A FIRST-ORDER ANALYSIS OF THE HEAT WAVE IN THE SOIL

A F I R S T - O R D E R A N A L Y S I S OF THE HEAT WAVE IN THE

SOIL

D A R I O C A M U F F O

CNR-Istituto di Chimica e Tecnologia dei Radioelementi, Corso Stati Uniti, /-35020 Camin Padova, Italy

and

S E R G I O V I N C E N Z I and L E O N A R D O P I L A N

CNR-Istituro per lo Studio della Dinamica delle Grandi Masse, S.Polo 1364,/-30125 Venezia, Italy

(Received January 2, 1984; revised February 29, 1984)

Abstract. On the basis of a realistic distribution of the net radiative flux density (composed of a half sinusoid for the shortwave contribution plus a term dependent on the soil surface temperature for the longwave contribution), the solutions regarding the propagation of both the diurnal thermal wave and the heat flux density in the soil are analyzed. The more relevant differences from the analytical solutions obtained under the classical hypothesis of pure sinusoidal forcing waves on the boundary are therefore pointed out.

List of Symbols

a albedo al , a2, a 3 coefficients in Equation (2) A absorbed fraction of R A 0 daylight amplitude of incoming shortwave radiation e~ Fourier coefficients D (KIo)) 112 g constant in Equation (1) G soil heat flux density h adimensional depth z/D x/~ H sensible heat flux density H(y) Heaviside function o f y k soil conductivity K soil diffusivity L$, L~" incoming and outcoming long, wave radiation LE latent heat flux density N daylight duration q 7D/k r Sun-Earth distance r* respondance R global solar radiation S solar constant t time t* retention-time T soil temperature x adimensional time nt/12 x o effective adimensional daylight duration nN/12 z depth Z Sun's zenith angle ~/ parameter defined in Equation (7) b solar declination

Wasp, Air, and SoiI Pollution 23 (1984) 441-454. 0049-6979/84/0234-0441501.90. © 1984 by D. Reidel Publishing Company.

4 4 2 D. C A M U F F O E T AL.

® ~o

O)

departure of T from the mean surface temperature latitude net radiation flux density angular speed of Earth rotation

1. Introduction

Soil thermal pollution can be due either to variations of external energy input or thermal diffusivity of the subsoil, which depends on both soil constituents and moisture content. Whatever is the cause, the study of the thermal pollution requires the knowledge of the thermal wave T(z , t) and heat flux density G(z, t) in the soil. The investigation of G(z, t) is obtained by means of sophisticated measurements or models. As an example, Lettau (1977) proposed a theoretically satisfactory model, known as the climatonomic method, to evaluate G(0, t) as a function of the temperature T(0, t) at the soil/air interface, as a rigorous consequence of the so-called 'First-Order Thermal Response', namely:

G(O, t) = r*(T(O, t) + t*(d/dt)T(O, t)) + g (1)

where r* is the 'respondance', t* the 'retention-time' and g a constant determined by

boundary conditions. Other researchers, however, attempted to relate T and G to the incoming solar

radiation and therefore to the net radiation flux qb. In fact 'the Sun provides about 99.97% of the heat energy required for the physical processes taking place in the Earth-atmosphere system' (Sellers, 1965). In this sense some empirical formulae were proposed to compute G from • which is easier to model than T. The more complete and recent is due to Camuffo and Bernardi (1982), i.e.:

G(0, t) = al~b(t) + a2(d/dt)rb(t) + a 3 . (2)

However, if ales s accurate analysis is required, the formula of Gadd and Keers (1970) i.e. G = 0.1 qb can be used. This formula, which can be viewed as a particular case of Equation (2) with a2 = a3 = 0, was described as (De Bruin and Holstag, 1982) 'a good estimate for G' and is very practical due to its simplicity. In practice, the Gadd and Keers formula states that in the Energy Balance Equation the heat fluxes in the atmosphere account for the 90~o of qb. This percentage is dependent upon soil characteristics and vegetation, so that it is better to derive an effective coefficient each time instead of 0.1. This formula may be used when qualitative considerations on the general description of G are needed, as in the present case. In fact it will be used in order to determine the constants entering in the solution of the heat equation. An extension to the more complete description with Equation (2) is not difficult, but more compli-

cated. The aim of this paper is to analyze the evolution with time t and depth z of the thermal

wave T(z , t) in the soil on the basis of realistic radiative input and boundary conditions. The radiative input • is described as the net balance between the downwelling minus the upwelling shortwave radiation and the longwave effective outgoing radiation.

FIRST-ORDER ANALYSIS OF THE HEAT WAVE IN THE SOIL 443

The shortwave contribution can be represented with a half Sinusoid during the daylight time and represents the absorbed fraction A of the shortwave solar global radiation R, i.e.

a = R ( 1 - a ) (3)

where a is the albedo. The longwave contribution is the net radiative transfer AL = L$ - LI" where the arrows respectively indicate the incoming and outcoming longwave radiation; AL causes the integral of • over the whole day to be near zero. Therefore, during the daylight, q5 is the sum of these two functions, i.e.

• = R(1 - a) + AL (4)

and during the night it is represented by the net loss AL which is negative at night. Of course, this approach is very practical when a is not strongly dependent on the solar height (e.g. over a water surface) or when R can be described by a pure sinusoid, i.e. when the absorption of the atmosphere does not vary too much.

Jaeger and Johnson (1953)proposed a first-order model assuming • = F cos Z + AL during the daytime and (I) = AL during the nighttime, where F and AL are constant and Z is the Sun's zenith distance. The model is valid for any latitude and season, since Z is expressed as a function of the latitude and Sun's declination, so that during the whole daylight period q~ is sinusoidally shaped. The model is an extension of the particular case of Brunt (1932) for the equinoxes, by assuming that the radiation absorbed by the ground consists of a cosine term during the daytime and a constant loss during day and night. In this hypothesis, Jaeger and Johnson described the soil surface temperature T(0, t) for three different daylight periods.

In the present paper we will discuss with a simple mathematical apparatus the solutions T(z, t) and G(z, t) of the heat equation in the soil for both the first and second order approximations with the aim of pointing out the main physical features of the heat propagation into the ground, with special reference to the more outstanding departures from the classical model. A comparison with the experimental data suggests to modify the daylight duration to lower effective values depending on the atmospheric conditions and albedo variations.

Of course, our analysis is sometimes limited by the assumptions made in order to simplify the mathematical apparatus; nevertheless, it points out many interesting results on the general problem.

2. Results and Discussion

The time distribution of the shortwave absorbed radiation A was computed by means of the formula:

A = A o cos (~t/N) H(cos (nt/N)) (5)

where A o is the maximal incoming shortwave radiation in the day considered and can be expressed as

A o = ( S / r 2) c o s ( ~ - q0) ( 6 )

444 D. CAMUFFO ET AL.

where S is the solar constant, r the Sun-Earth distance expressed in unit of the mean distance, Nis the daylight duration, t the time (hr) before or after noon and H ( c o s ( n t / N )

the Heaviside function which is 1 when the argument c o s ( n t / N ) is positive, i.e. during the daytime, and 0 when the argument is negative, i.e. during the nighttime.

In the present work, the net radiative flux density is therefore described by Equation (4) where for each day the time variation of the longwave radiative loss is approximated by the first-order relationships:

AL = AL + y(T(0, t) - T(0)) (7)

where the overbar denotes the mean daily value and y the sensitivity to the soil surface temperature variation. As it can be noted, AL is not formally represented with one of the classical formulae(i.e. Brunt, McDonald, Swinbank, Robizsch, Elsasser, and Loenquist, Anderson, Anderson and Budyko) reported by Sellers (1965), since it is referred to as an adjustment to the mean value AL which can be evaluated with one of the named formulae or experimentally observed. The main hypothesis is that soil and weather conditions do not vary during the day, so that the longwave balance is subjected only to the inevitable daily oscillation, thus the first-order approximation could be accepted. Higher order approximations could be used; however, the very low values of 7 discourage the use of more improved formulae. In practice, field studies showed that at Padova, over open grassland country, AL is practically constant during the night, except when the change of the breeze causes a substitution of the air masses in the lower atmosphere, thus changing the water vapor content and therefore the optical thickness and the back-scattered radiation. During the day the temperature variation of the soil surface is more pronounced, so that a time dependence of AL may be appreciable. The periodicity requires that

A L = NAo/12rc . (8)

The soil conductivity and diffusivity are supposed, as usual, to be constant. In the following we will express the above equations in terms of the adimensional time

x = nt /12 and x o = nN/12; of the adimensional depth h = z /D ~ (with D = x / ~ c o , K the diffusivity and cn the Earth's angular speed) and departure ® from the mean surface temperature, i.e.

O(h, t) = T(h, x) - T(O) . (9)

By expanding A in a Fourier series, the solution for the heat equation into the ground becomes:

c. exp ( - hxfn ) ®(h,x): j2×

× cos nx - - - h x / n + arctan (10) 4 q

F I R S T - O R D E R A N A L Y S I S O F T H E H E A T W A V E IN T H E SOIL 445

2

0.7

-0. 12

i i

0 6

t ( h r )

0.7

io.7 12 18 (] 6 12

t ( h r )

E ~ o

-0.7 12

1

2

I I i

18 O 6 12

t (hr)

Fig. 1. Adimensional temperature (adim. T = k®(h, x)/alAoD ) for: (a) winter solstice, (b) equinoxes, (c) summer solstice. The temperature is computed for q = 0 at four different depths h, i.e. h = 0 (1), 0.25 (2), 0.5 (3), 0.75 (4). In this Figure, as well as in the following, the Fourier expansion has been considered up

to the 50th harmonic.

where k is the conductivi ty, q = 7D/k,

c, = 2 (cos (nxo/2))/Xo((~/Xo) 2 - n 2) (11)

so that, at the equinoxes, c, = 1/2 and cn = (2/~(1 - n 2 ) ) c o s ( ~ n / 2 ) for n = 2, 3, . . . .

G r aph i c representa t ion of the solutions ®(h, x) in terms of the adimensional tempera-

ture k®(h , x)/alAo D are repor ted in Figure 1 for some typical cases, i.e. at both the

solstices and equinoxes, with q = 0. A dramat ic depar ture from a pure sinusoid is

evident, chiefly in the winter solstice. This agrees with the exper imental findings cited

by Sut ton (1953) who states that in clear summer weather in mid- la t i tudes ' T can be

closely approx imated by a single sine term' , 's ince the second harmonic of the Four ie r

expans ion is of less impor tance , and the diurnal t empera ture curve can be assumed to

446 D. C A M U F F O ET AL.

3c

~-

20

1o i i i i i i i i i I ( I i 12 18 0 6

t ( h r )

4 0

3 O

1-- 2 0

6 1 2 1 8 0 6

t (hr)

1.4

"I~ -

~ 1,0 -~

~ 0~ E

,~

z = 5 , 5 c m

Z = 0 . 5 c m

O ' ~ ~ - , ~ . - ~

0 , 4 I I I I I I I `) 4a s ~ 12,35 ~s~ 2`),35 o.~ 1.35 s~ ?2,,)5

7 Seplem~lel 8 September

Fig. 2. Qualitative comparison between: (a) computed T(z, t) with q = 0 and (b) field data (O'Neill, Nebraska for 7 September 1953; Figure (a) is from Equation (10). (c): Thermal conductivity k at two depths,

i.e. z = 0.5 and 5.5 cm, for the same case study.

follow a simple sinusoidal variation throughout the whole 24 hr without serious error'. As opposed, 'in winter the diurnal temperature curve cannot be represented as a single

sine wave with much accuracy'. A comparison with field data is shown in Figure 2 where the solutions of Equation

(10) are reported together with data of the O'Neill experimental compaign (Lettau and Davidson, 1957). As it can be seen, the qualitative agreement is quite good, especially for the night-time period. By day, in contrast, the smaller spread calculated is a consequence of having assumed the diffusivity to be independent of both time and depth and of having used in the present work the nocturnal value. In order to give an idea of the rough approximation made in assuming constant the soil characteristics, we reported in the same Figure the time and depth variations of the soil thermal conductivity k for the same case study. As it can be noted, k becomes double after the first 5 cm in depth.

F I R S T - O R D E R A N A L Y S I S O F T H E H E A T W A V E I N T H E S O I L 447

The general behavior of the soil characteristics has some implication also in the

evaluation of the soil heat flux, as it is stated in Lettau and Davidson 's (1957) book:

'in heat flow evaluations the problem is further complicated by the fact that temperature,

heat capacity and thermal conductivity all show large depth variations within the

micro-layer beneath the surface'.

An improvement to the above formula (and similarly for G), can be made by making

time and depth dependent the soil characteristics. This will be the object of further

investigations.

12 -12

_ - - q=O 3 - - - ~ . . . . . . 3

~ ~ ~ , . . . . , ~ ~ - ~ ~

~" ~ ~ q=0.5 ~ ~ - - ~ ~

~ ~ . ~ , . ~ - - ~ ~ ~ ~ ~

U_-2222- . . . . . . . I1=1 ~ ~

2--

I I I I I 0 8 1 0 1 2 1 4 1 6 1 8

N(hr)

~

Fig. 3. Left scale: Time lag (L) with reference to noon of the maximum of T(0, t) (broken lines) as a function of q and daylight duration N. Right scale: Time interval (I) (continuous lines) in which OT/~t is positive for the above variables. In the classical case of T sinusoidally shaped and q = 0, the corresponding broken line should be the horizontal straight-line L = 3. Similarly, for the corresponding continuous line,

I = 12.


6

I,-

I I I

1 2 3

Fig. 4. Time interval I in which c3T/at is positive for three different daylight durations (winter solstice: (1); equinoxes: (2); summer solstice: (3)) as a function of the dimensionless depth h. In the case of a pure

sinusoidal wave I = 12.

The parameter q reduces the amplitude of the thermal wave and outphases the

maxima and the minima by making them earlier but not in the same manner, so that

the warming time is slightly reduced when q increases. With reference to noon, the time lag of the maximum of T(0, t) as a function of q and

of the daylight duration is shown in Figure 3. The time lag appears to be strongly

dependent on q and very slightly on N. In the same Figure also the time interval I in

which OT/Ot at the surface is positive is reported against N for different values of q. In

this range of N, the named time intervals appear to be dependent on both N (linearly in a first approximation) and q. As evidenced in Figure 3, the differences from the

classical representation with a sine wave are noticeable.

In Figure 4, I is reported against h for deeper layers, when q = 0. In the classical

representation the three curves will reduce to the straight line I = 12, to which the curve

for the summer solstice is closer.

The heat flux G is expressed by:

x /n c. exp ( - h x /n) G(h'x)=a'A° n=l ~ (~-;-q % ~ ~

x cos (nx - h x/~ + arctan (12)


In the case of the first-order approximation, i.e. when q = 0, Equations (10) and (12) reduce to

AoD__ ~ c,, = e x p ( - hx fn)cos (nx - ~- - hx /n) (13) O=(h'x)=alk n = l x / n 4

G(h,x) = alA o ~ c. e x p ( - hx /n)cos (nx - h x / n ). (14) n = l

Graphic representations of the solutions G(h, x) in terms of the adimensional heat flux G(h, x)/alA o for some typical cases are reported in Figure 5. In this figure the solutions of Equation (14) for the same cases as in Figure 1 are shown. As mentioned before, the winter solstice shows the chief departure from a sinusoid. Also, one can note that the sine wave approximation can be reasonably assumed only in deep layers since at night

-1 12

r3

E

ro

3

A _ I I I 1 8 O 6 1 2 I I I I

12 18 0 6

, (hr) , (,.)

1

g2

[ _ I I

12 18 o 6 12

, ( . r )

Fig. 5. Adimensional heat flux density (adim. G = G/alAo) for : (a) winter solstice, (b) equinoxes, (c) summer solstlce. The f lux is computed for q = 0 at four different dimensionless depths h, i.e. h = 0 ( I ) , 0.25 (2), 0.5

(3), 0.75 (4).


1 1 00.5 - 1

l~ 16 20 O0 04 08 12

t(~)

Fig. 6. Adimensional surface heat flux density (adim. G = G/alAo) for the summer solstice with three values of q, i.e. q = 0 (1), q = 0.1 (2) and q = 0.2 (3). One can observe that by increasing q the amplitude of G is reduced and that, when q > 0, during the night G is damped but is seen to vary abruptly after sunrise.

m

I i 1

1 2. 3 4

12

v

Fig. 7. As in Figure 4 but for cqG/cSt > O.


in the surface layer G appears to be constant. This is true for the case with q = 0. A more realistic treatment requires positive values of q, as shown in Figure 6. By assuming

positive values for q, one obtains a damped decrease of the absolute value of G in the

course of night and this decrease is greatly marked the greater q is. A sophisticated analysis ofq alone is not always sufficient to improve the approxima-

tion. In practice, since there is more than one parameter, the best approximation is obtained when all the parameters are optimized.

Similarly to the case in Figure 4, the time intervals I for which ~?G/Ot is greater than zero are reported in Figure 7. One could note that the time intervals in which OG/Ot is positive are not equal to those in which OG/~t is negative, differently from the case of a sinusoidal forcing wave. This is due to the asymmetrical shape of the functions during

the warming and cooling periods. The present analysis shows that the departures from a sinusoidal forcing wave are

maximal for the shortest daylight duration (winter solstice) and become negligible in mid-summer, at our latitude ((p = 45 ° 20' ). The departures are reduced with depth so

0

I-- <~

c-

'I

2-

O/

/ /

,?

I /

/

/ /

//

/ /

i / / I .,~]

j f

/ •

0 1 2 3

h

Fig. 8. Amplitude attenuation ln(AT(O)/AT(h)) with depth of the thermal wave at the winter solstice (1), equinoxes (2), summer solstice (3). The classical case for a sinusoidal forcing wave (continuous line) is also

shown. (q = 0).

452 D, C A M U F F O ET AL.

that when h = 4 the curves for the different seasons tend to be practically undistinguish- able, since the error in computing them becomes comparable with the amplitudes.

By plotting the lorgarithm of the amplitude attenuation with the depth h, as in Figure 8, one can note plots very close'to straight lines, i.e. in a first-order approximation

in ( A t ( 0 ) / A T ( h ) ) = s h (15)

where the parameter s is dependent on the daylight duration and only slightly variable With depth. As an example, at the adimensional depth h = 2, s = 1.1, 1.04. 1.02 respectively at the winter solstice, at the equinoxes and in the summer solstice. The classical case for sinusoidal forcing function yields s = 1 independent of depth, reason-

18

1 5 -

1 2 -

i . t-

9

6

/ f i t . / ' l / i 3-/ .

/ i

i i i I I I

i i / ; / / /

/ I

i , . /

, / z / ., ¢1" / / J

/ / " j /

, . ¢ i /

z z

1 0 1 I [ 0 1 2 3

h

Fig. 9. Phase lag At with depth of the maximum of T after noon at the winter solstice ( l ) , equinoxes (2), summer solstice. The classical case for a sinusoidal forcing wave (continuous line) is also shown.

(q = 0).


20

15

10

el

'E I9

5

(9 o

5

-113 .... 4 6 8 10 12 14 16 18 20

t(~) Fig. 10. Heat flux density in the soil G measured with the method of the phase lag (dots) and of the

attenuation of the amplitude (squares) of the thermal wave with depth. Padova, 26 October, 1980.

ably close to the case for the summer solstice as stated by Sutton, and not for the equinoxes when the daylight period equals the night-time.

The phase lag of the maximum of T varies with season and depth as shown in Figure 9, reaching the maximal departure at depth between h = 1.2 and h = 2 depending on the daylight dm'ation in our latitude and tends asymptotically in the deep soil to the straight line corresponding to the sinusoidal case.

From the above figures it follows that the heat flux in the soil calculated with the amplitude attenuation generally differs from that calculated with the phase lag, as shown in Figure 10. Therefore, the best approximation could be obtained with the amplitude attenuation in the upper layers and with the phase lag in the deeper layers. However, 'in general the most consistent results are obtained with phase relation' (Sellers, 1965), since even in the layers closest to the surface the former method is largely invalidated by the assumption of soil characteristics independent of both time and depth. In fact, 'heat capacity and thermal conductivity all show large depth variations within the microlayer beneath the surface' (Lettau and Davidson, 1957), due to variations in both soil moisture content and homogeneity.

3. Conclusions

The analysis of the time evolution of temperature and heat flux in the soil has shown some relevant differences from the classical sinusoidal boundary conditions. Among


these, the time period in which T(z, t) and G(z, t) are increasing is not a half day; the departure increases with the shortening of the daytime duration but decreases with depth. The amplitude seems to be practically attenuated exponentially, as for a sinusoidal forcing wave. A noticeable influence is evidenced to be ascribed to the dependence of the effective outgoing longwave radiation on the surface temperature. Further work is needed to take into account the variability in time and depth of the soil characteristics.

Acknowledgments

The authors wish to express their gratitude to Dr A. Giommoni for his support at the computer. The present analysis is devoted to study the deterioration of marble buildings due to the diurnal heat wave and related effects. The study is supported by the Commission of the European Communities, Research and Development Programme in the Field of Environmental Protection. (ENV.757.I(SB)).

References

Brunt, D.: 1932, Quart. J. Roy. MeteoroL Soc. 28, 389. Camuffo, D. and Bernardi, A.: 1982, Boundary-Layer MeteoroL 23, 359. De Bruin, H. A. R. and Holstag, A. A. M.: 1982, J. Appl. Meteor. 21, 1610. Gadd, A. J. and Keers, J. F.: 1970, Quart. J. Roy. MeteoroL Soc. 96, 297. Jaeger, J. C. and Johnson, C. H.: 1953, Geofisica Pura e Appl. 24, 104. Lettau, H. H.: 1977, Boundary-Layer Meteorol. 12, 213. Lettau, H. H. and Davidson, B.: 1957, Exploring the Atmosphere's First Mile, 578 pp., Pergamon Press,

London. Sellers, W. D.: 1965, Physical Climatology, The University of Chicago Press, Chicago 272 pp. Sutton, O. G.: 1953, Micrometeorology, McGraw Hill, N.Y., 333 pp.

A FIRST-ORDER ANALYSIS OF THE HEAT WAVE IN THE SOIL

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