Intermediate complexity models

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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

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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

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• 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

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• Suppose there are 2 layers in the atmosphere and
  each layer absorbs the incident radiation on it ?
  infrared optical thickness 1

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• The principal absorber in the Earths 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

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• 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

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• 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.

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• Surface temperature can be obtained as
• 2sT24 sTg4 sT14
• ? Tg (3Te4)1/4

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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

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• 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 Stefan-Botlzmann 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 Earths surface.
• The top of the atmosphere has s 0 where the
  surface has always s 1.

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• 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

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• Model from CD ?

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• Simple case single cloud or aerosol layer is
  spread homogeneously over the surface

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• 1) Part of the incident S radiation is reflected
  (acS) part is absorbed (ac(1-ac)S and part is
  transmitted ((1-ac)(1-ac)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)(1-ac)S) which, in turn, is absorbed by
  the cloud (acag(1-ac)(1-ac)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)

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• Three main assumptions have been made
• No reflection of the upwelling shortwave
  radiation by the cloud
• The surface emissivity has been set to 1
• The cloud/dust absorption in the infrared region
  is equal to e

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• 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

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• 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)

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• From sTg4 ((1-ac)S)(2-ac)/(2-e)
• Consider S 343 W/m2
• 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

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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

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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 0.65Scosm
• Ra 0.35Scosm with m being the solar zenith
  angle

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Albedo

• The albedo of the clear atmosphere in the
  shortwave is subject to Rayleigh scattering and
  it is given by
• a0 min1, 0.085-0.247log10((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

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Albedo contd.

• spectral dependence must be introduced

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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 Earths surface, given by

Clear sky
Cloudy sky
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• Multiple reflections between sky and ground or
  between cloud base and ground are accounted for
  by the terms in the denominators
• For partly cloudy conditions
• Being N the fractional cloudiness of the sky

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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 Earths 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

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• When the sky is partially cloudy the total flux
  at level I is given by
• Rai NRai(1-N)Rai
• The part of the flux subject to absorption which
  is actually absorbed by the ground is
• Rag (1-ag)Ra4 ? clear sky
• Rag (1-ag)Ra4/(1-agac) ? cloudy
• Rag NRag(1-N)Rag

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• The total solar radiation absorbed by the ground
  is
• Rg RagRsg

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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?

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• The upward flux at z h for a radiation at
  wavelength l is
• With the first term being the infrared flux
  arriving at z h from the surface (z0), given
  by the surface flux BlT(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 zh.

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• 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

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• The downward infrared flux is composed only of
  atmospheric emission (as incoming infrared
  radiation from space is essentially 0)

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Heat balance at the ground

• Ground temperature is obtained from the heat
  balance at the ground
• RgF-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

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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

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• 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)

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• 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
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• 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

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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
• The temperature, Ti, of a given layer I, with
  height z and at time tDt 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.

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• There are 2 common methods of using RC models
• To gain an equilibrium solution after a
  perturbation
• To follow the time evolution of the radiative
  fluxes immediately following a perturbation

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• Sensitivity to humidity

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• Readings
• McGuffie and Henderson-Sellers
• Chapter 4, pp 117 - 163