Beyond the Tidal Bulge

[D H]

Contents
ToggleOverviewWhat is the tidal bulge?There is no tidal bulgeWhy isn’t there a bulge?What others sayBut there must be a tidal bulge!Tide generating forcesNewtonian...

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[pines-demon]

Is the Newtonian tidal bulge an approximation of Laplace dynamics theory of tides in some limit?

Edit: I mean if I just take the predominant mode or something like that does everything correspond with Newtonian theory?

Edit2: I felt that the article was missing a conclusion on key differences between the two models aside from Laplace's model being more realistic. Very interesting nonetheless!

 

[bob012345]

If the Earth were a perfect sphere covered with a uniform ocean, then would there be?

 

[pines-demon]

If the Earth were a perfect sphere covered with a uniform ocean, then would there be?

My take on this is that "no, at least not the bulge predicted by Newton", one of the key take aways is that the ocean is not deep enough. But as you, I still wonder if there would be any other difference aside from that. Would there be a bulge?

 

[Greg Bernhardt]

Thanks DH! Just in time to make the member awards!

 

[D H]

Is the Newtonian tidal bulge an approximation of Laplace dynamics theory of tides in some limit?

In the limit of a water planet tidally-locked to its principal moon, whose rotational angular velocity vector aligns with the moon's orbital angular velocity vector, and the orbital eccentricity is very small, then yes, there is no need for the dynamics theory of the tides (at least not for the moon. The only thing the dynamics theory carries over from the equilibrium model is the tidal forcing function, and then only the horizontal components. The dynamics theory is a component replacement for the equilibrium theory.

Edit: I mean if I just take the predominant mode or something like that does everything correspond with Newtonian theory?

The predominant mode of the dynamics theory is the one cycle per 12.421 hours M₂ tidal. The predominant mode of the equilibrium tide model is the tidal bulge moving at one cycle per 12.421 hours around the Earth. That however does not validate the Newtonian theory.

 

[D H]

If the Earth were a perfect sphere covered with a uniform ocean, then would there be?

We'll perhaps have a clearer picture of a tidally-locked terrestrial object with an ocean that covers the globe beginning in April 2030. That's when the Europa Clipper will arrive in orbit around Jupiter with a strong focus on developing a clearer picture of what's happening in Europa.

 

[pines-demon]

The predominant mode of the dynamics theory is the one cycle per 12.421 hours M₂ tidal. The predominant mode of the equilibrium tide model is the tidal bulge moving at one cycle per 12.421 hours around the Earth. That however does not validate the Newtonian theory.

Maybe does not validate Newtonian theory but makes it a good first approximation to learn (maybe with a warning "but it is more complicated due to continents"). I mean we still teach Newtonian gravitation and we do not go straight into general relativity for a reason. For many practical purposes Newtonian gravitation is enough and the same could be said about the Newtonian tide model.

Edit: the Newtonian tide model is not in the ensemble of obsolete theories that are clearly wrong, like for example the emission theory of vision.

 

[bob012345]

We'll perhaps have a clearer picture of a tidally-locked terrestrial object with an ocean that covers the globe beginning in April 2030. That's when the Europa Clipper will arrive in orbit around Jupiter with a strong focus on developing a clearer picture of what's happening in Europa.

I would be surprised if this hasn’t been simulated many times. BTW, what would we call that big wave in the movie Interstellar?

 

[pbuk]

Maybe does not validate Newtonian theory but makes it a good first approximation to learn (maybe with a warning "but it is more complicated due to continents").

In practice, this is exactly what is done. See for example NASA and the (US) National Oceanographic and Atmospheric Administration.

 

[pbuk]

BTW, what would we call that big wave in the movie Interstellar?

I don't know about you but I would call it "bollocks".

 

[Ken G]

It sounds to me like the Newtonian picture is really not of much value for understanding what gets called "ocean tides" (the way the ocean goes up and down a beach), quite frankly. Note that is quite a bit different from what we might call "tidal deformation", for which the Newtonian picture applies quite well for simple tidally locked systems. But a simple tidally locked system doesn't have "ocean tides" at all, so the only situation where the Newtonian picture is good is when you have nothing to understand in the first place (if you are a surfer or a boater).

 

[D H]

Maybe does not validate Newtonian theory but makes it a good first approximation to learn (maybe with a warning "but it is more complicated due to continents"). I mean we still teach Newtonian gravitation and we do not go straight into general relativity for a reason. For many practical purposes Newtonian gravitation is enough and the same could be said about the Newtonian tide model.

Not really. It is used as a "lie-to-children" as a first step in learning how the ocean tides work. The next step is to teach oceanography students that this model is indeed a "lie to children", that it doesn't describe the oceanic tides at all.

Edit: the Newtonian tide model is not in the ensemble of obsolete theories that are clearly wrong, like for example the emission theory of vision.

While the equilibrium tide model is not universally falsified, it is falsified as an explanation for oceanic tides on the Earth. The equilibrium tide model is a fairly good model for the solid body tides in the Earth and in the Moon. It is also presumably a fairly good model for tidally-locked moons.

I would be surprised if this hasn’t been simulated many times.

Not just many times, but many times over, at least from the perspectives of Europa Clipper failure detection & handling thereof, robustness against radiation, thermal control, GNC *guidance, navigation & control), and many other aspects. NASA has a very strong tendency to simulate things to death. Regarding what the Europa Clipper will observe -- that's a different question. The instruments aboard the vehicle were designed to detect phenomena based on predictions, but also detect where those predictions might be wrong. I didn't do scientific instrument design in my long aerospace career, but I suspect they too heavily rely on simulations.

BTW, what would we call that big wave in the movie Interstellar?

Something made up to make a movie more dramatic? Movies, sci-fi movies in particular, make stuff up all the time.

In practice, this is exactly what is done. See for example NASA and the (US) National Oceanographic and Atmospheric Administration.

That is the "lie-to-children" (and in the case of the NASA site, also a "lie-to-adults") I mentioned earlier (and in the article) regarding the modeling of the oceanic tides. Other than frequency (1 cycle per 12.421 hours), the oceanic tides do not follow the predictions of this lie-to-children. The frequency of 1 cycle per 12.421 hours is exactly the same as that of the M₂ tidal constituent. The M₂ tidal constituent from the dynamic theory of the tides does a very good job of predicting the tides while the equilibrium tide model does not.

It sounds to me like the Newtonian picture is really not of much value for understanding what gets called "ocean tides" (the way the ocean goes up and down a beach), quite frankly. Note that is quite a bit different from what we might call "tidal deformation", for which the Newtonian picture applies quite well for simple tidally locked systems.

I wholeheartedly agree. If one goes to the National Oceanic and Atmospheric Administration's tide prediction page and chooses a tide station locations (make sure to pick stations that are listed as "harmonic" rather than "subordinate") and looks at the harmonic constituents for those locations, one will see that the phase angle for the M₂ tidal constituent takes on all values between 0° and 360°. This phase angle would be close to the same value for all sites if the equilibrium tide model was anywhere close to correct.

But a simple tidally locked system doesn't have "ocean tides" at all, so the only situation where the Newtonian picture is good is when you have nothing to understand in the first place (if you are a surfer or a boater).

While not quite right, what you wrote is very close to right. None of the tidally locked moons of Jupiter have a perfectly circular orbit. Their orbits are instead very slightly elliptical. That slight ellipticity is enough to create time-varying tides. Those moons are more or less tidally locked, where "more or less" in this case means "on average". The tides on those moons can be viewed as small time-varying perturbations on top of a very large permanent tide. Those small variations are large enough to make Io the most volcanically active object in the solar system and to make Europa's supposedly thin ice layer on top of a global ocean geologically young. I wrote "supposedly" because that is the best scientific guess without actually going there. Europa Clipper is going there.

 

[pines-demon]

Not really. It is used as a "lie-to-children" as a first step in learning how the ocean tides work. The next step is to teach oceanography students that this model is indeed a "lie to children", that it doesn't describe the oceanic tides at all.

While the equilibrium tide model is not universally falsified, it is falsified as an explanation for oceanic tides on the Earth. The equilibrium tide model is a fairly good model for the solid body tides in the Earth and in the Moon. It is also presumably a fairly good model for tidally-locked moons.

I am trying to figure out how to classify the Newtonian water tidal theory. In science, there seem to be theories that are obsolete and we throw them to the trash as soon as we find a better one, and less accurate ones that we keep as first approximation. If I take the dynamic tidal model and reduce some couplings or average out some features do I get the Newtonian one? It seems so. In this sense, the Newtonian tide model is to the dynamic theory of tides what the ray optics is to wave optics or to quantum optics. You get one from the other in the eikonal limit.

The question asked early seems important here, if I have a perfect sphere covered with water (circular orbit, tidally-locked if needed) what is the prediction? If the prediction is different from the Newtonian model what is the key difference?

 

[pines-demon]

I would be surprised if this hasn’t been simulated many times. BTW, what would we call that big wave in the movie Interstellar?

This is Kip Thorne's explanation from the book on the science of Interstellar:

What could possibly produce the two gigantic water waves, 1.2 kilometers high, that bear down on the Ranger as it rests on Miller's planet (Figure 17.5)? I searched for a while, did various calculations with the laws of physics, and found two possible answers for my science interpretation of the movie. Both answers require that the planet be not quite locked to Gargantua. Instead it must rock back and forth relative to Gargantua by a small amount [snip Thorne's explanation of how Gargantua's tidal gravity will naturally provide a sort of restoring force back to its preferred orientation, explaining why the planet would rock this way] ... The result is a simple rocking of the planet, back and forth, if the tilts are small enough that the planet's mantle isn't pulverized. When I computed the period of this rocking, how long it takes to swing from left to right and back again, I got a joyous answer. About an hour. The same as the observed time between giant waves, a time chosen by Chris without knowing my science interpretation.

The first explanation for the giant waves, in my science interpretation, is a sloshing of the planet's oceans as the planet rocks under the influence of Gargantua's tidal gravity.

A similar sloshing, called "tidal bores," happens on Earth, on nearly flat rivers that empty into the sea. When the ocean tide rises, a wall of water can go rushing up the river; usually a tiny wall, but occasionally respectably big. ... But the moon's tidal gravity that drives this tidal bore is tiny—really tiny—compared to Gargantua's huge tidal gravity!

My second explanation is tsunamis. As Miller's planet rocks, Gargantua's tidal forces may not pulverize its crust, but they do deform the crust first this way and then that, once an hour, and those deformations could easily produce gigantic earthquakes (or "millerquakes," I suppose we should call them). And those millerquakes could generate tsunamis on the planet's oceans, far larger than any tsunami ever seen on Earth

 

[Ken G]

Not really. It is used as a "lie-to-children" as a first step in learning how the ocean tides work. The next step is to teach oceanography students that this model is indeed a "lie to children", that it doesn't describe the oceanic tides at all.

While the equilibrium tide model is not universally falsified, it is falsified as an explanation for oceanic tides on the Earth. The equilibrium tide model is a fairly good model for the solid body tides in the Earth and in the Moon. It is also presumably a fairly good model for tidally-locked moons.


Not just many times, but many times over, at least from the perspectives of Europa Clipper failure detection & handling thereof, robustness against radiation, thermal control, GNC *guidance, navigation & control), and many other aspects. NASA has a very strong tendency to simulate things to death. Regarding what the Europa Clipper will observe -- that's a different question. The instruments aboard the vehicle were designed to detect phenomena based on predictions, but also detect where those predictions might be wrong. I didn't do scientific instrument design in my long aerospace career, but I suspect they too heavily rely on simulations.

Something made up to make a movie more dramatic? Movies, sci-fi movies in particular, make stuff up all the time.


That is the "lie-to-children" (and in the case of the NASA site, also a "lie-to-adults") I mentioned earlier (and in the article) regarding the modeling of the oceanic tides. Other than frequency (1 cycle per 12.421 hours), the oceanic tides do not follow the predictions of this lie-to-children. The frequency of 1 cycle per 12.421 hours is exactly the same as that of the M₂ tidal constituent. The M₂ tidal constituent from the dynamic theory of the tides does a very good job of predicting the tides while the equilibrium tide model does not.


I wholeheartedly agree. If one goes to the National Oceanic and Atmospheric Administration's tide prediction page and chooses a tide station locations (make sure to pick stations that are listed as "harmonic" rather than "subordinate") and looks at the harmonic constituents for those locations, one will see that the phase angle for the M₂ tidal constituent takes on all values between 0° and 360°. This phase angle would be close to the same value for all sites if the equilibrium tide model was anywhere close to correct.

It certainly looks like this is not a "lie to children," it's just a "lie." I'm not a big fan of lying to students, it smacks of "truthiness," and science is supposed to be pretty close to the opposite of that.

While not quite right, what you wrote is very close to right. None of the tidally locked moons of Jupiter have a perfectly circular orbit.

That's what I meant by "simple" tidal locking. But then, if it's not simple, then the Newtonian picture suffers its own problems. So again, it really looks like we should just bite the bullet and either take on the Laplace approach, or steer clear of ocean tides altogether. One can do the Newtonian approach to tidal deformation if you want to understand Hill spheres and Lagrange points, and if you want to understand the two-a-day frequency of ocean tides, you can use Newton to get a concept of "tidal forcing", as you say, but stop short of talking about the response to tidal forcing as being like the ocean getting "rotated through the bulges" twice each day (that's the part that's clearly as wrong as wrong gets, given that phase information you mention).

But this raises two deeper questions. It is often said that the Moon is getting farther from the Earth because the closest "point of the football" is rotated (by the lag in moving the oceans around) so it leads the Moon a little. In the Laplace approach, there must be a similar kind of effect, so should we use the Newtonian tidal deformation argument to understand why the Moon is gaining orbital angular momentum, or is that just as wrong as the phase information you mention?

Another question I have wondered about is, I recall hearing that Io is volcanic because it is "kneaded like bread" by the changing tidal deformation of its (very slightly) elliptical orbit. But there are deeper issues here, because Io's orbit is less than 2 days, yet the orbit itself evolves on timescales longer than a year. So immediately we must wonder if the "kneading" is the 2-day effect of its eccentricity, or much longer timescale changes in the eccentricity. The reason the latter is relevant is that if we are dealing with longer variations than one orbit, how do we know the real heating that matter happens during much larger changes in its eccentricity over time, such that the current eccentricity plays almost no role at all in keeping Io volcanically active? (This paper: https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2019GL082691 suggests that volcanic eruptions do vary on the ~480 day period of orbital changes, but it leaves open the question of whether there are much longer period changes that are more important for keeping the moon's interior molten in the first place.)

 

[D H]

I am trying to figure out how to classify the Newtonian water tidal theory.

Aside: It's called the equilibrium tide model rather than Newtonian water tidal theory. There is at least one aspect that Newton did get right and is still used, which is the tidal forcing function. That said, it is best to discard the vertical component of that forcing function, as Laplace did. The response of the Earth's oceans to that forcing, the equilibrium tide: Newton got that completely wrong. So, falsified. It is nowhere close to being universally true, even in a Newtonian sense. Moreover, it has been falsified on the one planet in the entire universe that matters most to humankind, which is of course the Earth.

If you're looking for an analogy, I would classify it somewhere between the phlogiston and caloric theories of heat, both of which have been tossed onto the big pile of discarded scientific theories. Phlogiston theory is completely off. Caloric theory has some truth to it. For example, Carnot's heat engine concepts are deeply rooted in caloric theory. Modern thermodynamics has very little in common with caloric theory. Just because a model is wrong does not mean some aspects of that model are not useful.

 

[pines-demon]

Aside: It's called the equilibrium tide model rather than Newtonian water tidal theory. There is at least one aspect that Newton did get right and is still used, which is the tidal forcing function. That said, it is best to discard the vertical component of that forcing function, as Laplace did. The response of the Earth's oceans to that forcing, the equilibrium tide: Newton got that completely wrong. So, falsified. It is nowhere close to being universally true, even in a Newtonian sense. Moreover, it has been falsified on the one planet in the entire universe that matters most to humankind, which is of course the Earth.

If you're looking for an analogy, I would classify it somewhere between the phlogiston and caloric theories of heat, both of which have been tossed onto the big pile of discarded scientific theories. Phlogiston theory is completely off. Caloric theory has some truth to it. For example, Carnot's heat engine concepts are deeply rooted in caloric theory. Modern thermodynamics has very little in common with caloric theory. Just because a model is wrong does not mean some aspects of that model are not useful.

It certainly looks like this is not a "lie to children," it's just a "lie." I'm not a big fan of lying to students, it smacks of "truthiness," and science is supposed to be pretty close to the opposite of that.

Ok let's say that the whole equilibrium tide model is FALSE. What does the dynamic theory of tides predicts for perfect sphere covered with water, circular orbit, tidally-locked, and how does that compare to the equilibrium theory?

 

[Ken G]

Ok let's say that the whole equilibrium tide model is FALSE. What does the dynamic theory of tides predicts for perfect sphere covered with water, circular orbit, tidally-locked, and how does that compare to the equilibrium theory?

If we have only circular orbits and tidally locked bodies, then there are no ocean tides at all to understand, and also no dynamical tide model because there are no dynamics. That was my point, the Newtonian model works perfectly in that situation, but it explains nothing but the shape of the object. The situation where Newton's equilibrium tide model is correct, is also the situation where it has no explanatory value toward understanding ocean tides, because there are none to explain, and the dynamical model also has no usefulness there. It's just an irrelevant test, the idealization has removed the purpose of the explanation.

 

[D H]

It certainly looks like this is not a "lie to children," it's just a "lie." I'm not a big fan of lying to students, it smacks of "truthiness," and science is supposed to be pretty close to the opposite of that.

The equilibrium tide theory (I'm using theory here in the mathematical sense: a body of knowledge, as in string theory or knot theory) has some validity for some tides, but not in the Earth's oceans, where it has no validity. Educators consider it to be too big of a jump to go from the tidal forcing functions directly to the dynamic theory of the tides. Many students do not comprehend how there can be two bulges, one centered on the zenith point and the other on the antipodal (nadir) point. Those instructors tend to follow Newton's derivation, calling the outer bulge due to a centrifugal force that is constant all across the world. (That is not a centrifugal force. Also, it is not needed. I did not invoke that concept in the article.)

There are bodies where the equilibrium tide model is approximately correct. The solid body tides are one example. Tidally-locked bodies with a liquid covering are another. These are bodies where the perturbations on top of the zero frequency tides are small. Most frequency analyses will inevitably yield a zero frequency response. The zero frequency response in the Earth's oceanic tides is ridiculously small. It is considerably larger in the Earth's solid body tides, and is presumably ridiculously large in Europa's world spanning ocean that lies under the ice.

However, as used for Earth's oceans, the equilibrium tide theory simply does not apply.

But this raises two deeper questions. It is often said that the Moon is getting farther from the Earth because the closest "point of the football" is rotated (by the lag in moving the oceans around) so it leads the Moon a little. In the Laplace approach, there must be a similar kind of effect, so should we use the Newtonian tidal deformation argument to understand why the Moon is gaining orbital angular momentum, or is that just as wrong as the phase information you mention?

No. We should use the observed tidal energy dissipation that represents a transfer of angular momentum between the solid Earth and the oceans. These are quite observable and stand in as a proxy for the transfer of angular momentum between the oceans and the Moon. Or one can calculate the gravitational effects of all of the various amphidromic systems on the Moon. I don't know whether this has been done.

One thing that has been done is to look at tidal rhythmites, ancient tidal patterns now locked in rock. These tidal rhythmites provide yet another death by a million cuts for the equilibrium tide theory. The Moon's current recession rate extrapolated over the last billion years or so would result in a Moon much younger than the solar system. Tidal rhythmites show that this is not the case' the Moon's current recession rate is considerably higher than it has been on average over the last two billion years. The North Atlantic exhibits considerable resonances with regard to the tidal driving forces. These resonances are nicely explained by the dynamic theory of the tides -- but not by the equilibrium tides.

Another question I have wondered about is, I recall hearing that Io is volcanic because it is "kneaded like bread" by the changing tidal deformation of its (very slightly) elliptical orbit.

Io allegedly undergoes an interesting hysteresis loop; I'll try to find the seminal paper that describes this. The tidal forces and responses to it tend to make Io's orbit less eccentric. The resonant forces due to Io's orbital resonances with Ganymede and Europa tend to make Io's orbit more eccentric. Which predominates varies over time. the responses to the tidal forcing from Jupiter, or the response to the resonant forces? The answer is, it depends. When its eccentricity is very low, Io's inner core becomes more solid, thereby reducing the tidal energy losses, which are already low due to the low eccentricity. This enables Ganymede and Europa to dominate, making Io's orbit become more elliptic. At this point, the tidal forces start "kneading Io like bread", making the interior more liquid. Now the tidal forces dominate, making Io's orbit less eccentric and eventually making Io less active. Rinse and repeat. It's a very interesting hysteresis.

But there are deeper issues here, because Io's orbit is less than 2 days, yet the orbit itself evolves on timescales longer than a year. So immediately we must wonder if the "kneading" is the 2-day effect of its eccentricity, or much longer timescale changes in the eccentricity. The reason the latter is relevant is that if we are dealing with longer variations than one orbit, how do we know the real heating that matter happens during much larger changes in its eccentricity over time, such that the current eccentricity plays almost no role at all in keeping Io volcanically active? (This paper: https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2019GL082691 suggests that volcanic eruptions do vary on the ~480 day period of orbital changes, but it leaves open the question of whether there are much longer period changes that are more important for keeping the moon's interior molten in the first place.)