Why the Tidal Bulge Doesn’t Exist – Tidal Theory Explained

There is no tidal bulge raised by the Moon on the Earth’s oceans, despite its widespread use to explain why the Moon is receding from the Earth. The real explanation relies on the dynamic theory of the tides, developed by Pierre-Simon Laplace and later extended by William Thomson (Lord Kelvin), George Darwin, and A.T. Doodson, which accounts for ocean depth, continental barriers, and the Coriolis effect to produce oceanic tides without any equilibrium bulge.

Key Takeaways

• Newton’s equilibrium tide theory predicts two tidal bulges, but high tides in most of the North Sea do not occur at the same time everywhere, which falsifies that theory.
• A tidal bulge wave would need to travel at roughly the Earth’s equatorial rotational speed of 465 m/s, but a shallow-water wave moves at approximately $\sqrt{gd}$, giving only about 205 m/s at the mean ocean depth of 4267 m and about 330 m/s over the deepest oceanic trench.
• The North Atlantic accounts for 40% of the M₂ tidal dissipation, and the North Sea contains two full M₂ amphidromic points plus a partial one near the tip of Norway.
• A.T. Doodson, starting in 1919, applied a newer lunar orbital model and identified 388 harmonic tidal components, leading to the Doodson numbering scheme.
• Affholder and Valiron’s Descriptive Physical Oceanography (2001) states that an equilibrium tidal wave moving at 1600 km/h would require an ocean depth of 22 km, far more than the actual average depth of 3.9 km, so the equilibrium position required by Newton’s theory cannot be established.

What is the tidal bulge, and why is it invoked for the Moon’s recession?

Searches for explanations of why the Moon is gradually spiraling outward typically turn up a vague reference to tides raised by the Moon, or a more detailed picture built around the Earth’s tidal bulge.

Source: Wikipedia page on the tides

In this traditional picture, the Moon’s gravitation raises a pair of tidal bulges in the Earth’s oceans, one near the sublunar point and one on the opposite side of the Earth. The Earth’s rotation carries these bulges slightly ahead of and behind the Moon’s position. Both bulges exert gravitational pulls on the Moon: the closer, leading bulge accelerates the Moon, while the farther, trailing bulge decelerates it. Because the leading bulge is nearer, its pull dominates, and the net effect is to accelerate the Moon into a higher orbit.

There is only one problem with this simple picture: there is no tidal bulge.

Why doesn’t the tidal bulge exist?

Newton’s equilibrium theory is falsified by observation

Isaac Newton correctly derived the tidal forcing function but was wrong about how the oceans respond to it. His equilibrium theory, with its two tidal bulges, predicts that high tide should occur when the Moon is at zenith and again at nadir, spaced 12.421 hours apart. Most ocean locations do experience a high tide roughly every 12.421 hours, but whether that high tide coincides with the Moon’s zenith or nadir is essentially a matter of local circumstance, with a predictable, nonzero offset at most sites.

The North Sea, in Newton’s own backyard, is one of the clearest counterexamples. If equilibrium theory were correct, high and low tide would occur at roughly the same time across the entire North Sea. Instead, at any given moment, one location in the North Sea is experiencing high tide while another simultaneously experiences low tide. Tidal bulges are also not observed at coastal tide stations, from meteorological buoys, or from satellites.

Three physical reasons a bulge cannot exist

A tidal bulge spanning half the Earth’s circumference would form a wave far longer than the ocean is deep, making it a shallow-water wave. The speed of a shallow wave at a given location is approximately $\sqrt{gd}$, where $d$ is the local ocean depth. At the equator, this limits the tidal wave to about 330 m/s over the deepest oceanic trench and about 205 m/s at the mean ocean depth of 4267 m, both slower than the Earth’s 465 m/s equatorial rotational speed. The bulge wave simply cannot keep pace with the Earth’s rotation.

Second, the Earth is not fully covered by ocean. The Americas and Afro-Eurasia form continuous north-south barriers that block any continuous bulge wave. As one example, the tides on Panama’s Pacific coast differ greatly in timing and form from the tides just 100 kilometers away on Panama’s Caribbean coast.

Third, because the Earth rotates at a different rate than the Moon orbits, the Coriolis effect would shear any bulge wave apart even on a hypothetical ocean-covered Earth of uniform, very great depth.

What do other sources say?

Lecture notes archived from the Virginia Institute of Marine Science ask whether water confined to a basin can engage in wave motion like the sweeping tidal bulges of equilibrium theory, and answer plainly: “The answer is – it can’t.”

Affholder and Valiron’s Descriptive Physical Oceanography (2001) introduces Newton’s equilibrium tide and then states that for the tidal wave to move at 1600 km/h, the ocean would need to be 22 km deep; with the actual average depth of 3.9 km, the tidal elevation can only move at 700 km/h, so “the equilibrium position at any instant required by this theory cannot be established.”

Doesn’t the bulge model successfully predict the Moon’s acceleration?

The tidal bulge model can be fit to reproduce the Moon’s observed acceleration, but this fit does not validate the model. The procedure starts from the invalid assumption that the bulge exists, then calculates a lag angle that best matches lunar laser retroreflector observations. Because the bulge’s 12.421-hour period matches the dominant 12.421-hour M₂ tidal period that actually governs most ocean tides, the fit looks good even though the underlying assumption is wrong. An invalid assumption can still yield a good numerical fit; that does not make the assumption true.

What forces actually generate the tides?

Newtonian gravity on a test object

This section covers what Newton did get right: the underlying tide-generating forces. Newton originally presented this using synthetic geometry in the Principia, in part to avoid foregrounding his newly invented calculus. The derivation below instead uses modern vector notation and the gravitational constant $G$, both of which postdate the Principia by roughly two centuries.

For a gravitating body $B$ with mass $M_B$ acting as a point mass (or a body with a radial mass distribution) on a small test object $o$ with mass $m$, with $\vec \rho$ the vector from the test object to body $B$, Newton’s universal law of gravitation gives the gravitational force on the test object as:

$$\vec F_{o,B} = \frac{G M_B \, m}{||\vec \rho||^3}\vec \rho\tag{1}$$

Dividing both sides of equation (1) by the test object’s mass $m$ yields the acceleration of the test object toward body $B$:

$$\vec a_{o,B} = \frac{G M_B}{||\vec \rho||^3}\vec \rho\tag{2}$$

If $\vec p$ is measured relative to the center of mass of a primary body $A$, and $\vec r$ is the vector between the centers of mass of bodies $A$ and $B$, Newton’s law also gives the gravitational acceleration of body $A$ toward body $B$:

$$\vec a_{A,B} = \frac{G M_B}{||\vec r||^3}\vec r\tag{3}$$

The test object’s position may be expressed as a vector $\vec p$ relative to the origin of body $A$. A satellite orbiting the Earth, for instance, is naturally described in an Earth-centered inertial (ECI) frame, and is affected by gravitational acceleration toward the Earth, the Moon, the Sun, and in principle everything else. In this article, the test object is instead a small parcel of water at the ocean’s surface.

Expressing gravity in an accelerating, body-centered frame

An Earth-centered frame is not inertial; it is accelerating. To express the gravitational acceleration from equation (2) in a frame centered on body $A$, one subtracts the gravitational acceleration of body $A$ toward body $B$ (equation 3) from the acceleration of the test object toward body $B$ (equation 2):

$$\vec a_{o,B;A} = \frac{G M_B}{||\vec \rho||^3}\vec \rho \,-\, \frac{G M_B}{||\vec r||^3}\vec r\tag{4}$$

Since $\vec \rho = \vec r – \vec p$, substituting into equation (4) gives:

$$\vec a_{o,B;A} = \frac{G M_B}{||\vec r – \vec p||^3}(\vec r – \vec p) \,-\, \frac{G M_B}{||\vec r||^3}\vec r\tag{5}$$

Writing $\vec p = p(\cos\theta \,\hat r + \sin\theta \,\hat t\,)$ where $p = ||\vec p||$, $\theta$ is the angle between $\vec r$ and $\vec p$, $\hat r$ is the unit vector from body $A$ to body $B$, and $\hat t$ is the unit vector orthogonal to $\hat r$ in the $A-o-B$ plane, equation (5) becomes:

$$\vec a_{o,B;A} = \frac{G M_B}{||(r-p\cos \theta)\hat r – p\sin\theta\,\hat t||^3}((r-p\cos \theta)\hat r – p\sin\theta\,\hat t)) \,-\, \frac{G M_B}{||\vec r||^3}\vec r\tag{6}$$

For $p \ll r$, at values of $\theta$ that are integer multiples of 90°, this simplifies to first order to a form proportional to $GMp/r^3$. This is the basis for describing tidal forces as inverse-cube forces, in contrast to the inverse-square law of Newtonian gravity.

A graphical depiction of tidal acceleration

Source: Tidal force asymmetry @ mercer.edu

A note on computation

Equation (5), computed naively, involves significant cancellations and can lose precision in a computer’s native floating-point representation. This can be addressed; Richard Battin’s An Introduction to the Mathematics and Methods of Astrodynamics (1999) covers such methods, though applied to satellites orbiting a primary body rather than ocean parcels. The mathematics is the same. In aerospace engineering, these tidal gravitational accelerations are called third-body effects, distinguishing them from tidal gravitation, the term used for how the Moon and Sun distort the Earth’s shape, which in turn affects satellites orbiting the Earth.

What is the dynamic theory of the tides?

The tidal forces shown above appear to stretch the Earth along the Earth-Moon line and compress it orthogonally, which is exactly how Newton’s bulge picture arises so naturally. But as established above, this bulge cannot and does not exist. What Newton got wrong, Pierre-Simon Laplace later corrected with the dynamic theory of the tides, which accounts for water’s fluid flow, varying ocean depths and shapes, seafloor friction, and resonance effects. This theory explains why it is always high tide somewhere in the North Sea, in Patagonia, and around New Zealand’s coast.

Horizontal forcing and amphidromic systems

Because water flows, it is the horizontal component of the tidal forcing function, not the vertical component, that drives ocean tides. Here, “vertical” means in the direction of the local inward gravitational gradient rather than strictly toward the Earth’s center. Rivers such as the Mississippi and the Mekong (downstream of the Manwan Dam) flow “uphill” in the sense of increasing distance from the Earth’s center, while still flowing “downhill” toward lower gravitational potential energy, since a plumb bob’s “down,” owing to the Earth’s equatorial bulge, does not generally point exactly toward the Earth’s center.

The horizontal components of the tidal forcing, combined with the Earth’s rotation, basin depth, continental outlines, and resonance, produce amphidromic systems: regions where tidal responses rotate about central points, called amphidromic points, that experience no tide for a given frequency component.

Who developed the dynamic theory?

Frequency-domain analysis of the tides, developed decades after Laplace, extended the dynamic theory of the tides through several key contributors:

Contributor	Contribution
Pierre-Simon Laplace	Originated the dynamic theory of the tides, accounting for fluid flow, ocean depth and shape, and friction
William Thomson (Lord Kelvin)	Applied Fourier analysis to historical tidal data
George Darwin (son of Charles Darwin)	Extended Thomson’s work using then-available lunar orbit models; named harmonic components such as M2, the principal lunar semidiurnal tide
A.T. Doodson	From 1919, applied a newer lunar orbit model, identified 388 harmonic components, and devised the Doodson numbering scheme

What are the main tidal harmonic components?

Component	Period / frequency	Cause
M2	1 cycle per 12.421 hours	Dominant lunar tidal force at most locations
S2	1 cycle per 12 hours	Solar tidal force, about half as strong as the Moon’s
M1	1 cycle per 24.841 hours	Asymmetry from the Earth’s rotational axis not being orthogonal to the Earth-Moon line
S1	1 cycle per 24 hours	Corresponding solar asymmetry
Fortnightly/monthly	Two cycles and one cycle per anomalistic month	Elliptical lunar orbit (varying lunar influence between apogee and perigee)
Semiannual/annual	Varies with Earth’s orbital position	Elliptical Earth orbit (varying solar influence between aphelion and perihelion)
Spring/neap	Two cycles per synodic month	Combined lunar and solar alignment

Each of these frequency components has its own distinct set of amphidromic systems.

How do amphidromic systems explain regional tides?

What is an amphidromic system?

Each tidal harmonic component has its own set of amphidromic systems, each centered on an amphidromic point about which tides rotate. Focusing on the M₂ component, the North Sea region contains three M₂ amphidromic systems, which explains why its tides behave so unusually.

Source: Wikipedia article on tidal constituents

The M₂ component is the roughly twice-daily response to the Moon’s tidal forcing and is the dominant tidal component in most of the world. At M₂ amphidromic points, the M₂ tidal response is zero, yet these points remain essential to modeling the overall tidal response. White “cotidal lines” radiating from each amphidromic point mark where high and low M₂ tides occur simultaneously, and tidal amplitude increases with distance from the amphidromic point.

Source: archived from volkov.oce.orst.edu/tides/global.html.

Why are North Sea tides so unusual?

The North Atlantic accounts for 40% of the M₂ tidal dissipation, and the North Sea is a hub of that dissipation. Two full M₂ amphidromic points lie within the North Sea, with a partial M₂ amphidromic point near the tip of Norway, which together account for the region’s chaotic tidal behavior.

Archived source: Archived from geog.ucsb.edu/~dylan/ocean.html

How does M₂ tidal energy flow around the globe?

Tidal energy transfer, from sources to sinks, explains the unusual Patagonian tides, which largely result from energy transferred from the Pacific to the Atlantic. The same transfer accounts for the large flow of energy into the North Atlantic, where 40% of M₂ tidal dissipation occurs.

Source: Archived from altimetry.info/thematic-use-cases/ocean-applications/tides/

How does this tidal energy transfer make the Moon recede?

This energy transfer is closely tied to the exchange of angular momentum between the Earth’s mantle/crust and its oceans, one measurable result being the gradual slowing of the Earth’s rotation, so that a day was shorter hundreds of millions of years ago than it is now. The same process also transfers angular momentum between the oceans and the Moon’s orbit, causing the Moon to recede gradually from the Earth. Every amphidromic system within every tidal harmonic component contributes to this transfer, with the M₂ component making the largest single contribution.

This overall energy transfer can be described loosely as a “net tidal bulge,” though that framing is optional and somewhat misleading given everything established above.

How do ocean tides compare to tsunamis and solid-body tides?

Tsunamis versus tides

A tsunami is the response to a nearly impulsive driving event such as an earthquake, while tides are the response to a continuous, cyclical driving force. Even so, a tsunami’s impulse response is informative about the ocean’s response to cyclical tidal forcing.

Property	Tsunamis	Ocean tides
Wave type	Shallow wave, “feels” the ocean bottom regardless of depth	Shallow wave, same behavior
Role of topography	Strongly affects wave speed	Shapes the amphidromic systems
Coriolis effect	Small, becomes apparent only in the long-term oceanic response	Contributes to the shape and extent of amphidromic systems

An animation of the 2011 Tohoku, Japan earthquake tsunami is available on YouTube.

References: Dao, M. H., & Tkalich, P. (2007). Tsunami propagation modeling: a sensitivity study. Natural Hazards and Earth System Science, 7(6), 741-754. Eze, C. L., Uko, D. E., Gobo, A. E., Sigalo, F. B., & Israel-Cookey, C. (2009). Mathematical Modelling of Tsunami Propagation. Journal of Applied Sciences and Environmental Management, 13(3). Kowalik, Z., Knight, W., Logan, T., & Whitmore, P. (2005). Numerical modeling of the global tsunami: Indonesian tsunami of 26 December 2004. Science of Tsunami Hazards, 23(1), 40-56.

Solid-body (Earth) tides

The Moon and Sun also raise tides in the solid Earth, called Earth tides or solid body tides, the latter term applying equally to other terrestrial bodies. Because waves propagate differently in solids than in liquids, and the core-mantle boundary lies far deeper than any ocean, an equilibrium tide model fits the Earth’s solid body tides reasonably well, unlike the ocean case.

Solid body tides and oceanic tides can fall out of sync, adding to or subtracting from oceanic tide heights at particular locations. Many researchers hypothesize that the Earth’s solid body tides can trigger earthquakes, though this remains an open question.

How is the tidal bulge used in teaching today?

Oceanographers still teach Newton’s equilibrium tide theory for several reasons: it correctly describes the tidal forcing function, and many students, along with some instructors and textbook authors, struggle to understand how a location can experience two tides a day. Some oceanography texts still attribute the outer bulge to a centrifugal force, a claim that frustrates geophysicists and geodesists. Over the last twenty years or so, some oceanography courses and references have begun teaching that gravitation alone explains the tides.

Laplace’s dynamic theory of the tides gives a truer picture of oceanic behavior, and local astronomical tide predictions at coastal sites worldwide rely on historical data to derive empirical, location-specific values for each tidal harmonic component. Though empirical and locally derived, these predictions match observations well once weather effects are filtered out, and they also inform the shape of local amphidromic systems.

Summary

The Moon, and to a lesser extent the Sun, whose tidal influence is less than half the Moon’s, drive the oceanic tides. Their combined forcing is best understood in the frequency domain, which yields distinct harmonic components; translating these back into the spatial domain produces separate amphidromic systems for each component. The tidal bulge survives in oceanography only as a simplified teaching device, openly flagged as such to students, while the dynamic theory of the tides is taught as the accurate model. For the Moon’s recession from the Earth, however, the tidal bulge is still commonly presented as literal fact rather than a simplification, without the caveat it deserves.

Glossary

• Tidal bulge: The hypothesized pair of ocean bulges raised directly by the Moon’s gravity under Newton’s equilibrium tide theory; shown in this article not to exist.
• Dynamic theory of the tides: Laplace’s model of oceanic tides accounting for fluid flow, ocean depth and shape, bottom friction, and resonance.
• Amphidromic point: A location where a given tidal harmonic component has zero amplitude, around which that component’s tidal response rotates.
• Amphidromic system: The full pattern of cotidal lines and rotating tidal response surrounding an amphidromic point.
• M₂ tidal component: The principal lunar semidiurnal tide, with a period of 12.421 hours, the dominant tidal harmonic in most locations.
• Cotidal line: A curve along which high (or low) tide for a given component occurs at the same time.
• Doodson numbers: A numbering scheme devised by A.T. Doodson to catalog the 388 tidal harmonic components he identified starting in 1919.

Frequently Asked Questions

Does the Earth really have a tidal bulge raised by the Moon?

No. Observations such as the inconsistent timing of high tide relative to the Moon’s zenith and nadir, the simultaneous high and low tides across the North Sea, and the absence of any observed bulge from tide stations, buoys, or satellites all contradict Newton’s equilibrium tide theory.

Why can’t a tidal bulge wave keep up with the Earth’s rotation?

A tidal bulge would form a shallow-water wave limited to roughly 205 m/s at the mean ocean depth of 4267 m, or about 330 m/s over the deepest trench, both slower than the Earth’s 465 m/s equatorial rotation speed, so the bulge cannot track the Moon’s position.

What theory correctly explains the ocean tides?

Laplace’s dynamic theory of the tides explains ocean tides by accounting for water’s fluid flow, varying ocean depth and shape, seafloor friction, and resonance, producing amphidromic systems rather than a simple equilibrium bulge.

What is an amphidromic point?

An amphidromic point is a location where a particular tidal harmonic component, such as M₂, has zero amplitude, with the tidal response for that component rotating around it via radiating cotidal lines.

Why are the tides in the North Sea so unpredictable?

The North Sea contains three M₂ amphidromic systems, including two full amphidromic points and a partial one near the tip of Norway, which causes high and low tide to occur at different times across the region simultaneously.

How does the ocean tide cause the Moon to recede?

Angular momentum transfers between the Earth’s oceans and the Moon’s orbit through the combined effect of all tidal harmonic amphidromic systems, with the M₂ component contributing the most, gradually pushing the Moon to a higher orbit while slowing the Earth’s rotation.

Are ocean tides the same as tsunamis?

No. Tsunamis result from a nearly impulsive event like an earthquake, while tides respond to a continuous cyclical force, though both are shallow waves affected by ocean topography and the Coriolis effect.