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 oxida