Sunday, July 26, 2009

Weather Forecasting - By Numbers

One of the great modern breakthroughs in the study of the weather and climate of our planet has been the developing ability to simulate the behaviour of our atmosphere through mathematics.

The use of mathematics to predict the future is called an “initial value” process - that is if we know the state of a physical system at a particular time, and we know the mathematical equations that govern its motion, then we can predict the future state of the system by “solving” the equations.

A simple example that we are all familiar with is as follows. Let’s say we’re in a car moving down a freeway at 100 kph. We want to know where the car will be after 2 hours. Obviously the answer is 200 km away, but let’s step back and see how we arrived with this solution.

We have, perhaps unknowingly, used a mathematical formula


which in words says “distance travelled equals speed multiplied by time”

In this case S = 100 and T=2, so to obtain our answer we use the process of multiplication.

That is D = 100x2, giving us our answer of 200 km.

We have therefore been able to predict the future position of our car by knowing

(1) The initial conditions – the car is travelling at 100 kph
(2) A mathematical equation that describes the motion - D=SxT
(3) A method of “solving” the equation – in this case the process of multiplication.

In any sort of mathematical prediction of the future, we must be able to meet these conditions – the “Big Three”. The principles involved in numerical weather and climate prediction are just the same, although vastly more complicated. By knowing the three conditions we can predict the future state of the atmosphere and so produce a weather forecast.

The first scientist to suggest that weather forecasting could be considered an initial value problem was the Norwegian Vilhelm Bjerknes whose work in the early 20th century generated great interest in the issue. He developed a set of five mathematical equations (called the primitive equations) that describe atmospheric motion and suggested that if we could accurately describe the state of the atmosphere today, a solution of these equations could tell us how the atmosphere would look tomorrow. This process became known as numerical weather prediction (NWP).

Whilst only specialist mathematicians understand in detail what these equations mean, we can gain at least some insight into the complexities involved just by looking at them and the symbolism used

(1) dV/dt= -fkxV –∆ø +F + g

In words this means that motion in the atmosphere is the result of the spin of the Earth (the coriolis force), air pressure differences, friction and gravity.

The other four equations describe other conditions that must be met, including constraints on heat, moisture and air density. In symbolic form these are written

(2) ∂ø/∂P=0

(3) dµ/dt=-µ∆.v

(4) dT(p)/dt=T(p)Q/TCp

(5) dr/dt=Fr - Ω

This is known as a coupled system of non linear partial differential equations and forms one of the most intractable and difficult of all mathematical problems. It was only centuries of effort by mathematicians from all around the world that eventually produced ways of handling systems of equations of this type.

During the first World War, NWP was taken a great deal further by the English scientist Lewis Fry Richardson, an eccentric genius, who actually devised a system that performed this process. However the computational workload was so great that it took him around 6 weeks to prepare a forecast for the next day – an obviously impractical process!

Lewis Fry Richardson - an eccentric genius and father of numerical weather prediction
Image: Wikipedia Commons
(Click on image to enlarge)

Richardson suggested that to overcome this problem the weather office of the future should consist of a vast amphitheatre that held about 64,000 mathematicians, each responsible for a single calculation that applied to a small area of the Earth’s surface. Working together, and directed like an orchestra with a lead mathematician as conductor, it was thought that a global forecast could then be prepared within a useful time frame.

Incredibly, Richardson performed much of his research not from within the closeted confines of a university, but on the battlefields of World War One. A Quaker, he would not take up arms, but volunteered his support as an ambulance driver, and spent any spare time working on his thesis at night by candlelight within ruined farmhouses amid the mud and carnage of the Western Front.

His ideas were published in 1922 in a book called “Weather Prediction by Numerical Process”, which was really the first textbook ever written on numerical weather prediction. Warmly received in academic circles it was, however, thought to be an impractical way of predicting the weather because of three main problems that were directly related to the “Big Three” conditions.

These were

(1) Tremendous difficulties in rapidly gathering together weather observations from all around the world to describe the current state, or initial conditions, of the atmosphere. This was needed to provide the "starting point" on which all future calculations would be based.

(2) The mathematical equations describing the motion of the atmosphere were known but were highly complicated.

(3) The equations could not be solved directly, necessitating the use of what mathematicians call an “iterative process”, that employs multiple calculation repetitions that converge towards a solution. This produces a colossal computational workload and even Richardson’s idea of a “Forecasting Orchestra” was thought to be inadequate for this problem.

The idea of numerical weather prediction then lay dormant for about the next 20 years, until a surprising turn of events resulted in a rebirth of interest.

During the early 1940’s, the eminent mathematician John von Neumann had been recruited to work on Project Manhattan, the construction and testing of the world’s first atomic explosion, codenamed “Trinity” that finally took place on July 16 1945.

In the lead up to “Trinity”, von Neumann and his co-workers, a galaxy of scientific stars from all around the world, realised that they could not cope with the computational workload required. The scientists of the day used slide rules and mechanical calculators – resembling cash registers – to perform all their mathematical work, and even in the hands of skilled operators, these devices were too slow for the workload required.

To get round this problem they managed to construct one of the very first electronic computers by connecting several IBM business machines, including a tabulator, multiplier, collator and sorter. To von Neumann’s delight, the resulting computational speed was much faster than the slide rules and mechanical calculators, and the scientists realised that this represented almost as big of a breakthrough as the atomic bomb itself.

Von Neumann was well acquainted with Richardson’s attempts at numerical weather prediction and realised that with electronic computers likely to become faster and faster that the computational workload required was going to become manageable.

The other main stumbling block – being able to rapidly communicate weather observations from around the world to a central point at high speed was another problem that was showing signs of becoming increasingly tractable. Great strides in methods of international communication had occurred since the days of Richardson in the early 1920’s.

Soon after the war, in 1946, Von Neumann called a major meeting at the Institute of Advanced Studies at Princeton University to review the concept of numerical weather prediction and Richardson’s work of two decades before was “much discussed”.

ENIAC - an early electronic computer that became operational in 1946.
ENIAC stood for Electronic Numerical Integrator and Computer
Image: Wikipedia Commons
(Click on image to enlarge)

From that point on NWP rapidly expanded into one of the most useful tools at the meteorologist’s disposal. As computers became faster and more powerful, communication speeds increased and more weather observation stations came “online” around the world, weather simulations produced by numerical processing became increasingly accurate and today are an indispensable part of the weather forecasting process.

Massive NASA supercomputer.
Calculating power of this scale is necessary to produce accurate weather forecasts out to week ahead for any location around the world.
Image: Wikipedia Commons
(Click on image to enlarge)

Mathematical “models” as they are called, are now routinely used as part of the forecasting process by national meteorological services around the world, to produce forecasts of useful accuracy that typically run out to 7 days ahead.

Richardson's dream has been fulfilled in the most complete manner and the use of numerical weather prediction now forms one of the cornerstones of meteorology. Well ahead of his time, he received little contemporary recognition for his work but he is now acknowledged as one of the key figures in the history of weather forecasting.

Reference: “Understanding Climate Change”, Richard Whitaker, New Holland Publishers (Australia) 2008.

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