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

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The air right at the surface must follow the water since there is no slip, and ... Kelvin Helmholtz Theory. Effect of turbulent stress. Miles theory ... – PowerPoint PPT presentation

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


1
The Air-Sea Momentum ExchangeR.W. Stewart
1973Dahai Jeong - AMP
2
Outline
  • Background
  • Importance of the air-sea momentum transfer
  • Magnitude drag coefficient
  • Mechanism By pressure fluctuations
  • Conclusion

3
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

4
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

5
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

6
  • Factors Effecting Drag Coefficient
  • wind speed,
  • stability of the air column,
  • wind duration and fetch
  • other parameters

7
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.

8
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.

9
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.

10
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

11
  • 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

12
  • 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.

13
  • (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.

14
M-Type Theory
  • Studies the ways that displacement and pressure
    fluctuations get out of phase
  • Kelvin Helmholtz Theory
  • Effect of turbulent stress
  • Miles theory
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