## Global Entropy Budget

An important aspect that is not captured by the global energy balance is that many of the involved energy conversions are irreversible in their nature, that is, they proceed only in a certain direction. For instance, heat is transported from warm to cold regions, but not in the other direction. This aspect is captured by the global entropy budget, where the rates of entropy production tell us about the strength of energy degradation of the different processes that convert energy.

### Irreversibility of Processes

The entropy budget quantifies the irreversible nature of Earth system processes, that is, that these processes proceed in a unique direction. The entropy budget is given by dS/dt = a + div(Fe)

where dS/dt is the change in local entropy S with time t, a is the production of entropy within the system, and div(Fe) is the divergence of entropy fluxes Fe resulting from the exchange of energy and mass. The entropy S of the system characterizes its organization, with a lower value representing more organization, and a maximum value representing thermodynamic equilibrium. The rate of entropy production measures the degree of irreversibility (with a = 0 characterizing a reversible process).

### Planetary Entropy Budget

A steady state of the entropy budget is defined by dS/ dt = 0. Note that this is a different, less common definition of the climatic steady state that includes a representation of internal organization within the system. The entropy budget of the Earth can be estimated from this steady-state condition since the rate of entropy production a is then balanced by the difference of entropy fluxes across the Earth-space boundary. The entropy production by the flux of energy from a warm to a cold point can be expressed by a = Q_ (1/Tc - 1/Tw)

where Q is the flux of energy, and Tc and Tw are the temperatures. For instance, at the planetary level in steady state, solar radiation that was emitted at Tsun = 5760 K is absorbed and emitted by the Earth at a much lower temperature of TR = 255 K, resulting in an estimate for the planetary entropy production of aearth = Qw,abs(l/TR - l/Tsml) = 892 mWm-2 K-1

with the average amount of absorbed radiation of Qsw,abs = 238 Wm- . Note that mass fluxes do not contribute to the global entropy balance since the mass of the Earth is approximately conserved.

Entropy Production by Earth System Processes

The planetary rate of entropy production results from various irreversible processes (Table 3):

• Scattering of solar radiation. Scattering of solar radiation within the atmosphere and at the surface results in the irreversible conversion of a direct, focused beam with small solid angle to radiation being distributed over a wide solid angle. The resulting amount of entropy production is CTscatter = 26mWm~ .

• Absorption of solar radiation. Absorption of solar radiation at a temperature lower than the Sun's emission temperature is irreversible (i.e., it cannot be re-emitted as shortwave radiation). On Earth, absorption occurs within the atmosphere (e.g., ozone in the stratosphere) and at the surface. The two associated rates ofentropy production are estimated to be: (1) atmospheric absorption of ^iw,abs,atm = 68 W m~2 at a stratospheric temperature of about Tstrat = 252 K results

Table 3 Global entropy budget. The global entropy budget characterizes the irreversibility of various Earth system processes. The columns give typical values of the heat flux Q (see global energy balance) and the temperatures Tcold and Twarm at which the energy is being transformed. The entropy production a is then estimated by a = Q (1/Tcold - 1/Twarm) using the steady-state assumption

Table 3 Global entropy budget. The global entropy budget characterizes the irreversibility of various Earth system processes. The columns give typical values of the heat flux Q (see global energy balance) and the temperatures Tcold and Twarm at which the energy is being transformed. The entropy production a is then estimated by a = Q (1/Tcold - 1/Twarm) using the steady-state assumption

 Heat flux Tcold Twarm J (Wm2) (K) (K) (mWm2 K1) Scattering of solar radiation 103 n/aa n/aa 26 Atmospheric absorption of solar radiation 68 252 5760 258 Surface absorption of solar radiation 170 288 5760 561 Atmospheric absorption of terrestrial radiation 28 252 288 14 Moist convection (evaporation-precipitation) 79 266 288 23 Dry convection (sensible heat into boundary 24 280 288 2 layer) Frictional dissipation of large-scale circulation 10 255 300 6 Biotic activity 8 288 5760 5b Planetary 235 255 5760 881c

aEntropy produced by scattering originates from broadening of the solid angle, not from temperature differences. bTerm included in surface absorption of solar radiation.

cTotal does not balance individual contributions due to estimated nature of the budget.

aEntropy produced by scattering originates from broadening of the solid angle, not from temperature differences. bTerm included in surface absorption of solar radiation.

cTotal does not balance individual contributions due to estimated nature of the budget.

in CTsw,abs,atm = 258mWm_2K_1; (2) surface absorption of 170 Wm~2 at a global mean temperature of about Ts = 288 K results in CTsw,abssrf = 561 mW m~2 K_1.

• Absorption of terrestrial radiation. Of the net transfer of energy of 68 W m~2 from the surface to the atmosphere by terrestrial radiation, c. 40 W m~2 escape to space without absorption, while the remaining 28 W m~2 is absorbed at a lower temperature of about Ta = 252 K. This results in ^lw,abs = 14 mW m 2 K_1.

• Moist convection. Water is evaporated at the surface at Ts = 288 K and subsequently condenses within the atmosphere at a lower temperature of about Tc = 266 K (i.e., evaporation into an unsaturated atmosphere is irreversible). The associated global mean latent heat flux of .h = 79 W m~2 yields an estimate of the overall entropy production of ^moist = 23mWm~2K~\ This term includes various irreversible processes associated with moist convection, such as the phase transitions liquidgas, the mixing of air masses of different humidity, and dissipation of kinetic energy of falling raindrops.

• Dry convection. Under dry conditions, the sensible heat flux reflects the dominant form of heat transport by turbulent motion from the surface into the convective boundary layer. Using a typical temperature of Ty = 280 K and the global mean value of .h = 24 W m~2 results in an entropy production of cdry = 2 mW m~2 K_1.

• Frictional dissipation of large-scale motion. Motion in the atmosphere and ocean are generated from density differences. The associated physical work W performed to accelerate the atmosphere and ocean is balanced on average by the amount of friction dissipation D. Through the motion, heat is transported and counteracts the density differences. With an average amount of atmospheric heat transport of = 10 Wm~2 and typical temperatures of the tropics, Ttrop = 300 K, and of the poles, Tpole = 255 K, this yields an estimate of cht = 6 mW m~ K~ .

• Biotic activity. On the global scale, the biosphere with a gross primary production of 200 GtC yr_1 converts approximately 8 W m~2 of solar radiation into organic carbon compounds that are eventually respired into heat at Ts = 288 K. The resulting entropy production is Cbio = 5 mW m~2 K_1. Note that this term is already included in the above estimate of entropy production by absorption of solar radiation at the surface. Not included in this estimate is the additional work done by transpiring vegetation (included in the estimate of moist convection).

There are various other irreversible processes, such as seasonal freeze-thaw associated with sea ice and snow cover, seasonal storage and release of heat, wetting and drying of soils, etc., that are not included here. Yet these simple estimates provide an important additional component of the workings of the global energy balance. 