Dynamics of the coupled human–climate system resulting from closed-loop control of solar geoengineering

Abstract

If solar radiation management (SRM) were ever implemented, feedback of the observed climate state might be used to adjust the radiative forcing of SRM in order to compensate for uncertainty in either the forcing or the climate response. Feedback might also compensate for unexpected changes in the system, e.g. a nonlinear change in climate sensitivity. However, in addition to the intended response to greenhouse-gas induced changes, the use of feedback would also result in a geoengineering response to natural climate variability. We use a box-diffusion dynamic model of the climate system to understand how changing the properties of the feedback control affect the emergent dynamics of this coupled human–climate system, and evaluate these predictions using the HadCM3L general circulation model. In particular, some amplification of natural variability is unavoidable; any time delay (e.g., to average out natural variability, or due to decision-making) exacerbates this amplification, with oscillatory behavior possible if there is a desire for rapid correction (high feedback gain). This is a challenge for policy as a delayed response is needed for decision making. Conversely, the need for feedback to compensate for uncertainty, combined with a desire to avoid excessive amplification of natural variability, results in a limit on how rapidly SRM could respond to changes in the observed state of the climate system.

Appendices

Appendix 1: Time-domain calculations with box-diffusion model

Equations (1–2) can be solved using the Laplace transform. From (2), the temperature in the deep ocean satisfies

$$ T_d(s,z)=M(s)e^{-\sqrt{s/\kappa}z} $$

(14)

for some function M(s), where s is the Laplace variable. Subtituting this into the Laplace transform of (1) and solving for M yields (3).

From Laplace transform tables, the response of a semi-infinite diffusion model (Eq. (3) with C = 0) to a unit step change in radiative forcing at t = 0 is

$$ g_{sd}(t)=\frac{1}{\lambda}\left(1-e^{t/\tau}\hbox{erfc}(\sqrt{t/\tau})\right) $$

(15)

where erfc denotes the complementary error function. With the surface layer included, the step response can be obtained by first factoring G(s) in Eq. (3) as

$$ G(s)=\frac{1/\xi}{\sqrt{s}+b}-\frac{1/\xi}{\sqrt{s}+a} $$

(16)

similar to the derivation in Lebedoff (1988) or Morantine and Watts (1990), where we introduce

$$ \xi=\sqrt{\beta^2/\kappa-4C\lambda}=\lambda\sqrt{\tau-4C/\lambda} $$

(17)

and $a,b=(\beta/\sqrt{\kappa}\pm\xi)/(2C)$ satisfying (aξ)⁻¹ − (bξ)⁻¹ = λ ⁻¹. As in (15), the step response is then

$$ g_{bd}(t)=1/\lambda-1/(b\xi)e^{b^2t}\hbox{erfc} (b\sqrt{t})+1/(a\xi)e^{a^2t}\hbox{erfc}(a\sqrt{t}) $$

(18)

For 4C ≪ λ τ as here, then a ²≃ λ ² τ/C ² and b ²≃ 1/τ, and the first two terms in Eq. (18) are approximately the same as the step response of the semi-infinite diffusion model in Eq. (15), while the final term provides a correction for small t/τ. In calculating this final term, note that for $a\sqrt{t}\gg 1$ then

$$ e^{a^2t}\hbox{erfc}(a\sqrt{t})\simeq \frac{(2/\sqrt{\pi})}{(a\sqrt{t}+\sqrt{a^2t+2})} $$

(19)

The simulations here consider only the average temperature over each year. For semi-infinite diffusion, the average temperature in the nth year after a step change in radiative forcing is

$$ \int\limits_{t=n-1}^{t=n}g_{sd}(t)dt= \frac{1}{\lambda}-\frac{1}{\lambda}\left(2\sqrt{\tau t/\pi}+\tau e^{t/\tau}\hbox{erfc}(\sqrt{t/\tau})\right)_{n-1}^n $$

(20)

$$ =\frac{1}{\lambda}[1-q(\tau^2;n)] $$

(21)

where this defines the function q(a;n). Then for the box-diffusion model,

$$ h(n)=\int\limits_{t=n-1}^{t=n}g_{bd}(t)dt= \frac{1}{\lambda}+\frac{1}{\xi}\left[\frac{q(a;n)}{a}-\frac{q(b;n)}{b}\right] $$

(22)

Simulations here also assume that the radiative forcing is held constant over each year. Given a sequence of forcings f(k) applied during year k, then since the sequence h(n) gives the annual-average temperatures due to a unit step forcing starting at k = 0, the temperature response to the sequence f(k) can be expressed as

$$ T(n)=\sum_{k=1}^{n}h(k)f(n-k)=\sum_{k=0}^{n-1}h(n-k)f(k) $$

(23)

Appendix 2: Frequency-domain calculations

The dynamic response of the climate system with feedback, shown in Figs. 5, 6, 7 and 8, can be understood and an approximate prediction made using G(s) in Eq. (3), K(s) in Eq. (10), and approximating the effects of the N-year averaging with the Laplace transform of a pure time delay, e ^−Ns (obtained from the Laplace transform of $\tilde{y}(t)=y(t-N)$).

However, more accurate calculations of the frequency response requires taking into account that updates of solar forcing are only made at discrete time-intervals, not as a continuous function of time. The feedback system including this detail is shown in Fig. 14, where the block G(s) describes the continuous-time evolution of the climate response to forcing as before, A(s) represents the averaging of the output over the past N years, which is then sampled every N years, and Z(s) is a “zero-order hold” that describes the assumption that the solar reduction computed at every N-year decision point is held constant over the subsequent N years. The discrete-time PI control law

$$ u(k)=k_py(k)+k_i\sum_{n=0}^{n=k}y(n) $$

(24)

is represented by its z-transform, K(z). Analysis details can be found in any discrete-time controls textbook (e.g. Franklin et al. 1997); here we simply provide the required formulae used in computing the results herein.

Fig. 14

Block diagram of geoengineering feedback, as in Fig. 2, but with additional detail required for accurately predicting dynamics. The response of the climate system G(s) is averaged over the previous N years [A(s)], sampled, and the actual feedback law implemented in discrete-time [K(z)] rather than continuous-time. The desired SRM forcing at each discrete decision point in time is assumed to be held constant for the next N years, until the next sample is made. The system within the dashed box is sampled, resulting in aliasing

For frequencies less than the Nyquist frequency (half the sampling rate), then the Laplace transform of the discrete-time control law K(z) can be obtained by setting z = e ^Ns, yielding

$$ K(s)=k_p+k_i\frac{N}{1-e^{-Ns}} $$

(25)

The Laplace transform of the N-year averaging process is

$$ A(s)=\frac{1-e^{-Ns}}{Ns} $$

(26)

which is used in all calculations here, and at frequencies small compared to 1/N, behaves similarly to a pure time delay of N/2 years. Maintaining a constant value of the applied radiative forcing for N years (a zero-order-hold) yields Z(s) = A(s), so that A(s)Z(s) has an effect similar to that of an N-year time delay.

Finally, note that sampling the continuous-time system G _az (s) = A(s)G(s)Z(s) at N-year intervals results in aliasing. That is, temperature variations with frequency f and variations at frequency 1/N − f are indistinguishable in the sampled signal. (There are an infinite sequence of indistinguishable frequencies, but only the first is significant in predicting the response.) Thus at frequencies below the Nyquist frequency, the system G _s (s) within the dashed lines of Fig. 14 is approximately

$$ G_s(i\omega)=G_{az}(i\omega)+G_{az}(2\pi/N-i\omega) $$

(27)

$$ =G_{az}(i\omega)+G_{az}^*(i\omega-2\pi/N) $$

(28)

with $(\cdot)^*$ denoting complex-conjugate. The loop transfer function in Figs. 3 and 4 and the subsequent calculations of the sensitivity function are obtained using K(s) in (25) and G _s (s) in (28), where the latter depends not only on G(s) in (3) but also on A(s) and Z(s) in (26).