When a perturbation is applied to an active and complex system, its response will be accompanied by some feedback mechanisms.

Block diagram of system with feedback

In this diagram the variables are:

• F_GHG = Forcing resulting from the absorption of long wave radiations by greenhouse gase_s
• F_F  = Forcing resulting from feedback mechanisms [W m^-2]
• F_all = F_GHG + F_F
• ∆T= Surface temperature variation resulting from the total forcing [K]
• G_S  = Primary system response = ∆T/F_all   [K W^-1 m²]
• G_F  = System feedback response = F_F/∆T   [W m^-2 K^-1]

Applying the transfer functions and resolving for ∆T:

∆T = G_S· F_all

F_F = G_F· ∆T

F_all = F_GHG + F_F =F_GHG + G_F· ∆T

∆T/G_S = F_GHG + G_F·∆T

∆T·(1- G_F·G_S)= G_S· F_GHG

           (eq. 12)

According to the Stefan-Boltzmann equation, the radiation is related to the surface temperature of a grey body:

F = ε·σ·T⁴   or   T = (F/(ε·σ))^0.25 thus

                         (eq. 13)

Where F is a forcing imposed on the top of the underlying system radiation.

Feedback transfer function

The feedback transfer function G_F is a combination of all mechanisms that will react to the temperature increase caused by the primary forcing exerted by GHGs.

Five feedback mechanisms are considered: Planck, water vapour, lapse rate, albedo, and clouds.

The feedback factor λ associated with each mechanism is defined as the forcing that would result from a temperature change of one degree [W m^-2 K^-1].

Thirty published values of these feedback factors are presented in the latest IPCC report (AR5 WG1 Chapter 9). Uncertainty is given with 2x the standard deviation, a 90% confidence.

Planck

As temperature increases, the radiative energy emitted will also increase according to Planck law. While losing this energy, the temperature of the emitting material will decrease. Part of the temperature increase contributes therefore to eliminate the forcing… and to dampen the temperature rise. Thus, Planck response is a negative feedback:

λ_P = -3.21 ± 0.11 W m^-2 K^-1

Water Vapour

More water will evaporate when temperature is increasing. The higher air humidity will absorb more radiative energy and contribute to amplifying the forcing. This is a positive feedback:

λ_WV = 1.63 ± 0.33 W m^-2 K^-1

However, the evaporation-condensation of water, with its large latent heat, is not included in this parameter. Another model would be required to reflect this.

Lapse Rate

The lapse rate is described in chapter 2.3.3. When the overall temperature increases the structure of the atmosphere will change, convective force will move warmer air upwards and the gradient will change: this gradient will tend to be less pronounced in tropical latitudes (negative feedback) and more pronounced in mid-latitudes (positive feedback). Overall it results to be a negative feedback, with a large uncertainty:

λ_LR = -0.60 ± 0.46 W m^-2 K^-1

Albedo

The reflectivity of the sunlight depends on the type and on the extent of the surface that is reflecting it. In particular there is a large difference between ice and water. With higher temperature less ice or snow coverage is expected, resulting in less reflection. This is a positive feedback:

λ_A = 0.31 ± 0.11 W m^-2 K^-1

Clouds

Clouds show two different feedbacks. With higher temperature, water will evaporate and more clouds will be formed. With a larger reflective surface, cloud albedo will increase, more sunlight will be reflected back to the outer space, a negative feedback. But at higher altitude the clouds will contribute to more radiative forcing in the IR range. Overall the current estimate is that it is a positive feedback, albeit with a high uncertainty (300%):

λ_C = 0.28 ± 0.84 W m^-2 K^-1

Overall these factors are additive, resulting in a negative feedback:

λ = λ_P + λ_WV + λ_LR + λ_A + λ_C = -3.21 + 1.63 – 0.60 + 0.31 + 0.28

and by adding the error ranges:

G_F = λ = -1.59 [±1.85] W m^-2 K^-1

This negative feedback indicates a stabilization of the system in response to the primary forcing. A positive feedback would mean an amplification of the primary effect, up to instability if the feedback factor would approach 1.

Note: the feedback factors are estimates from various models that are summarized by the IPCC in its 5^th report. When remembering the sensitivity to basic parameters (albedo, emissivity, and cloudiness) one can only expect that these estimates may change. Also, it looks somehow absurd to evaluate a model with other models’ results, but this is all what we have at this time.

Model with feedback

The energy balance with two layers and the overall feedback can be put together to resolve Eq. 12:

Climate sensitivity to radiative forcing, before and after feedback.
^* the slight difference from the previous table is due to the linearization of G_S.

The same calculations can be made for tentative CO₂ concentrations at pre-industrial time, or in a possible future. In following diagram error margins mentioned above for the feedback factors are also represented.

This is incredibly not new, but astonishing!

Temperature response to radiative forcing by CO₂ alone with feedback.

Read like this:
with currently 400 ppm, CO₂ contribution to warming is 0.29 °C when including a feedback of -1.59 W m^-2 K^-1.
Or: when the historic CO₂ concentration will have doubled to reach 560 ppm, the temperature will have increased by an additional 0.27 °C (total “equilibrium temperature sensitivity” = 0.56 °C).
Or: if the feedback is weak (even slightly positive), the current CO₂ contribution to warming may be 0.40 °C