Interpreting Dropsonde Measurements of Turbulence in the Tropical Cyclone Boundary Layer

P1D.7

Jeffrey D. Kepert ∗ Centre for Australian Weather and Climate Research A partnership between the Australian Bureau of Meteorology and CSIRO

AMS Hurricanes Conference, Apr 28 - May 2, 2008

1. Introduction Dropsonde 981820069 90 Some of the most dangerous winds on earth occur in tropical cyclones. Their de- ) structive power depends as much on their −1 80 gustiness, or turbulence, as on their mean strength. Further, the efficiency with which 70 energy is extracted from the ocean to power the tropical cyclone depends strongly on 60 the boundary layer turbulence. Thus understanding turbulence in the tropical cy- 50 clone boundary layer is crucial. At one Azimuthal velocity (m s level, this understanding should be a simple 40 problem because stability effects become −50 −40 −30 −20 −10 0 10 20 small as the wind speed increases. At an- Radial velocity (m s−1) other, the problem is complex because the strong winds severely modify the air-sea in- Fig. 1. Measured wind hodograph from a terface. Finally, testing theoretical predic- dropsonde deployed into the eyewall of Hur- tions and extrapolations from lower wind ricane Georges on 19 September 1998. speeds is problematic due to the severe dif- ficulty in taking observations where it mat- ters, in close to the interface. layer (Franklin et al. 2003; Wroe and Barnes Over the last decade, U.S. hurricane re- 2003; Schneider and Barnes 2005; Kepert connaissance aircraft have deployed thou- 2006a,b; Montgomery et al. 2006; Bell and sands of dropsondes in and near the eye- Montgomery 2008; Schwendike and Kepert wall of tropical cyclones. These instruments 2008), but the high sampling rate implies parachute towards the surface, reporting that there is a considerable potential to il- back wind, temperature, humidity and pres- luminate the turbulent structure as well. sure at ½-second intervals, or about every Figure 1 shows the hodograph of a sin- 6 m vertically (Hock and Franklin 1999). gle dropsonde deployed into the eyewall of These measurements have had a consider- Hurricane Georges (1998). It is clear that able impact on our knowledge of the mean the measured wind profile is consistent with structure of the tropical cyclone boundary many of the characteristics of turbulence: it is random, spatially correlated, contains ∗Centre for Australian Weather and Climate Re- search, Bureau of Meteorology, 700 Collins St, Mel- multiple scales, and is vortical. bourne Vic 3000. Email: [email protected] This paper presents a theoretical frame-

1 work for interpreting the turbulence signal Charnock’s relation (3) in combination with within these measurements. Proper inter- (1) and (2) is quite consistent with a linear pretation requires close attention to the re- increase of CD over the wind speed range of sponse characteristics of the instrument, interest. careful quality control, and a suitable theoretical description of 3-dimensional surface b. Dropsonde dynamics layer turbulence. In this section, a theory is developed for Several useful results can be inferred from the mean and turbulent accelerations expe- the comparison of theory and measurement. rienced by a dropsonde as it falls through This paper will focus on one of these, the the atmospheric surface layer. The turbu- marine drag coefficient C . The main result D lent part of the theory is an extension of Kris- is that C does not increase indefinitely with D tensen (1993)’s analysis of cup anemome- wind speed, thus confirming recent direct ter dynamics to the case where the mean measurements (Powell et al. 2003; Black flow and the turbulence are not stationary et al. 2007; French et al. 2007) by a new in time. and independent technique, and extending The dropsonde is a first-order linear filter those results to much higher wind speeds. of the true wind (Hock and Franklin 1999): ˙ 2. Theory ˜(t) = s˜(t) + τ0s˜(t) (4) a. Air-sea momentum exchange where (,˜ ˜) is the instantaneous horizontal wind vector, (s˜, s˜y) is the horizontal compo- The surface stress τ may be written nent of the dropsonde velocity, τ0 = ƒ /g ≈ 1.2 s is a time scale, and ƒ is the dropsonde τ = ρC U2 = ρ2 (1) D 10 ∗ fall velocity. We adopt the usual surface- layer convention that the -axis is in the where ρ is the air density, U10 is the 10-m mean wind direction, y is to the left and z mean wind speed and CD the drag coeffi- is upwards. The solution of (4) is (Kristensen cient. Over the ocean, CD is thought to in- 1993) crease approx linearly with wind speed, as ∞ reflected in numerous empirical parameteri- 1 s˜ (t) = e−τ/τ0 ˜(t − τ)dτ (5) sations, for example Large and Pond (1981).  τ0 Z0 The drag coefficient can be related to the We decompose all variables into mean and surface roughness length z0 through the logarithmic wind profile, fluctuating components, ˜(t) = U(t) + (t), s˜(t) = S(t) + s(t), ∗ z U(z) = log (2) U(t) = S(t) + τ0S˙(t) (6) k z0 (t) = s(t) + τ0s˙(t) (7) with z = 10 m, where k is von Kärmän’s Here, the ergodic hypothesis is unavail- constant and the roughness length z over 0 able because the dropsondes are falling the ocean is frequently parameterised after through gradients of both mean and turbu- Charnock (1955) lence quantities, and so “mean” should be 2 interpreted as an ensemble-average follow- ∗ z0 = α . (3) ing the dropsonde trajectory. g In the case where the mean wind profile U(z) is logarithmic (2), an analytic solution Here α, the Charnock coefficient, is an em- is available for (6), pirical constant and g is the gravitational acceleration. The value α = 0.011 is widely ∗ t ( ) = ( ) − −t/τ0 used, see for example Fairall et al. (2003). S t U t e Ei , (8) k τ0

2 where Ei is the exponential integral func- With initial condition 〈s2(−∞)〉 = 0, the solu-  tion and the dropsonde fall velocity has been tion is used for the space-time conversion, t = −ƒ z. 〈s2(t)〉 An expression for the variance of the drop-  2 ∞ sonde acceleration is found by multiplying −2τ/τ0 = e 〈(t − τ)s(t − τ)〉dτ (16) (7) in turn by , s and s˙, ensemble- τ0 Z0 averaging, and a little manipulation which can be evaluated from (13) by a nu- 2 2 〈 〉 − 2〈s〉 + 〈s 〉 merical integration. 〈s˙ 2〉 =  (9)  2 Note that τ0 ∞ 4π2ƒ 2τ2 Angle brackets denote an ensemble average 2 0 〈 〉−〈s〉 = S(ƒ )dƒ = H at any fixed point in the dropsonde trajec- Z 2 2 2 −∞ 1 + 4π ƒ τ0 tory. We now proceed to evaluate the right- (17) hand side of (9). and Taking the fluctuating part of (5), multiplying by  and ensemble-averaging, 2 ∞ 〈s2〉−〈s 〉 = H − H e−2τ/τ0 dτ (18)     ∞ τ0 Z0 −τ/τ0 〈s〉 = R(−τ)e d(τ/τ0), (10) Z0 where H is a high-pass filtered wind velocity variance. Thus, from (9) the dropsonde where 2 -acceleration variance 〈s˙ 〉 depends only R(−τ) = 〈(t)(t − τ)〉 (11) on the high-frequency part of the turbulence is the autocovariance of the along-stream spectrum, consistent with the fact that dif- wind component with lag −τ. We represent ferentiation is a high-pass filtering opera- R by its spectrum S, tion. ∞ Dropsondes do not measure a simultane- −2πƒ τ R(−τ) = S(ƒ )e dƒ (12) ous wind profile, but rather a time-series Z−∞ along a slant trajectory. Thus the above the- Thus the covariance between the - ory requires the autocorrelation of the time- component of the dropsonde and wind ve- series along the dropsonde slant trajectory: locities is R(−τ) ≡ R(−Uτ, 0, ƒ τ, −τ) ∞ 1 (19) = R(0, 0, ƒ τ, 0), 〈s〉 = S(ƒ )dƒ (13) Z 2 2 2 −∞ 1 + 4π ƒ τ0 where the equality with the instantaneous Note that the wind velocity variance can also vertical profile follows by Taylor’s hypothe- be written in terms of S by virtue of Parse- sis. val’s theorem, Thus the one-dimensional line spectrum in 3 ∞ the vertical direction, F (k ), is required. 11 3 2 〈 〉 = S(ƒ )dƒ (14) There seem to be no published forms for Z−∞ such spectra in the atmospheric surface layer, but they may be derived from the full Multiplying (7) by s and ensemble- averaging gives another equation for a first- spectral velocity tensor, theoretical forms of order linear filter, which were given by Kristensen et al. (1989) and Mann (1994). Here, we shall adopt the τ ∂〈s2〉 2 2 0  former. 〈s〉 = 〈s 〉 + τ0〈ss˙〉 = 〈s 〉 + ,   2 ∂t The one-dimensional along-stream turbu- (15) lence line spectrum is defined as the Fourier

3 transform of the along-stream autocorrela- 0 10 tion 1 ∞ F1(k ) = e−k1r1 R1(r )dr (20) j 1 j 1 1 2π Z −2 −∞ 10 where the indices , j = 1, 2, 3 refer to the

three Cartesian axes. The spectral tensor is n S the extension of this transformation to three −4 dimensions, 10 1 −k·r j(k) = Rj(r)e dr (21) 3 3 (2π) ZZZR −6 10 −4 −2 0 2 Line spectra in any direction may be ob- 10 10 10 10 n tained from j(k) by integrating out the components of k normal to the desired di- Fig. 2. Atmospheric surface-layer velocity rection. In particular, vertical line spectra line spectra, for the  (blue),  (red) and  are obtained by (green) velocity components. The dashed ∞ ∞ curves are for line spectra in the along- F3(k ) =  (k , k , k )dk dk (22) j 3 j 1 2 3 1 2 stream direction, and the solid curves are for Z−∞ Z−∞ the vertical. Figure 2 shows the variance vertical line- spectra F3 (along-stream), F3 (cross- 11 22 the excellent fit of (23) to the observed PDF stream) and F3 (vertical) so found, com- 33 is apparent, along with it’s superiority to a pared to the well-known along-stream line- Gaussian. The fit is slightly less success- spectra of Kaimal et al. (1972). The differ- ful for the vertical component of accelera- ences include the expected 4/3 ratios in the tion, probably due to an unequal dropsonde inertial subrange. response to upwards and downwards turbu- Velocity differences  in turbulence over lent fluctuations leading to a slight skewing small time or space scales have a probability of the observed distribution. density function (PDF) (Castaing et al. 1990) In summary, this theory allows us to cal- culate the dropsonde acceleration variance p() due to turbulence within the atmospheric ∞ 1 2 2 2 2 surface layer, as a function of height and = e− /(2σ )e− log(σ/σ0) /(2λ )dσ 2 friction velocity only. The theory has been Z0 2πλσ (23) presented for the -component of acceleration variance, with the main changes for This distribution is equivalent to drawing the y-component being that the mean ac- data from an infinite ensemble of Gaussian celeration is 0 instead of (8) and the use of distributions with mean 0 and variance σ2, the appropriate spectrum. Given these vari- where log(σ) is drawn from a Gaussian with ances and the empirical λ = 0.6, Castaing 2 mean log(σ0) and variance λ . It has “fatter et al.’s distribution then allows calculation tails” than a Gaussian, and has been shown of the PDF of the acceleration components to fit laboratory (Castaing et al. 1990) and due to turbulence. The dropsonde will also atmospheric (Boettcher et al. 2003) turbu- experience an acceleration due to its falling lence data well. We found that λ = 0.6 gave through a sheared mean flow, given by dif- a good fit to tropical cyclone dropsonde ferentiating (8). Combining the mean and data. An example is shown in Fig 3, where turbulent components gives the PDF of the

4 Accelerations Hurricane dropsondes Raw 40−50 m/s

0 0 0 10 10 10

−1 −1 −1 10 10 10

−2 −2 −2 10 10 10 Frequency

−3 −3 −3 10 10 10

−4 −4 −4 10 10 10 −5 0 5 −5 0 5 −5 0 5 x−accel y−accel z−accel

Fig. 3. Comparison of acceleration PDF’s. Black curves: Observed acceleration from dropsondes at all heights and with BLM wind speeds in the range of 30 – 40 m s−1. Blue curves: Gaussian with the observational variance. Red curves: Castaing et al.’s PDF (23) with λ = 0.6 and the observational variance. In this ﬁgure only, the -axis is oriented eastwards rather than in the mean ﬂow direction.

total acceleration over the layer, which will BLM), as in Powell et al. (2003). The mean be compared to observations below. These wind direction is calculated over the lowest theoretical calculations were performed us- 100 m before splash, and the three accelera- ing a Monte Carlo technique for a range tion components in that reference frame are of friction velocities and 10-m wind speeds, calculated by finite differences. Histograms and the acceleration PDFs accumulated over of acceleration frequency are accumulated depths0 – 50 m, 0 – 100 m and 0 – 150 m. in 0.1 m s−2-wide bins over the same height ranges as for the theoretical calculation. To date, all 5248 available dropsondes up to 3. Observations the end of the 2002 hurricane season have been processed. The GPS dropsonde has been described by Hock and Franklin (1999). The filtering of the true wind by the inertia of the instrument 4. Results has been taken into account by the above a. Comparison to Charnock (1955) theory; thus we use raw soundings with no motion correction and no filtering applied. Figure 4 displays the theoretical and ob- The profiles are subjected to quality control served horizontal acceleration PDFs when checks for fall velocity consistency between ∗ is calculated from the 10-m mean wind GPS and pressure measurements, and for speed using Charnock’s relation (3) with co- valid splash point and launch detection. The efficient 0.011 (Fairall et al. 2003). For BLM soundings are stratified into 10 m s−1-wide wind speeds below 30 m s−1 (10-m mean bins according to the mean wind speed be- wind up to about 23 m s−1, the upper two low 600-m height (the boundary layer mean, panels), excellent agreement is obtained be-

5 tween theory and observations. As surface momentum transfer is generally believed to be well-understood in this wind speed range, this agreement provides conﬁrmation of the validity of the method. As the BLM wind speed increases, a dis- crepancy between theory and observations u =10.2, u =0.37 λ=0.6 u =18.0, u =0.74 λ=0.6 10 * 10 * appears and grows. Relative to the the- 0 0 10 10 ory, high accelerations are too rarely, and −1 −1 10 10 low accelerations too commonly, observed. −2 −2 Dropsondes tend to fail near the surface at 10 10 high wind speeds (Franklin et al. 2003; Pow- −3 −3 Frequency 10 10 ell et al. 2003), possibly due to a failure of −4 −4 10 10 the GPS sensor to maintain synchronisation 0 2 4 6 8 0 2 4 6 8 with the satellite under extreme accelera- u =27.5, u =1.28 λ=0.6 u =33.0, u =1.62 λ=0.6 10 * 10 * tion. The paucity of high accelerations ob- 0 0 10 10 served could be due to this failure; however, −1 −1 if the failures were due to this reason alone 10 10 then the relative gap between the theoret- −2 −2 10 10 ical and observed curves in Fig 4 would be −3 −3 Frequency 10 10 the same at any given acceleration in all −4 −4 10 10 panels. This is not so, so we conclude that 0 2 4 6 8 0 2 4 6 8 dropsonde failure is not the dominant rea- u =41.0, u =2.17 λ=0.6 u =50.5, u =2.90 λ=0.6 son for the poor agreement. 10 * 10 * 0 0 Rather, the paucity of high accelerations is 10 10 accompanied by an abundance of low ones. −1 −1 10 10 Reducing the surface roughness in the theo- −2 −2 10 10 retical calculation would lead to a decrease −3 −3 Frequency 10 10 in the turbulence intensity and hence in −4 −4 the relative frequency of high accelerations. 10 10 0 2 4 6 8 0 2 4 6 8 Hence we conclude that Charnock’s relation, acceleration (m s−2) acceleration (m s−2) within the theory presented here, overesti- mates the marine surface roughness at extreme wind speeds. Fig. 4. Theoretical (black) and observed Similar results were obtained when Large (blue) PDFs of dropsonde acceleration in the and Pond (1981)’s parameterisation of CD lowest 100 m of the atmosphere. The BLM was used in place of Charnock’s relation. wind speed ranges are, from top left, 10 – 20 m s−1, 20 – 30 m s−1, 30 – 40 m s−1, b. Comparison to Powell et al. (2003) 40 – 50 m s−1, 50 – 60 m s−1, and over Figure 5 shows a similar comparison to 60 m s−1. The observed mean 10-m wind Fig. 4, except that the roughness lengths de- speeds are as shown, together with friction rived by Powell et al. (2003) are used. Excel- velocities calculated using Charnock’s for- lent agreement is obtained between theory mula. and observation. We thus conclude, subject to the limitations of the dropsonde acceleration theory, that Powell et al. (2003)’s roughness lengths are consistent with the observations of dropsonde acceleration. Estimating error bars in a complex calcu-

6 u =27.5, u =1.20 λ=0.6 u =33.0, u =1.50 λ=0.6 erations and insufficient low ones, in com- 10 * 10 * parison to the observations. In contrast, 0 0 10 10 the drag coefficients determined by Powell −1 −1 10 10 et al. (2003) produce excellent agreement −2 −2 between theory and observations. 10 10 Note that the method presented here uses −3 −3 Frequency 10 10 a completely different part of the dropsonde −4 −4 10 10 signal to Powell et al. (2003), who fit loga- 0 2 4 6 8 0 2 4 6 8 rithmic profiles to the data, thereby effec- u =41.0, u =1.70 λ=0.6 u =50.5, u =1.90 λ=0.6 tively applying a low-pass filter. In contrast, 10 * 10 * 0 0 we differentiate the profile to obtain the ac- 10 10 celeration, a high-pass filtering operation. −1 −1 10 10 These results should be regarded as pre- −2 −2 10 10 liminary. Work is ongoing to

−3 −3 Frequency 10 10 • process dropsonde data from 2003 on- −4 −4 10 10 wards, 0 2 4 6 8 0 2 4 6 8 acceleration (m s−2) acceleration (m s−2) • use narrower velocity bins, • perform statistical significance testing, Fig. 5. As for Fig. 4 except for the highest four wind speed ranges only, and using the • compare results using the spectral ve- roughness lengths obtained by Powell et al. locity tensor of Mann (1994) in place of (2003) in place of Charnock’s. that of Kristensen et al. (1989), and • consider along-stream and cross-stream lation such as this is difficult and has not acceleration components separately. yet been attempted. However, increasing These results will be reported in due course.  by 10% significantly degrades the agree- ∗ Nevertheless, these preliminary results ment in Fig. 5 (not shown). Given that confirm earlier studies. The marine drag co- C = ( /U )2, we therefore argue that D ∗ 10 efficient does not increase indefinitely with drag coefficients derived from these  val- ∗ wind speed in tropical cyclones, but rather ues are accurate to within 20% or better. reaches a maximum at a 10-m wind speed in the vicinity of 30 – 35 m s−1. This confir- 5. Conclusions mation is independent of earlier studies that produced a similar result, as it uses a differ- We have developed a theory, with free pa- ent part of the data and relies upon a novel rameters ∗ and 10, for the mean and tur- and independent technique. bulent accelerations experienced by a dropsonde as it falls through the atmospheric surface layer. These free parameters may 6. Acknowledgement be adjusted to fit the theory to observa- I am immensely grateful to the NOAA tions, and crucial air-sea exchange parame- Hurricane Research Division for making the ters such as the roughness length and drag dropsonde data available that was used in coefficient thereby derived. this research. The results show that the widely-used Charnock relation, and similar empirical pa- References rameterisations of CD such as Large and Pond (1981), lead to a prediction at extreme Bell, M. M. and M. T. Montgomery, 2008: Ob- wind speeds of overly frequent high accel- served structure, evolution and potential

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