This work presents the Thermodynamic Diagnostic Tool (TheDiaTo), a novel diagnostic tool for investigating the thermodynamics of climate systems with a wide range of applications, from sensitivity studies to model tuning. It includes a number of modules for assessing the internal energy budget, the hydrological cycle, the Lorenz energy cycle and the material entropy production. The routine takes as inputs energy fluxes at the surface and at the top of the atmosphere (TOA), which allows for the computation of energy budgets at the TOA, the surface and in the atmosphere as a residual. Meridional enthalpy transports are also computed from the divergence of the zonal mean energy budget from which the location and intensity of the maxima in each hemisphere are calculated. Rainfall, snowfall and latent heat fluxes are received as inputs for computation of the water mass and latent energy budgets. If a land–sea mask is provided, the required quantities are separately computed over continents and oceans. The diagnostic tool also computes the annual Lorenz energy cycle (LEC) and its storage and conversion terms by hemisphere and as a global mean. This is computed from three-dimensional daily fields of horizontal wind velocity and temperature in the troposphere. Two methods have been implemented for the computation of the material entropy production: one relying on the convergence of radiative heat fluxes in the atmosphere (indirect method) and the other combining the irreversible processes occurring in the climate system, particularly heat fluxes in the boundary layer, the hydrological cycle and the kinetic energy dissipation as retrieved from the residuals of the LEC (direct method). A version of these diagnostics has been developed as part of the Earth System Model eValuation Tool (ESMValTool) v2.0a1 in order to assess the performances of CMIP6 model simulations, and it will be available in the next release. The aim of this software is to provide a comprehensive picture of the thermodynamics of the climate system, as reproduced in the state-of-the-art coupled general circulation models. This can prove useful for better understanding anthropogenic and natural climate change, paleoclimatic climate variability, and climatic tipping points.

The climate can be viewed as a forced, dissipative non-equilibrium system exchanging energy with the external environment. The inhomogeneous absorption of solar radiation is an ongoing source of available potential energy. The complex mixture of fluids is then set into motion by the conversion of available potential into mechanical energy via a vast range of nonlinear processes. The kinetic energy is eventually dissipated through viscous stress and converted back into heat. Such processes can be described by taking advantage of the theory of the non-equilibrium thermodynamics of continuous multiphase media and, in particular, of fluids. The presence of (possibly fluctuating) fluxes of matter, chemical species and energy is a key characteristic of a non-equilibrium system. The steady state is reached as a result of a potentially complex balance of positive and negative feedbacks and through the interplay of processes with very diverse timescales and physical underpinning mechanisms. The climate is a prime example of this, with observed variability extending over many orders of magnitude in terms of both spatial and temporal scales and with many extremely complex subdomains – the atmosphere, the ocean, the cryosphere, the biosphere, the active soil – which themselves have very diverse characteristic internal timescales and are nonlinearly coupled

It is a major endeavour of contemporary science to improve our understanding of the climate system in the context of the past, present and projected future conditions. This is key for understanding, as far as the past goes, the co-evolution of life and of the physicochemical properties of the ocean, soil and atmosphere, as well as for addressing the major challenge faced by our planet as a result of the current anthropogenic climate change. Improving climate models is key to nearing these goals, and, indeed, efforts aimed in this direction have been widely documented (see the related chapter on the last report of the

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

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

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

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

The climate system has long been recognized as featuring irreversible processes through dissipation and mixing in various forms, leading to the production of entropy

Given the multiscale properties of the climate system, accurate energy and entropy budgets are affected by subgrid-scale parametrizations (see also

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;

total, atmospheric and oceanic meridional enthalpy transports;

water mass and latent energy budget;

strength of the LEC by means of kinetic energy dissipation conversion terms; and

material entropy production by both direct and indirect methods.

Therefore, our goal is to equip climate modellers and developers with tools for better understanding the strong and weak points of the models of interest. The aim is to reduce the risk of a model having accurate reproductions of quantities of common interest, such as surface temperature or precipitation, but for the wrong dynamical reasons. This is a necessary first step in the direction of creating a suite of model diagnostics composed of process-oriented metrics.

The paper is structured as follows. The dataset requirements are described in Sect.

The diagnostic tool consists of a set of independent modules, each, except the first one, being triggered by a switch decided by the user: energy budgets and enthalpy transports, hydrological cycle, Lorenz energy cycle, and material entropy production via the direct or indirect method.

The software ingests all variables as fields on a regular longitude–latitude grid covering the whole globe. Therefore, the tool is suitable for the evaluation of any kind of gridded dataset, provided that it contains the necessary variables on a regular grid, including blends of observations and reanalyses. In our description of the software features, we focus on model evaluation and multi-model intercomparison.

For the LEC computation, 3-D fields are required, stored in pressure levels at a daily or finer temporal resolution. If the model does not store data where the surface pressure is lower than the respective pressure levels, daily mean or higher-resolution data of near-surface temperatures and horizontal velocity fields are also required for vertical interpolation. For all other computations, 2-D fields are required as monthly means at TOA and at the surface. Input variables are given as separate files in NetCDF format. For model intercomparisons (Sect.

Variables used in the modules of the diagnostic tool.

Energy budgets are computed from residuals of instantaneous radiative shortwave (SW) and longwave (LW) fluxes at TOA. At the surface, SW and LW fluxes are combined with instantaneous turbulent latent and sensible heat fluxes (see Sect.

For the LEC module, 3-D fields of the three components of velocity and temperatures are needed at the daily resolution. For the 3-D fields, there is no specific constraint on the number of pressure levels, although the programme has been tested on the standard pressure level vertical discretization used in CMIP5 outputs, consisting of 17 levels from 1000 to 1 hPa. The programme then subsets the troposphere between 900 and 100 hPa. LEC computation is performed on Fourier coefficients of the temperature and velocity fields. The 2-D fields of temperature and horizontal velocity are also required in order to fill gaps in the pressure level discretization by interpolating from the surface on a reference vertical profile. As further discussed later (Sect.

If explicitly required by the user, the programme is also able to perform computations of energy budgets, enthalpy transports and the hydrological cycle on oceans and continents separately, provided a land–sea mask is supplied. This can either be in the form of land area fraction or a binary mask.

The ESMValTool architecture is implemented as a Python package, and the latest version is available at

A stand-alone version of the software is maintained as well, utilizing Python bindings for Climate Data Operators

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

When globally averaged, Eq. (

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. (

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

The latent energy budget in the atmosphere

The calculation of the atmospheric LEC in the diagnostic tool follows directly from the general framework proposed by

The LEC is depicted diagrammatically in Fig.

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

Following the arguments by

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

The MEP computation with the direct method involves taking into account the viscous processes related to energy cascades toward the dissipative scales and the non-viscous processes related to sensible heat fluxes (i.e. heat diffusion in the boundary layer, mainly dry air convection;

Overall, the hydrological cycle accounts for about

Dealing with the direct computation of MEP in climate models has a number of additional implications. These processes are dealt with in climate models via subgrid-scale parametrizations. The fact that they should be energy conserving and entropy consistent is far from trivial

For the direct expression of the MEP, we first explicitly write the non-viscous terms in Eq. (

In order to express Eq. (

evaporation at working temperature

rain droplet formation through condensation at working temperature

snow droplet formation through condensation

snow melting at the ground at working temperature

The first term on the right-hand side of Eq. (

The boundary layer temperature

In Eq. (

The potential energy of the droplets in Eq. (

In conclusion, note that the MEP budget provided in Eq. (

For the indirect estimation of the entropy budget, we use a simplified expression of the entropy associated with radiative heat convergence, following

As already stressed by

Climatological annual mean maps of

Let us finally consider that both the direct and the indirect methods contain crucial approximations. The two-layer assumption reduces the estimated MEP both with the direct and indirect methods, as shown by

A 20-year extract of a CMIP5 model (CanESM2) simulation under pre-industrial conditions is analysed in order to demonstrate the capabilities of the diagnostic tool. The datasets are retrieved from the Earth System Grid Federation (ESGF) node at the Deutsches Klimarechenzentrum (DKRZ). The run used here for the analysis is the one denoted by “r1i1p1”. From the 995-year (2015–3010) run, the 2441–2460 period was used. The choice of the sub-period is motivated by the fact that it is the only part of the run for which a 20-year subsequent dataset of the needed variables is available in the repository.

Figure

The meridional sections of climatological annual mean total, atmospheric and oceanic northward meridional enthalpy transports are shown in Fig.

Climatological annual mean total (blue), atmospheric (orange) and oceanic (green) northward meridional enthalpy transports for 20 years of a pre-industrial CanESM2 model run (W).

Scatter plots of 20-year pre-industrial CanESM2 annual mean atmospheric vs. oceanic peak magnitudes (W) in the SH

Figure

Figure

Diagram of Lorenz energy cycle (LEC) annual mean production, dissipation, storage and conversion terms for 1 year of the pre-industrial CanESM2 model run. AZ denotes the APE reservoir in the zonal mean flow, and ASE and ATE denote the APE associated with stationary and transient eddies, respectively. KZ denotes the KE associated with the zonal mean flow, and KSE and KTE denote the KE associated with the stationary and transient eddies (see Appendix

The Lorenz energy cycle for 1 year of a CanESM2 model run (Fig.

Most of the energy is stored in the form of APE in the zonal mean flux and to a lesser extent in the zonal mean kinetic energy. The zonal mean APE is almost instantly converted into eddy potential energy (mainly through meridional advection of sensible heat) and then into eddy kinetic energy (through vertical motions in eddies) by means of mid-latitudinal baroclinic instability, so the two conversion terms are unsurprisingly qualitatively similar. We also notice that the eddy APE and KE reservoirs have similar magnitudes. As argued before

Compared to reanalysis datasets

Results obtained from the indirect method for MEP with the CanESM2 model are shown in Fig.

Annual mean values of a 20-year subset of control runs from 12 models participating in CMIP5 for TOA and surface energy budgets (

Climatological annual mean maps of

We now focus on comparing a seven-member multi-model ensemble from CMIP5 under three different scenarios: “piControl” (piC), denoting pre-industrial conditions, “historical” (hist), i.e. a realistic forcing evolution for the 1870–2005 period, and “RCP8.5”, representing the 2005–2100 evolution of greenhouse gas (GHG) forcing under a business-as-usual emission scenario (in other words,

As shown in the first two columns of Table

Multi-model ensemble scatter plots of annual mean averaged quantities vs. inter-annual variability in the piC scenario for

Same as in Table

Same as in Table

As a consequence of a time-varying GHG forcing, the TOA imbalance increases (see Tables

Figure

Annual mean land–ocean asymmetries for atmospheric, latent energy budget and the difference of the two, for 20 years of multi-model ensemble simulations under piC conditions. Values are in watts per square metre.

Table

Climatological annual mean

Latent heat transport between the Equator and 10

The mean meridional sections of total, atmospheric and oceanic enthalpy transports for each model are shown in Fig.

Evolution of land–ocean asymmetries for latent energy and for the residual of the atmospheric budget for 20 years of multi-model ensemble simulations under the two extremal scenarios (piC and RCP8.5). Values are in petawatts (PW) and are positive if they are directed toward land. Values in brackets are from spatial integration over land; those not in brackets are from integration over oceans.

Column 11 in Tables

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

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.

Annual mean values of APE and KE reservoirs and conversion terms in the LEC (reservoir values:

Table

Annual mean components of the material entropy production obtained with the direct and indirect methods. For each model, the first row denotes the estimates from piC, the second row from hist and the third row from RCP8.5 (values: mW m

The 20-year annual mean material entropy production associated with the hydrological cycle. Each component denotes a different process: (from left to right) evaporation, rainfall precipitation, snowfall precipitation, snow melting at the ground and the potential energy of the droplet. For each model, the first row denotes the estimates from piC, the second row from hist and the third row from RCP8.5 (values: mW m

Table

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.

In total, the entropy production is found to increase with increasing GHG forcing (see Tables

As a wrap-up of the various aspects touched on in this section, we introduce two metrics. The first is the “baroclinic efficiency”

The 20-year annual mean irreversibility and baroclinic efficiency for each model and each scenario. Baroclinic efficiency is rescaled as

Table

Multi-model ensemble scatter plots from the piC scenario for

Figure

Figure

Figure

Finally, Fig.

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.

We have presented TheDiaTo v1.0, a diagnostic tool for the study of different aspects of the thermodynamics of the climate system, with a focus on the atmosphere. The goal of this diagnostic tool is to support the development, evaluation and intercomparison of climate models, as well as to help the investigation of the properties of the climate in past, current and projected future conditions. The diagnostic tool is comprised of independent modules accounting for (1) the energy budgets and transports in the atmosphere, the oceans and in the system as a whole, (2) the water mass and latent energy budgets and transports, (3) the Lorenz energy cycle, and (4) material entropy production via the direct and indirect methods. Global metrics are provided for immediate comparison between different datasets.

We provide some examples of practical use of the diagnostic tool. We have presented results obtained from a 20-year subset of CMIP5 model runs under unforced pre-industrial conditions and results from a 20-year multi-model ensemble in three different scenarios: unforced pre-industrial conditions (piC), the end of the historical period (hist) and the last 2 decades of the 21st century with a business-as-usual scenario (RCP8.5). A summary of the metrics and of the comparisons between the results obtained across models and scenarios is given in Tables

The energy and water mass budgets are computed locally and from these the transports are inferred, providing information about the global-scale circulation. Similarly, the material entropy production has been decomposed in a component which essentially accounts for a local budget on the vertical and another one which accounts for the global meridional enthalpy transport. In other words, the metrics link the local features of the climate to the global energy and mass exchange, allowing for the evaluation of the global impact of localized changes.

We have shown how the tool can provide a comprehensive view of the dynamics of the climate system and its response to perturbations. It facilitates the evaluation of the spatial distributions of model biases and their impacts and the interpretation of the changing properties of the system with time in the reduced space defined by the considered metrics.

Apart from the specific – yet important – problem of looking into climate change scenarios, it is also straightforward to use the diagnostic tool to study paleoclimates, investigate tipping points and study the climate under varied astronomical factors or chemical compositions of the atmosphere. One can envision adapting the model for the analysis of the properties of Earth-like exoplanets.

The requirement of flexibility, which allows the tool to be easily applied to a large class of models, inevitably leads to some simplifying hypotheses. The most relevant are the following: (a) the quasi-steady-state assumption; (b) the hydrostatic assumptions as background to the LEC framework; and (c) the identification of the emission temperature as the characteristic temperature in the atmosphere, leading to the 2-D formulation of the material entropy production with the indirect method. Other assumptions involve the analysis of the hydrological cycle, in which the latent heat of evaporation and solidification has been assumed to be constant, even though its value depends on temperature and pressure. Further, it is worth noticing

Thus far, we have pointed out that the thermodynamic point of view can be linked to fundamental aspects of the atmospheric dynamics. We have related the idea of a baroclinic heat engine

Another open issue is assessing the relevance of coarse graining for the results, not only on the material entropy production (as discussed by

The diagnostics are part of the ESMValTool community diagnostics (v2.0). The latest release of the tool is available for download at

PZ:

EPE:

KZ:

EKE:

The contribution of eddies to APE (EPE) and KE (EKE) consists of two terms; the first one is the term accounting for stationary eddies (ASE and KSE in the diagrams in Fig.

The curly brackets in

The source and sink terms, i.e. the generation and dissipation of APE and KE, are computed as residuals of the conversion terms at each reservoir.

Valerio Lembo implemented the new direct method for the material entropy production, wrote the code and wrote the text of this paper. Frank Lunkeit implemented the first version of the LEC computation and supervised the whole code. Valerio Lucarini designed the diagnostic tool with its module partitioning and substantially contributed to the paper, in particular in the Introduction and Conclusions.

The authors declare that they have no conflict of interest.

We acknowledge the World Climate Research Programme's Working Group on Coupled Modelling, which is responsible for CMIP. For CMIP the U.S. Department of Energy's Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led the development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. We are also grateful to Goodwin Gibbins for a careful proofreading. Valerio Lembo was supported by the Collaborative Research Centre TRR181 “Energy Transfers in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project no. 274762653. Valerio Lucarini was partially supported by the SFB/Transregio TRR181 project, the EU Horizon 2020 Blue-Action Project (grant no. 727852) and the EU Horizon 2020 CRESCENDO Project (grant no. 641816).

This research has been supported by the DFG (grant no. 274762653).

This paper was edited by Carlos Sierra and reviewed by two anonymous referees.