A Study on the Relation between Terminal Velocity and VHF ...

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A Study on the Relation between Terminal Velocity and VHF Backscatter from Precipitation Particles Using the Chung-Li VHF Radar YEN-HSYANG CHU, SHUN-PENG SHIH, CHING-LUN SU, KAN-LIN LEE, TZER-HORNG LIN, AND WEI-CHUNG LIANG Institute of Space Science/Center for Space and Remote Sensing Research, National Central University, Chung-Li, Taiwan (Manuscript received 25 August 1997, in final form 5 January 1999) ABSTRACT The relationships between the mean Doppler terminal velocities V T of the hydrometeors with different phases and the range-corrected VHF backscatter P from the corresponding precipitation particles are investigated by using the Chung-Li VHF radar. The radar precipitation data employed for the analysis were taken from four independent experiments conducted on different weather conditions. They show that the observed a and b values in the power-law approximation V T 5 aP b above the melting layer are generally smaller than those below the layer, while in the bright band the values of b (a) are enormously smaller (greater) than those above and below the bright band. Theoretical analysis shows that the mathematical relationship between a and b can be approximated very well by a simple exponential function, which is in excellent agreement with the observations. A new method for estimating the coefficient A and exponent B in the fall speed–diameter relationship V(D ) 5 AD B with respect to still air on the basis of the theoretical relation between a and b is also proposed. A comparison of the estimations of A and B with those reported in the literature indicates that the former are smaller than the latter. The authors believe that the difference in the two is due to the different types of the clouds producing the precipitation. In addition, it is found that the value of A is proportional to the height of the 08C isotherm, implying that the air temperature plays a role in establishing the relationship between the fall speed and the diameter of the hydrometeors.

1. Introduction The capability of observing precipitation made with VHF/UHF Doppler radar (or wind profiler) has been recognized by the scientific community for many years. By using these kinds of radars, several important precipitation-related parameters and phenomena can be measured, such as the size distribution of precipitation particles (Wakasugi et al. 1986; Currier et al. 1992; Rajopadhyaya et al. 1993), the brightband structure (Chu et al. 1991; Gossard et al. 1992; Ralph 1995), the lightning stroke in a thunderstorm (Rottger et al. 1995), the severe depletion of the clear-air echo intensity associated with intense updrafts (Chu and Lin 1994; Chu and Song 1998), the nonfrozen property of hydrometeors in the background wind (Chu et al. 1997), and so on. Recently, the relationship between the mean Doppler terminal velocity of precipitation particles V T , which is obtained by removing the contribution of vertical air velocity from the mean Doppler velocity of an observed precipitation Doppler spectrum, and the range-corrected

Corresponding author address: Dr. Yen-Hsyang Chu, Institute of Space Science, National Central University, 32054 Chung-Li, Taiwan. E-mail: [email protected]

q 1999 American Meteorological Society

echo strength P (or radar reflectivity factor Z defined in section 2) was also studied by using UHF radar combined with VHF radar (Chilson et al. 1993; Ulbrich and Chilson 1994). Chilson et al. (1993) analyzed the relationship between V T and UHF (430 MHz) radar reflectivity Z for the hydrometeors above the melting level associated with a thunderstorm and found the relationships of V T 5 0.21Z 0.34 and V T 5 0.91Z 0.16 , where the former corresponds to the data taken during the early development of the storm and the latter to the convective region. However, their data show considerable scatter about the fitted line, and they attributed it to the effects of lightning and various phases of hydrometeors in the thunderstorm. This paper is an attempt to study the V T–P relationship not only for the solid hydrometeors above the freezing level but also for the melting and liquid particles occurring within and below the bright band (i.e., melting layer) by using the Chung-Li VHF radar. Because the Chung-Li VHF radar is not well calibrated and thus cannot measure the true radar reflectivity factor, the V T– P relationship, not the V T–Z relationship, is investigated in this study. In light of considerable changes in their reflectivities and particle size distributions (Battan 1973), the corresponding V T–P relationships for solid, melting, and liquid hydrometeors will be significantly

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different from each other (Gossard et al. 1992; Ralph 1995). To the authors’ knowledge, it seems that not only the V T –P relationship from VHF precipitation echoes has not been established yet, but the observation of the V T –P relationship for melting particles in the melting layer is also not well documented. Furthermore, it appears that correlation analysis of the coefficient a and exponent b in the power-law approximation V T 5 aP b for VHF precipitation echoes is not made as well. It is one of the major objectives of this article to quantitatively analyze the VHF precipitation backscatter and to find the corresponding V T –P relationships for the precipitation particles above, within, and below the melting region. The other purpose of this paper is to explore a new method for estimating the coefficient A and exponent B in the power-law approximation to the fall speed–diameter relationship of precipitation particles V 5 AD B . For that purpose, we investigate the theoretical relationship between a and b and find that their relation can be approximated perfectly by an exponential function. In order to achieve more general and convincing results, the radar data employed for the analysis in this article were taken from four separate experiments conducted on different days by using the Chung-Li VHF radar. The authors hope that through this investigation several important questions regarding the V T –P relationship can be clarified, such as the characteristics of the height variation of the V T –P relationship in the subtropical region, the difference of the V T –P relationships among solid, melting, and liquid hydrometeors, the validation of the power-law approximation to the relation between fall speed and diameter of precipitation particles with different phases, the influence of VHF backscatter from the melting particles at the height of 08C isotherm on a height-averaged V T –P relationship, and the differences of the magnitudes of A and B estimated in this article from those reported in the earlier literature. In section 2, a theoretical analysis of the V T –P relationship for VHF radar returns will be introduced. The method of estimating the values of A and B in V 5 AD B is also proposed in this section. The observational results and the discussions on the characteristics of the estimated A and B will be presented in section 3. The conclusions are stated in section 4. 2. Theoretical consideration When a VHF radar is employed to observe the precipitation, the echoing mechanism can be thought to be the Rayleigh scattering due to the fact that the radar wavelength is larger than the radius of precipitation particle by more than three orders of magnitude. In the case of a Gaussian antenna beam pattern, the radar equation that links the average echo power Pr from precipitation with the radar parameters and the reflectivity factor Z can be expressed as (Battan 1973; Rogers 1979)

Pr 5 (cP t G 2p 3tuw|K| 2 Z )/(1024 ln2R 2l 2 ) 5 C|K| 2 Z/R 2 ,

(1)

where P t is the peak transmitted power (W); c is light speed (m s21 ); t is the pulse length (s); u and w are, respectively, the 3-dB widths of the major and minor axes of the antenna beam; l is the radar wavelength (m); G is the antenna gain; C is a constant depending on the radar parameters (W m21 ); K 5 (m 2 2 1)/(m 2 1 1); m is the complex index of refraction of the hydrometeor; R is the range (m); and Z (cm 6 m23 ) is defined by Z5

O D 5 E N(D)D dD, 6 1

6 1

(2)

where D i (in cm) is the diameter of the ith precipitation particle, N(D) dD is the number of the precipitation particles with diameters between D and D 1 dD per unit volume, and S denotes the summation over unit volume. Note that from Eq. (1) it is apparent that the range-corrected echo power Pr R 2 is proportional to the reflectivity factor Z. Assume that N(D) takes the exponential form (Marshall and Palmer 1948) N(D) 5 N o exp(2dD),

(3)

where N o (cm m ) and d (cm ) determine the intensity and breadth of the particle size distribution, respectively. Substituting Eq. (3) into (2) yields (Atlas et al. 1973) 21

23

21

Z 5 6!N o /d 7 .

(4)

The mean Doppler terminal velocity of precipitation particles V T (m s21 ) is defined by VT 5

E

@E

V(D)N(D)D 6 dD

N(D)D 6 dD,

(5)

where V(D) (m s21 ) is the hydrometeor fall speed with respect to the still air. Assume that the hydrometeor fall speed can be approximated by a power law in diameter of the form (Atlas et al. 1973) V(D) 5 AD B ,

(6)

where A is in m s21 cm2B . Equation (6) has been employed by many researchers in investigating the hydrometeor size distribution, V T–P relationship, and other applications (Atlas et al. 1973; Leary and House 1979; Hauser and Amayenc 1981). Note that Eq. (6) cannot apply to the raindrop with a diameter greater than 5.8 mm. Beyond that, the raindrop will become unstable and eventually break up due to the vibration and deformation of the drop itself (Gunn and Kinzer 1949). Substituting Eq. (6) into (5) and eliminating d using Eq. (4), we have (Atlas et al. 1973) VT 5

AG(7 1 B)Z B/7 , [G(7)(720N o ) B/7 ]

(7)

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where G(x) is the complete gamma function. Because Z relates to Pr in accordance with Eq. (1), substituting Eq. (1) into (7) results in V T 5 aP b ,

(8)

where the expressions of a and b are given below:

a5

AG(7 1 B) [G(7)(720N o C|K| 2 ) B/7 ]

(9)

and

b 5 B/7

(10)

and P (5Pr R 2 ) in Eq. (8) is the range-corrected echo power (in W cm 6 m24 ). Note that Eqs. (2)–(7) derived above can also be found in the review of Atlas et al. (1973). Although Eq. (8) is obtained under the assumptions of the power-law approximation to the fall speed–diameter relation and the exponential form of the particle-size distribution, the power-law relation between V T and P can also be derived for other approximations to the fall speed–diameter relation and particlesize distribution (Ulbrich and Chilson 1994). It is worth pointing out that the power-law approximation to the relationship between V(D) and D is applicable near the ground surface. If this expression is employed at other altitudes, a correction factor should be multiplied to the right-hand side of Eq. (6). Although a number of different correction factors have been introduced by many investigators (Lin et al. 1983; Beard and Heymsfield 1988; Zrnic et al. 1993), the correction factor employed in this study will be (r 0 /r) 0.4 (Foote and du Toit 1969; Atlas et al. 1973), where r 0 and r are the air density at the ground and at the level of observation, respectively. It is obvious that the coefficient a and exponent b in Eq. (8) can be estimated through the best fit to the observed data, provided the precipitation echoes are measured accurately and the contribution of the vertical air velocity to the observed mean Doppler fall speed of hydrometeors is removed totally. Superficially, once a and b are obtained, the parameters A and B can be calculated in accordance with Eqs. (9) and (10). However, examining Eq. (9) shows that the calculation is complicated and the parameters C, K, and N o are involved in the calculation. Obviously, the value of A estimated in this way will be inaccurate if the radar is not well-calibrated and the exact drop size distribution is not known a priori. Therefore, it is not appropriate to use Eq. (9) to compute directly the A value, and we have to find another more accurate method to estimate the A value so that the effects of C, K, and N o can be avoided. Inspecting Eq. (9) in more detail shows that a will be identical to A if b is zero. Moreover, from Eq. (9) we also note that a as the function of A will be inversely proportional to B because the change rate of G(7 1 B)/G(7) with B in the range 0 , B , 1 is considerably sharper than that of (720N o C|K| 2 ) B/7 . On the basis of the general behavior of Eq. (9) as mentioned above, we find that a simple equation in the exponential

FIG. 1. Curves showing negative dependence of a on b for A 5 14.2 (cross), 16.9 (open circle), and 21.2 m s21 cm2B (plus sign) in Eq. (9), where the discrete curves are the realizations of Eq. (9) and the continuous curves are the best fitted curves to the discrete data points in accordance with Eq. (11). For the details of the parameters used for the calculation, see the text.

form can perfectly describe the mathematical relation between a and b as given below

a 5 A exp(2jb),

(11)

where the magnitude of j is the functions of C, K, and N o . Therefore, a more accurate value of A is thus obtained by best fitting Eq. (11) to the data and letting b be zero in the equation. Figure 1 shows the examples of how a changes with b (5B/7) for the case of a liquid hydrometeor, where the discrete data points marked with a cross, open circle, and plus sign correspond, respectively, to the data calculated from Eq. (9) with A values of 14.2, 16.9, and 21.2 m s21 cm2B (Atlas et al. 1973), and the continuous curves are the realizations of Eq. (11), which is best fitted to the respective discrete data points. Calculations show that the residual errors for the curve fitting are less than 0.03%. The common parameters adopted in the calculations are listed below: c 5 3 3 10 8 m s21 , l 5 5.77 m (corresponding to 52 MHz), u 5 w 5 0.129 rad (corresponding to 7.48), t 5 2 3 1026 s, P t 5 30 kW, G 5 10 3 (corresponding to 30 dB), |K| 2 5 0.93, and N o 5 8 3 10 4 m23 cm21 . As shown in Fig. 1, it is very clear that a indeed decreases exponentially with the increase of b, and the curve of Eq. (11) agrees perfectly with the data calculated from Eq. (9), where the value of j as the functions of C, K, and N o for these three cases is 28.8393. Because the magnitude of |K| 2 and the values of A and B vary significantly with the change in the phase of hydrometeor (Atlas et al. 1973; Battan 1973; Ulbrich and Chilson 1994), it is expected that the corresponding values of A, j, a, and b will be considerably different for the precipitation echoes from the hydrometeors above, with-

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TABLE 1. Radar parameters employed in the experiments.

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Date

t ( ms)

IPP ( ms)

T (h)

Ncoh

Nincoh

NFFT

Jun 1988 Dec 1994 Dec 1995 Feb 1996

2 2 2 2

500 500 400 350

2.55 5.35 4.65 1.5

500 300 300 200

7 6 5 6

128 64 64 128

in, and below the melting layer. The observational results will be presented later. 3. Experimental setup The precipitation data presented in this article were taken from four independent experiments conducted at the Chung-Li VHF radar using a vertically pointing radar beam. The radar parameters employed for these four experiments are listed in Table 1, where t is the radar pulse length, IPP is the interpulse period, T is the period of the data employed for the analysis, Ncoh is the number of coherent integration, Nincoh is the number of raw Doppler spectra averaged to produce a resultant spectrum, and NFFT is the number of data points employed to calculate a raw spectrum using the fast Fourier transform (FFT) algorithm. The operational frequency of the Chung-Li VHF radar is 52 MHz (corresponding to 5.77-m wavelength) and the nominal peak transmitted power is 180 kW (3 3 60). The maximum duty cycle is 2%, and the pulse width can be set arbitrarily from 1 to 999 ms. The detailed characteristics of the ChungLi VHF radar already have been introduced in Rottger et al. (1990) and will not be mentioned here again. In view of the complexity of the observed Doppler spectrum for the Chung-Li VHF radar, the spectral components of precipitation and clear-air turbulence can be discerned effectively only by eye. For a vertically pointing radar beam the observed Doppler spectral component from clear-air refractivity fluctuations usually locates around the zero Doppler frequency, while that from precipitation always occurs in the Doppler spectral domain with positive frequency shift (provided the updraft has a velocity smaller than the terminal velocity of hydrometeor). Note that the precipitation spectral component may be folded from the positive Doppler frequency domain into the negative Doppler frequency domain if the time resolution of the data points is not fine enough. The true spectrum should be restored by shifting the folded spectral component to its right lo-

cation before further analysis is performed. Once the spectral components of the radar echoes from precipitation and refractivity fluctuations are identified and separated, the echo powers, mean Doppler frequency shifts, and spectral widths of the respective echoes can be estimated with the least squares method, in which a double Gaussian curve is employed to best fit to the corresponding Doppler spectral component. The reason why the shape of precipitation can be taken in the Gaussian form is given below. It is generally believed that an observed precipitation Doppler spectrum is a result of the convolution of the Doppler spectrum of clear-air echoes with that of the hydrometeor size distribution, provided the precipitation particles are frozen in the background wind (Wakasugi et a1. 1986; Gossard et al. 1990). The shape of the Doppler spectrum from clearair turbulence is usually assumed to be Gaussian because of the beam broadening and turbulent broadening effects. Superficially, the shape of the Doppler spectrum from hydrometeor size distribution is not Gaussian due to the exponential or gamma form of the drop size distribution (Marshall and Palmer 1948; Ulbrich 1983). However, because the radar echo power from precipitation is proportional to the sixth power of the diameter of precipitation particle for the case of the Rayleigh scattering, the corresponding Doppler spectrum of the hydrometeor size distribution can be treated to be quasiGaussian (Atlas et al. 1973). As a result, through the manipulation of the convolution, the resultant Doppler spectrum of the precipitation echoes is approximated reasonably to the Gaussian form. 4. Observation and discussion The information about the experiments, such as the dates, the types of the clouds associated with the weather systems responsible for the precipitation, and the types of the precipitation particles observed by the Chung-Li VHF radar for the investigation of V T–P relationship are listed in Table 2. During the periods of the radar experiments, light to moderate rainfalls were recorded on the ground. By appropriately setting the radar parameters (as shown in Table 1) and carefully processing radar returns (as mentioned in the previous section), the spectral components from clear-air turbulence and precipitation can be identified in the observed Doppler spectrum for further analysis. Figure 2 shows an example of a height–frequency–intensity contour of the vertical Doppler spectrum observed on 2 June 1988. As

TABLE 2. Information on radar experiments for the investigation of VT–P relation. Date 2 Jun 1988 25 Dec 1994 18 Dec 1995 26 Feb 1996

Weather type

Cloud type

Typhoon

Stratiform

Cold front Cold front Cold front

Stratiform Stratiform Stratiform

Phase of observed hydrometeor Ice crystal with supercooled drops, melting particles, raindrops Melting particles, raindrops Melting particles, raindrops Melting particles, raindrops

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FIG. 2. Height–intensity contour of the observed Doppler spectrum calculated by using a 128-point FFT algorithm. The bright band located at the height of around 5.1 km is clearly discernible. Notice that the height of the 08C isotherm responsible for the formation of the melting layer, which is scaled from the rawinsonde temperature profile, is about 5.24 km.

indicated, the spectral component of precipitation located within the spectral domain with negative Doppler velocity (or positive Doppler frequency) can be distinguished unambiguously from that of refractivity fluctuations located around zero Doppler frequency shift, where the negative (positive) Doppler velocity corresponds to the target moving toward (away from) the radar. Inspecting in more detail the height variation of the precipitation spectral component, we find that there is a striking enhancement of the precipitation echoes (i.e., brightband structure) with a sharp gradient of the Doppler velocity in the height range 4.5–5.7 km. The temperature profile observed at 0800LT 2 June 1988, by the Pan-Chiao rawinsonde station (located at 25 km northeast from the Chung-Li VHF radar site) shows that the level of the 08C isotherm occurs at the height of 5.24 km, supporting the brightband structure presented in Fig. 2. It is obvious from Fig. 2 that the averaged Doppler frequency (or velocity) of the hydrometeors (primarily ice particles) above 5.7 km and that of the raindrops below 4.5 km are about 0.62 Hz (or 21.8 m s21 ) and 2.3 Hz (or 26.7 m s21 ), respectively, while the mean Doppler velocity of the melting particles within the melting layer increases abruptly with the decrease of height at the rate of about 1.32 Hz km21 (or 3.8 m s21 km21 ). It is noteworthy that supercooled water droplets may coexist with the solid precipitation particles

above the 08C isotherm, provided the ice nuclei (or aerosols) are sparse and intense updrafts supplying the sufficient water vapor to produce the supersaturation exist in this region. However, in the present case we believe that the VHF backscatter from supercooled droplets is minor compared with the contribution of solid ice particles (see below). It is well known that a supercooled droplet is in an unstable state, it will be soon frozen as the supercooled water droplet collides with an ice particle. Moreover, because the saturation water vapor pressure over a supercooled water droplet is higher than that over an ice particle, it is expected that the ice crystal will grow. In addition the supercooled droplet will be shrunk by transferring the water vapor from the latter to the former (Fletcher 1962) if the supersaturation with respect to the supercooled water droplet is not reached because of weak or downward vertical air motion. Under this environment, it appears that the growth of the small supercooled water droplets to the extent of being capable of producing the significant radar backscatter is impossible. Examining the Doppler velocity of the clearair turbulence spectral component above melting level presented in Fig. 2 reveals that the air below 8.0 km moves primarily downward at a velocity of smaller than 0.8 m s21 . This feature implies that 1) sufficient water vapor cannot be supplied by the weak and downward vertical air motion to maintain supersaturation and 2)

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FIG. 3. Scatter diagram of the terminal velocity (in m s21 ) vs rangecorrected VHF precipitation echo power (in dB), where the data points for ice crystals (open circle sign), melting particles (asterisk sign), and raindrops (plus sign) are shown.

the large supercooled droplets cannot be sustained and survive in this situation. Therefore, for these reasons we believe that the precipitation particles responsible for the radar returns above the 08C isotherm are mainly the solid hydrometeors. However, in the case of a strong updraft, one cannot rule out the possibility that the supercooled droplets coexist with the ice particles in the region above the 08C isotherm and contribute to significant radar backscatter. In establishing a correct V T–P relationship vertical air velocity should be subtracted from observed Doppler velocity of hydrometeor to obtain the true terminal velocity and a scaling factor, as mentioned in section 2, should be given in the right-hand side of Eq. (6) to correct the effect of air drag on the falling speed. As a result we obtain the scatter diagram of terminal velocity (in m s21 ) versus range-corrected VHF precipitation echo power (in dB), as shown in Fig. 3. As indicated, the scatter patterns for ice crystals (characterized by small terminal velocity and weak backscatter and marked with open circles), melting particles (characterized by the abrupt increase of fall velocity and enormously enhanced backscatter and marked with an asterisk), and raindrops (characterized by large terminal velocity and moderate backscatter and marked with a plus sign) can be distinguished on the whole from one another whether or not there are several data points with large scatter in the plot. It is clear from Fig. 3 that different hydrometeors have different scatter patterns, implying that the precipitation particles with different phases will have different V T–P relationships. The coefficient a and exponent b in the V T–P relationship for each group of the scatter pattern can thus be estimated through the best fit to the corresponding data in accordance with Eq. (8). Figure 4a shows the height variations of a (curve with open circles) and b (curve with as-

FIG. 4. Height variations of a (curves with open circles) and b (curves with asterisks) in power-law approximation V T 5 aP b for four different experiments, where the arrows shown in each panel denote the heights of the 08C isotherm.

terisks) for the data presented in Fig. 3, where the standard errors of the estimations are also given. The solid arrow in the plot marks the level of the 08C isotherm (about 5.24 km) scaled from the rawinsonde temperature profile. Figure 4a clearly demonstrates a negative correlation between a and b, which is in excellent agreement with the theoretical prediction, as shown in Fig. 1. The detailed discussion on the relation between lna and b will be given later. Figures 4b–d present the height variations of a (curves with open circles) and b (curves with asterisks) calculated from the data obtained by three other experiments. As shown in Fig. 4, dramatic changes in a and b with height are seen, especially around the height of the 08C isotherm. It is obvious that the magnitude of a (b) in the melting layer is enor-

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FIG. 5. Scatter diagrams of ln(a) vs b for the data presented in Fig. 4, where the datasets marked by 1 (asterisks) and 2 (open circles) in the upper-left panel are the data below and above the melting level. The straight line in each panel is the regression line in accordance with equation ln(a) 5 ln(A) 1 jb, and the correlation coefficient (marked by C, C1, or C2) for each dataset is also shown.

mously larger (smaller) than those above and below the 08C isotherm. These results imply that the height-averaged value of a (b) will be overestimated (underestimated) if the radar returns from melting particles are not excluded in establishing the height-integrated V T–P relationship for ice particles and raindrops. Therefore, great caution should be used in the investigation of the height-averaged V T–P relationship using the radar echoes from the precipitation particles in the height range including the melting level. As mentioned in section 2, the mathematical relation between a and b can be described accurately by the simple function in the exponential form as given by Eq. (11). The parameter A and j in the equation can thus be estimated through the best fit to observed data. Figure TABLE 3. Comparison of the estimated A and B in V(D) 5 ADB in this paper with those presented in the earlier literature. Source 2 Jun 1988 25 Dec 1994 18 Dec 1995 26 Feb 1996 Spilhaus (1948) Liu and Orville (1968) Sekhon and Srivastava (1971) Atlas and Ulbrich (1977)

A(m s21 cm2B )

B

10.32 8.115 9.3 7.932 14.20 21.15 16.90 17.67

0.495 0.407 0.477 0.513 0.5 0.8 0.6 0.67

5 presents the scatter diagrams of lna versus b for the data shown in Fig. 4, where the straight line in each panel is the regression line obtained by best fitting to the respective datasets in accordance with Eq. (11) and the symbol C represents the correlation coefficient between lna and b. Note that the data points in the melting layer are not included in Fig. 5 and the datasets labeled with 1 (asterisk) and 2 (open circle) in the upper-left panel of Fig. 5 correspond, respectively, to the data below and above the melting level. As indicated, the observed relationship between lna and b with tremendously high correlation is in excellent agreement with the theoretical prediction, implying that the assumptions of power-law approximation to the fall speed–diameter relationship of hydrometeor and exponential form of particle-size distribution are both valid for the present cases. Table 3 compares the magnitudes of A and B obtained in this paper for the raindrops in the powerlaw approximation V(D) 5 AD B with those published in the literature, where our B values are calculated from the height-averaged b values in accordance with Eq. (10). Note that the A value of Spilhaus (1948) as the functions of the deformation of the raindrop, the ratio of the density for air to raindrop, and the drag coefficient of the raindrop was obtained by fitting a theoretical equation to the experimental data of Laws (1941). Spilhaus’ B value is always equal to 0.5, regardless of the

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deformation of the raindrop. Recalculation shows that the A value of 14.2 obtained by Spilhaus (1948) corresponds to the raindrop with the fineness ratio y/x of 0.56, where x is the length of major axis and y is the length of minor axis of a deformed raindrop. The other parameters employed by Spilhaus (1948) to calculate the A value are the air density is 1.155 3 1023 g cm21 , and the drag coefficient is 0.6 for a flat plate, 0.21 for a sphere. The values of A and B for Atlas and Ulbrich (1977) were derived by fitting the power-law expression V(D) 5 AD B to the data of Gunn and Kinzer (1949) over the diameter range between 0.05 and 0.5 cm. Note that the experimental data of Gunn and Kinzer (1949) were obtained by using a water dropper to produce the electrically charged water droplets in the mass range from 0.2 to 100 000 mg to measure their terminal velocities in the laboratory. Atlas et al. (1973) found that the expression fitting Gunn and Kinzer’s data very well with the error less than 2% is V(D) 5 9.65 2 10.3 exp(26D)

(m s21 ),

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FIG. 6. Scatter diagram of the coefficient A in the relation V(D) 5 AD B vs the height of the 08C isotherm.

(12)

where D is in centimeters. The values of A and B obtained by Liu and Orville (1968) and Sekhon and Srivastava (1971) were estimated from the observed precipitation data. As shown in Table 3, our A and B values are systematically smaller than those obtained by other investigations. By employing A and B, we can calculate the terminal velocity of a raindrop with a certain diameter in accordance with Eq. (8). A comparison of the terminal velocities computed from our data listed in Table 3 with those from Eq. (12) reveals that the former agree with the latter in the diameter range from 0.05 to 0.08 cm with an error of less than 10%. With the same error, the diameter range in which the terminal velocities calculated from the data of Spilhaus (1948) agree with those of Eq. (12) is between 0.1 and 0.5 cm, while it is 0.035–0.06 cm and 0.17–0.36 cm for the data of Liu and Orville (1968) and 0.1–0.35 cm for the data of Sekhon and Srivastava (1971). Presumably, the differences of A and B values estimated in this paper from those obtained by Liu and Orville (1968) and Sekhon and Srivastava (1971) may be due to the different types of the precipitation. Remember that the precipitation in our cases is generated from the stratiform clouds. However, the clouds responsible for the precipitation for Liu and Orville (1968) and Sekhon and Srivastava (1971) are convective ones, including cumulus clouds and thunderstorms. Therefore, it suggests that the type of cloud generating the precipitation may be one of the crucial factors influencing the magnitudes of A and B. However, more observations and data analysis are needed before that conclusion is drawn. A comparison of the values of A shown in Fig. 5 and the heights of the 08C isotherm presented in Fig. 4 appears to show a positive correlation between the A value and 08C isotherm height (see Fig. 6). This feature implies that the height of the 08C isotherm (or, equivalently, the vertical extent of rain height), which is closely

related to the surface air temperature, seems to play a role in governing the behavior of parameter A. The connection between them is illustrated as follows. It is well known that the air drag exerted on a precipitation particle through viscous drag is the function of the product of air density and cross section (or diameter square) of the particle. Therefore, the hydrometeors with the same radius but falling in the environments with different air densities will be subject to different air drags and eventually have different fall velocities. Consequently, large air drag reduces the fall speed of hydrometeor and results in the small coefficient A in accordance with the relations of V(D) 5 AD B . Because air density as the function of temperature determines the air drag and the rain height is proportional to the air temperature, a connection between the air drag, the height of the 08C isotherm, and the magnitude of A is thus established. Namely, the higher the level of the 08C isotherm is, the weaker the air drag will be (due to the smaller air density), and the greater the value A to be expected. In fact, the above scenario can also be supported by Spilhaus (1948). He showed that the ratio of the terminal velocity square to the diameter of the raindrop will be inversely proportional to the product of the air density and the drag coefficient. Note that, according to the definition of Spilhaus (1948), this ratio is exactly equal to A 2 . Therefore, it is expected that the magnitude of A will be inversely proportional to the air density (or air drag), which confirms the above scenario. Examining Fig. 4 in more detail shows that for melting particles the value of b is usually smaller than or close to 0, indicating negative or no correlation between fall speed and radar reflectivity (or particle size). Obviously, this behavior violating the physical law is unrealistic and suggests that the power-law approximation to the V T–P relationship is not applicable to the hydrometeors in the melting layer. Several factors influencing

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the behavior of the melting particle can explain why the V T–P relationship breaks down in the melting layer. First of all, we note that the fall velocity of a melting particle is changing (or increasing) during its fall in the melting layer. In this case, it is questionable that a balance between the gravitational force exerted on the melting particle and the viscous drag experienced by the falling hydrometeor can be reached in the melting layer. Namely, the basic assumption that the melting particle is falling at the terminal velocity will be invalid if these two forces are not identical. Second, because there are many types of hydrometeors coexisting in the melting region, including small droplets, melting ice particles, and raindrops, the size distribution of the hydrometeors in the melting layer will be too complicated to be described in terms of the simple exponential function, as mentioned in section 2. Furthermore, the power-law approximation to the fall speed–diameter relation can be applied only to a precipitation particle with fixed phase. The question then becomes whether or not this approximation can be applicable to the phase-changing particles. In addition, the nonspherical shape and nonhomogeneous distribution of refractive index of the melting particle also invalidate the conventional radar equation employed to derive the V T–P relationship. On the basis of the reasons mentioned above, we conclude that the current model of the V T–P relationship established in this paper cannot be applied to the melting particles. Therefore, how to find an appropriate V T–P relationship for the melting particles will be a challenging work in the future. 5. Concluding remarks The theoretical relationship between coefficient a and exponent b in the power-law expression of the V T–P relationship is analyzed in this article. It shows that the mathematical relationship between a and b can be approximated perfectly by an exponential function, which is in excellent agreement with the observational data. This agreement justifies the assumptions of the exponential size distribution and power-law approximation to the fall speed–diameter relationship of hydrometeors employed in this study. The V T–P relationships for the hydrometeors above, within, and below the melting layer are also investigated. The results show that the powerlaw approximation to the fall speed and VHF backscatter is applicable to the hydrometeors above and below the melting layer, but it fails for the particles in the melting layer. In light of enormously large a and negative b for the melting particles, great caution should be used in establishing the height-averaged V T–P relationship if the precipitation echoes employed for the analysis contain the echoes from the melting particles. In this article, we also propose a method to estimate the coefficient A and exponent B for the expression V(D) 5 AD B on the basis of the theoretical relation between a and b. Comparison of the values of A and B estimated in this article with

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those published in the earlier literature shows a reasonable agreement, although the former are slightly smaller than the latter. We attribute the difference in the two to the different types of precipitation. In addition, we also find that the correlation between the magnitude of A and the height of melting layer is positive. A scenario connecting the level of the 08C isotherm, air drag, and air temperature to the coefficient A is proposed to illustrate the dependence of the magnitude of A on the altitude of the 08C isotherm. Acknowledgments. This work was partially supported by National Science Council of Taiwan under Grants NSC-87-2111-M-008-017-A10 and NSC-87-NSPO-AECP-008-01. REFERENCES Atlas, D., and C. W. Ulbrich, 1977: Path- and area-integrated rainfall measurement by microwave attenuation in the 1–3 cm band. J. Appl. Meteor., 16, 1322–1331. , R. C. Srivastava, and R. S. Sekhon, 1973: Doppler radar characteristics of precipitation at vertical incidence. Rev. Geophys. Space Phys., 11, 1–35. Battan, L. J., 1973: Radar Observations of the Atmosphere. The University of Chicago Press, 323 pp. Beard, K., and A. J. Heymsfield, 1988: Terminal velocity adjustment for plate-like crystals and graupel. J. Atmos. Sci., 45, 3515–3518. Chilson, P. B., C. W. Ulbrich, M. F. Larsen, P. Perillat, and J. E. Keener, 1993: Observations of a tropical thunderstorm using a vertically pointing, dual-frequency, collinear beam Doppler radar. J. Atmos. Oceanic Technol., 10, 663–673. Chu, Y. H., and C. H. Lin, 1994: The severe depletion of turbulent echo power in association with precipitation observed by using Chung-Li VHF radar. Radio Sci., 29, 1311–1320. , and J. S. Song, 1998: Observations of precipitation associated with a cold front using a VHF wind profiler and a ground-based optical rain gauge. J. Geophys. Res., 103, 11 401–11 409. , L. P. Chian, and C. H. Liu, 1991: The investigation of atmospheric precipitation by using Chung-Li VHF radar. Radio Sci., 26, 717–729. , T. Y. Chen, and T. H. Lin, 1997: An examination of wind-driven effect on drift of precipitation particles using Chung-Li VHF radar. Radio Sci., 32, 957–966. Currier, P. E., S. K. Avery, B. B. Balsley, K. S. Gage, and W. L. Ecklund, 1992: Use of two wind profilers in the estimation of raindrop size distribution. Geophys. Res. Lett., 19, 1017–1020. Fletcher, N. H., 1962: The Physics of Raincloud. Cambridge University Press, 386 pp. Foote, G. B., and P. S. du Toit, 1969: Terminal velocity of raindrops aloft. J. Appl. Meteor., 8, 585–591. Gossard, E. E., R. G. Strauch, and R. R. Rogers, 1990: Evaluation of drop-size distribution in liquid precipitation observed by ground-based Doppler radar. J. Atmos. Oceanic Technol., 7, 815– 828. , R. G. Strauch, D. C. Welsh, and S. Y. Matrosov, 1992: Cloud layers, particle identification, and rain-rate profiles from ZRV f measurements by clear-air Doppler radar. J. Atmos. Oceanic Technol., 9, 108–119. Gunn, R., and G. D. Kinzer, 1949: The terminal velocity of fall for water droplets in stagnant air. J. Meteor., 6, 243–248. Hauser, D., and P. Amayenc, 1981: A new method for deducing hydrometeor-size distributions and vertical air motions from Doppler radar measurements at vertical incidence. J. Appl. Meteor., 43, 547–555. Laws, J. O., 1941: Measurement of fall velocity of water-drops and raindrops. Trans. Amer. Geophys. Union, 22 (Part III), 709–721.

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