Survey of Oceanography
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28TidesTsunamisandLongWavese.pptxTides, Tsunamis and Long Waves
By: John Van Leer
November 2, 2020
Long waves travel in shallow water depths compared to their wave length. Tides have wave lengths of thousands of kilometers compared to water depths of 4 or 5 km so they are shallow waves. Tsunamis are also shallow water waves.
Surface Gravity Wave Dispersion Diagram The phase speed C of waves in shallow water is the square root of (g x h) where g is the acceleration of gravity = 9.8 meters per second squared, and h is the water depth). If h = 4km then C is about 200m/s. So Tidal and Tsunami waves will feel the bottom and will be refracted everywhere.
The Earth and the moon rotate around their common center of mass which is located within the earth. On the side of the earth farthest from the moon, the centrifugal force dominates while on the side closest to the moon, lunar attraction dominates.
It is the horizontal component of these forces which move the water at tidal frequencies.
In a simple model earth with a uniform oceanic depth great enough, there would be a bulge facing the moon and another bulge facing away from the moon.
The spring/neap cycle during a lunar month, produces two spring tides at the new and full moon plus two neap tides at first and last quarter.
Tide records of a month long clearly show a strong spring/neap tidal variation.
Depending on the geometry of the ocean basins, there can be a Diurnal tide, a Semidiurnal tide or a Mixed tide with both components.
The Principal Tidal Harmonics are determined by the earth/moon and solar forcing. These tidal periods are so close to each other that it takes a very long tide gage record to resolve one from the other.
The Principal Tidal Harmonics components of the tide can be determined from a long record from a tide gage mounted at a point of interest, like the RSMAS dock at the University of Miami. By computation of these components into the future, with the correct phases, one can compute the future tides.
The effects of earth rotation produce a standing wave around the edge of the basin producing a rotary tide within the basin.
Rotary tide of the sort you find on Georges Bank. After 24 hours, you are almost back where you started, after two complete loops.
Oceanic scale surface elevation record for the M2 tide shows several amphidromic points with no elevation change and a rotary tide, like the one east of Newfoundland.
Cotidal Lines are lines of equal phase as seen in Figure a and Corange Lines of equal amplitude are seen in Figure b.
An amphidromic point is seen in the North Atlantic and a progressive tide runs in the South Atlantic.
North Sea has two amphidromic points in the south making for complex rotary tides.
Global tidal ranges show considerable variability. Note: Tidal Range is the vertical distance between high tide and low tide.
This spring mass system illustrates the natural period of a resonant system. Note: Another resonant system is a child on a swing. If you push the child at the resonant period the amplitude of the swinging motion increases.
The seiche (resonant) period of the first mode of oscillation is determined by Miriam’s formula below.
A standing wave in a closed basin has a natural resonant period determined by the shallow water wave speed and the reflection from both ends. (Illustration below shows the first resonant mode.)
This illustration shows the second resonant mode of the same standing wave tank. Note: A music major describing an analogous sound wave would call this the second harmonic.
If the basin has an open end, connected to the sea, then the natural period doubles. Note: Our music major might say that this arrangement resembles an organ pipe with a natural frequency, which determines its tone. The longer the organ pipe, the lower the tone.
The Bay of Fundy has an open connecting it to the open sea which has a resonant period. The small n is the mode number as illustrated above. If the river flowing outward has the same speed as shallow water wave speed as the incoming tide, then a tidal bore results.
A couple of photos of tidal bores.
The power generated by a water mill follows Betz’s Law which increases with the third power of the water speed. The same formula works for a windmill with a suitable change of constants as seen on the left.
A bottom mounted water mill can produce power in a steady current.
By combining a dam and a turbine, one can change the power generation cycle.
Two different power cycles can shape the output to meet local demand. Because there is a solar component to the tide, it may more closely match peak demand.
In Sothern Florida, the strongest currents are in the shallow inlets, where the stars are located. There are strong currents in the Gulf Stream but these are harder to harness, because the Gulf Stream meanders east and west.
To function efficiently in shallow water, the geometry of the turbine needs to look more like an old-fashioned “reel type” lawn mower, with the axis of rotation perpendicular to the tidal current.
A hurricane is a cyclonic storm, which has low pressure at its center, and rotates counter clockwise in the Northern Hemisphere. So on the right side of the storm, looking in the direction of its motion, will have wind stress which is enhanced by the forward motion of the storm. A mound of water stands under the storm drawn up by the low atmospheric pressure at its center. As the mound enters shallower water, it behaves like a shoaling wave, with the wave length shrinking and the amplitude growing. Note: This effect is greater on the right front quadrant of the storm in the N.H.
A hurricane making landfall on the Texas coast. Note: The storm surge is strongest on the right front quadrant because of strong wind stress added to the shoaling wave (bulge).
Tide gages measure storm surge along the US Atlantic coast. Note: The resonances in Baltimore with the whole Chesapeake Bay Seiching, plus Sandy Hook’s record shows inertial period oscillations. Providence has a funnel shaped bay, which amplified the storm surge felt in downtown Providence.
An earthquake at 2am in the Aleutian Islands creates a tsunami wave which arrives in Hawaii at about 6:30am traveling at about 200 meters/sec.
In the local area of Hawaii, this wave amplifies in a funnel shaped bay at Hilo, reaching heights of 35’.
Tsunami comes ashore in Hilo Hawaii. Note: Waves don’t usually break this far up the beach in the middle of a grove of coconut palms.
Tsunami with about 15minute period an about 1ft amplitude is caught in a tide gage record, super imposed on the slower tidal variation. Note: The higher harmonics which arrive after the firs six cycles.
