Math, Models, and Climate Change How shaving cream moved a jet stream, and how mathematics can help us better understand why

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1 Math, Models, and Climate Change How shaving cream moved a jet stream, and how mathematics can help us better understand why Edwin P. Gerber Center for Atmosphere and Ocean Science Courant Institute of Mathematical Sciences New York University 9 October Wichita State University Support for this research was provided by the U.S. National Science Foundation.

2 Anthropogenic Climate Change Global Warming vs. Ozone Hole greenhouse gases CO2, CH4, N2O chlorofluorcarbons (CFCs such as freon)

3 Anthropogenic Climate Change Global Warming vs. Ozone Hole [IPCC 213 Assessment Report]

Assessment Report] 1 2 3 4 5 Total Ozone (Dobson units)

4 Anthropogenic Climate Change Global Warming vs. Ozone Hole Antarctic Ozone Hole 4 October 21 [IPCC 213 Assessment Report] Total Ozone (Dobson units) igure Q11-1. Antarctic ozone hole. Total ozone [WMO 26 Ozone Assessment]

5 How does anthropogenic forcing affect the atmospheric circulation? [Image from wikipedia]

6 But first, what drives the atmospheric circulation? r=6371 km troposphere ~ 1 km

7 But first, what drives the atmospheric circulation? (not drawn to scale)

8 But first, what drives the atmospheric circulation? Short answer: differential heating! solar radiation: heat from below, and more at the equator

9 But first, what drives the atmospheric circulation? Short answer: differential heating! atmosphere transports energy (heat and moisture) upwards and polewards

10 Early ideas: George C. Hadley (1735) a cell that transports heat in the meridional direction

11 Early ideas: George C. Hadley (1735) as low level winds approach equator, they will turn westward to conserve momentum: the trade winds

12 What about the upper level winds? (no one really worried about upper level winds until the 2th century)

13 What about the upper level winds? x but upper level winds are generate surface winds kept in check.. x o. x x o. strong eastward (westerly) jets, generating strong vertical shear by friction

14 An unstable situation... Hadley s single cell is unstable (baroclinic instability) [Charney, 1947; Eady 1949]

15 Flow is fundamentally not zonally symmetric Hadley s single cell is unstable (baroclinic instability) Generates Rossby waves, whose restoring force is the differential rotation of the planet. [Rossby et al. 1939]

16 Meridional structure of the atmospheric circulation Polar Cell Ferrel Cell Hadley Cell Instability breaks up the meridional circulation into three cells

17 Meridional structure of the atmospheric circulation Polar Cell Ferrel Cell Hadley Cell Instability breaks up the meridional circulation into three cells William Ferrel ( )

18 The eddies, or deviations from the zonal mean play a critical role in the circulation Polar Cell Ferrel Cell Hadley Cell Rossby waves and eddies transport heat and momentum - necessary to explain the zonal mean. [Lorenz, 1967]

19 The circulation in all its glory... The brightness (equivalent blackbody) temperature

20 The circulation in all its glory... The brightness (equivalent blackbody) temperature

21 The jet streams in austral summer (Dec.-Feb.) 2 km 1 km km pressure (hpa) ERA4 DJF zonal mean zonal wind [u] S 6S 4S 2S EQ 2N 4N 6N 8N Latitude =12 mph m/s

22 The jet streams in austral summer (Dec.-Feb.) SOUTHERN Recent HEMISPHERE trends CLIMATE DM7 2 km 1 km pressure (hpa) km [Son et al. 21]

23 DJF Trends in zonal mean zonal wind late 2th century reanalysis [Son et al. 28; Gerber et al. 211]

24 DJF Trends in zonal mean zonal wind late 2th century reanalysis models w/ghgs models w/ GHGs+O3 [Son et al. 28; Gerber et al. 211]

25 DJF Trends in zonal mean zonal wind late 2th century reanalysis models w/ghgs models w/ GHGs+O3 [Son et al. 28; Gerber et al. 211]

26 DJF Trends in zonal mean zonal wind late 2th century reanalysis models w/ghgs models w/ GHGs+O3 predictions? [Son et al. 28; Gerber et al. 211]

27 Questions What are the relative roles of greenhouse gases and ozone in forcing Southern Hemisphere circulation changes? What causes uncertainty in the circulation response? (That is, why is there such variance in model projections?) How can we reduce the uncertainty in the circulation response?

Coupled Models (CMIP3,5 Coupled Model Intercomparison Project, phases 3,5) simulate the atmosphere, ocean, and land surface (a coupled simulation between the key components of the climate system) our

28 Coupled Models (CMIP3,5 Coupled Model Intercomparison Project, phases 3,5) simulate the atmosphere, ocean, and land surface (a coupled simulation between the key components of the climate system) our best tool for quantitative prediction of climate change Chemistry Climate Models (CCMs) simulate interactive ozone chemistry in the stratosphere: can predict ozone hole and its recovery generally specify the surface ocean temperatures (not a coupled simulation) Idealized Atmospheric Models Century II Performing Arts Center (cast, in order of decreasing CPU time) primitive equation dynamics on the sphere (guts of an atmospheric model) simplified climate physics (no radiation, clouds, moisture)

29 Temperature Signature of Anthropogenic Forcing Temperature change, Temperature change, Models with fixed ozone hpa (a) (b) Models with varying ozone 1 hpa S 3S 3N 6N latitude 6S 3S 3N 6N latitude (c) 6S 3S 3N 6N latitude S 3S 3N 6N latitude C C (d)

30 Temperature Signature of Anthropogenic Forcing Temperature change, Temperature change, Models with fixed ozone hpa (a) (b) Models with varying ozone 1 hpa (c) 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude C (d)

31 Temperature Signature of Anthropogenic Forcing Temperature change, Temperature change, Models with fixed ozone hpa (a) (b) Models with varying ozone 1 hpa (c) 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude C (d)

32 Circulation responds to changes in temperature gradients 1 JULY 21 BUTLER ET AL. GHG-like warming Butler et al. 21

33 The circulation response to thermal forcing in an idealized, dry atmospheric model JULY 21 B U T L E R E T A L. BUTLER ET AL. GHG-like warming Butler et al. 21

34 The circulation response to thermal forcing in an idealized, dry atmospheric model JULY 21 B U T L E R E T A L. GHG-like warming 3484 JOURNAL OF CLIMATE BUTLER ET AL. J O U R N A L O F C L I M AV TOLUME E 23 ozone-like cooling Butler et al. 21

35 Which forcing has dominated to date?

36 A Simple Model of the Jet Response jet shift = ozone pull + GHG push U lat = r O3 T 3 + r GHG T GHG model simulations give us the forcings and response

37 Quantifying the temperature forcing Temperature change, Temperature change, ΔTO3 ΔTGHG Models with fixed ozone Models with varying ozone hpa hpa (a) (c) 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude C (b) (d)

38 A Simple Model of the Jet Response jet shift = ozone pull + GHG push U lat = r O3 T 3 + r GHG T GHG model simulations give us the forcings and response DM7 SON ET AL.: OZONE AND SOUTHERN HEMISPHERE CLIMATE DM7

39 A Simple Model of the Jet Response jet shift = ozone pull + GHG push U lat = r O3 T 3 + r GHG T GHG two unknowns

40 A Simple Model of the Jet Response jet shift = ozone pull + GHG push U lat = r O3 T 3 + r GHG T GHG two equations trends trends [Perlwitz et al. 28]

41 Regression Coefficients: Estimate of Sensitivity CCMVal2 Models strat. polar cap temp. (O 3 ) regression coefficients (deg./ K) tropical temp. (GHG) model U lat = r O3 T 3 + r GHG T GHG

42 Regression Coefficients: Estimate of Sensitivity CCMVal2 Models strat. polar cap temp. (O 3 ) regression coefficients (deg./ K) tropical temp. (GHG) model mean U lat = r O3 T 3 + r GHG T GHG

43 Regression Coefficients: Estimate of Sensitivity CCMVal2 Models CMIP3 Models strat. polar cap temp. (O 3 ) strat. polar cap temp. (O 3 ) regression coefficients (deg./ K) tropical temp. (GHG) regression coefficients (deg./ K) tropical temp. (GHG) model mean model mean U lat = r O3 T 3 + r GHG T GHG

44 Attribution of 2 Century Climate Trends CCMVal2 Models CMIP3 Models trends (deg./decade) trends (deg./decade) total model 1 mean total model mean U lat = r O3 T 3 + r GHG T GHG

45 Attribution of 2 Century Climate Trends CCMVal2 Models CMIP3 Models trends (deg./decade) trends (deg./decade) O 3 total GHG model.8 1 mean O 3 total GHG model mean U lat = r O3 T 3 + r GHG T GHG

46 Summary of Model Trends a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

47 Summary of Model Trends a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

48 Summary of Model Trends a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

49 Summary of Model Trends a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1.2 Shaving cream CCMVal2 CMIP3 CMIP5 U lat ( /decade) moved the jet stream! CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

50 Summary of Model Trends a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

51 Summary of Model Trends a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

52 But what about the black bars?

53 But what about the black bars? A Tale of Two Models (London and Paris) Princeton Changes in Jet Position 3 2 GFDL CM3 IPSL CM5A MR Jet shifts equatorward U lat ( latitude) Jet shifts poleward year

54 Two Sources of Model Spread: Differences in the thermal response to GHG, O3 a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

55 Returning to our case study... change in jet position 3 GFDL CM3 IPSL CM5A MR 2 U lat ( latitude) year

56 Returning to our case study... change in tropical temperature 12 1 GFDL CM3 IPSL CM5A MR 8 3 change in jet position GFDL CM3 IPSL CM5A MR T GHG (K) U lat ( latitude) year year

57 Returning to our case study... change in tropical temperature 12 1 GFDL CM3 IPSL CM5A MR 8 3 change in jet position GFDL CM3 IPSL CM5A MR T GHG (K) U lat ( latitude) year change in polar temperature GFDL CM3 IPSL CM5A MR year T 3 (K) year

58 Uncertainty in global warming a poor predictor a)! U lat vs.! T trop b)! U lat vs.! T polar! U lat ( /decade) CCMVal2, R=.18 CMIP3, R=.52 CMIP5, R= ! T tropical (K/decade) CCMVal2, R=.62 CMIP3, R=.81 CMIP5, R= ! T polar (K/decade)

59 Uncertainty in global warming a poor predictor... rather, key is what is happening over the pole!.3.2 a)! U lat vs.! T trop b)! U lat vs.! T polar! U lat ( /decade) CCMVal2, R=.18 CMIP3, R=.52 CMIP5, R= ! T tropical (K/decade) CCMVal2, R=.62 CMIP3, R=.81 CMIP5, R= ! T polar (K/decade)

60 Two Sources of Model Spread: Differences in the circulation response to temperature a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

61 Two Sources of Model Spread: Differences in the circulation response to temperature a) regression coef. b) T trends, c) jet shift, d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) CCMVal2 CMIP3 CMIP5 U lat ( /decade) CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG

62 Uncertain Forcing vs. Uncertain Dynamics Variability in modeled circulation response due to differences in thermal forcing by ozone and GHGs differences in circulation sensitivity

63 Connection between 21st Century Jet Shift and 2th Century Climatology 21 Century Jet Shift (degrees) Jet position in historical simulation (degrees) [ Kidston and Gerber 21]

64 Connection between 21st Century Jet Shift and 2th Century Climatology 21 Century Jet Shift (degrees) position of jet in reanalyses equatorward bias Jet position in historical simulation (degrees) [ Kidston and Gerber 21]

65 Connection between 21st Century Jet Shift and 2th Century Climatology 21 Century Jet Shift (degrees) larger jet shift position of jet in reanalyses equatorward bias Jet position in historical simulation (degrees) [ Kidston and Gerber 21]

66 Connection between the Climatological Jet Position and Time Scales of Internal Variability Annular Mode Time Scale (days) longer time scales equatorward bias Jet position in historical simulation (degrees) [ Kidston and Gerber 21]

67 What does this annular mode time scale represent? a) model w/ short time scales b) model w/ long time scales J 5 hpa geopotential height anomalies in two models D N O S A J time J M A [ Gerber et al. 21] January M F J latitude latitude

68 Connection between the Climatological Jet Position and Time Scales of Internal Variability Annular Mode Time Scale (days) longer time scales equatorward bias Jet position in historical simulation (degrees) [ Kidston and Gerber 21]

69 Internal Variability - Jet Shift Connection 21 Century Jet Shift (degrees) larger jet shift longer time scale Annular Mode Time Scale (days) [ Kidston and Gerber 21]

70 Similar Connections in CCMVal2 Models (2th Century) equatorward bias longer time scale longer time scale larger jet shift [ Son et al. 21]

71 What connects variability and change?

72 Springs: An (imperfect) analogy F = kx Hooke s Law

73 Springs: An (imperfect) analogy F = kx Hooke s Law it pulls x { back! F=kx pull spring down

74 Springs: An (imperfect) analogy F = kx Hooke s Law F = ma Newton s Second Law

75 Springs: An (imperfect) analogy F = kx Hooke s Law F = ma Newton s Second Law kx = m d2 x dt 2 k m x = d2 x dt 2 α 2 x = d2 x dt 2 let α = k m

76 Springs: An (imperfect) analogy F = kx Hooke s Law F = ma Newton s Second Law kx = m d2 x dt 2 k m x = d2 x dt 2 α 2 x = d2 x dt 2 Look for solution of form: then: x(t) =A cos(αt)+bsin(αt) d 2 x dt 2 = α2 x(t) So, period of oscillation is 2π α = 2π m k

77 Springs: An (imperfect) analogy Period of oscillation is 2π α = 2π m k The response to external forcing: in equilibrium, F spring = F external x { it pulls back Fspring=kx pull spring down F external

78 Springs: An (imperfect) analogy Period of oscillation is 2π α = 2π m k The response to external forcing: in equilibrium, F spring = F external { x it pulls back Fspring=kx kx = F external x = F external k pull spring down F external

79 Fluctuation-Dissipation Theory (in brief!) x t = B(x) = Lx + N(x) = Lx + Ẇ L is related to the time correlation structure of x, properties of the natural variability.

80 Fluctuation-Dissipation Theory (in brief!) x t = B(x) = Lx + N(x) = Lx + Ẇ +f +f +f L is related to the time correlation structure of x, properties of the natural variability. external perturbation

81 Fluctuation-Dissipation Theory (in brief!) x t = B(x) = Lx + N(x) = Lx + Ẇ x t = Lx + Ẇ + f = Lx ++f x = L 1 f +f +f +f L is related to the time correlation structure of x, properties of the natural variability. external perturbation

82 Fluctuation-Dissipation Theory (in brief!) x t = B(x) = Lx + N(x) = Lx + Ẇ x t = Lx + Ẇ + f = Lx ++f x = L 1 f +f +f +f L is related to the time correlation structure of x, properties of the natural variability. In most simple case, L 1 = ρ(τ)dτ ρ(τ) =x(t)x(t + τ)

83 Does it work?

84 Idealized Atmospheric Model Experiments zonal mean zonal wind, u 1 pressure (hpa) latitude

85 Idealized Atmospheric Model Experiments u and the annular mode 1 5 pressure (hpa) latitude

86 Apply torque that projects on internal variability u and the annular mode 1 5 pressure (hpa) latitude [Ring and Plumb, 28]

87 pressure System responds modally: strong projection on to internal variability shading: annular mode positive and negative torque 2 contours: response of model to the torque, u forced - u control (negative dashed) latitude [after Ring and Plumb 28]

88 Model with greater persistence more sensitive to external forcing 5 τ = 96 days projection of response 5 τ = 33 days m=89 m=17 NH, L2 NH, L4 SH, L2 SH, L projection of forcing [Gerber, Voronin, and Polvani 28]

89 Conclusions In austral summer, the Southern Hemisphere jet stream is pushed poleward by greenhouse gas induced tropical warming and pulled poleward by ozone induce cooling of the polar stratosphere. To date, ozone loss has been the most important driver. Uncertainty in climate forecasts stems from differences in the thermal response to anthropogenic forcing (primarily differences in ozone) and the circulation sensitivity to temperature changes. A model s ability to simulate today s climate and variability is an important measure for determining if its climate change projections are trustworthy.

Climate Change and Variability in the Southern Hemisphere: An Atmospheric Dynamics Perspective

Climate Change and Variability in the Southern Hemisphere: An Atmospheric Dynamics Perspective Edwin P. Gerber Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences New York University