# How to Calculate the Equilibrum Function for use in Tide Calculations using Harmonic Constituents

I have found a wealth of information using Harmonic Constituents to calculate tide levels. H(t) = Amplitude * cos ( t * harmonicSpeed + phase lag). Yet when I try to apply this to constituents from the NOAA the results never match the tables produced. I have since found that the NOAA constituents require an additional term, the Equilibrium value with the resulting constituent function looking more like

H(t) = Amplitude * cos ( t * harmonicSpeed + EquilibriumValue + phase lag).

I am looking for any hints on how this Equilibrium value is calculated or derived.

Addendum: The constituents are found here, this for San Diego, Ca Constituents for San Diego. Sorry cannot find the source of the Equilibrium value but will when I get home from work.

• Which tidal constituents are you including? Also, can you point to where you found the reference to the "Equilibrium value"? – Isopycnal Oscillation Jan 7 '15 at 20:15
• Added the link for constituents for San Diego. Cannot find the equilibrium value reference at this time. – user824 Jan 7 '15 at 22:42

The equilibrium tide is a theoretical concept developed by Newton in the 17th century that considers only the gravitational pull by the Moon and the Sun and the centrifugal forces, with no inertia, no friction and no land masses. From the NOAA definitions:

equilibrium theory — A model under which it is assumed that the waters covering the face of the Earth instantly respond to the tide-producing forces of the Moon and Sun to form a surface of equilibrium under the action of these forces. The model disregards friction, inertia, and the irregular distribution of the land masses of the Earth. The theoretical tide formed under these conditions is known as the equilibrium tide.

Lets consider as a first cut the two tide producing forces associated with just the Moon and Earth systems. The two are the centrifugal force of Earth from the center of mass of the Moon-Earth system and the gravitational attraction due to the Moon. For the point closest to the Moon:

$$Force = \frac {GM_1M_2}{(R-r)^2} - \frac{GM_1M_2}{R^2}$$ where R is the distance between Earth and Moon and r is Earth's radius. It gets a bit more complex for other points on the surface because instead of R you have to use $$R\pm r cos(lat)$$ for the gravitational force.

Ultimately, it comes down to a set of horizontal forces that tend to concentrate water in two theoretical points: one, the closest to the Moon and two, the farthest from the Moon. An equilibrium state would be reached, called equilibrium tide, that results in an ellipsoid with its two bulges directed toward and away from the Moon.

In practice, this equilibrium ellipsoid does not develop because of the rotation of Earth, the Coriolis acceleration and the fact that the tidal wave feels the ocean bottom and it is subject to friction. Going beyond the equilibrium theory of the tide, the dynamic theory of tides (developed by Euler, Laplace and Bernoulli) includes all these concepts (friction, inertia, Coriolis, land masses,...) and provides a better approximation to the observed tides.

My recommendation to understand tidal constituents is to use some decent tidal harmonic analysis package like t_tide for Matlab, which is likely to be the tool that NOAA is using to generate their harmonic constituents in the first place. If you really want to get into the details, then the best option is to go to the still likely best source available. That will the 1973 book "The Analysis of Tides" by G. Godin.

Pawlowicz, R., B. Beardsley, and S. Lentz, "Classical Tidal Harmonic Analysis Including Error Estimates in MATLAB using t_tide", Computers and Geosciences, 28, 929-937 (2002).

Godin, G. (1973). The analysis of tides., by Godin, G.. Liverpool (UK): Liverpool University Press, 264 p.