Modeling CO2 Decay with a Multi-Box Model

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Hello! Let's begin the final lesson in our module on the Global Carbon Cycle.

Introduction

In our previous lesson, we established a clear accounting of where anthropogenic CO₂ emissions go on an annual basis. We found that roughly half remains in the atmosphere (the "airborne fraction"), with the other half being absorbed by the ocean and land sinks.

This annual partitioning, however, doesn't tell the full story. A crucial question remains: once CO₂ is added to the atmosphere, how long does it stay there? Answering this requires moving from a static, annual budget to a dynamic model of the carbon cycle.

Today, we will explore the long-term fate of an atmospheric CO₂ pulse. You'll see that unlike a simple radioactive decay process described by a single half-life, the removal of CO₂ from the atmosphere is a multi-stage process governed by exchanges with different reservoirs, each operating on a different timescale.

Your background in differential equations and stochastic systems will be particularly useful here, as we'll be dealing with systems of reservoirs and fluxes. The core of today's lesson is to model the decay of an atmospheric CO₂ pulse using a multi-box model to understand its long residence time.

1. From Single-Box Residence Time to Multi-Box Adjustment Time

The simplest way to model a reservoir is with a one-box model. Let's consider the atmosphere as a single, well-mixed box. The rate of change of the mass of a substance in the box is the difference between sources and sinks.

CHAPTER 3. SIMPLE MODELS

To formalize this, please read the first part of the 'Simple Models' chapter from a textbook on atmospheric chemistry. It introduces the one-box model, the mass balance equation, and the concept of residence time.

Please read section 3.1 'ONE-BOX MODEL' (pages 23-25). Focus on the mass balance equation (3.7) and how it simplifies to equation (3.8) for a first-order loss process. Pay attention to the definition of lifetime (τ) in equation (3.1).

As the text explains, for a substance with a first-order loss process (where the loss rate is proportional to the mass present), the mass balance equation is:
$\frac{d m}{d t} = S - k \cdot m$
Here, $S$ is the source rate, $m$ is the mass in the box, and $k$ is the loss rate constant. The residence time (or lifetime) is defined as $τ = 1/ k$ . It represents the average time a molecule stays in the box.

We can make a quick, but ultimately misleading, calculation for the residence time of a CO₂ molecule in the atmosphere. From our previous lesson's diagram, the atmosphere holds about 890 GtC, and the total natural fluxes out of it (to the ocean and land) are about 130 + 80 = 210 GtC/yr.

$τ_{r es i d e n ce} = \frac{Total Mass}{Total Flux Out} = \frac{890 GtC}{210 GtC/yr} \approx 4.2 years$

This number is surprisingly short and represents the average time a single CO₂ molecule spends in the atmosphere before being swapped with one from the ocean or land. However, this is not the time it takes for the atmosphere to get rid of an excess amount of CO₂. Why? Because the ocean and land are not perfect, permanent sinks. They are themselves reservoirs that exchange CO₂ back with the atmosphere.

To capture this two-way exchange, we need a multi-box model.

2. The Multi-Box Model and the Impulse Response Function (IRF)

Imagine a system with multiple connected boxes: one for the atmosphere, one for the surface ocean, one for the deep ocean, one for the terrestrial biosphere, etc. An injection of CO₂ into the atmospheric box will be absorbed by the other boxes, but some of it will also be returned to the atmosphere later.

This system of exchanges can be described by a set of coupled first-order differential equations, one for each box.

CHAPTER 3. SIMPLE MODELS

The same textbook chapter extends the single-box concept to multiple boxes. This section provides the mathematical foundation for what we're about to discuss.

Please read the first part of section 3.2 'MULTI-BOX MODELS' (pages 28-29), up to equation (3.14). This shows how coupled differential equations arise from describing the mass balance for interconnected boxes.

When you solve this system of linear ODEs for the atmospheric box following an initial "pulse" of CO₂, the solution is not a single exponential decay, but a sum of exponential terms. Each term corresponds to a different timescale of exchange within the Earth system.

This decay profile is formally known as the Impulse Response Function (IRF). It describes the fraction of an initial emission pulse that remains in the atmosphere at time $t$ .

Carbon dioxide and climate impulse response functions for ...

Now let's look at how this concept is applied specifically to CO₂. This research paper by Joos et al. (2013) is a cornerstone study that synthesizes results from many carbon cycle models. It provides the precise mathematical form of the CO₂ IRF.

Please read the following sections: Section 2.1 (pages 2795-2796): Focus on the definition of the IRF and its representation as a sum of exponentials in Equation (5). This is the key formula for our lesson. Section 2.1 (page 2796, paragraphs starting 'CO2 is, unlike most other agents...'): This part explains why CO₂ requires this multi-timescale approach, linking it to the different carbon reservoirs.

As detailed in the paper, the IRF for CO₂ is well-approximated by the following function:
$IRF (t) = a_{0} + \sum_{i = 1}^{n} a_{i} \cdot exp (- \frac{t}{τ _{i}})$
Where:

$t$ is the time since the pulse.
$a_{0}$ is the fraction of the pulse that remains in the atmosphere after a very long time (approaching millennia).
Each $a_{i}$ is the fraction of the pulse that decays with a characteristic timescale $τ_{i}$ .
The sum of all coefficients ( $a_{0} + \sum a_{i}$ ) equals 1.

The different $τ_{i}$ terms represent distinct physical processes:

A fast timescale (years to decades) associated with uptake by the land biosphere and the ocean surface.
A medium timescale (decades to centuries) associated with circulation into the deeper ocean.
Very long timescales (millennia) associated with ocean-sediment interactions, which are needed to neutralize the remaining $a_{0}$ fraction.

This multi-exponential form is the reason for CO₂'s "long tail"—a significant portion of the gas persists in the atmosphere for centuries and millennia.

3. Quantifying the Decay of a CO₂ Pulse

Let's now use the concrete results from the Joos et al. (2013) paper to see this in action. The paper provides the best-fit parameters for the multi-model mean IRF, which synthesizes our best understanding of the carbon cycle.

Figure 1 from Joos et al. (2013)

The multi-model mean IRF shown in the figure above is fitted with the following parameters (from Table 5 in the paper):

$IRF (t) = 0.2173 + 0.2240 e^{- t /394.4} + 0.2824 e^{- t /36.54} + 0.2763 e^{- t /4.304}$

Exercise: Using this equation, please calculate the approximate fraction of a CO₂ pulse remaining in the atmosphere at three key milestones: 20 years, 100 years, and 1000 years.

Click to see the solution

At t = 20 years:
$IRF (20) = 0.2173 + 0.2240 e^{- 20/394.4} + 0.2824 e^{- 20/36.54} + 0.2763 e^{- 20/4.304}$
$IRF (20) \approx 0.2173 + 0.2131 + 0.1633 + 0.0027 \approx 0.596$
So, after two decades, about 60% of the CO₂ pulse is still in the atmosphere.
At t = 100 years:
$IRF (100) = 0.2173 + 0.2240 e^{- 100/394.4} + 0.2824 e^{- 100/36.54} + 0.2763 e^{- 100/4.304}$
$IRF (100) \approx 0.2173 + 0.1741 + 0.0183 + (\approx 0) \approx 0.4097$
After a century, about 41% remains.
At t = 1000 years:
$IRF (1000) = 0.2173 + 0.2240 e^{- 1000/394.4} + 0.2824 e^{- 1000/36.54} + 0.2763 e^{- 1000/4.304}$
$IRF (1000) \approx 0.2173 + 0.0177 + (\approx 0) + (\approx 0) \approx 0.235$
Even after a millennium, nearly a quarter (24%) of the initial pulse is still airborne. This is primarily the $a_{0}$ term plus the small residual from the longest-timescale exponential.

These calculations demonstrate the profound persistence of atmospheric CO₂. The initial rapid drop is misleading; the long-term adjustment is governed by much slower processes, ensuring that our emissions today will continue to affect the climate for many generations.

Conclusion

This lesson completes our journey through the global carbon cycle, from sources to sinks to long-term persistence.

Key Takeaways:

A simple one-box model gives a misleadingly short "residence time" for CO₂ because it ignores the return fluxes from the ocean and land reservoirs.
A multi-box model is required to accurately capture the dynamics of the carbon cycle. The decay of an atmospheric CO₂ pulse in such a model is described by an Impulse Response Function (IRF).
The CO₂ IRF is a sum of multiple exponential terms with different timescales ( $τ_{i}$ ), reflecting various uptake mechanisms (land biosphere, ocean surface, deep ocean).
This multi-exponential decay results in a "long tail": after an initial rapid decrease, a significant fraction of emitted CO₂ remains in the atmosphere for centuries to millennia.
Specifically, about 40% of a CO₂ pulse is still airborne after 100 years, and over 20% remains after 1000 years. This long atmospheric lifetime is fundamental to why CO₂ is such a potent long-term driver of climate change.

Preview of the Next Module:

We have now concluded our study of the foundational physics and carbon cycle dynamics. In the next module, "Evidence and Data Analysis," we will shift our focus to the observational data that underpins our understanding of climate change. We will begin by critically assessing how the global temperature record is constructed, examining the statistical adjustments required to create a reliable time series from disparate historical measurements.