Modeling Geochemical Cycles

Nelson Eby

Department of Environmental, Earth & Atmospheric Sciences

University of Massachusetts

Lowell, MA

Introduction

The use of box models to describe the cycling of various elements and chemical species through the lithosphere - hydrosphere - atmosphere - biosphere was pioneered by Garrels, McKenzie, and Hunt (1975). Over the past 30 years a variety of models have been developed to characterize these cycles. Such models are useful in assessing the impact of changes in the release or uptake of various species on the concentrations of these elements and species in various reservoirs.

1

A simple box model for the hydrologic cycle 2

The basic assumption in using box models is that the system is in a steady state, that is the rates of addition and removal (flux) of substances from the various reservoirs are in equilibrium - the concentration of the substance in the various reservoirs remains constant. Given a steady state one can calculate the residence time for a particular substance in a particular reservoir. Residence time is defined as the average length of time a particular substance will reside in a reservoir.

Residence time Amt of material reservoir

Rate of addition (removal)

3

EXAMPLE

Calculate the residence time for water in the atmosphericreservoir. Since this is a steady state model the rate of additionof water to the reservoir must equal the rate of removal. We canuse either set of fluxes. Using the rate of addition, 0.63 x 1017 kgof water are added each year by evaporation from lakes andrivers and 3.8 x 1017 kg are added each year by evaporationfrom the ocean. The total rate of addition is 4.43 x 1017 kg y-1.The residence time for water in the atmospheric reservoir is

The residence time of water vapor in the atmospheric reservoiris very short. This result suggests that changes in the rate ofaddition of water to the atmosphere (e.g., increases in the rate ofevaporation due to atmospheric warming) would lead to rapidincreases in water vapor in the atmosphere, and a correspondingincrease in the amount of precipitation.

Residence time Amt material reservoir

Rate of addition

0.13 x 1017kg

4.43 x 1017kg y 1

0.029 y 10.7 d

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5WHAT HAPPENS WHEN WE PERTURB A STEADY STATE

SYSTEM?

This problem can be investigated in several ways. The approach illustrated here is a first-orderkinetics model . In terms of geochemical cycles, we can write the following first-order kineticsequation

dAii/dt = Finputinput - Foutputoutput = Finputinput - kAii

where dA i/dt is the rate of change of the amount of substance A in reservoir i , F input is the rate ofaddition of substance A to the reservoir, Foutput is the rate of removal of substance A from thereservoir and k is the rate constant. When the system is in a steady state, Finput = Foutput. Solution ofthe preceding equation for the amount of substance A in reservoir i at some particular time gives

where Ao

i is the amount of substance A in the reservoir at time zero. When the system is in asteady state, the preceding equation can be written

dAii/dt = 0 = Finputinput - k A oi

and

k = Finputinput/ A oi

Note that in this model the new steady state for a particular reservoir is approachedexponentially.

Ai(t) Finput

k

Finput

k A o

i exp ( kt)

dAii/dt = 0 = Finputinput - k A oi

k = Finputinput/ A oi

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Pre-human cycle for mercury. Reservoir masses in 108 g. Fluxes in 108 g y-1. From Garrels et al. (1975)

7Let us evaluate the impact of the anthropogenic mercury vapor emissions on the mercury

content of the atmosphere using the first-order kinetic model. First we calculate the rate constant.In the pre-human mercury cycle, the total input of mercury vapor to the atmospheric reservoir is250 x 108 g y-1.

k = Finputinput/ A oi = 250 x 1088 g y-1-1/40 x 1088 g = 6.25 y-1-1

Adding the anthropogenic mercury vapor input of 102 x 108 g y-1 to the pre-human input gives,Finput = 352 x 108 g y-1. We can now calculate the concentration of mercury in the atmosphere forany time t. For example, when t = 1 year

We see that after 1 year the mercury content of the atmosphere has increased by 41%. Are wenear the final steady state value for the atmosphere? Since this is an exponential relationship, theapproach to the new equilibrium will be asymptotic. We can easily solve for multiple timeintervals using a spreadsheet. Such a solution is shown in the following figure. From the steadystate model we know that the mean residence time for mercury in the atmospheric reservoir is0.16 years. Given this short mean residence time we would expect that the atmosphere wouldquickly achieve a new steady state. From the following figure we see that the atmosphere haseffectively reached its new equilibrium value after one year. Note that in doing these calculationswe have assumed that the rate constant calculated from the steady state model remains constantwhen the system is perturbed. This is not necessarily true, and calculations of this type need to beevaluated in terms of the actual behavior of the system.

Ai(t) Finput

k

Finput

k A o

i exp ( kt) 352

6.25 352

6.25 40 exp [( 6.25)(1)]

56.29 x 108 g

k = Finputinput/ A oi = 250 x 1088 g y-1-1/40 x 1088 g = 6.25 y-1-1

38

42

46

50

54

58

0 0.2 0.4 0.6 0.8 1

Time (years)

Co

nc

en

tra

tio

n (

10

8 g

)

Exponential approach of mercury concentration in the atmospheric reservoir to a new equilibrium value calculated from a first-order kinetics model.

8

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In a sense the traditional steady-state box models are static, that is they do not show the various cause-and-effect feedbacks that can occur between the different reservoirs. Berner (1999) proposed a different approach using an interactive model that shows the various cause-and-effect feedbacks.

10

Traditional box model for the long-term carbon cycle. From Berner (1999)

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Cause-and-effect feedback diagram for the long-term carbon cycle. Arrows originate at causes and end at effects. Arrows with the small concentric circles represent inverse responses. Arrows without concentric circles represent direct responses. From Berner (1999)

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We can look at subcycles in the diagram to see if they have positive or negative feedbacks. If the subcycle contains an even number of concentric circles the the feedback is positive. If the subcycle contains an odd number of concentric circles the feedback is negative. Positive feedbacks lead to an amplification of an initial increase or decrease, and negative feedbacks lead to a dampening of the initial increase or decrease.

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For example, consider the subcycle B-L-G which has an odd number of concentric circles. In this cycle, increasing CO2 leads to a warmer and wetter climate with a concomitant increase in the weathering of Ca-Mg silicates. This increase in weathering leads to an increase in the uptake of CO2, a negative feedback. Hence, this cycle tends to dampen increases in CO2.

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Using the cause-and-effect feedback diagram for carbon, for each of the following determine if they are positive or negative feedback cycles.

1. N-S-G

2. A-H-Q-C

3. D-E-C

4. D-F-P-Q-C

5. D-F-M

6. B-J-P-Q-R

15

References

Berner, R. A. , 1999. A new look at the long-term carbon cycle. GSA Today 9, 11, 1-6.

Eby, G. N., 2004. Principles of Environmental Geochemistry. Pacific Grove, CA: Brooks/Cole, 510 pp.

Garrels, R. M., Mackenzie, F. T., and Hunt, C., 1975. Chemical Cycles and the Global Environment: Assessing Human Influences. Los Altos, CA: William Kaufmann, Inc., 206 pp.