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Today we will look at one of the software models from the Azimuth code project, which aims to produce educational software that is relevant to the study of climate. The program, which was contributed by Allan Erskine and Glyn Adgie, demonstrates the concept of “stochastic resonance.” This is a widely studied phenomenon, which has an application to the theory of ice-age cycles. The Azimuth models are implemented as interactive web pages, which have a responsive behavior because they run right in the browser, as javascript.
Start by opening the stochastic resonance model web page. It displays a sine wave, called the forcing signal, alongside an intricate, time-varying function, called the output signal. There are four sliders, labelled A, B, C and D.
One of the sliders controls the frequency of the sine wave, and another controls its amplitude. Try them.
The sine wave is input to a random process, which produces the output signal. Observe how the amplitude and frequency sliders affect, in a complex way, the generated output signal. Can you detect any patterns in the input-output relationship?
The noise slider controls the randomness of the process. Set it to zero, and verify that the output is completely smooth. See that the output becomes increasingly chaotic as you increase it.
Changing the Sample-Path parameter gives you different instances of the process.
The program runs a discrete simulation for a “stochastic differential equation” (SDE), which specifies the derivative of a function in terms of time, the current value of the function, and a noise process.
The program contains:
A general simulator for SDE’s, based on the Euler method
A function which defines the derivative for a particular SDE. The “stochastic resonance” is a description of the behavior of the solutions to this particular equation.
Helper functions used by this derivative function
Interactive controls to set parameters
Plot of the forcing signal (the sine curve)
Plot of the output signal
I would like everyone now to locate the source code, by the following method.
Open the web page for the model.
The code is now downloaded and running in your browser.
Now, while viewing the page, run your browser’s view-source function. For Firefox on the Mac, it’s Apple-U, for Firefox on the PC it’s … (TODO: fill in)
You should see the html for the web page itself.
Note these lines at the top of the file:
<script src='http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=default'></script> <script src='http://cdnjs.cloudflare.com/ajax/libs/jsxgraph/0.93/jsxgraphcore.js'></script> <script src='./StochasticResonanceEuler.js'></script> <script src='./normals.js'></script>
These lines refer to javascript program files at various locations on the web. They direct the browser to fetch the code and load it into its internal javascript interpreter. First, MathJax is an open-source formula rendering engine. Second, JSXGraph is a cross-platform library for interactive graphics, function plotting, and data visualization. Third, StochasticResonanceEuler.js is the main code for the model. Fourth, normals.js contains a table of random numbers, used in the calculation.
Now click on the link for StochasticResonance.js – and you’re there!
Now that you’re at StochasticResonanceEuler.js, your next challenge is to scan through it and try to make loose associations between its terms and clauses and the items of the program inventory given in the preceding section. Try to put a blur lens over any items that look obscure, since the goal here is just to form rough hypotheses about what might be going on.
In the stochastic resonance formula, the deterministic part of the derivative is set to a sinusoidal function of t, called the forcing function, plus a function of x that has two stable points of equilibrium:
DerivDeterministic(t, x) = SineWave(t, amplitude, frequency) + Bistable(x),
where Bistable(x) = x * (1 - x^2).
We consider now the effects of each of these terms, both separately and in combination.
Alone, the sinusoidal forcing function would cause the output signal to vary sinusoidally.
Now let’s analyze the effect of Bistable(x). The roots of Bistable(x) are -1, 0 and 1. This gives stable equilibrium points at -1 and 1, and an unstable equilibrium at 0. The basin of attraction for -1 is all the negative numbers, the basin for 1 is the positive numbers, and the unstable equilibrium point separates the basins.
We will view each basin of attraction as one of the states of a bistable system.
Now let’s consider the combined effect of the sine wave and the bistable polynomial. If the wave amplitude is below a threshold, the system will remain in one basin, forever oscillating around its attractor. But if the amplitude is large enough, it will pull the system back and forth between the two basins – it will resonate.
Now let’s complete the picture by adding in the noise. Suppose the sine wave were large enough to periodically pull the system close to zero, but not enough to cross over to the other side – so it would still remain in a single state. If we add in some noise, then a well-timed random event could push the system over to the other side. The noise, then, will introduce some random transitions between states, with higher probabilities of this occurring at certain phases of the sine wave. As the noise amplitude increases, transitions will occur during more and more cycles of the sine wave will lead to transitions, and the frequency of the flip-flopping will approach that of the sine wave. As noise increases even further, it will dominate the state transitions, which will themselves become noise.
Conclusion: a certain amount of noise will amplify the effect of the input signal.
A major hypothesis for the timing of the ice ages is based on a stochastic resonance model. In Didier Paillard’s model, the climate of the Earth is modeled as a tristable system, and the forcing function is produced by certain astronomical variations called Milankovich cycles, which produce periodic variation in the amount of solar energy received at the northern latitudes.
In the most simplified model, the climate has two stable states: a cold, “snowball Earth,” and a hot Earth with no ice. The snowball Earth is stable: because it is white, it doesn’t absorb much solar energy, which keeps it cold and frozen. The hot Earth is stable: because it is dark, it absorbs a lot of solar energy, which keeps it hot and melted.
Now in order to bring the Milankovitch cycles into the picture, we bring into the picture the fact that the glaciers are concentrated up north, and it is the temperatures there that control the state of the system. If it gets warm enough up north, the glaciers melt, and the state goes to hot. If it gets cold enough, the glaciers form, and the state goes to cold.
There are three Milankovitch cycles that have been identified:
Changing of the eccentricity (ovalness) of the Earth’s orbit, with a period of 100 kiloyears
Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 kiloyears
Precession of the Earth’s axis (rotation of the direction of the axis, like a wobbling top), with a period of 23 kiloyears
Now the amplitude of these temperature changes in not enough to move the climate from one state to another. But random temperature events may cause crossings if they occur at the right phase of a Milankovitch cycle.
Researchers have found correspondences (complex correlations) between the actual spacing of the ice ages and the timings of these three Milankovitch cycles.
For further information, see:
The program for the stochastic resonance model consists of seven functions. The top level function is initCharts, which dispatches to two other functions, initControls and initSrBoard. (Key: sr = stochastic resonance, board = graphical gizmo.) The job of initControls is to build the sliders.
The main logic of the application is contained in this second function, initSrBoard. Two curve objects are constructed there, positionCurve and forcingCurve. The forcing curve is given by the locally defined forcingFunction, which defines the sine wave. This function reads its values from the amplitude and frequency sliders.
Now a crucial function “mkSrPlot” gets attached to the “update—” method of the positionCurve object. This function is responsible for redrawing the curve, whenever its defining parameters get changed. It returns an object with a list of time values, and a corresponding list of values for X(t).
The main algorithm is spelled out by the function mkSrPlot. It first constructs a function that gives the deterministic part of the derivative:
deriv = Deriv(t,x) = SineCurve + BiStability,
Then a “stepper” function is constructed, by the call to Euler(deriv, tStep). A stepper function inputs the current point (t,x) and a noise sample, and outputs the next point (t’,x’). The Euler stepper maps (t,x) to (t + tStep, x + tStep * Deriv(t,x) + noiseSample).
The top level loop of the simulator is implemented by the function sdeLoop, which is passed the following arguments: (1) the stepper function, (2) the noise amplitude, or “dither”, (3) the initial point (t0,x0), (4) the randomization seed, and (5) the number of points to be generated. The loop initializes a currentPoint to (t0,x0), and then repeatedly applies the stepper to the current point and the next noise sample; the output returned is just this sequence of (t,x) values.
The noise samples are taken from a block of values from the (large) array normals(i), and scaling them by the dither value. The Sample-path variable controls which section of the array gets used.
How to post it.
Effect of frequency
Design a study of the effectiveness of signal transmission, as a function of noise amplitude and signal frequency. How you define the effectiveness measure?
How would you restructure the code for general, statistical studies of the output time series?
When the sliders are moved, an event must be fired, which causes the recalculation to take place. How is this mechanism implemented in the javascript / JSXGraph application library?
Modify to add an exponent slider
Modify to show graph of expected value (add slider for nTrials) (Not enough random numbers.)
Add a standard deviation plot
If you are a climate scientist, let us know of next steps
Begin to study this book —-, and think of how to write programs for some of the models. Simplify! The hierarchy of models. All models that you post here will be considered as candidates for the Azimuth Code Project page. This may be a way for programmers, ultimately, to give back to the Earth.