For our description of the surface–atmosphere system, we need two basic energy balance constraints: the energy balance of the surface and the energy balance of the whole system.

The surface energy balance is described in terms of heat storage changes that take place below the surface, dUs/dt; the absorbed solar radiation at the surface, Rs; the net cooling by longwave radiation, Rl,net; and the turbulent heat fluxes, J (the sum of the sensible and latent heat flux, which are combined here for simplicity): dUsdt=Rs-Rl,net-J. The energy balance of the whole column is described by dUtotdt=dUsdt+dUadt=Rs,toa-σTr4, where dUtot/dt represents the total change of heat storage within the surface–atmosphere system (consisting of heat storage changes below the surface, dUs/dt, and within the atmosphere, dUa/dt), Rs,toa is the total absorption of solar radiation (surface and atmosphere) and Tr is the radiative temperature by which radiation is emitted to space, and σ is the Stefan–Boltzmann constant. The radiative temperature is determined from the mean energy balance taken over a sufficiently long time so that Tr=Rs,toa,avgσ1/4, where Rs,toa,avg is the mean value of Rs,toa. We assume that Tr does not change at the diurnal scale. This effectively represents our assumption that the system has sufficient capacity to store heat to balance out the variations in solar radiation.

The total change in heat storage within the system can be determined from the approximation that this total heat storage does not change when averaged over the course of day and night. The total change in heat storage can then be inferred from the difference between the instantaneous and mean solar forcing, given by dUtotdt=Rs,toa-Rs,toa,avg. In the following, we assume for simplicity that all solar radiation is absorbed at the surface, so that Rs=Rs,toa. This assumption simplifies the following considerations, but does not affect the results, as discussed in Sect. . We then describe the ocean–atmosphere system as a system in which the heat storage changes take place below the surface (that is, in the surface ocean), so that dUs/dt=dUtot/dt. For the land–atmosphere system, we neglect the ground heat flux, which is typically small on a diurnal timescale (as discussed in the introduction), so that the change in heat storage needs to take place in the lower atmosphere to meet this diurnal energy balance constraint. As it turns out, this heat storage change does not enter the formulations explicitly so that the term dUa/dt does not appear in the equations below.

To describe the net cooling by terrestrial radiation at the surface, Rl,net, we use a simple, linearised parameterisation of net longwave radiation of the form Rl,net=Rl,0+kr(Ts-Tr). This formulation of net longwave radiation at the surface is similar to well-established empirical, linearised forms e.g., but it can also be derived from the grey atmosphere approximation of radiative transfer in combination with a linearisation of surface emission, as in . Net longwave radiation, Rl,net, is the difference between surface emission, Rl,u=σTs4≈Tr+(4σTr3)(Ts-Tr), and the downwelling flux of longwave radiation, Rl,d=(3/4)τ⋅Rs,toa. Here, τ is the longwave optical depth and Tr is described by Eq. (). This interpretation has the advantage that the sensitivity of the parameters to a change in greenhouse forcing can directly be identified. The parameters Rl,0 and kr are then related to the longwave optical depth, τ; the mean emission of terrestrial radiation to space, Rs,toa,avg; and the radiative temperature, Tr, by Rl,0=1-34τ⋅Rs,toa,avg and kr=4σTr3=4⋅Rs,toa,avgTr. Note that a change in the greenhouse effect is associated with a change Δτ, which alters the value of Rl,0, but not kr. A change in absorption of solar radiation, for instance due to enhanced reflectance by clouds or aerosols, affects both expressions if the total absorption, Rs,toa, is altered.

The turbulent fluxes J in the surface energy balance are derived from the assumption that these operate at the thermodynamic limit of maximum power . In this formulation, turbulent fluxes are seen as the driver of a convective, atmospheric heat engine that generates the power to sustain motion and the turbulent exchange between the surface and the atmosphere. This approach uses the common and well-accepted Carnot limit of a heat engine and combines it with the surface energy balance. The latter aspect plays a central role, because turbulent fluxes lower surface temperatures and thus affect the Carnot limit. The approach then assumes that natural systems evolve to and operate near their thermodynamic limit. This assumption falls into a broader range of thermodynamic optimality approaches. In particular, it relates closely to a general hypothesis of maximum entropy production MEP; e.g., which has been applied rather successfully in the past to describe atmospheric dynamics and to the mean climatological surface energy balance partitioning on land . As maximising the power of a heat engine results in maximum frictional dissipation, this is almost the same as maximising the associated entropy production of this process. The focus on maximising the power associated with turbulent heat fluxes, however, allows for a more specific application of thermodynamic optimality to the particular process of atmospheric turbulent heat transfer in relation to MEP, and it can be more easily explained using the well-established concept of a heat engine. In the following, this approach to estimate the magnitude of turbulent heat fluxes is briefly summarised and extended to include the effects of diurnal heat storage changes.

The power, G, or work per time, that a heat engine can provide is constrained by the Carnot limit, given by G=J⋅Ts-TrTs, where for the application to vertical heat transfer in the atmosphere the driving temperature difference is set to the difference between the surface temperature and the radiative temperature. The motivation for using the radiative temperature as the cold temperature of the heat engine is to not use the temperature at a specific height of the atmosphere but rather to use the temperature at which the entropy export by radiative cooling to space is at a maximum. This temperature is, by definition, the radiative temperature, as it is the temperature of a blackbody that emits radiation at the rate Rs,toa,avg.

To derive the maximum power limit from the Carnot limit, we combine this limit with a fundamental tradeoff by which a greater turbulent heat flux results in a lower surface temperature, so that the derived power has a maximum with an associated, optimum value of J and Ts. This tradeoff is obtained by combining the surface energy balance (Eq. ) and the expression for Rl,net (Eq. ) to express Ts in terms of the radiative forcing, the heat storage change dUs/dt, and the turbulent fluxes J, Ts=Tr+Rs-dUs/dt-Rl,0-Jkr. When used in the expression for the Carnot limit (Eq. ), we obtain G=J⋅Rs-dUs/dt-Rl,0-JkrTs. This expression has a maximum in power (i.e. maximum generation of turbulent kinetic energy), which can be derived analytically from dG/dJ=0. When neglecting the variation in Ts with J in the denominator, the maximisation yields an optimum heat flux, Jopt, and net longwave flux, Rl,net,opt, given by Jopt=Rs-dUs/dt-Rl,02Rl,net,opt=Rs-dUs/dt+Rl,02. Note how this formulation of surface energy balance partitioning depends on heat storage changes below the surface, dUs/dt, but not on heat storage changes that take place in the lower atmosphere, dUa/dt. We use these two contrasting cases of heat storage change to describe how this partitioning looks for ocean (day and night) and land surfaces (daytime only).

For the ocean surface, the dominant heat storage changes take place below the surface, so that dUs/dt≈dUtot/dt=Rs,toa-Rs,toa,avg (see Eq. ). With this expression for dUs/dt, the optimum surface energy partitioning is then given by Jopt,ocean=Rs,avg-Rl,02Rl,net,opt,ocean=Rs,avg+Rl,02. This partitioning describes no temporal changes during the course of the day, as the turbulent fluxes as well as net longwave radiation are described by the mean solar radiation at the surface, Rs,avg, rather than the instantaneous forcing, Rs.

For the land surface, we assume that the heat storage changes take place in the lower atmosphere and dUs/dt≈0. Then, the energy balance partitioning during the day is given by Jopt,land=Rs-Rl,02Rl,net,opt,land=Rs+Rl,02. Note how this partitioning includes the instantaneous rate of absorption of solar radiation, Rs, thus resulting in a pronounced diurnal variation in surface energy balance partitioning as it is commonly observed on land. During night where Rs=0 and J=0 due to the prevalent stable conditions, we assume Rl,net≈0. This simplification is reasonable as observations typically show that the cooling of the surface by net longwave radiation at night is less than 100 Wm-2 e.g. and thus much smaller than the peak heating rate by solar radiation during the day.

For the two contrasting cases of land and ocean surfaces, we can derive expressions for surface temperature by equating the optimum expressions for the net longwave radiative flux with Eq. ().

For the ocean surface, surface energy balance partitioning does not change over the course of day and night. Hence, surface temperature is constant and depends only on the mean absorption of solar radiation: Tocean=Tr+Rs,avg-Rl,02kr. For land, we split the surface energy balance partitioning into two parts of night and day. The nighttime temperature is derived directly from Rl,net≈0. This yields an equation for the nighttime temperature of Tnight=Tr-Rl,0kr. During the day, the mean absorption of solar radiation is about Rs,day=2⋅Rs,avg, which we use for Rs in Eq. (). The mean daytime surface temperature is then given by Tday=Tr+Rs,day-Rl,02kr. This yields a mean surface temperature over land, Tland=1/2⋅(Tnight+Tday), of Tland=Tr+Rs,avg-3/2⋅Rl,02kr. When both temperatures are combined, the global mean surface temperature, Tglobal, is described by Tglobal=(1-fland)⋅Tocean+fland⋅Tland, where fland=0.29 is the fraction of land area of the total surface area of the Earth.

Equations ()–() represent the key equations used in the following to evaluate the sensitivity of surface temperature to a change in radiative forcing. These estimates are then compared to the respective sensitivities derived from the CMIP5 climate model simulations , using the 4×CO2 and preindustrial control simulations (a list of models used is provided in Table ).

We first evaluate the energy balance partitioning and expressions for temperatures using global means. The forcing is described by observations of the total mean absorption of solar radiation of the surface–atmosphere system of about Rs,toa,avg=240 Wm-2 and the mean absorption at the surface of Rs,avg=165 Wm-2 . To obtain a global mean surface temperature of about 288 K, we choose a value of τ=1.74 for the longwave optical depth. This is the only parameter in the formulations that we adjusted to match observations. From this radiative forcing, the parameters Rl,0 and kr are derived for the surface energy balance partitioning. The resulting surface energy balance partitioning is illustrated in Fig.  and the respective values are provided in Table . The turbulent fluxes of 119 Wm-2 and net longwave radiation of 46 Wm-2 derived from the maximum power limit compare reasonably well to the estimates from observations of 112 and 53 Wm-2 . Note that the radiative properties as well as continental area show strong geographic variations that are not accounted for here, so that this evaluation merely shows the plausibility of the formulations.

The difference in diurnal energy balance partitioning between the ocean and land surface is illustrated for global mean conditions in Fig. . Note that these global mean conditions are hypothetical and used here to illustrate the difference in energy balance partitioning using the formulations described in the methods section. At the top of Fig. , the diurnal variation in the total heat storage within the surface–atmosphere system is shown (compare with Eq. ) for a mean absorption of solar radiation at the surface of 165 Wm-2 and the respective diurnal variation in Rs. The differences in energy balance partitioning for ocean and land surfaces, using Eqs. () and (), are then illustrated in Fig. b and c. This figure clearly illustrates that by buffering the diurnal variations in solar radiation below the surface, the ocean surface energy balance has no diurnal variation in turbulent fluxes. If the buffering takes place above the surface, as is mostly the case for land surfaces, this results in a pronounced diurnal variation in turbulent fluxes.

We next evaluate the case of global warming. An increase in the greenhouse effect is represented in our formulation by an increase in the longwave optical depth, Δτ>0. We used a value of Δτ=0.18 to obtain a global temperature increase of ΔTglobal=5.0 K. The increase in optical thickness then changes Rl,0 by ΔRl,0=-3/4⋅Rs,toa⋅Δτ<0. Using the grey atmosphere approximation, this change in τ implies an increase in the downwelling longwave radiation of about ΔRl,d=33 Wm-2. This increase compares fairly well to the range found in CMIP5 simulations used here, which range from 20 to 42 Wm-2 (global mean) and are associated with a warming of 2.9 to 6.0 K in surface temperatures (4×CO2 scenario – PI control; see Fig. ). The effect of this change in optical thickness on the diurnal course of surface energy balance partitioning is shown in Fig. b and c by the dotted lines. The consequences for mean ocean and land temperatures as well as for daytime and nighttime temperatures on land is illustrated in Fig. , with values given in Table . These sensitivities can be derived analytically, using the expressions derived in Sect. .

The warming of the ocean surface, ΔTocean, is then given by ΔTocean=-ΔRl,02kr. When the same change of optical thickness is applied to land, it results in a warming of the land surface, ΔTland, of ΔTland=-32ΔRl,02kr. When taking the ratio, ϕ, of these two changes, we obtain ϕ=ΔTlandΔTocean=32. Hence, the land surface is 50 % more sensitive to a change in longwave optical depth than the ocean surface. We can also translate this fixed ratio between land and ocean warming into respective expressions that relate land and ocean warming to the global temperature change: ΔTocean=11+0.5⋅fland⋅ΔTglobal≈0.87⋅ΔTglobalΔTland=32-fland⋅ΔTglobal≈1.31⋅ΔTglobal. When using the global mean values as shown in Table , a global mean temperature increase of 5 K due to an increased greenhouse effect translates into an increase by 6.6 K over land (or 31 % more than the global mean increase), while oceans only increase by 4.4 K (or 13 % less than the global mean increase). In Fig.  we compare the predicted ratio ϕ as well as the temperature differences ΔTland and ΔTocean to ΔTglobal to the respective simulated values of CMIP5 climate model simulations. Although some deviations can be seen, our estimates overall compare very well to the global mean changes found in the CMIP5 simulations.

As argued in the introduction, the difference in the climate sensitivity of land and ocean surfaces should be attributable to the different behaviour of the land surface at night than during the day. To evaluate this in our formulations, we also looked at the sensitivities of nighttime and daytime temperatures as proxies for minimum and maximum temperatures. The minimum temperature typically occurs at the end of the night, and we approximate it by the use of Tnight. The maximum temperature occurs at the end of the day, and for this temperature we use Tday. Using the above expressions for these temperatures, we obtain ΔTnight=-ΔRl,0krΔTday=-ΔRl,02kr=12ΔTnight. Hence, minimum temperatures increase at twice the rate of maximum temperatures in our formulation, thus reducing the diurnal temperature range. This is broadly consistent with observations, for which a range of about 1.6–2.4 is reported for most seasons , although in observations the ratio varies between hemispheres and seasons.

We did not perform an evaluation of the diurnal temperature range for the CMIP5 simulations for a few reasons. There are other effects, e.g. due to changes in the hydrologic cycle, as well as model biases that quite substantially affect the trend in the diurnal temperature range in the CMIP5 simulations so that this direct effect of an enhanced greenhouse forcing is not the dominant factor in the 4×CO2 simulations. These effects would need to be accounted for in our expressions before a more detailed comparison could be made. Yet, the well-established observation that the diurnal temperature range decreases with global warming is consistent with our interpretation why the climate sensitivity of land is higher than for ocean surfaces.

To illustrate that changes in solar radiation affect the temperature sensitivity quite differently, as described in , we consider a case in which the total absorption of solar radiation is unchanged, but more solar radiation is absorbed at the surface (i.e. ΔRs,avg=+10 Wm-2). This magnitude of change in solar absorption is comparable to observed changes associated with solar dimming and brightening over the last decades e.g., which in turn relates mostly to changes in aerosol concentrations in the atmosphere. For comparability, we use the same value for the longwave optical depth and considered the case of solar brightening to better compare the effects of solar changes to changes in greenhouse forcing. The sensitivity to absorbed solar radiation at the surface is shown in Fig.  by the red bars, with the values given in Table  in the column labelled “Solar Brightening”.

Our simple estimates partition the increase in surface solar absorption equally into increases in ΔJ and ΔRl,net (see Eqs.  and ). The change in ocean and land temperatures is given by (see Eqs.  and ) ΔTocean=ΔRs,avg2kr=ΔTland, and the land surface warms on average by the same amount as the ocean surface. This is quite different than the result from the change in the greenhouse effect, where the sensitivities were different. The effect on the diurnal temperature range on land is also markedly different. While the nighttime temperatures remain unchanged as they do not depend on Rs, the daytime temperatures are increased by twice the mean warming, with ΔTday=ΔRs/kr=2ΔTland. This effect of solar radiation on maximum temperatures is well known e.g. and has been used to infer solar radiation from the diurnal temperature range e.g..

Our formulations thus show that the temperature sensitivities of ocean and land surfaces, as well as sensitivities of minimum and maximum temperatures on land, and thus of the diurnal temperature range, are closely connected and react differently depending on the type of radiative change at the surface.

Despite its physical basis, our model has, obviously, several potential limitations due to its extremely simple nature. These potential limitations relate to the parameterisation of radiation and turbulent fluxes, as well as to how evaporation is treated in our formulations.

To start, the use of the grey atmosphere approximation for the downwelling flux of longwave radiation is an approximation. It represents a more mechanistic approach of parameterising longwave radiative transfer, with the main difference to earlier work being the additive constant Rl,0 in Eq. () that played here an important role. The use of the grey atmosphere approximation, however, is likely to overestimate the downward longwave flux for a given optical depth. Turbulent fluxes cause a lower surface temperature than in radiative equilibrium, which results in a colder lower atmosphere that is in radiative–convective equilibrium. This, in turn, should be associated with a lower downwelling flux of longwave radiation. By using the grey atmosphere approximation, we do not account for this effect, which is likely to result in some biases in our formulation. This is likely to result in an overestimation of the sensitivity of surface temperature to changes in the optical depth. However, as we do not calculate optical depths or use observations but rather adjust it to represent the global mean temperature or a given temperature change, the effect of this bias in the radiation parameterisation is likely to be small for our results.

We also did not specifically consider absorption of solar radiation within the atmosphere (which can be seen by comparing Rs,toa,avg to Rs,avg in Table ). This atmospheric absorption would result in some diurnal variation in heat storage within the atmosphere over oceans. However, since our expressions do not explicitly depend on changes in atmospheric heat storage, the effect of this should not change our results.

Another limitation relates to our representation of turbulent fluxes. We used the Carnot limit and the assumption that turbulent fluxes operate near the limit of maximised power. Yet, the diurnal variation in heat storage in the lower atmosphere over land may need to be accounted for in the derivation of thermodynamic limits, which may then result in a different partitioning of energy fluxes at the surface. However, as long as the turbulent fluxes on land are proportional to the instantaneous value of absorbed solar radiation at the surface (which is a good assumption as turbulent fluxes on land show a strong diurnal variation), turbulent fluxes must then be small at night. The basic reasoning would then still apply that nighttime temperatures are more sensitive to a change in greenhouse forcing, thus resulting in an altered climate sensitivity of land surfaces compared to ocean surfaces, although the specific ratio ϕ may differ from our value of 3/2.

Furthermore, we do not explicitly consider evaporation in our formulation, but include it in the turbulent fluxes J. Evaporation and the associated latent heat flux cools the surface, yet it only heats the atmosphere (and the surface) when it condenses. At the global scale in steady state, evaporation needs to balance precipitation, so that evaporation does not necessarily need to be represented as a separate term in the surface energy balance. Yet, spatiotemporal imbalances between evaporation and precipitation due to storage changes of water vapour and moisture transport can result in regional temperature variations due to evaporation . Furthermore, differences in radiative parameters during dry and wet periods may result in further modulations of surface temperatures that we did not account for and that could have an effect . Those effects would clearly need to be addressed when our formulations are applied to the regional scale, which could form a topic of future research.

Yet, overall, it would seem that despite these deficiencies, our simple representation is able to adequately illustrate our explanation from the introduction in a parsimonious way as it captures the difference in climate sensitivity of ocean and land surfaces, and connects this difference to the difference in sensitivity between minimum and maximum temperatures.

Our explanation for the difference in temperature sensitivity between oceans and land is quite different, yet not in contradiction to previous approaches, which we explain in the following. Previous attempts to explain the difference in temperature sensitivity typically start with the reduced water availability on land. In arid regions, water limits evaporation, so that the radiative heating results in an enhanced sensible heat flux and thermal emission, which is accomplished by a warmer surface temperature. When ocean and land are exposed to an equal increase in the downwelling flux of longwave radiation by an enhanced greenhouse effect, the land surface in arid regions should respond by a stronger warming than the ocean surface (which to some extent is found in climate model simulations). These effects then result in different levels of humidity and affect lapse rates in the lower atmosphere. This line of explanation was developed and extended by , and .

Our approach uses a systems approach to the surface–atmosphere system, so it neither focuses entirely on the surface energy balance nor on the atmosphere, but rather on the coupled system. This system is subjected to the energy balance constraint during the diurnal cycle, which we use to infer the buffering needed to level out the strong variation in solar radiation. While the buffering below the ocean surface is relatively straightforward and established for the ocean–atmosphere system, the buffering in the lower atmosphere for the land–atmosphere system is less established but central to the explanation presented here. This buffering implies the formation of a convective boundary layer on land during the day to heat the lower atmosphere and to accomplish the diurnal heat storage change. It also implies unstable conditions during the day that drives the sensible heat flux, dry convection, and boundary layer development. This likely results in a greater lapse rate that is closer to the dry adiabatic lapse rate as the heating of the lower atmosphere is primarily driven by the sensible heat flux and dry convection. These consequences link to the properties that were used before to explain the difference in temperature sensitivity, so our explanation is consistent with previous interpretations in this respect. Yet, what our approach shows is that these dynamics do not need to be resolved in detail to derive the difference in climate sensitivity, as these essentially follow from the energy balance closure assumption applied to the surface–atmosphere system over the whole day. It would thus seem that this energy balance closure and the way by which the land surface–atmosphere system accomplishes it are the primary cause for the difference in temperature sensitivity.

Our explanation can not only explain the difference in temperature sensitivity over land and ocean but also connects to the well-established difference in sensitivity of minimum and maximum temperatures as well as the distinctively different diurnal course of turbulent fluxes between land and ocean. In this sense, our explanation appears to be more general, as it is able to explain more phenomena by a less complex approach.

We can draw a few broader implications from these insights that open up possibilities for a range of further research.

First, our results explain why the diurnal dynamics of the surface energy balance of ocean and land surfaces are so distinctively different. While these differences are well established in observations and are described in textbooks e.g., we can explain these by the different means by which the diurnal variation in solar radiation is being buffered, with critical implications for the temperature sensitivity. For the ocean, variations in solar radiation are buffered by heat storage changes below the surface, so that turbulent fluxes do not show much of a diurnal variation. On land, however, it is well known that turbulent fluxes show a pronounced diurnal variation during the day and are practically absent during the night. We interpreted this different behaviour of land surfaces here as a result of the buffering taking place in the lower atmosphere above the surface, rather than below as in the case for the ocean, which results in stable conditions during nighttime that are more sensitive to changes in the greenhouse effect. As variations in solar radiation are buffered in the lower atmosphere over land, this symbolises the strong coupling between the land surface and the lower atmosphere, with this coupling nevertheless being constrained by the energy balance over day and night over the whole surface–atmosphere system. Our approach could thus serve as the basis for a parsimonious approach to better understand land surface–atmosphere interactions.

Our interpretation that diurnal heat storage variations explain the difference in climate sensitivity can, clearly, be analysed in much greater detail in observations, reanalyses, and climate models. The variations in heat content in the lower atmosphere should be relatively straightforward to analyse in observations and model output, and one would expect a noticeable difference in variations over oceans and land. However, such an analysis requires the ability to diagnose diurnal heat storage variations in the atmosphere at sufficient temporal resolution. Measurements by radiosoundings, for instance, are available only twice a day, which is insufficient to diagnose the magnitude of diurnal heat storage variations. For climate models, this would require a substantial amount of model output at the diurnal scale, which is typically not available. It would thus require measurements and output at higher temporal resolution to evaluate these heat storage variations in greater detail.

One can also evaluate and extend this approach with respect to some of the simplifying assumptions, for instance regarding evaporation (as already discussed above), the ground heat flux, and different land cover types as well as inland water bodies such as lakes and rivers. While the role of the ground heat flux has been neglected here for the land surface, observations show a noticeable magnitude of this flux, especially for non-forested surfaces. Our interpretation would suggest that regions with a greater ground heat flux would show diurnal temperature variations that are reduced for the given radiative forcing and somewhat more similar to ocean surfaces (resulting in a lower ratio ϕ). To evaluate this further and go into more regional variations in the diurnal temperature range would, however, require a more specific treatment of the different factors that vary geographically. For instance, the ground heat flux is typically larger in desert regions, which are dry, lack evaporative cooling, and typically have comparatively low optical thicknesses. To expand this approach to regional variations and different processes would thus require more spatial details in the forcing and may need to consider other relevant effects (such as evaporation and lateral heat transport), but would form interesting extensions for further research.

The ability of our rather simple formulation of the surface–atmosphere system to explain the difference in climate sensitivity suggests that diurnal variations in temperature contain a lot of information to learn from. When our approach is extended to derive analytical solutions of the whole diurnal cycle, with possible extensions regarding the role of the ground heat flux and evaporation, one may use observed temperature variations and invert these to infer the magnitude of turbulent fluxes at the land surface, evaporation, and other aspects of the land surface energy balance. This could provide an additional means to better understand the functioning of the highly coupled and interactive yet constrained land–atmosphere system that complements data-driven approaches and land surface modelling.

Last, but not least, our explanation should also be applicable to the different climate sensitivity of the seasonal cycle. It is well known that winter temperatures increase more strongly than summer temperatures with global warming, particularly at high latitudes. Our interpretation here would suggest that this can be explained by winter conditions being shaped by short hours of daylight. As turbulent fluxes would only play a role during daylight in winter, this should result in a longer period of the whole day which are shaped by stable conditions and which are more sensitive to changes in longwave radiation. This longer period of stable conditions, in turn, could result in values of ϕ that exceed ϕ=1.5. (Note that here we assumed an equal length of daytime and nighttime, which resulted in the amplification of 50 %.) In the extreme case of the polar night, one would actually expect a ratio of ϕ=2 in the absence of any absorption of solar radiation, so stable conditions prevail over the whole day. During the polar day, one would still expect a ratio of ϕ=1.5 because the variations in solar radiation are still buffered by the lower atmosphere, resulting in a heat gain over half of the day, while losing heat over the other half of the day. This, in turn, can explain why our value of ϕ=1.5 appears to set a lower bound in the comparison to climate models (as shown in Fig. ), and this can explain the greater sensitivity of high latitudes to global warming. However, a detailed analysis would be necessary in future research to substantiate this reasoning.