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ECMWF Introduction to Chaos by Tim Palmer

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ECMWF Introduction to chaos for: Weather prediction

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ECMWF Wilhelm Bjerknes ( ) Proposed weather forecasting as a deterministic initial value problem based on the laws of physics

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ECMWF Lewis Fry Richardson ( ) The first numerical weather forecast

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ECMWF

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Why have meteorologists such difficulty in predicting the weather with any certainty? Why is it that showers and even storms seem to come by chance... a tenth of a degree (C) more or less at any given point, and the cyclone will burst here and not there, and extend its ravages over districts that it would otherwise have spared. If (the meteorologists) had been aware of this tenth of a degree, they could have known (about the cyclone) beforehand, but the observations were neither sufficiently comprehensive nor sufficiently precise, and that is the reason why it all seems due to the intervention of chance Poincaré, 1909

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ECMWF Edward Lorenz (1917 – ) … one flap of a sea-gulls wing may forever change the future course of the weather (Lorenz, 1963)

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The Lorenz (1963) attractor, the prototype chaotic model…..

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ECMWF

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Isopleth of initial pdf Isopleth of forecast pdf Nb M * M symmetric operator therefore eigenvectors form a complete orthogonal basis Tangent propagator

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ECMWF Because M is not a normal operator (M*M MM *), Ms eigenmodes are not orthogonal Substantial perturbation growth is possible over finite time, even if M has no growing eigenmodes

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ECMWF Lothar: 08Z, 26 Dec Fatalities 400 million trees blown down 3.5 million electricity users affected for 20 days 3 million people without water Lothar +Martin

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Ensemble Initial Conditions 24 December 1999

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ECMWF Lothar (T+42 hours)

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ECMWF Thanks to Rob Hine Headlines: Oct

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Frost free Frosty Charlie loses L if concrete freezes. But Charlie has fixed (eg staff) costs There may be a penalty for late completion of this job. By delaying completion of this job, he will miss out on other jobs. These cost C Is Lp>C? If p > C/L dont lay concrete! Charlie is planning to lay concrete tomorrow. Consensus forecast: frost free YES Should he? NO ? Let p denote the probability of frost

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ECMWF Value of EPS over high-res deterministic forecast for financial weather-derivative trading based on Heathrow temperature (Roulston and Smith, London School of Economics, 2003)

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ECMWF Introduction to chaos for: Seasonal climate prediction Atmospheric predictability arises from slow variations in lower-boundary forcing

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ECMWF Edward Lorenz (1917 – ) What is the impact of f on the attractor?

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ECMWF The influence of f on the state vector probability function is itself predictable. f=0 f=2 f=3 f=4 Add external steady forcing f to the Lorenz (1963) equations

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ECMWF

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The tropical Pacific ocean/atmosphere

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ECMWF Introduction to chaos for: Stochastic parametrisation

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ECMWF Eg 2) Lorenz(1963) in an EOF basis 3 rd EOF only explains 4% of variance (Selten, 1995). Parametrise it?

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ECMWF Lorenz(1963) in a truncated EOF basis with parametrisation of a 3 Good as a short-range forecast model (using L63 as truth), but exhibits major systematic errors compared with L63, as, by Poincaré-Bendixon theorem, the system cannot exhibit chaotic variability – system collapses onto a point attractor.

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ECMWF Stochastic-Lorenz(1963) in a truncated EOF basis Stochastic noise

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ECMWF Lorenz attractor Truncated Stochastic- Lorenz attractor –weak noise Truncated Stochastic- Lorenz attractor Error in mean and variance Palmer, 2001 (acknowledgment to Frank Selten)

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ECMWF He believed in the primacy of doubt; not as a blemish upon our ability to know, but as the essence of knowing Gleick (1992) on Richard Feynmans philosophy of science.

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