The weather and climate as problems in physics: scale invariance and multifractals. S. Lovejoy McGill, Montreal

Transcription

1 The weather and climate as problems in physics: scale invariance and multifractals S. Lovejoy McGill, Montreal McGill April 13, 2012

2 Required reading for this course The Weather and Climate Emergent Laws and Multifractal Cascades SHAUN LOVEJOY and DANIEL SCHERTZER (in press, Oct. 2012)

3 The Weather 3

4 Meteorologists

5 The Emergence of physical laws Quantum mechanics Statistical mechanics General Relativity Special Relativity Large scales (usually) Large numbers of particles Low energy mass density Classical Mechanics Continuum mechanics, Fluid mechanics thermodynamics Special Relativity Classical (Galilean) Relativity Low level (fundamental) Velocities << speed of light high level (simpler if applicable)

6 Example: The emergence of Thermodynamics from Newton s laws Newton s laws: Thermodynamics: F=ma F=ma F=ma Large number of particles First law: conservation of energy Second law: increase in entropy ex.: Boyle s law: (pressure) x (volume) = constant Low level, (difficult to handle for many particles) High level (Valid when many particles are present)

7 Pioneers of turbulence Richardson Kolmogorov

8 Corrsin Obukhov Ralph Bolgiano, Jr

9 The emergence of turbulence dynamics (Classical) Spaghetti Vortices in strongly turbulent fluid (M. Wiczek, numerical simulation, 2010) Fluid mechanics Low level (fundamental) Strong stirring (nonlinearity) Laws of turbulence Classical: Richardson, Kolmogorov, Corrsin, Obukhov, Bolgiano High level

10 Emergent laws reduce seeming complexity to simplicity at another level

11 Mandelbrot

12 Complex? Blowing up gives the same type of shapes The Mandelbrot set ( self-similar, scale invariant, fractal)

13 Or simple? Generating the Mandelbrot set -Take a number. -Multiply it by itself. -Add a constant. -Repeat. (I forgot to mention: take a COMPLEX number)

14 Complex? Δx 20 steps: 1:1 20 steps: 5:1 200 steps: 1:1 200 steps: 5:1 Drunkard s walk 20,000 steps: 1:1 20,000 steps: 5:1

15 Or simple? (distance) x (distance) = number of bars visited From initial bar Average number of bars visited (or displacements made) (Brownian motion)

16 Complex? or simple? v(t) 1700 mm/s t Infra Red satellite effective temperatures, January 16, second of wind data (roof of Rutherford building, McGill) The Atmosphere

17 Brute force

18 Atmosphere: Laws of Fluid mechanics (low level) wind Earth angular velocity Gravitational potential Specific volume=1/ρ pressure Friction Specific heat temperature Heating rate density Gas constant Governing atmospheric laws

19 Brute force numerical solution of the equations (2) Discretization of the equations

20 Brute force numerical solution of the equations (3) Earth system modelling

21 Or simplicity?

22 Atmosphere: Emergent laws (high level) Power law Fluctuations (turbulent flux) x (scale) H Differences, tendencies, wavelet coefficients Cascading Turbulent flux Size: Anisotropic Space-time Scale function Fluctuation /conservation exponent These laws are scale Fluctuation = change in time and/or space Scale = size invariant Turbulent flux = strength of stirring

23 Which Richardson? The father of Numerical Weather Prediction

24 The father of numerical weather prediction 1922

25 The weather prediction factory (artist: Francois Schuiten)

26 Richardson s numerical grid for integrating Each column was divided into 5 vertical cells and defined 7 quantities: pressure, temperature, density, water content, 3 velocity components It took me the best part of six weeks to draw up the computing forms and to work out the new distribution in two vertical columns for the first time. My office was a heap of hay in cold rest billet. With practice the work of an average computer might go perhaps ten times faster. If the time-step were 3 hours, then 32 individuals could just compute two-points so as to keep up with the weather. -Richardson 1922

27 ... or the grandfather of cascades? Weather prediction by Numerical Process 1922, p.66

28 Scale by scale simplicity: cascades CASCADE LEVELS 0 -- l 0 x ε y 1 -- l 0 / λ 1 multiplication by 4 independent random (multiplicative) increments n -- 2 l 0 / λ n l 0 / λ multiplication by 16 independent random (multiplicative) increments

29 W(t) Does the wind have a velocity? Although at first sight strange, the question grows upon acquaintance - Richardson 1926 v(t) mm/s t second of wind data (roof of Rutherford building, McGill) t 2048 Richardson suggested that the trajectory of a particle could be like a Wierstrass function (1872)

30 Scale invariance and fractals

31 λ Η blowup Wierstrass function showing scale invariance under anisotropic blowup (H=1/3 in this example, λ=3) -1-2 λ λ=3

32 Cantor set Let us start with: and let us iterate: A small part is same as the whole if blown up by a factor 3 ( scale invariance, selfsimilarity ) x3

33 Sierpinski Triangle 1 2

34 Let us start with: Koch snowflake and let us iterate:

35 Sierpinski Pyramid First iteration: 10 th iteration:

36 Menger Sponge motif: iterations:

37 CASCADES Parent eddy Homogeneous Daughter eddies Grand-daughter eddies Intermittent

38 β-model Fractal set active calm

39 Cascades and Multifractals

40 Aircraft temperature transect (12km altitude) - 49 Temperature ( o C) x (km) Turbulent flux (gradient of the above) 1 σ

41 Temperature turbulent flux ε at 280m resolution High to low Resolution: degrading by factors of 4 20 ε ε ε ε ε ε ε x x x x x x km x

42 Cascades and Multifractals Simulations: adding small scale details (low resolution to high) ε 0 ε 1 ( α model )

43 BARE ENERGY 1. Cascade level 1 bare density DRESSED ENERGY FLUX FLUX DENSITY DENSITY 1. FIELDS FIELDS 1. Cascades Multifractal Cascade level 3 bare density 1. Cascade level 4 bare density Dressed density averaged over 16x16 1. Dressed density averaged over 8x Cascade level 5 bare density Dressed density averaged over 4x Cascade level 6 bare density y ε x 1. y Π Dressed density averaged over 2x2 x Cascade level 7 bare density

44 0.4 Log 10 M q 0.3 EW wind M q λ K ( q) q= Log 10 M q NS wind Cascades work!! q= km km Log 10 M q Temp q= km 200km Log 10 λ q=2 q= Log 10 λ 20000km 20000km Log 10 M q h s humidity 200km km q=1.5 q=2 q=1.5

45 Cloud liquid water (top) Cascade modeling: clouds and radiative transfer Cloud top visible Cloud top, infra red Cloud liquid water (side) Cloud bottom visible

46

47 Cascade Simulations

48 The Climate 48

49 The produc+on of maple syrup is affected by global warming... Instead... have you tried my delicious 100% Canadian syrup on your pancakes? Tar sands syrup

50 What is the climate? "Climate is what you expect, weather is what you get. -Farmers Almanac Climate is conventionally defined as the long-term statistics of the weather. -Committee on Radiative Forcing Effects on Climate, 2005 US National Academy of Science

51 3 Three regimes: three types of variability: not two! Temperature T/ 10 Fluctuations Growing Climate (10-30 yrs to 50,000 yrs) Low frequency weather (10 days to yrs) Fluctuations Decreasing 1 Century, Vostok, 20-92kyr BP 20 days, 75 o N,100 o W, Weather (up to 10 days) 0-2 Fluctuations Growing hour, Lander, 10 Feb.-12 March, 2005 t

52 ΔT (K) Today Holocene Vostok core (Antarctica 75 o S) 130 yr resolution (average) Age (kyr) yr resolution Last four Glacials 600yr resolution

53 18 O - 34 Paleotemperatures: GRIP (GReenland Ice core Project), summit location ( 75 o N), High (5.2 yr ) resolution section Dansgaard events K Holocene Last glacial NOW Age (kyr BP) 53

54 Briffa Mann Esper Jones -0.2 Crowley Ljundquist -0.4 Moburg T(K) 0.2 Multiproxy Temperatures Date Huang post 2003 (30 year average applied) Eight Reconstructions of the Northern hemisphere temperatures

55 T (K) Global temperature anomaly mean and spread from three different instrumental estimates Date -0.4

56 Do Global Climate models predict... The climate?...or low frequency weather? 56

57 average temperature change O C O C Data Atmosphere (3km) O C O C Mean surface IPSL 0.1 C 1 year 10 years 100 years GCM Log 10 Δt (yrs)

58 average temperature change o C 0.5 Global Temperature Vostok (Antarctic) ±3K ±2K days 1 year 100 years years Log 10 Δt (yrs) 1 o C yrs 20CR global scale -0.5 Global Surface 0.1 o C >2003 multiproxies, weather Low frequency weather climate GCM s Low frequency climate

59 Implications for global warming By comparing model and natural variability, we found that GCM s seem to be missing a long-time mechanism of internal variability such as land-ice. Anthropogenic contributions to 20th warming and 21st C warming scenarios may thus be either over - or under estimated. 59

60

61 Conclusions 1. Low level laws: complex (Fluid mechanics) High level laws simplicity (emergent turbulent laws) 2. Emergent Atmospheric laws are power laws Fluctuations are scaling, their exponents are scale invariant 3. There are three different regimes: Weather to 10 days, Low frequency weather to yrs, Climate to kyrs. 4. Without special forcing GCM s produce low frequency weather not climate type variability