Parameterization of Radiation

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IACETH Institute for Atmospheric and Climate Science

Parameterization of Radiation

Ulrike Lohmann (IACETH) Parameterization of Radiation June 7, 2007 1 / 41

Global mean considerations Param. of radiation SW radiation SW & clouds Longwave radiation

Flow chart from the ECHAM GCM

IACETH stitute for Atmospheric and Climate Science

Parameterization

flow chart

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Shortwave and longwave radiation

Figure: Shortwave (SW) = solar; longwave (LW) = terrestrial

Global mean considerations

I Balance of incoming and outgoing radiation:

F_net=F_SW−F_LW (1) whereF_net is the net radiative flux. The globally averaged absorbed solar energy is:

FSW =So/4(1−αp) = 342Wm⁻²·0.69 = 235Wm⁻² (2) whereS_o is the solar constant andα_p the

top-of-the-atmosphere (TOA) planetary albedo.

I The factor 4 arises because a surface area of 4πa²emits LW radiation, but only an area ofπa²intercepts SW radiation.

I In equilibrium:

Fnet= 0 (3)

Global mean considerations

I If the system is perturbed (e.g. by doubling CO2), then the initial balance will be perturbed:

∆F_net= ∆F_SW−∆F_LW (4)

where ∆ represents the perturbation to the system.

I For a balance to re-establish, it is assumed that the surface temperature changes by ∆Ts, thus:

∆Ts

∆FLW

∆Ts

−∆FSW

∆Ts

=−∆Fnet=G (5) whereG is called the direct radiative forcing to the climate system (G = 4 W m⁻²) for a doubling of CO₂.

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Climate sensitivity

I The relation between the change in globally averagedT_s and G is:

∆T_s=λG (6)

I λ, the climate sensitivity parameter, is defined as:

λ= 1

∆FLW

∆Ts −^∆F_∆T^SW_s (7)

I λdetermines the magnitude of the response of the climate system.

Climate sensitivity

I Furthermore, the dependence ofλon other climate processes is evident upon expanding the LW and SW factors in terms of a Taylor series:

∆F_LW

∆T_s =∂F_LW

∂T_s +∂F_LW

∂W

∆W

∆T_s +∂F_LW

∂A_c

∆Ac

∆T_s +... (8)

I

−∆FSW

∆T_s = So

4

∂αp

∂T_s + ∂αp

∂A_v

∆Av

∆T_s +∂αp

∂A_c

∆Ac

∆T_s +... (9)

I whereW,AcandAv are the globally averaged column water vapor amount, cloud amount and vegetation amount.

Climate sensitivity

I Without feedbacks,FLW is given by the Stefan-Boltzmann law:

FLW =σT_s⁴ (10)

I Neglecting all other feedback except the above, we obtain the no-feedback climate sensitivity parameter:

λ= 1

∆FLW

∆Ts

= Ts

4F_LW = 288K

4·235Wm⁻² = 0.3KW⁻¹m² (11)

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Feedbacks

Figure:Ruddiman, 2001, Figure 18-4

Water vapor feedback

I Because much of the Earth’s surface is wet, the humidity of air near the surface tends to remain close to the saturation vapor pressure of air (Clausius-Clapeyron equation)

I Thus the change in saturation specific humidity divided by the actual value of specific humidity is related to the change in temperature by the actual temperature:

dq_s qs

= de_s es

= L

RvT dT

T (12)

I For terrestrial conditions (L/R_v T)∼20, so that a 1% change in temperature, which is about 3^◦C, is associated with about a 20% change in saturation specific humidity.

Water vapor feedback

I It is observed that RH in the atmosphere tends to remain constant

I Estimates of the water vapor feedback on climate sensitivity yield that the outgoing longwave radiation (OLR) emitted from the planet increases much less rapidly with temperature than suggested by the Stefan-Boltzmann law.

I Moreover, the terrestrial emission increases linearly with surface temperature, rather than to the power 4

I This yieldsλ= 0.5 K W⁻¹m², i.e. adding the water vapor feedback nearly doubles the estimated sensitivity of the climate.

I Processes that increaseλthey are referred to as positive feedbacks.

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Figure:Hartmann: Global Physical Climatology, Figure 9-1

Cloud feedback

I Clouds double the albedo of the Earth from 15m⁻²) but reduce the OLR by∼30 W m⁻².

I Net TOA effect of clouds: F_cloud∼-20 W m⁻².

I Estimate the cloud effect on climate noting that the global area coverage of clouds,A_c, is∼60%:

∂F_cloud

∂Ac

∼ ∆F_cloud Ac

∼ −20Wm⁻²

0.6 =−33Wm⁻² (13)

I Thus, a 10% change in cloudiness has the same magnitude on the energy balance as a doubling of CO2. Decreasing the cloudiness by 10% would double the effect of CO₂doubling.

I However, the radiative effect of an individual cloud depends on its height and optical thickness, the insolation, and the characteristics of the underlying surface

Parameterization of atmospheric radiation

I GCMs require the net (upward and downward) radiative fluxes at the TOA and the surface, and internal atmospheric heating rates.

I The TOA fluxes are required for determining the overall energy budget of the surface-atmosphere system.

I The surface radiation fluxes are components of the surface energy balance (see lecture from May 3):

C_L∂T_s

∂t =F_net+LE+H+G (14) where

F_net= (1−α_s)F_SW^↓ +F_LW^↓ −σT_s⁴ (15)

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Parameterization of atmospheric radiation

The net radiative heatingQrad is needed in the dry static energy equation (see last lecture):

∂s

∂t+~v· ∇s+ω∂s

∂p =−∂ ω⁰s⁰

∂p +L(C−E) +Q_rad (16) with

Qrad=QSW+QLW (17) Obtain the heating terms (Q_SW,Q_LW) from the divergence of the net radiative flux (from the first law of thermodynamics with dp= 0):

Qrad =ρcp

dT

dt =−∇~F=−d

dz(F^↑−F^↓) (18) whereF^↑ andF^↓are the upward and downward radiative fluxes.

Radiative transfer equation (RTE)

I RTE is a conservation equation for radiant energy : cosΘdI

dτ =I−J (19)

whereI denotes the intensity at an angle Θ from the upward normalk,τ is the optical depth:

τ= Z ∞

z

ρkdz (20)

which is 0 at TOA and increases downward and J= 1

πB(T) (21)

is the blackbody source function with

B(T) =σT⁴ (22)

I Photons can either be absorbed or scattered as they propagate through the atmosphere.

Objectives of radiation parameterizations

I CalculateF_s^net andF_l^net for clear and/or cloudy conditions at each gridpoint of the model domain.

I The processes that need to be parameterized are essentially of molecular scale for gaseous absorption and micrometer scale for particulate scattering.

I Since the source of radiation is quite different for SW and LW radiation, these processes are considered separately

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Beer’s law

= Attenuation of solar radiation by absorbers and scatters in the atmosphere

wherek_λ = extinction coefficient (absorption and scattering/unit mass),I_λ0= insolation at TOA,I_λ= shortwave flux at heightz

Attenuation of solar radiation in the atmosphere

I Integration of Beer’s law:

Z z⁰=z

z⁰=0

dIλ

I_λdz⁰= Zz⁰=z

z⁰=0

−ρk_λdz⁰ (23)

I with boundary conditionsI_λ=I_λ0atz⁰= 0 andI_λ =I_λ at z⁰=z:

I_λ=I_λ0exp Zz⁰=z

z⁰=0

−ρk_λdz⁰

!

=I_λ0exp(−τ)≡I_λ0q_λ (24)

I withτ=Rz⁰=z

z⁰=0 ρk_λdz⁰= optical depth andq_λ=_I^I^λ

λ0 = transmission

I q= 0.7 (sun perpendicular, no clouds)

I q<0.65 (clouds are present)

Attenuation of SW radiation in the clear-sky atmosphere

I Optical depth: τ=Rz⁰=z z⁰=0 ρk_λdz⁰

I Under cloud-free conditions: Extinction coefficient k=kA+kR+kM, where

I k_A= gas absorption (H2O, CO2, O3, O2,....)

I k_R = molecular scattering (Rayleigh scattering,r_scatter >0.1λ)

∼λ⁻⁴

I kM = aerosol scattering (Mie scattering,rscatter = 0.1−25λ)

∼λ^−1.3

I Clear sky transmittanceq_clear: q_clear=e^(−τ) = e⁻^R^z

0=z

z0=0ρk_Adz⁰e⁻^R^z

0=z

z0=0ρk_Rdz⁰e⁻^R^z

0=z z0=0ρk_Mdz⁰

= q_Aq_Rq_M (25)

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Ozone absorption

Figure:Trenberth [Figure 10-9]

Two-stream method

I Obtain closed form solutions to the original RTE by dividing the radiation into a constant upward and downward stream of radiant energy (2-stream method) assuming horizontally isotropic radiation:

dF^↑

dτ = F^↑−B(T) (26)

−dF^↓

dτ = F^↓−B(T) (27)

I where

F^↑,↓= Z

2π

~I·~kdΩ^↑,↓ (28)

I Need multiple layers for the entire atmosphere to obtain fluxes and heating rates.

Shortwave clear-sky heating rates

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High clouds and shortwave radiation

Figure:Cloud albedo∼0.2

Low clouds and shortwave radiation

Figure:Cloud albedo∼0.6-0.7

Shortwave radiation in a cloudy atmosphere

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Solar radiation in cloudy atmosphere

I Ray tracing method: approach for including scattering in a GCM. It assumes that solar radiation can be represented by beams propagating through the atmosphere

I Consider the case of a single cloud layer in a non-absorbing clear-sky atmosphere

I α_c: cloud albedo,t_c: transmissivity of the cloud,α_s: surface albedo

I αa: part of the initial SW radiation beam that is reflected back to space

I The remaining fraction will pass through the cloud and be reflected from the surfacet_cα_s.

I This beam is then partially reflected and transmitted through the cloud.

Solar radiation

I The beam can be followed through a number of reflections and transmissions.

I The first few terms in this scattering process yield a planetary albedo of:

α_p=α_c+t_c²α_s+t_c²α²_sα_c+t_c²α³_sα²_c+t_c²α⁴_sα³_c+.... (29) which can be factored as:

α_p=α_c+t_c²α_s(1 +α_sα_c+ (α_sα_c)²+ (α_sα_c)³+...) (30) or

αp=αc+ t_c²αs

1−α_sα_c (31)

Solar radiation

I The amount of SW radiation absorbed in the presence of clouds is

SW_abs = S_o

4(1−α_p) (32)

I while the amount absorbed for clear sky conditions is SW_abs^cs = So

4(1−αs) (33)

I For a highly reflecting surface the combined albedo of cloud and surface is lower than the clear-sky albedo, i.e. more solar radiation is absorbed in the Earth-atmosphere system.

I Example: stratus clouds over snow and ice, such as in the northern and southern polar regions.

I Disadvantage: The ray tracing method is cumbersome for multiple cloud layer cases.

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Shortwave radiation

Shortwave radiation in a cloudy atmosphere

Schwarzschild’s law for LW (> 4 µm) radiation

I SW: only extinction, no emission: ^dI_dz^λ =−ρkλIλ I LW: both emission and absorption: ^dI_dz^λ =−ρkλ(I_λ−B_λ)

I B_λ(T) = blackbody emission of the material

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Longwave radiation

I The most important absorber of LW radiation in the Earth’s atmosphere is water vapor followed by CO2, which has dominant effects in the upper stratosphere, then followed by ozone.

I Methane, N2O and CFCs (the other trace gases) are also radiatively significant and thus need to be included in GCMs

I As in the case of SW radiation, including the absorption of radiant energy by all of the above mentioned gases would require integration over 10,000s individual narrow absorption lines.

I For computational efficiency, implicit integration of these lines is provided byband models.

Band models

I Band models calculate the spectral absorption Aν for relatively large (10-100 cm⁻¹) wavenumber interval widths, ∆ν.

I These models assume some analytic expression for the distribution of spectral lines, and a line shape function.

I Broad band models define an absorptivityαand emissivityas:

α(p,p⁰) = R∞

0 A_ν(p,p⁰)^dπB_dT^ν^(p⁰⁾dν

dπB dT

(34) (p,p⁰) =

R∞

0 Aν(p,p⁰)πBν(p⁰)dν

πB dν (35)

respectively, whereB=σT⁴.

Band models

I With that the flux equations have the simple form:

F⁻(p) = σT⁴(p₀)(p,p₀) + Zp

p0

α(p,p⁰)T⁴(p⁰)dσ F⁺(p) = σT⁴(p_s)−

Z ps

p

α(p,p⁰)T⁴(p⁰)dσ (36) The advantage of those 2 equations is the implicit integration overν.

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Longwave clear-sky cooling rates in the tropics

Longwave clear-sky cooling rates

I Cooling by H₂O dominates the troposphere with peak cooling rates near the surface of -3 K/d, whileCO2 and O3are most important for stratospheric cooling.

I The large H₂O cooling near the surface is due to continuum absorption.

I Thus, for moist environments (e.g., the near surface tropical atmosphere), the absorption and re-emission of LW radiation by the H2O continuum in the 8-12µm spectral region plays a significant role, and is responsible for the peak near-surface cooling in Figure 10-14.

Longwave radiation in the cloudy atmosphere

I Include clouds in the above equations by separating the total flux into a cloudyFo and a clear-sky fluxFclr:

F=F_clr(1−A_c) +F_oA_c (37)

I EvaluateF_o from (36) by replacing the surface and TOAT with cloud top and cloud baseT.

I LW processes tend to warm cloud base due to the net convergence of radiation from the warm surface into the cold cloud base

I LW processes cool cloud tops because the cloud top loses radiant energy to space

I The above equation and cloud overlap assumptions are very simplistic compared to real cloud-radiation interactions; yet they are extensively used in climate models.

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Radiative heating of clouds

Figure:Webster and Stephens [1980]

Cloud overlap

Figure:

www.met.reading.ac.uk/radar/research/cloudoverlap/index.html