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Equilibrium Model of Tides Highly Idealized, but Very Instructive, View of Tides

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Equilibrium Model of Tides Highly Idealized, but Very Instructive, View of Tides Tides Outline • Equilibrium Theory of Tides — diurnal, semidiurnal and mixed semidiurnal tides — spring and neap tides • Dynamic Theory of Tides — rotary tidal motion — larger tidal ranges in coastal versus open-ocean regions • Special Cases — Forcing ocean water into a narrow embayment — Tidal forcing that is in resonance with the tide wave Equilibrium Model of Tides Highly Idealized, but very instructive, View of Tides • Tide wave treated as a deep-water wave in equilibrium with lunar/solar forcing • No interference of tide wave propagation by continents Tidal Patterns for Various Locations 1 Looking Down on Top of the Earth The Earth’s Rotation Under the Tidal Bulge Produces the Rise moon and Fall of Tides over an Approximately 24h hour period Note: This is describing the ‘hypothetical’ condition of a 100% water planet Tidal Day = 24h + 50min It takes 50 minutes for the earth to rotate 12 degrees of longitude Earth & Moon Orbit Around Sun 2 R b a P Fa = Gravity Force on a small mass m at point a from the gravitational attraction between the small mass and the moon of mass M Fa = Centrifugal Force on a due to rotation about the center of mass of the two mass system using similar arguments The Main Point: Force at and point a and b are equal and opposite. It can be shown that the upward (normal the the earth’s surface) tidal force on a parcel of water produced by the moon’s gravitational attraction is small (1 part in 9 million) compared to the downward gravitation force on that parcel of water caused by earth’s on gravitational attraction. What matters most is the component of the moon’s gravity that is tangential to the earth’s surface since this is not balanced by the earth downward gravitational force. 3 Illustration of the combined effects of Gravitational and Centrifugal Forces on Earth Tides Looking Down on Top of the Earth So far we have shown how we get semi-dirual Tides that have a period of 24h hours and 50 minutes moon We still need to figure out why we get dirunal and mixed semi-dirunal tides… The moon’s orbit is inclined 28o relative to the earth’s equator and this has important consequences for differences on the daily pattern of tides depending on the latitude 4 Effect of Moon Orbit Declination on Tidal Pattern The Tidal Force Exerted on the Earth by the Sun is About 46% of the Tidal Force Exerted on the Earth by the Moon 5 Combined Effects of the Moon & Sun Induced Tides Spring Tides Occur When the Moon Pulls along the same Line as the Sun. (new and full moon). Neap Tides Occur When the Moon Pulls at 90o to the Sun (first and last quarter moon) 6 Equilibrium Tide Theory Roughly Explains: • diurnal, semidiurnal and mixed semidiurnal tides • spring/neap tide series Dynamic Model of Tides A More Realistic View of Tides • Tide wave treated as a forced shallow-water wave – not in equilibrium with lunar/solar forcing • Continents interfere with of tide wave propagation • Affected by Coriolis Force Tide Waves Are Shallow-Water Waves • The tide wave has wavelength (L) on the order of 1/2 the diameter of the earth or about 20,000 km • The tide wave can be considered a Shallow- Water Wave for depths < L/20 or for bottom depths < 1000km. • Since ocean bottom depths are typically only about 4 km, it is safe to assume that a tide wave is a Shallow-Water Wave 7 Tide Waves Are Actually Forced Shallow-Water Waves Under ideal conditions, tide wave speed (as a free wave) would be determined by ocean bottom depth alone. However, this would only be the case if tides were briefly generated and then allowed to propagate freely on their own - but this is not the case… Tidal waves are Forced shallow-water waves because tidal forces exerted on the ocean by the moon and sun constantly interfere with the free propagation of the shallow water wave. note that the wave speed for a shallow water wave -1 -1 in 4km of water is 200m s (400 miles h ). The speed that the earth rotates under the moon at the -1 -1 equator is 463m s (1044 miles h ). Since tidal waves are of the same length scale as the earth and motions are on the order of a day, Coriolis Force also has a very significant effect on the direction of tidal wave propagation. 8 Rotary motion of a tide wave in an ocean basin caused by Coriolis Force Amphidromic Point of a Rotary Tide Theoretical and Actual Rotary Tide Depicted using Cotidal Lines (solid) and Corange Lines (dashed) 9 Earth’s Rotary Tides Depicted with Cotidal Lines Earth’s Corange Lines Note that coastal 10 8 regions 6 generally 4 exhibit largest tidal ranges 2 Dynamic Theory of Tides Tide waves are forced shallow water waves that are subject to Coriolis force and constrained by the geometry of ocean basins • Explains why tidal bulge precedes the moon’s orbit • Explains the rotary motion of tides • Explains why coastal regions experience larger tides than open ocean regions 10 A Couple of Other Interesting Features of Tides… Tidal Forcing of Shallow Embayments can Enhance Tidal Ranges For certain coastal geometries tidal forces can be in Resonance with the tide wave and this can lead to exaggerated tidal variation 11 Summary Points • Equilibrium Theory Explains: • diurnal, semidiurnal and mixed semidiurnal tides • spring neap tide series • Dynamic Theory Explains: • rotary motion of tides • why coastal regions experience larger tides than open ocean regions • Tidal Ranges can be exaggerated by: • Forcing ocean water into a narrow embayment • Tidal forcing that is in resonance with the tide wave 12.
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