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
and Fall of Tides
over an
Approximately 24h
hour period
moon
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
b
R
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
So far we have
shown how we get
semi-dirual Tides
that have a period
of 24h hours and 50
minutes
Looking Down on Top of the Earth
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 ShallowWater 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
in 4km of water is 200m s-1 (400 miles h-1). The
speed that the earth rotates under the moon at the
equator is 463m s -1 (1044 miles h-1).
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
regions
generally
exhibit
largest tidal
ranges
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8
6
4
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
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