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Sea Surface Roughness and Drag Coefficient under free Convection Conditions
By C. H. Huang
ABSTRACT---This paper investigates the sea surface Minerals Management Service, New Orleans, LA roughness and the drag coefficient under free 70123, USA. E mail: [email protected] convection conditions. Two different approaches have hence the Charnock wind stress relation breaks down. been used to estimate the sea surface roughness. One Therefore, the traditional Monin Obukhov similarity approach is to assume that the sea surface roughness is theory (MOST) is inadequate. Thus, in the limit of free aerodynamically smooth (i.e., zo ~ ν/u*). In the other convection, the friction velocity is not a relevant approach, the sea surface roughness is related to the velocity scale; the relevant velocity scale is the convective velocity scale. The Businger Dyer formula convective velocity, w*. Recently, an alternative for the non dimensional wind shear and one other formulation for the sea surface roughness has been formula that has the non dimensional form of the wind proposed, that is to relate the sea surface roughness shear with the exponent of 1/3 are used to derive the length, zo, to the convective velocity scale, w* (zo ~ minimum stability length, the minimum friction w*2) rather than the friction velocity [1] [2]. This velocity, and the drag coefficient under free convective relationship can also be derived on the basis of the conditions. The results indicate that under free mixing length theory [2]. As a result, the singularity in convection conditions, the convective processes the Charnock relation can be avoided. enhance the surface roughness and also increase the locally induced wind stress in connection with the large Since in the limit of free convection the traditional scale eddies in the convective boundary layer. Monin Obukhov similarity theory (MOST) is inadequate, the concept of the wind gustiness has been Keywords Drag Coefficient, Free Convection, Minimum introduced to enhance the wind speed and to modify Stability Length, Sea Surface Roughness MOST such that the singularity is avoided. In the MOST, the non dimensional form of the velocity I. INTRODUCTION gradient is commonly used to study the behavior of the The sea surface roughness and the drag coefficient convective boundary layer. The Businger Dyer formula under free convection conditions are investigated in this [3] is the most widely used form of the non dimensional paper. It is rather interesting and important to wind shear. The other formula is also used in this study, understand the physical processes of air sea interaction which is to express the non dimensional form of wind under such conditions, for example with the application shear as a function of ( z/L) 1/3 with the exponent of to the climate change, since it sets the upper bound for 1/3. Businger [4] and Schuman [5] introduced the the drag coefficients in light wind conditions; concepts of the minimum friction velocity. Businger furthermore, this information is needed in other suggested that near the ground, one will find horizontal applications, such as in the fields of the environmental velocity fluctuations in connection with the large scale fluid mechanics and meteorology, evaporation from oil eddies in the convective boundary layer and hence there spills, and oil burning in the open sea. exists, locally, a wind stress. Sykes et al. [6] also studied this subject using the method of large eddy simulations In free convection or light winds, it is commonly (LES). assumed that the sea surface is aerodynamically smooth and the formula, zo ~ αν /u*, is used for the estimation In this paper, the Monin Obukhov stability lengths of surface roughness length. As for pure free associated with the minimum friction velocity are convection, as the wind speed and the friction velocity derived. Using the minimum Monin Obukhov stability vanish, the surface roughness length for the length L, the drag coefficients under the free convection aerodynamically smooth surface has singularity and are also derived in this study. The results obtained in this study for drag coefficient are also compared to that proposed by Zilinkevich [7]. Wu [8] considered the sea
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surface is aerodynamically rough even under light 2 2 * + γα wu * )( winds; he attributes this condition to the effect of zo = , (3) surface tension. In their experiments under light wind g conditions, Honda and Mitsuyasu [9] used pure water and aqueous surfactant solution; they obtained the same where α is the Charnock constant and γ is also an drag coefficient and roughness length under neutral and empirical constant, which are equal to 0.015 and 0.15, light wind conditions and did not find that the surface respectively (see Abdella and D’Alessio [1]). And w* is tension is a contributing factor for the increase of the the convective velocity length scale, which is defined as drag coefficient. This leads us to explore and propose (Deardorff [12]) another factor, i.e., the convective processes associated with the large scale convective circulations that could 1 influence and enhance the sea surface roughness. g 3 , (4) w* = zH io II. SEA SURFACE ROUGHNESS To
In this section, we focus on two different formulations where To is the temperature of air, Ho is the heat flux at for the surface roughness length. One is the well known the surface, and zi is the mixed layer height. In Eq. (3), Charnock wind stress relation [10], and another as the wind speed and the friction velocity approach formulation has been suggested by Abdella and zero, it becomes D’Alessio [1]. These two formulations will be discussed briefly in the following section. 2 αγw* zo = , (5) A. Charnock Wind Stress Relation g
In the ocean, the Charnock wind stress relation is well Eq. (3) for the surface roughness is not singular when known and has widely been used to calculate the surface the mean wind speed and the friction velocity vanish roughness, which relates the surface roughness length to under the conditions of free convection. the wind stress. The Charnock wind stress relation can be expressed in the following as Abdella and D’Alessio [1] conducted the numerical simulations using Eq. (3), the formula of the surface 2 roughness; they found that the simulation results are in αu* 11.0 ν zo += , (1) good agreement with the observed heat content and sea ug * surface temperature. where zo is the surface roughness length, u* the friction III. ALTERNATIVE DERIVATION OF SEA velocity, α the Charnock constant, and ν the kinematic SURFACE ROUGHNESS viscosity of air. The last term in Eq. (1) is suggested by Smith [11], which represents the surface roughness The exchange coefficient for momentum in the surface length for the flow over the aerodynamically smooth boundary layer suggested by (Zililtinkevich et al. [7]) surface. When the friction velocity vanishes, the surface can be used to derive the sea surface roughness length. roughness length in Eq. (1) becomes The exchange coefficient for momentum, Km, is expressed as 11.0 ν . (2) 1 zo = u* w z 3 * , (6) As the wind speed and the friction velocity approach m ukzK * += zero, Eqs. (1) and (2) have singularity and hence the kC ziu MOST is inadequate.
B. Wind Stress Relation in Free Convection where Cu = 1.7 is an empirical constant. Taking the square of Eq. (6), it becomes Recently, another approach for the formulation of surface roughness length under the forced and free convection has been proposed by Abdella and D’Alessio [1], which can be expressed as
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2 IV. MONIN OBUKHOV SIMILARITY THEORY 1 3 2 w z 2 * . (7) Traditionally, the Monin Obukhov similarity theory m () ukzK * += kC ziu [13] is commonly employed to estimate the transfer of momentum, heat, and moisture in the atmospheric boundary layer. In this paper, we focus on the Expanding Eq. (7) and neglecting the term with the momentum transfer in the MOST. The non dimensional product of u*w*, then, Eq. (7) can be simplified as wind shear in MOST can be expressed as
2 2 ( ) ( 2 += γwukzK 2 ) (8) kz ∂u z m * * = φ m u* ∂z L and , (14) 2 2 3 1 z In Eq. (14), φm is the non dimensional wind shear and L is γ = . (9) kC z the Monin Obukhov stability length, which is defined as iu
θ u 3 If we set z = 10 m and zi = 600 m, the value of γ is equal L −= v * '' to 0.47 and if we set the ratio of z/zi = 0.1, that is the wgk θ surface layer is 10 % of the mixed layer (see Schumann ( v ) (15a) [5]), the value of γ is equal to 0.14. or If we define the effective total momentum flux as 2 + )61.01( uqT * L −= kg θ + Tq )61.0[ 2 = kzKu )/( 2 (10) * * . (15b) *eff m Where T is the temperature, θ is the virtual temperature, θ Then, Eq. (8) becomes v * is the potential temperature scale, q is the humility and q* is the humility scale. 2 2 2 *eff * += γwuu * . (11) Two different forms are usually used to specify of the non Replacing the wind stress in the Charnock relation in Eq. dimensional wind shear. One is the Businger Dyer formula (1) by the effective total wind stress, we obtain the (see Dyer [3]), which can be written as following equation for the surface roughness length, zo, 1 − z 4 2 2 1−= γφ , (16) ( * + γα wu * ) m 1 zo = , (12) L g where γ1 is an empirical constant and is equal to16. The This equation has also been obtained by Huang [2]. In other non dimensional wind shear can be expressed as (Carl Eq. (12), the first term on the right hand side represents et al. [14]) the mechanical forces in the generation of the sea 1 − surface roughness, and the second term represents the z 3 1−= γφ , (17) buoyancy forces in the generation of intense turbulences m 2 L and hence in increasing the sea surface roughness. In the limit of free convection when the mean wind speed and the friction velocity vanish, Eq. (12) reduces to: where γ2 is an empirical constant and is equal to 15.
Integration of Eq. (16) or Eq. (17), the wind profile can αγw2 z = * . (13) be written as o g u z z z * o (18) That is, under free convections the buoyancy forces are u = ln − m +ψψ m dominant, which enhance the sea surface roughness. k zo L L
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where k is the von Karman constant and is equal to 0.4 1 3 and the function ψm is defined as kL β 3uw −= (22) y dx ** zo ψ m ()()y []1−= φm x , (19) ∫y o x where y = z/L. In Eq. (22), we have replaced the wind velocity u with
βw*, the convection induced horizontal velocity motion in connection with the large scale eddies in the convective boundary layer. Eq. (22) shows the
relationship between the induced wind stress, u*, and the V. MINIMUM STABILITY LENGTH convective velocity, w*, under free convection conditions. The concepts of the minimum friction velocity and wind gustiness were proposed by several researchers (Businger By definition of the convective velocity scale, from Eq. [4]; Schumann [5]; and Sykes et al. [6]). Businger [4] (22), we obtain the minimum stability length L as postulated that in the limit of free convection, when the mean wind speed and the friction velocity vanish, one still 3 finds, close enough to the ground, the fluctuations of wind 1 β 2 1 2 speed, the gustiness, resulting in developing a local wind LL min −== ()zz oi (23) profile. With this wind profile, there exists locally a wind k 3 stress and also a shear production of turbulent energy; this turbulent energy is generated by the buoyancy force. The For example, if we set β = 1.25, zi = 600 m, zo = 3.0x10 4 friction velocity associated with this local wind stress is , we obtain the minimum stability length, Lmin = called the minimum friction velocity, which is in 0.285. Once the minimum stability length is known, the connection with the large scale convective circulations. minimum friction can be calculated from the definition Corresponding to the minimum friction velocity is a of the Monin Obukhov stability length as specified in minimum stability length which is derived in the following Eq. (15a). section. B. Beljaars Formulation A. Present Formulation Under free convection, as the wind speed approaches Since under convective conditions, the buoyant forces zero, in connection with the large scale convective dominate the mechanical forces, the non dimensional wind circulations, close to the ground, there still exists a local shear can be written as wind profile due to wind gust and hence a local wind stress such that the horizontal velocity fluctuation is 4 1 replaced by βw* to enhance the wind speed, Thus, under − kz ∂u 3 zk 3 free convection, the MOST, Eq. (14) with the use of the −= , (20) Businger Dyer formula of the non dimensional wind u ∂z 2 L * α 3 shear, Eq. (16), can be written in the form (see Beljaars 1 [16]):
where α1 is an empirical constant. In obtaining Eq. (20), we have utilized the eddy exchange coefficient for βwk * u* = (24) momentum given by Priestley [15]) as follows: − 5.38 L z o ln +ψ m 1 γ 1 zo L 4 3 2 gH o 3 K m = α1 z , (21) θ This formulation has been successfully implemented in the ECMWF (European Center for Medium Range Weather Forecasts) and found significant improvement Eq. (20) with the exponent of 1/3 is considered in the numerical weather predictions. theoretically as the preferable form of non dimension wind shear in the limit of free convection. Here, we set For free convection with u =0 and using Eqs. (16) and α1 = 1 since α1 is close to one (see Priestley [15]). (19), the asymptotic form of function ψm takes the following expression (see Beljaars, 1994): Integrating Eq. (20), we obtain
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γ z 1 . (25) In this method, we use the Beljaars’ formula, Eq. (24). ψ m ≈ ln − 5.38 L From Eq. (24), the drag coefficient under free convection can be written as 2 Using the definition of the convective velocity scale, w*, k (28) Eq. (24) which relates the ratio of zi/zo to L/zo can be Cd = 2 written as (see Beljaars [16]): − 5.38 L z o ln +ψ m γ 1 zo L 3 z L −− 5.38 L z i o Since the value for the function, ψm (zo/L) is quite small. = ln +ψ m , (26) 32 Therefore, we can simplify the drag coefficient in Eq. zo β zk o γ 1.zo L (28) as follows:
Eq. (26) shows the relationship between the L/z and o 2 z /z , where L is the minimum Monin Obukhov stability k i o Cd ≈ (29) length, L = L , as defined in Eq. (26). Once the values 2 min − 5.38 L of zi and zo or the ratio of zi/zo are known, the minimum ln( ) stability length, L, can be calculated. For example, for γ 1 zo the values of α = 0.015, γ =0.15, and for a typical value of the convective velocity, w = 1 m/s, then from Eq. * Substituting the value of z /L = 1.1x103 by setting the (5), we obtain the surface roughness length, z = 3.0x10 o min o roughness length z = 3.0x10 4 and L = 0.336 in Eq. 4. The typical value of the mixed height, z , in the Gulf o min i (28), we obtain the drag coefficient, Cd = 2.6x10 3. of Mexico or the tropical ocean is 600 m. Thus, we have z /z = 5.0x10 7. Once the values of z and z or the ratio i o i o For aerodynamically smooth flow and for the mean of z /z are known, the minimum Monin Obukhov i o wind speed of u = 1 m/s, from Eq. (2) and the stability length, L , can be determined from Eq. (26). min logarithmic wind profile, we obtain the surface In this case, we obtain the value of the minimum roughness length of z = 5.0x10 5 and the neutral drag stability length, L = 0.336. o min coefficient of Cdn = 1.1x10 3. Bradley et al. [17] in his
field experiments obtained the neutral drag coefficient VI. COMPARISON OF DRAG COEFFICIENTS of Cdn = 2.6x10 3 for the mean wind speed of u = 1
m/s. Therefore, by comparison of the drag coefficients In this section, we make comparison of various drag with that of the smooth surface, it would appear that for coefficients calculated under free convection conditions. light wind conditions, the sea surface can be considered Three approaches have been used to estimate the drag as aerodynamically rough even under light wind coefficient in the limit of free convective conditions. conditions. One formula is obtained in this study; the other two formulas used for the calculation of the drag coefficient C. Method 3 are from Beljaars [16] and Zilitinkevich et al. [7]).
In this section, we use the Zilitinkevich et al.’s formula A. Method 1 for the estimate of the drag coefficient (Zilitinkevich et
al. [7]). In his study, Zilitinkevich proposed a From Eq. (22) in Section V, the drag coefficient for this formulation for the drag coefficient, which is expressed condition can be defined and written as as
2 2 − 3 u* 1 kLmin −2 Cd ≡ −= (27) βw* 9 zo zi 2 C z Using the formula of the minimum stability length, L , Cd = u1 ln ou + C , min β 3 u2 given in Eq. (23), once the values of zi, and zo are given, z ln i − C we obtain the minimum stability length, Lmin = 0.285 uo 3 z and the drag coefficient, Cd = 2.1x10 . ou (30) B. Method 2
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where Cu0 = 6, Cu1 =0.29, and Cu2 = 2.56 are empirical [1] K. Abdella, and S. J. D. D’Alessio, “A constants. Using the same parameters given in method parameterization of the roughness length for 2, the calculated drag coefficient from Eq. (30) is equal the air sea interaction in free convection,” Envir. 3 to Cd = 1.8x10 . Fluid Mechanics, 3, pp. 55 77, 2003 [2] C. H. Huang, “Parameterization of the roughness VII. CONCLUSION length over the sea in forced and free convection,” Environmental Fluid Mechanics, 9, pp. 359 366, The sea surface roughness and the drag coefficient 2009 under free convection conditions, as well as the [3] A. J. Dyer, “A review of flux profile relationships,” minimum stability length and the minimum friction Boundary-Layer Meteor., 7, pp. 363 372, 1974. velocity, are investigated in this paper. We also make [4] J. A. Businger, “A note on free convection,” the comparisons of the drag coefficients proposed by Boundary-Layer Meteor., 4, pp. 323–326, several researchers. Two different approaches have been 1973. used to estimate the sea surface roughness. One [5] U. Schumann, “Minimum friction velocity and heat approach is to assume that at low winds the sea surface transfer in the rough surface layer of a convective roughness is aerodynamically smooth (i.e., zo ~ ν/u*) boundary layer,” Boundary-Layer Meteor., 44, pp. (see Smith, 1988). In this formulation, when the mean 311 326, 1988. wind speed and the friction velocity vanish, the sea [6] R. I. Sykes, D. S. Henn, and W. S. Lewellen, surface roughness length has singularity and hence the “Surface layer description under free convection traditional Monin Obukhov similarity theory breaks conditions,” Q. J. R. Meteorol. Soc., 119, pp. 409 down. In this paper, we derive a formula for the sea 421, 1993. surface roughness length on the basis of the exchange [7] S. S. Zilitinkevich, J. C. R. Hunt, I. N. Esau, A. A. coefficient for momentum, which supports the Grachev, D. P. Lalas, E. Asylas, M. Tombrou, C. formulations obtained in previous studies (Abdella and W. Fairall, H. J. S. Fernando, A. A. Baklanov, and D’Alessio [1]; Huang [2]). In this formula, the sea S. M. Joffre, “The influence of large convective surface roughness is related to the convective velocity eddies on the surface layer turbulence,” Q. J. R. scale, w*. As a result, the singularity in the Monin Meteorol. Soc., 132, pp. 1423 1456, 2006. Obukhov similarity theory is avoided. The non [8] J. Wu, “The sea surface is aerodynamically rough dimensional wind shear can be used to derive the drag even under light winds,” Boundary-Layer coefficient. For instance, the Businger Dyer formula for Meteor., 69, pp. 149 158, 1994. the non dimensional wind shear and the other formula, [9] T. Honda, and H. Mitsuyasu, “An experimental used in this study, which has the non dimensional form study of the action of wind on water surface,” with the exponent of 1/3 are used to derive the Proc. Coastal Eng., JSCE, 27, pp. 90 93, 1980. minimum stability length, L, the minimum friction [10] H. Charnock, H. 1955. “Wind stress on a water velocity, u*, and the drag coefficient under free surface,” Q. J. R. Meteorol. Soc., 81, pp. 639 640, convective conditions. In addition, the formula of the 1955. drag coefficient proposed by Zilitinkevich at al. [7] is [11] S. D. Smith,”Coefficients for sea surface wind also used to estimate the value of the drag coefficient. stress, heat flux, and wind profiles as a The results indicate that under free convection function of wind speed and temperature,”. J. conditions, the convective processes due to buoyancy Geophys. Res., 93, pp.15467 15472, 1988. forces generate intense convective turbulence, which in [12] J. W. Deardorff, “Convective velocity and turn enhances the surface roughness and also increases temperature scales for the unstable boundary the locally induced wind stress in connection with the layer,” J. Atmos. Sci., 27, pp. 1211 1213, 1970. large scale eddies in the convective boundary layer. [13] A. S. Monin, and A. M. Obukhov, “Basic turbulence mixing laws in the atmospheric There is evidence that the sea surface can be considered surface layer,” Trudy Geofiz. Inst. AN SSSR., No. as aerodynamically rough in the limit of free convection 24 (151), pp. 163 187, 1954. and even under light winds. The experimental data and [14] M. D. Carl, T. C. Tarbell, and H. P. Panofsky. the present study show that under neutral and free 1973: “Profiles of wind and temperature convective conditions, the drag coefficients are much from towers over homogenous terrain,” J. Atmos. larger than that of the aerodynamically smooth flow. Sci., 30, pp. 788 794, 1973. [15] C. H. B. Priestley, “Convection from a large horizontal surface,” Australian Jour. Phys., 6, pp. 279 290, 1954. REFERENCES [16] A. C. M. Beljaars, “The parameterization of surface fluxes in large scale models under free
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convection,” Q. J. R. Meteorol. Soc., 121, pp. 255 the Western Equatorial Pacific Ocean,” J. 270, 1994. Goephys. Res. (Suppl.) 96, pp. 3375 3389, 1991. [17] E. F. Bradley, P. A. Coppin., and J. S. Godfrey, “Measurements of sensible and latent heat flux in
ISSN: 1792-4359 127 ISBN: 978-960-474-211-0