Gravitational Pull

In Chapter 12 of the book, we provide an empirical gravitational forcing term that can be applied to the Laplace’s Tidal Equation (LTE) solution for modeling ENSO. The inverse squared law is modified to a cubic law to take into account the differential pull from opposite sides of the earth.

excerpt from Mathematical Geoenergy (Wiley/2018)

The two main terms are the monthly anomalistic (Mm) cycle and the fortnightly tropical/draconic pair (Mf, Mf’ w/ a 18.6 year nodal modulation). Due to the inverse cube gravitational pull found in the denominator of F(t), faster harmonic periods are also created — with the 9-day (Mt) created from the monthly/fortnightly cross-term and the weekly (Mq) from the fortnightly crossed against itself. It’s amazing how few terms are needed to create a canonical fit to a tidally-forced ENSO model.

The recipe for the model is shown in the chart below (click to magnify), following sequentially steps (A) through (G) :

(A) Long-period fortnightly and anomalistic tidal terms as F(t) forcing
(B) The Fourier spectrum of F(t) revealing higher frequency cross terms
(C) An annual impulse modulates the forcing, reinforcing the amplitude
(D) The impulse is integrated producing a lagged quasi-periodic input
(E) Resulting Fourier spectrum is complex due to annual cycle aliasing
(F) Oceanic response is a Laplace’s Tidal Equation (LTE) modulation
(G) Final step is fit the LTE modulation to match the ENSO time-series

The tidal forcing is constrained by the known effects of the lunisolar gravitational torque on the earth’s length-of-day (LOD) variations. An essentially identical set of monthly, fortnightly, 9-day, and weekly terms are required for both a solid-body LOD model fit and a fluid-volume ENSO model fit.

Fitting tidal terms to the dLOD/dt data is only complicated by the aliasing of the annual cycle, making factors such as the weekly 7.095 and 6.83-day cycles difficult to distinguish.

If we apply the same tidal terms as forcing for matching dLOD data, we can use the fit below as a perturbed ENSO tidal forcing. Not a lot of difference here — the weekly harmonics are higher in magnitude.

Modified initial calibration of lunar terms for fitting ENSO

So the only real unknown in this process is guessing the LTE modulation of steps (F) and (G). That’s what differentiates the inertial response of a spinning solid such as the earth’s core and mantle from the response of a rotating liquid volume such as the equatorial Pacific ocean. The former is essentially linear, but the latter is non-linear, making it an infinitely harder problem to solve — as there are infinitely many non-linear transformations one can choose to apply. The only reason that I stumbled across this particular LTE modulation is that it comes directly from a clever solution of Laplace’s tidal equations.

for full derivation see Mathematical Geoenergy (Wiley/2018)

One thought on “Gravitational Pull

  1. Pingback: Complexity vs Simplicity in Geophysics | GeoEnergy Math

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