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 $varphi$ and longitude $lambda$ at distance $a$ from the center of mass, then point can written in cartesian coordinates as $mathbf\{p\}\; =\; amathbf\{x\}$ 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

:$mathbf\{F\}\_\{a\}=\; frac\{G\; M\_m\; (r\_mmathbf\{x\}\_\{m\}-amathbf\{x\})\}\{R^3\}.$

where $R\; =\; |r\_mmathbf\{x\}\_\{m\}-amathbf\{x\}|$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

:$mathbf\{F\}\_p=\; frac\{G\; M\_m\; (r\_mmathbf\{x\}\_\{m\}-amathbf\{x\})\}\{R^3\}\; -r\_\{cm\}omega^2mathbf\{x\}\_m.$

Substituting for center of mass acceleration, and reordering

:$mathbf\{F\}\_p\; =\; G\; M\_m\; r\_m\; left(\; frac\; \{1\}\{R^3\}\; -\; frac\; \{1\}\{r\_m^3\}\; ight)\; mathbf\{x\}\_m\; -frac\{\; (\; G\; M\_m\; amathbf\{x\})\}\{R^3\}.$

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

:$left(\; frac\; \{1\}\{R^3\}\; -\; frac\; \{1\}\{r\_m^3\}\; ight)\; approx-\; frac\{3varepsilon\; cos\; z\; \}\{r\_m^3\},$

if ε is small. If particle is on the surface of the planet then the local gravity is$g=frac\{\; G\; M\}\{a^2\}$ andset $mu=\; M\_a\; /\; M$.

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

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

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 [http://kiwi.atmos.colostate.edu/group/dave/pdf/LTE.frame.pdf] [http://siam.org/pdf/news/621.pdf] ::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

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.

References