Long run surface temperature dynamics of an A-OGCM: the HadCM3 4×CO₂ forcing experiment revisited

Abstract

The global mean surface temperature (GMST) response of HadCM3 to a 1,000 year 4×CO₂ forcing is analysed using a transfer function methodology. We identify a third order transfer function as being an appropriate characterisation of the dynamic relationship between the radiative forcing input and GMST output of this Atmosphere-Ocean General Circulation Model (A-OGCM). From this transfer function the equilibrium climate sensitivity is estimated as 4.62 (3.92–11.88) K which is significantly higher than previously estimated for HadCM3. The response is also characterised by time constants of 4.5 (3.2–6.4), 140 (78–191) and 1,476 (564–11,737) years. The fact that the longest time constant element is significantly longer than the 1,000 year simulation run makes estimation of this element of the response problematic, highlighting the need for significantly longer model runs to express A-OGCM behaviour fully. The transfer function is interpreted in relation to a three box global energy balance model. It was found that this interpretation gave rise to three fractions of ocean heat capacity with effective depths of 63.0 (46.7–85.4), 1291.7 (787.3–2,955.3) and 2,358.0 (661.3–17,283.8) meters of seawater, associated with three discrete time constants of 4.6 (3.2–6.5), 107.7 (68.9–144.3) and 537.1 (196.2–1,243.1) years. Given this accounts for approximately 94% of the ocean heat capacity in HadCM3, it appears HadCM3 could be significantly more well mixed than previously thought when viewed on the millennial timescale.

Appendices

Appendix A: The equivalence between transfer function, sum of exponential and ordinary differential equation representations of linear dynamic systems

In Eq. 1 we presented the generic linear, continuous time transfer function structure with zero initial conditions, i.e.

$$ H\left( s \right) = \frac{x\left( s \right)}{u\left( s \right)} = \frac{{b_{1} s^{m} + b_{2} s^{m - 1} + \cdots + b_{m - 1} s + b_{m} }}{{s^{n} + a_{1} s^{n - 1} + \cdots + a_{n - 1} s + a_{n} }} $$

(A1)

If we take s ^m,n = d ^m,n/dt ^m,n in this zero initial condition case then Eq. A1 can be re-arranged to give a generic linear, continuous time ODE of the form,

$$ \frac{{d^{n} x\left( t \right)}}{{dt^{n} }} + a_{1} \frac{{d^{n - 1} x\left( t \right)}}{{dt^{n - 1} }} + \cdots + a_{n} x\left( t \right) = b_{0} \frac{{d^{m} u\left( t \right)}}{{dt^{m} }} + \cdots + b_{m} u\left( t \right) $$

(A2)

For the m = n case, Eq. A1 can also be re-expressed in a partial fraction expansion form,

$$ H\left( s \right) = \frac{{r_{1} }}{{s - p_{1} }} + \frac{{r_{2} }}{{s - p_{2} }} + \cdots + \frac{{r_{n} }}{{s - p_{n} }} $$

(A3)

where p _i {i = 1…n} are the poles of H(s). Providing p _i  < 0 and real, then -p ⁻¹ _i are the time constants (or e-folding times) T _i of each first order element in Eq. A3. The equilibrium gains (or amplitudes) G _i of each first order element are simply -r _i /p _i .

Taking the inverse Laplace transform L ⁻¹ of Eq. A3 for this m = n case gives the SEs response function,

$$ L^{ - 1} \left\{ {H\left( s \right)} \right\} = \sum\limits_{i = 1}^{n} {r_{i} e^{{p_{i} t}} } $$

(A4)

Appendix B: The relationship between surface ocean mixed layer heat capacity (c _s) and coupled atmosphere-surface ocean mixed layer heat capacity (c ₁)

Assuming the thermal inertia of the atmosphere is approximately zero and ignoring any deep ocean feedbacks, the atmosphere and surface ocean mixed layer energy balances can be written in the following TF forms,

$$ x_{a} \left( s \right) = \frac{1}{{\lambda + f_{o} k_{as} }}\left\{ {u\left( s \right) + f_{o} k_{as} x_{s} \left( s \right)} \right\} $$

(B1a)

$$ c_{s} \cdot s \cdot x_{s} \left( s \right) = f_{o} k_{as} \left\{ {x_{a} \left( s \right) - x_{s} \left( s \right)} \right\} - k_{sd} x_{s} \left( s \right) $$

(B1b)

where x _a is the atmospheric temperature response. Inserting (B1a) into (B1b), one obtains

$$ x_{s} \left( s \right) = \frac{1}{\begin{gathered} \frac{{c_{s} }}{{\frac{{f_{o} k_{as} }}{{\lambda + f_{o} k_{as} }}}} \cdot s \hfill + \frac{{f_{o} k_{as} \left( {\lambda + k_{sd} } \right) + \lambda \cdot k_{sd} }}{{f_{o} k_{as} }} \hfill \\ \end{gathered} }u\left( s \right) $$

(B2)

Likewise, expressing the coupled atmosphere-surface ocean mixed layer energy balance in its TF form, again ignoring any deep ocean feedback (c.f. Eq. 5a) gives,

$$ x_{1} \left( s \right) = \frac{1}{{c_{1} s + \left( {k_{1} + \lambda } \right)}}u\left( s \right) $$

(B3)

Assuming x _a ∝x _s (Hoffert et al. 1980; Parker et al. 1995; Eickhout et al. 2004; Joshi et al. 2008) then x ₁ = βx _s , which holds for the HadCM3 experiment considered here. Then, equating βc ₁ in Eq. B3 with c _s /f ₀ k _as /(λ + f ₀ k _as ) in Eq. B2 gives,

$$ \frac{{c_{1} }}{{c_{s} }} = \frac{{d_{1} }}{{d_{s} }} = \frac{1}{\beta }\frac{{\lambda + f_{o} k_{as} }}{{f_{o} k_{as} }} $$

(B4)