Fig. 1. Every non-equilibrium (flow) component of the earth functions as an engine that drives a brake. The constructal law governs ‘how’ the system functions: by generating a flow architecture that distributes imperfections optimally to fill the flow space. The ‘engine’ part evolves in time toward generating maximum power (or minimum dissipation) and, as a consequence, the ‘brake’ part exhibits maximum dissipation. Evolution means that each flow system assures its persistence (survival) in time by freely morphing into easier and easier flow structures under finiteness constraints. The arrows proceed from left to right because this is the general drawing for a flow (non-equilibrium) system, in steady or unsteady state. When equilibrium is reached, all the flows cease, and the arrows disappear.

Fig. 2. Thick line: latitude pressure cross section of the meridional stream function (the contour interval is 21,010 kg s⁻¹ and 41,010 kg s⁻¹ for the bottom two panels). Thin line: zonally averaged zonal speed. From top to bottom: 18, 24, 72, 144, 360 h a day rotation period (reprinted from [18] with permission).

Fig. 3. Zonally averaged mean meridional stream function: (a) control, (b) 22HR, (c) 20HR, (d) 18HR, (e) 16HR and (f) 14HR. Units in kg s⁻¹ (reprinted from [19] with permission).

Fig. 4. Zonal mean zonal speed in m s⁻¹ (shaded contours) and mass stream function in 1010 kg s⁻¹ (lines) for (a) ΔT_R = 20 K, (b) 30 K, (c) 60 K, (d) 130 K and (e) 190 K (reprinted from [20] with permission).

Fig. 5. Earth model with equatorial (A_H) and polar (A_L) surfaces, and convective heat current between them.

Fig. 6. Natural convection loops in a fluid layer connecting the equatorial and polar surfaces.

Fig. 7. The heat current q at constant T_H, as function of the partitioning of the earth surface (x). Also shown is the T_L temperature corresponding to the maximum of each curve.

Fig. 8. The heat current q at constant T_L, as function of the partitioning of the earth surface (x). Also shown is the T_H temperature corresponding to the maximum of each curve.

Fig. 9. The temperatures T_H and T_L that maximize the heat current leaving the heat source (the part to the left of the intersection), and the temperatures that maximize the heat current entering the heat sink (the part to the right of the intersection).

Fig. 10. The two partitionings of the heat surface, which maximize the latitudinal heat current. The surface partitioning (x) is shown in both cases. The corresponding average temperatures of the heat source (T_H) and heat sink (T_L) are also shown.

Fig. 11. The atmospheric latitudinal circulation shown for the annual mean. The contour interval is 10 sverdrups (1 sverdrup = 10⁶ m³ s⁻¹) and the ordinate corresponds to the height above earth surface as measured in atmospheric pressure units (adapted from [15]).

Fig. 12. The heat current from the heat source (left branch) and the heat current to the heat sink (right branch) as function of the latitude. The latitudes at which q is globally maximized are indicated with dashed lines.

Fig. 13. The overall earth entropy generation as a function of the latitude of the boundary between the heat source and the heat sink.

Fig. 14. The average latitudinal wind speed as a function of the latitude of the boundary between the heat source and the heat sink.

Fig. 15. The average height of the friction layer as a function of the latitude of the boundary between the heat source and the heat sink.

Fig. 16. The average pressure difference between the heat source and the heat sink as a function of the latitude of the boundary between the heat source and the heat sink.

Fig. 17. The conductivity that corresponds to the heat current as a function of the latitude of the boundary between the heat source and the heat sink.

Fig. 18. The heat current q_d between the illuminated and the dark zones on earth. Air from the illuminated zone is entering the dark zone with the velocity component u relative to the earth surface.

Fig. 19. The average power intensity as a function of the Ekman number.

Fig. 20. The convective heat current as a function of the Ekman number.

Fig. 21. The temperatures of the illuminated and dark zones, average wind speed, pressure difference, and height of the friction layer, as a function of the Ekman number.

Fig. 22. The overall entropy generation rate as a function of the Ekman number.

Fig. 23. The conductivity as a function of the Ekman number.

The long-term values of the meridional heat flow between the equatorial and the polar zones that result from the optimization on the basis of the constructal law

The values of the dynamic variables that correspond to the optimized long-term meridional circulation

The values of the thermal and dynamic variables that correspond to the optimized diurnal heat transport between the illuminated and dark hemispheres