Spring & neap tides

Spring & neap tides • Tidal range often changes regularly, i.e. every fortnight (14 day period) • We see: • Spring tides - times of greater tidal range; coincide with full moon or new moon • Tidal range during spring tides is usually 20% higher than mean tidal range • Neap tides - times of lower tidal range; coincide with first or third quarter • Tidal range during neap tides is usually 20% lower than the mean tidal range

Pliny the Elder (23-79 A.D.): tides ‘follow’ the moon If that were strictly true, would have diurnal (daily) tides everywhere

Origin of tides - equilibrium model • Newton proposed the first explanation for semidiurnal tides with his equilibrium model • In Newton’s equilibrium model: • Earth & moon exist in isolation • Earth is a non-rotating sphere • Have a single ocean that encircles the globe • Ocean is static, i.e. has no currents • The only forces acting on ocean result from the movement of Earth & moon about their common center of mass

Equilibrium means ‘no net force’ • Earth & moon persist in their relative motion, so there are no net forces acting on their centers of mass - forces must sum to zero • Earth & moon exert gravitational attraction on each other, & so fall toward each other • Earth & moon never collide because they are moving past each other - each has inertia • Gravitational attraction is exactly sufficient to produce the centripetal acceleration required to move each body in a circular path about their common center of mass

Explain this situation by saying that the two bodies are forced outward by an apparent (or fictitious) force - the centrifugal force

But have local inequities in forces • At center of Earth, moon’s gravitational attraction balances exactly the centripetal acceleration for Earth’s circular path • Elsewhere on Earth, gravitational attraction need not balance centripetal acceleration exactly • Because all points on Earth trace out circles of equal radii, all points experience equal centripetal acceleration • Gravitational attraction exerted by moon diminishes with increasing distance according to the inverse square law • If gravitational & centripetal accelerations have different magnitudes at different points, forces cannot be equal

Net forces produce accelerations • On moonward side of Earth, gravitational attraction is greater, so particles are accelerated toward moon • On side of Earth facing away from moon, inertial effects related to earth’s circular path exceed gravitational attraction, so particles are accelerated away from the moon • Accelerations perpendicular to earth’s surface are miniscule - on the order of one millionth of Earth’s gravitational attraction • Accelerations parallel to Earth’s surface are more effective – they displace water into two bulges, one on side facing moon & one on side away from moon

Equilibrium tide • If Earth rotates beneath the two bulges, points experience semidiurnal tides • Call this the equilibrium tide • Refinements to equilibrium theory include: • Plane of moon’s orbit is inclined 61.5° to Earth’s rotation axis - expect daily inequity that varies with a monthly cycle • Moon’s orbit is elliptical - expect variation in magnitude of bulge with a 27.55 day cycle

Solar equilibrium tide • Earth-sun interactions produce ocean tides • Earth-moon distance = 59 x RE; Mass of moon 1/82 mass of earth • Earth-sun distance = 23,000 x RE; Mass of sun 330,00 x mass of earth • Solar tide is 47% of that generated by earth moon interactions • Solar tide has12 hr period; see effects of tilt of rotation axis & elliptical orbit, too • Add lunar & solar equilibrium tides together to get spring & neap tide

Shortcomings of the equilibrium model • It predicts semidiurnal tides at all locations - not observed • It predicts that high tides should occur when moon passes overhead or 12 hours 25 minutes later - rarely observed • Calculations suggest that tidal ranges should be 20-50 cm - observed tidal ranges are often much larger • It predicts values for daily inequities that are rarely observed

What is wrong with equilibrium model for the tides? It ignores the fact that ocean basins have irregular shapes, that bulges must respond to friction in moving through basins, & that water itself has inertia once it is moving

Dynamic theory for tides • Begins where equilibrium model ends, i.e. with two bulges created by gravitational/inertial interactions of Earth & moon • Explains real tides by envisioning tidal bulges as tidal wave • Tidal wave has small wave height - about 50 cm in open ocean • Tidal wave has very long wavelength, about one half earth’s circumference = 20,000 km

The tidal wave • Wavelength (L) of tidal wave = 1/2 Earth’s circumference = 20,000 km • Water depth of oceans = 4 km <<< L/2 • Tidal wave is a shallow water wave everywhere, i.e. it interacts continually with the ocean bottom • As a shallow water wave, tidal wave feels the bottom, slows, steepens, & sometimes breaks • Tidal wave reflects, refracts, & interferes with other waves or with reflections of itself

Standing waves • Have a wave with long wavelength (L) in a rectangular basin of uniform depth (D) • Wave advances across basin, reflects off boundary, & travels back across basin • Given enough time, a regular wave will interfere to produce a standing or stationary wave • Where water level does not change = nodal line (node) • Where water level changes the most = antinodal lines (antinodes) • Oscillation period for standing wave with one node = [2 x L]/ [g x D]1/2 (depends on basin length & depth) • Length & depth dependence holds for all standing waves

In a rectangular basin of uniform depth, perturbing wave advances across basin, reflects off boundary, & travels back across basin. In many cases, interference between inducing wave & reflected wave generates a standing wave. •Nodal line or node = where water level does not change •Antinodallines or antinodes = where water level changes the most •Oscillation period depends on basin length & depth •May see resonance if natural period of basin is close to that of perturbation

Tidal standing waves • Tidal standing waves are not free waves, where generating force acts only at the outset • Tidal standing waves are forced waves, where the wave generating force(s) continue(s) to act as the system responds • Each passage of the moon overhead is another disturbing force • Complicated wave motion in any basin is the sum of latest disturbance interacting with waves generated by numerous previous passes overhead • Depending on the shape of a basin, the net result may be a standing wave that corresponds to a diurnal tide, a semidiurnal tide, or a mixed tide

Kelvin waves • Special kind of standing wave seen in large basins where masses of water experience the Coriolis effect • Wave crest swings about the basin margin like water swirling in a glass • Nodal point, called an amphidromic point, occurs near the center of the basin • Co-tidal lines- connect points that have high tides at the same time, typically measured in hours after the moon crosses the Greenwich meridian • Co-range lines connect points that have equal tidal ranges

Amphidromic point (node) occurs near center of the basin • Co-tidal lines connect points that have high tides at the same time (denoted by hours after the moon crossed the Greenwich meridian) • Co-range lines connect points that have equal tidal ranges • Co-tidal lines are spokes; co-range lines are closed loops about node

Tidal currents • Can understand tidal currents by envisioning tide as one of three types of wave • Progressive wave tide - where coast is too irregular to give pronounced reflection, so no standing wave • High tide = wave crest; low tide = wave trough • Flood current coincides with high tide; ebb current coincides with low tide • Standing wave tide - in confined rectangular basin • Slack water coincides with high or low water level • Flood and ebb currents coincide with mean water level • Kelvin wave tide - azimuth & magnitude of currents may vary

Spring & neap tides