In 1776, 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 dynamics 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 ζ, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations:
where Ω is the angular frequency of the planets rotation, g is the planets gravitational acceleration at the mean ocean surface, a is the planetary radius, 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.