Mathematics of the Environment (Part 5)













See also “week302” of This Week’s Finds, where Nathan Urban tells us more about feedbacks and how big they’re likely to be.



• Gerald R. North, Simple Models of Global Climate.



\displaystyle{ A + B T }



Q c(T)




Given all this, we get




A + B T = Q c(T)






c_i = 1  - 0.65 =  0.35


c_f = 1 - 0.3 = 0.7








Temperature versus insolation.










\gamma = 0.00

\gamma = 0.01

\gamma = 0.02

\gamma = 0.03

\gamma = 0.04

\gamma = 0.05


from here:





17 Responses to Mathematics of the Environment (Part 5)

  1. I found it interesting that B is two orders of magnitude less than A. A quick and dirty numerical simulation shows that setting Q = A + B T + C T^2 with a C two orders of magnitude less than B yields the same qualitative behavior. This is no surprise if one has structural stability. However, if C is just one order of magnitude less than B, e.g. C = 0.1 than things seem to change. I am not quite sure whether that means something important, but maybe for real world problems structural stability is not enough. One might also have to argue that the linearization is indeed ‘good’ enough.

    One a second thought. C might well be zero for symmetry reasons and the physically ‘correct’ form should then be more like Q = A + B T + D T^3.

  2. amarashiki says:

    Dear John. I dedicated you this post, the 50th, in my young blog. I hope you will like it… My LOG 50th available here http://thespectrumofriemannium.wordpress.com/2012/11/07/log050-why-riemannium/

    • John Baez says:

      That’s very nice! Thank you! That’s a very beautiful, detailed post, so I’m flattered that you dedicated it to me…

      … especially because I always I wish I had the time and patience to understand more about the Riemann zeta function. I’ve always loved the ‘primon gas’ idea… but nowadays, I’m busy trying to be a bit more practical, so I don’t have time to think about this. Someday, when I feel in the mood to write another pure math paper, I’ll try to finish and publish this:

      • John Baez and James Dolan, Zeta functions.

      This emphasizes the combinatorics behind the Hasse–Weil zeta function.

  3. amarashiki says:

    No, John…You…I am in debt to you more…Your This Week Finds was an incredible blog, like this Azimuth blog! Believe me, I am sure many people started blogs reading your stuff. Some time ago, I had envy…And finally, I couldn’t stand anymore…I decided to begin my blog!!! And about this post…It is not as detailed as I wanted, but more stuff is coming soon…My draft dashboard is going to explode right now, but I must focus to create good logs or posts…And it carries time.

    Well, … I think that the zeta stuff is practical stuff!!!!! It is like the fiber bundle approach behind the gauge theories…It is real math behind our (ultra)quantum world… In my opinion, we have to understand the zeta stuff better in order to understand QFT, the vacuum and, likely, quantum gravity! Of course, this is a crazy idea!

    Don’t be flattered, John…You make people to be flattered as long as we read your posts, LOL… If you read my log 51, I also wrote a basic zeta function”zoological park” for beginners. I would like to extend that in the future…But I am so disturbed by the RH than sometimes I am seduced by the own formulae you, mathematicians, invent…Hehehe…

    Best wishes!

  4. amarashiki says:

    By the way… Zeta functions are everywhere, even they arise in graph theory, hypergraph theory and it can also appear, I presume, in network theory. How? I am not sure, since I began to learn about zeta seriously only some time ago…And it is hard to keep you updated with the zeta theory…

  5. Renato Iturriaga says:

    Hi John

    I have a couple of questions and comments. About the values of A= 218 \frac{W}{m^2} and B=1.9 \frac{W}{m^2 K} , the value of A is the emission of a body at -23, very close to the radiate equilibrium of -18, I guess this is not a coincidence. Or is it? The derivative of \sigma T^4 at this value is 3.5, B is in this order of magnitude, but not as close one could imagine. Is there some reason for that?

    To see the stability or instability of a particular equilibrium is convenient to look at the graph of \frac {dT}{dt}. In this case -A-BT+Qc(T). In particular the zeros give the equilibrium points. For large negative values of T , the function is positive, and for large positive temperatures the function is negative so there is a zero somewhere, an equilibrium. Almost surely there is an odd number of them: The first time you cross the zero (the coldest equilibrium) you cross it downwards, and if you go up and reach zero you will cross it upwards, afterwards you will have to come down again. So we have either one or three, the only exception is if the zero is double, meaning that the derivative is also zero. The equilibrium points where the derivative is negative, there is local negative feedback effect: if your are slightly above the equilibrium you tend to go down if you are slightly down you tend to go up, so the point is stable, in the points where you have positive derivative -when you cross upwards- there is a positive feedback: if you are a little bit up you tend to go upwards and if you are a little bit down you tend to downwards, hence unstable. So if you have three points, the coldest and the hottest are stable and the middle one is unstable.

    • John Baez says:

      Renato wrote:


      Good question! It would be nice to understand this in detail.

      The main reason the observed infrared power emission of the Earth doesn’t match the power emission of a blackbody is the greenhouse effect. Quite a bit of infrared doesn’t make it through the atmosphere. So, the simplest thing we can say is that we should have

      A + B T < \sigma T^4

      for temperatures in the range we see on Earth. But it would be nice to predict the (linearized) observed average power emission per square meter, A + B T, starting from a simple model.

      Could Simpson’s model in Part 3 be good enough to approximately calculate A and B? Or is it too simple?

      In its simplest form, this model amounts to saying that the fraction of infrared radiation that escapes to space is a function

      0 \le  F(\lambda) \le 1

      of the wavelength \lambda, given as follows. To a crude approximation, the atmosphere is:



      • completely transparent from 8.5 to 11 micrometers,



      Simpson’s function F is 0 where the atmosphere is ‘completely opaque’, 1 where it’s ‘completely transparent’, and it linearly interpolates between these values for wavelengths where the atmosphere is ‘partly transparent’.

      One could multiply the Planck distribution at temperature T by this function of wavelength, then integrate it over wavelengths, to estimate the power emitted per square meter as a function of T.

      This neglects many effects like ‘water vapor feedback’ (there’s more water vapor when it’s hotter), ‘lapse rate feedback’ and ‘cloud feedback’, as mentioned in this post. But it would still be worth doing!

      • Renato Iturriaga says:

        Hi John

        In the homepage of Kerry Emanuel (http://eaps4.mit.edu/faculty/Emanuel/) there are a lot of things. In particular there are presentations of a lot of courses. In one of them (Tropical meteorology) there is a model for the equilibrium radiation. The simplest model has two layers, the atmosphere and the surface. The atmosphere is completely transparent to visible light and completely opaque to infrared. Let T_e be the temperature of equilibrium (250 K), then the equilibrium is achieved at at T_e in the atmosphere and 2^{\frac{1}{4}} T_e in the surface.

        A 3 layer model predicts T_e in the outer part and $3^{\frac{1}{4}} T_e$ in the surface. This is way to hot and the explanation is partially because it is not completely opaque and because of convection. Something else should be wrong because if we do the same thing with n layers, we would get a surface temperature of n^{\frac{1}{4}} T_e. But it is funny that the outer part is always the equilibrium temperature. Finally there is a complete calculation, that unfortunately is not clear how it is done, that gives a vertical profile of the temperature if only radiation is taken into account. The profile is very hot in the surface, again the explanation is the lack of the convection in the model.

    • John Baez says:

      Renato wrote:


      Thanks for explaining this! I hope all the students in my class read your explanation, which is better than mine.

  6. Last time we saw a ‘bistable’ climate model, where the temperatures compatible with a given amount of sunshine can form an S-shaped curve like this: […]


  8. gymnosperm says:

    Very nice. When one extends the timeline back to our horizons, basically the Proterozoic for semi reliable temperature, it is clear that bistability applies not only to glacial and interglacial periods within a snowball earth, but to the alternation between snowball and greenhouse climates. At the larger scale the warm state appears more stable.

  9. John Roe says:

    I set up a version of this model on InsightMaker, you can find it here. If you want to play around, note that A, B are as above; C is rescaled to measure time in days; G is twice gamma. Please, let me know if I messed something up.

    The time scale is quicker than I had expected, this is because the heat capacity C is quite low. In particular, the latent heat of fusion of a 1cm layer of ice appears to be of the same order of magnitude as C… one might try to incorporate this in some way, e.g. with an “ice quantity” variable driving c, and the “ice quantity” evolving according to temperature.

  10. […] Part 5 – A model showing bistability of the Earth’s climate due to the ice albedo effect: statics. […]

  11. These cycles, for instance, don’t begin to consider the various feedback dynamics that contribute to the global climate, including the ice albedo effect and the rate of atmospheric carbon. Each of these feedback systems contribute something to the dynamics of the climate, but it is often extremely difficult to know how to integrate these factors into a single model.



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