The Air-Sea Momentum Exchange R.W. Stewart; 1973 Dahai Jeong - AMP

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The Air-Sea Momentum ExchangeR.W. Stewart
1973Dahai Jeong - AMP
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Outline

• Background
• Importance of the air-sea momentum transfer
• Magnitude drag coefficient
• Mechanism By pressure fluctuations
• Conclusion

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Background

• Real vs. Ideal fluids
• There can be no slip at a boundary in a real
  fluid as contrasted with the possibility of slip
  at a boundary of an idea fluid

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The nature of the mechanism for the transport of
momentum between the atmosphere and the surface
of water

• Why is this important?
• Parameterization of this process is important to
  understand the circulation of the atmosphere and
  the ocean.
• The nature of the process is intimately connected
  with wave generation

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The magnitude of momentum transfer

• Brocks and Krugermeyer (1970)
• (By dimensional analysis, t CD?u2 )
• The drag coefficient, defined as
• CD10t/?U210
• where, t the stress, or rate of momentum
  transfer ? the air density
• U10 the mean wind velocity at 10-m height

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• Factors Effecting Drag Coefficient
• wind speed,
• stability of the air column,
• wind duration and fetch
• other parameters

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Charnock relation

• The drag coefficient for the ocean surface is
  found to increase with wind speed.
• CD 1.1X10-3 ult 6 m s-1
• 103 CD 0.610.063u 6 m s-1lt u lt 22 m
  s-1 Smith(1980)
• Alternatively, the data can be fitted by a
  relationship obtained on dimensional grounds by
  Charnock. This creates a quantity called the
  roughness length z0 and friction velocity u,
  which can be obtained from t, ?, and g.
• u2 t/?
• z0 u2 /ga where, a is constant
• The drag coefficient is then given by
• CD k/ln(?gz/at)2
• In the neutral stability case, usual turbulent
  boundary-layer analysis then yields a logarithmic
  profile
• U(z) ulnz/z0 where z is the height above
  the surface
• With the wind-speed variation with height taken
  to be logarithmic, we get a relationship between
  drag coefficient and wind speed, indicating a
  significant increase in drag coefficient with
  wind speed.
• On the whole, most observations tend to indicate
  that there is an increase in drag coefficient
  with wind speed, but It is weaker than that
  predicted by charnock relation.
• surface tension seem to act make the drag
  coefficient less dependent on wind speed than
  that predicted in the charnock relation.

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Irrotational? Zero circulation?

• The momentum goes into the water by pressure
  fluctuations and the only kind of motion which
  pressure fluctuations are able to set up in a
  homogeneous fluid are irrotational ones.
• Assuming the deep water to be stationary, the
  motion we seek in the upper water must carry
  horizontal momentum.
• Since the motion is irrotationl, the line
  integral of velocity around the circuit, which is
  the surface integral of the vorticity over the
  enclosed area (Stokes theorem), must vanish.
• Any closed circuit entirely within the water
  like ABCD has zero circulation. But a closed
  circuit, like Á,B,C,D, has a net clockwise
  circulation and there is net momentum to the
  right in the neighborhood of the dashed line
  AB.

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Long wave VS. Short wave

• Dobsons (1971) measurements, interpretation of
  the JONSWAP (1973) observations, and simple
  calculations based on standard wave climate data
  (Stewart, 1961), show that a substantial
  proportion of momentum is transferred into rather
  long waves in the system.
• Non-linear wave interactions generate very short
  waves susceptible to rapid viscous dissipation.
  When a wave loses its energy, it must lose its
  momentum as well.

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Theories of wave generation P-Type (O. M.
Phillips) and M-Type (J. W. Miles)

• P-Type theory (wave generation)
• wave generation in terms of pressure
  fluctuations generated in a turbulent atmosphere
  and advected over the surface by the wind
• However, it cannot provide an important
  proportion of the transfer of momentum from the
  atmosphere to the water.
• Thus, one has to consider the P-type mechanism
  to be real, but not very important except perhaps
  in the very initial stages of the generation of
  waves on a smooth surface.
• M-type theory (wave growth or decay)
• non-linear, involving the interaction of the
  existing wave field with the shear flow in the
  atmosphere above it

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• The water is moving rapidly to the left and
  the wind at upper elevations is moving to the
  right. The air right at the surface must follow
  the water since there is no slip, and therefore
  at very low levels, the air is moving to the
  left. There must be some particular level at
  which the mean motion of the air is stationary.
  Above this level, air moves to the right and
  below it to the left.
• In order to conform with the wave profile, the
  air close to the surface must be subjected to
  vertical pressure gradients, which must fluctuate
  horizontally according to the phase of the wave

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• Make the assumption that a wave field represented
  by a single sinusoid induces
• a sinusoidal pressure fluctuation of p in the air
  
• a sinusoidal vertical displacement of the air
  flow, each with the same wavelength as the
  underlying wave.
• By Assuming there is no shear stress in the
  system and by ignoring hydrostatic effects and
  acceleration due to gravity, we can assume the
  right side of Bernoullie eq. to be constant.

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• (Bernoullis equation is an equation for energy,
  since it is formed form a line integral of a
  force equation. It provide an easy way to relate
  changes in p with changes in u, along a
  streamline. )
• As a result, the phase of the neighboring
  streamline differs from the phase of the original
  streamline. There are several important
  consequences if this.
• The upward flow is slower than the downward
  flow. Thus averaged over a horizon plane covering
  one full wavelength, the product uw is negative.
  That is a Reynolds shear stress transporting
  momentum downward exists.

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M-Type Theory

• Studies the ways that displacement and pressure
  fluctuations get out of phase
• Kelvin Helmholtz Theory
• Effect of turbulent stress
• Miles theory