## Definition of the climatic system

The climatic system (%) is the totality of the atmosphere, {si\ hydrosphere (Jf), cryosphere (#), lithosphere {<£), and biosphere (J1) interacting and exchanging energy and substance with each other, that is, where by the atmosphere is meant the Earth's gas shell; by the hydrosphere - the World Ocean with all marginal and internal seas; by the cryosphere -the continental ice sheets, sea ice, and land snow cover; by the lithosphere -the active land layer and everything that is on it, including lakes, rivers and underground waters; and, finally, by the biosphere is meant the land and sea vegetation and living organisms, including human beings.

The state of each of the subsystems of the climatic system is described by a finite number of determining parameters xu x2,..., x„. All other variables describing the state of one or another subsystem are functions of these parameters. Just as in thermodynamics, the determining parameters may be divided into extensive parameters that are proportional to the size and mass of the relevant subsystems, and intensive parameters that do not depend on size and mass. The former relate to volume, internal energy, enthalpy and entropy; the latter relate to temperature, pressure and concentration. Internal and external determining parameters are differentiated as follows: internal parameters determine the system state, the external ones determine the environment state. This separation of parameters into internal and external is very conditional in the sense that it depends on the time scales of the processes being investigated. Thus, for a time scale of the order of 103 years or less, the area and volume of the continental ice sheets can be identified by external parameters; for longer time scales these can be identified by the internal parameters of the climatic system. Respectively, in the first case the continental ice sheets represent a part of the environment, in the second case they represent the constituents of the climatic system.

Generally, whether one or another subsystem belongs to the environment or climatic system is determined by the ratio between the time scale of the process being investigated (changes of the subsystem state because of changes of its parameters) and the relaxation time of the subsystem. If this ratio is small, then the subsystem is defined as belonging to the environment; otherwise it is defined as belonging to the climatic system.

As an illustration of the above we consider the simplest formulation of the problem of the ocean-atmosphere system response to constant external forcing. According to Dickinson (1981), the temperature disturbances in the atmosphere (ST.d), in the upper mixed layer (STm) and the deep layer (STd ) of the ocean, created by the disturbance of heat influx (5Q), can be described by the following equations:

where Equations (1.1.1)—(1.1.3) describe variations in the heat budget in the atmosphere, in the upper mixed layer (UML) and in the deep layer (DL) of the ocean, and their separate components: mean disturbances of the heat content in separate subsystems (the first terms in (1.1.1)—(1.1.3)), disturbances in the resulting heat flux at the ocean-atmosphere interface (the second terms in (1.1.1) and (1.1.2)), disturbances in heat radiation sources and sinks of heat in the atmosphere (the third term in (1.1.1)) and disturbances in heat exchange between UML and DL (the third term in (1.1.2) and the second term in (1.1.3)). Here ca, cm and cd are the specific heat capacities of the atmosphere, UML and DL; Aam and Xm& are the heat exchange coefficients at the atmosphere-UML and UML-DL interfaces; t is time.

Let ca, cm and cd, respectively, be equal to 0.45, 10 and 100 W/m2 K/year, that is, equivalent to choosing the following mass values referring to the unit of area surface: 14 x 103 kg/m2 for the atmosphere, 81 x 103 kg/m2 for the UML and 807 x 103 kg/m2 for the DL. The latter restricts the analyses of relatively small (order of decades) time scales where not the whole ocean but, rather, its upper 500-1000 m layer takes part in heat exchange with the atmosphere. We will assign parameters Aa, Aam and lmi equal to 2.4, 45 and 2 W/m2 K and assume also that temperature disturbances are absent at the initial moment, that is, and the disturbance of heat influx is changed in jumps from 0 up to SQ at t = 0 and, further (at t > 0), is constant everywhere. Equations (1.1.1)—(1.1.3), forming a linear system of ordinary differential equations, allow an exact solution. But its features can be understood simply by using the asymptotic expansion procedure. Keeping this in mind we first find the stationary solution of Equations (1.1.1)—(1.1.3), representing the equilibrium response of the ocean-atmosphere system to external forcing.

ca dôTJdt + KJSf - ÔTJ + Aa<5ra = ÔQ, cm dôTJdt + Aam(ÔTm - ÔTJ + Xmd(ÔTm - ÔTd) = 0, ed dÔTJdt + Â.md(ÔTd - ÔTJ = 0,

At t => oo these equations take the form

X.m(ST.-STm) + X.ST. = SQ, Km(STm - ST.) + Xmi(STm - ÔT6) = 0, (1.1.4)

Let us assume that the specific heat capacity of the atmosphere is much less than that of the UML, and the specific heat capacity of the UML in its turn is much less than that of the DL, that is, ca/cm and cm/cd are small parameters, which means that adaptation of the atmosphere to external forcing occurs faster than that of the UML and faster still than that of the DL, and hence, when determining ST. from (1.1.1) the value STm can be assumed to be fixed to a first approximation. Using this fact we fix STm in (1.1.1) and rewrite the first equation of the system (1.1.1)—(1.1.3) in the following form:

dSTJdt + (X. + X.Jc;'ST. = (SQ + X.mSTm)/c.. (1.1.6)

Its integration, with an allowance for the initial condition for ST. from (1.1.4), yields

^a + Km where fa = c./(K + Km) « 4 days. When t = 0(fa) Equation (1.1.7) takes the form