In order to be in steady state, a non-equilibrium system in contact with an external environment must have a vanishing – on average – energy budget. Inconsistencies in the overall energy budget of long-term stationary simulations have been carefully pointed out , and various aspects of the radiative and heat transfers within the atmosphere and between the atmosphere and the oceans have been evaluated in order to constrain models to a realistic climate . A substantial bias in the energy budget of the atmosphere, in particular, has been identified in many global climate models (GCMs), resulting from either the imperfect closure of the kinetic energy budget or of the mass balance in the hydrological cycle . This picture is made even more complicated by the difficult task of having an accurate observational benchmark of the Earth's energy budget e.g.. Many authors have recently suggested that the improvement of climate models requires improving the energetic consistency of the modelled system . Rather than a proxy for a changing climate, surface temperatures and precipitation changes can be better viewed as a consequence of a non-equilibrium steady-state system which is responding to a radiative energy imbalance through complex interacting feedbacks. A changing climate, under the effect of an external transient forcing, can only be properly addressed if the energy imbalance, and the way it is transported within the system and converted into different forms, is taken into account. The skill of the models at representing historical energy and heat exchanges in the climate system has been assessed by comparing numerical simulations against available observations, where available , including the fundamental problem of ocean heat uptake .

A key element in defining the steady state of the climate system is the balance between the convergence of the horizontal (mostly meridional) enthalpy fluxes by the atmospheric and the oceanic circulations and the radiative imbalance at the top of the atmosphere (TOA). The net radiative imbalance is positive (in-bound) in the low latitudes and negative in the high latitudes, and a compensating horizontal transport must be present in order to ensure steady state. This transport dramatically reduces the meridional temperature gradient with respect to what would be set by radiative–convective equilibrium , i.e. in the absence of large-scale atmospheric and oceanic transport. Differences in the boundary conditions, in the forcing and dissipative processes, and in chemical and physical properties of the atmosphere and ocean lead to a specific partitioning of the enthalpy transport between the two geophysical fluids; a different partitioning associated with the same total transport would lead to significantly different climate conditions . The role of meridional heat transports in different paleoclimate scenarios and in relation to different forcing has been addressed in various studies see e.g.. The question of whether the current state-of-the-art climate models are able to correctly represent the global picture as well as the details of the atmospheric and oceanic heat transports has been analysed with mixed findings .

In order to understand how heat is transported by the geophysical fluids, one should clarify what sets them into motion. We focus here on the atmosphere. A comprehensive view of the energetics fuelling the general circulation is given by the Lorenz energy cycle (LEC) framework . This provides a picture of the various processes responsible for the conversion of available potential energy (APE) – the excess of potential energy with respect to a state of thermodynamic equilibrium seefor a review – into kinetic energy and dissipative heating. Under stationary conditions, the dissipative heating exactly equals the mechanical work performed by the atmosphere. Thus, the LEC formulation allows us to constrain the atmosphere to the first law of thermodynamics, and the system as a whole can be interpreted as a heat engine under dissipative non-equilibrium conditions . The strength of the LEC, or in other words the rate of the conversion of available potential energy (APE) into kinetic energy (KE), has been evaluated in observational-based datasets and climate models , with estimates generally ranging from 1.5 to 3.5 W m-2. A considerable source of uncertainty is the hydrostatic assumption, on which the LEC formulation relies, which can lead to significant underestimation of the kinetic energy dissipation . The usual formulation of the LEC can be seen as describing energy exchanges and transformation at a coarse-grained level, at which non-hydrostatic processes are not relevant and the system is accurately described by primitive equations. An efficiency can also be attributed to the atmosphere as a heat engine, assuming the system is analogous to an engine operating between a warmer and a colder temperature . This approach has been generalized by in order to account for the role of water vapour.

Water is an essential ingredient of the climate system, and the hydrological cycle plays an important role in the energy pathways of the climate system. Water vapour and clouds influence the radiative processes inside the system, and water phase exchanges are extremely energy intensive. As in the case of energy imbalances, a closed water-mass-conserving reproduction of the hydrological cycle is essential, not only because of the diverse implications of the hydrological cycle for energy balance and transports in the atmosphere, but also because of its sensitivity to climate change and the importance of the cloud and water vapour feedbacks . The energy budget is influenced by semi-empirical formulations of the water vapour spectrum , and, similarly, the energy budget influences the moisture budget by means of uncertainties in aerosol–cloud interactions and mechanisms of tropical deep convection . Water mass budget has been assessed in observations , as well as in climate models, focusing on the hydrological cycle alone or evaluating it in conjunction with the energy budgets . Therefore, a global-scale evaluation of the hydrological cycle, both from a moisture and from an energetic perspective, is considered an integral part of overall diagnostics for the thermodynamics of climate system.

The climate system has long been recognized as featuring irreversible processes through dissipation and mixing in various forms, leading to the production of entropy , which is a key characteristic of non-equilibrium systems . Several early attempts were made to understand the complex nature of irreversible climatic processes. Recent works have proposed an innovative approach by partitioning the system into a control volume made of matter and radiation, which exchanges energy with its surroundings (building on , early works). has described the entropy budget of an aggregated dry air + water vapour parcel. From a macroscopic point of view, “material entropy production” generally refers to the entropy produced by the geophysical fluids in the climate system, which is not related to the properties of the radiative fields but rather to the irreversible processes related to the motion of these fluids. Material entropy production is dominated by phase changes and water vapour diffusion, as outlined by . underlined the link between entropy production and efficiency of the climate engine, which were then applied to study climatic tipping points, in particular the snowball–warm Earth critical transition . This allowed for the definition of a wider class of climate response metrics that have been used to study planetary circulation regimes . A constraint has also been proposed to the entropy production of the atmospheric heat engine, given by the emerging importance of non-viscous processes in a warming climate .

Given the multiscale properties of the climate system, accurate energy and entropy budgets are affected by subgrid-scale parametrizations (see also ). These and the discretization of the numerical scheme are generally problematic in terms of conservation principles and can eventually lead to macroscopic model drifts . See for a theoretical analysis in a simplified setting.

We present TheDiaTo v1.0, a new software for diagnosing the mentioned aspects of the thermodynamics of climate systems (i.e. the energy and water mass budgets and meridional transports, the LEC strength, and the material entropy production) in a broad range of global-scale gridded datasets of the atmosphere. The diagnostic tool provides global metrics, allowing for straightforward comparison of different products. These include the following:

top-of-atmosphere, atmospheric and surface energy budgets;

By making the crucial assumption that the heat content of liquid and solid water in the atmosphere, the heat associated with the phase transitions in the atmosphere, and the effect of salinity and pressure in the oceans are negligible, we can write the total specific energy per unit mass for the subdomains constituting the climate system as ϵa=u2+cvT+ϕ+Lq for the atmosphere, ϵo=u2+cwT+ϕ for the ocean, and ϵs=csT+ϕ for solid earth and ice. Here u is the velocity vector; cv, cw and cs are the specific heat of the atmospheric mix at constant volume, water and the solid medium, respectively, L is the latent heat of vaporization of water, q the specific humidity and ϕ the gravitational potential. This leads to an equation for the evolution of the local specific energy in the atmosphere as such : 1∂ρϵ∂t=-∇⋅Jh+R+HS+HL-∇(τ⋅u), where Jh=(ρE+p)u is the specific enthalpy transport, R is the net radiative flux, HL and HS are the turbulent sensible and latent heat fluxes, respectively, and τ is the stress tensor. By neglecting the kinetic energy component and vertically integrating Eq. (), we find an equation for the energy tendencies at each latitude (ϕ), longitude (λ) and time for the whole climate system, for the atmosphere, and for the surface below it: 2E˙t(λ,ϕ)=St↓(λ,ϕ)-St↑(λ,ϕ)-Lt(λ,ϕ)-∇⋅Jt(λ,ϕ)=Rt(λ,ϕ)-∇⋅Jt(λ,ϕ)E˙a(λ,ϕ)=Rt(λ,ϕ)-Fs(λ,ϕ)-∇⋅Ja(λ,ϕ)=Fa(λ,ϕ)-∇⋅Ja(λ,ϕ)E˙s(λ,ϕ)=Ss(λ,ϕ)+Ls(λ,ϕ)-HS↑(λ,ϕ)-HL↑(λ,ϕ)-∇⋅Jo(λ,ϕ)=Fs(λ,ϕ)-∇⋅Jo(λ,ϕ), where E˙t, E˙a and E˙s denote the total, atmospheric and surface energy tendencies, respectively. Ss=Ss↑(λ,ϕ)-Ss↑(λ,ϕ) and Ls=Ls↑(λ,ϕ)-Ls↑(λ,ϕ) are the net SW and LW radiative fluxes at the surface, respectively, and St↓, Ss↓, St↑ and Ss↑ the TOA (t) and surface (s) upward (↑) and downward (↓) SW radiative fluxes, respectively (and similarly for LW radiative fluxes, denoted by L, provided that there is no downward LW flux at TOA). Rt, Fa and Fs are the total, atmospheric and surface net energy fluxes, respectively. Net fluxes are defined as positive when there is a net energy input and negative when there is a net output. Jt, Ja and Jo are the meridional total, atmospheric and oceanic enthalpy transports. Oceanic transports are assumed to be related to surface energy budgets because the long-term enthalpy transports through land are nearly negligible .

When globally averaged, Eq. () can be rewritten as 3E˙t=RtE˙a=Rt-Fs=FaE˙s=Fs. Given the small thermal inertia of the atmosphere compared to the oceans, the TOA energy imbalance is expected to be transferred for the most part to the ocean interior, with Fa much smaller than Fs . This is usually not reproduced in climate models e.g..

Under steady-state conditions, the tendency of the internal energy of the system will vanish when averaged over sufficiently long time periods. We can thus zonally average Eq. () and derive the long-term time means of the meridional heat transports or, more appropriately in this formulation, “meridional enthalpy transports”: 4Tt(ϕ)=2π∫ϕπ/2a2cos⁡ϕ′〈Rt(ϕ′)‾〉dϕ′Ta(ϕ)=2π∫ϕπ/2a2cos⁡ϕ′〈Fa(ϕ′)‾〉dϕ′To(ϕ)=2π∫ϕπ/2a2cos⁡ϕ′〈Fs(ϕ′)‾〉dϕ′, where Tt, Ta and To denote the steady-state total, atmospheric and oceanic meridional enthalpy transports, a the Earth's radius, 〈〉 long-term time averaging and overbars the zonal mean quantities. The stationary condition has to be achieved in the models if the system is unforced. If the system is forced, for instance by a transient greenhouse gas forcing, a correction is applied in order to avoid unphysical cross-polar transports. This correction can be applied as long as the weak non-stationarity condition holds ; i.e. the latitudinal variability of the net energy flux is much larger than the global mean imbalance see. The energy fluxes are modified as follows: 5〈B‾x(ϕ)〉co=〈B‾x(ϕ)〉-〈B‾x(ϕ)〉/2πa, where Bx refers to Rt, Fa or Fs.

Peak intensities and latitudes are computed as metrics for these transports.

The atmospheric moisture budget is obtained by globally averaging precipitation and evaporation fluxes. The latter are derived from surface latent heat fluxes as 6E=HLLv, where E from now on denotes the evaporation fluxes and Lv=2.5008×106 J kg-1 the latent heat of evaporation (assumed to be constant). If HL is given in units of watts per square metre, the implied evaporation flux is in units of kilometres per square metre per second. Rainfall fluxes are derived from total (P) and snowfall (Ps) precipitation as 7Pr=P-Ps. Under the stationarity assumption, the equation for the moisture budget is thus written as 8E‾-P‾=E‾-Pr‾-Ps‾, where the overbars denote global averages here.

The latent energy budget in the atmosphere RL is then 9RL‾=HL‾-LvPr‾-Lv+LfPs‾, where Lf=3.34×105 J kg-1 is the latent heat of fusion (which is assumed to be constant). Unlike the water mass budget of Eq. (), the latent energy budget is not closed because it does not include the heat captured by snow melting from the ground. The sublimation of ice is also not accounted for here for the sake of simplicity, although it plays a significant role over the Arctic region.

The calculation of the atmospheric LEC in the diagnostic tool follows directly from the general framework proposed by and revised by in order to provide a separation between different scales of motion. The LEC allows for the investigation of the general circulation of the atmosphere by looking comprehensively at the energy exchanges between eddies and zonal flow, at the conversion between available potential and kinetic energy, and at the dissipation due to viscous and mixing processes.

The LEC is depicted diagrammatically in Fig. , and the equations used to obtain the energy reservoirs and conversion terms are described in Appendices –. Energy is injected in the reservoirs of APE through differential diabatic heating, primarily impacting the zonal component but also the eddy component. The so-called baroclinic instability (mainly occurring in the mid-latitudinal baroclinic eddies at the synoptic scale) acts in two steps: first by transforming zonal mean APE into eddy APE and then by conversion into eddy kinetic energy (KE) via a lowering of the atmospheric centre of mass. The eddy KE is then converted into smaller scales through the kinetic energy cascade, eventually reaching the dissipative scale, at which it is transformed into frictional heating. A remaining part of the eddy KE is converted back into zonal mean flow through the barotropic governor, ensuring the maintenance of the tropospheric and stratospheric jet streams see. Other terms, i.e. the generation and dissipation of APE and KE, are computed as residuals of the conversion terms at each reservoir.

In the formulation proposed by Lorenz and widely adopted afterwards, motions are assumed to be quasi-hydrostatic. This assumption is correct as long as one considers sufficiently coarse-grained atmospheric fields. A detailed analysis of non-hydrostatic effects requires dealing with the exchange of available potential and kinetic energy taking place through accelerated vertical motions see. Furthermore, the approach used here refers to the dry atmosphere (see Appendix –). In order to include moisture effects, temperatures must be replaced by virtual temperatures. A comprehensive moist formulation of the LEC is left for future study.

Following the arguments by , we note that, under non-stationary conditions, internal energy conservation applies (see Eq. ), and the stationarity condition implies that APE and KE tendencies both vanish. Given that the tendency in KE is a balance between the APE–KE overall conversion (or in other words the mechanical work exerted by the LEC) and the dissipation of KE, we can write 10〈W〉=〈D〉=∫Ωρκ2dΩ′, where W and D denote the LEC work and the dissipation of KE, respectively, κ2 is the specific kinetic energy dissipation rate, ρ is the atmospheric density and Ω the volume of the atmosphere. The LEC can thus be used to obtain the kinetic energy dissipation of the atmosphere.

The total entropy production in the climate system is given by two qualitatively different kinds of processes: firstly, the irreversible thermalization of photons emitted from the Sun at the much lower Earth surface and atmospheric temperature and, secondly, the irreversible processes responsible for mixing and diffusion in the fluids and in the active soil of the climate system. The former accounts for roughly 95 % of the total entropy production; the latter is the material entropy production (MEP) and is the quantity of most interest in climate science because it involves the dynamics of the atmosphere and its interaction with the Earth's surface (see discussion in ).

In the long-term mean, assuming that the system is in a statistically steady-state condition, one can write an equation for the entropy rate of change in the system as 11∫ΩdΩ′q˙radT‾+S˙mat‾=0, with q˙rad denoting the local net radiative heating and S˙mat the global material entropy production. It is evident from Eq. () that the entropy rate of change can be computed indirectly from the net radiative heating or directly as the sum of the entropy production from all the irreversible processes, both viscous (such as the energy dissipation) and non-viscous (such as the hydrological cycle). These two equivalent methods are referred to here as the “indirect method” and “direct method”. As long as the volume integral in the first left-hand side member of Eq. () is performed on the whole atmospheric domain, the two methods are exactly equivalent (for the sake of simplicity we assume that this amounts to the MEP of the climate itself, since the oceanic contribution to the MEP is negligible, accounting for about 2 % of the budget; ).

As shown in the first two columns of Table , there is a large imbalance in the TOA energy budget under unforced piC conditions. Such an imbalance sums up with the atmospheric energy budget imbalance (not shown, see Eq. ), resulting in the surface energy budget estimates (the multi-model mean value for Rt being 0.21 W m-2; for Fs 0.73 W m-2). Some clear outliers are found, having either negative (BNU) or positive (MIR-C) values. This imbalance is the signature of the well-known model drift in climate models . The fact that it is larger on surface budgets is explained by the fact that models are generally tuned in order to achieve vanishing TOA budgets see, whereas surface energy budgets are often untuned. Panels a–c of Fig.  emphasize how these biases are relevant with respect to the inter-annual variability of the budgets (computed as the standard deviation of the annual mean values). The atmospheric inter-annual variability is roughly 1 order of magnitude smaller (about 0.1 W m-2) than the variability in the TOA and surface budgets, emphasizing that the changes in the overall energy imbalance are transferred to a large extent into the ocean (see also Fig. d). The inter-model spread on net energy fluxes is similar at the surface and at TOA, except for two models (BNU and MIR5) exhibiting a very large imbalance in the atmosphere, which is then reflected in TOA imbalance. There is limited correlation between surface and atmospheric imbalances. The inter-annual variability is roughly the same order of magnitude as the imbalances, both in the atmosphere and at the surface–TOA (about 0.20–0.25 W m-2).

As a consequence of a time-varying GHG forcing, the TOA imbalance increases (see Tables  and ). By the end of the historical period, most models agree on a positive imbalance (with respect to unforced biased conditions), ranging between 0.2 and 0.7 W m-2, although still in the range of the inter-annual variability (see Fig. a). The imbalance is much stronger by the end of the RCP8.5 period, peaking at 2.8 W m-2 (net of the bias, i.e. the value of the imbalance in the piC scenario) in MIR-C. The surface imbalance appears to increase consistently with the TOA imbalance.

Figure  shows the difference in the zonal mean meridional latent heat transport between the Equator and 10∘ N. As mentioned in Sect.  (see Fig. b), this is a measure of the moisture convergence toward the ITCZ. There is quite large uncertainty on its value in the piC scenario, ranging between 1 and 3 PW. In all models the convergence is found to increase by up to 1 PW between piC and RCP8.5. Even though it is beyond the scope of this work, this may be a robust estimate of the intensity of the uplifting branch and to some extent of the intensity of the Hadley circulation.

Table  denotes a discrepancy of several watts per square metre in the atmospheric budgets over ocean and land, with a positive imbalance over the former and negative over the latter. Such a well-known imbalance (see , for a review on the model perspective) is key to probing the model ability to reproduce the hydrological cycle, which mediates the convergence of latent energy (see Eq. ) from oceans (where water mass evaporates) toward the continents (where a large part of it precipitates). Atmospheric and latent energy land–ocean asymmetries are compared in the first two columns of Table . Models that are relatively well balanced (Can2, IPSL-M, MPI-LR and MPI-MR) also feature relatively similar asymmetries. These are translated into land–ocean transports if ocean and land fluxes are multiplied by their respective surface area. The two transports are theoretically required to be of equal magnitude but opposite sign (see Table ) and are estimated to be close to 2.8 PW . Few models comply with this basic energy conservation requirement (with Can2, IPSL-M, MPI-LR and MPI-MR performing better than the others); others feature differences up to 1.7 PW for BNU. For the better-performing models, note that a residual asymmetry holds, which is not attributable to the latent energy asymmetry (third column in Table ). Those transports are directed from land toward the oceans and are interpreted as the land–ocean transports related to asymmetries in the sensible heat fluxes at the surface. The ocean–land latent energy transport is found to increase in RCP8.5 by 0.4–0.8 PW. This can be interpreted as an increase in the strength of the hydrological cycle, in line with previous findings e.g.. Looking at individual components of the hydrological cycle, we find an increase in both evaporation over oceans and precipitation over land.

The mean meridional sections of total, atmospheric and oceanic enthalpy transports for each model are shown in Fig.  for the piC conditions alone (see Tables – for an overview of peak magnitude and position values). The choice of not showing hist and RCP8.5 is motivated by the insensitivity of enthalpy transports to the different forcing (in agreement with the theory by , and , confirmed by previous findings on CMIP model behaviours in disparate forcings representative of present-day climate or future emission scenarios; e.g. ). Models agree on the asymmetry in the atmospheric transport, being stronger in the Southern Hemisphere than in the Northern Hemisphere (see Sect. ). A significant source of uncertainty is the location of the zero crossing, which in some cases is very close to the Equator or even displaced in the SH. This is an important metric for the strength and shape of the mean meridional circulation. Compared to the atmosphere, the uncertainty on oceanic enthalpy transports is much larger, especially in the Southern Hemisphere, with some models featuring a counter-transport toward the Equator. As already mentioned (see Sect. ), a typical source of uncertainty here is the relative maximum in the Southern Hemisphere extratropics.

Column 11 in Tables – evidences that the intensity of the Lorenz energy cycle is relatively constant through the scenarios. Its value ranges between 2.2 W m-2 (Can2) and 1.1 W m-2 (MIR5 in the RCP8.5 scenario). As mentioned in Sect. , a major source of uncertainty is the way fields are vertically discretized in pressure levels across the different models. Some models (BNU, Can2, MPI-LR, MPI-MR) internally interpolate the fields to provide a value in each grid point at each level. Others have no values where the surface pressure is lower than the respective pressure level. This normally occurs over Antarctica and mountainous regions (most notably Himalaya and the Rocky Mountains), where the surface pressure reaches values even lower than 700 hPa. Since the LEC is computed on spectral fields, the original grid point fields have to be continuous, and any gap must be filled with a vertical interpolation. In order to do so, daily mean near-surface fields of zonal and meridional velocities as well as near-surface temperatures are used. Near-surface velocities replace the gaps, whereas near-surface temperatures are interpolated assuming a vertical profile of temperature reconstructed from barometric equations (see Sect. ). Despite the retrieved velocity and temperature profiles being qualitatively comparable with those from the internally interpolated models, the stationary eddy conversion terms are unreasonably weaker. This is not entirely surprising, since the interpolation mostly affects mountain regions over the mid-latitudinal continents, i.e. the regions where orographically driven stationary planetary waves are generated.

The intensity of the LEC is not very sensitive to the type of forcing. The result is partially in contrast with a previous version of MPI-LR and reanalyses , though the changes and trends in the APE–KE conversion (i.e. the intensity of the LEC) for both studies are not very strong. Table  provides values of the reservoirs and conversion terms for the seven models in the three scenarios, evidencing relevant changes in some of the reservoirs across the different scenarios. Note that the zonal mean APE is largely decreased from piC to RCP8.5. This is compensated for by an increase of similar magnitude in the zonal mean KE (and, to a lesser extent, in the eddy KE). In other words, in the wake of an increasing GHG forcing, the share of kinetic energy to available potential energy is changed in favour of the first. The total (APE + KE) energy contained in all storage terms remains approximately constant, in agreement with . These results are consistent with what was found by with a previous version of MPI-ESM-LR and with a state-of-the-art version of MPI-ESM-MR. The zonal mean APE reduction is consistent with a reduction of the meridional temperature gradient, which is predominantly a consequence of high-latitude warming amplification (see also ). The increase in zonal mean kinetic energy reflects a strengthening of the tropospheric mid-latitude jet stream (consistent with what was previously found about the SH tropospheric jet; e.g. ). The impact of climate change on the mid-latitudinal eddy activity is less clear. The (slight) increase in eddy kinetic energy may reflect an increased mid-latitudinal baroclinic eddy activity in the Pacific and Atlantic storm tracks (despite large model uncertainty on this aspect; see ). This, together with the rise of the tropical tropopause (mainly as a consequence of surface warming; see ), may contrast with the expected decrease in baroclinic eddy activity associated with a smaller meridional temperature gradient due to polar amplification. This approach allows us to straightforwardly associate the different response of the models to the increasing GHG with key aspects of the general circulation of the atmosphere.

The components of the material entropy production in the indirect and direct methods are closely related to each other and provide further insight into the interpretation of water mass, energy budgets and LEC results.

Table  summarizes the main components of the material entropy production in the two methods. Most of the MEP obtained with the direct method is related to the hydrological cycle. The vertical component overcomes the horizontal component in the indirect method by an order of magnitude. We have already commented on this in Sects.  and . The previous arguments about the changes in intensity of the hydrological cycle and the atmospheric circulation are reflected here as well. The entropy production increases with increasing GHG forcing in all models (except BNU, being strongly water mass and energy unbalanced). The increase in the MEP related to the hydrological cycle ranges between 2.4 mW m-2 K-1 (MIR5) and 4.7 mW m-2 K-1 (IPSL-M) from piC to RCP8.5, amounting to about a 10 % increase. The vertical component increase ranges between 5.6 mW m-2 K-1 (MPI-MR) and 7.3 mW m-2 K-1 (IPSL-M). Models showing larger increases in the hydrological-cycle-related MEP also feature a stronger increase in the vertical component. In other words, the vertical component, which is a signature of MEP related to vertical uplift, especially deep tropical convective activity, is closely relate to the strength of the hydrological cycle.

Table  provides more insight into the components of the MEP related to the hydrological cycle (see Eq. ). On the one hand is the reduction in MEP due to a decrease in snowfall precipitation (Ss). This reduction ranges between 3 mW m-2 K-1 (MIR5) and 8.5 mW m-2 K-1 (BNU) from piC to RCP8.5, to which a reduction of less than 1 mW m-2 K-1 must be added to reflect a reduction in snowmelt (Sm). On the other hand, a large increase in MEP due to rainfall precipitation (Sr) is found, ranging between 10.7 mW m-2 K-1 (MIR5) and 32.1 mW m-2 K-1 (IPSL-M) from piC to RCP8.5, generally overcoming the MEP reduction related to evaporation at the surface (Se). This increase can be interpreted in different ways: either as an increase in water mass which is precipitated or in terms of a lower working temperature for rain droplet formation (TC). As a consequence of the water mass balance, the latent heat associated with evaporation and precipitation must equal each other (less the changes in latent heat related to snowfall precipitation and the marginal contribution by snow melting at the ground). We thus attribute such an increase in Sr to changes in TC. This might also be indicative of a larger rate of convective precipitation on stratiform precipitation and could be investigated further.

Concerning the other terms of the MEP budget, the one related to the sensible heat fluxes at the surface is slightly reduced, whereas the kinetic energy dissipation term is increased. Given that the LEC intensity, from which the kinetic energy dissipation has been derived, is to a large extent stationary across the scenario, such change is not related to the intensification of the atmospheric circulation (see Sect. ); it is rather attributable to the near-surface warming. Finally, the decrease in the horizontal component is in line with the decrease in the APE terms of the LEC (especially the zonal mean term), denoting a weaker heat convergence toward the high latitudes (mainly as a consequence of the polar amplification).

In total, the entropy production is found to increase with increasing GHG forcing (see Tables –), both via the indirect and the direct method. This is consistent with previous findings and points to the role of latent heat release in convective processes in establishing the response of the climate system. The reduction of the meridional enthalpy transports is also consistent with previous comparisons between dry and moist entropy fluxes , suggesting that the hydrological cycle affects the efficiency of the atmospheric thermal engine .

As a wrap-up of the various aspects touched on in this section, we introduce two metrics. The first is the “baroclinic efficiency” : 25η=TE>-TE<TE>, where TE> and TE< are the emission temperatures averaged in the domains defined by TOA net energy gain and net energy loss (see Fig. ), respectively. The second is the “degree of irreversibility” : 26α=S˙dirmat‾-S˙kmat‾S˙kmat‾, i.e. the ratio of MEP from irreversible processes other than the kinetic energy dissipation to the MEP related to the kinetic energy dissipation alone. The first parameter accounts for the strength of the mean meridional circulation driven by the differential thermal gradient. This is a measure of the dry entropy fluxes related to the heat-to-work conversion associated with the existence of the LEC and constitutes an upper limit to the efficiency of the atmospheric thermal engine. The second parameter accounts for the relevance of viscous dissipation compared to other non-viscous irreversible processes. This parameter is closely related to the Bejan number, which is widely used in thermodynamics for the study of heat transfers in a fluid .

Table shows the results from the three scenarios for the seven models that have been considered. The baroclinic efficiency ranges between 0.051 (MIR-C) and 0.066 (IPSL-M) in piC. It undergoes a clear reduction in the wake of increasing forcing as a consequence of the already discussed reduction in the heat convergence at high latitudes. Remarkably, the last 20 years of hist do not seem to be significantly different from piC, contrary to RCP8.5. The irreversibility, in turn, is generally increased, especially between hist and RCP8.5. As noted before, the hydrological cycle plays an increasingly significant role in converting energy into the system in a warmer climate see also. This reflects a less efficient meridional enthalpy transport from low to high latitudes and consequently a larger irreversibility of the system, as already argued in .

Figure brings up some of the metrics discussed for the piC scenario, showing how they relate with each other. The TOA and atmospheric budgets (Fig. a) are strongly related, with positive–negative outliers determined by positive–negative biases in atmospheric budgets. The other models cluster around vanishing atmospheric energy imbalances and slightly positive TOA imbalances, likely reflecting the oceanic model drift.

Figure b shows that the uncertainty on the value of baroclinic efficiency is much smaller (10 %) than the one related to the LEC (about 50%, even though the treatment of fields in pressure levels is a critical issue here, as discussed in Sect. ). The two quantities are related through the meridional enthalpy transports see, which in mid-latitudes are mainly effected by baroclinic eddies. As mentioned, the strength of the baroclinic conversions in the LEC is itself a measure of the strength of the LEC. One might argue that the baroclinic efficiency peaks for certain values of the LEC intensity, but a larger ensemble would be necessary to prove or disprove such a hypothesis.

Figure c shows the relation between horizontal and vertical components of the MEP computed with the indirect methods see. It can be noticed that larger–smaller values of the vertical components correspond to smaller–larger values of the horizontal component. Indeed, the overall MEP ranges between about 57 and 60 mW m-2 K-1, while the vertical component alone has a 4 mW m-2 K-1 uncertainty range, suggesting that a compensation mechanism occurs somewhat between the vertical and horizontal component, i.e. between the local MEP on the vertical (especially where convection occurs) and the convergence of heat towards the high latitudes.

Finally, Fig. d shows the relation between MEP computed via the direct and indirect methods. Besides one outlier (BNU) all models have values for the direct method of about 40 mW m-2 K-1, whereas through the indirect method they cluster at about 57.5 mW m-2 K-1. Comparison with explicit computations by suggests that the discrepancy is mainly attributable to an insufficient representation of the MEP related to the hydrological cycle (which is expected, since some intermediate phase change processes are not taken into account) and to the kinetic energy dissipation (because unphysical processes cannot be accounted for here, given the different numerical schemes of the models, leading to up to a 50 % reduction of this term).

Across metrics, this analysis emphasizes that the distribution of different models is quite far from Gaussian. The multi-model ensemble mean and variances used here are certainly useful criteria for assessing the model uncertainty, but care should be taken on the choice of the members and the behaviour of each of them.