362
Eugene E. Clothiaux, Thomas P. Ackerman, David M. Babb
Solving for Dn,i in (13.101) and substituting the result into (13.102) yields
LWC j = CP~Nt) [ZJ!2Nt-l/2e-I8u2/2]e9U2/2.
(13.103)
Integrating the liquid water content over the depth of the cloud layer, we have the liquid water
path LWP:
(13.104)
Substituting the measured LW P and radar reflectivities Zj into (13.104), one can solve for Nt.
Once Nt is in hand, Dn,i can be obtained from (13.101) and the LWC i from (13.102).
Cloud Drop Size Distributions from Radar Doppler Spectra
Gossard (1994) presents a technique that uses the entire Doppler spectrum in a retrieval of the
cloud drop size distribution. The innovation in Gossard's retrieval is that the effects of turbulence and any mean vertical wind motion within the radar resolution volume on the Doppler
spectrum are explicity accounted for in the retrieval. Assuming that the radar wavelength is
much larger than the cloud drops, the density of the reflectivity Z as a function of diameter D
i.e., the integrand of (13.90a), is
(13.105a)
where the superscript q indicates that turbulence is not included in the definition of S2(D). In
Figure 13.23a we plot nmg(D) and S2(D) for gamma distribution parameters Nt = 6.54 X 10 6
m- 3 , v = 1, and Dn = 0.0233 mm. Since v is one, the drop size distribution in Figure 13.23a
is exponential. The choice of Dn = 0.0233 mm forces the maximum value of S%(D) in Figure
13.23a to occur at Dm = 0.14 mm since the modal value of Si(D) occurs at Dm = Dn(6+v-1).
Changing variables in (13.105a) from the drop diameter D to the drop fall speed v, where we
assume a known relationship D = D(v), we have
dD
SJ(v) = S%(D(v))Tv(v).
(13.105b)
For the spectrum S2(D) in Figure 13.23a, the corresponding fall velocity spectrum S;(v) is
illustrated in Figure 13.23b, where we use (Rogers et al., 1993)
(
)
-05
v(D) = 4:a . D(l - e- 12D ).
(13.106)
Again, the superscript q in S%( v) indicates a quiet air spectrum with particle fall velocities
that are not affected by turbulence. If turbulence is present, then the spectrum S2( v) must be
convolved with the turbulence spectrum to produce the final spectrum S;(v):
S!(v) = 1"0 S;(v') ke-[(VI-V)/V6J2dV',
(13.107)
where, like Gossard (1994), we have assumed Gaussian turbulence. The results of such a
convolution are illustrated in Figure 13.23b for Vu equal to 0.4 and 0.8. Apart from mean
vertical wind motions within the radar resolution volume, which shifts the spectra in Figure
13.23b either to the left or right, S; (v) corresponds to the observed radar Doppler spectrum and
is one of the outputs of a cloud radar. Gossard's technique attempts to retrieve the parameters
Nt and Dn given spectra S;(v) that are potentially influenced by turbulence and mean vertical
air motion within the radar resolution volume.
Eugene E. Clothiaux, Thomas P. Ackerman, David M. Babb
Solving for Dn,i in (13.101) and substituting the result into (13.102) yields
LWC j = CP~Nt) [ZJ!2Nt-l/2e-I8u2/2]e9U2/2.
(13.103)
Integrating the liquid water content over the depth of the cloud layer, we have the liquid water
path LWP:
(13.104)
Substituting the measured LW P and radar reflectivities Zj into (13.104), one can solve for Nt.
Once Nt is in hand, Dn,i can be obtained from (13.101) and the LWC i from (13.102).
Cloud Drop Size Distributions from Radar Doppler Spectra
Gossard (1994) presents a technique that uses the entire Doppler spectrum in a retrieval of the
cloud drop size distribution. The innovation in Gossard's retrieval is that the effects of turbulence and any mean vertical wind motion within the radar resolution volume on the Doppler
spectrum are explicity accounted for in the retrieval. Assuming that the radar wavelength is
much larger than the cloud drops, the density of the reflectivity Z as a function of diameter D
i.e., the integrand of (13.90a), is
(13.105a)
where the superscript q indicates that turbulence is not included in the definition of S2(D). In
Figure 13.23a we plot nmg(D) and S2(D) for gamma distribution parameters Nt = 6.54 X 10 6
m- 3 , v = 1, and Dn = 0.0233 mm. Since v is one, the drop size distribution in Figure 13.23a
is exponential. The choice of Dn = 0.0233 mm forces the maximum value of S%(D) in Figure
13.23a to occur at Dm = 0.14 mm since the modal value of Si(D) occurs at Dm = Dn(6+v-1).
Changing variables in (13.105a) from the drop diameter D to the drop fall speed v, where we
assume a known relationship D = D(v), we have
dD
SJ(v) = S%(D(v))Tv(v).
(13.105b)
For the spectrum S2(D) in Figure 13.23a, the corresponding fall velocity spectrum S;(v) is
illustrated in Figure 13.23b, where we use (Rogers et al., 1993)
(
)
-05
v(D) = 4:a . D(l - e- 12D ).
(13.106)
Again, the superscript q in S%( v) indicates a quiet air spectrum with particle fall velocities
that are not affected by turbulence. If turbulence is present, then the spectrum S2( v) must be
convolved with the turbulence spectrum to produce the final spectrum S;(v):
S!(v) = 1"0 S;(v') ke-[(VI-V)/V6J2dV',
(13.107)
where, like Gossard (1994), we have assumed Gaussian turbulence. The results of such a
convolution are illustrated in Figure 13.23b for Vu equal to 0.4 and 0.8. Apart from mean
vertical wind motions within the radar resolution volume, which shifts the spectra in Figure
13.23b either to the left or right, S; (v) corresponds to the observed radar Doppler spectrum and
is one of the outputs of a cloud radar. Gossard's technique attempts to retrieve the parameters
Nt and Dn given spectra S;(v) that are potentially influenced by turbulence and mean vertical
air motion within the radar resolution volume.