Continuum Mech. Thermodyn. 8 (1996) 37-51 @ Springer-Verlag 1996

Original Article

The balance of entropy on earth

W. Weiss

Institut fiir Thermodynamik und Reaktionstechnik, TU Berlin, Sekr. HF2, Fasanenstr. 90, D-10623 Berlin, Germany

Received July 1, 1994

Entropy is absorbed and emitted in the atmosphere and in the litho- and hydrosphere of the earth by absorption and emission of radiation. The value of the total production rate of entropy may be decomposed into the production rates of radiative entropy and of material entropy. Although the latter amounts to only 3,4 % of the total, the material entropy production rate is the one that is relevant to meteorology, life science and ecology. Civilization accounts for less than 1% of the material entropy production.

1 Introduction

The earth receives energy from the sun and emits energy into the depths of the universe. These effects proceed by absorption and emission of radiation. The absorption and emission of entropy is closely related to the transmission of energy. Indeed, if a body absorbs the energy r per unit volume, its absorption of entropy is equal to r/T, where T is the absolute temperature of the body.

Energy is conserved, so that, under stationary conditions, we must assume that the energy comes in and goes out in the same amount. This is not so for the entropy: the entropy production is positive so that, again for stationarity, the emission of entropy must surpass its absorption; entropy is produced. The entropy production may be decomposed into two parts: The entropy production of radiation and the entropy production of matter. In this paper, we calculate both for two examples: a heat-conducting slab and the naked earth, both under solar radiation. It turns out, in both cases, that the radiative contribution to the total entropy production is much bigger than the material one. This is explained in Sects. 2 and 3.

While the radiative entropy production rate may be relevant to the degree of order or disorder of the universe, the evaluation of processes on earth requires only the knowledge of the entropy of matter. Therefore Sect. 4, which comprises the main part of this paper, considers the material entropy production rate of the earth and its atmosphere.

Most values used in this paper are rough estimates, but we expect them to give correct orders of magnitude.

2 Basic formulae

2.1 Balance of energy and entropy

The equations of balance of energy of matter and radiation read, respectively

fv 0 1 2 fo ( } fv -~p(u+qo+-~v )dV + p(u+~p+lv2)vi-tijvj+qi nidF= rdV, v z

(2.1)

38 W. Weiss

and

Here we have introduced

p D

V i

N -

v2/2 -

fV 0ec,l f0 fV Z7 dV + p i n i dF = - r dV. v

mass density, velocity of matter, specific internal energy of matter, gravitational potential, specific kinetic energy,

(2.2)

and internal friction, i.e. viscosity, and 0 has a specific form, viz.

¢ = -~5 ~xi + - ~ \Oxj + Oxi 3 -~xk 00 ' (2.7)

t~ and # are the coefficients of heat conduction and viscosity. We shall see that 7 contributes much more to the total entropy production rate than ~b and yet ~b is more

interesting than % because we are usually concerned with matter rather than radiation. For a discussion of both, 0 and 7, see Sect. 3 below.

The total entropy production rates of a body of volume V read

k ~ = f v ~ d V , and F = f v ' y d V , (2.8)

e - energy density of radiation, Pi - r a d i a t i o n f l u x .

t/j is the stress tensor of matter and qi is the heat flux. r in both equations represents the balance of absorption and emission of radiation. The balance equation of total energy is the sum of (2.1) and (2.2)

p ( u + ~ + ~ v ) + e d V + p ( u + g ) + ~ v ) v i - t o v j + q i + p i n i d F = O , (2.3) v

and it represents a conservation law, since the right-hand sides cancel. The equation of balance of entropy of matter reads

dV + psvi T J ni dF = ~ dV + ~b dV (2.4) v

s is the specific entropy of matter and ~b is the density of material entropy production rate. T is the absolute temperature of matter absorbing and emitting radiation. The equation of balance of entropy of radiation reads

- ~ + Oini dF = - -- dV + 7 dV. (2.5) v T

h is the entropy density of radiation, Oi its entropy flux density, and 7 the radiative entropy production rate density. If the radiation is in near-equilibrium with the matter, Oi equals the radiative energy flux divided by T and multiplied by 4/3, see [1]. This is usually the case for emitted radiation, in particular that of the sun, but it is generally not the case for radiation absorbed, because usually the impinging radiation is far from being in equilibrium with the absorbing body. The sum of (2.4) and (2.5) is the balance of the total entropy

O (ps + h ) dV + p s v i + --~ + Oi ni d F = O dV + 7 dV. (2.6) v

According to the second law of thermodynamics the total entropy production rate, which is the sum of ¢ + 7, should be non-negative. We know little about the nature of 7, the rate of the radiative entropy production, except that for a body under solar radiation it is connected to the absorption of near parallel low-entropy- radiation from the sun and emission of near isotropic high-entropy-radiation from the body. Our knowledge about ~ is more specific. Indeed in most fluids, the entropy production of matter is due to heat-conduction

Balance of entropy on earth 39

for the material and radiative productions respectively. It is often impracticable, of course, to monitor the fields of temperature and velocity inside a body and

then calculate ~b, and hence !P. Therefore, we approach the problem in a different manner: We consider only stationary processes and express the productions by balancing fluxes.

2.2 Stationarity and piecewise homogeneity

We assume that the conditions are stationary and that the temperature is homogenous in certain subbodies of volume Vs. In that case, all time derivatives vanish and (2.4) reduces to

N ¢ { +qi~ dA -~1 ~v ~ = p s v i ni - E ~ r dV.

v T J ~=1

=:Re,

(2.9)

Moreover, from (2.2) we have

With the abbreviations introduced in (2.9), (2.10), we calculate k~ as

(2.10)

N N R~

k ~ = ~ - ~ - - ~ - or ~ = ~ + E Pa. c~=l o~=1 T~

(2.11)

This is a useful formula, since often we do have information about the fluxes ~ and P~, as we shall see. Hence follows gz.

3 Discussion of the entropy production of a concrete slab and a first discussion of the earth's temperature and entropy production

3.1 A simple example: a heat-conducting concrete slab

3.1.1 Description

We consider a slab of concrete of thickness L and cross section A subject solar radiation from the right (Fig. 1). The absorptivity of the material for solar radiation is a and the emission coefficient is e.

a - -

E-- A -

OT q=-~z-O- ~ =-~ - ~

- - a

2- ~ A ~ p~'n=( I'a)p°

~ = - a Po ] Po , Oo

p~, o;

) Fig. 1. A heat-conducting slab

40 W. Weiss

The incident radiation maintains the right hand side at temperature T1. The left hand side radiates energy to the left, its temperature is 7"2. The slab is heat-conducting, so that under stationary conditions, the temperature field inside the slab is linear, and the heat flux is equal to

Tl - - T2 (3 .1)

q = - ~ L

We chose the geometrical data as L = 0,1 m, A = 1 m 2 and pick out the constitutive data for concrete from [2] as a = 0,65, e = 0,97 and ~ = 0,74 W/InK.

3.1.2 The temperatures T1 and T2

Of the incoming radiation with density P0 = 1341 W/m 2, the fraction p~- = -apo is absorbed in a thin volume V1 just below the right surface, and that volume emits radiation at a density of p~- = s~rTa 4, according to the Stefan-Boltzmann law. ~r = 5,66 • 10 - s W/m 2 K 4 is the the Stefan-Boltzmann constant. A volume V2 close to the left surface emits the radiation at a density of p~" = sorT 4.

Since the incoming and outgoing energy must balance for the slab as a whole, as well as for V1 (or V2),

we obtain - apo + ecrT 4 + ecrT 4 = 0 (3.2)

and T1 - T2 _ 0, (3 .3 ) - aP° + scrT4 + n L

the latter by use of (3.1). These are two equations for the temperatures 7"1 and T2. They lead to an algebraic equation of order 16 (!) and a numerical solution provides only one real positive solution, which for the given

values reads T1 = 317,3 K = 44,2 ° C, T2 = 274,9 K = 1,7 ° C. (3.4)

Therefore, we know the temperature field inside the slab explicitly:

T1 - T2 424 K T = T2 + x = 274, 9 K + - - x . (3.5)

L m

3.1.3 Material entropy production rate (2 ways)

In this case, knowing the temperature field, we may calculate the material entropy production directly. Indeed, insertion of (3.5) into (2.7), of which only the first term matters, gives

foL t~ ( O T ) 2 A n-(T1- T2)z (3.6) k~= ~ ~xi A d x = L T1T2 "

We conclude that the material entropy production, just like T1 and T2, depends only on the geometrical and constitutive data of the slab and on the energy flux P0. It is independent of the entropy flux 0o of the incoming

radiation. We may also determine the entropy production rate # indirectly, i.e. by balancing fluxes, see (2.11). (Later,

for the earth, this will be the only possible method, because we do not know the fields of the temperature and motion in that case). We use (2.11), and since there is no material entropy flux on OV, we may write

pini dA. (3.7) ro jovo c~=l cz=l

The integration extends over the surfaces of V1 and V2, and since only the outer surfaces of both are traversed

by radiation, we have 1 1 4

gu = { ~7(-apo + c~T4) + ~scrT~ ) A , (3.8)

Balance of entropy on earth 41

or, by use of (3.2) and (3.3)

k~ = A n (T 1 - T2) 2 L T1T2 ' (3.9)

exactly as before in (3.6). This is as it should be, of course. Insertion of the known temperatures (3.4) and of the geometric and constitutive data into (3.6) or (3.9)

provides a value for the material entropy production

W gr = 0, 1 5 3 - - . (3.10)

K

3.1.4 Total entropy production of the slab

From (2.6), applied to a stationary case and to a surface OV, which surrounds the whole slab, we have

gt + -P = fOv OinidF', (3.11)

69

because there is no material entropy flux on OV. The emitted entropy flux on the left surface is

O~ = A 4 p~ - 43Ae~rT3, (3.12) 3T2

provided that the radiation emitted there is in equilibrium with the slab. On the right surface, there is a similar contribution, viz.

O~ = A 4 P~ 4 3 3 T1 - -3 AeCrTl" (3.13)

In addition, there is the entropy flux 0o of the incoming radiation, of which the fraction a is absorbed and the fraction (1 - a) is reflected. We assume that the reflection is ideal, so that the reflected fraction has the same entropy coming in and going out.

We now calculate the absorbed entropy flux @~-. The incoming energy flux P0 is not in equilibrium with the body, but when it was emitted by the sun, it was in equilibrium with the sun. We assume the sun to be a black body, so that the entropy flux density on the sun is equal to (4/3)~rT3sun and the total entropy flux leaving the sun is equal to (4/3)crT3sun&rRasu n. This flux passes also through a sphere with radius rse of the earth's orbit. Therefore, we obtain for the entropy absorbed by the right surface of the slab

4 3 R2sun (3.14) (91 = - ~ a A c r T s u n r~ e

Combining (3.12), (3.13) and (3.14) with (3.11), we obtain

~P + F = @-~ + @-~ + 0 1 5A~r e(T~ + T 3) - sun o3 = a .--2T--~sun , (3.15) rSE or

W +/7 = 3,664-~, (3.16)

where the numerical value has been calculated from the data and results given before and Tsun = 5740 K. With gz from (3.10), we conclude that the radiative entropy production rate of this concrete slab is given

by W

/~ = 3 ,511- - . (3.17) K

It amounts to about 96 % of the total entropy production rate.

42 W. Weiss

3.2 The case of the naked earth

3.2.1 Homogeneous temperature of the naked earth

For stationary conditions, the conservation law of energy (2.3) reads, when applied to the naked earth, a sphere of radius Re

~o dA = 0. (3.18) pini ve

The energy flux density Pi is the sum of the influx density p - coming from the sun and the radial efflux density p+. Therefore, (3.18) reads

7rRZ p - + 47rR~p + = 0. (3.19)

The earth receives the energy flux density P0 = 1341 W/m 2 from the sun, a value called the solar constant. But not all of this is absorbed, a part is reflected and thereby contributes equal but opposite amounts to both terms in (3.19). In the range of wavelengths of the main solar radiation, the absorption coefficient of the earth is aE = 0,72, which corresponds to an albedo of 0,28. This means that 28 % of incoming radiation is reflected and therefore does not contribute to the integral (3.18). We do not count it, and therefore we write

W 7rR2p- = - a e 7rR~po = -0 ,72 • 7rR~. 1341 ~ 5 = - 1,23.1017 W. (3.20)

The emission coefficient in the range of wavelengths of the earth's radiation is ee ~ 1. Therefore, with the Stefan-Boltzmann law, we may write

47vR~p+ 2 4 = 47rREcrT~ , (3.21)

TeA is the "radiation temperature" of the earth, supposed homogeneous in this section. With (3.20) and (3.21), we rewrite (3.19) as

- 1,23.1017 W + 47rRZ~TT4 R = O, (3.22)

and hence follows TeR = 255,4 K = -17 ,6 ° C.

This value is smaller than the measured mean temperature on the earth's surface, which is

(3.23)

TE = 287 K = 14 ° C. (3.24)

The reason for this difference lies in the atmosphere which has been neglected here. The atmosphere absorbs some of the earth's radiation and radiates it back to earth, an effect known as greenhouse effect. The above calculation can be found in textbooks on meteorology (e.g. see [3] ). It is not concerned with entropy.

Indeed, this discussion does not provide for a material entropy production, since absorption and emission take place at the same temperature. An obvious improvement should allow for the difference of temperature on the day- and nightsides of the earth.

3.2.2 Accounting for day-night-differences on the naked earth

If we make allowance for a temperature difference 2AT between the day- and night-sides of the earth, we obtain, instead of (3.21):

47rR~p + 47rR~ 47rR~ o- T = 2 cr (TER - - A T ) 4 + ~ ( ER q- A T ) 4 = 1,23 • 1017 W. (3.25)

Balance of entropy on earth 43

For AT = 10 K, we obtain 1 the new value

TeR = 254,8 K = -18 ,2 ° C, (3.26)

which is only insignificantly different from the value (3.23). And indeed, a careful calculation shows that AT does not affect TeR linearly; the small difference is due to a second order term O((BT/T)2) .

This is different with the material entropy production rate. Dividing p+ into parts p+ and p+ for night and d day, we obtain by (2.11) and (3.18)-(3.21)

1 , 2 3 • 1017 W ~Pe = TeR + AT + 2rcR~cr {(Tee + AT) 3 + (Tee - AT)3}, (3.27)

and by (3.25)

47rR~c~(T;~ R- A T ) 3 A T 2 2 k~e = + A T ~ 47rRE~rT~AT. (3.28)

For AT = 10 K and Tee = 254,8 K, the total material entropy production rate turns out to be

k~e = 1,606.10 ~3-~-W. (3.29) K

Dividing (3.29) by AE = 47rR 2, the material entropy production rate per m 2 then follows as

k~e W - - = 0 , 0 3 1 - - ( 3 . 3 0 ) AE m 2 K"

Note that, in contrast to TER, the entropy production rate depends linearly on the day-night difference AT. Note further that we have not determined ~P by integrating the entropy production rate from the conduction

and convection in the litho- and hydrosphere; indeed we have judged this production rate by the balance of incoming and outgoing fluxes. This will also be the modus operandi in the subsequent Sect. 4.

3.2.3 The radiative entropy production of the naked earth

Having calculated the material entropy production !Pe, we proceed to calculate the total entropy production rate, which is the sum of the material and the radiative entropy production rate. Like in (3.14), the entropy flux absorbed by the earth is

O~ -aeTrR2 00 a 7rR 2 4crT3 r2un = = - E E ' q sb ln . ~ - ,

- t S E

(3.31)

where ae = 0,72 as before. The outgoing entropy flux is @~: = 47rR 2 4 3 5eecrT~R. The total production rate is the balance of the two fluxes, viz.

O+ e _ te R ~ n2 4 ~3 RZun ~e + re = - - u E T r l ~ E ' q G I S l g n T " 3 ~ 'SE

A simpler form of this equation reads

1) P + , ~e + re = ~ rs .n

since P+ = 47rR~zecrT4R is equal to aETrR}po. Hence follows

~E + FE = 6,153 - I014w or ~E + f~E _ 1,204 m VV K AE K"

(3.32)

(3.33)

(3.34)

J In contrast to the arguments for the concrete slab the temperature difference between the two sides of the earth are not ca lcu la t ed

but a s s u m e d from reasonable expectations. A calculation would be difficult, even for the naked earth, because of the complex geometry of the problem and the rotation of the earth

44 W. Weiss

If we take day-night differences with AT = 10K into account, these values increase to 6,184. 1014W/K and 1,212 W/m 2 K respectively. The material entropy production rate (3.30) is then 2,6 % of the total.

While the material entropy production rate ~P~ depends only 2 on P0, the energy flux of the sun reaching the earth, the total entropy production rate kVE + Fe depends also on 0o, the entropy flux of the sun reaching the earth.

By use of gte from (3.30), we obtain the radiative entropy production for the case with night-day difference

re -- (1,212 - 0,031) m-~WK = 1,181 W (3.35)

Ae m2K

This result confirms the earlier remark, according to which the entropy production of radiat ion/~ by far exceeds the entropy production gt~ of matter. For meteorology and the life sciences, however, it is the smaller value that is significant, In the next section we calculate this more carefully than before, i.e. by accounting for the presence of the atmosphere on the earth.

Note, that in the case of constant temperature on earth, the entropy production rate of matter ~Pe is zero. In the case of night-day difference of the temperature on earth the entropy production rate of matter is not zero and given by (3.30). The total entropy production ~PE + Fe, however, is nearly the same with or without day-night differences.

The value (kV~ + F~)/Ae ~ 1 W / m 2 K is also given by Ebeling [4], Stahl [5], Stephens [6] and Aoki [7]. Peixoto [8] calculates 0,589W/m 2 K, his smaller value is due to differences in temperature in different altitudes of the atmosphere.

3.3 Relative significance of the radiative and material entropy production rates

For two examples, the concrete slab and the naked earth, we have calculated that the radiative entropy production rate was much bigger than the material entropy production rate.

Relevant for the thermodynamic processes on earth, e.g. meteorology, are the inhomogeneities in thermo- dynamic fields, viz. mass density, velocity, temperature, etc. If such inhomogeneities were absent, the material entropy production rate were zero. An illustration is furnished by the above consideration of the naked earth with and without day and night differences of temperature.

The inhomogeneities in the thermodynamical fields of matter are also important for life sciences and ecology. The material entropy production rate, as a function of such inhomogenities, is a quantity which gives the production of disorder of matter.

Therefore, it is the material entropy production rate, which is the important quantity for an assessment of processes on earth, instead of the total entropy production rate.

The physicist who is interested in cosmological problems may concentrate on the total entropy production rate. Concerning earth, his only result is quite simple: It is given in (3.33), stating that the total entropy production rate of earth equals the emitted energy times the difference of the reciprocal temperatures of earth and sun. While the material production rate contributes to the total production rate, its contribution is small and cannot be evaluate from the knowledge of the total production rate.

4 Balance of energy and entropy for the earth and its atmosphere

4.1 Litho- and hydrosphere, troposphere and stratosphere

A simple model of the earth and its atmosphere suitable for consideration of the radiation fluxes is shown in Fig. 2. The figure schematically shows the litho-, hydro-, tropo- and stratosphere, and it indicates their mean temperatures.

The total non-reflected solar energy flux

W 1017 0,72Po = 0,72- 7rR~ • 1314 - ~ = 1,23. W, (4.1)

2 TER is a function of po

Balance of entropy on earth 45

o,7Zeo o,7ZVo A ~ A

0,47e o,~2Po 0,64eo o, OSPo

Pu;

O,O~3P~ t'O, O3P o

~s e~

TS~-22 stratosphere (17-70 km)

l troposphere (0-17 kin) ~ Tr=250 g

P r / u ¢ ~ c O, OSPo

litho-hydrosphere 1,09Po TL=287 K Mot

Fig. 2. Model for the energy balance of the earth. Refer to [9], except of the temperatures

is split into the parts PZH, Pr- and Ps- that reach the surface of the earth or are absorbed by the troposphere and stratosphere, respectively. Conversely, the outgoing fluxes P/~H, P~" and P~- are emitted from the surface, the troposphere and stratosphere, respectively. Their sum is again equal to 0,72P0. There are also radiative energy fluxes Pr/LH and PLu/r between the surface and the troposphere and vice versa. Finally, there is the heat flux QL from the lithosphere to the adjacent atmosphere and the convective energy flux MLt/ due to the evaporation of water on the surface.

Note that Pr/Ltl is nearly equal to P0, and that PI, H/r is larger than Po. This is so because of multiple emission and absorption of radiation between the earth's surface and the lower troposphere. Pr/LH is refered to "counter-radiation"; this is the greenhouse effect of the atmosphere. It is helpful in picturing this structure, if we think of the levels at the temperatures Tr and Tc as the upper and lower surface of the clouds, respectively.

In the stratosphere, radiation is mostly absorbed and emitted by ozone, and Ts corresponds to the tem- perature at the altitude of the maximum ozone concentration. Tr = 250 K corresponds to the temperature at the altitude of 6000 m according to the standard atmosphere, and TL~/ = 287 K is the mean temperature of the earth's surface. Tc will be determined later.

It is obvious that the model in Fig. 2 is extremely simplified. It recognizes only four temperatures, and the litho- and hydrosphere are characterized by the same temperature. In addition, as we have seen in the previous section, it is necessary for the calculation of the material entropy production rate to account for the temperature differences between night and day. Also, we distinguish between land and sea, i.e. litho- and hydrosphere.

These extensions to the model are illustrated in Fig. 3. We note that the largest contribution to heating of air close to the ground is due to the lithosphere, occurring mainly during daytime. The evaporation of water from the litho- and hydrosphere is also a daytime effect.

4.2 Energy fluxes

We proceed to calculate all fluxes drawn in Fig. 3 from either the known value of Po = 1,71 • 1 0 1 7 W, or from the Stefan-Boltzmann law appropriate to the temperature of emission.

From Fig. 2 and Fig. 3, we have

P s = -0,03Po, P~- = -0,22Po, PL-H ---P/- + P~ = -0,47P0, (4.2)

and it is reasonable to assume that P~- and P~ contribute to P/-H in the proportion of the surface areas of land and water, viz. AL = 0,29AE and AH = 0,71Ae. Hence

PL-= -AL0 ,47 P o and Pff=-At to ,47Po, (4.3) AE AE

46 W. Weiss

e~

radiation from the sun

Fig. 3. Model with day and night differences and distinction between litho- and hydrosphere

and thus, by (4.2)1,2 and (4.3), all incoming fluxes are known. P~n, P~d characterize the fluxes emanating from the stratosphere, which has the emission coefficient es.

From Fig. 2 and Fig. 3 we have

P~ = Pin + P~d = 0,03Po and (4.4)

p.~n Ae ,~4 Ae ,',.4 = es~r-~-~s,,, P~a = escr~-~Sd" (4.5)

Given the temperature Ts as in Fig. 2, as well as the day-night variation 2ATs, these three equations determine the three variables Ps +, P~a and Zs. We obtain

4 2 10 ,03Po + r~. 0 ' ~ P (4.6)

AE Tsn+T~d ' Psn = ~s. Sd ,Ua O,

P~d = ~ O , 0 3 P o . (4.7)

P~,, P~a and er are calculated in an analogous manner from the flux P~ = 0,64Po, viz.

2 1 0,64Po 4 r4 +r4 0,64Po, ¢r = T E ~ , P?" = Tr"

n l a r e Td

P~a = v~ 0,64Po. (4.8)

Next, we calculate the energy fluxes Pr/Ln, Pr/un, Pr/~, Pr/Ha associated with the counter radiation that emanates from a troposphere layer of temperature Tc. From Fig. 2 and Fig. 3, we have the five equations

PT/LH = PT/Ln + PT/Ld + PT/Hn + PT/Hd = 0,96Po,

A t T4 AH ~4 PT/Ln = eTt7-~ Cn, PT/Hn = erc*--~-lc,, (4.9)

A L T4 AH ,~4 PT/Ld = 6TO- T Cd, PT/Hd = grO '~ - lCd ,

which yield an equation for the calculation of Tc, viz.

Balance of entropy on earth 47

2 T4, + T4d - - - 0,96P0 (4.10)

eT ~AE

and, by (4.9), the fluxes Pr/rn, Pr/za, PT/H~ and PT/Ha. It remains to calculate the fluxes emanating from the earth's surface. From Fig. 2 and Fig. 3, we have

and

P~H = P~,, + P ~ + P~4n + P~Id = O,05Po,

PLH/T = PL/Tn + PL/Td + PH/Tn + PH/Td = 1,09Po,

AL T4 + A_~ P[~ + PL/Tn = eLCr-~ Ln, PLd + PL/Ta = eLcr T 4 ,

AH T4 + Aft_ P{4,, +PH/Tn =eHCr~- Hn, PHd +PH/Td =ettO- THd.

We reasonably assume that all fluxes Pin, P/z , P~/n, PHd are related in the same way to PL/T., PL/rd, P~t/Tn, PU/Td as the total flux P[H is to Pu4/r , namely 0,05/1,09 = 0,046. Therefore, we have

x - P{n _ P ~ _ Phn _ P~ld =0,046. (4.11) PL/Tn PL/Td PH/r~ PH/Td

We consider CH to be equal to eL, and thus we have nine equations for nine unknowns. The solution reads

2 ~L = EH : ~LH = orAL (r4n + r 4 ) + ~TAH (T4~ + T4d) '

x AL T4 x AH ~4 PLn 1 + X + - - eLHC*-~ Ln, P~4n = 1 + x ~ L " ~ r 2 --*R~'

x AL T4 + x Att ,~4 P~d 1 +x - ~L~'~ T ~ , P.d = ]77x~LH~-g-~,

1 AL T4 1 AH ~4 PL/Tn 1 +xgLHff'--f Ln, PH/Tn 1 +X -- : E LH ~r T I Hn ,

1 AH ,~4 1 e ~ & T 4 , PH/vd= eLHa-~-~Hd. PL/ra - l + x 2 l + x

(4.12)

In this calculation, we have assumed that the hydrosphere has the same temperature day and night, and that its temperature is equal to the mean temperature of the lithosphere. All other temperatures are supposed to vary by the amount AT , different for the different layers, above and below the mean value.

With the mean temperatures of Fig. 2, we may use (4.6)1, (4.8)1 and (4.12)1 to calculate the emission coefficients. We obtain

¢s = 0,076, Cr = 0,968, CLH = 0,993, (4.13)

and these values are independent of AT to within second-order terms. The same is true for Tc according to (4.10), and we obtain

Tc = 276,7 K = 3,7 ° C, (4.14)

which corresponds to the temperature of the standard atmosphere at an altitude of 1660m. All radiative energy fluxes are now determined, and we may proceed to calculate the material entropy

production rate.

4.3 Material entropy production rate

4.3.1 Material entropy production rate of the whole earth

The flux term ~ in (2.11) vanishes if OV is assumed to be the spherical shell including the litho- and hydrosphere, as well as the whole atmosphere, because vi and qi vanish on the surface. Therefore, the material entropy production rate of the earth as a whole is equal to the sum over the nine shells - or partial shells - with the temperatures

Tsn, TSd, TTn, TTd, TLn, TLd, Tcn, Ted, THn = THd = TL.

48 W. Weiss

We obtain

= Ps + + P~a + P s P L P~a + P T - - - - + + - - + Ts, TSd ~ Tra Ten

P~4n + PH/Tn -- PT/Hn + P~ld + PH + PIUTd -- Pr/Ha

TL

PT/Ld + PT/Hd -- PL/Td -- PH/Td

Tca

P L + PL/Tn -- PT/Ln + P ~ + P L + PL/Td -- PT/La

T ~

PT/Ln + PT/Hn -- PL/Tn -- PH/Tn + Tc.

(4.15)

An expansion of (4.15) in AT to within terms of O ( ( A T / T ) 2) yields

= + + AE :It..]

Ts Tr Tc

+(PL~/r - Pr/LH) TL Tc + O {(aT)2}. (4.1.6)

We note that the difference A T c occurs only in second order terms so that no value has to be assumed for it. The fluxes in (4.15) may all be taken, in terms of the temperatures, from (4.7)-(4.12), and the temperatures

are read off from Fig. 2 and (4.14). It remains to choose ATs , ATT and ATc. These are strongly dependent on position and season, and at high altitudes there are considerable day-night fluctuations. In [10], the daily temperature fluctuations in the stratosphere are given as 4 K to 20 K. We calculate kOE for some choices:

W W ~P~(ATs = ATT = ATs = 0 K) = 3,414.1013 , or a~- = 0,0669 m2 K'

W K W (4.17) fftE(ATs = A T r = ATs = 5 K) = 3,910.1013 , or ~ = 0,0767 m2 K'

ffJE(ATs = ATT = A T s = 10 K) = 4,407. 1013~W K or ~-~ = 0,0864 mWK ' AE

The values in (4.17)1 follow, if the day-night differences are ignored and only the different temperatures at the different altitudes are taken into account. The corresponding value of kve is roughly twice as large as the value (3.29) or (3.30) for the naked earth which, however, takes only daily fluctuations into account. A value of 20 K for this fluctuation is already quite large and we may therefore conclude that ~Pe/Ae ~ 0,1 W/m 2 K is an upper bound for the material entropy production rate per m 2 of the earth.

4.3.2 Material entropy production rate of the troposphere, lithosphere, hydrosphere and stratosphere

The troposphere represents that part of the atmosphere where weather is generated, and it is reasonable to ask how much of the whole material entropy production rate tPe is due to weather.

Inspection of Fig. 2 shows that the litho- and hydrosphere absorbs more radiative energy than it emits. The surplus is given off by heat conduction from the lithosphere to the adjacent layer of the troposphere and, above all, by the litho- and hydrosphere through evaporation of water. Both "mechanisms" are only effective during the day. Therefore, the lower surface of the day-time troposphere is traversed by a convective entropy flux psvi and by a non-convective entropy flux qi /T , the latter occurring only on land.

From (2.9) and (2.11), we conclude

N

/o ( q i ) ~ @ _ ~ ~T = pSVi + "-~ ni dA + . (4.18) vr =

Here, OVr is a closed surface surrounding the troposphere and the sum extends over the four partial shells with temperatures

TTn , Trd , Tcn , Tcd .

The convective flux in (4.18) is due to the fact that water entering the troposphere from below is in the vapour phase, while the water leaving the atmosphere is liquid or solid, i.e., it comes as rain or snow. The

Balance of entropy on earth 49

mass flux of water from the troposphere to the litho- and hydrosphere is denoted by rhr = prvrAe. Under stationary conditions, this flux is equal to the mass flux from the litho- and hydrosphere to the troposphere, which we denote by rnL + rht/ = pLVLAL + pHVHAH. Thus the convective entropy flux is given by

fo dA = rhrST -- rhLSL -- mH sn. (4.19) psvini Vr

The specific entropies SL, SiC of the vapour are much larger than the specific entropy ST of liquid or solid water. Therefore, we shall neglect the first term on the right hand side of (4.19).

The non-convective entropy flux is due to the earth heating the adjacent air. This phenomenon occurs only at the lithosphere troposphere boundary and only during daytime. Therefore, we have

J£o qi dA - QL (4.20) Vr -~ni TL + Z~TL

The radiation term in (4.18) takes the form

@~ Pc~ P~n + P~d + PT PT/Ln + PT/Hn -- PL/Tn -- PH/Tn PT/Ld + PT/Hd -- PL/Td -- PH/Td = - - + + (4.21)

~ TTn rrd Tcn TCd

Insertion of (4.19), (4.20) and (4.21) into (4.18), and subsequent linearisation with respect to £1T, gives

Q L ( ATL'~ 1 { , ~ . + P T ( I _ A T r ) } Pr/Ln- -PLn/T (4.22) ~Z = --rhLSg -- rhHs. - --~L 1 - --~L J + ~r TT + Tc

In the same manner, we calculate the entropy material production rates of the lithosphere, the hydrosphere and the stratosphere and obtain

1 { ( ATL'~ AL ( ATL~ Ac ) + P [ . - - 1 - + (P[. +PLa/T - PT/LH) (4.23) (~L = mLSL + ~L L QL 1-- TL ] AE --~L J ~

k~H = mH SH + ~1 AHA___E (PL+H + PL. + Ptu / r -- PT/LH) , (4.24)

1 { , ~ + P s ( 1 - A T s ) } . (4.25) eS = -T-ss Ts

Numerical values for these can be obtained from the previously calculated values for the P 's . We also have QL = 0,05Po (Fig. 3). The specific entropy SH of water vapour is SH(TH = 287 K) = 8, 8 kJ/kg K, and sc for the lithosphere is taken to be SL(TL + ATL). For the mass flux rh of the evaporating water, the literature values differ; e.g., the volume flux of liquid water is given in [11] as 399000 km3/year and in [12] as 577000 km3/year. These values correspond to mass fluxes rh = 1,265 • 101° kg/s and rn = 1,831 • 10 w kg/s, respectively. We choose m = 1,667.101° kg/s because, with the specific heat of evaporation r(TL = 287 K) = 2467 kJ/kg, this value corresponds to the energy flux Men = 0,24Po given in Fig. 2. rhH equals 0,85rn, and accordingly rhL = 0,15rh; these values are taken from [11].

Table 1 provides some values of the material entropy production rates for different values of ZIT. In each case, of course, we have k~s + k~r + kOL + ~PH = kOe.

Table 1. Material entropy production rates

ATs Z3TT ATL ~PE/AE ff~s/AE ~PT/AE ~RL/AE g~H/AE g'S/ff~E k~T/~PE ~L/~E t~H/g~E inK inK inK W/m 2 W/m 2 W/m 2 W/m 2 W/m 2

0 0 0 0,0669 0 0,0597 0,0026 0,0047 0 0,892 0,039 0,069 0 0 10 0,0725 0 0,0628 0,0050 0,0047 0 0,866 0,070 0,064 5 5 5 0,0767 0,0010 0,0672 0,0038 0,0047 0,013 0,876 0,050 0,061 5 5 10 0,0795 0,0010 0,0687 0,0050 0,0047 0,013 0,865 0,063 0,059 10 10 10 0,0864 0,0020 0,0746 0,0050 0,0047 0,024 0,864 0,058 0,054

50 W. Weiss

Inspection of the table shows that the troposphere contributes the largest amount to the material entropy production of the earth. Thus, not surprisingly, the weather is by far the most dissipative phenomenon on earth.

In addition, the contributions of litho- and hydrosphere are considerably less than the material entropy production of the naked earth, calculated in Sect. 3. This is obviously due to the shielding of the surface from solar radiation by the atmosphere.

5 Direct calculation of the material entropy production by heat conduction in the lithosphere

So far, with the exception of the slab in Sect. 3.1.3., we have not calculated material entropy production rates directly by integration over the field of the material entropy production rate density (2.7). Instead, we have calculated the production by balancing entropy fluxes - and assuming stationarity. The reason is, of course, that the temperature and velocity fields in the atmosphere are much too complex to be known as functions of position and time.

Only in the lithosphere is it possible to estimate the temperature field roughly. Assuming a flat earth, we may solve the heat equation to give

/ p/77 T(x,t) = TL + ATL exp | - , / ' - - - - x / \ v ; cos _cot - /T2 x ) (5.5)

This is the solution appropriate to a periodic boundary condition with frequency co = 27r/24 h. Here, p, c and t~ are the density, specific heat and heat conductivity of the lithosphere.

The material entropy production rate may then by calculated from (2.7) as

fo ~ 1 ( O T ~ 2

This is a function of time, and its mean value over the period of one day is given by

ez4 = AL~;~ -~ \ ~ x / ax dt

or, by (5.5),

-- ALI~ p ~ (z~TL ~ 2 f ( ATL ~4"I~ \ rL/ +ol,\rL/ J

With t~ = 2,2W/m 2 K and pc = 2,96 • 106 J/m 3 K, (5.6) yields

~z~(ATL = 5 K) = 3,58- 1011 -~WK, or

~ ( A T L = 10 K) = 1,43 • 1012 - ~ , or

T~ =0,0007 W AE m 2 K '

W ~ = 0,0028 m2 K "

(5.6)

(5.7)

For AT = 10 K, we obtain, by comparison with Table 1, the value of (5.3)4 representing 56 % of the material entropy production of the lithosphere. We suppose that a large part of the remainder is due to the entropy production from chemical reactions in the animal and plant life on the lithosphere. Another part may be due to dissipation associated with tectonic movements.

6 Discussion of the entropy production of civilisation

According to [13], the total energy consumption of civilisation is 1013 W. If we assume that all of this were converted to heat at the temperature TL of the lithosphere, the corresponding entropy production rate would be

1013 W 101o W ~civ W gZciv - - - - 3,48 • ~ - 6,83 - 10 -5 - - (6.1) 287 K K AE m 2 K"

Balance of entropy on earth 51

This va lue can be c o m p a r e d wi th the total mate r ia l en t ropy p roduc t ion rate on ear th and in the l i t hosphe re

f rom Table 1, i.e., ~civ ffYci v

= 0 ,0008, = 0,014. (6.2) ~e ~L

Consequen t ly , c iv i l i sa t ion is r e spons ib l e for less than 0,08 % of the ea r t h ' s mater ia l en t ropy p r o d u c t i o n rate,

bu t n e v e r t h e l e s s 1,4 % of the l i t hosphe re ' s .

Acknowledgemems. I thank Ingo Mtiller for many valuable discussions and suggestions.

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