 
,
(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.
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