Modeling Biogeochemical Cycles

Henning Rodhe

4.1 Introductory Remarks spatial average (i.e., the physical size of the reservoir itself) often has a horizontal size To formulate a model is to put together pieces of approaching that of a continent, or larger, i.e., knowledge about a particular system into a > 1000 km. The time scales corresponding to this consistent pattern that can form the basis for (1) spatial average are months, or longer. This interpretation of the past history of the system means that day-to-day variations in weather and (2) prediction of the future of the system. To and ocean currents are not generally considered be credible and useful, any model of a physical, explicitly when modeling biogeochemical cycles. chemical or biological system must rely on both The advent of fast computers and the avail- scientific fundamentals and observations of the ability of detailed data on the occurrence of world around us. High-quality observational certain chemical species have made it possible data are the basis upon which our understand- to construct meaningful cycle models with a ing of the environment rests. However, observa- much smaller and faster spatial and temporal tions themselves are not very useful unless the resolution. These spatial and time scales corre- results can be interpreted in some kind of model. spond to those in weather forecast models, i.e. Thus observations and modeling go hand in down to 100 km and 1 h. Transport processes hand. (e.g., for CO2 and sulfur compounds) in the This chapter focuses on types of models used oceans and atmosphere can be explicitly to describe the functioning of biogeochemical described in such models. These are often cycles, i.e., reservoir or box models. Certain referred to as "tracer transport models." This fundamental concepts are introduced and some type of model will also be discussed briefly in examples are given of applications to biogeo- this chapter. chemical cycles. Further examples can be found in the chapters devoted to the various cycles. The chapter also contains a brief discussion of the nature and mathematical description of 4.2 Time Scales and Single Reservoir Systems exchange and transport processes that occur in the oceans and in the atmosphere. This chapter 4,2,1 Turnover Time assumes familiarity with the definitions and basic concepts listed in Section 1.5 of the intro- Consider the reservoir shown in Fig. 4-1. The duction such as reservoir, flux, cycle, etc. turnover time is the ratio between the content Modeling biogeochemical cycles normally (M) of a substance (tracer) in the reservoir and involves estimating the spatial and temporal the total flux out of it (S): averages for concentrations and fluxes in and out of reservoirs (i.e., reservoir modeling). The ^0 = -^ (1)

implies some departure from this simple relation. If material is removed from the reservoir by two or more separate processes, each with a flux S/, then turnover times with respect to each Fig. 4-1 Schematic illustration of a single reservoir process can be defined as: with source flux Q, sink flux S, and content M. M (2)

The turnover time may be thought of as the time Since Yl ^t — S/ these time scales are related to it would take to empty the reservoir if the sink the turnover time of the reservoir, TQ, by (S) remained constant while the sources were zero (TQS = M). This time scale is also sometimes To^^E^o;-' (3) referred to as "renewal time" or "flushing As an application of the turnover time con- time." In the common case when the sink is cept, let us consider the model of the carbon proportional to the reservoir content (S = kM), cycle shown in Fig. 4-3. This diagram is different the turnover time is the inverse of the propor- from the one used in the chapter on the carbon tionality constant (fc~^), which is analogous to cycle (Chapter 11), because it serves our pur- first-order chemical kinetics. poses better for this chapter. The values given In fluid reservoirs like the atmosphere or the for the various fluxes and burdens are very ocean, the turnover time of a tracer is also similar to the corresponding figure in Chapter related to the spatial and temporal variability of 11 (Fig. 11-1). its concentration within the reservoir; a long The turnover time of carbon in biota in turnover time corresponds to a small variability the ocean surface water is 3 x 10^^/(4 + 36) x and vice versa Qunge, 1974; Hamrud, 1983). 10^^ yr ;^ 1 month. The turnover time with Figure 4-2 shows a plot of measured trace gas respect to settling of detritus to deeper layers is variability in the atmosphere versus turnover considerably longer: 9 months. Faster removal time estimated by applying budget considera- processes in this case must determine the turn- tions as indicated by Equation (1). An inverse over time: respiration and decomposition. relation is obvious, but the scatter in the data The equation describing the rate of change of the content of a reservoir can be written as

I HpO (4) ^=° To CO CH4 If the reservoir is in a steady state (dM/df = 0) .^. then the sources (Q) and sinks (S) must balance. "> ^ N2O CD b In this case Q can replace S in Equation (1). •Eo 10-2 "D 2 ^v-T^a^/c =0.14yr C += CO2 \ ^^pq (/) o 10-4 He' 4,2.2 Residence Time (Transit Time) Op CO M- 0 O The residence time is the time spent in a reservoir DC 10-6 J I I L J L 10-2 IOO 102 104 106 108 by an individual atom or molecule. It is also the age of a molecule when it leaves the reservoir. If Turnover Time x (yr) the pathway of a tracer from the source to the sink is characterized by a physical transport, the Fig. 4-2 Inverse relationship between relative standard deviation of atmospheric concentration and word transit time can also be used. Even for a turnover time for important trace chemicals in the single chemical substance, different atoms and troposphere. (Modified from Junge (1974) with per- molecules will have different residence times in mission from the Swedish Geophysical Society.) a given reservoir. Let the probability density 64 Henning Rodhe

Atmosphere 725 Annual increase ~3)

Intermediate and Deep water Dissolved inorg. 36,700 Dissolved org. 975 (Annual increase •-2.5)

Fossil fuels oil, coal, gas 5,000-10,000

Fig. 4-3 Principal reservoirs and fluxes in the carbon cycle. Units are 10^^ g (Pg) C (burdens) and Pg C/yr (fluxes). (From Bolin (1986) v^ith permission from John Wiley and Sons.)

function of residence times be denoted by (/)(T), inlet to the outlet, i.e., x^- Very few molecules where 0(T)dT describes the fraction of the tracer will have a residence time much greater than or having a residence time in the interval to i to much less than ij., as illustrated in the 0 curve. T + dr. The average residence time (average transit Another example is a human population where time) Tr is defined by: most people live to attain mature age. The 0 curve in this case can be interpreted as the T(/)(T)dT (5) frequency function for the age at which people Jo die; few die very young and few survive the In many cases the word "average" is left out and average age of death by more than 50%. this quantity is simply referred to as "residence Figure 4-4b illustrates exponential decay. A time." simple example could be the reservoir of all ^"^^U The shape of the probability density function, on Earth. The half-life of this radionuclide is (j){x), depends on the system. Some examples are 4.5 X 10^ years. Since the Earth is approximately shown in Fig. 4-4. This figure also contains 4.5 X 10^ years old, this implies that the content probability density of age (see Section 4.2.3). of the ^^^U reservoir today is about half of what Figure 4-4a might correspond to a lake with it was when the Earth was formed. The prob- inlet and outlet on opposite sides of the lake. ability density function of residence time of the Most water molecules will then have a residence uranium atoms originally present is an expo- time in the lake roughly equal to the time it takes nential decay function. The average residence for the mean current to carry the water from the time is 6.5 x 10^ years. (The average value of Modeling Biogeochemical Cycles 65

residence times, the probability density function of ages, I/^(T), has a shape that depends on the 1 ^\ (P(x) situation. In a steady-state reservoir, however, 1 \A—-->^ 1 \I/{T) is always a nonincreasing function. The shapes of II/{T) corresponding to the three resi- 1 1 ^^^^ 1 1 dence time distributions discussed above are 1 ^ai S 2 included in Fig. 4-4. The average age of atoms in a reservoir is given by

/•OO = / T\l/{T)dT (6) Jo

3 T 4.2.4 Relations Between TQ, T^ and Ta

It can be shown that for a reservoir in steady state. To is equal to ir, i.e. the turnover time is equal to the average residence time spent in the reservoir by individual particles (Eriksson, 1971; Bolin and Rodhe, 1973). This may seem to be a trivial result but it is actually of great signifi- cance. For example, if TQ can b e estimated from budget considerations by comparing fluxes and burdens in Equation (1) and if the average transport velocity (V) within the reservoir is known, the average distance (L = VT^) over Fig. 4-4 The age frequency function II/{T) and the which the transport takes place in the reservoir residence time frequency function (J)(T) and the corre-can be estimated. sponding average values la and i^ for the three cases The relation between TQ and T^ is not as described in the text: (a) ig > ir; (b) Ta = ir; (c) la > Tr- (Adapted from Bolin and Rodhe (1973) with permis- simple. Ta may be larger or less than TQ depend- sion from the Swedish Geophysical Society.) ing on the shape of the age probability density function (as shown in Fig. 4-4). For a well-mixed reservoir, or one with a first-order removal time for an exponential decay function is the process, Ta = TQ (Fig. 4-4b). half-life divided by In 2.) In a well-mixed reser- In the case of a human population corre- voir all particles always have the same prob- sponding to Fig. 4-4a, Ta is only about half of ability of being removed. In such situations the TQ. This example applies to the average age of frequency function for the residence time is also all Swedes, which is around 40 years, whereas exponential. the average residence time, i.e., the average In the reservoir corresponding to Fig. 4-4c the length of life (average age at death) is almost removal is biased towards "young" particles. 80 years. This might occur when the sink is located close In the situation where most atoms leave the to the source (the "short circuit" case). reservoir soon and few of them remain very long (Fig. 4-4c), Ta is larger than TQ (the "short circuit" case). Some further examples of age 423 Age distributions and relations between Ta and TQ are given in Lerman (1979). The age of an atom or molecule in a reservoir is When equating TQ and Tr it must be made clear the time since it entered the reservoir. As with that the flux, S, which defines TQ (see Equation 66 Henning Rodhe

(1)) is the gross flux and not a net flux. For Mass example, removal of water from the atmosphere occurs both by precipitation and dry deposition (direct uptake by diffusion to the surface). Dry (Mi-Mo)«^ deposition is not normally explicitly evaluated but subtracted from the gross evaporation flux to yield the net evaporation from the surface. The turnover time of water in the atmosphere calculated as the ratio between the atmospheric content and the precipitation rate (about 10 days) is thus not equal to the average residence Fig. 4-5 Illustration of an exponential adjustment time of water molecules in the atmosphere. The process. In this case, the response time is equal to k"^. actual value of the average residence time of individual water molecules is substantially shorter. trated in Fig. 4-5. In this case, with an exponential adjustment, the response time is defined as the time it takes to reduce the imbalance to 4,2,5 Response Time e"^ = 37% of the initial imbalance. This time The response time (relaxation time, adjustment scale is sometimes referred to as ''e-folding time) of a reservoir is a time scale that charac- time.'' Thus, for a single reservoir with a sink terizes the adjustment to equilibrium after a proportional to its content, the response time sudden change in the system. A precise defini- equals the turnover time. tion is not easy to give except in special circum- As a specific example, consider oceanic sul- stances like in the following example. fate as the reservoir. Its main source is river Consider a single reservoir, like the one runoff (pre-industrial value: 100 Tg S/yr) and shown in Fig. 4-1, for which the sink is propor- the sink is probably incorporation into the tional to the content (S = kM) and which is lithosphere by hydrogeothermal circulation in initially in a steady state with fluxes Qo = So mid-ocean ridges (100 Tg S/yr, McDuff and and content MQ. The turnover time of this Morel, 1980). This is discussed more fully in reservoir is Chapter 13. The content of sulfate in the oceans is about 1.3 X lO^TgS. If we make the (un- Mo realistic) assumption that the present runoff, ^0 (7) which due to man-made activities has increased Suppose now that the source strength is sud- to 200 Tg S/yr, would continue indefinitely, denly changed to a new value Qi. How long how fast would the sulfate concentration in the would it take for the reservoir to reach a new ocean adjust to a new equilibrium value? The time scale characterizing the adjustment would steady state? The adjustment process IS described by the differential equation be To ^ 1.3 X 10^ Tg/(10^ Tg/yr) ^ 10^ years and the new equilibrium concentration even- dM ^ tually approached would be twice the original S = Qi-kM (8) value. A more detailed treatment of a similar with the initial condition M{t = 0) = Mo. problem can be found in Southam and Hay The solution (1976). M{t) - Ml - (Ml - Mo) exp{-kt) (9) approaches the new steady-state value 4.2,6 Reservoirs in Non-steady State (Ml = Qi/k) with a response time equal to k~^ or TQ. The change of the reservoir mass from the Let us analyze the situation when one observes a initial value MQ to the final value Mi is illus- change in reservoir content and wants to draw Modeling Biogeochemical Cycles 67 conclusions regarding the sources and sinks. We is a very stable molecule) the turnover time of rewrite Equation (8) as SFe is as large as 3000 years. The rate of increase is thus a reflection of an imbalance 1 dM_ Q 1 between sources and sinks rather than an M dt ~M~T^ increase in the source flux Q. where TQ = l/fc is the turnover time in the steady In situations where Tobs is comparable in state situation. Let us denote the left side of the magnitude to TQ, a more complex relation pre- equation (the observed rate of change of the vails between Q, S, and M. Atmospheric CO2 reservoir content) as lobs- If the mass were falls in this last category although its turnover observed to increase by, say, 1% per year, lobs time (3-4 years, cf. Fig. 4-3) is much shorter than would be 100 years. Two limiting cases can be Tobs (about 300 years). This is because the atmos- singled out: pheric CO2 reservoir is closely coupled to the 1- Tobs > 'z^o- In this case there has to be an carbon reservoir in the biota and in the surface approximate balance between the two terms layer of the oceans (Section 4.3). The effective on the right-hand side of the equation. turnover time of the combined system is actually several hundred years (Rodhe and Bjork- 1 M strom, 1979). (or) Q^ M ' -^0 ^0 This means that the observed change in M mainly reflects a change in the source flux Q 4.3 Coupled Reservoirs or the sink function. As an example we may take the methane concentration in the atmos- The treatment of time scales and dynamic be- phere, which in recent years has been increas- havior of single reservoirs given in the previous ing by about 0.5% per year. The turnover section can easily be generalized to systems of time is estimated to be about 10 years, i.e., two or more reservoirs. While the simple system much less than lobs (200 years). Conse- analyzed in the previous section illustrates quently, the observed rate of increase in many important characteristics of cycles, most atmospheric methane is a direct consequence natural cycles are more complex. The matrix of a similar rate of increase of emissions into method described in Section 4.3.1 provides an the atmosphere. (In fact, this is not quite true. approach to systems with very large numbers of A fraction of the observed increase is prob- reservoirs that is at least simple in notation. The ably due to a decrease in sink strength caused treatments in the preceding section and in Sec- by a decrease in the concentration of hydrox- tion 4.3.1 are still limited to linear systems. In yl radicals responsible for the decomposition many cases we assume linearity because our of methane in the atmosphere.) knowledge is not adequate to assume any other 2. Tobs < 'z^o- In this case dM/dt ^ Q which dependence and because the solution of linear means that there is an increase in reservoir systems is straightforward. There are, however, content about equal to the source flux with some important cases where non-linearities are little influence on the part of the sink. The reasonably well understood. In such cases ana- reservoir is then in an accumulative stage and lytical mathematical solutions, corresponding to its mass is increasing with time largely as a those given in Section 4.3.1, normally do not function of Q, irrespective of whether Q itself exist and the mathematical expressions describ- is increasing, decreasing or constant. A good ing the dynamics of the cycle have to be replaced example of this situation is sulfur hexafluo- by finite difference expressions that may be ride (SFe) whose concentration in the atmos- solved by computer. A few of these cases are phere is currently increasing by about 0.5% described in Section 4.3.2. per year (lobs = 200 years) as a result of As important as coupled reservoirs and non- various industrial emissions (IPCC, 1996). linear systems are, the less mathematically Because of inefficient removal processes (SF^ inclined may want to read this section only for Henning Rodhe its qualitative material. The treatment described here is not essential for understanding the mate- ^2^, rial later in the book. However, an excellent M, k2,M2 Mg overview of solving differential equations by the eigenvalue-eigenvector method used in the next section appears in Section 3.6 of Braun Fig. 4-6 A coupled two-reservoir system with fluxes (1983). proportional to the content of the emitting reservoirs.

4,3.1 Linear Systems can be generalized by adding an external forcing A linear system of reservoirs is one where the function on the right-hand side of Equations (11) fluxes between the reservoirs are linearly and (12). related to the reservoir contents. A special case, As an illustration of the concept introduced that is commonly assumed to apply, is one above, let us consider a coupled two-reservoir where the fluxes between reservoirs are propor- system with no external forcing (Fig. 4-6). The tional to the content of the reservoirs where dynamic behavior of this system is governed by they originate. Under this proportionality the two differential equations assumption the flux f^y from reservoir i to dMi = -kuMi + k2iM2 reservoir ; is given by dt (14) kijMi (10) dM2 = kuMi - /C21M2 The rate of change of the amount M, in reservoir ~dr / is thus the expression of conservation of mass dM, Ml + M2 = MT (15) for j^i (11) "dT and the initial condition

where n is the total number of reservoirs in the Mi(f = 0)=Mio (16) system. M2{t = 0) = MT - Mio This system of differential equations can be written in matrix form as Equations (14) can be written in matrix form as dM dM kM (12) kM (17) "dT "dT where the vector M is equal to (Mi,M2,... ,M„) where M is the vector (Mi,M2) describing the and the elements of matrix k are linear combina- contents of the two reservoirs and k the matrix: tions of the coefficients kij. The solution to ki\ Equation (12) describes the adjustment of all -fcl2 reservoirs to a steady state by a finite sum of ku -hi exponential decay functions (Lasaga, 1980; The eigenvalues of k are the solutions to the Chameides and Perdue, 1997). The time scales equation of the exponential decay factors correspond to the nonzero eigenvalues of the matrix k. The -ku - ^ k2i response time of the system, icycie/ niay be hi -k2i - X defined by {-h2-^^){-hi-^)-h2k2i^0 (18) '^cycle = TTTT (13) Ai = 0 and h = -(^12 + hi). The general solution to Equation (17) can be written as where £i is the nonzero eigenvalue with smal- lest absolute value (Lasaga, 1980). The treatment M(f) = (Ai exp(.^if) + xl/2 exp(220 (19) Modeling Biogeochemical Cycles 69 where ij/i and 1/^2 are the eigenvectors of the matrix k. In our case, we have

M{t) = lAi + ^A2exp(-(fci2 + k2i)t) (20) or, in component form and in terms of the initial conditions:

l^ll + l<^ll

+ I Mio - P ^ ^ ] exp[-(fci2 + fc2i)^] hi12 ^+ 1^21k

\Yl (21) M2{t) -MT \Y1 -t- /C21 kyiMi + MT - Mio hi + hi Fig. 4-7 Example of a coupled reservoir system X exp[-fci2-f fc2i)t] where the steady-state distribution of mass is not uniquely determined by the parameters describing It is seen that in the steady state the total mass the fluxes within the system but also by the initial is distributed between the two reservoirs in conditions (see text). proportion to the sink coefficients (in reverse proportion to the turnover times), independent of the initial distribution. depend on the amount initially located in the In this simple case there is only one time scale two upper reservoirs. characterizing the adjustment process, that is Before turning to nonlinear situations, let {hi+hi)~ ' This is also the response time, us consider two specific examples of coupled '^cycie/ as defined by Equation (13). linear systems. The first describes the dynamic 1 behavior of a multireservoir system; the second '^cycle = 7r~7ir~ v^^) represents a steady-state situation of an open hi+k 21 two-reservoir system. or, if expressed in terms of the turnover times of the two reservoirs: Example 7. As a specific example of a time- + T, (23) dependent linear system we may take the ''cycle ''01 02 model of the phosphorus cycle shown in Fig. 4- The response time in this simple model will 8. This is a duplicate of the figure shown in the depend on the turnover times of both reservoirs chapter on the phosphorous cycle (Fig. 14-7). and will always be shorter than the shortest of The authors used a computer to solve the system the two turnover times. If TQI is equal to T02, then of equations in Equation (11) with a time-depen- '^cycie will be equal to half of this value. dent source term added to represent the tran- An investigation of the dynamic behavior of a sient situation with an exponentially increasing coupled three-reservoir system using the tech- industrial mining input (7% increase per year). niques described above is included in the prob- Lasaga (1980) studied the same situation in a lems listed at the end of the chapter. more elegant way using matrix algebra. The It should be noted that the steady-state solu- evolution of the phosphorus content of the tion of Equation (12) is not necessarily unique. various reservoirs (except in sediments) during This can easily be seen in the case of the four- the first 70 years is shown in Table 4-1. Within reservoir system shown in Fig. 4-7. In the steady this time frame, the only noticeable change is state all material will end up in the two accumu- seen to occur in the land reservoir. Lasaga lating reservoirs at the bottom. However, the showed that the adjustment time scale of the distribution between these two reservoirs will system, icycie/ is 53000 years. This is much 70 Henning Rodhe

ATMOSPHERE LAND BIOTA 0.00009 96.9 0.14 0.02

OCEAN (2) (D SURFACE op o BIOTA ^ OCEAN "''^"^

0.39 0.60

DEEP OCEAN Figure 4-9 An open two-reservoir system.

® removed at a rate S2. The turnover times (aver- MINE- ABLE SEDIMENTS P age residence times) of the two reservoirs and of the combined reservoir (defined as the sum of Fig. 4-8 The global phosphorus cycle. Values shown the two reservoirs) are easily calculated to be are in Tmol and Tmol/yr. (Adapted from Lerman et Ml Ml al (1975) and modified to include atmospheric trans- L (24) ui — ) fers. The mass of P in each reservoir and rates of T + Si Q exchange are taken from Jahnke (1992), MacKenzie M2 M2 et al (1993) and FoUmi (1996).) T02 (25)

Ml + M2 M1+M 2 To = shorter than the turnover time of the sediment Si + S2 Q reservoir (2 x 10^ years) but much longer than MA/Tli M Mn2 S,S2o Mli M2Mo T the turnover times of all other reservoirs. This TOl -h CCT02 sT' Q cycle is described in greater detail in Chapter 14. (26) Example 2. As a much simpler example let us where (x = T/Q is the fraction of the material consider a system consisting of two connected passing through reservoir 1 that is transferred to reservoirs as depicted in Fig. 4-9. Steady state is reservoir 2. An example application of this two- assumed to prevail. Material is introduced at a reservoir model is that it has been used to study constant rate Q into reservoir 1. Some of this the oxidation of sulfur in the atmosphere where material is removed (Si) and the rest (T) is S02-sulfur was treated as one reservoir and transferred to reservoir 2, from which it is sulfate-sulfur as the other (Rodhe, 1978). In the

Table 4-1 Response of phosphorus cycle to mining output. Phosphorus amounts are given in TgP (1 Tg = 10^^ g). Initial contents and fluxes as in Fig. 4-7 (system at steady state). In addition, a perturbation is introduced by the flux from reservoir 7 (mineable phosphorus) to reservoir 2 (land phosphorus), which is given by 12 exp(0.070 in units of Tg P/yr

Time (years) Land Land biota Organic biota Surface ocean Deep ocean

0 200000 3000 138 2710 87100 10 200173 3000 138 2710 87100 20 200522 3001 138 2710 87100 30 201224 3003 138 2710 87100 40 202636 3008 138 2710 87100 50 205481 3018 138 2710 87100 60 211208 3018 138 2711 87100 70 222 741 3078 138 2712 87101 Modeling Biogeochemical Cycles 71 special case where Si = 0, T = Q and a = 1 (all off. M eventually reaches a steady state equal to material introduced in reservoir 1 is transferred y/QjB but the response time scale is not as to reservoir 2) the turnover time of the combined easily defined as in the linear case. Relative to a reservoir equals the sum of the turnover times of simple exponential relaxation process the the individual reservoirs. adjustment given by Equation (28) is more rapid initially, and slower as time progresses. In general, if the removal flux is dependent 4J,2 Non-linear Systems upon the reservoir content raised to the power a (a 7^ 1), i.e., S = BM", the adjustment process In many situations the assumption about linear will be faster or slower than the steady-state relations between removal rates and reservoir turnover time depending on whether a is larger contents is invalid and more complex relations or smaller than unity (Rodhe and Bjorkstrom, must be assumed. No simple theory exists for 1979). treating the various non-linear situations that A similar simple non-linear adjustment pro- are possible. The following discussion will be cess is described by the equation limited to a few examples of non-linear reser- (29) voir/flux relations and cycles. For a more com- dt prehensive discussion, see the review by Lasaga which is a common model for the growth of (1980). biological systems (it is called logistical growth). Consider a single reservoir with a constant The term AM represents exponential growth rate of supply and a removal rate proportional (unlimited supply of space and nutrients) and to the square of the reservoir content. The the term BNf is a removal term, a negative equation governing the rate of change of the feedback effect of "crowdedness." Initially reservoir content is (where AMQ > BM§), the growth will be close dM to exponential and will then gradually level off Q-BM^ (27) to the equilibrium value A/B (Fig. 4-11). If M(0) = 0, the solution to this equation is A sink flux that has a weaker than proportional dependence on the content M of the JQ l-exp(-2^/QBI) emitting reservoir is often described by the M: (28) B ' 1 + exp{-2^/QBi) Michaelis-Menten equation: This is graphically illustrated in Fig. 4-10. Ini- S = - ^ ^ (30) tially, the mass increases almost linearly with {M + D) ^ ^ time. After the time {2^/QB) ~ the removal term where C is a rate coefficient and D a term becomes effective and the mass begins to level

Mass 4 Mass

Time Fig. 4-11 Shape of 'logistical growth.'' The rate of Time change increases slowly initially. The rate of growth Fig. 4-10 The shape of the function given in Equa- reaches a maximum and eventually drops to zero as tion (28). the mass levels off, approaching the value A/B. 72 Henning Rodhe (Michaelis constant) which determines the col ). The ocean to atmosphere flux, which is deviation from proportionality; a small D (com- dependent on the concentration of the dissolved pared to M) corresponds to a weak dependence species C02(aq), is related to the total carbon of S on M and a large D corresponds to a near content in the surface layer (Ms) by proportional dependence. An important example of non-linearity in a f SA = ksAMl'- (31) biogeochemical cycle is the exchange of carbon where the exponent asA (the buffer, or Revelle dioxide between the ocean surface water and the factor) is about 9. The buffer factor results from atmosphere and between the atmosphere and the equilibrium between C02(g) and the more the terrestrial system. To illustrate some effects prevalent forms of dissolved carbon. This effect of these non-linearities, let us consider the sim- is discussed further in Chapter 11. As a conse- plified model of the carbon cycle shown in Fig. quence of this strong dependence of f SA on Ms, 4-12. Ms represents the sum of all forms of a substantial increase in CO2 in the atmosphere dissolved carbon (CO2, H2CO3, HCO3 and is balanced by a small increase of Ms. Similarly, the flux from the atmosphere to the terrestrial system may be represented by the ^AT expression Atmosphere Terrestrial System M A M-r FAT = kAjMr (32) FTA i I TA The exponent aAx is considerably less than unity FSA FAS owing to the fact that CO2 generally is not the •' limiting factor for vegetation growth. This Ocean surface layer means that even a substantial increase in M A 1 Ms does not produce a corresponding increase in .k FAT- Assuming that the carbon cycle of Fig. 4-12 f^DS will remain a closed system over several thou- r sands of years, we can ask how the equilibrium Deep layers of ocean distribution within the system would change M , after the introduction of a certain amount of fossil carbon. Table 4-2 contains the answer for Fig. 4-12 Simplified model of the biogeochemical two different assumptions about the total input. carbon cycle. (Adapted from Rodhe and Bjorkstrom The first 1000 Pg corresponds to the total input (1979) with the permission of the Swedish Geophysi- from fossil fuel up to about the year 2000; the cal Society.) second (6000 Pg) is roughly equal to the now

Table 4-2 Steady-state carbon contents (unit: Pg = 10^^ g) for the four-reservoir model of Fig. 4-11: (a) during the unperturbed (pre-industrial) situation; (b) after the introduction of 1000 Pg carbon; and (c) after the introduction of 6000 Pg carbon

After 1000 Pg After 6000 Pg

Pre-industrial content (Pg) Content (Pg) % increase Content (Pg) % increase

Atmosphere 700 840 20 1880 170 Terrestrial system 3000 3110 4 3655 22 Ocean surface layer 1000 1020 2 1115 12 Deep ocean 35000 35 730 2 39050 12 Modeling Biogeochemical Cycles 73 known accessible reserves of fossil carbon (Keel- problem with an expression like this is that ing and Bacastow, 1977). mathematically speaking, the flux FAT and the If all fluxes are proportional to the reservoir mass MTB nnay grow without bounds; the larger contents, the percentage change in reservoir MTB/ the larger the flux to it, i.e., exponential content will be equal for all the reservoirs. The growth. To avoid such a mathematical explo- non-linear relations discussed above give rise to sion, Williams (1987) suggested that the factor substantial variations between the reservoirs. MTB in Equation (33) be replaced by Note that the atmospheric reservoir is much MTB(MTBâx -MTB) more significantly perturbed than any of the other three reservoirs. Even in the case with a 6000 Pg input, the carbon content of the oceans where MTBâx i^ ^^ upper limit to the size of MTB- does not increase by more than 12% at steady Once MTB approaches the value MTBâx/ the flux state. diminishes to zero and MTB is kept limited. However, with "only" 1000 Pg emitted into the system, i.e. less than 3% of the total amount of carbon in the four reservoirs, the atmospheric reservoir would still remain significantly 4.5 Coupled Cycles affected (20%) at steady state. In this case the change in oceanic carbon would be only 2% and An important class of cycles with non-linear hardly noticeable. The steady-state distributions behavior is represented by situations when are independent of where the addition occurs. If coupling occurs between cycles of different ele- the CO2 from fossil fuel combustion were col- ments. The behavior of coupled systems of this lected and dumped into the ocean, the final type has been studied in detail by Prigogine distribution would still be the same. (1967) and others. In these systems, multiple If all fluxes were proportional to the reservoir equilibria are sometimes possible and oscilla- content, i.e., if asA and aAx were unity, all tory behavior can occur. There have been sug- reservoirs would be equally affected; 15% in the gestions that atmospheric systems of chemical 6000 Pg case and 2.5% in the 1000 Pg case. species, coupled by chemical reactions, could exhibit multiple equilibria under realistic ranges of concentration (Fox et ah, 1982; White, 4.4 Fluxes Influenced by the Receiving 1984). However, no such situations have been Reservoir confirmed by measurements. The cycles of carbon and the other main plant There are some important situations in which a nutrients are coupled in a fundamental way by flux between two reservoirs is determined not the involvement of these elements in photosyn- only by the mass of the emitting reservoir but thetic assimilation and plant growth. Redfield also by the mass of the receptor. Uptake of CO2, (1934) and several others have shown that there or indeed any other nutrient by a plant commu- are approximately constant proportions of C, N, nity depends also on the magnitude of its bio- S, and P in marine plankton and land plants mass because that determines the size of the ("Redfield ratios"); see Chapter 10. This implies surfaces where photosynthesis take place. Con- that the exchange flux of one of these elements sider, for example, the uptake of atmospheric between the biota reservoir and the atmosphere CO2 by terrestrial biota. A reasonable parame- - or ocean - must be strongly influenced by the terization of this flux would be flux of the others. Williams (1987) has pointed out that there are ÂT^MAAITB AT (33) two main approaches to the treatment of such M A + D couplings. The first is to apply flux expressions where the notations follow those in Fig. 4-12 like the one described for CO2 in Section 4.4, in with MTB being the content of carbon in terres- Equation (33), and let both the rate coefficient {k) trial biota and D, a Michaelis constant. One and the upper limit of the biota reservoir size 74 Henning Rodhe

(MTB^^J be explicit functions of the available methyl chloroform (CH3CCI3) from a global concentrations of the other nutrients. This network and information about its emission approach allows for a pronounced interdepen- rate to estimate the removal rate of this gas in dence between the fluxes of the different nutri- the atmosphere. Since the only removal process ents but it does not ensure that the Redfield for methyl chloroform is the reaction with ratios are maintained. In the second approach hydroxyl radical and the rate of this chemical the contents of the nutrients in the biota reser- reaction is well established, their result also voir are forced to remain close to the Redfield provided an estimate of the concentration of ratios. This method was used by Mackenzie et al hydroxyl radical in the atmosphere. This esti- (1993) in their study of the global cycles of C, N, mate of the atmospheric concentration of hy- P, and S and their interactions. They were able to droxyl radical has been very useful in demonstrate how a human perturbation in one connection with later (forward) modeling of of these element cycles could influence the many other chemical compounds that are also cycles of the other elements. removed by reaction with the hydroxyl radical, e.g., CO, CH4, SO2, etc.

4.6 Forward and Inverse Modeling 4.7 High-Resolution Models In most cases models describing biogeochemical cycles are used to estimate the concentration (or So far the focus of this chapter has been on total mass) in the various reservoirs based on relatively simple box models with each box information about source and sink processes, as representing a reservoir with well-defined in the examples given in Section 4.4. This is often boundaries in terms of its physical, chemical or called forward modeling. If direct measurements biological characteristics, e.g., the ocean surface of the concentration are available, they can be layer, the atmosphere, the terrestrial biota, etc. compared to the model estimates. This process In many situations it is also important to model is referred to as model testing. If there are sig- the distribution of a species within such a nificant differences between observations and reservoir, especially the fluid reservoirs (atmos- model simulations, improvements in the model phere and water bodies) where transport pro- are necessary. A natural step is then to recon- cesses are rapid. A common tool for modeling sider the specification of the sources and/or the such processes is a gridpoint model in which the sinks and perform additional simulations. fluid space is divided into smaller boxes, each Inverse modeling represents a situation when a one of them represented mathematically in the model is used, in a systematic fashion, to esti- model by a single gridpoint. The physical trans- mate the magnitude of sources or sinks from port of the species in question, by mean motions observed concentrations (mass). In the simplest as well as by turbulence, can then be described case with a single reservoir (Fig. 4-1) in steady according to Equations (40) and (41) in Section state, the formal solution of the inverse problem 4.8.1 if the spatial derivatives are approximated follows directly from the relations Q — kM or by finite differences between adjacent grid- k = Q/M, depending on whether Q or fc is the points. quantity being sought. In a situation when con- The simplest kind of gridpoint model is one centrations vary in space and time and the where only one spatial dimension is considered, model consists of several reservoirs the inverse most often the vertical. Such one-dimensional modeling becomes more complex and statistical models are particularly useful when the condi- methods, such as minimizing the squares of tions are horizontally homogeneous and the deviations between observations and simula- main transport occurs in the vertical direction. tions, have to be employed. For example, Prinn Examples of such situations are the vertical et al (1992) used an eight-box model of the distribution of CO2 within the ocean (except for atmosphere (four latitude bands and two the downwelling regions in high latitudes, Sie- height layers) together with observations of genthaler, 1983) and the vertical distribution of Modeling Biogeochemical Cycles 75 ozone and other trace gases in the stratosphere variables. For example, convective motion sys- (Ko et fl/., 1989). tems in the oceans and in the atmosphere occur Two-dimensional gridpoint models enable on spatial scales of a few km or less and there- more realistic descriptions of transport pro- fore cannot be resolved explicitly in large-scale cesses in that circulation cells like the Hadley models with a grid size of 100 km or more. cell in the tropical and subtropical troposphere However, the combined effect of such motions (cf. Section 10.5.2) can be explicitly modeled. is of fimdamental importance for vertical Models of this kind, with height vs. latitude energy transport. Parameterization schemes dimensions, have been very useful for improv- therefore have to be used to describe how ing our understanding of the global-scale dis- convection develops under certain conditions tributions of long-lived gaseous components in and how they influence the large-scale flow in the atmosphere. For species with an atmos- the ocean and in the atmosphere. pheric lifetime that is short compared to the Figure 4-13 shows an example from a three- characteristic time for mixing longitudinally dimensional model simulation of the global around the globe, i.e., several weeks, two- atmospheric sulfur balance (Feichter et ah, dimensional (height/latitude) models are less 1996). The model had a grid resolution of about useful. This is the case for reactive gases like 500 km in the horizontal and on average 1 km in SO2 and NOx as well as for species carried by the vertical. The chemical scheme of the model aerosol particles, which have average atmos- included emissions of dimethyl sulfide (DMS) pheric lifetimes of only a few days. For such from the oceans and SO2 from industrial pro- species the concentration also exhibits large cesses and volcanoes. Atmospheric DMS is oxi- variations in the longitudinal direction, so dized by the hydroxyl radical to form SO2, three-dimensional models (height / latitude / which, in turn, is further oxidized to sulfuric longitude) are required. acid and sulfates by reaction with either hydrox- In three-dimensional models many more yl radical in the gas phase or with hydrogen oceanic and atmospheric motion systems can be peroxide or ozone in cloud droplets. Both SO2 explicitly described, including ocean currents and aerosol sulfate are removed from the atmos- and monsoon circulations and extratropical phere by dry and wet deposition processes. The cyclones in the atmosphere. The drawback is reasonable agreement between the simulated that these models quickly become very complex and observed wet deposition of sulfate indicates and require large computer facilities. This lim- that the most important processes affecting the itation has recently become less of a problem atmospheric sulfur balance have been ade- because of the rapid development of computers quately treated in the model. and the increasing number of high-quality In gridpoint models, transport processes such observations for model testing. Indeed, regional as speed and direction of wind and ocean and global scale three-dimensional models have currents, and turbulent diffusivities (see Section become a standard tool for studies of biogeo- 4.8.1) normally have to be prescribed. Informa- chemical cycles, especially their atmospheric tion on these physical quantities may come from component. observations or from other (dynamic) models, Although many important features of oceanic which calculate the flow patterns from basic and atmospheric circulation can be explicitly hydrodynamic equations. Tracer transport resolved in three-dimensional gridpoint models, in which the transport processes are models, there will always be many processes prescribed in this way, are often referred to as that occur on the sub-gridscale level that off-line models. An on-line model, on the other cannot. The effects of these sub-gridscale pro- hand, is one where the tracers have been incor- cesses must be parameterized, i.e., summarized porated directly into a dynamic model such that in a statistical fashion in a way related to the the tracer concentrations and the motions are large-scale flow. The purpose of parameteriza- calculated simultaneously. A major advantage tion is to describe the combined effect of sub- of an on-line model is that feedbacks of the gridscale processes on the larger-scale tracer on the energy balance can be described 76 Henning Rodhe

T—I—I—I—r 180 -1-20 -60 0 60 120 180 Longitude Fig. 4-13 Calculated and observed annual wet deposition of sulfur in mg S/m^ per year. (Reprinted from "Atmospheric Environment," Volume 30, Feichter, J., Kjellstrom, E., Rodhe, H., Dentener, P., Lelieveld, J., and Roelofs, G.-J., Simulation of the tropospheric sulfur cycle in a global climate model, pp. 1693-1707, Copyright © 1996, with permission from Elsevier Science.)

realistically. Examples of such dynamically schematic of the components considered in a interactive tracers that are important for the GCM. climate system are CO2 and the other green- house gases, as well as aerosols. Diie to the great complexity of global models that combine in an 4.8 Transport Processes interactive fashion tracer transport and explicit hydrodynamics, few such simulations have yet So far we have not gone in-depth into the nature been carried out. of the transport processes responsible for fluxes Dynamic models based on gridpoint or spec- of material between and within reservoirs. This tral discretization have been used by meteorol- section includes a very brief discussion of some ogists to forecast weather for several decades. of the processes that are important in the context Today, the climate issue has stimulated rapid of global biogeochemical cycles. More compre- development of models .designed for simula- hensive treatments can be found in textbooks on tions of climate and its variations on time geology, oceanography and meteorology and in scales of decades and centuries. Unlike weather^ reviews such as Lerman (1979) and Liss and forecast models, global-scale climate models - Slinn (1983). general circulation models (GCMs) - must include descriptions of ocean circulation, the distribution of ice, snow and vegetation, and 4.8.1 Advection^ Turbulent Flux and Molecular especially the exchange of heat, water and Diffusion momentum (friction) between the atmosphere and the underlying surface. Figure 4-14 is a Let us consider a fluid in which a tracer / is