Conceptual models on spectral radiative forcing and spectral cloud radiative effect

Radiative forcing and CRE (cloud radiative effect, a.k.a. cloud radiative forcing) are important concepts for delineating climate changes and feedbacks. Usually they are represented and studied in terms of energy flux (i.e. Wm^-2). However, if the spectral compositions of two radiative forcings are distinctively different from each other, even identical magnitude of radiative forcings can cause different atmospheric and surface responses. Similarly, two similar broadband CRE could imply very different cloud feedbacks for the same argument of the spectral composition. Motivated by this line of reasoning, we use two simple models to understand the spectral radiative forcing and spectral CRE. The first model is a one-dimensional radiative-convective equilibrium model developed at Virtual Planetary Laboratory. For an US 1976 standard atmospheric profile, doubling of CO₂ causes 2.05K change of the surface temperature, 2.78K change of the tropospheric mean temperature, and -2.58K change of the stratospheric mean temperature, respectively. A change of CH₄ by 12.56 times can cause the same amount of radiative forcing at the tropopause as the doubling of CO₂. However, the surface temperature change is 4.4% less than that caused by the doubling of CO₂ and the tropospheric and stratospheric temperature changes are different by 10.23% and 133.11%, respectively. These differences can be understood by the different radiative heating structure of CO₂ and CH₄ in the atmosphere. The second model is a 2-layer energy balance model, of which the longwave consists of five bands, i.e. H₂O band, CO₂ band, the 1st window region, O₃ band, and the 2nd window region. Cloud is assumed to reside in the lower layer only. Three scenarios are investigated: (1) constant emissivity across the entire longwave spectrum for both the upper layer and the lower layer; (2) emissivity in the lower layer is a stepwise function from band to band; (3) same as (2) but cloud fraction is ad hoc parameterized as a function of temperature. For each scenario, lower-layer emissivity is perturbed in a way to make the instantaneous change of OLR nearly identical (i.e. identical TOA radiative forcing), the new equilibrium temperature are then calculated. The surface temperature changes in response to such perturbation could differ as much as 100%, in spite of the identical radiative forcing. Such large difference is due to the fact that surface response to downward radiative flux from different band is different. The sensitivity to model parameters is further explored. Next, using global mean band-by-band greenhouse efficiency and CRE from three GCMs (GFDL AM2, NASA GEOS-5, and CCCma CanAM4) as constraining parameters, we solve the 2-layer energy balance model for the clear-sky band-by-band emissivity and temperature for both layers. With a parameterization of CO₂ band emissivity, we then can solve the temperature change due to a doubling of CO₂ in this simple model. As expected, when no feedback is included, the surface temperature changes for three sets of GCM parameters are similar to each other. A simple parameterization of emissivity of water vapor band with temperature is included to further explore the effect of water vapor feedback.