Internal waves and internal tides 1

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Internal waves and internal tides 1

• Last time looked at
• Potential energy demand for supplying energy to
  turbulent dissipation
• Examples of turbulence research
• Today
• Internal waves,
• why and how they differ from surface waves
• Two layer internal waves
• Internal waves in a stratified fluid
• Internal tides internal waves of a particular
  frequency

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• Two-layer internal waves
• The simplest internal waves are like those
  executive toys you can buy in the shops. There
  are two immiscible fluids of different densities
  and you can set up an internal wave along the
  interface by tilting the box back and forth.
• In the ocean analogy is a wave much like surface
  gravity waves but which acts on the pycnocline.
• In reality, the physics is identical -- surface
  waves can be viewed as internal waves on the
  interface between two largely immiscible fluids
  (air and water) with a very large density
  contrast.

?
? ? ?
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• Imagine raising a parcel of water across the
  interface between the layers. The restoring force
  that will cause the parcel to return to its
  equilibrium position (actually, overshoot its
  equilibrium position) is equal to gravity times
  the difference in density between the water
  parcel and its surroundings.
• B -g ?' -g?? is called the buoyancy force.
• B gt 0, if ?' (the density of a water parcel
  minus the density of its surroundings) is
  negative, resulting in an upward force.
• In general, the frequency of a wave increases
  with strength of the restoring force. Thus waves
  on the thermocline will have longer periods
  (typically hours) than surface waves on the
  air-sea interface.
• For gravity waves, the wave speed, as well as the
  frequency, is proportional to the buoyant
  restoring force. An internal wave on the
  thermocline propagates at a phase speed,
• c ?(g' H), where g' g ??/?, H is depth of
  thermocline, g' is reduced gravity.
• For surface waves, g' g(?Ocean -?Atm)/ ?Ocean
  ? g c ?(g H)
• Across the ocean thermocline ??/? 1 g'
  0.01g
• The phase speed of an internal wave on the
  thermocline is at least 100 times slower than the
  speed of a wave on the sea surface. Amplitude can
  be tens of metres.

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A first order internal seiche.
Note the node in the centre (where the interface
between the layers does not move up or down), and
that water under and above the node moves only
horizontally, while water at both ends of the
basin moves vertically.
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A second order internal seiche.
Note the two nodes at 1/4 and 3/4 of the basin
length (where the interface between the layers
does not move up or down), and that water under
and above the nodes moves only horizontally,
while water at both ends and in the centre of the
basin moves vertically.
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An internal wave propagating on the interface
between two layers

• By watching a yellow dot you can see how a water
  particle in the middle of the water column moves
  up and down, but does not move horizontally, as
  the wave passes through.
• By selecting a particular magenta dot at the
  bottom of the water column and watching it you
  can see how a water particle moves back and forth
  horizontally as the wave passes. By comparing it
  with the movement of a particular dot above it
  you can see that at any one location, particles
  at the top and bottom of the water column move
  always in opposite direction.
• By watching groups of magenta dots you can see
  that convergences (where water particles cluster
  together) and divergences (where particles are
  spread out) follow the wave, and that
  convergences are always located where the
  respective layer is thickest, while divergences
  are found where the layers are thinnest.

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• As internal waves propagate, the motion induced
  in the water can sometimes be detected at the
  surface. One visualization is due to the
  formation of bands of calm water, referred to as
  "slicks", between regions of normal gravity wave
  motion.
• The slicks are formed when the water motion near
  the surface alternately stretches and bunches up
  naturally occurring surfactants and biological
  material. In regions where the material is
  bunched up the modification to the surface
  tension is enough to damp out the gravity waves,
  thus causing a strip of calm water.

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Internal waves in a stratified fluid BUT - in
reality the ocean is not made up of two layers
but is stratified what does this mean for
internal waves? For continuous stratification
internal waves can propagate vertically as well
as horizontally. When these waves break they
generate vertical mixing ? vertical eddy
diffusivity The restoring force in a
continuously stratified fluid is the
Brunt-Vaisala frequency or buoyancy
frequency More strongly stratified fluids have
higher frequency oscillations.
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• Internal waves in a stratified fluid dynamical
  description
• Internal waves have some counter-intuitive
  properties
• The wave number vector points in the same
  direction as the phase velocity
• However energy (carried by the group velocity)
  propagates at 90o to this direction

Define ? as the angle between x-axis and
direction of group velocity,
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• Energy travels at right angles to the
• direction that wave crests propagate.
• Water parcels move back and forth along
• wave crests and troughs which are also
• perpendicular to the wavenumber vector.
• ? is also the angle that the direction of
• particle oscillation makes with the
• horizontal.

Animation of an internal plane wave The parallel
lines are lines of constant phase. They are blue
when moving up to the left, red when moving down
to the right. The tilted sinusoidal curve shows
the propagating phase. It represents a line of
particles which lay in a straight line prior to
the waves arrival.
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Internal waves in a continuously stratified fluid
the dispersion relation
One consequence of the relation is that internal
waves only exist in a band of frequencies between
f, the inertial frequency and N, the buoyancy
frequency.
1) m0, energy propagates nearly vertically
tan2? inf ? ?? Ni.e., oscillations with the
highest frequency have water parcels that move
nearlyvertically. Rotation is unimportant
because the frequency is high and there is
littlehorizontal motion.2) k0, energy
propagates nearly horizontally tan2? 0 ? ?
? fi.e., oscillations with the lowest frequency
have water parcels that move nearly
horizontally. Rotation becomes important because
the frequency is low and there is littlevertical
motion. INERTIAL OSCILLATIONS
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• CLASS PROBLEM
• At a Latitude of 36o N what if the lowest
  frequency of internal wave allowed? What is the
  period (in hours) of this wave?
• Given a 100m water column with the following
  density stratification
• what is the highest frequency internal wave that
  can exist?
• What is the period (in minutes) of this wave?

0 10 20 30 40 50 60 70 80 90 100
1025.0
1025.5
1026.0
1026.5
1027.0
Depth (m)
1027.5
1028.0
1028.5
1029.0
1029.5
1030.0
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• Internal waves
• So far today
• Two layer internal waves
• Travel much SLOWER than surface gravity waves
  (100 times slower)
• Amplitudes can be much larger than surface
  gravity waves (tens of metres)
• Internal waves in a stratified fluid
• Can propagate at any angle
• Have a frequency ? such that f lt ? lt N
• Surface expressions of internal waves can
  sometimes be seen
• After the break
• Internal tides internal waves of a particular
  frequency

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The Internal Tide.
Stratification can support waves within the
ocean. These internal waves are often set up by
the interaction between flow and topography, an
important example being the interaction between
the barotropic tidal currents and the shelf
edge. In a simple two-layer system
Ebb tide
Flow of ebb tide off shelf drags thermocline down
at the shelf edge.
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3.
2.
1.
A.
The initial perturbation (1) in the thermocline
propagates both onto and away from the shelf as
an internal wave (2). Propagation speed is
approximately
reduced gravity
This wave propagates as a temperature (strictly,
a density) disturbance, so imagine what you would
see in a record of temperature at position A.
temp.
time
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Example temperature time series influenced by the
passage on internal tidal waves. Sample period
1 minute.
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(No Transcript)
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14-hour time series of temperature profiles from
just North of the Rodrigues Ridge in the Indian
Ocean
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Predicted ray paths
Observed displacements
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Like any other shallow water wave moving into
shoaling water, the internal tidal wave can
steepen and break, producing series of smaller
scale internal waves (solitons). Observations
of internal solitons in the central Bay of Biscay
(New Pingree, 1992, DSR, 39, p1521) showed
that internal tide beams, originating from the
shelf break, upon encountering the thermocline
after their first reflection at the ocean bottom,
were found to generate packets of internal
solitons. These solitons quickly faded away after
their generation (within about 60km). So, every
12.5 hours we can get a repeating sequence of
internal tidal waves and associated
solitons. Internal tide wavelength ? 20 - 40 km,
12.5 hr period. Soliton wavelength ? 1 - 3 km, 10
- 30 minute period. Internal tidal wave energy
? solitons ? mixing
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Photograph taken from space shuttle Challenger in
June, 1983. The centre of the image is located
southeast of Hainan Island, China on the
northern continental shelf of SCS. The water
depth varies from 30 m near the coast to 200 m
on the offshore side.
The alternative dark-bright, groups of lines are
images of ocean internal waves. One can see 3
major internal wave packets. The numbers of
solitons visible in a packet vary from 3 to 8.
The average maximum wavelength, (the distance
between the first two solitons, is 0.68 km). The
length of wave crests is as long as 60 km
implying a strong internal wave system. A
shoreward curvature of crest lines indicates
that the waves propagated towards the coast.
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Estimates of M2 tidal energy dissipation from
TOPEX/Poseidon altimeter data Egbert, G.D. and
R.D. Ray (2001) J. Geophys. Res. 106 (C10)
22,475,22502
? Can we observe the rate of energy dissipation
of the internal tidal wave from oceanographic
measurements?