click to enlarge

“*Nonlinear Differential Equations with External Forcing*” presentation starts at slide 2020 http://iclr2020deepdiffeq.rice.edu/ — citation

click to enlarge

“*Nonlinear Differential Equations with External Forcing*” presentation starts at slide 2020 http://iclr2020deepdiffeq.rice.edu/ — citation

This is a continuation from the previous Length of Day post showing how closely the ENSO forcing aligns to the dLOD forcing.

Ding & Chao apply an AR-z technique as a supplement to Fourier and Max Entropy spectral techniques to isolate the tidal factors in dLOD

- H. Ding and B. F. Chao, “Application of stabilized AR‐z spectrum in harmonic analysis for geophysics,” Journal of Geophysical Research: Solid Earth, vol. 123, no. 9, pp. 8249–8259, 2018.

The red data points are the spectral values used in the ENSO model fit.

The top panel below is the LTE modulated tidal forcing fitted against the ENSO time series. The lower panel below is the tidal forcing model over a short interval overlaid on the dLOD/dt data.

That’s all there is to it — it’s all geophysical fluid dynamics. Essentially the same tidal forcing impacts both the rotating solid earth and the equatorial ocean, but the ocean shows a lagged nonlinear response as described in Chapter 12 of the book. In contrast, the solid earth shows an apparently direct linear inertial response. Bottom line is that if one doesn’t know how to do the proper GFD, one will never be able to fit ENSO to a known forcing.

In Chapter 12 of the book, we discuss tropical instability waves (TIW) of the equatorial Pacific as the higher wavenumber (and higher frequency) companion to the lower wavenumber ENSO (El Nino /Southern Oscillation) behavior. Sutherland *et al* have already published several papers this year that appear to add some valuable insight to the mathematical underpinnings to the fluid-mechanical relationship.

Continue reading“It is estimated that globally 1 TW of power is transferred from the lunisolar tides to internal tides[1]. The action of the barotropic tide over bottom topography can generate vertically propagating beams near the source. While some fraction of that energy is dissipated in the near field (as observed, for example, near the Hawaiian Ridge [2]), most of the energy becomes manifest as low-mode internal tides in the far field where they may then propagate thousands of kilometers from the source [3]. An outstanding question asks how the energy from these waves ultimately cascades from large to small scale where it may be dissipated, thus closing this branch of the oceanic energy budget. Several possibilities have been explored, including dissipation when the internal tide interacts with rough bottom topography, with the continental slopes and shelves, and with mean flows and eddies (for a recent review, see MacKinnon et al. [4]). It has also been suggested that, away from topography and background flows, internal modes may be dissipated due to nonlinear wave-wave interactions including the case of triadic resonant instability (hereafter TRI), in which a pair of “sibling” waves grow out of the background noise field through resonant interactions with the “parent” wave”

see reference [2]

You really don’t need a supercomputer or petabytes of memory or a high bandwidth network to do climate science.

*If you want to learn how to build a climate model, build a climate model. Don’t ask anybody, just build a climate model.*

One of the frustrating aspects of climatology as a science is in the cavalier treatment of data that is often shown, and in particular through the potential loss of information through filtering. A group of scientists at NASA JPL (Perigaud *et al*) and elsewhere have pointed out how constraining it is to remove what are considered errors (or *nuisance parameters*) in time-series by assuming that they relate to known tidal or seasonal factors and so can be safely filtered out and ignored. The problem is that this is only appropriate **IF** those factors relate to an independent process and don’t also cause non-linear interactions with the rest of the data. So if a model predicts both a linear component and non-linear component, it’s not helpful to eliminate portions of the data that can help distinguish the two.

As an example, this extends to the pre-mature filtering of annual data. If you dig enough you will find that NINO3.4 data is filtered to remove the annual data, and that the filtering is over-zealous in that it removes all annual harmonics as well. Worse yet, the weighting of these harmonics changes over time, which means that they are removing other parts of the spectrum not related to the annual signal. Found in an “ensostuff” subdirectory on the NOAA.gov site:

Continue readingChapter 5 of the book describes a model of the production of oil based on discoveries followed by a sequence of lags relating to decisions made and physical constraints governing the flow of that oil. As it turns out, this so-named Oil Shock Model is mathematically similar to the * compartmental models* used to model contagion growth in epidemiology, pharmaceutical/drug deliver systems, and other applications as demonstrated in Appendix E of the book.

One aspect of the 2020 pandemic is that everyone with any math acumen is becoming aware of contagion models such as the SIR compartmental model, where **S I R** stands for **S**usceptible, **I**nfectious, and **R**ecovered individuals. The Infectious part of the time progression within a population resembles a bell curve that peaks at a particular point indicating maximum contagiousness. The hope is that this either peaks quickly or that it doesn’t peak at too high a level.

Appendix E of the book contains information on compartmental models, of which resource depletion models, contagion growth models, drug delivery models, and population growth models belong to.

One compartmental population growth model, that specified by the Lotka-Volterra-type predator-prey equations, can be manipulated to match a cyclic wildlife population in a fashion approximating that of observations. The cyclic variation is typically explained as a nonlinear resonance period arising from the competition between the predators and their prey. However, a more realistic model may take into account seasonal and climate variations that control populations directly. The following is a recent paper by wildlife ecologist H. L. Archibald who has long been working on the thesis that seasonal/tidal cycles play a role (one paper that he wrote on the topic dates back to 1977! ).

Continue readingOur 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):

- Laplace’s Tidal Equation Analytic Solution.

(Ch 11, 12) 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. - 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. - 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. - 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. - 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. - The Dispersive Discovery Model.

(Ch 4) A probabilistic model of resource discovery which accounts for technological advancement and a finite search volume. - 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. - 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. - Origin of Tropical Instability Waves (TIW).

(Ch 12) As the ENSO model was developed, a higher harmonic component was found which matches TIW - 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. - 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. - 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. - 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. - 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. - 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. - 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. - Solving the Reserve Growth “enigma”.

(Ch 6) An application of dispersive discovery on a localized level which models the hyperbolic reserve growth characteristics observed. - Shocklets.

(Ch 7) A kernel approach to characterizing production from individual oil fields. - 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. - The Hubbert Peak Logistic Curve explained.

(Ch 7) The Logistic curve is trivially explained by dispersive discovery with exponential technology advancement. - Laplace Transform Analysis of Dispersive Discovery.

(Ch 7) Dispersion curves are solved by looking up the Laplace transform of the spatial uncertainty profile. - Gompertz Decline Model.

(Ch 7) Exponentially increasing extraction rates lead to steep production decline. - 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. - 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. - The Overshoot Point (TOP) and the Oil Production Plateau.

(Ch 8) How increases in extraction rate can maintain production levels. - 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). - 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. - Oil Recovery Factor Model.

(Ch 6) A model of oil recovery which takes into account reservoir size. - 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. - Particle and Crystal Growth Statistics.

(Ch 20) Detailed model of ice crystal size distribution in high-altitude cirrus clouds. - Rainfall Amount Dispersion.

(Ch 15) Explanation of rainfall variation based on dispersion in rate of cloud build-up along with dispersion in critical size. - Earthquake Magnitude Distribution.

(Ch 13) Distribution of earthquake magnitudes based on dispersion of energy buildup and critical threshold. - IceBox Earth Setpoint Calculation.

(Ch 17) Simple model for determining the earth’s setpoint temperature extremes — current and low-CO2 icebox earth. - 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. - GPS Acquisition Time Analysis.

(Ch 21) Engineering analysis of GPS cold-start acquisition times. Using Maximum Entropy in EMI clutter statistics. - 1/f Noise Model

(Ch 21) Deriving a random noise spectrum from maximum entropy statistics. - Stochastic Aquatic Waves

(Ch 12) Maximum Entropy Analysis of wave height distribution of surface gravity waves. - 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. - 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.

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):

**Laplace’s Tidal Equation Analytic Solution**.

**(Ch 11, 12)**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.**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.**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.**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.**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.**The Dispersive Discovery Model.**

**(Ch 4)**A probabilistic model of resource discovery which accounts for technological advancement and a finite search volume.**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.**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.**Origin of Tropical Instability Waves (TIW)**.

**(Ch 12)**As the ENSO model was developed, a higher harmonic component was found which matches TIW**Characterization of Battery Charging and Dischargin**g.

**(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.**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.**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.**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.**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.**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.**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.**Solving the Reserve Growth “enigma”.**

**(Ch 6)**An application of dispersive discovery on a localized level which models the hyperbolic reserve growth characteristics observed.**Shocklets.**

**(Ch 7)**A kernel approach to characterizing production from individual oil fields.**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.**The Hubbert Peak Logistic Curve explained.**

**(Ch 7)**The Logistic curve is trivially explained by dispersive discovery with exponential technology advancement.**Laplace Transform Analysis of Dispersive Discovery.**

**(Ch 7)**Dispersion curves are solved by looking up the Laplace transform of the spatial uncertainty profile.**Gompertz Decline Model.**

**(Ch 7)**Exponentially increasing extraction rates lead to steep production decline.**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.**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.**The Overshoot Point (TOP) and the Oil Production Plateau.**

**(Ch 8)**How increases in extraction rate can maintain production levels.**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).**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.**Oil Recovery Factor Model.**

**(Ch 6)**A model of oil recovery which takes into account reservoir size.**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.**Particle and Crystal Growth Statistics.**

**(Ch 20)**Detailed model of ice crystal size distribution in high-altitude cirrus clouds.**Rainfall Amount Dispersion.**

**(Ch 15)**Explanation of rainfall variation based on dispersion in rate of cloud build-up along with dispersion in critical size.**Earthquake Magnitude Distribution.**

**(Ch 13)**Distribution of earthquake magnitudes based on dispersion of energy buildup and critical threshold.**IceBox Earth Setpoint Calculation.**

**(Ch 17)**Simple model for determining the earth’s setpoint temperature extremes — current and low-CO2 icebox earth.**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.**GPS Acquisition Time Analysis**.

**(Ch 21)**Engineering analysis of GPS cold-start acquisition times. Using Maximum Entropy in EMI clutter statistics.**1/f Noise****Model**

**(Ch 21)**Deriving a random noise spectrum from maximum entropy statistics.**Stochastic Aquatic Waves**

**(Ch 12)**Maximum Entropy Analysis of wave height distribution of surface gravity waves.**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.**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.

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):

Continue reading