There is always a symmetrical ebb and flow.(It isn’t always obvious.) Movement of water expresses energy.
Willis, I thank you. I hadn’t realized this thought needed to be brought to light.
There is one point, though, I’m unsure of.
“The Constructal Law ensures that they are up against the stops, so to speak, always going flat out.” (I believe this properly paraphrases the law.)
It seems to me, (and I believe this is what many warmists hang their hat on) is that when unnaturally altered, nature seems to have an urgency to correct, or seek a balance. Mind you, it is simply a judgment based on observations that may be tinted, yet, it seems that way to me. Of course, the door swings both ways.
Again, Willis,
thanks.
James

Grey Lensman

Has Willis enabled me to define energy?
I think he has!
Energy is the evolution of time and space.
Can any maths wizzes get their heads around that and write a formula.
It ties in with all of the above

If you look at the surges in the atmospheric circulation caused by the interactions of the 18.6 year period of lunar tidal variations, as being similar to the erosion caused by periods of increased flow rate in spring or monsoonal periods. Why would this effect not be understandable as the driver of the positions of the jet streams?
http://research.aerology.com/uncategorized/blogstuff-comments-by-myself-and-others/
Above is a compilation of threads where lunar declinational tides in the atmosphere are discussed as to applicable uses in increasing the validity of long range forecasting methods. On the site are daily forecast maps with a lead time now of 35 months, still seem to be working ok.

Greg Cavanagh

I get the impression that any fluid flow through any other medius acts pretty much the same way every time. Water through earth, jet streams, ocean currents, hot air rising through cool air.
It appears to be a fundamental of fluid mechanics, as applied to any fluid.

Thanks Willis. This goes a long long way in explaining what a lot of geology is all about. I can remember being a GSA and AAPG meetings in the 80’s trying to convince these modelers they were on the wrong track. I was not alone either. Thanks too to all the fine comments and suggestions from the other commentators.

Steve Koch

Willis,
Great post about a fundamental concept that is intuitively obvious once it is stated. It applies to so many different things, not just energy. You can see it in roads or electricity or neighborhoods or politics or human relationships or football or investing or on and on.

Barbee

If a society could contain lava flows in this manner it may actually save lives.

Grey Lensman

Willis is getting frustrated with the hand waving. Dont blame him. Come on guys, put thinking caps on, this makes real sense. I have even quantified it for you. Now you can ascertain values and appoint symbols and get calculating. My logic even defines entropy.
This will only be resolved by discussion not hand waving.
In space, the earth can be considered a closed system in balance but what are the total inputs and total outputs to achieve that balance and more importantly, how.

Michael D Smith

Rivers oscillate to balance the flow rate with the sediment load. Rivers can be direct and straight when the load to move is more than the capability of the flow rate to do work. This is seen in straight, deep ravines where high flows have a high erosion impact. In high flow situations, more sediment is carried, thereby carving away new material, which falls into the river whenever sediment is left higher than the angle of repose. In low flow, sediment is carried at a much lower rate, with light materials being moved downstream. Meandering rivers have a load that is lower, usually very lightweight silt and organic matter. The river balances this by producing a lower slope, reducing the work done per unit length. The only way to reduce slope is to meander producing a longer distance between inlet and outlet. This can be seen readily in the difference between the Missouri river, which has a high slope (and a rocky bottom), and the Lower Mississippi river, which has a very low slope, lightweight load, and huge meanders. Both effects can be seen in braided rivers, where the sediment load is fairly high, with a gravel bottom. You’ll get straight areas that can carry a heavy load, but if the velocity increases too much, stones are carried downstream and deposit in deltas a few meters away. This lowers the speed by widening the flow, smaller particles and sand will deposit, and this backs up until the slope is too small. Another channel will be carved through this delta and the rocks will be moved further downstream as a result, forming a new delta that is soon also carried away. Here’s an example: http://www.teara.govt.nz/files/p18195gns.jpg
So in a river, the equilibrium that is never reached is getting the slope and sediment load to match the power delivered by the water flow.

Jim says:
November 15, 2010 at 7:24 pm

I’m wondering if the curves are carved during periods of high flow. The river breaches a bank and the turbulent waters carve a new curve, following generally the land. Then during periods of low flow, the river straightens itself out again. So it would be an interaction of water volume, the shape of the bank, and the lie of the land.

During periods of low flow, the water isn’t moving fast enough to rearrange the banks.
A river with a steep vertical gradient tends to erode downward and the river stays stright or follows some guide, like a fault line. (Their rocks are often easy to erode, hence they’ll capture a river.) When a river has a very shallow gradient, then the forces to erode downward no longer exist and erosion acts on the river banks.
One interesting effect is if a meandering river is uplifted and the vertical gradient increases. Then an “incised meander” is formed, sometimes with a deep canyon.
http://en.wikipedia.org/wiki/Meander seems pretty decent.
Apparently meanders can have a climatic signature,
http://www.sciencemag.org/cgi/content/abstract/327/5972/1497 says:

Climate controls landscape evolution, but quantitative signatures of climatic drivers have yet to be found in topography on a broad scale. Here we describe how a topographic signature of typhoon rainfall is recorded in the meandering of incising mountain rivers in the western North Pacific. Spatially averaged river sinuosity generated from digital elevation data peaks in the typhoon-dominated subtropics, where extreme rainfall and flood events are common, and decreases toward the equatorial tropics and mid-latitudes, where such extremes are rare. Once climatic trends are removed, the primary control on sinuosity is rock weakness. Our results indicate that the weakness of bedrock channel walls and their weakening by heavy rainfall together modulate rates of meander propagation and sinuosity development in incising rivers.

John Day

@Willis:
> In that case the Second Law of Thermodynamics is also “all bun, no meat”.
> It contains no more math than the Constructal Law.
Not quite.
http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Mathematical_Statement
Is there a similar, fundamental truth that can be stated mathematically for Constructual Theory?

Grey Lensman

That should read ” total inputs and total outputs to achieve” , thats what happens when you get excited and think on the hoof

Jim D

Let’s follow this through.
So if the energy from the sun to the earth and radiated back out is the flow, and adding CO2 is adding a resistance to the flow, which it is, then the river level goes up, that being the temperature. Do I have that right? Sounds like the theory works.

David A. Evans

I have vague memories of this from years gone by.
As I always understood it, the law is about optimising entropy, not maximising.
If entropy is optimised, there is a sustainable loss. If, (in the case of a river>, entropy were maximised, all rivers would be straight, they’re not and whenever straightening occurs, it’s corrected at a later date.
I think the same sort of thing happens with climate and that’s where constructal law comes in. difficult if not impossible to quantify however.
DaveE.

I may be out of my depth here, but Prof Lindzen has written that the earth’s temperature at the equator has not varied by more than 1°C over the past billion years. So the large variations at the higher latitudes might correspond to oscillations above and below that temperature?
Also, just for fun, reversing entropy.

Dave Springer

Willis asked for a mathematical description of the law of entropy in reply to someone asking for a mathematical description for constructal law.
Okay. Here it is for entropy:
http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Mathematical_statement
Now Willis, cough up the mathematical statement for constructal law.

Rational Debate

I think the modeler’s have an even more fundamental flaw – they’ve left a central, vital, principle out of the mix. In their application of the scientific method to the entire question, they’ve utterly failed to control for confounding factors. That, and the flip side as well – they’ve similarly failed to include all of the major influential factors of the system in question.