Climate Modeling using a Leaky Integrator

[VillagePlank]

VP, absolutely keep going. this thread has inspired me to think up other instances where you can use the leaky integrator model, such as life expectancy.

Life is not like a box of chocolates, but a leaky bucket...

Keep up the good work. It's a fresh new perspective.

I also now think I understand why you can't combine the models - neither model assumes a non-linear system, rather they both are modeled with linear differential equations, whereby the initial linear bias is then adjusted accodding to variables, feedbacks and such like.

The LI is simply a method of introducting hysteresis into an otherwise straightforward model. It does produce non-linearity in that 'y' is not linearly dependent on 'x' (if you know what I mean) in such a manner that a previous value is plugged into the next value - a very simple example of this is the logistic equation, which is the absolute starting point for studying chaos.

Your 'hunch' (if that's what it is - it sounds like a more thought out idea, to me) that life-expentancy seems good to me. After all the logistic equation studies populations within an environment, and life expentancy, whilst an often difficult and emotive subject, can almost certainly be studied as a semi-linear, or even linear phenomena. In my opinion, of course, I've never done any work on that - but hysteresis requires previous values to be 'plugged' into 'next' values - that's how we get the time lag, or latency effect.

Of course, my interest in this started with nothing at all to do with climate. It started out with a construction of a neural network. Briefly, neurons have a membrane on them that acquires charge (or potential) and once a certain value is reached the neuron fires - the so called threshold value. However, many neurons can be linked to many neurons, so you might have a lot of soma/dendrites all connecting to the same membrane. How do you cumulative add the inputs of all of these connections over time whilst modelling discharge? You use the leaky integrator. Of course, in reality, the dynamics are much more complicated .

One of the next steps will be to scale the output of the LI with the sigmoid function (a technique, once again borrowed from neural networks) to a value between 0 and 1 (can't you tell I am a computer programmer?) We can then construct multiple models (without worrying about scaling) and 'evolve' the constants to find the best fit. That way, the model emerges without human intervention. The sigmoid formally makes this model semi-linear.

We might even 'preprocess' some variables with neural networks. Neural networks are ideal at two primary things: recognising patterns that they haven't seen before, and making linear sense of non-linear systems.

However, a bit at a time. Fourier analysis is next

Edited August 17, 2009 by VillagePlank

[VillagePlank]

We've played with the Fourier Transform, here, and I've asked the mods if it's possible to retrieve the images. I will try the code I posted all that time ago to see if still works.

Note the mention of, perhaps, that Fourier Analysis may well be a suspect method of analysis of this sort of data, and it is probably better to do it with Wavelet analysis.

Many thanks to wysysyg

Edited August 17, 2009 by VillagePlank

[Chris Knight]

recognising patterns that they haven't seen before

That's a strange one - re (again) co (with) gnoscere (to become aquainted) - recognising requires deja vu. I suppose they could identify novel patterns by internal matching - reflection, symmetry, chirality etc., tho. :lol:

[VillagePlank]

That's a strange one - re (again) co (with) gnoscere (to become aquainted) - recognising requires deja vu. I suppose they could identify novel patterns by internal matching - reflection, symmetry, chirality etc., tho. :lol:

Yeah - maybe a bit of a mistype by me. There are good at classifying patterns, that they haven't seen before, according to some internal classification scheme. For instance, your handwriting almost certainly hasn't been seen by, say, my neural network, but I bet it would do a good job at reading it.

I'll post an example, later.

Edited August 18, 2009 by VillagePlank

[Chris Knight]

The leaky integrator has been linked to climate, before: book

From that, a better theoretical basis for the LI model might be Torricelli's Law which is specific formulation (from that wiki) of Bernoulli's Principle although, of course, these are about fluid dynamics and not thermodynamics.

Might be time to do the hard work and link it formally.

Yikes.

The author of that book, Walter A Robinson, sounds like a really experienced professor of atmospheric science. I wonder if it would be possible to get him to comment on your work, VP? His CV is here

Yeah - maybe a bit of a mistype by me. There are good at classifying patterns, that they haven't seen before, according to some internal classification scheme. For instance, your handwriting almost certainly hasn't been seen by, say, my neural network, but I bet it would do a good job at reading it.

I'll post an example, later.

Ha Ha - not even I can read my handwriting!! :lol:

[VillagePlank]

The author of that book, Walter A Robinson, sounds like a really experienced professor of atmospheric science. I wonder if it would be possible to get him to comment on your work, VP? His CV is here

I've sent an email. I guess the guy is extremely busy, being a professor of climate et al, so not really expecting a reply.

I've started the formal work (ie PDF that might be publishable through peer-review) Mucho repsect to those who have already been published, it's a very difficult task to raise the quality level to that which we'd all expect.

Any further thoughts or problems, please post them as soon as you can.

(CB - PM me, if you can, I've lost your email address)

Edited August 18, 2009 by VillagePlank

[Hiya]

I've sent an email. I guess the guy is extremely busy, being a professor of climate et al, so not really expecting a reply.

I've started the formal work (ie PDF that might be publishable through peer-review) Mucho repsect to those who have already been published, it's a very difficult task to raise the quality level to that which we'd all expect.

Any further thoughts or problems, please post them as soon as you can.

(CB - PM me, if you can, I've lost your email address)

I haven't really been following your work on the LI in depth, only the basics. If it is your intention to publish this work then I'd probably be careful what you make publically available and what you give away via private correspondence, certainly keep some cards up your selve. Get it published then talk away as much as possible.

Drop me a PM wrt to publishing and reading if you want, I might be able to help you.

[VillagePlank]

(Still working on the write up ...)

Here's an interesting story. The authors state that "[it] cannot be used to explain recent global warming because of the trend over the past 30 years" but they've still made a link to a climatological phenomenon using sun spots. They seem to have been heavily criticised by their peers, too.

Critics, of course, will suggest that this is confirmation bias (in me) - I suspect they're right.

Edited August 31, 2009 by VillagePlank

[parmenides3]

Let's see then if my brief perusal has got this right...

'Solar activity has been at a higher level over the past fifty years than previously'. Is this the case?

'There is a strong relationship between Solar activity and global surface temperature'. Seems reasonable.

'There is a physical mechanism which allows a proportion of the suface temperature of the Earth to be retained around the Earth rather than radiated out into space'. Yep.

'In the past thirty-odd years, though there is no measured correlative increase in solar activity, there has been a (positive) change in the balance of temperature retained in the earth's atmosphere and at the surface'. fine.

'A physical mechanism exists, which has not previously been apparent, which accounts for the change in temperature'. Yes...

'It is possible to mathematically model a number of variables into a system such that it replicates the change in global average temperature'. This is easy...

This is by no means meant to be read as a hostile or aggressive response to what is clearly a serious attempt to understand what is going on with our climate, but if C-Bob's summary elsewhere is accurate, the hope is to establish that it is mathematically possible that GW is a consequence of an unspecified Solar forcing, and not something else; is this correct?

As I understand it,whether or not there is a lag in the relationship between Solar activity and global temp., there would still need to be some physical mechanism or process which accounts for the change in the global temperature balance.

There has been some very interesting work done on the idea that the oceans, soaking up the majority of the solar energy, have a lag associated with release of the energy absorbed from the Sun(which, BTW, is different for NH and SH), but the bad news is that this lag implies that the change in global temps has been constrained, not exaggerated, so far, and that the future release of the energy, combined with the heightened temps. in situ, is likely to result in an acceleration of GW.

Just having fun...

:lol: P

[VillagePlank]

.. but if C-Bob's summary elsewhere is accurate, the hope is to establish that it is mathematically possible that GW is a consequence of an unspecified Solar forcing, and not something else; is this correct?

Sort of. See if I can summarise it slightly differently (as a set of assertions) ...

(i) Hysteresis exists in temperature measurements of the world's climate

(ii) The extent of hysteresis is dependent on how much temperature is already in the system

(iii) If you model (i) and (ii), using a leaky integrator, it correlates at above 90% against the HadCru temperature measurements. Using a normal distribution it is significant enough to exclude the alternate hypothesis but, of course, that doesn't prove that the LI hypothesis to be a statement of fact, simply that it is significant enough to consider.

Why this might be the case is unknown. Serious questions might be such that ...

(i) If insolation is directly related to sunspots how come insolation continues to occur when there are none?

(ii) Assertion (i) almost certainly (way way above the 95% certainty band) exists in any energy transport/transfer system, and the earth is no exception. In terms of the Earth's climate, what is, or are, the physical manifestations?

(iii) EDIT: is assertion (ii) observable in the lab? It should be easily measurable if it is the case, being that it should be a perfectly demonstrable and linearly dependent?

Given the assertions and the obvious questions, then, yes, it would seem that if sunspots are causally affecting temperature there seems to be no known mechanism at work. The GCR theories are probably the closest thing we have that's been published so far, and serious lab work needs to be done to verify (or otherwise), particularly, assertion (ii).

Edited September 19, 2009 by VillagePlank

[Admiral_Bobski]

Sort of. See if I can summarise it slightly differently (as a set of assertions) ...

(i) Hysteresis exists in temperature measurements of the world's climate

(ii) The extent of hysteresis is dependent on how much temperature is already in the system

(iii) If you model (i) and (ii), using a leaky integrator, it correlates at above 90% against the HadCru temperature measurements. Using a normal distribution it is significant enough to exclude the alternate hypothesis but, of course, that doesn't prove that the LI hypothesis to be a statement of fact, simply that it is significant enough to consider.

Why this might be the case is unknown. Serious questions might be such that ...

(i) If insolation is directly related to sunspots how come insolation continues to occur when there are none?

(ii) Assertion (i) almost certainly (way way above the 95% certainty band) exists in any energy transport/transfer system, and the earth is no exception. In terms of the Earth's climate, what is, or are, the physical manifestations?

Given the assertions and the obvious questions, then, yes, it would seem that if sunspots are causally affecting temperature there seems to be no known mechanism at work. The GCR theories are probably the closest thing we have that's been published so far.

I think it would be fairer to say that insolation may be directly related to solar activity rather than to sunspots - sunspots are just a useful visible indicator of underlying changes in solar output and, as they have been recorded visually for hundreds of years, there is a long record of sunspot activity that can be used to infer changes in solar activity (and hence solar output). Obviously the Sun is still burning when there are no sunspots, so there is a constant output from the Sun, but modulations can, to a large extent, be tracked by visible surface changes (sunspots).

As I said on another thread, I am leaning towards the idea that the mechanism that is being sought is nothing more than the long-established greenhouse effect - no need to change the basic principles there. The change to AGW theory would be that the greenhouse effect is not significantly altered by small changes in CO2 concentration - rather, the greenhouse effect's basic ability to retain heat, and how it reacts to changes in incoming energy, is sufficient to explain global warming in conjunction with the sun's varying output.

Does that make sense, or should I try describing my thoughts differently?

CB

EDIT - A quick point of clarification for P3 - it's not that GW is a consequence of an unspecified solar forcing. GW is a consequence of variations in solar output.

Another quick clarification; the average solar activity (sunspots) since around 1950 has been higher than the average solar activity over any other same-length time period since records began (around 1650-ish).

Edited September 19, 2009 by Captain_Bobski

[VillagePlank]

As I said on another thread, I am leaning towards the idea that the mechanism that is being sought is nothing more than the long-established greenhouse effect - no need to change the basic principles there. The change to AGW theory would be that the greenhouse effect is not significantly altered by small changes in CO2 concentration - rather, the greenhouse effect's basic ability to retain heat, and how it reacts to changes in incoming energy, is sufficient to explain global warming in conjunction with the sun's varying output.

Yes, the greenhouse climate can be directly modelled using a leaky integrator, by tweaking the knobs, a bit. Sort of.

Indeed you can show, on an Excel spreadsheet that increases in GHG lead to increases in temperature, on the basis of a postulated dT/dCO₂ relationship of which I have assumed to be the Power Law given that this link is assumed to be logarithmic (and linear)

As always, and as discussed on the general thread - the key is the qualification of criteria, and their associated quantities, not necessarily the model. I should add, for balance, that I haven't been able to use a leaky intergrator style model to replicate the HadCru dataset, from the dT/dCO₂ postulate, whilst it seems much more easily possible using sunspots - which, of course, leads to Occams Razor ... but also confirmation bias, which is why open discussion is imperative.

Bottom line is, and agreed with plenty of published authors, the leaky integrator (or leaky bucket if you want to Google it) is a very important (basic) model for most things natural, and climate is no exception.

EDIT: Of course, given a dCO₂/dT (the other way around) so that CO₂ increases with temperature, that is sufficient to explain assertion (ii) and also a mechanism to show hysteresis (it takes time to soak up the excess CO2 in a cooling climate) But that requires the general theme of AGW to be dead and buried, or, at least, an agreement that man really isn't adding that much to the GHG total and, if he is, then it's a marginal effect.

A very big (scary and frightening) assertion to make on the basis of assertion and no lab work, but, as you say, a natural consequence of this line of reasoning such that a reasonable stable greenhouse effect (in terms of W/m2) should produce the same output as the leaky integrator model with varying solar 'input'

Edited September 19, 2009 by VillagePlank

[VillagePlank]

What is the percentage of man's contribution to overall CO₂ I presume we can measure it on the basis of carbon isotopes?

[parmenides3]

VP; starting from the start: what do you mean by (i)?

Hysteresis; are we presuming this refers to rate-independent ('memory') hysteresis?

I can understand how GSTs can exhibit hysteresis - in the sense that the present state is (partially) past-state dependent, and for this, the application of a leaky integrator might be fun. But how can the temperature record be affected? Surely it merely measures the consequences of the hysteresis, rather than being subject to hysteresis itself?

Is the leaky integrator another way of expressing the non-linear nature of climate?

Once we've sorted this out, we'll move to the next item...

C-Bob:

As I said on another thread, I am leaning towards the idea that the mechanism that is being sought is nothing more than the long-established greenhouse effect - no need to change the basic principles there. The change to AGW theory would be that the greenhouse effect is not significantly altered by small changes in CO2 concentration - rather, the greenhouse effect's basic ability to retain heat, and how it reacts to changes in incoming energy, is sufficient to explain global warming in conjunction with the sun's varying output.

we need to be much clearer about this; 'greenhouse effect' is no more than a (dodgy) name used to provide a convenient analogy to the global climate system. You seem to be suggesting that the system's record of temperature change might be disconnected from the CO2 content - to a greater or lesser degree - if it can be shown that 'heat memory' from the Sun is in some way the cause of changes to the temperature record. But why would this phenomenon suddenly come into being since the 1950's, and not before?

If the system is retaining more heat than it used to, and the rate of retention is increasing, there must be some physical cause. Thermometers measure current conditions (+/- some short-term hysteresis); other than that, there is no 'memory' in the measurement tools, surely?

P

[VillagePlank]

VP; starting from the start: what do you mean by (i)?

Hysteresis; are we presuming this refers to rate-independent ('memory') hysteresis?

Hysteresis is simply a fancy word for a time lag. It's everywhere. For instance when you press a button, it takes time to return to it's initial 'before P3 pressed it' state. With neural networks, which is where I learnt the term, and first come across the leaky integrator, it is where the soma membrane retains a charge, and it takes time for that charge to dissipate.

We see it in the weather. It doesn't instantly return to a very cold temperature the minute the sun goes down, there is a significant lag, assuming identical daytime, and nighttime synoptics, until the temperature gets to it's night-time norm.

Capacitance exploits that effect, so do harddrives.

If the system is retaining more heat than it used to, and the rate of retention is increasing, there must be some physical cause. Thermometers measure current conditions (+/- some short-term hysteresis); other than that, there is no 'memory' in the measurement tools, surely?

Absolutely - it is retained in the system, not in the tools that measure the system; that's hystersis. The hypothesis says that if there is more heat in the system, two things are true: firstly, that it becomes harder to heat (the rate of energy exchange slows down) and, secondly, it will lose heat much more rapidly - that is the proportion of the leak is directly related to how much 'water' is in the bucket. Convenient, this leads us onto assertion (ii) and Stefan Boltzmann, Planck et al ...

Surely it merely measures the consequences of the hysteresis, rather than being subject to hysteresis itself?

Yes, that's true. But we can easily measure hysteresis by calculating what it should have done, if there were no hysteresis, and comparing the difference. Similar to measuring the greenhouse effect.

Edited September 19, 2009 by VillagePlank

[Admiral_Bobski]

C-Bob:

As I said on another thread, I am leaning towards the idea that the mechanism that is being sought is nothing more than the long-established greenhouse effect - no need to change the basic principles there. The change to AGW theory would be that the greenhouse effect is not significantly altered by small changes in CO2 concentration - rather, the greenhouse effect's basic ability to retain heat, and how it reacts to changes in incoming energy, is sufficient to explain global warming in conjunction with the sun's varying output.

we need to be much clearer about this; 'greenhouse effect' is no more than a (dodgy) name used to provide a convenient analogy to the global climate system. You seem to be suggesting that the system's record of temperature change might be disconnected from the CO2 content - to a greater or lesser degree - if it can be shown that 'heat memory' from the Sun is in some way the cause of changes to the temperature record. But why would this phenomenon suddenly come into being since the 1950's, and not before?

If the system is retaining more heat than it used to, and the rate of retention is increasing, there must be some physical cause. Thermometers measure current conditions (+/- some short-term hysteresis); other than that, there is no 'memory' in the measurement tools, surely?

P

Hi P3,

I shall skip the hysteresis bit, since VP has answered it already and knows far more about it than I do! So, in response to that quoted above...

In this sense I am referring to the "greenhouse effect" as the effect by which the atmosphere helps to maintain a comfortable temperature for life on Earth. It blocks a certain amount of incoming energy, thereby preventing the daytime temperature from soaring to around 120C. It also blocks a certain amount of outgoing energy, thereby preventing the nighttime temperature from plummeting to around -230C. (These figures are the night- and daytime temperatures of the Moon - since the Earth doesn't stay illuminated on one side for as long as the Moon then these extremes may not be reached in the event of there being no atmosphere, but it illustrates the point.)

So, if you like, the greenhouse effect is the term used for the atmosphere's ability to even out the Earth's surface temperature.

We know that the greenhouse effect stops all the energy (heat) from rushing away from the Earth all at once by, if you like, cushioning the release of energy. (There's more to it than that, of course, because the ground absorbs radiation, the oceans absorb radiation and every object on the Earth absorb radiation, in addition to the atmospheric absorption.)

The Stefan-Boltzmann law tells us that as something heats up it radiates more energy, but its temperature at equilibrium will increase as well (since the emissivity does not increase quite as rapidly as the absorptivity).

So, my thinking which led to this Leaky Integrator idea is that if solar activity remains above a certain level then the absorptivity of the Earth increases a bit more than its emissivity, meaning that the Earth will warm even if the Sun's output remains constant until it reaches its new equilibrium.

As I say, the Sun's average output over the past 60 years or so has been higher than any other 60-year period since records began.

As I have said before (though you may have missed it since you've been away, P3) the temperature increase over the last 100 years or so would require the Earth to retain no more than an extra 1/1000th of a degree Celsius every month, which is not a great deal. Temperatures can still give us peaks and troughs as they always would, but those peaks and troughs would all be 1/1000th of a degree per month higher than they would otherwise have been, giving an increasing temperature trend.

CB

EDIT - I'm just about to start a Leaky Integrator Discussion thread, so that we don't clutter up this pinned thread with discussion that will take it off-course.

Edited September 19, 2009 by Captain_Bobski

[parmenides3]

'As I have said before (though you may have missed it since you've been away, P3) the temperature increase over the last 100 years or so would require the Earth to retain no more than an extra 1/1000th of a degree Celsius every month, which is not a great deal.'

If the System retains 'extra' heat, this suggests that something has changed, either to the system, or the external forces working on it.

But consider a tropical desert at night, or a temperate location through the seasons; to what extent do these places exhibit hysteresis?

If I recall rightly, for example, temperatures in the UK respond to our relationship to the Sun with an approximate six-week lag (variable), so that we get our warmest months after the period of greatest insolation, and vice versa.

Where the global surface temperature may well exhibit hysteresis is in the case of our old friend the Arctic, sea ice and sea surface temperature. This has to do with the way in which the measurements works today compared to 30 years ago. The reason why the lower Arctic waters show such a high anomaly is at least partly because, 30 years ago, areas which were perennially ice-covered are no longer; therefore there is a large anomaly recorded. It is also important to understand how much of the long-term global trend is affected by these anomalies (though also important to recognise that models account for the phenomenon to some degree).

So, is the sea ice reducing at 8% per decade because of warming oceans, or are the oceans showing warming because the sea ice is reduced? The two are obviously demonstrating the same overall pattern - one of warming - but establishing cause and effect is less obvious.

So, a suggestion here might be that the leaky integrator model could work quite effectively in relation to the Arctic Ice; think of single-year and multi-year ice as the 'new' and 'carried over' elements of the system. For a year like 2007 to happen, there has to be a loss of multi-year ice as well as the expected seasonal variation in the single year ice. This is, in part, what happened on 07; because of the currents, the winds and the Summer synoptics, multi year ice was pushed out of the 'bowl' of the Arctic Ocean, via the Fram Strait, out into the warmer Greenland Sea, whence it melted.

If the system is 'in balance', then, over time and allowing for variability, the proportion of multi-year and single year ice remains broadly similar. This is because the steady build-up of the hysteretic element is countered by the occasional 'burst' of the system, leaking what has been gained out into the oceans surrounding.

Is this suggesting anything?...

[Admiral_Bobski]

'As I have said before (though you may have missed it since you've been away, P3) the temperature increase over the last 100 years or so would require the Earth to retain no more than an extra 1/1000th of a degree Celsius every month, which is not a great deal.'

If the System retains 'extra' heat, this suggests that something has changed, either to the system, or the external forces working on it.

I can't say too much at the moment, but I think it's worth pointing out that in the Earth Climate system something is always changing - that is the nature of dynamic equilibrium.

More later...

CB

[parmenides3]

I can't say too much at the moment, but I think it's worth pointing out that in the Earth Climate system something is always changing - that is the nature of dynamic equilibrium.

More later...

CB

Yes. Here we need to find something which is/has changed which has led to an observed trend; a long-term positive forcing which did not previously exist. P

[Admiral_Bobski]

Yes. Here we need to find something which is/has changed which has led to an observed trend; a long-term positive forcing which did not previously exist. P

Why does it need to have not existed previously...?

CB

[parmenides3]

Why does it need to have not existed previously...?

CB

Well, it can have existed, perhaps at a relatively stable level, for a long time, but then an increase or decrease over an extended period would have to come in to alter the response of the system to said forcing. Volcanoes are an interesting short-term example; Pinatubo changed several elements of the system, resulting in short term fluctuations which would not have other happened. There is also a degree of hysteresis in the system response, but, once it stopped erupting, the system was eventually allowed to restabilise at close to its previous state. We have to find a mechanism to account for the persistent trend.

Actually, we have found a lot of mechanisms, with various degrees of influence; the question still remains which mechanisms matter, and which if any we (humans) can do anything about.

Without the change to the system, your solar hysteresis idea would suggest that, over time, the earth would warm indefinitely (I think) P

[Admiral_Bobski]

Well, it can have existed, perhaps at a relatively stable level, for a long time, but then an increase or decrease over an extended period would have to come in to alter the response of the system to said forcing. Volcanoes are an interesting short-term example; Pinatubo changed several elements of the system, resulting in short term fluctuations which would not have other happened. There is also a degree of hysteresis in the system response, but, once it stopped erupting, the system was eventually allowed to restabilise at close to its previous state. We have to find a mechanism to account for the persistent trend.

Actually, we have found a lot of mechanisms, with various degrees of influence; the question still remains which mechanisms matter, and which if any we (humans) can do anything about.

Without the change to the system, your solar hysteresis idea would suggest that, over time, the earth would warm indefinitely (I think) P

Not indefinitely, because the Sun goes through periods of high and low activity. That's kind of the crux of the matter.

CB

(Sorry for the brevity of my response - lots to do!)

[VillagePlank]

As I've started to rewrite this thing - for the umpteenth time - I thought it would be good to show what the beginnings of one of the very first drafts looked like.

Unfortunately, there are spelling/grammar and other errors in it, but it should be OK to get the gist. For reference, this sort of stuff will appear in the final paper, but a couple of the ideas in this lot have turned out to be sufficiently large to become research subjects in their own right without any reference to climate.

Here you go, and as always constructive criticism is always helpful.

li3.pdf

I am deeply indebted to Captain Bobski for endless encouragement and for first determining the idea that hysteresis (it wasn't called that back then) might be a critical player in climate.

[Admiral_Bobski]

As I've started to rewrite this thing - for the umpteenth time - I thought it would be good to show what the beginnings of one of the very first drafts looked like.

Unfortunately, there are spelling/grammar and other errors in it, but it should be OK to get the gist. For reference, this sort of stuff will appear in the final paper, but a couple of the ideas in this lot have turned out to be sufficiently large to become research subjects in their own right without any reference to climate.

Here you go, and as always constructive criticism is always helpful.

li3.pdf

I am deeply indebted to Captain Bobski for endless encouragement and for first determining the idea that hysteresis (it wasn't called that back then) might be a critical player in climate.

Thank you, VP. And, at the risk of making this into some kind of love-in (!), I am deeply indebted to you for taking the idea and running with it. When I first proposed the concept I had no idea that it would lead to anything of any interest (though I had hoped!). I wasn't sure that I was barking up the right tree. In fact I wasn't sure that the tree was in the right forest, or even whether or not it was actually a tree!

Back on-topic, though, I would encourage everyone with any interest in the idea to read VP's document - he has done, as usual, an excellent job of simply explaining a concept that can be quite complicated. Any comments, questions or suggestions would be greatly appreciated.

CB

[VillagePlank]

I should note - this is a 'cut and paste' edit to protect the work. As a consequence I've managed to leave in some references that, ahem, are not referenced!

(They are all a good read, though )

Edited December 7, 2009 by VillagePlank