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.
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).