## Onedimensional models

We distinguish the zonal belt of the unit meridional extension in the climatic system. The respective heat budget equation for annual mean conditions has the form

where the first term on the left-hand side describes, as before, the outgoing long-wave emission, the second term describes the divergence of the meridional heat transport integrated over the thickness of the atmosphere and the ocean; the term on the right-hand side describes the short-wave solar radiation absorbed by the climatic system; a(x) = [1 — a(x)] is the local absorption coefficient; other symbols are the same.

We supplement Equation (5.4.1) with representation of each function in terms of the annual mean zonally averaged surface air temperature T(x) and appropriate boundary conditions at the poles. We assume that the meridional heat transport is zero at the equator and the poles. To determine I(x) and a(x) we use the approximations proposed by Budyko (1969), that is, we assume that I(x) = A + BT(x), a(x, xs) = at at T(x) < Ts and a(x, xs) = cij at T(x) > Ts, where, as in the preceding section, A and B are numerical constants, at and aj are absorption coefficients for the short-wave solar radiation in ice-covered and ice-free latitude zones; Ts= —10 °C is the annual mean surface air temperature at the boundary (x = xs) between these latitude zones.

Before dealing with parametrization of the meridional heat transport divergence, we consider, according to North et al. (1981), two extreme cases where MHT(x) is equal to infinity and zero. In the first case the surface air temperature at all latitudes must remain identical, that is, the one-dimensional model will reproduce a situation similar to the one described by the zero-dimensional model. The respective solution complying with the one-dimensional model has to coincide with that shown in Figure 5.1, the only difference being that in the range of temperature changes from —15 °C to +15 °C the solid curve in Figure 5.1 is now replaced by the dashed lines which are an extension of the solid lines representing solutions in the regime of the ice-free (dark) and ice-covered (white) Earth. The physical interpretation of the broken curve at the phase plane (T, S0/Sq) is apparent: for a decrease in the solar constant there is a reduction of the surface air temperature of the isothermal planet. The latter remains dark up to the point where the surface air temperature reaches —10 °C, following which the planet suddenly becomes white. The new (appropriate for the regime of the white planet) surface air temperature is —44 °C.

In the second limiting case (zero meridional heat transport) Equation (5.4.1) reduces to the form

If a(x, xs) at x = xs takes an average value between at and aj7 then, applying Equation (5.4.2) to the boundary between ice-covered and ice-free latitude zones, we obtain S0(xs) = 4(A + BTs)/S(xs)a, where a = (af + a})/2.

This relation defines the value of the solar constant S0 providing the setting of the southern ice boundary at one fixed latitude or another. Calculated results presented in Figure 5.2 demonstrate quite a curious situation: if we agree with current theories on the Sun's evolution and, in accordance with these, assume that the solar constant at the early stages of the Earth's geological history was less by 20-40% than its present-day value, then the question arises as to why the whole Earth was not covered by ice as predicted, at present, by all the available one-dimensional climatic models? Several explanations can be proposed for this paradox, among them water shortage for creation of the ice cover, and the existence of unconsidered negative feedbacks determined, for example, by a change in cloudiness and gaseous composition of the atmosphere, etc.

We note in this connection that the results of the calculation were obtained

with an implicit assumption of the universality of the numerical constants A and B which, as a matter of fact, have to be changed on the geological time scale at least, due to changes in the content of absorbing substances in the atmosphere. It is also necessary to take into account the fact that one-dimensional models with zero meridional heat transport are not able to reproduce the real climatic system. Indeed, it may be seen from Figure 5.2 that the present position of the southern boundary of the polar glaciation (xs = 0.95) is met by the solar constant which exceeds its contemporary value by 70%. Moreover, for the given values S0/Sq and xs the global average surface air temperature turns out to be equal to 93 °C, and at x = xs a temperature discontinuity of the order of 50 °C occurs (see (5.4.2)). It is clear that without taking the meridional heat transport into account it is impossible to eliminate discrepancies in the observed data.

Two methods for parametrization of the meridional heat transport divergence within the framework of one-dimensional models are known. The first one (the so-called transport approximation) was proposed by Budyko (1969) (see also Budyko, 1980), the second one (the diffusive approximation) was proposed by Faegre (1972) and North (1975). We discuss these parametri-zations at length in order to establish their peculiarities and general features.

Transport approximation. For annual mean conditions the meridional heat transport in the climatic system is determined only by the net radiation flux at the upper atmospheric boundary. On the other hand, it has to be correlated with the average (over the atmospheric thickness) air temperature. But the vertical variability of the average air temperature is small compared to the horizontal variability, so that the vertical average temperature has to be correlated with the surface air temperature. In other words, there should be a relationship between the meridional heat transport and the surface air temperature distribution. These considerations serve as the basis for approximation of the meridional heat transport divergence in the form

where, as before, T0 is the global average surface air temperature; /? = 3.75 x 104 W/m2 °C is an empirical constant.

Substitution of (5.4.3), and the definition of 7(x), into (5.4.1) yields

A + BT(x) + p[T(x) - To] = iS0S(x)a(x, xs). (5.4.4)

From here, using the expression

found by integration of (5.4.4) over x from 0 to 1, we obtain the relationship for the determination of the annual mean zonally averaged temperature in the surface atmospheric layer. This has the form

T(x) = —AB~1 + ±S0[S(x)a(x, xs) + + ISB"1)"1. (5.4.6)

It remains to find the location xs of the southern boundary of the polar glaciation. Using the condition T(x) = Ts at x = xs we obtain

Ts(x) = — AB~l + 5S0[S(xs)a(xs, xs) + f!B~1 H0(xsy]B~ 1(l + /JB-1)"1.

where a(xs, xs) = (at + a,-)/2 is the average absorption coefficient.

The dependence of xs on S0/S% derived from (5.4.7) is shown in Figure 5.2. We pay attention to the jump-like change of the southern boundary of the polar glaciation after it reaches latitude 50°. The following explains this fact: at the boundary of the polar glaciation (x = xs) the flux of the outgoing long-wave radiation remains constant for fixed Ts. Because of this, when S0/Sq decreases, the southern boundary moves to the equator (xs decreases) and, therefore, S(x) increases. Simultaneously, a decrease in the global average surface air temperature occurs and, following this an increase occurs in the meridional heat transport divergence. But as soon as S0/S% becomes sufficiently small the meridional heat transport divergence increases more than the flux of the absorbed short-wave solar radiation and the polar glaciation propagates rapidly to the equator.

The reverse sequence of events takes place with an increase in S0/Sq. Nevertheless, the rapid transition from the regime of partial glaciation to the limiting regime (ice-free Earth) is preserved in this case as well.

Diffusive approximation. When identifying the heat content with a passive scalar characteristic the annual mean meridional heat transport will be proportional to the temperature gradient, or, in terms of T(x), to — (1 — x2)1/2 dT(x)/dx with a numerical factor called the phenomenological coefficient of heat diffusion. Accordingly, the meridional heat transport divergence will be equal to j AT1

V ■ MHT(x) =--D{ 1 - x2) —, dx dx where the factor D represents the product of the heat diffusion coefficient by l/a2.

We take advantage of this approximation and rewrite Equation (5.4.1) in the form

--D( 1 - x2) — + A + BT = |S0S(x)a(x, xs). (5.4.8)

dx dx

Let the conditions of vanishing of the meridional heat transport be boundary conditions at the equator and the pole, that is, dx

The first of these conditions is equivalent to an assumption of the symmetry of the surface air temperature field in both hemispheres.

We will find a solution of Equation (5.4.8) with D = const in the form of a series in Legendre polynomials P„(x). First we recollect that the Legendre polynomials are eigenfunctions of the diffusive operator

dx dx and that only those that have even index n satisfy the boundary conditions (5.4.9). Hence, the solution of (5.4.8), (5.4.9) can be presented in the form

T(x) = TnPn(x), n = 0, 2,____We substitute this expansion into (5.4.8), then multiply it by P„(x) and integrate over x from 0 to 1. Then, using the orthogonality condition jo P„(x)Pm(x) dx = (2n + l)_1<5„m, where <5„m is the Kronecker symbol, we obtain the following relationship for the expansion coefficient T„:

where Hn = H„(xs) = (2n + 1) j¿ P„(x)S(x)a(x, xs) dx is the polynomial function of the argument xs tabulated for a(x, xs) = a0 + a2P2(x) by North (1975).

Specifically, when n = 0 the relationship (5.4.10) reduces to (5.4.5), from where, for the above-mentioned values of the numerical constants A and B and function H0(s), T0 — 14.97 °C follows. For n = 2 the value of the function H„(xs) is equal to H2 « —0.5. Assuming, according to North et al. (1981), that DB~1 = 0.310 we have T2= -28 °C from (5.4.10). If we restrict ourselves to the first two terms of the expansion T k T0 + T2P2{x) - the two-mode approximation - and substitute the values of T0 and T2, then the agreement between calculated and observed meridional distributions of the annual mean surface air temperature will be good. Of course, the fact that the diffusive parametrization is in close agreement with observed data is impressive. But it should be remembered that this is the result of a lucky choice of parameters of the two-mode approximation rather than of the advantages of the parametrization adopted. By the way, the diffusive parametrization of the meridional heat transport within the limits of the two-mode approximation coincides with that proposed by Budyko (1969). Indeed, substituting the definition of T into the expression for the diffusive operator we obtain with D = const and n = 2

d dx

= 6DT2P2(x) = 6D[_T(x) - To], from here and from the definition of T it is found that j? = 6D.

To determine xs we follow North (1975). Instead of T(x) we introduce a new dependent variable - the outgoing long-wave emission flux /(x).Then Equation (5.4.8) is rewritten in the form

Next, we present I in the form of an even Legendre polynomial series and make the same transformations as in the derivation of the expression for T, that is, we substitute the expansion into (5.4.11), and then multiply the equation obtained by P„(x) and integrate over x from 0 to 1. As a result we will have [n(n + 1 )£>' + 1 ]/„ = (S0/4)Hn(xs). Substitution of this relationship into I = InP„(x), n = 0, 2,..., yields, at x = xs,

from which the position of the southern boundary of the polar glaciation is found.

The dependence of xs on Sq is shown in Figure 5.3. Under its construction in expansion (5.4.12) the terms with subscript n from 0 to 6 inclusive have been retained. It can be seen that in the vicinity of S0/Sq = 1 the function xs has several stationary solutions. In particular, there are five solutions for S0 ~ 0.98, which agree, respectively, with the ice-free Earth (xs = 1), small (xs % 0.99), intermediate (xs % 0.88), and large (xs « 0.30) area of the polar glaciation, and with the ice-covered Earth (xs ss 0). The stability of these solutions can be verified similarly as for the zero-dimensional model, that is, by means of the addition of the term c dl(x, t)/dt with the heat capacity coefficient c to the left-hand side of Equation (5.4.11) and linearization of the non-stationary equation with respect to small disturbances of the stationary solution. Having performed such transformations North et al. (1981) showed that the climatic regime corresponding to the intermediate area of the polar glaciation, as well x* 1.0

Ice-free Earth

Ice-free Earth

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