The Air-Sea Momentum ExchangeThe Air-Sea Momentum ExchangeR.W. Stewart; 1973R.W. Stewart; 1973

Dahai Jeong - AMPDahai Jeong - AMP

OutlineOutline

• BackgroundBackground• Importance of the air-sea momentum Importance of the air-sea momentum

transfertransfer

• Magnitude : drag coefficientMagnitude : drag coefficient

• Mechanism : By pressure fluctuationsMechanism : By pressure fluctuations

• ConclusionConclusion

BackgroundBackground

• Real vs. Ideal fluidsReal vs. Ideal fluids– There can be no slip at a boundary in a There can be no slip at a boundary in a

real fluid as contrasted with the real fluid as contrasted with the possibility of slip at a boundary of an possibility of slip at a boundary of an idea fluid idea fluid

The nature of the mechanism for the The nature of the mechanism for the transport of momentum between the transport of momentum between the

atmosphere and the surface of wateratmosphere and the surface of water

Why is this important?Why is this important?

• ParameterizationParameterization of this process is important of this process is important to understand the circulation of the atmosphere to understand the circulation of the atmosphere and the ocean.and the ocean.

• The nature of the process is intimately The nature of the process is intimately connected with connected with wave generationwave generation

The magnitude of momentum transferThe magnitude of momentum transfer

Brocks and Krugermeyer (1970)Brocks and Krugermeyer (1970)

(By dimensional analysis, τ = C(By dimensional analysis, τ = CDDρuρu22 ) )

The drag coefficient, defined as The drag coefficient, defined as

CCD10D10=τ/ρU=τ/ρU221010

where, τ = the stress, or rate of momentum transfer where, τ = the stress, or rate of momentum transfer ρ = the air densityρ = the air density

UU1010 = the mean wind velocity at 10-m height = the mean wind velocity at 10-m height

Factors Effecting Drag Factors Effecting Drag Coefficient :Coefficient :

• wind speed, wind speed,

• stability of the air stability of the air column, column,

• wind duration and wind duration and fetchfetch

• other parameters other parameters

Charnock relationCharnock 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. CCDD ~ 1.1X10 ~ 1.1X10-3 -3 u< 6 m su< 6 m s-1-1

101033 C CDD = 0.61+0.063u 6 m s = 0.61+0.063u 6 m s-1-1< u < 22 m s< u < 22 m s-1 -1 Smith(1980) Smith(1980)

• Alternatively, the data can be fitted by a relationship obtained on dimensional grounds by ChAlternatively, the data can be fitted by a relationship obtained on dimensional grounds by Charnock. This creates a quantity called the roughness length zarnock. This creates a quantity called the roughness length z00 and friction velocity u and friction velocity u**, which , which can be obtained from τ, ρ, and g. can be obtained from τ, ρ, and g.

uu**2 2 = τ/ρ = τ/ρ

zz00 = u = u**22 /ga where, a is constant /ga where, a is constant

The drag coefficient is then given byThe drag coefficient is then given byCCDD = [k/ln(ρgz/aτ)] = [k/ln(ρgz/aτ)]22

• In the neutral stability case, usual turbulent boundary-layer analysis then yields a logarithmic In the neutral stability case, usual turbulent boundary-layer analysis then yields a logarithmic profileprofile

U(z) ~ uU(z) ~ u**lnz/zlnz/z00 where z is the height above the surface where z is the height above the surface

With the wind-speed variation with height taken to be logarithmic, we get a relationship betwWith 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 een drag coefficient and wind speed, indicating a significant increase in drag coefficient with wind speed.wind speed.

On the whole, most observations tend to indicate that there is an increase in drag coefficienOn 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. t 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 tsurface tension seem to act make the drag coefficient less dependent on wind speed than that predicted in the charnock relation. hat predicted in the charnock relation.

Irrotational? Zero circulation?Irrotational? Zero circulation?

The momentum goes into the water by pressure fluctuatioThe momentum goes into the water by pressure fluctuations and the only kind of motion which pressure fluctuationns and the only kind of motion which pressure fluctuations are able to set up in a homogeneous fluid are irrotations are able to set up in a homogeneous fluid are irrotational ones.al ones.

• Assuming the deep water to be stationary, the motion we Assuming the deep water to be stationary, the motion we seek in the upper water must carry horizontal momentum.seek in the upper water must carry horizontal momentum.

Since the motion is irrotationl, the line integral of velocity Since the motion is irrotationl, the line integral of velocity around the circuit, which is the surface integral of the vortaround the circuit, which is the surface integral of the vorticity over the enclosed area (Stokes theorem), must vanisicity over the enclosed area (Stokes theorem), must vanish.h.

Any closed circuit entirely within the water like ABCD has Any closed circuit entirely within the water like ABCD has zero circulation. But a closed circuit, like Á’,B’,C’,D’, has zero circulation. But a closed circuit, like Á’,B’,C’,D’, has a a net clockwise circulationnet clockwise circulation and there is and there is net momentum to net momentum to the rightthe right in the neighborhood of the dashed line A’B’. in the neighborhood of the dashed line A’B’.

v x

u yxy

Long wave VS. Short waveLong wave VS. Short wave

• Dobson’s (1971) measurements, interpretation of the Dobson’s (1971) measurements, interpretation of the JONSWAP (1973) observations, and simple calculations JONSWAP (1973) observations, and simple calculations based on standard wave climate data (Stewart, 1961), based on standard wave climate data (Stewart, 1961), show that a substantial proportion of momentum is show that a substantial proportion of momentum is transferred into rather long waves in the system.transferred into rather long waves in the system.

• Non-linear wave interactions generate very short waves Non-linear wave interactions generate very short waves susceptible to rapid viscous dissipation. When a wave loses susceptible to rapid viscous dissipation. When a wave loses its energy, it must lose its momentum as well.its energy, it must lose its momentum as well.

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

• P-Type theory (wave generation):P-Type theory (wave generation):

wave generation in terms of pressure fluctuations generated in a turwave generation in terms of pressure fluctuations generated in a turbulent atmosphere and advected over the surface by the windbulent atmosphere and advected over the surface by the windHowever, it cannot provide an important proportion of the transfer of However, it cannot provide an important proportion of the transfer of momentum from the atmosphere to the water.momentum from the atmosphere to the water.Thus, one has to consider the P-type mechanism to be real, but not vThus, one has to consider the P-type mechanism to be real, but not very important except perhaps in the very initial stages of the generatiery important except perhaps in the very initial stages of the generation of waves on a smooth surface.on 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 non-linear, involving the interaction of the existing wave field with the shear flow in the atmosphere above itshear flow in the atmosphere above it

The water is moving rapidly to the left and the wind at upper elevations is moving to 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 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 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 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.moves to the right and below it to the left.

In order to conform with the wave profile, the air In order to conform with the wave profile, the air close to the surface must be subjected to vertical close to the surface must be subjected to vertical pressure gradients, which must fluctuate pressure gradients, which must fluctuate horizontally according to the phase of the wavehorizontally according to the phase of the wave

Make the assumption that a wave Make the assumption that a wave field represented by a single sifield represented by a single sinusoid induces:nusoid induces:

• a sinusoidal pressure fluctuation oa sinusoidal pressure fluctuation of p in the air f p in the air

• a sinusoidal vertical displacement a sinusoidal vertical displacement of the air flow, each with the same of the air flow, each with the same wavelength as the underlying wave.wavelength as the underlying wave.

By Assuming there is no shear stress By Assuming there is no shear stress in the system and by ignoring hydrin the system and by ignoring hydrostatic effects and acceleration duostatic effects and acceleration due to gravity, we can assume the rige to gravity, we can assume the right side of Bernoullie eq. to be conht side of Bernoullie eq. to be constant.stant.

p12

U u2 const gz pzstream function

u

y, v

x

((Bernoullis equation is an equation for Bernoullis equation is an equation for energy, since it is formed form a linenergy, since it is formed form a line integral of a force equation. It prove integral of a force equation. It provide an easy way to relate changes in ide an easy way to relate changes in p with changes in u, along a streamlp with changes in u, along a streamline.ine. ) )

As a result, the phase of the neighbAs a result, the phase of the neighboring streamline differs from the phoring streamline differs from the phase of the original streamline. There ase of the original streamline. There are several important consequences are several important consequences if this. if this.

The upward flow is slower than the The upward flow is slower than the downward flow. Thus averaged over downward flow. Thus averaged over a horizon plane covering one full waa horizon plane covering one full wavelength, the product uw is negativvelength, the product uw is negative. That is a Reynold’s shear stress te. That is a Reynold’s shear stress transporting momentum downward eransporting momentum downward exists.xists.

M-Type TheoryM-Type Theory

Studies the ways that diStudies the ways that displacement and presssplacement and pressure fluctuations get ouure fluctuations get out of phaset of phase– Kelvin Helmholtz TheorKelvin Helmholtz Theor

yy– Effect of turbulent streEffect of turbulent stre

ssss– Miles theoryMiles theory