All climate models use parameterizations and tuning in order to be accurate. The different parameterizations and tuning processes are the primary source of difference between models. Because models are tuned with present observations of Earth, they may not accurately simulate climates of other planets or palaeoclimate. A model with no adjustable parameter that happens to fit today's observations is probably more universal and should be more appropriate to model palaeoclimate. However, to our knowledge, such a model does not exist or is yet to be developed. This paper aims to improve a parameter-free radiative–convective model that computes a realistic temperature vertical profile to compute the water cycle, giving a value on average tropical precipitation. Although it is known that the radiative transfer constrains the order of magnitude of precipitation, no parameter-free model has yet been able to compute precipitation. Our model finds a precipitation value closer to observations than similar radiative–convective models or some general circulation models (GCMs).

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)

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 CO

Indeed, the atmospheric part of GCMs or ESMs is based on the Navier–Stokes equations, whose length scales range from

Looking at global mean evaporation in the tropics, GCMs from the atmospheric model intercomparison project have a mean positive bias of 20

The efficiency of radiative–convective models in reproducing accurate surface temperature or climate sensitivity comes from three ingredients (

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,

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

In climate, it has been used to predict oceanic or atmospheric horizontal heat fluxes

The radiative code used here is the one of

Usually, when computing the radiative energy budget, one uses a local description of the radiative energy fluxes. In the net exchange formulation (NEF) framework

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

This constraint was used alone in

Scheme of the 1D vertical box model, with the box 0 being infinitely small and other boxes separated by equal level of pressure.

Air convection was studied in

Now, take a mass flow rate

The model's novelty described here comes from the addition of water transport and precipitation. In the formulation above, water vapour is a function of the temperature only and thus has no reason to be conserved when transported by the air masses. Infinite levels of water vapour could be created or disappear. In this study, we add a constraint on conserving water vapour that is supposed to mimic precipitation. We impose that water vapour cannot appear when transported, but it can disappear, and we call this phenomenon precipitation as if water vapour was transformed into liquid water. The flow rate

The variational problems (Eqs.

We solve the problems with 21 boxes (1 surface boundary layer with albedo

Results for Eqs. (

The use of energy conservation only has already been studied in

The use of energy and mass conservation has already been studied in

Three different cases: (1) Eq. (

Adding the constraint on precipitation (Eq.

The computed precipitation is equal to

Another local maximum of entropy production can give, for example, 2.2

To the author's knowledge, it is the first time precipitation is computed with a model using maximizing entropy production and without any data-tuned parameter. At least, the fact that the good order of magnitude of precipitation can be computed with little knowledge is of prime theoretical importance for climate scientists because it means the radiative transfer, or greenhouse gases, mainly drives atmospheric precipitation. This statement is not new and can also be deducted from

When looking at absolute values, the MEP approach seems to provide a good order of magnitude for climate variables. It is interesting to test if it can capture small changes in external forcing. A classic test for climate models is to look at the climate sensitivity, which is the difference of temperature at 1.50 m between conditions where CO

Nevertheless, the model of

Differences of temperatures profiles between CO

The vertical temperature distribution within the atmosphere is plotted for Eqs. (

Moreover, when interpreting these results, one should remember that depending on the resolution method of the variational problem (or the choice of initial conditions), results may differ by about 1 K. They differ even more by choosing an arbitrary local maximum of entropy production instead of the “supposedly” global maximum. Indeed, we note that because we changed the resolution method and found a new result with higher entropy production, the climate sensitivity of the problem (Eq.

In the above simulations, in the energy equation (Eq.

Values of maximum entropy production

When

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

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

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

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

One could wonder if there is a need to add a constraint on water vapour conservation to compute precipitation. Indeed, Eqs. (

The MEP model could be improved by exploring several approaches. First, the specific humidity of water vapour

Since

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.

Two different cases: (1) a global maximum of entropy production

The code used to produce the results can be found at

DP is the brains with overall understanding. KW implemented many possibilities in the resolution code. QP also worked on the code, used it to get the article's results, and wrote the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank Bérengère Dubrulle for insightful discussions.

This paper was edited by Yongze Song and reviewed by three anonymous referees.