Moist static energy budget
Boundary-layer clouds exert a radiative cooling on the troposphere, which can be quantified through the so-called Atmospheric Cloud Radiative Forcing (ACRF). The ACRF is defined as the difference between the CRF at TOA and at the surface or, equivalently, as the vertically-integrated cloud perturbation of the tropospheric radiative cooling (defined as the all-sky minus clear-sky radiative heating rates) [R]−[R0]:
$$ [ACRF] = [R] - [R_0] = \int\limits_{P_{SFC}}^{P_{toa}} (R-R_0) {\frac{dP} {g}} = CRF_{TOA} - CRF_{SFC} $$
(1)
The change in ACRF induced by different perturbations being strongly correlated with the change in SW CRF at the top of the atmosphere (Fig. 10, Table 3) and with the change in low-level cloud fraction (not shown), it may be used as a proxy for the cloud-radiative response that we aim to interpret.
To understand what controls the cloud-radiative response to a given perturbation, and interpret in particular the strong sensitivity of low-level clouds to changes in the vertical stratification of the atmosphere, we analyze the tropospheric moist static energy (MSE) defined as h = c
p
T + g
z + L
q where T is the temperature, c
p
is the specific heat at constant pressure, z is height, g is the gravitational acceleration, L is the latent heat of vaporization at 0°C, and q is the specific humidity. The vertically integrated budget of MSE (brackets refer to vertical averages) may be expressed as:
$$ (LH + SH) + [R] - \left[\omega {\frac{\partial h}{\partial P}}\right] - [\overrightarrow{V}.\overrightarrow{\nabla} h] = 0 $$
(2)
where LH and SH are surface turbulent fluxes of latent and sensible heat, respectively, \(\overrightarrow{V}\) is the horizontal wind, and ω the large-scale vertical velocity. The ACRF may then be expressed as:
$$ [ACRF] = -[R_0] - (LH + SH) + \left[\omega {\frac{\partial h}{\partial P}}\right] + [\overrightarrow{V}.\overrightarrow{\nabla} h] $$
(3)
Through this equation, the dimensionality of the cloud-feedback problem may be reduced to a problem of four components. In regimes of large-scale subsidence, the MSE of the planetary boundary-layer is increased by surface turbulent fluxes, and decreased by the emission of clear-sky radiation and by the downward advection of low-MSE from the free troposphere (Eq. 2). The presence of clouds also contributes to lower the PBL MSE through radiative cooling (ACRF), as well as the horizontal MSE advection. For a given horizontal advection of MSE, Eq. 3 shows that the radiative effects of clouds and the downward advection of low MSE into the PBL both contribute and eventually compete to balance the combined effect of surface fluxes and clear-sky radiative cooling on the PBL energy budget. It also shows that a change in the vertical profiles of large-scale subsidence and atmospheric stratification may change the magnitude of the vertically-integrated downward advection term of MSE \([\omega{\frac{\partial h}{\partial P}}]\).
Figure 11 compares the perturbations of the different terms of Eq. 3 in SCM experiments (G, J, K, L and M) in which a given vertically-averaged clear-sky radiative cooling is applied to the model with different vertical distributions (Sect. 2). In response to an increased [R
0], surface fluxes always increase, all the more that the radiative cooling is applied near the surface (experiment M). However, the vertical advection term of MSE substantially depends on the vertical distribution of the clear-sky radiative perturbation and appears to be primarily responsible for differences in the cloud response among the different experiments: when the radiative cooling perturbation is applied in the upper troposphere, the change in the vertical stratification of MSE (the vertical velocity profile remains unchanged in this experiment but the MSE strongly decreases above the PBL) induces a strong negative anomaly of the vertical advection term which is not compensated by the increase in surface fluxes and is associated with a decreased low-cloud cover and a weakened ACRF (positive anomaly) to ensure energy conservation. At the other extreme, when the increased clear-sky radiative cooling perturbation is applied within the low-cloud layer, the vertical gradient of MSE in the lower troposphere weakens, which makes the vertical advection of MSE less negative in the PBL and leads to an enhanced low-level cloud cover and cloud radiative cooling. These experiments suggest that the impact of an external perturbation on the low-cloud cover strongly depends on how this perturbation affects the MSE vertical gradient within the PBL.
Physical understanding of the relationship between MSE advection term and low-level clouds
To understand physically the correlation between changes in the vertical gradient of MSE and changes in the low-level cloud cover, we examine the vertical profile of MSE normalized by the near-surface MSE value (Fig. 12) in SCM and GCM experiments. In the subsidence regimes of the tropics, the atmosphere exhibits a minimum MSE above the PBL (around 700 hPa) and thus a negative vertical advection term of MSE (−ω ∂h/∂P) below this minimum and a positive term above. In a warmer climate (SCM experiment A), the PBL deepens, the minimum MSE occurs higher in altitude and the MSE contrast between the near-surface and minimum MSE values increases: this induces a change in the MSE vertical advection term which maximizes between 900 and 700 hPa. A similar behaviour is found in GCM experiments, both in realistic (AMIP) and aqua-planet configurations.
Equation 3 and SCM sensitivity experiments (Fig. 11) suggest some correlation between the low-cloud radiative response and the change in vertical MSE advection. Since the sensitivity of the latter to climate change perturbations is maximum at the top of the PBL, we consider the vertically-integrated MSE vertical advection term between 900 and 700 hPa, an index hereafter referred to as boundary-layer vertical advection term or BVA (BVA = \(\int_{900\,{\rm hPa}}^{700\,{\rm hPa}} -\omega {\frac{\partial h}{\partial P}} {\frac{dP} {g}}\)). Figure 13 shows that across the range of SCM and GCM experiments, the change in low-level cloudiness (characterized by the change in PBL cloud fraction at the vertical level where the cloud fraction is maximum, which typically occurs around 950 hPa) is well correlated with the change in BVA (R
2 = 0.55 with point M and R
2 = 0.81 without). In response to a large range of perturbations (including changes in SST, CO2 or large-scale subsidence), the change in BVA thus appears to be the term of Eq. 3 that correlates best with the change in ACRF, both in SCM and GCM experiments (Fig. 13).
The vertical advection of MSE being dependent on both the vertical velocity profile and the vertical gradient in MSE, it may be perturbed both by local (e.g. surface temperature changes) and remote changes. Those latters may be associated with a change in the large-scale atmospheric dynamics (change in ω) or with a change in the free-tropospheric temperature profile, which is mainly controlled by deep convective processes. To clarify the origin of the change in low-level clouds, we thus examine in the next section the reasons for the change in BVA in GCM experiments.
Interpretation of low-cloud changes in GCM experiments
GCM experiments associated with a uniform (4 K) SST increase exhibit a decrease of low-level clouds while those associated with a 4xCO2 radiative forcing exhibit an increase of low-level clouds (Sect. 3.1, Fig. 4). These opposite responses are also associated with opposite changes in the vertical advection term of MSE in the PBL (Fig. 13). To interpret these different changes in BVA, we decompose the change in the MSE vertical advection term in three components as following:
$$ \Updelta \left[-\omega {\frac{\partial h}{\partial P}}\right] = \left[-{\frac{\partial h}{\partial P}}\Updelta \omega\right] + \left[-\omega \Updelta {\frac{\partial h}{\partial P}}\right] + \left[-\Updelta \omega \Updelta {\frac{\partial h}{\partial P}}\right] $$
(4)
Both in +4 K and 4xCO2 experiments, the second right-hand-term quantifying the contribution of changes in the MSE vertical gradient represents more than 75 % of the total change in the two atmospheric models. The impact of ω changes on BVA is thus of secondary importance in modulating BVA in these experiments.
The robust change in the MSE vertical gradient and in BVA in surface warming experiments (Fig. 12) results from two factors. On the one hand, the deepening of the PBL, which is consistent with the expected growth of a marine shallow cumulus boundary layer in response to increased surface turbulent fluxes (Medeiros et al. 2005; Stevens 2007), rises the height of minimum MSE and then makes the vertical advection term of MSE more negative around the top of the PBL. However, a second and even more robust explanation is related to the non-linearity of the thermodynamic relationship of Clausius-Clapeyron, which increases the specific humidity (and thus MSE) with temperature at a larger rate near the surface than at altitude (changes in relative humidity play a secondary role, Fig. 12b). This enhances the MSE vertical gradient between the surface and the height of minimum MSE and then strengthens the import by large-scale subsidence of low-MSE from the free troposphere down to the surface. This effect, together with the deepening of the PBL, make BVA more negative and decreases the low-cloud fraction.