Transcript PowerPoint Presentation - CHAOS Verhulst’ population model

Water Waves
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Herein we begin with a general review of waves. Recognize that in terms of their oceanic spatial and
temporal scales, all waves are represented on a portion of this graph. It is not an ordinary linear x-y plot.
It is three-dimensional in length [L], time [P], and has relative energy expressed topographically in the
vertical direction. The horizontal plane is logarithmic [base 10] in both directions and has the virtue of
covering an enormous range of distance, in centimeters, and range of time, in seconds. Note that the
longest distance one can measure, point-to-point, is bounded by the size of the Earth. The time scale runs
from less than a minute to thousands of years. Astronomical tides with lengths spanning ocean basin
widths are represented by the high energy ridges at 12 and 24 hours. Adjacent to the tidal terms is the
ocean current realm and the annual variations associated with seasonal climatological differences.
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A wave definition general enough for us is: a disturbance that propagates energy as it
travels along an interface or within a medium.
Oceanographers define, analyze, and describe ocean waves, processes, and their
effects by using specific spatial and temporal characteristics of an ideal mathematical
entity; a simple harmonic wave form. These are the characteristics:
• amplitude : the vertical distance between the highest [or lowest] part of a wave
and an horizontal reference level [we use the dashed, horizontal line above ]
• height : the vertical distance between the apex of the crest and the lowest point
of the trough [H]
• wavelength : the horizontal distance between two equivalent dynamic points on
consecutive waves [L]
• steepness : H / L : the ratio of wave height to wavelength
• period : the time it takes for a wavelength-worth to pass a fixed point [P]
• speed [celerity] : L / P : the ratio of wavelength to period
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There are a number of ways to classify waves, here are a few : and an example for
each
• sources of energy that cause disturbances : atmosphere
• forces that dispel and eventually eliminate disturbances : gravity
• the manner in which energy is transmitted : progressive
Progressive waves are the most common and they can be categorized further. The ocean
developes wind waves that are superimposed on an existing wave state when the wind begins to
blow over the ocean. The effect of wind can be estimated in terms of its speed, duration and fetch.
Fetch needs definition : it is the sea surface area directly under the influence of the wind.
The named categories of wind waves are capillary, chop and swell. Capillary waves have periods
of less than a second. Chop have periods of one to ten seconds and are within the fetch. Swell
have periods greater than ten seconds and develop outside the fetch.
Suppose the wind begins to gust and blow over an ocean initially at rest, a state practically
impossible to find. At first, there appear what we normal folk call cat’s paws, that is, capillary
waves, that can be tiny and short-lived. Unless the wind persists, capillary waves die off as the
ocean surface is restored flat by the water’s stickiness, surface tension. The cat’s paws, small
bumps on the ocean surface, cause the wind to move unevenly over the surface and push against
the surface unevenly. So there are fetch areas that have a little higher or a little lower generating
force associated with them and this tends to amplify a local wave’s height and length, that is, there
is uneven but increasing wave growth on the ocean’s surface. As growth progresses, the restoring force that tends to smooth out the waves gradually shifts in influence from surface tension
to gravity and the wave nomenclature by which they are known shifts from capillary waves to
gravity waves, that is, the chop and swell. And then the wind wave’s energy continues to grow by
further transfer of momentum from wind to water. At any given moment, there are many different
waves with different wavelengths, periods, amplitudes and energies. We call this result a wave
spectrum.
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There is more in the way of thinking about waves :
• ocean waves have energy of two types : kinetic due to movement and potential
due to the wave’s surface position in the earth’s gravitational field relative to a flat
sea surface [ the dashed line introduced in slide 3 ]
• while ocean waves travel the fetch, over relatively deep water, their passage
does not leave the water permanently displaced; what shape are the water orbits?
• during a wave’s passage, the water involved in its movement describes an
orbital path that is not completely closed in shallow depths, kind of like in loops,
each of which moves a small amount in the direction of the wave disturbance’s
propagation
• the above mass transport is inconsequential in very deep water but becomes
significant in extent as waves progress into shallower water; what shape are the
orbits then?
So what is “deep” water as far as waves are concerned ? That depends on the ratio of
the water depth to the wave’s wavelength :
• deep water waves : waves for which the ratio is greater than half; short wavelengths
• shallow water waves : ratio is less than 5 one-hundreths; long wavelengths
Remember : not all shallow water waves occur near the shore, there are very long
wavelength entities on the ocean : tsunami for instance, and tides as well.
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Resonance …….
When there Is a match between the an imposed force
and the natural modes of oscillation of an object the
result can produce a response that far exceeds
ordinary expectations. As an example, a television
commercial thirty-or-so years ago asked “is it actually
Ella or Ella on memorex tape ?”. A crystal wine glass
was subjected to Ella Fitzgerald holding a single
note; she had perfect pitch and an incredible diaphram,
the note being a natural mode of the glass. In the
response to her sustained note and power, the wine glass walls began to vibrate and, since the situation was a
resonance one, the vibrations continued to grow in amplitude until the glass shattered. A number of singers have done
this including Mythbusters’ Jamie Vendera.
Galloping Gertie does the twist
Bridge builders met resonance head-on in Dec 1940 when the
first Tacoma Narrows suspension bridge tore itself apart in
response to a sustained wind system. The amplitude of the
torsion that twisted the road bed increased to a point that the
cables failed and the road bed fell into the Narrows. The
second bridge was built after wind tunnel tests suggested
wider roads and an air space that ran the length of the bridge
between the lanes. These structural changes altered the
bridge’s natural modes of oscillation and reduced the potential
for resonance.
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We can examine a geophysical resonance phenomena that has been
repeated time and time again in Lake
Michigan at Chicago. Continental air
masses that pass from Canada over
Illinois can take the form of squall
lines, rapidly moving atmospheric
pressure jumps that can sweep out
over the lake from west to east. When
the speed of such a system over the
lake is above a depth contour that
yields an equivalent shallow water
wave speed [square root of the
product of the acceleration of gravity
and the depth], resonance is
achieved. A water wave is generated
that continues to grow in height as
long as the pressure jump is traveling
over the depth contour. Pressure
jump and wave are “locked” together
while momentum is transferred to the
lake; a resonance situation. When the
squall and wave reach the opposite
eastern shore at Michigan City, the
squall continues east and the wave is
reflected back toward Chicago. The
water level record taken at a breakwater there shows a four-foot
excursion
crest-to-trough
that
repeatedly traversed the lake, westeast and east-west.
The two-foot
amplitude following an initial drawdown has taken the unwary walking
the lake bottom for offshore swims
several times in the last few decades
despite warnings.
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in the Arabian Sea …. 90,000 tons of diplomacy
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Right whale and calf … for us to understand and conserve
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