Sensitivity to Initial Conditions: Characterizing the Accuracy of the Weather Forecast

Kate Snow

AMSI Summer Project



The atmosphere is a forced dissipative system, the fluid of the atmosphere is forced into motion by thermal heating while thermal and mechanical dissipative forces act on it. It is hence described by physical laws derived from fluid dynamical and thermo dynamical equations. These equations need to take into account movement of air such as convection, heating and cooling, moisture, gravitational effect, interaction between the air and oceans, etc. Equations for such a purpose as weather forecasting which must incorporate many of these factors then are inherently complex, too complex for a suitable project to be based upon them in the time frame. Hence a simplified model using the Lorenz equations will be considered.

The problem of forecasting the weather was identified by Bjerknes as the problem of solving the equations representing the basics laws governing the atmosphere using initial conditions obtained from current observations of the weather [1]. The first numerical solution was produced by Richardson in 1922 in his work "Weather Prediction by Numerical Process" however practical solutions could not be obtained until the development of computer technology. One of the first computer simulations of an idealized set of ordinary differential equations describing the weather was by Edward Lorenz, one of the first pioneers of chaos theory [2], was his computer the Royal McBee.

Lorenz, an American mathematician and meteorologist, started out his career as a weather forecaster for the Army Air Corps during WWII, after which he was able to focus more on the mathematical theories of meteorology. At the time, weather forecasting was considered barely more than guesswork, yet Newtonian theory led to the idea of the universe and hence the weather being deterministic. This led to the belief that if approximate initial conditions could be found for the weather system, then the approximate behavior of the system could be determined with the appropriate defining laws. Lorenz sought to find a system of equations simple enough to simulate a suitably long period of weather forecast while maintaining a reasonable amount of computation. Finally he settled upon a system of twelve equations relating temperature, pressure and wind speed with aperiodic solutions. This system was input into Lorenz's Royal McBee computer to produce a numerical weather prediction. While his predictions illustrated trends over time, there were never exact repetition. "There was pattern, with disturbances. An orderly disorder" [2]. This illustrated the chaotic nature of they system.

The real breakthrough came when Lorenz created graphical printouts of his system (taking approximately one second per iteration [3]) and taking a shortcut of an output previously produced once before, reduced the number of decimal places being used for the initial conditions thinking one part in a thousand inconsequential [2]. Yet the results were far from equivalent, while initially the results seemed almost the same, as time progressed all resemblance between the two graphs was lost. Small errors caused dramatic changes in the non-periodic system and this, Lorenz discovered, was the reason long-term forecasting would be so unpredictable.

With his discovery, Lorenz turned his attention to the mathematics of the aperiodic system. He was able to reason that aperiodicity and sensitive dependence on initial conditions was necessary for the system to be unpredictable. This is seen since the variables of his system were bounded in a limited range hence near repetitions of previous values were unavoidable. If the system had been stable, then near repetitions of values would produce similar outputs giving periodic behavior. However this was not the case illustrating that the instability of the system lead to the lack of periodicity.

Lorenz wished to better appreciate the problem by simplifying his set of equations further. He was unable to do this until he discovered a set of seven ordinary differential equations on thermal convection by Dr. Barry Saltzman. Non-periodic solutions occurred for a set of conditions in which four of the variables approached zero, allowing Lorenz to simplify the equations into the remaining three variables. These equations are the famous Lorenz equations first appearing in his paper "Deterministic Nonperiodic Flow," Journal of the Atmospheric Sciences 20(1963), pp. 130-41. The importance of these equations was the discovery that relatively simple systems could display complex and chaotic behavior. The chaotic aspect of this system demonstrates the property that small perturbations in the initial conditions will be amplified to great extents as time progresses so that sufficiently advanced states of the system are become unpredictable, this is the Butterfly Effect. This in turn led to the important discovery that other chaotic systems such at the weather and atmosphere will be inherently unpredictable over larger periods as initial conditions will never be obtained to exact degrees of accuracy. Lorenz's paper "Deterministic Nonperiodic Flow" also paved the way for the now popular and fast growing area of chaos theory with applications ranging from circuit theory to fractal theory.


  1. Lorenz, E. 1979. On the Prevalence of Aperiodicity in Simple Systems. Springer-Verlag, Berlin; New York.
  2. Gleick, J. 1987. Chaos; Making a New Science, Penguin Books, New York.
  3. Lorenz, E. Deterministic Nonperiodic Flow. 1963. Journal of the Atmospheric Sciences, 20, pp. 130-40.

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