Radiative Equilibrium and the Lapse Ratechem.atmos.colostate.edu/AT620/Sonia_uploads/Radiation_Climate.pdf · George Mason University / COLA 1 ... Radiative Equilibrium and the Lapse

David StrausGeorge Mason University / COLA

1

Atmospheric Radiative Transfer and Climate: Introduction

Radiative Equilibrium and the Lapse Rate

Let us return to the simple radiative model of a previous section

(see next page):

The atmosphere and the surface are each characterized by a single temperature, TA and Ts respectively.

The atmosphere is considered a single layer, which emits radiation upward and downward, in each case with the flux density given by the Stefan-Boltzmann law.

We will make one refinement, namely to let the atmosphere transmit some of the surface radiation directly to space. This is accomplished by letting the single layer representing the atmosphere absorb radiation from the ground with a < 1

Sonia Kreidenweis

www.iges.org/straus/CLIM_710/Radiation_Climate.pdf

Sonia Kreidenweis


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Figure 2.3 Hartmann

An extremely simple model of the radiation transfer:

Think of the atmosphere as a finite layer of air above the ground


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Atmospheric Radiative Transfer and Climate

List of assumptions:(1) The incoming flux density (1-!) S / 4 is absorbed at the surface

(2) The surface radiates as a black body with brightness temperature

of Ts: The upward flux density at the surface is ! Ts4

(3) The atmosphere layer radiates upward and downward as a black

body, with brightness temperature Ta, giving flux density ! Ta4

both upward and downward

(4) The atmosphere absorbs a ! Ts4 of the upward surface flux, and

the remainder (1-a) ! Ts4 is transmitted and escapes into outer

space

Number 4 is a new assumption – It seems to be inconsistent with 3. How can the atmosphere emit as a perfect black body, but not

asborb as one? Isn’t this inconsistent with Kirchoff’s Law?


4


It turns out that these assumptions are not contradictory, in part because the atmospheric temperature decreases with height, so that the radiation field upwelling from the ground has a higher temperature associated with it than the absorbing atmosphere. (Thus the conditions under which Kirchoff’s Law was derived are violated.) Local equilibrium is possible by assuming a<1 for absorption only

The brightness temperature of the whole earth-atmosphere system is

Te = (1-")S/4, while the energy balances at the surface, and for the

atmosphere are:

()44441/42assaSTTaTT!""""#+==

(1−!)S/4+"T4

a="T

4

s

a"T4

s=2"T 4

a


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These equations can be easily solved:

As before the overall brightness temperature Te is 255 K

We thus get these solutions for the atmospheric and surface temperatures, and their difference:

1/41/4222seaeTTaaTTa!"=#$%&'!"=#$%&'

a Ta Ts Ts-Ta

1.00 255 303 48

0.77 227 288 59

Ts=�

2

2−a

�1/4

Te

Ta=�

a

2−a

�1/4

Te


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• With a less opaque atmosphere (less absorption), both the atmospheric temperature and the surface temperature decrease (in fact now the surface temperature equals the observed global mean)

• The difference between the surface and atmospheric temperatures, which is a crude measure of the lapse rate, also increases

• This simple calculation gives some insight into the poleward energy transport by the atmosphere and oceans:

– The brightness temperature Te associated with the north pole is ~205 K

– The brightness temperature Te associated with the equator is ~290 K

– Converting this 85 K difference to a surface temperature (Ts) difference

using the solution on the previous page yields ~96 K difference in Ts

– But the observed equator-to-pole temperature difference is only about 35 K

• Clearly the energy transport by the atmosphere and oceans is very effective in reducing the surface temperature difference !


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Vertical Heat Transport

• Radiative cooling to space takes place mainly from the atmosphere

• Radiative heating takes place mainly at the ground

• Therefore in equilibrium the atmospheric motions must transport heat vertically!

• “Radiative Equilibrium” – this refers to the dependence of temperature with height, taking into account

– The radiative properties of water vapor, carbon dioxide, aerosols and ozone

– The surface albedo

– The radiative effects of high, medium and low clouds

Figure 3.16H shown on the next page gives an example


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Example of Radiative Equilibrium T Profile

(From Hartmann)


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• Note the very steep lapse rate, certainly larger than the observed value of ~ 6.5 K per km

• This is consistent with the overall radiation balance we saw earlier, which showed that sensible and latent heat transport from the surface to the atmosphere was an important part of the surface budget

• In fact, the radiative equilibrium temperature profile is statically unstable and cannot exist in nature


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For an ideal gas, the entropy (per unit mass) s is related to the

potential temperature # as:

Where the potential temperature is given by:

Here $ = R/Cp, and the specific

heat at constant pressure Cp is

(for an ideal gas) 7/2 R. Also, p0

= 1000 mb .

0ln()PsCpTp!="#$"=%&'(

Entropy and Static Stability

(summary of notes)

s=Cp log(!)

! =�p0

p

�"

T


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The condition that convection is absent

(see notes)

Basic argument:

5) For any given vertical profile of the atmosphere, the environmental pressure and entropy (p,s) are a function of altitude z

6) Consider a motionless parcel of air in equilibrium at level z1, where

this level is characterized by pressure and entropy (p1,s1)

7) Displace this parcel adiabatically (no change in entropy, so no heating) upward to a new level z2 characterized by (p2,s2)

8) The pressure of the parcel adjusts very rapidly to the new environment (assumes hydrostatic balance). But no heat has been added to the parcel, so its entropy has not changed. Thus p=p2, but

s=s1


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Basic argument (con’t)

The parcel now has a specific volume ! (volume per unit mass) of

!(p2,s1)

If this volume is greater than that of its new environment, namely if

!(p2,s1) > !(p2,s2), the parcel is lighter than its environment and will

keep rising. The profile is statically unstable – convection will develop

But if this volume is less than that of its new environment, namely if

!(p2,s1) < !(p2,s2), the parcel is heavier than its environment and will

return to its original level. The profile is statically stable

If !(p2,s1) = !(p2,s2) the profile is neutral


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We have:

The derivative

can be shown to be positive definite in general. For an ideal gas, this is

easily calculated:

Since "z >0,

!(p2,s1)−!(p2,s2)=�"!

"s

�

p

(s1− s2)

=−�"!

"s

�

p

"s

"z#z

�!"

!s

�

p

=

��!s

!"

�

p

�−1

�!"

!s

�

p

=pCp

RT


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In terms of the entropy s, the profile stability criteria become:

! ds/dz < 0 for statically unstable profile

! ds/dz = 0 for neutral profile

! ds/dz > 0 for statically stable profile

In terms of potential temperate %, the criteria become:

! d%/dz < 0 for statically unstable profile

! d%/dz = 0 for neutral profile

! d%/dz > 0 for statically stable profile


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In terms of the static stability, & = -dT/dz this becomes:

! & > g/Cp for statically unstable profile

! & = g/Cp for neutral profile

! & < g/Cp for statically stable profile

The ratio g/Cp is known as the dry adiabatic lapse rate &d


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Returning to the radiative equilibrium calculation shown in Figure 3.16H:

• It is clear that the profile is statically unstable

• Convection must take place to transport heat upward to reduce the lapse rate

•

• This is one of the mechanisms for sensible heat transfer from the surface to the atmosphere

• Concept of Convective Adjustment: Assume that whenever the radiative equilibrium lapse rate exceeds some pre-specified value (either the dry adiabatic lapse rate or an empirically defined rate such as 6.5 degrees / km), upward heat transport occurs to restore the lapse rate to the pre-specified value, while conserving energy


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Example of Radiative-Convective Equilibrium T Profile

From Hartmann


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The temperature profile computed from radiative equilibrium with convective adjustment is called radiative-convective equilibrium. This is basically given by the specified lapse rate in the lower troposphere and radiative equilibrium further aloft.

The effects of water, carbon dioxide and ozone on the radiative-dry convective equilibrium profile are indicated in Figure 3.17H.

• The basic profile in the troposphere with all three gases present is nearly achieved with water alone – in the lower troposphere the adjustment yields 6.5 K / km

• Adding carbon dioxide raises the temperature by about 10 K

• Ozone is necessary to achieve the increasing temperature with height characteristic of the stratosphere


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The contribution of individual gases to the actual heating rate in pure radiative equilibrium is shown in Figure 3.18H

• In the stratosphere, ozone heating is balanced by cooling due to long-wave carbon dioxide emission

• In the troposphere, net radiation leads to cooling – this cooling is partly balanced by heating due to sensible heat transfer from the surface

• In the troposphere, the cooling is partly balanced by latent heating due to rainfall


22From Hartmann


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Radiative Effects of Clouds

Clouds reflect incoming solar radiation, preventing it from reaching the surface

Clouds also emit long-wave radiation, both upward and downward

Some simple estimates of the effects of different types of clouds on the radiative-convective equilibrium temperature profile are given in Figure 3.19H

• Low Clouds are highly reflective, and emit long-wave radiation at a T nearly that of the surface. They tend to decrease T (net cooling)

• High clouds radiate to space with a much lower brightness T, so they tend to increase T (net heating)


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Radiative-Convective Equilibrium with Clouds

From Hartmann


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A simple model of the net radiative effects of clouds

The energy balance at the top of the atmosphere is just:

RTOA = SO/4 (1-") – F

F is the outgoing energy flux due to outgoing long-wave radiation (OLR)

RTOA will vanish if averaged over a year and over the earth

The difference between RTOA for cloudy and clear situations can be

written:

' RTOA = Rcloud – Rclear = - SO/4 ("cloud - "clear)

– (Fcloud - Fclear)


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Now defining "cloud-"clear = '" and using the black body assumption

for the emission of clouds to space: Fcloud = ! Tc4, where Tc is the

temperature of the cloud top, we obtain:

'RTOA = -SO/4 '" + Fclear – !Tc4

where we have assumed that the cloud top is above the long-wave absorbers which are not related to clouds. This makes sense above about 4 or 5 km altitude, since most of the absorber (water vapor) is in the lower troposphere


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Making the simple assumption that the cloud top temperature varies

linearly with height, following a constant lapse rate &:

Tc = Ts – & zc

where Ts is the surface temperature and zc the cloud top height, we

can plot 'RTOA as a function of '" and zc, as in Figure 3.20H


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Note that

• High clouds lead to warming, unless there is a large difference in albedo (strong reflection)

• Intermediate clouds lead mostly to cooling

• Low clouds are not handled by this simple model


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The following table from Hartmann estimates the global and annual average radiation budget of the earth-atmosphere system measured by satellite.

What are given are the radiative flux densities (in W/m2), and the albedo (in percent). The “average” column gives the total, and the “Cloud-free” using only clear areas of the atmosphere. The “Cloud Forcing” is estimated as the difference

Average Cloud-Free Cloud Forcing

OLR 234 266 +31

Solar 239 288 -48

Net +5 +22 -17

Albedo (%) 30 15 -15


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Geographical Variation of Clouds

Maps of the annual average cloud coverage (measured by fractional area coverage) are shown in Figure 3.21H. Panel (a) shows high cloud (tops with pressure less than 440 hPa). Panel (b) shows low cloud (tops with pressure greater than 680 hPa). (c) shows all clouds

• High clouds are concentrated in the major tropical convection zones over South America, Africa, Indonesia

• Low clouds are seen most in subtropical eastern oceans, where they are associated with lower than average SST, and consist of stratocumulus clouds trapped below an inversion


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Cloud Fractional Coverage(in percent)

High Cloudsp(top) < 440 hPa

Low Cloudsp(top) > 680 hPa

All Clouds


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Geographical Variation of Clouds (con’t)

• Low clouds are concentrated over oceans

• Total cloudiness shows a preference for mid-latitudes, associated with the “storm tracks”

• Minima in total cloudiness are seen over deserts in particular, but over the subtropics in general


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The annual averaged radiative cloud forcing is shown in map form in Figure 3.22H.

• Panel (a) shows the reduction of OLR caused by clouds, which is greatest in precisely those tropical convection regions where the high clouds are present

• Panel (b) shows the increase in solar absorption by clouds. It is negative (decreased absorption) over the tropical convection regions, where the high clouds are very reflective

• Low clouds at high latitudes are also highly effective in reducing solar absorption, partly because the albedo increases with zenith angle

• Since OLR and solar absorption effects tend to cancel, the net effect of clouds is smaller than either component

• The net effect is largest near the poles (low clouds) and in the stratus cloud regimes in the eastern subtropical oceans


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Cloud ForcingAnnual Average

(W / m2)

Reduction of OLR

Increase in Absorbed Solar Radiation

Net Increase in net radiationdue to clouds

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Radiative Equilibrium and the Lapse Ratechem.atmos.colostate.edu/AT620/Sonia_uploads/Radiation_Climate.pdf · George Mason University / COLA 1 ... Radiative Equilibrium and the Lapse