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

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

Background Real vs. Ideal fluids 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. 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 nature of the process is intimately connected with wave generation

The magnitude of momentum transfer Brocks and Krugermeyer (1970) (By dimensional analysis, τ = C D ρu 2 ) The drag coefficient, defined as C D10 =τ/ρU 2 10 where, τ = the stress, or rate of momentum transfer ρ = the air density U 10 = the mean wind velocity at 10-m height

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

Charnock relation The drag coefficient for the ocean surface is found to increase with wind speed. The drag coefficient for the ocean surface is found to increase with wind speed. C D ~ 1.1X10 -3 u< 6 m s -1 C D ~ 1.1X10 -3 u< 6 m s C D = u 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. 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 z 0 = u * 2 /ga where, a is constant The drag coefficient is then given by C D = [k/ln(ρgz/aτ)] 2 In the neutral stability case, usual turbulent boundary-layer analysis then yields a logarithmic profile In the neutral stability case, usual turbulent boundary-layer analysis then yields a logarithmic profile U(z) ~ u * lnz/z 0 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. 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 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. 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. 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): 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 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): 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. 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 of p in the air 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. 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