The Chaos in Weather and the Lorenz Model Using Mathematica Software.

This PPT describes the basics of Lorenz model and chaos in weather and the simulation o Lorenz attractor using Wolfram Mathematica software

1. Chaos in weather : the Lorenz MODEL PRESENTED BY : RUFOS S SURESH DEPARTMENT OF PHYSICS St ALOSIYOUS COLLEGE ,JABALPUR

2. CONTENTS DETERMINISTIC SYSTEM DISCOVERY OF CHAOS INTRODUCTION TO CHAOS CHAOS IN WEATHER LORENZ EQUATION LORENZ ATTRACTOR CHAOS IN REAL WORLD (EXAMPLES) CONCLUSION

3. DETERMINISTIC SYSTEM In mathematics and physics, a deterministic system is a system in which no randomness is involved in the development of future state of system A deterministic model will always produce the same output from a given starting condition or initial state or initial condition Example: Most of the basic laws of nature are deterministic. They allow us to determine what will happen next from the present condition Predicting the amount of money in a bank account

4. DISCOVERY OF CHAOS The first true experimenter in chaos was a metrologist, Edward Lorenz, who in 1960 discovered it while working on the problem of weather prediction However, the term “ chaos ” was introduced by Tien- Yien and James A

5. INTRODUCTION TO CHAOS The unpredictable behaviour of a Deterministic system is called as chaos chaos is often marked by small changes at the start leading to big differences later (Sensitive to initial conditions)

6. Sensitivity to initial condition In deterministic system the output pattern of the motion or representation remains the same for different initial conditions In chaotic system the output pattern will change for different initial condition The figure shows the phase portraits of a chaotic system on changing the initial condition

7. Chaos in weather

8. Contd… The above figure shows three different weather forecasts, all started from very similar initial conditions . The differences among the three initial conditions were smaller than estimated analysis errors, and each of the three initial conditions could be considered as an equally probable estimate of the “true” initial state of the atmosphere. After 5 days, the three forecasts evolved into very different atmospheric situations. The first forecast indicated two areas of weak cyclonic circulation west and south. The second forecast positioned a more intense cyclone southwest. The third forecast kept the cyclone in the open seas. The fourth is a typical example of orbits initially close together and then diverging during time evolutions.

9. Contd… The atmosphere is an intricate dynamical system with many degrees of freedom. The state of the atmosphere is described by the spatial distribution of wind, temperature, and other weather variables The mathematical differential equations describing the system time evolution include Newton's laws of motion used in the form ‘acceleration equals force divided by mass’, and the laws of thermodynamics which describe the behavior of temperature and the other weather variables. There is a set of differential equations that describe the weather evolution, at least, in an approximate form. These set of differential equations is known as Lorenz equation

10. Lorenz equation In March 1963, Lorenz wrote that he wanted to introduce, “ordinary differential equations. Whose solutions afford the simplest example of deterministic non periodic flow and finite amplitude convection. In his paper, he examines the work of meteorologist Barry Saltzman and physicist John Rayleigh while incorporating several physical phenomena (Bradley, Viswanath ). Lorenz found that when applying the Fourier Series to one of Rayleigh’s convection equations that , all except three variables tended to zero, and that these three variables underwent irregular, apparently non periodic functions. He then used these variables to construct a simple model based on the 2-dimensional representation of the earth’s atmosphere this is known as Lorenz equation

11. Contd…. Here x, y, z do not refer to coordinates in space . In fact, x represents the convective overturning turning on the plane, while y and z are the horizontal and vertical temperature variation respectively . The parameters of this model are σ , which represents the Prandtl number , or the ratio between the fluid viscosity to its thermal conductivity, ρ , which represents the difference in temperature between the top and bottom of the atmosphere plane , and β, which is the ratio of the width to the height of the plane Lorenz found the values of σ = 10 and β = 8/3 , and initial conditions of (x0, y0, z0) = (0, 1, 0) to be the best representation of the earth's atmosphere . For this project, we assume these values

12. Contd… While Lorenz found chaos to be a large factor in meteorology, the equation he created does not exhibit chaos for all parameters . In fact, there are many parameter values where the function is stable and contain fixed points . We will now explore how Lorenz came to realize fixed points of his system, as well as for what values the equation exhibits chaos . ρ Fixed point [0-1] (0,0,0) (24.74-30.1) None, Chaos Occur 30.1-∞ Not proven

13. LORENZ ATTRACTOR We see from the above figure that the trajectories around one lobe of the attractor. After the number of revolutions around one lobe of the attractor the trajectory then switches to the other lobe. It then spirals around the other lobe before switching back to the first. The number of revolutions that the trajectory will make around each lobe , before returning to the other is unpredictable.

14. CHAOS IN REAL WORLD DISEASE :- An outbreak of deadly disease which has no cure. POLITICAL UNREST :- Can cause revolt ,overthrow of government and vast war WAR :- Lives of many people can be ruined in no time STOCK MARKET CHECMICAL REACTIONS

15. CONCLUSION The Lorenz model, with its simple yet profound representation of chaotic systems, has had a transformative impact across a wide range of fields . From improving weather and climate predictions to enhancing the design of engineering systems, understanding economic dynamics, and advancing medical research, the applications of the Lorenz model are vast and varied . Its contribution to chaos theory has deepened our understanding of complex systems and the unpredictable nature of the world around us . As we continue to explore and apply the principles of the Lorenz model, we can expect to unlock new insights and innovations that will shape the future of science and technology .