On the parameterization of surface momentum transport via drag coefficient in low-wind conditions


Corresponding author: P. Zhu, Department of Earth and Environment, Florida International University, Miami, FL 33199, USA. (zhup@fiu.edu)


[1] The subgrid-scale surface momentum transport, which plays an important role in determining the exchange between the atmosphere and the underlying surface, is often parameterized in terms of the surface mean wind speed via drag coefficient (CD), a parameter that needs to be determined externally often through the Monin-Obukhov Similarity (MOS) Theory. However, some characteristics of CD derived from observations for overland conditions, particularly the substantial increase of CD with a decrease in wind speed in low-wind conditions, cannot be explained by MOS. This issue is investigated using data collected by a portable meteorological tower. By analyzing the turbulent kinetic energy budget, a novel parameterization framework for momentum fluxes is proposed. The new parameterization not only appropriately describes the observed variation of CD but also can be simplified to MOS with certain assumptions. Moreover, the effect of stability, which traditionally has to be determined empirically, can now be determined internally within the new framework.

1 Introduction

[2] Quantifying momentum transport resulting from turbulent mixing requires turbulent eddies to be explicitly resolved. Thus, in numerical prediction of weather and climate and many other applications in geosciences, the surface momentum fluxes are often determined parametrically in terms of mean wind speed

display math(1)

where u and U10 are the friction velocity and 10-m wind speed, respectively. CD is a dimensionless coefficient often called the drag coefficient in the literature [Deardorff, 1968]. Despite the simplicity of the concept, CD cannot be determined by the bulk transfer model itself. In practice, the Monin-Obukhov Similarity (MOS) Theory [Monin and Obukhov, 1954] is often used to close the system to yield

display math(2)

where z, z0, and κ represent the height, aerodynamic surface roughness, and Von Karman constant, respectively. inline image is the dimensionless stability parameter, and L is the Obukhov length defined as inline image, where θ0, g, and inline image are the surface mean potential temperature, gravity, and kinematic surface buoyancy flux, respectively. In this study, L is computed directly based on its definition from the surface fluxes determined by the eddy correlation method. Ψm(ζ) is an empirical stability function. CDN is the drag coefficient in neutral conditions. Since the aerodynamic surface roughness is estimated within the same framework, to avoid redundancy, CDN is often estimated directly from CD corrected by stability as [Grachev et al., 1998]

display math(3)

[3] To appropriately use equations (1) and (2) to parameterize surface momentum fluxes, there are issues that need be addressed. Over the ocean, the surface roughness depends on wind speed. Combining equation (2) with the Charnock formula [Charnock, 1955] for oceanic roughness, it can be shown that CDN increases with wind speed. However, observations show that CDN does not increase unlimitedly with wind speed; instead, it starts to level off at certain wind speeds [Powell et al., 2003; Black et al., 2007; Donelan et al., 2004; Drennan and Graber, 1999; Drennan et al., 2005]. Such a behavior of CDN reflects the response of the ocean surface to the exerted winds. The aerodynamic roughness may attain a “saturated” state at a certain large wind speed since the ocean surface simply cannot be any rougher in an aerodynamic sense [Donelan et al., 2004]. Although there exists a disagreement on the wind speed at which CDN starts to level off, the change in CDN from moderate to high wind speed can be well explained by MOS as long as the oceanic surface roughness can be accurately determined.

[4] For overland conditions, a large change in surface roughness with wind speed is not expected because of the static nature of the surface roughness elements. Equation (2) predicts a constant CDN independent of wind speed for a fixed z0, but this is not supported by observations. Figure 1 shows CDN as a function of wind speed obtained from multiple independent field experiments for overland conditions [Grachev et al., 2006; Al-Jiboori, 2010; Mitsuta and Tsukamoto, 1978; Rao et al., 1996; Mahrt et al., 2001; Rao, 2004] where CDN, in some cases, is calculated based on equation (3), whereas in others, it is computed under near-neutral conditions. CDN obtained from different experiments shows a large spread possibly due to the different land surface conditions in these studies. For example, the data in Al-Jiboori [2010] were collected in a complicated urban area. The large z0 leads to a large CDN. Despite the large spread of CDN, all the data show a clear trend of CDN increasing with a decrease in wind speed. There have been attempts to explain the observed variation of CDN with wind speed within the MOS framework. For example, Mahrt et al. [2001] argued that lower wind speeds enhance the viscous effects and reduce the streamlining of surface obstacles. The combined effect results in an increase in aerodynamic surface roughness as wind speed decreases. But, such an argument of attributing the observed variation of CDN solely to the change in z0 with wind speed may lack legitimacy. From equation (2), one can readily calculate z0 required to obtain the values of CDN represented by the best fit curve in Figure 1. There is a sharp increase of z0 as wind speed decreases, and it can reach up to 10 m for extremely low wind speeds, which is impossible for most of the overland conditions. Therefore, the objectives of this paper are to provide a physically sound explanation for the observed variations of CD and CDN and attempt to extend the classic MOS framework into the low-wind regime.

Figure 1.

CDN against surface wind speed obtained from multiple field experiments over land. The thick blue curve is the best fit curve of all the data presented in the figure. The thick dashed curve (scaled to the right) indicates the surface roughness that is required to obtain CDN represented by the best fit curve based on equation (2). Note that the nonlinear fitting function used here is inline image, where u is the surface mean wind speed. The coefficients b1, b2, and b3 are then determined using the least squares regression method.

2 Observational Data

[5] The turbulent data used in this study were collected by the Florida International University (FIU) International Hurricane Research Center 10 m portable tower, which was originally built during the Florida Coastal Monitoring Program [Reinhold et al., 2000]. In 2012, two sets of Gill WindMaster Pro (WMP) sonic anemometers and LI-COR LI-7200 closed path CO2/H2O gas analyzers were installed at 5 m and 10 m to replace the 3-D Gill propeller anemometers and RM Young wind vane anemometers originally installed on the tower. The new sets of sensors measure the 3-D wind components, sonic temperature, and water vapor mixing ratio at 20 Hz in all weather conditions.

[6] In the summer and fall of 2012, the tower was deployed at several coastal locations including Everglades National Park (ENP), Biscayne Bay, and Naples Municipal Airport in Florida and Waveland in Mississippi. The ENP deployment took place from 12 to 16 October 2012 in a small parking lot at Lake Chekika, which was almost entirely under water during the experimental period. The site is surrounded in all other directions by a nearly homogeneous covering of saw grass wetland except for a small patch of plants to the north of the tower. The second deployment was at the FIU Biscayne Bay Campus (BBC) from 21 to 27 October. The tower was deployed on a slightly elevated grass-covered field, which is surrounded by a variety of roughness elements including water to the south and southeast and small bushes, shrubs, and some taller trees in all other directions. The third and fourth tower deployments were associated with Hurricane Isaac. The tower was first deployed at the Naples Municipal Airport from 19:45 EDT on 25 August to 17:30 EDT on 26 August, during which Tropical Storm Isaac passed over Key West, FL. The tower was set in a flat landscape (a mix of concrete and grass) for several hundred meters in all directions. After that, the tower was towed to Waveland, MS, and deployed as Isaac made landfall near the mouth of the Mississippi River. The site contains no structures or vegetation in any direction for approximately 200 m. The data collection started at 11:00 CDT on 28 August and terminated at 16:00 CDT on 30 August.

[7] The data postprocessing includes several data quality control procedures. The method proposed by Schmid et al. [2000] is used to remove spikes caused by insects, feces, and other unknown reasons. The double rotation method [Tanner and Thurtell, 1969] is used to correct the sensor tilting. To obtain statistically meaningful momentum, heat, and moisture fluxes, the data are split into 15 min length segments. Data quality assurance for individual segments includes inspection of linear cumulative summation of covariance, examination of power spectra of wind components, and analysis of ogives.

3 Data Analyses

[8] Using the data collected at 10 m, CD and CDN can be determined based on equations (2) and (3), where momentum and buoyancy fluxes are computed directly via the eddy correlation method. Despite the large scattering of CD and CDN, the estimated CDN falls well within the range of previous studies (Figure 1) and shows a clear increasing trend with a decrease in wind speed for low wind speeds (not shown here). The causes for the large scattering are complex. Different surface roughness at different sites and different weather conditions can all lead to variations in CDN, and CD can be further affected by stability.

[9] To enhance understanding of the large scattering of CD, several analyses were carried out by grouping CD into different categories of wind speed and stability, as well as normalizing CD using different higher-order moments such as turbulent kinetic energy (TKE). It turns out that there is an excellent relationship among CD, TKE, and wind speed. Figure 2a shows CD against TKE for all data collected at the four sites. The data are scattered; however, the scattering can be well explained by different wind speeds. In each wind speed category, there exists a strong linear relationship between CD and TKE. For a better illustration, Figure 2b shows the ratio of CD to TKE against wind speed in a logarithmic coordinate. The same data from the four sites fall nicely along a common line. The slope of the line appears to be universal considering that the data were collected under different surface and weather conditions. To further examine this relationship, data collected during Hurricane Ivan (2004) [Zhu et al., 2010] are also plotted in Figure 2b, which simply extend the line into the high-wind regime along the same slope. Note that the data from Hurricane Ivan were collected by a different instrument (Gill 3-D propeller anemometer) and under different surface and weather conditions. Given that CD is a parameter that relates momentum fluxes to mean wind speed, the strong linear relationship shown in Figure 2b suggests that momentum fluxes are linearly correlated to TKE, and this is confirmed by Figure 2c. Since TKE provides a measure of turbulent intensity, this result suggests that an appropriate parameterization of turbulent momentum fluxes via CD should take the turbulent intensity into consideration.

Figure 2.

(a) CD against TKE for different wind speeds of the data collected at the ENP, BBC, Naples, and Waveland sites. (b) Ratio of CD to TKE against 10-m wind speed in a logarithmic coordinate for the four sites as well as the data collected during Hurricane Ivan (2004). (c) Momentum flux against TKE in a logarithmic coordinate. The data are the same as those in Figure 2b.

4 A New Parameterization for the Drag Coefficient

[10] The strong correlation between momentum fluxes and TKE suggests that the parameterization of momentum fluxes may be enlightened by analyzing the TKE budget. Assuming horizontal homogeneity and aligning the x axis along the mean wind direction, the TKE budget equation may be written as [Stull, 1988]

display math(4)

where e represents TKE, p is pressure, and g is gravity. θ0 and ρ0 are the ambient potential temperature and air density, respectively. The terms in equation (4) from left to right are the local storage, shear production, buoyancy production, vertical transport, pressure correlation, and dissipation of TKE, respectively. The TKE vertical transport and pressure correlation terms are negligible since the surface layer is considered to be the constant flux layer, which means that ideally, fluxes inline image and inline image should be independent of height. The data collected at 5 m and 10 m in this study allow us to compute inline image at two levels, and indeed, they are nearly identical. In the following, we will analyze the TKE budget under steady neutral and nonneutral conditions separately.

[11] Under steady neutral conditions, the TKE budget simply becomes a balance between the TKE shear production and TKE dissipation. In higher-order turbulent closures [Mellor and Yamada, 1974; Deardorff, 1973], ϵe is often parameterized as inline image, where Λis an empirical dissipation length scale. In the surface layer, Λ is often considered as a function of height; i.e., Λ=c2z, where c2 is an empirical coefficient [Stull, 1988]. The momentum fluxes may be written in terms of friction velocity (i.e., inline image). In the previous section, we demonstrated that there exists a robust linear relationship between TKE and u; i.e., inline image, where c1 is a linear regression coefficient. Inserting these relationships into the simplified TKE budget, it yields

display math(5)

Assuming inline imageis constant (equivalent to the Von Karman constant), then, equation ((5)) simply becomes the classic MOS relationship in neutral conditions.

[12] Although c1 appears to be a universal constant as supported by Figure 2c, the strong dependence of CDN on wind speed for low wind speeds suggests that c2 may not be a constant as assumed. In fact, c2 can be directly quantified from Kolmogorov's energy spectrum law [Kolmogorov, 1941], which states that in the inertial subrange, the energy density per unit wave number S(ν) depends only on the angular wave number ν and the rate of energy dissipation ϵe; i.e., inline image, where α is the universal Kolmogorov constant. The turbulence spectra of all the data collected in this study are carefully examined. As an example, Figure 3a shows the power spectra of u, v, and w of an arbitrary spectral lag (15 min) from the ENP data. In the inertial subrange, the spectra of u, v, and w closely follow the inline imagepower law consistent with Kolmogorov's power law. From the obtained spectra, ϵe averaged over the inertial subrange is considered as the mean ϵe of a spectrum lag (15 min). Once ϵe is determined, c2 is estimated from inline image. The spectra of both horizontal and vertical wind components are used to estimate ϵe.

Figure 3.

(a) Power spectra of u, v, and w for an arbitrary 15 min spectral lag from the ENP data. (b) Estimated coefficient inline image against 10-m wind speed for all the spectral lags in near-neutral conditions inline image (black dots), the best fit curve of all black dots (blue thick curve), bin-averaged values of inline image with the velocity interval of 0.5  m s−1 (green circles), and CDN derived from equations (5) and (6) for z0=0.02 m (thick red dashed curve scaled to the right).

[13] Figure 3b shows inline image against wind speed for data in near-neutral conditions inline image. The inline imageincreases substantially with a decrease in wind speed at low wind speeds. The best fit curve of inline image may be written as

display math(6)

where the coefficients are obtained through the least squares regression. The result shown in Figure 3 suggests that the TKE dissipation slows down significantly as wind speed decreases. Physically, this may reflect the change of eddy energy cascade processes in response to a change in wind speed. As wind speed increases and becomes sufficiently large, Hunt and Carlotti [2001] showed that the traditional eddy-to-eddy “staircase”-like energy cascade of the surface layer flow is shortcut and replaced by a much more efficient “elevator”-like energy cascade in which there is direct energy transport from each large eddy beyond a critical scale toward small-scale eddies. Thus, the energy dissipation has to speed up in order to prevent energy accumulation at the small-scale eddies. Conversely, as wind speed becomes sufficiently low, the slower eddy-to-eddy energy cascade process results in smaller dissipation rates. Despite the less efficient energy cascade and possible lack of large eddies at low wind speeds, the efficiency of momentum transport may not be affected since small-scale eddies, such as bursts and sweeps, are the efficient carriers of momentum fluxes [Shaw and Businger, 1985; Mahrt and Gibson, 1992; Högström and Bergström, 1996]. Combining equations (5) and (6), CDN can be determined. The dashed red curve in Figure 3b indicates CDN predicted by equation (6) for z0=0.02 m, which adequately captures the observed variation pattern of CDN (Figure 1).

[14] In steady nonneutral conditions, the TKE buoyancy production needs to be considered. The TKE budget equation then becomes

display math(7)

Normalizing equation ((7) by the shear production term and using the same method that we treat the TKE dissipation term and momentum fluxes as stated previously, it is easy to show that equation (7) can be rewritten as

display math(8)

Comparing equation (8) with the classic MOS formula, it is easy to see that the two equations have a similar format. One advantage of equation (8) is that the effect of stability is directly determined by the TKE budget itself. This contrasts MOS, in which the stability function φm(ζ) has to be determined empirically [e.g., Businger et al., 1970]. Since ϵe has to be positive, equation (8) is valid only for inline image. This is logical since under stable conditions, the TKE shear production must be greater than the buoyancy suppression in order to maintain a steady equilibrium turbulent state; otherwise, turbulence will eventually die away. The MOS formula does not have a limit for stability in stable conditions most likely because the stability function is determined empirically. However, it remains to be a question for MOS as to how a steady equilibrium turbulent state can be maintained when the buoyancy suppression dominates shear production. Nonetheless, the effect of stability in our derivation is consistent with that obtained empirically in MOS. With equations  (6)(8), CD can be readily derived:

display math(9)

5 Summary

[15] One of the important applications of MOS is to provide an analytical expression for CD, a key parameter that relates turbulent momentum fluxes to surface mean wind speed. Although widely used in numerical simulations of weather and climate and many other applications in geosciences and environmental sciences, some characteristics of CD predicted by MOS are not supported by observations. For example, CDN reported from previous observational studies for overland conditions (Figure 1) shows a significant increase with a decrease in wind speed for low wind speeds. However, such a behavior of CDN cannot be explained within the MOS framework, suggesting some assumptions that lead to MOS may not be applicable in calm wind conditions. The underestimated surface wind stress due to the inappropriate determination of CD by MOS for low wind speeds could be an important source of error in many applications that involve the exchange between the atmosphere and the underlying land surface.

[16] In this study, using data collected by a 10 m portable tower, we revisited issues involving CD. Our analyses show that momentum fluxes are well linearly correlated with TKE, indicating that momentum transport in the surface layer is not only controlled by the ambient meteorological properties but also closely tied to turbulent intensity. This also suggests that an improvement of the bulk transfer parameterization of momentum fluxes may be enlightened from analyzing the TKE budget. From the result of the TKE budget analyses, a novel but more generalized parameterization framework for momentum fluxes is proposed. With certain assumptions, the new parameterization can be reduced to the classic MOS in neutral conditions. Most importantly, it well explains the observed variation of CD with wind speed, particularly for low wind speeds. Moreover, the effect of stability, which has to be determined empirically in the MOS framework, can now be determined internally within the system. Lastly, we note that the data used in this study were collected under relatively homogeneous surface conditions. How the inhomogeneous terrain and surface conditions affect the momentum transport in the surface layer and whether the parameterization framework developed in this study can be extended to such conditions are the questions that need to be further investigated.

[17] This study focuses only on CD. In fact, the temperature and moisture data collected in this study can be used to directly compute heat and moisture fluxes, which will allow us to quantify and characterize the exchange coefficients for heat and moisture transport. This will be the focus of our future research.


[18] This work is supported by the National Science Foundation under grant AGS-0847332 and BP/The Gulf of Mexico Research Initiative. We wish to thank the International Hurricane Research Center and Roy Liu and Walter Conklin at the College of Engineering and Computing, Florida International University, for providing help on the installation of the instruments, the tower deployment, and other technical issues. We are very grateful to the two anonymous reviewers for their constructive comments. Their helpful suggestions lead to the substantial improvements of this paper.

[19] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.