Tides

1
Tides
Lecture 16
OEAS-306
March 24, 2009

• Outline
• Gravitational Forces on the Ocean
• Equilibrium Theory of Tides and Astronomical
  Forcing
• Semidiurnal, Mixed and Diurnal Tides
• Tidal Propagation in Basins
• Reflection, convergence and resonance
• Tidal bores

2
Tides are the result of the imbalances in
gravitational forces acting on the earth.
Gravitational Force between two objects (F)
G 6.610-11 Newton m2/kg2 M1 mass object
1 M2 mass object 2 r distance between them
Considering only the moon for now
Earth
Common axis of revolution
Moon
r
3
The tide generating force (T) is equal to
difference between the gravitational force at the
center of mass of the earth and the gravitational
force at other points
Gravitational force between earth and moon
r
x
x
At center of the earth
a
At side of the earth facing moon
So tide generating force is inversely
proportional to the cube of the distance between
earth and moon.
What about the Sun?
Sun is 27 million times more massive than moon,
but 390 times farther away.
Moon over twice as important for creating tides.
4
A planets orbit is in balance between gravity
and inertia
(a) If the planet is not moving, gravity will
pull it into the sun. (b) If the planet is
moving, the inertia of the planet will keep it
moving in a straight line. (c) In a stable orbit,
gravity and inertia together cause the planet to
travel in a fixed path around the sun.
5
Rotation of the Earth-moon system creates tidal
bulges
The moon does not rotate around the center of
Earth. Earth and moon together rotate around a
common center of mass about 1,650 kilometers
(1,023 miles) beneath Earths surface.
The moons gravity attracts the ocean toward it.
The motion of Earth around the center of mass of
the Earth moon system throws up a bulge on the
side of Earth opposite the moon. The combination
of the two effects creates two tidal bulges.
6
Equilibrium Theory of Tides
Developed based on the assumption that the Earth
is completely covered in water.
The action of gravity and inertia on particles at
five different locations on Earth. At points (1)
and (2), the gravitations attraction of the moon
slightly exceeds the outward-moving tendency of
inertia the imbalance of forces causes water to
move along Earths surface, converging at a point
toward the moon. At points (3) and (4), inertia
exceeds gravitational force, so water moves along
Earths surface to converge at a point opposite
the moon. Forces are balanced only at the center
of Earth (point CE).
7
Rotation of the Earth Relative to the
Gravitational Bulge causes the Tides
(a) How Earths rotation beneath the tidal bulges
produces high and low tides. Notice that the
tidal cycle is 24 hrs 50 minutes long because the
moon rises 50 minutes later each day. (b) A graph
of the tides at the island in (a).
8
Lunar Day is 24 hours 50 minutes
This gives two high and two low tides, each 12.42
hours apart. So tidal period is 12.42 hours (the
tide is a very long wave).
A lunar day is longer than a solar day. A lunar
day is the time that elapses between the time the
moon is highest in the sky and the next time it
is highest in the sky. In a 24-hour solar day,
the moon moves eastward about 12.2. Earth must
rotate another 12.2 - 50 minutes to again
place the moon at the highest position overhead.
A lunar day is therefore 24 hours 50 minutes
long. Because Earth must turn an additional 50
minutes for the same tidal alignment, lunar tides
usually arrive 50 minutes later each day.
9
Both Sun and Moon Create Tides
Remember, moons tide generating force is 2.2
times bigger than suns.
Relative positions of the sun, moon, and Earth
during spring and neap tides. (a) At the new and
full moons, the solar and lunar tides reinforce
each other, making spring tides, the highest high
and lowest low tides. (b) At the first-and
third-quarter moons, the sun, Earth, and moon
form a right angle, creating neap tides, the
lowest high and the highest low tides.
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Just like surface waves, tidal waves can
constructively and destructively interfere with
each other.
Wave period due to moon 12.42 hours. This
called the M2 tide.
Wave period due to sun 12.00 hours. This
called the S2 tide.
Tides that occur twice a day (2 highs and 2 lows)
are called semi-diurnal tides.
Spring Tides
Neap Tides
Neap Tides
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Earths axis is tilted 23.5 degrees relative to
its rotation about the Sun
12
Moons orbit around Earth is Nearly aligned (5)
with the Ecliptic (Path of Sun).
23.5 North of Equator
moon
moon
23.5 South of Equator
Moon orbits Earth every 27.3 days
declination
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Declination of Earths Axis Leads to Diurnal and
Mixed Tides
C
B
A
13.6 days
Observer at point A sees 2 high tides of equal
size (semidiurnal). Observer at point B sees 2
high tides of different size (mixed
tides). Observer at point C see only 1 high tide
(diurnal tides).
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Mixed Tides have 2 unequal highs and lows during
the day.
Small/No Diurnal Inequality
Big Diurnal Inequality
Diurnal Inequality When two tides have
different amplitudes.
15
Moons rotation about Earth is an ellipse.
Apogee
Perigee
Moons orbit around Earth is elliptical not
circular so the size of the bulge will vary with
the moons distance from earth. This has period
of 27.55 days. So there are also small variations
of the tides over monthly time scales.
16
Earths rotation about sun is also an ellipse.
January 3
July 7
aphelion
Perihelion
Perihelion to aphelion occurs in one anomalistic
year (365.2596 days).
Tide generating forces are greater in the
northern hemisphere winter.
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Equilibrium Theory Assumes That Earth is
Completely Covered In Water.
This theory provides a lot of insight into what
creates the tides.
But, the Earth is not totally covered in water,
which makes tides a little more complicated.
Tides are really long waves, so they are
considered shallow water waves (L gtgt 20d)
Wave period is so long (12 hours), that wave
motion is influenced by the Earths rotation
(Coriolis force is important to tides.)
For very long waves like tides, orbital
velocities become flat.
18
As a result of the Earths rotation, tides
propagate around ocean basins.
19
Tidal Bulge moves like a wave, that is turned by
rotation of the Earth. Wave turns right in the
Northern Hemisphere and left in the Southern
Hemisphere.
When it encounters the coastline, it becomes a
coastally-trapped Kelvin Wave.
20
Tides Propagate Around Ocean Basins with the
Coastline on their Right in the Northern
Hemisphere and on their Left in the Southern
Hemisphere.
21
Along open coast lines, tides are usually
progressive waves.
Maximum velocity magnitude occurs at high and low
water.
In some embayments or estuaries, the tide is a
standing wave.
Velocity is zero is zero at high and low water.
22
There are several different mechanisms that can
cause a tide to be a standing wave.
In an enclosed basin, the simplest explanation is
the super-position of an incident and reflected
wave
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Standing Waves can Resonate.
resonance
In physics, resonance is the tendency of a system
to oscillate at maximum amplitude at certain
frequencies. At these frequencies, even small
periodic driving forces can produce large
amplitude vibrations, because the system stores
vibrational energy.
Tides can produce resonance, when the basin
length is ¼ the tidal wave length
Bay of Fundy has 15 meter tides (over 50 feet)
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LOW TIDE
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HIGH TIDE
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Tides can also increase due to convergence
Remember the continuity equation
Delaware Bay
Converging channel can increase the water
elevation and tides get bigger.
27
When the tidal range is big and convergence is
rapid, a tidal bore can develop.
tidal bore surfing
28

• Summary
• Tides are huge shallow-water waves--the largest
  waves in the ocean. Tides are caused by a
  combination of the gravitational force of the
  moon and sun and the motion of Earth.
• The moon's influence on tides is about twice that
  of the sun's.
• Gravity and inertia cause the ocean surface to
  bulge. Tides occur as Earth rotates beneath the
  bulges.
• Tides that occur twice a day (2 highs and 2 lows)
  are called semi-diurnal tides.
• The tilting of the Earths axis causes the bulge
  of water to be distorted, leading to mixed and
  diurnal (once a day) tides.
• The equilibrium theory of tides deals primarily
  with the position and attraction of the Earth,
  moon, and sun. It assumes that the ocean conforms
  instantly to the forces that affect the position
  of its surface, and only approximately predicts
  the behavior of the tides.
• In reality, tides propagate as Kelvin waves,
  rotating around the ocean basins with the coast
  to the right in the Northern Hemisphere and to
  the left in the Southern Hemisphere because of
  the Coriolis force.
• Tides on open coastlines tend to be progressive
  waves (high tide max velocity), but tides in
  embayments can be standing waves (high tide
  zero velocity).
• Standing waves can resonate leading to large
  amplification of tidal range.
• Channel convergence can also contribute to
  increasing tides. In the extreme case, this can
  lead to the formation of a tidal bore.