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Fucking Tides, How Do They Work?

I’ve always wondered what causes the tides, and I’ve always found the traditional explanations unsatisfying.

This is the explanation most commonly provided: the point on earth closest to the moon feels the moon’s gravity more strongly, so the moon creates a bulge as it pulls the ocean towards itself at this point (and, uhh… on the opposite side of the planet too for some reason…). These bulges are high tide.

But this explanation raises more questions than it answers.

  • If that force is strong enough to pull the ocean up several meters, why can’t I feel it manifested any other way?

  • Where does all that water come from? It seems that there’s a several-meter-thick layer of water that’s switching between opposite sides of the planet twice a day. If anything, shouldn’t that create massive currents in the straits connecting the various oceans?

  • Polar regions would never experience a high tide (as the angle to the sun/moon is always low against the horizon), even though they in fact have some of the highest tides.

  • Why do the actual times of high tide not line up with when the moon is directly overhead? Is the delay somehow due to the continents plowing through the bulge as the earth rotates?

Quantifying the tidal force

What kind of forces are we actually talking about here?

The tidal force is computed as (which is the first-order approximation of Newton’s law of gravitation ), where:

r
radius of object subject to tidal forces
M
mass of object causing tidal forces
R
distance between center of two objects
G
universal gravitational constant

The moon (r=6,371 km, R=363,300 km at its closest, M=7.348×1022 kg) yields a maximum tidal force of 1.30×10−6 m/s2.

The sun (r=6,371 km, R=147,100,000 km at its closest, M=1.989×1030 kg) yields a maximum tidal force of 5.31×10−7 m/s2. (Observe that nearly a full third of the tidal forces on earth are solely due to the effect of the sun, not the moon)

So we can see that the tidal force is actually ridiculously small – less than one-millionth the normal force of gravity (9.81 m/s2).

For kicks, lets calculate the tidal force for a person standing on the surface of the earth (r=88 cm – half of average human height, R=6,371 km, M=5.972×1024 kg). The tidal force is 2.71×10−6 m/s2 – greater than the combined force of the sun and moon on the earth! Let me repeat – your own body feels more tidal force from the earth than the entire earth does from the sun and moon.

The strength of gravity weakens with altitude, and the full tidal force on the earth at its maximum strength is gravitationally equivalent to rising up a mere 59 cm. Even assuming the weak tidal force could raise the seas up to this height in the face of a rapidly rotating earth, observed tides are often much higher.

Clearly, something doesn’t add up with the literal bulge explanation.

What’s Really Going On

The ‘tidal bulge’ theory is wrong, but it is actually an oversimplification of the correct depiction of tidal forces.

Before going any further, lets resolve the first confusion of why there’s a second bulge (in tidal force, not actual tide itself) on the far side of the planet. Although intuitively the moon pulls stronger on the ocean nearest to it, it is also exerting a pull on the rigid body of the earth itself, and this pull is stronger than the pull of the moon on the ocean on the far side of the planet. The stronger pull on the close side follows our intuition – a slight pull away from the earth towards the moon. The weaker pull on the far side of the body means anything here is essentially being ‘left behind’ as the earth itself is being pulled more strongly towards the moon. This manifests – again – as a slight pull away from the earth, now in the opposite direction. Mystery solved.

To really understand what’s happening with the tides you see, you need to soak in this image:

Ignore the colors; the important part is the white lines.

Each white line marks an area that experiences high tide simultaneously. The current high tide moves from one white line to the next at a rate of one per hour. Points where the lines converge experience no tidal variation; instead, high and low tide lines meet here from opposite directions, and slowly rotate around the point*.

* This is again an oversimplification – the tides are composed of many different frequency components, of which this map visualizes only the most dominant one. Each frequency component would have its own map like above, with different lines and ‘no-tide’ points. A ‘no-tide’ point on the above map may still experience tidal variation from the other less significant tidal components.

Now we start to be able to understand the true nature of the tides. Every point on the earth is subject to the earth’s constant gravity. But every point is then also subject to very slight perturbations due to the tidal forces. These perturbations vary over time in a complex but cyclical fashion due to the orbits and rotation of the sun, moon, and earth.

Now imagine a basin of water. The constantly varying tidal force is like a slight nudging or tipping of this basin of water. Done in isolation, each nudge causes a miniscule ripple but nothing more. But as any kid discovers during bathtime, even very small movements can eventually create huge waves if done repeatedly and with the right timing such that the ripples build upon each other, i.e., resonance.

Applying these recurring tidal nudges to our basin (the world’s oceans) will over a long time build up a complex but predictable pattern of recurring waves and ripples. Depending on the natural resonant frequencies, some of these waves will be seemingly much larger than the forces we’re putting it. The shape of the basin – coastlines and ocean depths – further has a profound effect on the resultant wave pattern.

At last, we realize: tides are the sloshing of water around ocean basins due to the gradual but cyclical nudges of the tidal force!

Fallout of the Epiphany

This revised understanding of the tidal mechanism resolves all the pesky questions we had before.

  • We now understand the tidal force is exceptionally weak, but that the large tides we see are due to wave resonance. That the force is constantly changing is more important than its strength.

  • The current tide has only the most casual relationship to the current tidal pull of the sun and moon. Just as the ripples in our basin would continue on for some time even after we stopped moving it, if the sun and moon were to suddenly disappear, the tides would continue on (for days? years? hundreds of years? an interesting question…) until dissipated by friction and turbulence.

  • The above is why the high tide at a given place may occur many hours after when the moon is actually overhead. That place will be located at some point in the sloshing cycle basically at random.

  • Polar regions can still experience strong tides because they still feel the nudging of the tidal force. In fact, they predominantly feel the tidal force horizontally, which is intuitively ideal for creating the sloshing effect*.

    * That’s not to discount the effect of the up/down forces though. When pulling up in one part of the basin and pushing down in the other, this is effectively tipping it. What was once a flat surface is now sloped, and anything at the now higher part will feel compelled to move towards the lower part. These opposing forces occur quite far away from each other though, so the basin must be very large for any appreciable effect.

  • Where does the water come from? Well, as our metaphorical basin has large dividers in it (continents), the sloshing is mostly confined to each ocean basin. A high tide at some part of the coastline generally means a low tide somewhere else on that same body of water. There is no water moving from one side of the earth to the other.

Sadly, in the rather comprehensive Wikipedia article for Tides, this key insight to tidal behavior is reduced to one arcane sentence buried in the middle of the article:

The tide-generating force (or its corresponding potential) is still relevant to tidal theory, but as an intermediate quantity (forcing function) rather than as a final result; theory must also consider the Earth’s accumulated dynamic tidal response to the applied forces, which response is influenced by bathymetry, Earth’s rotation, and other factors.

Aren’t you glad you read this post instead?

Food for Thought

Return of the Bulge

So is the bulge explanation completely useless?

Although the bulge is deceptive when considering the tides we experience day-to-day, it may still be useful when considering the average tide for the earth as a whole. That is, considering all the various waves of ripples that make up the current global tide, it may still be the case that there is more water on average in the areas predicted by the bulge. (And how high might that average bulge be? Around 59 cm, perhaps?) The bulge, after all, is important to the phenomenon of tidal locking*, which is (very, very, very) gradually slowing the rotation of the earth to match the orbital period of the moon. It is for this reason the day is getting longer by ~2 milliseconds per century (necessitating leap seconds), and the moon is moving farther from the earth by ~4 cm per year. Yes, physics is weird.

* a careful reading of that article, however, will also show you that the impossibility of the idealized bulge is equally important to tidal locking. Did science just tell you to go fuck yourself? Quite probably.

On the other hand, the rigid body of the earth also experiences tidal deformation, and many bodies undergo tidal locking without the help of oceans, including… the moon, which is tidally locked to the earth. How much of the tidal locking of the earth is attributable to ocean tides? (I really don’t know but I’d guess it’s a lot)

Resonance by Coincidence?

The formation of sizable tides is due to the resonance of variations in the tidal force. But resonance requires that the cycles of variation line up with the natural resonant frequencies already inherent in the system. The tidal frequencies are predominantly 12- and 24-hour cycles, while the resonant frequency of the ocean (time it takes for a deep-sea wave to travel half-way around the world) is approximately 30 hours. The former period is simply astronomical happenstance, while the latter is a factor of earth’s gravity, size, and the depth and viscosity of the oceans. How much of a coincidence is it that these two factors happen to be on the same scale so as to produce tides? Is it common for oceanic planets to have tides as earth does?

The question becomes even more intriguing when you consider the theory that tides were a key part of the initial formation of life on earth. The theory goes that tidal pools provided a safe place for the precursor chemicals of life to concentrate and stew, while the tides themselves provided regular pumping and mixing. Does the earth have tides because if it didn’t, no one would be here to observe their absence?


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