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 213 - Wichita State University Support for this research was provided by the U.S. National Science Foundation.
Anthropogenic Climate Change Global Warming vs. Ozone Hole greenhouse gases CO2, CH4, N2O chlorofluorcarbons (CFCs such as freon)
Anthropogenic Climate Change Global Warming vs. Ozone Hole [IPCC 213 Assessment Report]
Anthropogenic Climate Change Global Warming vs. Ozone Hole Antarctic Ozone Hole 4 October 21 [IPCC 213 Assessment Report] 1 2 3 4 5 Total Ozone (Dobson units) igure Q11-1. Antarctic ozone hole. Total ozone [WMO 26 Ozone Assessment]
How does anthropogenic forcing affect the atmospheric circulation? [Image from wikipedia]
But first, what drives the atmospheric circulation? r=6371 km troposphere ~ 1 km
But first, what drives the atmospheric circulation? (not drawn to scale)
But first, what drives the atmospheric circulation? Short answer: differential heating! solar radiation: heat from below, and more at the equator
But first, what drives the atmospheric circulation? Short answer: differential heating! atmosphere transports energy (heat and moisture) upwards and polewards
Early ideas: George C. Hadley (1735) a cell that transports heat in the meridional direction
Early ideas: George C. Hadley (1735) as low level winds approach equator, they will turn westward to conserve momentum: the trade winds
What about the upper level winds? (no one really worried about upper level winds until the 2th century)
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
An unstable situation... Hadley s single cell is unstable (baroclinic instability) [Charney, 1947; Eady 1949]
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]
Meridional structure of the atmospheric circulation Polar Cell Ferrel Cell Hadley Cell Instability breaks up the meridional circulation into three cells
Meridional structure of the atmospheric circulation Polar Cell Ferrel Cell Hadley Cell Instability breaks up the meridional circulation into three cells William Ferrel (1817-1891)
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]
The circulation in all its glory... The brightness (equivalent blackbody) temperature
The circulation in all its glory... The brightness (equivalent blackbody) temperature
The jet streams in austral summer (Dec.-Feb.) 2 km 1 km km pressure (hpa) ERA4 DJF zonal mean zonal wind [u] 2 55 5 45 4 5 35 3 25 2 15 2 1 5 5 5 1 85 15 1 2 8S 6S 4S 2S EQ 2N 4N 6N 8N Latitude =12 mph m/s
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]
DJF Trends in zonal mean zonal wind late 2th century reanalysis [Son et al. 28; Gerber et al. 211]
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]
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]
DJF Trends in zonal mean zonal wind late 2th century reanalysis models w/ghgs models w/ GHGs+O3 predictions? 2-279 [Son et al. 28; Gerber et al. 211]
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 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)
Temperature Signature of Anthropogenic Forcing Temperature change, 196 1999 Temperature change, 2 279 Models with fixed ozone hpa 2 4 6 8 (a) (b) Models with varying ozone 1 hpa 2 4 6 8 1 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude (c) 6S 3S 3N 6N latitude 1 8 6 4 2 2 4 6 8 6S 3S 3N 6N latitude 1 8 6 4 2 2 4 6 8 C C (d)
Temperature Signature of Anthropogenic Forcing Temperature change, 196 1999 Temperature change, 2 279 Models with fixed ozone hpa 2 4 6 8 (a) (b) Models with varying ozone 1 hpa 2 4 6 8 1 (c) 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude 1 8 6 4 2 2 4 6 8 C (d)
Temperature Signature of Anthropogenic Forcing Temperature change, 196 1999 Temperature change, 2 279 Models with fixed ozone hpa 2 4 6 8 (a) (b) Models with varying ozone 1 hpa 2 4 6 8 1 (c) 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude 1 8 6 4 2 2 4 6 8 C (d)
Circulation responds to changes in temperature gradients 1 JULY 21 BUTLER ET AL. GHG-like warming Butler et al. 21
The circulation response to thermal forcing in an idealized, dry atmospheric model 3481 1 JULY 21 B U T L E R E T A L. BUTLER ET AL. GHG-like warming Butler et al. 21
The circulation response to thermal forcing in an idealized, dry atmospheric model 3481 1 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
Which forcing has dominated to date?
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
Quantifying the temperature forcing Temperature change, 196 1999 Temperature change, 2 279 ΔTO3 ΔTGHG Models with fixed ozone Models with varying ozone hpa 2 4 6 8 1 hpa 2 4 6 8 1 (a) (c) 6S 3S 3N 6N latitude 6S 3S 3N 6N latitude 1 8 6 4 2 2 4 6 8 C (b) (d)
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
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
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 196-1999 trends 2-279 trends [Perlwitz et al. 28]
Regression Coefficients: Estimate of Sensitivity CCMVal2 Models strat. polar cap temp. (O 3 ) regression coefficients (deg./ K).6.4.2.2.4 tropical temp. (GHG) 1 2 3 4 5 6 7 8 9 1 model U lat = r O3 T 3 + r GHG T GHG
Regression Coefficients: Estimate of Sensitivity CCMVal2 Models strat. polar cap temp. (O 3 ) regression coefficients (deg./ K).6.4.2.2.4 tropical temp. (GHG) 1 2 3 4 5 6 7 8 9 1 model mean U lat = r O3 T 3 + r GHG T GHG
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).6.4.2.2.4 tropical temp. (GHG) regression coefficients (deg./ K).6.4.2.2.4 tropical temp. (GHG) 1 2 3 4 5 6 7 8 9 1 model mean 1 2 3 4 5 6 7 8 9 1 model mean U lat = r O3 T 3 + r GHG T GHG
Attribution of 2 Century Climate Trends CCMVal2 Models CMIP3 Models trends (deg./decade).2.4.6 trends (deg./decade).2.4.6.8.8 1 total 1 2 3 4 5 6 7 8 9 1 model 1 mean total 1 2 3 4 5 6 7 8 9 1 model mean U lat = r O3 T 3 + r GHG T GHG
Attribution of 2 Century Climate Trends CCMVal2 Models CMIP3 Models trends (deg./decade).2.4.6 trends (deg./decade).2.4.6.8 1 O 3 total GHG 1 2 3 4 5 6 7 8 9 1 model.8 1 mean O 3 total GHG 1 2 3 4 5 6 7 8 9 1 model mean U lat = r O3 T 3 + r GHG T GHG
Summary of Model Trends a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Summary of Model Trends a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Summary of Model Trends a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Summary of Model Trends a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1.2 Shaving cream.2 1.4 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade) moved the jet stream!.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Summary of Model Trends a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Summary of Model Trends a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
But what about the black bars?
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) 1 1 2 3 Jet shifts poleward 21 22 23 24 25 26 27 28 year
Two Sources of Model Spread: Differences in the thermal response to GHG, O3 a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Returning to our case study... change in jet position 3 GFDL CM3 IPSL CM5A MR 2 U lat ( latitude) 1 1 2 3 21 22 23 24 25 year 26 27 28
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) 6 4 2 U lat ( latitude) 2 1 1 2 3 21 22 23 24 25 year 26 27 28 2 21 22 23 24 25 year 26 27 28
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) 6 4 2 U lat ( latitude) 2 1 1 2 2 21 22 23 24 25 year 26 27 28 change in polar temperature 12 1 8 GFDL CM3 IPSL CM5A MR 3 21 22 23 24 25 26 27 28 year T 3 (K) 6 4 2 2 21 22 23 24 25 year 26 27 28
Uncertainty in global warming a poor predictor....3.2 a)! U lat vs.! T trop b)! U lat vs.! T polar! U lat ( /decade).1.1.2.3.4 CCMVal2, R=.18 CMIP3, R=.52 CMIP5, R=.24.2.4.6.8! T tropical (K/decade) CCMVal2, R=.62 CMIP3, R=.81 CMIP5, R=.73.5.5 1 1.5! T polar (K/decade)
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).1.1.2.3.4 CCMVal2, R=.18 CMIP3, R=.52 CMIP5, R=.24.2.4.6.8! T tropical (K/decade) CCMVal2, R=.62 CMIP3, R=.81 CMIP5, R=.73.5.5 1 1.5! T polar (K/decade)
Two Sources of Model Spread: Differences in the circulation response to temperature a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Two Sources of Model Spread: Differences in the circulation response to temperature a) regression coef. b) T trends, 196 99 c) jet shift, 196 99 d) T trends, 2 79 e) jet shift, 2 79 r 3 and r GHG ( /K).4.2.2.4.6 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 T3 and TGHG (K/decade) 1 1 2 3 CCMVal2 CMIP3 CMIP5 U lat ( /decade).2.2.4.6.8 1 CCMVal2 CMIP3 CMIP5 U lat = r O3 T 3 + r GHG T GHG
Uncertain Forcing vs. Uncertain Dynamics Variability in modeled circulation response due to differences in thermal forcing by ozone and GHGs differences in circulation sensitivity
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]
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]
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]
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]
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 8 6 4 2 latitude 8 6 4 2 latitude 7 3 1 1 3 7
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]
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]
Similar Connections in CCMVal2 Models (2th Century) equatorward bias longer time scale longer time scale larger jet shift [ Son et al. 21]
What connects variability and change?
Springs: An (imperfect) analogy F = kx Hooke s Law
Springs: An (imperfect) analogy F = kx Hooke s Law it pulls x { back! F=kx pull spring down
Springs: An (imperfect) analogy F = kx Hooke s Law F = ma Newton s Second Law
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
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
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
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
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.
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
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
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)
Does it work?
Idealized Atmospheric Model Experiments zonal mean zonal wind, u 1 pressure (hpa) 2 3 4 5 6 7 3 5 8 9 5 8 6 4 2 latitude
Idealized Atmospheric Model Experiments u and the annular mode 1 5 pressure (hpa) 2 3 4 5 6 7 8 3 1 1 3 9 5 8 6 4 2 latitude
Apply torque that projects on internal variability u and the annular mode 1 5 pressure (hpa) 2 3 4 5 6 7 8 3 1 1 3 9 5 8 6 4 2 latitude [Ring and Plumb, 28]
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 3 45 6 75 15 8 6 4 [after Ring and Plumb 28]
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, L4.1.5.5.1.15 projection of forcing [Gerber, Voronin, and Polvani 28]
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.