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De-Leon and Paldor (2011)
  • De-Leon and Paldor (2011)

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Solutions for alpha=1e-4

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Published By

Dr. Dick van der Wateren


Physics, Fluid Dynamics on a Sphere, Physical oceanography, Schrödinger equation, Tidal wave

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Solution to longstanding tidal equation problem

04.01.2012, Age: 3583 days

Israeli scientists have improved solutions for the famous Tidal Equations first formulated by Pierre-Simon Laplace in 1776. Their new contribution for this longstanding problem involves a new formulation of an approximate equation that, suprisingly, yields highly accurate solutions of Laplace's Equations.

Yair De-Leon and Nathan Paldor from the Institute of Earth Sciences, The Hebrew University, Jerusalem, Israel, published their results in 2011 in the journal Tellus.

Summary by the authors:

By constructing an approximate Schrödinger equation we derived new zonally propagating wave solutions of Laplace's Tidal Equations on the rotating spherical Earth. The energy levels of this equation yield the dispersion relation of these waves via a simple algebraic relationship while its eigenfunctions yield the meridional structure of the amplitudes of the zonal velocity of these waves.

The new theory is a modern and simpler version of an eigenvalue approach that was suggested in the 1960s by Longuet-Higgins. In a baroclinic ocean on an aqua-planet (similar to a thin ocean that completely covers a continent-free planet like Earth) the equation reduces to the well known equation of Harmonic oscillator in Quantum Mechanics.

In this case the solutions yield Planetary (Rossby) waves as well as Inertia-Gravity (Poincare) waves as particular roots of the general relationship between the energy levels (of the Schrödinger equation) and the waves' phase speeds. In the baroclinic case (as is the case in a 4-kilometer deep ocean) the energy levels yield the Planetary wave phase speed and a first approximation only to the phase speed of inertia-gravity waves. This is a major progress in a problem that dates back 250 years ago when Pierre Simon Laplace first formulated the problem correctly but due to its complexity no explicit solutions were obtained for either of the two waves on a spherical Earth.

The theoretical advances are only half-way done and more work is required for its completion but the finding obtained so far associate the problem of wave in Geophysical Fluid Dynamics in a broader context of a Schrödinger eigenvalue problem, which is probably one of the most fundamental problems of 20th century Physics.

Yair De-Leon and Nathan Paldor (2011). Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet. Tellus, 63A, 348-353. doi: 10.1111/j.1600-0870.2010.00490.x Link to article>>

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