Newton s Second Law of Motion. Isaac Newton (in 1689) The Universe is Regular and Predictable. The Foundation of Science

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1 Key Concepts and Fundamental Theorems Isaac Newton (in 689) of Atmospheric Science This is the earliest portrait of Newton to survive. The artist was Godfrey Kneller, perhaps the greatest portrait painter of his day. Part III Atmospheric Predictability Newton was 46 years old and Principia had been published two years previously, even though it had been completed before his 5th birthday. John A. Dutton Meteo 485 Spring 4 The Foundation of Science And because the universe is regular and predictable Assumption or Principle Sufficient study of a physical process will reveal the laws that govern its evolution The Universe is Regular and Predictable We can then make predictions about its future The Three-Body Problem Newton s Second Law of Motion Asteroid Jupiter Force equals mass times acceleration (F = ma) m dv =F dt dx =v dt Oscar II King of Sweden and Norway Sun Gösta Mittag-Leffler Acta Mathematica 885

2 Edvard Phragmén Henri Poincaré noticed some problems with the prize proof. Einstein s Clocks, Poincaré s Maps Peter Galison,, Norton Poincaré soon understood that.something was deeply wrong with his work. Chaos in the Universe The years go by The theory of relativity World War I The great depression World War II Henri Poincaré I shall not even try to draw it. The atomic bomb Computers A New Era Begins ( t + u g H )u g + Ro( + λ)w u g z = k u' π ' z θ ' = ( t + u )θ '+ w'σ = Q' g H H u'+ + λ ρ z (ρ w') βv = g u g = k H π ' Jule Charney John von Neumann Copyright Carl Rossby John A. Dutton 4 Electronic Numerical Integrator and Calculator ENIAC First numerical weather forecast--march 95 4 hours Quasi-geostgrophic barotropic vorticity equation

3 A Rotating Fluid in Action A Rotating Fluid in (Numerical) Action Pierre Welander, Tellus,955 Pierre Welander, Tellus,955 And then the trouble begins an old crisis reappears The general circulation of the atmosphere: a numerical experiment Norman A. Phillips QJRMS 956 The Lorenz Model A Simple Model of Convection dx/dt = - P (X - Y) dy/dt = - XZ + R X - Y.5 Temperature.5 T = Y(t) cos x sin z Z(t) sin(z) P - Prandtl number (ratio of the fluid viscosity to its thermal conductivity) R - temperature difference between the top and bottom of the system B - ratio of width to height of the box used to hold the system. 6.5 Stream function.5 ψ ( x, y,t) = X(t) sin x sin z dz/dt = XY - BY The values Lorenz used are P =, R = 8, B = 8/

4 Integrating the Spectral Model of Convection Sensitivity to Initial Conditions Dr. Lorenz ran two simulations One simulation produced the value.5667 at the halfway point and continued 499 Started next simulation from that point but used.56 Edward N. Lorenz MIT The solutions diverged rapidly The Lorenz Attractor Chaos Theory 96 Y-Z plane Edward N Lorenz Professor of Meteorology Massachusetts Institute of Technology X-Z plane To the Demo was the first to recognize that the atmosphere is a chaotic system A New View of Science A New Assumption or Principle: Part of the universe may be regular and predictable, but part of it is chaotic and only predictable for limited periods 4

5 Performance of Several Forecast Systems Although the study of chaos and predictability lead to very profound mathematics chaos really is a child of the computer age and it is computer simulation and prediction that make it intensely relevant. Years of Forecast Improvement Error Growth in the ECMWF Forecast System Dec 99 - Feb 994 Error in 5 mb height (gpm)^ Simmons, A.J., R. Mureau, T. Petroliagis Error growth and estimates of predictability from the ECMWF Forecasting System QJRMS Figure 5c for Dec 99-Feb 994 Error Doubling Time-ECMWF Dec99 - Feb 994 Loss of Predictability in Nonlinear Systems WN 4 8 Information flows from right to left in the numbers DAYS a^n b^n c^n Days to Double WN 4 Mean WN WN 8 y = (x/.87) _8_ x.xxxxxxxx a = b = a(.) c = a(.99) Days of Integration Steps: Every minutes for 5 days

6 Distribution of Atmospheric Energy by Wavenumber - -5/ Slopes of Spectra -- Turbulence KE= E( κ )dκ E(κ ) = m / s / m If the flux of energy depends only the dissipation rate ε E = const ε a κ b (s) : a = / (m) : b = ( / ) = 5 / ε = m / s s An Implication of the Quasi-Geostrophic Invariants E M,N = λ n a n = const F M,N = λ n a n = const λ n < λ n+ n= M E M, λ M λ a n n n= M n= M = F M, F (), as M λ M M λ M F, M = λ n a n λ M λ n a n = λ M E, M n= E M, E, M <= λ M λ M F M, F, M M n= Forbidden Possible Slopes of Spectra -- Quasi-Geostrophic Flow KE= E( κ )dκ E(κ ) = m / s / m If the flux of energy depends only the rate η of enstrophy flux past wavenumber κ η = s E = const η a κ b (s) : a = / (m) : b = ( / ) = Implications of Spectral Slopes Stability in Fluid Flow Quasi-Geostrophic (-D) Flow -D Turbulence E(κ ) ~ η / κ V L ~ η / L V ~ η / L T ~ η / A ~ η / L A V ~ η/ = const E(κ ) ~ ε / κ 5/ V L ~ ε / L 5/ V ~ ε / L / T ~ ε / L / A ~ ε / L / A V ~ ε/ L / Intensity of Forcing Chaos, Turbulence Periodic Flow Symmetric Steady Flow No Motion Structural Parameter (Rotation, Prandtl number, ) 6

7 The Third Fundamental Theorem of Atmospheric Science THEOREM (Conjectured). For sufficiently strong solar heating I > I * and sufficiently weak dissipation and thermal conductivity µ < µ*,κ <κ * large-scale atmospheric flow will be chaotic with a maximum predictability period P of days. P = P(I, µ,κ ) Proof. Start by defining defining predictability period and then ascertain what is actually sufficient. Long-Range Prediction If useful predictability of weather events is restricted to time periods of two weeks or less, then how we can expect to succeed at prediction of seasonal anomalies?. We predict averages or average anomalies rather than events. (Here s a forecast:the average temperature in State College in July in 7 will be 7.9 F. We try to take advantage of parts of the system that have greater predictability than weather events. Perhaps a prize awaits you. Active Thermal Mass A Fundamental Theorem of Earth System Science..8 Days Years Ocean+Land Total THEOREM (Conjectured). For sufficiently strong solar heating I > I * and sufficiently weak dissipation and heat conduction in the atmosphere, ocean, and land, µ a < µ a *, κ a <κ a *, µ o < µ o *, κ o <κ o *, κ l <κ l * Ratio.6 large-scale variability will be chaotic with a maximum.4. Atmosphere Total Atmosphere + Ocean + Land = Total predictability period P of P = P(I, µ a,κ a, µ o,κ o,κ l ) days..e-.e+.e+.e+.e+.e+4.e+5.e+6 Proof. Proceed as with the Third Fundamental Theorem of Atmospheric Science. Time Period (Days) Perhaps an even more prestigious prize awaits you. 7

8 The origin of uncertainty Chaos Theory The computer models stimulated hopes that weather prediction in a Newtonian world would eventually demonstrate amazing accuracy But the same computer power that would make that accuracy possible revealed an astounding problem foreseen by Poincare at the beginning of the th century. We arrived at an entirely new view of science: Part of the universe may be regular and predictable, but part of it is chaotic and only predictable for limited periods Tropical and Polar Air Currents Robert Fitzroy, The Weather, Edward N Lorenz Professor of Meteorology Massachusetts Institute of Technology was the first to recognize that the atmosphere is a chaotic system NOAA Library A New View of Science A New Assumption or Principle: Everything is connected to everything else Global change : Atmosphere, ocean, land, cryosphere Biosphere Energy and economics Politics -- global and local 8