Doc Martyn gets the boots (with box model)

This is Eli's humble contribution to our first mob post. Tamino at Open Mind is posting on the latest trends in atmospheric CO2 concentrations (buy, they are going up) and Simon Donner at Maribo wants to tell you about where all the carbon goes.

To get back to the shoe store, for those of you who don't know, Doc Martins are steel toed boots favored by the British version of football hooligans who want to get up close and personal. It is also the name of a climate hooligan on Real Climate who pretends he knows something about the carbon cycle and flims the flam. Eli ran into this character when he proposed to calculate the atmospheric lifetime of CO2

To do this I needed an estimate of the amount of CO2 released by Humans per year. . . I also used the Hawaiian data from 1959 to 2003 (averaging May and November).

The steady state equation for atmospheric [CO2] ppm is as follows:-

[CO2] ppm = (NCO2 + ACO2)/K(efflux)

Where:-
NCO2 is the release of CO2 into the atmosphere from non-human sources
ACO2 is man-made CO2 released from all activities
and K(efflux) is the rate the CO2 is removed from the atmosphere by all mechanisms.

You can calculate that NCO2 is 21 GT per year, ACO2 in 2003 was 7.303 GT and K(efflux) is 0.076 per year. This last figure gives a half-life for a molecule of CO2 in the atmosphere of 9.12 years.

The Rabett's reply was

Flow of carbon (in the form of CO2, plant and soils, etc) into and out of the atmosphere is treated in what are called box models. There are three boxes which can rapidly (5-10 years) interchange carbon, the atmosphere, the upper oceans, and the land. The annual cycle seen in the Mauna Loa record (and elsewhere) is a flow of CO2 into the land (plants) in the summer and out of the atmosphere as the Northern Hemisphere blooms (the South is pretty much green all year long) and in reverse in the winter as plants decay. Think of it as tossing the carbon ball back and forth, but not dropping it into the drain. Thus Doc Martyn's model says nothing about how long it would take for an increase in CO2 in the atmosphere to be reduced to its original value.

To find that, we have to have a place to "hide" the carbon for long times, e.g. boxes where there is a much slower interchange of carbon with the first three. The first is the deep ocean. Carbon is carried into the deep ocean by the sinking of dead animals and plants from the upper ocean (the biological pump). This deep ocean reservoir exchanges carbon with the surface on time scales of hundreds of years. Moreover the amount of carbon in the deep ocean is more than ten times greater than that of the three surface reservoirs.

The second is the incorporation of carbonates (from shells and such) into the lithosphere at deep ocean ridges. That carbon is REALLY lost for a long long time.

A good picture of the process can be found at
http://earthobservatory.nasa.gov/Library/CarbonCycle/Images/carbon_cycle_diagram.jpg

A simple discussions of box models can be found at
http://www.nd.edu/~enviro/pdf/Carbon_Cycle.pdf

And David Archer has provided a box model that can be run online at
http://geosci.uchicago.edu/~archer/cgimodels/isam.html

and here is a homework assignment
http://shadow.eas.gatech.edu/~jean/paleo/sets/undergrads1.pdf

There ensued a great deal of posing on Doc's part and confusion ensued about rather simple kinetics and similar things. Thus this post and some which will follow.

Eli has created a spreadsheet, called Box for Doc's boots, and the anonymice are free to play with it and improve thereon. You can download it at Rabett Labs, a new Google Group. The file walks you through a number of simple models. Eli does not have the space and you the patience to go through this in detail, but we can start with the simple stuff which will give you an idea how to roll your own. There are lots of better ways to do this. Symbolic algebra (and more) packages such as Mathematica, Maple and Mathcad spring to mind, as do such things as Origin and Igor on the spreadsheet/graphing side, but lots of people have Excel, and the folks that have the other packages, probably don't need this.

On the first sheet called first order decay, you get to play with a simple, first order decay where the rate of change of [A] at any time t is simply proportional to the the amount of A. The rate equation for this is

d[A]/dt = -k[A]

and if you don't speak calculus you can just read that as the rate of change of the concentration of A with time is equal to some number (the rate constant, ka) times the concentration of A. You can change the rate constant, and the initial concentrations. Eli, being a simple Rabett used Excel, and a simple differential equation solver called the Euler method. You can Google it, but the idea is that if you measure A at some time (t), and then later at (t+dt) then

[A(t+dt)] = [A(t) ]- k[A(t)]dt

This works if dt is small. In this case there is an exact solution A(t) = A(0) exp(-kt). Eli set the spreadsheet up so that you enter negative numbers (-k) for the rate constants.

This, basically is the idea that DocMartyn was pushing. The problem, of course, is that you move the carbon from the fossil fuel deposit into the atmosphere, and from there it goes into the land (soils/vegetation/rabetts) and the upper ocean, but it does not get lost, it comes right back into the atmosphere in a few years.

To model this, we have a simple two box model called "Opposing Reactions" You have box A and box B. And not only can you move carbon from box A to box B, but also from box B to box A. That looks like this

The rate constants are kab and kba respectively. After some time, the system will come to equilibrium. The rate equations are

d[A]/dt = - kab [A] + kba [B] and d[B]/dt = - kba [B] + kab [A]

if you add the two you find that d([A]+ [B])/dt = 0 , in other words the total amount of carbon just moves between the two boxes. There is an exact solution of the system of equation, but here we simply use Euler integration. The ratio of the equilibrium values of [A] and [B] are simply given by

kab / kba = [B]eq/[A]eq

You can play with this by changing the initial amounts of [A] and [B] or the rate constants at the top of the spreadsheet. Remember this is a toy you can play with to get some feel for the system. If things start oscillating wildly or diverge in strange ways you probably have to decrease either the step size or the rate constants. There are a couple of other simple spreadsheets. The next, CO2 pulse, shows what happens if you push a pulse of CO2 into the atmosphere. This is followed by "fossil fuel, where a constant amount of CO2 enters the two box system each time step.

We are now in position to put a more realistic carbon box model together. We can use the figure up towards the top to estimate how many Gt of carbon there are in each reservoir. We will for now exclude the geological reservoirs. Carbon moves very, very slowly into and out of rocks and sediments and we can treat them as being roughly constant. The first thing that one observes is that the deep ocean, with ~38,000 Gt C is MUCH bigger than the Atmosphere ( ~750) and the upper or surface ocean (~ 1000) and the land (~2000). The land includes both C in soils and in biosystems.

Us bunnies like to keep things simple and Mom Nature has helped us out. To a first approximation there is no flow of carbon between the land and the upper or deep ocean and between the atmosphere and the deep ocean for sure. Think about that for a moment, they really don't touch much. But Mom ain't that nice. We have been using first order rates to describe the flows between the various boxes. What that means is the change per unit time is equal to -kxy Mx where Mx is the mass of carbon in an box and kxy is the rate constant. You can see what happens when everything is linear in the spreadsheet called linear box model. One of the things you have to do in this model and the more realistic one below is adjust the rates so that with no fossil carbon flowing into the system they balance each other and the flow into each reservoir equals the flow out. Eli has done this in the spreadsheet called equilibria.

The linear model does not work because two of the flows are actually highly non-linear, the flow from the surface of the (upper) ocean into the atmosphere, and the one from the atmosphere into the land/soils. The former is governed by a series of chemical equilibria between CO2, H2CO3, and the negative ions HCO3(-1) and CO3(2-) as well as other ionic species dissolved in sea water. Roger Revelle's major contribution was giving us an understanding of these complex equilibria. The bottom line is that the flow from the upper ocean to the atmosphere is proportional to the ninth power of the mass of carbon in the upper ocean (Mu^9).

The rate at which CO2 flows from the atmosphere into the land is controlled by photosynthesis. This is much slower than linear, proportional to Ma^0.2. This flow depends more on biological and solar factors than the amount of CO2 in the atmosphere.

Rabett Labs created the final two worksheets which include the correct functionality for the fluxes, between the land, air, and upper and deep oceans. In the first, the emission of CO2 from fossil fuel is constant over time. That is called "Constant Fossil". In the last one, called "Stop Emitting" you can enter your own scenerio and see what the effects are. In the example shown below Eli let emissions of CO2 from fossil fuel continue for 200 years at a bit more than today's rate and then cut it off. Notice that the decay takes hundreds of years. The

Anyhow, have fun with the new toy. David Archer has a more realistic model, Shodor has one,

the Maryland Virtual High School has one, here is yet another and there are more out there. The purpose of the Rabett Lab model is to simplify things as much as possible for the mice to play.