TIDES: Astronomical Forcing & Tidal Constituents Parker MacCready, January 2011 Reference: Admiralty Manual of Tides, Doodson & Warburg, 1941, His Majesty’s Stationery Office

Orbital Force Balance Tides are forced by the sun and moon, with similar physics. The Moon’s forcing is about twice that of the sun.

Perturbations to the Force Balance There is an average force which keeps the moon in orbit, equal to about 3.4x10 -6 g. BUT, there are perturbations to this due to distance and angle.

Forcing of Tides For forcing tides, it is only the component of the perturbation that is parallel to Earth’s surface that matters. This is called the “tractive force.”

But the Earth is also rotating… The Earth rotates under this pattern of tractive forces. Note: lh = “lunar hour” = (12.42/12) x (1 hour). It takes a point on Earth 12 lunar hours to get halfway around (with respect to the Moon’s forcing) because the Moon is moving too.

Pattern of Tractive Forces The pattern of tractive forces experienced by a point is different at different at different places. Here is an example at latitude 30° N, with a lunar declination of 15°. This forcing can be represented as a sum of: Semidiurnal + Diurnal + Steady Forces (the steady part doesn’t matter) Lunar: h (M 2 ) h Solar: 12 h (S 2 ) 24 h And note that we have started to assign names (M 2, S 2 ) to important frequencies. These are called “constituents” with subscripts 1 and 2 for roughly once or twice a day.

Notes Diurnal forcing requires declination, so it goes to zero twice a month for lunar forcing (not in phase with the new or full moon), or twice a year for solar forcing (at the autumnal and vernal equinox). If the Earth was fully covered with ocean, and the water surface was in perfect equilibrium with the tractive forces, it would form a “bulge” pattern:

More notes But… the ocean basins have irregular shapes, and the water sloshes around in wave patterns, influenced by bottom friction and Earth’s rotation. Often high tide travels counterclockwise around an ocean basin (northern hemisphere) like water sloshing around a bowl. Even though the tides are constantly forced by the Sun and Moon, friction keeps them in balance. Tidal waves move at speed (gH) 1/2 where H is the water depth. So the speed = 200 m s -1 in the deep ocean, just like Tsunamis.

Global Patterns For eachconstituent weknow what thepatterns ofamplitude andphase look like.“Phase” isalways relativeto theconstituent, sofor this figure360° of Phase =12.42 hours.

Estuarine Tides Tides in estuaries and bays are NOT primarily driven by the local tractive forces. Instead they are forced by changing ocean sea level at their mouths.

Mathematical Representation At a given place the tidal height (and currents) are the result of all these different influences. It is customary to represent the signal as the sum of a bunch of cosines with different amplitude, frequency, and phase lag:

Secondary Constituents The confusing part is that many astronomical forcings are NOT perfect cosines, mainly due to changing orbital distance (ellipticity) and the angle of the orbital plane relative to Earth’s rotational axis (declination). For example, the semidiurnal lunar forcing at a location over about a month looks like:

More Secondary Constituents For declinational changes as they affect the DIURNAL forcing the pattern is like: This is expressed as the sum of only 2 secondaries (no primary) Solar forcing: the analysis is the same as for the Moon, except that the forcing is ~half as big, the ellipticity is small, and the declination varies over a year.

The most important constituents Periods are in hours

The Spring-Neap Cycle Caused by the sum of M 2 and S 2 This is typically the BIGGEST source of variation of tidal amplitude, and it happens twice per ~month