Climate stability and sensitivity in some simple conceptual models

Abstract

A theoretical investigation of climate stability and sensitivity is carried out using three simple linearized models based on the top-of-the-atmosphere energy budget. The simplest is the zero-dimensional model (ZDM) commonly used as a conceptual basis for climate sensitivity and feedback studies. The others are two-zone models with tropics and extratropics of equal area; in the first of these (Model A), the dynamical heat transport (DHT) between the zones is implicit, in the second (Model B) it is explicitly parameterized. It is found that the stability and sensitivity properties of the ZDM and Model A are very similar, both depending only on the global-mean radiative response coefficient and the global-mean forcing. The corresponding properties of Model B are more complex, depending asymmetrically on the separate tropical and extratropical values of these quantities, as well as on the DHT coefficient. Adopting Model B as a benchmark, conditions are found under which the validity of the ZDM and Model A as climate sensitivity models holds. It is shown that parameter ranges of physical interest exist for which such validity may not hold. The 2 × CO₂ sensitivities of the simple models are studied and compared. Possible implications of the results for sensitivities derived from GCMs and palaeoclimate data are suggested. Sensitivities for more general scenarios that include negative forcing in the tropics (due to aerosols, inadvertent or geoengineered) are also studied. Some unexpected outcomes are found in this case. These include the possibility of a negative global-mean temperature response to a positive global-mean forcing, and vice versa.

Appendices

Appendix 1: Symbols and abbreviations

$\bar{x}$ :

global-mean value of x in the unperturbed equilibrium climate

$x^{\prime}$ :

$x - \bar{x}$

a :

Earth’s radius (6,370 km)

AMIP:

Atmospheric Model Intercomparison Project

ASR:

absorbed solar radiation (by the earth-atmosphere system)

b :

unit-area TOA radiative response coefficient [≡$- \{ d\left( {F_{\rm TOA}^{ \downarrow } } \right)/dT\} /4\pi a^{2}$]

b ₁ :

unit-area TOA radiative response coefficient in Zone 1 (the tropics) [≡ $\{ - d\left( {F_{TOA}^{ \downarrow } } \right)_{1} /dT_{1} \} /2\pi a^{2}$]

b ₂ :

unit-area TOA radiative response coefficient in Zone 2 (the extratropics) [≡ $\{ - d\left( {F_{\rm TOA}^{ \downarrow } } \right)_{2} /dT_{2} \} /2\pi a^{2}$]

B99:

Bates (1999)

B07:

Bates (2007)

c _O :

mixed-layer heat capacity per unit area

$\hat{d}$ :

dynamical heat transport coefficient in Eq. (39)

d :

dynamical heat transport (DHT) coefficient $\left\lfloor { \equiv \hat{d}/\left( {2\pi a^{2} } \right)} \right\rfloor$ [Eq. (40]

dFlux/dSST:

rate of change with increasing SST of total (LW + SW) outgoing radiation per unit area at TOA in the tropics

$D \left[ { \equiv F_{E} + F_{\rm OH} } \right]$ :

dynamical heat transport (oceanic plus atmospheric) from Zone 1 to Zone 2

DHT:

dynamical heat transport (D)

$\tilde{D}^{\prime}$ :

$D^{\prime}/2\pi a^{2}$

$\tilde{D}^{\prime}_{A}$ :

$\tilde{D}^{\prime}$ for Model A

$\tilde{D}^{\prime}_{B}$ :

$\tilde{D}^{\prime}$ for Model B

E :

average heat content of the land-ocean system per unit area

E _1,2 :

average heat content of the land-ocean system per unit area in Zone (1,2)

ERBE:

Earth Radiation Budget Experiment

F _OH :

heat transport by ocean currents from Zone 1 to Zone 2

F _E :

transport of moist static energy by atmospheric motions from Zone 1 to Zone 2

$F_{\rm SFC}^{ \downarrow }$ :

net downward global energy flux at the surface

$F_{\rm TOA} {}^{ \downarrow }$ :

net downward global energy flux at TOA

$\left( {F_{\rm SFC}^{ \downarrow } } \right)_{1,2}$ :

value of $F_{\rm SFC}^{ \downarrow }$ in Zone (1,2)

$\left( {F_{\rm TOA}^{ \downarrow } } \right)_{1,2}$ :

value of $F_{\rm TOA}^{ \downarrow }$ in Zone (1,2)

GCM:

general circulation model

GHG:

greenhouse gas

LCH01:

Lindzen et al. (2001)

LC09:

Lindzen and Choi (2009)

LC10:

Lindzen and Choi (2010)

LW:

longwave

OLR:

outgoing LW radiation (at TOA)

PW:

petawatt (10¹⁵ W)

$Q^{\prime}$ :

global-mean external (perturbation) forcing per unit area

$Q^{\prime}_{1,2}$ :

global-mean external (perturbation) forcing per unit area in Zone (1,2)

R :

characteristic rate of decay of an impulsively forced perturbation in the ZDM or Model A

RH:

relative humidity

R _F,S :

fast and slow characteristic rates of decay of an impulsively forced perturbation in Model B

S :

b ₁ + b ₂

$S_{1} \equiv c_{O} (R_{F} + R_{S})$ :

$b_{1} + {\text{b}}_{ 2} + 2 {\text{d }}$ [Eq. (43) and Appendix 2]

$S_{2} \equiv c_{O}^{2} R_{F} R_{S}$ :

$b_{1} b_{2} + d(b_{1} + b_{2} )$ [Eq. (44) and Appendix 2]

SST:

sea surface temperature

SW:

shortwave

t :

time

TOA:

top-of-the-atmosphere

T :

globally averaged surface temperature [=$\left( {T_{1} + T_{2} } \right)/2$]

T ₁ :

average surface temperature in Zone 1 (the tropics)

T ₂ :

average temperature in Zone 2 (the extratropics)

ZDM:

zero-dimensional model

ΔD :

$D^{\prime}(\infty )$ = equilibrium value of perturbation DHT

$\Updelta \tilde{D}$ :

$\Updelta D/(2\pi a^{2} )$

$\Updelta \tilde{D}_{A}$ :

$\Updelta \tilde{D}$ for Model A

$\Updelta \tilde{D}_{B}$ :

$\Updelta \tilde{D}$ for Model B

ΔQ :

global-mean TOA forcing per unit area in the ZDM (used as the amplitude of an impulsive or step-function forcing)

ΔQ _1,2 :

mean forcing per unit area in Zone (1,2) of the two-zone models (used as the amplitude of an impulsive or step-function forcing)

$\left( {\Updelta Q_{1} } \right)_{C}$ :

critical value of ∆Q ₁ separating Sensitivity Regions I and II (Fig. 2)

ΔT :

equilibrium temperature increment for a step-function external forcing in the ZDM

ΔT _A :

equilibrium temperature increment for a step-function external forcing in Model A

ΔT _B :

equilibrium global-mean temperature increment for a step-function external forcing in Model B $[\equiv\left( {\Updelta T_{1} + \Updelta T_{2} } \right)/2]$

ΔT ₁ :

$T^{\prime}_{1} (\infty )$ (Model B)

ΔT ₂ :

$T^{\prime}_{2} (\infty )$ (Model B)

Appendix 2: Solution to Model B for impulsive forcing

To obtain the solution to the perturbation equations (41) and (42) of Model B, it is convenient to rewrite them in the form

$$ c_{O} {\frac{{dT_{1}^{\prime } }}{dt}} = - \alpha_{1} T_{1}^{\prime } - \alpha_{2} T_{2}^{\prime } + Q_{1}^{\prime } $$

(70)

$$ c_{O} {\frac{{dT_{2}^{\prime } }}{dt}} = - \alpha_{3} T_{1}^{\prime } - \alpha_{4} T_{2}^{\prime } + Q_{2}^{\prime } $$

(71)

where

$$ \left( {\alpha_{1} ,\alpha_{2} ,\alpha_{3} ,\alpha_{4} } \right) = \left( {b_{1} + d, - d, - d,b_{2} + d} \right) $$

(72)

These equations are of the same form as those governing the TOA models considered in B99 and B07. Their solution for the impulsive forcing (29) (see B07 for details) is

$$ T_{1}^{\prime } = {\frac{1}{{c_{O}^{2} \left( {R_{F} - R_{S} } \right)}}}\left[ {\left\{ {\left( {c_{O} R_{F} - \alpha_{4} } \right)\Updelta Q_{1} + \alpha_{2} \Updelta Q_{2} } \right\}\exp \left( { - R_{F} t} \right) - \left\{ {\left( {c_{O} R_{S} - \alpha_{4} } \right)\Updelta Q_{1} + \alpha_{2} \Updelta Q_{2} } \right\}\exp \left( { - R_{S} t} \right)} \right] $$

(73)

$$ T_{2}^{\prime } = {\frac{1}{{c_{O}^{2} \left( {R_{F} - R_{S} } \right)}}}\left[ {\left\{ {\left( {c_{O} R_{F} - \alpha_{1} } \right)\Updelta Q_{2} + \alpha_{3} \Updelta Q_{1} } \right\}\exp \left( { - R_{F} t} \right) - \left\{ {\left( {c_{O} R_{S} - \alpha_{1} } \right)\Updelta Q_{2} + \alpha_{3} \Updelta Q_{1} } \right\}\exp \left( { - R_{S} t} \right)} \right] $$

(74)

where the characteristic rates of decay, R _F and R _S , are given by

$$ R_{F} = {\frac{1}{{c_{O} }}}\;{\frac{{\alpha_{1} + \alpha_{4} }}{2}}\left( {1 + \sqrt {1 - x} } \right) $$

(75)

$$ R_{S} = {\frac{1}{{c_{O} }}}\;{\frac{{\alpha_{1} + \alpha_{4} }}{2}}\left( {1 - \sqrt {1 - x} } \right) $$

(76)

with

$$ x = 4{\frac{{\alpha_{1} \alpha_{4} - \alpha_{2} \alpha_{3} }}{{\left( {\alpha_{1} + \alpha_{4} } \right)^{2} }}} $$

(77)

The above solution satisfies the initial condition $\left( {T_{1}^{\prime } ,T_{2}^{\prime } } \right) = \left( {\Updelta Q_{1} ,\Updelta Q_{2} } \right)/c_{O}$ at t = 0₊, and it can easily be verified by substitution that it satisfies the governing equations (70) and (71) for t > 0.

The conditions for global stability of Model B are that the real parts of R_F and R_S be positive. These conditions are satisfied if

$$ S_{1} \equiv c_{O} (R_{F} + R_{S} ) = \alpha_{1} + \alpha_{4} > 0 $$

(78)

and

$$ S_{2} \equiv c_{O}^{2} R_{F} R_{S} = \alpha_{1} \alpha_{4} - \alpha_{2} \alpha_{3} > 0 $$

(79)

Using (72), these criteria can be rewritten in the forms (43) and (44).

Appendix 3: Local instability in both zones implies global instability of Model B

We show here that if Model B is locally unstable in both zones (i.e., b ₁ < 0 and b ₂ < 0), the global stability criteria S ₁ > 0 and S ₂ > 0 cannot both be satisfied.

Using (43), we see that under the conditions assumed, satisfying S ₁ > 0 requires

$$ d > {\frac{{\left| {b_{1} } \right| + \left| {b_{2} } \right|}}{2}} $$

(80)

Using (44), we see that, under the same conditions, satisfying S₂ > 0 requires

$$ d < {\frac{{\left| {b_{1} } \right|\left| {b_{2} } \right|}}{{\left| {b_{1} } \right| + \left| {b_{2} } \right|}}} $$

(81)

The inequalities (80) and (81) can be satisfied simultaneously only if

$$ {\frac{{\left| {b_{1} } \right| + \left| {b_{2} } \right|}}{2}} < {\frac{{\left| {b_{1} } \right|\left| {b_{2} } \right|}}{{\left| {b_{1} } \right| + \left| {b_{2} } \right|}}} $$

(82)

i.e., only if

$$ \left| {b_{1} } \right|^{2} + \left| {b_{2} } \right|^{2} < 0 $$

(83)

It is impossible to satisfy this condition when b ₁ < 0 and b ₂ < 0. Thus, it is impossible to satisfy both of the stability criteria S ₁ > 0 and S ₂ > 0 under these conditions and the model must be globally unstable.

Appendix 4: Conditions determining the sign of ∆T _B in sensitivity Regions II and III

To specify the conditions determining the sign of ∆T _B in Regions II and III more precisely, we rewrite (50) in the form

$$ \Updelta T_{B} = \left( {{\frac{{b_{2} + 2d}}{{2S_{2} }}}} \right)\left[ {\left( {\Updelta Q_{1} + \Updelta Q_{2} } \right) - \mu } \right] $$

(84)

where

$$ \mu \equiv \left( {{\frac{{b_{2} - b_{1} }}{{b_{2} + 2d}}}} \right)\Updelta Q_{2} $$

(85)

4.1 (a) Sensitivity Region II

Here, ∆Q ₁ is defined to lie in the range (58) and, as already noted, this implies ∆Q ₁ + ∆Q ₂ > 0. From (84) and (85) it is clear that the conditions for ∆T _B  > 0 are then

$$ b_{1} > b_{2} $$

(86)

or

$$ b_{1} < b_{2}\quad \text{and} \quad \Updelta Q_{1} + \Updelta Q_{2} > \mu $$

(87)

If (86) is satisfied, ∆T _B  > 0 throughout Region II. If (87) is satisfied, Region II is divided into two sub-regions, the crossover point between them being defined by

$$ \Updelta Q_{1} + \Updelta Q_{2} = \mu $$

(88)

In the sub-region to the left, ∆T _B  < 0, while in the sub-region to the right, ∆T _B  > 0. We have already seen [from (55)] that if b ₁ is assigned values approaching its lower limit for global stability (in which case $S_{2} \to 0_{ + }$), $\Updelta T_{B} \to -\infty$ throughout Region II; in this case the crossover point defined by (88) approaches the right boundary of Sensitivity Region II.

4.2 (b) Sensitivity Region III

Here, ∆Q ₁ is defined to lie in the range (59) and, as already noted, this implies ∆Q ₁ + ∆Q ₂ < 0. In this case, we see from (84) and (85) that ∆T _B  > 0 only if

$$ b_{1} > b_{2}\; \text{and}\; \left| {\Updelta Q_{1} + \Updelta Q_{2} } \right| < \left| \mu \right| $$

(89)

If (89) is satisfied, ∆T _B  > 0 in a small sub-region of Region III adjoining its right boundary, with ∆T _B  < 0 elsewhere in the region. If (89) is not satisfied, ∆T _B  < 0 everywhere in Region III.