DRAG COEFFICIENT REDUCTION AT VERY HIGH WIND SPEEDS

John A.T. Bye1 and Alastair D. Jenkins2

(1) School of Ear th Sciences, The University of Melbourne, Victor ia 3010, Australia

(2) Bjerknes Centre for Climate Research, Geophysical Institute, The University of

Bergen, Allégaten 70 , 5007 Bergen, Norway

Abstract

The correct representation of the 10m drag coefficient for momentum (K10) at extreme

wind speeds is very important for modeling the development of tropical depressions and

may also be relevant to the understanding of other intense marine meteorological

phenomena. We present a unified model for K10 , which takes account of both the wave

field and spray production, and asymptotes to the growing wind wave state in the absence

of spray. A feature of the results, is the prediction of a broad maximum in K10 . For a

spray velocity of 9 m s-1 , a maximum of K10 ∼ 2.0 × 10-3 occurs for a 10 m wind speed,

u10 ∼ 40 m s-1 , in agreement with recent GPS sonde data in tropical cyclones. Thus, K10

is "capped" at its maximum value for all higher wind speeds expected. The effect of

spray is also shown to flatten the sea surface by transferring energy to longer wavelengths.

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1. Introduction

It is of importance to be able to accurately parameterize air-sea exchange processes at

extreme wind speeds in order to understand the mechanisms which control the evolution of

tropical cyclones (Emanuel, 2003). There are also indications that rapid increases in wind

speed may tend to depress the height of surface waves and thus perhaps reduce the drag

coefficient by the flattening of sea-surface roughness elements (Jenkins, 2001). Here, we

consider momentum exchange, and present a seamless formulation which predicts the drag

coefficient over the complete range of wind speeds. An important aspect of the physics is

the momentum used in the production of spray. The results are calibrated against the

data set of Powell et al (2003), obtained by Global Positioning System dropwind-sonde

(GPS sonde) releases in tropical cyclones .

The basis of the analysis is to apply a general expression for the drag coefficient ( K10 ),

that has been derived from the inertial coupling relations (Bye, 1995), which take account

of the wave field (Bye et al, 2001), to the wave boundary layer (Bye, 1988) in the situation

occurring under hurricane winds, when spray plays a significant role in the air-sea

momentum transfer. The inertial coupling relation may be regarded as a parameterization

of the of the dynamical effect of ocean waves within the coupled system containing the

atmospheric and oceanic near-surface turbulent boundary layers (Jenkins 1989, 1992).

We will outline the derivation of the general expression for the drag coefficient, and then

introduce a simple formulation, which takes account of spray production.

2. A general expression for the 10m drag coefficient ( K10 )

In the wave boundary layer (Bye, 1988),

2

u10 = u1 − u* /κ ln ( zB/10 ) (1)

in which u10 is the wind velocity at 10 m, and u1 is the wind velocity at the height, zB =

1/(2 k0) where k0 is the peak wavenumber of the wave spectrum (which will be called

the surface wind), u* is the friction velocity and κ is von Karman's constant. On

introducing the inertial coupling relationships (Bye, 1995) in which,

u* = KI1/2 ( u1 − u2 / ε ) (2)

and

εuL = ½ (εu1 + u2) (3)

where KI is the inertial drag coefficient, ε = (ρ1/ρ2)1/2, in which ρ1 and ρ2 are

respectively the densities of air and water, u2 is the current velocity at the depth, zB

(which will be called the surface current), εuL is the surface Stokes drift velocity, and the

reference velocity has been set equal to zero for convenience, and also the relation (Bye

and Wolff, 2001),

εuL = r (−u2) (4)

which partitions the Stokes shear and the Eulerian shear in the water in the ratio (r), we

obtain the expression,

u1 = R u* / √KI (5)

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in which R = ½ (1 + 2r)/(1 + r). Hence on substituting for u1 in (1), we obtain,

1/√K10 = R/√KI -1/κ ln (1/ (20 k0) ) (6)

where K10 = u*2/u10

2 . Finally, on substituting into (6), the relation,

c0/u1 = B (7)

where B is the ratio of the phase speed of the peak wave, c0 = (g/k0)1/2 , in which g is

the acceleration of gravity, and the surface wind, we obtain,

1/√K10 = R/√KI − 1/κ ln ( B2 u*2 R2 / (20 g KI) ) (8)

Equation (8) is a general expression for K10 in terms of KI , κ , g , u* and R. In Bye and

Wolff (2001), it was shown that in a fully developed growing wind wave sea, r = ± ∞ (R =

1), such that the Stokes shear dominates the Eulerian shear. For finite values of r , r > 0

implies that the Eulerian shear and the Stokes shear are of the same sign, giving rise to an

augmented transfer of momentum from the air to the water, whereas for r < 0, the Eulerian

shear and the Stokes shear are of opposite sign such that the transfer of momentum from

the air to the water is reduced.

For the fully developed growing wind wave sea (R = 1), the inverse wave age, u* /c0 =

A, where A = 0.029 (Toba, 1973), and hence B = √KI / A, and the Charnock constant

(Charnock, 1955),

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α = 1/(2A2) exp(-κ/√KI) (9)

For agreement with observational estimates of α = 0.018 (Wu, 1980) with κ = 0.4, (9) is

satisfied by an inertial drag coefficient, KI = 1.5×10-3, from which B = 1.3.

The calibration for the fully developed growing wind wave sea, which yields the

parameters, KI = 1.5×10-3 , B = 1.3 and R = 1, is applicable for all wind speeds ( u10 ) on

the assumption that similar dynamical processes control the momentum transfer for all

values of the surface wind ( u1 ). At hurricane wind speeds this assumption is unlikely to

be valid. The main thrust of the paper is to introduce a model to be applied in (8), that

takes account of spray production, which almost certainly is an important factor at extreme

wind speeds.

3. The spray model

As pointed out in Section 2, (8) is a similarity expression applicable at all wind speeds

( u10 ). Allowance for spray production can be made by replacing the constant parameter

( R = 1 ), by a wind speed dependent relationship which accommodates a dynamical

system in which the relative importance of the Stokes shear to the Eulerian shear decreases

as the surface wind speed ( u1 ) increases, due to the greater relative importance of

turbulent transfer processes in comparison with wave growth. From (4), this moderation

is brought about by assuming that at higher surface wind speeds, r < -∞ , (R > 1),

indicating a reduction in the momentum transfer from the air to the water. A simple linear

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expression for R (which we show below has the correct asymptotic behaviour) is,

R = R0 + u1 / q0 (10)

in which R0 = 1, is applicable to the growing wind wave sea environment, and q0 is a

velocity which characterizes the spray production see Section 4. On using (5), we obtain,

R = 1 / (1−u*/ (q0√KI) ) (11)

and on eliminating R between (10) and (11), we have,

u* = u1√KI / (1+ u1/q0) (12)

which in the limit of u1/q0 << 1 (small surface wind velocities) reduces to (5), and in the

limit of u1/q0 >> 1 (large surface wind velocities) yields,

u* = q0 √KI (13)

in which R→ ∞ (r = −1), and q0 is the sole velocity which determines u* , and hence u*

tends to a constant.

On substituting (11) in (8), we obtain,

u10/u* = 1/√KI ( 1/ [1-u* /(q0√KI) ] ) − 1/κ ln(B2u*2 /(20gKI [1-u* /(q0√KI) ]

2 ) (14)

Equation (14) is an implicit expression for K10 as a function of u10, for a specified

velocity (q0). On differentiating (14) with respect to u10 , it is found that a maximum

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(K10) occurs at the velocity,

u10 = q0√KI [( 2 − ln( KIB2q0

2/[5gκ2] ) / (1 + κ/(2√KI) ) ]/κ (15)

and is given by the expression,

K10)max = KI ( 1 / ( u10/q0 [ 1 + κ/(2√KI) ] ) )2 (16)

Fig. 1a shows how q0 determines the 10 m wind velocity (u10) at which the maximum

drag coefficient (K10)max ) occurs. At this maximum, R = 1.19 ( r = -3.58), in contrast

to the fully developed growing wind wave sea limit of R = 1 It is apparent that the

observations of Powell et al (2003), which are reproduced in Fig. 1b are well represented

by q0 = 300 m s-1 for which K10)max = 1.99×10-3 at u10 = 42 m s-1 with u* =

1.88 m s−1.

It is interesting that q0 has a similar value to the velocity of sound, but there is no clear

physical interpretation of this coincidence.

Fig 2 indicates that the drag coefficient for q0 = 300 m s-1 has a broad maximum. This

behavior appears to be similar to the observations, although there is a considerable

variability between the estimates for different height ranges. The profile for q0 =

100 m s-1 shows the gradual approach to the the high surface wind limit, which occurs for

u* = 3.87 m s-1, at which K10 → 0, and u10 → ∞. It is also clear that the effects of

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spray (according to the model (10) ) begin to be felt at relatively low wind speeds. In

reality, it is likely that q0 would depend on meteorological conditions. Different wind

regimes would be characterized by different values of q0 . We speculate that the near

cyclostrophic balance in hurricanes may lead to a larger q0 than the near geostrophic

balance in extra-tropical cyclones, in which conditions most observations of air-sea

momentum transfer have been made.

4. Physical proper ties of the spray model

4.1 Flattening of the sea state

A characteristic of the sea state in hurricane winds is that the waves appear to be

flattened by the wind. This effect can be quantified using the spray model. We adopt the

Toba wave spectrum for the growing wind wave sea, truncated at the peak wavenumber

(k0), for which,

E = 1/3 γ0 u* c03/g2 (16)

where E = <ζ2> is the root mean square wave height, and γ0 is Toba's constant. On

substituting for u* , we obtain,

E = 1/3 γ √KI c04 / (g2B) (17)

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in which γ = γ0 ( 1 − u* / (q0√KI) ). Hence, the reduction in wave energy, due to spray,

can be interpreted in terms of a reduced Toba constant (γ). In the limit, u*→ q0√KI , γ→

0, indicating a totally flattened sea state. At K10)max , which occurs at u10 = 42 m s-1 ,

γ/γ0 = 0.84 indicating a mild flattening in which the wave height is reduced by about 8%.

The peak wave speed,

c0 = Bu* / (√KI [1 − u* /(q0√KI) ]) (18)

such that c0→ ∞ for u*→ q0√KI , and at u10 = 42 m s-1, c0 increases by about 20% due to

the spray effect. Thus, the production of spray tends to increase the wave speed of the

peak wave, i.e. transfer energy to longer wavelengths.

4.2 The similarity profile at extreme wind speeds

The key result of Section 3 is that the drag coefficient passes through a maximum

(K10)max), with wind speed, and then is almost constant over a wide range of higher

speeds, see Fig. 2. Hence, for the purposes of hurricane dynamics, where K10)max

occurs at about 40 m s-1, the drag coefficient is "capped" at its maximum value over the

full range of extreme wind speeds that are likely to occur.

The physical processes which bring about this approximate similarity regime for extreme

wind speeds are a dilation of the wave boundary layer, in which its thickness (zB) and non-

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dimensional velocity scale (u1/u*) both increase, but without a significant change in K10 ,

see (1). The dynamical process which is occurring, is that as the friction velocity

increases, there is a progressive increase in the return flow of momentum from the ocean

to the atmosphere due to the oceanic (Eulerian) shear in comparison with that from the

atmosphere to the ocean due to the atmospheric shear. This two-way momentum

exchange across the air-sea interface is represented by the two terms on the right hand side

of (2), the first of which arises from the atmospheric shear, and the second from the

oceanic shear. Using (3) and (4), the ratio of the two shears,

(u2/ε)/u1 = −1/(2r+1) (19)

For the growing wind wave sea, u2/(εu1) = 0, whereas with the inclusion of spray

production, u2/(εu1) increases with u* (Fig.3). The increase over the range in u10 from

about 30 - 60 m s-1 gives rise to an almost constant, K10 over this range through

corresponding changes in zB and u1/u* .

4.3 The spray velocity

We look now at the energetics of spray formation, making use of the expression for the

rate of working on the wave field,

W = ρ1 u*2 uL (20)

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where uL is the velocity at which the transfer of momentum to the wave field is centered

(Bye and Wolff, 2001). On substituting for uL we obtain,

W = ½ ρ1 u*3 ( [1 + u* /(q0√KI)] / [1 − u* /(q0√KI)] ) /√KI (21)

such that as u*→ q0√KI , W→ ∞. The rate of working (W) can be usefully partitioned

into the two components,

W = W0 + WS (22)

where W0 = ½ ρ1 u*3/√KI is the rate of working on the growing wind wave field, and,

WS = ρ1 u*2 q (23)

is the rate of working which generates the spray, where q is the spray velocity. For W0

= WS , we find that the friction velocity (u* ) is 3.9 m s-1, which is very similar to that of

4.2 m s-1, predicted by Andreas and Emanuel (2001) for the condition that the spray stress

and the interfacial stress are equal. This gives independent support for the choice of q0 =

300 m s-1 in the spray model. At K10)max (for q0 = 300 m s-1), q = 9.2 m s-1, and

WS/W0 = 0.38, hence just over ¼ of the rate of working is used for spray production, and

¾ for wave growth.

This partitioning of the rate of working highlights that the changes occurring in the wave

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field, described in Subsection 4.1, are due to spray production.

4.4 Property transfer across the sea surface

The implications of the partitioning of the rate of working into a wave (W0) and a spray

(WS) component are apposite. The wave component (W0) has no significance for

property transfers across the sea surface; these are encompassed (at least in part) by the

spray component (WS). In the event that processes other than spray production are

unimportant at extreme wind speeds, as proposed by Emanuel (2003), heat and momentum

transfer should be governed by the same physics. Thus, on expressing the surface shear

stress (τS = ρ1 u*2 ) in terms of the spray velocity, we have,

τS = ρ1 CS q2 (24)

where CS is a drag coefficient appropriate to the spray production, and the net upward heat

flux is,

F = ρ1 Cp CS q (TS − TW) (25)

where the drag coefficients (CS) in (24) and (25) are identical, TS is the surface water

temperature, TW is the wet bulb temperature of the descending spray particles, and Cp is

the specific heat of water at constant pressure (Emanuel, 2003). Equation (25) is of the

same form as that applicable for heat exchange due to rainfall, in which q is replaced by

the precipitation velocity (P), see for example, Bye (1996). We note, however, that whilst

P is a vertical velocity, q is a horizontal velocity, and hence it is presupposed in (25) that

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the spray production occurs through an isotropic velocity field in which the vertical and

the horizontal components are equal. Allowance for evaporative heat exchange can also

be made, and it is found that the drag coefficient for enthalpy transfer at the temperatures

occurring in hurricanes is similar to that for heat (Emanuel, 2003).

In summary, at extreme wind speeds in which property transfers across the sea surface

are dominated by spray production, the drag coefficients (CS) for momentum and heat

transfer, relative to the spray velocity (q), and hence also the drag coefficients (K10)

relative to u10 are identical, and since the momentum drag coefficient (K10) is "capped",

as discussed in Section 4.2, that for heat transfer is also capped.

5. Conclusion

We have presented a unified model for predicting the drag coefficient (K10) for

momentum exchange at the sea surface, which takes account of wave growth and also

spray production. It is found that K10 passes through a broad maximum due primarily to

the return flow of momentum from the ocean to the atmosphere, which increases with

friction velocity (u* ). The physical processes, which become evident in this extreme

wind speed "similarity range" are the flattening of the sea surface with the transfer of

energy to longer wavelengths, together with the production of spray. On the assumption

that heat transfer across the sea surface at extreme wind speeds is mainly due to spray

production (Emanuel, 2003), it is argued that the drag coefficient for heat should be similar

to that for momentum, and also "capped" at extreme wind speeds,.

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The analysis uses a simple expression (10) to model spray production, which asymptotes

to a flat sea surface for wind speeds well beyond those expected in nature. Equation (10)

is essentially a linear expansion about the classical growing wind wave state, which takes

account of spray production, and is appropriate for an open ocean environment. A

similar expansion can be made about the wave state applicable in wave tanks by a suitable

choice of R0 in (10). Since there are two parameters in (10), R0 and q0, both of which

depend on the specification of the sea state, it seems unlikely that a predictive relationship

for K10 can be obtained, which is of universal applicability.

Acknowledgments

This work was begun whilst JATB was a Visiting fellow at the Bjerknes Centre for

Climate Research, The University of Bergen in September 2003. ADJ is supported by the

Research Council of Norway under project no. 155923/700.

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References

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457-472

Bye, J.A.T. 1995 Inertial coupling of fluids with large density contrast. Physics Letters A

202 222-224

Bye, J.A.T. 1996 Coupling ocean-atmosphere models Earth-Science Reviews 40 149-

162.

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Francisco, CA, volume 1, pages 494-500, ASCE.

15

Emanuel, K. 2003 A similarity hypothesis for air-sea exchange at extreme wind speeds.

J. Phys. Oceanogr. 60 1420-1428

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wind speeds in tropical cyclones Nature 422 279-283

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L ist of Figures

1. (a) The maximum drag coefficient (K10)max ) , and the 10 m wind speed (u10) of its

occurrence as a function of q0

(b) Observations of K10 as a function of u10 adapted from Fig. 3c of Powell et al

(2003). Circles: 10-100 m; squares: 10-150 m; triangles: 20-100 m; diamonds: 20-150

m

2. The drag coefficient (K10) as a function of u10 for (a) q0 = 100 m s-1 , (b) q0 = 300

m s-1 , and (c) q0 → ∞

3. The ratio (u2/(εu1)) as a function of u* , for q0 = 300 m s-1

17

(Figure 1a)

18

K10 u10 / m s-1

80

40

u10

K10

0.000

0.002

0.004

0 200 400 600 800 1000

q0 / m s-1

(Figure 1b)

19

K10

0.000

0.002

0.004

20 40

u10 / m s-1

(Figure 2)

20

K10

q0 = 100 m s-1

q0 = 300 m s-1

q0 → ∞

0.000

0.002

0.004

0.006

0 20 40 60 80 100

u10 / m s-1

(Figure 3)

21

u2/( u1)ε

u10 = 42 m s-1

↓

0.0

0.1

0.2

0.3

0 1 2 3

u* / m s-1