3.1 Previous setting: energy conservation

This constraint was used alone in Herbert et al. (2013). The atmosphere is divided into n+1 layers on the vertical axis. Each layer i has a temperature T_i, a variable of the variational problem. Between layer i and i+1, there is a non-radiative energy flux F_i, whose nature is not explicit yet (with no additional constraints, it may be interpreted as conduction). When considering a steady state, the total energy balance reads

where F₀ and F_n+1 are taken equal to zero as if no energy (other than radiative) goes to the space or comes from the ground. The entropy production associated with these energy fluxes is written as

At stationary state, the total input of radiative energy must be equal to zero:

The entropy production (Eq. 3) is maximized under the constraint of energy conservation (Eq. 4), leading to the following variational problem:

It is exactly the same problem as Eq. (24) of Herbert et al. (2013). The predicted lapse rate then has the correct magnitude in the lower troposphere but not in the upper layers (see blue square points in Fig. 2a).

Figure 1Scheme of the 1D vertical box model, with the box 0 being infinitely small and other boxes separated by equal level of pressure. ℛ_i is the radiative energy balance and depends on all temperatures. F_i is the energy flux between adjacent boxes. m_i is the air mass flux. e_i is the specific energy (and depends on the local temperature), T_i is the temperature, and $q_{{\mathrm{S}}_{\mathrm{i}}}$ is the relative humidity taken at saturation (depends on the local temperature).

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3.2 Previous setting: air convection

Air convection was studied in Labarre et al. (2019). So far in the article, the atmosphere's internal energy or air mass fluxes have not been expressed. It might give more physical results to put a constraint on how air masses transport the energy in the atmosphere in the variational problem. By taking into account the sensible heat, the gravitational energy, and the latent heat, the specific energy (the energy per unit mass) is written as

where C_p is the heat capacity of the air, g the standard acceleration due to gravity, z_i the elevation of layer i, L the latent heat of vaporization, and q_i the specific humidity of water vapour ($m_{\mathrm{water}}/m_{\mathrm{air}}$). The elevation z_i is expressed as a function of the temperatures below (see Appendix A of Labarre et al., 2019), and the specific humidity q_i is taken equal to its value at saturation, q_i=q_s(T). Consequently, the specific energy e_i(T) is, like ℛ_i(T), a functional of temperature.

Now, take a mass flow rate m_i between layer i−1 and i. Suppose the air is transported adiabatically and then thermalizes once in layer i. It means that the energy in the air mass taken from layer i−1 is entirely transported to layer i. Thus, the upward energy transported equals m_ie_i−1. To easily conserve the air in all boxes, the same amount of air m_i is adiabatically taken from layer i to layer i−1, transporting downward energy equal to m_ie_i. Thereby, the total energy flux between layer i−1 and i is written as

Of course, for the reasoning to be consistent, we must have m_i≥0, which gives a new constraint on the energy fluxes and the temperatures. The following equation summarizes the variational problem:

It is the same variational problem as Eq. (11) of Labarre et al. (2019) (although expressed as a function of temperature instead of energy fluxes). The additional constraint allows for a much better simulation of the upper atmosphere (see orange diamonds Fig. 2).

3.3 New setting: water transport and precipitation

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 m_i transports upward between layer i−1 and i an amount of water equal to m_iq_i−1, where q_i is the specific humidity of water vapour in the air, and it transports downward an amount of water equal to m_iq_i. Then the water flux between layers i−1 and i is written as (similarly to Eq. 7)

The amount of water vapour that disappears in layer i is written as

where $F_{\mathrm{0}}^{\mathrm{w}}$ and $F_{n+\mathrm{1}}^{\mathrm{w}}$ are taken equal to zero as if no water comes from underground or goes to space. The layer i=0 is a surface boundary layer, supposed to be very thin, and plays the role of the surface. For $i=\mathrm{1},\mathrm{\dots },n$, we impose that P_i≥0, and P_i is called precipitation. On the layer i=0, because ${\sum }_{\mathrm{i}}{P}_{\mathrm{i}}=\mathrm{0}$ we have

where −P₀ is the evaporation in layer i=0 and is equal to the total precipitation. The specific humidity q_i is considered equal to its value at saturation q_s(T) and then depends only on the temperature. The variational problem can be summarized by

Although it looks like a complex equation, it is, in fact, very simple when one considers that absolutely all the physics of the model is contained in it.

5.1 With only energy conservation (blue square points)

The use of energy conservation only has already been studied in Herbert et al. (2013). Although the numerical resolution has been improved, the results here are very similar to Herbert et al. (2013), but we show them for comparison with the next ones. The only variables present in Eq. (5) are temperatures T and energy fluxes F. However, afterwards it is possible to compute variables like energy e (Eq. 6), mass fluxes m (Eq. 7), or precipitation P (Eq. 10), but of course, there is no reason for m or P to obey the constraint of positivity, and they do not. When looking at the entropy production equation (Eq. 3), one sees two terms: an energy flux and a gradient of inverse temperature. To maximize the entropy production, variables find a balance between a state where energy fluxes are very high but temperatures homogeneous (so σ=0) and a state where temperature gradients are high but energy fluxes are equal to zero (also σ=0). Between the two, a balance is found where entropy production is maximized. Results are plotted in Fig. 2a for temperature and Fig. 2b for energy fluxes. As expected, temperatures are higher near the ground because the atmosphere is essentially heated by the solar radiation intercepted at z=0. The energy fluxes compensate a bit for the temperature difference by going upwards (i.e. are positive Fig. 2b). The specific energy (Eq. 6, Fig. 2c) decreases at the bottom because sensible heat C_pT is the dominant term and increases at the top because gravity gz becomes the dominant term. “Mass fluxes” are computed with Eq. (7). Because F is always positive, the sign of m is always opposite to that of the energy gradients ∇e. When these gradients are negative, m is positive, but higher in the atmosphere when energy gradients become positive, the computed m is then negative, which has no physical meaning regarding Eq. (7). Then, imposing m>0 becomes natural, as done in the following paragraphs.

5.2 With a pattern of convection (orange diamond points)

The use of energy and mass conservation has already been studied in Labarre et al. (2019). Because the numerical resolution has been improved, results here differ slightly from Labarre et al. (2019). Adding the constraint of convection (Eq. 7) forces the energy fluxes (Fig. 2b) to be opposed to the energy gradients. Thus, energy (Fig. 2c) gradients must remain negative to keep positive upward energy fluxes (energy must decrease with height). To do so, temperatures (Fig. 2a) adapt themselves to counteract the gravity, leading to a zone in the middle of the atmosphere where energy gradients equal zero and mass fluxes (Fig. 2d) equal +∞; the atmosphere is perfectly mixed in this area, energy follows the moist adiabatic lapse rate (Fig. 2c, a vertical line). Thus, precipitation (Fig. 2e) is infinitely positive or negative. Higher in the sky, geopotential becomes too strong, energy gradients become positive, and energy fluxes equal 0. This region defines the stratosphere at about P≈250 hPa and z≈9 km, where no convection occurs. By adding a convection pattern and maximizing entropy production, we see a moist adiabatic lapse rate in the middle of the atmosphere, and a stratosphere naturally emerges. This model already gives suitable results for temperatures as the profile is similar to the IPSL-CM6A-LR model (light blue triangles) or the McClatchey (1972) standard tropical atmosphere (circles). The temperature profile remains close when adding a new constraint to compute precipitation.

Figure 2Three different cases: (1) Eq. (5), (2) Eq. (8), and (3) Eq. (12). (1) σ=53.917, (2) 44.304, and (3) 41.108 $\mathrm{mW}{\mathrm{m}}^{-\mathrm{2}}{\mathrm{K}}^{-\mathrm{1}}$.

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5.3 With a constraint on precipitation (grey triangle points)

Adding the constraint on precipitation (Eq. 10) is the article's novelty. Nevertheless, the philosophy remains the same: because the atmosphere is heated from below, energy fluxes (Fig. 2b) want to go upward, but they have to be opposed to the energy gradients (Fig. 2c), so these gradients want to be negative or to go to zero, leading to infinite mass fluxes (Fig. 2d) in the middle of the atmosphere. However, saying that no water vapour can be created prevents infinite mass fluxes and nil energy gradients. Consequently, energy decreases (while energy fluxes are positive) below a zone we call the tropopause; above, energy increases (while energy fluxes are equal to zero). Regarding precipitation (Fig. 2e), some water vapour is taken in box number 0 and continues going up to maximize mass fluxes. Then it reaches the tropopause, where mass fluxes become nil because there is not enough energy to go up, and then water vapour disappears; i.e. it precipitates.

The computed precipitation is equal to 1.2 m yr⁻¹, which is the correct order of magnitude of tropical precipitation. Comparison of this result with real-world or modelled tropical precipitation depends on the box size chosen for the tropical area. For example, the average Earth system model IPSL-CM6-LR shows that precipitation between −23 and +23° of latitude is 1.4 m yr⁻¹. For real-world data, average precipitation between 30° S and 30° N between 1980 and 1994 is 1.3 m yr⁻¹ (Fig. 8 of Xie and Arkin, 1997), and zonally averaged precipitation between 1979 and 2001 lies between 0.6 and 2 m yr⁻¹ (Fig. 5 of Adler et al., 2003). Note that the MEP model gives less precipitation, and the other model gives more precipitation than observations. We will see that it is a common bias for models to overestimate precipitation.

Another local maximum of entropy production can give, for example, 2.2 m yr⁻¹, though with less overall entropy production. This “uncertainty” in computed precipitation is probably due to the low resolution of only 20 boxes because choosing one box or another to precipitate leads to a different value of precipitation. To test this hypothesis, it is possible to simplify the problem by imposing the precipitation value in only one box. Then, the problem is numerically easily solved, and precipitation converges when the number of boxes increases. With 81 boxes, the computed precipitation is 1.15 m yr⁻¹, and the other local maximum of entropy production leads to precipitation of 1.21 m yr⁻¹. Thus, the 1.2 m yr⁻¹ precipitation is a robust value.

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 Pierrehumbert (2002), Rennó et al. (1994), or Eq. (9) of O’Gorman and Schneider (2008). Still, variation in surface sensible heat flux or how the surface radiative transfer depends on other processes is a source of discordance between models, leading to slightly different computed surface latent heat flux (or precipitation). Our model has less bias than radiative–convective models or the one from the atmospheric model intercomparison project, and further investigation is needed to understand if it is by chance or if there is some profoundly hidden physics in the MEP hypothesis or the imposed constraints.

5.4 Doubling the CO₂ concentration

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 CO₂ is at pre-industrial level (280 ppm) and conditions where CO₂ is doubled (560 ppm). The only feedbacks present here are the water vapour feedback incorporated by fixing the relative humidity in the radiative code, and the lapse rate feedback. Because box number 0 is a thin surface boundary layer, we take the temperature of layer number 1, whose middle is at 988 hPa (∼220 m). The climate sensitivity is 1.1 K for Eq. (5), 0.7 K for Eq. (8), and 1.0 K for Eq. (12) (see Fig. 3). For comparison, the climate sensitivity of the state-of-the-art IPSL-CM6A-LR model is 3.0 K, which is much more. The limited sensitivity to CO₂ is not surprising since our model only represents a few amplifying phenomena. For instance, the albedo is fixed, there is no deep atmospheric convection, and the ground has no heat capacity. The ground is an infinite water reservoir, the model is 1D, and no clouds are in the atmosphere. How to integrate (some of) these additional processes in the MEP procedure remains to be investigated, as well as how this would affect climate sensitivity.

Nevertheless, the model of Manabe and Wetherald (1967) contains physics similar to the MEP model and predicts 2.9 K (see Table 5). The result looks better because the convective adjustment method used in Manabe and Wetherald (1967) is very efficient in transferring a rise of temperature from the top of the atmosphere into the bottom of the atmosphere, and that is why the climate sensitivity, which only regards the temperature at the bottom of the atmosphere, is so well predicted. In the MEP case, the temperature at the bottom of the atmosphere is much less constrained by radiative transfer, which is why the result seems less good. However, looking not at one temperature but at temperature vertical profiles, the Manabe and Wetherald (1967) model computes a constant elevation of temperature in the whole troposphere (see their Fig. 16), which is not at all consistent with the vertical profile computed by the state-of-the-art IPSL-CM6A-LR model (SSP245-CTRL Fig. 3). Interestingly, radiative–convective models that do not use a mean moist convective adjustment (of 6.5 K m⁻¹) but a local moist adiabatic convective adjustment or a cumulus model (Lindzen et al., 1982) predict both a better temperature profile and a considerably reduced climate sensitivity. For example, depending on the imposed parameters, Lindzen et al. (1982) find a climate sensitivity of 1.4 and 1.7 K with a moist adiabatic convective adjustment and 1.6, 2.2, 2.3, or 0.9 K with a cumulus model (see Tables 1 and 2 of Lindzen et al., 1982).

Figure 3Differences of temperatures profiles between CO₂ at 180 ppm (or 560 ppm) and CO₂ at 280 ppm.

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The vertical temperature distribution within the atmosphere is plotted for Eqs. (5), (8), and (12) (Fig. 3), that is the difference of temperatures between a case with 560 ppm (or 180 ppm corresponding to the last glacial maximum) and a case with 280 ppm. The difference between 560 and 280 ppm is also plotted for the IPSL-CM6A-LR model (SSP245-CTRL), with a temperature average taken in the tropical region (between ±23°). When looking at the shape of temperature distribution and comparing it to the SSP245-CTRL, the best model seems to be the problem (Eq. 8) since the temperature difference increases with height and then decreases in the stratosphere. For the Eq. (12) model, it just decreases in the entire atmosphere. The less-constrained CONV experiment provides a better sensitivity of the temperature profile to CO₂ doubling and, in that sense, does better than the Eq. (12) case or Manabe and Wetherald (1967) model. Results from more evoluted radiative–convective models are similar to Eq. (8) (see, for example, Figs. 3 and 8 of Lindzen et al., 1982). Considering the simplicity of the model and the absence of parameterizations, it was expected that correct values for all model outputs would not be obtained. The fact that Eq. (8) is better and compares well with similar parameterized models suggests that the Eq. (12) experiment is over-constraining the water cycle, something that could be relaxed by introducing new degrees of freedom. For example, the convection pattern could allow mass fluxes between layers that are not adjacent, allowing deep convection to occur, or water vapour could be allowed to vary between zero and saturation, or the setup could be extended from 1D to 2D. We think that once this is done, the sensitivity of the temperature profile to CO₂ should get close to the Eq. (8) experiment. Unfortunately, this complexifies the optimization problem, and we have yet to be able to test this idea.

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. 8) is a bit different than in Fig. 6 of Labarre et al. (2019). However, the purpose of this model was not to produce precise or reliable predictions. More modestly, this model aims to demonstrate that some relevant elements of the climate system (temperature and precipitation) can be computed with only a minimal set of hypotheses, i.e. conservation laws only, without any parameter tuning.

5.5 Fixing the relative humidity

In the above simulations, in the energy equation (Eq. 6), the specific humidity q has been fixed to its value at saturation q_s(T). In other words, the relative humidity is fixed to $h=q/q_{\mathrm{s}}\left(T\right)=\mathrm{1}$. Relaxing this constraint by letting the relative humidity vary between 0 and 1 poses a numerical problem that has yet to be solved. However, one may wonder how fixing the relative humidity by hand would change the results. In this section, we solve the problem with a constraint on precipitation (Eq. 12) and a relative humidity constant in the entire column but chosen between 0.1 and 1. h=1 corresponds to the Eq. (12) setting. As before, a possible method is to test different initial conditions and see which results give the maximum entropy production. However, this method seems to fail when h<0.5, and the results are not robust anymore. Fortunately, phenomenology shows that in all our simulations, the case where precipitation happens in only one box seems to always correspond to the maximum entropy production. With this knowledge, we use a simpler way to obtain the MEP solution. We use a method where the place and value of precipitation are imposed to obtain the result more directly. We find the maximum entropy production by exploring the phase space by changing the value of P. We use the following protocol: relative humidity (h), precipitation (P), and the box where precipitation occurs (n_p) are fixed. A value of entropy production σ is found for each set of $(h,P,n_{\mathrm{p}})$. The explored phase space is $h\in \mathit{\{}\mathrm{0.1},\mathrm{0.2},\mathrm{\dots },\mathrm{1}\mathit{\}}$, $P\in \mathit{\{}\mathrm{0.1},\mathrm{0.2},\mathrm{\dots },\mathrm{10}\mathit{\}}$, $n_{\mathrm{p}}\in \mathit{\{}\mathrm{13},\mathrm{14},\mathrm{\dots },\mathrm{19}\mathit{\}}$. The number of atmospheric boxes is fixed at n=20. The results are shown in Fig. 4, with the obtained values of maximum entropy production σ and precipitation P. For all values of h, the maximum of entropy production corresponds to precipitation in box number n_p=15, as in the Eq. (12) problem (corresponding to h=1 here).

Figure 4Values of maximum entropy production σ and corresponding precipitation P for each constant profile of relative humidity h. The maximum only corresponds to precipitation in box number 15 (with n=20). The maximum between all h is h=1, with σ=41.108 $\mathrm{mW}{\mathrm{m}}^{-\mathrm{2}}{\mathrm{K}}^{-\mathrm{1}}$ and P=1.2 m yr⁻¹.

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When h tends to zero, precipitation seems to go to infinity. We stopped at P=10 m yr⁻¹, so the true value of P is greater than 10 m yr⁻¹ for h≤0.3. Further analysis shows that when h tends to zero, the function σ(P) has less and less of a clear maximum and becomes very flat, so very different values of precipitation can give slightly different values of entropy production. This explains why results were not robust when using the classic convergence algorithm when h is low: the phase space becomes very flat in the region of the maximum. The important result arising from Fig. 4 is that the maximum of entropy production corresponds to h=1, and P=1.2 m yr⁻¹. This means that in this 1D vertical and stationary case, the system is the most efficient to create entropy when water vapour is saturated. We did not test all possible relative humidity profiles, but we expect that the MEP solution is close to h=1 in each box.