Tidal Harmonics and Stream Turbine

What is modelled

Tides are driven by the Moon and the Sun, and the sea level at a place is very well described as a sum of cosines, one for each astronomical frequency. M2 and S2 are the main twice-daily lunar and solar tides; N2 allows for the elliptic lunar orbit; K1 and O1 are once-daily tides. A turbine on the sea bed in a channel feels a current that follows the same frequencies, and its power depends on the cube of the speed.

Equations

• Sea level η(t) = Σ ai cos(ωi t + φi) with speeds 28.984, 30.000, 28.440, 15.041 and 13.943 degrees per hour.
• Spring-neap beat T = 2π / (ωM2 − ωS2) = 14.77 days; a tidal day is 24.84 hours.
• Current in a standing-wave channel u = k Σ ai sin(ωi t + φi), turbine power P = ½ ρ A Cp |u|3 between cut-in (0.7 m/s) and rated speed (3 m/s).

Checks run in the page

• The spring-neap period found by scanning the high-water envelope against the beat formula.
• A least-squares harmonic analysis of a 59-day hourly record that recovers the amplitudes you set.
• The 50-minute daily delay of the M2 tide.
• The mean cubic power of a single constituent against the closed form with the factor 4 / 3 pi.

Honest simplifications

Only five constituents are used, with the nodal corrections and the shallow-water terms left out, and phases are fixed apart from the S2 shift. The current is taken as exactly proportional to the quarter-period-shifted tide, a single power coefficient and a single rotor stand for a real turbine, and wake effects are ignored. This is a teaching model, not a tide table or a yield forecast.

• Press Semidiurnal coast and watch the monthly sea-level curve swell and shrink every 14.8 days.
• Set S2 to zero and see the spring-neap cycle disappear.
• Press Diurnal tide to see one high water a day and the turbine idle in slack water.