SCOPE 13 - The Global Carbon Cycle

ABSTRACT

A mathematical model of the global carbon cycle is developed. The model includes atmospheric, oceanic, and terrestrial organic carbon as the principal reservoirs. The latter two are subdivided in a way suitable for analysing the distribution of excess carbon, released in the period 1860-1970, from fossil fuel burning and forest clearing. An attempt is made to model quantitatively the dynamic characteristics of natural transfers. Numerical experiments are performed using the model.

It is demonstrated that several quite different pictures of the distribution of this excess carbon are obtainable within the ranges of present quantitative uncertainty regarding the carbon cycle. Results for the still-airborne fraction of the accumulated man-made release vary between 25% and more than 70%. The major causes of these variations are shown to be the uncertainty regarding the atmospheric CO₂ concentration prevailing before 1860 and the potential of the terrestrial biomass to accumulate carbon in organic material on land. The assumptions made on the circulation in the `intermediate' ocean layer, between roughly 75 and 1000 m, also exert in some cases a considerable modifying influence on the picture. On the whole, however, the results indicate that the uncertainty related to biological processes in the ocean is of minor importance in the present context.

15.1 INTRODUCTION

In order to estimate correctly what increase in atmospheric CO₂ content we may expect as a result of man's burning of fossil fuels and other CO₂-releasing activities, we must know which natural reservoirs represent the major sinks of airborne CO₂. We must also estimate at what rate these will absorb the atmospheric excess in the future. Closely related to this question is the problem of how the CO₂ release until now has been distributed between the same sinks. Qualitatively, oceans and, probably to a minor extent, terrestrial plants have taken up the excess CO₂ that no longer remains in the atmosphere.

When we discuss the global carbon cycle with focus on this aspect, we want to examine the dynamics of the interrelationships between atmosphere and plants. Our model of the cycle must thus contain these three major compartments. We have to investigate both how the flux of carbon between them would be in a state unaffected by man, and how the fluxes (and, with them, their contents) change with time as the atmospheric CO₂ content grows. In this chapter, we shall indicate a way of describing the structure of the ocean more realistically. We shall also suggest a more detailed way of modelling the terrestrial plants than has been customary so far.

Early models of the global carbon cycle usually treated the world ocean as a system of a few well-mixed boxes. Examples of this type of representation are given by Craig (1957), Revelle and Suess (1957), Bolin and Eriksson (1959), and Broecker et al. (1971). In theirs and other models of this type, the fluxes of carbon between the reservoirs are assumed to be first-order processes. The properties of the model can then be completely described by a set of transfer coefficients, k_ij denoting the proportionality between the transfer, F_ij, from box i to j, and the carbon content of i, N_i. The response of the model to an input of man-made CO₂ into the atmosphere can then be studied for various assumptions on the coefficients k_ij. A thorough analysis of the properties of this type of model is given by Keeling (1973a). 

From simple 'few-box models' such as these, one can formulate conclusions that assuming the oceanic circulation is similar to water exchange between a few internally well-mixed reservoirs one or the other set of exchange coefficients will give the best description of, for example, the atmospheric CO₂ increase. However, this ideal picture of the real ocean cannot be confirmed within the scope of the box model itself. In principle, it is impossible to arrive at a confirmation of the few-box model within a framework where one has a priori assumed it to be an acceptable simplification of reality. This requires a more refined model of the structure of the ocean. A realistic picture of oceanic circulation is probably necessary for a correct estimate of how the atmospheric CO₂ level will develop in the future. For this reason, we shall here formulate a model that takes into consideration what is known about the overall vertical circulation in the ocean. With this model, we can test the validity of the usual assumption that the deep sea is a so-called well-mixed reservoir.

It is well established that the formation of intermediate and deep water takes place in very limited areas of the world oceans. Along the polar sides of the large horizontal ocean currents, convergence implies rather intensive mixing in the vertical. This involves the water volume to a depth many times greater than the average depth of the surface layer, but it is probably of much less importance below approximately 1000 m. The intermediate water volume between approximately 75 m and 1000 m is, therefore, of special interest as a possible sink for the CO₂ released from the beginning of the industrial era until the present day. Because of convective mixing at the circumpolar convergence zones, this volume has a significantly more rapid exchange with the surface layer (and, indirectly, with the atmosphere) than the deeper strata. The total amount of carbon in the intermediate water is more than ten times that in the atmosphere, it has thus a considerably larger storage capacity for excess carbon than the surface layer. Over a time span of one or two centuries, the intermediate layer may, therefore, be a more powerful sink for excess CO₂ than can be deduced from two-box models of the ocean, where this layer is not treated separately from the rest of the deep ocean.

Renewal of the deep water below the intermediate levels is caused by sinking of surface water in areas where it is subject to cooling. The main region of this sinking surrounds the Antarctic continent, but it also occurs in some other parts of the ocean at high latitudes, for example in the North Atlantic. For reasons of mass continuity, the downward convection of water in these areas must be compensated for by a slow upward motion of the water in the rest of the ocean. An upward velocity of a few metres per year has been estimated.

This penetrative convection is a rather irregular phenomenon. Some years, the sinking water may never be dense enough to penetrate to the bottom of the ocean. The total amount of penetrating water may vary from year to year. However, in this model we shall regard this process as if it took place at a constant rate, and with a constant vertical distribution of the penetrating water.

In addition to advective exchange, carbon is transported from the surface to intermediate and deep water by the sedimentation of organic particles. The concentration of dissolved carbon in the water below the surface layer is approximately 10% higher than in the water near the surface. The sedimentation of carbon out of the surface water is, therefore, not likely to be more than about 10% of the amount transported from the surface by water motions. Most of the biological material produced in the ocean is remineralized within the photic zone or immediately below it. However, oxidation in the intermediate water proceeds at a sufficient rate to create an oxygen minimum. The level of the oxygen minimum, as well as the minimum value and the general shape of the oxygen profile near this level, vary considerably between different regions of the ocean, but a meaningful average oxygen profile may still be constructed. Since the oxygen budget is a balance of advective processes and biological consumption, the average oxygen profile provides information of great relevance to the oceanic carbon cycle. As discussed in more detail in Section 15.7, the oxygen profile will prove to be a valuable source of information regarding the intensity of convective exchange between the surface and the intermediate water.

15.2 FORMULATION OF AN OCEAN MODEL

Let the oceanic surface layer be modelled by two boxes, WSW and CSW, both of 75 m depth. WSW represents the warm surface water in those areas of the world oceans that have a well-defined annual thermocline below. This is roughly the ocean area between 40^° N and 40^° S. CSW represents the remaining surface water, to the north and to the south of this area. Let the intermediate water mass, between 75 m and 1000 m below sea level, be represented by two well-mixed boxes of equal depth, denoted by 1 and 2, and let these boxes be interconnected by water fluxes as shown in Figure 15.1.

Figure 15.1 Division of the world ocean, and water transports between the compartments. CSW = cold surface water, WSW = warm surface water

Let the deep water below 1000 m be described as a continuous medium where vertical circulation is given by a function f(z). The quantity f(z)dz denotes the rate of inflow of cold surface water (expressed as volume per unit time) between the levels z and z + dz. It is convenient here to count z from the boundary between the deep water and the intermediate water. If W_ij denotes water transport from box i to box j, the following fluxes occur in the model (see Figure 15.1).

W_cw and W_cw: the transport from CSW to WSW and vice versa by horizontal ocean currents and turbulent exchange.

W_c1 and W_1c, W_c2 and W_2c: the transport from CSW to the boxes 1 and 2 and vice versa, due to zones of convergence along the horizontal currents.

P: the transport of water from CSW to the deep sea by penetrative convection and from the deep sea into box number 2 by slow upward motion.

W₂₁ , and W_1w: the transport from box 2 into box 1 and from box 1 into WSW, resulting from slow upward motion.

From the definition of f(z) we have:

(15.2.1)

where D is the vertical extent of the deep sea. For reasons of mass continuity we must also have

 For mixing at the convergence zones we assume

 which gives

The horizontal area of each of these boxes is given in Table 15.4. In the deep sea, we denote the horizontal area at level z by A(z). We denote the sedimentation of organic carbon out of CSW by B_c and out of WSW by B_w, and the rate of dissolution in box number i by B_i, i =1, 2.

For the deep sea, we denote sedimentation through level z by g(z). The rate of dissolution in the volume between two levels, z₁ and z₂, is thus

B =g(z1 )g(z2)

(15.2.7)

For reasons of continuity, we have

We shall return later to the modelling of sedimentation and dissolution of calcium carbonate particles.

15.3 ORGANIC CARBON ON LAND

The subcycle of carbon from the atmosphere to organic matter on land and back to the atmosphere begins with photosynthesis in green plants. A large fraction of the carbon is transferred back to CO₂ by the respiration of the plants themselves. The rest will become dead organic material, detritus, after some time. A substantial part of all detritus is decomposed within a few years, when the carbon is oxidized to atmospheric CO₂. Estimates of the total amount of detritus in the world, a considerable part of which is found as organic compounds in the soil, have been summarized by Bramryd (Chapter 6, this volume). They may vary by a factor of two (Bohn, 1976; Schlesinger, 1977) but this value appears to be at least three times greater than the amount of living plants, expressed in terms of carbon. The global turnover time of detritus is, therefore, at least several decades. A small part of the detritus formed during a one-year period will remain as organic carbon in the soil for approximately 1000 years before being completely oxidized. The amount of carbon in animals (including man) is far less than in plants, and is, in the present context, negligible.

Considering the large variation in the time during which different carbon atoms stay in organic compounds, it seems unsatisfactory to model organic carbon as one well-mixed reservoir, Craig (1957) suggests a division between living and dead matter and models the biota and humus in principally the same way as we have indicated in Figure 15.2. By assuming that the transfer from organic compounds to the atmosphere depends not only on the total amount of organic carbon, but also on its distribution between living and dead matter, the dynamic properties of the organic carbon reservoir can probably be modelled better. Also, different carbon atoms have very differing residence times in the reservoir of living matter. Keeling (1973a) proposes a separation between `short-lived' and `long-lived' biota. The short-lived biota consists of annuals and of short-lived parts of perennials, such as leaves and needles. Most of these are decomposed within a few years after their formation. By contrast, carbon in tree trunks, for example, may remain in organic form for several decades. It is useful, therefore, to think of the return flux from the living biota directly to the atmosphere as being made up not only of respiration by autotrophs, but also of decomposition, of the remains of short-lived biota.

Terrestrial ecosystems are quite variable, and characterized by different time scales for a life cycle. Thus, organic carbon transit times are variable from system to system, as well as within each individual system. It seems necessary to account in some way for the difference in turnover time between tropical rain forest, for example, and coniferous forests of the boreal zone. We can do this by connecting the atmospheric box in our model to an arbitrary number of pairs of boxes as shown in Figure 15.2.

Figure 15.2 Principle of modelling the carbon flux through a terrestrial vegetation system

Figure 15.3 Some examples of modelling the water transport from the surface layer to intermediate layers (volume W_i) and deep layers (volume W_d). Boxes indicate wellmixed reservoirs except in case (iv), where lifting without vertical mixing is assumed

This type of refined division of land biota will be discussed and used in a later study. For the present experiments we have used two pairs of boxes. However, these are not intended to depict different vegetation types. Instead, one represents the natural system of biota and soil, and the other is introduced to illustrate more clearly human impact on the size of biota and the rate of exchange between biota and the atmosphere.

The probable magnitude of man's reduction of terrestrial biota over the last hundred years has been estimated by Bolin (1977) to be between 40 and 100 x 10¹⁵ g carbon. Stuiver (1978) gives 100 x 10¹⁵ g C as the most probable value, based on ¹³C analyses of tree rings, with annual transfers peaking around 1900. Stuiver's estimate is for the period up to 1950, and the total amount of carbon transferred during the last 100 years could, therefore, be even more than 100 x 10¹⁵ g.

In the following computations, the rather low value of 60 x 10¹⁵ g C has been used. The reason for this low choice will be made clear when we discuss the results (Section 15.7.4).

In the model, man-made transfer could be represented simply by subtracting an amount of carbon from the biota reservoir each year, and adding the same amount to the atmosphere. However, if we want to allow the biota to increase in response to a CO₂-richer atmosphere, its net reduction over the period will be less than 60 x 10¹⁵ g C. Parallel to forest clearing and fuel wood burning, accumulation of organic carbon may take place in regions not affected by man. It is instructive to consider these two effects separately. We have, therefore, introduced a second pair of biota-soil-boxes, and we assume the man-made input to be from the biota reservoir of this pair. This compartment will thus have contained 60 x 10¹⁵ g C in 1860, and have shrunk essentially to zero in 1970. Over the same period, there may have been growth in the reservoir representing the major part of the biota.

15.4 SOME PROPERTIES OF THE MODEL OCEAN

The degree to which the ocean below the mixed layer functions as a sink for CO₂ depends mainly on exchange processes with the surface water. The ratio of volume to rate of exchange (the so-called turnover time) is one parameter of interest, but it does not describe completely how the excess carbon in this part of the sea increases with time. For example, in Figure 15.3 four different models for water exchange between the surface layer and the intermediate and deep layers are indicated schematically. In case (i), the entire volume below the surface layer is assumed to be one rapidly mixed reservoir. The amount of carbon leaving it per unit time is proportional to its total carbon content. In cases (ii) and (iii), the intermediate and deep water are treated separately as well-mixed reservoirs. These two cases differ from each other in the assumptions made regarding the pattern of water circulation between deep water, intermediate water, and surface water. The circulation is indicated by arrows for both cases. In case (iv), the replenishing of water in the deep and intermediate ocean is assumed to be by injection of surface water at the ocean bottom. The water is assumed to move upwards without vertical mixing. If the volumes W_i and W_d, and the fluxes F_i and F_d remain the same throughout the four cases, the turnover time _o will also be the same:

	Wi+Wd
o=		(15.4.1)
	Fi + Fd

Let us assume that an arbitrary passive tracer substance which initially is not present in the ocean is added, at a constant rate, to the water entering these two reservoirs. The amount of tracer substance in W_i and W_d will then increase at a different rate in each of the four cases. The increase with time is shown in Figure 15.4. In case (i) the curve can be shown to be of an exponential form:

M(t) = M_o(1 ^_ e^-t/_o)

In case (ii), part of the influx is to the reservoir W_i, which has a more rapid turnover rate. This part will have a shorter transit time than in case (i). This is to some extent counterbalanced by a longer characteristic transit time for matter entering reservoir W_d, and the resulting time-dependent increase of tracer concentration in the reservoir W_i + W_d differs only slightly from case (i). In case (iii) we assume all influx to be to the smaller reservoir W_i. The deep-ocean reservoir W_d is only in contact with W_i and has no direct exchange with the surface water. In response to a tracer input to W_i, the concentration will, therefore, increase more rapidly in W_i than in the ocean, and there will thus be a rather rapid build-up of a return flow to the surface water. This is a different situation from case (i), where we have no possibility of describing any vertical concentration differences in the water below the surface layer. Because of the larger return flux, the increase of tracer in W_i + W_d is considerably slower in case (iii) than in case (i). Case (iv) represents a situation where no material leaves the reservoir W_i + W_d before residing in it time _o. Since we assume the reservoirs to contain no tracer substance before the influx begins, and since the influx rate is constant, the content of tracer will initially increase linearly in this case. At a time _o, an outflow will begin at the same rate as the influx, and the concentration will remain constant thereafter. It can be demonstrated that the total amount of tracer in W_i + W_d will also tend toward the same equilibrium concentration for any of the other three circulation patterns, but case (iv) represents the fastest possible way to reach it.

Figure 15.4 Increase with time of content of an arbitrary passive tracer in the intermediate and deep water volume, corresponding to the circulations of Figure 15.3. For all cases, W_i = 303 x 10¹⁵ m³, W_d= 975 x 10¹⁵ m³, Fd = 700 x 10¹² m³ /year, F_d = 736 x 10¹² m³ /year, giving o = 890 years

We have thus seen that the assumption of a well-mixed ocean below the surface layer leads to a prediction of its future uptake rate, but this prediction is not the only one possible. The uptake may proceed at a considerably slower or more rapid rate. Based on present knowledge about the oceanic vertical circulation it is not possible to determine the precise form of the graph in Figure 15.4 that corresponds to the real ocean. Considering that deep water exchanges with surface water by convection, and with intermediate water by vertical mixing, it seems plausible that the response of the ocean would fall somewhere between cases (i) and (iii).

It is of interest that, even though the four graphs differ significantly from one another in the long term, their initial developments over the first few tenths of a turnover time are very similar. The turnover time of the oceans is in the area of one thousand years, and the part of the industrial era so far elapsed is about as long as the interval along the time axis, over which the developments are nearly the same. This may explain why the customary assumption of a well-mixed deep sea can make model calculations fit with observations of the CO₂ increase until now. It also demonstrates that it is by no means certain that this will remain a satisfactory assumption when predicting the future behaviour of the ocean as a sink for excess CO₂.

The transit-time distribution functions (T) suitably summarize the oceanic properties relevant to the present problem. This signifies the statistical distribution function which denotes the fraction of the inflow to the water below the mixed layer that returns per unit time after a time T. It can be shown (Eriksson, 1961) that the customary assumption that a reservoir is well mixed has the consequence that

	1
()=		e/o	(15.4.2)
	o

where _o is the average transit time for matter in the reservoir. This corresponds to case (i) of Figure 15.3

Since the present model offers a better description of oceanic circulation, we can now examine the assumption that (_o) is exponential. The greater the depth to which a water parcel sinks, the longer it takes to leave the deep sea. The transit-time distribution function _d(T) for the deep sea (the volume below the surface and intermediate layers) is thus a transform of the distribution function f(z), depicting the statistical distribution of the penetration depth for convection. By applying this transform, we can compute which functions  _d(T) result as a consequence of certain simple assumptions about f(z). We will, therefore, devote the next paragraph to a formal derivation of this transform. The first question is: at what depth must a water parcel be injected in the deep sea to leave it after a time t? Denote this depth z(t)..

Since the upward water transport through level z is

(15.4.3)

and since the upward velocity w at level z is

 where A(z) is the horizontal area of the ocean at the depth z, we can write the equation of continuity for a rising water parcel:

(15.4.5)

 Equation (15.4.5) is a first-order differential equation, involving z and its first derivative dz/dt. The function z(t) must satisfy the differential equation (15.4.5), together with the initial condition z = 0 for t = 0. The function z(t) is thereby determined. The fraction of the total inflow that has a transit-time shorter than must then be equal to the fraction that penetrates to a depth of, at most z()

(15.4.6)

where P is total penetrating inflow, 

P =		D	f(z)dz   and
		o

(15.4.7)

Differentiation gives

(15.4.8)

or with (15.4.5):

(15.4.9)

For simple functions f(z) and A(z), the systems (15.4.5) and (15.4.9) can be solved explicitly. For example, if f is proportional to A, so that f(z) = C • A(z) for all z, equation (15.4.9) gives

Since (d/d)_d() = _d(), differentiation of the above expression gives 

that is, if the rate of inflow is proportional to the horizontal ocean area at each depth, the deep sea has the same transit-time distribution as a well-mixed reservoir. 

If f(z) is not proportional to A(z) we must generally solve equations (15.4.5) and (15.4.8) by numerical methods rather than by deducing an analytical expression.

From the above, it follows that the function f(z) is a key to the description of the transport processes in the deep sea. Radiocarbon measurements provide a useful tool for obtaining information about f(z).

For the sake of simplicity and as an illustration, let us assume that there are no vertical gradients in total inorganic carbon concentration. This is the case if sedimentation is negligible, which is probably a reasonably justified assumption below 1000 m. The radiocarbon balance for a small interval from z to z + z is then

f(z) • z • Rc + F(z + Az)R (z + z) = F(z)R (z) + A(z) • z • Rz •

(15.4.10)

 where = radiocarbon decay constant, R_c = ratio ¹⁴ C/C in the cold surface water, and R(z) = ratio ¹⁴C/C at depth z. F(z) and A (z) are the same as before.

As z 0 we obtain, with (z) = R(z)/R_c 

	d
f(z) +		(F) A(z)(z)=0
	dz

Since  

we have

	d
f(z) • (1 (z) + F(z)		A(z)(z)=0	(15.4.11)
	dz

which provides a way of determining f(z), given (z).

To determine the average profile of (z), a series of measurements from the Atlantic and Pacific Oceans was considered (Östlund et al., 1976; Bien et al., 1960). There is considerable scatter in the observations and it is of interest to note the higher apparent ¹⁴ C ages in samples from the Pacific Ocean compared with those from similar depths in the Atlantic. This difference indicates inhomogeneity in the vertical circulation of the oceans. It is possible that, by averaging samples from different oceans of the world, features of the circulation could be eliminated which might influence the picture of the deep ocean as a sink for CO₂ .

If we assume (z) to be constant below 1000 m we have (z) = o and d/dz = 0 in equation (15.4.11):

f(z) • (1- o) - A(z)o = 0

or

As demonstrated above, proportionality between f and A implies that the deep sea has the characteristics of a well-mixed reservoir. In this case the time constant is 

	1o
o=
	o

Assuming the apparent age profile to be constant with depth below a certain level is thus equivalent to treating this part of the water volume as a well-mixed reservoir. 

To the extent that (z) is not constant with z, it seems reasonable from the observations to prescribe d/dz < 0 for all z.  Since F(z) > 0, equation (15.4.11) gives

	A(z)(z)
f(z)>
	1 (z)

for all z.

Since is a decreasing function of z, 

(z)		(D)
	>
1 (z)		1 (D)

so that

The total penetrating flux P must then be

If the apparent radiocarbon age at the ocean bottom is taken to be 1500 years (which is probably an overestimate), and if the volume of the ocean below 1000 m is 0.98 x 10¹⁸ m³, we have the result

P > 580 x 10¹² m³ /year.

For an apparent age at the bottom of 1200 years we obtain 

P > 760 x 10¹² m³ /year.

Thus we have a lower limit for the rate of water flux through the deep sea.

The reasoning leading to this estimate is based on the assumption that the biological influxes of carbon and radiocarbon to the deep sea are negligible compared to the advective mechanisms. Results (Section 15.7.2) show that sedimentation accounts for less than 10% of the total carbon transport out of the surface layer, and that roughly half of this must dissolve in the intermediate water above 1000 m below sea level.

Even though sedimentation may play a minor role in the carbon budget of the deep ocean, its importance for the radiocarbon balance could be greater. The successive enrichment in ¹⁴C derived from the dissolution of `young' organic particles settling from above could reduce the apparent radiocarbon age below the value it would have if only advection and radioactive decay controlled the budget, as was assumed in the derivation of equation (15.4.11).

A radiocarbon-balance equation corresponding to equation (15.4.10) can be derived for a case where biological transport is included. This is more complex than adding an extra term to equation (15.4.10), since a vertical gradient in total carbon concentration must also be introduced.

Using this type of equation, the radiocarbon age profile required for steady state can be computed, given a distribution function f(z) and the biological flux. Having determined the statistical distribution function f(z) from equation (15.4.11), we introduced a reasonable biological flux (see Section 15.7.2) and computed the radiocarbon age profile. For the ocean below 1000 m, the resulting ages agreed, to within a few years, with the original ¹⁴C profile. We therefore conclude that biological transport has only a slight effect on the average radiocarbon profile in the ocean, and that the limits arrived at for P are valid.

15.5 A COMPUTER VERSION OF THE MODEL

15.5.1 The Model Equations

In order to obtain a set of equations that can be the basis for a computer program we shall now make some further simplifications of the picture of the ocean and specify numerical values for the parameters involved. Let the ocean consist of twelve well-mixed boxes as shown in the central part of Figure 15.5. Let the surface water have an area of 360 x 10¹² m² and a depth of 75 m. The warm surface water covers 240 x 10¹² m² and the cold surface water covers the remaining third. Below this layer there are ten boxes of successively decreasing horizontal area, representing the intermediate and deep water. The intermediate interval from 75 to 1000 m is divided equally between boxes 1 and 2, and the eight boxes below are given a depth of 500 m each, so that the bottom of the model ocean is at a depth of 5000 m. The horizontal area of each box is determined with the aid of the hypsographic curve. We shall use index i = 1, . . . , 10 for quantities referring to the ten intermediate and deep water boxes, c for cold surface water, w for warm surface water, and a for the atmosphere. The circulation systems mentioned above are represented by fluxes of water W_ij between the boxes. The rate of sedimentation of carbon out of CSW is denoted B_i, and out of WSW, B_w The corresponding rate of decomposition in box i is denoted by B_i, i = 1, . . . , 10.

The land biota and soil organic matter are represented by two reservoirs, N_b1 and N_s1 , respectively.

The complete model is then as shown in Figure 15.5, N_i denotes the amount of carbon in ocean reservoir i, and F_ij the flux from reservoir i to reservoir j. fos denotes the input of carbon to the atmosphere from fossil fuel burning and bio the man-made transfer of carbon from land biota to the atmosphere. The system in Figure 15.5 is then governed by the following set of equations:

	(15.5.1)
	(15.5.2)
	(15.5.3)
	(15.5.4)
	(15.5.5)
	(15.5.6)
	(15.5.7)
	(15.5.8)
	(15.5.9)
	(15.5.10)

Figure 15.5 Model of the global carbon cycle, used for numerical computations

15.5.2 Functional Relationships between Contents and Fluxes

The flux rates are functions of the box contents N_i , and we shall use the following expressions:

A. F_ac and F_aw

The atmosphere can be regarded as a well-mixed reservoir, which implies that the flux of carbon into the ocean is proportional to N_a. If the residence time for CO₂ in the atmosphere prior to dissolution in the ocean is T_a, we have F_ac +F_aw =Na/Ta. Assuming the flux from the atmosphere to an ocean region to be proportional to its area, and letting the boundary between `cold' and `warm' surface water be at 40^° N and 40^° S, it seems reasonable to take

(15.5.11)

B. F_ca and F_wa

These fluxes vary with N_c and N_w in a more complex way. We can put

where P_c denotes the partial pressure of dissolved CO₂ in the cold surface water, and

  We need to determine the proportionality constants k_ca and k_wa and express the variations of P_c and P_w with N_c  and N_w.

Table 15.1 Parameters relating to the atmosphere

Determination of k_ca and k_wa. We shall assume that the CO₂ flux from an arbitrary region A of the ocean to the atmosphere is the product of the area of A and the average CO₂ pressure in the region, times a constant K independent of the area chosen. Thus

If we take A =A_o = area of the entire ocean, and assume that steady state prevails at N_a =N_ao and P_a = P_o, we have

By definition we obtain

Functional dependence of P_i on  N_i, i = c, w. We can put

[H+] [HCO3 ]		[H+] [CO23]
	= K1		K2	(15.5.18)
[CO2]		[HCO3 ]

	[ H+ ]2
[CO2] =		C	(15.5.19)
	EC

                        =[H⁺] ² + [H⁺] K₁ + K₁ K₂

	Nc
C= [CO2] + [HCO3 ] + [HCO23 ] =		(15.5.20)
	Wc

where W_c is the volume of the cold surface water. When C varies, [H⁺] will change also, and following Keeling (1973a) we assume the variation to be such that A remains constant, where A denotes the carbonate-borate alkalinity, defined by 

[H+] +A = [OH] + [H2BO3] + [HCO3 ] +2 [CO23]

(15.5.21)

Using the equilibrium conditions

together with an expression for the sum of the dissociated and undissociated borate:

[H3 BO3] + [H2 BO3] = B

(15.5.23)

 equation (15.5.22) can be written

	Kw		KBB		[H+] K1 + 2K1K2		Nc
[H+] +A =		+		+				(15.5.24)
	[H+]		[H+] + KB				Wc

  The present computer program contains a subroutine that solves this fifth-degree equation for [H⁺] , given N_c. The partial pressure P_c is then computed via (15.5.20), (15.5.19), and (15.5.17). In the same way, P_w is determined as a function of N_w. 

It should be mentioned here that this model of the chemistry of sea-water has recently been brought into question (Rebello and Wagener, 1977). If it is not valid, the time-dependent increase of the CO₂ pressure in the surface layer of the ocean deduced from it may be too rapid. In Section 15.7.5, we perform some computations to explore the sensitivity of the model to this type of uncertainty.

C. Advective fluxes

All advective transports, F_ij, where i and j are reservoirs within the sea, are assumed to be proportional to the amount of carbon in the reservoirs where the fluxes originate:

where

 with W_i denoting the volume of water in box i, and W_ij denoting the water flux, expressed as volume per unit time, from box i to box j.

Determination of W_ij. Having determined the function f(z) from consideration of radiocarbon data (see equation (15.4.11)), we obtain for the penetrative convection 

15.5.27

Table 15.2 Parameters relating to the surface water. Index i is c for CSW, w for WSW. 

			Value
Symbol	Definition		for CSW	for WSW	Comments
Wi	volume of water		9 x 1015 m3	18 x 1015 m3	equation (5.27)
Ci	concentration of dissolved inorganic		2.139 mol/m3	2.125 mol/m3	value for CSW taken from Keeling and
		carbon				Bolin (1968)
					value for WSW computed in step 7 (see
						text)
Ni	total amount of dissolved inorganic		231 x 1015 g	459 x 1015 g	from Wi and Ci
		carbon
K0i	CO2 solubility		4.78 x 10-2	2.87 x 10-2	*
K1i	first dissociation constant of		7.6 x 10-7	10.7 x 10-7	*
		H2CO3
K2i	second dissociation constant of		5.5 x 10-10	8.9 x 10-10	*
		H2CO3
[CO2] i	concentration of dissolved CO2		13.9 mmol/m3	8.3 mmol/m3	computed in step 8 (see text)
Pi	partial pressure of CO2 at the surface		9.2 µatm	285 µatm	estimated value for 1860
k	see equation (15.5.14)		1.18 g/(µatm m2 year)		computed in step 2 (see text)
kia	see equation (15.5.12), (15.5.13)		0.14	0.28	in units of 1015 g C/(year atm)
					computed in step 3 (see text)
Wwc	water transport from WSW to CSW			2640 x 1012 m3/year	computed in step 1 (see text)
Wwc	water transport from CSW to WSW		1320 x 1012 m3/		computed in step 1 (see text)
			year
kwc	coefficient of transfer from WSW			0.08/year	computed in step 1 (see text)
		to CSW
kcw	coefficient of transfer from CSW to		0.08/year		computed in step 1 (see text)
		WSW
Fwc	carbon transport from WSW to CSW			37 x 1015 g C/year	computed in step 7 (see text)
Fcw	carbon transport from CSW to WSW		18 x 1015 g C/year		computed in step 7.(see text)
Bi	gravitational sinking of carbon		0.6 x 1015 g C/year	1.8 x 1015 g C/year	computed in step 4 (see text)
Kw	dissociation constant of water		10-14	10-14	*
KB	dissociation constant of borate acid		2 x 10-9	2 x 10-9	*
B	total borate concentration		0.4 mol/m3	0.4 mol/m3	*
Value within the range of values are given by Keeling (1973a).

For reasons of continuity we have 

	10
Wi+1, i =		Wcj, i = 2, . . . , 9	(15.5.28)
	j=i+1

 The volumes W_i are computed from the equations

Wi = Ai (zi+1 - zi), i = 1, . . . , 10

(15.5.29)

 where z_i denotes the depth of the top of box i, counted from the top of box 1 (and z_{1 1} = 4925 m).

The values of W_c and W_c2 are probably more significant for the short- and medium-term response of the ocean to rising CO₂ levels in the atmosphere than are W_ci for i = 3, . . . , 10. One of the objectives of the present computations was to assess reasonable limits for W_c1 and W_c2. We shall, therefore, return in Section 15.7.3 to the question of determining W_c1 and W_c2, and only point out here that the condition

                        W_ci = W_ic

has been assumed to be valid for i = 1,2. For reasons of continuity we have then, similarly to equation (15.2.6),

 The terms W_cw and W_wc depict the exchange of surface water across 40^° latitude. With our assumptions on the circulation pattern, mass continuity requires that W_wc dominates W_cw by the same amount as is transported into WSW from below:

Estimates in the oceanographic literature (e.g. Defant, 1961) indicate that the water transport of a major ocean current, such as the Gulf Stream or the Kuroshio Current, amounts to several tens of millions of cubic metres per second. Adding to these currents their counterparts in the southern hemisphere, and the possible effect of smaller-scale motion systems, the total water transport across 40^° latitude appears to be of the order of magnitude of a few hundred million cubic metres per second.

With the present assumptions on the volume of the surface water reservoir, the exchange of 100 x 10⁶ m³ /s implies that a little more than 10% of the surface water is transferred from one of the two reservoirs to the other during a one-year period.

The effect of the parameters W_cw and W_wc on the computed distribution of excess CO₂ was tested. It was found that when W_cw takes on very high values, the resulting atmospheric CO₂ content in 1970 tends towards a value about 1 ppm below the result obtained when W_cw = 0. Values up to 500 x 10⁶ m³ /s were used in this study.

Table 15.3 Parameters relating to the intermediate and deep ocean. Index i runs from 1 through 10

Symbol	Definition		Value		Comments
Ai	horizontal section area of ocean box i		see Table 15.4		from the hypsographic curve
zi	distance from top of ocean box 1 to top		see Table 15.4		see text
		of ocean box i
Wi	volume of ocean box i		see Table 15.4		from equation (15.5.25)
Ci	carbon concentration in ocean box i		see Table 15.4		computed in step 7 (see text)
Wij	water transport from box i to box j		see Table 15.4		computed in step 1 (see text)
kij	see equation (15.5.26)		see Table 15.4		computed in step 1 (see text)
Fij	advective carbon transport from box i		variable between the		computed in step 7 (see text,)
		to box j		experiments
Ds	characteristic length for sinking of organic		2000 m		computed in step 4 (see text)
		material before dissolution
Gi	gravitational flux of particulate carbon		see Table 15.4		computed in step 4 (see text)
		into box i
Bi	dissolution of particulate carbon in		see Table 15.4		computed in step 4 (see text)
		reservoir i

Table 15.4 Parameters relating to the ten boxes representing the intermediate (1-2) and deep (3-10) ocean. The amounts of carbon initially in the boxes were varied slightly between the experiments. The values given by W_ij and k_ij refer to a `young ocean' case (see Section 15.7.1)

i	Ai	zi	Wi	ci	Wi, i-1	ki, i - 1	Wci	kci	Gi	Bi
unit	1012 m2	m	1015 m3	mol/m3	1012 m3/	per 1000	1012 m3/	per 1000	Tmol/	Tmol/
					year	years	year	years	year	year
1	337	0	155	2.28	1323	8.5			200	50
2	318	460	148	2.25	1323	8.9			150	34
3	310	925	155	2.23	1323	8.5	283	31	116	29
4	300	1425	150	2.22	1040	6.9	217	24	87	22
5	290	1925	145	2.22	823	5.7	198	22	66	16
6	280	2425	140	2.22	625	4.5	167	19	49	15
7	250	2925	125	2.21	458	3.7	149	17	34	8.7
8	240	3425	120	2.22	310	2.6	143	16	26	12
9	160	3925	80	2.22	167	2.1	95	11	13	5.6
10	120	4425	60	2.25	71	1.2	71	7.9	7.8	7.8

Since this is a minor variation compared to those arising when varying other parameters, as discussed in Section 15.7, it was decided to set

and to refrain from investigating the effects of varying these ratios throughout the rest of this study.

Numerical values for W_i and W_ij are summarized in Table 15.4. 

D. Sedimentation of organic matter

To parameterize the sedimentation in a convenient way, we assume the flux below a unit area of the ocean to decrease exponentially with depth. Denoting the influx of organic matter into box i by G_i and the total outflow of organic material from the surface water by B_T, we have

G1= BT	(15.5.34)
	Ai
Gi=		Gi-1• e-(zi-zi-1)/Ds,  i =2,...,10	(15.5.35)
	Ai-1

The factor A_ij/A_j-1 expresses the fact that each box has a smaller horizontal area than the box above it. The quantity D_s denotes a characteristic sedimentation time for an organic particle before it is decomposed. We have assumed D_s to be constant with depth. The rate of decomposition in each box is

                          Bi⁼G_iG_i+1, i=1,...,9

 It should be mentioned that equation (15.5.36) is valid for a stationary state without accumulation of organic material on the ocean bottom. The equation is an expression of conservation of mass of organic material in the water and on the bottom within a reservoir. When B_i and G_i are constant in time, and the net accumulation is zero, the total mass of organic carbon remains constant, and equation (15.5.36) is valid. However, the assumption of no accumulation does not imply that there is immediate dissolution of all organic particles setting on to the ocean bottom. If the influx G_i were enhanced, a net accumulation of material could, therefore, well be the result, and equation (15.5.36) would have to be modified to read

                            B_i +A_i = G_i - G_i+1

where A_i denotes the accumulation of organic carbon on the ocean bottom within reservoir i.

We denote by B_c and B_w the rate of outflow of organic material from CSW and WSW respectively. Obviously

        B_c +B_w = B_T 

so that we have

        B_c = B_T

and

        B_w = (1 )B_T

for some between 0 and 1.

The biological productivity in the sea is very variable, as discussed by de Vooys (Chapter 10, this volume) and Mopper and Degens (Chapter 11, this volume). It is well known that the productivity is largest in areas where deep water wells up and causes a rich supply of nutrients. However this fact gives no immediate information about any systematic variations in productivity between CSW and WSW, since areas of upwelling are present north and south of 40^° latitude. If the average productivity is the same in CSW as in WSW, the ratio B_c/B_w should be equal to the ratio of the areas of these, or, with our assumptions:

or =0.33.

Computer experiments indicate, however, that may take any value between 0 and 1 without affecting the results of the computations that will be accounted for in Section 15.7. For these experiments, the value = 0.25 was used, based on the assumption that the warm surface waters are 50% more productive per unit area than are the cold waters. Although this may prove to be an incorrect assumption, the error is without consequences in the present context.

The numbers B_c and B_w, and the profile B_i are thus completely described by the two numbers B_T and D_s, the determination of which we shall return to in Section 15.7.2.

E. F_{a, b i,} F_{b i a}, F_{b i, si,} F_{si„ a}

The gross primary production F_a,bi of vegetation system N_bi is assumed to vary with the CO₂ content of the atmosphere and also with the size of the biota pool itself, according to the formula

(15.5.37)

where the `growth parameter' is a measure of the ability of the vegetation system to respond to increased atmospheric CO₂ levels with an increase in gross assimilation. The mathematical formulation of equation (15.5.37) is based on observations under controlled conditions. Its applicability to vegetation under natural conditions has been discussed by Keeling (1973 a).

It is advantageous to introduce the two fluxes F_bi,a and F_a,bi and not only the net of these. This separation enables us to set the sum of the rate of respiration by the autotrophs, the rate of decomposition of short-lived detritus and the rates of other rapid mechanisms for a return transfer to the atmosphere proportional to the amount of living matter at every time:

 where 1/T_bi,a denotes the fraction of the total organic carbon undergoing one of these processes annually.

The assimilated carbon not reconverted rapidly from living matter to CO₂ is sooner or later transferred into long-lived detritus. We assume that the rate of this transfer is proportional to the amount of living matter, with a characteristic time T_bi,si:

The transit time for different carbon atoms in the reservoir of soil organic carbon exhibits a great degree of scatter. Transit times of approximately 1000 years occur. The dynamic characteristcs of this reservoir would therefore be most completely described by a transit time distribution function () of the type discussed in Section 15.4 for the below-surface ocean. However, the general effect of the soil reservoir as a delay before oxidation can probably be reasonably well described by applying a first-order relation, involving a turnover time of appropriate length:

where T_si, a is the turnover time for organic carbon in the soil. At equilibrium we must have

where index 0 denotes a steady-state value.

The length of time during which a carbon atom remains withdrawn from the atmosphere is obviously largely dependent on whether or not it enters the soil reservoir. The ratio between the two fluxes F_bi,a and F_bi,si is, therefore, a parameter of interest, since, given the gross assimilation, this ratio determines the rate of transfer into the soil reservoir.

For the respiration of autotrophs, Whittaker and Likens (1973) give values ranging from 20% to 75% of the gross assimilation. There is, of course, considerable variation, for example, between the rapid turnover in a tropical rain forest, where little formation of long-lived detritus takes place, and a boreal forest, where microbial decomposition is sometimes very slow. The ratio of the amount of long-lived biota to short-lived biota is also variable from one ecosystem to the other, which also affects the ratio F_bi,a/F_bi,si, as we have defined it. This kind of difference will be modelled in more detail in a later study. For the present computations, we have generally assumed an equal partitioning of the gross assimilation between the two fluxes F_bi,a and F_bi,si, that is, we have taken

 The sensitivity of the computed results to assumptions on the ratio T_bi,a/T_bi,si need of course be explored by numerical experiments. We shall return to this point in a later study.

In summary, we have assumed the gross assimilation to be twice the net primary production in a year. The gross assimilation is divided equally between a return transfer to the atmosphere and a transfer to dead organic material.

F Human impact: _fos and _bio

We have used values estimated by Keeling (1973b) for the production of CO₂ from fuel burning and other activities involving a release of fossil carbon to the atmosphere. His estimates are for the period 18601969. The computations in the present study were not pursued beyond the year 1970, and there has not been a need for data from this later period. The accumulated release of fossil carbon since 1970 is estimated to be 114 x 10¹⁵ g C. The release of CO₂ of biological origin has been taken to be 60 x 10¹⁵ g C in the period 1860-1975, in accordance with the deductions given in Sections 15.3 and 15.7.4. Estimates of the production of biological CO₂ for individual years were obtained by assuming proportionality to the production of fossil CO₂ in the same year. This is admittedly a very crude assumption. Although it is true that man's large-scale impact on the terrestrial biota in the last century has been made possible, to a large extent, by the growing access to energy, it is a dubious conclusion that the variations in biota-affecting activities should have followed the same pattern as the fluctuations in fossil fuel consumption on a year-to-year time scale. Recently, Stuiver (1978) has put forward results indicating a maximum in annual production of biological CO₂ around 1900, followed by a decrease over the period until 1950. With our assumption of proportionality between fossil fuel consumption and release of biological carbon each year, the annual transfer of biological carbon in 1970 is 2 x 10¹⁵ g C/year. As  comparison Hampicke (Chapter 7, this volume) concludes that the correct value most likely is in the range 1.54.5 x 10¹⁵ g C/year.

Assume that Stuiver's results are valid regarding the time of maximal biological CO₂ production, and that the accumulated release of fossil and biological carbon is correct as we have taken it. Annual releases were then, in reality, larger in the early part of the period than we have assumed here, while for later years the present model underestimates the total release by a corresponding amount. A computation of the atmospheric CO₂ content in 1970 using our assumption of proportionality is then likely to be an overestimate since, in reality, the releases of excess carbon have had more time to penetrate into the ocean and into the biota than we have assumed.

Using a better assumption than proportionality would thus probably yield lower values for the CO₂ content in 1970. On the other hand, Stuiver's results apply only to the period until 1950, and therefore do not fully account for the rapid acceleration of tropical forest clearing and use of fuel wood that has taken place since then. Including these releases would, therefore, increase the resulting values, and perhaps even overcompensate for the reduction discussed.

In a later study, the present model will be expanded to include the cycle of  ¹³C separately from the ¹²C cycle. Observations of isotopic ratios in tree rings can then be interpreted to give information about the annual releases of biological carbon, and _bio can be adjusted accordingly.

However, although the assumption of proportionality between fossil and biological releases is crude, it permits us to see some general effects of adding a biological source to the fossil source in model computations.

15.5.3 Description of a Steady State

Before an experimental integration was commenced, an internally consistent picture of the system at steady state was deduced. Since some dynamic parameters were varied between the experiments, the picture of the steady state has been varied slightly from one experiment to the other. For example, in a case where the oceanic circulation was assumed to be rapid, the carbon gradient between surface water and deep water at steady state had to be taken somewhat smaller than the same gradient in an experiment with a very slow oceanic circulation.

The picture of the steady state was constructed step by step with the aid of the functional relationships given above. Typical values are summarized in Tables 15.115.5. The procedure was:

Step 1.    From radiocarbon data, values for W_ci were chosen for i = 3, . . . , 10, by equations (15.4.11) and (15.5.27). Transfer coefficients k_ij were computed from equation (15.5.26). W_cw and W_wc were determined from (15.5.31)(15.5.33).

Step 2.    Values for N_ao and T_a were chosen. k was determined from equation (15.5.15).

Step 3.    F_ac and F_aw were determined from equation (15.5.11); k_ca and k_wa were determined from equations (15.5.12) and (15.5.13) and the assumptions 
F_ac ⁼F_ca, F_aw ⁼F_wa ^.

Step 4.    Values for W_{e 1}, W_{c 2} , and B_i, i = 1, . . . , 10, were selected. The choice was preceded by an analysis of the range of reasonable combinations of these parameters using the vertical oxygen profile for comparison (see Section 15.7.2).

Step 6.    k_{c 1} , k_{c 2}, k_{2 c}, and k_{c 2} were determined from equation (15.5.26).

Step 8.    Values for K_{o i}, K_{1 i} , and K_{2 i} (see equation (15.5.18)) were chosen from Keeling (1973a). The initial value of [CO₂] _i was computed from equation (15.5.17) to give P_oi = 285 ppm. The initial value of [H⁺] _i was computed from equations (15.5.19)(15.5.20), and for A_i from (15.5.24), i = c, w.

Step 10.   A value for the growth factor (see equation (15.5.37)) was selected.

Step 12 .   k_bi was determined from equation (15.5.41).

15.6 THE RADIOCARBON EQUATIONS

All fluxes transporting carbon between the reservoirs also transport radiocarbon. The equations governing the rate of change of ¹⁴C in the system are therefore similar to those for inactive carbon, although additional terms enter for the radioactive decay and the production in the atmosphere from cosmic radiation. If we denote by N*_i the amount of ¹⁴C in reservoir i and by F_ij the rate of transport of ¹⁴C from reservoir i to reservoir j, we obtain the equation system

dNa*		2
	=Q*Fa*cFa*w+Fc*a+Fw*a+		(Fb*i,aFa*,bi,+Fs*i,a)+b*io Na*	(15.6.1)
dt		i=1

dNw*
	=Fa*wFw*a+Fc*wFw*c+F1*wBw*Nw*	(15.6.2)
dt

	(15.6.3)
	(15.6.4)
	(15.6.5)
	(15.6.6)
	(15.6.7)
	(15.6.8)
	(15.6.9)
	(15.6.10)

The fluxes are functions of the N *_i in a similar way as the fluxes of inactive carbon are functions of N i. There are a few important cases where F_i^*_j is a function, not only of N^*_i , but also of N_i.

A.  F_a*_c and F_a*_w

If there were no differences between the carbon isotopes in regard to uptake of CO₂ by the ocean surface , the fluxes F_a*_c and F_ac would be proportional to the atmosphere content of ¹⁴CO₂ and CO₂ :

         
In reality, there is a preference for dissolution of the lighter CO₂ molecule ¹²CO₂ , and the flux F_a*_c is somewhat smaller than proportionality indicates:
         

(15.6.11)

where _as has a value of 0.972 (Keeling, 1973a).

(15.6.12)

B.  F_c^*_a  and F_w^*_a

(15.6.13)

Since F_ca varies non-linearly with N_c, there is no simplification of equation (15.6.13) similar to the last step of (15.6.11) or (15.6.12). The flux F_c*_a is independent of the amount of inactive CO₂ in the atmosphere, but the flux F_a*_c will be affected by an increase in inactive carbon in the surface water.

(15.6.14)

C. Adjective fluxes

(15.6.15)

D. Sedimentation of organic matter

_ba.

	(15.6.16)
	(15.6.17)
	(15.6.18)

(15.6.19)

(15.6.20)

	(15.6.21)
	(15.6.22)

(15.6.23)

(15.6.24)

F. Human impact

The carbon released from fossil fuels is free from ¹⁴ C, and burning thus provides no ¹⁴C to the atmosphere. However, as the amount of inactive carbon in the surface water increases, the fluxes F_i*_a, i = c,w are also affected, as seen from equations (15.6.13) and (15.6.14). For this reason, the dilution of atmospheric ¹⁴ C is less than it would have been if the buffering effect of sea-water had not been present. When the content of inactive carbon in the ocean surface layer increases, a redistribution of radiocarbon between the atmosphere and the surface ocean will occur, as follows from equations (15.6.13) and (15.6.14). With the aid of equations (15.6.1)(15.6.10), the change of ¹⁴C content in the carbon reservoirs can be studied parallel to the increase of total carbon content in response to man-made inputs to the atmosphere. The computed dilution of atmospheric ¹⁴ C can then be compared to the value that was observed until 1954, when bomb tests produced for the first time large amounts of atmospheric ¹⁴ C.

The transfer of carbon from the living biomass to the atmosphere, due to man's impact on the world's forests, is independent of isotopic differences:

(15.6.25)

G. Production and decay of ¹⁴C

The average lifetime of a ¹⁴ C atom is

-1 = 8267 years

(15.6.26)

which determines the value of in the system (15.6.1)(15.6.10). The production of ¹⁴C in the atmosphere by cosmic rays can, in the present model, be assigned an arbitrary value, since we have not specified the units for ¹⁴ C and all the equations in this section in principle only relate ratios F_ij /N_i to the corresponding ratios for non radiogenic carbon.

15.7 RESULTS AND DISCUSSION 

15.7.1 The Ocean below 1000 m

On average, the observations of ¹⁴ C activity indicate a continuous increase in apparent radiocarbon age with increasing depth in all oceans. Taking the radiocarbon activity of surface water to correspond to `zero age', we find an apparent age at the bottom of the Atlantic of around 800 years, while samples from the Pacific show charcteristically lower activities, with an average age of near 1500 years. This difference causes uncertainty as to how the radiocarbon profile for an average ocean, such as that in the model, should best be drawn. As formulated more rigorously in Section 15.4, a high value of the radiocarbon age indicates a slow rate of water circulation, while lower values indicate a more rapid turnover. It is to be expected, therefore, that the choice of radiocarbon profile for the model ocean has some influence regarding the rate at which the deep ocean acts as a sink for atmospheric excess carbon, and thus, also on the picture obtained of the atmosphere-biota-ocean distribution of the man-made carbon released until now. In order to see how sensitive the present model system is to this uncertainty, most of the following computations were performed both for an extremely high-age profile and for an extremely low-age profile. In Figure 15.6, these two profiles are given, together with their corresponding distribution functions f(z), as computed from a finite-difference form of equation (15.4.11). The total flux of water through the volume below 1000 m is 740 x 10<sup>12</sup> m<sup>3</sup> /year for the `old' case, and 1320 x 10¹² m³/year for the `young' case. Since the total volume of the cold surface water is 9000 x 10¹² m³, the two situations imply that 8% and 15%, respectively, of the water in this part of the ocean take part in deep-water formation during a one-year period. It will be shown below that there is only a slight difference between the two cases in the resulting partitioning of excess carbon released until now. It can, therefore, be expected that the average rate of deep-water formation is a parameter of relatively minor importance for determining the capacity of the ocean as a sink for CO₂,. over time spans of one or two centuries. However, when we consider the eventual decline of excess CO₂ over a period of 1000 years or more, this parameter will exert a significant influence on the resulting picture.

Figure 15.6 Alternative forms of apparent ¹⁴ C age profiles below 1000 m (solid lines), and corresponding distribution functions f(z) (dashed lines), for the `young' and `old' cases. Units for f(z) are 10¹² m³ per year and per 100 m vertical extent

15.7.2 The Intermediate Water 

In Section 15.5, the turbulent exchange between the cold surface water and the intermediate water was modelled by the two fluxes, W_{c 1} and W_{c 2}, but no way to estimate these two was devised.

It is natural to assume that the intensity of exchange decreases with depth in the intermediate water, so that the inequality W_c1 > W_c2 holds. To give the simplest possible description of this feature, we assume the exchange rate per metre to decrease linearly with depth, becoming zero at 1000 m below sea level. Since box 1 and box 2 (see Figure 15.5) extend over equal vertical intervals, this assumption gives (as shown in Figure 15.7):

        W_c1 =3W_c2

Figure 15.7 A motivation for the choice W_{c 1} = 3W_{c 2}. The exchange rate, expressed as water volume per unit time and per metre along the vertical, is demonstrated. The amount of water exchanged with CSW in a one-year period, for a given depth interval, is proportional to the integral of the curve over that interval. When the curve is linear, and the two boxes 1 and 2 extend over equal vertical intervals, the area W_{c 1} is three times the area W_c2

so that if the total exchange rate W_c1 + W_c2 is denoted by T: 

 For the ocean below 1000 m, radiocarbon activity observations were used for information about the natural rate of replenishment, as discussed in Section 15.4. In principle, similar considerations could be used to determine the circulation system for the intermediate water. However, as tritium observations indicate, the radiocarbon activity in this layer has been affected by penetration of ¹⁴C from nuclear bomb tests. Conclusions regarding the average circulation rates in this part of the ocean, based on measurements of the radiocarbon activity here, may therefore be misleading. Instead, some other tracer compound should be employed for deriving information about T.

As mentioned in the introduction, the oxygen profile is of great interest in this context. The oxygen budget is a balance of advective as well as biological processes, and the equation of continuity for oxygen is, therefore, similar to that for carbon (equation (15.5.6)):

(15.7.2)

where k_ic, and k_ci are dependent on T for i = 1, 2. We shall make use of this equation to arrive at an estimate of the order of magnitude of T. The term B_i is the same as in equation (15.5.36). It enters equation (15.7.2) with a negative sign, because for each mole of inorganic carbon liberated by decomposition, one mole of oxygen is consumed. We recall from equations (15.5.34)(15.5.36) that the numbers B_i, i = 1, . . . , 10, are completely determined by the two parameters, B_T and Ds. At steady state, the left side of equation (15.7.2) is zero, and, given a set of values for T, B_T, and D_s, the resulting oxygen profile can be computed from (15.7.2). The result depends not only on T, but also on B_T and D_s. With the aid of (15.7.2) we can thus not only investigate the limits for T, but also the limits for possible combinations of T, B_T, and D_s.

Estimates of the oceanic net primary production vary within a wide range. De Vooys (Chapter 10, this volume) summarizes estimates ranging from 15 x 10¹⁵ to over 100 x 10¹⁵ g C/year. In his opinion, the best estimate is 43.5 x 10¹⁵ g C/year. However, the value of B_T in the present calculations has to be much smaller than these estimates. A large fraction of the particulate matter formed consists of easily degradable substances that are oxidized before leaving the photic zone. Only a small percentage of the primary production can be assumed to leave the photic zone, let alone out of the lower boundary of the surface water, at a depth of several tens of metres. It was, therefore, decided to choose the value 200 Tmol/year as an upper limit for values of B_T which were of interest to investigate. This corresponds to about 5% of the net primary production in the world ocean, as estimated by De Vooys. The variable D_s is a function of average size and lifetime of a sinking particle. These characteristics are dependent on chemical composition. Although most organic particulate material is rapidly decomposed, a small fraction consists of long-lived compounds. In the present model, only those particles sinking below 75 m will have an effect in transporting carbon from one reservoir to another. It is possible that the average lifetime for this kind of particle is long enough to reach the required D_s value of approximately a few kilometres, although the average settling distance for all particulate material in the ocean may perhaps be a few orders of magnitude smaller. A natural upper limit for D_s is the depth of the model ocean, 5000 m. As for a lower limit, distances shorter than 20 m were not investigated. The reason for omitting smaller values is that with D_s = 20 m, only around 2% of the material leaving the photic zone would sink 75 m and enter the intermediate water. To have an influx here large enough to have an appreciable effect on the oxygen budget, an unrealistic flux of particles from the photic zone would be required. We recall, however, that the assumption of an exponential distribution for the distances travelled by settling particles (as implied by equation (15.5.35)) is only intended as a first approximation. The possibility exists that D_s could be variable with depth. However, although there are some chemical differences between the surface water and the water in the main thermocline, it seems unlikely that these should be of such a kind as to initiate a sudden decomposition when particles sink into the intermediate water.

Within these ranges for B_T and D_s, several combinations of T, B_T, and D_s were tested for steady state by means of equation (15.7.2).

Wyrtki (1962) has summarized some measurements of the vertical oxygen profile from different parts of the ocean. Assuming these observations to be representative for the ocean as a whole, maximum and minimum values for the computed oxygen concentrations in each of the boxes 1, . . . , 10 were selected as shown in Table 15.6. A given combination of T, B_T, and D_s was considered `possible' if the computed profile fell within the limits for all ten boxes.

It is naturally very uncertain to draw bounds for the possible oxygen values from such a restricted set of data. Nevertheless, even if the absolute values of the limits . given in Table 15.6 may have to be adjusted, the accuracy is sufficient to decide at least on an upper limit for T. Since the general shape of the average oxygen profile hardly needs to be questioned, the following analysis is probably at least qualitatively correct.

For very large values of T, biological transport mechanisms become negligible compared with turbulent transport. The oxygen concentration in the intermediate water will then be almost equal to the concentration at the surface. This contradicts our norm, because the maximal value for box number 2 is smaller than the value assumed at the surface. Numerical experiments show that, in order to reduce the oxygen concentration in box 2 to within the norm, we must have (with B_T < 200 Tmol/year

        T 10 000 x 10¹² m³ /year

This is a very crude limit for the turbulent exchange. Since the volume of the cold surface water is taken to be 9000 x 10¹² m³, it implies that more than the volume of the cold surface water is involved in the turbulent exchange during a one-year period. Neverthless, combinations of B_T and D_s exist, for which T = 10 000 yields an oxygen profile in accordance with the norm of Table 15.6. For various values of  T between the extremes T = 0 and T = 10 000, the `possible' combinations of B_T and D_s can be illustrated in diagrammatical form. Figure 15.8 indicates how the range of possible combinations changes its area and position as T varies.

To understand the effects of variations in B_T and D_s we consider the case T = 1000x 10¹² m³ /year. Firstly we recall that B_T denotes the total inflow of organic material in the deep sea, while D_s is related to the vertical distribution of the oxidation activity. Increasing Ds implies a redistribution of the main region of decomposition towards greater depths.

For the low value D_s = 200 m, we find that almost 90% of the biocarbon is dissolved in box 1, and that resulting oxygen concentrations of the deeper volumes are too high to fall within the norm. Increasing B_T raises the dissolution rates in each of the boxes, thus lowering the oxygen concentration, but we need a value of 

Table 15.6 Lower and upper limits for dissolved oxygen concentrations, to consider acceptable as global averages for the intermediate (1-2) and deep (3-10) ocean boxes (see text). The average oxygen concentration in the surface water was estimated to be 0.2 mol/m³

Box	Oxygen concentration
number	mol/m3
1	0.0580.219
2	0.0710.174
3	0.0170.152
4	0.0440.148
5	0.0710.148
6	0.0930.152
7	0.1240.165
8	0.1240.183
9	0.1240.223
10	0.1240.223

Figure 15.8 Indications of `possible' combinations of B<sub>T</sub> and D<sub>s</sub>, for various assumptions about the oceanic circulation: `Old ocean' case: (a) T= 100; (b) T= 1000; (c) T= 10 000. `Young ocean' case: (d) T= 100; (e) T= 1000; (f) T= 10 000

B_T = 300 Tmol/year before any of the boxes 2 through 8 will acquire an oxygen concentration within the norm. As mentioned above, we require

B_T < 200 Tmol/year

We must increase D_s to at least 1200 m before we can make all computed oxygen concentrations fall below their maximal permissible values. This requires the high value B_T = 200 Tmol/year. For smaller values of B_T, D_s must be larger than 1200 m to avoid too great values for the oxygen concentration. D_s = 1200 m is thus the smallest possible value for this parameter. For B_T as low as 50 Tmol/year, we can only make all oxygen values fall below their upper limits if a value of D_s near 4000 m or more is assumed. In summary, we have here a condition of the form

D_s >D_min (B_T)

where D_min is a decreasing function of B_T.

As B_T increases, the range of possible D_s values widens. However, with an increasing input of biocarbon to the below-surface ocean, there is the risk that the computed oxygen concentrations fall below their accepted lower bounds.

Since an increase of D_s means a redistribution of the remineralization towards lower levels in the ocean, we must require D_s <D_max for a given B_T in order to avoid oxygen concentrations falling below their postulated minima in the lower parts of the ocean. Already with B_T as low as 50 Tmol/year, the oxygen concentration near the ocean bottom will fall below its minimal value if D_s = 2400 m or larger is used. The larger the input of biocarbon, the smaller is the largest permissible D_s before remineralization becomes too high in one of the deep-sea boxes. Thus, like D_min , D_max is also a decreasing function of B_T. Any `possible' value for B_T must, therefore, satisfy the condition

D_min(B_T) < D_max(B_T)

For the present value of T= 1000 x 10¹² m³ /year, and an `old' ocean circulation, this occurs in the interval

80 < B_T < 180 Tmol/year

with the corresponding D_s varying between 

1400 < D_s < 1800 m

The assumptions made about the deep ocean circulation are relevant here, as shown in Figure 15.8. If we shift to the young-ocean case, we increase the rate of inflow of surface water to the deeper boxes. This yields a more rapid replenishment of oxygen here, and, for a given B_T, the value of D_s permissible before exceeding D_max thus increases. The curve for D_min is also shifted upwards, and the effect on the region of possible B_TD_s combinations is thus a shift towards higher values of B_T as well as of D_s. The largest value for T that gives acceptable oxygen profiles is about 10 000 x 10¹² m³ /year. However, this maximal value is sensitive to the chosen limits for the oxygen profile in intermediate water.

It should be pointed out, therefore, that the limits chosen here probably admit an unnecessarily wide scatter for the global average oxygen profile. Limit values were obtained from consideration of profiles given in Wyrtki (1962). Oxygen concentrations were read off for depths corresponding to the contours of the boxes 1 to 10. For the intermediate water boxes, the highest and lowest values read off were used as limits. The extreme values thus obtained admit a wide scatter for the computed oxygen profiles, as shown in Table 15.6. It can be expected that considerably narrower margins could be drawn for a global average profile if more data was considered. It is a probable consequence that the upper bound for T can, then, also be considerably reduced.

To summarize, all values of T between 0 and 10 000 x 10¹² m³ /year for both a young and an old ocean yield reasonable combinations of B_T and D_s. The values occurring are

        80 < B_T < 200 Tmol/year 

        1200 < D_s < 2000 m

We are now in a position to estimate the ratio of the biological carbon transport into the deep ocean (below 1000 m) to the advective transport into it. Since the depth of the intermediate layer is roughly 1000 m, the rate of biological transport out of it is approximately

        B_T • e^-1000/D_s  Tmol/year

The advective transport of carbon is P • C_cTmol/year,where P is 740 x 10¹² m³ /year for the old case and 1320 x 10¹² m³ /year for the young case (see Section 15.7.1), and C_c is 2.139 mol/m^s (see Table 15.1). The ratio is thus 

Inspection of the graphs in Figure 15.8 reveals that this ratio never exceeds 5%. The maximal value occurs for the case B_T = 180, D_s = 1200, and an old deep-ocean circulation.

These results support the assumptions made in Section 15.4, where the effect of sedimentation on the radiocarbon age distribution in the deep ocean was neglected. Since the biological transfer is of minor importance in transporting excess carbon out of the surface layer, we shall fix B_T at 200 Tmol/year and D_s at 2000 m and refrain from varying them throughout the rest of the present study.

15.7.3 Experiments with an Anthropogenic Source

Although some of the very high values for T tested in the previous section may seem rather unlikely, we cannot, on the basis of the above arguments, exclude them from the range of possible values. Our next step was, therefore, to investigate the implications of this uncertainty for calculations of the response of the ocean to increasing levels of atmospheric CO₂.

The model was integrated stepwise from 1860 to 1969, and an amount of fossil and biological carbon was injected each year into the atmosphere, as described in Section 15.5.2. In a series of experiments, the turbulent exchange rate T (defined in equation (15.7.1)) was varied between 100 and 10 000 x 10¹² m³ /year, and the deep-ocean circulation was alternated between the `young' and `old' cases (see Section 15.7.1). The parameter for biota growth (see equation (15.5.37)) was adjusted to give a value of the atmospheric CO₂ content in 1970 near 320 ppm for T= 1000. For parameters related to terrestrial plants and soil organic carbon, values were chosen ad hoc from within the ranges of customary estimates, and were kept constant, since the only purpose of this experimental series was to illustrate the sensitivity of the model system to uncertainties relative to the oceanic circulation. The initial values used for the biota parameters and for the chemical constants regulating the oceanic uptake are given in Tables 15.1-15.5. Considering subsequent experiments, it should be pointed out that an initial value for the atmospheric CO₂ content of 285 ppm was used for this series.

The results are summarized in Table 15.7. The most striking feature of this table is the very large fraction computed as having been taken up by the biota and soil; in no case less than twice what has gone into the ocean. We shall comment on the  realism of this result in the next section, but first another observation from Table 15.7 will be made here. It is seen that the airborne fraction varies between 32% and 37%. Since the atmospheric CO₂ content in 1970 was 320 ppm in all experiments and since the assumptions made about the total man-made release of CO₂ in the period 1860-1970 were the same throughout the experiments, it follows that the resulting airborne fraction must increase linearly if we lower the assumed preindustrial atmospheric CO₂ content. A variation from 32% to 37% corresponds to a difference of about 5 ppm in the preindustrial content. However, the present uncertainty concerning the atmospheric CO₂ level before 1860 is several times larger than 5 ppm. The results in Table 15.7 thus suggest that variations in the assumed initial CO₂ content have a greater potential for affecting the distribution picture of excess carbon. Nevertheless, the variations arising as the assumptions regarding the oceanic circulation characteristics are altered should not be underemphasized. We see, for example, that the percentage computed to have been transferred into the deep and intermediate ocean varies from less than 7% to almost 20%

Table 15.7 Percentage distribution of excess carbon between the atmosphere, the surface water, the intermediate and deep water, and the biota, for various values of the intermediate water. Values are given for a `young ocean' and an `old ocean' circulation (see Section 15.7.1). Numbers in brackets refer to the `old ocean' case. The biota is assumed to be the `average biota' (see Table 15.5)

15.7.4 The Possibility of a Biota Growth. Reasonable Values for the Preindustrial CO₂ Level

The results obtained in the previous section give a general impression of the degree of uncertainty introduced by the lack of precise knowledge on oceanic circulation. It was indicated that this uncertainty, although it may not lack significant effects, is probably not as important as the possibility of an error in the assumed preindustrial CO₂ content in the atmosphere. We shall therefore look into the effect of commencing the time integration from a lower atmospheric CO₂  content. The accumulated airborne fraction must then obviously be larger, and its increase is achieved primarily by reducing the uptake capacity of the biota, i.e. by selecting a smaller .

As for the parameters regulating the uptake by the biota, we do not know what value to expect for . For the computations in Table 15.7, the value 0.25 was needed to give a correct value for the atmospheric CO₂ content in 1970. By selecting a smaller value, it is possible to decrease the amount of carbon taken up by the biota. However, this also has the consequence of yielding an atmospheric content in 1970 of considerably more then 320 ppm. It is possible that a more realistic partitioning could be obtained if a more rapid transfer from the living biota to the large reservoir of dead organic matter is assumed. Recently, Revelle and Munk (1977) have suggested a division of the terrestrial organic carbon between assimilating and non-assimilating material rather than between `living' and `dead', which may provide a realistic way to account for a larger uptake by the terrestrial organic carbon. However, if we do not wish to change the characteristics of either the ocean or the biota as they have been depicted in the present model, the most natural approach to take at this point seems to be to question the assumed preindustrial atmospheric CO₂ content of 285 ppm. If this value was in reality lower, a larger fraction of the man-made CO₂ would obviously still be in the atmosphere.

A series of experiments was performed, where the preindustrial CO₂ level, P_o, and the growth parameter, , were varied. The values for other biota and soil variables and for the oceanic variables were kept constant at the values mentioned in Section 15.7.3. The atmospheric CO₂ level in 1970 was computed and is illustrated in Figure 15.9. It was noted, regarding these computations, that the assumption of no biota growth ( = 0) requires P_o less than 250 ppm. The corresponding percentage distribution of excess carbon gives 29% for the ocean, and an airborne fraction of 71%. Increasing the value of P_o, the value of required for a CO₂ level of 320 ppm in 1970 increases as indicated by the solid line. In Figure 15.10 the corresponding changes in the percentage distribution between the main reservoirs are shown. We can see that the airborne fraction decreases linearly from 66%, corresponding to P_o = 250 ppm, to 29%, when P_o is 290 ppm. If CO₂ is not the main limiting factor for the natural vegetation in the world, as discussed by Goudriaan and Ajtay (Chapter 8, this volume), it is probable that the uptake by the biota has been small, and that a value of P_o around 260 ppm may seem plausible. It is interesting to note the close agreement between this result and that arrived at by Stuiver (1978), based on isotope analyses of tree-ring carbon.

Figure 15.9 Combinations of preindustrial CO₂ partial pressure, P_o, and biota growth factor, , yielding an atmospheric CO₂ pressure in 1970 of 320 ppm. The figure is for an `average' biota (see Table 15.5)

Figure 15.10 Distribution of the released excess CO₂ between the atmosphere (A), biota and soil (B and S), and the ocean (O), for the cases corresponding to Figure 15.9

So far, the amounts of carbon in the living biomass and the soil have been assumed to be 830 and 2000 x 10¹⁵ g C, respectively. However, estimates of the total biomass in the world span a wide range. Since the amount of carbon enters here into the equation governing the assimilation rate (equation 15.5.37), this scatter implies an equally large uncertainty in the resulting estimates of the assimilation rates. It can thus affect the dynamic properties of the biological part of the carbon cycle and modify the picture of the terrestrial biota as a sink for excess CO₂. It is therefore necessary to explore the importance of this uncertainty for the graphs of Figures 15.9 and 15.10. The estimates of soil carbon vary even more, as mentioned in Section 15.3. Although this carbon takes no part in the gross assimilation, its magnitude and exchange rate may well have an influence on atmospheric CO₂ developments, not least in modifying the Suess effect.

Figure 15.11 Same as Figure 15.9, but for a `small and fast' biota (see Table 15.5)

The uncertainties relating to the biota and soil are not only a question of their magnitude as carbon reservoirs. Their dynamic properties are also uncertain. To express this in terms of model variables, not only can N_b1 and N_s1, be subject to sensitivity analyses, but also T_b1,s1 and T_b1,a (see equations (15.5.38)(15.5.39)). The two types of uncertainty are not entirely independent, naturally, since N_b1 , N_s1, T_b1,s1, and T_b1,_a are related to one another by steady-state conditions such as in equation (15.5.42). For the present study, we consider the dynamic characteristics to be the most significant. Therefore, estimates from different authors were combined to yield, in one case, the slowest possible biota and soil system (T_b1,s1 and T_b1,a maximal) and, in the other case, the fastest possible system (T_b1,s1 and T_b1,a minimal). The same series of experiments as above was performed for these two `extreme' cases and the results are shown in Figures 15.11, 15.12, 15.13 and 15.14.

Figure 15.12 Same as Figure 15.10, but for a 'small and fast' biota (see Table 15.5)

Figure 15.13 Same as Figure 15.9, but for a `slow and large' biota (see Table 15.5)

Sensitivity analyses of the dependence of the results on uncertainties in the oceanic circulation parameters, similar to those accounted for in Section 15.7.3, were performed for the `slow' and `fast' biota cases. The results are indicated in Tables 15.8 and 15.9. We can see the same general pattern as in Table 15.7. The fraction remaining in the atmosphere is subject to much smaller variations than it would be expected to show in response to alterations in the assumed preindustrial CO₂ content. The amount of carbon taken up by the intermediate and deep ocean, on the other hand, may vary by a factor of about five. The importance of this reservoir as a sink for the excess CO₂ released until now is thus largely dependent on circulation characteristics.

Figure 15.14 Same as Figure 15.10. but for a ^,slow and large' biota (see Table 15.5)

Table 15.8 Same as Table 15.7, but for the case of a `small and fast' biota (see Table 15.5)

It should be pointed out, however, that the experiments in Tables 15.715.9 were performed using the assumption P_o = 285 ppm. As illustrated in Figures 15.10, 15.12 and 15.14, the relative importance of the oceans as a sink for CO₂ increases as we decrease our assumed P_o. It can be expected, therefore, that the picture adopted for oceanic circulation becomes of greater importance to the accumulated airborne fraction, the smaller the chosen value for the preindustrial CO₂ content. To illustrate this effect, the same computations as in Table 15.9 were repeated for a case with P_o as low as 250 ppm, which corresponds to B 0 (see Figure 15.14). The result is shown in Table 15.10. It can be seen that as T varies from 100 to 10 000 x 10¹² m³ /year, the fraction of the total release remaining airborne varies between 76% and 58%. In absolute units for carbon, these variations are four times the corresponding variations in Table 15.9, where we started from P_o = 285 ppm.

Table 15.9 Same as Table 15.7, but for the case of a `large and slow' biota (see Table 15.5)

                  Table 15.10 Same as Table 15.9, but for the case P_o = 250 ppm, = 0. The deep ocean circulation is the `young' case

15.7.5 Sensitivity to Uncertainty in the Buffer Factor

In simple projections of the future development of the atmospheric CO₂ level, a quantity of large interest is the so-called buffer factor, B. It is defined as the ratio of the percentage increase of carbon in the atmosphere to the percentage increase of carbon in the surface water, when the total amount of carbon in these two reservoirs increases by an incremental amount, and chemical equilibrium remains. Formally,

(15.7.4)

 where we have made use of the fact that the amount of carbon in the atmosphere at equilibrium is proportional to the concentration of dissolved CO₂ in the surface water. Since this factor is of importance for the time scale of the decline in a perturbation of the atmospheric CO₂ content, its significance for simple considerations of the dynamics of the carbon cycle is obvious. Most investigators have used a value of between 9 and 12 for this quantity (e.g., Bolin and Eriksson, 1959; Keeling, 1973a). These estimates were based on chemical considerations similar to those discussed in Section 15.5.2. With the present model and with values of chemical constants as given in Table 15.2, a value of about 10 can be computed for the cold surface water, and about 9 for the warm surface water. (Since we have used the same chemical constants as Keeling (1973a), and applied the same model for the chemistry of the surface water, our value for the buffer factor is naturally the same as that which he obtained). Recent results (Rebello and Wagener, 1977) indicate that this rather simple model of the chemistry of sea-water may not be realistic. These authors measured the CO₂ uptake from the air for sea-water under controlled conditions. Generally, it was found that the buffer factor has a considerably lower value of approximately 7.

If the correct value of B is 7 rather than 9 or 10, the picture of the ocean as a sink for excess carbon changes. The degree to which the uncertainty about the buffer factor can affect the resulting picture of the distribution of the released carbon is dependent on the values chosen for other model parameters and the assumptions made on the initial state.

In order to obtain a general impression of the magnitude of the effects involved here, we consider first the partitioning of excess carbon between the atmosphere and the surface water, as regulated by the condition of chemical equilibrium. Let us denote by N_a and N_s the increases in carbon content in the atmosphere and in the surface water that result from a total increase of OM units of carbon in these two reservoirs. For small values of M we obtain

	(15.7.5)
	(15.7.6)

where N_ao and N_so denote initial values. In the present model calculations, N_so 680 x 10¹⁵ g C. With B between 7 and 10, and with N_ao between 530 and 615 x 10¹⁵ g C (corresponding to a partial pressure of between 250 and 290 ppm), we obtain N_s/M between 10% and 15%. The ratio N_a /M thus varies between 85% and 90%. Although this is only a 6% reduction of the amount remaining airborne, it is a 50% increase of the amount taken up by the ocean. The uncertainty about the correct value for the buffer factor may thus have a significant effect or the picture of the ocean as a sink for the excess carbon released until now.

It deserves mention, however, that equations (15.7.5) and (15.7.6) should not be used to estimate the accumulated increase of the carbon content in the atmosphere and the surface water in the period 18601970. As we have seen, a significant fraction of the man-made CO₂ may have gone into organic compounds or into the below-surface ocean.

Since M is defined as the sum N + N_s, there is thus considerable uncertainty about the value of M for the period as a whole. Because of the scatter of possible values for M, an assessment of the effect of buffer factor variations in absolute units of carbon appears to be an uncertain undertaking.

To see an example of the effect of buffer factor variations on the more complete system, including the organic carbon and the deep sea, we can take advantage of a flexibility in the present computer version of the model. The increasing CO₂ escape from the ocean surface can be remodelled to be determined by the condition of a constant buffer factor with a prescribed value. This is achieved by substituting the functional relationships defined by equations (15.5.17)(15.5.24) by another equation. Denoting by F_ia the fluxes from the cold surface water (i = c) and the warm surface water (i = w), the condition for a constant buffer factor B is

(15.7.7)

Table 15.11 Variations of the computed partitioning of excess carbon with variations in the buffer factor. The table illustrates a case with no biota uptake, a rapid oceanic turnover and an initial P_o = 250 ppm. As illustrated in Figure 15.10, the role of the oceans as a sink for excess carbon is larger for this combination of parameters than for any other. The sensitivity to variations in the buffer factor was therefore assumed to be largest in this case.

From Figures 15.10, 15.12, and 15.14 it follows that the role of the ocean as a sink for excess CO₂ is greatest when we assume a rapid oceanic turnover, no uptake by the biota, and an initial CO₂ pressure of 250 ppm. The sensitivity to buffer factor variations was therefore assumed to be larger in this case than for any other combination of model parameters and initial state assumptions. Applying these conditions, an experiment was performed using equation (15.7.7) instead of the previous assumptions about the chemistry of the oceanic carbonate system. The value B = 7 was assigned to the buffer factor for the entire ocean surface.

The result is illustrated in Table 15.11. We find for the accumulated airborne fraction a difference corresponding to 7% of the total release. Comparing this result to Tables 15.715.9, we find that the effect of buffer factor uncertainty is similar in sire to the effect of the uncertainty about oceanic circulation characteristics. As was mentioned before, this is a significant modification of the picture in some cases.

15.8 SUMMARY

Without contradicting present knowledge, several different pictures of the distribution of the excess CO₂ released until now can be constructed by varying certain critical factors in the computations. An investigation of the sensitivity of the resulting picture to variations in these factors must, therefore, be a primary and important task. The experiments so far performed have provided information about the relative importance of some of the critical factors.

The sources of uncertainty regarding the picture of the distribution of excess carbon are of two different kinds. Firstly, the natural processes that bring about the transfers of carbon from one reservoir to another cannot always be quantitatively described with accuracy. For example, estimates of the amount of carbon taken up by the intermediate water vary by a factor of almost 3, owing to insufficient knowledge about the exchange rate of water between the surface layers and intermediate depths (Tables 15.715.9). Such uncertainty makes it impossible to give a precise formulation of the relationship between the carbon content of a reservoir and the flux out of it. The model variables have a margin of uncertainty.

Secondly, our knowledge about the carbon cycle before human impact began is incomplete, and several rather different pictures of it are conceivable. For example, the value of the atmospheric CO₂ content from which to start the time integration has been varied over a range of 40 ppm in this study. Naturally, this has proven to have a decisive influence on the computed distribution of the excess carbon.

The factors with greatest potential for affecting the distribution of the released excess carbon appear to be the assumed preindustrial CO₂ content in the atmosphere, P_o, and the coefficient, (, describing the potential of the biota for responding to an increased atmospheric CO₂ content with an increase in photosynthesis. Figures 15.915.14 illustrate the importance of P_o and in this regard.

As can be seen in Tables 15.7-15.9, the assumptions made about oceanic circulation characteristics may also have a considerable modifying influence, although less profound than that exerted by P_o or . The extent to which oceanic circulation assumptions affect the resulting picture seems to be greater, the smaller the value chosen for P_o. (Inspection of Figure 15.10 provides support for this observation, since it shows that the percentage of the excess carbon taken up by the ocean increases with decreasing P_o). This example illustrates that the importance of one parameter may change as the values of other parameters are varied. A more complete sensitivity analysis must take features of this kind into account.

As for the importance of the ocean, the circulation characteristcs of the `intermediate layer' between roughly 75 m and 1000 m appear to be of special interest when determining the uptake of excess CO₂ in the next few centuries. The discussion in connection with Figures 15.3 and 15.4 show how different model assumptions regarding the water exchange between this volume and the surface layer may yield quite different results for the course of oceanic uptake of excess carbon.

As discussed in the previous section, uncertainty about the buffer mechanism of airsea exchange also appears to be great enough to permit variations of about the same size as those due to the uncertainty about the circulation.

It is to be expected that both physical and chemical properties of the ocean will increase in importance. Not only the equilibrium distribution of a growing amount of man-made CO₂, but also the time scale required to approach this distribution will then depend on a realistic description of the oceanic processes for a proper treatment.

As shown by Keeling and Bacastow (1977), and by Broecker and Takahashi (1977), dissolution of carbonates on the ocean floor may have a decisive effect on the development of the atmospheric CO₂ content in the future. Their results show that, especially for the long time perspective of more than a 1000 years, this process may be of overwhelming importance for the eventual rate of decline of the atmospheric CO₂ concentration. The process of sediment dissolution has not been discussed in the present study, since we have been concerned almost exclusively with the distribution of the excess carbon input between 1860 and 1970. However, the fact that some shift in the carbonate sediment balance has already taken place in certain regions should not be excluded.

It should again be emphasized that the estimates made on the relative significance of various model parameters refer only to their impact on the distribution of CO₂ released until 1970. Turning to the problem of predicting the CO₂ development in the next few centuries, or the final decline of the atmospheric CO₂ content over perhaps one or two millenia, other natural processes than those discussed in this study will become of interest, while some of those described here will be of reduced importance. It seems logical to expect that in order to approach these problems a modification of the carbon cycle model developed in the present study will be necessary. The inclusion of the interaction between the sediments and the dissolved carbonate appears to be an especially necessary extension of the model.

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