These two characteristics of differential equations often go hand in hand. A tidal response is fundamentally a forced problem, not a free oscillation problem. In differential-equation language, the primacy of forcing makes it both non-homogeneous and non-autonomous. In other familiar calculus terms, tidal responses are particular solutions of non-homogeneous, non-autonomous dynamical systems, forced by explicitly time-dependent lunisolar potentials, superposed on the system’s autonomous homogeneous modes. The following post will make it clear that (i) tides are not free modes, (ii) the forcing is external, and (iii) the external forcing introduces explicit time dependence beyond the internal state evolution. We then ask the question: Can this be extended beyond conventional tides?

Consider a generic dynamical system:: $\frac{dx}{dt} = A(t) x + f(t)$

where:
A(t) represents the internal dynamics,
f(t) represents external forcing.

The homogeneous system satisfies that differential equation without the forcing term, a classic formulation that can describe the eigenvalues and resonant frequencies of a system

However, tides are never described this way because the gravitational potential of the Moon and Sun enters explicitly as a source term: $f(t) = F_{tide}(t)$

The tidal potential acts continuously. The observed response is therefore the sum of

• transient free modes
• particular solutions driven by the forcing.

The free modes usually decay through friction, turbulent mixing, radiation losses, etc., leaving the particular forced solution dominant. This is why major tidal constituents of known frequencies (diurnal, semi-diurnal, etc) are regarded as forced responses rather than natural eigenmodes.

So that tides are primarily a non-homogeneous system, or a forced response differential equation. Mathematically, the full response decomposes into a sum of homogeneous and particular pieces, of which we keep only the latter
$$\text{solution}=\text{free modes}+\text{forced tide},$$

Characteristically, an autonomous system has no explicit time dependence: $\frac{dx}{dt} = F(x)$. Tidal systems are generally non-autonomous because the forcing potential depends explicitly on time through the orbital motions: $\frac{dx}{dt} = F(x,t)$

So, it is both accurate and useful to describe tidal responses as non-homogeneous and non-autonomous, and the two labels emphasize different but complementary aspects of the same forced-dynamics picture. To re-iterate, in the standard ODE/PDE framing, a non-homogeneous equation is one with an explicit forcing term on the right-hand side, e.g.

$$\mathcal{L}[\mathbf{q}(t,\mathbf{x})]={\mathcal{F}}_{\text{tide}}(t,\mathbf{x}),$$

where $\mathcal{L}$ encodes the internal dynamics (gravity, rotation, elasticity, stratification, damping) and ${\mathcal{F}}_{\text{tide}}$​ is the external body force or boundary stress from the lunisolar potential. The tidal response we care about is the particular solution associated with ${\mathcal{F}}_{\text{tide}}$​, and the usual statement that “tides are a forced response” is exactly the assertion that we are dealing with a non-homogeneous problem, not a free eigenmode problem.

In other words, “non-homogeneous” is pointing at the presence of an explicit source term and the accompanying decomposition

$$\text{solution}=\text{homogeneous (free)}+\text{particular (tidal)}\mathrm{.}$$

The subtlety is that a system is non-autonomous when the governing equation depends explicitly on time, typically through a forcing term $F(t)$ such as $F(t)=A\sin (\omega t)$. Tidal forcing is literally of this form: the astronomical potential, and any derived body force or boundary pressure, has explicit quasi-periodic time dependence through the changing Earth–Moon–Sun geometry.

The lunar and solar orbital frequencies continually modulate the forcing. For example, the forcing contains frequencies associated with:

• annual orbital motion,
• tropical month,
• draconic month,
• anomalistic month,
• nodal cycle,
• apsidal cycle.

The forcing therefore lives on a multi-frequency torus rather than a single periodic cycle. It’s referred to as a torus because the periodic boundary conditions and incommensurate frequencies when plotted in phase space ends up looking like a donut when traced long enough. This can become a complicated bounding manifold, foreshadowing what is to come, hint: it doesn’t take much to get completely lost.

So, if we write a simple linear tidal oscillator as

$$\ddot{\eta }+2\gamma \dot{\eta }+{\omega }_0^2\eta =F_{\text{tide}}(t),$$the presence of $F_{\text{tide}}(t)$​ makes the equation both non-homogeneous (right-hand side non-zero) and non-autonomous (explicit $t$ in the coefficients/forcing). In general, geophysical models, the same logic holds: even if $\mathcal{L}$ is time-independent, the full equation $\mathcal{L}[\mathbf{q}]={\mathcal{F}}_{\text{tide}}(t,\mathbf{x})$ is non-autonomous because the forcing injects explicit time dependence.

Non-homogeneous and non-autonomous emphasize slightly different conceptual features:

• Non-homogeneous: “The system is being driven by an external source term.” This is about the structure of the equation and the decomposition into homogeneous vs particular solutions.
• Non-autonomous: “The external driving depends explicitly on time (and, for tides, in a quasi-periodic way).” This highlights that the system’s evolution cannot be described purely by its current state; the phase of the astronomical forcing matters as an independent variable.

For tidal problems, both are essentially unavoidable: if you remove the non-homogeneous forcing, there is no tide; if you remove the explicit time dependence and somehow “average” the forcing, you lose the defining phase-locked signal.

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Now consider the Laplace Tidal Equations viewpoint — first in conventional terms, then nonlinear.

In the linearized form of the shallow-water equations underlying the Laplace Tidal Equations, where $\Phi(t)$ is the tidal potential :

$\frac{\partial \eta}{\partial t} = L(\eta, u, v) + \Phi(t)$

The solution naturally decomposes into

$\eta(t) =\eta_{free}(t) +\eta_{forced}(t).$

After sufficient time,

$\eta(t) \approx \eta_{forced}(t).$

This is one reason that fitting observed tides by harmonic constituents works so well: the ocean is largely tracking the forcing spectrum.

BUT … at some point the tidal forcing can go nonlinear, which I describe in detail in Mathematical Geoenergy¹

The perturbation to linearity occurs via a sinusoidal modulation of the forcing manifold — a peculiar looking non-autonomous transfer function:

$\sin(k F(t) + \epsilon \sin(k F(t) + \theta))$

This formulation also features a self-referential factor that is scaled by an $\epsilon$ factor. Upon differentiation, this will not add new irreducible terms but will in fact allow the differential equation solution to balance after application of the chain rule.

Which leads to the question: Does this actually occur in tidal measurements? The measurements of monthly mean-sea-level at coastal tidal gauge stations (maintained at PSMSL.org) definitely show erratic behavior that doesn’t exhibit the clean beat-frequency envelope of the faster diurnal tides. No convincing explanation exists either, other than atmospheric disturbances playing a role. Now recall the multi-frequency torus mentioned earlier and consider how a nonlinear modulation as shown above and applied to that tidal force trajectory will completely distort any obvious symmetries and beating relationships.

Yet, that doesn’t mean that we can’t try and extract the underlying forcing — after all we do have a good idea of what the candidate tidal forcing is: the same forcing that paces the cycles of the Earth’s length-of-day (LOD) variations.² Shown below, the envelope of dLOD/dt is very clean and shows the beats of the competing lunar cycles:

Given the dLOD/dt as a forcing component, can’t stress enough that the key to matching an output is to maintain the amplitudes of the tidal factors but allow the phases to subtly change. This is critical because we only have monthly resolution to all these measurements and can’t guarantee that the various time series precisely align. But by modifying the phases slightly, one can essentially calibrate an alignment. This phase alignment can be extremely slight as shown below, where the red mode nearly overlays the original dLOD/dt used as a starting point.

As an example using the above adjustment, the following is a nonlinear tidal analysis to the monthly mean sea level (MSL) at the Swedish coastal site of Ystad (#72 at PSMSL)

Note the k modulation signature, after the $\epsilon$ modulation has been applied, shown above and in the inset to the right. This appears in every fitted model, whether a MSL time series or climate index such as NAO, AMO, or ENSO. What guides the emergence of this signature is the optimization of a metric such as correlation coefficient across a training interval. In the above fit, the interval 1945-1955 was excluded as a test/validation. To further sharpen the fit, an additional harmonic to k can be added. There are many of these MSL sites to choose from in the Baltic region and they all show a similar signature — once the parameters are locked in, difficult not to observe a sinusoidal non-autonomous modulation. In short, this is a non-homogeneous, non-autonomous forced system that selectively amplifies/modulates the combinations, aliases (via annual strobosocopic sampling), holds, folds, and sidebands of the underlying forcing frequencies.

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In terms of tidal gauge data, the scientific curiosity has always been in understanding the disparity between the easily modeled diurnal and semidiurnal tidal cycles and the inexplicable erratic excursions at monthly and longer intervals. There are clearly seasonal influences in the variations, yet there is no distinctive set of cycles apart from that to match any pattern. The approach I am advocating in the post above and elsewhere has been to assume the premise that the longer period tidal forces have to be in there somewhere, and this tidal forcing shouldn’t simply disappear at a longer time-scale, especially considering how strong it is on a daily basis.

From the perspective of the LTE model, the latent forcing manifolds are then not close to an autonomous oscillator at all. This is in contradiction to other models that claim a natural eigenvalue-based free resonance oscillation. In fact, the manifold is continuously driven by a quasiperiodic astronomical forcing torus, and the observed climate indices are the projected response of that forced manifold. The “periods” that appear in the observations are therefore properties of the forcing-response operator, not necessarily properties of the unforced system itself. This is why annual, semiannual, nodal, draconic, and beat frequencies can appear as organizing structures without invoking corresponding free resonances.

If instead most of these behaviors are a forced response, the relevant question becomes: What frequencies emerge from the interaction of the forcing torus with the system’s transfer function?

In that framework:

• QBO need not possess a natural 28-month eigenperiod.
• Chandler wobble need not possess a perfectly tuned 433-day resonance.
• ENSO need not possess a unique intrinsic 3–7 year oscillation.

This is much closer to signal-processing language: the Earth system behaves as a set of filters and nonlinear modulators acting on astronomical inputs.

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REFERENCES

1. Mathematical Geoenergy, Pukite et al (Wiley/AGU, 2019).
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2. Pukite, P. R., “Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models”, AGU Fall Meeting 2017.
   (As the lunar forcing consists of three fundamental periods (draconic, anomalistic, synodic), we used the measured Earth’s length-of-day (LOD) decomposed and resolved at a monthly time-scale to align the amplitude and phase precisely. Even slight variations from the known values of the long-period tides will degrade the fit, so a high-resolution calibration is possible.) ↩︎