A mdCm P KKmLCmK KmJ S CmCd

Let us take advantage of the smallness of parameter s and introduce a slow time t related to a fast time t by the ratio t = et. Then on a fast time scale Equation (1.1.9) takes the form dSTJdt + (a + P)STm = aSTd + (P/oi)ÔQ, (1.1.11)

and on a slow time scale it takes the form

STm( t) = (1 + a//?) ~1 [(a//?) <5 (t) + 5Q/AJ. (1.1.12)

Similarly, on a fast time scale Equation (1.1.10) takes the form ddTJdt = 0 (1.1.13)

and on a slow time scale it takes the form dSTJdr + <x{STd - STJ = 0 (1.1.14)

or, after substitution of an expression for 3Tm from (1.1.12),

KddTJdx + 5Td = dQ/ila, (1.1.15)y where k = a'1 + jT1.

The solutions of Equations (1.1.10) and (1.1.13) with regard to the initial conditions for STm and 5Td from (1.1.4) take the form

Thus, the quasi-stationary regime of UML is established at a time of the order of several years, and on time scales t = 0(tm) the response of UML to external forcing is less, by (1 + a//?), than its limiting value determined by Equation (1.1.5). Rewriting the factor (1 + aIff) as [1 + (/imd/Aam)(l + Aam//a)] and substituting the above-mentioned values of parameters Aam and <imd, we draw the conclusion that on a time scale of the order of tm the response of UML to external forcing is about half its limiting value.

On the other hand, when t » tm the solution of Equation (1.1.15) after transition from t to t is written as

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