Laplace's tidal equations

In 1776, Pierre-Simon Laplace published a paper in which he analyzed the tides of a body of fluid with consideration as to the rotation of the body that fluid exists on.

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Daily Visual Balance Check

Given a sphere of radius r and angular velocity ω, covered in a fluid of depth D with velocity vector relative to the moving surface of the sphere (u, v), and with co-latitude and east longitude co-ordinates (θ, φ), and fluid displacement ζ, we have

$\frac{\partial}{\partial \theta} \left(vD \sin( \theta )\right) + \frac{\partial}{\partial \phi} (uD) + r \sin( \theta ) \frac{\partial \zeta}{\partial t} = 0.$