Tutorial – coupling the realms

Many slides from

Peter Franks, Marc Spigelman, Eli Tziperman

Hezi Gildor

Ruddiman, 2001

Conclusions• “All Models are wrong, but some are useful”

(Box,1979)

Conclusions• “All Models are wrong, but some are useful”

(Box,1979)

• Be skeptic

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/

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.

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

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

What is a model?

Modeling is simple book-keeping

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).

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.

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 don’t know we don't know.”

Defense Secretary Donald Rumsfeld’s winning entry for "Foot in Mouth” award, London, 2003.

We want to identify the unknown uknown….

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 doesn’t necessarily represent the real system more accurately. More complex models contain more (often uncertain) parameters.

How to build a model?• Therefore:

–Be problem driven–Keep it simple–Model only what you can understand

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

Nonlinearities and surprises

Introduction to bifurcationsA qualitative change in the solutions of an equation in response to a parameter change

Looking for steady-states (fixed point)

Example:

Stability states

There are stable and unstable steady states

Transitions can be induced by:•Gradual changes•Abrupt changes; catastrophes, “tipping points”•Stochastic resonance

Introduction to bifurcationsA qualitative change in the solutions of an equation in response to a parameter change

Looking for steady-states (fixed point)

Example:

Introduction to bifurcationsA 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

Simplest climate model

S,

Hysteresis, Jumps

Prediction of snowball earth?

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

MITgcm (GCM-general circulation model), ROMS (regional

ocean model system), MICOM (Miami isopycnal coordinates

ocean model), Ecopath/Ecosim….

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

Numerical methods

• If we keep all the terms, this will be an exact approximation. However, this isn’t 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

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

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:

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+2x• And so on, ad infinitum…

– We need to set a boundary condition

Numerical methods• There are many schemes for solving

equations in this way (forward, backward, leapfrog, etc.)

e.g.

• How do we choose among them?

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

QuizDefine

We can derive a recursive formula:

Let try it:

QuizDefine

We can derive a recursive formula:

Let try it:

It works fine in the other way….

Why?

Toward a Synthesis of the Newtonian and Darwinian Worldviews

John Harte, Physics Today, October 2002

Use and abuse of models• Models can be manipulated

• Harte 2002:

“ …Often, one could predict the result from the politics of the modeler”

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)

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

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

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. Ptolemy’s epycycle theory (140 AD) predicted the positions of the sun, moon, and planets, using combinations of circular motion called epicycles.

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.

“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

“Always remember, a bad day at sea is better than running models in the office”, Confucius.

Thank you!