Mathematical Geoenergy

Our book Mathematical Geoenergy presents a number of novel approaches that each deserve a research paper on their own. Here is the list, ordered roughly by importance (IMHO):

  1. Laplace’s Tidal Equation Analytic Solution.
    (Ch 1112) A solution of a Navier-Stokes variant along the equator. Laplace’s Tidal Equations are a simplified version of Navier-Stokes and the equatorial topology allows an exact closed-form analytic solution. This could classify for the Clay Institute Millenium Prize if the practical implications are considered, but it’s a lower-dimensional solution than a complete 3-D Navier-Stokes formulation requires.
  2. Model of El Nino/Southern Oscillation (ENSO).
    (Ch 12) A tidally forced model of the equatorial Pacific’s thermocline sloshing (the ENSO dipole) which assumes a strong annual interaction. Not surprisingly this uses the Laplace’s Tidal Equation solution described above, otherwise the tidal pattern connection would have been discovered long ago.
  3. Model of Quasi-Biennial Oscillation (QBO).
    (Ch 11) A model of the equatorial stratospheric winds which cycle by reversing direction ~28 months. This incorporates the idea of amplified cycling of the sun and moon nodal declination pattern on the atmosphere’s tidal response.
  4. Origin of the Chandler Wobble.
    (Ch 13) An explanation for the ~433 day cycle of the Earth’s Chandler wobble. Finding this is a fairly obvious consequence of modeling the QBO.
  5. The Oil Shock Model.
    (Ch 5) A data flow model of oil extraction and production which allows for perturbations. We are seeing this in action with the recession caused by oil supply perturbations due to the Corona Virus pandemic.
  6. The Dispersive Discovery Model.
    (Ch 4) A probabilistic model of resource discovery which accounts for technological advancement and a finite search volume.
  7. Ornstein-Uhlenbeck Diffusion Model
    (Ch 6) Applying Ornstein-Uhlenbeck diffusion to describe the decline and asymptotic limiting flow from volumes such as occur in fracked shale oil reservoirs.
  8. The Reservoir Size Dispersive Aggregation Model.
    (Ch 4) A first-principles model that explains and describes the size distribution of oil reservoirs and fields around the world.
  9. Origin of Tropical Instability Waves (TIW).
    (Ch 12) As the ENSO model was developed, a higher harmonic component was found which matches TIW
  10. Characterization of Battery Charging and Discharging.
    (Ch 18) Simplified expressions for modeling Li-ion battery charging and discharging profiles by applying dispersion on the diffusion equation, which reflects the disorder within the ion matrix.
  11. Anomalous Behavior in Dispersive Transport explained.
    (Ch 18) Photovoltaic (PV) material made from disordered and amorphous semiconductor material shows poor photoresponse characteristics. Solution to simple entropic dispersion relations or the more general Fokker-Planck leads to good agreement with the data over orders of magnitude in current and response times.
  12. Framework for understanding Breakthrough Curves and Solute Transport in Porous Materials.
    (Ch 20) The same disordered Fokker-Planck construction explains the dispersive transport of solute in groundwater or liquids flowing in porous materials.
  13. Wind Energy Analysis.
    (Ch 11) Universality of wind energy probability distribution by applying maximum entropy to the mean energy observed. Data from Canada and Germany. Found a universal BesselK distribution which improves on the conventional Rayleigh distribution.
  14. Terrain Slope Distribution Analysis.
    (Ch 16) Explanation and derivation of the topographic slope distribution across the USA. This uses mean energy and maximum entropy principle.
  15. Thermal Entropic Dispersion Analysis.
    (Ch 14) Solving the Fokker-Planck equation or Fourier’s Law for thermal diffusion in a disordered environment. A subtle effect but the result is a simplified expression not involving complex errf transcendental functions. Useful in ocean heat content (OHC) studies.
  16. The Maximum Entropy Principle and the Entropic Dispersion Framework.
    (Ch 10) The generalized math framework applied to many models of disorder, natural or man-made. Explains the origin of the entroplet.
  17. Solving the Reserve Growth “enigma”.
    (Ch 6) An application of dispersive discovery on a localized level which models the hyperbolic reserve growth characteristics observed.
  18. Shocklets.
    (Ch 7) A kernel approach to characterizing production from individual oil fields.
  19. Reserve Growth, Creaming Curve, and Size Distribution Linearization.
    (Ch 6) An obvious linearization of this family of curves, related to Hubbert Linearization but more useful since it stems from first principles.
  20. The Hubbert Peak Logistic Curve explained.
    (Ch 7) The Logistic curve is trivially explained by dispersive discovery with exponential technology advancement.
  21. Laplace Transform Analysis of Dispersive Discovery.
    (Ch 7) Dispersion curves are solved by looking up the Laplace transform of the spatial uncertainty profile.
  22. Gompertz Decline Model.
    (Ch 7) Exponentially increasing extraction rates lead to steep production decline.
  23. The Dynamics of Atmospheric CO2 buildup and Extrapolation.
    (Ch 9) Convolving a fat-tailed CO2 residence time impulse response function with a fossil-fuel emissions stimulus. This shows the long latency of CO2 buildup very straightforwardly.
  24. Reliability Analysis and Understanding the “Bathtub Curve”.
    (Ch 19) Using a dispersion in failure rates to generate the characteristic bathtub curves of failure occurrences in parts and components.
  25. The Overshoot Point (TOP) and the Oil Production Plateau.
    (Ch 8) How increases in extraction rate can maintain production levels.
  26. Lake Size Distribution.
    (Ch 15) Analogous to explaining reservoir size distribution, uses similar arguments to derive the distribution of freshwater lake sizes. This provides a good feel for how often super-giant reservoirs and Great Lakes occur (by comparison).
  27. The Quandary of Infinite Reserves due to Fat-Tail Statistics.
    (Ch 9) Demonstrated that even infinite reserves can lead to limited resource production in the face of maximum extraction constraints.
  28. Oil Recovery Factor Model.
    (Ch 6) A model of oil recovery which takes into account reservoir size.
  29. Network Transit Time Statistics.
    (Ch 21) Dispersion in TCP/IP transport rates leads to the measured fat-tails in round-trip time statistics on loaded networks.
  30. Particle and Crystal Growth Statistics.
    (Ch 20) Detailed model of ice crystal size distribution in high-altitude cirrus clouds.
  31. Rainfall Amount Dispersion.
    (Ch 15) Explanation of rainfall variation based on dispersion in rate of cloud build-up along with dispersion in critical size.
  32. Earthquake Magnitude Distribution.
    (Ch 13) Distribution of earthquake magnitudes based on dispersion of energy buildup and critical threshold.
  33. IceBox Earth Setpoint Calculation.
    (Ch 17) Simple model for determining the earth’s setpoint temperature extremes — current and low-CO2 icebox earth.
  34. Global Temperature Multiple Linear Regression Model
    (Ch 17) The global surface temperature records show variability that is largely due to the GHG rise along with fluctuating changes due to ocean dipoles such as ENSO (via the SOI measure and also AAM) and sporadic volcanic eruptions impacting the atmospheric aerosol concentrations.
  35. GPS Acquisition Time Analysis.
    (Ch 21) Engineering analysis of GPS cold-start acquisition times. Using Maximum Entropy in EMI clutter statistics.
  36. 1/f Noise Model
    (Ch 21) Deriving a random noise spectrum from maximum entropy statistics.
  37. Stochastic Aquatic Waves
    (Ch 12) Maximum Entropy Analysis of wave height distribution of surface gravity waves.
  38. The Stochastic Model of Popcorn Popping.
    (Appx C) The novel explanation of why popcorn popping follows the same bell-shaped curve of the Hubbert Peak in oil production. Can use this to model epidemics, etc.
  39. Dispersion Analysis of Human Transportation Statistics.
    (Appx C) Alternate take on the empirical distribution of travel times between geographical points. This uses a maximum entropy approximation to the mean speed and mean distance across all the data points.

Rating of Climate Change blogs

Scientific blogging is on a decline and that is especially evident with respect to climate change blogs. Nothing really good left apart from forums such as (which allows equation markup, image posting, and freedom to create threaded discussions).

Here is a grading of blogs that I have on my RSS feed:

  • WUWT :  F-
    A horrible AGW denier blog that pretends to be fair & balanced. RSS feed does not work with Owl.
  • Tallbloke’s Talkshop : F-
    A horrible AGW denier blog that specializes in numerology
  • Real Climate : C
    Sparse postings and comment moderation has long latencies so the discussion is glacially paced
  • Open Mind : C
    Not very interesting, mostly from a statistical angle, which is not where progress in analysis occurs
  • Science of Doom : D
    I don’t understand this site, seems to be run by a thinly veiled skeptic. Might as well read books by Pierrehumbert to gain an understanding of the physics instead of struggling along with the topics.
  • And Then There’s Physics : D+
    The moderators are control freaks, and the discussions are safe as milk
  • Peak Oil Barrel : A
    A very good blog that allows both fossil fuel discussion and climate change discussion, separated in distinct threads.  Moderated slightly and images allowed, along with short-term editing.
  • The Blackboard : F-
    An awful blog run by a mechanical engineer which at one point had climate science discussion but now consists of pro-Trump cheerleading.
  • Clive Best : D+
    The moderator tries hard but then stumbles as he desperately tries to debunk the AGW consensus. Marginally better than Science of Doom because at least the scientific ideas are creative.
  • Climate Audit : F-
    Awful conspiracy-laden blog run by a former Canadian mining executive.
  • Climate Etc : F-
    Pointless blog stressing climate science uncertainty run by a now-retired climate science professor J. Curry with a comment section that seems infested with Australian AGW deniers. 
  • Moyhu : B
    Halfway-decent posts by a retired fluid dynamics researcher but an ugly and unstable comment-entry system. 
  • Hotwhopper : C+
    Well-thought out counter-attacks to nonsense at sites such as WUWT, but nothing really about discussions of science
  • Robert Scribbler : C
    The American version of HotWhopper, with probably too much doom & gloom.
  • Skeptical Science: D
    Nothing interesting here as they never seem to veer from the consensus.  The comments seem to be overly moderated and at one time the RSS feed was broken, but that has recently been fixed.
  • Roy Spencer, PhD : F-
    Horrible blog by a religious zealot with comments infested by AGW deniers
  • Rabbet Run : B-
    Below Moyhu because nothing really innovative but occasional insight
  • More Grumbine Science : B
    By a NASA guy,  posts very rarely

This is a previous summary I had written two years ago (I had forgotten I had saved it in a draft folder, and so you can see how little has changed)

Blog Grade Rationale
Real Climate C Too long turnaround for comments
And Then There’s Physics D Too much ClimateBall
Skeptical Science D Too insular, won’t discuss cutting edge
Science of Doom D- Inflitrated by deniers
Open Mind C Too much on statistics, which does a disservice to unlocking deterministic aspects such as ENSO
Moyhu B Worst comment entry, but the research is quality
This Week in Science: DailyKos C+ Nothing in depth
Watt’s Up With That F Garbage
Tallbloke’s Talkshop F Loony bin
The Blackboard F Nasty people
Climate Etc F Clueless (mainly Aussies) lead by a clueless
Roy Spencer F Zero real science
Rabbet Run C Too much inside posturing
Hot Whopper C Good if you want to see deniers get debunked
Robert Scribbler C Verges on hysterical but who knows
Azimuth Project A A true forum. Allows everyone to create markup and add charts

Commenting at PubPeer

For our Mathematical GeoEnergy book, there is an entry at for comments (one can also comment at, but you need to be a verified purchaser of the book to be able to comment there)

PubPeer provides a good way to debunk poorly researched work as shown in the recent comments pertaining to the Zharkova paper published in Nature’s Scientific Reports journal.

An issue with the comment policy at Amazon is that one can easily evaluate the contents of a book via the “Look Inside” feature or through the Table of Contents. Often there is enough evidence to provide a critical book review just through this feature — in a sense, a statistical sampling of the contents — yet Amazon requires a full purchase before a review is possible. Even if one can check the book out at a university library this is not allowable. Therefore it favors profiting by the potential fraudster because they will get royalties in spite of damaging reviews by critics that are willing to sink money into a purchase.

In the good old days at Amazon, one could actually warn people about pseudo-scientific research. This is exemplified by Curry’s Bose-Einstein statistics debacle, where unfortunately political cronies and acolytes of Curry’s have since purchased her book and have used the comments to do damage control. No further negative comments are possible since smart people have not bought her book and therefore can no longer comment.

PubPeer does away with this Catch-22 situation.

Mathematical GeoEnergy

Book will be out next year published by Wiley

This blog will be ramped up for the book, but contains all the research leading up to the book.

Two papers at AGU 2017:



Dynamic Context Server

Tropics, poles and reefs

Diagram Monkey

2014, 2015 and 2016 played a recurring theme of El Nino. A tentative El Nino in late 2014 and early 2015 segued with a stutter into a strong El Nino in 2015/2016 dragging global temperatures in train. Temperatures in the tropical Pacific dropped a bit after that and may or may not have slipped into La Nina depending on which agency you listen to, but now, it looks like El Nino might be coming back: surface water temperatures in the eastern Pacific, off the coast of South America, have risen to four or more degrees above average although they’ve not spread further west and a number of seasonal forecasting centres are suggesting that temperatures might continue to rise. No one’s called it an El Nino, yet, but the effects of the elevated sea-surface temperatures are sadly plain to see. Heavy rain in Peru has already led to flooding and all the…

View original post 261 more words


Please refer to my new WordPress blog Context/Earth for future posts.

WordPress has better commenting options such as provisions for pictures and charts.

The scope of the blog is also more comprehensive, as it will include all environmental and energy topics tied together in a semantic web framework. 

Onward and upward as they say.

Expansion of atmosphere and ocean

This is a short tutorial together with some observational evidence explaining how the atmosphere and ocean is expanding measurably in the face of global warming.

Ocean thermal expansion

The ocean absorbs heat per area according to its heat capacity

$$ \Delta E = C_p \cdot \Delta T \cdot {Depth}$$

The linear coefficient of thermal expansion is assumed constant over a temperature range. Multiplying this over a depth:

$$ \Delta Z = \epsilon \cdot \Delta T \cdot Depth $$
But now we can substitute the total energy gained from the first equation:
$$ \Delta Z = \epsilon \frac{\Delta E}{C_p} $$
Assume that the linear coefficient of thermal expansion is 0.000207 per °C, and specific heat capacity is 4,000,000 J/m^3/°C.

If an excess forcing of 0.6 W/m^2 occurs over one year (see the OHC model), then the increase in the level of the ocean is

0.000207  * 0.6*(365*24*60*60) / 4,000,000 = 0.98 mm

This is called the steric sea-level rise, and it is just one component of the sea-level rise over time (the others have to do with melting ice). From the figure below, one can see that the thermal rise is close to 1mm/year over the last decade.

The red line shows the thermal expansion from the ocean heating. Thermal rise is 1 mm/year over the 3 mm/year total sea-level rise.

Atmosphere thermal expansion

The trick here is to infer the atmosphere expansion by looking at an equal pressure point as a function of altitude (a geopotential height isobar) and determine how much that point has increased over time.  I was able to find one piece of data from The Weather Channel’s senior meteorologist Stu Ostro.

The geopotential height anomaly is shown below for the 500 mb (1/2 atmosphere)

Geopotential height anomaly @ 500 mb plotted alongside global temperature anomaly.

In absolute terms it is charted as follows:

Global average geopotential height @ 500 mb plotted alongside global temperature.

To understand how the altitude has changed, consider the classical barometric formula:
$$ P(H) = P_0 e^{-mgH/RT} $$

We take the point at which we reached 1/2 the STP of 1 atmosphere at sea-level:

$$ P(H)/P_0 = 0.5 = e^{-mgH/RT} $$

$$ H = RT/mg \cdot ln(2) $$
Assuming the average molecular weight of the atmospheric gas constituents does not change, the change in altitude (H) should be related to the change in temperature (T) by:
$$ \Delta H = R \Delta T / mg \cdot ln(2) $$
$$ \frac{ \Delta H}{\Delta T} =  \frac{R}{mg} ln(2) $$
For R = 8314, m=29, and g=9.8, the slope should be 29.25 *ln(2) = 20.3 m/°C.

From the linear regression agreement between the two, we get a value of 25.7 m/°C.

Linear regression between the geopotential height change and temperature change

Why is this geopotential height change about 26% higher than it should be from the theoretical value considering that the height should track the temperature according to the barometric formula?

If we use the polytropic approximation (equation 1 in the barometric formula), the altitudinal difference between the low temperature and high temperature 500 mb pressure values remains the same as when we use the classic exponential damped barometric formula:

If we apply the polytropic barometric formula instead of the exponential, we still show a real height change that is higher than theoretically predicted by ~25% at the 500 mb isobar.

This discrepancy could be due to measurement error, as the readings are taken by weather balloons and the accuracy could have drifted over the years.

It is also possible that the composition of the atmosphere could have changed slightly at altitude. What happens if the moisture increased slightly? This shouldn’t make much difference.

Most likely is the possibility that the baseline sea-level pressure has changed, which shifted the 500 mbar point artificially. See

“ Sea Level Pressure and Atmospheric Circulation

As a basic component of the mean atmospheric circulations and weather patterns, projections of the mean sea level pressure for the medium scenario A1B are considered. Seasonal mean changes for DJF and JJA are shown in Figure 10.9 (matching results in Wang and Swail, 2006b). Sea level pressure differences show decreases at high latitudes in both seasons in both hemispheres. The compensating increases are predominantly over the mid-latitude and subtropical ocean regions, extending across South America, Australia and southern Asia in JJA, and the Mediterranean in DJF. Many of these increases are consistent across the models. This pattern of change, discussed further in Section, has been linked to an expansion of the Hadley Circulation and a poleward shift of the mid-latitude storm tracks (Yin, 2005). This helps explain, in part, the increases in precipitation at high latitudes and decreases in the subtropics and parts of the mid-latitudes. Further analysis of the regional details of these changes is given in Chapter 11. The pattern of pressure change implies increased westerly flows across the western parts of the continents. These contribute to increases in mean precipitation (Figure 10.9) and increased precipitation intensity (Meehl et al., 2005a). “

I will have to figure out the mean sea-level pressure change over this time period to verify this hypothesis.

The following recent research article looks into the shifts in geopotential height over shorter time durations:

Y. Y. Hafez and M. Almazroui, “The Role Played by Blocking Systems over Europe in Abnormal Weather over Kingdom of Saudi Arabia in Summer 2010,” Advances in Meteorology, vol. 2013, p. 20, 2013.


                   ADDED  7/8/2013                

One possibility for the larger-than-expected altitude change is that the average lapse rate has changed slightly.  We can use the lapse rate variation of the barometric formula, and perturb the lapse rate, L, around its average value:

$$ P(H) = P(0) (1- \frac{LH}{T_0})^{\frac{gm}{LR}} $$

Granted, the error bars on this calculation are significant but we can see how subtle the effect is.

Sea-level pressure, P(0) = 1013.25 mb
Gas constant,  R = 8.31446 J/K/mol
Earth’s gravity, g = 9.807 m/s^2
Avg molecular weight, m = 0.02896 kg/m

In 1960, the temperature was about 14.65°C and 500 mb altitude was 5647 m.
In 2010, the temperature was about 15.4°C and 500 mb altitude was 5667 m.

All we need to do is invert the P(H) formula for each pair of values, modifying L slightly.

If we select a lapse rate, L, for 1960 of 0.005C/m, we calculate H = 5647.9m for P(H)=500 mb.
If we select a lapse rate, L, for 2010 of 0.0050°C/m, we calculate H = 5668.4m for P(H)=500 mb.

The difference in the two altitudes for a change in L of -0.0001°C/m is 20.5m, about what the geopotential height chart shows.

If we leave the L at 0.005C/m for both 1960 and 2010, the difference of the 500mb altitudes is only 14.7m.

To the extent that we can trust the numbers on the charts from Ostro, the change in geopotential height is suggesting a feedback effect in the lapse rate due to global warming.  The lapse rate is decreasing over time, which implies that the heat capacity of the atmosphere is increasing (likely due to higher specific humidity), thereby buffering changes in temperature with altitude.

This means that a given temperature increase at a particular altitude (where the CO2 IR window can achieve a radiative balance) will be reflected as a scale-modified temperature at sea level

In 1960, the temperature difference at 500mb altitude is 0.005C/m * 5647.9m = 28.8C
In 2010, the temperature difference at 500mb altitude is 0.0050°C/m * 5668.4m = 28.34°C

The difference at sea-level from the chart is 15.4°C-14.65°C = 0.75°C whereas the difference at the 500mb altitude assuming the modified lapse rate is 0.75°C – 0.46°C = 0.29°C.  If the lapse rate didn’t change then this sea-level difference would maintain at a constant atmospheric pressure isobar in altitude.

For implications in the interpretation, see page 24 of National Research Council Panel on Climate Change Feedbacks (2003). Understanding climate change feedbacks : National Academies Press. ISBN 978-0-309-09072-8.  They caution that the measurements require some precision, otherwise the errors can multiply due to the differences between two large numbers.

Characterization of Battery Charging and Discharging

I had the good fortune of taking a week long Society of Automotive Engineers (SAE) Academy class on hybrid/electric vehicles. The take-home message behind HEV and EV technology is to remember that a quality battery plus optimizing the management of battery cycling remain the keys to success.   That is not surprising —  we all know that gasoline has long been “king”, and since current battery technology has nowhere near the energy density of gasoline, the battery has turned into a “diva”.  In other words, it will perform as long as it is in charge (so to speak) and the battery is well maintained.

I can report that much class time was devoted to the electrochemistry of Lithium-ion batteries. Lithium is an ideal elemental material due to its position in the upper left-hand corner of the periodic table — in other words a very lightweight material with a potentially high energy density.

What was surprising to me was the sparseness of detailed characterization of the material properties. One instructor stated that the lack of measured diffusion properties for battery cell specifications was a pet peeve of his.  Having these properties at hand allows a design engineer to better model the charging and discharging characteristics of the battery, and thus to perhaps to develop better battery management schemes. Coming from the semiconductor world, it is almost unheard of to do design without adequate device characteristics such as mobility and diffusivity.

From my perspective, this is not necessarily bad. Any time I see an anomalous behavior or missing piece, it opens the possibility I can fill a modeling niche.


Modern rechargeable battery technology still relies on the principles of electro-chemistry and a reversible process, which hasn’t changed in fundamental terms since the first lead-acid battery came to market in the early 1900’s. What has changed is the combination of materials that make a low-cost, lightweight, and energy-efficient battery which will serve the needs of demanding applications such as electric and hybrid-electric vehicles (EV/HEV).

As energy efficient operation is dependent on the properties of the materials being combined, it is well understood that characterizing the materials is important to advancing the state-of-the-art (and in increasing EV acceptance).

Of vital importance is the characterization of diffusion in the electrode materials, as that is the rate-limiting factor in determining the absolute charging and discharging speed of the material-specific battery technology. Unfortunately, because of the competitive nature of battery producers, many of the characteristics are well-guarded and treated as trade secrets. For example, it is very rare to find diffusion coefficient characteristics on commercial battery specification sheets, even though this kind of information is vital for optimizing battery management schemes [7].

In comparison to the relatively simple diffusional mechanisms of silicon oxide growth, the engineered structure of well-designed battery cell presents a significant constraint to the diffusional behavior. In Figure 1 below we show a schematic of a single lithium-ion cell and the storage particles that charge and discharge. The disordered nature of the storage particles in Figure 2 is often described by what is referred to as a tortuosity measure.

Figure 1: Exaggerated three-dimensional view of a lithium-ion battery cell and the direction of current flow during charging and discharging
Figure 2 : Realistic view of the heavily disordered nature of the LiFePO4 storage particles [1].

The constraints on the diffusion is that it is limited in scale to that of the radius of the storage particle. The length scale is limited essentially to the values L to Lmax shown in Figure 3 below.

Figure 3 : Diffusion of ions takes place through the radial shell of the LiFePO4 spherical particle [1]. During the discharge phase, the ions need to migrate outward through shell and through the SEI barrier before reaching the electrolyte. At this point they can contribute to current flow.

The size of the particles also varies as shown in Figure 4 below.  The two Lithium-ion materials under consideration, LiFePO4 and LiFeSO4F, have different materials properties but are structurally very similar (matrixed particles of mixed size) so that we can use a common analysis approach.  This essentially allows us to apply uncertainty in the diffusion coefficient and uncertainties in the particle size to establish a common diffusional behavior formulation.

Figure 4 : Particle size distribution of FeSO4F spherical granules [2]. The variation in lengths and material diffusivities opens the possibility of applying uncertainty quantification to a model of diffusive growth.

Dispersive Diffusion Analysis of Discharging

The diffusion of ions through the volume of a spherical particle does have similarity to classical regimes such as the diffusion of silicon through silicon dioxide.  That process in fact leads to the familiar Fick’s law of diffusion, whereby the growing layer of oxide follows a parabolic growth law (in fact this is a square root with time, but was named parabolic by the semiconductor technology industry for historical reasons, see for example here).

The model that we can use for Li+ diffusion derives from the classic solution to the Fokker-Planck equation of continuity (neglecting any field driven drift).
$$ \frac{\partial C}{\partial t} – D \nabla^2 C=0 $$ where C is a concentration and D is the diffusion coefficient.  Ignoring the spherical orientation, we can just assume a solution along a one dimensional radially outward axis, x:

$$ \large C(t,x|D) = \frac{1}{\sqrt{4 \pi D t}} e^{-x^2/{4 D t}} $$
This is a marginal probability which depends on the diffusion coefficient. Since we do not know the variance of the diffusivity, we can apply a maximum entropy distribution across D.
$$ \large p_d(D) = \frac{1}{D_0} e^{-D/D_0} $$
This simplifies the representation to the following workable formulation.
$$ \large C(t,x) = \frac{1}{2 \sqrt{D_0 t}} e^{-x/{\sqrt{D_0 t}}} $$
We now have what is called a kernel solution (i.e. Green’s function) that we can apply to specific sets of initial conditions and forcing functions, the latter solved via convolution.

Fully Charged Initial Conditions
Assume the spherical particle is uniformly distributed with a charge density C(0, x) at time t=0.

Discharging Model
For every point along the dimensions of the particle of size L, we calculate the time it takes to diffuse to the outer edge, where it can enter the electrolytic medium.  This is simply an integral of the C(t, x) term for all points starting from x’ = d to L, where d is the inner core radius.
$$  \large C(t) = \int_{d}^L C(t,L-x) dx $$
this integrates straightforwardly to this concise representation:
$$ \large C(t) = C_0 \frac{ 1 – e^{-(L-d)/{\sqrt{D_0 t}}} } {L – d} $$
The voltage of the cell is essentially the amount of charge available, so as this charge depletes, the voltage decreases in proportion.

We can test the model on two data sets corresponding to a LiPO4 cell [1] and a LiSO4F cell [2]. Figure 5 below shows the model fit for LiPO4 as the red dotted line, and which should be level-compared to the solid black line labelled 1. The other curves labelled 2,3,4,5 are alternative diffusional model approximations applied by Churikov et al that clearly do not work as well as the dispersive diffusion formulation derived above.

Figure 5 : Discharge profile of LiFePO4 battery cell [1], with the red dotted line showing the parameterized dispersive diffusion model.  The curves labelled 1 through 5 show alternative models that the authors applied to fit the data. Only the dispersive diffusion model duplicates the fast drop-off and long-time scale decline.

Figure 6 below shows the fit to voltage characteristics of a LiSO4F cell, drawn as a red dotted line above the light gray data points. In this case the diffusional model by Delacourt shown in solid black is well outside acceptable agreement.

Figure 6 : Discharge profile of LiFeSO4F battery cell [2], with the red dotted line showing the parameterized dispersive diffusion model.  The black curve shows the model that the authors applied to fit the data. 

The question is why does this simple formulation work so well? As with many similar cases of characterizing disordered material, the fundamentally derived solution needs to be adjusted to take into account the uncertainty in the parameter space.  However, this step is not routinely performed and by adding modeling details (see [4]) to try to make up for a poor fit works only as a cosmetic heuristic.   In contrast, by performing the uncertainty quantification, like we did with the diffusion coefficient, the first-order solution works surprisingly well with no need for additional detail.

Constant Current Discharge
Instead of assuming that the particle size is L, we can say that the L is an average and apply the same maximum entropy spread in values.
$$  \large C(t) = \int_{0}^{\infty} C(t,x) \frac{1}{L} e^{-x/L} dx $$
this integrates straightforwardly to this concise representation:
$$ \large C(t) = C_0 \frac{1}{ L + {\sqrt{D_0 t}} } $$
The reason we do this is to allow us to recursively define the change in charge to a current. In  this case, to get current we need to differentiate the charge with respect to time.
$$ \large I(t) = \frac{dC(t)}{dt}  $$
This differentiates to the following expression
$$ \large I(t) = – \frac{C_0}{ (L + {\sqrt{D_0 t}})^2} \frac{1}{2 \sqrt{t}} $$
But note that we can insert C(t) back in to the expression
$$ \large I(t) = \frac{C(t)}{(L + {\sqrt{D_0 t}}) 2 \sqrt{t}} $$
Finally, since I(t) is a constant and we can set that to a value of I_constant. Then the charge has the following profile
$$ C(t) = C(0) – k_c I_{constant} (L + \sqrt{Dt}) 2 \sqrt{t} $$
or as a voltage decline
$$ V(t) = V(0) – k_v I_{constant} (L + \sqrt{Dt}) 2 \sqrt{t} $$
For a set of constant current values, we can compare this formulation against experimental data for LiFePO4 (shown as gray open circles) shown in Figure 7 below. A slight constant current offset (which may arise from unspecified shunting and/or series elements) was required to allow for the curves to align proportionally.  Even with that, it is clear that the dispersive diffusion formulation works better than the conventional model (solid black lines) except where the discharge is nearing completion.

Figure 7 : Constant current discharge profile [6]. Superimposed as dotted lines are the set of model fits which use the current value as a fixed parameter.

We can also model battery charging but the lack of information on the charging profile makes the discharge behavior a simpler study.

Related Diffusion Topics


A. Churikov, A. Ivanishchev, I. Ivanishcheva, V. Sycheva, N. Khasanova, and E. Antipov, “Determination of lithium diffusion coefficient in LiFePO 4 electrode by galvanostatic and potentiostatic intermittent titration techniques,” Electrochimica Acta, vol. 55, no. 8, pp. 2939–2950, 2010.

C. Delacourt, M. Ati, and J. Tarascon, “Measurement of Lithium Diffusion Coefficient in Li y FeSO4F,” Journal of The Electrochemical Society, vol. 158, no. 6, pp. A741–A749, 2011.

M. Park, X. Zhang, M. Chung, G. B. Less, and A. M. Sastry, “A review of conduction phenomena in Li-ion batteries,” Journal of Power Sources, vol. 195, no. 24, pp. 7904–7929, Dec. 2010.

J. Christensen and J. Newman, “A mathematical model for the lithium-ion negative electrode solid electrolyte interphase,” Journal of The Electrochemical Society, vol. 151, no. 11, pp. A1977–A1988, 2004.

Q. Wang, H. Li, X. Huang, and L. Chen, “Determination of chemical diffusion coefficient of lithium ion in graphitized mesocarbon microbeads with potential relaxation technique,” Journal of The Electrochemical Society, vol. 148, no. 7, pp. A737–A741, 2001.

P. M. Gomadam, J. W. Weidner, R. A. Dougal, and R. E. White, “Mathematical modeling of lithium-ion and nickel battery systems,” Journal of Power Sources, vol. 110, no. 2, pp. 267–284, 2002.

“Hybrid and Electric Vehicle Engineering Academy.” [Online]. Available: [Accessed: 09-Jun-2013].

The homework problem to end all homework problems

This is a problem that has driven anyone that has studied climate science up the wall.

Premise: Venus has an adiabatic index γ (gamma) and a temperature lapse rate λ (lambda). Earth also has an adiabatic index and temperature lapse rate.  These have been measured, and for the Earth a standard atmospheric profile has been established. The general relationship is based on thermodynamic principles but the shape of the profile diverges from simple applications of adiabatic principles.  In other words, a heuristic is applied to allow it to match the empirical observations, both for Venus and Earth. See this link for more background

Assigned Problem: Derive the adiabatic index and lapse rate for both planets, Venus and Earth, using only the planetary gravitational constant, the molar composition of atmospheric constituents, and any laws of physics that you can apply.  The answer has to be right on the mark with respect to the empirically-established standards.

Caveat: Reminder that this is a tough nut to crack.

Solution:  The approach to use is concise but somewhat twisty.  We work along two paths, the initial path uses basic physics and equations of continuity;  while the subsequent path ties the loose ends together using thermodynamic relationships which result in the familiar barometric formula and lapse rate formula.  The initial assumption that we make is to start with a sphere that forms a continuum from the origin; this forms the basis of a polytrope, a useful abstraction to infer the generic properties of planetary objects.

An abstracted planetary atmosphere

The atmosphere has a density ρ, that decreases outward from the origin. The basic laws we work with are the following:

Mass Conservation

$$ \frac{dm(r)}{dr} = 4 \pi r^2 \rho $$

Hydrostatic Equilibrium

$$ \frac{dP(r)}{dr} = – \rho g = – \frac{Gm(r)}{r^2}\rho $$

To convert to purely thermodynamic terms, we first integrate the hydrostatic equilibrium relationship over the volume of the sphere
$$ \int_0^R \frac{dP(r)}{dr} 4 \pi r^3 dr = 4 \pi R^3 P(R) – \int_0^R 12 P(r) \pi r^2 dr $$
on the right side we have integrated by parts, and eliminate the first term as P(R) goes to zero (note: upon review, the zeroing of P(R) is an approximation if we do not let R extend to the deep pressure vacuum of space, as we recover the differential form later — right now we just assume P(R) decreases much faster than R^3 increases ). We then reduce the second term using the mass conservation relationship, while recovering the gravitational part:
$$ – 3 \int_0^M\frac{P}{\rho}dm = -\int_0^R 4 \pi r^3 \frac{G m(r)}{r^2} dr $$
again we apply the mass conservation
$$ – 3 \int_0^M\frac{P}{\rho}dm = -\int_0^M  \frac{G m(r)}{r} dm $$
The  right hand side is simply the total gravitational potential energy Ω while the left side reduces to a pressure to volume relationship:
 $$ – 3 \int_0^V P dV = \Omega$$
This becomes a variation of the Virial Theorem relating internal energy to potential energy.

Now we bring in the thermodynamic relationships, starting with the ideal gas law with its three independent variables. 

Ideal Gas Law

$$ PV = nRT $$

Gibbs Free Energy

$$ E = U – TS + PV $$

Specific Heat (in terms of molecular degrees of freedom)

$$ c_p = c_v + R = (N/2 + 1) R $$

On this path, we make the assertion that the Gibbs free energy will be minimized with respect to perturbations. i.e. a variational approach.

$$ dE = 0 = dU – d(TS) + d(PV) = dU – TdS – SdT + PdV + VdP $$

Noting that the system is closed with respect to entropy changes (an adiabatic or isentropic process) and substituting the ideal gas law featuring a molar gas constant for the last term.

$$ 0 = dU – SdT + PdV + VdP = dU – SdT + PdV + R_n dT$$

At constant pressure (dP=0) the temperature terms reduce to the specific heat at constant pressure:

$$ – S dT + nR dT = (c_v +R_n) dT = c_p dT $$

Rewriting the equation

$$ 0 = dU + c_p dT + P dV $$

Now we can recover the differential virial relationship derived earlier:

$$ – 3 P dV = d \Omega $$

and replace the unknown PdV term

$$ 0 = dU + c_p dT – d \Omega / 3 $$

but dU is the same potential energy term as dΩ, so

$$ 0 = 2/3 d \Omega+ c_p dT $$

Linearizing the potential gravitational energy change with respect to radius

$$ 0 = \frac{2 m g}{3} dr + c_p dT $$

Rearranging this term we have derived the lapse formula

$$ \frac{dT}{dr} = – \frac{mg}{3/2 c_p} $$

Reducing this in terms of the ideal gas constant and molecular degrees of freedom N

$$ \frac{dT}{dr} = – \frac{mg}{3/2 (N/2+1) R_n} $$

We still need to derive the adiabatic index, by coupling the lapse rate formula back to the hydrostatic equilibrium formulation.

Recall that the perfect adiabatic relationship (the Poisson’s equation result describing the potential temperature) does not adequately describe a standard atmosphere — being 50% off in lapse rate —  and so we must use a more general polytropic process approach.

Combining the Mass Conservation with the Hydrostatic Equilibrium:

$$ \frac{1}{r^2} \frac{d}{dr} (\frac{r^2}{\rho} \frac{dP}{dr}) = -4 \pi G \rho $$

if we make the substitution
$$ \rho = \rho_c \theta^n $$
where n is the polytropic index.  In terms of pressure via the ideal gas law
$$ P = P_c \theta^{n+1} $$
if we scale r as the dimensionless ξ :

$$ \frac{1}{\xi^2} \frac{d}{d\xi} (\frac{\xi^2}{\rho} \frac{dP}{d\xi}) = – \theta^n $$

This formulation is known as the Lane-Emden equation and is notable for resolving to a polytropic term. A solution for n=5 is
$$ \theta = ({1 + \xi^2/3})^{-1/2} $$

We now have a link to the polytropic process equation
$$ P V^\gamma = {constant} $$
$$ P^{1-\gamma} T^{\gamma} = {constant} $$
$$ P = P_0 (\frac{T}{T_0})^{\frac{\gamma}{1-\gamma}} $$
Tieing together the loose ends, we take our lapse rate gradient
$$ \frac{dT}{dr} = \frac{mg}{3/2 (N/2+1) R} $$
and convert that into an altitude profile, where r = z
$$ T = T_0 (1 – \frac{z}{f z_0}) $$
$$ z_0 = \frac{R T_0}{m g} $$
$$ f = 3/2 (1 + N/2) $$
and the temperature gradient, aka lapse rate
$$ \lambda=   \frac{m g}{ 3/2 (1 + N/2) R } $$
To generate a polytropic process equation from this, we merely have to raise the lapse rate to a power, so that we recreate the power law version of the barometric formula:
$$  P = P_0 (1 – \frac{z}{f z_0})^f $$
which essentially reduces to Poisson’s equation on substitution:
$$ P = P_0 (T/T_0)^f $$
where the equivalent adiabatic exponent is
$$ f =  \frac{\gamma}{1-\gamma} $$

Now we have both the lapse rate, barometric formula, and Poisson’s equation derived based only on the gravitational constant g, the gas law constant R, the average molar molecular weight of the atmospheric constituents m, and the average degrees of freedom N.

Answer: Now we want to check the results against the observed values for the two planets


Object Main Gas N m g
Earth N2, O2 5 28.96 9.807
Venus CO2 6 43.44 8.87


Object Lapse Rate observed f observed
Earth 6.506 C/km 6.5 21/4 5.25
Venus 7.72 7.72 6 6

All the numbers are spot on with respect to the empirical data recorded for both Earth and Venus, with supporting figures available here.


The rough derivation that I previously posted to explain the empirical data was not very satisfying in its thoroughness.  The more comprehensive derivation in this post serves to shore up the mystery behind the deviation from the adiabatic derivation.  The key seems to be correctly accounting for the internal energy necessary to maintain the gravitational hydrostatic equilibrium. Since the polytropic expansion describes a process, the actual atmosphere can accommodate these constraints (while minimizing Gibbs free energy under constant entropy conditions) by selecting the appropriate polytropic index.   The mystery of the profile seems not so mysterious anymore.

Criticisms welcome as I have not run across anything like this derivation to explain the Earth’s standard atmosphere profile nor the stable Venus data (not to mention the less stable Martian atmosphere).  The other big outer planets filled with hydrogen are still an issue, as they seem to follow the conventional adiabatic profile, according to the few charts I have access to.  The moon of Saturn, Titan, is an exception as it has a nitrogen atmosphere with methane as a greenhouse gas.

BTW, this post is definitely not dedicated to Ferenc Miskolczi. Please shoot me if I ever drift in that direction. It’s a tough slog laying everything out methodically but worthwhile in the long run.

Added Fig 1 : Lapse Rate on Earth versus Latitude. From

D. J. Lorenz and E. T. DeWeaver, “Tropopause height and zonal wind response to global warming in the IPCC scenario integrations,” Journal of Geophysical Research: Atmospheres (1984–2012), vol. 112, no. D10, 2007.

Added Fig 2 : Lapse Rate on Earth versus Latitude. The average was calculated by integrating
with effective cross-sectional area weighting of (sin(Latitude+2.5)-sin(Latitude-2.5)) . Adapted from

J. P. Syvitski, S. D. Peckham, R. Hilberman, and T. Mulder, “Predicting the terrestrial flux of sediment to the global ocean: a planetary perspective,” Sedimentary Geology, vol. 162, no. 1, pp. 5–24, 2003.

Added Fig 3: This study also suggests an average lapse rate of 6.1C/km over the northern hemisphere.

I. Mokhov and M. Akperov, “Tropospheric lapse rate and its relation to surface temperature from reanalysis data,” Izvestiya, Atmospheric and Oceanic Physics, vol. 42, no. 4, pp. 430–438, 2006.

Since I posted this derivation, I received feedback from several other blogs which I attached as comments below this post.  In the original post I concluded by saying that I was satisfied with my alternate derivation, but after receiving the feedback, there is still the nagging issue of why the Venus lapse rate profile can be so linear in the lower atmosphere even though we know that the heat capacity of CO2 varies with temperature (particularly in the high temperature range of greater than 500 Kelvin).

If  we go back and look at the hydrostatic relation derived earlier, we see an interesting identity:
$$ – 3 \int_0^M\frac{P}{\rho}dm = -\int_0^M  \frac{G M}{r} dm $$
If I pull out the differential from the integral
$$ 3 \frac{P}{\rho} = \frac{G M}{r} $$
and then realize that the left-hand side is just the Ideal Gas law
$$ 3RT/m = \frac{G M}{r} $$
This is internal energy due to  gravitational potential energy.
If we take the derivative with respect to r, or altitude:
$$ 3R \frac{dT}{dr} = – \frac{G M m}{r^2} $$
The right side is just the gravitational force on an average particle. So we essentially can derive a lapse rate directly:
$$  \frac{dT}{dr} = – \frac{g m}{3 R} $$
This will generate a linear lapse rate profile of temperature that decreases with increasing altitude. Note however that this does not depend on the specific heat of the constituent atmospheric molecules. That is not surprising since it only uses the Ideal Gas law, with no application of the variational Gibbs Free Energy approach used earlier.

What this gives us is a universal lapse rate that does not depend on the specific heat capacity of the constituent gases, only the mean molar molecular weight, m.   This is of course an interesting turn of events in that it could explain the highly linear lapse  profile of Venus.  However, plugging in numbers for the gravity of Venus and the mean molecular weight (CO2 plus trace gases), we get a lapse rate that is precisely twice that which is observed.

The “obvious”  temptation is to suggest that half of the value of this derived hydrodynamic lapse rate would position it as the mean of the lapse rate gradient and an isothermal lapse rate (i.e. slope of zero).
$$  \frac{dT}{dr} = – \frac{g m}{6 R} $$
The rationale for this is that most of the planetary atmospheres are not any kind of equilibrium with energy flow and are constantly swinging between an insolating phase during daylight hours, and then a outward radiating phase at night.   The uncertainty is essentially describing fluctuations between when an atmosphere is isothermal (little change of temperature with altitude producing a MaxEnt outcome in distribution of pressures, leading to the classic barometric formula) or isentropic (where no heat is exchanged with the surroundings, but the temperature can vary as rapid convection occurs).

In keeping with the Bayesian decision making, the uncertainty is reflected by equal an weighting between isothermal (zero lapse rate gradient) and an isentropic (adiabatic derivation shown).  This puts the mean lapse rate at half the isentropic value. For Earth, the value of g*m/3R is 11.4 C/km.  Half of this value is 5.7 C/km, which is a value closer to actual mean value than the US Standard Atmosphere of 6.5 C/km

J. Levine, The Photochemistry of Atmospheres. Elsevier Science, 1985.
“The value chosen for the convective adjustment also influences the calculated surface temperature. In lower latitudes, the actual temperature decrease with height approximates the moist adiabatic rate. Convection transports H2O to higher elevations where condensation occurs, releasing latent heat to the atmosphere; this lapse rate, although variable, has an average annual value of 5.7 K/km in the troposphere. In mid and high latitudes, the actual lapse rates are more stable; the vertical temperature profile is controlled by eddies that are driven by horizontal temperature gradients and by topography. These so-called baroclinic processes produce an average lapse rate of 5.2 K/km – It is interesting to note that most radiative convective models have used a lapse rate of 6.5 K km – which was based on date sets extending back to 1933. We know now that a better hemispherical annual lapse rate is closer to 5.2 K/km, although there may be significant seasonal variations.

BTW, the following references are very interesting presentations on the polytropic approach.


“Polytropes.” [Online]. Available: [Accessed: 19-May-2013].

B. Davies, “Stars Lecture.” [Online]. Available: . [Accessed: 28-May-2013].

                      Even More Recent Research                    

A number of Chinese academics [3,4] are attacking the polytropic atmosphere problem from an angle that I hinted at in the original Standard Atmosphere Model and Uncertainty in Entropy post.    The gist of their approach is to assume that the atmosphere is not under thermodynamic equilibrium (which it isn’t as it continuously exchanges heat with the sun and outer space in a stationary steady-state) and therefore use some ideas of non-extensible thermodynamics.  Specifically they invoke Tsallis entropy and a generalized Maxwell-Boltzmann distribution to model the behavioral move toward an equilibrium.  This is all in the context of self-gravitational systems, which is the theme of this post.  Why I think it is intriguing, is that they seem to tie the entropy considerations together with the polytropic process and arrive at some very simple relations (at least they appear somewhat simple to me).

In the non-extensive entropy approach, the original Maxwell-Boltzmann (MB) exponential velocity distribution is replaced with the Tsallis-derived generalized distribution — which looks like the following power-law equation:

$$ f_q(v)=n_q B_q (\frac{m}{2 \pi k T})^{3/2} (1-(1-q) \frac{m v^2}{2 k T})^{\frac{1}{1-q}}$$

The so-called q-factor is a non-extensivity parameter which indicates how much the distribution deviates from MB statistics. As q approaches 1, the expression gradually trasforms into the familiar MB exponentially damped v^2 profile.

When q is slightly less than 1, all the thermodynamic gas equations change slightly in character.  In particular, the scientist Du postulated that the lapse rate follows the familiar linear profile, but scaled by the (1-q) factor:

$$ \frac{dT}{dr} = \frac{(1-q)g m}{R} $$

Note that this again has no dependence on the specific heat of the constituent gases, and only assumes an average molecular weight.  If q=7/6 or Q = 1-q = -1/6, we can model the f=6 lapse rate curve that we fit to earlier.

There is nothing special about the value of f=6 other than the claim that this polytropic exponent is on the borderline for maintaining a self-gravitational system [5].

Note that as q approaches unity, the thermodynamic equilibrium value, the lapse rate goes to zero, which is of course the maximum entropy condition of uniform temperature.

The Tsallis entropy approach is suspiciously close to solving the problem of the polytropic standard atmosphere. Read Zheng’s paper for their take [3] and also Plastino [6]. 

The cut-off in the polytropic distribution (5) is an example of what is known, within the field of non extensive thermostatistics, as “Tsallis cut-off prescription”, which affects the q-maximum entropy distributions when q < 1. In the case of stellar polytropic distributions this cut-off arises naturally, and has a clear physical meaning. The cut-off corresponds, for each value of the radial coordinate r, to the corresponding gravitational escape velocity.

This has implications for the derivation of the homework problem that we solved at the top of this post, where we eliminated one term of the integration-by-parts solution. Obviously, the generalized MB formulation does have a limit to the velocity of a gas particle in comparison to the classical MB view. The tail in the statistics is actually cut-off as velocities greater than a certain value are not allowed, depending on the value of q.  As q approaches unity, the velocities allowed (i.e. escape velocity) approach infinity.

As Plastino states [6]:

Polytropic distributions happen to exhibit the form of q-MaxEnt distributions, that is, they constitute distribution functions in the (x,v) space that maximize the entropic functional Sq under the natural constraints imposed by the conservation of mass and energy.

The enduring question is does this describe our atmosphere adequately enough? Zheng and company certainly open it up to another interpretation.

Y. Zheng, W. Luo, Q. Li, and J. Li, “The polytropic index and adiabatic limit: Another interpretation to the convection stability criterion,” EPL (Europhysics Letters), vol. 102, no. 1, p. 10007, 2013.

Z. Liu, L. Guo, and J. Du, “Nonextensivity and the q-distribution of a relativistic gas under an external electromagnetic field,” Chinese Science Bulletin, vol. 56, no. 34, pp. 3689–3692, Dec. 2011.

M. V. Medvedev and G. Rybicki, “The Structure of Self-gravitating Polytropic Systems with n around 5,” The Astrophysical Journal, vol. 555, no. 2, p. 863, 2001.

A. Plastino, “Sq entropy and selfgravitating systems,” europhysics news, vol. 36, no. 6, pp. 208–210, 2005.

Airborne fraction of CO2 explained by sequestering model

As acknowledgement of the atmospheric levels of CO2 reaching 400 PPM, this post is meant to clear up one important misconception (suggested prerequisite reading on fat-tail CO2 sequestration here and the significance of the fat-tail here)

A recently active skeptic meme is that the amount of CO2 as an airborne fraction is decreasing over time.

“If we look at the data since Mauna Loa started, we see that the percentage of the CO2 emitted by humans that “remains” in the atmosphere has averaged around half, but that it has diminished over time, by around 1% per decade.
Over the 30 year period 1959-1989 it was around 55%; over the following 20+ years it was just over 50%.
Why is this?”

What the befuddled fellow is talking about are the charts being shown below. These are being shown without much context and no supporting documentation, which puts the burden on the climate scientists to explain. Note that the airborne fraction does seem to decrease slightly over the past 50 years, even though the carbon emissions are increasing.

This obviously needs some explaining.  The following figure illustrates what the CO2 sequestration model actually does.

Figure 1:  Model airborne fraction of CO2 against actual data

On the left is the data plotted together with the model of the yearly fraction not sequestered out. The model is less noisy than the data but it does clearly decline as well.  No big surprise as this is a response function, and responses are known to vary depending on the temporal profile of the input and the fat-tail in the adjustment time impulse response function.

On the right is the model with the incorporation of a temperature-dependent outgassed fraction. In this case the model is more noisy than the data, as it includes outgassing of CO2 depending on the global temperature for that year. Since the temperature is noisy, the CO2 fraction picks up all of that noise.  Still, the airborne fraction shows a small yet perceptible decline, and the model matches the data well, especially in recent years where the temperature fluctuations are reduced.

Amazing that over 50 years, the mean fraction has not varied much from 55%. That has a lot to do with the math of diffusional physics. Essentially a random walk moving into and out of sequestering sites is a 50/50 proposition. That’s the way to intuit the behavior, but the math really does the heavy lifting in predicting the fraction sequestered out.

It looks like the theory matches the data once again. The skeptics provide a knee-jerk view that this behavior is not well understood, but not having done the analysis themselves, they lose out — the skeptic meme is simply one of further propagating fear, uncertainty, and doubt (FUD) without concern for the underlying science.