Options

Towards a climate model

Since I've written several times about the status of my pet project, I finally created a Wiki page about it: Towards a climate model.

In its final state it could be a how-to for implementing climate models, a step-by-step introduction. (1. Step: Learn how to program, 2. Step: program a numerical solution to..., 17.Step: predict the climate of a region of your choice for the next 80 years.)

Comments

  • 1.

    It's a small page about a huge project, but everything has to start somewhere!

    I would offer to help you with this Burgers equation simulation but right now I need to catch up with Eric Forgy on the Milankovitch cycle stuff!

    Comment Source:It's a small page about a huge project, but everything has to start somewhere! I would offer to help you with this Burgers equation simulation but right now I need to catch up with Eric Forgy on the Milankovitch cycle stuff!
  • 2.

    The Burgers' equation stuff is an exercise in programming numerical algorithms, it's about:

    -> Implement the necessary algorithms to evaluate the solution in closed form, which entails the evaluation of infinite sums and Bessel functions,

    -> Implement the spectral approximation, for which one needs a robust implementation of complex number arithmetic, a FFT and numerical integration,

    -> implement an online visualisation, for which one needs a client-server architecture where the client displays both solutions and updates itself with a certain clock frequency.

    Of course "implement" does not mean "implement everything from scratch", it means choose appropriate libraries, integrate these libraries into your programming environment/your program, reprogram existing implementations if those cannot be integrated, choose appropriate algorithms and implement those if there aren't any good examples available.

    After all of this is done, we're about 20% along the way to a turbulence simulation of the Navier-Stokes equations in a 3D cube.

    Comment Source:The Burgers' equation stuff is an exercise in programming numerical algorithms, it's about: -> Implement the necessary algorithms to evaluate the solution in closed form, which entails the evaluation of infinite sums and Bessel functions, -> Implement the spectral approximation, for which one needs a robust implementation of complex number arithmetic, a FFT and numerical integration, -> implement an online visualisation, for which one needs a client-server architecture where the client displays both solutions and updates itself with a certain clock frequency. Of course "implement" does not mean "implement everything from scratch", it means choose appropriate libraries, integrate these libraries into your programming environment/your program, reprogram existing implementations if those cannot be integrated, choose appropriate algorithms and implement those if there aren't any good examples available. After all of this is done, we're about 20% along the way to a turbulence simulation of the Navier-Stokes equations in a 3D cube.
  • 3.

    I'm happy to help, but I should warn that I'm not a big believer in closed form solutions. Thy only exist for the most simple systems. I prefer to go fully numerical, but numerical "done right"

    Comment Source:I'm happy to help, but I should warn that I'm not a big believer in closed form solutions. Thy only exist for the most simple systems. I prefer to go fully numerical, but numerical "done right"
  • 4.
    edited April 2011

    As I understand it, Tim is doing the "numerically solve a problem with a known solution and check they're the same in order to validate the numerical solution machinery" approach to building the foundations.

    Comment Source:As I understand it, Tim is doing the "numerically solve a problem with a known solution and check they're the same in order to validate the numerical solution machinery" approach to building the foundations.
  • 5.

    That's right, I looked for a 1D problem with simple boundary conditions where a solution in closed form exists, so that it is possible to compare the approximate solution obtained from a spectral method - with varying order - to the solution that can be evaluated from an analytic formula.

    Comment Source:That's right, I looked for a 1D problem with simple boundary conditions where a solution in closed form exists, so that it is possible to compare the approximate solution obtained from a spectral method - with varying order - to the solution that can be evaluated from an analytic formula.
  • 6.

    Ah. That I am all for :)

    Comment Source:Ah. That I am all for :)
  • 7.

    You can see the formula for the initial+boundary value problem I'm talking about on the Burgers' equation page ("An Exact Solution of a Boundary Value Problem") and the description of the numerical approximation under "Approximation via Fourier-Galerkin Spectral Method". The wiki is back up.

    Comment Source:You can see the formula for the initial+boundary value problem I'm talking about on the [[Burgers' equation]] page ("An Exact Solution of a Boundary Value Problem") and the description of the numerical approximation under "Approximation via Fourier-Galerkin Spectral Method". The wiki is back up.
  • 8.

    This is a good example where having a super genius like John around to help with the math can help :)

    Is there a coordinate independent way to write the Burgers' equation? Expressing things in a coordinate free manner is usually a good step toward robust numerical implementations.

    Oh! You can transform Burgers' equation to the heat equation? In that case there is a relation to stochastic calculus somehow. In that case, there is a relation to noncommutative geometry and discrete calculus somehow. That is the stuff I love :)

    Comment Source:This is a good example where having a super genius like John around to help with the math can help :) Is there a coordinate independent way to write the Burgers' equation? Expressing things in a coordinate free manner is usually a good step toward robust numerical implementations. Oh! You can transform Burgers' equation to the heat equation? In that case there is a relation to stochastic calculus somehow. In that case, there is a relation to noncommutative geometry and discrete calculus somehow. That is the stuff I love :)
  • 9.
    edited April 2011

    I knew I was getting a faint feeling of deja vu...

    See Section 2 of:

    The deformed noncommutative calculus they use is precisely the same I used (with references to Dimakis & Meuller-Hoissen) in

    or here if the nlab is down.

    Since there is a correspondence between graphs and calculi, this implies the "mesh" you should use to solve the Burger equation numerically is a binary tree. To see how that works, see

    The meta algorithm would be to follow the first paper above, turn the crank using my discrete stochastic calculus paper and it will pop out the discrete Burger equation.

    Following this prescription to its fruition would likely be worth publishing in an academic journal.

    Edit: Sorry to go on and on, but this is cool. The Burger equation provides the first example I've ever seen "in the wild" of a stochastic 2-form.

    Comment Source:I knew I was getting a faint feeling of deja vu... See Section 2 of: * [Soliton equations and the zero curvature condition in noncommutative geometry](http://arxiv.org/abs/hep-th/9608009) The deformed noncommutative calculus they use is precisely the same I used (with references to Dimakis & Meuller-Hoissen) in * [Noncommutative Geometry and Stochastic Calculus](http://ncatlab.org/ericforgy/show/Noncommutative+Geometry+and+Stochastic+Calculus) or [here](http://phorgyphynance.files.wordpress.com/2008/06/blackscholes.pdf) if the nlab is down. Since there is a correspondence between graphs and calculi, this implies the "mesh" you should use to solve the Burger equation numerically is a binary tree. To see how that works, see * [Financial Modelling Using Discrete Stochastic Calculus](http://phorgyphynance.files.wordpress.com/2008/06/discretesc.pdf) The meta algorithm would be to follow the first paper above, turn the crank using my discrete stochastic calculus paper and it will pop out the discrete Burger equation. Following this prescription to its fruition would likely be worth publishing in an academic journal. Edit: Sorry to go on and on, but this is cool. The Burger equation provides the first example I've ever seen "in the wild" of a stochastic 2-form.
  • 10.

    Burgers' equation, in the limit of zero viscosity, develops shock fronts. This can make it difficult to numerically solve accurately when the viscosity is small, because of the Gibbs phenomenon; you can get a lot of spurious wiggles in the solution when you try to force a Fourier series to approximate a sharp edge. Have you thought about how to deal with this? I'm not sure to what extent issues like this become very important in climate modeling. You can always just include enough terms in the Fourier series that you're not too uncomfortable with the result. Another approach is to use some sort of adaptive procedure that allows higher spatial resolution in the region where things change sharply. I played with wavelet methods for numerically solving Burgers' equation that did this, many years ago, and I think it worked fairly well, although I've long since forgotten the details.

    I've had several things come up to swallow up the part of my free time that I was hoping to use to contribute here, at least for the next several weeks, but hopefully by summer I'll be able to help out more substantially. The Milankovitch project particularly interests me, so I'm trying to drop in occasionally to see how things are progressing.

    Comment Source:Burgers' equation, in the limit of zero viscosity, develops shock fronts. This can make it difficult to numerically solve accurately when the viscosity is small, because of the [Gibbs phenomenon](https://secure.wikimedia.org/wikipedia/en/wiki/Gibbs_phenomenon); you can get a lot of spurious wiggles in the solution when you try to force a Fourier series to approximate a sharp edge. Have you thought about how to deal with this? I'm not sure to what extent issues like this become very important in climate modeling. You can always just include enough terms in the Fourier series that you're not too uncomfortable with the result. Another approach is to use some sort of adaptive procedure that allows higher spatial resolution in the region where things change sharply. I played with wavelet methods for numerically solving Burgers' equation that did this, many years ago, and I think it worked fairly well, although I've long since forgotten the details. I've had several things come up to swallow up the part of my free time that I was hoping to use to contribute here, at least for the next several weeks, but hopefully by summer I'll be able to help out more substantially. The Milankovitch project particularly interests me, so I'm trying to drop in occasionally to see how things are progressing.
  • 11.
    edited April 2011

    Hey Matt,

    Interesting comment! So you have experience with the Burgers equation. Very cool.

    If my suspicion is correct (and I know it probably sounds like gibberish at the moment), then I believe we can solve Burgers in a way that is guaranteed to converge in the continuum limit by construction. The trick is to use "discrete calculus". Since the calculus itself converges, any solution obtained using the discrete calculus is guaranteed to converge as well.

    Your comments about Gibbs make perfect sense and was one of my thoughts when Tim mentioned spectral methods.

    Comment Source:Hey Matt, Interesting comment! So you have experience with the Burgers equation. Very cool. If my suspicion is correct (and I know it probably sounds like gibberish at the moment), then I believe we can solve Burgers in a way that is guaranteed to converge in the continuum limit _by construction_. The trick is to use "discrete calculus". Since the calculus itself converges, any solution obtained using the discrete calculus is guaranteed to converge as well. Your comments about Gibbs make perfect sense and was one of my thoughts when Tim mentioned spectral methods.
  • 12.

    Eric, the inventor was Mr Burgers, so it is the Burgers equation, or Burgers' equation.

    This is important, unless you like to search and search for the one Mr Burger who came up with this stuff...(and no, it was not Mr. Ham Burg).

    I'd like to use a spectral method because these are the methods that are popular for global atmosphere models, and Burgers' equation is very interesting because it has shock like solutions, just like the Navier-Stokes equations, and thus can be a simple - very simple - model to illustrate these effects. The basic idea behind using spectral methods for global models is that they naturally suppress effects below a chosen length scale, which is precisely what people want in their models. The question is, of course, if that is a valid approach to approximation. I haven't seen any systematic study of this question (in the books that discuss these approaches to climate models).

    @Matt: That's really cool that you already know this stuff!

    Comment Source:Eric, the inventor was Mr Burger<b>s</b>, so it is the Burgers equation, or Burgers<b>'</b> equation. This is important, unless you like to search and search for the one Mr Burger who came up with this stuff...(and no, it was not Mr. Ham Burg). I'd like to use a spectral method because these are the methods that are popular for global atmosphere models, and Burgers' equation is very interesting <i>because</i> it has shock like solutions, just like the Navier-Stokes equations, and thus can be a simple - very simple - model to illustrate these effects. The basic idea behind using spectral methods for global models is that they naturally suppress effects below a chosen length scale, which is precisely what people want in their models. The question is, of course, if that is a valid approach to approximation. I haven't seen any systematic study of this question (in the books that discuss these approaches to climate models). @Matt: That's really cool that you already know this stuff!
  • 13.
    edited April 2011

    Tim wrote:

    Eric, the inventor was Mr Burgers, so it is the Burgers equation, or Burgers' equation.

    Whoops - by the time I read this I'd already corrected Eric's slip.

    Eric wrote:

    This is a good example where having a super genius like John around to help with the math can help :)

    Umm, I don't know much about this stuff, and I'm working on a lot of projects so someone will have to pose me a specific puzzle if they want me to apply my superpowers to this subject. For example, if you need me to prove existence and uniqueness of solutions of an equation, just ask:

    But frankly, it sounds like Matt Reece is the superhero you need for help with Burgers' equation.

    Comment Source:Tim wrote: > Eric, the inventor was Mr Burger<b>s</b>, so it is the Burgers equation, or Burgers<b>'</b> equation. Whoops - by the time I read this I'd already corrected Eric's slip. Eric wrote: > This is a good example where having a super genius like John around to help with the math can help :) Umm, I don't know much about this stuff, and I'm working on a lot of projects so someone will have to pose me a specific puzzle if they want me to apply my superpowers to this subject. For example, if you need me to prove existence and uniqueness of solutions of an equation, just ask: <img width = "500" src = "http://math.ucr.edu/home/baez/superbaez.jpg" alt = ""/> But frankly, it sounds like Matt Reece is the superhero you need for help with Burgers' equation.
  • 14.

    Ok, two specific questions: Where did you get that graphic?! And what is the meaning of the logo?!

    (And shouldn't it be a 3D background of the whole world?)

    But seriously: The little Burgers' project is about the implementation of a specific numerical approximation and the comparision to the solution in closed form, the main objectives are:

    a) create a necessary part of the software infrastructure, including the implementation of numerical algorithms, that are needed for the Navier-Stokes equations,

    b) gain some insights into spectral methods, like: How do I handle millions of coupled ordinary differential equations? What happens when the solution develops singular behaviour like shock waves?

    Of course we won't need millions of ODE to get a good approximation, I think, but you get millions of coupled ODE when you use spectral methods for the globe, if you'd like to get a decent approximation of the synoptic scale.

    I'm sorry but I don't think there is anything of interest for a pure mathematician here.

    Comment Source:Ok, two specific questions: Where did you get <i>that</i> graphic?! And what is the meaning of the logo?! (And shouldn't it be a 3D background of the <i>whole</i> world?) But seriously: The little Burgers' project is about the <i>implementation</i> of a specific numerical approximation and the comparision to the solution in closed form, the main objectives are: a) create a necessary part of the software infrastructure, including the implementation of numerical algorithms, that are needed for the Navier-Stokes equations, b) gain some insights into spectral methods, like: How do I handle millions of coupled ordinary differential equations? What happens when the solution develops singular behaviour like shock waves? Of course we won't need millions of ODE to get a good approximation, I think, but you get millions of coupled ODE when you use spectral methods for the globe, if you'd like to get a decent approximation of the synoptic scale. I'm sorry but I don't think there is anything of interest for a pure mathematician here.
  • 15.

    Tim wrote:

    Where did you get that graphic?!

    When I told my student Alex Hoffnung that I was trying to get more mathematicians interested in "saving the planet", he joked that I needed a cape. And then he got a friend of his, an artist named John Vance, to draw this picture.

    And what is the meaning of the logo?!

    I'm not completely sure, but the $\exists !$ ("there exists a unique") symbol is an allusion to math, and the map is an allusion to saving the planet.

    I'm sorry but I don't think there is anything of interest for a pure mathematician here.

    I think it's always good to have people thinking about things from different angles and talking to each other. The (1+1)-dimensional Burgers' equation is exactly solvable so it hooks up to the world of completely integrable systems, which pure mathematicians like. As you can see, Eric is fascinated by the connections to the heat equation, the Black-Scholes equation, discretizations of these, and gauge theory. You have your own reasons for being interested in it. The overlap in these interests may be small but it's not zero. I think you tend towards pessimism in this sort of matter, while I tend towards optimism. True, I'm not going to help you write your programs (Eric might, I don't know). But you might have more fun writing them if you talk to us.

    Comment Source:Tim wrote: > Where did you get <i>that</i> graphic?! When I told my student Alex Hoffnung that I was trying to get more mathematicians interested in "saving the planet", he joked that I needed a cape. And then he got a friend of his, an artist named [John Vance](http://www.johnnyterrific.com/bio), to draw this picture. > And what is the meaning of the logo?! I'm not completely sure, but the $\exists !$ ("there exists a unique") symbol is an allusion to math, and the map is an allusion to saving the planet. > I'm sorry but I don't think there is anything of interest for a pure mathematician here. I think it's always good to have people thinking about things from different angles and talking to each other. The (1+1)-dimensional Burgers' equation is exactly solvable so it hooks up to the world of completely integrable systems, which pure mathematicians like. As you can see, Eric is fascinated by the connections to the heat equation, the Black-Scholes equation, discretizations of these, and gauge theory. You have your own reasons for being interested in it. The overlap in these interests may be small but it's not zero. I think you tend towards pessimism in this sort of matter, while I tend towards optimism. True, I'm not going to help you write your programs (Eric might, I don't know). But you might have more fun writing them if you talk to us.
  • 16.

    John wrote:

    I'm not completely sure, but the ∃! ("there exists a unique") symbol is an allusion to math, and the map is an allusion to saving the planet.

    I suggest "protector of the unique earth" (since most people will be familiar with the existence of earths, the uniqueness is the part that should be emphasized).

    I think you tend towards pessimism in this sort of matter, while I tend towards optimism.

    I've been working for 10 years with people that don't care about math, and, well, part of my reason to leave academia before that was because I did not find a topic, position and mentor where I could do mathematical physics the way I liked to (with a strong emphasis on formulating physics in terms of precise mathematics), so my pessimism is a result of my biography.

    Well, maybe I can come up with some questions for pure mathematicians, with nonzero overlap with the numerics project: I'll ask them in a new thread, however.

    Comment Source:John wrote: <blockquote> <p> I'm not completely sure, but the ∃! ("there exists a unique") symbol is an allusion to math, and the map is an allusion to saving the planet. </p> </blockquote> I suggest "protector of the unique earth" (since most people will be familiar with the existence of earths, the uniqueness is the part that should be emphasized). <blockquote> <p> I think you tend towards pessimism in this sort of matter, while I tend towards optimism. </p> </blockquote> I've been working for 10 years with people that don't care about math, and, well, part of my reason to leave academia before that was because I did not find a topic, position and mentor where I could do <i>mathematical</i> physics the way I liked to (with a strong emphasis on formulating physics in terms of precise mathematics), so my pessimism is a result of my biography. Well, maybe I <i>can</i> come up with some questions for pure mathematicians, with nonzero overlap with the numerics project: I'll ask them in a new thread, however.
  • 17.

    I've been working for 10 years with people that don't care about math [...] so my pessimism is a result of my biography.

    I understand. But now you've got friends here.

    Comment Source:> I've been working for 10 years with people that don't care about math [...] so my pessimism is a result of my biography. I understand. But now you've got friends here.
  • 18.

    I understand. But now you've got friends here.

    Hear hear! :)

    Comment Source:>I understand. But now you've got friends here. Hear hear! :)
  • 19.

    ∃!

    Great logo on the superhero cartoon! Mortals might think/hope John is into saving the planet by inventing a new sort of ∃nergy...

    Another great picture of John, very much to the point:

    Comment Source:> ∃! Great logo on the superhero cartoon! Mortals might think/hope John is into saving the planet by inventing a new sort of ∃nergy... Another great picture of John, very much to the point: <img src="http://math.ucr.edu/home/baez/diary/dagomir/1244209638_qwJH6-S-2.jpg" alt=""/>
  • 20.

    so it hooks up to the world of completely integrable systems, which pure mathematicians like. As you can see, Eric is fascinated by the connections to the heat equation, the Black-Scholes equation, discretizations of these, and gauge theory. You have your own reasons for being interested in it.

    I thought Black-Scholes was dead after they received the nobel price :-) and bankrupted their initial company !

    Eric, ayway William Stein gave a course SIMUW2008 using Sage for finance people/investors on Sage, multifractals, hidden markov models and how to combine them

    Comment Source:> so it hooks up to the world of completely integrable systems, which pure mathematicians like. As you can see, Eric is fascinated by the connections to the heat equation, the Black-Scholes equation, discretizations of these, and gauge theory. You have your own reasons for being interested in it. I thought Black-Scholes was dead after they received the nobel price :-) and bankrupted their initial company ! Eric, ayway William Stein gave a course [SIMUW2008](http://wiki.wstein.org/2008/simuw) using Sage for finance people/investors on Sage, [[multifractal]]s, hidden markov models and how to combine them
Sign In or Register to comment.