Theory of tides

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Theory of tides

The theory of tides is the application of continuum mechanics

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids ....
to interpret and predict the tidal

Tide

Tides are the rising of Earth's ocean surface caused by the tidal forces of the Moon and the Sun acting on the oceans. Tides cause changes in the depth of the marine and estuary water bodies and produce oscillating currents known as tidal streams, making prediction of tides important for coastal navigation ....
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

Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
's ocean

Ocean

An ocean is a major body of Seawater, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a World Ocean that is customarily divided into several principal oceans and smaller seas....
s.

Tidal physics
Tidal forcing
The forces discussed here apply to body (Earth tides), oceanic and atmospheric tides.

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Encyclopedia

The theory of tides is the application of continuum mechanics

Continuum mechanics

Tide

Earth

Ocean

Tidal physics

Tidal forcing

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.

On the planet (or satellite) experiencing tidal motion consider a point at latitude and longitude at distance from the center of mass, then point can written in cartesian coordinates as where

Let be the declination

Declination

In astronomy, declination is one of the two coordinates of the equatorial coordinate system, the other being either right ascension or hour angle....
and be the right ascension

Right ascension

Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system....
of the deforming body, the Moon

Moon

The Moon is Earth's only natural satellite and the List of natural satellites by diameter satellite in the Solar System. The average centre-to-centre distance from the Earth to the Moon is km, about thirty times the diameter of the Earth....
for example, then the vector direction is

and be the orbital distance between the center of masses and the mass of the body. Then the force on the point is

where For a circular orbit the angular momentum centripetal acceleration balances gravity at the planetary center of mass

where is the distance between the center of mass for the orbit and planet and 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

Substituting for center of mass acceleration, and reordering

In ocean tidal forcing, the radial force is not significant, the next step is to rewrite the coefficient. Let then

where is the inner product determining the angle z of the deforming body or Moon from the zenith. This means that

if ε is small. If particle is on the surface of the planet then the local gravity is and set .

which is a small fraction of . Note also that force is attractive toward the Moon when the and repulsive when .

This can also be used to derive a tidal potential.

Laplace's tidal equations

in 1776, Pierre-Simon Laplace

Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace was a France mathematician and astronomer whose work was pivotal to the development of astronomy and statistics....
formulated a single set of linear partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s, for tidal flow described as a barotropic

Barotropic

In meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that Isobaric process surfaces are also isopycnic surfaces ....
two-dimensional sheet flow. Coriolis effect

Coriolis effect

In physics, the Coriolis effect is an apparent deflection of moving objects when they are viewed from a rotating reference frame.Newton's laws of motion govern the motion of an object in an inertial frame of reference....
s 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

Average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "Expected value" value of the data set....
thickness D, the vertical tidal elevation ?, as well as the horizontal velocity components u and v (in the latitude

Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
f and longitude

Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
? directions, respectively) satisfy Laplace's tidal equations:

where O is the angular frequency

Angular frequency

In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....
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

Potential

*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds. This mathematical formulation arises from the fact that, in physics, the scalar potential is irrotational, and thus has a vanishing Laplacian ? the very definition of a harmonic function....
.

William Thomson (Lord Kelvin)

William Thomson, 1st Baron Kelvin

William Thomson, 1st Baron Kelvin , Order of Merit , Royal Victorian Order, Privy Council of the United Kingdom, Presidents of the Royal Society, Royal Society of Edinburgh, was an Ireland-born United Kingdom of Great Britain and Ireland Mathematical physics and engineer....
rewrote Laplace's momentum terms using the curl to find an equation for vorticity

Vorticity

Vorticity is a mathematical concept used in fluid dynamics. It can be related to the amount of "Circulation " or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the Circulation around the boundary of the small region, divided by the area A of the small region....
. Under certain conditions this can be further rewritten as a conservation of vorticity.

Tidal analysis and prediction

Harmonic analysis

There are about 62 constituents that could be used, but many less are needed to predict tides accurately.

Tidal constituents

Amplitudes are given 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	?₂	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	?₂	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	?	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	M_m	661.3111655	0.5443747	0	1		-1	65.455			0.7	1.9			20
Solar semiannual	S_sa	4383.076325	0.0821373	0		2		57.555	1.6		2.1	1.5	3.9		21
Solar annual	S_a	8766.15265	0.0410686	0		1		56.555			5.5	7.8	3.8	4.3	22
Lunisolar synodic fortnightly	M_sf	354.3670666	1.0158958	0	2	-2		73.555				1.5			23
Lunisolar fortnightly	M_f	327.8599387	1.0980331	0	2			75.555			1.4	2.0		0.7	24

Theory of tides

Tidal physics

Tidal forcing

Laplace's tidal equations

Tidal analysis and prediction

Harmonic analysis

Tidal constituents

Higher harmonics

Semi-diurnal

Diurnal

Long period