How Long Does it Take for the Earth's Climate System to Reach Equilibrium?

Little Big Econ State Forest (just because it's pretty)

Any given climate forcing affects the Earth's Energy Imbalance (EEI) - that is, it affects the amount of energy entering the climate system vs energy escaping into space. When a forcing occurs, such as from an increase in CO2, the imbalance is created and surface temperature increases until EEI returns to equilibrium (that is, EEI = 0). But how long does this take? That is, what is the lag between a climate forcing and the new equilibrium temperature? It turns out, this depends on sensitivity and the rate of temperature increase, but it is possible estimate the time to achieve this equilibrium with a forcing. But this is going to be rather math-intensive. Apologies in advance!

We can begin with a simple energy balance equation:

(1) ΔF = ΔT/λ + EEI, where

λ is sensitivity (ECS/F2x)

ECS is Equilibrium Climate Sensitivity

F2x is the forcing for doubling CO2

EEI is the Earth's Energy Imbalance

ΔF is the change in radiative forcings

ΔT is the change in temperature from a change in forcing

Now ΔT occurs in response to a change in EEI created from a change in radiative forcing, ΔF. Initially, EEI = ΔF and ΔT = 0. But at equilibrium EEI = 0 and ΔT = λ*ΔF. Using the Earth's effective heat capacity (C) we can estimate the lag time to equilibrium (L). The relationship between EEI and ΔT is complex, but it can be modeled simply as:

ΔF = C*(dT/dt + ΔT/L), so,

(2) C = ΔF/(dT/dt + ΔT/L), where

C is the Earth's effective heat capacity

dT/dt is the rate of temperature change in years

L is time in years to reach equilibrium to a given forcing

We can combine and rearrange terms in (1) and (2) above to get:

ΔT/λ + EEI = C*(dT/dt + ΔT/L)

EEI = C*dT/dt + C*ΔT/L - ΔT/λ

Initially, at the beginning of a climate forcing, ΔT = 0, and EEI = ΔF, so the above formula reduces to:

EEI = C*dT/dt

(3) C = EEI/(dT/dt)

When equilibrium is reached, then EEI returns to 0, so equations (1) and (2) reduce to

ΔF = ΔT/λ from (1), and

ΔF = C*ΔT/L from (2), so

L/C = λ

(4) L = λ*C

With formula (3), we can calculate the Earth's effective heat capacity (C).  For this I used a recent estimate of EEI to be 0.77 W/m^2 (see a previous post). For the rate of temperature increase (dT/dt), all GMST datasets show warming at 0.2°C/decade or faster. I chose a lower estimate. From the equation, C increases as EEI increases or as dT/dt decreases. So,

C = EEI/(dT/dt), where

EEI = 0.77 W/m^2, and

dT/dt = 0.02°C/year, so

C = 0.77/0.02 = 38.5 Wy/Cm^2

With formula (4), we can estimate the length of time (L) to reach equilibrium with a climate forcing. Interestingly, L increases with sensitivity (λ) which is ECS/F2x. In a previous post, I calculated ECS to be about 3.4°C, but I chose to use the more frequently cited estimate of 3°C here. This means that λ = 3/3.71 = 0.81 C/W/m^2.

 

L = λ*C = 0.81*38.5 = 31 years

 

In other words, we can expect that we'll reach equilibrium with current emissions about 30 years from now, or in the early to mid 2040s. Even if we were to level off CO2 concentrations at current levels, climate will continue to warm over the next 30 years or so until a new equilibrium temperature is reached. If we continue to add CO2 to the atmosphere, we add to these forcings, and so we push back the year we will reach an equilibrium temperature, and when it does, it will happen at a higher temperature. If instead we lower CO2 levels, we may be able to avoid reaching this equilibrium temperature.

But if we were to somehow hold steady at 420 ppm CO2 and keep all radiative forcings from other GHGs and aerosols constant, then we can expect that over the next 30 years, the additional change in temperature would be

ΔT = λ*EEI = 0.81*0.77 = 0.624°C

Since we have already exceeded 1.2°C above the 1850-1900 mean, that suggests that about 1.9°C is already baked into current CO2 levels. This suggests that the 1.5°C threshold from the IPCC is all but lost to us, and we have dwindling hope of achieving 2°C without both achieving net-zero emissions  and lowering CO2 concentrations below current levels within the next 30 years or so.