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Fast–slow climate dynamics and peak global warming

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Abstract

The dynamics of a linear two-box energy balance climate model is analyzed as a fast–slow system, where the atmosphere, land, and near-surface ocean taken together respond within few years to external forcing whereas the deep-ocean responds much more slowly. Solutions to this system are approximated by estimating the system’s time-constants using a first-order expansion of the system’s eigenvalue problem in a perturbation parameter, which is the ratio of heat capacities of upper and lower boxes. The solution naturally admits an interpretation in terms of a fast response that depends approximately on radiative forcing and a slow response depending on integrals of radiative forcing with respect to time. The slow response is inversely proportional to the “damping-timescale”, the timescale with which deep-ocean warming influences global warming. Applications of approximate solutions are discussed: conditions for a warming peak, effects of an individual pulse emission of carbon dioxide (CO\(_{\mathrm{2}}\)), and metrics for estimating and comparing contributions of different climate forcers to maximum global warming.

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Notes

  1. This can be seen by writing Eq. (1) as \(c_{\mathrm{s}}\frac{{\text {d}}T_{\mathrm{s}}}{{\text {d}}t}=-B\left( T_{\mathrm{s}},T_{\mathrm{d}}\right) -H\left( T_{\mathrm{s}},T_{\mathrm{d}}\right) +F(t)\), where \(B\left( T_{\mathrm{s}},T_{\mathrm{d}}\right) =\beta T_{\mathrm{s}}+\left( \eta -1\right) \gamma \left( T_{\mathrm{s}}-T_{\mathrm{d}}\right)\) describes change in radiative flux at the top of the atmosphere (Held et al. 2010) and \(H\left( T_{\mathrm{s}},T_{\mathrm{d}}\right) =\gamma \left( T_{\mathrm{s}}-T_{\mathrm{d}}\right)\). As the system approaches equilibrium, change in radiative flux varies only with \(T_{\mathrm{s}}\) as \(\beta T_{\mathrm{s}}\); but in the beginning of the response where \(T_{\mathrm{d}}\approx 0\), the sensitivity is higher, i.e. \(\left( \beta +\left( \eta -1\right) \gamma \right) T_{\mathrm{s}}\). The magnitude of this difference depends on the value of \(\eta\), with generally \(\eta >1\) (Held et al. 2010; Winton et al. 2010).

  2. This is based on heat capacity of water of 4180 J kg\(^{{-1 }}\)K\(^{{-1}}\), density of seawater of 1030 kg m\(^{{-3}}\), and approximating that the oceans cover 70 % of Earth’s surface, following Geoffroy et al. (2013a).

  3. We use estimates of \({\text {BC}}_{{\text {PI}}}=32\times 10^{6}\) kg and \(\tau _{{\text {BC}}}=10\) days, based on results in Skeie et al. (2011a); see Seshadri (2015) for details on estimation of these quantities. Present emissions and mean atmospheric burden of BC (for 2014) are estimated as \(8.6\times 10^{9}\) kg and \(200\times 10^{6}\) kg respectively, based on Skeie et al. (2011b) and following Seshadri (2015).

  4. For N\(_{\mathrm{2}}\)O, preindustrial and present concentrations are taken as 0.27 ppmv and 0.32 ppmv respectively (Myhre et al. 2013), present emissions are 16.3 Tg N\(_{\mathrm{2}}\)O–N per year (Davidson and Kanter 2014), and atmospheric lifetime is 114 years (Pierrehumbert 2014). For CH\(_{\mathrm{4}}\), preindustrial and present concentrations are taken as 0.80 ppmv and 1.80 ppmv respectively (Myhre et al. 2013), present emissions are 350 Tg per year (Myhre et al. 2013), and atmospheric lifetime is 12 years (Pierrehumbert 2014).

  5. \(B_{T_{\mathrm{s}}}=\beta +\left( \eta -1\right) \gamma\), \(B_{T_{\mathrm{d}}}=-\left( \eta -1\right) \gamma\), \(H_{T_{\mathrm{s}}}=\gamma\), and \(H_{T_{\mathrm{d}}}=-\gamma\), and the tilde-variables are simply these aforementioned variables divided by \(c_{\mathrm{s}}\).

  6. These differ from eigenvalues estimated by applying the small-\(\varepsilon\) approximation to the characteristic polynomial of A, as can be shown. The perturbation approach above is essential to correctly estimate eigenvalues, because it alone preserves continuity of the eigenvectors’ components in the zero-\(\varepsilon\) limit.

  7. One might also view this as the condition for “adjusted forcing” \(F_{{\text {adj}}}(t)=\eta \gamma T_{\mathrm{d}}(t)+F(t)\) to peak and then decline [compare with Eq. (1)]. This can be verified by substituting the approximation for \(T_{\mathrm{d}}(t)\) and comparing results with the aforementioned discussion. This clarifies the origin of the effect that is approximately characterized by the damping-timescale. When F(t) begins to decrease the deep-ocean temperature \(T_{\mathrm{d}}(t)\) is still increasing, and the rate of decrease in radiative forcing must be large enough to compensate so that \(F_{{\text {adj}}}(t)\) can peak. The introduction of \(F_{{\text {adj}}}(t)\) decouples the upper box from the deep-ocean response, and peaking of the corresponding forcing \(F_{{\text {adj}}}(t)\) leads the fast subsystem forced by it, i.e. the EBM’s upper box temperature, to also peak.

  8. We treat \(m_{Pulse}\) and \(\triangle {\text {CO}}_{2,{\text {Pulse}}}(t)\) as having the same units, for example mass can be described in concentration units or concentration of CO\(_{\mathrm{2}}\) can be described by corresponding mass in the atmosphere.

  9. It can be shown that at the critical point \(t=t_{wp}\) where \(\dot{T}_{\mathrm{s}}\left( t_{wp}\right) =0\), maximum warming from the pulse follows the simplified equation \(\frac{c_{\mathrm{s}}T_{\mathrm{s}}\left( t_{wp}\right) }{\tau _{2}}\cong F\left( t_{wp}\right) +\frac{1}{\tau _{D}}F_{1}\left( t_{wp}\right)\).

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Acknowledgments

This research has been supported by Divecha Centre for Climate Change, Indian Institute of Science. The author is grateful to several colleagues for helpful discussion. Two reviewers made suggestions that substantially improved the paper.

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Correspondence to Ashwin K. Seshadri.

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Appendices

Appendix 1: Solution of differential equations

The equation to be solved is

$$\begin{aligned} \dot{\vec {u}}=A\vec {u}+\vec {f}(t) \end{aligned}$$
(48)

with initial condition \(\vec {u}\left( 0\right) =\vec {0}\). To solve we write the above equation as \(\dot{\vec {u}}-A\vec {u}=\vec {f}\), and multiply on the left by matrix exponential \({\text {e}}^{-At}\), yielding \({\text {e}}^{-At}\dot{\vec {u}}-{\text {e}}^{-At}A\vec {u}={\text {e}}^{-At}\vec {f}\), or

$$\frac{{\text {d}}\left( {\text {e}}^{-At}\vec {u}\right) }{{\text {d}}t}={\text {e}}^{-At}\vec {f}$$
(49)

using identity \(\frac{{\text {d}}}{{\text {d}}t}{\text {e}}^{-At}=-Ae^{-At}=-{\text {e}}^{-At}A\) [see, for example, Hirsch and Smale (1974)]. The last equation is integrated for

$$\begin{aligned} \vec {u}(t)={\text {e}}^{At}\int _{0}^{t}{\text {e}}^{-Az}\vec {f}(z){{\text {d}}z} \end{aligned}$$
(50)

using initial condition \(\vec {u}\left( 0\right) =\vec {0}\) and relation \(\left( {\text {e}}^{-At}\right) ^{-1}={\text {e}}^{At}\), which follows from the series expansion of \({\text {e}}^{At}\), i.e. \(I+At+\frac{\left( At\right) ^{2}}{2!}+\cdots\), where I is the identity matrix.

The matrix exponential in the solution above is simplified further by writing \(A=\varPhi \varLambda \varPhi ^{-1}\) with \(\varLambda\) the diagonal matrix of eigenvalues and \(\varPhi =\left[ \begin{array}{cc} \vec {\phi }_{1}&\vec {\phi }_{2}\end{array}\right]\) the \(2\times 2\) matrix containing the eigenvectors of A as columns. We know that \({\text {e}}^{At}=\varPhi {\text {e}}^{\varLambda t}\varPhi ^{-1}\) and \({\text {e}}^{-Az}=\varPhi {\text {e}}^{-\varLambda z}\varPhi ^{-1}\), which follow the series expansion of matrix \({\text {e}}^{At}\). Substituting this into Eq. (50) we obtain

$$\begin{aligned} \vec {u}(t)=\varPhi {\text {e}}^{\varLambda t}\int _{0}^{t}{\text {e}}^{-\varLambda z}\varPhi ^{-1}\vec {f}(z){{\text {d}}z}.\end{aligned}$$
(51)

Appendix 2: Approximation of fast contribution to global warming

Consider continuous function f(s) that is positive, and which can be approximated by a countable sequence of linear segments, so that \(\partial f/\partial s\) is constant in each segment, and with \(f\left( 0\right) =0\). Individual segments are indexed by \(i=0,1,....\) and \(\left[ s_{i},s_{i+1}\right]\) are their domains, with \(s_{0}=0\). Integration by parts shows that for coefficient \(c>0\)

$$\begin{aligned} \int _{0}^{s}{\text {e}}^{cz}f(z){{\text {d}}z} \end{aligned}$$
(52)

equals

$$\begin{aligned} \frac{{\text {e}}^{{\text {cs}}}f(s)}{c}-\frac{1}{c^{2}}\sum _{i}\left( \frac{\partial f}{\partial z}\right) _{i}\left( {\text {e}}^{{\text {cs}}_{i+1}}-{\text {e}}^{{\text {cs}}_{i}}\right) \end{aligned}$$
(53)

where \(\left( \frac{\partial f}{\partial z}\right) _{i}\) is the constant slope in segment i. Let \(\dot{f}_{{\text {sup}}}\equiv \sup _{i}\left( \frac{\partial f}{\partial z}\right) _{i}\) be the least-upper-bound of the segments’ slopes. Then

$$\begin{aligned} \frac{1}{c^{2}}\sum _{i}\left( \frac{\partial f}{\partial z}\right) _{i}\left( {\text {e}}^{{\text {cs}}_{i+1}}-{\text {e}}^{{\text {cs}}_{i}}\right) <\frac{1}{c^{2}}\dot{f}_{{\text {sup}}}\left( {\text {e}}^{{\text {cs}}}-1\right) \end{aligned}$$
(54)

If, furthermore, \(s\gg 1/c\), then \({\text {e}}^{{\text {cs}}}-1\cong {\text {e}}^{{\text {cs}}}\). Under the condition that fluctuations in f(s) are slow compared to cf(s), so that \(\dot{f}_{{\text {sup}}}/c\ll f(s)\), we obtain

$$\begin{aligned} \frac{1}{c^{2}}\sum _{i}\left( \frac{\partial f}{\partial z}\right) _{i}\left( {\text {e}}^{{\text {cs}}_{i+1}}-{\text {e}}^{{\text {cs}}_{i}}\right) \ll \frac{{\text {e}}^{{\text {cs}}}f(s)}{c} \end{aligned}$$
(55)

so that finally

$$\begin{aligned} \int _{0}^{s}{\text {e}}^{cz}f(z){{\text {d}}z}\cong \frac{{\text {e}}^{{\text {cs}}}f(s)}{c} \end{aligned}$$
(56)

If z denotes time, then c has units of inverse time. Integrating a positive function f(z) weighted by \({\text {e}}^{cz}\) is approximated by Eq. (56) if integration time s is large compared to 1 / c and, furthermore, growth of fluctuations over duration 1 / c is small compared to final value f(s). The analogy with physical systems is that linear systems close to equilibrium remain in equilibrium if forcing changes slowly. It is sufficient for the approximation above that the least-upper-bound of \(\partial f/\partial z\) be small enough, but in general the approximation would also be valid if fluctuations were small almost everywhere.

Returning to our problem, we treat forcing \(\tilde{F}(z)\) as a special case of f(z); and the negative of fast eigenvalue, \(-\lambda _{2}\), as a corresponding example of c above. Integration time is longer than 250 years, i.e. between simulation years 1765-2014, whereas \(-\lambda _{2}\, \gtrapprox \, 0.1\). Hence the first condition is met and \({\text {e}}^{-\lambda _{2}s}\gg 1\). The second condition is met if the change in radiative forcing within any 10-year period is much smaller than radiative forcing at the time of interest. With our focus on peak warming, where radiative forcing is still quite large, this condition is also met. Therefore, approximately \(\int _{0}^{t}{\text {e}}^{-\lambda _{2}z}\tilde{F}(z){{\text {d}}z}\cong \frac{{\text {e}}^{-\lambda _{2}t}\tilde{F}(t)}{-\lambda _{2}}\).

Appendix 3: Nonzero initial conditions

The solutions for the EBM in Sect. 3 are derived after assuming that initial conditions are zero, corresponding to preindustrial equilibrium. Here we relax this assumption, which is relevant to the case of pulse emissions. We start from Eq. (49)

$$\begin{aligned} \frac{{\text {d}}\left( {\text {e}}^{-At}\vec {u}\right) }{{\text {d}}t}={\text {e}}^{-At}\vec {f} \end{aligned}$$
(57)

but integrated from some time \(t_{0}\) where the state \(\vec {u}_{0}\ne \vec {0}\). Integrating

$$\begin{aligned} {\text {e}}^{-At}\vec {u}(t)-{\text {e}}^{-At_{0}}\vec {u}_{0}=\int _{t_{0}}^{t}{\text {e}}^{-Az}\vec {f}(z){{\text {d}}z} \end{aligned}$$
(58)

this is solved for

$$\begin{aligned} \vec {u}(t)={\text {e}}^{A\left( t-t_{0}\right) }\vec {u}_{0}+{\text {e}}^{At}\int _{t_{0}}^{t}{\text {e}}^{-Az}\vec {f}(z){{\text {d}}z} \end{aligned}$$
(59)

The effect of the forcing beginning at \(t=t_{0}\) is simply \({\text {e}}^{At}\int _{t_{0}}^{t}{\text {e}}^{-Az}\vec {f}(z){{\text {d}}z}\) or \(\varPhi {\text {e}}^{\varLambda t}\int _{t_{0}}^{t}{\text {e}}^{-\varLambda z}\varPhi ^{-1}\vec {f}(z){{\text {d}}z}\), which results in solutions derived in Sect. 3, with the only difference being that integration starts at \(t=t_{0}\). The EBM is linear, so its response to forcing is independent of the system’s state.

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Seshadri, A.K. Fast–slow climate dynamics and peak global warming. Clim Dyn 48, 2235–2253 (2017). https://doi.org/10.1007/s00382-016-3202-8

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