Transcript Intermediate complexity models

Intermediate complexity models
• Models which are like
comprehensive models in
aspiration but their
developers make specific
decisions to parametrize
interactions so that the
models can simulate tens to
hundreds of thousands of
years
1D radiative-convective models
• Greenhouse absorbers do not only affect
the surface temperature but also they
modify atmospheric temperature by their
absorption and emission.
• Radiative-convective (RC) were developed
to study these effects
• RC models are one-dimensional models like the
EBMs with many vertical layers
• These models resolve many layers in the
atmosphere and seek to compute atmospheric
and surface temperatures.
• They can be used for sensitivity tests and they
can offer the opportunity to incorporate more
complex radiation treatments than can be
afforded in GCMs
• Suppose there are 2 layers in the atmosphere and each layer absorbs
the incident radiation on it  infrared optical thickness =1
• The principal absorber in the Earth’s atmosphere
is water vapor which is contained almost entirely
within the first few kilometers.
• The 2 layers are therefore assume to be
centered at 0.5 and 3 Km.
• Both layers radiate above and below as black
bodies and the ground radiates upwards
• All radiation from the
planet must be
absorbed by the top
layer 
T1 = Te
• The energy balance of
the lower atmospheric
layer is
sT24 = 2sT14 = 2sTe4
• It can be shown that for n layers , the
temperature of layer n can be related to the
effective temperature Te by:
Tn4 = ttotal(n)* Te4
with ttotal(n) being the total infrared optical
thickness from the top of the atmosphere to the
layer n
In the previous case, as each layer has t = 1 then
ttotal(n) = n.
• Surface temperature can
be obtained as
2sT24 = sTg4+ sT14
 Tg = (3Te4)1/4
The structure of global
radiative-convective models
• The RC model can be
seen as a single column
containing the
atmosphere and
bounded beneath by the
surface
• The radiation scheme is
detailed and occupies
the majority of the total
computation time while
the ‘convection’ is
accomplished by
numerical adjustment of
the temperature profile
at the end of each
timestep
•
•
The atmosphere is
divided into a number of
layers not necessarily of
equal thickness.
Layering can be defined
with respect to height or
pressure but it is more
common to introduce the
non-dimensional vertical
coordinate s , not to be
confused with the StefanBotlzmann constant)
s = (p-pT)/(ps-pT)
With p being the pressure, pT
the (constant) top of the
atmosphere pressure and
ps the (variable) pressure
at the Earth’s surface.
• The top of the
atmosphere has s = 0
where the surface has
always s = 1.
• Compared with the standard
lapse rate (6.5 K/Km) the
computed radiative temperature
profile is unstable
• If a small parcel of air were
disturbed from a location close to
the surface it would tend to rise
because it would be warmer than
the surrounding air. Its
temperature would decrease at
roughly the observed lapse rate
so that at a given height H its
temperature would be higher than
that of the atmosphere and it
would continue to rise.
• This air would carry energy
upwards and the resulting
convection currents would mix
the atmosphere
• This convective adjustment of
a radiatively produced profile
is the essence of RC
• Model from CD ?
• Simple case: single cloud or aerosol layer is spread
homogeneously over the surface
• 1) Part of the incident
S radiation is reflected
(acS) part is absorbed
(ac(1-ac)S and part is
transmitted ((1-ac)(1ac)S)
• 2) The transmitted radiation interacts with the surface. Part of it
is absorbed ((1-ag)(1-ac)(1-ac)S), part is reflected (ag(1-ac)(1ac)S) which, in turn, is absorbed by the cloud (acag(1-ac)(1ac)S) and transmitted ((1-ac)ag(1-ac)(1-ac)S)
• 3) the cloud emits as well (esTc4) as well as the surface (sTg4)
,which is partially absorbed by the cloud (esTg4) and
transmitted ((1-e)sTg4)
•
Three main assumptions have been
made:
1) No reflection of the upwelling shortwave
radiation by the cloud
2) The surface emissivity has been set to 1
3) The cloud/dust absorption in the infrared
region is equal to e
• When equating absorbed, emitted and reflected radiation at
each level we have:
Eq. 1) S = acS + ag(1-ac)(1-ac)S + esTc4 + ((1-e)sTg4)
Eq. 2) ac(1-ac)S + acag(1-ac)(1-ac)S + esTg4 = 2 esTc4
Eq. 3) (1-ac)ag(1-ac)(1-ac)S + esTc4 = sTg4
Eq. 1) S = acS + ag(1-ac)(1-ac)S + esTc4 + ((1-e)sTg4)
Eq. 2) ac(1-ac)S + acag(1-ac)(1-ac)S + esTg4 = 2 esTc4
Eq. 3) (1-ac)ag(1-ac)(1-ac)S + esTc4 = sTg4
The above equations can be solved directly by giving
values for the dust/cloud shortwave absorption,
albedo, infrared emissivity and surface albedo.
In alternative, the surface albedo term can be
eliminated leaving an expression for Tg:
sTg4 = ((1-ac)S)*(2-ac)/(2-e)
•
From sTg4 = ((1-ac)S)*(2-ac)/(2-e)
Consider S = 343 W/m2
1) Cloudless case
ac = 0.08 (scattering by atmospheric molecules alone),
ac = 0.15 and e = 0.4  Tg = 283 K
2) Cloudy skies
Volcanic aerosol
ac = 0.12, ac = 0.18 and e = 0.43 Tg = 280 K 
Cooling !
Water droplet cloud
ac = 0.3, ac = 0.2 and e = 0.9 Tg = 288 K  Warming !
The Greenhouse effect of the cloud is greater than the
albedo effect
More on radiation
• We saw that: solar radiation is absorbed
and infrared radiation is emitted, with
these two terms balancing over the globe
when averaged over a few years
• We also saw that RC models pay a lot of
attention to the radiative component
• Let us see how these models attack the
problem
Shortwave radiation
• Shortwave incoming radiation is simply divided into two
parts, depending on wavelength, with the division being
somewhere around 0.7 – 0.9 mm.
• The 2 wavelength regions can either be treated
identically or absorption and scattering can be
partitioned by wavelength.
• Rs stands for the shortwave part where Ra stands for
the near infrared part
• Rs is ~ 65 % of the total
•  Ra is ~ 35 % of the total
• Therefore:
Rs = 0.65*S*cosm
Ra = 0.35*S*cosm with m being the solar zenith angle
Albedo
The albedo of the clear atmosphere in the shortwave is subject to
Rayleigh scattering and it is given by:
a0 = min[1, 0.085-0.247*log10((p0/ps)*cosm))
For overcast atmosphere the albedo for the scattered part of the
radiation is composed of the contribution of Rayeigh scattering
(atmosphere molecules) and of Mie scattering (water droplets). The
simplest used formulation is:
aac = 1-(1- a0)(1- ac)
Where ac is the cloud albedo for both Rs and Ra
Albedo contd.
• spectral dependence must be introduced
Shortwave radiation subject to
scattering
• The part of solar radiation that is assumed to
be scattered does not interact with the
atmosphere.
• Thus, the only contribute to which we are
interested in is the amount that reaches, and
is absorbed by, the Earth’s surface, given
by:
R' sg = Rs
(1  a g )(1  a 0 )
Clear sky
(1  a ga 0 )
R' ' sg = Rs
(1  a g )(1  a ac )
(1  a ga ac )
Cloudy sky
• Multiple reflections between sky and ground or between
cloud base and ground are accounted for by the terms in
the denominators
R' sg = Rs
(1  a g )(1  a 0 )
(1  a ga 0 )
R' ' sg = Rs
(1  a g )(1  a ac )
• For partly cloudy conditions:
Rsg = R' ' sg * N  R' sg * (1  N )
Being N the fractional cloudiness of the sky
(1  a ga ac )
Shortwave radiation subject to
absorption
• The solar radiation subject to absorption is
distributed as heat to the various layers in the
atmosphere and to the Earth’s surface.
• The absorption depends on upon the effective
water vapor content as well as the ozone and
carbon dioxide amounts
• Generally, for cloudy skies, the absorption in a
cloud is prescribed as a function of cloud type
only
• When the sky is partially cloudy the total
flux at level I is given by:
Rai = NR’’ai+(1-N)R’ai
The part of the flux subject to absorption
which is actually absorbed by the ground
is:
R’ag = (1-ag)R’a4  clear sky
R’’ag = (1-ag)R’a4/(1-agac)  cloudy
Rag = NR’’ag+(1-N)R’ag
• The total solar radiation absorbed by the
ground is
Rg = Rag+Rsg
Longwave radiation
• The calculation og longwave radiation (as
the shortwave) is based on an empirical
transmission function mainly depending on
the amount of water vapor
• The net longwave radition at any level can
be expressed as:
F(net) = F↓-F↑
• The upward flux at z = h for a radiation at wavelength
h
l is:
Fl  (h) = Bl [T (0)]t l (h,0)   Bl [T ( z )](d / dz)t l (h, z )dz
0
With the first term being the infrared flux arriving at z = h
from the surface (z=0), given by the surface flux
Bl[T(0)] times the infrared trasmittance of the
atmosphere, tl. Bl is the Planck function
The second term in the equation is the contribution of the
total upward flux from the emission of infrared
radiation by atmospheric gases below the level z=h.
h
Fl  (h) = Bl [T (0)]t l (h,0)   Bl [T ( z )](d / dz)t l (h, z )dz
0
• Note that, unlike the surface emission, the
atmospheric emission is highly wavelength dependent,
as a consequence of the selective absorption by CO2
or H2O in certain spectral regions
• The downward infrared flux is composed only
of atmospheric emission (as incoming infrared
radiation from space is essentially 0)

Fl  (h) =  Bl [T ( z )](d / dz)t l (h, z )dz
h
Heat balance at the ground
• Ground temperature is obtained from the heat balance at
the ground
Rg+F-esTg4-HL-HS = stored energy
With HL and HS being, respectively, the sensible heat flux
from the surface and the flux of latent heat due to
evaporation from the surface, and Rg being the solar
radiation absorbed by the ground and
F the downwelling longwave radiation at the surface
Convective adjustment
• The computational scheme analyzed so far defines a
radiative temperature profile, T(z), only determined by
the vertical divergence of the net radiative fluxes.
• Globally computed averaged vertical radiative
temperature profiles for clear sky and with either a fixed
distribution of relative humidity or a fixed distribution of
absolute humidity yields very high surface temperatures
and a temperature profile that decreases extremely rapid
with altitude
• By the mid 60s, it was realized that it was necessary to
modify the unstable profiles.
• This modification was termed ‘convective adjustment’,
though it is not really a computation of convection but
rather a numerical re-adjustment
• Temperature
vertical profiles
when (a) the
lapse rate is 6.5
Km/K, (b) the
moist adiabatic
lapse rate, (c)
no convective
adjustment and
(d) the U.S.
standard
atmosphere
(1976)
• The temperature difference between vertical layers is
adjusted to the critical lapse rate (LRc) by changing
the temperature with time according to the integrated
rate of heat addiction.
The flow continues until the
atmospheric temperature
converges to a final,
equilibrium state
• An example of convergence is shown in the figure.
• The left and right figures show, respectively, the approach to
states of pure radiative (left) and RC equilibrium (right). The
solid and dashed lines show the approach from a warm and cold
isothermal atmosphere respectively
Sensitivity experiments with RC
models
• The RC model can be summarized by saying that the vertical
temperature profile of the atmosphere plus surface system,
expressed as a vertical temperature set Ti, is calculated in a
time-stepping procedure such that:
Dt
Ti ( z, t  Dt ) = Ti ( z, t ) 
rc p
 dFr dFc

 dz
dz 
The temperature, Ti, of a given layer I, with height z and at time
t+Dt is a function of the temperature of that layer at the
previous time t and the combined effects of the net radiative
and ‘convective’ energy fluxes deposited at height z. In the
equation, cp is the heat capacity at constant pressure and r is
the atmospheric density.
•
There are 2 common methods of using
RC models:
1) To gain an equilibrium solution after a
perturbation
2) To follow the time evolution of the
radiative fluxes immediately following a
perturbation
• Sensitivity to humidity
• Readings:
McGuffie and Henderson-Sellers
Chapter 4, pp 117 - 163