Time scales in atmospheric chemistry: Theory, GWPs for CH4 and CO, and runaway growth

. Atmospheric CH 4 perturbations, caused directly by CH 4 emissions or indirectly by those of CO are enhanced by chemical feedbacks. They can be diagnosed in terms of the natural modes of atmospheric chemistry that are general solutions of the continuity equations. Each mode is a pattern in the global distribution of all chemical species, and each has a single time-constant that accurately describes its exponential decay about a given atmospheric state. This mathematical theory extends earlier work and is general for A formal proof relates the steady-state distribution and its lifetime to the integral of the true time-dependent response (properly included in the recent IPCC assessment). Changes in CO are also known to perturb CH4; however, the impact of CO emissions on climate has not been formally assessed in part because the short lifetime of CO (months) relative to that of CH 4 (decade) was believed to limit the integrated impact. Using the IPCC model studies, this theory predicts that adding 5 CO molecules to today's atmosphere is equivalent to adding 1 CH 4 molecule with the same decadal duration as direct CH 4 addition. Extrapolating these results, CH 4 sources would have to triple before runaway growth, wherein CH 4 emissions exceed the oxidizing capacity of the troposphere.


Introduction
The abundance of trace gases and aerosols in Earth's atmosphere can determine the habitability of the planet. Human activities have significantly altered the atmospheric cycles of most trace gases, leading to their rapid increase over the past century. Methane is a greenhouse gas whose growth since the pre-industrial era has been documented in ice-core bubbles and recently in atmospheric measurements (Etheridge et al., 1992;Steele et al., 1992). Certain human activities (e.g., raising cattle, growing rice, use of natural gas) lead to direct release of CH 4 into the atmosphere. A major goal of methane-related research is to identify and quantify these sources and then to predict how CH 4 and other trace gases are affected (Cicerone and Oremland, 1988). Increases in CH 4 directly enhance the trapping of terrestrial infrared radiation, but they also perturb tropospheric chemistry: make 03 in the upper troposphere (another greenhouse gas), reduce hydroxyl radical (OH) concentrations, increase carbon monoxide (CO). CH 4 increases also reach into the stratosphere: raise stratospheric H20 levels, interfere with CFC-induced ozone depletion, and thence alter the amount of solar ultraviolet driving tropospheric photo-Copyright 1996 by the American Geophysical Union.

Paper number 96GL02371
0094-8534/96/96GL-02371 $05.00 chemistry. The most difficult problem of atmospheric chemistry today is evaluating the cumulative and collective environmental impacts of all such chemical feedback loops associated with a single action, i.e. emission of one gas at one location.
Theoretical studies have long noted unusual behavior when CH 4 or CO were perturbed in tropospheric chemistry models (Chameides et al., 1976). Sze (1977) found CO perturbations to last several decades even though the CO lifetime was only a few months. Later studies identified the OH-CH 4 feedback as causing greater-than-proportional increases in steady-state CH 4 concentrations (Isaksen and Hov, 1987), e.g., +10% in emission yields +15% in concentration. Fisher (1993) found that small CH 4 pulses had e-fold times greater than the lifetime as defined by the budget (abundance/loss), contrary to the view that the average loss frequency of large reservoirs (e.g., tropospheric CH4) should represent a time scale for change (the turn-over time or This pap?r develops a general mathematical theory of eigenvalue•rlethods for atmospheric chemistry in Section 2 that applies generally to multi-dimensional chemistry and transport models. A formal relationship between the natural modes and a steady-state distribution and lifetime is derived. Section 3 re-examines the one-box {CH 4, CO, OH)-system of Prather (1994) with this formalism using multi-dimensional model studies to constrain the chemical feedbacks. IPCC's (1995) use of an extended'lifetime to assess small CH 4 perturbations is shown to be rigorously correct; however, generation of long-lived, CH4-1ike perturbations from CO and other short-lived gases, predicted here, has not yet been addressed. The degree of non-linearity as a function of CH 4 source is examined, as well as the potential for run-away CH 4 growth. Section 4 summarizes the advantages of diagnosing action-and-effect in terms of natural modes.

Eigenvalue Theory -Atmospheric Chemistry
The continuity equation (1) for the concentration of each atmospheric species at a given spatial location, x k, can be expressed in terms of its local net chemical production, Pk, and its transport tendency, VOk, a flux divergence. These individual equations can be written as a single vector equation (21._X being a vector of species concentrations x k where k=i+m(j-1), i--l:m is the species index and j = l:n is the spatial location. The vector P represents the individual Pk, (function of different species at the same location); and the vector VO, the individual V• (like species at neighboring locations). The nmxnm Jacobian matrix J is defined (3) as the partial derivative of each equation (1) with respect to 2597 each independent variable, x k. Let the vector of chemical species Xø(t) be a time-dependent solution to (2), then solve for a perturbation, X ø + D, by expanding equation (2) in a Taylor series (4). Only the chemical terms are non-linear and appear in (4) as second-order terms. To first-order, the perturbation vector D satisfies the matrix equation (5). If a perturbation A k is an eigenvector of J with eigenvalue -c k, then the vector solution to (5) decays, maintaining its pattern, as a simple exponential (6).
The natural modes in atmospheric chemistry are the eigenvectors A k (k--l:nm) whose set spans the nm-dimensional space of chemical species. Any perturbation D has a unique expansion with coefficients d k and decays with an ensemble of decay frequencies corresponding to the eigenvalues (7). These modes are basic properties of the atmospheric state, not of the perturbation. The vectors A k are independent of the perturbation D provided that the Jacobian varies slowly: , meaning the terms of order D 2 in (4) are negligible. In a fully linear system J is constant, and any state, not just a small perturbation, is described by natural modes (7). A real, asymmetric matrix like J can have positive, negative, or complex eigenvalues. Provided that all eigenvalues are negative (-c < 0), any perturbation decays. If there were a single positive eigenvalue of J[Xø], then the system would be locally unstable about X ø since any realistic perturbation would likely excite the unstable eigenvector. Although complex eigenvalues in atmospheric chemical systems are rare, such oscillating "clocks" are known (Nitzan and Ross, 1973).

Jik = 3(dX/dt)i / 3Xk -' •)(P)i/•)Xk -•)(VO)i/•)Xk (3) d(Xø(t)+D)(dt = (P[Xø(t)+D] -VcD[Xø(t)+D])• (4) = (P[Xø(t)] -VO[Xø(t)])• + •']k=l:nm{Oq(P)i/OqXk -3(VcD)i/3x k }(D) k + Order(32p/3x 2 D 2) = d(Xø(t))i/dt + •k=l:nmJik(D)k + Order(oq2p/oqX 2 D 2) dD/dt = J D (5) dAk/dt = -c k A k • Ak(t) = Ak(0) exp[-ckt ] (6) D(0)--•k=l:nmdk Ak :=:> D(t)--•k=l:nmdk Ak exp[-Ckt ] (7)
A steady-state distribution and its lifetime can be related to the natural modes. Continuous forcing of the system at the same rate over an infinite time (i.e., integrating (2)) builds up a steady-state distribution SS(z), where the spatial variable z is explicitly noted. Consider the natural modes A•'(z) of a single-species system (m=l) where each mode is a spatial pattern (with n degrees of freedom). The rate of emission of a gas (kg/sec) integrated over a brief period (1 sec) produces a spatial pattern of abundance F(z) (kg), which maps onto the natural modes (8) (e.g., surface emissions place a certain amount in the lowest layer). The modes A k are dimensionless, and the coefficients fk have units of abundance (kg). The individual modes represented in each burst of emissions, F(z), decay according to their eigenvalue frequencies c k. The steady-state distribution (kg) is com-I•rised of an infinite history of emissions, F(z,t), from all previous times (9). The normalization factor, u (=l sec here), converts the sum to an integral. The steady-state lifetime Tss is the total burden (kg) divided by emissions (or loss rate, kg/s), and can be derived (10), recognizing that the steady state is a mix of modes (9) each with its own loss frequency. It is simply the sum of the lifetimes of the individual modes, 1/Ck, weighted by the abundance in that mode, fkak , where a k --IAk(z)dz (10). Tss is often mistakenly assumed to be a fundamental time constant of the system; whereas, the true system time scales for perturbations are the inverse eigenvalues of the Jacobian.
The net environmental impact, I, of trace-gas emissions is measured by integrating over the resulting atmospheric perturbations (e.g., kg-sec or ppt-yr). This integral is usually assumed (IPCC, 1995;WMO, 1995) to be the product of the steady-state distribution and the corresponding mean lifetime, viz I(z) = SS(z) Tss. Using natural-mode decomposition, the cumulative impact of a scaled pulse, s F(z), is calculated by explicit integration (11) and can be related to a steady-state distribution with the same global content by selecting the appropriate scale factor, s, (12). Thus equations ( however, for more realistic 2-D and 3-D models with many chemicals (nm>106), calculation of the modes will need a different approach. The long-lived modes may be found numerically by following out the tail of a perturbation and fitting it to a sum of exponential decays; and some information about the matrix A may be found by noting how different perturbations excite these principal modes.

Non-linearity in the {CH4, CO, OH}-system
The chemical cycles of CH 4, CO, and OH in the global atmosphere are coupled (Levy, 1972). They also involve other species (e.g., NO, C2H6) and transport that connects different photochemical regimes (e.g., Isaksen and Hov, 1987;Cicerone and Oremland, 1988). Nevertheless, this system is usefully studied with a one-box model (Prather, 1994) as defined in Table I. Here the rate coefficients, air density, and source terms are selected to constrain the one box to represent a "global mean": CH4=1700 ppb, CO=100 ppb, CH 4 budget litbtime = 9.6 yr. The single remaining free variable is diagnosed t¾om the CH 4 feedback factors, R=l/(l-151nTc•n/151nCH4), reported by multi-dimensional models in IPCC (Prather et al., 1995). In this formalism, R

= -J t t/c t, the time scale of the primary mode divided by the lifetime of CH 4. In the one-box model, R sums up complexities not included such as the partial recycling of OH by ROO+NO reactions during oxidation of CH 4.
For typical feedbacks, R=l.6 (IPCC range 1.2-1.7), about 50% of OH production is lost to CH 4 and CO. The lifetimes of CH 4, CO, and OH are 9.6 yr, 88 d, and 0.71 s, respectively. The modes are linearly independent but not orthogonal; and thus an isolated perturbation to CH4, CO, or OH becomes a unique mix of all three modes (see Table 1
The coupling of CO perturbations to CH 4 is not new (Sze, 1977); however, the tbrmalism of natural modes allows us to write explicitly the magnitude and time dependence of the coupling as given in Table 1 1.4-1.6, IPCC, 1995), the previous history of CH 4 is mainly linear; however, the thture may be less so. A doubling of the current source is predicted to lower OH concentrations so much that CH 4 reaches 10,000 ppb, a factor-of-6 increase. If CH 4 sources treble, there is runaway growth, and no steady-state solution exists. In the case R-2.7 (unlikely from IPCC results), the change since pre-industrial is already non-linear, and the preindustrial source was more than half as large as that today. In this case, a runaway system occurs with only a 50% increase in today's CH 4 emissions. This instability needs to be reexamined with multi-dimensional models that couple varied photochemical regimes (tropics/poles, tropo/stratosphere).
The Short-lived gases like CO have can affect indirectly the trends of Other gases like CH 4 and tropospheric 0 3 (e.g., Thompson and CicerOne, 1986). These "indirect" perturbations are shown here to have the same pattern and time scale as those caused by direct addition of the long-lived species.
Although extremely difficult to evaluate with confidence, the emissions of very short-lived gases (e.g., CO, NO, C2H6) should be treated as having a long-term environmental impact and given an ODP (ozone depletion potential) or GWP in proportion to the amplitude of the induced longlived natural modes. Individual perturbations couple across all modes of tropospheric and stratospheric chemistry. We can expect emissions of a short-lived gas such as CO (months) to lead to a long-lived perturbation in CH 4 (decade) as shown here and perhaps even a longer-lived perturbation in N20 (century). Extension of this method to examples with transport (CH3Br in the stratosphere, troposphere and ocean) and uv radiative coupling (N20, NOy, and O 3 in the stratosphere) is presented in a following paper.