ThatsMaths

The Earth’s climate is changing, and the consequences may be very grave. This week, That’s Maths in The Irish Times ( TM040  ) is about computer models for simulating and predicting the future climate.

Liffey Bursts its Banks: St. Stephen’s Green Flooded Again

The above is an improbable but not entirely impossible future headline. Sea levels are rising steadily and, if the polar ice-caps melt, the results could be catastrophic. The changing climate is now beyond doubt. A vast body of solid scientific evidence shows how human activities, mainly burning of coal and oil, are warming the Earth.

Of course, there is always room for debate and considered dissent in science. But most of those who try to rebut or discredit the consensus view are lacking objectivity. There are frequent attacks on climate science by groups with vested interests in maintaining the status quo. The resulting confusion is closely analogous to the obfuscation of the link between smoking and cancer stirred up by the tobacco industry.

The atmosphere is measured and monitored in minute detail by a global observing network, so we have excellent knowledge of what has occurred and what is happening now to the climate. But what of the future? We have no observations for 2050 or 2100. Instead, we must simulate future climate change using computer models of the atmosphere and ocean.

Two recent reports on future climate conditions in Ireland [both available online at Met Eireann’s website: http://www.met.ie/publications/ ]

The climate is governed by well-understood physical laws, including Newton’s laws of motion, conservation of mass and energy and Boyle’s Law relating pressure and volume. The laws are expressed in terms of mathematical equations. These equations can be solved, starting from today’s climate and including factors like increasing CO₂ which alter the temperature.

The Primitive Equations

The panel below shows the primitive equations. The coordinates (x,y,z) are distances eastward, northwards and upward on the sphere, the wind components are (u,v,w), pressure, temperature and density are p, T and ρ, and  ρ_w represents humidity. The first two equations are horizontal components of Newton’s equation of motion. The next two are the gas law and the equation of hydrostatic balance. Then come the thermodynamic equation (conservation of energy) and continuity equation (conservation of mass), and finally conservation of water substance.

The Primitive Equations: a coupled system of non-linear partial differential equations that govern the dynamics of the atmosphere.

The equations are of a type that mathematicians call nonlinear partial differential equations. They are reduced to a manageable form by a process called discretisation: we replace continuous fields like temperature by values at a finite set of points called a computational grid. In essence, this reduces a ‘hard’ problem in calculus to an ‘easy’ problem in algebra.

The algebraic problem can be solved using a computer. The number of unknown quantities is vast and the most powerful computer machinery is required. The result of the calculation is a description of climate variables at a future time. Rising temperatures and rising sea-levels are consistently found, although the time-scales vary from model to model. The seasonal and regional variations and the patterns of extreme weather events are more problematical, and there are significant disagreements between different simulations.

Uncertainties

It is important to recognise that there are substantial uncertainties at every stage. To address these, we run not one but several simulations of future climate. This collection of predictions is called an “ensemble”. The spread of the ensemble gives us an indication of the range of possibilities and a measure of the probability of any particular outcome.

Although computer simulations inevitably involve errors and all climate projections are uncertain, they are the best means we have of anticipating future conditions. To those who deny that they have any validity, I can only urge them to enter into dialogue with the astrological community. Both groups ignore scientific reality and both appear to be more concerned with financial gain than with seeking the truth.

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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/