““Monday is an awful way to spendMonday is an awful way to spend1/7 of your week1/7 of your week…”…”““A clear conscience is usually a signA clear conscience is usually a signof a bad memoryof a bad memory””““I used to have an open mind,I used to have an open mind,but my brains kept falling outbut my brains kept falling out””OCNG 251: OceanographyOCNG 251: OceanographyTuesday,Tuesday, Nov. 18, 2008 Nov. 18, 2008The surface of the Ocean: 2) TidesThe surface of the Ocean: 2) TidesTides:Tides:a.a.Equilibrium tidal theoryEquilibrium tidal theoryb.b.Dynamical tidal theoryDynamical tidal theorya.a. Equilibrium tidal theory Equilibrium tidal theoryEarth covered with a uniform and ideal layer of water (geoid) Earth covered with a uniform and ideal layer of water (geoid) !!simplification of the relationships between the Oceans and the tide-simplification of the relationships between the Oceans and the tide-rising bodies, the Moon and the Sun.rising bodies, the Moon and the Sun.The Surface of the Ocean: 2) TidesThe Surface of the Ocean: 2) TidesThe Earth-Moon SystemThe Earth-Moon SystemThe system orbits the SunThe system orbits the Sunaround a common center of mass:around a common center of mass:the the barycenterbarycenter((!! within the Earth within the Earth’’s mantle, !s mantle, !radius from center)radius from center)a.a. Equilibrium tidal theory Equilibrium tidal theoryThe rotation of the Earth-Moon System creates a motion that is equalThe rotation of the Earth-Moon System creates a motion that is equalat all points within and upon the Earth. Each point will also have theat all points within and upon the Earth. Each point will also have thesame angular velocity and hence a similar same angular velocity and hence a similar centrifugal force (CF)centrifugal force (CF)!!a.a.Equilibrium tidal theoryEquilibrium tidal theoryThe total centrifugal force within the Earth-Moon system exactlyThe total centrifugal force within the Earth-Moon system exactlybalances the forces of gravitational attraction between the twobalances the forces of gravitational attraction between the twobodies so that the system as a whole is in equilibriumbodies so that the system as a whole is in equilibrium!! We should neither lose the Moon We should neither lose the Moon!! nor collide with it! nor collide with it!ThereforeTherefore! Fg=GMmD2= CFWhere:Where:G: Universal gravity cstG: Universal gravity cstM: Mass of the EarthM: Mass of the Earthm: Mass of the Moonm: Mass of the MoonD: Distance between M-ED: Distance between M-ELetLet’’s work it out!s work it out!Tidal Producing Force (TPF)Tidal Producing Force (TPF)The tidal producing force at A is equal to the TPF at B but of opposite sign!The tidal producing force at A is equal to the TPF at B but of opposite sign!! TPFA=GMm2rD3= "TPFBHow does it compare to the EarthHow does it compare to the Earth’’s gravitational force?s gravitational force?! TPFFg=GMm2rD3gM=Gm2rgD3= 10"7The Tidal Force is 10 million times smaller than EarthThe Tidal Force is 10 million times smaller than Earth’’s gravitationals gravitationalpull! There is, therefore, no possible vertical movement of water!pull! There is, therefore, no possible vertical movement of water!! TPFx=GMm(D ± r)2"GMmD2Which is approximately equal toWhich is approximately equal toTidal Producing Force (TPF)Tidal Producing Force (TPF)Gravitational attraction and centrifugal force produce two tidalGravitational attraction and centrifugal force produce two tidal““bulgesbulges”” of water of approximately the same size, positioned on of water of approximately the same size, positioned onopposite sides of the Earth.opposite sides of the Earth.Tidal Producing Force (TPF)Tidal Producing Force (TPF)The effects of tides are due to horizontal components (The effects of tides are due to horizontal components (tractive forcestractive forces))of the tidal producing force: It is the tractive forces that cause theof the tidal producing force: It is the tractive forces that cause thewater to move, because this horizontal component is water to move, because this horizontal component is unopposedunopposed by any by anyother lateral force.other lateral force.Formation of the tidal Formation of the tidal ““bulgebulge”” !! Production of unequal tides! Production of unequal tides!Tidal Producing Force (TPF)Tidal Producing Force (TPF)What about the Sun?What about the Sun?Thus the sun has 46% the tide-generating force of the Moon!Thus the sun has 46% the tide-generating force of the Moon!46.010.5910.27)390(10.27)Distance(66363==!"!"=#DSunMSunMassTPFTide inequalitiesTide inequalities1)1) ““Tidal DayTidal Day””While the Earth rotates upon its axis, the Moon is moving in the sameWhile the Earth rotates upon its axis, the Moon is moving in the samedirection along its orbit about the Earth. After 24 hours, the Earthdirection along its orbit about the Earth. After 24 hours, the Earthpoint that began directly under the Moon is no longer point that began directly under the Moon is no longer underunder the Moon! the Moon!The Earth must turn an additional 50 minutes to bring the startingThe Earth must turn an additional 50 minutes to bring the startingpoint back in line with the Moon.point back in line with the Moon.Therefore, a Therefore, a tidal daytidal day is not 24 hours but 24hrs and 50 min! is not 24 hours but 24hrs and 50 min!(Tides arrive at locations an hour later each day)(Tides arrive at locations an hour later each day)Tide inequalitiesTide inequalities““Spring TidesSpring Tides”” and and ““Neap TidesNeap Tides””The Moon orbits the Earth (29" days) and the Earth-Moon systemThe Moon orbits the Earth (29" days) and the Earth-Moon systemorbits the Sun (365 days).orbits the Sun (365 days).Tide inequalitiesTide inequalities2) 2) ““Spring TidesSpring Tides”” and and ““Neap TidesNeap Tides””. The Moon. The Moonorbits the Earth (29" days) and the Earth-orbits the Earth (29" days) and the Earth-Moon system orbits the Sun (365 days).Moon system orbits the Sun (365 days).Tide inequalitiesTide inequalities3) 3) Additional Tidal variationsAdditional Tidal variations!!The MoonThe Moon’’s declinations declinationTide inequalitiesTide inequalitiesThe MoonThe Moon’’s and Earths and Earth’’s elliptical orbits (REM: TPF s elliptical orbits (REM: TPF !! 1/D 1/D33))!!Perigee vs. Apogee (13% difference) (27.5 days)Perigee vs. Apogee (13%