Historically, climate models have evolved from elementary conceptual models to energy balance models (EBMs) and then to radiative–convective models (RCMs) and, after that, general circulation models (GCMs) and, finally, the state-of-the-art Earth system models (ESMs) , with a constant increase in complexity and calculation rate.

Researchers generally consider GCMs and ESMs “the best” models because they account for many phenomena. Indeed, if some specific part of the model does not fit well enough with observations, it is always possible to spend time adding more complexity and making it fit better. There is a hope that if we put enough work into it, GCMs or ESMs will be very close to observations. Furthermore, these models cover the entire Earth, accurately describing the position of oceans and continents, the orography, the cryosphere, and the vegetation with a resolution now below a hundred kilometres. Thus, they are beneficial in answering specific questions, for example, how crop yield would evolve in a particular area within this century. Today's ESMs predict very robust temperature changes for increasing levels of CO2 (see Fig. 4.19) for most regions of the globe. On the contrary, they do not predict robust changes in precipitation ( Fig. 4.24). Individual models show opposite signs of precipitation changes in some regions (Fig. 4.42d of ). Even when looking for global mean changes when increasing CO2, temperature shows much less uncertainty than precipitation ( Fig. 4.2a–b). The reason for such uncertainty lies in the difficulty of parameterizing the equations for water fluxes.

Indeed, the atmospheric part of GCMs or ESMs is based on the Navier–Stokes equations, whose length scales range from L=103 km to less than η=10 mm, the viscous or the Kolmogorov scale of the atmosphere. The number of modes required to model every scale is N=(L/η)3≈1024, by far unreachable by today's computers (we would need 108 times the full storage capacity of today's supercomputer to store one time step). To deal with this problem, climate computers only integrate large scales with the Navier–Stokes equation, and sub-grid processes are parameterized differently for every model, leading to different results, although the physics of every model is the same. How much the parameterizations affect our ability to truly predict climate is a deep and open question . Today's ESMs have hundreds of adjustable parameters.

Looking at global mean evaporation in the tropics, GCMs from the atmospheric model intercomparison project have a mean positive bias of 20 Wm-2 (). Radiative–convective models compute similar values of mean evaporation to GCMs (). While radiative–convective models approximate spatial mean quantities well, they are sometimes close to observations at finer scales (). showed that a GCM gives similar output to a radiative–convective model (using the same simple radiative code and adequate parameterization of surface wind speed for the radiative–convective model); for example, a similar relationship between temperature and precipitation is found (Figs. 3 and 4 of ). Equation (19) of gives an upper bound for the surface latent heat flux (and so to evaporation, equal to precipitation in a stationary setting). The exact evaporation value depends on the sensible heat flux and surface radiative transfer. However, if the same drag coefficient and wind speed are used to compute surface sensible and latent heat flux, then because of the exponential dependence of mixing ratio with temperature, latent heat flux dominates sensible heat flux (see Eqs. 7 and 8 of ). Thus, only the knowledge of the surface radiative transfer gives an upper limit to the evaporation. Therefore, in our goal to compute precipitation with a parameter-free model, it is relevant to use a radiative–convective model.

The efficiency of radiative–convective models in reproducing accurate surface temperature or climate sensitivity comes from three ingredients (): (1) a realistic computation of CO2, O3, and water vapour radiative transfer; (2) the convective adjustment, where all unstable lapse rates are adjusted to the moist adiabatic lapse rate of -6.5 Km-1; and (3) a fixed relative humidity in the radiative code to account for the water vapour feedback. introduced the convective adjustment and the fixed relative humidity. Cloud microphysics was already present in the model. added a flux between the atmosphere and the ocean and found that in a model with no clouds and a surface with no heat capacity, the absence of the ocean compensates for the neglect of clouds. explicitly computed the moisture profile with a cumulus convection scheme, leading to a wider model sensitivity to parameter changes. Recent developments in radiative–convective modelling focus on improving the cloud parameterization, creating a 3D radiative–convective model, or using it in a GCM (). Radiative–convective models mainly follow the same tendency as GCM and ESM models. More and more complexity is added, and more and more parameters are present. All models are tuned with observations. Because of this tuning, we do not know whether these models are accurate in climates very far from the present Earth condition, such as palaeoclimates. The first “parameter” is the convection scheme in a radiative–convective model with a convective adjustment: the algorithm by which the lapse rate adjusts to the moist adiabatic can change the results. Also, small diffusion of temperature and moisture is used to stabilize the numerical scheme. Most importantly, the surface drag coefficient and wind speed are prescribed to values unknown in palaeoclimates, though they impact the surface energy budget. Indeed, the sensible and latent heat flux is usually calculated from gradients using a Fickian law, with a coefficient adjusted on observations. Since the physical (conservation) laws are the same for all climate models, the main differences between them are mostly linked to the tuning of these parameters: differences are therefore linked to different model settings, different choices of parametric formula, different tuning strategies, or different tuning data sets. It is probably one of the main problems in climate modelling.

To overcome this issue, we build a radiative–convective model with zero adjustable parameters in this study. Because the method used is completely new and has been little investigated, the model is built from scratch and appears to be a jump back into the 1970s. Many issues could be raised: the model is cloud-free (and we do not know how to compute clouds without parameters), it has a ground with no heat capacity and an infinite water reservoir, relative humidity is fixed to climatology in the radiative code and to 1 in the energy fluxes computation, albedo and solar constant are fixed, stationarity is assumed, and the model is 1D radiative–convective. However, the article's purpose is not to build a code as accurate as today's complex and highly tuned codes but to build a model as accurate as equivalent radiative–convective models built in the 1970s and 1980s. If it is possible to obtain similar or better results with a parameter-free model than a similar but tuned model, it would pave the way for creating a full climate model, as has been done for tuned models since the 1970s but this time not using tuning.

To get rid of parameters, the unknown variables are determined with a variational problem, the maximum of entropy production (MEP). The idea is to express the variational problem (entropy production and constraints) as a function of the unknown variables, like energy fluxes or water vapour fluxes, so they will adjust themselves to maximize entropy production. Therefore, we do not parameterize them. One could argue that MEP is just another method of parameterization. However, it is very different from the usual data assimilation techniques (see, for example, , for precipitation and clouds) because it does not use any observational data about the variable of interest to predict it (like temperature or precipitation). A MEP model was first used for climate by to predict meridional fluxes and showed good agreement with observations. Unfortunately, it also had some parameterizations.

MEP is only a hypothesis and lacks rigorous mathematical proof. However, it seems very general and is used in many domains, like crystal growth, electric charge transfer, and biological evolution . First, the MEP hypothesis must not be mistaken for the second law of thermodynamics, which only states that entropy production is positive (σ≥0) but not necessarily maximized. Next, when equilibrium thermodynamics and entropy are reinterpreted with the formalism of information theory, it is possible to obtain the main results of equilibrium thermodynamics using the maximum entropy principle (MaxEnt). A possible way to understand the maximum entropy production hypothesis is to see it as the non-equilibrium or time derivative “equivalent” of MaxEnt. Following this idea, tried to prove the MEP hypothesis using MaxEnt. However, to quote , “Dewar's argument not only involves a number of non-obvious fundamental assumptions but also is non-rigorous and erroneous in a number of points”. The MEP hypothesis has also been related to other variational approaches in fluid dynamics or climate , like the maximum dissipation rate when temperatures are fixed or the maximum generation of available potential energy when in a steady state . We do not try to demonstrate the MEP hypothesis here, but we prefer modelling the climate and the water cycle using it. Suppose results happen to be close to observations. In that case, the explanation as to why it works in our particular case will be left as an open question for theoreticians (see , for a recent general review of the MEP hypothesis).

In climate, it has been used to predict oceanic or atmospheric horizontal heat fluxes , as well as vertical heat fluxes or both horizontal and vertical , where always the entropy production of heat transfers due to atmospheric turbulence is maximized under some constraints. A limitation of using the MEP hypothesis (and variational formulations in general) is that all the variables of the variational problem need to be solved at once, making it difficult to add new phenomena. Another limitation is that to get meaningful results, we have to put on the variational problem constraints that are physically relevant and represent the main processes of the atmosphere . This requires a lot of thinking and talent and is only sometimes easily solvable. Moreover, as the driver for heat transfers is the input of radiative energy in the system, the radiative code must be accurate to have results close to observations. Otherwise, we are limited to qualitative but not quantitative agreement with data. In previous studies, MEP has generally been used for straightforward cases or with additional parameters. For 1D vertical models, a grey atmosphere was used, and gravity was not considered. Recently, a new radiative–convective model was created with a more realistic radiative code . Moreover, geopotential heat and latent heat were added . This model already produces reasonable temperatures. This model is not based on a convective adjustment like but has the same crucial physical ingredients, allowing it to give similar results. However, the dynamical part (i.e. non-radiative) is treated without any adjustable parameter, giving the MEP model more universality because no one has ever told the model which values to take in any standard configuration. If the results are close to observations, it alerts us that there might be something powerful in the MEP hypothesis or the imposed constraints. In this study, the same model was used, and a new constraint on the water cycle was added, leading to a prediction of precipitation. The fact that this procedure can produce suitable water vapour fluxes has never been shown before.

The radiative code used here is the one of (see their supplemental material for details) and is based on the net exchange formulation (). It is more advanced than the grey atmosphere models used in previous studies , thus leading to results closer to observations. Let us consider an atmosphere divided into n+1 layers on the vertical axis (see Fig. ). The net radiative energy budget Ri (i.e. the input of energy thanks to radiation) is written as 1Ri(T,q,O3,CO2,α)=SWi(q,O3,α)+LWi(T,q,CO2), where SWi and LWi are the solar and infrared net energy budgets. q=Mwater/Mair, O3, and CO2 are prescribed vertical profiles of specific humidity and ozone and carbon dioxide concentrations, given on page 3 of , corresponding to a standard atmosphere, based on averaging observations. To take into account the water vapour feedback with temperature, the relative humidity profile h=q/qs(T) is fixed for the computation of Ri; therefore q varies with T. α is the albedo of the surface. T is the temperature profile. The parameters h, O3, CO2, and α are fixed, so the energy budget Ri is a function of the temperatures only. Note that T={Tj}j=0,…,n, where Tj is the temperature in box j; hence Ri(T) is a functional of the temperatures, i.e. a non-local function of T. That will be important for the variational problem.

Usually, when computing the radiative energy budget, one uses a local description of the radiative energy fluxes. In the net exchange formulation (NEF) framework , the description of energy transfer is global. Each radiative energy input in the layer i is broken down into the individual contributions of all atmospheric layers. The net energy exchange rate between layer i and j is ψij. At this point, this formulation is strictly equivalent to the usual one. However, it makes it easier to develop approximations that reduce computational time while automatically satisfying the fundamental laws of physics: the energy exchange rate is antisymmetric (ψij=-ψji). Thus, the total energy is conserved (∑i,jψij=0), and the second law of thermodynamics is also satisfied by keeping ψij the same sign as Tj-Ti. Yet, because the resolution of the variational problem is strongly sensitive to the constraints imposed, it is of utmost importance that the laws of physics are rigorously satisfied.

To approximate the radiative transfer, the infrared spectrum is divided into 22 narrow bands, and the absorption coefficient for water vapour and carbon dioxide is calculated with the statistical model and the data from . The diffusive approximation is made with the diffusion factor μ=1.66 for spatial integration. In the visible domain, water vapour and ozone absorption are computed with the parameterization of .

The variational problems (Eqs. , , and ) are solved using a sequential quadratic programming algorithm . The basic principle is that it takes some initial conditions and performs a gradient descent until it finds a local maximum. Such an algorithm mathematically converges to a global maximum for a convex problem; however, for a non-convex problem, there is no guarantee that the maximum found is global. Several techniques that are not detailed here are used to get better results. One possibility is to manually test different initial conditions and see which gives the highest entropy production. For a given problem, every result presented here is the one with the highest entropy production found. Given that we tested many different initial conditions and found only a few local maxima, we are confident our results represent a global maximum. Nevertheless, it is still possible that another better result mathematically exists.

We solve the problems with 21 boxes (1 surface boundary layer with albedo α and 20 atmospheric boxes; see Fig. ), with prescribed standard vertical profiles of O3 and CO2 concentrations and relative humidity taken from , corresponding to a tropical atmosphere. Indeed, because it is a 1D vertical model, taking midlatitude or boreal profiles would not make much sense, as horizontal fluxes become significant there. In the radiative code, the relative humidity h=q/qS(T) is fixed; therefore more H2O is present when temperature increases, which is an important positive feedback for climate sensitivity. However, in the variational problem, qi is still equal to qs(T) (Eq. ). This decoupling may be inconsistent, but it is common in the climate community or GCMs to examine models decoupled before studying fully coupled model configurations. We also choose to proceed in this way to keep the problem simpler and to be able to interpret results more easily. As in , incoming solar radiation is fixed to 342 Wm-2, and albedo is equal to α=0.1.

Results for Eqs. (), (), and () are shown Fig. . For temperature and energy, the lowest point corresponds to the surface. For comparison, standard temperatures for a tropical atmosphere (based on observations) are plotted in Fig. a, taken from , as well as mean temperatures from the IPSL-CM6A-LR model between 23° S and 23° N. The observations are not taken in a specific area of the tropics. However, if we knew a place matching the requirements of the model, that is to say, where horizontal fluxes are negligible, and where we know the greenhouse gas concentration, it could be used and compared to observations at this place. The comparison is made here for spatially average tropical conditions, using standard greenhouse gas concentrations and temperatures, as is customary for radiative–convective models. Also, we only consider a stationary atmosphere, so the results should be interpreted only in a climatological context, not a meteorological one. The comparison should remain qualitative, as our model's purpose is not to fit the observations precisely but to give a “proof of concept” that obtaining relevant results with the MEP hypothesis is possible. First, when putting more and more constraints into the variational problem, it is expected that the entropy production found decreases because the space of possibilities wanes. We obtain the following: the entropy production is equal to σ=53.9 mWm-2K-1 for Eq. () where only energy is conserved, σ=44.3 mWm-2K-1 for Eq. () where a specific pattern of mass fluxes transports energy, and σ=41.1 mWm-2K-1 for Eq. () where water vapour is not allowed to appear.

The state-of-the-art GCMs and ESMs and MEP models are based on the same conservation laws. In a GCM, the local conservation laws lead to partial derivative equations that are true in the limit of infinitely small differentials. The momentum conservation is the Navier–Stokes equation (present only to the horizontal), the energy conservation is the thermal energy equation, and the mass conservation is ∇⋅u=0. These equations were demonstrated for infinitely small increments. However, in GCMs and ESMs, they are integrated under a grid that needs to be smaller. Indeed, because of non-linearity, small scales do have an impact on large scales. Therefore, something more than the first conservation equations is needed for consistent results. Additional equations involving tunable parameters are added and are sometimes called “closure equations”. In the MEP framework, the energy conservation is Eq. (), and the mass conservation is immediately imposed by the convection pattern (Eq. ) and constraint m≥0. The water conservation is imposed by P≥0 in Eq. (). So, with MEP, the conservation laws are defined as constraints of an optimization problem, and unknown variables are resolved simultaneously to reach the maximum entropy production.

Everything else in our MEP model is similar to what is done in a GCM. The radiative code is based on integrating Planck's law on different wavelength bands corresponding to different constant extinction coefficients (see supplementary materials of ). The air is considered an ideal gas, and the hypothesis of hydrostatic equilibrium is made (dp=-ρgdz). Of course, well-known parameters like heat capacity Cp or the relation between qs and T (see Appendix B of ) are not variables of the optimization problem but just taken equal to well-established values used in GCMs and ESMs.

Still, there are many reasons why our MEP model could give different results than an ESM like the IPSL-CM6A-LR. Our MEP model has no continental surface and no energy flux between the ground and the atmosphere. Because there is no constraint on evaporation at the surface level, the ground can be seen as an infinite water reservoir, like an ocean. The ground heat flux is crucial on daily or seasonal timescales. However, it must equal zero in a stationary state without sub-surface fluxes (neglecting oceanic horizontal fluxes for ocean surfaces and geothermal fluxes on continental ones). The ground heat flux will be important if we add time to the problem to see a daily or seasonal cycle. However, this is a work beyond the scope of this article.

Also, no clouds (i.e. liquid water) are present in the air, although they are known to have an important impact on the radiative forcing. However, according to , neglecting the clouds introduces a false cooling of the surface that compensates for the false cooling by neglecting the heat transfer in the ocean. Moreover, the model is only 1D vertical and works well only for a tropical column where vertical fluxes dominate. However, showed that the radiative–convective approximation is good for computing average precipitation. To refine the results, it would be easy to use fixed prescribed horizontal fluxes, but this differs from our goal. Also, there is no theoretical difficulty in extending this model to 2D or 3D and obtaining horizontal fluxes, but it is currently very technically challenging.

Finally, a reason for getting different results could be the possible lack of validity of the MEP hypothesis. That said, obtaining precipitation values close to the IPSL-CM6A-LR model or observations by , 1.2 myr-1 compared to 1.4 and 1.3 myr-1, is very encouraging. Intuitively, given the fact that the model is built in a fully non-parametric fashion and is so minimalistic, we could have expected much more discrepancy. However, showed that radiative transfer strongly constrains precipitation (see their Fig. 3). For example, the world's mean precipitation cannot exceed 2.1 myr-1 (Eq. 6 of ). Indeed, there is a balance at the surface between the net radiative energy budget, sensible heat flux, and latent heat flux. In the tropics, the sensible heat flux is 1 order of magnitude less than the latent heat flux and can be neglected. The simplified energy budget allows us to give a precipitation value, given a temperature; see Fig. 3 of . If the surface temperature is well constrained, which is the case in a tuned parameterized model, precipitation is also well constrained. The MEP model reproduces this simplicity: both a dominant latent heat flux and a good surface temperature. At 300 K, find precipitation of about 1.8 myr-1. The difference with us comes from the grey atmosphere radiative code they use that is too simple. Interestingly, they prove that the same results regarding precipitation obtained from a GCM can be obtained with a radiative–convective model (with the surface wind speed parameter chosen from the mean value of the GCM). This means the radiative–convective approach is a reasonable first approximation to investigate precipitation.

One could wonder if there is a need to add a constraint on water vapour conservation to compute precipitation. Indeed, Eqs. (), (), and () have similar surface net radiative energy budget (84, 82, and 98 Wm-2), and saying sensible heat flux is negligible, one could already deduce the surface latent heat flux thus evaporation (i.e. precipitation). However, in the three models, the computed sensible heat flux is not always negligible: for Eq. (), the surface sensible heat flux is 19 Wm-2, and the surface latent heat flux is 65 Wm-2. For Eq. (), it is 15 and 67 Wm-2. And for Eq. (), it is 2 and 96 Wm-2. Evaporation is higher in the Eq. () model. Most importantly, Eqs. () and () do not conserve water (precipitation defined as the convergence of water fluxes is either negative or infinite): evaluating the water fluxes not at the surface but at another model level would give completely arbitrary results.

The MEP model could be improved by exploring several approaches. First, the specific humidity of water vapour q could be chosen not equal to saturation. Then, it needs to be clarified if an additional constraint on precipitation should be imposed, for example, saying that precipitation can occur only if q=qs, that is (q-qs)P=0, where P is precipitation. Moreover, this constraint is highly non-convex and numerically very difficult to solve. Second, convection is not allowed between every layer because air masses are compelled to move to adjacent layers. However, in the tropics, there is a phenomenon called deep convection, where an air mass can go adiabatically from the bottom to the top of the troposphere. This phenomenon is not authorized with the convection pattern imposed here. However, this could be implemented by changing Eq. () and adding fluxes for non-adjacent layers.

Since , many improvements have been made. Taking a more realistic radiative code leads to a stable atmosphere: the potential temperature increases with altitude. Adding a convection pattern gives much more realistic results in terms of temperature and reveals a stratosphere up to ≈250 hPa where no convection occurs. Then, imposing a constraint on water vapour conservation leads to precipitation as close as or closer to observations than a GCM or an ESM would find. These results look great in absolute value, but results seem less satisfying when performing a sensitivity experiment such as looking at the temperature difference when doubling CO2. We hope that future work that adds degrees of freedom in the model should solve this issue. Nonetheless, for the first time, this article demonstrates that it is possible to compute tropical precipitation in a radiative–convective model only from conservation laws, replacing the usual atmospheric parameterizations with an optimization procedure (MEP hypothesis).

Several approaches, such as changing the convection pattern and letting relative humidity vary, need to be explored in the future. Along these lines, it might be possible to build a 2D or 3D climate model, with a representation of clouds, vegetation, and oceans, whose “closure equations” would not at all be based on parameters that are “tuned towards observations”. The way clouds could be added without using any parameter is complicated. However, letting the relative humidity vary between zero and saturation, adding fluxes between non-adjacent boxes (i.e. deep convection), or making the problem 2D or 3D is mathematically straightforward. However, in such more complex settings, we have to solve a non-linear and non-convex optimization problem with many more variables. Currently, technical difficulties appear in the algorithm's convergence: the less convex the problem is, the more variables and non-linearities there are, and the harder it is to find the absolute global maximum. This technical problem might be solved using more specific optimization algorithms or finding a more straightforward mathematical but equivalent formulation.

Such a model could be used in contexts where tuning is impossible, such as other planets or palaeoclimates. The only mandatory knowledge is the atmosphere's chemical composition, the value of the solar radiation, and the ground albedo. This simplicity opens new and exciting perspectives in palaeoclimate or exoplanet climate modelling.