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GREB Decomposition: Long Wave Radiation

2012 December 30

Diving once again into Conceptual understanding of climate change with a globally
resolved energy balance model
(Dommenget and Floter, 2011), this time to flesh out the emissivity and long wave radiation within the model.

3.2 Thermal radiation
The thermal radiation forcing to Tsurf is due to the black body emission of the surface and due to the atmospheric downward thermal radiation, which depends on the atmospheric temperature, Tatmos, the CO2 concentration, the vertical integrated atmospheric water vapor concentration, viwvatmos, and the cloud cover. It is the only way the greenhouse gas CO2 influences the climate system, but it also represents the most fundamental negative feedback to increasing Tsurf and the most important positive feedback due to atmospheric water vapor response. Simple climate models (e.g. Harvey and Schneider 1985; Weaver et al. 2001) use different approaches to simulate the long wave radiation, which mostly include some parameterizations of the surface outgoing and atmospheric downward radiation (e.g. Ramanathan 1977; Ramanathan et al. 1979; Fanning and Weaver 1996). We try to keep the model as simple as possible with the least number of parameters and base our thermal radiation model on the slab atmosphere greenhouse model (Bohren and Clothiaux 2006).

Two take away points here. First OLR depends on Tatmos, CO2, water vapor, and cloud cover. Second, this is a slab atmosphere greenhouse model.

The net thermal radiation is due to a loss by outgoing thermal radiation and a gain by atmospheric thermal radiation:

Fthermal = -sigma*T4surf + eatmos*sigma*T4atmos-rad

The atmosphere is radiating with the temperature Tatmos-rad and an effective emissivity, eatmos. Tatmos-rad will be defined in the context of the atmospheric temperature, Tatmos, in Sect. 3.4. The effective emissivity, eatmos, is depending on the CO2 concentration, the vertical integrated atmospheric water vapor concentration, viwvatmos, and the cloud cover. As a simple approximation of the dependency on CO2 and viwvatmos we use a log-function approach as in Myhre et al. (1998). We also need to consider that the absorption and emission of thermal radiation at different spectral bands are overlapping for CO2 and viwvatmos (e.g. Kiehl and Ramanathan 1982). This can be approximated by:

e0 =
pe4 * log [pe1 * CO2topo + pe2 * viwvatmos + pe3]
+ pe5 * log [pe1 * CO2topo + pe3]
+ pe6 * log [pe2 * viwvatmos + pe3]

e0 is the emissivity without considering clouds first. CO2topo is the atmospheric concentration of CO2 scaled by changes in surface pressure due to the topographic height, ztopo: CO2topo = e-ztopo/zatmos * CO2. The scaling height ztopo = 8,400 m is a measure of the thickness of the atmosphere. The first term RHS (right hand side) in Eq. 4 is the emissivity due to CO2, viwvatmos and some residual component, pe3, in spectral bands where the components thermal emissivity overlaps. The second and third term RHS are the non-overlapping spectral bands CO2 and viwvatmos terms (they still overlap with pe3). The parameters pe4–6 give the relative importance of each absorption band, pe13 are the greenhouse gas species scaling concentration, which we assume to be the same for each absorption band to simplify the approximation. The log-function approach is a simple approximation to consider the saturation effective. Note that this e-function can be 1, but in all simulations discussed in this study such values are not reached. It needs to be noted that the slab greenhouse is an approximation. If greenhouse gasses increase, a multi layer model would be better, which effectively means that eatmos can become larger than one (Bohren and Clothiaux 2006). If no greenhouse gasses exist eatmos should be zero. It is an extreme case (e.g. no CO2, VIWV and cloud cover), but should be considered.

Two main take aways here as well. First is the thermal forcing component defined in this equation:

Fthermal = -sigma*T4surf + eatmos*sigma*T4atmos-rad

The second is the emissivity calculation which is divided into three parts: a CO2 component, a water vapor component, and a mixed CO2-water vapor component. Each component is weighted by the values p4,5,6. The CO2 and water vapor terms are scaled by pe1 and pe2 respectively. Emissivity due to “other” green house gasses (pe3)is distributed equally through the three components. This 3 bin treatment of emissivity is designed to separate those portions of the spectrum which is opaque to CO2 from those that H2O opacity dominates from those with significant overlap in the two. See the figure below. This model doesn’t try to calculate the opacity line by line but uses a simple weighting of these bins instead.

Both the previous equations, as well as the following, can be identified in the ‘LWradiation’ subroutine shown below.

There is one more piece to this puzzle – cloud cover.

Cloud cover is considered by:

eatmos = ( (pe8 – CLD) / pe9 ) * (e0 – pe10) + pe10

Thus cloud cover, CLD, scales the effective emissivity, eatmos, by shifting it up or down and by diluting the effects of the trace gasses. So in the presence of clouds emissivity is larger and the effect of trace gas concentrations is reduced. This also reduces the sensitivity to changes in trace gasses concentration. The parameters of the model are fitted to literature values constraining the effective emissivity as function of CO2, viwvatmos, and CLD with an iterative numerical fitting routine minimizing a cost-function (see Appendix 2 for details).

My final take-away here is that all of these equations are heavily parameterized.

subroutine LWradiation(Tsurf, Tair, q, CO2, LWsurf, LWair_up, LWair_down, em)
! new approach with LW atmos

  USE mo_numerics,    ONLY: xdim, ydim
  USE mo_physics,     ONLY: sig, eps, qclim, cldclim, z_topo, jday, ityr,         &
&                           r_qviwv, z_air, z_vapor, dTrad, p_emi, log_exp

! declare temporary fields
  real, dimension(xdim,ydim)  :: Tsurf, Tair, q, LWsurf, LWair, e_co2, e_cloud,   &
&                                LWair_up, LWair_down, e_vapor, em

  e_co2   = exp(-z_topo/z_air)*CO2         ! CO2
  e_vapor = exp(-z_topo/z_air)*r_qviwv*q   ! water vapor
  e_cloud = cldclim(:,:,ityr)              ! clouds

  if(log_exp == 11) e_vapor = exp(-z_topo/z_air)*r_qviwv*qclim(:,:,ityr)     ! sens. exp. linear-function

! total  
  em      = p_emi(4)*log( p_emi(1)*e_co2 +p_emi(2)*e_vapor +p_emi(3) ) +p_emi(7)   &
&          +p_emi(5)*log( p_emi(1)*e_co2   +p_emi(3) )                             &
&          +p_emi(6)*log( p_emi(2)*e_vapor +p_emi(3) )
  em      = (p_emi(8)-e_cloud)/p_emi(9)*(em-p_emi(10))+p_emi(10) 
  if(log_exp == 11)  em = em +0.022/(0.15*24.)*r_qviwv*(q-qclim(:,:,ityr)) ! sens. exp. linear-function
  LWsurf      = -sig*Tsurf**4
  LWair_down  = -em*sig*(Tair+dTrad(:,:,ityr))**4 
  LWair_up    = LWair_down 

end subroutine LWradiation

So grabbing the emissivity knob firmly, it is easy to twist emissivity up to 1 and down to 0 as seen in the chart above. Dommenget and Floter often cite Bohren and Clothiaux 2006 which is an atmospheric radiation text. Their back-of-the envelope for e=1 is a Te of 303K and for e=0 is Te of 255K. (I have only sampled the B&C text and am not sure Te is a surface temperature.) GREB runs a little cooler with an e=1 leading Tsurf of 293K which is a bit cooler than B&C’s 303K. But the real divergence is on the low end, where an e=0 leads to a Tsurf of 238K. This is quite a bit lower than most “no greenhouse” estimates I have seen and is a cautionary note regarding GREB’s emissivity model and parameterization.


Dommenget, D., and J. Floeter 2011: Conceptual Understanding of Climate Change with a Globally Resolved Energy Balance Model. Climate dynamics, 2011, 37, 2143-2165.

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