Quantification of temperature response to CO₂ forcing in atmosphere–ocean general circulation models

Abstract

The present study establishes a general formulation to represent the behavior and variation of an ensemble of complex climate models in terms of the global mean surface temperature response to atmospheric CO₂ increase. The response parameters of this formulation provide a set of metrics that extends the conventional concept of climate sensitivity and quantifies transient temperature changes with sufficient simplicity and transparency to serve studies on climate change mitigation. Two commonly used metrics for transient and equilibrium climate sensitivity are analytically derived from the formulation, such that conventional estimates of equilibrium climate sensitivity based on standard numerical experiments for quadrupling CO₂ increase are properly scaled down to the reference level of doubling CO₂. The characteristics and variations of a specific ensemble of complex climate models can be simulated with a statistical model built using the principal components (PCs) of the response parameters. This approach is applied to the probabilistic assessment of temperature changes as well as to the diagnosis of the base ensemble. In current complex climate models, the ratio of transient-to-equilibrium sensitivity decreases with an increase of equilibrium sensitivity, as identified in variations associated with two specific PCs that characterize coherence between transient temperature response and properties of heat uptake by the ocean.

Appendix A. Relationship between IR and EB formulations

IR(n) is derived from its equivalent n-layer energy balance equations, termed EB(n), through Laplace transform (Li and Jarvis 2009). EB(2) is given by

$$ \frac{d}{dt}\left(\frac{T_S}{T_1}\right)=\left(\begin{array}{ll}-\left(\lambda +{\lambda}_1\right)/{\zeta}_S\hfill & {\lambda}_1/{\zeta}_S\hfill \\ {}{\lambda}_1/\kern-2em {\zeta}_1\hfill & -{\lambda}_1/{\zeta}_1\hfill \end{array}\right)\left(\frac{T_S}{T_1}\right)+\left(\begin{array}{l}F/{\zeta}_S\hfill \\ {}0\hfill \end{array}\right) $$

(A1)

in a matrix form, where ${T}_1$ is the temperature anomaly in the sub-surface layer, ζ with subscript ‘S’ and ‘1’ is the heat content of the surface and sub-surface layers, respectively, and ${\lambda}_1$ is the heat exchange coefficient between the two layers. Taking the Laplace transform of this equation with zero initial conditions yields

$$ \left(\begin{array}{ll}s+\left(\lambda +{\lambda}_1\right)/{\zeta}_S\hfill & -\lambda /{\zeta}_S\hfill \\ {}\kern1.8em -\kern-2em {\lambda}_1/\kern-2em {\zeta}_1\hfill & s+{\lambda}_1/{\zeta}_1\hfill \end{array}\right)\left(\begin{array}{l}{T}_S(s)\hfill \\ {}{T}_1(s)\hfill \end{array}\right)=\left(\begin{array}{l}F(s)/{\zeta}_S\hfill \\ {}0\hfill \end{array}\right) $$

(A2)

in the s domain, and solved for ${T}_S(s)$ in a transform function form as

$$ \frac{T_S(s)}{F(s)}=\frac{\left(1/{\zeta}_S\right)\left(s+{\lambda}_1/{\zeta}_1\right)}{\left[s+\left(\lambda +{\lambda}_1\right)/{\zeta}_S\right]\left[s+{\lambda}_1/{\zeta}_1\right]-{\lambda}_1^2\left({\zeta}_S{\zeta}_1\right)}. $$

(A3)

This equation is arranged in a partial fraction expansion form

$$ \frac{T_S(S)}{F(S)}=\frac{a_1}{s-{b}_1}+\frac{a_2}{s-{b}_2}, $$

(A4)

and the comparison of the above two equations gives the following relations

$$ {a}_1+{a}_2=1/{\zeta}_S $$

(A5)

$$ -{a}_1{b}_2-{a}_2{b}_1={\lambda}_1/\left({\zeta}_S{\zeta}_1\right) $$

(A6)

$$ -{b}_1-{b}_2=\left(\lambda +{\lambda}_1\right)/{\zeta}_S+{\lambda}_1/{\zeta}_1 $$

(A7)

$$ {b}_1{b}_2=\left(\lambda {\lambda}_1\right)/\left({\zeta}_S{\zeta}_1\right). $$

(A8)

Considering that the inverse Laplace transform of equation (A4) is the impulse response function with a sum of two exponentials, equations (A5)–(A8) are transformed by substituting ${a}_i$ and ${b}_i$ with $\left(1/\uplambda \right){A}_i/{\tau}_i$ and $-1/{\tau}_i$ into

$$ {A}_1/{\tau}_1+A/{\tau}_2=\lambda /{\zeta}_S $$

(A9)

$$ {A}_1+{A}_2=1 $$

(A10)

$$ {A}_1{\tau}_1+{A}_2{\tau}_2=\left({\zeta}_S+{\zeta}_1\right)/\lambda $$

(A11)

$$ {\tau}_1{\tau}_2=\left({\zeta}_S{\zeta}_1\right)/\left(\lambda {\lambda}_1\right) $$

(A12)

Similarly, EB(3) having the third layer denoted with subscript ‘2’

$$ \left(\begin{array}{lll}s+\left(\lambda +{\lambda}_1\right)/{\zeta}_S\hfill & \kern2em -\kern-1em {\lambda}_1/\kern-2em {\zeta}_S\hfill & \kern1.3em 0\hfill \\ {}-{\lambda}_1/\kern-1.5em {\zeta}_1\hfill & s+\left({\lambda}_1+{\lambda}_2\right)/{\zeta}_1\hfill & -{\lambda}_2/{\zeta}_1\hfill \\ {}\kern3em 0\hfill & \kern2em -\kern-1em {\lambda}_2/\kern-2em {\zeta}_2\hfill & s+{\lambda}_2/{\zeta}_2\hfill \end{array}\right)\left(\begin{array}{l}{T}_S(s)\hfill \\ {}{T}_1(s)\hfill \\ {}{T}_2(s)\hfill \end{array}\right)=\left(\begin{array}{l}F(s)/{\zeta}_S\hfill \\ {}\kern1.3em 0\hfill \\ {}\kern1.3em 0\hfill \end{array}\right) $$

(A13)

is transformed into

$$ \begin{array}{c}\hfill \frac{T_S(s)}{F(s)}=\frac{1}{\zeta_S}\left[{S}^2+\left(\frac{\lambda_1+{\lambda}_2}{\zeta_1}+\frac{\lambda_2}{\zeta_2}\right)s+\frac{\lambda_1{\lambda}_2}{\zeta_1{\zeta}_2}\right]\hfill \\ {}\hfill /\left[{S}^3+\left(\frac{\lambda +{\lambda}_1}{\zeta_S}+\frac{\lambda_1+{\lambda}_2}{\zeta_1}+\frac{\lambda_2}{\zeta_2}\right){S}^2\right.\hfill \\ {}\hfill \left.+\left(\frac{\lambda {\lambda}_1+{\lambda}_1{\lambda}_2+{\lambda}_2\lambda }{\zeta_S{\zeta}_1}+\frac{\lambda_1{\lambda}_2}{\zeta_1{\zeta}_2}+\lambda {\lambda}_2+\frac{\lambda_1{\lambda}_2}{\zeta_2{\zeta}_S}\right)S+\frac{\lambda {\lambda}_1{\lambda}_2}{\zeta_S{\zeta}_1{\zeta}_2}\right].\hfill \end{array} $$

(A14)

Again, comparing this with its partial fraction expansion form

$$ \frac{T_S(s)}{F(s)}=\frac{a_1}{s-{b}_1}+\frac{a_2}{s-{b}_2}+\frac{a_3}{s-{b}_3} $$

(A15)

the following relations are obtained

$$ {a}_1+{a}_2+{a}_3=1/{\zeta}_S $$

(A16)

$$ {a}_1\left({b}_2+{b}_3\right)+{a}_2\left({b}_3+{b}_1\right)+{a}_3\left({b}_1+{b}_2\right)=\frac{1}{\zeta_S}\left(\frac{\lambda_1+{\lambda}_2}{\zeta_1}+\frac{\lambda_2}{\zeta_2}\right) $$

(A17)

$$ {a}_1{b}_2{b}_3+{a}_2{b}_3{b}_1+{a}_3{b}_1{b}_2=\frac{\lambda_1{\lambda}_2}{\zeta_S{\zeta}_1{\zeta}_2} $$

(A18)

$$ {b}_1+{b}_2+{b}_3=\frac{\lambda +{\lambda}_1}{\zeta_S}+\frac{\lambda_1+{\lambda}_2}{\zeta_1}+\frac{\lambda_2}{\zeta_2} $$

(A19)

$$ {b}_1{b}_2+{b}_2{b}_3+{b}_3{b}_1=\frac{\lambda {\lambda}_1+{\lambda}_1{\lambda}_2+{\lambda}_2\lambda }{\zeta_S{\zeta}_1}+\frac{\lambda_1{\lambda}_2}{\zeta_1{\zeta}_2}+\frac{\lambda {\lambda}_2+{\lambda}_1{\lambda}_2}{\zeta_2{\zeta}_S} $$

(A20)

$$ {b}_1{b}_2{b}_3=\frac{\lambda {\lambda}_1{\lambda}_2}{\zeta_S{\zeta}_1{\zeta}_2}. $$

(A21)

These equations are transformed with ${A}_i$ and ${\tau}_i$ into

$$ \frac{A_1}{\tau_1}+\frac{A_2}{\tau_2}+\frac{A_3}{\tau_3}=\frac{\lambda }{\zeta_S} $$

(A22)

$$ {A}_1{\tau}_1+{A}_2{\tau}_2+{A}_3{\tau}_3=\frac{\zeta_S+{\zeta}_1+{\zeta}_2}{\lambda } $$

(A23)

$$ {A}_1+{A}_2+{A}_3=1 $$

(A24)

$$ \frac{A_1}{\tau_1^2}+\frac{A_2}{\tau_2^2}+\frac{A_3}{\tau_3^2}=\frac{\lambda \left(\lambda +{\lambda}_1\right)}{\zeta_S^2} $$

(A25)

$$ {\tau}_1+{\tau}_2+{\tau}_3=\frac{\zeta_S+{\zeta}_1+{\zeta}_2}{\lambda }+\frac{\zeta_1+{\zeta}_2}{\lambda_1}+\frac{\zeta_2}{\lambda_2} $$

(A26)

$$ {\tau}_1{\tau}_2{\tau}_3=\frac{\zeta_S{\zeta}_1{\zeta}_2}{\lambda {\lambda}_1{\lambda}_2}. $$

(A27)

Note that ${a}_i$ and ${b}_i$ in the partial fraction expansion form are associated with the eigenvalues and the eigenvectors of the system of the differential equations, where ${b}_i$ corresponds to the eigenvalues, and ${a}_i$ corresponds to the inverse of the matrix that contains the eigenvectors in its columns. When numerical computation is applied, differential equations in a diagonalized form

$$ \frac{d\left({v}_{jk}^{-1}{T}_k\right)}{dt}=-\frac{1}{\tau_j}\left({v}_{jk}^{-1}{T}_k\right)+\left({v}_{j1}^{-1}F/{\zeta}_S\right) $$

(A28)

are efficiently integrated, where ${v}_{jk}^{-1}$ is an element of the inverse eigenvectors matrix, and subscript k is the layer index denoting each of ‘S’ and numbers from 1 to $n-1$. Assuming a linear change of forcing $\Delta F(t)$ during a time step from t to $t+\Delta t$, the time integration is calculated as

$$ \begin{array}{c}\hfill {v}_{jk}^{-1}{T}_k\left(t+\varDelta t\right)={v}_{jk}^{-1}{T}_k(t){\mathrm{e}}^{-\varDelta t/{\tau}_j}+\left[{v}_{j1}^{-1}F(t)/{\zeta}_S\right]{\uptau}_j\left(1-{\mathrm{e}}^{-\varDelta t/{\tau}_j}\right)\hfill \\ {}\hfill +\frac{v_{j1}^{-1}\varDelta F(t)/{\zeta}_S}{\varDelta t}\left[t-{\tau}_j\left(1-{\mathrm{e}}^{-\varDelta t/{\tau}_j}\right)\right].\hfill \end{array} $$

(A29)

The heat content ${\zeta}_k$ is converted into effective ocean depth ${h}_k$ for reference. The conversion factor used in the present study is the product of the density of sea water (1.03 × 10³ kg m⁻³), the specific heat of sea water (4.18 × 10³ J kg⁻¹ K⁻¹), and the fraction of the earth surface covered by the ocean (0.71).