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# 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.

## 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 φ and longitude λ at distance a from the center of mass, then point can written in cartesian coordinates as $\mathbf{p} = a\mathbf{x}$ where

$\mathbf{x} = (\cos \lambda \cos \phi, \sin \lambda \cos \phi, \sin \phi).$

Let δ be the declination and α 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 rm be the orbital distance between the center of masses and Mm 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 ω centripetal acceleration balances gravity at the planetary center of mass

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

where rcm 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 $\varepsilon= a / r_m$ 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 μ = Ma / M.

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

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

This can also be used to derive a tidal potential.

### Laplace tidal equations

Laplace summarized the work to his time with a single set of linear partial differential equations simplified from the fluid dynamic equations, but can also be derived via Lagranges equation from energy integrals. It summarizes tidal flow as a barotropic two-dimensional sheet flow, where Coriolis effects are introduced a fictious lateral force.

Thomson rewote 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

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

### Tidal constituents

#### Higher harmonics

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude (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 (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 (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 (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

Example amplitudes from Eastport, ME; Biloxi, MS; San Juan, PR; Kodiak, AK; San Francisco, CA; and Hilo HI;.