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Tutorial – coupling the realms Many slides from Peter Franks, Marc Spigelman, Eli Tziperman Hezi Gildor Ruddiman, 2001

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Conclusions All Models are wrong, but some are useful (Box,1979)

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Conclusions All Models are wrong, but some are useful (Box,1979) Be skeptic

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Sources and recommended reading: 1. Presentation by Peter Franks at the HAB workshop, summer 2003. Check: www.obs-vlfr.fr/habwatch (go to proceedings- >lectures). 2. Excellent site of a course on modelling and numerical methods by Marc Spigelman, Columbia University. http://www.ldeo.columbia.edu/~mspieg/mmm/ http://www.ldeo.columbia.edu/~mspieg/mmm/ 3. John Harte published a book called Consider a spherical cow and also an interesting article in Physics Today, Toward synthesis of the Newtonian and Darwinian world views, Oct. 2002, 29-34. 4. Rules of thumb for judging ecological theories, Ginzburg and Jensen, Trends in Ecology and evolution, 2004. This lecture will not be an exhaustive treatment; will only introduce basic concepts.

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Outline What is a model? Why to build/use models? How to build a model? Introduction to bifurcations Numerical methods Use and abuse of models Summary

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What is a model? Merriam-Webster: 13 entries! Entry 4: a usually miniature representation of something Entry 12: a system of postulates, data, and inferences presented as a mathematical description of an entity or state of affairs

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What is a model? Modeling is simple book-keeping

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What is a model? Governing equations (often conservation equations); processes and relationship between forcing and variables. Initial conditions. Boundary conditions. External forcing (e.g. radiation, nutrients supply).

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Why to build/use models? Hypothesis testing (e.g. functional forms). Identifying data gaps. Sensitivity analysis; what-if scenarios -> identify research priorities. Can be used to recover information from a data set –Use statistical techniques to fit model to data to obtain model parameters from best fit Manager: quantitative tool for decision making.

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Why to build/use models? … as we know, there are known knowns; … there are known unknowns; … But there are also unknown unknowns the ones we dont know we don't know. Defense Secretary Donald Rumsfelds winning entry for "Foot in Mouth award, London, 2003. We want to identify the unknown uknown….

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How to build a model? The kitchen sink approach: Throw everything into it and hope something useful comes out (BAD IDEA). The Model Problem Approach: Gain insight by developing simple model problems that balance interesting behavior with comprehensibility. More complex model doesnt necessarily represent the real system more accurately. More complex models contain more (often uncertain) parameters.

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How to build a model? Therefore: –Be problem driven –Keep it simple –Model only what you can understand

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How to build a model? Therefore: –Be problem driven –Keep it simple –Model only what you can understand There is no right model for all problems. Depends on question: –What are the dominant physical forcings? –What space and time scales are important? –What kinds of data are available? Sensitivity tests Think before you model, think while you model, think after you model

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Nonlinearities and surprises

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Introduction to bifurcations A qualitative change in the solutions of an equation in response to a parameter change Looking for steady-states (fixed point) Example:

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Stability states There are stable and unstable steady states Transitions can be induced by: Gradual changes Abrupt changes; catastrophes, tipping points Stochastic resonance

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Introduction to bifurcations A qualitative change in the solutions of an equation in response to a parameter change Looking for steady-states (fixed point) Example:

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Introduction to bifurcations A qualitative change in the solutions of an equation in response to a parameter change Combining all values of into one plot Stable state Unstable state

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Simplest climate model S, Hysteresis, Jumps Prediction of snowball earth?

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How to build a model? Choose an algorithms and discretization (finite differences, spectral methods, etc.). Search for available code (Numerical Recipes, NAG, etc) or suitable software (Stella, NetLogo, Matlab, etc.) Sometimes it is simpler to apply and refine an existing model. Luckily, other people have done a lot of the work for you: http://stommel.tamu.edu/~baum/ocean_models.html http://stommel.tamu.edu/~baum/ocean_models.html MITgcm (GCM-general circulation model), ROMS (regional ocean model system), MICOM (Miami isopycnal coordinates ocean model), Ecopath/Ecosim….

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Numerical methods Solving an equation numerically involves making approximations to the actual equation, and iterating these approximations in space and time For example, a simplified one-dimensional form of the u momentum equation is

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Numerical methods If we keep all the terms, this will be an exact approximation. However, this isnt practical For now, assume that t and x are small, so any term multiplied by them is small and can be neglected… Definition of a derivative To define a gradient (u/t or u/x) we need at least 2 points for each Define the points as t and t+ t, x and x+ x Then approximate the derivatives u/t and u/x by using a Taylor series expansion

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Numerical methods If we let t and x tend to 0, these equations are exact. But if we let t and x remain finite, these equations are approximations: a finite difference This is the essence of numerical models Definition of a derivative

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Numerical methods Re-write the original equation, making substitutions for u/x and u/t: Now rearrange this so that all the new times (t+ t) are on the left-hand side:

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Numerical methods This equation says that the new velocity at the point x is given by the old velocity at that location plus some measure of the velocity difference between x and x+ x, times the time step t. This means we need to know the velocity at x+ x But to calculate the new velocity at x+ x, we need to know the velocity at x+2 x And so on, ad infinitum… –We need to set a boundary condition

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Numerical methods There are many schemes for solving equations in this way (forward, backward, leapfrog, etc.) e.g. How do we choose among them?

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Numerical models Accuracy: how close is our approximation to the true solution? Smaller t, x is better, but takes more computer time Stability: how do the errors in our approximation change with time? If errors increase exponentially, model will blow up

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Quiz Define We can derive a recursive formula: Let try it:

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Quiz Define We can derive a recursive formula: Let try it: It works fine in the other way…. Why?

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Toward a Synthesis of the Newtonian and Darwinian Worldviews John Harte, Physics Today, October 2002

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Use and abuse of models Models can be manipulated Harte 2002: …Often, one could predict the result from the politics of the modeler

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Use and abuse of models Models can be manipulated Harte 2002: …Often, one could predict the result from the politics of the modeler We always miss something; Parameterizations (model within model)

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Use and abuse of models Models can be manipulated Harte 2002: …Often, one could predict the result from the politics of the modeler We always miss something; Parameterizations (model within model) Be careful from tuning exercises

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We first tune the model to fit the observations, use the model results to validate the model and then use the model to predict future climate

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Use and abuse of models Models can be manipulated Harte 2002: …Often, one could predict the result from the politics of the modeler We always miss something; Parameterizations (model within model) Be careful from tuning exercises Over-fitting. Ptolemys epycycle theory (140 AD) predicted the positions of the sun, moon, and planets, using combinations of circular motion called epicycles.

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Summary Models are excellent tools!! Can synthesize data, test hypotheses, forecast dynamics (with suitable input) Easier and less harmful to manipulate models than nature Defining the question is the most important first step in constructing a model Models and field work should be conducted together, with the results from both being compared regularly.

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as the efficient use of tool depends on the worker, so the efficient use of a model will depend on its user, probably more on his physical sense than on his mathematical capabilities: a good mathematician without physical sense is a very bad modeler, a good physicist or engineer, with just a basic knowledge of mathematics, can be a very good modeler! Groundwater Pollution, by J.J. Fried, 1975

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Always remember, a bad day at sea is better than running models in the office, Confucius. Thank you!

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