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

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

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

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).

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