Radiative controls by clouds and thermodynamics shape surface temperatures and turbulent fluxes over land

Significance Land surface temperatures are a key characteristic of climate. Yet, understanding the main factors that shape them remains challenging because of the apparent dependence on many factors, such as radiation, turbulence, water availability, and vegetation. We use a fundamental, physical approach starting with radiation as the main forcing and constraining turbulent fluxes by their ability to perform maximum work to generate convective motion. This approach works very well in predicting observed climatological variations in surface temperatures, showing that arid regions are typically warmer due to the stronger solar heating in the absence of clouds. The implication is that the climatological variations of surface temperatures are predominantly shaped by radiation, clouds, and thermodynamic limits.


Figures S1 to S9
Table T1 to T2

SI References
Text A1: Thermodynamically constrained surface energy balance model:

A1.1: Model Conceptualization
The Earth's surface is continuously heated by incoming solar radiation, which makes it warmer.This energy is then released back by the earth into the atmosphere.However, this emission takes place at top of the atmosphere at a much lower temperature than Earth's surface.This temperature difference between the surface and the atmosphere drives the exchange of heat and mass (turbulent fluxes) as the surface-atmosphere system tries to achieve a state of thermal equilibrium.
We conceptualize this transfer as a result of a heat engine (Figure 1) operating between the warmer earth's surface and the cooler atmosphere.We then explicitly considered the second law of thermodynamics to quantify and constrain this transfer following the approach shown in (Kleidon & Renner, 2013;Dhara et al., 2016;Kleidon et al., 2018) and briefly described below.

A1.2 Deriving the thermodynamic limit
We start by applying the first law of thermodynamics to the conceptualized atmospheric heat engine, which is given by equation 1.
Where   represents the heat added into the system from the hot source (surface) through the exchange of turbulent fluxes,   represents the total heat exported out of the heat engine at the cold sink (atmosphere).G denotes the total power generated by the engine to sustain vertical mixing while D denotes the energy associated with the frictional dissipative heating.We assumed a steady state where the total power generated balances the frictional dissipation (G = D).dU/dt denotes the seasonal heat storage and heat transport changes within the system (Kleidon et al., 2018) and is expressed as in equation 2.

𝑑𝑈
Rs and Rl,toa in equation 2 are the absorbed solar radiation and outgoing longwave radiation respectively.The second step is to write the entropy budget for the system (second law of thermodynamics).It includes the entropy added into the system by turbulent fluxes (Jin) at hot source temperature, entropy exported out by radiative cooling (Jout) at cold sink temperature, entropy associated with heat storage changes, and entropy generated by the frictional dissipation.We consider an idealized case where no entropy is produced from any other irreversible processes besides frictional dissipation.The change in entropy of the system is then given by equation 3.
The source and sink temperatures were defined as the temperature of the earth's surface (Ts) and the radiative temperature of the atmosphere (Tr) respectively.It is to note that dissipation D primarily occurs in the lower atmosphere where mixing happens.The entropy associated with this term should ideally correspond to the potential temperature of lower atmosphere.As surface is closer to the lower atmosphere, we make an assumption to use surface temperature instead.
Similar assumption has been made in previous studies (Kleidon & Renner 2018;Conte et al., 2019).The surface and radiative temperatures were derived from the upwelling longwave radiation (Rl,up) and outgoing longwave radiation (Rl,toa) respectively from equations 4 and 5.
= (  ,  ) 1 4 (4) By replacing   from equation 1 and combining it with equation 3, we then get the expression for power (G) generated by the atmosphere which is given by equation 6.

𝐺 = (𝐽
The resulting expression in absence of atmospheric heat storage change is very similar to the widely known Carnot limit and have been referred to as the thermodynamic limit for cold heat engine (Kleidon et al., 2018).The equation 6 can be rewritten in terms of turbulent flux (Jin) using the surface energy balance and equation ( 4) as in equation 7.

A1.3: Maximum power trade-off
Based on equation 3, the convective power generated by the atmosphere to sustain vertical motion depends on the turbulent flux exchange (  ), heat storage changes (dU/dt), and the difference between the surface and radiative temperature (Ts -Tr).However, this temperature difference is not a fixed property of the system as there exists a covariation between the terms of turbulent flux exchange (   ) and the temperature difference (Ts -Tr).On one hand, a higher temperature difference between the surface and atmosphere will increase the turbulent flux exchange.On the other, increased turbulent fluxes will imply more evaporative cooling at the surface and condensational heating in the atmosphere which will deplete the driving temperature difference (Ts -Tr).This trade-off leads to a maximum in power for an optimum turbulent flux (Jopt) and is also reflected in equation 6.This is referred to as the maximum power limit.This optimum flux was calculated at the maximum power limit by numerically solving equation 8.
Jin was varied within the limits of heat engine from Jin = 0 (no surface cooling by convection) to Jin = Jmax = Rs + Rld -  4 .Jmax represents a case, where Ts-Tr = 0 and there is no thermal disequilibrium within the heat engine anymore to drive J and J reaches its theoretical maximum value.Also, it is to note that dU/dt is not zero in the solution of equation 8, It results in an offset and thereby does not affect the maximum power trade-off but affects the magnitude of optimized turbulent flux (Jopt).

A1.4 Estimation of surface Temperatures
The surface temperatures at maximum power were then calculated using the surface energy balance together with the optimised turbulent fluxes using equation 9.
max  = ( Here Rs is the absorbed solar radiation, Rld is the downward longwave radiation, Jopt is the optimal turbulent flux that maximizes the convective power in equation 6 and  is the Stefan -Boltzmann constant with the value of 5.67 * 10 -8 Wm -2 K -4 .

A1.5 Removing the cloud radiative effects from surface temperatures
To remove the cloud radiative effects from surface temperatures, we used the "clear-sky" fluxes from the NASA-CERES dataset as forcing to our thermodynamically constrained formulation of surface energy balance."Clear-sky" fluxes from NASA -CERES are diagnosed by removing the clouds from the radiative transfer.More details can be found here (Loeb et al., 2018;Kato et al., 2018).
We first numerically calculated the maximum convective power generated from the clear-sky fluxes by solving equation 8 and then use it to estimate the "Clear-sky" temperatures using equation 10.

Text A2: Surface Energy partitioning
To partition the optimized turbulent fluxes estimated from the maximum power limit into sensible and latent heat, we used the equilibrium energy partitioning approach (Slayter & McIlroy, 1961;Priestley & Taylor, 1972) and also described in Kleidon & Renner (2013).This framework however assumes a saturated surface with no water limitation.To apply it at a global scale, we account for water limitation by introducing a limitation factor (fw).This factor was calculated as the ratio of actual to potential evaporation using GLEAM V3.6b data (Martens et al., 2017).Latent heat and sensible heat were then calculated from equations 11 and 12.

𝐿𝐸 = 𝑓
Where s and  are the slope of the saturation vapor pressure curve and Psychometric constant respectively.Jopt is the optimized turbulent flux estimated from maximum power limit.

Text A3: Decomposition of Downwelling longwave radiation
Downwelling longwave radiation largely depends on how hot and black the atmosphere is.On one hand, a hotter atmosphere will emit more radiation back to earth as a result of higher radiative temperature.On the other hand, the increase in emissivity of the atmosphere will lead to enhanced absorption and re-emission of downward longwave radiation.The former is likely to increase with enhanced heat transport while the latter largely depends on the amount of water vapor and clouds in the atmospheric column.To decompose these two effects, we used the semiempirical formulation of downwelling longwave radiation proposed by Brutsaert (1975) and Crawford & Duchon (1999).Downwelling radiation can then be described by the following equation: Where  is the Steffan Boltsman constant with the value of 5.67 x 10 -8 Wm -2 K -4 .Ta is the nearsurface air temperature and  is the emissivity of the atmosphere which is a function of cloud area fraction and vapor pressure as described in equation 14.

𝜀 = (𝑓
Here, fc is the cloud fraction (0 -1) which was derived using NASA-CERES EBAF ed4,1 dataset and eo denotes the actual vapor pressure.We first compared the estimated downwelling longwave radiation calculated from equation 14 with the observations from NASA-CERES (Loeb et al., 2018;Kato et al., 2018).We find strong agreement over global land with the R 2 value of 0.97.The differential form of equation 13 was then used to decompose the downwelling longwave radiation as shown in equation 15.
∆  =   ̅̅̅ 4 ∆ + 4̅   ̅̅̅ 3 ∆ The first term in equation 15 shows the variation in Rld due to changes in the emissivity of the atmosphere (blue line in figure S6) while the second term shows the changes in Rld due to changes in the atmospheric temperature (Red line in figure S6).

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Figure S2

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Figure S4

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Figure S7