## 1. Introduction

Thermal convection has long been examined since an early investigation of the atmosphere by Hadley (1735), early laboratory experiments by Bénard (1901), and a numerical solution of nonlinear equations by Lorenz (1963). Yet there is no solid physical theory that is capable of expressing the complete process of thermal convection. Understanding of the convection process may be of urgent necessity since all living creatures, including human beings, are distributed at the earth’s surface where convective transport of sensible heat and latent heat is largest. If convection were more active (inactive), surface temperature would decrease (increase). Yet we have no solid understanding how convection would change with a future increase of, for instance, carbon dioxide. Figure 1 shows global-mean (surface-area mean) shortwave and longwave radiation at the bottom and top of the atmosphere. The values for the earth’s surface are based on direct measurements evaluated for the globe (Ohmura and Gilgen 1993), and those at the top of the atmosphere are based on satellite measurements (Barkstrom et al. 1990). The energy gain at the surface is transported to the atmosphere to balance its loss. This is done through the convective heat transport (102 W m^{−2}), and this flow rate is 72% of the total shortwave absorption (142 W m^{−2}) at the surface. The rest of the energy, 28% (40 W m^{−2}), is transported upward by net longwave radiation. Yet we do not have a reasonable explanation for this partitioning of energies at the surface of the earth.

One may expect general circulation models to represent the convective process. However, GCMs contain an artificial device for convection. The convective adjustment was first introduced by Manabe and Strickler (1964) in order to adjust the vertical temperature profile to observations. The adjustment was necessary for GCMs since it was not possible to treat vertical instability of the atmosphere by a grid-scale dynamic motion; thus the calculation diverged during time integration (Manabe et al. 1965). Even current versions of GCMs contain a sort of convective adjustment whose parameters are tuned to reproduce observations (e.g., Tiedtke 1988). We do not in any way deny valuable results obtained by GCMs. Instead, we shall try to approach the convective problem from a different viewpoint.

The postulate for convection considered here is the one that has been proposed in the field of nonequilibrium thermodynamics. Onsager (1931) implicitly suggested a variational principle of maximum dissipation of energy in his original treatise on nonequilibrium thermodynamics. Félici (1974) suggested maximum production of entropy as a facilitation of the convection of electromagnetic fluid. Paltridge (1975, 1978) used the postulate of a maximum rate of entropy production through horizontal heat transport from the equator to the poles and, thereby, reproduced a global format of earth climate that resembles that of the present earth. Sawada (1981) stated that entropy of thermal reservoirs connected through a nonlinear system increases along a path of evolution, with a maximum rate of entropy increase among a manifold of dynamically possible paths. The atmosphere can be seen as a nonlinear system that transports heat from the earth’s surfaces (a hot reservoir) to outer space (a cold reservoir) through the convective motions of the air mass containing moisture. We therefore suspect that the postulate might also be valid for the vertical heat transport process and could be an alternative to the convective adjustment.

## 2. Method of entropy calculation

*σ*is the Stefan–Boltzmann constant,

*τ*is longwave optical depth ranging from zero at the top to

*τ*

_{1}at the bottom of the atmosphere,

*T*(

*τ*) is temperature at the optical depth

*τ*, and

*F*

_{L}(

*τ*) is net longwave radiation flux density (total energy flowing across a horizontal unit area per unit time) at the optical depth

*τ*and is defined as positive upward. For shortwave radiation, Beer’s law of absorption, with an effective shortwave optical depth, is assumed (Goody and Yung 1989, 51) such that

*F*

_{S}

*τ*

_{S}

*F*

_{S}

*τ*

_{S}

*F*

_{S}(

*τ*

_{S}) is net shortwave radiation flux density defined as positive downward, and

*τ*

_{S}is the effective optical depth ranging from zero (top) to

*τ*

_{S 1}(bottom). From Fig. 1, the net shortwave flux is 240 W m

^{−2}at the top of the atmosphere and 142 W m

^{−2}at the bottom. Then the total shortwave optical depth is calculated to be

*τ*

_{S 1}= 0.53. The net shortwave flux at the surface (142 W m

^{−2}) is already the result of the absorption and scattering in the atmosphere, the reflection and absorption by clouds, and the reflection by the surface. If the shortwave optical depth is assumed to be proportional to longwave optical depth, then the shortwave flux distribution can be written as a function of

*τ*:

*F*

_{S}(

*τ*) =

*F*

_{S}(0) exp(−

*α*

*τ*), where

*α*= 0.53/

*τ*

_{1}. This assumption may be crude, but it is practical for a simple treatment with a gray atmosphere (e.g., Goody and Yung 1989, pp. 393–396). The physical justification for this assumption is the fact that the variability of the optical depths of short- and longwave radiation is to great extent influenced by the precipitable water vapor.

*F*

_{S}

*τ*

*F*

_{C}

*τ*

*F*

_{L}

*τ*

*F*

_{C}(

*τ*) is convective heat flux at the optical depth

*τ*and is defined as positive upward;

*F*

_{C}(

*τ*) is unknown since we do not know the physics of convection. However, once the distribution

*F*

_{C}(

*τ*) is arbitrarily given, then by the energy balance requirement (3) with the shortwave radiation (2), net longwave radiation

*F*

_{L}(

*τ*) can be obtained. Then by a numerical solution of the differential equation (1) with an appropriate upper-boundary condition such as

*σ*

*T*(0)

^{4}=

*F*

_{L}(0)/2 − (

*dF*

_{L}(

*τ*)/

*dτ*)

_{τ=0}/4 (Goody and Yung 1989, 60), we can evaluate the temperature distribution as a function of

*τ*. In short, the vertical temperature distribution can be evaluated once a convective heat flux distribution is given.

*dS*

_{C}/

*d*t is the rate of entropy increase in the whole system through the convective heat transport process, and

*T*

_{g}is the global-mean earth surface temperature and is evaluated with an appropriate boundary condition such that

*σ*

*T*

_{g}

^{4}=

*σ*

*T*(

*τ*

_{1})

^{4}+

*F*

_{L}(

*τ*

_{1})/2 + (

*dF*

_{L}(

*τ*)/

*d*

*τ*)

_{τ=τ1}/4

*T*

_{a}can be assumed, the integration results in

*F*

_{C}(

*τ*

_{1})/

*T*

_{a}; then the rate is

*dS*

_{C}/

*d*t ≈

*F*

_{C}(

*τ*

_{1})(

*T*

_{g}−

*T*

_{a}) (

*T*

_{g}

*T*

_{a})

^{−1}. In this simplest case the rate must have a maximum because the rate starts to increase from zero with increasing

*F*

_{C}(

*τ*

_{1}) from zero, but if

*F*

_{C}(

*τ*

_{1}) becomes infinitely large, then the extreme mixing makes the temperature difference (

*T*

_{g}−

*T*

_{a}) negligible. Thus, the product of

*F*

_{C}(

*τ*

_{1}) and (

*T*

_{g}−

*T*

_{a}) should have a maximum in between. An attempt has been made to seek such a maximum value for a more realistic atmosphere. To this end, the atmosphere is divided into 20 layers of equal optical depth (

*τ*

_{1}/20). First, we assume an arbitrary distribution of convective heat flux,

*F*

_{C}(i) at the i-th layer and then evaluate the corresponding temperature distribution by solving the radiation transfer equation (1) with the energy balance requirement (3). Second, the rate of entropy increase is calculated by numerical integration (4). Third, the convective heat flux at the i-th layer is modified by a finite small amount,

*F*

_{C}(i) =

*F*

_{C}(i) ± Δ, so that the rate tends to increase. The process is applied to successive layers and repeated until the rate of entropy increase reaches a maximum value. It is found that, starting from any initial distribution of

*F*

_{C}(i), the calculation converges into a single maximum point. The result suggests that the global-mean state of atmosphere will evolve into a final steady state that seems to be a thermodynamically optimum state and that is independent of initial conditions.

## 3. Results

Results of temperature distributions and convective heat flux deduced from the state of maximum entropy increase are shown in Fig. 2, and the numerical values at the surface are listed in Table 1. The optical depths chosen for the figure are 2, 3, and 4, and those for the table are 1 to 5. In Fig. 2, the optical depth is replaced with the height *z* (m) for easy interpretation using the relation *τ* = *τ*_{1} exp(−*zH*), where *H* is a scale height and is approximately 2000 m for water vapor in the air (Goody and Yung 1989, 392). For a comparison, temperature distributions without convection—that is, temperatures at pure radiation balance—are shown in Fig. 2b. It is shown that, owing to the convective heat transport, the temperature gradient in the atmosphere is moderated. The predicted temperature profile (Fig. 2a) is somewhat different from observations in that there is a smaller temperature gradient in the upper troposphere and a higher temperature gradient in the lower troposphere. We attribute the reason to our simple assumption of a gray atmosphere. What is not clearly known is an appropriate value of longwave optical depth for the present atmosphere. According to the report by London (1957), approximately 6% of longwave radiation emitted from the surface is directly transported into space through the atmosphere. A rough estimation using the Beer’s law of absorption [0.06 = exp(−*τ*_{1}), cf. (2)] gives a value of *τ*_{1} = 2.8. We shall assume a mean optical depth of 3 for simplicity. Table 1 shows a considerable agreement between the surface temperature and energy flux components estimated with the optical depth of 3 and those of observations. The largest error is 16 W m^{−2}. Figure 2d shows convective heat convergence (−*dF*_{C}/*dz*). The heating rate has a peak at altitude around 4000 m, which is consistent with observations (e.g., Newell et al. 1970). It is suggested from our calculation that the convective heat flux distribution in the atmosphere is determined through a total entropy generation process in the entire atmosphere. A nearly constant partition ratio of energy fluxes at the earth’s surface can be seen (last column in Table 1). The physical meaning of the partition of energy fluxes (convection/radiation) will be discussed in a separate paper.

It should be noted that the method developed here contains no dynamics of the atmosphere, but rather treats a convective heat flux that may be transported by the dynamic motions of the atmosphere. Presumably, a nonlinear system with large degrees of freedom, such as the atmosphere, may behave in a way that can hardly be handled by the dynamic equation alone. Lorenz (1963), for instance, solved a set of nonlinear equations for a dissipative atmospheric system and found unstable solutions whose behavior is nonperiodic and is almost unpredictable. In contrast to the unstable solutions of dynamic equations, we found a thermodynamically stable solution for convection. Our reasoning for convection is that, despite nonlinearity and the consequent unpredictability of the individual motions of convection, the global-mean state is controlled so as to increase entropy in the whole system at a possible maximum rate. The stabilization at the state of maximum entropy increase has also been confirmed by numerical simulations of Bénard-type thermal convection (e.g., Suzuki and Sawada 1983; Chen and Wang 1983). When we consider the universe, the isolated system including the earth–atmosphere system, space, and the sun, the entropy in the universe is increasing by a contribution of the thermal convection (the general circulation) in the earth system. It is therefore suggested that the initiation and evolution of convection in our planet, as well as in other planets and stars, have been controlled by a universal requirement of entropy increase in the universe.

## Acknowledgments

This work was carried out during one of the authors’ (H.O.) stay at the Swiss Federal Institute of Technology. We wish to express our gratitude to the Canon Foundation in Europe and the Japan Society for Promotion of Science for financial support. Acknowledgment is also given to Dr. K. Masuda of Tokyo Metropolitan University for stimulating the author’s interest in thermodynamics of climate; to Dr. B. T. Alt of Balanced Environments Associates, Canada, for valuable comments on the early version of the manuscript; and to Dr. K. Goto-Azuma of the National Research Institute for Earth Science and Disaster Prevention for encouragement. Reevaluation of the earth’s surface energy balance was carried out by research grants from the Swiss National Science Foundation (2100-037698.93 and 20-43542.95).

## REFERENCES

Barkstrom, B., E. F. Harrison, and R. B. Lee, 1990: Earth radiation budget experiment.

*Eos, Trans. Amer. Geophys. Union,***71,**297–305.Bénard, H., 1901: Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. (Cellular vortex in a liquid layer transporting heat through convection in a stationary state.)

*Ann. Chim. Phys.,***23,**62–144.Chen, S., and Y. Wang, 1983: The phenomenon of wave-length increase in Rayleigh–Bénard convection and criterion of maximum entropy production.

*Chin. Phys.,***3,**595–601.Félici, N. J., 1974: Thermodynamique—Instabilité, transport, et production d’entropie. (Thermodynamics—Instability, transport and production of entropy).

*C. R. Seances Acad. Sci. Ser. 2,***278B,**151–153.Goody, R. M., and Y. L. Yung, 1989:

*Atmospheric Radiation, Theoretical Basis.*2d ed. Oxford University Press, 519 pp.Hadley, G., 1735: Concerning the cause of the general trade-wind.

*Philos. Trans. Roy. Soc. London,***29,**58–62.London, J., 1957: A study of the atmospheric heat balance. Final Rep. on Contract No. AF (122)-165, New York University, 99 pp. [Available from Prof. J. London, Department of Astrophysical Planetary and Atmospheric Physics, University of Colorado, Boulder, CO 80302.].

Lorenz, E. N., 1963: Deterministic nonperiodic flow.

*J. Atmos. Sci.,***20,**130–141.Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment.

*J. Atmos. Sci.,***21,**361–385.——, J. Smagorinsky, and R. F. Strickler, 1965: Simulated climatology of a general circulation model with a hydrologic cycle.

*Mon. Wea. Rev.,***93,**769–798.Newell, R. E., D. G. Vincent, T. G. Dopplick, D. Ferruza, and J. W. Kidson, 1970: The energy balance of the global atmosphere.

*The Global Circulation of the Atmosphere,*G. A. Corby, Ed., Royal Meteorological Society, 42–90.Ohmura, A., and H. Gilgen, 1993: Re-evaluation of the global energy balance.

*Interaction Between Global Climate Subsystems, The Legacy of Hann, Geophys. Monogr.,*No. 75, Amer. Geophys. Union, 93–110.Onsager, L., 1931: Reciprocal relations in irreversible processes I.

*Phys. Rev.,***37,**405–426.Paltridge, G. W., 1975: Global dynamics and climate—A system of minimum entropy exchange.

*Quart. J. Roy. Meteor. Soc.,***101,**475–484.——, 1978: The steady-state format of global climate.

*Quart. J. Roy. Meteor. Soc.,***104,**927–945.Sawada, Y., 1981: A thermodynamic variational principle in nonlinear non-equilibrium phenomena.

*Prog. Theor. Phys.,***66,**68–76.Suzuki, M., and Y. Sawada, 1983: Relative stabilities of metastable states of convecting charged-fluid systems by computer simulation.

*Phys. Rev.,***A27,**478–489.Tiedtke, M., 1988: Parameterization of cumulus convection in large-scale models.

*Physically-Based Modelling and Simulation of Climate and Climatic Change,*M. E. Schlesinger, Ed., Kluwer Academic Publishers, 375–425.

Vertical distributions of (a) air temperature, (c) convective heat flux in the upward direction, and (d) convective heating (convergence of the convective heat flux) per unit volume for different optical depths for longwave radiation (*τ*_{1} = 2, 3, and 4) estimated by a condition of maximum entropy increase [(4)]. For a comparison, air temperature distributions at pure radiation balance (no convective heat flux and, therefore, no convective entropy increase) are shown in (b).

Citation: Journal of Climate 10, 3; 10.1175/1520-0442(1997)010<0441:TOAGMS>2.0.CO;2

Vertical distributions of (a) air temperature, (c) convective heat flux in the upward direction, and (d) convective heating (convergence of the convective heat flux) per unit volume for different optical depths for longwave radiation (*τ*_{1} = 2, 3, and 4) estimated by a condition of maximum entropy increase [(4)]. For a comparison, air temperature distributions at pure radiation balance (no convective heat flux and, therefore, no convective entropy increase) are shown in (b).

Citation: Journal of Climate 10, 3; 10.1175/1520-0442(1997)010<0441:TOAGMS>2.0.CO;2

Vertical distributions of (a) air temperature, (c) convective heat flux in the upward direction, and (d) convective heating (convergence of the convective heat flux) per unit volume for different optical depths for longwave radiation (*τ*_{1} = 2, 3, and 4) estimated by a condition of maximum entropy increase [(4)]. For a comparison, air temperature distributions at pure radiation balance (no convective heat flux and, therefore, no convective entropy increase) are shown in (b).

Citation: Journal of Climate 10, 3; 10.1175/1520-0442(1997)010<0441:TOAGMS>2.0.CO;2

Global-mean (surface-area mean) temperature and energy flux components at the surface estimated by the entropy maximum condition.