,

(2)

where t is the constant of thermal inertia of the planet;

In order to solve this equation, it is necessary to ex-

plicitly specify a function f(

), which should correspond

to possibly real processes as well as allow analytical de-

scription thus making possible to get analytical solution.

If one uses dimensionless solution in the form F(

) =

/

m, the problem comes down to determining the time

lag between the functions F(

) and f(

). The value of

can be derived from the expression:

f

F

Fto

00

(3)

We have examined four variants of the function f(

)

and corresponding F(

) obtained by solving the Equation

(2) with initial condition

.

1) A hypothetical case of jump discontinuity of Q:

1e t

xpF

1f

with (4)

This case corresponds to the catastrophe like rapid

change of Bond’s albedo or atmospheric transparency.

2) A hypothetical case of linear dependence of the ab-

sorbed power, when o

f

ttt

. Here to is the heat-

ing/cooling time, 0

to.

1exp

1; ;

o

Ftt

(5)

This case correspond to the possible scenario of the

Bond’s albedo change as a result of the ice cover change

or the cloud cover change.

3) A hypothetical case of exponential change of the

absorbed power with time,

1exp o

f

kt

:

12

12

1expexp ;

;,

o

o

oo

FMkM

tt

tkt

MM

tkt tkt

(6)

where k is the coefficient determining the gradient of Q.

This case corresponds to Bond’s albedo change during

the process of the system ocean-atmosphere transition

from one quasi-equilibrium state to another.

4) The case of periodic variations of the solar irradi-

ance with, for example, bicentennial period can be de-

scribed by the harmonic function

sin 2π

o

ft

(7)

The function F(

) derived from solving the Equation

(1) has a form:

1

2

1sin2π;

cos 2πexp

2π

o

o

o

F

btbg

gt t

t

bt

(8)

In the quasi-steady state regime, when the duration of

the process is long enough (as for the system Sun-Earth)

the latter term is close to zero. Then one can derive the

time lag between the functions F(

+

) and f(

) from

the distance between their nearest nodes equating both

functions (f and F) to zero.

can be evaluated using the expression:

arctg

2π

o

tb

(9)

To estimate the value of t we have adopted the values

of initial parameters which are realized nowadays and

determined in our previous papers [8-10]. Then the value

of t can be defined according to the following depend-

ence on the depth of the ocean’s active layer:

0.095 10.42tH (10)

where t is given in years, H—in meters.

It is worthwhile to estimate the minimum value of the

depth of the ocean’s active layer assuming an absence of

Copyright © 2012 SciRes. JGIS

H. I. ABDUSSAMATOV ET AL. 481

vertical convection, that is when the process of heat

transfer is determined only by the heat conductivity of

the sea-water. Applying the second Fourier law one can

evaluate the depth of the layer in which the temperature

increment from the ocean’s surface (corresponding to

vertical coordinate z = 0) down to the depth z changes

from от

о to

z. This depth is determined according to

expression:

0ln

π;o

z

at

zKK

tH

(11)

where a is the temperature transfer coefficient of the

sea-water, K is a relative decrease of the temperature

increment at the depth z with respect to the surface value.

Adopting the values K = 10 and K = 100 for the bi-

centennial cycle we get z = 300 m and z = 600 m, respec-

tively. The influence of convection can be accounted for

using the convection coefficient, however its global val-

ue is difficult to determine. It is known only that it is

greater than unity and is multiplied by the temperature

transfer coefficient in the Equation (11), and therefore,

the real value of the depth of the ocean’s active layer

exceeds the values mentioned above (H > z). It is neces-

sary to note that the notion of the depth of the ocean’s

active layer is rather conventional since the value of H

(which is proportional to z) depends on an arbitrary taken

value of K. It is important to emphasize that both the

results of calculations for the case of harmonic fluctua-

tions of the temperature of the ocean’s surface layer

(Figure 1) and estimations on the basis of formula (11)

with the adequate choice of the criterion for attenuation

of temperature variations K, the time lag ranges from 15

to 20 years.

3. Calculation Results and Discussions

In case of abrupt change of Q the transition time be-

tween two steady states is totally and unambiguously

determined by the constant of thermal inertia,

(3/4)t,

with an exact value being determined by a given error.

In case of linear and exponential changes of absorbed

energy the lag of the temperature increment increases

with time. On the final stage

= t for the linear law, and

=

/М2 for the exponential law.

Dependences , calculated according to expres-

sion (10), and

H

determined from (9) are pre-

sented in Figure 1 Our calculations have shown that in

case of harmonic fluctuations the time lag

arctg b

is small-

er than the constant of thermal inertia. This can be ex-

plained from the physical point of view analyzing ex-

pressions (9) and (10). Expression (10) allows conclud-

ing that in case of infinite growth of H (in a purely hy-

pothetical case) the value of thermal inertia should in-

crease infinitely almost proportionally to the value of H.

However, with the growth of H and, correspondingly, t,

the value of b which is proportional to t (see Equation (8))

increases infinitely as well. But is limited to

/2. Hence the value of

in the expression (9) is lim-

ited to to/4. For the bicentennial period and all real and

even hypothetical values of H the time lag cannot exceed

50 years, and the function

H

is growing with sa-

turation (line 2 in Figure 1).

4. Conclusions

The formulation of the equation of the Earth energy bal-

ance for the case of small increments of the solar power

absorbed by the ocean and the atmosphere leads to the

analytical formulas making it possible to calculate the

planetary temperature change in time.

Using the obtained solutions, the time lag between the

planetary temperature change and the planetary absorbed

power change was investigated for four cases of the

change of power absorbed by the ocean and the atmos-

phere in time.

The thermal inertia constant is defined for the wide

range of the temperature change which corresponds to

the Equation (1). Also, the thermal inertia constant is

defined for the small increment of the absorbed power

and the temperature, respectively, which corresponds to

the Equation (2). The calculation results for the

non-linear Equation (1) and the linearized Equation (2)

are in agreement with ±10 К accuracy. The linearization

error of the Equation (2) raises with the increase of the

temperature variations.

Figure 1. Dependence of the thermal inertia constant t cal-

culated using the expression (9), line (1), and of the time lag

calculated using the expression (8), line (2), on the depth

of the ocean’s active layer H for the case of harmonic bi-

centennial variations of the absorbed thermal energy de-

fined by expression (6).

Copyright © 2012 SciRes. JGIS

H. I. ABDUSSAMATOV ET AL.

Copyright © 2012 SciRes. JGIS

482

f

The time lag dependence on the thermal inertia value

is estimated for each case of the

where the ther-

mal inertia is defined by the ocean active layer depth

(10).

The formula (11) is offered to estimate the ocean ac-

tive layer depth. The formula (11) leads to the conclusion

that in case of harmonic form of

f

the time lag

would range from 15 to 20 years.

Since the real bicentennial cyclic variations of the total

solar irradiance are not purely harmonic and contain dips

and glitches of 11-year cycles, the real time lag is deter-

mined by more complicated dependences. The changes

of can be caused also by long-term variations of

the Bond albedo. The character of its variations is poorly

understood so far, therefore both the linear and exponen-

tial presentations of the function

Q

f

can be used for

approximation of real variations during certain time in-

tervals.

REFERENCES

[1] M. I. Budyko, “The Effect of Solar Radiation Variations

on the Climate of Earth,” Tellus, Vol. 21, No. 5, 1969, pp.

611-619. doi:10.3402/2Ftellusa.v21i5.10109

[2] W. D. Sellers, “A Climate Model Based on the Energy

Balance of the Earth-Atmosphere System,” Journal of

Applied Meteorology and Climatology, Vol. 8, 1969, pp.

392-400.

doi:10.1175/2F1520-0450/281969%29008%3C0392%3A

AGCMBO%3E2.0.CO%3B2

[3] G. R. North, R. F. Cahalan and J. A. Coakley Jr., “Energy

Balance Climate Models,” Reviews of Geophysics, Vol.

19, No. 1, 1981, pp. 91-121.

doi:10.1029/2FRG019i001p00091

[4] C. E. Graves, W. Lee and G. R. North, “New Parameteri-

zation and Sensetivities for Simple Climate Models,”

Journal of Geophysical Research, Vol. 98, No. D3, 1993,

pp. 5025-5036. doi:10.1029/2F92JD02666

[5] J. I. Diaz, G. Hetzer and L. Tello, “An Energy Balance

Climate Model with Hysteresis,” Nonlinear Analysis, Vol.

64, No. 9, 2006, pp. 2053-2074.

doi:10.1016/2Fj.na.2005.07.038

[6] A. V. Karnaukhov, “Role of the Biosphere in the Forma-

tion of The Earth’s Climate: The Greenhouse Catastro-

phe,” Biophysics, Vol. 46, No. 6, 2001, pp. 1078-1088.

[7] B. G. Sherstyukov, “Thermal Inertia of the Ocean and the

Green-House Effect in the Modern Climate Variations,”

Meteorology and Hydrology, No. 7, 2006, pp. 66-72.

[8] H. I. Abdussamatov, A. Bogoyavlenskii, Y. V. Lapovok

and S. I. Khankov, “The Influence of the Atmospheric

Parameters Determining the Transparency of Solar and

Earth’s Radiation on the Climate,” Proceedings of the

All-Russian Annual Conference “Solar and Solar-Terre-

strial Physics—2010”, Saint-Petersburg, 2010, pp. 7-10.

[9] H. I. Abdussamatov, A. I. Bogoyavlenskii, Y.V. Lapovok

and S. I. Khankov, “Modeling of the Earth’s Planetary

Heat Balance with Electrical Circuit Analogy,” Journal of

Electromagnetic Analysis and Applications, Vol. 2, No. 3,

2010, pp. 133-138. doi:10.4236/jemaa.2010.23020

[10] H. I. Abdussamatov, A. I. Bogoyavlenskii, Y. V. Lapo-

vok and S. I. Khankov, “The Influence of the Atmos-

pheric Transmission for the Solar Radiation and Earth’s

Surface Radiation on the Earth’s Climate,” Journal of

Geographic Information System, Vol. 2, No. 10, 2010, pp.

194-200. doi:10.4236/jgis.2010.24027