Surface Layer Drag Coefficient at Different Radius Ranges in Tropical Cyclones

Abstract

Using dropsonde data and a flux-profile method, this study investigates the drag coefficient ($C_d$)–wind speed relationship within different radius ranges. The results show a systematic decrease of friction velocity $u_{*}$ from the range of R/RMW > 1.05 to that of R/RMW < 0.95 (R is the radial location of a dropsonde profile, and RMW is the radius of maximum wind), and the reduction is 5~25% for different wind speeds. Further, within the ranges of either R/RMW > 1.05 or R/RMW < 1.05, a clear feature of “roll-off” at about 35 m s⁻¹ can be obtained. However, the roll feature becomes vague in the ranges of R/RMW < 0.95, R/RMW < 0.85, and R/RMW < 0.75, indicating the TC dynamics within and near RMW play a role in affecting the flux-profile relationship. Even more, $C_d$ of R < 0.75RMW deviates significantly from the $C_d$ of R < 0.85RMW and R < 0.95RMW, while the deviation between R < 0.85RMW and R < 0.95RMW is much smaller. Especially when 10 m winds exceed 40 m s⁻¹, $u_{*}$ of R < 0.75RMW is significantly larger than that of R < 0.85RMW. This phenomenon is also linked to the TC dynamics (e.g., the large radial gradients of winds and the drastic vertical variation of the bulk Richardson number), but the speculation needs to be verified in future study.

1. Introduction

Tropical cyclone (TC) is one of the most devastating weather systems that cause huge losses of lives and properties [1,2,3,4]. Its enormous energy mostly comes from and dissipates near the surface, which is expressed as enthalpy flux and surface drag [5,6], and the theory is well accepted that the intensity development of TC strongly depends on the ratio of the drag coefficient ($C_d$) and enthalpy flux transfer coefficient ($C_k$) (e.g., [7,8,9,10]).

With field dropsondes observation and the flux-profile method, Powell et al. [11] first found that $C_d$, under strong wind conditions, peaked at a 10-m wind speed of around 38 m s⁻¹. Donelan et al. [12] also found a similar variation pattern with laboratory measurements, only that $C_d$ tended to level off rather than decline after it reached the peak value. While, by using ocean current velocity profiles, Jarosz et al. [13] calculated the drag coefficient with a bottom-up method and also found $C_d$ first increased, but later when the 10-m wind speed was larger than about 32 m s⁻¹, it declined with wind speed. Bi et al. [14] used offshore tower observation and found $C_d$ peaked at 18 m s⁻¹ and decreased afterward, but leveled off when larger than 27 m s⁻¹. Other studies with various methods generally also found the same feature that $C_d$ decreased or leveled off with wind speed when it was larger than around 30 m s⁻¹ (e.g., [15,16,17,18,19,20,21,22,23,24]).

The observation of hurricanes by dropsondes has been carried out for several decades and thousands of profiles have been obtained (e.g., [25,26]), which leads to the hope to thoroughly understand the relationship between drag coefficient and wind speed. Nevertheless, Richter et al. [27] concluded from virtual dropsondes of a numerical simulation that, caused by the sensitivity of this method to the regression procedure, the calculation of $C_d$ with dropsondes observation and the flux-profile method was accurate to within approximately 50%. Richter et al. [28] further showed that the flux-profile method had an inherent underestimation inclination of $C_d$ at hurricane-force winds, which was a result of the uncertainty in the vertical position of the sonde near the surface and the non-monotonic profile of wind speed with height, but these might be mitigated by selecting the regression profile within about 20–150 m. Through virtual dropsondes, Richter et al. [28] also showed that within the radius of maximum wind (RMW), where the traditional Monin-Obukhov similarity theory was not applicable anymore, $C_d$ was drastically underestimated. However, this characteristic was not observed in their study with real dropsondes.

Based on the findings of Richter et al. [28], this study takes a further step to understand how the $C_d$–wind speed relationship is within the RMW based on the real dropsondes. Especially, under the same 10-m wind speed, how the $C_d$ calculated from the flux-profile method varies with RMW. This paper is organized as follows. Section 2 describes the data and the methods used in this paper. Section 3 presents the results, including the variation of friction velocity ($u_{*}$) against R/RMW in the real dropsonde observations. And this is followed by a summary and conclusions in Section 4.

2. Data and Methods

2.1. Data

Dropsonde data used in this study are from the Long-Term NOAA Dropsonde Hurricane Archive released by NOAA’s National Hurricane Center and Hurricane Research Division (https://data.eol.ucar.edu/dataset/542.001 (accessed on 5 December 2021)). The data in this archive include over 13,600 Global Positioning System (GPS) dropsonde profiles from 120 TCs in the Atlantic and East Pacific basins from 1996 to 2012 [26]. The GPS dropsondes are released from aircrafts flying into the TC system, and aircraft types include P3 (flying at 1–5 km altitude in the inner and outer core of TC) and G-IV (flying at 14–15 km altitude in the outside environment of TC). The data quality control of the sounding data is processed by Atmospheric Sounding Processing Environment (ASPEN) software, and various visualization tools and statistical methods are used to evaluate data products to identify and correct data quality problems caused by various errors and deviations. The wind speed is observed at 4 Hz frequency, corresponding to 3–8 m vertical resolution, but prior to 2010, the frequency of wind data was measured at 2 Hz. More detailed information about the dataset can be found in the data archive and Wang et al. [26].

In this work, the radial locations of the profiles (R) are also given by the Dropsonde Hurricane Archive introduced above, while the RMW data are from the Extended Best Track (EBT) dataset [29], which includes 6-h interval RMW data of the corresponding TCs in the Dropsonde Hurricane Archive. The EBT RMW data are then interpolated linearly to fit the R data of the dropsonde profiles.

2.2. Method

Following Powell et al. [11], Holthuijsen et al. [16], and Richter et al. [27], we use the flux-profile method to estimate the drag coefficient $C_d$, which is based on the logarithmic profile of wind speed within the hurricane boundary layer. According to Monin-Obukhov (MO) similarity theory, the wind profile in the neutral condition can be described as

$$U=\frac{u_{*}}{k}\ln (\frac{z}{z_0}),$$

(1)

$$lnz=\left(\frac{k}{u_{*}}\right)U+ln{z}_0,$$

(2)

where $U$ is the wind speed at altitude z and the Von Karman constant $k$ is taken as 0.4. The friction velocity $u_{*}$ and the roughness length $z_0$ can be estimated through logarithmic linear regression. Then $C_d$ can be derived from

$$\tau =\rho u_{*}^2=\rho C_d{U}_{10}^2,$$

(3)

where $U_{10}$ is the 10-m wind speed and $\rho$ is air density.

Following the approach used in Powell et al. [11] and Richter et al. [28], the vertical profiles of wind speed are analyzed in a composite sense, as a function of the mean boundary layer (MBL, here defined as the height of 10 to 500 m) wind speed. That is, the dropsonde profiles are grouped by their MBL mean wind speeds. This study considers only dropsonde profiles with MBL wind speed larger than 20 m s⁻¹, and 3661 dropsonde profiles meet this criterion. Figure 1 presents the number of profiles in each 5 m s⁻¹ MBL mean wind speed bin, which shows a decreasing trend with increasing wind speed. The data are also categorized by R/RMW to analyze the $C_d$–wind speed relationship at different R/RMW ranges. Figure 1 also presents the number of profiles in the ranges of R > 1.05RMW, R < 1.05RMW, and R < 0.75RMW. Here, the value of 1.05 of R/RMW is chosen to divide the profiles within and outside of the RMW, while the selection of 0.75 of R/RMW as a dividing point is determined by the fact that in the range of R < 0.75RMW there are still relatively large quantity observations. It also can be seen from Figure 1 that the sample numbers are less in high wind speed 5 m s⁻¹ bins, and to maintain a relatively large quantity of samples in each MBL mean wind speed group, the wind speed intervals are chosen as 20–25 m s⁻¹, 25–30 m s⁻¹, 30–35 m s⁻¹, 35–40 m s⁻¹, 40–45 m s⁻¹, 45–50 m s⁻¹, 50–60 m s⁻¹, 60–70 m s⁻¹, and 70–90 m s⁻¹ as in

Table 1. The profile number in each MBL mean wind speed group for different R/RMW ranges.

MBL WS Group (Unit: m s−1)	R/RMW Range
	All	>1.05	<1.05	<1.00	<0.95	<0.90	<0.85	<0.80	<0.75
20–25	820	722	98	92	85	77	73	70	64
25–30	619	491	128	117	113	107	100	88	80
30–35	450	355	95	92	87	83	73	64	53
35–40	336	245	91	86	79	76	71	67	59
40–45	251	143	108	92	82	75	68	57	44
45–50	196	104	92	84	74	65	52	39	29
50–60	310	135	175	160	144	128	113	96	83
60–70	213	67	146	139	127	117	107	87	70
70–90	127	35	92	79	70	61	51	41	35
Sum	3322	2297	1025	941	861	789	708	609	517

. Richter et al. [28] used both real and virtual dropsondes to examine the flux-profile method, and they found an underestimation of $C_d$ through the flux-profile method with virtual dropsondes, but not with real dropsondes. By using the virtual dropsondes from a full physics simulation, a large number of profiles can be obtained, and Richter et al. [28] divided these virtual samples by an interval of 0.26RMW (Figures 10 and 11 in Richter et al. [28]). However, for the real dropsondes, they used an interval of 1 RMW due to the limited samples (Figure 3 in Richter et al. [28]). Here, we follow Richter et al. [28] using 1.05RMW as a watershed, and 0.75RMW as the lowest limit to guarantee relatively enough samples (more than ~30 for each 10 m s⁻¹ bin), and between 1.05RMW to 0.75RMW we use an interval of 0.05RMW to see how the flux-profile relationship varied from 0.75 to 1.05 RMW (Table 1).

Next, each measurement in each profile is further binned into 10-m height intervals, and the measurements within each height bin are collected and averaged together. Variability associated with mesoscale, convective, and under-sampled turbulent scales is then removed by averaging all profiles in a given wind speed group [11,30]. Finally, we choose the 20 to 160 m height range over which the fit to the mean profiles will be made, based on the suggestion of Powell [31] that 20–160 m surface layer is more representative for the lowest levels. Meanwhile, Richter et al. [28] and Jiang et al. [32] also indicated that under 20 m height, the vertical position of dropsondes might be contaminated; besides, the regressed $C_d$ began to decrease monotonically once the upper height bound exceeded roughly 150 m. Byrne and Zhang [33] found that there was a transition of the flow from 3-D to 2-D turbulence in the hurricane boundary layer occurring around 150 m, suggesting that above 150 m MO similarity theory does not apply.

The 20–160 m of each MBL mean wind speed group are plotted in Figure 2 with a semi-log coordinate, and a least-squares linear regression fit is used to estimate the roughness length $z_0$ (as intercept), friction velocity $u_{*}$ ($k/u_{*}$ as slope), and 10-m wind speed $U_{10}$, then drag coefficient $C_d$ was computed from Equation (3). It can be seen that the regression coefficient of each group is above 99%, indicating the sample size of each MBL mean wind speed group is large enough, and this guarantees the validity of the fitting results.

3. Results

Figure 3 depicts the $u_{*}$ obtained in this study, but within different R/RMW ranges, and also the $u_{*}$ derived by previous studies. It can be found that the $u_{*}$ obtained from the various studies generally shows the same increasing trend when the 10-m wind speed is less than 40 m s⁻¹, but levels off (or presents a smaller increasing trend) at higher wind speeds. Specifically, the data points obtained from the range of R/RMW > 1.05 and R/RMW < 1.05 bear very little difference, except that only at the 10-m wind speed of about 38 m s⁻¹ does the R/RMW < 1.05 group show an evidently smaller value of $u_{*}$ than the R/RMW > 1.05 group, which is even smaller in the group of R/RMW < 0.75. By using virtual dropsondes, Richter et al. [28] showed that with a decreasing R/RMW, the $u_{*}$ obtained by the flux-profile method tends to be underestimated (Figures 10 and 11 in Richter et al. [28]), which are seen here in the 10-m wind speed of about 38 m s⁻¹. At other wind speed ranges, either no significant difference is observed (e.g., at a 10-m wind speed range of 40–50 m s⁻¹), or the results from the group of R/RMW < 0.75 are even larger than from the group of R/RMW > 1.05 (e.g., at a 10-m wind speed ranges of 50–70 m s⁻¹).

Richter et al. [28] also draw a similar conclusion that the different R/RMW ranges generally do not affect the $C_d$–10-m wind speed relationship (Figure 3), which is a result of the various profiles from many TCs with different sizes and intensities.

To further look into the $u_{*}$–$U_{10}$ relationship under different R/RMW conditions, we plot the variation of $u_{*}$ with R/RMW in the range of >1.05, <1.05, and later on at 0.05 interval to <0.75 (Figure 4). When comparing the $u_{*}$ in the ranges of R/RMW > 1.05 and <0.95, a systematic decrease of $u_{*}$ with R/RMW is generally found for all MBL mean wind speed groups, except the 20–25 m s⁻¹ and 35–40 m s⁻¹ MBL mean wind speed groups. Richter et al. [28] showed by virtual dropsondes, that with a $U_{10}$ of about 65 m s⁻¹ and R/RMW of about 0.85 (and $U_{10}$ of about 40 m s⁻¹ and R/RMW of about 0.7), $u_{*}$ was underestimated by about 10~20% percent (Figure 10c,d in Richter et al. [28]). Here, although the real dropsondes do not repeat these values exactly, a reduction of $u_{*}$ by 5~25% from R/RMW > 1.05 to <0.95 is observed in the 7 out of 9 MBL mean wind speed groups.

Nonetheless, an unexpected increase of $u_{*}$ with R/RMW from R/RMW < 0.85 to <0.75 is found in the groups of an MBL mean wind speed larger than 50 m s⁻¹. This contradicts the findings of Richter et al. [28]. To examine whether this is caused by some extreme values that could bias the mean of the distribution when sample sizes are small, the wind profiles of R < 0.85RMW and R < 0.75RMW in all MBL mean wind speed groups are presented here as Figure 5. It can be found that for all the wind profiles, a logarithmic profile is generally kept, but when the mean MBL mean wind speed is larger than 50 m s⁻¹, an increase of $u_{*}$ (i.e., decrease of the slope of the profile) is found from R < 0.85RMW to R < 0.75RMW. This phenomenon may be caused by the drastic radial gradient of winds and temperature inside the RMW, as shown in previous studies (e.g., Kepert and Wang [34]; Bell and Montgomery [35]; Zhang et al. [36]). Zhang et al. [36] also found that the depth of the non-strongly stable layer (i.e., the layer with a bulk Richardson number < 0.25) decreased significantly from ~700 m at the RMW to ~200 m at 0.5 RMW (Figure 9 in Zhang et al. [36]), which is less than half of our 500 m MBL layer depth. A bulk Richardson number is an important parameter to indicate the turbulent mixing intensity, and in the atmospheric models, 0.25 is usually used as the marker of the boundary layer height. Combining these findings, it is speculated that when it comes to about < 0.75RMW, the logarithmic wind profile is affected by the TC dynamics (e.g., the large radial gradients of winds and the drastic vertical variation of the bulk Richardson number). This phenomenon will be further investigated in the future through a high-resolution simulation study.

Figure 6 shows the $C_d$–10-m wind speed relationship under different R/RMW ranges. Generally, when considering all the data or the data with R/RMW > 1.05, a clear feature of “roll-off” at about 35 m s⁻¹ can be obtained. Even, if considering the whole range of R/RMW < 1.05, such a feature may still roughly exist. However, at the ranges of R/RMW < 0.95, R/RMW < 0.85, or R/RMW < 0.75, the roll feature becomes vague, indicating the TC dynamics within and near the RMW play a role in affecting the flux-profile relationship. On one hand, the drift of dropsondes with the tangential and radial winds may play a role. Real dropsondes are advected with the horizontal wind, and, in the region near the RMW, that advection can move the sondes 10–20 km downwind as the sondes descend to the surface, violating the flux-profile assumptions. Richter et al. [28] concluded that the effects of a dropsonde drift were negligible, and the features of $C_d–U_{10}$ relationship with and without drift effects were similar to each other. Nevertheless, some differences are still seen in the $C_d–U_{10}$ with and without drift effects, especially, the $C_d–U_{10}$ at 0.78 < R/RMW ≤ 1.04 is significantly different: with/without drift effect, $C_d$ is about 0.0018/0.0012, and $U_{10}$ is about 65/52 m s⁻¹ (Figures 10 and 11 in Richter et al. [28]). Therefore, the drift of dropsondes within and near RMW indeed plays a role in the $C_d–U_{10}$ relationship. On the other hand, the assumption of the constant flux layer may not be valid anymore within and near the RMW, as pointed by Richter et al. [28], even though a logarithmic layer is persistent throughout all the different ranges of R/RMW. The violation of the constant flux layer may be caused by the large radial gradient of winds and the transport of the momentum flux (e.g., Kepert [37]; Zhang et al., [38]; Bryan et al., [39]).

In addition, in Figure 6b, the $C_d$ of R < 0.75RMW (i.e., green square) deviates significantly from the $C_d$ of R < 0.85RMW and R < 0.95RMW (i.e., brown and pink squares), while the deviation between R < 0.85RMW (brown squares) and R < 0.95RMW (pink squares) is much smaller, especially when 10 m winds exceed 40 m s⁻¹. As discussed previously, when it comes to about < 0.75RMW, the logarithmic wind profile may be altered by the drift, the large radial gradients, and the advection of the momentum flux here.

4. Summary and Conclusions

Using dropsonde data and flux-profile method, in this study, we investigated the $C_d$–wind speed relationship within different R/RMW ranges, and also the variation of $C_d$ with R/RMW under the same wind speed bins. The main findings and conclusions are as follows:

(1)

From the range of R/RMW > 1.05 to that of R/RMW < 0.95, a systematic decrease of 5~25% in friction velocity $u_{*}$ is found for 7 out of the 9 MBL mean wind speed groups.

(2)

A clear feature of “roll-off” at about 35 m s⁻¹ can be obtained for the ranges of either R/RMW > 1.05 or R/RMW < 1.05. However, in the ranges of R/RMW < 0.95, R/RMW < 0.85, and R/RMW < 0.75, the roll feature becomes vague, indicating the TC dynamics within and near the RMW play a role in affecting the flux-profile relationship.

(3)

When $U_{10}$ exceeds 40 m s⁻¹, $u_{*}$ of R < 0.75RMW is significantly larger than that of R < 0.85RMW. This phenomenon is not caused by the extreme values in the relatively small sample size, and it is speculated to link with the TC dynamics (e.g., the large radial gradients of winds and the drastic vertical variation of the bulk Richardson number).

At last, these results may be affected by the limited data quantity in this study, and future work will be carried out with more observations and simulations to further understand how the TC dynamics within and near the RMW affect the wind profiles in TCs.