State dependent feedback literature

[znicholls]

State dependent feedback is discussed here

https://link.springer.com/article/10.1007/s00382-019-04686-4

Could also do a nice notebook about stability, given the paper above says, "Stability depends additionally on the radiative forcing F and the quadratic coefficient a, and we get a quadratic runaway in the relationship between N and T in the case that
𝐹>𝛬24𝑎F>Λ24a ."

[znicholls]

https://link.springer.com/article/10.1007/s00382-019-04686-4 appears to follow https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2015GL064240.

Best name is still probably two-layer model though, then people can use efficacy (although doesn't change the maths) or state dependent feedback as they wish.

[znicholls]

From Rohrschneider et al. 2019 (https://link.springer.com/article/10.1007/s00382-019-04686-4). Important distinction between time-dependent and state-dependent feedback variations:

Hasselmann (1979) already recognized the multiple timescale structure of the climate response, but at this time the global feedback was assumed to depend on the global mean surface temperature only. Recently, these characteristic adjustment timescales have been explicitly linked to nonconstant global feedback associating them with different state-variables each of which influences the radiative response (e.g. Held et al. 2010; Winton et al. 2010; Geoffroy et al. 2013a, b; Armour et al. 2013; Andrews et al. 2015; Hedemann et al. 2019). Held et al. (2010) and Geoffroy et al. (2013a, b) describe a model based on two distinct timescales to reproduce the transient global mean temperature response of a wide range of AOGCMs. They introduce a globally averaged two-layer ocean model with an efficacy parameter modifying the upper ocean response to heat uptake. Hereafter it is referred to as a two-layer model with efficacy, or just the two-layer model. Two or more adjustment timescales can also be explicitly linked to a one-layer model with two regions. In contrast to Held et al. (2010) and Geoffroy et al. (2013a), Armour et al. (2013) suggest a geographic approach in which constant regional feedbacks are weighted by an evolving pattern of surface warming. This approach is motivated by the idea that the adjustment over parts of the Earth’s surface which are weakly coupled to the state of the deep-ocean is more rapid than over regions which are more strongly coupled to the state of the deep-ocean. Consequently, this can be called as pattern effect (Stevens et al. 2016). In addition to time-varying sensitivity, two-region models have also been used to analyze the effect of perturbation heat transport on stability properties and equilibrium sensitivity (e.g. Bates 2012, 2016) or to explore polar amplification (e.g. Langen and Alexeev 2007). In all of these models, the time-variation of 𝜆(𝑡) does not depend on the strength of the forcing– in this sense these models are linear in forcing. The tendency of such models to have a value of 𝜆(𝑡) that varies with time is thus often paraphrased as a time-dependent feedback though it arises from a dependence of the radiative response on an additional state-variable with its own distinct temporal evolution. Using AOGCM output, e.g. Senior and Mitchell (2000) and Hedemann et al. (2019) discuss time-dependent feedback as a way to understand the radiative response in more comprehensive climate models.

A different approach to representing variation in the radiative response has been to relax the assumption that a temperature perturbation would be proportional to the associated radiative forcing F (e.g. Roe and Baker 2007; Zaliapin and Ghil 2010; Roe and Armour 2011; Meraner et al. 2013; Bloch-Johnson et al. 2015). Studies adopting this approach investigate the long-term response on centennial to millennial timescales as well as the nonlinear change in equilibrium sensitivity between different forcing strengths. Conceptual models on nonlinear feedbacks of the climate system can be traced back at least to Budyko (1969) and Sellers (1969). In a recent study, Bloch-Johnson et al. (2015) extend the linear forcing-feedback framework Eq. (1) by a quadratic coefficient a representing second-order feedback temperature dependence and solve for stationary solutions, −𝐹=𝜆𝑇+𝑎𝑇2. They estimate the range of likely feedback temperature dependencies for various GCM simulations and find −0.04≤𝑎≤0.06 W m−2 K−2. This range is in line with Roe and Armour (2011) who find −0.058≤𝑎≤0.06 W m−2 K−2. Positive feedback temperature dependence makes the feedback of the climate system continuously less effective as the surface temperature increases, while the opposite is true for negative feedback temperature dependence. In this connection, the radiative response depends explicitly on the climate state; i.e., feedback temperature dependence gives rise to state-dependent feedback such that the variation of 𝜆(𝑡) is changed as the strength of the forcing is altered.