A WUWT post (with comments invoking yours truly) is arguing over the merits of the BERN model to describe the sequestration of CO2.  In the past, I have described the sum of multiple exponentials of varying time constants in the BERN model as a heuristic approximation to the full diffusional model (see Figure 1) in several places — see the book “The Oil ConunDrum”, the “Diffusive Growth” paper  (both available in the menu), and several blog posts here.

Fig. 1: Impulse Response of the sequestering of Carbon Dioxide to a normalized stimulus. The solid blue curve represents the generally accepted model, while the dashed and dotted curves represent the dispersive diffusion model

No question that the fat-tails of the CO2 adjustment time is due to diffusional sequestration.  If it wasn’t for the slow uptake of CO2 into sequestering sites, CO2 as a GHG would be a non-issue (see the profile labeled Segalstad in Figure 1, which describes the non-operable fast dynamics residence time).  But as it is, the fat-tails prevent the CO2 from sequestering rapidly and completely, so the gradual buildup over time is what is causing the excess atmospheric CO2.

Interestingly, the same diffusional dynamics is operational for thermal diffusion into the oceans, and the example of applying simple first-order thermal response dynamics is a poor approximation to the full diffusional treatment. For example, when climate scientist Isaac Held invokes first-order lagged (exponential) responses to thermal impulses, such as here, be careful on the interpretation. This is not meant to describe the real dynamics of a thermal response. In contrast, see these blog posts by Held on “Playing with a diffusive energy balance model” and “A diffusive model of atmospheric heat transport”. These are the diffusional models which reflect reality to a much better approximation.

The replacement of full diffusional models with too-simple first-order damped response models is a pet peeve of mine, having worked in the semiconductor industry where characterizing process is important. If one tries to model diffusion as a first-order damped exponential process, all the process formulas (such as for predicting oxide growth, dopant concentration, etc) become useless. So what happens when climate science tries to dumb down the math, it just doesn’t cut it — in much the same way that semiconductor process engineering would fail. In other words, none of the dynamics will come out correctly.  Ultimately this turns into an intuition gap where the desire is to come up with first-order calculations with perhaps some compartmentalization, while the reality is that finite difference slab numerical calculations are closer to the truth.  The latter is often what it takes to solve diffusional formulations correctly.

Our paper “Characterizing Diffusive Growth by Uncertainty Quantification” provides several examples where simple analytical expressions are done correctly. When a full uncertainty quantification via the MaxEnt principle is applied, we call this model dispersive diffusion.  This is used in many applications, from estimated Bakken crude oil yields to estimating the charge and discharge kinetics on Lithium ion batteries — essentially any process that features disorder and diffusional dynamics.  The key finding is that in many of these cases closed-form analytical expressions are indeed practical and useful.