Month: September 2024

AMO and the Mt tide

/ Paul Pukite (@whut) / 1 Comment

Geophysics is a challenging endeavor — one part geology as historical evidence and the other part current measurements revealing how the forces of nature are continuously shaping the behavior of the Earth. Yet, how much faith can we place on any one interpretation, or on any past consensus?

"But for geophysics on such a massive scale, nothing can be checked via a lab controlled experiment."Someone has described geology!I prefer my own description (though taken from someone else and modified): Geology is like trying to learn about barley by studying beer.

— Compton Scatterbrained (@stommepoes.bsky.social) 2024-09-18T15:13:20.671Z

This goes double for the study of geophysical fluid dynamics — volumes and forces so large that controlled experiments are impossible. So that when a semi-controlled impulse comes about, say from the calving of a glacier, the empirical results may be surprising:

The subsequent mega-tsunami — one of the highest in recent history — set off a wave which became trapped in the bendy, narrow fjord for more than a week, sloshing back and forth every 90 seconds.

The phenomenon, called a “seiche,” refers to the rhythmic movement of a wave in an enclosed space, similar to water splashing backwards and forwards in a bathtub or cup. One of the scientists even tried (and failed) to recreate the impact in their own bathtub.

While seiches are well-known, scientists previously had no idea they could last so long.

“Had I suggested a year ago that a seiche could persist for nine days, people would shake their heads and say that’s impossible,” said Svennevig, who likened the discovery to suddenly finding a new colour in a rainbow. — http://www.msn.com/en-in/news/techandscience/what-caused-the-mysterious-9-day-earth-vibration-scientists-point-to-650-ft-mega-tsunami-that-rocked-greenland/ar-AA1qwIqk

This took place in a large Greenland fjord — massive but not as massive as the ocean. And that’s where massive geophysical forces also take place, such as from tidal pull.

So what about the Atlantic Multidecadal Oscillation (AMO)? The Mt long-period tide is an interference between the Mf nodal tidal cycle (13.66 days) and the Mm anomalistic tidal cycle (27.55 days). When this is synchronized to an annual impulse, a multidecadal response results.

It is much too easy to model the 50-60 year AMO cycle via nonlinear triad frequency-doubling mechanisms, and also simulate the faster cycling from the predominate Mf and Mm interactions with the annual cycle. This is essentially a sloshing behavior operating on the ocean’s thermocline as described in Chapter 12 of Mathematical Geoenergy, applying the nonlinear geophysical dynamics embedded within Laplace’s Tidal Equations.

New colors of the rainbow are waiting to be revealed. No excuse for others to reproduce the tidal forcing models as described and perform cross-validated model fits to replicate the results above. Work smart not hard.

NAO and the Median Filter

/ Paul Pukite (@whut) / Leave a comment

a random quote

“LLMs run a median filter on the corpus”

The North Atlantic Oscillation (NAO) time-series has always been intimidating to analyze. It appears outwardly like white-noise, but enough scientists refer to long periods of positive or negative-leaning intervals that there must be more of an underlying autocorrelation embedded in it. To reduce the white noise and to extract the signal requires a clever filtering trick. The desire is that the filter preserve or retain the edges of the waveform while reducing the noise level. The 5-point median filter does exactly that at a minimal complexity level, as the algorithm is simply expressed. It will leave edges and steep slopes alone as the median will naturally occur on the slope. That’s just what we want to retain the underlying signal.

Once applied, the NAO time-series still appears erratic, yet the problematic monthly extremes are reduced with the median filter suppressing many of them completely. If we then use the LTE model to fit the NAO time-series (with a starting point the annual-impulsed tidal forcing used for the AMO), a clear correlation emerges. The standing-wave modes of course are all high, in contrast to the one low mode for the AMO, so the higher frequency cycling is expected yet the fit is surprisingly good for the number of peaks and valleys it must traverse in the 140+ year historical interval.

Non-autonomous mathematical models

/ Paul Pukite (@whut) / Leave a comment

Non-autonomous mathematical formulations differ from autonomous ones in that their governing equations explicitly depend on time or another external variable. In natural systems, certain behaviors or processes are better modeled with non-autonomous formulations because they are influenced by external, time-dependent factors. Some examples of natural behaviors that qualify as non-autonomous include:

1. Seasonal Climate Variation:
Climate patterns, such as temperature changes or monsoon cycles, are influenced by external factors like the Earth’s orbit, axial tilt, and solar radiation, all of which vary over time. These changes make non-autonomous systems suitable for modeling long-term climate behavior.

2. Tidal Forces:
Tidal movements are driven by the gravitational pull of the Moon and the Sun, which vary as the positions of these celestial bodies change relative to Earth. Tidal equations thus have time-dependent forcing terms, making them non-autonomous.

3. Biological Rhythms:
Circadian rhythms in living organisms, which regulate daily cycles such as sleep and feeding, are influenced by the 24-hour light-dark cycle. These external light variations necessitate non-autonomous models.

4. Astronomical Geophysical Cycles:
Systems like the Chandler wobble (the irregular movement of Earth’s rotation axis) or the Quasi-Biennial Oscillation (QBO) in the equatorial stratosphere are influenced by periodic external factors, such as lunar cycles, making them non-autonomous. This also includes systems where lunar or Draconic cycles interact with annual cycles in non-linear ways, as explored in studies of Earth’s rotational dynamics.

5. Oceanographic Dynamical Phenomena:
Non-autonomous formulations are needed to model phenomena such as El Niño, which is influenced by complex interactions between atmospheric and oceanic conditions, themselves driven by seasonal and longer-term climatic variations.

6. Planetary Motion in a Varying Gravitational Field:
In astrophysical systems where a planet moves in the gravitational field of other bodies, such as a multi-body problem where external forces vary in time, non-autonomous dynamics become essential to account for these influences.

In contrast, autonomous systems are self-contained and their behavior depends only on their internal state variables, independent of any external time-varying influence. So that non-autonomous systems often better capture the complexity and variability introduced by time-dependent external factors.

However, many still want to find connections to autonomous formulations as they often coincide with resonant conditions or some natural damping rate.

Autonomous mathematical formulations are characterized by the fact that their governing equations do not explicitly depend on time or other external variables (they can be implicit via time derivatives though). These systems evolve based solely on their internal state variables. Many natural behaviors can be modeled using autonomous systems when external influences are either negligible or can be ignored. Here are some examples of natural behaviors that qualify as autonomous:

1. Radioactive Decay:
The decay of radioactive isotopes is governed by an internal process where the rate of decay depends only on the amount of the substance present at a given moment. The decay equation does not depend on time explicitly, making it an autonomous system.

2. Epidemiological Models (without external intervention):
Simplified models of disease spread, such as the SIR (Susceptible-Infected-Recovered) model, can be autonomous if no external factors (like seasonal effects or interventions) are considered. The evolution of the system depends only on the current number of susceptible, infected, and recovered individuals.

3. Predator-Prey Dynamics (Lotka-Volterra Model):
In the absence of external influences like seasonal changes or human intervention, predator-prey relationships, such as those described by the Lotka-Volterra equations, can be modeled as autonomous systems. The population changes depend solely on the interaction between predators and prey.

4. Chemical Reactions (closed systems):
In a closed system with no external input or removal of substances, the kinetics of chemical reactions can be modeled as autonomous. The rate of reaction depends only on the concentrations of reactants and products at any given time.

5. Newtonian Mechanics of Isolated Systems:
For an isolated mechanical system (e.g., a simple pendulum or two-body orbital system), the equations of motion can be autonomous. The system evolves based solely on the internal energy and forces within the system, without any external time-dependent influences. This relates to general oscillatory systems or harmonic oscillators — the simple harmonic oscillator (such as a mass on a spring) can be modeled autonomously if no external time-varying forces are acting on the system. The system’s behavior depends only on its position and velocity at any point in time. In the classical gravitational two-body problem in celestial mechanics, where two bodies interact only through their mutual gravitational attraction, the motion can be described autonomously. The positions and velocities of the two bodies determine their future motion, independent of any external time-dependent factors.

6. Thermodynamics of Isolated Systems:
In an isolated thermodynamic system, where there is no exchange of energy or matter with the surroundings, the internal state (e.g., pressure, temperature, volume) evolves autonomously based on the system’s internal conditions.

These examples illustrate systems where internal dynamics govern the evolution of the system, and time or external influences do not explicitly appear in the equations. However, in many real-world cases, external factors often come into play, making non-autonomous formulations more appropriate for capturing the full complexity of natural behaviors. A pendulum that is periodically synchronized as for example a child pushed on a swing set,  may be either formulated as a forced response in an autonomous set of equations or a non-autonomous description if the swing pusher carefully guides the cycle.

This is where the distinctions between autonomous vs non-autonomous and forced vs natural responses should be elaborated.

Understanding the Structure of the General Solution

In the case of a forced linear second-order dynamical system, the general solution to the system is typically the sum of two components:

Homogeneous (natural) solution: This is the solution to the system when there is no external forcing (i.e., the forcing term is zero).

Particular solution: This is the solution driven by the external forcing.

The homogeneous solution depends only on the internal properties of the system (such as natural frequency, damping, etc.) and is the solution when F(t) = 0.

The particular solution is directly related to the forcing function F(t), which can be time-dependent in the case of a non-autonomous system.

So let’s  consider the autonomous vs non-autonomous context.

Autonomous System: In an autonomous system, even though the system is subject to forcing, the forcing term does not explicitly depend on time but rather on internal state variables (such as x or dx/dt). Here, the particular solution would also be state-dependent and would not explicitly involve time as an independent variable.

Non-Autonomous System: In a non-autonomous system, the forcing term explicitly depends on time, such as F(t) = A sin(w t). This external time-dependent forcing drives the particular solution. While the homogeneous solution remains autonomous (since it’s based on the system’s internal properties), the particular solution reflects the non-autonomous nature of the system.

The key insight is that of the non-autonomous particular solution. Even though a system’s response can have components from the homogeneous solution (which are autonomous in nature), the particular solution in a non-autonomous system will be time-dependent and follow the time-dependence of the external forcing.

So consider the transition from autonomous to non-autonomous: when you introduce a periodic forcing function F(t), the particular solution becomes non-autonomous, even though the overall system response still includes the autonomous homogeneous solution. This results in the system being classified as non-autonomous, as the particular solution carries the time-dependent behavior, despite the autonomous structure of the homogeneous solution.

Summary: A forced response in a linear second-order system can include both autonomous and non-autonomous components. Even though the homogeneous solution remains autonomous, the particular solution introduces non-autonomous characteristics when the forcing term depends explicitly on time. In non-autonomous systems, the forcing introduces time dependence in the particular solution, making the overall system non-autonomous, even though part of the response (the homogeneous solution) is autonomous.