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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.
Additional recommended knowledge
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 where
Let δ be the declination and α be the right ascension of the deforming body, the Moon for example, then the vector direction is
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
where For a circular orbit the angular momentum ω centripetal acceleration balances gravity at the planetary center of mass
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
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 μ = Ma / M.
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
There are about 62 constituents that could be used, but many less are needed to predict tides accurately.
Example amplitudes from Eastport, ME; Biloxi, MS; San Juan, PR; Kodiak, AK; San Francisco, CA; and Hilo HI;.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Theory_of_tides". A list of authors is available in Wikipedia.|