y y = (1,0, S9\ F\), and B and ky = [(3T/dy)] are constants. Thus, the GCM is a major component of this system, providing not only the three-dimensional atmospheric and surficial climate (denoted by X in Chapter 5) in equilibrium with the slow-response state (Y) of any geologic period, but also by systematic experiments, providing the sensitivity functions (e.g., B, ky) by which closure of the slow-response dynamical system can be achieved (see Fig. 5-4). This determination of the sensitivity functions constitutes Problem 1 posed in Section 5.2.

A schematic diagram of the complete feedback system implied by these equations is shown in Fig. 12-1, emphasizing the potential destabilizing role of the carbon cycle discussed in Chapter 10, the interrelations of the ice-sheet variables (/, D, C\, Wb, and <Si) described in Chapter 9, and the connection of the main slow-response variables (fi, /, 0, Sip) with the fast-response variables representing the surface climatic state and the oceanic circulations governed by a coupled AGCM and OGCM. The composite surface climatic state is denoted by the symbol "E," which includes all atmospheric and surface state variables (e.g., T, V, P, E, (p\,...). As a rough categorization, large values of £ represent a warm climate characterized by high temperature and snowline solar radiation tectonics solar radiation tectonics

Figure 12-1 Representation of the dynamical system described in Section 12.1 in the form of a feedbackloop diagram showing the main couplings and interactions between the ice, ocean, and atmospheric variables under the influence of external forcing. A wiggly arrow denotes a time-delayed (inertial) influence of changes in one variable on another, in the direction of the arrow. See text (Section 12.2) for further discussion.

Figure 12-1 Representation of the dynamical system described in Section 12.1 in the form of a feedbackloop diagram showing the main couplings and interactions between the ice, ocean, and atmospheric variables under the influence of external forcing. A wiggly arrow denotes a time-delayed (inertial) influence of changes in one variable on another, in the direction of the arrow. See text (Section 12.2) for further discussion.

latitude <p\, weak temperature gradients and winds, and a strong hydrologic cycle [e.g., large spatial variance of (E — P)], as distinct from a cold climate in which opposite conditions prevail. In addition, the strength of the ocean circulation is denoted by Z, which includes the thermohaline circulation (i/0, the gyre circulation (0), and more localized convective and baroclinic circulations (/), i.e., Z = (i/r, <p, /). In general, a "warm climate" (high E) tends to equilibrate with a weak oceanic circulation (low Z).

In these feedback diagrams a barbed link connecting two variables signifies that a change in one variable leads causally, in the direction of the arrow, to a change in the other variable. This change may be either of the same sign or opposite sign (denoted by a minus). A wiggle in the link signifies an inertial phase lag (or delay) in the response, representing a "prognostic" relationship; otherwise the response is essentially "instantaneous," representing a "diagnostic" relationship. Any closed set of links, which can be followed around in the direction of the arrows, constitutes a feedback loop. If there is an odd number of negative links in a loop it represents a negative feedback; this means that a change of a given sign in any variable in the loop will be opposed, tending to produce oscillatory behavior that can, however, be unstable as well as stable. On the other hand, if there is an even number of negative links (or none) the loop represents positive feedback that tends to reinforce a change in any variable in the loop, in the direction of the change. This can lead either to some larger finite value of the variable or to an unstable growth of the variable to an infinite value.

In general there are competing positive and negative feedback loops involving any prognostic variable. To form a mathematical dynamic model in the absence of quantitative measures of the strength of these loops a qualitative judgment must be made concerning the dominance of one over the other. A test of the validity of the judgment is the agreement of the output with the observational evidence, though even in this case the right answer might be achieved for the wrong reason. In the case of dominant positive feedbacks leading to a first-order instability, it is plausible from conservation requirements to demand that the negative feedbacks become dominant in the higher orders as the system departs markedly from the unstable equilibrium. This was discussed in more detail in Chapter 6.

The presence of damping due to fast-response negative feedback dissipative processes is assumed for each variable, though not shown in the figure; in our equations these possibilities are represented formally by vy (y = /, D, Wq , ¡1,6). The two main sources of external forcing of the system are radiative forcing (R), which includes both solar constant changes R and Earth-orbital (Milankovitch) forcing 8R; and tectonic forcing, which alters the continent-ocean distribution h, CO2 weather-ability W, and volcanic outgassing V^.

Note that the basic energetical drive of the system is provided by the influence of external solar radiative forcing on the fast-response climatic system (the physics of which is represented, for example, by a GCM). As a further step, we have assumed that most of the relevant fast-response behavior, including the hydrologic cycle, can be associated with the induced surface temperature field, T, and (E — P) field, through the full GCM solutions.

Figure 12-2 Box diagram showing the air-sea carbon flux process that determines Q^ in Fig. 12-1 based on the system [Eqs. (10.19)-(10.26)] described in Chapter 10.

Figure 12-2 Box diagram showing the air-sea carbon flux process that determines Q^ in Fig. 12-1 based on the system [Eqs. (10.19)-(10.26)] described in Chapter 10.

A more detailed representation of the air-sea carbon flux processes contained within the box, q\, based on the system given by Eqs. (10.19)-(10.26), is shown in Fig. 12-2. Whereas Fig. 12-1 is a feedback-loop diagram possessing the properties described above, Fig. 12-2 is more in the nature of a conventional "box-diagram" showing the fluxes of carbon between various reservoirs in the ocean (related by the processes BiC) and B(G)) as they affect the surface pC02 and hence £)}+, under the influence of all the other physical factors that were represented in Fig. 12-1. Another discussion of the distinction between these two types of diagrams in the context of the carbon cycle is given by Berner (1999).

12.3 ELIMINATION OF THE FAST-RESPONSE VARIABLES: THE CENTER MANIFOLD

By substituting the diagnostic equations [Eqs. (12.8a) and (12.8b)] into the dynamical equations for ice-sheet mass [Eq. (12.1)], deep ocean temperature [Eq. (12.6)], and the salinity equation [Eq. (12.7)] in the manner of Eq. (5.14), we obtain the following alternate forms of these two equations, in which all explicit reference to the fast-response variables is removed [i.e., the so-called "adiabatic elimination," Haken (1983)]:

where ao = a0 - a, [7^(0) - ^^(O) - k^'Oifi)], and Ky = (vj + a^y,

where I = £ Vy, c0 = {veT sw(0) - T%(0) - £i'v • [7(0), 0(0)]} (v = _/, 9), = (y\ink{Jv) - vek{?'\ Cfl = (yi/zxS^ - veB^), and Ke = [vfl(l - k?s)) + yiMT^'land

Note that due to the ice-albedo feedback we have fcj*^ < 0, and we can also expect

that kg > 0 and k0 ''' >0, affording the possibility that, as in the case of CO2 (see Section 10.1), positive feedbacks may dominate the behavior of and/or 9 as well as of Sv for some ranges of their values. The ramifications of these possibilities will be discussed in the next section. When these three prognostic equations are coupled with the other prognostic equations governing ice sheet basal processes and carbon dioxide, Eqs. (12.2)—(12.5), we arrive at an alternate set that constitutes our proposed "slow" or "center" manifold of the global climate system, to which all the faster response (e.g., atmospheric) climatic fields are attracted (as governed by a GCM). Note that in developing the carbon dioxide equation [Eq. (12.5)] in accordance with Eq. (10.28), we implicitly incorporated the elimination of the fast-response variables [e.g., of Ts in Eq. (10.16)].

In essence, these dynamical statements are our proposed "equations of motion" of the slow-response climatic trajectory, placing in a formal dynamical structure some of the leading hypotheses regarding the cause and behavior of the ice ages: (1) the orbital hypothesis (Croll, 1864; Milankovitch, 1930), (2) the carbon dioxide hypothesis (Tyn-dall, 1861; Arrhenius, 1896; Chamberlin, 1899; Plass, 1956), and (3) the bedrock depression/calving catastrophe hypothesis (Ramsay, 1925; Pollard, 1982), (4) the basal sliding hypothesis (Wilson, 1964; Weertman, 1969; Budd, 1975; Oerlemans, 1982a,b), and (5) the thermohaline circulation hypothesis (Chamberlin, 1906; Stommel, 1961; Weyl, 1968). In addition to these individual hypotheses, the full set of equations, taken as a whole, essentially represents a new hypothesis: namely, that ice variations may be the consequence of a dynamical interaction between all of these physical influences, especially if positive feedbacks lead to instability of the system. This possibility will be discussed next.

12.4 SOURCES OF INSTABILITY: THE DISSIPATIVE RATE CONSTANTS

We have just shown that after substituting for the fast-response variables (e.g., T) the intrinsic dissipative processes, measured by vy appearing in the dynamical equations, are modified by the sensitivity functions B and ky. As noted above, for both variables (4>, 9) the new damping rate constants Ky are smaller than vy as a result of positive

temperature feedbacks represented by k\ < 0 and (B, ke , ke v ) > 0, tending therefore to destabilize the system. Such a possible destabilization was already incorporated in the CO2 equation, Eq. (12.5), and the salinity equation, Eq. (12.7), by expressing the effective rate constants as K^ = (/3i — + PlU2) and Ks = (72 — 73Sy + J4S2). Similarly, the modified rate constants K^ for ice might also require a nonlinear form that admits a range of values of /, within which positive feedbacks can dominate (Kf < 0), but which are constrained by conservation requirements to remain bounded. That is, at the cost of additional free parameters (01, 4>2, $3) one could postulate the same generic form as was suggested for /x and S^, i.e.,

The cubic nature of damping, Kyy, implied by this coefficient, arises in many areas of physics, usually called the Landau (1944) form when only one variable is involved, but generalizes to the so-called Landau-Hopf form when more than one variable is involved, thereby permitting a bifurcation to oscillatory behavior. A discussion of the mathematical-physical foundations and consequences of this general cubic form of damping was given in Chapter 6 as part of a broad overview of relevant aspects of dynamical systems analysis. We now remark briefly on the physical nature of the positive feedback processes involved for ice sheets and the deep ocean, recalling that a more detailed discussion of such processes for the carbon cycle was already given in Chapter 10.

In the case of the ice sheets (y = for example, in addition to the need to allow for an increasing glacial response time as ice sheets grow (that should ultimately be limited by the "ice desert effect" resulting from reduced snowfall and lower temperatures at higher ice elevations and by an increased discharge creep velocity), there is an added ice-albedo effect represented in K\ due to the added expanse of snow and sea-ice cover associated with increased ice-sheet mass. Many ice-age models have placed a main potential source of instability in the ice equation operating through this positive ice-albedo feedback; that is, greater ice coverage causes higher surface shortwave solar reflectivity, which causes colder surface conditions and more extensive ice coverage. This effect is probably amplified by the high dust loads during glacial pri-ods. Thus, the positive ice-albedo feedback, embodied in Ki, lengthens the response time of the ice sheets, prolonging the existence of ice sheets in spite of the dissipa-tive process measured by i>i. This illustrates the general role of positive feedback: if dominant, it can be a source of instability; if it is large, but not dominant over other dis-sipative processes (negative feedbacks), it can still be very important by significantly increasing the response-time to a larger value than would be implied by the explicit dissipative processes alone.

Similar arguments have also been made for the behavior of the ocean state. In this regard we have noted that the dissipative rate constant for salinity Ks may give rise to a positive feedback that can destabilize the system. For example, S<p 0 will probably imply lower than average ocean surface salinity in higher latitudes, which in turn would imply a weaker thermohaline circulation. This would then tend to reduce the transport of salt poleward, further decreasing S,p and weakening the TH circulation. The net result would be an increase of the thermally direct part of the thermohaline circulation, which would tend to cool the deep ocean. These feedbacks are illustrated in Fig. 12-1 (see box labeled S<f). The simplified physics underlying this scenario was discussed in Section 11.4 and is the basis for several models of the deep ocean, containing the possibility for instability and bimodality (possible end-member states being similar to the ones shown in Fig. 8-3), e.g., Manabe and Stouffer (1988).

Thus, we must recognize, that some positive feedback exists for ice variability alone (e.g., the ice-albedo feedback), and possibly for the salinity gradient and thermohaline circulation leading in some models to instability and multiple equilibrium states; both of these sources of instability might be important in driving a natural oscillation of a form exhibited by the proxy data. However, as was discussed more fully in Chapter 10, because of the many possible sources of positive feedback in the carbon cycle, especially when /x is small and ice is more prevalent in the climate system [see Section 10.1 and Saltzman (1987b) and Saltzman and Maasch (1988, 1991)], we have guessed that the behavior of CO2 is the most likely source of major instability in the slow-response climate system, as was suggested by Plass (1956), and we will expand on this possibility in the illustrative model to be described in Chapter 14. Thus, in this illustrative case, we shall explicitly employ the form, Eq. (12.12), only for CO2, as already expressed in Eq. (12.5), treating Ky and Ks as constants; we remain aware, however, of the possibility that these qualities might also be better represented in the form given by Eq. (12.12), implying new potential sources of instability.

As noted in Section 12.2 regarding the feedback-loop structure shown in Fig. 12-1, in addition to the possible instability that can arise due to the physics of each individual slow-response variable [e.g., (/ T /), (p. T ->• /x)|, there are other sources for instability due to positive feedback loops between the individual variables that can dominate over dissipative processes within some ranges. Thus, for example, in Fig. 12-1 we find the positive carbon cycle loop (¡1 —> T —> I —> Qj^ —> /¿). Other possible sources of instability are introduced by the potential for ice sheet calving (/ Db C\) and basal melting-surge processes (/ -»• Wb -» <Si) that can result in a catastrophic collapse of an ice sheet when critical thresholds are reached.

12.5 FORMAL SEPARATION INTO TECTONIC EQUILIBRIUM AND DEPARTURE EQUATIONS

Following the approach suggested in Section 5.4 we now resolve the PDM system [Eqs. (12.2)—(12.5), (12.9), (12.10), and (12.11)] in accordance with Eq. (5.5), i.e., y = y+ Ay where >-' = (/, ¡1, 0; /•",), y is the ultra-long-term, roughly 10-My average, state wherein the climatic variables (/, jl, 6) are in equilibrium with geologic tectonic rock processes and slow variations in external forcing F, i.e., dy/dt = 0. Thus the set of equations governing the tectonic-mean ice mass I, carbon dioxide /z, and deep ocean temperature 9, respectively, is as follows:

1. Ice mass (/ = I]^), j a0 - fllB(*) + " £«>)] - = 0 (12-13)

where K^ is an augmented damping constant that includes the dissipative effects of C\ and Si, and over the long 10-My time scale we neglect the details of basal ice-sheet behavior represented by Db and Wq.

2. Carbon dioxide, ft,

As discussed in Section 10.6, because is difficult to estimate it is desirable to combine this atmospheric equation with the corresponding ocean carbon balance, Eq. (10.39), thereby eliminating this term. This leads to the fundamental Eq. (10.40) for the GEOCARB model (Berner, 1994), w£(T) + wl + v£ + v£-B^-B^ =0 (12.14b)

which, when coupled with Eqs. (10.41)—(10.50) and (7.34), forms a closed system for £ from which a solution, jl(t), can be obtained. Note that it follows from Eqs. (12.14) and (10.28) that fttfQ = {Q3, -J2Jl + + M~ ^[yj - wj[(£)]}.

3. Themohaline ocean state, 9, S(p,

12.5 Formal Separation into Tectonic Equilibrium and Departure Equations 245

where K^iS^) = (jj — j^S^ + JaS2). If, in accordance, with the suggestion made in the last subsection, we chose to assume a diminished role for a salinity-driven instability, we could set Ks = a constant. In this case — K^1 [ jo + n (y) ], which upon expansion via GCM experiments in the manner of Eq. (12.8) and substitution in Eq. (12.5) would lead to modified values of cq, ci, c,- and K$.

To recover the fast-response fields that are in equilibrium with y, we must invoke the GCM sensitivity relationships embodied in the relations given by Eqs. (12.8a) and (12.8b), where T and Ttp are determined with reference to an arbitrary fixed state that we may take to be the present-day values denoted by T(Q) and 7^(0), i.e.,

(f, Tv) = [f(0), 7^(0)] + In(^) + Y,kfJv)[y- y(0)] (12.17)

where y = (I, 6\ F,). In accordance with Eq. (12.8) and the discussion in Section 7.9, the ultra-long-term variations of T and Tv that accompany the changes in /, '¡1, 0, and S<p are also functions of all the processes included as external forcing, F;. That is, e.g., f = T(J, Ft), where Fi = {R(S), £2, UK h, G^, V*, W}. This tectonic-mean component comprising Problem 3 posed in Section 5.2 will be discussed more fully in Chapter 13.

If we subtract Eqs. (12.11)-(12.13) from the full set [Eqs. (12.2)-(12.5), (12.8), and (12.9)], we obtain a dynamical system governing the departures AY, which we can view as "corrections" due to all the internally driven phenomena (e.g., fluxes within and between the ocean, atmosphere, and ice masses). To simplify the notation we have already set A/x = £ and AO = d (see Section 10.5), and we shall henceforth further set AVj = C, (A/ = 0)' and ASv =

Concerning the dependence of A Y on external forcing F, , we shall assume at this stage that the variations of £2 are neglectable (A£2 = 0). However, due to the special importance of Earth-orbital insolation changes on temperature we shall explicitly represent only this insolation forcing in the ice-sheet and deep ocean temperature equations to follow. Similarly, because of the possible importance of CO2 outgassing (V'1) and weathering (W^), even on glacial-interglacial time scales, and because of our previous inclusion of thermal effects in the parameterization [Eqs. (10.27) and (10.28), we shall explicitly include only A Vt and AW1 in the CO2 equation governing f [see Eq. (10.37)]. Thus, the set governing A Y can be written in the following form, which includes the equation for carbon dioxide already derived in Section 10.5 as well as the approximation ln(yu. fix) = £ / Ji, valid near ft:

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