So we must bear in mind the great inertia of the system, in order to find effective policy to avoid dangerous climate change.

Timescale of each process

Most other greenhouse gases are eventually destroyed in the atmosphere, their lifetime ranging from about a decade for methane, to many millenia for some CFCs and SF6. As the atmosphere mixes globally within a few months, these gases can be considered to be almost uniformly distributed.

Aerosols of sulphate or soot, by-products of fossil fuel combustion, biomass burning, are washed out of the atmosphere by rain and so survive at most few weeks. Tropospheric ozone also has a short lifetime as it is highly reactive. Consequently these are concentrated in more polluted regions.

The radiative forcing combines the effect of all these gases and aerosols, plus solar variability. RF shows instantaneous heating power rather than accumulated heat, so it is measured in Watts (per m2).

The land surface responds quickly to changes in radiative forcing (both the land and the atmosphere have such a low heat capacity, that these are neglected in this simple energy-balance climate model). The ocean surface however, lags behind due to the slow exchange of heat with the deep ocean.

If you choose the "expert" complexity level, you can compare the land and ocean temperature changes -notice how the oscillations due to solar variability (you can adjust this from the radiative forcing plot) affect the land more than the ocean.

The surface ocean is only mildly influenced by the warming of the deep ocean, hence if greenhouse gas concentrations are stabilised the temperature rises only slowly (note however, that dramatic changes in the thermohaline circulation, not included in this model, might alter this conclusion!).

However the deep ocean warming determines the thermal expansion of seawater which is the largest contributor to the sea-level rise. You can see that this only begins to slow down, even centuries after the surface temperature has stabilised.

Sea-level is also influenced by ice-melt. Some mountain glaciers may melt within a few decades, however it requires thousands of years to melt the polar icecaps. Indeed they are still responding to the warming at the end of the last ice-age.

Note the different timescales of the climate system are also discussed in IPCC Synthesis report Q5

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Note also that there are many more physical and biogeochemical feedback processes which are not yet included in this model, such as the response of permafrost, ocean phytoplankton, or the thermohaline circulation. Although these are generally slow, a combination of such feedback processes may lead to dramatic surprises on passing critical thresholds.

Calculus of climate change

For the mathematically minded, we can say:
∫ represents the time integral
E = Emissions, S = Sinks,
C =Concentration
RF = Radiative Forcing
T_s = Surface temperature
T_d = Deep ocean temperature
I = Ice Melt
S = Sea-level rise

then to a first approximation:

If we consider also that emissions reductions depend on cumulative policy actions,

Then you can see that we have a triple time-integral

in going from climate policy to impacts such as sea-level rise. Hence it is so difficult to calculate in inverse mode (differentiate) to find the best policy to avoid dangerous impacts. Even with a double integral, a kink in the target curve implies an infinite jump in the policy!

An alternative approach is to devise "fuzzy control" strateges for deliberate climate-policy feedback (as used for the stabilise temperature option), but such formulae tend to cause oscillations, which may not be so unrealistic!

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