Simple models of forced warming

Caldeira and Myrhvold [1] attempted the obvious by compiling various global warming models to try to extract simple thermal behavioral patterns, see their online paper.
They succeeded in my opinion.

What is most important about their work is that they placed diffusional models of warming, as pioneered by James Hansen [2], on an equal footing with first-order kinetics model. The first-order models use damped exponentials and are favored by analysts that want to keep the math simple. Caldeira and Myrhvold understand this and provide mixed exponential models that will piece-wise map to the diffusional responses normally seen in the numerical simulations.

The Figure 1 below shows the various responses to a 4× step input of CO2 applied to a steady-state climate. The CO2 generates a greenhouse gas forcing to the earth’s surface, which is partially absorbed by  the vast expanse of the ocean acting as a thermal heat sink.   The depths of the ocean gradually equilibriate with the surface as the heat diffuses downward — but until the layers reach a steady state, the temperature will continue to creep up.   Note the fast transient, the Transient Climate Response (TCR), followed by a gradual increase to the Equilibrium Climate Sensitivity (ECS) .

C&M fits to warming models

Fig. 1 : Most global warming models show a fast transient followed by a gradual Fickian rise characteristic of thermal diffusuion. The curves in green are naive first-order damped exponentials that map poorly to the actual behavior. The key to analyzing these profiles is to apply uncertainty to the diffusivity, which captures the range in diffusional path response times.

The foundational methods that I employ in models of thermal diffusion are described in a paper Diffusive Growth (also linked by the menu item at the top of this blog). This model assumes a dispersion of thermal diffusivities and uncertainty in the impulse depth.  Figure 2 below provides a general generic view of how this model is applied.

Fig 2 : The thermal step response measured in borehole experiments is composed of the transient response followed by a steady-state response. The red curve shows a dispersive diffusion fit described in the Diffusive Growth paper.

If one compares the shape of the step response of Figure 2 with the responses shown in Figure 1, the overall agreement is apparent.  This is not unexpected as diffusional flow is the solution to the heat equation, which is quite general in its applicability. The differences arise mainly in the abstraction needed to describe the effective diffusion of heat in the ocean — this is a combination of various eddy diffusivities at different scales.  I describe this also here in terms of a Ocean Heat Content model.

The DCS has a few interactive screens for evaluating the thermal diffusion curves, see for an example.

This diffusional response can be added to the CSALT model of global warming. Consider Caldeira and Myrhvold’s Figure 5 reproduced below as Figure 3, which gradually adds 1% additional CO2 per year.   The fast TCR intuitively tracks the growth of CO2 so that it is difficult to distinguish the CO2 growth from the transient response as long as the CO2 accumulation continues.   (This is very similar to the Red Queen growth in Bakken reserves described elsewhere, where the diffusional production is difficult to separate from the well growth as long as wells are accelerating).

Fig 3 : The addition of 1% CO2 per year in Caldeira & Myrhwold’s simplified model. The fast transient (TCR) keeps up with the CO2 growth, up to where the growth stops or slows down and then the equilibrium climate sensitivity (ECS) takes over and provides an asymptotic relaxation.

The full diffusional response will eventually be added to the CSALT model as a convolution response term. Right now the land temperature warming is considered instantaneous as it reaches its full ECS response quickly, while the ocean warming reaches an average TCR representative of the most recent accumulative CO2 warming value (with a slight lag).  It is not an absolutely critical addition yet it will add the extra fidelity that a comprehensive model should include, especially in the out-years as the diffusional effects accumulate and the ocean TCR inches closer to the ECS.


[1] K. Caldeira and N. Myhrvold, “Projections of the pace of warming following an abrupt increase in atmospheric carbon dioxide concentration,” Environmental Research Letters, vol. 8, no. 3, p. 034039, 2013.
[2] J. Hansen, D. Johnson, A. Lacis, S. Lebedeff, P. Lee, D. Rind, and G. Russell, “Climate impact of increasing atmospheric carbon dioxide,” Science, vol. 213 (4511), pp. 957–966, 1981.

4 thoughts on “Simple models of forced warming

    • StefanTheDenier, John Cleese starred in a TV show called Fawlty Towers. Fawlty’s excuse for everything that Basil does is “He’s from Barcelona”. For you, it’s because you’re from Australia. Nothing else to say.


  1. This is from a comment at Climate Etc and my response
    Marler said:

    They do not answer my question either: If the equilibrium warming effect is 3.18K, then how long after the doubling occurs will the surface temperature meet or exceed a specified value, such as 2.88K of warming? Notice that they say “heat either mixes downward into the ocean or radiates outward to space”, but there must be some non-negligible amount of heat that is transferred into evaporation, and then convection from the surface to the upper troposphere.

    This was the 4th time in one thread that you wrote about a lot of other stuff without answering the question or admitting that you don’t know the answer. “

    I know the answer.
    (1) On LAND, the equilibrium will be reached quite quickly, within years of the forcing, moderated very slowly by gradual ocean changes

    This is the relevant passage by Caldeira and Myrhvold

    We speculate that the 3-exp fits were better able to represent the rapid adjustment of land-surface (and land-plant) temperatures. Indeed, substantial temperature changes over land are observed to occur within days in climate model simulations of step-function changes in radiative forcing [20]. The median value of the shortest time constant in the 3-exp fits was 0.6 years, which is less than the annual resolution used in this analysis.

    (2) In the OCEAN, the equilibrium is reached asymptotically as a fat-tail. This means that at the surface, a fast transient to the TCR is reached quickly according to diffusion kinetics, followed by a gradual climb to the ECS. This could easily take hundreds of years partly because that is the way that Fickian fat tails work and mainly in consideration of how long it will take for the ocean to sink all the heat necessary for the temperature of the bulk to rise.

    This is another relevant passage by Caldeira and Myrhvold

    For many purposes, it may be a sufficient approximation to use a one-dimensional heat-diffusion ocean model having just one degree of freedom—in effect, to approximate warming as a simple heat-diffusion process.

    Why you think I am being evasive, I don’t know. I have worked out process diffusion equations my entire career. The SiO2 that is grown on the MOSFET devices that constitute your computer’s RAM and CPU is grown according to the Fickian diffusion kinetics that former Intel CEO Andy Grove wrote up in his PhD thesis in the early 1960’s. You wouldn’t ask a semiconductor engineer how long it would take to grow a thickness of an oxide unless you were being very specific. To grow a micron thick oxide it doesn’t take too long, but to grow a millimeter thick oxide thick oxide will likely take millions of times longer using conventional techniques. It is actually insane to even think about that once you realize how diffusion and bulk effects work.

    The ocean is a huge heat sink and it will equilibrate very slowly to an external forcing.


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