## Zerodimensional models

The simplest model is a model with one parameter - the temperature T{ of the outgoing terrestrial emission. It is defined by the condition of radiative equilibrium of the planet characterizing a balance between the absorbed short-wave solar radiation flux na2S0(l — ap) and the outgoing long-wave emission flux 4na2faTr4, that is,

where, along with the known symbols, ap is the mean weighted planetary albedo defined by the expression ap = j JLj S(x)a(x) dx; x = sin (p; a(x) is the local albedo at latitude (p (or at fixed x); S(x) is the annual mean distribution of solar radiation at the upper atmospheric boundary normalized so that the integral of S(x) from 0 to 1 is equal to 1. The following formula serves as a good approximation for S(x) (see North et al, 1981)

where S2 = —0.477 is a numerical constant; P2(x) = (3x2 — l)/2 is the second Legendre polynomial. In accordance with (5.3.2) the annual mean distribution of solar radiation at the upper atmospheric boundary is a parabolic function of x, having zero derivative at the equator (x = 0) and taken to be 0.523 at the pole (x = 1).

From (5.3.1) it follows for ap = 0.30 that Tr = 254.6 K, that is, the temperature of the outgoing terrestrial emission turns out to be much less than the observed global average temperature of the surface atmospheric layer, amounting to for 287.4 K. The cause of this discrepancy is clear: it is the greenhouse effect of the atmosphere which is not taken into account, and which is formed by absorption of the long-wave emission of the Earth's surface and of the adjoining air layer primarily by water vapour and carbon dioxide. To describe the feedback between the flux I of the outgoing long-wave emission and the average global air surface temperature T0 we take advantage of the approximation I = A + BT0 (see Budyko, 1969) where A = 203.3 W/m2, B = 2.09 W/m2 °C are empirical constants which implicitly take into account the effects of cloudiness and active admixtures in the atmosphere of the Northern Hemisphere. Equating the flux / to the flux of the absorbed solar radiation we have

from which with ap = 0.30 and the above-mentioned values of the numerical constants A and B we obtain the estimate T0 = 14.97 °C which in practice coincides with the observed average (for the Northern Hemisphere) surface air temperature T0 = 14.9 °C.

We now consider the influence of the feedback between the planetary albedo and surface air temperature. In doing so we assume that the planetary albedo is equal to 0.30 if the planet has no snow and ice, and 0.62 if it is covered with snow and ice. After substitution of these values into (5.3.3) we find that T0 = 15 °C in the first, and T0 = —36.4 °C in the second case. The Earth occupies an intermediate position between these two extreme cases. We assume that the southern boundary of the ice-snow cover is consistent with x = xs, and that xs = 1 at T0> 15 °C, xs = 0 at T0 < -15 °C, and xs = 1 + (T0 - 15)/30 at -15 °C < T0 < 15 °C. Then designating the local albedo as aj for 0 < x < xs we obtain from the definition of ap and (5.3.3)

= 1 - a, + (a, - a,.)[xs - ¿S2(xs - xs3)], (5.3.4)

where H0(xs(T0)) is the global coefficient of the short-wave solar radiation absorption (the planetary co-albedo).

For the present-day value of the solar constant S0 Equation (5.3.3) has three solutions. The first one complies with the present climate, the second one -with the climate when about 30% of the planet's area is covered by ice, the third one - with the climate when the planet is completely covered by ice (the case of the white Earth).

We examine the linear stability of all the above-mentioned solutions of Equation (5.3.3). First we write down a proper non-stationary heat budget

Figure 5.1 Dependence of the global average surface air temperature on S0/Sq for the zero-dimensional model, according to North et al. (1981).

equation. Considering (5.3.4), this takes the form c ^ + I(T0) = iS0H0(xs(T0)) (5.3.5)

di and in the limit (as t => oo) reduces to an algebraic (with respect to T0) equation which for the above-mentioned approximations of I and H0(xs(T0)) describes the dependence of T0 on S0/Sq presented in Figure 5.1. Here is the present-day value of the solar constant, c is the heat capacity of the climatic system in question.

We define T0 as + ST0, where 7q is the solution of the stationary equation; ST0 is a small departure from it. Then we express I(T0) and H(xs(T0)) as a Taylor series expansion keeping only the first terms containing ST0. Then after a substitution T0 = T°0 + ST0; I(T0) = I(T°0) + (dI/8T0)Tg 5T0; H(xs(T0) = Hq(xs(To)) + (dH0/8T0)To ST0 into (5.3.5) and subtraction of the stationary equation

we have

We transform the expression in square brackets and to do this we differentiate (5.3.6) with respect to T0. As a result we have whence l(dI/dT0)-(S0/4)(dH0/dT0)']To = (H0/4)(dS0/8T0)Too. Substitution of this relationship into (5.3.7) yields

This ordinary first-order differential equation has an exponential solution with an index (H0/4c)(dS0/dT0)To. Hence, the tendency for 5T0 towards zero (stability), or to recede from it (instability), should be determined exclusively by the sign of the combination (H0/4c)(dS0/dT0)To. But because c and H0 are always positive then the stability or instability of the stationary solution has to depend on the sign of the derivative (dS0/8T0)Tg. Thus the following theorem applies (see Cahalan and North, 1979): the stationary state of the climate system described by Equation (5.3.3) will be stable when (dS0/dT0)To > 0, and unstable when (dS0/dT0)To < 0. The last inequality is valid if the negative feedback between the outgoing long-wave emission and surface air temperature is weaker than the positive feedback between the planetary albedo and surface air temperature, that is, if (dI/dT0)To < iS0(8H0/dT0)To (see Equation (5.3.7)).

So we arrive at the following two conclusions on the basis of the considerations mentioned above: (1) the climatic regime complying with point 2 in Figure 5.1 is unstable in the sense that for small changes in the solar constant it converts into the regime of the ice-free Earth or into the regime of the ice-covered (white) Earth; (2) the climatic regimes which have global average surface air temperature above 10 °C or below —15 °C are stable with respect to small variations in the solar constant. All this is applied to both the present-day climate which is characterized by partial glaciation of the Earth and climates of the ice-covered and ice-free Earth.

One-dimensional models of a latitudinal structure where only one dependent variable - surface air temperature - is considered to be a function of latitude are close, in the ideological sense, to the zero-dimensional models. Therefore, to cover the succession of ideas we now discuss models, and

only after this do we consider the richer, from the viewpoint of their physical content, 0.5-dimensional (box) models.

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