Transcript AIM: Introduce you to scientific study of the world's

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 has 12 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 CLOSED BASIN
Antinode
Node
IN A BASIN WITH ONE ARM
OPEN TO THE SEA
Antinode Maximum
Maximum lateral movement of water,
and minimum vertical movement
Length of container = l
half the wavelength of the standing wave =
L/2
Antinode
W avelength = L
vertical
movement
and
minimum
lateral
movement
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
•Antinodal lines 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
IN THE NORTHERN HEMISPHERE:
FLOOD TIDE
CO-TIDAL LINES &
AMPHIDROMIC POINT
EBB TIDE
(3/12)T
(4/12)T
(5/12)T
(1/12)T
(6/12)T
t=0
(7/12)T
(11/12)T
(8/12)T
(10/12)T
(9/12)T
t=0
cross-section
t = (6/12)T
cross-section
AMPHIDROMIC
POINT
Co-range lines are concentric curves
about the amphidromic point
(2/12)T
MOTION OF CREST
• Amphidromic point (node)
occurs near center of the basin
CO-TIDAL LINES &
• Co-tidal lines connect points
AMPHIDROMIC
that have high tides at the
POINT
same time (denoted by hours
(3/12)T
(2/12)T
(4/12)T
after the moon crossed the
Greenwich meridian)
(5/12)T
(1/12)T
• Co-range lines connect points (6/12)T
t = 0
that have equal tidal ranges
(7/12)T
(11/12)T
• Co-tidal lines are spokes; co(8/12)T
range lines are closed loops
(10/12)T
about node
(9/12)T
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