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  T2 4 = 2  T1 4 = 2  Te 4

It can be shown that for n layers, the temperature of layer n can be related to the effective temperature Te by: Tn 4 =  total (n)* Te 4 with  total (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  then  total (n) = n.

Surface temperature can be obtained as 2  T2 4 =  Tg 4+  T1 4  Tg = (  Te 4 ) 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 , not to be confused with the Stefan- Botlzmann constant)  (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  where the surface has always 

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 (  c S) part is absorbed (a c (1-  c )S and part is transmitted ((1-a c )(1-  c )S) 2) The transmitted radiation interacts with the surface. Part of it is absorbed ((1-  g )(1-a c )(1-  c )S), part is reflected (  g (1-a c )(1-  c )S) which, in turn, is absorbed by the cloud (a c  g (1-a c )(1-  c )S) and transmitted ((1-a c )  g (1-a c )(1-  c )S) 3) the cloud emits as well (  Tc 4 ) as well as the surface (  Tg 4 ),which is partially absorbed by the cloud (  Tg 4 ) and transmitted ((1-  Tg 4 )

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 

When equating absorbed, emitted and reflected radiation at each level we have: Eq. 1) S =  c S +  g (1-a c )(1-  c )S +  Tc 4 + ((1-  Tg 4 ) Eq. 2) a c (1-  c )S + a c  g (1-a c )(1-  c )S +  Tg 4 = 2  Tc 4 Eq. 3) (1-a c )  g (1-a c )(1-  c )S +  Tc 4 =  Tg 4

Eq. 1) S =  c S +  g (1-a c )(1-  c )S +  Tc 4 + ((1-  Tg 4 ) Eq. 2) a c (1-  c )S + a c  g (1-a c )(1-  c )S +  Tg 4 = 2  Tc 4 Eq. 3) (1-a c )  g (1-a c )(1-  c )S +  Tc 4 =  Tg 4 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:  Tg 4 = ((1-  c )S)*(2-a c )/(2-  )

From  Tg 4 = ((1-  c )S)*(2-a c )/(2-  ) Consider S = 343 W/m2 1)Cloudless case  c = 0.08 (scattering by atmospheric molecules alone), a c = 0.15 and  = 0.4  Tg = 283 K 2) Cloudy skies Volcanic aerosol  c = 0.12, a c = 0.18 and  = 0.43  Tg = 280 K  Cooling ! Water droplet cloud  c = 0.3, a c = 0.2 and  = 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  m. 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*cos  Ra = 0.35*S*cos  with  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:  0 = min[1, *log 10 ((p 0 /p s )*cos  )) 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:  ac = 1-(1-  0 )(1-  c ) Where  c 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: Clear sky Cloudy sky

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

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-  g )R’a4  clear sky R’’ag = (1-  g )R’a4/(1-  g  c )  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 is: With the first term being the infrared flux arriving at z = h from the surface (z=0), given by the surface flux B [T(0)] times the infrared trasmittance of the atmosphere, . B  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.

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)

Heat balance at the ground Ground temperature is obtained from the heat balance at the ground Rg+F-  Tg 4 -H L -H S = stored energy With H L and H S 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: The temperature, Ti, of a given layer I, with height z and at time t+  t 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, c p is the heat capacity at constant pressure and  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