AT760 Global Carbon Cycle. Assignment #3 Due Friday, May 4, 2007 Atmospheric Transport and Inverse Modeling of CO 2

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1 AT760 Global Carbon Cycle Assignment 3 Due Friday, May 4, 2007 Atmospheric Transport and Inverse Modeling of CO 2 In this exercise you will develop a very simplified model of the mixing of the global atmosphere. Sources and sinks of CO 2 at the surface of the model will be specified, and a global distribution of timemean CO 2 will be obtained. Then you will sample the CO 2 from your model, and try to estimate the original map of sources and sinks using the Bayesian synthesis inversion method discussed in class. Problem 1: The forward transport model Calculate [CO 2 ] concentrations from fluxes Consider an atmosphere consisting of six wellmixed boxes. The boxes divide the Earth by zones: Northern, Tropical, and Southern; and by altitude: lower and upper troposphere. Divide the zones at 30 N and 30 S latitude. In the vertical, the lower boxes will represent the air from the surface to 800 mb (about 2 km), and the upper box will span from there to 200 mb (which we ll think of as the tropopause). We ll ignore the stratosphere. Transport: Pretend the atmospheric transport is just by mixing, rather than organized advection, convection, and turbulence. Each zone mixes vertically, and adjacent zones mix horizontally in the upper troposphere. We ll ignore horizontal mixing in the lower troposphere. Predict the rate of change of concentration in each box due to surface fluxes (sources and sinks) and mixing with each adjacent box. Because the masses of the upper and lower boxes differ, you ll also have to take this into account. So, for each box M!C!t N = F + " M (C C) i i i=1 $ i 1

2 AT760 Assignment 3 Due May 4, 2007 where M is the mass of the current box, C is the concentration of CO 2, F is the surface flux of CO 2 into the box, C i is the mixng ratio of CO 2 in the i th neighboring box, and τ i is the time scale for mixing between box i and the current box. There are N neighboring boxes with which to mix. The timescale for vertical mixing in the tropics is 10 days, because of the vigorous convective clouds there, but 20 days outside the tropics. Horizontal mixing is slower: use 50 days for mixing between the tropics and the higher latitudes of both hemispheres, but only in the upper boxes. We ll assume that horizontal mixing between the lower boxes is negligible. Units: You can work this problem in a variety of ways, but here s my suggestion. Predict the mass mixing ratio of carbon (not CO 2 ) in each box, so that C is kg C per kg air. That way M is just the mass of each box in kg. (Don t worry about the extra mass of C, which is negligible compared to M.) Surface fluxes S are then to be expressed in kg s 1, and mixing times are in s. I ll give you fluxes in Gt C yr 1, and mixing times in days, but you can just convert them. 1 GtC is kg or g. CO 2 concentrations are usually written in ppm by volume, which is the same as mole fraction or µmol mol 1. This can be a bit confusing, because you ve predicted kg C kg air 1. The molecular weight of carbon (not CO 2 ) is 12 g mol 1, and the molecular weight of dry air is 29 g mol 1. The mass of air in each box is M = A Δp/g where A = the area of the surface of the box (m 2 ), Δp is the pressure depth (in Pascals, 1 mb = 100 Pa), and g = 9.81 m s 2. As it turns out, exactly half the area of the Earth s surface lies between 30 N and 30 S latitude. The radius of the Earth is 6371 km. Let the surface pressure be mb. Sources and Sinks: Prescribe the following surface fluxes for your model = 2 Gt C yr 1 (uptake by the Southern Ocean) = +1 Gt C yr 1 (deforestation plus Equatorial upwelling minus Cfertilization) = +5 Gt C yr 1 (fossil fuel combustion minus a big land sink) Numerics: I suggest using an explicit timedifferencing scheme because it s the easiest to code. This is a diffusion problem, and explicit schemes are unstable for diffusion, but it should be OK if your model time step is much shorter than the fastest mixing time in the problem. I suggest a time step of 1 day. Procedure: Start the model from an initial condition with 375 ppm of CO 2 everywhere, and run it for 3 years. Make a plot of the concentration in each box over time, using different linestyles or colors for each box. 2

3 AT760 Assignment 3 Due May 4, 2007 Problem 2: The inverse model Estimate fluxes from [CO 2 ] concentrations Now suppose you don t know the surface fluxes,, and, but have a set of observations of [CO 2 ] and want to estimate the fluxes using the transport model you developed in problem 1. Consider an observing system in which we make measurements of the mean CO 2 concentration in each of the lower boxes in the a given year. Each measured concentration results from the influence of the three fluxes after mixing by atmospheric transport. There is also some background or offset concentration that all three observations share. So the concentration at the observing stations (C NL, C TL, C SL ) can be represented in terms of the three unknown fluxes (,, ) as a linear system of three equations in three unknowns (four if you count C 0 ): C NL =!C NL! +!C NL! +!C NL! C TL =!C TL! +!C TL! +!C TL!. C SL =!C SL! +!C SL! +!C SL! For the very simple problem here, it can be shown the time variation is irrelevant after equilibrium gradients have been established, and you can therefore just substitute the final value of your simulated concentrations for annual means. This linear system can be rewritten in matrix form as where d = G m! d = " C NL C TL C SL $!!!,!!G = % ( ) ' ' ' ' ' ' ' ' ' +!!,!!!m = ", $. % 3

4 AT760 Assignment 3 Due May 4, 2007 Solution: To estimate the fluxes, solve the matrix equation d = G m for the vector m, subject to prior constraints on the fluxes, and considering uncertainties in both the fluxes and the concentration data. The inversion should be done using the weighted leastsquares method discussed by Tarantola (1987) and Gurney et al (2002), as presented in class. Observations: Let there be a single observation in each surface grid box, representing conditions at the end of your run. The madeup observations d = [7.345, 4.750, 2.803] represent changes from the initial condition of 375 ppmv at the beginning of the simulation in each surface box at the end of 3 years. d is a vector with 3 elements. Jacobian matrix G: The partial derivatives in each term are simply the annual mean response of [CO 2 ] at each site to a flux of unit strength in each region. These can easily be found by just running your transport model from problem 1 to equilibrium three times, with a flux of 1 GtC yr 1 in each of the three zones and zero in the other two. Take the final concentration in each box and subtract the initial 375 to obtain each element of G. G is a 3x3 matrix. Error covariance in the observations are formally given by the matrix C d, whose main diagonal contains σ 2 of each observation of CO 2. Traditionally this would be thought of as measurement error, but for the CO 2 problem it contains primarily representation error (degree to which individual observations differ from the zonal annual mean in the model) and transport error. Use σ=0.2 ppm (σ 2 =0.04 ppm 2 ) for each of these, and assume they are uncorrelated. That means the offdiagonal elements will all be exactly zero, and that the inverse C d 1 is found by replacing the elements on the main diagonal of C d with their reciprocals. C d 1 is a 3x3 matrix. For prior constraints, assume that the fluxes in each box are the ones given in Problem 1: m prior = [+5, +1, 2] Gt C yr 1. Let uncertainty in these prior estimates have a standard deviation of 1.0 Gt C yr 1 and no correlation in the errors. This corresponds to C m matrix that has 1.0 down the diagonal and 0.0 elsewhere. As for the error covariance of the observations, the inverse of C m is therefore the same as C m itself. C m 1 is a 3x3 matrix. Remember (or look up) the rules for multiplying and adding matrices, and for finding their inverse. You will find that the biggest matrix you have to work with in this problem is a 3x3, so it would be easy to do this whole problem with pencil and paper (no coding). You might have to remind yourself how to do Gaussian elimination to get the final answers. You may you a computer for this if you prefer, but in either case, please show your work. Don t just hand in a final answer. 4

5 AT760 Assignment 3 Due May 4, 2007 The solution is found from m est = m prior + [G T C 1 d G + C 1 m ] 1 G T C 1 d [d G m prior ] with uncertainty in the final estimate given by C m = [G T C 1 d G + C 1 m ] 1 Here m prior is your prior estimate of the fluxes C m 1 is the reciprocal of the covariance matrix of prior flux estimates C d 1 is the reciprocal of the error covariance matrix of the data C m is the a posteriori covariance matrix. Uncertainty (1 σ) in your final estimates (in Gt C yr 1 ) is given by the square root of each element of the main diagonal of C m. Make a table of your results as follows: NH Tropics SH Prior Estimate Uncertainty 5