Theory of tides

The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans, under the gravitational loading of another astronomical body or bodies. It commonly refers to the fluid dynamic motions for the Earth's oceans.

Origin of theory

In 1616, Galileo Galilei wrote Discourse on the Tides (in Italian: Discorso del flusso e reflusso del mare), a paper in which he tried to explain the occurrence of the tides as the result of the Earth's rotation around the Sun. However, Galileo's theory was, in the later Newtonian terms, an error. Later analysis over the centuries had led to the current tidal physics.

Tidal physics

Tidal forcing

A. Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere.

B. This view shows same potential from 180° from view A. Viewed from above the Northern Hemisphere. Red up, blue down.

The forces discussed here apply to body (Earth tides), oceanic and atmospheric tides. Atmospheric tides on Earth, however, tend to be dominated by forcing due to solar heating. Oceanic tides act according to the Dynamic theory of tides.

On the planet (or satellite) experiencing tidal motion consider a point at latitude \(\varphi\) and longitude \(\lambda\) at distance \(a\) from the center of mass, then this point can be written in cartesian coordinates as \(\mathbf{p} = a\mathbf{x}\) where

\[ \mathbf{x} = (\cos \lambda \cos \varphi, \sin \lambda \cos \varphi, \sin \varphi).\]

Let \(\delta\) be the declination and \(\alpha\) be the right ascension of the deforming body, the Moon for example, then the vector direction is

\[ \mathbf{x}_m = (\cos \alpha \cos \delta, \sin \alpha \cos \delta, \sin \delta),\]

and \(r_m\) be the orbital distance between the center of masses and \(M_m\) the mass of the body. Then the force on the point is

\[ \mathbf{F}_{a}= \frac{G M_m (r_m\mathbf{x}_{m}-a\mathbf{x})}{R^3}.\]

where \(R = \|r_m\mathbf{x}_{m}-a\mathbf{x}\| \)

For a circular orbit the angular momentum \(\omega\) centripetal acceleration balances gravity at the planetary center of mass

\[ Mr_{cm}\omega^2= \frac{G M M_m }{r_m^2}.\]

where \(r_{cm}\) is the distance between the center of mass for the orbit and planet and \(M\) is the planetary mass. Consider the point in the reference fixed without rotation, but translating at a fixed translation with respect to the center of mass of the planet. The body's centripetal force acts on the point so that the total force is

\[\mathbf{F}_p= \frac{G M_m (r_m\mathbf{x}_{m}-a\mathbf{x})}{R^3} -r_{cm}\omega^2\mathbf{x}_m. \]

Substituting for center of mass acceleration, and reordering

\[\mathbf{F}_p = G M_m r_m \left( \frac {1}{R^3} - \frac {1}{r_m^3} \right) \mathbf{x}_m -\frac{ ( G M_m a\mathbf{x})}{R^3}. \] In ocean tidal forcing, the radial force is not significant, the next step is to rewrite the \(\mathbf{x}_m\) coefficient. Let \(\varepsilon= \frac {a} {r_m}\) then

\[R = r_m \sqrt{ 1+ \varepsilon ^2-2 \varepsilon ( \mathbf{x}_m,\mathbf{x} ) } \]

where \(( \mathbf{x}_m,\mathbf{x} )= \cos z \) is the inner product determining the angle z of the deforming body or Moon from the zenith. This means that

\[\left( \frac {1}{R^3} - \frac {1}{r_m^3} \right) \approx \frac{3\varepsilon \cos z }{r_m^3}, \]

if ε is small. If particle is on the surface of the planet then the local gravity is \(g=\frac{ G M}{a^2}\) and set \(\mu= \frac{M_m} {M}\).

\[\mathbf{F}_p = 3 g \mu \varepsilon^3 \cos z \mathbf{x}_m - \frac{ ( g \mu a^3\mathbf{x})}{R^3} + O(\varepsilon^4), \]

which is a small fraction of \(g\). Note also that force is attractive toward the Moon when the \(z<\pi/2\) and repulsive when \(z > \pi/2\).

This can also be used to derive a tidal potential.

Laplace's tidal equations

in 1776, Pierre-Simon Laplace formulated a single set of linear partial differential equations, for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from energy integrals via Lagrange's equation.

For a fluid sheet of average thickness D, the vertical tidal elevation ς, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations: \[ \begin{align} \frac{\partial \zeta}{\partial t} &+ \frac{1}{a \cos( \varphi )} \left[ \frac{\partial}{\partial \lambda} (uD) + \frac{\partial}{\partial \varphi} \left(vD \cos( \varphi )\right) \right] = 0, \\[2ex] \frac{\partial u}{\partial t} &- v \left( 2 \Omega \sin( \varphi ) \right) + \frac{1}{a \cos( \varphi )} \frac{\partial}{\partial \lambda} \left( g \zeta + U \right) =0 \qquad \text{and} \\[2ex] \frac{\partial v}{\partial t} &+ u \left( 2 \Omega \sin( \varphi ) \right) + \frac{1}{a} \frac{\partial}{\partial \varphi} \left( g \zeta + U \right) =0, \end{align} \] where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, and U is the external gravitational tidal-forcing potential.

William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the curl to find an equation for vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity.

Tidal analysis and prediction

Harmonic analysis

Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by William Thomson's application of Fourier analysis to the tidal motions. Thomson's work in this field was then further developed and extended by George Darwin: Darwin's work was based on the lunar theory current in his time. His symbols for the tidal harmonic constituents are still used. Darwin's harmonic developments of the tide-generating forces were later brought up to date with modern developments by A T Doodson whose development of the tide generating potential (TGP) in harmonic form was carried out and published in 1921: Doodson distinguished 388 tidal frequencies. Doodson's analysis of 1921 was based on the then-latest lunar theory of E W Brown.

Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson Numbers, a system still in use.

Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many less even than that can predict tides to useful accuracy. The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tide-predicting machine, a special-purpose form of analog computer now superseded in this work by digital electronic computers that can be programmed to carry out the same computations.

Tidal constituents

Tidal constituents combine to give an endlessly-varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an animation of the American Mathematical Society illustrating the way in which the components used to be mechanically combined in the tide-predicting machine. Amplitudes of tidal constituents are given below for the following example locations:

ME Eastport,

MS Biloxi,

PR San Juan,

AK Kodiak,

CA San Francisco, and

HI Hilo.

Higher harmonics	Darwin	Period	Phase	Doodson coefs				Doodson	Amplitude at example location (cm)						NOAA
Species	Symbol	(hr)	rate(°/hr)	n1 (L)	n2 (m)	n3 (y)	n4 (mp)	number	ME	MS	PR	AK	CA	HI	order
Shallow water overtides of principal lunar	M4	6.210300601	57.9682084	4				455.555	6.0	0.6		0.9	2.3		5
Shallow water overtides of principal lunar	M6	4.140200401	86.9523127	6				655.555	5.1	0.1		1.0			7
Shallow water terdiurnal	MK3	8.177140247	44.0251729	3	1			365.555				0.5	1.9		8
Shallow water overtides of principal solar	S4	6	60	4	4	-4		491.555		0.1					9
Shallow water quarter diurnal	MN4	6.269173724	57.4238337	4	-1		1	445.655	2.3			0.3	0.9		10
Shallow water overtides of principal solar	S6	4	90	6	6	-6		*		0.1					12
Lunar terdiurnal	M3	8.280400802	43.4761563	3				355.555					0.5		32
Shallow water terdiurnal	2"MK3	8.38630265	42.9271398	3	-1			345.555	0.5			0.5	1.4		34
Shallow water eighth diurnal	M8	3.105150301	115.9364166	8				855.555	0.5	0.1					36
Shallow water quarter diurnal	MS4	6.103339275	58.9841042	4	2	-2		473.555	1.8			0.6	1.0		37
Semi-diurnal	Darwin	Period	Phase	Doodson coefs				Doodson	Amplitude at example location (cm)						NOAA
Species	Symbol	(hr)	(°/hr)	n1 (L)	n2 (m)	n3 (y)	n4 (mp)	number	ME	MS	PR	AK	CA	HI	order
Principal lunar semidiurnal	M2	12.4206012	28.9841042	2				255.555	268.7	3.9	15.9	97.3	58.0	23.0	1
Principal solar semidiurnal	S2	12	30	2	2	-2		273.555	42.0	3.3	2.1	32.5	13.7	9.2	2
Larger lunar elliptic semidiurnal	N2	12.65834751	28.4397295	2	-1		1	245.655	54.3	1.1	3.7	20.1	12.3	4.4	3
Larger lunar evectional	ν2	12.62600509	28.5125831	2	-1	2	-1	247.455	12.6	0.2	0.8	3.9	2.6	0.9	11
Variational	MU2	12.8717576	27.9682084	2	-2	2		237.555	2.0	0.1	0.5	2.2	0.7	0.8	13
Lunar elliptical semidiurnal second-order	2"N2	12.90537297	27.8953548	2	-2		2	235.755	6.5	0.1	0.5	2.4	1.4	0.6	14
Smaller lunar evectional	λ2	12.22177348	29.4556253	2	1	-2	1	263.655	5.3		0.1	0.7	0.6	0.2	16
Larger solar elliptic	T2	12.01644934	29.9589333	2	2	-3		272.555	3.7	0.2	0.1	1.9	0.9	0.6	27
Smaller solar elliptic	R2	11.98359564	30.0410667	2	2	-1		274.555	0.9			0.2	0.1	0.1	28
Shallow water semidiurnal	2SM2	11.60695157	31.0158958	2	4	-4		291.555	0.5						31
Smaller lunar elliptic semidiurnal	L2	12.19162085	29.5284789	2	1		-1	265.455	13.5	0.1	0.5	2.4	1.6	0.5	33
Lunisolar semidiurnal	K2	11.96723606	30.0821373	2	2			275.555	11.6	0.9	0.6	9.0	4.0	2.8	35
Diurnal	Darwin	Period	Phase	Doodson coefs				Doodson	Amplitude at example location (cm)						NOAA
Species	Symbol	(hr)	(°/hr)	n1 (L)	n2 (m)	n3 (y)	n4 (mp)	number	ME	MS	PR	AK	CA	HI	order
Lunar diurnal	K1	23.93447213	15.0410686	1	1			165.555	15.6	16.2	9.0	39.8	36.8	16.7	'4
Lunar diurnal	O1	25.81933871	13.9430356	1	-1			145.555	11.9	16.9	7.7	25.9	23.0	9.2	6
Lunar diurnal	OO1	22.30608083	16.1391017	1	3			185.555	0.5	0.7	0.4	1.2	1.1	0.7	15
Solar diurnal	S1	24	15	1	1	-1		164.555	1.0		0.5	1.2	0.7	0.3	17
Smaller lunar elliptic diurnal	M1	24.84120241	14.4920521	1				155.555	0.6	1.2	0.5	1.4	1.1	0.5	18
Smaller lunar elliptic diurnal	J1	23.09848146	15.5854433	1	2		-1	175.455	0.9	1.3	0.6	2.3	1.9	1.1	19
Larger lunar evectional diurnal	ρ	26.72305326	13.4715145	1	-2	2	-1	137.455	0.3	0.6	0.3	0.9	0.9	0.3	25
Larger lunar elliptic diurnal	Q1	26.868350	13.3986609	1	-2		1	135.655	2.0	3.3	1.4	4.7	4.0	1.6	26
Larger elliptic diurnal	2Q1	28.00621204	12.8542862	1	-3		2	125.755	0.3	0.4	0.2	0.7	0.4	0.2	29
Solar diurnal	P1	24.06588766	14.9589314	1	1	-2		163.555	5.2	5.4	2.9	12.6	11.6	5.1	30
Long period	Darwin	Period	Phase	Doodson coefs				Doodson	Amplitude at example location (cm)						NOAA
Species	Symbol	(hr)	(°/hr)	n1 (L)	n2 (m)	n3 (y)	n4 (mp)	number	ME	MS	PR	AK	CA	HI	order
Lunar monthly	Mm	661.3111655	0.5443747	0	1		-1	65.455			0.7	1.9			20
Solar semiannual	Ssa	4383.076325	0.0821373	0		2		57.555	1.6		2.1	1.5	3.9		21
Solar annual	Sa	8766.15265	0.0410686	0		1		56.555			5.5	7.8	3.8	4.3	22
Lunisolar synodic fortnightly	Msf	354.3670666	1.0158958	0	2	-2		73.555				1.5			23
Lunisolar fortnightly	Mf	327.8599387	1.0980331	0	2			75.555			1.4	2.0		0.7	24

References

ar:نظرية المد والجزر