Andrews, Jim

Marsh, Robert

Oliver, Kevin

Shepherd, John

English

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SOES 6006 Lecture 51-D Energy Balance ModelsBob MarshSchool of Ocean & Earth ScienceUniversity of SouthamptonOverview : Lectures 4,51) Basics• The global energy balance• Zero-dimensional (globally-averaged) Energy Balance Models (EBMs)2) Introducing latitudinal effects• 1D (meridional) EBMs• Energy transportThemes (Lecture 5):• Some spherical geometry, for ISWR as function of latitude• A (slightly) more sophisticated OLWR formula• Simplest representation of ocean/atmosphere heat transport• Modelling zonally-averaged surface temperature• The ice-albedo feedback & hysteresis1) Zonal radiation imbalanceWhat factors govern the surplus & deficit of radiation?Schematic, for precise diagram go to: Kump, L.R., Arthur, M.A., Patzkowsky, M.E., Gibbs, M.T., Pinkus, D.S., Sheehan, P.M., (1999)A weathering hypothesis for glaciation at high atmospheric pCO2 in the late Ordovician. Palaeoclimatol.Palaeoecol. Palaeogeogr. v, 152, p. 173-1872) Meridional heat transport & meridionaltemperature gradientHow should we represent meridionaltransport of heat in a 1-D EBM?Reproduced by permission of American Geophysical Union: North, G.R., Cahalan, R.F., Coakley, J.A, jr. Energy Balance Climate Models. Rev. Geopys. Space Phys, v. 19, no. 1, p. 91-121. February 1981. Copyright [1981] American Geophysical Union.3) The Faint Young Sun Paradox How can 1-D EBMshelp explain liquid water on the early Earth?Courtesy of University of Oregon4) Snowball Earth EpisodesWhy were these episodes so long?http://www.snowballearth.org/images/snwbltvst.gifDiagrams courtesy of Snowball Earth (Paul F Hoffmann).5) Meridional heat transport in the Ocean & AtmosphereWhat is the difference between the transport represented in these two images, and how might it matter?The Global Ocean Conveyor Belt(http://www.cmmap.org/images/learn/climate/ferrel.jpg)Courtesy of Center for Multiscale Modelling of Atmospheric Processes Courtesy of LDEOMeridional Non-uniformity Insolation varies with latitude- allowing for the effect of obliquity Geometrical effects- projected and actual areas of latitude bands Pole-Equator temperature differences- moderated (reduced) by atmospheric & oceanic transports of heat (and also water, etc...)- so need to include advection & diffusion , i.e. transport by mean circulation & eddies (recall end of lecture 3) We will build a more complete 1-D EBM …1D (meridional) EBMs Incoming SW solar radiation - varies as a function of latitude - and of albedo (may be a function of temperature...) Outgoing LW infra-red radiation- now a function of local (not global) surface temperature Heat transport, by mixing (and maybe advection)- needs to be parameterized Heat capacity of land/sea surface (mixed layer?)- needed for time-dependent calculations only See review by North et al (1981)- North GR, Cahalan RF & Coakley JA, Rev Geophys & Space Physics, 19, 91-121 (1981)ISWR at different latitudesAngle of incidence affects the distribution of energy over the surface area (and also reflectivity of surface)Present-day obliquity23.5OUniversity of SouthamptonProjected Areas (Zones) for ISWR Within latitude bands θ −  Δθ / 2( ) and θ +  Δθ / 2( )projected area :dAp = 2R cosθ R dθ cosθ = 2R2 cos2 θ dθdAp = R2(1+ cos 2θ )dθ∴ Ap = dAp =∫ R2 θ + 12 sin 2θ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ θ − Δθ / 2θ +  Δθ / 2= R2 Δθ + 12sin 2θ + Δθ( )− sin 2θ − Δθ( ){ }⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ → π R2 / 2  for a hemisphere :  OKObliquity & annual-mean insolation~430 Wm-2 at equator• For zero obliquity, ISWR varies as cos2θ • But non-zero Angle of Obliquity strongly alters the variation of annual-mean ISWR with latitude:EquatorPoleReproduced by permission of American Geophysical Union: Ward, W.R., Climatic variations on Mars, 1. Astronomical Theory of Isolation. Journal of Geophysical Research v 79, no 24. p, 3375. 20 August 1974. Copyright [1974] American Geophysical UnionAnnual-mean ISWR for present obliquity The annual-mean of seasonally-varying insolation, as a function of latitude, is complicated! But from observations (for present-day obliquity):S(sinθ) ≈ Sbar{1 - 0.477 P2(sinθ)}    (by G.R.North)where P2(x) = (3x2 -1)/2(a second Legendre polynomial) From which:S(Eq) = 340 × {1-(0.477×(-0.5))} = 340 × 1.239 = 421 Wm-2vs. 433 W m-2 for zero obliquityS(pole) = 340 × {1-(0.477 × (-1))} = 340 × 0.523 = 178 Wm-2>> 0 for zero obliquity So the main effect of obliquity is polar warmingOLWR re-visited Budyko’s Linear Approximation for OLWR:• F ≈ 204 + 2.17 Ts W m-2• already (implicitly) incorporates the greenhouse effect due to water vapour• good approximation to data for 0 < Ts < 30 ºC• but may be poor if extrapolated outside that range A better approximation is the logistic form: F ≈ 0.9 × σ (Ts+273)4 × [1-logistic{(Ts-50)/100}]• allows for saturation of F (asymptoting to maximum of ~320 Wm-2) due to absorption by water vapour• recall last lecture (L4)Outgoing Long-Wave Radiation andthe Water Vapour Greenhouse EffectNote: Logistic form asymptoting towards  320 Wm-2 at high TsJohn Shepherd, University of Southampton1) Zonal radiation imbalance (explained)Present-Day Obliquity & projected area effect ensure that considerably more ISWR received at low latitudes; OLWR largely determined by “surface temperature”Schematic, for precise diagram go to: Kump, L.R., Arthur, M.A., Patzkowsky, M.E., Gibbs, M.T., Pinkus, D.S., Sheehan, P.M., (1999)A weathering hypothesis for glaciation at high atmospheric pCO2 in the late Ordovician. Palaeoclimatol.Palaeoecol. Palaeogeogr. v, 152, p. 173-187Atmospheric & Oceanic heat transport Latent & sensible heat transport• by advection (mean flow)• and/or by mixing (eddies), due to baroclinic instability Parameterize most simply (Budyko 1969):Q = K′ (Ts - Tbar)- relaxation of local temperature, Ts to global mean temperature• useful for a simplest 1-D EBM, but results are peculiar …Heat Balance Equation - general form C∂Ts θ( )∂t = A θ( ) (1− α )S θ( )− F Ts( )[ ]− ∂Q∂θC = heat capacity;A θ( ) =  area of zone centred on latitude θ ;Q =  advective-diffusive heat transport ISWR OLWR Meridionaldivergence of transportNet heating• Evolution of temperature field (1D, meridional)One-dimensional (meridional) EBM schematicN.B. mixing between boxes (double-headed arrows) plusuni-directional transport (single-headed arrows) …(Sketch courtesy John Shepherd)Building a simple 1-D EBM Consider only the steady-state (∂Ts/∂t = 0) Use the logistic formulation for F(θ) - LHS below Use North’s approximation for S(θ) Optionally allow albedo to vary with latitude- i.e., ice-albedo feedback, via Ts Use Budyko’s (simplest) formulation for mixing- therefore a local balance for each latitude band Put it all together as OLWR = ISWR + mixing: 0.9σTs4 1− logistic Ts - 50{ } 100[ ]≈ 1− α θ( )( )S 1− 0.477P2 sinθ( ){ }− ′ K Ts − T ( )1-D EBM solutions - with Ice-Albedo and Water Vapour Feedbacks : Mixing = 0ISWR, OLWRISWR, OLWRTemperatureSingle curve = OLWR(Ts)         [LHS]Multiple curves = absorbed ISWR(θ)   [RHS]High albedo(ice-covered)Low albedo(ice-free)Albedo transitional betweenIce-covered & ice-freeJohn Shepherd, University of Southampton- with Ice-Albedo & Water Vapour Feedbacks, plus Budyko-type mixing, Q = K′ (T - Tbar) [illustrative]ISWR + mixing, OLWRISWR + mixing, OLWRTemperatureSingle curve = OLWR [LHS]Multiple curves = ISWR(θ) + mixing   [RHS]Mixing warms cold latitudesMixing cools hot latitudesJohn Shepherd, University of Southampton1-D EBM : Without Ice-Albedo Effect ; No Atmospheric/Ocean Mixing♦ “Solid” surface = ISWR + mixing (= 0); “Hatched” surface = OLWR♦ Surfaces intersect  (i.e., equlibria as function of latitude) where: Outgoing heat flux = Incoming heat fluxHeat FluxHeat FluxOLWR surfaceISWR + mixingsurfaceJohn Shepherd, University of Southampton1-D EBM : Without Ice-Albedo Effect ; Mixing Diffusivity = 2 x 103 mks♦ Mixing reduces Equator-Pole temperature differenceJohn Shepherd, University of Southampton1-D EBM : With Ice-Albedo & Water Vapour Feedbacks ; Mixing = 0♦ “Kink” in the surface reveals 2 stable states (e.g.,   ) possible across mid-latitudes (~15-40º) - supports possibility of “Snowball Earth”***John Shepherd, University of Southampton1-D EBM : With Ice-Albedo and Water Vapour Feedbacks ;Mixing Diffusivity = 2 x 103 mksIce**♦ Kink (hence bistability) reduced by meridional mixing♦ Range of latitude with 2 stable states limited to ~50-60°John Shepherd, University of Southampton1-D EBM : With Ice-Albedo and Water Vapour Feedbacks ; (larger) Mixing Diffusivity = 3 x 103 mks♦ Kink almost eliminated under strong meridional mixing John Shepherd, University of SouthamptonPlan view: With Ice-Albedo and Water Vapour Feedbacks ; Diffusivity = 2 x 103 mksBistablezoneLarge (~30°C) temperature difference (between the two stable states)John Shepherd, University of SouthamptonIncorporating proper mixing into 1-D EBMs Fickian diffusion, relating meridional mixing of heat to local temperature gradient:Q = K ∂TS/∂y Several methods are available:(a) Analytical solutions, using expansion in Legendre polynomials in sin(θ) (North 1975, 1981)(b) Numerical solutions• steady-state : iterative solution of linear equations• time-dependent : solving first-order ordinary or partial differential equations And K may be constant or variable …- e.g., parameterization of atmospheric eddy mixing (Peter Stone, MIT):- K ∝ k (∂T/∂y)n   where 0.5 < n < 3 (varies with lat.)2) Meridional heat transport & meridionaltemperature gradient (explained)A sensible parameterization is necessary, to reproduce the observed temp. gradient - e.g., Fickian mixing with appropriate choice of KReproduced by permission of American Geophysical Union: North, G.R., Cahalan, R.F., Coakley, J.A, jr. Energy Balance Climate Models. Rev. Geopys. Space Phys, v. 19, no. 1, p. 91-121. February 1981. Copyright [1981] American Geophysical Union.Pole-Equator Temperature Gradient :vs. Solar Input & for constant / variable mixing• Climate less sensitive to changing solar constant with mixing that can adjust3) The Faint Young Sun Paradox: Explained? Could mixing help to keep poles (hence whole Earth) warm at lower luminosity? More likely associated with changing atmospheric compositionhttp://zebu.uoregon.edu/~imamura/122/images/faint_young_sun.jpgSimple EBM: the effect of (Fickian) mixingincreasing mixingincreasing mixingJohn Shepherd, University of Southampton1-D EBM : with Ice-Albedo feedback ;variable mixing in the range 0  to  6 x 104 m2s-1(decreasing, then increasing, in increments of 1 x 104 m2s-1)-10°CJohn Shepherd, University of SouthamptonGlobal-mean temperature, for each expt (from previous slide) - reveals limited hysteresis** Once glaciated, the Earth cannot so easily escape the glacial state“start”hereNote: expts proceeded with Mixing rate = 6,5,4,3,2,1,0,1,2,3,4 x 104John Shepherd, University of Southampton1-D EBM : with Ice-Albedo effect ; fixed mixing 2 x 104 m2s-1 ; varying solar constant(decreasing, then increasing, in increments of 100 Wm-2)SnowballEarths (< -10°C at all latitudes)-10°CJohn Shepherd, University of SouthamptonGlobal-mean temperature, for each expt (from previous slide) - reveals more extensive hysteresisSolar ConstantSnowball statesJohn Shepherd, University of Southampton4) Snowball Earth Episodes (explained)Once the Earth became glaciated, the higher ice albedo reflected more ISWR, prolonging Snowballhttp://www.snowballearth.org/images/snwbltvst.gifDiagrams courtesy of Snowball Earth (Paul F Hoffmann).Adding advective ocean transport Assume same temperature for atmosphere & surface ocean (tight coupling = rapid exchange) Diffusive treatment for mixing (in both)• actual location is irrelevant Add advective transport (overturning) in ocean• can introduce asymmetry between N & S hemispheres• may be modelled (THC) or specified (stream function)• first approximation: additional single box for deep ocean …1-D meridional EBM with specified oceanic MOC (THC) & inter-hemispheric asymmetry of heat fluxUpper layer (moves northwards)Deep layer (moves southwards)N.B. upwelling & downwelling (“sinking”) link surface & deep ocean(Sketch courtesy John Shepherd)John Shepherd, University of SouthamptonMeridional temperature distribution for varying THC rates(0 to 60 Sv and back down) : with ice-albedo & water vapour feedbacks, K = 2 x 104 m2/s• Northward surface flow, Southward deep flow (Northward heat transport)• Northern Hemisphere becomes (and remains) ice-freeNH high latitudes warm as northward ocean heat transport increasesJohn Shepherd, University of SouthamptonGlobal mean temperature for varying THC rates(from previous slide)Earth avoids Snowball THC rate (Sv) GlobalGlobal--mean Tmean TNote: expts proceeded with THC rate = 0, 10, … 50, 60, 50, … 10, 0 SvJohn Shepherd, University of SouthamptonMeridional temperature distribution for varying THC rates(0 to 60 Sv and back down) : with ice-albedo and water vapour feedbacks, K = 2 x 104 m2/s ; Ice albedo = 0.75 (increased from 0.7)• Southern Hemisphere glaciation spreads to the whole globe: Snowball state• Harder to escape Snowball state, but possible, with strong enough THCJohn Shepherd, University of Southampton5) Meridional heat transport in the Ocean & AtmosphereAtmosphere mixes heat, similarly in both hemispheres; advection more dominant mode of ocean heat transport, typically asymmetric about the Equator, perhaps influencing Snowball stateThe Global Ocean Conveyor Belt(http://www.cmmap.org/images/learn/climate/ferrel.jpg)Courtesy of LDEOCourtesy of Center for Multiscale Modelling of Atmospheric ProcessesClimate sensitivity re-visited Values of λ (°K per W m-2):(e.g. for varying solar constant)• black body (S-B law) 0.30• + water vapour (Budyko) 0.46• + ice-albedo feedback 0.64 (polar ice only)• if snowball state 0.24 (more ISWR reflects)Summary (1)¾ Introducing latitudinal effects• Net gain (loss) of heat at low (high) latitudes (depending on angle of obliquity)• Equator-Pole temperature gradient - net heating/cooling, moderated by meridional heat transport¾ 1-D (meridional) EBMs• Extension of 0-D EBMs (a connected set of such models)• Prescribing insolation as a function of latitude• Predicting local (rather than global) temperature• Allowing meridional (temperature) variation in albedo• Allowing meridional exchanges of heatSummary (2)¾ Introducing a heat balance with more terms• Steady State balance of ISWR, OLWR, heat transport• Parameterize heat transport with Newtonian relaxation (simple) or Fickian diffusion (more appropriate)¾ 1-D EBM predictions of temperature vs.latitude• With/Without Ice-Albedo feedback• With/Without Heat Mixing (and varying strength thereof)¾Prediction of multiple equilibria• Bistability - two stable states - within a range of latitude• The extent of which is controlled by mixing (stronger mixing = reduced Equator-pole temperature gradients, narrower or eliminated zone of bistability)Summary (3)¾ 1-D EBM experiments with Fickian diffusion further reveal “Snowball Earth” state• Under sufficiently small solar constant• And reduced mixing• Once in the Snowball state, can’t escape (hysteresis -incrementally increase & decrease control parameter)• Also under variable ice albedo (0.70 to 0.75)¾Add explicit ocean heat transport (advective)• Preferentially transporting heat northwards (at present)• Raising temperature in high northern latitudes• Alternative Snowball state also evident under varying ocean heat transport♦ This resource was created by the University of Southampton and released as an open educational resource through the 'C-change in GEES' project exploring the open licensing of climate change and sustainability resources in the Geography, Earth and Environmental Sciences.  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