Internal Wave Lecture Series

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Internal WaveLecture Series

• Lecture 1
• Interfacial Waves
• March 20, 2008.
• Matthew Murphy
• www.physics.mun.ca/murphy

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Interfacial Waves

• Analogous to surface waves
• Owe their existence to gravity and density
  stratification
• Also known as internal gravity waves
• Occur in stratified fluids
• Oceans
• Atmosphere
• Estuaries and rivers
• Lakes
• Gravitational clarifiers
• Your car or bathroom
• Difficult to observe directly
• But, may have a surface signature
• Currents generated by the wave interact with
  capillary waves, altering surface roughness
• Can be detected by satellite, shipboard radar, etc

From http//envisat.esa.int/handbooks/asar/CNTR1-
1-6.htm
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Interfacial Waves
Notice vectors decrease as you move away from
the interface (i.e. the origin)
The sine curve represents the physical
displacement of the interface
? is the displacement from equilibrium whereas a
is the amplitude, i.e. the maximum displacement
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Interfacial Waves

• Consider two semi-infinite fluids
• Inviscid
• Incompressible
• Irrotational (wx - uz 0)
• Small amplitude waves
• Semi-infinite having one defined edge but going
  to infinity in other directions (Recall heat
  transfer, i.e. conduction in a semi-infinite
  plate, cylinder, etc.)
• We will only consider solutions with an
  exponential decay term so that the velocity goes
  to zero as z goes to /- infinity
• This leaves us needing to solve the Laplace
  equation for the velocity potential in both
  layers subject to continuous P and w

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Interfacial Waves

• Laplace Equation
• 2 M . M ?2M/?x2 ?2M/?y2 ?2M/?z2
• Where M is some variable
• ?2?1/?x2 ?2?1 /?z2 0
• ?2?2/?x2 ?2?2 /?z2 0
• u ? (for irrotational flow)
• With boundary conditions
• ?1 0 z 8 (I1)
• ?2 0 z -8 (I2)
• ??1/?z ??2/?z ??/?z z0 Dynamic boundary
  condition where the vertical velocity on both
  sides of the interface is the same (I3)

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Interfacial Waves

• For pressure
• Du/Dt -1/? P gk
• ?u/?t (u. )(u) -1/? P gk
• ?/?t ? ( ?. ) ? -1/? P gk
• ??/?t u2/2 P/? gz 0

The Bernouli Function, which is constant along
streamlines.
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Interfacial Waves

• Clarifying the math behind the Bernoulli
  equation
• We assumed the only body force acting is gravity,
  thus
• body force -del(gz)
• Which allows us to re-write the vector term gk as
  
• -del(gz)
• del solid white triangles ?/?x ?/?y ?/?z
  ?/?xj
• The nonlinear term of the material derivative can
  be re-written in terms of vorticity, or
• uj(?ui/?xj)
• uj(?ui/?xj - ?uj/?xi) uj(?uj/?xi)
• But we assumed irrotational fluids, leaving us
  with
• uj(?ui/?xj) uj(?uj/?xi)
• ?/?xi(1/2 ujuj) del(u2/2)
• In the end, this leaves all terms with a del
  operator. We can then factor this out, leaving us
  with the form shown.

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Interfacial Waves

• For small amplitude fluctuations at the interface
• ??/?t u2/2 P/? gz 0 ??/?t P/? z0 g?
• Note that P1 must equal P2 at the interface since
  it represents a streamline common to both fluids
• This implies that
• ?1u12/2 ?2u22/2
• The dynamic boundary condition then becomes
• ?1??1/?t ?1g? ?2??2/?t ?2g? z0 (I4)

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Interfacial Waves

• We look for a solution in the form
• ? aei(kx-?t)
• Where k is the wavenumber and ? is the angular
  frequency.
• (I1) and (I2) (Recall velocity goes to zero away
  from the interface) demand solutions in the form
• ?1 Ae-kzei(kx-?t)
• ?2 Bekzei(kx-?t)
• ?1 is not valid for z lt 0 and ?2 is not valid for
  z gt 0. This will be realised through complex
  forms for A.

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Interfacial Waves

• Check to verify forms of ? satisfy Laplace
  equation. Look at fluid 1
• ?2?/?x2 ?2? /?z2 0
• ??/?z -Ake-kzei(kx-?t)
• ?2? /?z2 Ak2e-kzei(kx-?t)
• ??/?x Ae-kzikei(kx-?t)
• ?2?/?x2 -Ak2e-kzei(kx-?t)
• ?2?/?x2 ?2? /?z2
• -Ak2e-kzei(kx-?t) Ak2e-kzei(kx-?t)
• 0
• The same exercise for fluid 2 yields similar
  results, except the z derivatives do not change
  sign theyre always positive.

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Interfacial Waves

• Substituting into the kinematic boundary
  condition (I3) (Recall ??1/?z ??2/?z ??/?z
  evaluated at z 0)
• -kAe-kzei(kx-?t) kBekzei(kx-?t) -i?aei(kx-?t)
• A -B i?a/k
• We can now find the dispersion relation from the
  dynamic boundary condition (Recall ?1??1/?t
  ?1g? ?2??2/?t ?2g? ,evaluate at z0)

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Interfacial Waves

• ?1i?ak-1e-kz(-i?)ei(kx-?t) ?1gaei(kx-?t)
• ?2(-i?ak-1)ekz(-i?)ei(kx-?t) ?2gaei(kx-?t)
• Evaluate at z 0
• ?1i?ak-1(-i?)ei(kx-?t) ?1gaei(kx-?t)
• ?2(-i?ak-1)(-i?)ei(kx-?t) ?2gaei(kx-?t)
• Simplifying yields
• ?2?1k-1 ?1g -?2?2k-1 ?2g

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Interfacial Waves

• From the previous
• ?2 (?2 - ?1)(?1 ?2)-1gk
• ? (?2 - ?1)(?1 ?2)-11/2(gk)1/2
• If we define
• e (?2 - ?1)(?1 ?2)-11/2
• Then
• This is the dispersion relation, and is an
  important result!
• Waves generated with the same frequency but
  different wavelengths will travel at different
  velocities

? e(gk)1/2
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Interfacial Waves

• Kinetic Energy
• As with surface waves, time average the depth
  integration of ½ ?(u2 w2) as with surface waves
  to find.
• Ek ¼ (?2 - ?1)ga2
• Potential Energy
• Consider the work done in deforming the interface
• In case A, fluid 2 is lifted up to displace fluid
  1
• In case B, fluid 1 has been pushed down to
  displace fluid 2
• Integrate work over half the wavelength and
  double it because of symmetry

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Interfacial Waves
½?
½?

• Ep 2?-1 ?0 ½ ?2 g ?2 dx - 2?-1 ?0 ½ ?1 g ?2
  dx
• Ep g (?2 - ?1) ?-1 ?0 ?2 dx
• Ep ¼ (?2 - ?1) ga2
• E Ep Ek
• E ½ (?2 - ?1) ga2
• Compare this result to that for surface waves
• E ½ ?ga2
• Esurface / Einternal ? / (?1 - ?2)
• For any given energy, internal waves will have a
  much larger amplitude.

½?
E ½(?2 - ?1)ga2
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Interfacial Waves

• Consider velocity
• u1 ??1/?x -a?e-kzei(kx-?t)
• u2 ??2/?x a?ekzei(kx-?t)
• Horizontal velocity flips sign when crossing the
  interface.
• Recall the Bernoulli function where we found
• ?1u12/2 ?2u22/2
• Although the direction can differ, at the
  interface the velocity of fluid 1 must match that
  of fluid 2
• This constitutes a vortex sheet.
• Vortex sheet A surface across which the
  tangential velocity is discontinuous (Kundu)
• Extrapolating from this, any fluid where density
  continuously varies with depth must be
  rotational, and therefore does not satisfy the
  Laplace equation.

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Interfacial Waves

• Things to Note
• If the density difference between the fluids is
  small, then e2
• (Recall e (?2 - ?1)(?1 ?2)-1 1/2 )
• is small.
• For example, a 10oC change in temperature causes
  a drop in surface layer density of 0.3. This
  means that, although interfacial waves travel
  like deep water surface waves with ? proportional
  to (gk)1/2, they have a much lower frequency and
  therefore a slower phase speed.

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Interfacial Waves

• Things to note contd
• As with surface waves, there is an even split
  between kinetic and potential energy.
• These equations neglect diffusion and viscosity
• Diffusion of heat and salt between layers blurs
  the interface by establishing an intermediate
  layer
• Viscosity exacerbates this situation
• This means that the interface is actually a
  continuously stratified fluid, which influences
  wave dynamics
• The theory of internal waves is incomplete and
  needs further refinement
• These waves are linear. In reality, a large
  proportion of interfacial waves are non-linear,
  solitary-like waves

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Interfacial Waves

• Measuring interfacial waves can be tricky,
  because of their internal nature. However, they
  effect water quality and properties, so they are
  detectable
• Echosounder
• Doppler current profiler
• Shipboard X-band radar with high speed signal
  processing
• Satellite
• Camera/visually
• Chain thermistors
• Depending upon the situation, sometimes even pH
  and dissolved oxygen sensors

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Interfacial Waves

• Why do we care about interfacial waves, and more
  generically internal waves?
• Physical Oceanography
• These waves transfer energy around the oceans,
  sometimes across large distances
• May contribute significantly to boundary mixing
  the observed mixing in the ocean is significantly
  less then the predicted value. However, no one
  has been able to find a problem with the
  predictions. To reconcile predictions with
  observations, it has been proposed that mixing is
  not uniform in the ocean with hot spots along
  boundaries. The average mixing will hopefully
  match predicted values (See the work of Munk for
  more details).
• They are a mesoscale phenomena that are not
  completely understood. For instance, no models
  exist to predict the amount of mixing associated
  with shoaling internal waves. Another gap in our
  understanding is what happens when these strike a
  boundary at an angle how much mixing and
  reflection occur?
• They help explain some odd phenomena

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Interfacial Waves

• Dead Water
• First observed in Norwegian fjords
• Ships entering fjords (and some estuaries)
  experience unusually high drag.
• Prior to fluid dynamics, this was attributed to
  dead water
• Bjerknes came up with the internal wave
  explanation
• Motion of the ship would generate internal waves
  at the interface.

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Interfacial Waves

• Internal waves transport mass and momentum
• When they impact something, it feels it.
• Ocean engineering interests
• Increase stress to submarines, offshore platforms
  and ships
• May play a role in submarine detection
• Increased turbulence around submarine by breaking
  internal wave
• Loading on offshore wind turbines (also of
  interest to environmental engineers!)

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Interfacial Waves

• Environmental engineering interests
• Selective withdrawal
• Role of internal waves in mixing stratified
  systems
• Ex. Heat and mass transfer between layers
• Sediment resuspension
• Contaminants may be reintroduced into the water
  column
• Transport of bottom water to the surface
• Ex. Bolus formation during shoaling
• Bottom water tends to contain more nutrients
  implications for ecosystems
• Suspended and dissolved contaminants in the
  bottom layer may be reintroduced at boundaries
• Increased vertical mixing

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Interfacial Waves

• Ecology/Biology interests
• Transportation of nutrients, increasing
  biological productivity above what would
  otherwise be seen
• Transportation of larvae
• Some data indicates that larvae can be trapped in
  the surface signature of internal waves and
  transported on/off shore
• Shoaling waves can create boluses which travel
  upslope, suspending and transporting benthic
  organisms (ex. Florida reef).
• Tidal bores generated from shoaling internal
  waves play a significant role in near shore
  population structure. In addition to transporting
  and concentrating drifting larvae and other
  free-floating organisms, bores transport cold,
  nutrient-rich water into shallow zones, promoting
  biological activity. These waves are also thought
  to transport and concentrate phytoplankton in
  coastal regions, providing an additional food
  source for many species.
• As with offshore structures, internal waves are
  thought to add stress to rooted organisms, such
  as corals and seaweeds.

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Surface manifestation of internal waves in the
St. Lawrence Estuary
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http//ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-
Planetary-Sciences/12-820Spring-2007/LectureNotes/
detail/lec21.htm