3.2 Greenhouse feedback with latitude
Now, the greenhouse component of the total climate feedback with latitude, Eq. (8), is evaluated using,
$${\lambda }_{greenhouse}\left(\varphi \right)=\sigma {T}_{S}^{4}\left(\varphi \right)\frac{\partial {\epsilon }_{AllSky}}{\partial {T}_{s}}\left(\varphi \right)$$
(10)
The value of \(\sigma {T}_{S}^{4}\left(\varphi \right)\) is constrained from observations of \({T}_{s}\) (Jones et al., 1999; Fig. 1a). The aim here is then to constrain \(\frac{\partial {\epsilon }_{AllSky}}{\partial {T}_{s}}\left(\varphi \right)\) with observations of the climatological mean state. This is complicated because \({\epsilon }_{AllSky}\left(\varphi \right)\) is affected by both the water vapour-lapse rate feedback and by clouds (e.g. Sherwood et al., 2020; IPCC, 2021), and the response of clouds to forcing is notoriously difficult to constrain.
A model is adopted where the all-sky emissivity at latitude \(\varphi\), \({\epsilon }_{AllSky}\left(\varphi \right)\), is assumed to be a linear combination of the clear-sky emissivity, \({\epsilon }_{ClearSky}\left(\varphi \right)\), and the emissivity of cloudy-sky, \({\epsilon }_{CloudySky}\left(\varphi \right)\), which is defined as the emissivity at latitude \(\varphi\) when latitudinal cloud amount fraction \({f}_{CA}\left(\varphi \right)=1\),
$${\epsilon }_{AllSky}\left(\varphi \right)=\left[1-{f}_{CA}\left(\varphi \right)\right]{\epsilon }_{ClearSky}\left(\varphi \right)+{f}_{CA}\left(\varphi \right){\epsilon }_{CloudySky}\left(\varphi \right)$$
(11)
This expression, (11), is first used with climatological observations of cloud amount (Stubenrauch et al., 2013), averaged here to form the annual- and zonal-mean \({f}_{CA}\left(\varphi \right)\) (Fig. 1e), to evaluate the annual- and zonal-mean emissivity of cloudy sky at latitude \(\varphi\), \({\epsilon }_{CloudySky}\left(\varphi \right)\) (Fig. 1c, blue)
By comparing the clear-sky emissivity to the cloudy-sky emissivity (Fig. 1c), now consider how clouds affect the annual- and zonal-mean emissivity with latitude (Fig. 1c, blue and orange). A cloudy sky, where local cloud amount \({f}_{CA}\left(\varphi \right)=1\), acts to decrease the emissivity relative to clear-sky conditions, but by an amount that varies with latitude (Fig. 1c, varying distance between blue and orange lines). However, if we consider the emitted longwave absorptivity, \(a\), defined here such that emissivity plus absorptivity must equal 1, \(a=1-\epsilon\), then we can express the fraction by which cloudy skies increase absorptivity relative to clear skies with latitude, \({c}_{\epsilon }\left(\varphi \right),\) as,
$${c}_{\epsilon }\left(\varphi \right)=\frac{{a}_{CloudySky}\left(\varphi \right)}{{a}_{ClearSky}\left(\varphi \right)}=\frac{1-{\epsilon }_{CloudySky}\left(\varphi \right)}{1-{\epsilon }_{ClearSky}\left(\varphi \right)}$$
(12)
Formally, \({c}_{\epsilon }\left(\varphi \right)\) is the annual mean fractional increase in longwave absorptivity of the atmosphere per unit cloud amount due to the presence of cloud at latitude \(\varphi\); where absorptivity is 1 minus the emissivity.
The resulting values of \({c}_{\epsilon }\left(\varphi \right)\) (12) are near-uniform with latitude (Fig. 1g): the distribution of values of \({c}_{\epsilon }\) at 5° latitude intervals (Fig. 1e) has mean and standard deviation \({c}_{\epsilon }\left(\varphi \right)=1.37\pm 0.06\), with the small variation with latitude illustrating the consistent effect of clouds on longwave atmospheric absorptivity per unit cloud amount for climate states ranging from polar to tropical.
This near-uniform value of \({c}_{\epsilon }\) with latitude reveals information about the nature of the impact of clouds on absorptivity of emitted longwave radiation (Fig. 1g): clouds act to increase the absorptivity of a clear-sky atmosphere by a set fraction per unit cloud amount across a range of conditions, from tropical (clear sky emissivity \({\epsilon }_{ClearSky}\left(\varphi \right)\tilde0.6\), surface temperature \({T}_{s}\tilde300\) K) to polar (clear sky emissivity \({\epsilon }_{ClearSky}\left(\varphi \right)\tilde0.9\), surface temperature \({T}_{s}\tilde230\) K). At all latitudes the all-sky absorptivity, \({a}_{AllSky}=1-{\epsilon }_{AllSky}\), is increased by \(0.37\pm 0.06\) per unit cloud amount relative to the clear-sky absorptivity,
$${a}_{AllSky}\left(\varphi \right)={a}_{ClearSky}\left(\varphi \right)\left[1+\left({c}_{\epsilon }\right(\varphi )-1){f}_{CA}\left(\varphi \right)\right]$$
(13)
Substituting Eq. (13), using \({a}_{i}=1-{\epsilon }_{i}\), into Eq. (11) and differentiating with respect to surface temperature reveals the sensitivity of local emissivity in all-sky conditions to local surface temperature;
$$\frac{\partial {\epsilon }_{AllSky}}{\partial {T}_{s}}=\left[1-{f}_{CA}+{c}_{\epsilon }{f}_{CA}\right]\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}-\left[{c}_{\epsilon }-1\right]\left[{1-\epsilon }_{ClearSky}\right]\frac{\partial {f}_{CA}}{\partial {T}_{s}}-{f}_{CA}\left[1-{\epsilon }_{ClearSky}\right]\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}$$
(14)
noting that while we remove the explicit \(\varphi\) dependence in this equation for simplicity, all terms do alter with \(\varphi\).
In this relationship for \(\frac{\partial {\epsilon }_{AllSky}}{\partial {T}_{s}}\), the right-hand side contains three terms different aspects of the system’s sensitivity of all-sky emissivity to surface temperature:
1. \(\left[1-{f}_{CA}+{c}_{\epsilon }{f}_{CA}\right]\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\) represents the local sensitivity of all-sky emissivity to surface temperature if local cloud amount and type remain constant;
2. \(-\left[{c}_{\epsilon }-1\right]\left[1-{\epsilon }_{ClearSky}\right]\frac{\partial {f}_{CA}}{\partial {T}_{s}}\) represents the local sensitivity of all-sky emissivity to surface temperature due to changes in local cloud amount; and
3. \(-{f}_{CA}\left[1-{\epsilon }_{ClearSky}\right]\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\) represents the local sensitivity of all-sky emissivity to surface temperature due to changes in local cloud type at fixed cloud amount.
It is not currently possible to use the current climatological state to constrain how cloud amount fraction, \({f}_{CA}\), or the fractional increase in absorptivity due to clouds, \({c}_{\epsilon }\), will vary with changes in surface temperature, \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\) or \(\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\), since there are no identifiable relationships between \({f}_{CA}\) or \({c}_{\epsilon }\) with \({T}_{s}\) (e.g. Figure 1). Therefore, to make some progress in evaluating greenhouse feedback, \({\lambda }_{greenhouse}\left(\varphi \right)\), an initial simplifying assumption is made in this section: that cloud type and cloud amount are insensitive to perturbation, \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}=\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}} =0\). Here, \({\lambda }_{greenhouse}\left(\varphi \right)\) is first evaluated under this simplifying assumption, and the impacts of this simplifying assumption are evaluated later in section 3.4.2 below.
Assuming that cloud type and amount are insensitive to perturbation, \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}=\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}} =0\), we may write the greenhouse climate feedback, by substituting Eq. (14) into Eq. (10), as,
$${\lambda }_{greenhouse,ConstCloud}\left(\varphi \right)=\sigma {T}_{S}^{4}\left(\varphi \right)\left[1-{f}_{CA}\left(\varphi \right)+{c}_{\epsilon }\left(\varphi \right){f}_{CA}\left(\varphi \right)\right]\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)$$
(15)
where this is termed here the ‘constant cloud’ assumption.
To evaluate the greenhouse climate feedback under the constant cloud assumption, the values of \({f}_{CA}\left(\varphi \right)\), \({c}_{\epsilon }\left(\varphi \right)\) and \(\sigma {T}_{S}^{4}\left(\varphi \right)\) and \(\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\) must be diagnosed. The values of \({f}_{CA}\left(\varphi \right)\), \({c}_{\epsilon }\left(\varphi \right)\) and \(\sigma {T}_{S}^{4}\left(\varphi \right)\) with latitude are relatively easy to calculate from observational reconstructions (Fig. 1). How can \(\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\) be evaluated?
Consider how annual- and zonal-mean clear sky emissivity and annual- and zonal-mean surface temperature are related, when evaluated at 5°-latitude intervals from observations of the recent climatological mean state (Fig. 3a, black dots). A clear and robust relationship emerges between annual- and zonal-mean surface temperature and annual- and zonal-mean clear-sky emissivity, approximated here by quadratic in surface temperature (Fig. 3a, grey line and black dots). Noting that this relationship spans considerable variation in local climate state, from polar to equatorial, and applies over both hemispheres (Fig. 3a), we assume here that the observed relationship for latitudinal variation in the current climate state will also apply in a new similar climate state following small perturbation. Therefore, an observational estimate of the value of \(\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\) is derived from the gradient of the line of best fit between \({\epsilon }_{ClearSky}\left(\varphi \right)\) and \({T}_{s}\left(\varphi \right)\) (Fig. 3a, gradient of grey line).
Thus, by evaluating \(\sigma {T}_{S}^{4}\left(\varphi \right)\) (Fig. 1a), \({f}_{CA}\left(\varphi \right)\) (Fig. 1c), \({c}_{\epsilon }\left(\varphi \right)\) (Fig. 1d), and \(\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\) (Fig. 3a, derivative of line of best fit), the greenhouse feedback to surface warming with latitude, \({\lambda }_{greenhouse,ConstCloud}\left(\varphi \right)\) is evaluated here (Fig. 2, orange solid line) from observations of the recent climatological mean state.
Greenhouse climate feedback varies from a minimum of \({\lambda }_{greenhouse,ConstCloud}\left(\varphi \right)=-2.05\) Wm− 2K− 1 at 7.5°N, rising to \(-0.786\) Wm− 2K− 1 toward the southern polar region and \(-1.15\) Wm− 2K− 1 in the northern polar region (Fig. 2, solid orange line). The global spatial mean greenhouse feedback under the constant cloud assumption is found to be \(\stackrel{-}{{\lambda }_{greenhouse,ConstCloud}}=-1.77\) Wm− 2K− 1 (Fig. 3, solid orange line). This quantifies \({\lambda }_{greenhouse,ConstCloud}\left(\varphi \right)\) using the constant cloud assumption, \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}=\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}} =0\). Uncertainties introduced by this constant cloud assumption are discussed below (Section 3.4.2). The next sub-section considers the albedo feedback.
3.3 Albedo feedback with latitude
The albedo feedback is written, by isolating the appropriate term from Eq. (8),
$${\lambda }_{albedo}\left(\varphi \right)={S}_{in}\left(\varphi \right)\frac{\partial {\alpha }_{AllSky}}{\partial {T}_{s}}\left(\varphi \right)$$
(16)
To evaluate \({\lambda }_{albedo}\left(\varphi \right)\), the aim here is to constrain \(\frac{\partial {\alpha }_{AllSky}}{\partial {T}_{s}}\left(\varphi \right)\) using observations of the climatological mean. Similar to the corresponding sensitivity of all-sky emissivity to surface temperature, evaluating \(\frac{\partial {\alpha }_{AllSky}}{\partial {T}_{s}}\left(\varphi \right)\) is complicated by the impact of clouds and the associated difficulties of predicting the responses of clouds to perturbation. A model is assumed whereby all-sky albedo is considered to be linearly composed of cloudy-sky albedo, \({\alpha }_{CloudySky}\left(\varphi \right)\), multiplied by the fraction of incident solar radiation incident on clouds, \({f}_{CI}\left(\varphi \right)\), and the clear-sky albedo, \({\alpha }_{ClearSky}\left(\varphi \right)\), multiplied by the fraction of solar insolation not incident on clouds, \(\left[1-{f}_{CI}\left(\varphi \right)\right]\), via,
$${\alpha }_{AllSky}\left(\varphi \right)=\left[1-{f}_{CI}\left(\varphi \right)\right]{\alpha }_{ClearSky}\left(\varphi \right)+{f}_{CI}\left(\varphi \right){\alpha }_{CloudySky}\left(\varphi \right)$$
(17)
The annual- and zonal-mean fraction of solar insolation incident on clouds, \({f}_{CI}\left(\varphi \right)\), is subtly different from the fraction of cloud amount, \({f}_{CA}\left(\varphi \right)\) (Fig. 1f, compare black to grey lines), because climatological mean cloud amount varies seasonally (e.g. Stubenrauch et al., 2013) and so too does incident solar insolation, \({S}_{in}\left(\varphi \right)\). Here, \({f}_{CI}\left(\varphi \right)\) (Fig. 1f, black line) is calculated from monthly-mean values of the product of \({f}_{CA}(\varphi ,m)\) and \({S}_{in}(\varphi ,m)\), and then taking the annual time-average,
$${f}_{CI}\left(\varphi \right)=\frac{\sum {f}_{CA}\left(\varphi ,m\right){S}_{in}\left(\varphi ,m\right){\Delta }{t}_{m}}{\sum {S}_{in}\left(\varphi ,m\right){\Delta }{t}_{m}}$$
(18)
where \({f}_{CA}\left(\varphi ,m\right)\) and \({S}_{in}\left(\varphi ,m\right)\) represent monthly mean values during month \(m\), \({\Delta }{t}_{m}\) is the duration of month \(m\), and the summation occurs over all 12 months.
Relations (17) and (18) are then used to calculate \({\alpha }_{CloudySky}\left(\varphi \right)\) with latitude (Fig. 1d, blue line), showing that cloudy skies have greater albedo than clear-skies at all latitudes, but that the difference between clear and cloudy sky albedo varies with latitude (Fig. 1d, compare blue to orange).
It is now advantageous to separate the albedo of cloudy-sky into terms representing the albedo of cloud itself, \({\alpha }_{Cloud}\left(\varphi \right)\), and the albedo of clear-skies, \({\alpha }_{ClearSky}\left(\varphi \right)\). To perform this separation, we assume that clouds themselves are not a significant source of atmospheric absorption of solar shortwave radiation. Out of ~ 340 Wm− 2 of incoming solar radiation, around 78 Wm− 2 is absorbed in the atmosphere (Trenberth et al., 2009). We assume that the majority of this shortwave atmospheric absorption is not due to the presence of clouds, but other agents such as aerosols, ozone and/or water vapour. This assumption allows us to write the outgoing solar radiation for a cloudy sky, \({S}_{out,CloudySky}\left(\varphi \right)={S}_{in}\left(\varphi \right){\alpha }_{CloudySky}\left(\varphi \right)\), in terms of the radiation reflected by the cloud, \({S}_{in}\left(\varphi \right){\alpha }_{Cloud}\left(\varphi \right)\), and the radiation that passes through the cloud on the way down, \({S}_{in}\left(\varphi \right)\left[1-{\alpha }_{Cloud}\left(\varphi \right)\right]\), is reflected off the surface reducing by factor \({\alpha }_{ClearSky}\left(\varphi \right)\), and then passed through the cloud again on the way up reducing by further factor \(1-{\alpha }_{Cloud}\left(\varphi \right)\), to result in outgoing radiation of \({S}_{in}\left(\varphi \right){\alpha }_{ClearSky}\left(\varphi \right){\left[1-{\alpha }_{Cloud}\left(\varphi \right)\right]}^{2}\). Adding the outgoing radiation from direct cloud reflection, \({S}_{in}\left(\varphi \right){\alpha }_{Cloud}\left(\varphi \right)\), to the outgoing radiation reflection from the surface having passed through the cloud in the way down and up, \({S}_{in}\left(\varphi \right){\alpha }_{ClearSky}\left(\varphi \right){\left[1-{\alpha }_{Cloud}\left(\varphi \right)\right]}^{2}\), gives total outgoing reflected shortwave above a cloudy sky,
$${S}_{out,CloudySky}\left(\varphi \right)={S}_{in}\left(\varphi \right){\alpha }_{CloudySky}\left(\varphi \right)={S}_{in}\left(\varphi \right){\alpha }_{Cloud}\left(\varphi \right)+{S}_{in}\left(\varphi \right){\alpha }_{ClearSky}\left(\varphi \right){\left[1-{\alpha }_{Cloud}\left(\varphi \right)\right]}^{2}$$
(19)
where \({\alpha }_{Cloud}\left(\varphi \right)\) is the albedo of cloud at latitude \(\varphi\). The albedo for cloudy-sky conditions is then related to the albedo of cloud and the albedo of clear-sky conditions via,
$${\alpha }_{CloudySky}\left(\varphi \right)={\alpha }_{Cloud}\left(\varphi \right)+{\alpha }_{ClearSky}\left(\varphi \right){\left[1-{\alpha }_{Cloud}\left(\varphi \right)\right]}^{2}$$
(20)
This relation will be accurate provided that the presence of the cloud is not significantly increasing local atmospheric absorption of shortwave radiation. Applying (20), along with the observation-derived estimates of \({\alpha }_{ClearSky}\left(\varphi \right)\), \({\alpha }_{CloudySky}\left(\varphi \right)\) and \({f}_{CI}\left(\varphi \right)\) (Fig. 1d, f), we find that the albedo of clouds ranges from \({\alpha }_{Cloud}=0.23\) at low latitudes up to ~ 0.5 at high northern latitudes and ~ 0.6 at high southern latitudes. The increasing \({\alpha }_{Cloud}\) at high latitudes is expected since the reflectance of solar radiation by clouds increases with the zenith angle of incident radiation (e.g. Shupe & Intrieri, 2004).
We may now substitute (20) into (17) and differentiate with respect to surface temperature, which finds the sensitivity of local albedo in all-sky conditions to local surface temperature;
$$\frac{\partial {\alpha }_{AllSky}}{\partial {T}_{s}}=\left[1-{{\alpha }_{Cloud}f}_{CI}\left[2-{\alpha }_{Cloud}\right]\right]\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}+\left[{\alpha }_{Cloud}\left[1-{2\alpha }_{ClearSky}+{\alpha }_{ClearSky}{\alpha }_{Cloud}\right]\right]\frac{\partial {f}_{CI}}{\partial {T}_{s}}+\left[{f}_{CI}\left[1-{2\alpha }_{ClearSky}\left[1-{\alpha }_{Cloud}\right]\right]\right]\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}$$
(21)
Noting that we remove the explicit \(\varphi\) dependency from Eq. (21) for clarity, but that all terms do alter with latitude.
In this relation for \(\frac{\partial {\alpha }_{AllSky}}{\partial {T}_{s}}\), (21), the right-hand-side contains three terms representing different aspects of the system’s sensitivity of all-sky albedo to surface temperature:
1. \(\left[1-{{\alpha }_{Cloud}f}_{CI}\left[2-{\alpha }_{Cloud}\right]\right]\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\) is the sensitivity of albedo to local surface temperature if local cloud amount and type remain constant;
2. \(\left[{\alpha }_{Cloud}\left[1-{2\alpha }_{ClearSky}+{\alpha }_{ClearSky}{\alpha }_{Cloud}\right]\right]\frac{\partial {f}_{CI}}{\partial {T}_{s}}\) is the sensitivity of all-sky albedo to local surface temperature due to changes in local cloud amount;
3. \(\left[{f}_{CI}\left[1-{2\alpha }_{ClearSky}\left[1-{\alpha }_{Cloud}\right]\right]\right]\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\) is the sensitivity of all-sky albedo to local surface temperature due to changes in local cloud type at fixed cloud amount and fixed latitude.
It is not currently possible to use the current climatological state to constrain how cloud incident insolation fraction, \({f}_{CI}\), or the albedo of cloud at fixed latitude, \({\alpha }_{Cloud}\), will vary with changes in surface temperature, \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\) or \(\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\), since there is no identifiable relationship between \({f}_{CI}\), with \({T}_{s}\) (e.g. Figure 1), and the apparent relationship between \({\alpha }_{Cloud}\) and \({T}_{s}\) likely reflects a functional relationship between \({\alpha }_{Cloud}\) and zenith angle (e.g. Shupe and Intrieri, 2004). Therefore, to make initial progress evaluating \({\lambda }_{albedo}\left(\varphi \right)\), a simplifying constant cloud assumption is made. The uncertainties introduced by this assumption are then assessed later in section 3.4.2. Assuming that cloud type (affecting \({\alpha }_{Cloud}\)) and cloud-incident insolation amount (\({f}_{CI}\)) are insensitive to perturbation, \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}=\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}} =0\), we may write the albedo climate feedback as,
$${\lambda }_{albedo,ConstCloud}\left(\varphi \right)={S}_{in}\left(\varphi \right)\left[1-{{\alpha }_{Cloud}\left(\varphi \right)f}_{CI}\left(\varphi \right)\left[2-{\alpha }_{Cloud}\left(\varphi \right)\right]\right]\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}$$
(22)
noting that this relation expresses the albedo feedback with latitude under the ‘constant cloud’ assumption.
\({\lambda }_{albedo,ConstCloud}\left(\varphi \right)\) may therefore be diagnosed by evaluating \({f}_{CI}\left(\varphi \right)\), \({\alpha }_{Cloud}\left(\varphi \right)\) and \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\), (Eq. 22). The latitudinal values of \({f}_{CI}\left(\varphi \right)\) (Fig. 1f), and \({\alpha }_{Cloud}\left(\varphi \right)\) (Fig. 1h) have been evaluated (Fig. 1f,h and Eq. 17), but how can the clear-skies albedo sensitivity to surface temperature be evaluated, \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\)?
Consider how the observational-based estimates of the climatological annual- and zonal-mean clear skies albedo vary with annual- and zonal-mean surface temperature (Fig. 3b, black dots). A relationship is revealed whereby albedo is high for temperatures under ~ 230 K, low for temperatures above ~ 290 K and is intermediate for temperatures between 230 and 290 K. A best estimate for \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\) is calculated assuming \({\alpha }_{ClearSky}\) is a function of \({T}_{s}\) (Fig. 3b, grey line). First, a cubic-form best fit line is generated from all data: \({\alpha }_{ClearSky}=a{T}_{s}^{3}+b{T}_{s}^{2}+cT{}_{S}+d\). This cubic form is used for the section \({230\le T}_{s}\le 290\) K, where the cubic has \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}<0\). However, where the best-fit cubic has \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}>0\) (i.e. for all \({T}_{s}>290\) K and for all \({T}_{s}<230\) K) the gradient is assumed to be zero, \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}=0\), giving two regions with constant values of \({\alpha }_{ClearSky}\) for small changes in\({T}_{s}\) (Fig. 3b, grey line).
Physically, this curve (Fig. 3b) is interpreted as representing the stages of snow and ice cover affecting surface albedo depending on annual average temperature. There is assumed no snow cover affecting annual average albedo when \({T}_{s}>290\) K. Then for annual average temperature from \({230\le T}_{s}\le 290\) K a latitudinal region ranges from having some small winter snow cover increasing albedo a small amount, \({T}_{s}\to 290\) K, to having year-round very high albedo ice cover, \({T}_{s}\to 230\) K. By assuming \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}=0\) for all \({T}_{s}>290\) K and for all \({T}_{s}<230\) K, we are making the assumption that surface temperature can only affect surface albedo through snow and ice cover, and not through other factors such as temperature-linked vegetation changes.
Therefore, an observational estimate of the value of \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)\) is derived from the gradient of the line of best fit between \({\alpha }_{ClearSky}\left(\varphi \right)\) and \({T}_{s}\left(\varphi \right)\) (Fig. 3b, gradient of grey line). This assumes that following small climate perturbation, the current climatological mean-state relationship between clear-skies albedo and surface temperature (Fig. 3b) will apply.
Using these constraints on \({f}_{CI}\left(\varphi \right)\), \({\alpha }_{Cloud}\left(\varphi \right)\) and \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\) from observations and mean-state climatology to Eq. (21), we find the albedo climate feedback under constant cloud approximation, \({\lambda }_{albedo,ConstCloud}\left(\varphi \right)\) (Fig. 3, solid red line). Climate feedback from albedo is zero in the low latitudes, where there is no snow cover to melt, and is also zero at extreme southern high latitudes, where it is so cold that warming has no instant impact on albedo (Fig. 2, solid red line).
The magnitude of albedo climate feedback is larger in the Northern hemisphere, where the minimum value of \({\lambda }_{albedo,ConstCloud}\left(\varphi \right)=-1.67\) Wm− 2K− 1 occurs at 67.5 °N, implying that changes in albedo reduce outgoing radiation at the top of the atmosphere by 1.67 Wm− 2 for every 1K surface warming. In the Southern hemisphere there is a smaller minimum of \({\lambda }_{albedo,ConstCloud}\left(\varphi \right)=-1.25\) Wm− 2K− 1 at 67.5 °S. The global spatial average albedo feedback under the constant cloud assumption is \(\stackrel{-}{{\lambda }_{albedo,ConstCloud}}=-0.51\) Wm− 2K− 1 (Fig. 2, solid red line). This represents the global surface albedo feedback in the presence of the current cloud conditions.
3.4 Climate feedback with latitude
We now evaluate the overall climate feedback with latitude, first under the constant cloud assumption and then considering uncertainties introduced by variations in cloud conditions in response to surface temperature changes.
3.4.1 Climate feedback with latitude under constant cloud approximation
Combining the relationships for \({\lambda }_{Planck}\left(\varphi \right)\), \({\lambda }_{greenhouse,ConstCloud}\left(\varphi \right)\), and \({\lambda }_{albedo,ConstCloud}\left(\varphi \right)\), Eqs. (9), (15) and (22), gives a relation for the overall climate feedback with latitude, \({\lambda }_{ConstCloud}\left(\varphi \right)=\) \({\lambda }_{Planck}\left(\varphi \right)+{\lambda }_{greenhouse,ConstCloud}\left(\varphi \right)+{\lambda }_{albedo,ConstCloud}\left(\varphi \right)\), when cloud type, and cloud amount are assumed to be insensitive to perturbation,
$${\lambda }_{ConstCloud}\left(\varphi \right)=4\sigma {\epsilon }_{AllSky}\left(\varphi \right){T}_{s}^{3}\left(\varphi \right)+\sigma {T}_{S}^{4}\left(\varphi \right)\left[1-{f}_{CA}\left(\varphi \right)+{c}_{\epsilon }\left(\varphi \right){f}_{CA}\left(\varphi \right)\right]\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\left(\varphi \right)+{S}_{in}\left(\varphi \right)\left[1-{{\alpha }_{Cloud}\left(\varphi \right)f}_{CI}\left(\varphi \right)\left[2-{\alpha }_{Cloud}\left(\varphi \right)\right]\right]\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}$$
(23)
Assuming local cloud properties remain constant in response to perturbation, we find local climate feedback (Eq. 23) has hemispheric maxima at ± 22.5° latitude (Fig. 2, solid black line), representing the areas with most damping feedbacks. Climate feedback reaches \({\lambda }_{ConstCloud}\left(\varphi \right)=1.83\) Wm− 2K− 1 at 22.5°N and \(1.84\) Wm− 2K− 1 at 22.5°S. The hemispheric minima in climate feedback at mid-latitudes, indicating the locations with most amplifying feedback: a global and hemispheric minimum of \({\lambda }_{ConstCloud}\left(\varphi \right)=-0.10\) Wm− 2K− 1 is reached at 62.5°N and a smaller hemispheric minimum of \({\lambda }_{ConstCloud}\left(\varphi \right)=0.15\) Wm− 2K− 1 is reached at 57.5°S (Fig. 2, solid black line). The overall area-weighted climate feedback with latitude is then \(\stackrel{-}{\lambda \left(\varphi \right)}=1.05\) Wm−2K−1 (Fig. 2, solid black line), in good agreement with the IPCC (2021) evaluation of effective global climate feedback of 1.16 ± 0.4 Wm−2K−1.
3.4.2 Impact of cloud sensitivity to perturbation
The overall relation for annual- and zonal-mean climate feedback at latitude \(\varphi\), \(\lambda \left(\varphi \right)\), is found by substituting for \(\frac{\partial {\alpha }_{AllSky}}{\partial {T}_{s}}\), (Eq. 21), and \(\frac{\partial {\epsilon }_{AllSky}}{\partial {T}_{s}}\), (Eq. 14) into the relation for \(\lambda \left(\varphi \right)\), Eq. (8). Under the constant cloud assumption \(\lambda \left(\varphi \right)\) (Fig. 2, solid black line) depends on the values of \(\sigma\), \({\epsilon }_{AllSky}\), \({T}_{s}^{3}\), \({T}_{s}^{4}\), \({f}_{CA}\), \({c}_{\epsilon }\), \({S}_{in}\), \({f}_{CI}\), \({\alpha }_{Cloud}\), \(\frac{\partial {\epsilon }_{ClearSky}}{\partial {T}_{s}}\) and \(\frac{\partial {\alpha }_{ClearSky}}{\partial {T}_{s}}\). However, the full relation for \(\lambda \left(\varphi \right)\) also depends on the values with latitude of: the sensitivity of cloud-incident fraction of solar radiation to temperature, \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\); the sensitivity of cloud amount to surface temperature, \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\); the sensitivity of cloud-impact on longwave absorptivity to surface temperature, \(\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\); and the sensitivity of cloud albedo to surface temperature, \(\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\); Eqs. (8), (14) and (21).
Both \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\) and \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\) reflect how cloudiness responds to climate perturbation. \(\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\) reflects how cloud longwave absorption properties change, at fixed measured cloud amount, in response to climate perturbation. The \(\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\) term reflects how cloud shortwave reflectance properties change, at fixed cloud amount and fixed latitude, in response to climate perturbation.
We cannot use observations of the current climatological state to constrain how cloud amount, or cloud shortwave reflectance or longwave absorption properties at fixed cloud amount, will respond to surface temperature change. This is because no relationships have yet been found that can relate these local cloud properties to local or global surface temperature. For example, there appear to be no identifiable relationships in the current climate state between the fraction of cloud-incident insolation, \({f}_{CI}\), the fraction of cloud amount, \({f}_{CA}\) or the fractional increase in longwave absorptivity due to clouds, \({c}_{\epsilon }\) (Fig. 1) with surface temperature. While \({\alpha }_{Cloud}\left(\varphi \right)\) is higher in colder polar regions and lower in the tropics (Fig. 1h), the apparent statistical relationship between \({\alpha }_{Cloud}\left(\varphi \right)\) and surface temperature may be better explained by changes in solar zenith angle affecting cloud albedo (e.g. Shupe and Intrieri, 2004), since both surface temperature and cloud albedo are affected by solar zenith angle. It should be noted that the cloud temperature and other cloud properties could affect reflectance of shortwave radiation e.g. via changing cloud-phase composition from water to ice (Murray et al., 2021).
Here, we estimate the uncertainty introduced from non-zero values of \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\) and \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\). Firstly, we assume to have no prior knowledge of the signs of \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\) and \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\), such that the best estimate for climate feedback terms remains the ‘constant cloud’ approximations (Fig. 2, solid lines). Secondly, we assume that from the initial climate state to a very similar final climate state, any small changes in \({f}_{CA}\) with latitude are equal to the small changes in \({f}_{CI}\) with latitude, \({\delta f}_{CI}=\delta {f}_{CA}\), since these are very similar and related quantities (Eq. 18; Fig. 1f). Thirdly, we assume that the uncertainty distribution for possible values of \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\) is normally distributed, and has standard deviation equal to the present-day standard deviation in \({f}_{CA}\) divided by the present-day standard deviation in \({T}_{S}\),
where present-day \({f}_{CA}\) and \({T}_{S}\) are both evaluated at 5° latitudinal intervals. Note that these three assumptions result in the standard deviation of \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\) being precisely equal to the standard deviation of \(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\), \({\sigma }_{\left(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\right)}={\sigma }_{\left(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\right)}\). Thus, we assume that the uncertainty in our knowledge of the sensitivity of cloudiness to surface temperature change due to a small climate perturbation, \({\sigma }_{\left(\frac{\partial {f}_{CI}}{\partial {T}_{s}}\right)}={\sigma }_{\left(\frac{\partial {f}_{CA}}{\partial {T}_{s}}\right)}\), scales with the variation in present-day cloudiness, \({\sigma }_{{f}_{CA}}\), divided by the variation in present-day surface temperature, \({\sigma }_{{T}_{S}}\). Alternatively, that future temporal changes in cloudiness with temperature change scale with current spatial variation in cloudiness per unit current spatial variation in temperature. This assumption allows a tractable estimate of the uncertainty in cloudiness sensitivity to temperature during climate change to be produced.
Using this approach (Eqs. 8, 14, 20, 23), we generate uncertainty estimates around our best estimates of albedo feedback, \({\lambda }_{albedo}\left(\varphi \right)\) (Fig. 2, red dashed lines to solid line), greenhouse feedback, \({\lambda }_{greenhouse}\left(\varphi \right)\) (Eqs. 10, Fig. 2, orange dashed lines to solid line), and in overall climate feedback, \(\lambda \left(\varphi \right)\) (Fig. 2, black dashed lines to solid line)
We find the global spatial average value for albedo feedback, including uncertainty from cloud amount changes with surfaced temperature, to be \(\stackrel{-}{{\lambda }_{albedo}}=-0.51\pm 0.35\) Wm−2K−1, while the global spatial average greenhouse feedback becomes \(\stackrel{-}{{\lambda }_{greenhouse}}=-1.77\pm 0.22\) Wm−2K−1 including the uncertainty estimate. However, these uncertainties combine to give an estimate for global spatial average for the overall climate feedback of \(\stackrel{-}{\lambda \left(\varphi \right)}=1.05\pm 0.13\) Wm− 2K− 1, with uncertainty in overall climate feedback reduced.
This reduction in uncertainty for the overall climate feedback, relative to uncertainty in albedo and greenhouse feedback contributions (Fig. 2), occurs because the uncertainty in greenhouse feedback and albedo feedback are perfectly anticorrelated: If \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}=\frac{\partial {f}_{CA}}{\partial {T}_{s}}\) is positive then albedo feedback becomes more positive and greenhouse feedback becomes more negative. Conversely, if \(\frac{\partial {f}_{CI}}{\partial {T}_{s}}=\frac{\partial {f}_{CA}}{\partial {T}_{s}}\) is negative, then albedo feedback becomes more negative and greenhouse feedback becomes more positive. Since albedo feedback is changed by more than greenhouse feedback (Fig. 2, orange and red dashed lines), then if cloudiness increases with warming (\(\frac{\partial {f}_{CI}}{\partial {T}_{s}}=\frac{\partial {f}_{CA}}{\partial {T}_{s}}>0\)) the climate feedback increases in magnitude.
This study does not attempt to quantify uncertainty through non-zero values of the cloud fractional longwave absorptivity sensitivity to surface temperature, \(\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\), or the cloud shortwave albedo sensitivity to surface temperature, \(\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\), terms (Eqs. 8, 14 and 21), which is reserved for future study. It is noted that current \({c}_{\epsilon }\) values are very stable across different surface temperatures from equatorial to polar (Fig. 1e), and that it is inherently difficult to separate the variation in \({\alpha }_{Cloud}\) caused by temperature from those caused by latitude (zenith angle) in the current distribution (Fig. 1f). The terms \(\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\) and \(\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\) represent the sensitivities of optical properties of clouds to climate perturbation to longwave and shortwave radiation respectively, but both at fixed measured cloud amount. Since changes in cloud type, at fixed measured cloud amount, are likely to affect both shortwave and longwave radiation in some way, there is the potential for \(\frac{\partial {c}_{\epsilon }}{\partial {T}_{s}}\) and \(\frac{\partial {\alpha }_{Cloud}}{\partial {T}_{s}}\) to have some correlation. Key to future quantification of the uncertainty introduced by these terms is likely the understanding of the correlations between them.