The Air-Sea Momentum Exchange
R.W. Stewart; 1973
Dahai Jeong - AMP
Outline
• Background
•
•
•
•
Importance of the air-sea momentum transfer
Magnitude : drag coefficient
Mechanism : By pressure fluctuations
Conclusion
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
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
The magnitude of momentum transfer
Brocks and Krugermeyer (1970)
(By dimensional analysis, τ = CDρu2 )
The drag coefficient, defined as
CD10=τ/ρU210
where, τ = the stress, or rate of momentum transfer
ρ = the air density
U10 = the mean wind velocity at 10-m height
Factors Effecting Drag
Coefficient :
• wind speed,
• stability of the air
column,
• wind duration and
fetch
• other parameters
Charnock relation
•
The drag coefficient for the ocean surface is found to increase with wind speed.
CD ~ 1.1X10-3
u< 6 m s-1
103 CD = 0.61+0.063u
6 m s-1< u < 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 z 0 and
friction velocity u*, which can be obtained from τ, ρ, and g.
u*2 = τ/ρ
z0 = u*2 /ga where, a is constant
The drag coefficient is then given by
CD = [k/ln(ρgz/aτ)]2
In the neutral stability case, usual turbulent boundary-layer analysis then yields a
logarithmic profile
U(z) ~ u*lnz/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.
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.
v
u
x
y
x y
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 A’B’.
Long wave VS. Short wave
• Dobson’s (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.
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
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
Make the assumption that a
wave field represented by a
single sinusoid induces:
• a sinusoidal pressure fluctuation
•
p
1
U
2
u
2
const
gz
stream function
u
y
, v
x
p z
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.
(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 Reynold’s shear
stress transporting momentum
downward exists.
M-Type Theory
Studies the ways that
displacement and
pressure fluctuations
get out of phase
– Kelvin Helmholtz
Theory
– Effect of turbulent
stress
– Miles theory