Let s make a simple climate model for Earth.

Let s make a simple climate model for Earth. What is the energy balance of the Earth? How is it controlled? ó How is it affected by humans? Energy balance (radiant energy) Greenhouse Effect (absorption and reradiation of longwave radiation by clouds and gases in the atmosphere) Albedo (refection of shortwave radiation by clouds, land surface)

Outline of this lecture Longwave (thermal) and shortwave (UV/visible/infrared) radiation Planck function and Stefan Boltzmann law Kirchhoff s law thermodynamics and thermal radiation The energy balance of the earth effective temperature The greenhouse radiation A climate model anyone can use, from an iphone. Radiative Forcing change in greenhouse radiation since 1700

If we could take a snapshot of a light wave as it traveled for 1 s, it would be 3 10 8 m long, and would look like the sine wave shown in the figure. The distance between two successive crests on the wave is called the wavelength (denoted l). The frequency (denoted n) is the number of wave cycles (wavelengths) that pass a reference point per unit time, and since our snapshot shows exactly the number of peaks that passed in one second, n is also the number of peaks in the picture, i.e. n =c/l. Alternatively, 1/n is the time it takes the wave to travel one wavelength at speed c. l n = c

R o radius earth s orbit = 940 10 6 km R S radius of the sun = 0.695 10 6 km (R S /R o ) 2 = 0.54 10-6 10-6 solar (shortwave) radiation terrestrial (longwave) radiation

Planck s Law of Radiation gives the distribution of wavelengths emitted from a black body. A black body is an idealized surface that absorbs all incoming radiation. Most objects are well-approximated as black bodies for emission. x 2p for all angles Planck s law is a function of l that varies only with T! It is often denoted B(l,T)

Stefan-Boltzmann Law (Planck function integrated over all l x 2p) The fundamental physics of the Earth s energy balance is based on a very simple physical principle, which may be described by the Stefan-Boltzmann law of radiation The amount of energy radiated by a black body, in Joules s -1 m -2, (W m -2 ), is given by: st 4 s = 5.67 10-8 W m -2 K -4, the Stefan-Boltzmann constant

EMISSION OF RADIATION Radiation is energy transmitted by electromagnetic waves; all objects emit radiation One can measure the radiation flux spectrum emitted by a unit surface area of object: Here ΔΦ is the radiation flux emitted in [λ, λ + δλ] is the flux distribution function characteristic of the object Total radiation flux emitted by object: F = òfldl 0

Kirchoff s Law of Radiation (1859) states that the emissivity of a black body is equal to its absorptivity. The amount of radiation absorbed by the atmosphere is equal to the amount of radiation emitted by it.

KIRCHHOFF S LAW: Emissivity e(l,t) = Absorptivity For any object: very useful! Illustrative example: Kirchhoff s law allows determination of the emission spectrum of any object solely from knowledge of its absorption spectrum and temperature (thermodynamics)

ABSORPTION OF RADIATION BY GAS MOLECULES requires quantum transition in internal energy of molecule. THREE TYPES OF TRANSITION Electronic transition: UV radiation (<0.4 µm) Jump of electron from valence shell to higher-energy shell, sometimes results in dissociation (example: O 3 +hn go 2 +O) Vibrational transition: near-ir (0.7-20 µm) Increase in vibrational frequency of a given bond requires change in dipole moment of molecule Rotational transition: far-ir (20-100 µm) Increase in angular momentum around rotation axis (requires a dipole moment) Gases that absorb radiation near the spectral maximum of terrestrial emission (10 µm) are called greenhouse gases; IR absorption (usually) requires vibrational or vibrational-rotational transitions

GREENHOUSE EFFECT: absorption of terrestrial radiation by the atmosphere Radiative Flux Thermal absorption occurs outside the atmospheric window Absorptivity Major greenhouse gases: H 2 O, CO 2, CH 4, O 3, N 2 O, CFCs, Not greenhouse gases: N 2, O 2, Ar,

The total solar energy striking by the Earth per second can be calculated by multiplying Fs by the shadow area (not the total surface area!) of the Earth, i.e. the area of solar beam intersected the earth. SUN The amount of energy striking the earth is given by: [shadow area (black circle) the solar flux] =pr e2 F s. (R e = radius of earth). The total energy flux (Watts) striking the surface of the earth is therefore F s pr e 2

Energy INPUT to the Earth from the Sun Not all solar radiation intercepted by the Earth is absorbed. The fraction of incident solar radiation reflected is defined as the albedo, A, and the fraction absorbed is therefore (1-A). The total energy input to Earth (Joules per second) is thus INPUT E abs = F s pr e 2 (1 - A). Energy OUTPUT from earth by thermal radiation The total energy emitted per unit area is given by st 4, and the emitting area is the surface area of the earth, 4pR e2. The total energy emitted by the planet per second is therefore E emit = 4pR e2 st 4. OUTPUT

Earth's albedo for March, 2005 (CERES satellite) snow, ice, and clouds ALBEDO The term has its origins from a Latin word albus, meaning white. It is quantified as the fraction of incident solar radiation of all wavelengths reflected by a body or surface. snow, ice, and clouds are variable features of the earth

T e : the Effective Temperature of the Earth a consequence of Energy Balance Energy balance requires that input=output, when averaged over a longenough period of time, i.e. on average E emit = E abs. Thus 4pR e2 st 4 = F s pr e2 (1 - A). (This is the Energy Balance Equation). σt 4 = F s /4 (1 A) F s /4 = 340 W m -2 Mean emission m -2 = mean absorbed E m -2 A ~ 0.3 T of earth as viewed from space. T e = [ F s (1 A) / (4σ) ] ¼

RADIATIVE EQUILIBRIUM FOR THE EARTH Solar radiation flux intercepted by Earth = solar constant F S = 1370 W m -2 Radiative balance c effective temperature of the Earth: = 255 K where A is the albedo (fraction reflected= 0.3) of the Earth

Radiant energy input and output for the Earth (with atmospheric absorption) Solar radiation shortwave Thermal radiation longwave Atmosphere

BASIC MODEL OF THE GREENHOUSE EFFECT VISIBLE Incoming solar F S /4 Reflected solar FA/4 S Top of the atmosphere abs. 0 for solar (VIS) F S /4 FA/4 S

BASIC MODEL OF THE GREENHOUSE EFFECT IR Emitted &Transmitted from the surface (1 - f ) sto 4 Emitted by the atmosphere 4 fst 1 Top of the atmosphere Emitted by the atmosphere Emitted by the atmosphere 4 fst 1 4 fst 1 Surface emission 4 st o Atmosphere (T 1 ) abs f for terr. (IR) Emits in both directions! Earth surface (T o ) Absorption efficiency 1 in IR

BASIC MODEL OF THE GREENHOUSE EFFECT VISIBLE IR Incoming solar F S /4 Reflected solar FA/4 S Emitted &Transmitted from the surface (1 - f ) sto 4 Emitted by the atmosphere 4 fst 1 Top of the atmosphere Emitted by the atmosphere Emitted by the atmosphere 4 fst 1 4 fst 1 Atmosphere (T 1 ) abs. 0 for solar (VIS) f for terr. (IR) F S /4 FA/4 S Surface emission 4 st o Earth surface (T o ) Absorption efficiency 1-A in VISIBLE 1 in IR

Energy balance equations: Earth system Atmospheric layer Solution: T o F (1 - A) / 4 = (1 - f) st + fst S 4 2 4 o 1 fst = fst é ù ê FS (1 - A) ú = ê f ú ê4(1 - ) s ú ë 2 û 1 4 4 4 o 1 A=0.3 ; Fs = 1361 T o =287 K e f=0.76 T 1 = 241 K (effective atm T)

Atmospheric Radiation: The Earth receives energy from the sun (on average 344 W/m 2 ) and emits the same amount to space Source: http://www.learner.org/resources/series209.html

Chapter 5, section 4 Our equation is: F c = (1-0.5 f) s T o4, thus we identify greenhouse radiation G a = 0.5 f s T o 4 (When discussing climate change, radiative forcing is defined as the change in Ga since 1700)

The energy budget of the earth with climate forcing from CO 2 (and H 2 O feedback): Inclusion of heat capacity of the earth A Complete Climate Model Solar forcing Albedo IR absorption dh/dt= C p dt/dt = F s (1 - A)/4 - (1-0.5 f) st o 4 C p = heat capacity of seawater 4.184x10 6 x depth (m) Greenhouse radiation Human Feedback?? Feedback Interference And Feedback

Greenhouse effect (simple model): how do changes in G a relate to climate change? G a = 0.5 f s T o4 ; NOTE G a depends on T o, a physical requirement. At any instant, the earth is not at exactly its steady-state T. And (1-0.5 f) st o4 does not exactly equal F s (1 - A)/4 But the system is stable, dt o /dt à 0 (with fluctuations in clouds etc, seasons, orbital changes, )

Greenhouse effect (simple model): How changes in G a relate to climate change: two components. G a = 0.5 f s T o4 ; dg a /dt o G a response to surface T o dg a /dx climate response to factor x in addition to direct change with T o. e.g. df/dco 2 humans add CO 2 RADIATIVE FORCING è DG a = dg a /df df/dx Dx FEEDBACK è { RF à DTo à Df } Satellite data tell us how Ga changes with T o

Changes in G a vs. T o from satellite dg a /dt o = 2f st o3 + 0.5 st o4 df/dt o = 3.53 ç empirical (satellite)

Greenhouse effect (simple model): water vapor feedback estimated using observed change in G a with surface T o G a = 0.5 f s T o4 ; dg a /dt o = 2f st o3 + 0.5 st o4 df/dt o = 3.53 ç empirical (satellite) 1 2! 2.04 = 2f st o3 change in G a due only to higher surface T (downwelling IR response to increasing T o ); 3.53 2.04 = 1.5! 1.5 = 0.5 st 4 o dln((h 2 O))/dT o df/d(ln(h 2 O)) 0.5 x 384.79 x 0.065 x df/d(ln(h 2 O)) Clausius- Clapeyron Eq. èdf/d(ln(h 2 O))= 0.12 effective change layer abs. from H 2 O è Check: 7% D H 2 O (4K Ts) gives DG a = 12 W m -2, Fig. 5.11; Ours: 6.5% in H2O è 3.5 W m -2 (1.5 due to df, 2 from dt o ) Note 4K Ts è 30% change in H2O so 5.11 not const RH è Is there also a cloud feedback (higher T or H 2 O, more high clouds?)

Δ(Radiant Energy Flux) Climate forcing due to human caused changes in concentrations of greenhouse gases, atmospheric aerosols, and clouds, since 1850 (Hansen, 2001). NET=1.6 Longwave forcing: change thermal emission Shortwave forcing: change Albedo

IPCC [2007]

This slide and the following 4 slides are for background only. Not part of the lecture or the course. The global average G a is 131 W m 2 or the normalized g a is 0.33, i.e., the atmosphere reduces the energy escaping to space by 131 W m 2 (or by a factor of 1/3). The ocean regions have a slightly larger greenhouse effect (0.35 for ocean vs. 0.33 for land) compared with the land (Ramanathan Fig.5.7b). In order to get another perspective on the results shown in Figure 5.7, we note that a doubling of CO 2 (holding the surface and atmospheric temperature fixed) will enhance G a by about 4W m 2. We should note that the g a shown in Figure 5.7 includes the greenhouse effect of water vapor and all other greenhouse gases including CO 2, O 3, and several trace gases. [not clouds?] Kiehl and Ramanthan

Background only. Not part of the lecture or the course. Units: W m -2

Background only. Not part of the lecture or the course.

Results from our simple model: Climate change in the industrial era

Results from our simple model, compare to real world temperature rise

Simple Model: Contribution of GHG changes over time to Delta-T during glacial cycles 800Kyr BP Present

l 25 16.7 12.5 10 8.3 7.1 µm The actual thermal emission of the Earth IRIS on Nimbus 4 ATM window R. Hanel et al., Applied Optics, 1971

THE GRAY ATMOSPHERE MODEL In a purely radiative equilibrium atmosphere T decreases exponentially with z, resulting in unstable conditions in the lower atmosphere; convection then redistributes heat vertically following the adiabatic lapse rate Integrate over z dz Absorption ~ ρ(z)dz σt o 4 surface

GENERAL CIRCULATION MODELS (GCMs) Standard research tools for studying the climate of the Earth Solve conservation equations for momentum, heat, and water on global 3-D atmospheric domain Horizontal resolution ~100 km Include coupling to ocean, land,biogeochemistry, atmospheric chemistry to various degrees Solution to equations of motion is chaotic, so that a GCM cannot simulate an observed meterorological year; it can only simulate climate statistics including interannual variability A GCM can be tested by its ability to simulate presentday climate statistics in a repeatable manner when run in radiative equilibrium (equilibrium climate simulation) A radiative imbalance (such as changing concentrations of greenhouse gases) will result in warming or cooling in the GCM

RADIATIVE FORCING OF CLIMATE CHANGE F in F out Incoming solar radiation Reflected solar radiation (surface, air, aerosols, clouds) IR terrestrial radiation ~ T 4 ; absorbed/reemitted by greenhouse gases, clouds, absorbing aerosols EARTH SURFACE Stable climate is defined by radiative equilibrium: F in = F out Instantaneous perturbation e Radiative forcing DF = F in F out Increasing greenhouse gases g DF > 0 positive forcing The radiative forcing changes the heat content H of the Earth system: dh dt DT o l D T = ldf = DF - eventually leading to steady state o where T o is the surface temperature and l is a climate sensitivity parameter IPCC GCMs give l = 0.3-1.4 K m 2 W -1, insensitive to nature of forcing; differences between models reflect different treatments of feedbacks. Layer model (no feedback) gives slightly less than 0.3).