The Emergence of Numerical Weather Prediction:

Transcription

1 The Emergence of Numerical Weather Prediction: Fulfilment of a Dream & Realization of a Fantasy Peter Lynch School of Mathematics & Statistics University College Dublin Mathematics of Planet Earth Jamboree Reading University, March 2016.

2 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

3 23 March: World Meteorological Day WMO: Hotter, Drier, Wetter. Face the Future

4 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

5 A Recent Paper in Nature A recent review of NWP by scientists from ECMWF. Nature, 3 September 2015 Vol 525 p.47

6 The Quiet Revolution of NWP [Abstract] Advances in NWP represent a quiet revolution. Steady accumulation of technological advances. Among the greatest impacts of physical science. NWP is a computational problem comparable to: Modelling the behaviour of the human brain Simulating the evolution of the early universe.

7 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

8 Pioneers of Scientific Forecasting Cleveland Abbe, Vilhelm Bjerknes, Lewis Fry Richardson

9 Cleveland Abbe By 1890, the American meteorologist Cleveland Abbe had recognized that: Meteorology is essentially the application of hydrodynamics and thermodynamics to the atmosphere. Abbe proposed a mathematical approach to forecasting.

10 Vilhelm Bjerknes A more explicit analysis of weather prediction was undertaken by the Norwegian scientist Vilhelm Bjerknes He identified the two crucial components of a scientific forecasting system: Analysis Integration

11 Vilhelm Bjerknes ( )

12 Lewis Fry Richardson The English Quaker scientist Lewis Fry Richardson attempted a direct solution of the equations of motion. He dreamed that numerical forecasting would become a reality one day in the distant future.

13 Lewis Fry Richardson The English Quaker scientist Lewis Fry Richardson attempted a direct solution of the equations of motion. He dreamed that numerical forecasting would become a reality one day in the distant future. Today, forecasts are prepared routinely using his method... his dream has indeed come true.

14 Lewis Fry Richardson, During WW I, Richardson computed by hand the pressure change at a single point. It took him two years!

15 Lewis Fry Richardson, During WW I, Richardson computed by hand the pressure change at a single point. It took him two years! His forecast was a catastrophic failure: p = 145 hpa in 6 hrs But Richardson s method was scientifically sound.

16 Initialization of Richardson s Forecast Richardson s Forecast was repeated on a computer. The atmospheric observations for 20 May, 1910, were recovered from original sources. ORIGINAL: p s t = +145 hpa/6 h

17 Initialization of Richardson s Forecast Richardson s Forecast was repeated on a computer. The atmospheric observations for 20 May, 1910, were recovered from original sources. ORIGINAL: p s t = +145 hpa/6 h INITIALIZED: p s t = 0.9 hpa/6 h Observations: The barometer was steady!

18 Full Account of the Forecast Richardson s Forecast and the Emergence of NWP are described in this book. Cambridge Univ. Press, 2006 [Recently issued in paperback form]

19 Richardson s Forecast Factory c François Schuiten 64,000 Computers: the first Massively Parallel Processor

20 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

21 Crucial Advances, Dynamic Meteorology Quasi-geostrophic Theory Numerical Analysis CFL Criterion Atmopsheric Observations Radiosondes Electronic Computing ENIAC

22 The ENIAC

23 The ENIAC The ENIAC was the first multipurpose programmable electronic digital computer: 18,000 vacuum tubes 70,000 resistors 10,000 capacitors 6,000 switches Power: 140 kwatts

24 Charney Fjørtoft von Neumann Numerical integration of the barotropic vorticity equation Tellus, 2, (1950).

25 Charney, et al., Tellus, The atmosphere is treated as a single layer. The flow is assumed to be nondivergent. Absolute vorticity is conserved. d(ζ + f) dt = 0.

26 The ENIAC Algorithm: Flow-chart G. W. Platzman: The ENIAC Computations of 1950 Gateway to Numerical Weather Prediction (BAMS, April, 1979).

27 ENIAC Forecast for Jan 5, 1949 Intro Pioneers ENIAC NWP Today ECMWF Factory

28 NWP Operations The Joint Numerical Weather Prediction Unit was established on July 1, 1954: Air Weather Service of US Air Force The US Weather Bureau The Naval Weather Service. Operational numerical weather forecasting began in May 1955 using a 3-level quasi-geostrophic model.

29 An Order of Magnitude every 5 Years

30 An Order of Magnitude every 5 Years

31 Forecasts by PHONIAC Peter Lynch & Owen Lynch A modern hand-held mobile phone has far greater power than the ENIAC had. We therefore decided to repeat the ENIAC integrations using a programmable mobile phone. We wrote a program PHONIAC.JAR, a J2ME application, and implemented it on a mobile phone.

32 PHONIAC: Portable Hand Operated Numerical Integrator and Computer

33 Notices of the AMS ISSN (print) ISSN (online) of the American Mathematical Society Volume 60, Number 8 September 2013 Forecasts by PHONIAC: Weather, Nov The Calculus Concept Inventory Measurement of the Effect of Teaching Methodology in Mathematics Cover of Sept Notices of the American Mathematical Society. page 1018 DML-CZ: The Experience of a MediumSized Digital Mathematics Library page 1028 Fingerprint Databases for Theorems page 1034 A History of the Arf-Kervaire Invariant Problem page 1040 About the cover: 63 years since ENIAC broke the ice (see page 1113) This technology has great potential for generation and delivery of operational weather forecast products. Intro Pioneers ENIAC NWP Today ECMWF Factory

34 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

35 Princeton Companion to Applied Maths Article available on my website

36 Reasons for Progress in Weather Forecasting Enhancements in model resolution; Faster computers; Better numerical schemes; More comprehensive physical processes; New observational data from satellites; More sophisticated methods of data assimilation; Paradigm shift to probabilistic forecasting.

37 The Equations of the Atmosphere GAS LAW (Boyle s Law and Charles Law.) Relates the pressure, temperature and density CONTINUITY EQUATION Conservation of mass WATER CONTINUITY EQUATION Conservation of water (liquid, solid and gas) EQUATIONS OF MOTION: Navier-Stokes Equations Describe how the change of velocity is determined by the pressure gradient, Coriolis force and friction THERMODYNAMIC EQUATION Determines changes of temperature due to heating or cooling, compression or rarefaction, etc. Seven equations; seven variables (u, v, w, ρ, p, T, q).

38 Seven equations; seven variables (u, v, w, p, T, ρ, ρ ). The Primitive Equations du dt dv dt + ρ w t ( f + u tan φ ) v + 1 a ρ ( f + u tan φ a dt dt p x + F x = 0 ) u + 1 p ρ y + F y = 0 p z + gρ = 0 p = RρT + (γ 1)T V = Q c p ρ t + ρv = 0 + ρ w V = [Sources Sinks]

39 Scientific Forecasting in a Nut-Shell The atmosphere is a physical system Its behaviour is governed by the laws of physics These laws are expressed quantitatively in the form of mathematical equations Using observations, we can specify the atmospheric state at a given initial time: Today s Weather Using the equations, we can calculate how this state will change over time: Tomorrow s Weather

40 Scientific Forecasting in a Nut-Shell Problems: The equations are very complicated (non-linear): Powerful computer required to solve them. The accuracy decreases as the range increases; There is an inherent limit of predictibility.

41 Physical Processes in the Atmosphere

42 Time stepping schemes Replace continuous time by {0, t, 2 t,..., n t}: dq dt = F(Q). Approximate the time derivative by Q n+1 Q n 1 2 t = F n, Compute the forecast value Q n+1 from Q n+1 = Q n t F n. Repeat until desired forecast range is reached.

43 Spatial finite differencing Consider the simple 1D wave equation Q t + c Q x = 0, Centered differences in space and time: ( Qm n+1 Qm n 1 Q n + c m+1 Q n ) m 1 = 0, 2 t 2 x Condition for stability of the solution: c t x 1. This is the Courant-Friedrichs-Lewy criterion (1928).

44 Spectral method Fields expanded in series of spherical harmonics: Q(λ, φ, t) = N n Qn m (t)yn m (λ, φ), n=0 m= n Coefficients Q m n (t) depend only on time. Fourier analysis on the sphere. The model partial differential equations become a coupled set of nonlinear ODEs for Q m n (t).

45 Variational assimilation The model state is a high-dimensional vector X. The cost function for 3D-Var is J = J B + J O. Background error term (Distance from first guess): J B = 1 2 (X X B) T B 1 (X X B ) Observational error term (Distance from obs): J O = 1 2 (Y HX)T R 1 (Y HX)

46 The minimum of J is attained at X = X A where X J = 0. Computing this gradient, we get X J = B 1 (X X B ) + H T R 1 (Y HX).

47 The minimum of J is attained at X = X A where X J = 0. Computing this gradient, we get X J = B 1 (X X B ) + H T R 1 (Y HX). Setting this to zero we can deduce the expression X = X B + K[Y HX B ] where the gain matrix, is given by K = BH T (R + HBH T ) 1.

48 Including the Time Dimension Satellite data are distributed continuously in time. Four-dimensional variational assimilation (4D-Var) uses all the observations in an interval t 0 t t N.

49 Including the Time Dimension Satellite data are distributed continuously in time. Four-dimensional variational assimilation (4D-Var) uses all the observations in an interval t 0 t t N. The cost function now includes terms measuring the distance to observations at each time step t n : J = J B + where J O (t n ) is given by N J O (t n ) n=0 J O (t n ) = (Y n H n X n ) T R 1 n (Y n H n X n )

50 Figure : 4D Variational Assimilation Scheme

51 Some Technicalities State vector X n at time t n generated by integrating the forecast model: X n = M n (X 0 ). Model operator M n linearized about trajectory from background field gives tangent linear model M n. Minimization in 4D-Var involves transposition. The transpose of the tangent linear model, M T n, is called the adjoint model. The control variable for the minimization of the cost function is X 0. Model used as a strong constraint.

52 Benefits of 4D-Var In OI and 3D-Var, all observations within a fixed time window typically of six hours assumed valid at the analysis time. 4D-Var finds initial conditions X 0 such that forecast best fits the observations within the assimilation window. At ECMWF, 4D-Var has led to substantial improvements in operational forecasts.

53 Growth in Forecast Skill Figure : Anomaly correlation of 500 hpa geopotential height

54 Operational Forecasting: Suite of Models Operational forecasting based on output from a suite of computer models. Global models are used for predictions of several days ahead Shorter-range forecasts are based on regional or limited-area models. At many European NMSs: Short Range (LAM): HARMONIE Model Medium Range (GLOBAL): ECMWF Model

55 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

56 European Centre for Medium-Range Weather Forecasts (ECMWF, Reading, UK) As an example of a global model, we consider the Integrated Forecast System (IFS) of ECMWF.

57 Forecast of Hurricane Sandy Figure : Landfall, New Jersey, 30 October 2012

58 Resolution of the IFS System

59 Resolution of IFS System: 10 March 2016

60 Table : ECMWF IFS Resolution Spectral Truncation T511 T799 T1279 T CO 1279 Effective Resolution 39 km 25 km 16 km 10 km Model Levels

61 Resolution of the IFS System

62 Some Details The basis of ECMWF operations is the IFS. Spectral representation of meteorological fields. The ENS system runs with a horizontal resolution half that of the deterministic model. Model has about a billion degrees of freedom. The computational task is formidable. The Centre has a Cray XC30 High Performance Computer, comprising some 160,000 processors, Sustained performance of over 200 Teraflops.

63 The OpenIFS Project

64

65 Outline Introduction Pioneers of NWP: The Dream ENIAC Integrations NWP Today ECMWF IFS System Forecast Factory: The Fantasy

66 Richardson s Forecast Factory c Stephen Conlin, 1986

67 Zoom: Richardson Directing the Forecast Lewis Fry Richardson conducting the forecast

68 Zoom: Historical Figures in Computing Napier / Babbage / Pascal / Peurbach

69 Key to the Historical Figures

70 Table of Historical Figures

71 Zoom: Communications & Computing The Tube Room

72 Zoom: Communications & Computing The Computer Laboratory

73 Zoom: Experimentation & Research Dish Pan Experiment (G. I. Taylor presiding)

74 Zoom: Experimentation & Research Babbage s Analytical Engine

75 Richardson s Forecast Factory 64,000 Computers: the first Massively Parallel Processor

76 The Fantastic Forecast Factory An Artist s Impression of Richardson s Fantastic Forecast Factory. Weather, 71, [Reprint on my website] High-res Image with Zoom on website of European Meteorological Society:

77 Thank you

78 Growth in Forecast Skill Figure : Anomaly correlation of 500 hpa geopotential height